Radiation forces on a dielectric
sphere in the Rayleigh and Mie
scattering regime
Yong-Gu Lee
Reference: 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, McGrawHill Book Company Inc. 1941
Akira Ishimaru, Electromagnetic wave propagation,
radiationa and scattering, Prentice-Hall Inc. 1991
Electromagnetic forces on
charges and currents
 J  nda  s
volt =
watt

ampere
 E  ds  
c
d
 dv
dt V
ampere
coulomb
1
coulomb
2
3

meter


meter


ampere=
meter 2
meter 3
second
second
meter
1

meter

2
second 2
second  kilogram  meter
2
1
coulomb

second
coulomb 
second
weber
kilogram  meter 2
volt=
 weber  volt  second=
second
coulomb  second
kilogram×
d
B  nda

dt s
coulombs volts
kilogram


 Fe  qE
3
2
2
2
meter
meter
second  meter
amperes webers
kilogram
[J  B] 


 dFm  Ids  B
meter 2 meter 2 second 2  meter 2
[  E] 
Julius Adams Stratton, “Electromagnetic thoery,” pp 96-97, McGraw-Hill Book Company, 1941
Wave optics crash course
• Wave equation
• Helmholz eqn.
• Elementary waves
 2U 
c
1  2U
0
c 2 t 2
c0
n

2

 k 2 U (r )  0
U (r, t )  U (r )e j 2 t  a(r)e j r e j 2 t
2 
k

c
c
– Spherical wave
– Paraboloidal wave
– Paraxial wave
U (r ) 
A  jkr
e
r
x2  y 2
r   z 
z
x2  y 2
, r  z
2z
x y
 jk
A
A
2z
U (r )  e  jkr   e  jkz e

r
z
2
U (r)  A(r)e jkz
• Paraxial Helmholz eqn.
T2 A  j 2k
A
0
z
T2 
2
2

x 2 y 2
2
 2 A  jkz  2 A  jkz
 jkz A
2  jkz
 jkz A
 jkz  A
e

e

jke

Ak
e

jke

e
 k 2 Ae  jkz
2
2
2
x
y
z
z
z
 2 A  jkz  2 A  jkz
A  jkz  jkz  2 A
 2 e  2 e  2 jk
e e
x
y
z
z 2
2
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.
x2  y 2
A  jkz  jk 2 q ( z )
U (r ) 
e e
q( z )
1
1


j
q( z ) R( z )
W 2 ( z)
2
wavefront radius of curvature and beam width
2
W0 W 2  z   jkz  jk 2 R z   j  z 
U (r )  A0
e
e
W z
1
  z 2  2
W  z   W0 1    
  z0  
  z0  2 
R  z   z 1    
  z  
z
  z   tan 1
z0
1
  z 2
W0   0 
  
Electric-field vector within zeroth-order approximation
in a paraxial Gaussian beam

x
E  r   E1  xˆ 
z  jz0


x
 E1  xˆ 
z  jz0


zˆ  U  r 

x2  y 2
 A  jkz  jk 2 q ( z )
ˆz 
e e
 q( z )

 A W0
x
 E1  xˆ 
zˆ 
e
z

jz
jz
W
z


0

 0
1
1


j
q( z ) R( z )
W 2 ( z)
  z 2 
W  z   W0 1    
  z0  
 z  
R  z   z 1   0  
  z  
z
  z   tan 1
z0
2
1
  z0  2
W0  

  
jkW02
e jkz
2
jkW0  2 z
E  r   xˆ E0

x2  y 2
W 2  z
j
e
kW02
e
 jkz  jk
x2  y 2
 j ( z )
2R( z)
H  r   zˆ 
1
2
where Z 0 
E r 
Z0
2



2
z

  
2 kz x 2  y 2

e


2
kW02 x 2  y 2

2
kW02 
2

 2 z 2
yˆ n2 0 cE  r   yˆ H  r 
 2 is the intrinsic impedance of the medium for plane waves.
The inherent relations of  2
 0 n22 and 2
0 for non-conducting and non-magnetic
medium are used in the previous formulation.
H  r   zˆ 
E  r, t   Re E  r  exp  ift   ,
where Z 0 
E r 
Z0
2
yˆ n2 0 cE  r   yˆ H  r 
 2 is the intrinsic impedance of the medium for plane waves.
The inherent relations of  2
H  r, t   Re  H  r  exp  ift   ,
 0 n22 and 2
0 for non-conducting and non-magneti
medium are used in the previous formulation.
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 corresponding to the Poynting vector is given by
S  r, t   E  r , t   H  r , t   Re E  r  exp  ift    Re  H  r  exp  ift  

 

1
1
E  r  exp  ift   E*  r  exp  ift   H  r  exp  ift   H *  r  exp  ift 
2
2
1
 E  r   H*  r   E*  r   H  r   E  r   H  r  exp  ift   E*  r   H *  r  exp  ift 
4
1
1
 E  r   H*  r   E*  r   H  r   E  r   H  r  exp  ift   E*  r   H*  r  exp  ift 
4
4
1
1
 S  r   S*  r   E  r   H  r  exp  ift   E*  r   H *  r  exp  ift 
2
4
An important and measurable physical quantity in evaluating the radiation force of the
light is the beam intensity or the irradiance at the position r  ( x, y, z ). This is defined
as a time-averaged version of the Poynting vector and is given by
1
I (r )  S  r, t  T  Re E  r   H *  r    zˆ I (r ),
2
where







 2P  1 
I r   
e
2 
2

w
1

z
0







2 x2  y 2
1 z
2

P   w02n20cE02 / 4, and  x, y, z    x / w0 , y / w0 , z / w0 
Eqn given in the paper is incorrect
I (r )  S  r, t 
T

1
Re E  r   H*  r    zˆ I (r ),
2
2 kz  x 2  y 2 
kW02  x 2  y 2   
2 kz  x 2  y 2 
kW02  x 2  y 2  


j

j

2
2
2
2
2
2
2
2
2
2
2
2
2 2






jkW0
 jkW0
1
 kW0   2 z 
 kW0   2 z 
 kW0   2 z 
 kW0   2 z 
 jkz
jkz
 zˆ Re   E0
e

e

e
E
e

e

e
/
z




0
0
2
2
2
  jkW0  2 z

   jkW0  2 z



 

kW02  x 2  y 2   
kW02  x 2  y 2  




2
2
2
2


jkW02
 jkW02
kW02    2 z   
kW02    2 z  
1


 zˆ Re   E0

e
E

e
/
z
 0
 0
2
2
2

jkW

2
z
0
  jkW0  2 z





 

 zˆ
2
0

kW
1E
z0
2
kW02

2
0

2
   2z 
2
2
2
e


kW02 x 2  y 2

2
kW02 

 2 z 2
H  r   zˆ 
where Z 0 
E r 
Z0
2
2
yˆ n2 0cE  r   yˆ H  r 
=
0
0 0
=
is the intrinsic impedance of the medium for plane wav
 0 n22
 0 0 n22
The inherent relations of  2
 0 n22 and 2
0 for non-conducting and non-magnetic
medium are used in the previous formulation.

 2P  1
I r   
e
2 
2
  w0  1  z


2 x2  y 2
1 z
2


 n  cE 2  1
 2 0 0 
e
2
2

1 z

2 x2  y 2

1 z 2
P   w02n20cE02 / 4, and  x, y, z    x / w0 , y / w0 , z / w0 

2 x2  y 2

 2 P  1  1 z 2
I r   
e
(Saleh eq. 3.1-15)
2 
2

w
1

z
 0
P   w02 n2 0 cE02 / 4, and  x, y, z    x / w0 , y / w0 , z / w0 
22
 2 2
 2 P  w02
 2 P  w02
w0  z
 0  0 I  r  d d    0  0   w02  w02  z 2 e  d d     w02  w02  z 2
2
2

2

1
2
2
w02  z 2
 2 P  w02 w02  z 2

2  P
2 
2
2

w
w

z
4
 0 0
H  r   zˆ 
where Z 0 
E r 
Z0
2
yˆ n2 0 cE  r   yˆ H  r 
 2 is the intrinsic impedance of the medium for plane waves.
The inherent relations of  2
 0 n22 and 2
0 for non-conducting and non-magnetic
medium are used in the previous formulation.
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 one give by
1

s=

.
kw 0 2 w 0
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
Taken from the course notes of Radar Metrology by Prof. Bob Rauber (UIUC)
http://www.atmos.uiuc.edu/courses/atmos410-fa04/presentations.html
Einc
incident
plane
wave
p
Dielectric
Sphere
(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.
Taken from the course notes of Radar Metrology by Prof. Bob Rauber (UIUC)
http://www.atmos.uiuc.edu/courses/atmos410-fa04/presentations.html
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Metal spheres
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/
Slides taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov
https://www.icfo.es/courses/biophotonics2006/html/