chapter 4. the intensity-dependent refractive index
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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.
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4.1 Description of the Intensity-Dependent Refractive Index
Refractive index of many materials can be described by
......~220 tEnnn
0n
2n
where, : weak-field refractive index
: 2nd order index of refraction
..)(~ cceEtE ti 2*2 )(2)()(2~ EEEtE
220 2 Ennn : optical Kerr effect (3rd order nonlinear effect)
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Polarization :
EEEEP eff 2)3()1( 3)(
In Gaussian units, effn 412
2)3()1(2220 12412 EEnn
2)3()1(422
220
20 124144 EEnEnnn
21)1(0 41 n
0)3(
2 3 nn
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Appendix A. Systems of Units in Nonlinear Optics (Gaussian units and MKS units)
1) Gaussian units ;
....~~~)(~ 3)3(2)2()1( tEtEtEtP
2cm
bstatcoulomcm
statvolt~~ EP
statvolt
cm~1# )2(
E
2
2
2)3(
statvoltcm
~1#
E
essdimensionl# )1(
The units of nonlinear susceptibilities arenot usually stated explicitly ; rather one simply states that the value is given in “esu” (electrostatic units).
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2) MKS units ;
....~~~)(~ 3)3(2)2()1(0 tEtEtEtP
2
~mCP
mVE ~
: MKS 1
....~~~)(~ 3)3(2)2()1(0 tEtEtEtP : MKS 2
essdimensionl)1( Vm
E
~
1)2( 2
2)3(
Vm
2)2(
VC
3)3(
VCm
[C/V][F] F/m,1085.8# 120
In MKS 1,
In MKS 2, essdimensionl)1(
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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)
10341) (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(
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Alternative way of defining the intensity-dependent refractive index
Innn 20 20 )(
2
EcnI
(4.1.4) 2
20 )(2 Ennn
)3(20
2
0
)3(
02
02
12344 cnncn
ncn
n
?)3(2 n
esu0395.0esu1012W
cm )3(20
)3(720
22
2 ncn
n
),:( esu12101.9)3(2
CS ,/1 2cmMWI ?n,58.10n
Wcm
esun
n
/103
109.158.10395.00395.0
214
122
)3(20
2
8146
2 103103101 Inn
Example)
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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
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4.2 Tensor Nature of the 3rd Susceptibility
rrrrF ~)~~(~~ 20 mbmres
Centrosymmetric media
Equation of motion :
mteb /)(~~)~~(~~2~ 20 Errrrrr
Solution :
cceEeEeEt tititi .)(~321
321 E n
tin
neE )(
Perturbation expansion method ;
...)(~)(~)(~)(~ )3(3)2(2)1( tttt rrrr
mte /)(~~~2~ )1(20
)1()1( Errr
0~~2~ )2(20
)2()2( rrr
0~~~~~2~ )1()1()1()3(20
)3()3( rrrrrr b
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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 witheven numberof index
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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
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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 BA21
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In some purpose, it is useful to describe the nonlinear polarization by in terms of an effective linear susceptibility,
j
jiji EP
ijas
jijiijij EEEEBA 21 EE
12211122 3621 BAA
12216 BB
where,
Physical mechanisms ;
ictionelectrostr : 0,0response electronict nonresonan : 2,1
norientatiomolecular : 3'/,6/
B'/A'B/AB'/A'B/A
ABAB
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4.3 Nonresonant Electronic Nonlinearities
# The most fast response : s10/2 160
va [a0(Bohr radius)~0.5x10-8cm, v(electron velocity)~c/137]
Classical, Anharmonic Oscillator Model of Electronic Nonlinearities
Approximated Potential : 4220 4
121 rrr mbmU
)()()()(3
),,,( 3
4)3(
pnmq
jklijlikklijpnmqijkl DDDDm
Nbe
DDm
Nbe jklijlikklijijkl 33
4)3(
3)(
(1.4.52)
where, iD 2220
According to the notation of Maker and Terhune,
DDmNbeBA 33
42
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Far off-resonant case, 22
0200 /,)( dbD
260
3
4)3(
dmNe
esu102~
g101.9 ,rad/s107
esu108.4 ,cm103 ,cm104
14)3(
28150
108322
m
edN
Typical value of )3(
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4.4 Nonlinearities due to Molecular Orientation
The torque exerted on the molecule when an electric field is applied :
EPτinduced dipole moment
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Second order index of refraction
Change of potential energy :
1133 dEpdEpdpdU E 111333 dEEdEE
221
223
211
233 sincos
21
21 EEEEU
2213
21 cos
21
21 EE
Optical field (orientational relaxation time ~ ps order) : 222 ~~ EtEE
1) With no local-field correction
Nn 412
Mean polarization :
2131
21
23 cos)(sincos
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kTUd
kTUd
/exp
/expcoscos
22
Defining intensity parameter, kTEJ /~21 2
13
0
2
0
222
sincosexp
sincosexpcoscos
dJ
dJ
i) J 0
31
sin
sincoscos
0
0
2
0
2
d
d
130 32
31
13
20 3
23141 Nn : linear refractive index
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ii) J 0
2131
2 cos41 Nn
3
2131
20
2
31cos
314 Nnn
31cos4 2
13 N
20
20
2 nnn
31cos2 2
130
0 n
Nnnn
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0
2
0
222
sincosexp
sincosexpcoscos
dJ
dJ
....9458
454
31 2
JJ
kTE
nNJ
nN
n
22
130
130
~
454
454
24 2
2~En
Second-order index of refraction :
kTn
Nn2
13
02 45
4
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2) With local-field correction
PEE ~34~~ loc
locp E~NpP~ and
PEP ~
34~~ N
EP ~
341
~
N
N E~)1(
N
N
341
)1(
)1()1( 41 N
34
21
)1(
)1(
N
nn
34
21or 2
2
: Lorentz-Lorenz law
kT
nnNn
213
420
02 3
2454
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8.2 Electrostriction
: Tendency of materials to become compressed in the presence of an electric field
Molecular potential energy : 2
00 21
21 EddpU
EE EEEEE
Force acting on the molecule :
221 EU F
density change
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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 pVVpw
So, electrostrictive pressure :
88
22 EEp est
where,
e : electrostrictive constant
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stst CPP
P
1
Density change :
PC
1where, : compressibility
8
2EC e
For optical field,
8
~2EC e
41
4
41
8
~2
E
C e2
2
~32
1 EC e
e
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..~ ccet ti EE EE2~2E
EE2216
1eC
<Classification according to the Maker and Terhune’s notation>
Nonlinear polarization : EP
EEP 22216
1eC
EE 2)3( )(3
22
)3(
481)( eC
0,16
2
2
is,That BCA eT
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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