Chapter 4. The Intensity-Dependent Refractive...

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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.


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)


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


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).


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(


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(


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)


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 ~)~~(~~ 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


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


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

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


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


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(


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

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


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


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


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


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


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


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


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


..~

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


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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