Nonlinear Optics Lab. Hanyang Univ. Chapter 4. The Intensity-Dependent Refractive Index - Third...

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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

......~2

20 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(22

20 12412 EEnn

2)3()1(422

2

2020 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

bstatcoulom

cm

statvolt~~ EP

statvolt

cm~1

# )2(

E

2

2

2

)3(

statvolt

cm~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

~

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# 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)

103

41) (MKS )2(

4)2(

(Gaussian)103

42) (MKS )2(

40)2(

(Gaussian))103(

41) (MKS )3(

24)3(

(Gaussian))103(

42) (MKS )3(

240)3(


Alternative way of defining the intensity-dependent refractive index

Innn 20 20 )(

2

E

cnI

(4.1.4) 2

20 )(2 Ennn

)3(20

2

0

)3(

02

02

12344 cnncn

ncn

n

?)3(2 n

esu0395.0

esu1012

W

cm )3(20

)3(720

22

2 ncn

n

),:( esu12101.9)3(2

CS ,/1 2cmMWI ?n,58.10n

Wcm

esun

n

/103

109.158.1

0395.00395.0

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~ 2

0 Errrrrr

Solution :

cceEeEeEt tititi .)(~

321321 E

n

tin

neE )(

Perturbation expansion method ;

...)(~)(~)(~)(~ )3(3)2(2)1( tttt rrrr

mte /)(~~~2~ )1(2

0)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 BA2

1


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 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


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

1

2

1rrr 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,

DDm

NbeBA

33

42


Far off-resonant case, 22

0200 /,)( dbD

260

3

4)3(

dm

Ne

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 :

EPτ

induced dipole moment


Second order index of refraction

Change of potential energy :

1133 dEpdEpdpdU E 111333 dEEdEE

221

223

211

233 sincos

2

1

2

1EEEEU

2213

21 cos

2

1

2

1EE

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

2

2

Defining intensity parameter, kTEJ /~

2

1 213

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

20 3

2

3

141 Nn : linear refractive index


ii) J 0

2131

2 cos41 Nn

3

2131

20

2

3

1cos

3

14 Nnn

3

1cos4 2

13 N

20

20

2 nnn

3

1cos

2 213

00

n

Nnnn


0

2

0

22

2

sincosexp

sincosexpcoscos

dJ

dJ

....945

8

45

4

3

1 2

JJ

kT

E

n

NJ

n

Nn

22

130

130

~

45

4

45

4

2

4 22

~En

Second-order index of refraction :

kTn

Nn

213

02 45

4


2) With local-field correction

PEE~

3

4~~ loc

locp E~NpP

~and

PEP

~

3

4~~ N

EP~

34

1

~

N

NE~)1(

N

N

34

1

)1(

)1()1( 41

N3

4

21

)1(

)1(

N

n

n

3

4

21

or 2

2

: Lorentz-Lorenz law

kT

n

n

Nn

213

420

02 3

2

45

4


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 :

22

1EU 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 pV

Vpw

So, electrostrictive pressure :

88

22 EEp est

where,

e : electrostrictive constant


stst CPPP

1

Density change :

PC

1where, : compressibility

8

2EC e

For optical field,

8

~2EC e

4

1

4

4

1

8

~2

E

C e2

2

~

32

1EC e

e


..

~ccet ti EE EE2

~2E

EE2216

1eC

<Classification according to the Maker and Terhune’s notation>

Nonlinear polarization : EP

EEP22

216

1eC

EE

2)3( )(3

22

)3(

48

1)( eC

0,16

2

2

is,That BC

A 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

Nonlinear Optics Lab. Hanyang Univ. Chapter 4. The Intensity-Dependent Refractive Index - Third...

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