chapter 4. the intensity -dependent refractive indexoptics.hanyang.ac.kr/~choh/degree/[2013-1]...
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Nonlinear Optics Lab. Hanyang Univ.
Chapter 4. The Intensity-Dependent Refractive Index
- Third order nonlinear effect
- Mathematical description of the nonlinear refractive index
- Physical processes that give rise to this effect
Reference :
R.W. Boyd, “Nonlinear Optics”, Academic Press, INC.
Nonlinear Optics Lab. Hanyang Univ.
4.1 Description of the Intensity-Dependent Refractive Index
Refractive index of many materials can be described by
......~2
20 tEnnn
0n
2n
where, : weak-field refractive index
: 2nd order index of refraction
..)(~
cceEtE ti 2*2 )(2)()(2~
EEEtE
2
20 2 Ennn : optical Kerr effect (3rd order nonlinear effect)
Nonlinear Optics Lab. Hanyang Univ.
Polarization :
EEEEP eff2)3()1( 3)(
In Gaussian units, effn 412
2)3()1(22
20 12412 EEnn
2)3()1(42
2
2
20
2
0 124144 EEnEnnn
21)1(
0 41 n
0
)3(
2 3 nn
Nonlinear Optics Lab. Hanyang Univ.
Appendix A. Systems of Units in Nonlinear Optics (Gaussian units and MKS units)
1) Gaussian units ;
....~~~
)(~ 3)3(2)2()1( tEtEtEtP
2cm
bstatcoulom
cm
statvolt~~ EP
statvolt
cm~1
# )2(
E
2
2
2
)3(
statvolt
cm~1
#
E
essdimensionl# )1(
The units of nonlinear susceptibilities are
not usually stated explicitly ; rather one
simply states that the value is given in
“esu” (electrostatic units).
Nonlinear Optics Lab. Hanyang Univ.
2) MKS units ;
....~~~
)(~ 3)3(2)2()1(
0 tEtEtEtP
2
~
m
CP
m
VE ~
: MKS 1
....~~~
)(~ 3)3(2)2()1(
0 tEtEtEtP : MKS 2
essdimensionl)1( V
m
E
~
1)2( 2
2)3(
V
m
2
)2(
V
C
3
)3(
V
Cm
[C/V][F] F/m,1085.8# 12
0
In MKS 1,
In MKS 2, essdimensionl)1(
Nonlinear Optics Lab. Hanyang Univ.
3) Conversion among the systems
)1(41~~
4~~
)2( EPED
)1(
00 1~~~~
EPED
(Gaussian) E~
103 (MKS) E~
V 300statvolt 1 )1( 4
(Gaussian)
(MKS) (Gaussian)4(MKS) )1()1(
)3((Gaussian)
103
41) (MKS )2(
4
)2(
(Gaussian)103
42) (MKS )2(
4
0)2(
(Gaussian))103(
41) (MKS )3(
24
)3(
(Gaussian))103(
42) (MKS )3(
24
0)3(
Nonlinear Optics Lab. Hanyang Univ.
Alternative way of defining the intensity-dependent refractive index
Innn 20
20 )(2
Ecn
I
(4.1.4) 2
20 )(2 Ennn
)3(
2
0
2
0
)3(
0
2
0
2
12344
cnncnn
cnn
?)3(
2 n
esu0395.0
esu1012
W
cm )3(
2
0
)3(7
2
0
22
2
ncnn
),:( esu12
101.9)3(
2
CS ,/1 2cmMWI ?n,58.10n
Wcm
esun
n
/103
109.158.1
0395.00395.0
214
12
2
)3(
2
0
2
8146
2 103103101 Inn
Example)
Nonlinear Optics Lab. Hanyang Univ.
Nonlinear Optics Lab. Hanyang Univ.
Physical processes producing the nonlinear change in the refractive index
1) Electronic polarization : Electronic charge redistribution
2) Molecular orientation : Molecular alignment due to the induced dipole
3) Electrostriction : Density change by optical field
4) Saturated absorption : Intensity-dependent absorption
5) Thermal effect : Temperature change due to the optical field
6) Photorefractive effect : Induced redistribution of electrons and holes
Refractive index change due to the local field inside the medium
Nonlinear Optics Lab. Hanyang Univ.
Nonlinear Optics Lab. Hanyang Univ.
4.2 Tensor Nature of the 3rd Susceptibility
rrrrF ~)~~(~~ 2
0 mbmres
Centrosymmetric media
Equation of motion :
mteb /)(~~)~~(~~2~ 2
0 Errrrrr
Solution :
cceEeEeEttititi
.)(~
321
321
E
n
ti
nneE
)(
Perturbation expansion method ;
...)(~)(~)(~)(~ )3(3)2(2)1( tttt rrrr
mte /)(~~~2~ )1(2
0
)1()1(Errr
0~~2~ )2(2
0
)2()2( rrr
0~~~~~2~ )1()1()1()3(2
0
)3()3( rrrrrr b
Nonlinear Optics Lab. Hanyang Univ.
3rd order polarization :
)()( )3()3(
qq Ne rP
jkl mnp
plnkmjpnmqijklqi EEE)(
)3()3( ),,,()( P
jkl
plnkmjpnmqijkl EEED ),,,()3(
where, D : Degeneracy factor
(The number of distinct permutations of the frequency m, n, p)
4th-rank tensor : 81 elements
Let’s consider the 3rd order susceptibility for the case of an isotropic material.
333322221111
332233112233221111332211
323231312121232313131212
322331132332211213311221
1221121211221111
21 nonzero
elements : and,
(Report) (Report)
Element with
even number
of index
Nonlinear Optics Lab. Hanyang Univ.
Expression for the nonlinear susceptibility in the compact form :
jkiljlikklijijkl 122112121122
Example) Third-harmonic generation : )3( ijkl
122112121122
)()3()3( 1122 jkiljlikklijijkl
Example) Intensity-dependent refractive index : )( ijkl
122112121122
jkiljlikklijijkl )()()()3( 12211122
Nonlinear Optics Lab. Hanyang Univ.
Nonlinear polarization for Intensity-dependent refractive index
jkl
lkjijkli EEE )(3)(P
EEEEP
iii EE 12211122 36)( EEEEEEP 12211122 36 in vector form
Defining the coefficients, A and B as
12211122 6,6 BA (Maker and Terhune’s notation)
EEEEEEP BA2
1
Nonlinear Optics Lab. Hanyang Univ.
In some purpose, it is useful to describe the nonlinear polarization by in terms of an
effective linear susceptibility,
j
jiji EP
ij as
jijiijij EEEEBA 2
1 EE
12211122 362
1 BAA
12216 BB
where,
Physical mechanisms ;
ictionelectrostr : 0,0
response electronict nonresonan : 2,1
norientatiomolecular : 3'/,6/
B'/A'B/A
B'/A'B/A
ABAB
Nonlinear Optics Lab. Hanyang Univ.
4.3 Nonresonant Electronic Nonlinearities
# The most fast response : s10/2 16
0
va [a0(Bohr radius)~0.5x10-8cm, v(electron velocity)~c/137]
Classical, Anharmonic Oscillator Model of Electronic Nonlinearities
Approximated Potential : 422
04
1
2
1rrr mbmU
)()()()(3
),,,(3
4
)3(
pnmq
jklijlikklij
pnmqijklDDDDm
Nbe
DDm
Nbe jklijlikklij
ijkl 33
4
)3(
3)(
(1.4.52)
where, iD 222
0
According to the notation of Maker and Terhune,
DDm
NbeBA
33
42
Nonlinear Optics Lab. Hanyang Univ.
Far off-resonant case, 22
0
2
00 /,)( dbD
26
0
3
4)3(
dm
Ne
esu102~
g101.9 ,rad/s107
esu108.4 ,cm103 ,cm104
14)3(
2815
0
108322
m
edN
Typical value of )3(
Nonlinear Optics Lab. Hanyang Univ.
4.4 Nonlinearities due to Molecular Orientation
The torque exerted on the molecule when an electric field is applied :
EPτ
induced dipole moment
Nonlinear Optics Lab. Hanyang Univ.
Second order index of refraction
Change of potential energy :
1133 dEpdEpdpdU E 111333 dEEdEE
22
1
22
3
2
11
2
33 sincos2
1
2
1EEEEU
22
13
2
1 cos2
1
2
1EE
* Optical field (orientational relaxation time ~ ps order) : 222 ~~EtEE
Mean polarization :
2
131
2
1
2
3 cos)(sincos
Nn 412
2
2
1EU
1) With no local-field correction :
*
Nonlinear Optics Lab. Hanyang Univ.
kTUd
kTUd
/exp
/expcoscos
2
2
Defining intensity parameter, kTEJ /~
2
1 2
13
0
2
0
22
2
sincosexp
sincosexpcoscos
dJ
dJ
i) J 0
3
1
sin
sincoscos
0
0
2
0
2
d
d
130 3
2
3
1
13
2
03
2
3
141 Nn : linear refractive index
Nonlinear Optics Lab. Hanyang Univ.
ii) J 0
2
131
2 cos41 Nn
3
2
131
2
0
2
3
1cos
3
14 Nnn
3
1cos4 2
13 N
2
0
2
0
2 nnn
3
1cos
2 2
13
0
0
n
Nnnn
Nonlinear Optics Lab. Hanyang Univ.
0
2
0
22
2
sincosexp
sincosexpcoscos
dJ
dJ
....945
8
45
4
3
1 2
JJ
kT
E
n
NJ
n
Nn
22
13
0
13
0
~
45
4
45
4
2
4
2
2
~En
Second-order index of refraction :
kTn
Nn
2
13
0
245
4
Nonlinear Optics Lab. Hanyang Univ.
2) With local-field correction
PEE~
3
4~~loc
locp E~
NpP~
and
PEP
~
3
4~~ N
EP~
3
41
~
N
NE~)1(
N
N
3
41
)1(
)1()1( 41
N
3
4
2
1)1(
)1(
N
n
n
3
4
2
1or
2
2
: Lorentz-Lorenz law
kT
n
n
Nn
2
13
42
0
0
23
2
45
4
Nonlinear Optics Lab. Hanyang Univ.
8.2 Electrostriction
: Tendency of materials to become compressed in the presence of an electric field
Molecular potential energy :
2
00 2
1
2
1EddpU
EE
EEEEE
Force acting on the molecule :
2
2
1EU F
density change
Nonlinear Optics Lab. Hanyang Univ.
Increase in electric permittivity due to the density change of the material :
Field energy density change in the material = Work density performed in compressing the material
88
22 EEu
stst p
V
Vpw
So, electrostrictive pressure :
88
22 EEp est
where,
e : electrostrictive constant
Nonlinear Optics Lab. Hanyang Univ.
stst CPPP
1
Density change :
PC
1where, : compressibility
8
2EC e
For optical field,
8
~2EC e
4
1
4
4
1
8
~2
EC e
2
2
~
32
1EC e
e
Nonlinear Optics Lab. Hanyang Univ.
..
~ccet ti
EE EE2
~2E
EE2
216
1eC
<Classification according to the Maker and Terhune’s notation>
Nonlinear polarization : EP
EEP22
216
1eC
EE
2)3( )(3
2
2
)3(
48
1)( eC
0,16
2
2
is,That BC
A eT
Nonlinear Optics Lab. Hanyang Univ.
e
12 ne
3/21 22 nne
: For dilute gas
: For condensed matter (Lorentz-Lorenz law)
Example) Cs2, C ~10-10 cm2/dyne, e ~1 (3) ~ 2x10-13 esu
Ideal gas, C ~10-6 cm2/dyne (1 atm), e =n2-1 ~ 6x10-4 (3) ~ 1x10-15 esu