Post on 06-Jan-2020
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.
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.
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
1) With no local-field correction
Nn 412
Mean polarization :
2
131
2
1
2
3 cos)(sincos
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