Focusing of light

Colin Sheppard

Division of Bioengineering

and

Department of Biological Sciences

National University of Singapore

E-mail: colin@nus.edu.sg

Tight focusing of light

•Microscopy

•Laser micromachining and microprocessing

•Optical data storage

•Optical lithography

•Laser trapping and cooling

•Physics of light/atom interactions

•Cavity QED

Overview

•Complete spherical focusing

•Bessel beams

•Gaussian beams

•High numerical aperture focusing

•Pupil masks (super-resolving filters)

•Polarization in focusing

•4Pi geometry

•Moments

Complete spherical focusing

Complete spherical (4 ) scalar focusing

Same as field of a point source and a point sink

From scalar form of Richards and Wolf

A plane-polarized wave after focusing

• px electric dipole alongx axis• my magnetic dipole alongy axis• C is nearly linear polarization• A is polarization singularity of order 2• Richards & Wolf (Ignatovsky) polarization

Direction of propagation

A plane polarized wave after focusing:

Polarization on reference sphere

direction of propagation

C

• Polarization is same as that of • px (electric dipole along x axis)• my (magnetic dipole along y axis)

• C is nearly linear polarization• Richards & Wolf polarization

Bessel beams

Bessel Beam

Annular mask(Linfoot & Wolf, 1953

Axicon (McLeod, 1954)

Diffractive axicon (Dyson, 1958)

Bessel beams

J0 beam propagates without spreading:

Also higher order beams Jn(v) exp (in ) with a phase singularity (vortex)

Sometimes called “diffraction-free” beams

Bessel-Gauss beam

= fractional Fourier (Hankel) order

transverse coordinate

Bessel-Gauss beam

a = 0.1

annular beam

Non-paraxial Bessel beam

(plane polarized illumination)

Time-averaged electric energy density:

30 90

60 120doublespot

x-polarized illumination:

Intensity along x, y axes: NA = 1.4

Circular pupil Annular pupilBroad along x axis because oflongitudinal field component

Radial polarization TM0

Annulus at high NA:

circular polarization or TM0

•High NA, circular polarized annulus: ~same width as Airy•High NA, TM0 annulus (radially polarized illumination): similar to paraxial (Dorn, Quabis and Leuchs, PRL 91, 233901, 2003)

Widths of Bessel beams

Solid immersion lens

Gaussian beams

Highly convergent Gaussian beams,

Complex source/sink theory:

Electric + magnetic dipoles at complex location

r R

Complex source point model:

Amplitude is the same as that for a source at

the point z = iz0, where z0 is the confocal parameter

Deschamps El. Lett. 7, 684 (1971)

Couture and Bélanger, Phys. Rev A 24, 355 (1981)

Intensity in waist, LP01

Time-averaged electric energy density:

•Double-spot•Caused by magnetic dipole component

Intensity and phase along axis, LP01

Gouy phase shift

Far field

Electric field:

Amplitude:

•axially symmetric

•more directional than the scalar case

Radiation pattern in far field

•Directional even for kz0 = 0

a2( )

Complex source/sink Gaussian beam,

TM01 and TE01modes

Transverse magnetic (axial electric dipole, radial illumination):

Transverse electric (axial magnetic dipole, azimuthal):

•Surface-emitting semiconductor lasers•also in gas, solid state and dye lasers•components of TEM01* (doughnut mode)

After focusing, not radial, as axial component

Intensity in waist, TM01 and

TE01modes Transverse magnetic(axial electric dipole)

Transverse electric (axial magnetic dipole)

E

H

E

Hzero on axis

non-zero on axis (longitudinal field)

Azimuthal

Radial + longitudinal component

Highly convergent focusing

Model for focusing by high numerical

aperture lens

(Debye approximation)

Equivalent refractive locus (sphere for aplanatic system)

f f

Front focal plane

E1( , )

E(r)

Black Box

Richards and Wolf, 1959 Angular spectrum of plane waves

I2: cross-polarization component (ey)I1: longitudinally-polarized component

Aplanatic factor

Ignatovsky, 1919

Focal field as integral over angular spectrum:

Aplanatic factor

I3: cross-polarization componentI1: longitudinally-polarized component

Pupil masks

Performance parameters, paraxial

t 2

Transverse gain

Axial gain (Radius of Gyration2 about Centre of Gravity)

Moments of pupil

Centre of Gravity

Radius of Gyration squared

For real-valued pupils:

(Zero order moment)2

• Calculate performance directly from pupil

High numerical aperture scalar systems

Filter performance parameters for high-aperture focusing

F =I0

2

2 Q (c)2

dc1

1

FI =2I0

2

Q (c)2

cdc

0

1

GT

=3

21

I2

I0

GA

= 3I

2

I0

I1

I0

2

GP

=1

32G

T+ G

A( ) =1I

1

I0

2

c = cosU( , z) = ikf0

P( )J0 k sin( )exp ikzcos( )sin d

Total power or integrated intensity in axial sidelobes

Integrated energy in outer rings

3D polar gain (1st moment)

Interpretation in terms of Moment of Inertia (2nd moment) of generalized 3D pupil (cap of sphere)

(zero order moment)2

Vectorial electromagnetic case

qn = Q (c)1

1

cndc

GA = 3q

0q

2q

1

2

q0

2

Gx =3

4

10q0q

13q

0q

23q

0

24q

1

2

q0

2

Gy =3

4

3q0

22q

0q

1q

0q

2

q0

2

GT =3

4

4q0q

12q

0q

22q

1

2

q0

2

GP = (Gx +Gy +GA )/ 3

GP =2q1(q0 q1)

q0

2

In =1

1

Q(c)1 c

1+ c

n / 2

Jn k 1 c2( )exp ikzc( )dc.

Q(c) = c1/ 2 (1+ c)Aplanatic (sine condition):

Richards and Wolf (equivalent form):

c= cos

Circular polarized or unpolarized:

Only zero and first moment (Centre of Gravity)

Gains for electromagnetic case, plane

polarized input

negative means double spot

Intensity at the focus

Mixed dipole apodization (p + m) gives greatest intensity at the focus

F

Electric dipole polarization

Electric dipole wave:

Ratio of focal intensity to power input

C. J. R. Sheppard and P. Török, "Electromagnetic field in the focal region of an electric dipole wave," Optik 104, 175-177 (1997).

electric dipole ED

mixed dipole(plane polarized)

TM0 (radial polarization)

ED is highest

90o 180o

Radial polarization (TM0):

polarization on reference sphere

direction of propagationred: electric field

blue: magnetic field

Radial polarization with phase mask

Wang HF, Shi LP, Luk’yanchuk B, Sheppard C, Chong CT (2008)

Creation of a needle of longitudinally polarized light in vacuum

using binary optics,

Nature Photonics, 2, 501-505, 22

Electric dipole: Polarization on

reference sphere

Mixed = ED + MD

direction of propagation

red: electric field

blue: magnetic field

Polarization on reference sphere:

TE1, TM1

Polarization of

input wave

Bessel beams: TE1 polarization Mixed

ED TE1

Mixed

ED TE1

Mixed

ED TE1

Mixed

ED TE1

30° 60°

90° 150°

High NA: Intensity at the focus for

different polarizations

1

1

p+m

TM0p

p

p+m

Gains for different polarizations

Area of focal spot

NA = 0.89NA = 0.91

Focal volume

NA = 0.98TE1 smallest

Rotationally symmetric beams

•TM0 = radial polarized input (longitudinal field in focus)

•TE0 = azimuthal polarization

•x polarized + i y polarized = circular polarized

•TE1x + i TE1y = azimuthal polarization with a phase singularity

(bright centre)

•EDx + i EDy = elliptical polarization with a phase singularity

(bright centre) (ellipticity increases with angle from axis)

•(TM1x + i TM1y = radial polarization with a phase singularity)

•Same GT as for average over

Normalized width for rotationally

symmetric

TE1 = azimuthal polarization with phase singularity (vortex)

TE annulus narrowestTE narrowest for NA<0.98

1

1

1

1

Bessel beams:

Transverse behaviour for rotationally

symmetric (also average over )

30o 60o

90omixed

mixed

MD

mixed,

TE1 narrowest

1

1

1

1

Bessel beams for rotationally

symmetric Side lobes

Eccentricity

Transverse gain

Rad

TE1 is narrowest

ED has weakest sidelobes

Rad has weakest sidelobes

NA=0.83

1

1

1

1

1

1

Points to note

•Focusing plane polarized light results in a large focal

spot

•Focusing is improved using radially polarized

illumination

-Strong longitudinal field on axis

•Electric dipole polarization gives higher electric energy

density at focus

•Transverse electric (TE1) polarization gives smallest

central lobe

(smaller than radially polarized for Bessel beam)

•TE1 is asymmetric: symmetric version is azimuthal

polarization with a phase singularity (vortex)

4 Pi microscope (Hell)

Hell, S. Europäisches Patent EP 0 491 281 B1 (18.12.1990) "Doppelkonfokales Rastermikroskop".

4Pi (a) Fluorescence Microscope

Resolution in 4 Pi

• Axial resolution is improved

• Longitudinal field components from counter-propagating beams

cancel out, so transverse resolution is also improved

Performance

parameters for

4 Pi

1.46NA

GP = 1As GT increases, GA decreases

1

1

1

1

1

1

Micron, to be published

4 Pi

transverse

axial

1

1

Micron, to be published

spherical spot

4 Pi:

Need to match electric dipole polarization

Electric field in input plane

radial

4 Pi

Table 1. Values of the parameters F, GT and M for NA = 1.46. “% increase in resolution” is for 4Pi compared with the single lens case.

1

Micron, to be published

Localization in terms of pupil moments

Localization in terms of pupil moments, μn (scalar)

Propagation of second moments (scalar)

(Transverse width)2

(Axial width)2

Axial position of centre of gravity: (Central second moment width)2

Thanks to: Silvia Ledesma, Buenos Aires, Argentina

Juan Campos, Barcelona

Juan-Carlos Escalera, Barcelona

Manuel Martinez-Corral, Valencia

Miguel Alonso, Rochester

Nicole Carlson, Rochester

Steven van Enk, Bell Labs

Gerd Leuchs, Erlangen

Susanne Quabis, Erlangen

Rolf Dorn, Erlangen

Silvania Pereira, Delft

Peter Török, Imperial College

Kieran Larkin, Sydney

Peeter Saari, Estonia

Amar Choudhury, Gauhati

Naveen Balla (SMA)Shakil Rehman (SERI)Tang Wai Teng (SMA)Elijah Yew (SMART)