Chapter 4. The Intensity-Dependent Refractive Index

$Page 1: Chapter 4. The Intensity-Dependent Refractive Index$
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.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.


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)


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


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).


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(


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(


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)



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



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


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


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 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


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


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


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(


4.4 Nonlinearities due to Molecular Orientation

The torque exerted on the molecule when an electric field is applied :

EPτinduced dipole moment


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


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


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


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


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


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


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


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


..~ 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


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

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Chapter 4. The Intensity-Dependent Refractive Index

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