Radiation forces on a dielectric sphere in the Rayleigh and Mie scattering regime

Radiation forces on a dielectric sphere in the

Rayleigh and Mie scattering regime

Yong-Gu LeeReference: Yasuhiro Harada et al. Radiation forces on a dielectric sphere in the Rayleigh scattering regime, Optics communications Vol 124. pp 529-541 (1996)Julius Adams Stratton, Electromagnetic theory, McGraw-Hill Book Company Inc. 1941Akira Ishimaru, Electromagnetic wave propagation, radiationa and scattering, Prentice-Hall Inc. 1991

Electromagnetic forces on charges and currents

Julius Adams Stratton, “Electromagnetic thoery,” pp 96-97, McGraw-Hill Book Company, 1941

2 32 3

22

2

ampere coulomb 1 coulomb- meter meter ampere=

meter meter second second

meter 1kilogram× meterwatt kilogram metersecond secondvolt =

1ampere coulomb secondcoulombsecond

s V

c

dda dv

dt

dds

J n

E2

3 2 2 2

2 2 2 2

weber kilogram meter volt= weber volt second=

second coulomb second

coulombs volts kilogram[ ]

meter meter second meteramperes webers kilogram

[ ]meter meter second meter

s

e

m

dadt

q

d Id

B n

E F E

J B F sB

Wave optics crash course

• Wave equation• Helmholz eqn.• Elementary waves

– Spherical wave– Paraboloidal wave– Paraxial wave

• Paraxial Helmholz eqn.

22

2 2

0

10

UU

c tc

cn

2 2

2 2

( ) 0

( , ) ( ) ( )

2

jj t j t

k U

U t U e a e e

kc c

r

r

r r r

( ) jkrAU e

rr

2 2

2 2

2 2

2

,2

( )x y

jkjkr jkz z

x y z

x yr z r z

z

A AU e e e

r z

r

( ) ( ) jkzU A er r

2 22 2

2 2

2 2 22 2

2 2 2

2 2 2

2 2 2

2 0

2

T T

jkz jkz jkz jkz jkz jkz jkz

jkz jkz jkz jkz

AA j k

z x y

A A A A Ae e jke Ak e jke e k Ae

x y z z z

A A A Ae e jk e e

x y z z

Gaussian beam

• One simple solution to the paraxial Helmholtz equation provides the paraboloidal wave

• Another solution of the paraxial Helmholtz equation provides the Gaussian beam.

2 2

2 2

2

12

2 ( )

2

200

2

00

2

0

1

0

00

( )( )

1 1 wavefront radius of curvature and beam width

( ) ( ) ( )

( )

1

1

tan

x yjk

jkz q z

jkz jk j zW z R z

AU e e

q z

jq z R z W z

WU A e e

W z

zW z W

z

zR z z

z

zz

z

zW

r

r

12

Electric-field vector within zeroth-order approximation in a paraxial Gaussian beam

2 2

2 2 2 2

2

12

10

2 ( )1

0

( )2 ( )0

10 0

2

2

00

2

0

ˆ ˆ

ˆ ˆ( )

ˆ ˆ

1 1

( ) ( ) ( )

1

1

x yjk

jkz q z

x y x yjkz jk j z

W z R z

xE U

z jz

x AE e e

z jz q z

Wx AE e e

z jz jz W z

jq z R z W z

zW z W

z

zR z z

z

E r x z r

x z

x z

12

1

0

00

tanz

zz

zW

2 2

2 220

2 2 20

2 220

20

0 20

2

2

2

ˆ2

jkz

kz x yjkW z

kW x y

kW z

jkWE ejkW z

e

e

E r x

2 00

20

2

22 0 2 2 0

ˆ ˆ ˆ

where is the intrinsic impedance of the medium for plane waves.

The inherent relations of and for non-conducting and non-magnetic

medium are used in the p

n cE r H rZ

Z

n

E rH r z y y

revious formulation.

, Re exp ,

, Re exp ,

where f is the temporal angular frequency of the light.

An instataneous energy flux crossing a unit area per unit time in the beam propagation

direction correspon

t ift

t ift

E r E r

H r H r

* *

* * * *

* *

ding to the Poynting vector is given by

, , , Re exp Re exp

1 1exp exp exp exp

2 21

exp exp41 1

exp4 4

t t t ift ift

ift ift ift ift

ift ift

ift

S r E r H r E r H r

E r E r H r H r

E r H r E r H r E r H r E r H r

E r H r E r H r E r H r

* *

* *

exp

1 1exp exp

2 4An important and measurable physical quantity in evaluating the radiation force of the

light is the beam intensity or the irradiance at the positio

ift

ift ift

*

E r H r

S r S r E r H r E r H r

2 2

2

*

2

12 20

n ( , , ). This is defined

as a time-averaged version of the Poynting vector and is given by

1ˆ( ) , Re ( ),

2where

2 1I

1

T

x y

z

x y z

t I

Pe

w z

r

I r S r E r H r z r

r

2 20 2 0 0 0 0 0/ 4, and , , / , / , /P w n cE x y z x w y w z w

Eqn given in the paper is incorrect

2 00

20

2

22 0 2 2 0

ˆ ˆ ˆ




n cE r H rZ

Z

n

E rH r z y y


2 2 2 2 2 2 2 2 2 20 0

2 2 2 22 2 2 22 2 2 20 0 0 0

*

2 2

2 22 2 2 20 0

0 0 02 20 0

1ˆ( ) , Re ( ),

2

1ˆ Re /

2 2 2

T

kz x y kW x y kz x y kW x yj jkW z kW z kW z kW zjkz jkz

t I

jkW jkWE e e e E e e e zjkW z jkW z

I r S r E r H r z r

z

2 2 2 2 2 20 0

2 22 22 20 0

2 2 20

2 220

2 22 20 0

0 0 02 20 0

2 222

200

2 2200

1ˆ Re /

2 2 2

1ˆ

2 2

kW x y kW x y

kW z kW z

kW x y

kW z

jkW jkWE e E e zjkW z jkW z

kWE ez kW z

z

z

2 00

0 0 020 2 2

2 0 2 0 0 2

22 0 2 2 0

ˆ ˆ ˆ

where = = is the intrinsic impedance of the medium for plane waves.


medi

n cE r H rZ

Zn n

n

E rH r z y y

um are used in the previous formulation.

2 2 2 2

2 2

2 222 0 01 1

2 2 20

2 1 1I

1 2 1

x y x y

z zn cEP

e ew z z

r

2 20 2 0 0 0 0 0/ 4, and , , / , / , /P w n cE x y z x w y w z w

2 2

2

2

2 20

2

12 20

2 20 2 0 0 0 0 0

22 22 2

0 02 2 2 2 2 20 0 0 00 0 0 0

2 20

2 1I (Saleh eq. 3.1-15)

1

/ 4, and , , / , / , /

2 2 1I 2

22

x y

z

w z

Pe

w z

P w n cE x y z x w y w z w

w wP Pd d e d d

w w z w w zw z

r

r

2 2 20 0

2 2 20 0

22

4

w w zPP

w w z

2 00

20

2

22 0 2 2 0

ˆ ˆ ˆ




n cE r H rZ

Z

n

E rH r z y y


In[2]:=0

xcxxx

Out[2]= IfRec 0,1

2c,

Integratecx2 x,x, 0, , Assumptions Rec 0

It must be noted that previous expressions can not describe a rigorous behavior

of the Gaussian laser beam, especially a tightly focused beam. The important

parameter in this context is a nondimensional

0 0

one give by

1s= .

kw 2 w

Previous descriptions are based on paraxial approximations to the scalar wave

equation of the Gaussian beam and correspond to a zeroth-order approach in s.

Thus, as far as s 1,

these descripts are quite accurate. However for other values

the percentage of erros are 0.817% for s=0.02, and 4.37% for s = 0.1.

What is the fundamental difference between the Rayleigh, Mie, and Optical regimes?

With Rayleigh scattering, the electric field is assumed to be invariant in the vicinity of the particle

Taken from the course notes of Radar Metrology by Prof. Bob Rauber (UIUC)http://www.atmos.uiuc.edu/courses/atmos410-fa04/presentations.html

The angular patterns of the scattered intensity from particles of three sizes: (a) small particles, (b) large particles, and (c) larger particles

Rayleigh scattering pattern


Einc

incidentplanewave

DielectricSphere

(water drop)

A plane wave with electric field Einc induces an electric dipole p in a small sphere. The induced dipole is parallel to the direction of Einc which is also the direction of polarization of the incident wave.

p


Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov https://www.icfo.es/courses/biophotonics2006/html/

Metal spheres


Radiation forces on a dielectric sphere in the Rayleigh and Mie scattering regime

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