Far-field and near-field optical trapping...Current optical trapping models based on ray optics...

Far-field and near-field opticaltrapping

A thesis submitted for the degree of

Doctor of Philosophy

by

Djenan Ganic

Centre for Micro-Photonics

Faculty of Engineering and Industrial Sciences

Swinburne University of Technology

Melbourne, Australia

for Dalila, Denis , and Elma.

In science the credit goes to the man who convinces the world, not the man

to whom the idea first occurs.

— Sir Francis Darwin (1848 - 1925)

I was born not knowing and have had only a little time to change that here

and there.

— Richard Feynman (1918 - 1988)

Declaration

I, Djenan Ganic, declare that this thesis entitled:

“Far-field and near-field optical trapping”

is my own work and has not been submitted previously, in whole or in part,in respect of any other academic award.

Djenan Ganic

Centre for Micro-PhotonicsFaculty of Engineering and Industrial SciencesSwinburne University of TechnologyAustralia

Dated this day, February 25, 2005

Abstract

Optical trapping techniques have become an important and irreplaceable tool

in many research disciplines for reaching non-invasively into the microscopic

world and to manipulate, cut, assemble and transform micro-objects with

nanometer precision and sub-micrometer resolution. Further advances in

optical trapping techniques promise to bridge the gap and bring together the

macroscopic world and experimental techniques and applications of micro-

systems in areas of physics, chemistry and biology.

In order to understand the optical trapping process and to improve and

tailor experimental techniques and applications in a variety of scientific

disciplines, an accurate knowledge of trapping forces exerted on particles

and their dependency on environmental and morphological factors is of

crucial importance. Furthermore, the recent trend in novel laser trapping

experiments sees the use of complex laser beams in trapping arrangements for

achieving more controllable laser trapping techniques. Focusing of such beams

with a high numerical aperture (NA) objective required for efficient trapping

leads to a complicated amplitude, phase and polarisation distributions of an

electromagnetic field in the focal region. Current optical trapping models

based on ray optics theory and the Gaussian beam approximation are

inadequate to deal with such a focal complexity.

Novel applications of the laser trapping such as the particle-trapped

i

ABSTRACT

scanning near field optical microscopy (SNOM) and optical-trap nanometry

techniques are currently investigated largely in the experimental sense or with

approximated theoretical models. These applications are implemented using

the efficient laser trapping with high NA and evanescent wave illumination of

the sample for high resolution sensing. The proper study of these novel laser

trapping applications and the potential benefits of implementation of these

applications with complex laser beams requires an exact physical model for

the laser trapping process and a nanometric sensing model for detection of

evanescent wave scattering.

This thesis is concerned with comprehensive and rigorous modelling

and characterisation of the trapping process of spherical dielectric particles

implemented using far-field and near-field optical trapping modalities. Two

types of incident illuminations are considered, the plane wave illumination

and the doughnut beam illumination of various topological charges. The

doughnut beams represent one class of complex laser beams. However, our

optical trapping model presented in this thesis is in no way restricted to

this type of incident illumination, but is equally applicable to other types

of complex laser beam illuminations. Furthermore, the thesis is concerned

with development of a physical model for nanometric sensing, which is of

great importance for optical trapping systems that utilise evanescent field

illumination for achieving high resolution position monitoring and imaging.

The nanometric sensing model, describing the conversion of evanescent

photons into propagating photons, is realised using an analytical approach

to evanescent wave scattering by a microscopic particle. The effects of

an interface at which the evanescent wave is generated are included by

considering the scattered field reflection from the interface. Collection and

imaging of the resultant scattered field by a high numerical aperture objective

is described using vectorial diffraction theory. Using our sensing model, we

PhD thesis: Far-field and near-field optical trapping ii

ABSTRACT

have investigated the dependance of the scattering on the particle size and

refractive index, the effects of the interface on the scattering cross-section,

morphology dependent resonance effects associated with the scattering

process, and the effects of the incident angle of a laser beam undergoing

total internal reflection to generate an evanescent field. Furthermore, we

have studied the detectability of the scattered signal using a wide area

detector and a pinhole detector. A good agreement between our experimental

measurements of the focal intensity distribution in the back focal region of

the collecting objective and the theoretical predictions confirm the validity of

our approach.

The optical trapping model is implemented using a rigorous vectorial

diffraction theory for characterisation of the electromagnetic field distribution

in the focal region of a high NA objective. It is an exact model capable

of considering arbitrary amplitude, phase and polarisation of the incident

laser beam as well as apodisation functions of the focusing objective. The

interaction of a particle with the complex focused field is described by an

extension of the classical plane wave Lorentz-Mie theory with the expansion

of the incident field requiring numerical integration of finite surface integrals

only. The net force exerted on the particle is then determined using the

Maxwell stress tensor approach. Using the optical trapping model one can

consider the laser trapping process in the far-field of the focusing objective,

also known as the far-field trapping, and the laser trapping achieved by

focused evanescent field, i.e. near-field optical trapping.

Investigations of far-field laser trapping show that spherical aberration

plays a significant role in the trapping process if a refractive index mismatch

exists between the objective immersion and particle suspension media. An

optical trap efficiency is severely degraded under the presence of spherical

aberration. However, our study shows that the spherical aberration effect

PhD thesis: Far-field and near-field optical trapping iii

ABSTRACT

can be successfully dealt with using our optical trapping model. Theoretical

investigations of the trapping process achieved using an obstructed laser

beam indicate that the transverse trapping efficiency decreases rapidly with

increasing size of the obstruction, unlike the trend predicted using a ray

optics model. These theoretical investigations are in a good agreement with

our experimentally observed results.

Far-field optical trapping with complex doughnut laser beams leads to

reduced lifting force for small dielectric particles, compared with plane wave

illumination, while for large particles it is relatively unchanged. A slight

advantage of using a doughnut laser beam over plane wave illumination for far-

field trapping of large dielectric particles manifests in a higher forward axial

trapping efficiency, which increases for increasing doughnut beam topological

charge. It is indicated that the maximal transverse trapping efficiency

decreases for reducing particle size and that the rate of decrease is higher for

doughnut beam illumination, compared with plane wave illumination, which

has been confirmed by experimental measurements.

A near-field trapping modality is investigated by considering a central

obstruction placed before the focusing objective so that the obstruction size

corresponds to the minimum convergence angle larger than the critical angle.

This implies that the portion of the incident wave that is passed through

the high numerical aperture objective satisfies the total internal reflection

condition at the surface of the coverslip, so that only a focused evanescent

field is present in the particle suspension medium. Interaction of this focused

near-field with a dielectric micro-particle is described and investigated using

our optical trapping model with a central obstruction. Our investigation

shows that the maximal backward axial trapping efficiency or the lifting

force is comparable to that achieved by the far-field trapping under similar

conditions for either plane wave illumination or complex doughnut beam

PhD thesis: Far-field and near-field optical trapping iv

ABSTRACT

illumination. The dependence of the maximal axial trapping efficiency on the

particle size is nearly linear for near-field trapping with focused evanescent

wave illumination in the Mie size regime, unlike that achieved using the far-

field trapping technique.

PhD thesis: Far-field and near-field optical trapping v

Acknowledgements

In March 2000, after several discussions with Prof. Min Gu and Dr. Xiaosong

Gan, I undertook a PhD research project at the Centre for Micro-Photonics

(CMP), which forms an integral part of the Swinburne Optronics and Laser

Laboratories at Swinburne University of Technology in Hawthorn. These

discussions were very fruitful and resulted in a framework of a research project

dealing with optical trapping technology. I was very thrilled and excited about

this new project, the feeling which has not subsided ever since. Therefore, first

and foremost I would like to thank my principal supervisor Prof. Min Gu and

my associate supervisor Dr. Xiaosong Gan for the remarkable opportunity

to embark upon a journey of research and discovery into the wonderful world

of physics. Their relentless encouragements, advices and guidance made the

completion of my research work possible, for which I am immensely grateful.

I would also like to thank Swinburne University of Technology and the

Centre for Micro-Photonics for their financial support throughout the dura-

tion of my candidature, with a Swinburne Chancellor’s Research Scholarship

and a Micro-Photonics Postgraduate Scholarship, which made the research

work so much easier.

My sincerest gratitude goes to Dr. Daniel Day for introducing me to

the field of the experimental physics during my Honors project at Victoria

University. Thank you Dr. James Chon for valuable discussions on the

vi

ACKNOWLEDGEMENTS

vectorial diffraction theory and near-field trapping. To the other current

members of the CMP staff, Dr. Martin Straub, Dr. Charles Cranfield, Dr.

Guangyong Zhou, and Ms. Shuhui Wu, I thank you for your help and advices

during our meetings. I would also like to thank the previous members of the

CMP staff, Dr. Ross Ashman, Dr. Ming Gun Xu, Dr. Xiaoyuan Deng, and

Dr. Nguyen Le Huong, for their help and company during the first years of

my research.

Thank you Mr. Dru Morrish for helping and assisting me with the

laboratory equipment and for helpful discussions on many problems that

have risen not only throughout the candidature but since our first day

as undergraduate students. Mr. Dennis McPhail, I thank you for your

constructive advices on various aspects of experimental research, and for your

company since our undergraduate days. To the other fellow students of the

CMP, Mr. Michael James Ventura, Mr. Mujahid Ashraf, Ms. Baohua Jia,

Mrs. Smitha Kuriakose, and Ms. Fu Ling, I thank you all for the help and

discussions during our meetings and day to day business. A former CMP

student Dr. Damian Bird, I thank you for all your help, company and in

particular for convincing me to write my thesis using LaTeX, which makes

the writing and updating so much simpler. Thank you Mr. Mark Kivinen for

making the stages and optical mounts for my experimental setups. I would

also like to thank the CMP administrative assistants Ms. Benita Hutchinson-

Reade and Mrs. Anna Buzescu for their help with paperwork and general

office tasks.

Also, I would like to thank Prof. Theo Tschudi from the Institute of

Applied Physics at Darmstadt Technical University (Germany) for accepting

me to undertake a part of my research on doughnut beam generation using

a liquid crystal cell in his group. Many thanks go to his students Mr.

Svetomir Stankovic, Mr. Mathias Hain, and Mr. Somakanthan Somalingam,

PhD thesis: Far-field and near-field optical trapping vii

ACKNOWLEDGEMENTS

for teaching me how to make liquid crystal cells, and for their help during the

experimental work.

Finally, I would like to acknowledge the support and encouragement from

my family. To my father, Sefik Ganic, my mother, Mirjana Ganic, my

grandmother, Muruveta Topcagic, my sister, Djelila, her husband, Mirza,

my niece, Melina and my nephew, Tarik Zahirovic thank you. You have

been a great source of strength through the hard times and a fountain of joy

through the fun times. Last, but certainly not least, I would like to thank

the people to whom this thesis is dedicated; my fiancee Dalila Deronic, for

being an inspiration and a source of constant encouragement, support and

motivation, and to my children Denis and Elma Ganic, for their patience and

understanding.

Djenan Ganic

Melbourne, Australia

February 25, 2005

PhD thesis: Far-field and near-field optical trapping viii

Contents

Declaration

Abstract i

Acknowledgements vi

Contents ix

List of Figures xiii

List of Tables xxv

1 Introduction 1

1.1 Introduction to optical trapping . . . . . . . . . . . . . . . . . 1

1.2 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Preview of the thesis . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature review 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Far-field trapping . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Optical-trap nanometry . . . . . . . . . . . . . . . . . 11

2.2.2 Particle-trapped SNOM . . . . . . . . . . . . . . . . . 13

2.3 Trapping force calculation . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Ray optics method . . . . . . . . . . . . . . . . . . . . 16

ix

CONTENTS

2.3.2 Electromagnetic method . . . . . . . . . . . . . . . . . 20

2.4 Near-field Mie scattering . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Geometric optics method . . . . . . . . . . . . . . . . . 25

2.4.2 Generalised Mie theory . . . . . . . . . . . . . . . . . . 27

2.4.3 Discrete sources method . . . . . . . . . . . . . . . . . 31

2.5 Vectorial diffraction of light . . . . . . . . . . . . . . . . . . . 33

2.5.1 Focusing through mismatched refractive index materials 34

2.5.2 Focal spot splitting with high NA objective . . . . . . 39

2.5.3 Spectral splitting near phase singularities of focused waves 42

2.6 Near-field trapping . . . . . . . . . . . . . . . . . . . . . . . . 44

2.6.1 Near-field trapping using a nano-aperture . . . . . . . . 45

2.6.2 Near-field trapping using a metallic tip . . . . . . . . . 46

2.6.3 Near-field trapping using an apertureless probe . . . . 48

2.6.4 Near-field trapping with evanescent field generated

under TIR illumination . . . . . . . . . . . . . . . . . . 50

2.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Three dimensional near-field Mie scattering by a small

particle 57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Mathematical description of 3-D near-field Mie scattering . . . 59

3.2.1 Particles far from the interface . . . . . . . . . . . . . . 61

3.2.2 Particles near the interface . . . . . . . . . . . . . . . . 63

3.3 3-D scattered intensity distribution around dielectric particles 64

3.3.1 Dielectric particle situated far from the interface . . . . 65

3.3.2 Dielectric particle situated near the interface . . . . . . 69

3.4 Effects of the interface on the morphology dependent resonance 70

PhD thesis: Far-field and near-field optical trapping x

CONTENTS

3.5 Mechanism for conversion of evanescent photons into propa-

gating photons . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5.1 Physical model . . . . . . . . . . . . . . . . . . . . . . 76

3.5.2 Theoretical results and morphology dependent resonances 79

3.5.3 Experimental setup and results . . . . . . . . . . . . . 81

3.6 Pinhole detection of the scattered near-field signal . . . . . . . 84

3.7 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Trapping force with a high numerical aperture objective 90

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3 Vectorial diffraction - Gaussian approximation comparison . . 96

4.4 Model applicability . . . . . . . . . . . . . . . . . . . . . . . . 98


5 Far-field optical trapping 101

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2 Trapping force with plane wave illumination . . . . . . . . . . 103

5.2.1 Force mapping . . . . . . . . . . . . . . . . . . . . . . 103

5.2.2 Spherical aberration . . . . . . . . . . . . . . . . . . . 105

5.2.3 Trapping efficiency with centrally obstructed plane wave 108

5.3 Trapping force with doughnut beam illumination . . . . . . . 114

5.3.1 Doughnut beam generation . . . . . . . . . . . . . . . 115

5.3.2 Vectorial diffraction of doughnut beam illumination . . 121

5.3.3 Trapping efficiency . . . . . . . . . . . . . . . . . . . . 128


PhD thesis: Far-field and near-field optical trapping xi

CONTENTS

6 Near-field optical trapping 136

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2 Optical forces on microparticles in a wide area evanescent field 139

6.3 Near-field trapping with focused evanescent illumination -

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.4 Near-field trapping with focused evanescent illumination -

Experimental results . . . . . . . . . . . . . . . . . . . . . . . 149

6.4.1 Plane wave illumination . . . . . . . . . . . . . . . . . 150

6.4.2 Doughnut beam illumination . . . . . . . . . . . . . . . 151


7 Conclusion 154

7.1 Thesis conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.2 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.2.1 Optical trapping of metallic particles . . . . . . . . . . 159

7.2.2 Near-field micromanipulation system . . . . . . . . . . 160

Bibliography 164

Author’s Publications 178

PhD thesis: Far-field and near-field optical trapping xii

List of Figures

1.1 Principle of optical tweezers trapping [1]. . . . . . . . . . . . . . 3

2.1 Optical trap nanometry for mechanical measurements of myosin [3]. 12

2.2 Schematic diagram of particle-trapped SNOM setup. . . . . . . . 15

2.3 Qualitative description of the trapping of dielectric spheres accord-

ing to Ashkin’s RO model [23]. The total force F, given by the

refraction of a typical pair of rays a and b of the trapping beam,

is always restoring for axial and transverse displacements of the

sphere from the trap focus. . . . . . . . . . . . . . . . . . . . . . 17

2.4 (a) Gradient (Fg) and scattering (Fs) forces for an incident ray.

(b) The total (Qtotal), scattering (Qs) and gradient (Qg) trapping

efficiencies for a single ray hitting a dielectric sphere of relative

index of refraction 1.2 at an angle θ. . . . . . . . . . . . . . . . . 18

2.5 Gaussian optical trapping model. . . . . . . . . . . . . . . . . . 22

2.6 (a) Geometrical optics model of scattering of evanescent waves. (b)

Typical ray paths through a dielectric particle. The integers p and

w denote the number of contacts made with the inner surface of

the dielectric particle and the substrate, respectively [71]. . . . . . 26

2.7 Geometry of the scattering system in discrete sources method. . . 32

xiii

LIST OF FIGURES

2.8 A schematic diagram of a light beam being refracted on an interface

between two media (n2 > n1). . . . . . . . . . . . . . . . . . . . 35

2.9 Contour plots of the intensity near the focus of a high NA objective

(NA = 1). (a) |E|2 for ε = 0.0, (b) |E|2 for ε = 0.98 [93]. . . . . . 40

2.10 Contour plots of the intensity near the focus of a high NA objective

focused on an interface between two media (n1 = 1.78 and n2 =

1.0), with NA = 1.65 and ε = 0.6 [97]. . . . . . . . . . . . . . . . 41

2.11 The normalised spectrum S[u1(ω0, ω]/S0 at the first axial zero-

intensity point (u1 = 4π) of the central frequency component ω0

for different values of NA: (a) NA = 0.025, (b) NA = 0.1, (c)

NA = 0.3, (d) NA = 0.4, (e) NA = 0.6, (f) NA = 0.9. . . . . . . 43

2.12 The normalised spectrum S(ω)/S0 at the first x and y zero-intensity

points of the central frequency component ω0 for different values of

NA: (a) NA = 0.025, (b) NA = 0.05, (c) NA = 0.1, (d) NA = 0.3,

(e) NA = 0.6, (f) NA = 0.9. Full lines show the variations in the

x direction, while dotted lines represent those in the y direction. . 44

2.13 The geometry of the near-field trapping model. Incident light

is x polarised and propagates along the z axis. (a) and (b)

Radiation force spatial distribution for a dielectric sphere near a

nano-aperture. The origin of each arrow represents the center of

the sphere and vectors represent the direction and the magnitude

of forces [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.14 Near field of a gold tip in water illuminated by two different

monochromatic waves at λ = 810 nm, indicated by the k and E

vectors. The numbers in the figure give the scaling in the multiples

of the exciting field (E2), with a factor of 2 between the successive

lines. (a) No field enhancement; (b) Field enhancement of ≈ 3000 [13]. 47

PhD thesis: Far-field and near-field optical trapping xiv

LIST OF FIGURES

2.15 A dielectric sphere situated on a flat dielectric surface is illuminated

by near-field generated under total internal reflection. A tungsten

probe is used to create an optical trap [12]. . . . . . . . . . . . . 49

2.16 Force experienced by the sphere as a function of the lateral position

of the probe. Thick lines denote θ = 43, while thin lines denote

θ = 50. The probe tip is either 20 nm (solid lines) or 31 nm (dashed

lines) above the substrate. (a) TM polarisation z direction; (b) TE

polarisation z direction; (c) TM polarisation x direction; (d) TE

polarisation x direction. According to Ref. [12]. . . . . . . . . . . 50

2.17 Schematic diagram of particle movement in an evanescent field.

Velocity of the moving particle versus the incident angle of the

beam undergoing TIR [67]. . . . . . . . . . . . . . . . . . . . . . 51

2.18 Near-field trapping under focused evanescent illumination. Density

plots (a) and (b) represent the calculated modulus squared of the

electric field at wavelength 532 nm in the focal region of an objective

of NA = 1.65 at the interface between the cover slip (n = 1.78) and

water (n = 1.33). (a) No central obstruction, i.e. ε = 0; (b) With

central obstruction ε = 0.8, whose size satisfies the TIR condition [15]. 53

3.1 Illustration of the scattering model including the effect of an

interface at which an evanescent wave is generated. n, n′ and n1

denote refractive indices of the substrate, surrounding medium and

the particle, respectively. d is the distance from the center of a

particle to the interface. . . . . . . . . . . . . . . . . . . . . . . 59

PhD thesis: Far-field and near-field optical trapping xv

LIST OF FIGURES

3.2 Three-dimensional far-field distribution of the scattered intensity

around a 2 µm dielectric particle situated far away from the

interface (a) in the XZ-plane, (b) in the plane containing the X-

axis at 45o anti-clockwise from the XZ-plane, (c) in the XY -plane

and (d) in the Y Z-plane. The solid and dotted curves correspond

to the TE and TM polarisation states of the illumination wave,

respectively. n1=1.6, n′=1.0, n=1.51, λ=632.8 nm and α=45. . . 66

3.3 A qualitative interpretation of the confined intensity regions in the

scattering of an evanescent wave by a dielectric particle of radius 2

µm. The relative intensity of rays is denoted by the arrow length. . 67

3.4 Dependence of the scattered intensity distribution in the XZ plane

on the radius of a particle, when the particle is situated far away

from the interface: (a) a = 0.05 µm, (b) a = 0.1 µm, (c) a = 0.5 µm

and (d) a = 1 µm. The plots in the left and the right columns show

the intensity distributions scattered by an evanescent wave and a

plane wave respectively. The solid and dotted curves correspond

to the TE and TM polarisation states of the illumination wave,

respectively. n1 = 1.6, n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45. 68

3.5 Dependence of the scattered intensity distribution in the XZ plane

on the radius of a particle, when particle is situated on the interface:

(a) a = 0.05 µm, (b) a = 0.1 µm, (c) a = 0.5 µm and (d) a = 1

µm. The solid and dotted curves correspond to the TE and TM

polarisation states of the illumination wave, respectively. n1 = 1.6,

n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45. . . . . . . . . . . . 70

PhD thesis: Far-field and near-field optical trapping xvi

LIST OF FIGURES

3.6 Dependence of the half-space scattered intensity on the particle

radius for the TE (solid line) and TM (dotted line) polarisation

illumination, when the particle is situated on the interface. n′ =

1.0, n = 1.51, λ = 632.8 nm and α = 45. Top: n1 = 1.1, middle:

n1 = 1.3 and bottom: n1 = 1.6. . . . . . . . . . . . . . . . . . . 72

3.7 Dependence of the half-space scattered intensity on the particle

radius for the TE (solid line) and TM (dotted line) polarisation

illumination, when the particle is situated on the interface. n′ =

1.0, n = 1.51, n1 = 1.6 and λ = 632.8 nm. Top: α = 42, middle:

α = 43 and bottom: α = 45. . . . . . . . . . . . . . . . . . . . 73

3.8 A comparison of the calculated half space scattered intensity

with the intensity measured by a NA=1.3 objective in particle-

trapped near-field microscopy [50]. The solid and dotted curves

represent TE and TM incident polarisation states, respectively.

α is the incident angle and a polystyrene particle of radius 0.25

µm immersed in water is placed on the interface between the

surrounding medium and the substrate. . . . . . . . . . . . . . . 75

3.9 (a) Schematic of our theoretical model for evanescent photon con-

version. (b) Representation of the lens transformation process. (c)

Experimental setup for recording the FID of converted evanescent

photons, collected by a high NA objective O. . . . . . . . . . . . 77

3.10 Calculated FID in the image focal plane of a 0.8 NA objective. TE

(left colomn) and TM (right colomn) incident illumination. (a) and

(e) a = 100 nm. (b) and (f) a = 500 nm. (c) and (g) a = 1000 nm.

(d) and (h) a = 2000 nm. All figures are normalised to 100. . . . . 80

PhD thesis: Far-field and near-field optical trapping xvii

LIST OF FIGURES

3.11 Maximum intensity in the FID as a function of the particle radius

near MDR for TE (a) and TM (b) illumination. Insets show the full

FID representing off and on resonance cases. The particle refractive

index is 1.59 and the illumination wavelength is 633 nm. . . . . . 81

3.12 Calculated (top) and observed (bottom) FID in image focal plane

of a 0.8 NA objective collecting propagating photons converted by

a=240 nm polystyrene particle under TE (left column) and TM

(right column) incident illumination. . . . . . . . . . . . . . . . . 82

3.13 Calculated and observed y axis scan through x=0, in image

focal plane of a 0.8 NA objective collecting propagating photons

converted by 1000 nm (radius) polystyrene particle under TE

incident illumination. (a) Calculated results. (b) Observed results

(full line) where the dotted line represents the convolution of the

calculated results and the PSF of the imaging lens. Insets show the

calculated and observed FID. . . . . . . . . . . . . . . . . . . . . 83

3.14 (a) A schematic diagram of a pinhole detection process. Only

the rays coming from the front focal region are detected. (b)

Detected signal intensity as a function of a pinhole radius, in

optical coordinates, for uniformly illuminated objective. Assumed

objective NA = 0.8 in the front focal region, aperture size ρa = 3

mm and the back focal length of the objective f = 160 mm. . . . . 85

3.15 Detected scattered intensity as a function of the pinhole size (in an

optical coordinate) of a polystyrene particle for TE illumination

(left column) and TM illumination (right column). Assumed

objective NA = 0.8 in the front focal region, aperture size ρa = 3

mm and the back focal length of the objective f = 160 mm. (a)

and (d) Particle radius 0.1 µm. (b) and (e) Particle radius 0.5 µm.

(c) and (f) Particle radius 1.0 µm. . . . . . . . . . . . . . . . . . 86

PhD thesis: Far-field and near-field optical trapping xviii

LIST OF FIGURES

4.1 Schematic diagram of our trapping model. . . . . . . . . . . . . . 92

4.2 Intensity distributions in (a) axial and (b) transversal directions

(blue-X axis, red-Y axis) for the fifth-order Gaussian approximation

(dashed line) and the vectorial diffraction theory (solid line). . . . 97

4.3 Comparison between the fifth-order Gaussian approximation (empty

symbols) and the vectorial diffraction theory (filled symbols) for the

calculation of the maximal TTE (triangles) and the backward ATE

(circles) of polystyrene particles suspended in water. λ0 = 1.064

µm, ω0 = 0.4 µm and NA = 1.2. . . . . . . . . . . . . . . . . . . 97

5.1 Magnitude and direction of the trapping efficiency for various

geometrical focus positions around a polystyrene particle suspended

in water and illuminated by a λ0 = 1.064µm laser beam focused by

a NA = 1.25 water immersion objective. (a) particle radius of 2

µm. (b) Particle radius of 200 nm. . . . . . . . . . . . . . . . . . 104

5.2 Maximal backward ATE of glass particles suspended in water,

illuminated by a laser beam (λ0 = 1.064 µm), focused by an oil

immersion microscope objective (NA = 1.3). The effect of SA is

considered at a depth of 9 µm from the cover glass. . . . . . . . . 106

5.3 Maximal backward ATE and TTE of a particle illuminated by a

laser (λ0 = 1.064 µm) focused by an oil immersion microscope

objective (NA = 1.3) as a function of the distance from the cover

glass. (a) A glass particle of diameter D = 2.7 µm in water. (b)

A polystyrene particle of diameter D = 1.02 µm suspended in 60%

glycerol solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4 Focus intensity distribution for a plane wave focused by a high NA

objective immersed in water (NA = 1.25). Top row - unobstructed

plane wave (ε = 0.0). Bottom row - obstructed plane wave (ε = 0.8). 109

PhD thesis: Far-field and near-field optical trapping xix

LIST OF FIGURES

5.5 Theoretical calculation (RO model and vectorial diffraction model)

of the maximal TTE as function of obstruction size for a polystyrene

particle of radius 1 µm immersed in water. NA = 1.25 and λ = 532

nm. The maximal TTE for the two models are normalised to start

from the same point (at ε = 0.0). . . . . . . . . . . . . . . . . . . 110

5.6 Focus intensity along a transverse direction for various obstruction

sizes. (a) Polarisation direction X and (b) Perpendicular to

polarisation direction Y. NA = 1.25 and λ = 532 nm. . . . . . . . 111

5.7 A schematic diagram of the experimental setup. . . . . . . . . . . 112

5.8 Experimental measurement of the maximal TTE as a function of

obstruction size for a polystyrene particle of radius 1 µm immersed

in water. Theoretical values are normalised by the experimental P

value at ε = 0.0. NA = 1.2 and λ = 532 nm. . . . . . . . . . . . . 113

5.9 Phase distribution of a doughnut beam. (a) The theoretical phase

distribution of a doughnut beam of charge 1 according to 16 phase

steps. (b) The electrode structure of the liquid crystal cell with 16

pie slices. (c) The phase wavefront of the doughnut beam of charge

1, measured using phase shifting interferometry. . . . . . . . . . . 116

5.10 Experimental setup for generation of a doughnut beam through

the liquid crystal cell and interference measurement of its phase

distribution (P: polariser; BS1 and BS2: beam splitters; LC: liquid

crystal cell; O: objective; L: lens; M1 and M2: mirrors; PH: pinhole;

S: screen). (a) The voltage variation as a function of the slice

position of the liquid crystal cell. (b) The unwrapped phase shift

of the liquid crystal cell as a function of applied voltage. . . . . . 117

PhD thesis: Far-field and near-field optical trapping xx

LIST OF FIGURES

5.11 The intensity distributions (a, b and c) of laser beams transmitted

through a liquid crystal cell and the corresponding interference

patterns (d, e and f). (a) and (d) plane wave. (b) and (e) Doughnut

beam of charge 1. (c) and (f) Doughnut beam of charge 2. . . . . 118

5.12 Variation of the voltage between the two contact points (see

Fig. 5.9(b)) as a function of the wavelength for the generation of a

doughnut beam of charge 1. . . . . . . . . . . . . . . . . . . . . 120

5.13 Doughnut beam of charge 1 generated using a computer controlled

SPM. (a) Applied phase-ramp pattern with 256 levels. (b) Intensity

profile. (c) Interference pattern. . . . . . . . . . . . . . . . . . . 121

5.14 Calculated intensity distribution in the focal region of a doughnut

beam focused by an objective with NA = 1 ((a)-(c)) and NA = 0.2

((d)-(f)): (a) and (d) Topological charge 1; (b) and (e) Topological

charge 2; (c) and (f) Topological charge 3. . . . . . . . . . . . . . 123

5.15 Contour plots of the intensity distribution in the focal region of

an objective with NA = 1, illuminated by a doughnut beam of

topological charge 1. (a) |Ex|2;(b) |Ey|2;(c) |Ez|2;(d) |E|2. . . . . . 124







5.18 Dependence of the peak ratio of |Ez|2/|Ex|2 on the numerical

aperture (a) and on the obstruction radius ε (b). . . . . . . . . . 127

PhD thesis: Far-field and near-field optical trapping xxi

LIST OF FIGURES

5.19 Maximal backward ATE of polystyrene particles suspended in

water and illuminated by a highly focused plane wave, doughnut

beam of topological charge 1 and obstructed plane wave with

ε = 0.8 as a function of particle size. NA = 1.2 and λ0 = 1.064 µm. 129

5.20 ATE of a polystyrene particle (a = 2 µm) suspended in water and

illuminated by a highly focused plane wave and doughnut beams of

different topological charges. NA = 1.2 and λ0 = 1.064 µm. . . . . 130

5.21 TTE in the polarisation (X) and perpendicular to the polarisation

(Y) directions of a polystyrene particle (a = 2 µm) suspended in

water and illuminated by a highly focused plane wave and doughnut

beams of different topological charges. NA = 1.2 and λ0 = 1.064 µm.131

6.1 Focused evanescent field produced at the coverslip interface by

high NA focusing of an obstructed plane wave polarised in the X

direction. Oil immersion and coverslip refractive index n1 = 1.78

(index matched), particle suspension medium n2 = 1.33, objective

NA = 1.65, obstruction size ε = 0.8, and illumination wavelength

λ0 = 1.064 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2 Non-dimensional horizontal Qy and vertical Qz near-field forces

exerted on polystyrene particle under TE and TM incident polarisa-

tion states as a function of the particle size parameter k′a = 2πa/λ.

The evanescent field is generated on a prism surface (refractive

index 1.722) by a plane wave incident at 51. . . . . . . . . . . . 140

6.3 Near-field trapping model. . . . . . . . . . . . . . . . . . . . . . 142

PhD thesis: Far-field and near-field optical trapping xxii

LIST OF FIGURES

6.4 Trapping efficiency mapping for a small and a large polystyrene

particle of radius a, scanned in the X direction (light polarisation

direction) across the focused evanescent field generated by a plane

wave (top) and doughnut beam illumination (bottom). NA=1.65,

λ=532 nm, ε=0.85, n1=1.78 and n2=1.33. . . . . . . . . . . . . . 143

6.5 Trapping efficiency mapping for a small and a large polystyrene

particle of radius a, scanned in the Y direction (perpendicular to the

polarisation direction) across the focused evanescent field generated

by a plane wave (top) and doughnut beam illumination (bottom).

NA=1.65, λ=532 nm, ε=0.85, n1=1.78 and n2=1.33. . . . . . . . 144

6.6 Theoretical calculations of the maximal TTE of a polystyrene

particle of 1 µm in radius as a function of the obstruction size

ε. The other conditions are the same as in Fig. 6.4. . . . . . . . . 145

6.7 Theoretical calculations of the maximal ATE of a polystyrene

particle of 1 µm in radius as a function of the obstruction size

ε. The other conditions are the same as in Fig. 6.4. . . . . . . . . 146

6.8 Dependence of the ATE on the virtual focus position d for a small

and large polystyrene particle (ε = 0.85). The other conditions are

the same as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . 147

6.9 Phase introduced by the Fresnel transmission coefficients as a

function of the incident angle. The refractive index of the incident

medium is 1.78, while the refractive index of the transmitting

medium is 1.33. . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.10 The maximal ATE as a function of a polystyrene particle size (ε =

0.85). The inset shows a schematic relation between the interaction

cross-section area and the particle size. The other conditions are

the same as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . 148

PhD thesis: Far-field and near-field optical trapping xxiii

LIST OF FIGURES

6.11 The magnitudes of the axial force for a plane wave of power 10

µW and the gravity force for different particle sizes. The other

conditions are the same as in Fig. 6.4. . . . . . . . . . . . . . . . 149

6.12 The measured and calculated plane wave illumination maximal

TTE as a function of obstruction size with a NA-1.65 objective

for both S and P scanning directions. The other conditions are the

same as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.13 The measured and calculated doughnut beam illumination maximal

TTE as a function of obstruction size with a NA-1.65 objective for

both S and P scanning directions. The other conditions are the

same as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.1 Operational near-field trapping system. . . . . . . . . . . . . . . 161

7.2 Trapping efficiency mapping in the XY plane for a polystyrene

particle of radius a = 1 µm, (light polarisation is in the X direction)

across the focused evanescent field generated at a coverslip interface

by a plane wave (Charge 0) and doughnut beam illumination

(Charge 1, 2, and 3). NA=1.65, λ=532 nm, ε=0.85, n1=1.78 and

n2=1.33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

PhD thesis: Far-field and near-field optical trapping xxiv

List of Tables

5.1 The maximal TTE for plane wave and doughnut beam illumi-

nation. Ch0 denotes plane wave input, while Ch1 denotes a

doughnut beam of topological charge 1. exp. - experimentally

measured result, th. - theoretically calculated result. . . . . . 132

5.2 The maximal TTE for a centrally obstructed plane wave and

a doughnut beam illumination. Ch0+ε denotes a centrally

obstructed plane wave input, while Ch1 denotes a doughnut

beam of topological charge 1. exp. - experimentally measured

result, th. - theoretically calculated result. Obstruction size

ε = 0.78. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xxv

Chapter 1

Introduction

1.1 Introduction to optical trapping

Roots of an idea that light can be used to move matter date as far back

as 1609, the year when Johannes Kepler the father of the laws of planetary

motions, proposed an extraordinary sailing trip from the earth to the moon

on light itself [1]. This idea may belong to the realm of science fiction, but

Kepler himself came to it when he realised that the sunlight has an effect

on comets and that the ”sunlight pressure” turns the tails of comets away

from the sun. Today, nearly 400 years later, his ideas are becoming a reality

and matter is moved using light through optical trapping and manipulation

techniques, commonly known as optical tweezers.

Optical tweezers provide a tool for non-invasive manipulation of micro-

scopic matter. An optical tweezer system consists of a laser beam tightly

focused into a very small region, generating an extremely large electric field

gradient, using a microscope objective. When such a tightly focused laser

beam interacts with a mesoscopic particle, piconewton forces are exerted

on the particle and it is attracted toward the highest intensity region by

1

CHAPTER 1. Introduction

the so called gradient force, while the radiation force, also known as the

scattering force, acts in the direction of the light propagation, analogous to

the Kepler’s ”sunlight pressure”. Under the conditions when the gradient

force dominates, the particle with a refractive index larger than that of the

surrounding medium, is trapped in three-dimensions.

The origin of these forces can be perceived from Newton’s laws. A light

ray that is refracted through a dielectric particle changes its direction due

to the refraction process. Since light carries momentum, a change in light

direction implies that there must exist a force associated with that change.

The resulting force, manifested as a recoil action due to the momentum

redirection, draws mesoscopic particles toward the highest photon flux in the

focal region. This recoil is unnoticeable for refraction by macro objects such

as lenses, but it has a substantial and measurable influence on mesoscopic

refractive object such as small dielectric particles. An illustration of the

optical tweezers trapping principle is shown in Fig. 1.1.

Optical tweezers techniques are finding an increasing use in various

scientific disciplines such as a rapidly expanding field of single-molecule

biophysics [2–4], cell biology [5, 6], studying of molecular motors [7, 8],

manipulation of atoms and particles [9], and optical near-field imaging [10].

Optical tweezers are also useful purely as manipulators and positioning

devices, owing to its ability to confine, organise and assemble micro-objects.

An extensive resource paper on the versatility of use of optical tweezers

technology is presented in Ref. [11].

A new trapping modality based on a near-field trapping process [12–15]

is recently gaining momentum, due to its ability to reduce the trapping

volume and minimise the background noise affecting single-molecule dynamics

measurements. Furthermore, complex laser beams, such as Laguerre-

Gaussian laser beams, are finding its way into the novel optical tweezers

PhD thesis: Far-field and near-field optical trapping 2


Spherepusheslight to left

Lightpushessphere to right

Spherepusheslightdown

Lightpushessphereup

Intensity profile

Fig. 1.1: Principle of optical tweezers trapping [1].

technology [16,17], enabling additional controllability of the particle trapping.

A large number of these optical trapping techniques, such as particle-

trapped scanning near-field microscopy (SNOM) and optical-trap nanometry,

use an evanescent wave illumination of laser trapped particle or sample,

which increases the resolution of the measuring system [3, 10, 18, 19]. Higher

resolution derives from the scattered near-field signal [20,21], which otherwise

would not propagate and reach a detector. In order to study and fully

understand such trapping systems, an appropriate model for detection of

evanescent wave scattering is required.

Despite its versatility, theoretical treatment of optical tweezers has been

lagging behind the experimental work in this field. The main reason for this

lag is in complexity of the trapping process combined with a high numerical

aperture (NA) objective used for trapping. The tight focusing achieved by a

high NA objective precludes scalar diffraction theory to describe the trapping



process, and necessitates the full vectorial diffraction approach [22]. To date,

there are a limited number of methods for description of optical trapping

covering a certain range of particle sizes. Particles in the Mie regime (a λ),

where λ is the wavelength of the incident illumination, are successfully treated

using the ray optics approach [23]. The ray optics approach indicates that the

trapping force is independent of the particle size. For particles in the Rayleigh

size regime with radius a < 0.2λ electromagnetic method for trapping force

characterisation, based on the Gaussian beam approximation [24], is generally

used. The electromagnetic model, on the other hand, specifies that particle

size has an influence on the trapping force exerted on the particle. Any

general theory of optical trapping needs to bridge these two models and

extend its validity into both of these size regimes. Additionally, it requires

to be capable of treating complex laser fields used in novel and future laser

trapping systems. Such an optical trapping theory would be extremely useful

for optical trapping community, as it would enhance our understanding of

the trapping process under various types of novel laser beams and it would

pave the way for predictions of new physical phenomena associated with such

beams.

1.2 Thesis objectives

Popularity of the optical trapping technique in various research disciplines,

briefly mentioned in the previous section, and a lack of an appropriate

theoretical model to understand the trapping process by focused plane wave

illumination or complex laser beams prompt for a necessity of a proper

physical model for optical trapping. Such a model would find a great use in

characterising trapping force for complex laser beams under realistic trapping

conditions and would give an important contribution to the optical trapping

community, while not being constrained to a certain particle range.



The objective of our research is to develop such a physical model to

investigate optical trapping, achieved by the use of an arbitrary laser beam

focused by a high NA objective, in both far and near fields. The optical

trapping model needs to include two physical processes, vectorial diffraction

by a high NA objective and scattering by a small spherical particle to

characterise the resulting field on the particle surface. The particle and

the complex focused field interaction is described by an extension of the

classical plane wave Lorentz-Mie theory. The Maxwell stress tensor can then

be applied to determine the force exerted on the particle.

Such an optical trapping model would provide an exact electromagnetic

field distribution in the focal region of a high NA objective. Due to the

inherent nature of the vectorial diffraction process, an arbitrary wavefront,

incident at the entrance pupil of the high NA objective could be considered.

In other words, any apodisation function of the focusing objective could

be considered and its effects examined. Furthermore, an arbitrary input

phase modulation of the incident beam could be included and its trapping

properties studied, which would provide a tool for studying the trapping

efficiencies of Laguerre-Gaussian or doughnut beams for example. Effects

of spherical aberration (SA), nearly always present in trapping experiments,

could be considered using such a model by including vectorial diffraction of

focusing through an interface between two media with mismatched refractive

indices. Near-field trapping force exerted on a microscopic particle by a

focused evanescent field, recently experimentally demonstrated [15], could

be examined by considering a central obstruction placed perpendicularly to

the beam axis. The size of such an obstruction needs to be selected so that

it allows only the portion of the incident wave that satisfies the total internal

reflection condition at the surface of the coverslip to pass through the high NA

objective, thus generating a focused evanescent field in the particle suspension

medium.



In addition to such an optical trapping model, our objective is to develop

an appropriate nanometric sensing model for detection of evanescent wave

scattering, which would enhance our understanding of dependence of the

collected signal on particle morphology in optical trapping systems that use

evanescent wave illumination for high resolution position monitoring and

imaging. Most important of such applications being the particle-trapped

SNOM and optical-trap nanometry systems. Such a nanometric sensing

model would include near-field Mie scattering process to determine the three-

dimensional scattered field distribution in the vicinity of a plane interface

and vectorial diffraction of the scattered signal for its transformation to the

detection plane.

1.3 Preview of the thesis

The research work presented in this thesis deals with characterisation of

optical trapping in far and near fields, based on our vectorial diffraction

model. Furthermore, it is concerned with evanescent wave scattering, which

is essential in order to understand novel optical trapping systems for high

resolution nanometry and imaging.

To introduce a foundation on which the research presented in this thesis is

built, a review of optical trapping modalities and evanescent wave scattering

is given in Chapter 2. A distinction between the far-field and near-field

trapping modalities and its technologies is outlined as well as different

theoretical treatments associated with a particular technique. Two most

important applications of optical trapping technology, optical-trap nanometry

and particle-trapped SNOM are briefly reviewed. The theoretical treatments

available for far-field trapping are classified into two groups based on the

physical characteristics of its approach. So, we have a ray optics and



electromagnetic approaches for optical trapping evaluation reviewed. The

different theoretical treatments of near-field Mie scattering process, which is

fundamental for developing our nanometric sensing model are discussed. This

chapter also includes a review of vectorial diffraction, which is essential for

proper understanding of our optical trapping model. A particular attention

is paid to recent vectorial diffraction phenomena, linked to the depolarisation

effects, in focal and spectral splitting of focused waves. Near-field optical

trapping methods and its theoretical treatments are outlined on the basis of

its trapping technique.

Chapter 3 presents our nanometric sensing model based on three-

dimensional near-field Mie scattering process which includes the effects of

an interface at which an evanescent field is generated. This model provides a

basis for studying optical trapping applications such as the particle-trapped

SNOM and optical-trap nanometry for single molecule detection. Near-field

scattering properties of dielectric particles of various sizes, ranging from very

small to large, situated close and far from the interface are investigated

using our theoretical approach. Interface influence on morphology dependent

resonance effects associated with large dielectric particles are studied. A

physical model for conversion of evanescent photons into propagating photons

detectable by an imaging system is described, together with the theoretical

predictions and experimentally measured results with a wide area detector

such as a CCD camera. Detection of the near-field scattered signal using a

pinhole detector is also discussed.

Optical trapping model based on vectorial diffraction approach for

trapping force calculations with high NA objective is presented in Chapter 4.

This model forms a foundation for understanding and investigation of both

the far-field trapping and the near-field trapping with focused evanescent field

illumination. The theoretical approach, based on the extension of the classical



plane wave Lorentz-Mie theory for the particle and the complex focused field

interaction and the Maxwell stress tensor approach for force evaluation is

described. A comparison of the vectorial diffraction model and a fifth order

Gaussian beam approximation is also given.

Far-field optical trapping is investigated in Chapter 5 using our optical

trapping model. Trapping efficiency of a focused plane wave is studied, while

a particular attention is paid to the trapping efficiency for small and large

dielectric particles and the effect of spherical aberration. Numerical and

experimental results for far-field trapping efficiencies achieved with centrally

obstructed laser beams are presented. Trapping efficiencies of complex

phase modulated doughnut beams are researched both theoretically and

experimentally. Furthermore, efficient methods for generation of doughnut

beams for experimental purposes are discussed, as well as the vectorial

diffraction effects associated with focusing of doughnut beams with a high

NA objective.

Near-field trapping, achieved with evanescent field generated under total

internal reflection condition, is studied in Chapter 6. Optical trapping with

a wide area evanescent field is discussed. The theoretical results of the near-

field optical trapping achieved with focused evanescent wave illumination are

presented, while the experimentally measured results and their comparison

with the theoretical results are also given. Both plane wave and doughnut

beam incident illuminations are considered.

Chapter 7 gives the conclusions drawn from the research work undertaken

in this thesis and includes a discussion of future work in this field.


Chapter 2

Literature review

2.1 Introduction

Ever since researchers realised that light can move matter because photons

carry momentum, optical micromanipulation and assembly at the microscopic

scale was the ultimate goal. Back in 1970’s first optical traps were built

by Arthur Ashkin at AT&T Bell Labs in the US. His ”Levitation traps”

were based on the upward-pointing radiation pressure from a photon stream

to counter-affect the gravitational pulling force. Then, in 1986, Ashkin

and colleagues realised that a single tightly focused laser beam generates

a sufficient gradient force to trap small particles in three dimensions [25].

Thus a first ”optical tweezer” was born. Trapping forces in optical tweezers

arise from the optical momentum transfer to a transparent particle with a

refractive index slightly greater than that of the surrounding medium. The

net radiation force is directed towards the highest intensity region near the

beam focus.

Most common optical tweezers setups include a high numerical aperture

(NA) objective to tightly focus a laser beam. Such an optical trapping

9

CHAPTER 2. Literature review

modality is known as the far-field trapping scheme, because the trapping

is performed by the propagating far-field component of a focused laser

beam. Recently, a new trapping mechanism that utilises the evanescent wave

illumination, also called near-field illumination, has been proposed [12–15]

and demonstrated [15]. In this near-field trapping modality, trapping

is performed by the non-propagating evanescent field. In either of the

two trapping modalities, the trapping process and the forces involved are

described by the scattering of the incident illumination by a microscopic

particle.

This chapter is a review of available methods to physically model and

understand the far-field and the near-field trapping methods, as well as

the vectorial diffraction necessary for a proper modelling of either trapping

modality. The chapter is organised as follows. Section 2.2 looks at the far-field

trapping technique and its applications in microcopy, as well as the models

available to provide a physical insight into this trapping process. Different

models for optical trapping force calculations are reviewed in Section 2.3. In

Section 2.4, physical models for description of the near-field Mie scattering

are reviewed. Vectorial diffraction is examined in Section 2.5, while the last

Section 2.6 discusses the near-field trapping techniques.

2.2 Far-field trapping

Optical far-field traps have found a wide use in many disciplines including

physics, chemistry and biological sciences [26]. Experiments have been

performed in the far-field trapping of bacteria and viruses [27]; yeast, blood

and plant cells, protozoa and various algae [26]; and internal cell surgery [28].

Optical techniques have also been used for cell sorting [29].

Recently, spatial phase modulators [30, 31], and computer-generated



holograms [32] are widely used for generating complex laser beams, such

as Laguerre-Gaussian beams, for novel laser trapping experiments [16, 31,

32]. Focusing of such beams with a high NA objective required for

efficient trapping leads to a complicated amplitude, phase and polarisation

distributions of an electromagnetic field in the focal region [33]. Interaction

of such a field with a micro-particle results in the controllable laser trapping

technique [16,17].

Beside these mainstream scientific fields, optical far-field trapping is

gaining momentum in the field of microscopy and particularly in the area

of the optical-trap nanometry [3] and near-field imaging (NFI) based on the

scanning near-field optical microscopy (SNOM) technique [34]. Applying

SNOM techniques researchers have achieved optical superresolution in the

range of 1-10 nm [35,36]. SNOM technique that utilises a trapped particle as

a near-field scatterer is termed particle-trapped SNOM [10,18,19].

2.2.1 Optical-trap nanometry

Optical-trap nanometry is a technique to measure nanometric movements of

biomolecules via optical means. An ordinary optical-trap nanometry system

consists of a one or two far-field laser traps with stably trapped microscopic

particles (Fig. 2.1). A biomolecule is attached to the trapped particles and

the whole system is usually illuminated by an evanescent field (near-field)

generated by a total internal reflection (TIR) [3]. One of the trapped particles

is used as an anchor to fix the one end of the molecule, while the other is

sensing its movements. The movements of the biomolecule are monitored by

observing the scattered evanescent field signal. The monitoring is usually

performed by a quadrant photodiode capable of determining particle position

in three dimensions. The quadrant photodiode is calibrated experimentally,

however it would be beneficial for the reliable high resolution position



Laser

Glass slide

Myosin filament

Cy3-ATP Evanescent field Bead

Optical trap

Actin filament

Fluorescenceof Cy3-ATP

Nanometry

Fig. 2.1: Optical trap nanometry for mechanical measurements of myosin [3].

determination to be able to calibrate the diode based on a theoretical model

for the scattered evanescent field. Such a theoretical model would need to

incorporate the near-field Mie scattering and the vectorial diffraction of the

scattered field by a high NA lens. Both of these research areas are reviewed in

sections 2.4 and 2.5 in this chapter, and will be addressed in more details in

Chapters 3 and 4. On the other hand a reliable far-field trapping model would

provide a physical insight into the trapping performance and lead to the most

stable trap designs, which is addressed in details in Chapters 4 and 5.

Using this technique, researchers have achieved movement resolutions of

a kinesin molecule of sub-8 nm [37] and even smaller movements of a single

myosin molecule [38]. These simple movement measurements can further

be interpolated to study the complex mechanics of biomolecules [39]. More

intricate molecular dynamics, such as the DNA transcription processes by

RNA polymerase, the rotary motion of molecular motors or the enzymatic

reactions, have been also investigated using the optical-trap systems [3].


HeWolff

Rectangle

HeWolff

Text Box

Image not available - see printed version


2.2.2 Particle-trapped SNOM

Optical near-field imaging has become a rich research field in recent years

due to its ability to achieve optical resolution well below the classical

diffraction limit of approximately half the wavelength of illuminating light.

The principle of this imaging technique is to probe the near field of an object

under investigation, and thereby extract high-resolution information about

the object which otherwise does not reach a far-field detector. The most

widely applied near-field imaging (NFI) technique is SNOM [34]. The SNOM

technique is utilised in various high-resolution applications such as single-

particle plasmons observations [40], single molecule detection experiments [41,

42], light confinement studies [43] and trapping and manipulation of nano-

scale objects [13,14].

Near-field imaging, implemented using SNOM techniques, can generally

be classified into three categories: scanning aperture type [44,45], frustrated

total internal refection (FTIR) illumination with fibre tip collection [46] and

TIR with scattering collection [47]. The scanning aperture type NFI utilises

a subwavelength-diameter aperture as a localised evanescent field source.

The aperture is scanned along a sample in a close proximity, typically a

few nanometers, and the transmitted signal is collected by a conventional

optical system. The theoretical treatment of image formation of this type of

NFI has been well dealt with [48]. NFI implemented with FTIR illumination

and fibre tip collection probes a sample modulated by a localised evanescent

field and collects evanescent photons through the photon tunnelling process.

The physical model for describing the signal collection with this type of

NFI is analogous to the electron tunnelling process in scanning tunnelling

microscopy [49]. The third NFI category employs TIR illumination to

generate evanescent field. The field is probed using a small scatterer, such as

a microscopic metallic or dielectric particle or a metallic needle, to convert



evanescent photons into propagating photons [18,19].

Optical imaging with laser trapped particle illuminated by a near-field

generated by a TIR offers higher resolution performance compared with the

classical imaging techniques. A laser trapped particle is used to probe the

near field of an object under investigation [10]. The sample image is built

by scanning the sample in two dimensions and collecting the signal scattered

by the trapped particle. Compared with a conventional SNOM, particle-

trapped SNOM enables non-invasive access to the sample, the axial pushing

force acting on the trapped particle eliminates the requirement of distance

control, and the probe deterioration problems are avoided since plenty of

particles are available for replacement.

Figure 2.2 shows a typical experimental setup used for particle-trapped

SNOM imaging. A particle is trapped with a laser beam focused by a high

NA microscope objective. The particle is laterally pulled toward the optical

axis of the focused beam by the transverse trapping force. At the same time

it is vertically pushed down or pulled up, depending on the focus position, by

the axial trapping force.

In addition to the trapping laser beam, another laser beam is incident

at the interface between the sample and its substrate satisfying the TIR

condition. Thus generated evanescent field illuminates the sample and is

scattered by the trapped particle. The scattered signal is collected by the

same high NA microscope objective and imaged onto a detector, which

is usually a photon-multiplier-tube (PMT) placed in the far-field region.

The image of the sample is reconstructed by scanning the particle in two

dimensions across the sample [10].

In relation to the other types of SNOM, the particle-trapped SNOM is

quite unique in that it relies on collecting evanescent waves scattered by a



CCD

PMT

FilterPinhole

Beam splitter

Beam splitter

Trapping laser

High NA objective

SampleTrapped particle

Prism Illumination beamunder TIR

Scanning

Fig. 2.2: Schematic diagram of particle-trapped SNOM setup.

trapped particle for image formation [10,18,19]. The strength of the scattered

evanescent waves is dependent on the scattering efficiency of the trapped

particle and the illumination intensity [50]. The evanescent wave strength can

be further optimised by coating the substrate surface at which an evanescent

field is generated by a thin film stack [51]. The speed of image acquisition

and image resolution are determined by the laser trapping performance

and stability, i.e. trapping efficiency [52]. All of these issues involving

evanescent wave scattering, vectorial diffraction and trapping performance

are investigated in more details in Chapters 3, 4, and 5.



2.3 Trapping force calculation

An accurate knowledge of the trapping force acting on a particle is essential for

describing and improving the trapping performance. This force depends on

many parameters of which most important ones are the particle morphology,

characteristics of the suspension medium, and the trapping illumination

distribution in the focal region. In this section, the theoretical approaches

for the calculation of trapping force on a dielectric particle are reviewed.

These approaches can be broadly classified into the ray optics (RO) approach

and the electromagnetic (EM) approach. Both of these models provide a

reasonable guidance for the evaluation of the far-field trapping force on small

spherical particles in their own region of applicability, except for the range

of λ-10λ, where λ is the wavelength of a trapping beam [53]. In Chapter 4

we present our model based on vectorial diffraction by a high NA lens for

trapping force evaluation, which is an exact model that does not suffer from

these limitations.

2.3.1 Ray optics method

In the RO method, applicable for trapping force evaluation of large particles,

a trapping laser beam focused by a high NA lens is simply decomposed into

individual rays and the ray density on the focusing lens is assumed to be

the same as that of the power density. In this method, the wave nature of a

trapping beam cannot be dealt with at all. The far-field optical trap action

on a dielectric particle, according to the RO model, can be described in terms

of the total force arising from a typical pair of rays a and b of a converging

beam [23]. The direction of the total force is dependent on the geometrical

focus position (Fig. 2.3). In this model the forces due to the refraction (Fa

and Fb) are pointing in the direction of the momentum change.



Laser beam

Microscope objective

Particle

O

FFaFb

Particle

O

F

Fa Fb

Particle

O

FFa

Fb

ab

a

a

b

b

b

b

a

a

a

b

(a) (b) (c)

Fig. 2.3: Qualitative description of the trapping of dielectric spheres according toAshkin’s RO model [23]. The total force F, given by the refraction of a typical pairof rays a and b of the trapping beam, is always restoring for axial and transversedisplacements of the sphere from the trap focus.

A single ray of power P incident on a dielectric particle gives a rise to a

series of reflected and refracted rays. As a result, the particle experiences a

force due to the net change in momentum. This force can be resolved into

the scattering force Fs (parallel to the incident ray) and the gradient force Fg

(perpendicular to the incident ray) (Fig. 2.4(a)). The gradient and scattering

forces can be mathematically expressed as [23]

Fg =nP

c

(

1 +R cos 2θ − T 2[cos(2θ − 2θ′) +R cos 2θ]

1 +R2 + 2R cos 2θ′

)

,

Fs =nP

c

(

R sin 2θ − T 2[sin(2θ − 2θ′) +R sin 2θ]

1 +R2 + 2R cos 2θ′

)

, (2.1)

where R and T are the Fresnel reflectance and transmittance at the incident

angle θ [54], n is the refractive index of the particle relative to the surrounding



0 10 20 30 40 50 60 70 80 90

q (DEGREES)

Q

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Qs

Qg

Qtotal

O

Z

f

ray

rmax

beam axis

n

v

q

f

ray

R=1

Fs

Fg

(a)

(b)

Fig. 2.4: (a) Gradient (Fg) and scattering (Fs) forces for an incident ray. (b) Thetotal (Qtotal), scattering (Qs) and gradient (Qg) trapping efficiencies for a singleray hitting a dielectric sphere of relative index of refraction 1.2 at an angle θ.

medium, c is the speed of light in vacuum and θ′ is the refractive angle of a

single ray incident on the particle.

If the light intensity distribution over the aperture of an objective lens

is denoted I(ρ), then the total trapping force on a dielctric particle can be

expressed as

Ft = FG + FS =

∫ 2π

0

∫ ρmax

0Fg · I(ρ) · ρdρdϕ

∫ 2π

0

∫ ρmax

0I(ρ) · ρdρdϕ

+

∫ 2π

0

∫ ρmax

0Fs · I(ρ) · ρdρdϕ

∫ 2π

0

∫ ρmax

0I(ρ) · ρdρdϕ

(2.2)

where ρ is the radial position over the trapping objective, ρmax is the

maximum radius of the objective aperture, ϕ is the azimuthal angle, Fg and

Fs are the vectorial gradient and scattering forces of a single ray, and FG and

FS are the vectorial gradient and scattering forces on a particle respectively.

The trapping efficiency Qj, a parameter independent of trapping power P



for the evaluation of trapping force Fj, is defined as

Qj =Fjc

nP, j = g, s. (2.3)

Here Qg and Qs are termed the gradient and scattering efficiency of a single

ray respectively.

Figure 2.4(b) shows the values of the scattering efficiency Qs, gradient

efficiency Qg and the magnitude of the total trapping efficiency Qtotal for a

ray incident at an angle θ on a dielectric sphere of a relative refractive index

of n = 1.2. One can see that the maximum gradient trapping efficiency as

high as 0.5 is generated for rays at angles of θ ∼= 70.

The RO model can be further extended to include Mie metallic particles,

and determine the trapping force exerted on metallic particles. This approach

is termed a modified RO model [55]. For metallic particles the energy density

of the transmitted light falls to 1/e of the original value after the light travels

through a skin depth that is usually of several or tens of nanometers. The

optical force exerted on a metallic Mie particle is determined mainly by the

reflection at the surface of the particle. Thus the optical force, caused by the

multiple reflection on the inner surface of the particle, can be neglected for a

metallic Mie particle. As a result the gradient and scattering forces due to a

single ray incident on an Mie metallic particle at an angle θ can be expressed

as [55]

Fg =nPR

csin 2θ, Fs =

nP

c(1 +R cos 2θ) . (2.4)

A variation of the RO model for the evaluation of the axial force exerted

on a dielectric particle by the optical pressure of a focused Gaussian beam

is given in Ref. [56]. The axial force is obtained by numerical integration of



the local force on the spherical surface with respect to the angle around the

center of the sphere. This approach does not require complicated equations

and one can easily calculate the axial force. The drawback, obviously, is that

this approach cannot give an indication of the transverse force acting on the

particle.

Other theoretical treatments for trapping force calculation that employ

the geometrical approximation as equivalent to the large-argument approxi-

mation in the Riccati-Bessel functions were published at around the same time

or little earlier as the Ashkin’s approach [23]. These theoretical treatments are

valid for large particles, with 2πa/λ ≥ 100, where a is the particle radius and

λ is the illumination wavelength. The theory for the interaction of the laser

beam and a dielectric particle was first derived by Roosen and co-workers [57],

while the extension of the theory to the tightly focused laser beams is given

by Gussgard et al. [58]. However, even though these treatments are fairly

complex, they lack the elegance and power of the Ashkin’s model [23].

2.3.2 Electromagnetic method

The EM field model for evaluation of the trapping force exerted on a

microscopic particle is based on the wave optics approach, i.e. based on

the wave nature of the incident illumination. In the EM method, the field

expansion coefficients are derived for an infinite representation of the electric

and magnetic fields external to the spherical particle. These expansion

coefficients describe the incident and scattered fields, which then can be used

to determine the force exerted on the particle using the Maxwell’s stress tensor

as

〈FN〉 = 〈∮

S

n · ←→T dS〉 , (2.5)



where 〈〉 represents a time average, FN is the net radiation force, n is an

outward normal unit vector,←→T is Maxwell’s stress tensor, and S is the surface

area of the particle. However, in order to correctly determine the trapping

force, the EM model requires an accurate expression for the radial component

of an incident field [24]. This method indicates that the magnitude of trapping

force is particle-size dependent and it is applicable for small particles (∼ λ

order). For the steady state optical condition,the Maxwell’s stress tensor in

the traditional Minkowski form is given as [24]

←→T =

1

4π(ε0EE + HH− 1

2(ε0E

2 +H2)←→I ) , (2.6)

where E and H are the electric and magnetic fields at the surface of the

particle, while ε0 is the permittivity of the particle and←→I is a unit tensor.

The trapping force can be further expressed as

〈FN〉 =1

4π

∫ 2π

0

∫ π

0

⟨(

εextErE+HrH−1

2(εextE

2+H2)r

)⟩

r2 sin θdθdφ

∣

∣

∣

∣

∣

r>a

,

(2.7)

where εext is the permittivity of the surrounding medium, Er and Hr are

the radial field components, r is the outward radial vector, a is the particle

radius, and r, θ and φ are spherical coordinates. In this method an accurate

knowledge of the EM field distribution in the focal region is of crucial

importance.

A simple Gaussian model (Fig. 2.5) for the incident field description is

not adequate due to its inability to represent a highly focused laser beam

required for efficient trapping. Therefore, researchers have developed fifth

order corrections [59] for the incident field components of a Gaussian laser

beam to deal with highly focused laser beams. However, even a fifth-order

Gaussian beam ignores the effect of diffraction by a high NA objective and

thus does not correctly represent the EM phase and polarisation distributions



FN

Intensity profileIntensity profile

Fig. 2.5: Gaussian optical trapping model.

near the focus of a high NA objective. In addition, the fifth order Gaussian

beam method is not able to model the effect of spherical aberration (SA)

usually presents during laser trapping experiments due to the refractive index

mismatch between the immersion oil (or the coverslip) and aqueous solution

in which particles are situated. It is one of the aims of this thesis to deal with

trapping force under focused illumination by a high NA objective (Chapter 4)

in the far-field region (Chapter 5) and in the near-field region (Chapter 6).

Recently, Rohrbach et al. [60,61] have developed a method for calculation

of trapping forces of an EM wave on an arbitrary-shaped dielectric particle,

based on the extension of Rayleigh-Debye theory to include second-order

scattering, which considers a stronger interaction between the incident field

and the particle. This method does not use the Maxwell stress tensor to

determine the force exerted on a particle, but splits the total optical force into

two components. These components are commonly known as the scattering

force, which is due to the momentum transfer of photons, and the gradient

force, which draws dipoles toward the highest amplitude of an EM field. One

can obtain the total EM force F(r) by integrating over all volume elements



dV ′ inside the scatterer:

F(r) = Fgrad(r) + Fsca(r) =

∫ ∫ ∫

V

αnm

2c∇Im(r′)dV ′ +

nm

cImCscag/kn ,

(2.8)

where α is the polarisability, nm is the refractive index of the medium in which

the particle is suspended, Im is the intensity distribution in the suspension

media, Csca is the scattering cross section and g is the transfer vector.

The gradient force Fgrad is determined by computing the gradient in x, y,

and z of the intensity distribution Im(x, y, z) for each volume element and

by summing all the gradient values. This is performed separately for the

intensities generated by the x and z components of the electric field. The

scattering force Fsca is obtained by the change of momentum that is due

to the scattering of the incident wave at the particle. The power of the

scattered light, Psca = ImCsca is defined by the intensity in the focus of the

incident beam Im = ε0c|Em(0, 0, 0)|2, while the scattering cross-section Csca is

calculated by the intensity scattered in the positive half-space and in negative

half-space.

However, the two components method [60,61] is limited to the case when

the maximum phase shift k0(ns − nm)2a produced by the particle of radius

a, is smaller than π/3. This is valid, for example, for a polystyrene particle

(ns = 1.57) in water illuminated by highly focused laser (λ0 = 1.064 µm) of

radius a = 370 nm or less, but for larger particles it would be inaccurate.

Generalised Lorenz-Mie theory (GLMT) is another approach that can be

used to determine the trapping force in the presence of spherical aberration,

but to date it has only been used to determine the on-axis trapping force [62,

63]. The GLMT formula for the on-axis trapping force is expressed as an

infinite series of partial wave contributions Σ as

Fz = (nm/c)(nmE20/µ0c)(π/n

2mk

2)Σ . (2.9)



E0 denotes the field strength and k is the free-space wave number. The

incident illumination with this GLMT method is a tightly focused, truncated

and aberrated Gaussian beam and a plane wave [62]. The GLMT model is

used to calculate the on-axis trapping force on a spherical particle whose size

can range from the Rayleigh scattering limit to the ray theory limit [63].

2.4 Near-field Mie scattering

Mie theory and its later extensions and modifications are extensively used

to determine the size, shape, and orientation of small particles in vacuum

or in gaseous, liquid, or solid media, since its first formulation by Gustav

Mie [64] in 1908 and independently in 1909 by Debye [65]. Researchers from a

large variety of disciplines such as physics, electrical engineering, meteorology,

chemistry, biophysics, and astronomy are interested and concerned with this

theory. The basic formulation of the Mie scattering of evanescent waves, i.e.

near-field is an analytical continuation of the standard case of plane-wave

excitation.

As mentioned earlier in this chapter, the thorough understanding and an

appropriate physical model of near-field Mie scattering is required for proper

investigation of a SNOM technique which uses a laser trapped particle as a

near-field probe. Such a physical understanding is also needed in optical trap

nanometry for single molecule detection, in which case a single molecule is

attached to a laser trapped microscopic particle immersed into evanescent

field, and is monitored by measuring the scattered field [3].

While the evanescent photon conversion mechanism induced by other

SNOM techniques has been studied [48, 49, 66], the underlying physical

principle of the mechanism for the evanescent photon conversion by a

microscopic particle probe has not been dealt with. Furthermore, such



physical model can be used to investigate the near-field force exerted on small

particles situated in the evanescent field [67–70]. Near-field Mie scattering

has been researched using a variety of theoretical methods, most popular of

which are the geometric optics method [71], generalised Mie theory [72–75],

and the discrete sources method [76,77]. A numerical model based on multiple

multipole (MMP) method [78] is generally applied for comparison of the

theoretically calculated results.

2.4.1 Geometric optics method

This method [71] assumes that the evanescent field is generated by a TIR of an

incident beam at an interface between two media. A particle is situated close

to the interface so that the evanescent wave propagating along the interface

is scattered (Fig. 2.6(a)). The intensity of the scattered light is a function of

the particle-interface separation due to the exponential nature of evanescent

illumination. Geometric or ray optics is an asymptotic solution to the light-

scattering problem, which is valid in the case when particle size is much

larger than the wavelength of incident illumination. Although not as rigorous

as proper Mie theory, geometric optics is mathematically simpler. This

technique can provide good approximations to the total scattering solution.

In geometric optics the incident EM wave is considered as a set of parallel

rays each of which can follow an independent path. When a given ray strikes

the particle it is divided into a reflected and a transmitted ray. The direction

of these rays is determined by the Snell’s law, while the amplitude and phase

are given by the Fresnel coefficients. This process is repeated when these rays

strike another surface, such as the particle or a substrate (Fig. 2.6(b)). The

objective of this method is to track the propagation path and intensity of each

ray as it travels within the particle or between the particle and the substrate.

It is necessary to follow only a small number of ray reflections due to reduction



Glass

y

q

Water

ParticleEvanescentField

IncidentField

Incident ray ofevanescent wavew=2, p=0

w=0, p=1

w=1, p=1

w=1, p=2

Glass

(a) (b)

Fig. 2.6: (a) Geometrical optics model of scattering of evanescent waves. (b)Typical ray paths through a dielectric particle. The integers p and w denote thenumber of contacts made with the inner surface of the dielectric particle and thesubstrate, respectively [71].

of energy of a reflected of transmitted ray compared to the incident energy.

Integers w denotes the number of times a ray contacts the substrate surface,

while integer p denotes the number of internal contacts inside the sphere

(Fig. 2.6(b)). The total path of any ray can be calculated if the azimuthal

angle φinc, a contact angle τ , and integers w and p are given. Given these

variables, the scattering direction and all necessary optical calculations can

be performed on this ray. Following this methodology the amplitude Esca of

any scattered ray can be determined from [71]

Esca = EincCF

√

cos τ sin τdτdφinc

sin θscadθscadφsca

exp(iσt) . (2.10)

Here Einc is the amplitude of the incident ray striking the particle, CF denotes

the cumulative Fresnel coefficient for the scattered ray, θsca is a scattering

angle, while σt gives the total phase shift. The total phase shift constitutes of



the phase change at reflection, the phase change caused by a different optical

path length and the phase change caused by the crossing of focal lines.

2.4.2 Generalised Mie theory

As early as 1979 Chew et al. [72] started to discuss elastic scattering of

evanescent waves by spherical particles. Their theory was recently slightly

corrected by Liu et al. [79]. Chew et al. [72] outlined the theory for scattering

of evanescent waves by spherical particles, which essentially consists in the

analytical continuation of the case of plane-wave excitation to the complex

angles of incidence. However, these authors were only interested in calculating

the differential scattering cross sections of relatively large dielectric particles.

Furthermore, their calculations involve an approximation, which is valid only

for two principal planes at a large distance from the scattering particle.

According to their approach the scattered field for an incident wave polarised

perpendicular to the plane of incidence is given by [72]

Esc(r) =∑

lm

icβE(l,m)

n′2ω∇×[h

(1)l (k′r)Yllm(r)]+βM(l,m)h

(1)l (k′r)Yllm(r)

large r −→ exp(ik′r)

k′r

∑

lm

(−i)l−1

βE(l,m)

n′r×Yllm(r)−βM(l,m)Yllm(r)

,

(2.11)

where l = 1 to∞ and m = -l to +l. n′ is the index of refraction of the medium

in which the particle is situated, k′ is the wave number in the same medium,

c is the speed of light in vacuum and ω denotes the angular frequency of the

incident light. The function Yllm is the vector spherical harmonics, while

h(1)l (k′r) is the spherical Hankel function of the first kind and the functions

βE(l,m) and βM(l,m) are the expansion coefficients of the scattered fields



given by

βE(l,m) =ε′jl(k′a)[k1ajl(k1a)]

′ − ε1jl(k1a)[k′ajl(k

′a)]′ exp(−βd)αE(l,m)

ε1jl(k1a)[k′ah(1)l (k′a)]′ − ε′h(1)

l (k′a)[k1ajl(k1a)]′,

(2.12)

and

βM(l,m) =µ1jl(k1a)[k

′ajl(k′a)]′ − µ′jl(k

′a)[k1ajl(k1a)]′ exp(−βd)αM(l,m)

µ′h(1)l (k′a)[k1ajl(k1a)]′ − µ1jl(k1a)[k′ah

(1)l (k′a)]′

,

(2.13)

where ε′, µ′, k′ and ε1, µ1, k1 denote the dielectric constant, magnetic perme-

ability, and the wave number in the medium surrounding the sphere and inside

the sphere respectively. jl is the spherical Bessel function of the l-th order, a

is the radius of the sphere, d is the distance of the centre of the sphere to the

interface at which evanescent wave is generated, β = k′(n2 sin2 α/n′2 − 1)1/2,

where α is the angle of incidence at the interface. αE(l,m) and αM(l,m) are

the expansion coefficients of the incident fields given by

αE(l,m) = 2miln′

[

π(2l + 1)(l −m)!

l(l + 1)(l +m)!

]1/2

[Pml (cos θk′)/ sin θk′ ]E ′

0 , (2.14)

and

αM(l,m) = −2il+1

[

π(2l + 1)(l −m)!

l(l + 1)(l +m)!

]1/2[

d

dθk′

Pml (cos θk′)

]

E ′

0 , (2.15)

where Pml are Legendre functions, E ′

0 = 2/(1 + µ tanα/µ′ tanα′) is the

amplitude of the refracted wave, while the angle θk′ is related to the angle of

refraction α′ through sin θk′ = cosα′ and cos θk′ = sinα′.

In the case of the polarisation state parallel to the plane of incidence, the

scattered electric field is also given by Eq. 2.11 with expansion coefficients

βE(l,m) and βM(l,m) substituted by βE(l,m) and βM(l,m), respectively [72].

These scattered fields expansion coefficients are given by Eqs. 2.12 and 2.13



with αE,M(l,m) replaced by αE,M(l,m) given by

αE(l,m) = 2il−1n′

[

π(2l + 1)(l −m)!

l(l + 1)(l +m)!

]1/2[

d

dθk′

Pml (cos θk′)

]

E ′

0 , (2.16)

and

αM(l,m) = −2mil

[

π(2l + 1)(l −m)!

l(l + 1)(l +m)!

]1/2

[Pml (cos θk′)/ sin θk′ ]E ′

0 , (2.17)

with E ′

0 = 2 sinα′ cosα/[sinα′ cosα′ + (µ/µ′) sinα cosα]. This method forms

the basis of the investigations of near-field Mie scattering described in this

thesis and is evaluated in more details in Chapter 3.

Quinten et al. [73–75] have extended the Chew et al. [72] method to

calculate the total cross sections for evanescent-wave excitation and discuss

their dependence on wavelength, angle of incidence, and particle sizes. In their

work they have not used the complex conjugates of the associated Legendre

polynomials but their actual values. The necessity of this correction was

already pointed out by Liu et al. [79]. Quinten et al. [73] have obtained

the total cross sections for extinction and scattering of evanescent waves by

applying Poynting’s law for the absorbed power density in the stationary case.

These cross sections for S-polarised light are given as

σsext =

2π

k′2N−1Re

∞∑

l=1

(2l + 1)(alΠl + blTl) ,

σssca =

2π

k′2N−1

∞∑

l=1

(2l + 1)(|al|2Πl + |bl|2Tl) , (2.18)



and for P-polarised light as

σpext =

2π

k′2N−1Re

∞∑

l=1

(2l + 1)(alTl + blΠl) ,

σpsca =

2π

k′2N−1

∞∑

l=1

(2l + 1)(|al|2Tl + |bl|2Πl) . (2.19)

al are absolute values of the Mie coefficients for electric multipoles, while bl

denote the absolute values of Mie coefficients for magnetic multipoles which

are defined in [74]. The normalisation factorN is equal to one for plane waves,

but assumes different values for evanescent waves as discussed in Ref. [73].

The definitions of Πl and Tl are

Πl(θk) =2

n(n+ 1)

l∑

m=−l

(l −m)!

(l +m)!

∣

∣

∣

∣

∣

mPlm(cos θk)

sin θk

∣

∣

∣

∣

∣

2

,

Tl(θk) =2

n(n+ 1)

l∑

m=−l

(l −m)!

(l +m)!

∣

∣

∣

∣

∣

dPlm(cos θk)

dθk

∣

∣

∣

∣

∣

2

. (2.20)

θk represents the complex refraction angle of the incident wave and Plm are

the associated Legendre polynomials. Eqs. 2.18 and 2.19 are valid not only

for homogeneous, but also for coated spherical particles with an arbitrary

number of layers. Only the expressions for al and bl differ in these cases.

In the case of an incident plane wave, it can be proved that Πl = Tl = 1

for all multipolar orders l and angels θk with cos θk ≤ 1. In this case the

cross-sections do not differ for S and P polarisation and are the well known

results from the standard Mie theory.

Using this method Quinten et al. [73–75] have calculated scattering

and extinction of evanescent waves by small metallic, dielectric and coated

particles. They have found that the scattering and extinction results

differ strongly for evanescent and plane-wave excitation, with much stronger

morphology dependent resonances (MDR) in the former case. The special



properties of the evanescent wave lead to increased contributions of multipolar

orders (l,m) with l > 1, which in turn leads to increased contributions of

orders l > 1 in the expansion of scattered wave. This results in polarisation-

dependent cross sections, which are much larger for P-polarised light than for

S-polarisation. Close proximity of the prism surface at which evanescent field

is generated reduce the resonant cross sections significantly. In addition, a

broadening and a redshift occur, when the particle is in contact with the prism

surface. However, the effect of multiple reflections caused by the interface has

not been dealt with. We will present a physical model to investigate the effect

of interface reflections in Chapter 3.

2.4.3 Discrete sources method

Geometry of the scattering problem considered with the discrete sources

method (DSM) is depicted in Fig. 2.7 [77]. An axisymmetric particle with

smooth boundary S and interior domain Di is situated on a plane surface. Its

symmetry axis coincides with the normal to the plane surface. The ambient

domain half space is denoted by D0, while the bottom half space consisting of

a glass prism is D1. The basis of discrete source method is in approximately

representing the electromagnetic fields as a finite linear combination of fields

of multipoles [76, 77]. Their treatment of evanescent wave scattering is an

extension of their earlier work on plane wave scattering by a small particle

near a surface [80]. Maxwell equations are satisfied in this approximation

in domains D0, D1 and Di, the radiation condition is satisfied in D0 and

D1, and transmission condition at the plane interface. Thus essentially, the

scattering problem is simplified to the approximation problem of the external

excitation on the particle surface. In this section we will not go into details

of the mathematical treatment of the DSM but instead will summarise the

main points of this technique:



Surface

q

qk

Z

X

D0

D1

Di

O

S

Fig. 2.7: Geometry of the scattering system in discrete sources method.

• The scattered field representation is based on Green’s tensor for the half

space. As a consequence the vector potentials of the multipole fields are

expressed as Weyl-Sommerfield integrals providing the continuity of the

tangential component of the scattered fields at the plane interface [81].

• Vector potentials of regular multipole fields are used for the internal

field representation.

• The multipoles are distributed along the axis od symmetry inside

the domain Di or in the image domain of the complex plane. The

approximate solution is represented as a linear combination of Fourier

harmonics with respect to the azimuthal angle. Thus the computational

effort is significantly reduced by simplifying the approximation problem

of the external excitation to a problem of a sequence of one dimensional

approximation along a scatterer generator.

• The convergence of the approximate solution to the exact solution in

closed domain D0 is guaranteed by the completeness of the system of



distributed multipoles [82].

The transmission condition at the particle surface is used to determine the

amplitudes of discrete sources. After the amplitudes of discrete sources have

been determined, one can calculate the far-field pattern of the scattered field

by using the asymptotic representation for the Sommerfield integrals [82].

Using DSM, Doicu et al. [77] have investigated evanescent wave differential

scattering cross sections for dielectric and metallic spherical particles. They

compare their results with the results of Liu et al. [79] which does not

include the interface effects and conclude that the effects of the interface

are significant, in particular for particles with high index of refraction.

These effects need to be taken into consideration to adequately describe the

scattering of evanescent waves by small particles near an interface.

2.5 Vectorial diffraction of light

Evaluation of trapping force on a small particle illuminated by a focused laser

beam depends, as we have pointed out in Section 2.2, on an accurate modelling

of the field distribution in the focused region. An exact model for the trapping

force calculation needs to be able to determine the field distribution of the

focused laser beam by a high NA objective precisely. Such an exact model

needs to include the vectorial diffraction of light by a high NA objective,

determining the focal distribution which interacts with a particle. As one

of the aims of this thesis, the vectorial diffraction model for laser trapping

is presented in greater details in Chapter 4. In addition to the particle

trapping discipline, focusing of light by a high NA system is attracting an ever-

increasing interest because of increasing demand of microscopy applications

in biological and material sciences [83]. Better knowledge of the diffracted-



light distribution by a lens has also helped in the design of better objective

lenses, especially those with high NA.

The structure of the focused EM distribution has been studied by a

number of authors. The high NA focusing of EM waves in a single

homogeneous material is dealt with by E. Wolf in an early paper in 1959 [84].

The staring point of his approach was the representation of the angular

spectrum of plane waves, from which an integral representation of the image

field, similar to the Debye [85] integral were obtained. An aplanatic system

was dealt with in a subsequent paper [86]. Wolf and Li [87] later showed that

the approach based on the Debye integral is valid for systems that satisfy the

high aperture condition.

In practice, however, the focusing is performed through an interface

between two materials, such as through a coverslip. If these two materials

have different optical properties, i.e. refractive index, then the spherical

aberration occur. Microscope objectives are usually manufactured with

aberration correction for certain but fixed penetration depths. Practically, for

biological applications, this depth is compensated by an appropriate coverslip.

2.5.1 Focusing through mismatched refractive index

materials

The first work on the focusing of EM waves through mismatched refractive

index materials is that of Gasper et al. [88], who consider an arbitrary

EM wave traversing a planar interface. However, due to the complexity of

their approach, the use of their formulas for the calculation of the EM field

distribution near focus was not practical in most cases.

The most straight forward approach is that of Gu [89] and Torok et al. [90].



Their approach is based on the extension of the Wolf’s [84] treatment of the

diffraction problem when the light is focused through a single homogeneous

material, to the case when light is focused by a high NA lens into a medium

of different refractive index to that of the medium of propagation and

introduces a considerable amount of spherical aberration. The difference in

the two treatments is that Torok et al. [90] use the matrix formalism in their

derivations.

Fig. 2.8: A schematic diagram of a light beam being refracted on an interfacebetween two media (n2 > n1).

A schematic diagram showing a linearly polarised plane wave focused by

a high NA lens into two media separated by a planar interface is shown in

Fig. 2.8. When a spherical wave is focused through an interface between

media of mismatched refractive indices, its wave front becomes distorted.

This distortion can be described by a phase function, called the aberration

function, which depends on the focus depth, the refractive indices n1 and n2,



and the azimuthal angle [90–92]. The aberration function can be written as

Ψ(φ1, φ2,−d) = −d(n1 cosφ1 − n2 cosφ2) , (2.21)

were d denotes the focus depth, φ1 is the angle of incidence on the interface,

φ2 is the angle of refraction, which are linked by the Snell’s law. Thus the

defined aberration function has a significant effect on the energy distribution

of light focused into the second medium, especially at deep depths. As either

the NA of the focusing objective, the focusing depth, or the refractive index

of the second material (refractive index n2) increases, the main peak of the

focus distribution shifts and the energy distributions become asymmetrical

and less concentrated [91].

The electric and magnetic fields components that describe the field near

the focal region in the second material (n2), at an arbitrary point P (rp, φp, θp)

(Fig. 2.8.), can be expressed as [90]

E2x =iK

2π

∫ α

0

∫ 2π

0

(cosφ1)1/2(sinφ1)[(τp cosφ2 + τs) + (cos 2θ)

×(τp cosφ2 − τs)] expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ ,

E2y =iK

2π

∫ α

0

∫ 2π

0

(cosφ1)1/2(sinφ1)(sin 2θ)(τp cosφ2 − τs)

× expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ ,

E2z = − iKπ

∫ α

0

∫ 2π

0

(cosφ1)1/2(sinφ1)τp sinφ2 cos θ

× expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ , (2.22)



and

H2x =iKn2

2π

∫ α

0

∫ 2π

0

(cosφ1)1/2(sinφ1)(sin 2θ)(τs cosφ2 − τp)

× expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ ,

H2y =iKn2

2π

∫ α

0

∫ 2π

0

(cosφ1)1/2(sinφ1)[(τs cosφ2 + τp) + (cos 2θ)

×(τp − τs cosφ2)] expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ ,

H2z = − iKn2

π

∫ α

0

∫ 2π

0

(cosφ1)1/2(sinφ1)τs sinφ2 sin θ

× expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ , (2.23)

where κ and the constant K are defined as

K =k1flo

2, (2.24)

and

κ = n1 sinφ1 sinφp cos(θ − θp) + n2 cosφ2 cosφp . (2.25)

Here θp is the spherical coordinate of the point P , k0 indicates the wave

number in vacuo, k1 is the wave number in the first medium, f is the focal

length of the lens in vacuo and l0 is an amplitude factor.

If the incident field E(0) is independent on angle θ, such as the case for a

plane wave for example, the integration in Eqs. 2.22 and 2.23 can be expressed

as the combination of two sets of three integrals, I0, I1, and I2 as derived by

Torok et al. [90] as

E2x = −iK[I(e)0 + I

(e)2 cos(2θp)] ,

E2y = −iKI(e)2 sin(2θp) ,

E2z = −2KI(e)1 cos(θp) , (2.26)



and

H2x = −iKn2I(h)2 sin(2θp) ,

H2y = −iKn2[I(h)0 − I

(h)2 cos(2θp)] ,

H2z = −2Kn2I(h)1 sin(θp) . (2.27)

After we substitute the normalised optical coordinates

v = k1rp sinφp sinα ,

u = k2rp cosφp sin2 α (2.28)

the integrals I0, I1, and I2 are given by

I(e)0 =

∫ α

0

(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)](τs + τp cosφ2)

×J0

(

v sinφ1

sinα

)

exp

(

iu cosφ2

sin2 α

)

dφ1 ,

I(e)1 =

∫ α

0

(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)]τp sinφ2

×J1

(

v sinφ1

sinα

)

exp

(

iu cosφ2

sin2 α

)

dφ1 ,

I(e)2 =

∫ α

0

(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)](τs − τp cosφ2)

×J2

(

v sinφ1

sinα

)

exp

(

iu cosφ2

sin2 α

)

dφ1 , (2.29)



and

I(h)0 =

∫ α

0

(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)](τp + τs cosφ2)

×J0

(

v sinφ1

sinα

)

exp

(

iu cosφ2

sin2 α

)

dφ1 ,

I(h)1 =

∫ α

0

(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)]τs sinφ2

×J1

(

v sinφ1

sinα

)

exp

(

iu cosφ2

sin2 α

)

dφ1 ,

I(h)2 =

∫ α

0

(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)](τp − τs cosφ2)

×J2

(

v sinφ1

sinα

)

exp

(

iu cosφ2

sin2 α

)

dφ1 . (2.30)

Here α denotes the maximum convergence angle, Jn is the Bessel function of

the first kind, of order n. τs and τp are the Fresnel coefficients given by [54]

τs =2 sinφ2 cosφ1

sin(φ1 + φ2),

τp =2 sinφ2 cosφ1

sin(φ1 + φ2) cos(φ1 − φ2). (2.31)

Using the vectorial diffraction approach given in Eqs. 2.22, 2.23, 2.26,

and 2.27 the EM field distribution in the region of the focus of a high NA

microscope objective, including the effects of spherical aberration, phase

modulation, and apodisation function can be determined, which would

provide a powerfull tool for the trapping force evaluation under a complex

beam illumination.

2.5.2 Focal spot splitting with high NA objective

Vectorial diffraction effects can lead to the focal splitting for objectives of

high NA. When the NA of an objective becomes large, a linearly polarised



beam becomes depolarised in the focal region after the refraction by the

objective. In other words, if the incident electric field is polarised along the

5

0.5

0.5

0.5

0.5

1

1

1

2.5

2.5

2.5

10

203550658095

1

2

3

4

5

6

7

-7

-6

-5

-4

-3

-2

-1

0vy

1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0

vx

1

2

3

4

5

6

7

-7

-6

-5

-4

-3

-2

-1

0vy

1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0

vx

(a) (b)

Fig. 2.9: Contour plots of the intensity near the focus of a high NA objective(NA = 1). (a) |E|2 for ε = 0.0, (b) |E|2 for ε = 0.98 [93].

x direction, the diffracted field contains an x component as well as y and

z components (Eqs. 2.26 and 2.27). The effect of depolarisation becomes

stronger for increasing convergence angle of the refracted wave. As a result

of depolarisation, the focal spot of high NA objective becomes elongated

(elliptical) along the polarisation direction (Fig. 2.9(a)) [89,94–96].

Illuminated by an annular beam (also known as a ring beam), an objective

exhibits a complicated nature in the focal region. For low NA objectives the

use of an annular beam produces a reduced but circular focal spot due to

the suppression of the waves of small convergence angles [89,94–96], which is

utilised in superresolution imaging [98]. However, for objectives with high NA

and a large inner radius of an annular beam, the waves of high convergence

angles become dominant. As a consequence the diffracted pattern may not

necessarily give a single focal spot (Fig. 2.9(b)) [93,99]. Chon et al. [93] have



Fig. 2.10: Contour plots of the intensity near the focus of a high NA objectivefocused on an interface between two media (n1 = 1.78 and n2 = 1.0), with NA =1.65 and ε = 0.6 [97].

shown that when the NA of an objective is between 0.9 and 1 in free space, the

longitudinal component of the electric field Ez at the focus becomes relatively

stronger as the obstruction of the annular beam becomes larger and results

in a split two-peak focus. The obstruction size ε is defined as the ratio of the

inner radius to the outer radius of an annular beam. The focal spot splitting

occurs only if the size of the central obstruction reaches a threshold depending

on the numerical aperture of an objective. The creation of the two-peak focus

by an objective illuminated by an annular beam may prove advantageous for

producing controllable torque [100] in laser trapping [101] systems.

Practically, the focal splitting is also observable with very high NA

objectives focused through an interface between two media, with moderate

obstruction size (Fig. 2.10) [97]. The focus-splitting effect is stronger for the

interface between the two materials with a smaller index difference, provided

the obstruction radius is chosen correctly.



2.5.3 Spectral splitting near phase singularities of fo-

cused waves

When the incident field is spatially fully coherent, but is polychromatic

rather than monochromatic, as with ultrashort-pulsed laser illumination, the

spectrum at the zero-intensity points, called phase singularities, exhibits the

anomalous behavior that causes the splitting of the spectrum [102,103]. This

spectral splitting occurs when the focusing is performed by a lens with an

aperture that has a small maximum angle of convergence, which is called the

paraxial approximation. Such a diffraction system corresponds to a low NA

objective.

However, when the NA becomes large, the focusing process involves

depolarisation. In other words, a linearly polarised incident beam Ex exhibits

two extra components, one in the orthogonal direction, Ey , and the other

one in the longitudinal direction, Ez (Eqs. 2.26 and 2.27). As a result the

spectral splitting phenomenon may not necessarily appear in the focal region

of a high-NA objective.

The anomalous behavior of the spectral intensity, calculated at the axial

zero-intensity points under the paraxial approximation, occurs because of

the condition that the spectral intensity at frequency ω0 has a zero value

when un(ω0) = 4πn is valid for low NA lenses. These zero-intensity points

disappear for lenses with high NA as a result of the contribution of the

depolarised longitudinal component Ez [103]. The normalised spectrum at

the first axial minimum intensity point of the central frequency component

ω0 for different values of the focusing objective NA is shown in Fig. 2.11.

The normalised spectrum is split into two lines of equal intensity for a

low NA lens (Fig. 2.11(a)) for which the paraxial approximation applies.

However, when the NA of the focusing lens increases, the spectrum at the



Fig. 2.11: The normalised spectrum S[u1(ω0, ω]/S0 at the first axial zero-intensitypoint (u1 = 4π) of the central frequency component ω0 for different values of NA:(a) NA = 0.025, (b) NA = 0.1, (c) NA = 0.3, (d) NA = 0.4, (e) NA = 0.6, (f)NA = 0.9.

first axial minimum intensity point undergoes a noticeable change. When the

NA increases (Fig. 2.11(b) and 2.11(c)), the spectrum is still split into two

lines, but with a shallow dip in the center. When the NA is larger than 0.4

(Fig. 2.11(d)), the dip in the spectrum disappears and the spectrum does not

split. In the case of a high-NA lens, the spectrum distribution is almost the

same as the input spectral distribution (Fig. 2.11(e) and 2.11(f)).

Due to the depolarisation effect, the focal spot is elongated along the

polarisation direction, which leads to the non-zero value of the first minimum

intensity point in the x direction (polarisation direction), while the first

zero-intensity point is retained in the y direction. As a consequence, the

spectral splitting in the y direction is independent of the NA value, while

the spectral behavior in the incident polarisation direction is NA dependent

(Fig. 2.12). In the paraxial case (Fig. 2.12(a)) the spectrum is split in the



Fig. 2.12: The normalised spectrum S(ω)/S0 at the first x and y zero-intensitypoints of the central frequency component ω0 for different values of NA: (a) NA =0.025, (b) NA = 0.05, (c) NA = 0.1, (d) NA = 0.3, (e) NA = 0.6, (f) NA = 0.9.Full lines show the variations in the x direction, while dotted lines represent thosein the y direction.

x direction but the spectrum already shows a nonzero dip at the central

frequency component ω0, while it completely disappears for NA = 0.05. The

spectrum in x direction approaches that of the incident beam quickly as the

NA value increases (Fig. 2.12(c)- 2.12(f)).

2.6 Near-field trapping

The trapping volume of the far-field laser trapping geometry is approximately

three times larger in the axial direction than that in the transverse direction.

Such trapping volume elongation leads to a significant background and

poses difficulties in the observations of the single-molecule dynamics. In

recent times, a new trapping modality based on the evanescent wave



illumination, also called near-field illumination, has been proposed [12–15]

and demonstrated [15]. This trapping technique results in a significantly

reduced trapping volume due to the fact that the strength of an evanescent

wave decays rapidly with the distance from the place at which the field is

generated. In this section, the near-field trapping mechanism based on the

different ways to generate a localised near-field is reviewed.

2.6.1 Near-field trapping using a nano-aperture

Okamoto and Kawata [14] have proposed a near-field trapping technique that

utilises a circular nano-aperture as a localised field source. The trapping is

achieved by interaction between the aperture and the dielectric sphere via

evanescent photons. The near-field trapping model geometry is shown in

Fig. 2.13. A glass substrate is coated with a metallic layer of thickness of

λ/5, where λ is the illumination wavelength in vacuum. The metallic layer

is in contact with water. A circular aperture is made in this layer with a

diameter of λ/4, and a small dielectric particle, with a diameter of λ/2 is

situated in water near the aperture.

Okamoto and Kawata [14] have made a series of numerical calculations

of the radiation force exerted on a such particle. The particle position

near the aperture is constantly changed in this calculation, thus a spatial

distribution of the radiation force, also known as a force mapping, is

determined (Fig. 2.13(a) and (b)). The EM field distribution is calculated

using using the finite-difference time-domain (FDTD) method. From each

calculated EM field distribution, the radiation force was obtained from the

Maxwell stress tensor on the surface of the sphere.

Using this methodology, it was confirmed that optical near-field trapping

using a nano-aperture configuration can be achieved. The result indicates



polarisation polarisation

Fig. 2.13: The geometry of the near-field trapping model. Incident light is xpolarised and propagates along the z axis. (a) and (b) Radiation force spatialdistribution for a dielectric sphere near a nano-aperture. The origin of each arrowrepresents the center of the sphere and vectors represent the direction and themagnitude of forces [14].

that a particle is attracted towards the aperture. The near-field radiation

force is found to be larger than the forces due to thermal fluctuations and

to gravity. Furthermore, they have found that if two particles are near the

aperture, the first particle is trapped and the second one is also attracted to

the first one.

2.6.2 Near-field trapping using a metallic tip

Novotny et al. [13] have investigated the possibility of trapping a nano-particle

using the field enhancements close to a laser illuminated sharply pointed metal

tip. The near-field close to the tip mainly consists of evanescent components

which decay rapidly with distance from the tip. Their analysis is based

on characterising the field enhancements near the tip using the numerical

MMP method. The effect of the particle in close proximity of the tip is


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also included. Once the field distribution on the surface of the particle is

determined, the Maxwell stress tensor approach is used to determine the

near-field force exerted on the particle [13].

Fig. 2.14: Near field of a gold tip in water illuminated by two differentmonochromatic waves at λ = 810 nm, indicated by the k and E vectors. Thenumbers in the figure give the scaling in the multiples of the exciting field (E2),with a factor of 2 between the successive lines. (a) No field enhancement; (b) Fieldenhancement of ≈ 3000 [13].

Figure 2.14 shows the three dimensional MMP calculation of a field

enhancement of a gold tip with radius of 5 nm in water for two different

monochromatic plane-wave excitations. The wavelength of the illuminating

light is λ = 810 nm, which does not match the surface plasmon resonance.

When the illumination field is incident from the bottom with the polarisation

perpendicular to the tip axis (Fig. 2.14(a)), there is no field enhancement

beneath the tip. In the case when the tip is illuminated from the side

with the polarisation parallel to the tip axis (Fig. 2.14(b)), a large intensity

enhancement at the foremost part of the tip occurs. This intensity at the

foremost part of the tip is approximately 3000 times stronger than the

illuminating intensity.


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Utilising this methodology, Novotny et al. [13] have shown that nano-

particles can be trapped using the field enhancements near a metallic tip when

the polarisation of the incident light is directed along the tip axis. Based on

their findings they propose a near-field trapping scheme in which a particle

is first trapped by a conventional far-field trapping means and then a sharp

metal tip is brought to the focus. A polarisation component along the tip

axis enables trapping of the particle to the near field of the tip. The trapped

particle can be moved within the focal region of the illuminating light by

translating the tip and can be released by turning off the laser illumination.

2.6.3 Near-field trapping using an apertureless probe

Another novel approach to trap small particles using a near-field is given by

Chaumet et al. [12]. Their approach is based on the use of a combination of

evanescent illumination and light scattering at the probe apex to shape the

optical field into a localised, three-dimensional optical trap. The schematic

representation of their configuration is shown in Fig. 2.15.

The particle under consideration has a radius of 10 nm and is situated

in air. It is illuminated (λ = 500 nm) by two by two counter-propagating

evanescent waves generated by the TIR of plane waves the substrate/air

interface. These two waves have the same polarisation and a random phase

relation This symmetric illumination ensures that the sphere is not pushed

away from the tip when the sphere is just below the tip. A tungsten probe

is placed in close proximity of the dielectric sphere. As the probe tip moves

closer to the sphere, a force is exerted on the particle.

Chaumet et al. [12] have used the coupled-dipole method and the Maxwell

stress tensor to determine the near-field force exerted on the particle. They

have calculated the force experienced by the particle as the probe is moved



Fig. 2.15: A dielectric sphere situated on a flat dielectric surface is illuminated bynear-field generated under total internal reflection. A tungsten probe is used tocreate an optical trap [12].

laterally, above the particle. The force components in axial z and lateral x

directions acting on the sphere are plotted for two distances between the tip

and the substrate (20 and 31 nm), and for two angles of illumination (43 and

50)(Fig. 2.16). It is shown that small objects can be selectively captured

and manipulated by the near-field force generated using an apertureless

probe with the trapping configuration shown in Fig. 2.15. Furthermore, the

magnitude and direction of trapping force greatly depend on the polarisation

state of the incident illumination.

For TM illumination the axial trapping force is positive, i.e. the particle is

attracted towards the probe tip. This occurs because of a large enhancement

of the field near the apex of the probe for TM illumination. This result


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(a)

(b)

(c)

(d)

Fig. 2.16: Force experienced by the sphere as a function of the lateral position ofthe probe. Thick lines denote θ = 43, while thin lines denote θ = 50. The probetip is either 20 nm (solid lines) or 31 nm (dashed lines) above the substrate. (a)TM polarisation z direction; (b) TE polarisation z direction; (c) TM polarisationx direction; (d) TE polarisation x direction. According to Ref. [12].

indicates that by using the TM illumination with this trapping configuration

one can achieve the particle lifting. In the case of the TE illumination,

however, the axial trapping force is directed away from the probe tip

as the tip gets closer to the particle. This prevents any lifting of the

particle for this polarisation state. Using this trapping configuration and its

polarisation dependence one can selectively trap a particle using TM polarised

illumination, lift it and manipulate it by moving the probe tip, and finally

eject the particle by switching to the TE polarised illumination.

2.6.4 Near-field trapping with evanescent field gener-

ated under TIR illumination

An evanescent field generated under the condition of the TIR can also trap

and move small microparticles [15, 67]. This near-field configuration can be

divided into two categories: near-field trapping with a wide area evanescent

field and near-field trapping with a focused evanescent field.


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The wide area evanescent field is usually generated at a surface of a

prism under the TIR illumination condition. Kawata and Sugiura [67]

have demonstrated a microparticle translation using the wide area near-field

configuration. The focused evanescent field is generated using an annular

beam, produced by a high NA objective that is centrally obstructed, which

satisfies the TIR condition at the objective coverslip surface. Gu et al. [15]

have recently demonstrated this type of near-field trapping. In this section,

both categories of the near-field trapping with evanescent field are reviewed.

2.6.4.1 Near-field trapping with a wide area evanescent field

The near-field trapping geometry for the wide area evanescent field trapping

is depicted in Fig. 2.17(left). It is called the wide area because the area at

which evanescent field is generated is much larger than the microparticles.

The laser beam is incident on the prism surface at an angle larger than the

Fig. 2.17: Schematic diagram of particle movement in an evanescent field. Velocityof the moving particle versus the incident angle of the beam undergoing TIR [67].

critical angle, generating a fast decaying evanescent field near the surface.

If a small dielectric particle is in the vicinity of the surface it can convert

evanescent photons into propagating photons through a photon tunneling


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Text Box



process [104, 105]. The tunneled photons scatter from the particle through

the process of multiple internal reflections inside the particle. Consequently,

a part of the momentum of photons of the incident laser beam is transferred

to the particle, driving the moving particle.

The velocity of the evanescent field driven polystyrene particles of a

diameter of 6.8 µm as a function of the incident angle of the beam undergoing

the TIR is shown in Fig. 2.17(right). Due to the evanescent field depth

decrease with increasing incident angle, the particle velocity for both P and S

incident polarisation is decreasing. It is found that the particles driven by S

polarisation moved much faster than those driven by P polarisation for each

incident angle.

The theoretical treatment of this type of near-field trapping is given by

Almaas and Brevik [68], while the effects of the prism surface are discussed

by Lester and Vesperinas [69]. Both of these approaches are using an EM

model for evanescent wave illumination and its interaction with a particle,

while the force exerted on the particle is determined using the Maxwell stress

tensor approach. Almaas and Brevik [68] are using an analytical expression

for evanescent wave generated at the prism surface and interacting with a

particle far-away from the surface, so that its effects are neglected. Lester

and Vesperinas [69], on the other hand, consider the effects of the interface

through a multiple-scattering numerical simulation. They show that when

the particle approaches the surface the axial trapping force changes from

attractive (Fz < 0) to strongly repulsive (Fz > 0).

2.6.4.2 Near-field trapping with a focused evanescent field

The geometry of the near-field trapping with a focused evanescent field is

shown in Fig. 2.18. The central obstruction has such a size that the minimum



angle of convergence of a ray is larger than the critical angle determined by

the inference between two media. This condition ensures that each ray is

incident on the interface at an angle that satisfies the TIR condition and

thus results in a focused evanescent wave on the interface. Due to the

Fig. 2.18: Near-field trapping under focused evanescent illumination. Densityplots (a) and (b) represent the calculated modulus squared of the electric fieldat wavelength 532 nm in the focal region of an objective of NA = 1.65 at theinterface between the cover slip (n = 1.78) and water (n = 1.33). (a) No centralobstruction, i.e. ε = 0; (b) With central obstruction ε = 0.8, whose size satisfiesthe TIR condition [15].

circular symmetric nature of this illumination, the resulting evanescent wave

constructively interferes at the center of the focus, enhancing the strength of

the evanescent field and reducing the lateral trapping size.

Gu et al. [15] have shown experimentally that such focused evanescent

field can trap microscopic dielectric particles due to the fast decaying nature

of the evanescent field. Using vectorial diffraction theory [97], one can show

that the axial size of the trapping volume is reduced to approximately 60

nm, while the lateral trapping size is reduced by 10% (Fig. 2.18(a) and

(b)). Additional advantage of this method compared to the other near-field

trapping methods is that the distance between the trapping site and the

objective is sufficiently large for micro-manipulation. Furthermore, there is



no heating problems associated with this near-field trapping method. The

strength of the evanescent field under the total internal condition can be

further increased by the use of a dielectric double-layer structure coated on

the cover slip, which can result in the enhancement of the evanescent field by

approximately three orders of magnitude [51].

As far as we are aware, there has been no theoretical treatment of

this type of near-field trapping. It is one of the aims of this thesis to

present a theoretical model for this promising near-field trapping method.

Our theoretical treatment is based on the vectorial diffraction and the

Maxwell stress tensor approach, given in greater details in Chapter 4,

while the treatment of the near-field trapping under focused evanescent field

illumination is undertaken in Chapter 6.

2.7 Chapter summary

One of the aims of this thesis is to develop an exact and rigorous model

for optical trapping by a focused laser beam based on vectorial diffraction

theory. The laser beam is focused by a high NA objective, while a dielectric

particle is suspended in a medium which in general differs from the immersion

medium of the objective. Two optical trapping modalities are considered, the

far-field laser trapping and the near-field trapping under focused evanescent

field illumination. In this chapter we have reviewed the theoretical models

currently available for determining the far-field trapping force exerted on a

small dielectric particle. These approximate theoretical models cannot deal

with focal distribution complexities of laser beams focused using a high NA

objective. The physical effects such as spherical aberration and objective

apodisation, as well as an arbitrary complex phase of the incident illumination

can not be considered using these approximate models. The different near-



field trapping techniques are also reviewed based on the method of generating

a localised near-field. The theoretical approaches for these different near-field

trapping schemes are pointed out. However, the near-field trapping technique

based on the focused evanescent field illumination has not been theoretically

dealt with. Our optical trapping model based on vectorial diffraction by a high

NA objective can be applied to provide an appropriate theoretical model for

such a trapping modality as well. In order to gain a physical insight into our

modelling of both far-field and near-field trapping modality in later chapters,

a review of vectorial diffraction theory, extremely important when dealing

with the focusing by high NA objectives, is also included. The details of our

optical trapping model are given in Chapter 4, while its applications in the

far-field and the near-field trapping are presented in Chapter 5 and Chapter 6

respectively.

The other aim of this thesis is to develop an appropriate nanometric

sensing model based on near-field (also known as evanescent field) Mie

scattering and vectorial diffraction of the scattered field. The applications

of the far-field trapping such as optical trap nanometry and particle trapped

SNOM, are based on the near-field scattering by a small trapped particle for

high resolution imaging and position measurements. The evanescent field

is generated at an interface between two different media under the TIR

condition, while the scattering particle is situated in a close proximity of

the interface. The current theoretical models of near-field Mie scattering

are dealing with differential scattering cross-sections or the asymptotic

representation of the scattered field and are reviewed in this chapter. In order

to develop an appropriate nanometric sensing model, the three-dimensional

distribution of the scattered field is required. Such a sensing model needs

to include the effects of the interface at which the near-field is generated,

which is of great importance when the particle is situated in its vicinity. The

detectability of the collected signal is addressed by considering a wide-area



and a pinhole detector. Our nanometric sensing model is described in greater

details in Chapter 3.


Chapter 3

Three dimensional near-field

Mie scattering by a small

particle

3.1 Introduction

Far-field trapping modality is often used with high resolution imaging, based

on evanescent (near-field) illumination of the sample under investigation. Two

of those high resolution imaging techniques that utilise a laser trapped particle

(far-field trapping) are reviewed in Sections 2.2.1 and 2.2.2 of Chapter 2. It is

indicated that for a proper model of near-field Mie scattering, when a particle

is near the interface at which the near-field is generated, the interface effects

need to be included (Section 2.4). Thus, this chapter investigates near-field

Mie scattering by microscopic dielectric particles situated in close proximity

of the surface at which an evanescent wave is generated and presents our

physical model for the detection of the scattered near-field signal.

Near-field Mie scattering is a scattering process caused by interaction of a

57

CHAPTER 3. Three dimensional near-field Mie scattering by a small particle

Mie particle with an evanescent wave rather than a plane wave (Section 2.4).

An evanescent wave occurs when an electromagnetic wave is incident from a

high refractive index medium into a low refractive index medium, under total

internal reflection. The strength of an evanescent wave decays quickly in the

low refractive index media; the larger the incident angle the larger the decay

constant β. Such an evanescent wave can be converted into a propagating

wave by interaction with a small particle situated in a low refractive index

medium, provided that the distance between the particle and the interface

separating the two media, d, is within a few illumination wavelengths in

length. At first the scattering of near-field is studied by neglecting the surface

effects, which corresponds to the case when the particle is far away from the

interface. Subsequently, the surface influence is included and its effects on

the scattering properties are determined.

Once the three-dimensional (3-D) electric vector field distribution in the

far-field is known a collection objective can be introduced, using the vectorial

diffraction theory, to determine the collected signal onto an optical detector.

The scalar diffraction theory would not be appropriate for this problem, due

to the vectorial nature of the scattered field.

This chapter is structured as follows. Section 3.2 gives the mathematical

treatment of near-field Mie scattering, while the 3-D scattered intensity

distribution around dielectric particles near and far from the interface is

examined in Section 3.3. Morphology dependent resonance effects are

investigated in Section 3.4. A physical model, theoretical and experimental

results for the conversion of evanescent photons into propagating photons are

given in Section 3.5. Pinhole detection of the scattered signal is examined in

Section 3.6, while the main conclusions are drawn in Section 3.7.



3.2 Mathematical description of 3-D near-

field Mie scattering

A schematic diagram of the near-field Mie scattering model, which includes

the effect of an interface where the evanescent wave is generated, is presented

in Fig. 3.1. The incident angle of the electromagnetic wave is represented

d

P

P’

n

n’

an

n’

surrounding

medium

substrate

z

x

Y

Fig. 3.1: Illustration of the scattering model including the effect of an interface atwhich an evanescent wave is generated. n, n′ and n1 denote refractive indices ofthe substrate, surrounding medium and the particle, respectively. d is the distancefrom the center of a particle to the interface.

by α. The refractive index of the scattering particle is denoted by n1, while

it is immersed into a medium of refractive index n′. The refractive index

of the substrate is n. Particle is situated at a distance d from the interface



at which an evanescent wave is generated. The incident electric wave for

the polarisation state of E0 perpendicular to the plane of incidence (i.e. TE

polarisation) is given by [72]

Einc(r) =∑

lm

ic

n′2ωαE(l,m)∇× [jl(k

′r)Yllm(r)]

+ αM(l,m)jl(k′r)Yllm(r)

, (3.1)

with the magnetic wave

Hinc(r) =∑

lm

αE(l,m)jl(k′r)Yllm(r)

− ic

ωαM(l,m)∇× [jl(k

′r)Yllm(r)]

. (3.2)

l = 1 to ∞ and m = -l to +l. c is the speed of light in vacuum and ω

is the angular frequency of the incident light. The functions αE(l,m) and

αM(l,m) are the expansion coefficients for the incident illumination field

given by Eqs. 2.14 and 2.15, while Yllm is the vector spherical harmonics.

jl is the spherical Bessel function of the l-th order. In the case of the

polarisation state E0 parallel to the plane of incidence (i.e. TM polarisation),

the scattered electric and magnetic fields are also given by Eq. 3.1 and Eq. 3.2

with expansion coefficients αE(l,m) and αM(l,m) substituted by αE(l,m) and

αM(l,m), respectively [72], given by Eqs. 2.16 and 2.17.

For particles which are situated far from the interface, the interaction

between the particle and the interface can be neglected. However, if the

particle is very close to the interface, then the interaction between the particle

and the interface is not negligible and it needs to be incorporated into the

scattering model.



3.2.1 Particles far from the interface

In the geometry shown in Fig. 3.1, when the incident angle α of the

illumination electric wave E0 is larger than the critical angle, the scattered

electric field by the Mie particle can be expressed, for the TE polarisation,

as [72]

Esc(r) =∑

lm

ic

n′2ωβE(l,m)∇× [h

(1)l (k′r)Yllm(r)]

+ βM(l,m)h(1)l (k′r)Yllm(r)

, (3.3)

while the magnetic field is given by

Hsc(r) =∑

lm

βE(l,m)h(1)l (k′r)Yllm(r)

− ic

ωβM(l,m)∇× [h

(1)l (k′r)Yllm(r)]

. (3.4)

When expressed in a spherical coordinate system, Eq. 3.3 and Eq. 3.4 can be

reduced to

Esc(r) =∑

lm

cβE(l,m)

n′2ω√

l(l + 1)

h(1)l (k′r)

r sin θ

[

∂

∂θ

(

∂Ylm sin θ

∂θ

)

+1

sin θ

∂2Ylm

∂ϕ2

]

r1

+

[

(−1)βM(l,m)h(1)l (k′r)

i sin θ√

l(l + 1)

∂Ylm

∂ϕ− cβE(l,m)

n′2ω√

l(l + 1)

1

r

∂Ylm

∂θ

∂

∂r(rh

(1)l (k′r))

]

θ1

+

[

βM(l,m)h(1)l (k′r)

i√

l(l + 1)

∂Ylm

∂θ− cβE(l,m)

n′2ω√

l(l + 1)

1

r sin θ

∂Ylm

∂ϕ

∂

∂r(rh

(1)l (k′r))

]

ϕ1

,

(3.5)



Hsc(r) =∑

lm

−cβM(l,m)

ω√

l(l + 1)

h(1)l (k′r)

r sin θ

[

∂

∂θ

(

∂Ylm sin θ

∂θ

)

+1

sin θ

∂2Ylm

∂ϕ2

]

r1

+

[

(−1)βE(l,m)h(1)l (k′r)

i sin θ√

l(l + 1)

∂Ylm

∂ϕ+

cβM(l,m)

ω√

l(l + 1)

1

r

∂Ylm

∂θ

∂

∂r(rh

(1)l (k′r))

]

θ1

+

[

βE(l,m)h(1)l (k′r)

i√

l(l + 1)

∂Ylm

∂θ+

cβM(l,m)

ω√

l(l + 1)

1

r sin θ

∂Ylm

∂ϕ

∂

∂r(rh

(1)l (k′r))

]

ϕ1

,

(3.6)

where r1, θ1 and ϕ1 are the unit vectors in the spherical coordinates. θ and

ϕ are the variables of the scalar spherical harmonics Ylm.

In the case of the polarisation state E0 parallel to the plane of incidence

(i.e. TM polarisation), the scattered electric and magnetic fields are also

given by Eq. 3.5 and Eq. 3.6 with expansion coefficients βE(l,m) and βM(l,m)

(Eqs. 2.12 and 2.13) substituted by βE(l,m) and βM(l,m), respectively [72]

as discussed in Chapter 2 Section 2.4.2.

Eq. 3.5 gives the electric field resulting from scattering of an evanescent

wave by a Mie particle without using any approximation. In the previous

calculation [72], an approximated form of Eq. 3.5, which is valid only in two

principal planes, was used. In this thesis, Eq. 3.5 is used to calculate the 3-D

distribution of the scattered field around a particle. The 3-D distribution of

the scattered field is dependent on the effective refractive index nef = n1/n′

and the size parameter q = k′a, where a is the radius of a scattering Mie

particle and k′ is the propagation constant of the electromagnetic wave in the

medium surrounding the particle. Another factor affecting the strength and

the distribution of the scattered field is the decay constant β of the evanescent

wave, defined as [72]



β = k′

√

n2 sin2 α

n′2− 1 . (3.7)

The larger the decay constant of the near-field wave, the faster the

evanescent wave decays in the depth direction (i.e. in the X-direction).

Depending on the particle size and the decay constant, a scattering particle

may be immersed into an evanescent wave completely or partially, which

greatly affects the scattered field distribution around the particle.

3.2.2 Particles near the interface

When the scattering particle is situated close to the interface on which

an evanescent wave is generated, interaction between the particle and the

interface needs to be taken into account. Although the effect of the

interface is discussed by other researchers, only the scattering cross-sections

of evanescent wave scattering by small particles are calculated [73–75, 79].

Prieve and Walz [71] have considered interface effects but only for large

particles using a ray optics approach. Doicu et al. [76, 77] have investigated

the differential scattering cross section and the integral response of evanescent

wave scattering by small particles and a sensor tip of up to 100 nm in radius.

These numerical calculations include the interface effects in the context of

the discrete sources method and the T-matrix method, and the far-field is

determined by extrapolation of the scattered field as r →∞. However, these

calculations have not discussed scattering properties of larger particles, most

commonly used in particle trapped SNOM, whose scattering characteristics

are markedly different from scattering properties of small particles due to the

MDR effects [79,106–111].

To include the effects of the interface, the following approach is used.

The scattered electric field Etotal at an arbitrary point P in the surrounding



medium can be expressed as a sum of the field that would be scattered

into point P if the interface between two media was not present, and the

contribution that is reflected into point P by the interface. Mathematically

it can be expressed as

Etotal(r) = Eupper(r) + rfEbottom(r) , (3.8)

where Eupper is scattered field into point P (upper space) and Ebottom is

scattered field into point P ′ (bottom space) without considering the interface,

while rf is the Fresnel amplitude reflection coefficient for a given incident

polarisation state [54]. P ′ is the mirror-reflection point of the point P .

The amount of the scattered field at point P ′ that is reflected to point P

is determined by the Fresnel amplitude reflection coefficients under given

conditions. Eupper(r) and Ebottom(r) can be determined from Eq. 3.5 without

considering the effect of the interface.

3.3 3-D scattered intensity distribution around

dielectric particles

According to the mathematical model described by Eq. 3.5, a 3-D intensity

distribution of the scattered field can be numerically calculated at any

distance from the scattering particle. To understand the effect of the interface

on scattering distribution, let us first consider that a particle is situated far

away from the interface, and then we will consider the case when the particle

is near the interface.



3.3.1 Dielectric particle situated far from the interface

Let us consider a dielectric particle of a radius of 2 µm immersed in air.

The scattered intensity distribution produced by such a particle is shown

in Fig. 3.2. To demonstrate the 3-D distribution of the scattered field, the

scattered intensity in the XZ plane, in the plane containing the X axis at 45

anticlockwise from the XZ plane, in the XY plane and in the YZ plane are

displayed in Fig. 3.2(a), 3.2(b), 3.2(c) and 3.2(d), respectively. The scattered

intensity distribution is asymmetric in the XZ plane and in the plane at 45

from the XZ plane because the illumination field is asymmetric in these planes

while the evanescent wave propagates in the Z direction.

Fig. 3.2 shows that the evanescent wave is most intensely scattered into

certain regions around the particle. This effect is due to the fact that the

interaction of the evanescent wave with the particle is confined to the bottom

of the particle because particle size is so large compared with the decay depth

of the evanescent wave (in this case the decay depth is 186.5 nm). This

phenomenon can be qualitatively explained using Snell’s law and the Fresnel

amplitude coefficients for reflection and transmission [54] (Fig. 3.3). Three

rays representing the evanescent wave propagating along the Z direction are

chosen for demonstration. The length of the vectors in Fig. 3.3 represents the

relative strength of the intensity. Ray 1, the highest intensity ray of the three

selected rays, interacts at the very bottom of the particle (point A) with

a large incident angle. Consequently, most of the ray intensity is reflected

rather than refracted because of the high incident angle with respect to the

normal of the particle’s surface. The refracted ray, after traversing through

the particle, interacts with the particle-medium boundary at point B and the

large amount of its intensity emerges into the medium with a small portion

reflected. This reflected portion of intensity traverses through the particle

again, interacts with the boundary (point C) and emerges into the medium



Fig. 3.2: Three-dimensional far-field distribution of the scattered intensity around a2 µm dielectric particle situated far away from the interface (a) in the XZ-plane, (b)in the plane containing the X-axis at 45o anti-clockwise from the XZ-plane, (c) inthe XY -plane and (d) in the Y Z-plane. The solid and dotted curves correspond tothe TE and TM polarisation states of the illumination wave, respectively. n1=1.6,n′=1.0, n=1.51, λ=632.8 nm and α=45.

while a negligible amount is reflected again. Rays 2 and 3, whose intensity

is weaker and determined by the decay constant of the evanescent wave

experience a similar process. As a result, the relative intensity distribution

of the three rays after three refraction processes indicates that the scattered

intensity profile is confined to certain regions around the particle and that

the highest intensity region is located in the region bellow 0 (see Fig. 3.2(a)).

The location of these intensity regions depends on the refractive indices of



A

A

A

B

B

C

C

B

C

1 1

1

1

2

2

2

2

3

3

3

3

n’=1.0n1=1.6

Fig. 3.3: A qualitative interpretation of the confined intensity regions in thescattering of an evanescent wave by a dielectric particle of radius 2 µm. Therelative intensity of rays is denoted by the arrow length.

the dielectric particle and its immersion medium, the particle size and the

decay constant of the near-field wave.

For particles far away from the interface the dependence of the asymmetric

scattered intensity distribution on the particle size is depicted in Fig. 3.4. For

small particle sizes (Figs. 3.4(a) and 3.4(b)), the scattered intensity profiles

are similar to those under plane wave illumination [54]. This similarity to

the plane wave Mie scattering occurs because when the particle is small,

it is completely immersed into the evanescent field and the difference of

the evanescent intensity between the top and the bottom of the particle

is negligible. When the particle size becomes large the difference of

the evanescent wave intensity between the top and the bottom of the

particle becomes pronounced, resulting in an asymmetric scattered intensity

distribution as shown in the plane of incidence (i.e., in the XZ plane).



(d)

(c)

(b)

(a)

0.00

1.50x10 -3

3.00x10 -3

4.50x10 -3

6.00x10 -3

7.50x10 -3

9.00x10 -3

0

30

60

90

120

150

180

210

240

270

300

330

0.00

1.50x10 -3

3.00x10 -3

4.50x10 -3

6.00x10 -3

7.50x10 -3

9.00x10 -3

0.0

1.0x10 -5

2.0x10 -5

3.0x10 -5

4.0x10 -5

5.0x10 -5

6.0x10 -5

7.0x10 -5

0

30

60

90

120

150

180

210

240

270

300

330

0.0

1.0x10 -5

2.0x10 -5

3.0x10 -5

4.0x10 -5

5.0x10 -5

6.0x10 -5

7.0x10 -5

0.0

2.0x10 -4

4.0x10 -4

6.0x10 -4

8.0x10 -4

1.0x10 -3

0

30

60

90

120

150

180

210

240

270

300

330

0.0

2.0x10 -4

4.0x10 -4

6.0x10 -4

8.0x10 -4

1.0x10 -3

0.0

1.0x10 -5

2.0x10 -5

3.0x10 -5

4.0x10 -5

5.0x10 -5

6.0x10 -5

0

30

60

90

120

150

180

210

240

270

300

330

0.0

1.0x10 -5

2.0x10 -5

3.0x10 -5

4.0x10 -5

5.0x10 -5

6.0x10 -5

0.0

5.0x10 -8

1.0x10 -7

1.5x10 -7

2.0x10 -7

2.5x10 -7

3.0x10 -7

3.5x10 -7

4.0x10 -7

0

30

60

90

120

150

180

210

240

270

300

330

0.0

5.0x10 -8

1.0x10 -7

1.5x10 -7

2.0x10 -7

2.5x10 -7

3.0x10 -7

3.5x10 -7

4.0x10 -7

0.0

2.0x10 -8

4.0x10 -8

6.0x10 -8

8.0x10 -8

1.0x10 -7

1.2x10 -7

1.4x10 -7

0

30

60

90

120

150

180

210

240

270

300

330

0.0

2.0x10 -8

4.0x10 -8

6.0x10 -8

8.0x10 -8

1.0x10 -7

1.2x10 -7

1.4x10 -7

0.0

1.0x10 -9

2.0x10 -9

3.0x10 -9

4.0x10 -9

5.0x10 -9

0

30

60

90

120

150

180

210

240

270

300

330

0.0

1.0x10 -9

2.0x10 -9

3.0x10 -9

4.0x10 -9

5.0x10 -9

0.0

5.0x10 -10

1.0x10 -9

1.5x10 -9

2.0x10 -9

2.5x10 -9

0

30

60

90

120

150

180

210

240

270

300

330

0.0

5.0x10 -10

1.0x10 -9

1.5x10 -9

2.0x10 -9

2.5x10 -9

Fig. 3.4: Dependence of the scattered intensity distribution in the XZ plane on theradius of a particle, when the particle is situated far away from the interface: (a)a = 0.05 µm, (b) a = 0.1 µm, (c) a = 0.5 µm and (d) a = 1 µm. The plots in the leftand the right columns show the intensity distributions scattered by an evanescentwave and a plane wave respectively. The solid and dotted curves correspond to theTE and TM polarisation states of the illumination wave, respectively. n1 = 1.6,n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45.



Fig. 3.4(c) can also be qualitatively explained using the same method as

for Fig. 3.2(a). In this case, most of the evanescent intensity on the particle

is incident with a small incident angle. As the Fresnel reflection coefficient

on the particle surface is small for small incident angles [54], the final result

is that the refracted rays lead to an intensity distribution in the region from

0 to 60.

For the 1 µm particle (Fig. 3.4(d)), the scattered intensity distribution

for the TE polarisation illumination is mostly confined to a region below the

Z axis but that for the TM polarisation illumination is mostly above the Z

axis. According to the particle size, the major contribution to the scattered

intensity distribution is from the rays with an incident angle between 70

to 80 near the bottom of the particle. The Fresnel reflection coefficients at

these incident angles for TE polarisation illumination are larger than that for

TM polarisation illumination (for example, the former is 4 times larger for

the incidence at 75). As a result, the majority of the intensity is reflected

in the case of TE polarisation, while a majority of the intensity is refracted

and transmitted through the particle in the case of TM polarisation (see

Fig. 3.4(d)).

3.3.2 Dielectric particle situated near the interface

When a particle is brought close to the interface on which the evanescent

wave is generated, the scattered intensity distribution, given by Eq. 3.8,

is drastically changed. Fig. 3.5 shows the XZ plane scattered intensity

distribution for small and large dielectric particles situated on the boundary

between the surrounding medium and the substrate. It can be seen that

not only has the scattered intensity distribution drastically changed for both

incident polarisations, but also the scattering profile for the TM polarisation

is much stronger at smaller scattering angles (0-30 from either +Z or -Z



directions) when compared with the TE polarisation. For larger particles,

the scattering profile for the TE polarisation state exhibits similar behavior.

0.0

0.2

0.4

0.6

0.8

1.0

0

30

60

90

120

150

180

210

240

270

300

330

0.0

0.2

0.4

0.6

0.8

1.0

(a)

(c)

0.0

0.2

0.4

0.6

0.8

1.0

0

30

60

90

120

150

180

210

240

270

300

330

0.0

0.2

0.4

0.6

0.8

1.0

(b)

0.0

0.2

0.4

0.6

0.8

1.0

0

30

60

90

120

150

180

210

240

270

300

330

0.0

0.2

0.4

0.6

0.8

1.0

(d)0.0

0.2

0.4

0.6

0.8

1.0

0

30

60

90

120

150

180

210

240

270

300

330

0.0

0.2

0.4

0.6

0.8

1.0

(c)

Fig. 3.5: Dependence of the scattered intensity distribution in the XZ plane onthe radius of a particle, when particle is situated on the interface: (a) a = 0.05µm, (b) a = 0.1 µm, (c) a = 0.5 µm and (d) a = 1 µm. The solid and dottedcurves correspond to the TE and TM polarisation states of the illumination wave,respectively. n1 = 1.6, n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45.

Such a drastic change in the scattered intensity profile caused by the

particle-interface interaction, indicates that for a proper study of near-field

Mie scattering or for the modelling of the actual experiments interface effects

need to be taken into consideration.

3.4 Effects of the interface on the morphology

dependent resonance

Morphology dependent resonance (MDR) is caused by the interference of a

light beam propagating inside a dielectric particle confined by total internal

reflection [106]. As a beam of light propagating inside the particle returns



to its starting position in phase, the constructive interference effect leads to

a series of peaks in the scattered field for given an appropriate particle size.

Due to this process, a large energy density can be created inside a particle

[106]. MDR has been observed in plane-wave Mie scattering [107–111] and

predicted in near-field Mie scattering [73–75].

Based on our method for determination of the field scattered by a

small particle situated near an interface, the intensity integrated over the

upper half-space of the particle, can be calculated. This parameter gives

a measure for the maximum signal strength in particle-trapped SNOM

and therefore can be used to study the MDR caused by near-field Mie

scattering. Fig. 3.6 shows the intensity integrated over the upper half-space

of a particle with varying refractive index immersed in air and situated on

the boundary between the surrounding medium and the substrate for TE and

TM polarisation illumination. MDR becomes evident as the particle radius

increases; the larger the particle size the sharper the resonance peaks. This

is understandable because a scattering particle can be considered to be a

micro-cavity. For a large particle, the evanescent wave interacts significantly

with the particle near its bottom. Therefore, the beam refracted into the

particle is incident on the particle boundary with a refraction angle close to

the critical angle inside the particle, which results in a high reflectance or a

large coefficient of finesse of such a cavity. Consequently, sharper resonance

peaks emerge in the scattered field.

According to Fig. 3.6-bottom, the difference of the radius between two

resonance peaks is 71.2 ± 2.0 nm, which agrees well with the result of 72.2

nm estimated by

∆a =λ

2π

arctan(√

n21 − 1)

√

n21 − 1

, (3.9)

for plane wave Mie scattering [108]. The dependence of MDR on the effective

refractive index nef = n1/n′ is shown in Fig. 3.6. As expected from Eq. 3.9,



( m)m

0.0 0.5 1.0 1.5 2.0 2.50.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

Inte

nsi

ty(a

.u.)

Particle radius

0.0 0.5 1.0 1.5 2.0 2.50.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

Inte

nsi

ty(a

.u.)

0.0 0.5 1.0 1.5 2.0 2.50.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

Inte

nsi

ty(a

.u.)

Fig. 3.6: Dependence of the half-space scattered intensity on the particle radiusfor the TE (solid line) and TM (dotted line) polarisation illumination, when theparticle is situated on the interface. n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45.Top: n1 = 1.1, middle: n1 = 1.3 and bottom: n1 = 1.6.



( m)m

0.0 0.5 1.0 1.5 2.0 2.50.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

Inte

nsi

ty(a

.u.)

Particle radius

0.0 0.5 1.0 1.5 2.0 2.50.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

Inte

nsi

ty(a

.u.)

0.0 0.5 1.0 1.5 2.0 2.50.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

Inte

nsi

ty(a

.u.)

Fig. 3.7: Dependence of the half-space scattered intensity on the particle radiusfor the TE (solid line) and TM (dotted line) polarisation illumination, when theparticle is situated on the interface. n′ = 1.0, n = 1.51, n1 = 1.6 and λ = 632.8nm. Top: α = 42, middle: α = 43 and bottom: α = 45.



the spacing between two adjacent peaks increases as n1 decreases. As the

effective refractive index of the particle increases, MDR effects become more

pronounced with sharper resonance peaks because the coefficient of finesse

of the cavity (i.e. the reflectance of the cavity) becomes larger. It can

be seen from the Fig. 3.7 that MDR peak positions are independent of the

incident angle α of the illumination wave. This feature indicates that MDR

is mainly caused by the wave interacting at the bottom of the particle. As

the decay constant β of the evanescent wave increases with the incident angle

α, the scattering of the evanescent wave with a particle is mainly confined

to the bottom of the particle. As a result, the coefficients of finesse of the

cavity are effectively increased, resulting in the shaper peaks (see Fig. 3.7-

bottom). The calculation step in Fig. 3.7 is chosen to be 5 nm. It is found

from our calculation that resonances are found much more precisely and

more significant in strength if smaller step is used. However, smaller step

calculation also requires significantly more computational time.

Signal strength of the light scattered by a polystyrene particle of radius

0.25 µm, and illuminated by an evanescent wave produced by a He Ne laser

incident at an angle larger than the critical angle, is shown in Fig. 3.8. The

particle is immersed in water and placed on the interface. Both calculated

and experimental results [50] agree that the scattered intensity decreases

with the incident angle α for both TE and TM polarisation states of the

incident illumination. Furthermore, it shows that the rate of decrease of

the scattered intensity under TE polarisation is slower than that under TM

polarised illumination. Our model also confirms the experimental finding

that the scattered intensity for TE and TM polarisation states of incident

illumination becomes equal at an incident angle, α, of approximately 58.

Provided that the refractive index of the particle does not change

appreciably when the illumination wavelength varies, the dependence of MDR



Fig. 3.8: A comparison of the calculated half space scattered intensity withthe intensity measured by a NA=1.3 objective in particle-trapped near-fieldmicroscopy [50]. The solid and dotted curves represent TE and TM incidentpolarisation states, respectively. α is the incident angle and a polystyrene particleof radius 0.25 µm immersed in water is placed on the interface between thesurrounding medium and the substrate.

described in Figs. 3.6 and 3.7 changes only by a scaling factor because MDR

is dependent on the size parameter q = k′a. In other words MDR in near-

field Mie scattering can be demonstrated from the fluorescence spectrum of

a fluorescent particle excited by an evanescent wave.

3.5 Mechanism for conversion of evanescent

photons into propagating photons

In the previous section, near-field (evanescent wave) scattering is investigated

in details for cases when a dielectric particle is situated near and far from the

interface. Using this treatment one can determine the scattered field distribu-



tion at the trapping/collecting objective entrance pupil, for a given particle

and conditions. Subsequently, the effect of the trapping/collecting objective

is included by investigating the vectorial diffraction process. Consequently

the focal intensity distribution (FID) in the image space focal region of the

collecting high NA objective can be determined. Such a physical process

describes a mechanism for conversion of evanescent photons into propagating

photons.

3.5.1 Physical model

Our physical model of the evanescent photon conversion mechanism is

based on the near-field Mie scattering enhanced by the MDR and vectorial

diffraction by a high NA lens. The analytical expression for the 3-D vectorial

field distribution around a microscopic particle immersed in an evanescent

field is given by Eq. 3.5, with the inclusion of the interface effects as described

in Eq. 3.8. Subsequently, we include the effect of the trapping/collecting

objective by investigating the vectorial diffraction process, to determine the

focal intensity distribution in the image space focal region of the collecting

high NA objective. The trapping/collecting objective is one objective which

is used for both trapping of a microscopic particle and for collecting the

scattered signal.

Let us consider a microscopic particle in a close proximity of the interface

at which an evanescent field is generated by the TIR (α > αc) under either

TE or TM incident illumination (Fig. 3.9(a)). The origin of the coordinate

system is located at the center of the particle with a coordinate systems

defined in Fig. 3.9(a). The particle is observed by a high NA objective whose

focal point coincides with the particle center. The evanescent wave generated

by TIR propagates in the Y1 direction and decays exponentially in the Z1

direction, while interacting with the microscopic particle. This interaction



j1

j2

q1

q2

r2

r1

objective

Reference spherein object space

Particle

a

a> ca

z1

y1

x1

Reference spherein image space

O1

O2

x2

z2

y2

(a)

C1 C3C2

(b)

x1

y1

z1 z2

x2

y2

r2r1

q2q1

j2j1

Laser

L2

PHL1

L3 L4

ICCD

Prism

O

BS

M1

M2 M3

(c)

Fig. 3.9: (a) Schematic of our theoretical model for evanescent photon conversion.(b) Representation of the lens transformation process. (c) Experimental setupfor recording the FID of converted evanescent photons, collected by a high NAobjective O.

can be physically described in terms of superposition of the field scattered

by the microscopic particle into upper space (space above the prism surface)

and partial reflection of the scattered field into the bottom space (space below

the prism surface)as given by Eq. 3.8. Using this method one can calculate

the superposed scattered field on the reference sphere in the object space.

The center of this reference sphere (also known as the entrance pupil of the

collecting objective) overlaps with the particle center, i.e. the origin of the

coordinate system O1.



The precise transformation of the field from the reference sphere in object

space to the field on the reference sphere in image space requires a detailed

model of the lens. Recent studies of imaging dipole emitters [112] and

microscopic particles [113] through a high NA objective have indicated that

we can assume that the imaging objective transforms a diverging spherical

wave with its origin O1 in the center of the particle into a converging spherical

wave whose origin O2 is in the center of the focal region in image space.

Therefore, the lens effect can physically be modelled as a retardation effect

affecting the wave traversing two different dielectric media (air and glass).

Consider the spherical wavefront C1 originating from the particle center O1

(the origin of the coordinate system X1Y1Z1), just before the collecting

lens (Fig. 3.9(b)). Its curvature corresponds exactly to the curvature of

the collecting lens in object space. After traversing the lens front surface,

the wavefront becomes the plane wavefront C2. All points on the spherical

wavefront C1 arrive at the plane wavefront C2 at the same time. The plane

wavefront is then similarly transformed to the converging spherical wavefront

C3, after traversing the lens back surface. The center of the spherical

wavefront C3 is at O2 (the origin of the coordinate system X2Y2Z2). Such

transformation further indicates that the lens imparts a scaling effect and a

vector rotation. If we consider scattered field vector components, described

by its unit vectors r1, θ1, and ϕ1 in the coordinate system X1Y1Z1, they are

transformed into −r2, θ2, and ϕ2 in the coordinate system X2Y2Z2.

Considering such a transformation process of the field from the entrance

pupil to the exit pupil, the focal field distribution in image space can be

derived by the vectorial diffraction process as given by Richards and Wolf [86],

E(r2, ψ, z2) =i

λ

∫ ∫

Ω

(−Er1r2+Eθ1θ2+Eϕ1ϕ2) exp[−ikr2 sin θ2 cos(ϕ2−ψ)]

× exp(−ikz2 cos θ2) sin θ2dθ2dϕ2 , (3.10)



where Er1, Eθ1 and Eϕ1 are given by Eq. 3.8, while r2, ψ and z2 are cylindrical

coordinates of a point in the image space with a coordinate system shown in

Fig. 3.9(a).

3.5.2 Theoretical results and morphology dependent

resonances

Applied to evanescent photon conversion by a small dielectric particle probe

for either TE or TM incident illumination, our model leads to the FID in the

far-field of the collecting lens, similar to the one obtained for imaging a dipole

emitter (Fig. 3.10(a) and 3.10(e)). This is because the probe in this case is

much smaller than the wavelength of illuminating light, and the particle is

completely immersed into the evanescent field, so the dipole approximation

applies. However, when the particle radius approaches and exceeds the

wavelength of the illuminating light, the FID shows a complex interference-

like structure (Fig. 3.10(b) - 3.10(d) and 3.10(f) - 3.10(h)). Furthermore, our

model indicates that the conversion and collection of TE evanescent photons

is somewhat different from TM evanescent photons (Fig. 3.10). The FID in

image space of the collecting lens shows a similar interference-like structure

for the conversion of either TE or TM localised photons by large particles.

However, when the conversion is performed by a small particle, this similarity

in the FID is less pronounced. The complex interference-like pattern shown

in Fig. 3.10 arises due to the enhancement of MDR and higher multipoles,

scattering properties of large particles and the effects of the interface on which

the evanescent field is generated. One would expect to see the MDR effects

in the FID of the collecting objective, due to the increase in the collected

energy. These effects are indeed manifested in the FID of our model. Two

particular MDR for TE and TM polarised incident illumination are shown

in Fig. 3.11. However, it seems that the effect is not just a mere increase in



Fig. 3.10: Calculated FID in the image focal plane of a 0.8 NA objective. TE (leftcolomn) and TM (right colomn) incident illumination. (a) and (e) a = 100 nm.(b) and (f) a = 500 nm. (c) and (g) a = 1000 nm. (d) and (h) a = 2000 nm. Allfigures are normalised to 100.



1445 1450 1455 1460 1465

1464 nm 1445 nm

1454 nm

(b) TM wave

a (nm)

1400 1405 1410 1415 1420

0

10

20

30

40

50

60

1419 nm 1403 nm

(a) 1411 nm

TE wave

Inte

nsity

(arb

.uni

ts)

a (nm)

Fig. 3.11: Maximum intensity in the FID as a function of the particle radius nearMDR for TE (a) and TM (b) illumination. Insets show the full FID representing offand on resonance cases. The particle refractive index is 1.59 and the illuminationwavelength is 633 nm.

collected energy due to particular MDR, but also leads to different energy

distribution for on and off resonance positions (Fig. 3.11 insets). These

different energy distributions outline the importance of detector selection for

systems operating at MDR positions, because it would be an advantage to

operate particle trapped SNOM and optical trap nanometry systems at MDR

positions to enhance the signal-to-noise ratio.

3.5.3 Experimental setup and results

To confirm the conversion mechanism given by our model we have conducted

an experiment. The experimental setup is depicted in Fig. 3.9(c). A helium-

neon laser beam was expanded and filtered using lenses L1 (microscope

objective, NA=0.2), L2 (focal length 50 mm) and a pinhole (PH). It was then

directed onto the prism-air surface by mirror M1 to form an incident angle of

51.4 ± 0.3, which in combination with a very low divergence of the helium-



neon laser ensured that the incident beam was well above the critical angle

(αc). The prism used in the experiment had a refractive index of 1.722 and

the particles under investigation were polystyrene particles diluted in water

and dried on the prism surface. The particles, immersed into the created

localised field, were imaged using a dry 0.8 NA objective (Olympus IC 100,

infinite tube length) and projected onto an intensified CCD camera (PicoStar

HR 12 from LaVision) via lenses L3 (focal length 70 mm) and L4 (focal length

100 mm). The TIR portion of the incident beam was re-routed via mirrors

M2 and M3 and a beam-splitter (BS) into the back aperture of the collecting

objective (O), to enable us to locate the prism surface and thus the center of

the particle under consideration.

-60 -40 -20 0 20 40 60 -60

-40

-20

0

20

40

60

y (10 -6 m)

x (1

0 -6

m)

-60 -40 -20 0 20 40 60 -60

-40

-20

0

20

40

60

y (10 -6 m)

x (1

0 -6

m)

-60 -40 -20 0 20 40 60 -60

-40

-20

0

20

40

60

y (10 -6 m)

x (1

0 -6

m)

-60 -40 -20 0 20 40 60 -60

-40

-20

0

20

40

60

y (10 -6 m)

x (1

0 -6

m)

Fig. 3.12: Calculated (top) and observed (bottom) FID in image focal plane of a 0.8NA objective collecting propagating photons converted by a=240 nm polystyreneparticle under TE (left column) and TM (right column) incident illumination.



(a)

-60 -40 -20 0 20 40 60

0.0

0.2

0.4

0.6

0.8

1.0 In

tens

ity (a

rb.u

nits

)

y (10 -6 m)

-60 -40 -20 0 20 40 60

(b)

y (10 -6 m)

Fig. 3.13: Calculated and observed y axis scan through x=0, in image focal plane ofa 0.8 NA objective collecting propagating photons converted by 1000 nm (radius)polystyrene particle under TE incident illumination. (a) Calculated results. (b)Observed results (full line) where the dotted line represents the convolution of thecalculated results and the PSF of the imaging lens. Insets show the calculated andobserved FID.

The calculated and observed results of the FID of the collecting objective,

for the conversion of both TE and TM evanescent photons by a polystyrene

particle of 240 nm in radius, are shown in Fig. 3.12. It can be seen

from Fig. 3.12 that the FID structure predicted by our model is in good

agreement with the experimentally observed results. Furthermore, the

predicted difference in the FID structure for TE and TM evanescent photons

converted by this small dielectric probe is observable in the experiment. The

interference-like FID structure for evanescent photon conversion by a large

dielectric particle probe can also be experimentally observed. Figure 3.13(a)

shows our calculated result of the FID for evanescent photon conversion by a 1

µm (radius) polystyrene particle. The corresponding experimentally observed

result is shown in Fig. 3.13(b). The measurement was repeated 10 times

and the error was estimated to be approximately 5%. Image resolution of

the observed result is somewhat degraded due to the imaging properties of



lens L4. The observed structure is a result of the convolution of the point-

spread function (PSF) of the imaging lens L4 and the calculated result shown

in Fig. 3.13(a). The agreement between calculated and experimental results

confirm that the conversion of evanescent photons is the result of two physical

processes, near-field Mie scattering and vectorial diffraction.

3.6 Pinhole detection of the scattered near-

field signal

In this section we will look at the detectability of the scattered near-field

signal, determined using our physical model described in previous section, by

a pinhole detector. Such a detector is utilised in trapped particle SNOM to

discriminate against the out of focus signal [18, 19]. A pinhole detector is

essentially a small circular opening of a few to several tens of micrometers, in

an otherwise opaque screen, placed perpendicularly to the optical axis at the

back focal plane of the imaging objective. The back focal plane focus coincides

with the center of the pinhole. Only the signal that can pass through this

opening is detected, and it constitutes of the signal coming from the front

focal region of the imaging objective (Fig. 3.14(a)).

Mathematically, the detected signal level η of a pinhole detector can be

expressed as

η =

∫ R

0

∫ 2π

0I(r, φ)rdrdφ

∫

∞

0

∫ 2π

0I(r, φ)rdrdφ

, (3.11)

where R denotes the pinhole radius and I(r, φ) is the intensity at a point

within the pinhole detector determined by distance r from the center of the

pinhole and an angle φ. If we express the pinhole radius R in an optical

coordinate VR defined as VR = 2πρaR/(λf), where ρa denotes objective

aperture radius and f is the back focal length of the objective, then the



rf

Pinholedetector

Objective

Front focalregion

Back focalregion

in focus rays

out of focus rays

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

Uniformly filled aperture

Sig

nall

eve

l

Pinhole radius VR

(a)

(b)

Fig. 3.14: (a) A schematic diagram of a pinhole detection process. Only the rayscoming from the front focal region are detected. (b) Detected signal intensity asa function of a pinhole radius, in optical coordinates, for uniformly illuminatedobjective. Assumed objective NA = 0.8 in the front focal region, aperture sizeρa = 3 mm and the back focal length of the objective f = 160 mm.

detected signal level as a function of the pinhole size for a uniformly filled

aperture is shown in Fig. 3.14(b). This result is essentially the same as that

given by Born and Wolf [54] for the fraction of the total energy contained

within circles of prescribed radii (varying pinhole size), in the Fraunhofer

diffraction pattern of a circular aperture. Eq. 3.11 can be used with the

intensity I(r, φ) determined by our scattering model (Eq. 3.10) for any

point within the pinhole detector. Performing the appropriate integration

in Eq. 3.11 determined by the pinhole size, the detected signal level can be

evaluated for the two polarisation states of the incident illumination and a

range of pinhole sizes and scattering particles.

If we consider a polystyrene scattering particle, the scattered signal

intensity detected by a pinhole detector is shown in Fig. 3.15. It can be

seen that for very small particles (Figs. 3.15(a) and (d)) the signal is similar

to the signal for a uniformly filled aperture (Fig. 3.14(b)). This is because a

very small particle behaves as a point source and at a far-field distance the



Fig. 3.15: Detected scattered intensity as a function of the pinhole size (in anoptical coordinate) of a polystyrene particle for TE illumination (left column) andTM illumination (right column). Assumed objective NA = 0.8 in the front focalregion, aperture size ρa = 3 mm and the back focal length of the objective f = 160mm. (a) and (d) Particle radius 0.1 µm. (b) and (e) Particle radius 0.5 µm. (c)and (f) Particle radius 1.0 µm.



scattered signal fills the entrance pupil of the objective nearly uniformly. For

the wavelength-size particles (Figs. 3.15(b) and (e)) a much larger pinhole size

is required to collect the signal completely. For the typical conditions given in

Fig. 3.15 and an illumination wavelength of 633 nm (helium-neon laser), it can

be estimated that a pinhole of a radius of 80 µm is required to collect the total

signal for either TE or TM incident illumination. The scattering properties of

large particles, manifested mainly through the MDR and the interaction cross-

section effects, result in the much larger pinhole size (≈ 200 µm) required for

signal collection ( Figs. 3.15(c) and (f)). As we have seen in Figs. 3.10((d)

and (h)), the FID of large particles shows the spreading of the scattered

signal in the forward direction, which is in agreement with our qualitative

interpretation of the evanescent wave scattering by large particles (Fig. 3.3),

thus the requirement for a large pinhole size for total signal collection of the

field scattered by large dielectric particles.

3.7 Chapter conclusions

3-D scattered field distribution in near-field Mie scattering is determined by

using the theory of the elastic scattering of evanescent electromagnetic waves

with the inclusion of the effects of the interface at which the evanescent wave

is generated. It is found that the scattered intensity profile is similar to

that obtained for plane wave Mie scattering, for small dielectric particles

but becomes asymmetric when dielectric particles are larger, for particles

situated far away from the interface. For particles close to the interface

scattered intensity profiles are markedly different from those expected when

the interface is not considered for both small and large dielectric particles.

MDR is evident in the scattered field of the evanescent waves and its

period is the same as that for the plane wave scattering. The peak position,



sharpness and separation of MDR depend on the size parameter and the

effective refractive index of the particle. As the size parameter increases

resonance peaks become more pronounced and sharper. Increasing the

effective refractive index causes sharper resonant peaks but reduces the

separation between neighboring resonance peaks. It is also shown that the

main contribution to MDR results from the wave at the bottom region of

the particle because the MDR peak positions are independent of the incident

angle α.

Based on the mathematical analysis of the 3-D near-field Mie scattering,

we have developed a physical model to describe the small particle conversion

of evanescent photons into propagating photons by a small scattering particle.

This model consists of two physical processes, near-field Mie scattering

enhanced by MDR and vectorial diffraction. As a result, the far-field intensity

pattern of the collecting objective shows an interference-like pattern for a

large dielectric particle probe, while it is similar to dipole radiation for small

dielectric particles. Due to the MDR effect the energy distribution in the

detection plane is different for on and off resonance conditions. This model

provides an understanding of the evanescent photon conversion in trapped

particle SNOM and a detailed physical picture of the energy distribution in

the far-field region of a collecting objective. The theoretical predictions of our

model have been experimentally confirmed by measuring the FID of small and

large polystyrene particles.

When the detection of the scattered signal is performed by a pinhole

detector, the detector size needs to be carefully selected depending on the

scattering particle size. A very small particles act as point sources and fill the

objective entrance pupil nearly uniformly. In that case, a very small pinhole

(≈ 20-30 µm in radius) is sufficient to collect the total signal. For large

particles, on the other hand, much larger pinholes are required to completely



collect the scattered signal using a typical imaging lens. This effect is mainly

caused by the MDR and larger interaction cross-section of the particle and

the evanescent field, which leads to the spreading of the scattered signal in

the forward direction and thus to the spreading of the signal in the imaging

plane.

Furthermore, the model is applicable for determination of the near-field

force exerted on a small particle situated in the evanescent field, which will

be explored in more details in Chapter 6. It also provides a tool for designing

novel detection arrangements in the fields of NFI, optical trap nanometry and

near-field metrology with high accuracy and resolution.


Chapter 4

Trapping force with a high

numerical aperture objective

4.1 Introduction

Trapping of small particles in the far-field of a focused laser beams has been

a rich research area ever since the first ”optical tweezer” was introduced by

Ashkin et al. [25]. As we have seen in Chapter 2 (Section 2.3), there are in

principle two methods for calculating trapping forces exerted on a spherical

micro-particle, each with certain limitations and not adequate to deal with

the focal distribution complexity. Inadequacy of these methods often lies

in the severity of the approximations used in their derivations. To gain an

accurate physical insight into the trapping force a proper physical model for

particle trapping using a focused laser beam is required.

In order to generate an efficient laser trap, incident field needs to be

tightly focused using a high numerical aperture (NA) microscope objective.

Tight focusing leads to a large field intensity gradient necessary for trapping

microscopic particles. The theoretical treatment of the focusing with high

90

CHAPTER 4. Trapping force with a high numerical aperture objective

convergence angles is provided by the vectorial diffraction theory, reviewed in

Section 2.5 of Chapter 2. Thus a proper physical model for particle trapping

using a high NA objective needs to include vectorial diffraction of the incident

light by an objective. Once the exact electromagnetic (EM) field distribution

in the focal region is known its interaction with a microscopic particle can be

calculated and the field distribution on the particle surface determined [114].

The Maxwell stress tensor can then be applied to evaluate the trapping

force [24].

The aim of this chapter is to present the implementation of our vectorial

diffraction [89, 90] approach to calculate the radiation trapping forces on

a micro-particle. Such an approach enables one to consider the vectorial

properties of the EM field distribution in the focal region of a high NA

microscope objective. Effects such as the complex phase modulation on the

entrance pupil of an objective, the refractive index mismatch, i.e. spherical

aberration (SA), the polarisation dependence, and the objective apodisation

can be considered using our model without the loss of generality. Such

an exact model can deal with complex laser beams, such as the Laguerre-

Gaussian laser beams, used in the novel laser trapping experiments [16,17].

The structure of the chapter is divided into four sections beginning with

this introductory section. Our physical model for small particle trapping using

a high NA objective, including the effect of SA, is presented in Section 4.2.

Section 4.3 discusses a comparison of the vectorial diffraction approach and

another EM model based on the fifth order Gaussian beam approximation [24,

59] which is often used to estimate the trapping force exerted on a small

particle [53]. The applicability of the optical trapping model is discussed in

Section 4.4. A chapter conclusion is drawn in Section 4.5.



4.2 Model

A schematic diagram of our model that includes the SA effect is shown

in Fig. 4.1, where the geometrical focus position defines the origin of our

coordinate system, while the position of the particle is defined as the position

of its center with respect to the origin. Such a situation usually occurs when

a dry or an oil immersion objective is used to trap particles suspended in

water. A linearly polarised laser beam with the electric field vector E0 is

E0

E2

E1

n1 n2

x

y zo

Interface z=-d

Objective

Laser beam

Particle

Fig. 4.1: Schematic diagram of our trapping model.

focused by a high NA objective through an interface between two media with

the refractive indices n1 and n2. The incident laser beam is thus brought to

a sharp focus in the medium n2, in which a small particle of the refractive

index n3 is suspended. Due to the high intensity gradients induced by the

laser focusing, the particle is attracted towards the focus provided that its

refractive index is higher than that of the surrounding medium. If the particle

refractive index is lower than that of the surrounding medium, the particle is

repelled from the focal region.

Using the vectorial Debye theory and considering a linearly polarised



monochromatic plane wave focused into two media separated by a planar

interface, one can express the electric and magnetic field distributions in the

focal region of a high NA objective, if the polarisation direction is along the

X direction, as [90]

E2(rp,−d) = − ik1

2π

∫ ∫

Ω1

c(φ1, φ2, θ) expik0[rpκ+ Ψ(φ1, φ2,−d)]

sinφ1dφ1dθ (4.1)

H2(rp,−d) = − ik1

2π

∫ ∫

Ω1

d(φ1, φ2, θ) expik0[rpκ+ Ψ(φ1, φ2,−d)]

sinφ1dφ1dθ . (4.2)

Eqs. 4.1 and 4.2 are given in spherical polar coordinates where indices 1

and 2 refer to the first medium (refractive index n1) and the second medium

(refractive index n2), respectively. φ1 is the angle of incidence on the planar

interface, while φ2 is the angle of refraction. rp is the position vector while k0

and k1 are wave vectors in vacuum, and the first medium, respectively. The

focus depth is denoted by d, while functions c(φ1, φ2, θ) and d(φ1, φ2, θ) are

defined in ref [90]. Function Ψ(φ1, φ2,−d) is the so called spherical aberration

function caused by the refractive index mismatching [90] and is given by

Eq. 2.21, while κ is given by Eq. 2.25. Eqs. 4.1 and 4.2 can be further

expanded to give Eqs. 2.22 and 2.23 respectively. Any other polarisation

state can be resolved into two orthogonal states each of which satisfies Eqs. 4.1

and 4.2. If the incident field E(0) is independent on angle θ, such as the case

for a plane wave illumination, the integration in Eqs. 4.1 and 4.2 can be

reduced to Eqs. 2.26 and 2.27.



If we consider a homogeneous microsphere situated in the second medium

n2 illuminated by a monochromatic EM field described by Eqs. 4.1 and 4.2,

the net radiation force on the microsphere according to the steady-state

Maxwell stress tensor analysis is given by [24]

〈F〉 =1

4π

∫ 2π

0

∫ π

0

⟨(

ε2ErE + HrH −1

2(ε2E

2 + H2)r

)⟩

r2 sinφdφdθ ,

(4.3)

where r, φ and θ are spherical polar coordinates, Er and Hr are the radial

parts of the resulting electric and magnetic fields evaluated on the spherical

surface enclosing the particle. The net force can be further expressed as

a series over the incident and scattered field coefficients Alm, Blm, alm and

blm [24],

〈Fx〉+ i〈Fy〉a2E2

0

= +i(k2a)

2

16π

∞∑

l=1

l∑

m=−l

(√

(l +m+ 2)(l +m+ 1)

(2l + 1)(2l + 3)l(l + 2)

× (2ε2alma∗

l+1,m+1 + ε2almA∗

l+1,m+1 + ε2Alma∗

l+1,m+1 + 2blmb∗

l+1,m+1

+ blmB∗

l+1,m+1 +Blmb∗

l+1,m+1) +

√

(l −m+ 1)(l −m+ 2)

(2l + 1)(2l + 3)l(l + 2)

× (2ε2al+1,m−1a∗

lm + ε2al+1,m−1A∗

lm + ε2Al+1,m−1a∗

lm + 2bl+1,m−1b∗

lm

+ bl+1,m−1B∗

lm +Bl+1,m−1b∗

lm)−√

(l +m+ 1)(l −m)ε2(−2almb∗

l,m+1

+ 2blma∗

l,m+1 − almB∗

l,m+1 + blmA∗

l,m+1 +Blma∗

l,m+1 − Almb∗

l,m+1)

)

, (4.4)

and



〈Fz〉a2E2

0

= −(k2a)2

8π

∞∑

l=1

l∑

m=−l

Im

(√

(l −m+ 1)(l +m+ 1)

(2l + 1)(2l + 3)l(l + 2)

× (2ε2al+1,ma∗

lm + ε2al+1,mA∗

lm + ε2Al+1,ma∗

lm + 2bl+1,mb∗

lm

+ bl+1,mB∗

lm +Bl+1,mb∗

lm) +√ε2m(2almb

∗

lm + almB∗

lm + Almb∗

lm)

)

, (4.5)

.

The incident and scattered field coefficients are defined as

Alm =1

l(l + 1)ψl(k2a)

∫ 2π

0

∫ π

0

sin θE2r(a, θ, φ)Y ∗

lm(θ, φ)dθdφ , (4.6)

Blm =1

l(l + 1)ψl(k2a)

∫ 2π

0

∫ π

0

sin θH2r(a, θ, φ)Y ∗

lm(θ, φ)dθdφ , (4.7)

alm =ψ′

l(nk2a)ψl(k2a)− nψl(nk2a)ψ′

l(k2a)

nψl(nk2a)ξ(1)′

l (k2a)− ψ′

l(nk2a)ξ(1)l (k2a)

Alm , (4.8)

and

blm =nψ′

l(nk2a)ψl(k2a)− ψl(nk2a)ψ′

l(k2a)

ψl(nk2a)ξ(1)′

l (k2a)− nψ′

l(nk2a)ξ(1)l (k2a)

Blm . (4.9)

ξ(1)l = ψl − iχl, where ψl and χl are the Riccati-Bessel functions. Ylm is the

spherical harmonic function. n is defined as n = (ε3/ε2)1/2.

All calculations are performed using standard computational methods for

integral evaluations. The trapping efficiency defined as a dimensionless factor



Q is given by

Q =cF

n2P, (4.10)

where c denotes the speed of light in vacuum, F is the trapping force and P

is the incident laser power at the focus. When trapping efficiency is evaluated

in the transverse direction it is known as the transverse trapping efficiency

(TTE), while when evaluated in the axial direction it is known as the axial

trapping efficiency (ATE). One can distinguish between two ATE; the forward

ATE (positive value) corresponding to the inverted microscope configuration

(particle pushing) and the backward ATE (negative value) corresponding to

the upright microscope configuration (particle lifting).

4.3 Vectorial diffraction - Gaussian approxi-

mation comparison

Using the methodology described in the previous section, one can incorporate

various input field characteristics, such as the apodisation function, complex

amplitude and the aberration function into the model using vectorial

diffraction (Eqs. 4.1 and 4.2). The incident field is thus represented exactly,

without approximations, resulting in the precise calculation of the trapping

force.

Since the incident illumination on a microsphere, given in Eqs. 4.1 and 4.2,

differs from the fifth-order corrected Gaussian approximation used by Barton

et al. [24](Fig. 4.2), it can be expected that the respective trapping efficiencies

predicted by our model are different. In Fig. 4.3 a comparison between

the fifth-order Gaussian and vectorial diffraction approaches in the case of

polystyrene particles suspended in water is presented. The Gaussian beam

waist is assumed to be ω0 = 0.4 µm while the vectorial diffraction method



(b)(a)

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsi

ty(a

.u.)

Z distance (mm)

Fig. 4.2: Intensity distributions in (a) axial and (b) transversal directions (blue-Xaxis, red-Y axis) for the fifth-order Gaussian approximation (dashed line) and thevectorial diffraction theory (solid line).

Fig. 4.3: Comparison between the fifth-order Gaussian approximation (emptysymbols) and the vectorial diffraction theory (filled symbols) for the calculationof the maximal TTE (triangles) and the backward ATE (circles) of polystyreneparticles suspended in water. λ0 = 1.064 µm, ω0 = 0.4 µm and NA = 1.2.



assumes the numerical aperture NA = 1.2, which gives approximately the

same focal spot size (Fig. 4.2). For small particles both methods give nearly

the same TTE and ATE, which shows an r3 dependence as expected for

Rayleigh sized particles [53]. When their size approaches the illumination

wavelength the two methods differ significantly. However, in the case of very

large particles (r = 100 µm,) the extrapolation of the vectorial diffraction

method (dotted lines in Fig. 4.3) approaches the RO prediction.

4.4 Model applicability

Using our optical trapping model based on the vectorial diffraction theory,

the far-field optical trapping process with a high NA objective can be

investigated without approximations. Various types of incident illumination

employed in laser trapping experiments, such as incident plane wave and

doughnut beam illuminations with and without annular masks [23] can be

considered. These types of incident illuminations cannot be considered

using the current approximate methods such as the fifth-order Gaussian

approximation method. While the RO method is able to give approximate

solutions for these types of incident illuminations for very large particles, it is

completely inadequate to consider small particles which are most commonly

used in laser trapping experiments. The investigation of the far-field optical

trapping will be presented in more details in Chapter 5.

Our optical trapping model can be applied to investigate the near-field

trapping with a focused evanescent illumination as well. This can be achieved

by considering an annular mask (also called a central obstruction) of a

sufficient size to ensure that the minimum angle of convergence of each

incident ray is larger than the critical angle determined by the total internal

reflection condition between two media (objective immersion and particle



suspension media). This arrangement results in a focused near-field generated

at the interface between the two media [15], and our model can calculate

the force exerted on a small particle due to its interaction with such a

field. Similar to the far-field trapping, our model can consider an incident

illumination with arbitrary complex input phase, polarisation and apodisation

with this type of near-field trapping. Such a near-field trapping method has

not been considered previously using any other theoretical method and will

be dealt with in greater details in Chapter 6.

The model presented in this chapter is not limited to plane wave and

doughnut incident laser beam illuminations, considered in greater details in

this thesis. It is actually applicable for an arbitrary field incident at a high

NA objective entrance pupil.


This chapter presents an exact method for the radiation trapping force

calculation. The EM field distribution in the focal region of a microscope

objective is determined using the vectorial diffraction theory and the optical

trapping force is evaluated using the Maxwell stress tensor approach.

The vectorial diffraction method offers a number of advantages over an

approximate method such as the fifth-order Gaussian beam incident field

approximation. Firstly, the vectorial diffraction model approaches the RO

predictions at a large particle limit, to a better degree than it can be achieved

by the approximated Gaussian beam model. Secondly, and most importantly

is that it provides the appropriate treatment of the incident illumination

phase modulation, polarisation and apodisation as well as the SA occurred

in trapping experiments for both small and large particles.



Thus, the model enables the appropriate modelling of various far-field

trapping arrangements, focused near-field trapping or the trapping systems

implemented using the novel spatial phase modulation techniques, such as

the trapping systems with focused doughnut beam illumination.


Chapter 5

Far-field optical trapping

5.1 Introduction

In the previous chapter (Chapter 4) the vectorial diffraction method is

presented for calculating the optical trapping force with a high numerical

aperture (NA) objective. We have seen that our optical trapping model can be

used to calculate the far-field trapping force exerted on small particles without

using any approximations. The model incorporates the full representation

of the field in the focused region, and can be used with any particle size,

given a sufficient computing power to calculate the scattered field. Inherently,

the model can deal with a complex phase modulation of the incident field,

apodisation of an objective and different polarisation states of incident

illumination. In this chapter, we will apply our optical trapping model

to investigate the far-field trapping process under the presence of spherical

aberration (SA). The effect of a central obstruction in the beam path, which

causes an elongation of the focused spot in the axial direction, will be studied.

Furthermore, the optical trapping efficiencies of doughnut laser beams, such

as the ones used in the novel laser trapping arrangements [16, 31, 32], are

101

CHAPTER 5. Far-field optical trapping

also investigated. The comparison between a centrally obstructed plane wave

and a doughnut beam is presented and the discrepancy between our optical

trapping model and the ray optics (RO) model is discussed.

Other theoretical models for the far-field optical trapping, reviewed in

Section 2.3 of Chapter 2, are inadequate to deal with complex laser fields,

such as the focused laser beam under the influence of spherical aberration,

phase modulation, and objective apodisation. Thus, to our knowledge, there

exist no comprehensive electromagnetic (EM) theoretical treatment of the far-

field optical trapping with doughnut beam illumination. The only theoretical

treatment of such illumination type is the one given by Ashkin et al. [23]

under the RO approximation, which is inadequate because it ignores the EM

field distribution in the focal region.

The influence of a central obstruction was previously considered using

a RO approach. However, the RO model does not consider the EM field

distribution in the focal region, i.e. the elongation of the focal spot,

which limits the RO approach severely. Dependence of trapping force on

the obstruction size of a centrally obstructed laser beam is studied both

theoretically, using our optical trapping model, and experimentally in the

case of plane wave and doughnut beam illuminations. The trapping efficiency

under such conditions is compared with the RO model, and it is found that

such a model is completely inadequate. The vectorial diffraction method, on

the other hand, agrees very well with the experimental results.

This chapter is organised as follows. Section 5.2 investigates the trapping

force with plane wave illumination focused by a high NA objective. The

effects of the refractive index mismatch between the objective immersion and

particle suspension media, i.e. SA, on the trapping force are investigated

theoretically and compared with the experimental results of Felgner et

al. [115] in Section 5.2.2. Trapping force with a centrally obstructed plane



wave is examined in Section 5.2.3, both numerically and experimentally.

Section 5.3 studies the trapping efficiency obtained with doughnut beam

illumination, while the efficient generation of such an illumination and

the effects of vectorial diffraction of doughnut beams are presented in

Sections 5.3.1 and 5.3.2, in order to confirm the predictions experimentally

and gain a physical insight into the trapping process with doughnut beam

incident illumination. Section 5.4 presents the chapter conclusions.

5.2 Trapping force with plane wave illumina-

tion

Plane wave illumination is a most common type of illumination used in the

optical tweezers experiments. However, only the vectorial diffraction model

can deal with this case exactly in the wave optics regime. This section deals

with the force mapping of small and large dielectric particles, the SA effects

and the influence of a central obstruction on the trapping performance.

5.2.1 Force mapping

According to our analysis of the dependence of the maximal trapping

efficiency on particle size (Fig. 4.3), both the maximal transverse trapping

efficiency (TTE) and the backward axial trapping efficiency (ATE), are

greatly reduced when a particle becomes small. However, even though such

a presentation is related to the effect that is measurable in the experiments,

it does not give a clear physical picture of how the force depends on the

relative position of the geometrical focus and the particle. Such a physical

picture was presented by Ashkin et al. in the case of very large particles

and was investigated using the RO model [23], which is not applicable for



particles whose size is comparable to the illumination wavelength or smaller,

and is particle size invariant. However, it could be expected that the

force dependence of the relative position of the geometrical focus and the

particle is markedly different for small and large particles due to the EM field

distribution in the focal region.

0 0.50 0.5

(a)

00 0.08

(b)

Fig. 5.1: Magnitude and direction of the trapping efficiency for various geometricalfocus positions around a polystyrene particle suspended in water and illuminatedby a λ0 = 1.064µm laser beam focused by a NA = 1.25 water immersion objective.(a) particle radius of 2 µm. (b) Particle radius of 200 nm.

Two polystyrene particle sizes (a = 2 µm and a = 200 nm) suspended in

water illuminated by a λ0 = 1.064 µm laser beam focused using NA = 1.25

water immersion objective (Fig. 5.1(a) and 5.1(b)) are considered. For large

particles, the magnitude and direction of the trapping force is similar to the

one given by the RO model and it is seen that the particle is most strongly

influenced when its boundary is situated near the geometrical focus, while

away from the boundary the trapping force falls rapidly (Fig. 5.1(a)). Such a



rapid decrease in the trapping force magnitude is not present when one deals

with small particles (Fig. 5.1(b)). Even at a distance of twice the particle

radius, the magnitude of the trapping efficiency is relatively unchanged. This

is because the particle is much smaller than the focal field distribution so

that even at a geometrical focus distance of 2a, the particle-field interaction

is significant.

5.2.2 Spherical aberration

SA plays an important role in laser trapping because most of the trapping

experiments are performed under conditions where the refractive index

mismatch occurs. Usually a high NA oil immersion objective is used for

trapping while microparticles are suspended in water. The refractive index

difference between the immersion and the suspending media leads to the SA

when a trapping beam is focused deep into the suspending medium, which

are manifested as focal spot distortions, and degrade the trapping efficiency

of an optical trap [116].

Using the approach given in Eqs. 4.1 and 4.2, one can incorporate the

effect of SA on the trapping force at an arbitrary depth in the suspending

medium. Without considering the effect of SA produced when a refractive

index mismatch exists, as it does in most of the experimental measurements,

one not only overestimates the value of the trapping efficiency but also lose

its physical dependence on other factors such as the particle size. As it can

be seen in Fig. 5.2, by including the effect of SA the calculated backward

ATE agrees with the experimental data given by Felgner et al. [115] under

the same conditions. Furthermore, the effect of the morphology dependent

resonance (MDR) is pronounced for the particles whose size is of the order of

wavelength in the case when SA is included. The stronger MDR effect is due

to the better coupling of the incident field into the microsphere when SA is



present, because the focal region is larger and the field interaction with the

edge of the sphere is more pronounced.

Fig. 5.2: Maximal backward ATE of glass particles suspended in water, illuminatedby a laser beam (λ0 = 1.064 µm), focused by an oil immersion microscope objective(NA = 1.3). The effect of SA is considered at a depth of 9 µm from the cover glass.

A dependence of the trapping efficiency on the trapping distance from the

cover glass, for both axial (Fig. 5.3(a)) and transverse (Fig. 5.3(b)) directions

are investigated and compared with the experimental results given by Felgner

et al. [115]. Note that the magnitude of the error bars of the experimental

results depends on the trapping distance from the cover glass because the

measurement of the trapping force close to the cover glass is more uncertain

than that deeper into the suspending medium. In the calculation, an oil

immersion objective with NA = 1.3 and an illumination wavelength λ0 =

1.064 µm are assumed, while the refractive indices are assumed to be 1.52

for oil and cover glass (index matched), 1.57 for polystyrene, 1.51 for glass,

1.33 for water and 1.41 for 60% glycerol. The calculated maximal ATE as

a function of the distance from the coverslip for the trapping of a spherical

glass particle of diameter D = 2.7 µm and suspended in water agrees (within



Fig. 5.3: Maximal backward ATE and TTE of a particle illuminated by a laser(λ0 = 1.064 µm) focused by an oil immersion microscope objective (NA = 1.3) asa function of the distance from the cover glass. (a) A glass particle of diameterD = 2.7 µm in water. (b) A polystyrene particle of diameter D = 1.02 µmsuspended in 60% glycerol solution.

its error bars) with the measured results (Fig. 5.3(a)). This agreement at a

large distance from the cover glass (deeper into the suspending medium) is

better than that near the surface.

Due to the difference in the transverse EM field distribution in the focal

region of a high NA objective (Fig. 4.2), the maximal TTE in the polarisation

direction (Qx) and in the direction perpendicular to the polarisation direction

(Qy), calculated for a polystyrene particle of D = 1.02 µm suspended in a

60% glycerol solution, is different (Fig. 5.3(b)). The TTE in the Qx direction

is generally smaller than that in the Qy direction. This difference is due to

the elongation of the focal spot in the X direction. Such a difference of the

TTE is larger near the surface. Although the absolute value of the TTE

is somewhat larger than the experimental result, which may be caused by

our assumptions about the microscope objective and the suspending medium

characteristics (60% glycerol solution for which it is difficult to estimate

the viscosity and the refractive index precisely), the maximal TTE trend is



consistent with the experimentally measured trend if the former is normalised

by the experimental result obtained at a deep distance.

5.2.3 Trapping efficiency with centrally obstructed plane

wave

5.2.3.1 Numerical results

It has been predicted by the early researchers in the particle trapping using a

highly focused laser beam that uses of an obstructed laser beam decreases the

transverse trapping force exerted on dielectric particles [23]. Their treatment

of this problem was based on the RO approach, and they show that the

projection of the net trapping force in the transverse direction decreases with

the angle of a ray of convergence. The obstruction (ε) is achieved by centering

an opaque disk perpendicular to the beam propagation axis and it is defined as

the ratio of the radius of the obstructed part to the radius of the unobstructed

part of the beam.

The decrease in the maximal TTE for a large polystyrene particle,

predicted by the RO model, is approximately 20% for the case when the laser

beam is not obstructed and when it is nearly completely obstructed. However,

the inherent nature of the RO approach is to treat the focal distribution of

a highly focused laser beam as a geometric point. Such an approach may be

appropriate when the particle is several times larger than the wavelength of

the incident light and the incident beam is not centrally obstructed. However,

due to the fact that the focus distribution of a highly convergent beam

is significantly affected by obstructing the low angle rays, the RO model

approach is not appropriate for a centrally obstructed incident beam. The

presence of the obstruction leads to more depolarised rays reaching the focus.

Inclusion of only the high angle rays leads to the focus distribution that has a



sharper focus but enhanced airy rings in the transverse plane and is elongated

in the axial direction (Fig. 5.4). For larger obstructions this elongation can

become significant enough to lead to the drastically incorrect TTE prediction

if one relays on the RO model.

Fig. 5.4: Focus intensity distribution for a plane wave focused by a high NAobjective immersed in water (NA = 1.25). Top row - unobstructed plane wave(ε = 0.0). Bottom row - obstructed plane wave (ε = 0.8).

Figure 5.5 shows the theoretical results of the RO model and the vectorial

diffraction model for a polystyrene particle of 2 µm diameter immersed in

water and illuminated by a highly convergent laser beam with NA of 1.25 and

the wavelength λ = 532 nm. Immediately, two features are clearly evident.

Firstly, the two models predict a very different behavior of the maximal TTE

for increasing obstruction size. Secondly, the RO model predicts a small

difference in the maximal TTE for the two polarisation states of incident

illumination, with the S polarisation resulting in a slightly larger transverse

force, while the vectorial diffraction model predicts a substantial difference for



the two polarisation states and predicts a considerably larger transverse force

for the P polarisation when no obstruction is present. For larger obstructions

the difference in the maximal TTE for the two polarisation states diminishes,

due to different depolarisation behaviors between the S and P directions.

Fig. 5.5: Theoretical calculation (RO model and vectorial diffraction model) of themaximal TTE as function of obstruction size for a polystyrene particle of radius 1µm immersed in water. NA = 1.25 and λ = 532 nm. The maximal TTE for thetwo models are normalised to start from the same point (at ε = 0.0).

The depolarisation feature in a focal region of an objective can be

described by the vectorial diffraction theory reviewed in Section 2.5. Fig. 5.6

shows the transverse intensity distribution in a focal plane of an objective

for different obstruction sizes. The diminishing difference in the maximal

TTE of the two polarisation states for larger obstructions is caused by two

processes. The first process that determines the trapping force is the overall

field distribution incident on a particle (Fig. 5.4). Since the incident field

in the transverse focal plane is elongated, it leads to a different maximal

TTE for S and P polarisations (or for S and P scanning directions for a

given linear polarisation). The second process is the intensity gradient along



Fig. 5.6: Focus intensity along a transverse direction for various obstruction sizes.(a) Polarisation direction X and (b) Perpendicular to polarisation direction Y.NA = 1.25 and λ = 532 nm.

a particular direction (Fig. 5.6). For large obstructions the change in the

overall focal field distribution causes the reduction in the trapping efficiency.

However, the intensity gradient along the polarisation direction (Fig. 5.6(a)) is

only slightly reduced, while in the direction perpendicular to the polarisation

direction (Fig. 5.6(b)) it increases significantly. The combined effect is that

the maximal TTE decreases for both directions, S and P, due to the change in

the overall incident field distribution. However, due to the increased intensity

gradient along the S direction the decrease in this direction is slower, which

results in the diminishing difference between the two polarisation directions

(Fig. 5.5).

5.2.3.2 Experimental results

To confirm the theoretical results predicted by the two models, we have

measured the maximal TTE of polystyrene beads of 2 µm in diameter

suspended in water (Polybead polystyrene microspheres, diameter=2.134 µm

with standard deviation δ=0.039 µm. However, from now on we will refer



CCD

StageController

La

se

r

PC

Monitor

Phasemodulator

Objectivehigh NA

Controller

Scanning stage

Immersionoil

Particlecell

Obstruction

Coverslip

Dichroicbeam-splitter

L1

L2

L3

Fig. 5.7: A schematic diagram of the experimental setup.

to these beads as 2 µm beads). A schematic diagram of the experimental

system is depicted in Fig. 5.7. The illumination light beam from a 532

nm continuous-wave laser is expanded to a parallel beam by the lenses L1

(microscope objective with NA = 0.2) and L2 (focal length 50 mm). The

beam is then reflected by a phase modulator and directed into a high NA

objective lens (NA = 1.2, water immersion objective). In the plane wave case

the phase modulator was replaced by a mirror. The incident beam on the

high NA objective is focused into a sample cell containing polystyrene micro-

spheres of 2 µm diameter, suspended in water. The sample cell is mounted

on a PC-controlled scanning stage while the trapping process is monitored

using a CCD camera. A circular obstruction disk is coaxially inserted in the



beam path and the maximal TTE is measured using the standard Stokes law

method.

The measurement based on the Stokes method is performed in such a

way that a particle is trapped with a laser beam of fixed power and then the

particle is transversally scanned across the sample cell. The scanning speed

is increased until the particle falls out of the trap. The maximum transverse

trapping force F on a trapped particle is then calculated using the Stokes law

F = 6πavµ, where a is the radius of the trapped particle, v is the maximum

translation speed and µ is the viscosity of the surrounding medium [117]. The

presented results are averaged over three measurements and the uncertainties

are estimated from the laser power fluctuations.

The maximal TTE for various obstruction sizes, experimentally measured

using the setup in Fig. 5.7, is shown in Fig. 5.8. It can be seen that the TTE

Fig. 5.8: Experimental measurement of the maximal TTE as a function ofobstruction size for a polystyrene particle of radius 1 µm immersed in water.Theoretical values are normalised by the experimental P value at ε = 0.0. NA = 1.2and λ = 532 nm.



decreases rapidly for increasing obstruction size, unlike the trend predicted by

the RO model. The experiment confirms that the vectorial diffraction model

agrees well with the measured results. Furthermore, the difference in the

maximal TTE for the two polarisation states is verified in the experiment,

with the maximal TTE for the S polarisation being approximately 85% of

that for the P polarisation state when no obstruction is present (ε = 0.0).

Similar to the theoretically predicted trend, the difference in the maximal

TTE for the two polarisation states diminishes for larger obstruction sizes.

5.3 Trapping force with doughnut beam illu-

mination

Doughnut laser beams have got their name because of their characteristic

intensity profile. The electric field of a doughnut beam can be expressed as

E = E exp(imθ), where θ is the polar coordinate in the plane perpendicular

to the beam axis and m is called the topological charge. Such beams are

also known as Laguerre-Gaussian (LG) beams and are becoming increasingly

popular in the novel laser trapping arrangements [16,31,32]. To theoretically

investigate the trapping force with doughnut beam illumination focused by a

high NA objective, vectorial diffraction of doughnut beam illumination needs

to be studied. For experimental studies, however, an efficient generation of

doughnut beams is required.

In this section, a novel method to generate doughnut beams with 100%

efficiency is described, the diffraction effects when focusing doughnut beams

by a high NA objective are discussed, as well as the trapping efficiency of

doughnut beams.



5.3.1 Doughnut beam generation

In this section we demonstrate a novel and convenient method for producing

a doughnut laser beam using a liquid crystal (LC) cell which is capable of

dynamically controlling the phase distribution. We also demonstrate the

use of a similar method for doughnut beam generation using a spatial phase

modulator (Hamamatsu PPM X8267 Series).

5.3.1.1 Generation of doughnut beams using a liquid crystal cell

The LC-based photonic devices are increasingly used in many applications in-

cluding light focusing [118,119], phase modulation [120], beam steering [121],

and filtering [122]. A LC based element works by applying an electric field

between two walls of a cell containing appropriately oriented liquid crystals.

The applied electric field causes LC molecules to tilt, which results in a

change in refractive index. By controlling this refractive-index change one

can provide an appropriate phase shift to the incoming wavefront in order to

produce a doughnut laser beam.

If one can make a phase mask by controlling LC molecules in such a

way to produce, for example, a gradual phase change from 0π to 2πm in

a circular fashion across the incoming beam wavefront (Fig. 5.9(a)), then

a helical wavefront of topological charge m would result. To achieve this

conversion of a plane wave (m = 0) into a doughnut beam (m 6= 0) we have

made a LC cell with an indium-tin-oxide (ITO) structure consisting of 16 pie

slices on the front side of the cell (Fig. 5.9(b)). The cell diameter was 1 cm.

The structure was made by a laser lithography process using a chrome mask.

The two contact points were connected to the first and the last pies slices,

while all the slices were connected by a narrow (≈ 10µm) strip of ITO. When

a voltage was supplied to the contact points this narrow strip of ITO had high



0p

(a) (b) (c)

Contact points

NarrowITO layer

Pie slice

216

15

14

12

11

109 8

7

6

5

4

3

1

+10 V0 V

Fig. 5.9: Phase distribution of a doughnut beam. (a) The theoretical phasedistribution of a doughnut beam of charge 1 according to 16 phase steps. (b)The electrode structure of the liquid crystal cell with 16 pie slices. (c) Thephase wavefront of the doughnut beam of charge 1, measured using phase shiftinginterferometry.

resistance and gave a linear voltage drop from the first to the last pie slices,

as shown in Fig. 5.10(a). On the other hand, the pie slices themselves were

much wider and offered very low resistance. The back side of the cell was

made of a uniformly coated layer of ITO on a glass substrate. This uniform

ITO layer was connected to the ground. The cell was then filled with LC

molecules and sealed.

When an appropriate voltage is applied to the connection points on the

front side of the cell, it then re-distributes across each of the pie slices. LC

molecules inside the cell tilt according to the electric field strength between

the pie slices and the grounded back wall of the cell. The amount of the

tilt depends on the voltage on each pie slice, giving rise to a corresponding

change in refractive index for the light polarised along the long axis of the

liquid crystals. For a given voltage such as 10 V , one can select a proper



He-Ne Laser

P

BS1 LC BS2

M1 M2

O

L

S

PH

0 2 4 6 8 10

0

1

2

3

4

5

6

Phase

shift

(p)

Voltage (V)

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

Volta

ge

(V)

Pie Slice Number

(a) (b)

Fig. 5.10: Experimental setup for generation of a doughnut beam through the liquidcrystal cell and interference measurement of its phase distribution (P: polariser;BS1 and BS2: beam splitters; LC: liquid crystal cell; O: objective; L: lens; M1 andM2: mirrors; PH: pinhole; S: screen). (a) The voltage variation as a function ofthe slice position of the liquid crystal cell. (b) The unwrapped phase shift of theliquid crystal cell as a function of applied voltage.

thickness of the LC cell to produce a phase change from 0π at the first slice

to 2π at the last slice, so that the outgoing wavefront has a helical shape with

topological charge 1.

To characterise the dependence of the phase change on the voltage, a

LC cell of thickness 9.5 µm was placed between two crossed polarisers and

the intensity variation after the analyser as a function of the applied voltage

was measured. After the phase unwrapping, this dependence gives a direct

relation between the applied voltage and the phase shift of the beam (λ =



632.8 nm), which is shown in Fig. 5.10(b). It is clear that the LC cell can

produce a maximum of a 4π phase shift, although there exist a saturated

response caused by the nonlinear response of the liquid crystals. To extend

the linear response region, one can increase the thickness of the cell or the

concentration of the liquid crystals.

(a) (b) (c)

(e)(d) (f)

Fig. 5.11: The intensity distributions (a, b and c) of laser beams transmittedthrough a liquid crystal cell and the corresponding interference patterns (d, e andf). (a) and (d) plane wave. (b) and (e) Doughnut beam of charge 1. (c) and (f)Doughnut beam of charge 2.

To demonstrate the dynamically switching nature of the LC cell, the

LC cell is placed in an optical setup shown in Fig. 5.10. A He-Ne laser

beam (632.8 nm) of output power 5 mW was used for illumination and was

linearly polarised by a polariser (P) in the direction of the liquid crystals. It

passed through the first beam splitter (BS1) to create a reflection arm and

a transmission arm. The transmitted part of the beam passed through the

LC cell that modified the wavefront to create a doughnut beam on a screen

(S). For the interference pattern measurement, the reflected beam at BS1

was recombined with the transmitted part at BS2 to create an interference



pattern. The resulting beam was filtered using the microscope objective O

(NA = 0.2) and projected onto the screen using the lens L (focal length 100

mm).

When the reflection arm is blocked, the recorded patterns are displayed in

Figs. 5.11(a), 5.11(b) and 5.11(c). If there is no voltage applied to the LC cell,

the incoming laser beam wavefront is not changed (Fig. 5.11(a)). Applying an

appropriate voltage to the cell according to Fig. 5.10(b) results in a phase shift

of 2π or 4π, so that the wavefront after the cell can be dynamically converted

into a doughnut beam of charges 1 and 2 (Figs. 5.11(b) and 5.11(c)). A slight

distortion of the circular symmetry in Fig. 5.11(c) is caused by the saturated

response in Fig. 5.10(b). It was observed that the power of the generated

doughnut beams was almost the same as that of the plane wave, which leads

to a conversion efficiency near 100%.

To confirm the helical nature of the generated wavefront, we introduced

the interference arm as shown in Fig. 5.10. The measured interference

patterns corresponding to Figs. 5.11(a), 5.11(b) and 5.11(c) are shown in

Figs. 5.11(d), 5.11(e) and 5.11(f), respectively. When the LC is switched off,

the interference pattern shows the interference fringes of an equal spacing,

resulting from two plane waves (Fig. 5.11(d)). The fringe splitting in

Figs. 5.11(e) and 5.11(f) indicates that the LC modulated beam becomes

a doughnut beam and the number of splitting fringes gives the number of

topological charges [123]. A direct wavefront test was also performed using

phase shifting interferometry method [124]. The reconstructed wavefront

of the charge-one doughnut beam (Fig. 5.9(c)) is similar to the theoretical

wavefront in Fig. 5.9(a).

Another feature of the designed LC cell is tunable over a range of

wavelengths. By simply changing the applied voltage, a doughnut beam

(m = 1) at wavelengths 488 nm and 800 nm was produced, as depicted



450 500 550 600 650 700 750 800

2.3

2.4

2.5

2.6

2.7

2.8

2.9

DV

(V)

l (nm)

Fig. 5.12: Variation of the voltage between the two contact points (see Fig. 5.9(b))as a function of the wavelength for the generation of a doughnut beam of charge 1.

in Fig. 5.12. The dependence of the voltage change on the illumination

wavelength shown in Fig. 5.12 indicates a reduced efficiency at wavelength

800 nm, implying that the liquid crystals used in this experiment exhibits a

certain amount of dispersion and a nonlinear response near this wavelength.

5.3.1.2 Generation of doughnut beams using a spatial phase

modulator

A phase-ramp pattern, similar to the one described in the previous subsection,

can be applied to a spatial phase modulator (SPM). A computer controlled

SPM is able to modulate the phase of an incoming plane wave, according to

the pattern loaded into its RGB port. Such device is increasingly used in

the novel laser trapping and manipulation experiments [31]. However, to our

knowledge, it was not yet used with the phase-ramp method described in the

subsection 5.3.1.1.



The advantage of using an SPM to modulate the phase according to the

phase-ramp pattern is in its ease of use and ability to use a greater number

of phase levels. With the Hamamatsu SPM we can achieve 256 phase levels

(Fig. 5.13), compared to the 16 levels with the LC cell. The LC cell is in

principle also capable to achieve a greater number of levels, but at a cost of

a much greater complexity in its design and manufacture.

(b)(a) (c)

Fig. 5.13: Doughnut beam of charge 1 generated using a computer controlledSPM. (a) Applied phase-ramp pattern with 256 levels. (b) Intensity profile. (c)Interference pattern.

Figure 5.13 shows a doughnut beam of topological charge 1, achieved

using a reflection type SPM (Hamamatsu PPM X8267 Series) with a 256

levels phase-ramp. The interference pattern of a such generated doughnut

beam and a plane wave reveals a characteristic fringe splitting confirming

that the generated beam is of topological charge 1.

5.3.2 Vectorial diffraction of doughnut beam illumina-

tion

It is well known that due to the helical phase distribution of a doughnut

beam its intensity distribution gives zero on the axis, when it is focused by

a low NA lens [89]. Physically, when a high NA lens is used the electric

field in the focal region exhibits a component along the incident polarisation



direction as well as an orthogonal and a longitudinal component, which is

called depolarisation.

Using the vectorial Debye theory, reviewed in Chapter 2, one can express

the electric field distribution in the focal region of a linearly polarised

monochromatic doughnut beam focused by a high NA objective satisfying

the sine condition, if the polarisation is along the x direction, as [86,89]

E2(r2, ψ, z2) =i

λ

∫ ∫

Ω

√

cosφ exp(imθ) exp[−ikr2 sinφ cos(θ − ψ)]

exp(−ikz2 cosφ)[cosφ+ sin2 θ(1− cosφ)]i + cos θ sin θ(cosφ− 1)j

+ cos θ sinφk sinφdφdθ (5.1)

where i, j, and k are unit vectors in the x, y, and z directions respectively.

Variable r2, ψ, and z2 are the cylindrical coordinates of an observation point.

Any other polarisation state can be resolved in two orthogonal directions each

of which satisfies Eq. 5.1.

The intensity is proportional to the modulus squared of Eq. 5.1 and is

shown in Fig. 5.14 for doughnut beams of different topological charges and

numerical aperture. When the numerical aperture of the focusing objective is

low, the focal intensity distribution shows a well-known doughnut shape [89],

which depends on topological charges. If, on the other hand, the numerical

aperture of the focusing objective is high, the intensity distribution in the

focal region becomes distorted and loses singularity for certain topological

charges. It can be seen from Fig. 5.14 that the intensity in the doughnut

ring increases along the direction perpendicular to the incident polarisation

state and that the focal spot is elongated along the polarisation direction.

Furthermore, the doughnut beam of topological charges 1 and 2 (Fig. 5.14(a)

and Fig. 5.14(b)) looses its zero intensity on the beam axis when focused

by an objective of NA = 1 in air. For a beam of topological charge ±1,



Fig. 5.14: Calculated intensity distribution in the focal region of a doughnut beamfocused by an objective with NA = 1 ((a)-(c)) and NA = 0.2 ((d)-(f)): (a) and(d) Topological charge 1; (b) and (e) Topological charge 2; (c) and (f) Topologicalcharge 3.

its intensity in the center of the focal region equals 48.8% of the maximum

intensity (Fig. 5.14(a)), while for a beam of topological charge ±2 it drops

down to 13.5% of the maximum intensity (Fig. 5.14(b)). When the topological

charge becomes −3 ≥ m ≥ 3, the intensity in the center of the focal region

regains its zero value (Fig. 5.14(c)). The zero intensity in the center of the

focal region has been observed by calculating focal intensity distributions

for higher doughnut beam topological charges (m = 4 and m = 5). This

topological charge dependence of the intensity in the central focal region can

be understood from Eq. 5.1.

Considering the point at the focus at r2 = 0 and z2 = 0, we find

that |Ex|2 = 0, |Ey|2 = 0, and |Ez|2 6= 0 when m = ±1 because∫ 2π

0cos(mθ) cos θdθ = 0 for all m, except for m = ±1. In the case of

m = ±2, the electric field components in the focal region give |Ex|2 6= 0,



|Ey|2 6= 0, and |Ez|2 = 0 because∫ 2π

0cos(mθ) sin2 θdθ in the Ex component

and∫ 2π

0cos θ sin θ sin(mθ)dθ in the Ey component are non-zero for m = ±2.

For −3 ≥ m ≥ 3, all integrals over θ give zero, which leads to the zero values

of all the E field components at the center of the focal region.

A more detailed picture of the intensity |E|2 normalised to 100 and its

components |Ex|2, |Ey|2 and |Ez|2 near the focal region of an objective (NA =

1 in air) illuminated by a charge 1 doughnut beam polarised in the x direction

is shown in Fig. 5.15. The contour plots are presented in terms of transverse

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(a)

0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

Vx

Vy

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(b)

0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

Vx

Vy

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(c)

0

6.3

12.5

18.8

25.0

31.3

37.5

43.8

50.0

Vx

Vy

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(d)

0

12.5

25.0

37.5

50.0

62.5

75.0

87.5

100.0

Vx

Vy

Fig. 5.15: Contour plots of the intensity distribution in the focal region of anobjective with NA = 1, illuminated by a doughnut beam of topological charge 1.(a) |Ex|2;(b) |Ey|2;(c) |Ez|2;(d) |E|2.

optical coordinates Vx and Vy, which are defined as Vx,y = k[x, y] sinαmax,

where k is the wave number and αmax is the maximum angle of convergence. It

is evident that due to the high convergence angle, |Ex|2 and |Ez|2 components



play a dominant role in shaping the overall intensity |E|2. The high intensity

of |Ez|2 in the central region of the geometrical focus causes the non-zero value

in the central region of the overall intensity distribution, and together with

|Ex|2 component leads to a ”two-peak focus” in the Vy direction (Fig. 5.15(d)).

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(a)

0

8.8

17.5

26.3

35.0

43.8

52.5

61.3

70.0

Vx

Vy

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(b)

0

0.9

1.8

2.6

3.5

4.4

5.3

6.1

7.0

Vx

Vy

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(c)

0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

Vx

Vy

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(d)

0

12.5

25.0

37.5

50.0

62.5

75.0

87.5

100.0

Vx

Vy


Fig. 5.16 gives the similar contour plots for a doughnut beam of topological

charge 2. It is interesting to note that in this case |Ey|2 component is

comparable to |Ex|2 in the central focal region, and that these two components

result in the non-zero value in the center of the focal region. The central focal

intensity for charge 2 is much smaller than that for charge 1. This is because

the relative strength of |Ez|2, which contributes to the central intensity for

charge 1, is much larger than the relative strength of |Ey|2, which contributes



to the central intensity for charge 2. The combination of |Ex|2 and |Ez|2

components produces higher intensity peaks in the doughnut ring along the

Vy direction (Fig. 5.16(d)).

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(a)

0

8.8

17.5

26.3

35.0

43.8

52.5

61.3

70.0

Vx

Vy

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(b)

0

0.4

0.9

1.3

1.8

2.2

2.6

3.1

3.5

VxV

y

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(c)

0

5.6

11.3

16.9

22.5

28.1

33.8

39.4

45.0

Vx

Vy

6 4 2 0 -2 -4 -66

4

2

0

-2

-4

-6

(d)

0

12.5

25.0

37.5

50.0

62.4

74.9

87.4

100.0

Vx

Vy


The electric field components |Ex|2, |Ey|2 and |Ez|2 all give zero field

intensity in the central region of a doughnut beam of charge 3 (Fig. 5.17).

Component |Ex|2 gives a ring of equal intensity around the singularity

(Fig. 5.17(a)), while component produces two high intensity peaks on

either side of the singularity in the Vy direction (Fig. 5.17(c)). These two

components are comparable in strength and are dominant in determining the

overall intensity in the focal region.

The peak intensity ratio of |Ez|2/|Ex|2, gives the relative strength of the



transversal |Ex|2 and the longitudinal |Ez|2 field components. The peak

intensity ratio as a function of NA for topological charges 1, 2 and 3 is

shown in Fig. 5.18(a). It is seen that the peak intensity ratio increases

Fig. 5.18: Dependence of the peak ratio of |Ez|2/|Ex|2 on the numerical aperture(a) and on the obstruction radius ε (b).

rapidly with NA. The longitudinal component |Ez|2 of the doughnut beam

attains approximately a half of the strength of the |Ex|2 component even for

NA = 0.9. The increase is more rapid for topological charge 1 than that

exhibited by the charges 2 and 3 beams of lower numerical aperture. As NA

becomes larger, the most rapid increase is manifested by charge 3.

Let us turn to the effect of a central opaque disk on the phenomenon shown

in Figs. 5.14- 5.17. Such a centrally obstructed doughnut beam is analogous

to using an axicon to extend the focal depth of the doughnut beam, which

has been used in laser trapping with a high NA objective [125]. Fig. 5.18(b)

presents the dependence of the peak intensity ratio on the central obstruction

ε for doughnut beams of charges 1, 2 and 3 for NA = 1 in air. Here, ε is

defined as the ratio of the radius of the obstructed part to the radius of

the unobstructed part of the beam. Presence of obstruction leads to more

depolarised rays reaching the focus, which enhances the contribution from



the |Ez|2 component to the total intensity [89]. For example, for obstructions

of ε = 0.83 and ε = 0.87 for a doughnut beam of charge 1 and charge 2,

respectively, the longitudinal contribution equals the transverse one. For the

limiting case of ε→ 1.0 the longitudinal contribution for the doughnut beam

of charges 1, 2 and 3 is approximately 1.47, 1.64 and 2.25 times larger than

that from the transverse component, respectively.

5.3.3 Trapping efficiency

The consistent comparison between theory and experiments described in

section 5.2.3 exhibits the applicability of the vectorial diffraction method

for the trapping force evaluation of the complex laser beams generated using

various spatial phase modulation techniques, by including the appropriate

phase modulation in the incident illumination. Based on the vectorial

diffraction analysis of the incident doughnut beams, the trapping efficiency

can be evaluated using the method described in Section 4.2.

A comparison of the maximal backward ATE of a highly focused doughnut

beam of charge 1 with an ordinary plane wave and a centrally obstructed plane

wave incident on polystyrene particles is shown in Fig. 5.19. The difference

between an unobstructed plane wave and a doughnut beam illumination types

becomes larger for smaller particles due to the reduced interaction of small

particles with the low-intensity central field region of the focused charge 1

doughnut beam (Fig. 5.14(a)). However, according to the RO model the

backward ATE achieved with a centrally obstructed plane wave is larger

than that achieved by either an unobstructed plane wave or a doughnut

beam of charge 1 [23]. The RO model indicates that in the case of a very

large obstruction, the backward ATE is approximately 1.6 times larger than

that achieved by an unobstructed plane wave, and 1.2 times larger than

that achieved by a doughnut beam of charge 1. The RO optics model



Fig. 5.19: Maximal backward ATE of polystyrene particles suspended in water andilluminated by a highly focused plane wave, doughnut beam of topological charge1 and obstructed plane wave with ε = 0.8 as a function of particle size. NA = 1.2and λ0 = 1.064 µm.

however, completely ignores the EM field distribution in the focal region

of an obstructed laser beam which leads to the focal spot elongation in the

axial direction. As it can be seen from the results obtained by our trapping

model in Fig. 5.19, the maximal backward ATE is actually reduced when a

plane wave is centrally obstructed compared to either an unobstructed plane

wave or a doughnut beam of charge 1. The reduction is due to the focal spot

elongation in the axial direction which leads to a reduced intensity gradient

in this direction.

The dependence of the ATE on the topological charge of a doughnut

beam is given in Fig. 5.20 for a polystyrene particle of radius a = 2 µm

and suspended in water. The ATE in the forward direction Qf is larger for



higher topological charge, while the ATE in the backward direction Qb is

relatively unchanged. This result is consistent with experimental findings of

Fig. 5.20: ATE of a polystyrene particle (a = 2 µm) suspended in water andilluminated by a highly focused plane wave and doughnut beams of differenttopological charges. NA = 1.2 and λ0 = 1.064 µm.

the backward ATE [126] and the forward ATE [127] of large microparticles.

The TTE, on the other hand, is reduced for higher topological charges in

either scanning direction (Fig. 5.21).

To investigate the validity of the theoretically determined results we have

conducted experimental measurements of the TTE. The measurement is

performed in the same manner as described in Section 5.2.3.2 without the

central obstruction. For plane wave illumination the reflection type SPM is set

to uniform phase distribution, which then operates as a simple mirror, while

for the doughnut beam illumination it is loaded with a 256 levels phase-ramp

distribution described in Fig. 5.13(a). Table 5.1 shows the experimentally

measured TTE for both S (Qs) and P (Qp) scanning directions with plane



Fig. 5.21: TTE in the polarisation (X) and perpendicular to the polarisation (Y)directions of a polystyrene particle (a = 2 µm) suspended in water and illuminatedby a highly focused plane wave and doughnut beams of different topological charges.NA = 1.2 and λ0 = 1.064 µm.

wave and doughnut beam illuminations for three different polystyrene particle

sizes with radii of 1 µm (Polybead polystyrene microspheres, radius=1.067

µm with standard deviation δ=0.020 µm), 0.5 µm (Polybead polystyrene

microspheres, radius=0.496 µm with standard deviation δ=0.013 µm) and

0.25 µm (Polybead polystyrene microspheres, radius=0.242 µm with standard

deviation δ=0.005 µm). The large particle has a radius of twice the

wavelength size, and the radius of the medium particle is approximately the

same as the illumination wavelength, while the smallest particle radius is half

of the illumination wavelength.

It can be seen from Table 5.1 that the maximal TTE decreases for smaller

particle sizes, with the decrease being more rapid for the doughnut beam

illumination. The ratios of the experimentally measured maximal TTE of

the plane wave and doughnut beam illuminations agree with the theoretically

calculated results within the experimental error. Furthermore the trend that

the ratio is higher for smaller particles is experimentally confirmed.

Similar to the maximal backward ATE in Fig. 5.19, the maximal TTE



a = 1µm a = 0.5µm a = 0.25µm

Qp(Ch0) exp. 0.091 ± 0.005 0.087 ± 0.002 0.040 ± 0.002Qp(Ch1) exp. 0.079 ± 0.005 0.048 ± 0.002 0.019 ± 0.002Qs(Ch0) exp. 0.079 ± 0.005 0.068 ± 0.002 0.040 ± 0.002Qs(Ch1) exp. 0.057 ± 0.005 0.043 ± 0.002 0.015 ± 0.002Qp(Ch0)/ Qp(Ch1) exp. 1.15 ± 0.15 1.81 ± 0.12 2.11 ± 0.36Qp(Ch0)/ Qp(Ch1) th. 1.29 1.55 2.15Qs(Ch0)/ Qs(Ch1) exp. 1.38 ± 0.23 1.58 ± 0.12 2.67 ± 0.56Qs(Ch0)/ Qs(Ch1) th. 1.25 1.50 2.30

Table 5.1: The maximal TTE for plane wave and doughnut beam illumination.Ch0 denotes plane wave input, while Ch1 denotes a doughnut beam of topologicalcharge 1. exp. - experimentally measured result, th. - theoretically calculatedresult.

a = 1µm

Qp(Ch0+ε) exp. 0.018 ± 0.002Qp(Ch1) exp. 0.079 ± 0.005Qs(Ch0+ε) exp. 0.017 ± 0.002Qs(Ch1) exp. 0.057 ± 0.005Qp(Ch0+ε)/ Qp(Ch1) exp. 0.23 ± 0.04Qp(Ch0+ε)/ Qp(Ch1) th. 0.257Qs(Ch0+ε)/ Qs(Ch1) exp. 0.30 ± 0.07Qs(Ch0+ε)/ Qs(Ch1) th. 0.260

Table 5.2: The maximal TTE for a centrally obstructed plane wave and a doughnutbeam illumination. Ch0+ε denotes a centrally obstructed plane wave input, whileCh1 denotes a doughnut beam of topological charge 1. exp. - experimentallymeasured result, th. - theoretically calculated result. Obstruction size ε = 0.78.

of an obstructed plane wave is smaller than that achieved with either an

unobstructed plane wave or a doughnut beam of charge 1. The RO model

indicates that the ratio of the maximal TTE of a highly obstructed plane

wave to the one achieved by a doughnut beam of charge 1 is approximately

0.8 [23]. Our optical trapping model, which considers the exact EM field

distribution in the focal region, gives this ratio as approximately 0.26, which

agrees well with the experimentally measured results (Table 5.2). This result



further indicates that if one deals with complex laser beams one needs to take

into account the complexity of the focal field distribution and its interaction

with a micro-particle to properly investigate the trapping force exerted on

such a particle, which is achieved using our optical trapping model based on

the vectorial diffraction theory.


Far-field optical trapping with high NA objective is investigated using the

vectorial diffraction model described in the previous Chapter 4 with two types

of incident illumination; plane wave and doughnut beam illumination.

It is found that the refractive index mismatch between the objective

immersion and particle suspension media, that leads to SA, severely affects

the trapping performance of an optical trap, due to the focal distribution

distortions. SA generally leads to the degradation of the trapping efficiency

for both the ATE and the TTE. The rate of degradation depends on the

particle and the environment parameters. Our optical trapping model based

on the vectorial diffraction theory, however, can successfully deal with this

issue, which has been confirmed by comparing the theoretically calculated

results with the experimental measurements of Felgner et al. [115].

When the incident plane wave is centrally obstructed the TTE of large

dielectric particles falls rapidly with the increasing size of the obstruction.

The decrease predicted by the vectorial diffraction model is much faster than

the one given by the RO model. We have experimentally measured the

dependence of the TTE on the obstruction size for large polystyrene particle

for both S and P polarisation states of the incident illuminations. It is found

that the TTE decrease occurs at a much faster rate than the one predicted

by the RO model, for both polarisations, and that it matches well the rate



predicted by the vectorial diffraction model. Furthermore, the difference in

the maximal TTE for the two polarisation states predicted by the vectorial

diffraction model is verified in the experiment.

Efficient generation and vectorial diffraction of doughnut beams is also

investigated in order to study the trapping force exerted on a dielectric

particle by focused doughnut laser beams. The doughnut beam generation

efficiency of 100% can be achieved when plane wave is converted into

doughnut beam using a LC cell or the reflection mode phase modulator. It

is found that the doughnut beams focused by a high NA objective yield focal

intensity distributions markedly different from those obtained when the same

beam is focused by a low NA objective. The central zero intensity points

disappear for doughnut beams of topological charges ±1 and ±2 due to the

depolarisation effect of a high NA objective. The focused distribution of a

doughnut beam of a given charge shows the increased ring intensity along the

direction perpendicular to the incident polarisation direction and the focal

spot becomes elongated in the polarisation direction. These effects are more

pronounced when such beams are centrally obstructed and may affect the

laser trapping performance.

The maximal backward ATE (lifting force) of small particles is greatly

reduced when the trapping is performed with focused doughnut beam

illumination, compared to the plane wave illumination, due to the reduced

central intensity distribution in the focal region of such an illumination.

Furthermore, when plane wave is centrally obstructed the lifting force is

reduced even further compared to either an unobstructed plane wave or

doughnut beam illumination.

For large particles, the forward ATE is increased when doughnut beam

illumination is used, while the backward ATE is relatively unchanged. On the

other hand, the TTE, under the same conditions, is reduced when doughnut



beams are used for trapping. It is shown theoretically and experimentally

that the TTE decreases for reducing particle size for both a plane wave and

a doughnut beam of charge 1 incident illumination. However, the rate of

decrease is faster in the case of doughnut beam illumination. The comparison

of the maximal TTE of an obstructed plane wave with ε ≈ 0.8 and a doughnut

beam of charge 1 reveals that the efficiency available with the obstructed

beam is only approximately 0.26 of that achieved by a doughnut beam,

which has been confirmed by an experimental measurement. This result

contradicts the RO model prediction which indicates that the two efficiencies

should be relatively comparable. The RO model is inadequate to describe

such a trapping process because it considers the highly obstructed beam

by calculating the trapping force produced by highly convergent rays only

(rays close to the maximum convergent angle), while completely ignoring

the diffraction effects and its influence on the focal field distribution. The

focal field distribution of a highly obstructed beam exhibits an elongation in

the axial direction and increased rings intensities in the transverse direction,

which greatly affects the trapping performance.


Chapter 6

Near-field optical trapping

6.1 Introduction

The trapping force generated by a conventional high numerical aperture

(NA) optical tweezers setup, also known as the far-field trapping discussed

in Chapter 5, acts toward the high intensity focal region due to the large

field gradients. The trapping volume of the far-field laser trapping technique

is diffraction limited, leads to a significant background signal, and poses

difficulties in the single-molecule experiments. The trapping modality based

on the particle trapping using a highly localised near-field, can overcome these

problems. A review of the near-field trapping is undertaken in Section 2.6

of Chapter 2. It can be seen from this review that all of the proposed near-

field trapping techniques have been theoretically investigated, except the one

implemented using a focused evanescent illumination. Our physical model

for the trapping force evaluation, based on vectorial diffraction introduced in

Chapter 4 and implemented with the far-field optical trapping discussed in

Chapter 5, can be applied to provide a theoretical model for the near-field

trapping with focused evanescent illumination.

136

CHAPTER 6. Near-field optical trapping

The focused evanescent field is produced by placing a central obstruction

in the laser beam path, before it enters a high NA objective. The size

of the obstruction is large enough to ensure that the minimum angle of

convergence of each incident ray is larger than the critical angle determined

by the total internal reflection (TIR) condition between two media (Fig. 6.1).

The circular symmetric nature of such an illumination enhances the strength

Fig. 6.1: Focused evanescent field produced at the coverslip interface by high NAfocusing of an obstructed plane wave polarised in the X direction. Oil immersionand coverslip refractive index n1 = 1.78 (index matched), particle suspensionmedium n2 = 1.33, objective NA = 1.65, obstruction size ε = 0.8, and illuminationwavelength λ0 = 1.064 µm.

of the evanescent field and reduces its lateral size, due to the constructive

interference of the resulting evanescent field near the center of the focus. The

focal splitting effect due to the relatively large obstruction size is evident

(Fig. 6.1 XY plane). In addition to the focused evanescent field generated

by plane wave illumination, we have considered the focused evanescent

illumination obtained by centrally obstructing a doughnut beam. This

chapter studies the distribution of the near-field trapping force with focused

evanescent illumination and investigates the dependence of the near-field force

on particle size. An important question of the capability of the near-field

trapping with focused evanescent illumination to trap small particles in three



dimensions is also dealt with in more details. In addition to the theoretical

study of the near-field trapping with focused evanescent illumination, we have

conducted experimental investigations of the transverse trapping efficiency

(TTE) of this type of near-field trapping. A good agreement between the

theoretical and experimental results confirms the validity of our approach.

This chapter also briefly deals with the near-field forces exerted on dielec-

tric microparticles situated in the wide area evanescent field illumination. The

wide area evanescent field is considered according to the three-dimensional

consideration of the evanescent field scattering for particles situated far from

the interface at which evanescent wave is generated, described in details

in Chapter 3. The Maxwell stress tensor approach is used for the force

calculation. This method is not suitable to deal with a focused evanescent

field because the field representation is that of a wide area plane wave incident

under the TIR condition at the prism interface.

The structure of the chapter is as follows. The near-field forces exerted

on small dielectric particles by a wide area evanescent illumination, whose

scattering properties were investigated earlier, are examined in Section 6.2.

Section 6.3 gives the numerical results of the near-field trapping with a focused

evanescent illumination under both plane wave and doughnut beam incident

illumination. The near-field force distribution for a small and a large dielectric

particle is presented as well as the trapping efficiency dependence on particle

size. Experimental results are presented in Section 6.4, for the plane wave

(Section 6.4.1) and doughnut beam (Section 6.4.2) incident illumination. The

chapter conclusions are summarised in Section 6.5.



6.2 Optical forces on microparticles in a wide

area evanescent field

The fast decaying wide area evanescent field, whose scattering properties

by small dielectric particles were studied in Chapter 3, exerts a force on

microparticles situated in such field, due to the high intensity gradient.

The effect of such force was investigated experimentally by Kawata and

Sugiura [67] where they have observed particle movements induced by the

evanescent field, while the theoretical treatment was described by Almaas

and Brevik [68]. Even though this configuration of the near-field trapping

has been dealt with theoretically in general, we include this section for the

sake of completeness, with some specific parameters of the evanescent field

corresponding to the ones used in studies of the evanescent field scattering in

Chapter 3. If we consider a homogeneous microsphere situated in the particle

immersion medium n′ illuminated by a monochromatic electromagnetic (EM)

field described by Eq. 3.1, the resulting scattered field by Eq. 3.3, and their

corresponding magnetic fields, the net radiation force on the microsphere can

be determined using the steady-state Maxwell stress tensor analysis given by

Eq. 2.7.

Figure 6.2 shows the vertical (Qz) and the horizontal (Qy) near-field

radiation forces exerted on a polystyrene particle situated in water and

illuminated by an evanescent field generated by a TIR of the Helium-Neon

laser beam. The TIR occurs at a prism surface. The prism refractive index is

assumed to be 1.722 and the laser beam is incident at an angle of 51. This

corresponds to the values assumed in Chapter 3, when scattering properties

of microparticles were investigated. The forces are given, similarly to Almaas



and Brevik [68], in their non-dimensional form, defined as

Qy =Fy

ε0E20a

2, Qz =

Fz

ε0E20a

2. (6.1)

Fig. 6.2: Non-dimensional horizontal Qy and vertical Qz near-field forces exertedon polystyrene particle under TE and TM incident polarisation states as a functionof the particle size parameter k′a = 2πa/λ. The evanescent field is generated on aprism surface (refractive index 1.722) by a plane wave incident at 51.

The results in Fig. 6.2 are similar to those of Almaas and Brevik [68],

and they indicate that the vertical force is negative, i.e. directed towards

the prism surface, while the horizontal force is always positive, indicating

particle pushing in this direction. Slight modulations on the force curves

is due to the morphology dependent resonance (MDR) effects described in

more details earlier in Chapter 3, which become more pronounced for larger

particles. The force strength can be selected and maximised by tuning the

particle size, or more practically the illumination wavelength. However, we

can conclude that the force in either direction is larger for the P polarised

incident illumination (TM illumination). For example, it can be seen from

Fig. 6.2 that a small polystyrene particle of 1 µm in diameter, and under the



influence of an evanescent field generated by a 150 mW Helium-Neon laser

beam, spread over an interaction cross-section area of 100 µm, experiences a

force of 0.028 pN and 0.044 pN in the horizontal direction for S (TE) and P

(TM) incident polarisation states respectively. The vertical force under the

same conditions is slightly larger; being 0.041 pN for S polarisation and 0.070

pN for P polarisation.

6.3 Near-field trapping with focused evanes-

cent illumination - Numerical results

To investigate near-field trapping implemented using a focused evanescent

field we have developed a physical model based on the theoretical treatment

described in Chapter 4 (Fig. 6.3). This model includes two physical processes,

vectorial diffraction by a high NA objective under the TIR condition and

scattering by a small particle with a focused evanescent wave. The limits

of integration in Eqs. 4.1 and 4.2 are modified to incorporate the central

obstruction. The lower limit is determined by the maximum obstruction

angle (α1), while the upper limit is given by the maximum convergence angle

(α2). Thus the achieved EM distribution on the coverslip interface represents

a focused evanescent field. Based on the scattering process of a small particle

with an evanescent focal spot, one can determine the EM field inside and

outside a particle and thus the trapping force exerted on the particle.

Now let us consider that a small particle interacts with focused evanescent

illumination generated at the coverslip interface of a centrally obstructed

high NA objective (NA=1.65). An axial force caused by the fast decaying

nature of such illumination and a transverse gradient force resulting from

the focal shape are exerted on the particle. Figs. 6.4 and 6.5 shows the

trapping efficiency mapping, when a small and large polystyrene particle



Laser beam

Obstruction

Objective

n1

n2

Fz

Fx Fx

a1

a2

Particle

Focusedevanescentfield

Fig. 6.3: Near-field trapping model.

is transversally scanned in the X and Y directions across the focused

evanescent field distribution, generated by placing a central obstruction

(ε=0.85) perpendicularly to the path of an incoming laser beam. The

focused evanescent field is produced for ε > 0.8 when the refractive indices of

the coverslip and immersion water are n1=1.78 and n2=1.33. The laser beam

(λ=532 nm) propagates in the Z direction and two cases are considered; the

first one being an ordinary plane wave [15], while the second one is a phase

modulated doughnut beam of charge 1. For both small and large particles

the ATE for plane wave illumination is slightly larger than that for doughnut

beam illumination. However, in the case of small particles the ATE also

decreases slightly faster with plane wave illumination when the particle is

scanned in either direction. Not only is the ATE for large particles stronger,

the force mapping structure in the case of large particles is markedly different

from the small particle case due to the larger interaction cross section of the



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

a=0.2

5mm

X (mm)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035T

rappin

geffic

iency

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

a=1

mm

X (mm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Tra

ppin

geffic

iency

Laser beam

Obstruction

Objective

n1

n2

Laser beam

Obstruction

Objective

n1

n2

Laser beam

Obstruction

Objective

n1

n2

Phase modulatorLaser beam

Obstruction

Objective

n1

n2

Phase modulator

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

a=0.2

5mm

X (mm)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Tra

ppin

geffic

iency

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

a=1

mm

X (mm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Tra

ppin

geffic

iency

Fig. 6.4: Trapping efficiency mapping for a small and a large polystyrene particle ofradius a, scanned in the X direction (light polarisation direction) across the focusedevanescent field generated by a plane wave (top) and doughnut beam illumination(bottom). NA=1.65, λ=532 nm, ε=0.85, n1=1.78 and n2=1.33.

small particle with the focused evanescent field. The maximal TTE for small

particles is much larger relative to the ATE than that for large particles. The

maximal TTE in the X direction, for example, constitutes 16.8% and 14.8%

of the maximal ATE for small particles in the case of the plane wave and

doughnut beam illuminations respectively, compared to the 7.4% and 7.2% for

large particles. The trapping efficiency decrease for small and large particles is

also slightly faster in the direction perpendicular to the polarisation direction

than that in the polarisation direction. This asymmetry is caused by the

focused spot elongation due to the focusing by a high NA objective.



Laser beam

Obstruction

Objective

n1

n2

Laser beam

Obstruction

Objective

n1

n2

Laser beam

Obstruction

Objective

n1

n2

Phase modulatorLaser beam

Obstruction

Objective

n1

n2

Phase modulator

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Y (mm)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

a=0.2

5mm

Tra

ppin

geffic

iency

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Y (mm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

a=1

mm

Tra

ppin

geffic

iency

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Y (mm)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

a=0.2

5mm

Tra

ppin

geffic

iency

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Y (mm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

a=1

mm

Tra

ppin

geffic

iency

Fig. 6.5: Trapping efficiency mapping for a small and a large polystyrene particleof radius a, scanned in the Y direction (perpendicular to the polarisation direction)across the focused evanescent field generated by a plane wave (top) and doughnutbeam illumination (bottom). NA=1.65, λ=532 nm, ε=0.85, n1=1.78 and n2=1.33.

The dependence of the maximal TTE on the obstruction size ε is critical

to capture a particle in the evanescent focal spot and is shown in Fig. 6.6 for

both plane wave and doughnut beam illumination for polystyrene particles

of diameter 2 µm. The maximal TTE decreases with increasing the size of

the beam obstruction due to the reduced contribution of the propagating

component to the transverse force and because the high angle rays are less

efficient in the transverse trapping of a dielectric particle [52]. The maximal

TTE for the plane wave case is generally slightly larger than that obtained

with doughnut beam illumination for both P and S directions. The difference



Fig. 6.6: Theoretical calculations of the maximal TTE of a polystyrene particle of1 µm in radius as a function of the obstruction size ε. The other conditions are thesame as in Fig. 6.4.

in the maximal TTE for the plane wave and doughnut beam cases is the

largest when no obstruction is present (plane wave TTE being 1.4 times

larger) and becomes smaller for the increasing obstruction size. In the near-

field trapping regime (ε > 0.8) this difference nearly vanishes.

The dependence of the maximal ATE on the obstruction size ε is at first

relatively unchanged until ε ∼ 0.6, at which point the maximal ATE decreases

for increasing ε (Fig. 6.7). For certain values of ε in this range it even shows

a slightly larger ATE compared to the case without the obstruction. At the

focused evanescent field condition, i.e. when ε > 0.8, the maximal ATE is

still approximately 43% of the far-field case when no obstruction is present.

So far only the case when the obstructed beam with ε corresponding to

the minimum convergence angle larger than the critical angle is focused onto

the interface between two media is considered, i.e. the geometrical focus

position is on the interface. When this geometrical focus position is brought

into the suspension medium, the fast decaying evanescent field is slightly

defocused at the interface. Since there is no propagating component, the



Fig. 6.7: Theoretical calculations of the maximal ATE of a polystyrene particle of1 µm in radius as a function of the obstruction size ε. The other conditions are thesame as in Fig. 6.4.

field is localised at the interface and the geometrical focus becomes virtual.

The relationship between the depth d of such a virtual focus, which describes

the spherical aberration function Ψ (Eq. 2.21) and the ATE for both small

and large dielectric particles is revealed in Fig. 6.8. Both plane wave and

doughnut beam illumination cases follow a similar trend with plane wave

illumination showing generally a slightly higher ATE, and converge at a larger

depth. It is interesting to note, however, that the highest ATE does not

occur when the geometric focus is at the interface, but it occurs for a slightly

defocused evanescent field, focused at a virtual depth of ∼ 100 nm into the

suspension medium. The physics of this phenomenon originates from phase

shifts under TIR (Fig. 6.9) [54]. According to Eqs. 2.22 and 2.23 these phase

shifts act as an equivalent spherical aberration. Consequently, the balance

between the spherical aberration function Ψ and these phase shifts determines

the maximum field at the interface. It can be shown that for the given



Fig. 6.8: Dependence of the ATE on the virtual focus position d for a small andlarge polystyrene particle (ε = 0.85). The other conditions are the same as inFig. 6.4.

maximum convergence angle in our case, the total phase accumulated gives

the maximum field at the interface for the depth d = 62 nm.

The dependence of the maximal near-field ATE on the polystyrene particle

size is illustrated in Fig. 6.10. The relationship between the maximal

near-field ATE is nearly linear in nature for both types of illumination

and the particle size range considered, which is different from the far-field

force relationship that depends on the cubic of the particle size. This

linear relationship can be qualitatively explained by examining the particle

interaction cross-section area (A). If the evanescent field depth is denoted

by h, it can be shown that the interaction cross-section area is given by

A = π(4ah − h2), where a is the particle radius (Fig. 6.10 inset). It can be

estimated from Fig. 6.10 that an input laser power of 10 µW is sufficient to

overcome the gravity (including the buoyancy) force and lift a polystyrene



Fig. 6.9: Phase introduced by the Fresnel transmission coefficients as a function ofthe incident angle. The refractive index of the incident medium is 1.78, while therefractive index of the transmitting medium is 1.33.

a a-h

h n1

n2

0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

0.10

0.12

(a)

AT

E

a (mm)

Plane waveDoughnut beam

Fig. 6.10: The maximal ATE as a function of a polystyrene particle size (ε = 0.85).The inset shows a schematic relation between the interaction cross-section area andthe particle size. The other conditions are the same as in Fig. 6.4.



particle of 800 nm in radius or smaller (Fig. 6.11) in an upright trapping

system.

Fig. 6.11: The magnitudes of the axial force for a plane wave of power 10 µW andthe gravity force for different particle sizes. The other conditions are the same asin Fig. 6.4.

6.4 Near-field trapping with focused evanes-

cent illumination - Experimental results

The same experimental configuration as the one used for the far-field trapping

measurements (Fig. 5.7) is used for the near-field trapping measurements.

The exception is that the water immersion high NA objective (NA = 1.2) is

replaced with the Olympus (Apo 100x oil HR) high NA objective (NA =

1.65). A special coverslip glass and the immersion oil for the NA-1.65

objective have a refractive index of 1.78. Particles are suspended in water with

refractive index of 1.33 (at the room temperature of 20 C), for which the NA-

1.65 objective generates focused evanescent field for ε > 0.8. This corresponds

to the cut-off angle of 48 for the TIR in water. The illumination source



was a 532 nm diode-pumped cw laser (Spectra-Physics Millenia II). The

trapping efficiency measurements were performed using the Stokes method

described in Section 5.2.3.2. The presented results represent the average of

three measurements with the uncertainties estimated from the laser power

fluctuations.

6.4.1 Plane wave illumination

The measurement of the maximal TTE with plane wave illumination as a

function of the central obstruction size is shown in Fig. 6.12. In the plane wave

case the phase modulator was replaced by a mirror. Under these experimental

Fig. 6.12: The measured and calculated plane wave illumination maximal TTE asa function of obstruction size with a NA-1.65 objective for both S and P scanningdirections. The other conditions are the same as in Fig. 6.4.

conditions the evanescent field is generated for the obstruction size ε > 0.8,

thus resulting in the near-field trapping.

First of all, it can be seen from Fig. 6.12 that our model agrees well with



the experimental measurements for both S and P scanning directions and that

the maximal TTE decreases with the obstruction size at the rate predicted by

our model. Furthermore, the maximal TTE when no obstruction is present

is ∼ 12% higher in the polarisation direction (P direction) compared to the

direction perpendicular to the polarisation direction (S direction). However,

in the near-field case the situation is reversed, the maximal TTE in the S

direction is higher than that in the P direction. For example in the case of

a central obstruction ε = 0.9 the maximal TTE in the S direction is ∼ 12%

higher than that in the P direction. The maximal TTE in the two directions

is nearly the same for ε ∼ 0.5. All of these findings are confirmed by the

experimental measurements (Fig. 6.12).

6.4.2 Doughnut beam illumination

The doughnut beam illumination used in the experimental measurements was

generated in the same manner as for the far-field measurements described in

the previous chapter (Chapter 5). The dependence of the maximal TTE

on the central obstruction size, for doughnut beam illumination is shown in

Fig. 6.13. Similar to the measurement with the plane wave illumination,

the maximal TTE decreases for increasing obstruction size. The maximal

TTE without the obstruction is ∼ 13% higher in the polarisation P direction

compared to the S direction. However, unlike the measurement with the plane

wave illumination, the maximal TTE for the near-field trapping case with

doughnut beam illumination is nearly the same in either scanning direction.

The near-field maximal TTE with doughnut beam illumination is nearly

the same as that for plane wave illumination for either polarisation direction.

However, in the far-field case when no obstruction is present, the maximal

TTE for doughnut beam illumination is ∼ 28% lower than that achieved with

plane wave illumination.



Fig. 6.13: The measured and calculated doughnut beam illumination maximal TTEas a function of obstruction size with a NA-1.65 objective for both S and P scanningdirections. The other conditions are the same as in Fig. 6.4.


The force mapping for the near-field trapping with a focused evanescent field

in the case of large polystyrene particles shows markedly different structure

from that obtained for small particles due to the larger interaction cross-

section. The ration of the maximal TTE to the maximal ATE for small

particles is much larger than that for large particles. However, the maximal

ATE is stronger for larger particles. The trapping efficiency is slightly larger

for the focused evanescent field generated by a plane wave, compared to the

one generated by a doughnut beam.

The maximal ATE for large polystyrene particles is relatively unchanged

for obstruction sizes of ε < 0.6. For certain values of ε in this range it even

shows a slightly larger ATE compared to the case without the obstruction.

For ε > 0.6, the ATE decreases rapidly for increasing ε. However, in the near-

field trapping case it still constitutes ∼ 43% of that for the far-field case. It



is also determined that the highest ATE does not occur when the geometric

focus is at the interface, but it occurs for a slightly defocused evanescent

field, when the geometrical focus is at a depth of ∼ 62 nm. It is found that

physically this phenomenon originates from phase shifts under TIR.

Unlike the far-field trapping, the dependence of the maximal ATE on the

particle size under focused evanescent wave illumination is nearly linear. It

is predicted that an input laser power of 10 µW is sufficient to overcome the

gravity force acting on a polystyrene particle of 800 nm in radius or smaller,

thus enabling the particle trapping in three dimensions, i.e. the particle

lifting, in an upright system.

The calculated trapping force in an evanescent focal spot for both plane

wave and doughnut beam illumination has been demonstrated to be in a good

agreement with experimental results.


Chapter 7

Conclusion

7.1 Thesis conclusion

The main research work in this thesis presents an optical trapping model

based on two physical processes, vectorial diffraction of incident laser beam by

a high numerical aperture (NA) objective and scattering by a small particle

(Chapter 4). The interaction of the small particle and the focused laser

beam is described using an extension of the classical plane wave Lorentz-

Mie theory, while the force is determined using the Maxwell stress tensor

approach. An important advantage of our approach over other optical

trapping models reviewed in Chapter 2 is that it provides an exact description

of the electromagnetic (EM) field distribution in the focal region of a high

NA objective. As opposed to the other optical trapping models, our model

is capable of considering an arbitrary wavefront incident at the objective

entrance pupil. In other words, our model is capable of including any

apodisation function of the focusing objective, phase modulation of the

incident beam, polarisation effects and the effects of spherical aberration (SA)

usually present in optical trapping experiments. As a consequence we are able

154

CHAPTER 7. Conclusion

to consider the trapping process under doughnut beam illumination, which

represents one class of phase modulated laser beams (Chapter 5). We have

shown that our optical trapping model can deal with the far-field trapping

modality, as well as with the near-field trapping modality implemented with

a focused evanescent field generated by using a central obstruction in the

incident beam path. Our optical trapping model provides a first theoretical

treatment of such a near-field trapping modality (Chapter 6).

In addition to the optical trapping model, this thesis presents a nanometric

sensing model for detection of scattered evanescent waves (Chapter 3).

The sensing model, which describes the conversion of evanescent photons

into propagating photons, is implemented by considering the near-field Mie

scattering process enhanced by morphology dependent resonance (MDR) by

a small particle in the vicinity of a plane interface. The transformation

of the three-dimensional scattered field distribution to the detector plane

is achieved using vectorial diffraction of the scattered signal. The vectorial

diffraction approach for the transformation of the scattered signal is necessary

due to the vectorial nature of the scattered field. Such a sensing model

is essential for understanding optical trapping systems that use evanescent

field illumination for high resolution position monitoring and imaging, such

as optical trap nanometry and particle-trapped scanning near-field optical

microscopy (SNOM).

The nanometric sensing model reveals that the intensity pattern generated

in the far-field region of the collecting objective shows an interference-like

pattern when large dielectric particles are employed as scatterers. For small

particles the intensity pattern is similar to that given by the dipole radiation.

Our research has shown that MDR effects are evident in near-field Mie

scattering of large dielectric particles and that the resonance peaks become

sharper and less separated with increasing the effective refractive index. The



MDR effects are present in the image forming field collected by a high NA

objective and it results in different energy distributions in the detection plane

for on and off resonance positions. These different energy distributions outline

the importance of detector selection for systems operating at MDR positions,

because it would be an advantage to operate particle-trapped SNOM and

optical trap nanometry systems at MDR positions to enhance signal-to-noise

ratio. If the detection of the collected scattered signal is performed with a

pinhole detector, the detector size needs to be carefully selected depending

on the scattering particle size. This is because small particles act as dipole

sources and fill the objective entrance pupil uniformly, which requires a

relatively small pinhole. For large particles much larger pinholes are required

to collect the signal completely with a typical imaging lens. This effect is

contributed to larger interaction cross-section and the MDR effect of large

particles, which result in signal spreading in forward direction in the imaging

plane. The theoretical predictions of our model have been experimentally

verified by a good agreement of predicted and measured intensity distribution

of small and large polystyrene particles in the focal plane of the collecting

objective.

Applying our optical trapping model for force determination based on

the vectorial diffraction approach for light focusing under a high convergence

angle we have shown that it is superior to the commonly used Gaussian beam

model in that it exactly represents the field in the focal region of a high NA

objective and that the results obtained by our vectorial diffraction model at

large particle limit are approaching the ray optics (RO) model. The RO model

is an approximate model valid only for very large particles, and our optical

trapping model can bridge the gap between small particle case for which the

focal field distribution is crucial and large particle case for which the RO

model gives approximate solutions. Furthermore, the vectorial diffraction

model can treat arbitrary incident wavefronts and simulate real experimental



conditions. Investigations of the SA effects on the far-field trapping force

show that the refractive index mismatch between the objective immersion

and particle suspension media severely affects the trapping performance of an

optical trap. The rate of degradation of the trapping performance depends

on the particle and environment parameters. Our research, however, has

demonstrated that the trapping force evaluated using our optical trapping

model agrees well with the experimental results, for both axial and transverse

trapping forces.

The maximal transverse trapping efficiency (TTE) of a centrally ob-

structed plane wave focused by a high NA objective, in the case of large

dielectric particle decreases rapidly for increasing the obstruction size, unlike

the trend predicted by the RO model. The rate of decrease and the trapping

efficiency for most obstruction sizes is higher for the incident P polarisation,

compared to the incident S polarisation. At very large obstruction sizes

(ε ≈ 0.9), however, the trapping force of the P polarisation approaches that of

the S polarisation, due to the different depolarisation behaviors between the

S and P polarisation states. These findings are confirmed by experimental

measurements.

In order to study the trapping performance under a doughnut beam

illumination, we have explored efficient methods for generating such a beam.

A near 100% efficient generation of doughnut laser beams from an incident

plane wave is achievable using a liquid crystal cell or a spatial phase

modulator. It is found that the focus field distribution of doughnut beams

focused by a high NA objective is substantially different from that given by

low NA objectives. Depolarisation effects destroy the central zero intensity

points for doughnut beams of topological charge ±1 and ±2. The focused

field distribution of doughnut beams shows an increased ring intensity along

the direction perpendicular to the polarisation direction, while the focus



becomes elongated in the polarisation direction. These diffraction effects

are enhanced when doughnut beams are centrally obstructed and affect the

trapping efficiency of such beams. It has been found that the lifting force of

small particles is greatly reduced when focused doughnut beam is used for

trapping, compared to the plane wave, due to the field intensity reduction in

the central focal region of such an illumination. For large particles, the axial

trapping efficiency (ATE) in forward direction is increased when doughnut

beam illumination is used, while the backward ATE is relatively unchanged.

It is shown, both theoretically and experimentally, that the maximal TTE


doughnut beam illumination, compared with the plane wave illumination.

Near-field trapping modality based on particle interaction with evanescent

field generated under the total internal reflection (TIR) has been investigated

in two different configurations: wide area evanescent field and focused

evanescent field. Our research in near-field trapping force exerted on dielectric

particles by a wide area evanescent field shows that the particle is attracted

towards the interface at which the evanescent field is generated and it is

pushed in the forward scattering direction. Both of these attractive and

pushing forces are stronger for the incident P polarised illumination.

The near-field trapping force achieved by focused evanescent field attracts

particles toward the interface at which the field is generated and confines

them in the focal region. The trapping efficiency is slightly larger for the

focused evanescent field generated by a plane wave, compared to the one

generated by a doughnut beam. The maximal ATE, for large dielectric

particles, constitutes∼43% of that achieved by the far-field trapping modality

under the same conditions. Furthermore, it is determined that the highest

ATE does not occur for the geometric focus at the interface, but it occurs for

a slightly defocused evanescent field with the geometrical focus at a depth of



∼62 nm. Our research indicates that unlike the far-field trapping modality,

the dependence of the maximal ATE on the particle size is nearly linear for

the near-field trapping with focused evanescent wave illumination. We predict

that an input laser power of ∼10 µW suffice to overcome the gravity force

acting on a polystyrene particle of 800 nm in radius or smaller, enabling the

particle three-dimensional trapping in an upright near-field trapping system.

7.2 Future prospects

The research investigations and methodology described in this thesis can

be further extended to include the following research key areas: trapping

force determination and study of far-field and near-field trapping of metallic

particles, investigations of torque exerted on microparticles for complete near-

field manipulation, including particle rotation, and analysis of the interface

effects on the near-field particle manipulation using a focused evanescent field.

7.2.1 Optical trapping of metallic particles

7.2.1.1 Far-field trapping

For certain experimental purposes it would be more advantageous to use

metallic particles as handle points for optical trapping manipulation. Such

experiments range from a biological cell manipulation using infused metallic

nano-particles to particle-trapped SNOM with metallic particle offering much

higher near-field scattering efficiency.

Our physical methodology described in Chapter 4 is also valid for

considering metallic particle trapping in both far and near fields. Research

into metallic particles trapping efficiency would indicate and determine an



optimum particle parameters, such as its size and material, for a particular

optical trapping experiment. Analogous to the MDR effects with dielectric

particles, using our methodology with optical trapping of metallic particles

would also provide a physical insight into the influence of the surface plasmon

effect, associated with small metallic particles, on the optical trapping

process.

7.2.1.2 Near-field trapping

The effects of the interface at which the focused evanescent field is generated

may play a significant role for studying the near-field trapping process with

metallic particles. Due to the relatively high reflectivity of metallic particles

it could be expected that the close proximity of a dielectric interface will

affect the trapping efficiency of such particles.

This issue can be dealt with an extension of our vectorial diffraction model,

presented in Chapter 4, to include the multiple field reflection between a

metallic particle and the interface. The multiple field reflections can be

implemented using the plane wave decompositions approach of Inami and

Kawata [128]. This approach would provide a comprehensive and rigorous

method for characterising the optical near-field trapping force for highly

reflective particles.

7.2.2 Near-field micromanipulation system

On the basis of our research into the near-field trapping process using a fo-

cused evanescent field illumination described in Chapter 6 a fully operational

near-field trapping system for single molecule detection experiments can be

envisioned.



(a) (b)

Fig. 7.1: Operational near-field trapping system.

A far-field optical trapping is utilised for reaching deep into the par-

ticle cell and selecting an appropriate particle, possibly a particle with a

biomolecule attached to it. The particle is lifted to the top surface using

a relatively large backward axial trapping force exerted with such a far-

trapping technique (Fig. 7.1(a)). Once the particle is brought to the surface,

the incident illumination is centrally obstructed, applying an appropriate

pattern on a spatial light modulator (SLM) operating in the intensity mode

for example, which switches the optical trapping modality to the near-field

trapping (Fig. 7.1(b)). The particle is then manipulated using the localised

near-field trapping volume.

Another interesting capability of the particle manipulation using near field

trapping with focused evanescent illumination lies in the area of particle

rotation. A particle can be induced to rotate by such a near-field using

two methods. The first method is based on rotating the polarisation state

of the incident beam, which rotates the focused field distribution and the

particle or particles trapped with it. The second method relies on using

a higher charge doughnut beam to rotate the particle. Due to the helical



wavefront of doughnut beam illumination, it can be expected that the near-

field distribution of a highly obstructed doughnut beam becomes asymmetric,

which can lead to an interesting force distribution (Fig. 7.2). Such a force

distribution facilitates particle rotation , provided that the trapping power

is sufficient to overcome the friction force due to the viscosity of the particle

suspension medium. Near-field particle rotation, based on highly obstructed

Fig. 7.2: Trapping efficiency mapping in the XY plane for a polystyrene particleof radius a = 1 µm, (light polarisation is in the X direction) across the focusedevanescent field generated at a coverslip interface by a plane wave (Charge 0) anddoughnut beam illumination (Charge 1, 2, and 3). NA=1.65, λ=532 nm, ε=0.85,n1=1.78 and n2=1.33.



doughnut beams, is certainly a very rich and broad research field which

provides immense opportunities in the future work. However, due to the

limited candidature time, we have not considered this novel particle rotation

mechanism in greater details.


Chapter 7

Conclusion

7.1 Thesis conclusion

The main research work in this thesis presents an optical trapping model

based on two physical processes, vectorial diffraction of incident laser beam by

a high numerical aperture (NA) objective and scattering by a small particle

(Chapter 4). The interaction of the small particle and the focused laser

beam is described using an extension of the classical plane wave Lorentz-

Mie theory, while the force is determined using the Maxwell stress tensor

approach. An important advantage of our approach over other optical

trapping models reviewed in Chapter 2 is that it provides an exact description

of the electromagnetic (EM) field distribution in the focal region of a high

NA objective. As opposed to the other optical trapping models, our model

is capable of considering an arbitrary wavefront incident at the objective

entrance pupil. In other words, our model is capable of including any

apodisation function of the focusing objective, phase modulation of the

incident beam, polarisation effects and the effects of spherical aberration (SA)

usually present in optical trapping experiments. As a consequence we are able

154


to consider the trapping process under doughnut beam illumination, which

represents one class of phase modulated laser beams (Chapter 5). We have

shown that our optical trapping model can deal with the far-field trapping

modality, as well as with the near-field trapping modality implemented with

a focused evanescent field generated by using a central obstruction in the

incident beam path. Our optical trapping model provides a first theoretical

treatment of such a near-field trapping modality (Chapter 6).

In addition to the optical trapping model, this thesis presents a nanometric

sensing model for detection of scattered evanescent waves (Chapter 3).

The sensing model, which describes the conversion of evanescent photons

into propagating photons, is implemented by considering the near-field Mie

scattering process enhanced by morphology dependent resonance (MDR) by

a small particle in the vicinity of a plane interface. The transformation

of the three-dimensional scattered field distribution to the detector plane

is achieved using vectorial diffraction of the scattered signal. The vectorial

diffraction approach for the transformation of the scattered signal is necessary

due to the vectorial nature of the scattered field. Such a sensing model

is essential for understanding optical trapping systems that use evanescent

field illumination for high resolution position monitoring and imaging, such

as optical trap nanometry and particle-trapped scanning near-field optical

microscopy (SNOM).

The nanometric sensing model reveals that the intensity pattern generated

in the far-field region of the collecting objective shows an interference-like

pattern when large dielectric particles are employed as scatterers. For small

particles the intensity pattern is similar to that given by the dipole radiation.

Our research has shown that MDR effects are evident in near-field Mie

scattering of large dielectric particles and that the resonance peaks become

sharper and less separated with increasing the effective refractive index. The
