Post on 01-Feb-2016
description
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
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.
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.
The inherent relations of and for non-conducting and non-magnetic
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
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.
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
Taken from the course notes of Radar Metrology by Prof. Bob Rauber (UIUC)http://www.atmos.uiuc.edu/courses/atmos410-fa04/presentations.html
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
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/