Chapter 4. The Intensity-Dependent Refractive Index

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

2

0  

 n n E t n

n0

n2

where, : weak-field refractive index : 2^nd order index of refraction

 

~ t E e cc

E   ^it  ^{ E~2}

 

 

2

0

2 n E 

n n  



Nonlinear Optics Lab . Hanyang Univ.

Polarization :

             





P( )  ⁽¹⁾ 3 ^{(3) 2}  _eff In Gaussian units, n²

 1  4 



 



^{ }



 

 

 

2 0 2

0 4n n E











n     





0

 1  4 



0 ) 3 (

2 3 n

n 



Nonlinear Optics Lab . Hanyang Univ.

Appendix A. Systems of Units in Nonlinear Optics (Gaussian units and MKS units) 1) Gaussian units ;

  ~   ~   ....

) ~

~ (









P

  

   

cm 2

b statcoulom cm

statvolt

~

~ 

 

P

 

~ cm

# ⁽²⁾ 1 





 E



 

statvolt

~1 cm

# 





 E



ess dimensionl

# 



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 ;

     





)

~( _{(1) (2) 2 (3) 3}

0   

 E t E t E t

t

P

   

 

m

P  C

 

m E~  V

: MKS 1

  ^~   ^~   ^....

) ~

~ (

0

  



t

P

   

ess dimensionl

) 1

(



  

V m E 





 1~

) 2

(

 

V

 m



 

V

 C



 

V

 Cm

 [C/V]

[F]

F/m, 10

85 . 8

# ₀  ^12 

In MKS 1,

In MKS 2,



 dimensionl ess

Nonlinear Optics Lab . Hanyang Univ.

3) Conversion among the systems





~ 4 ~

~ ~

) 2

( DE









0

0~ ~ ~1

~ 







(Gaussian)

E~ 10 3 (MKS)

E~ V 300 statvolt

1 ) 1

(     ⁴

(Gaussian) (MKS)

(Gaussian) 4

(MKS) ⁽¹⁾

) 1

(







) 3 (

(Gaussian) 10

3 1) 4

(MKS ₄ ⁽²⁾

) 2

(

 



(Gaussian) 10

3 2) 4

(MKS ⁰₄ ⁽²⁾

) 2

(

 



(Gaussian) )

10 3 ( 1) 4

(MKS _{4 2} ⁽³⁾

) 3

(

 



(Gaussian) )

10 3 ( 2) 4

(MKS ₄⁰ ₂ ⁽³⁾

) 3

(

 



Nonlinear Optics Lab . Hanyang Univ.

Alternative way of defining the intensity-dependent refractive index

n n

n





0 ( )2

2 

 ^E c In

(4.1.4)

2 2

0 2n E() n

n 

) 3 ( 2

0 2

0 ) 3 (

0 2 0 2

12 3

4

4

    

c n n

c n n

c

n n  



)

?

3 (

2

 

n

 

 

12 10 W

cm ₍₃₎

2 0 )

3 ( 7 2

0 2 2

2

  

n c

n  n 



 





), :

( 12esu

10

) 1.9

3 ( 2

 



CS ⁿ₀^{1.58, I1MW/cm2, n?}

   

W cm n esu n

/ 10

3

10 9 . 58 1 . 1

0395 .

0 0395

. 0

2 14

12 2

) 3 ( 2

0 2

















8 14

6

2 110 310^ 310^



n n I Example)

Nonlinear Optics Lab . Hanyang Univ.

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.

Nonlinear Optics Lab . Hanyang Univ.

4.2 Tensor Nature of the 3

Susceptibility

r r r r

F~ ₂~ (~ ~)~

0  



 m mb

res 

Centrosymmetric media

Equation of motion :

m t e

b(~ ~)~ ~( )/

~ 2 ~

~ ²

0r r r r E

r

r^ ^    

 Solution :

c c e

E e

E e

E

t) ^{i t i t i t} .

~( _{1 2 3}

3 2

1   

 ^{  }

E ^



n

t i n

e n

E( ) ^

Perturbation expansion method ;

...

)

~ ( )

~ ( )

~ ( )

~r(t  r⁽¹⁾ t ²r⁽²⁾ t ³r⁽³⁾ t  m t e~( )/

~ 2 ~

~ ^{2 (1)}

0 ) 1 ( )

1

( r r E

r^  ^  



~ 0 2 ~

~ ^{2 (2)}

0 ) 2 ( )

2

(  r  r 

r^ ^ 







~ 2 ~

~ ^{2 (3) (1) (1) (1)}

0 ) 3 ( )

3

(  r  r  r r r 

r^ ^  b





Nonlinear Optics Lab . Hanyang Univ.

3

order polarization :

) ( )

( ⁽³⁾

) 3 (

q

q Ne 

 r

P 

     







jkl mnp

p l n k m j p n m q ijkl q

i E E E

) (

) 3 ( )

3

( ( )  ( , , , )   

P

     





jkl

p l n k m j p n m q

ijkl E E E

D ⁽³⁾( , , , )   

where, D : Degeneracy factor

(The number of distinct permutations of the frequency _m, _n, _p) 4^th-rank tensor : 81 elements

Let’s consider the 3^rd order susceptibility for the case of an isotropic material.

3333 2222

1111  

  

3322 3311

2233 2211

1133 22

11     

     

3232 3131

2121 2323

1313

1212     

     

3223 3113

2332 2112

1331

1221     

     

1221 1212

1122

1111   

   

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 :

jk il jl

ik kl

ij

ijkl

        



Example) Third-harmonic generation :











1221 1212

1122  

  

) (

) 3

( )

3

( _{1122 ij kl ik jl il jk}

ijkl

              





Example) Intensity-dependent refractive index :











1221 1212

1122  

  

jk il jl

ik kl ij

ijkl

                   





Nonlinear Optics Lab . Hanyang Univ.

Nonlinear polarization for Intensity-dependent refractive index

     





jkl

l k

j ijkl

i() 3  (   )E  E  E 

P

 

 

P    

 _i() 6₁₁₂₂E_i ^ 3₁₂₂₁E_i^{  P}61122

 

^{ }

Defining the coefficients, A and B as

1221

1122, 6

6





 B

A (Maker and Terhune’s notation)

 ^

 ^{   }

 E E E E E E

P A B

2

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

j ij

i E

P





 





ijA  ^  B E E^E^E

 2





1221

1122 3

2 6

1    



  A B A

61221



 B B where,

Physical mechanisms ;

iction electrostr

: 0 ,

0

response

electronic t

nonresonan

: 2 ,

1

n orientatio molecular

: 3 '/

, 6 /















B'/A' B/A

B'/A' B/A

A B A

B

Nonlinear Optics Lab . Hanyang Univ.

4.3 Nonresonant Electronic Nonlinearities

# The most fast response :

  2  a

/ v  10

s

Classical, Anharmonic Oscillator Model of Electronic Nonlinearities

Approximated Potential :

 

4 1 2

1 r r

r m mb

U   

 

) ( ) ( ) ( ) ( ) 3

, , ,

( ₃

4 )

3 (

p n

m q

jk l i jl ik kl ij p

n m q

ijkl m D D D D

Nbe



















 







  ^{ }

 

   



 







 



 







 m D D

Nbe _{ij kl ik jl il jk}

ijkl 3 3

4 )

3 (

) 3 (

(1.4.52)

where, D

 

According to the notation of Maker and Terhune,

    ^

^



 m D D

B Nbe

A _{3 3}

2 4

Nonlinear Optics Lab . Hanyang Univ.

Far off-resonant case, _{0  D() 0}²_{, b0}²_/d²

2 6 0 3

4 )

3 (

d m

Ne

  

esu 10

2

~

g 10 1 . 9 , rad/s 10

7

esu 10

8 . 4 , cm 10

3 , cm 10

4

14 )

3 (

28 15

0

10 8

3 22







































e d

N

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 :

E P 

 τ

induced dipole moment

Nonlinear Optics Lab . Hanyang Univ.

Second order index of refraction

Change of potential energy :

1 1

3

3 dE p dE

p d

p

dU    E    ^





 ^{ }   ^{  }

^ 

2 2

3 2

1 1 2

3

3 cos sin

2 1 2

1 E E E E

U      

    



cos

2

1 2

1

 





* Optical field (orientational relaxation time ~ ps order) : ^E2

^

^~

 

^

^~

Mean polarization :

















  ₃ cos²  ₁ sin²  ₁( ₃ ₁) cos²





2

2

1 E

U  



1) With no local-field correction :

*

Nonlinear Optics Lab . Hanyang Univ.

 

 

   

 







 

kT U

d

kT U

d

/ exp

/ exp

cos cos

2 2





 

Defining intensity parameter, J

 

1 ₂

1

3 

 



 

 





















 

0

2 0

2 2

2

sin cos

exp

sin cos

exp cos

cos

d J

d J

i) J  0

3 1 sin

sin cos

cos

0 0

2

0

2  















 

d d

1

0 3 3

2 3

1 

  



 

 



 







3 2 3

4 1

1 

 

n : linear refractive index

Nonlinear Optics Lab . Hanyang Univ.

ii) J 0 

 

 ^{   } 



2 14 N   cos

n

 

   





 ² ₀² _{1 3 1} ² ₃

3 cos 1

3

4



    

n n

 

 



 

 



 3

cos 1 4 







2 0 2

0

2 n n

n  

 

 



 

 









 3

cos 1

2

1 3 0

0

   

 n

n N

n

n

Nonlinear Optics Lab . Hanyang Univ.

 

 





















 

0

2 0

2 2

2

sin cos

exp

sin cos

exp cos

cos

d J

d J

945 ....

8 45 4 3

1 ²







 J J

   

kT E n

N J

n N

n

2 2 1 3 0 1

3 0

~ 45

4 45 4 2

4

     



 ₂

~

E

 n

Second-order index of refraction :

 

kT n

n N

2 1 3

0

2

45

4    



Nonlinear Optics Lab . Hanyang Univ.

2) With local-field correction

P E

E ~

3 4

~

~_loc   

p



Np P~

and 



 

 



 P E P~

3 4

~

~ N

 

E

P ~

3 1 4

~



 

 





 



N

 

~



 

 





 

 

N N

3 1 4

) 1 (

) 1 ( )

1

(

1 4 

   _{ }





3 4 2 1

) 1 (

) 1

( 



  N 

n n

3 4 2 or ₂ 1

2 



 : Lorentz-Lorenz law

 

kT n

n n N

2 1 3 2 4

0 0

2 3

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 0

0 2

1 2

1 E

d d

p

U  





 

2

1 E

U  



 

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

 



 

 ^_^

 







 





 8 8

2

2 E

u E





 

 





 _st p_st

V p V w

So, electrostrictive pressure :

 





 

8 8

2

2 E

p_st E  _e



 



where,



 

 

 

e : electrostrictive constant

Nonlinear Optics Lab . Hanyang Univ.

st

st CP

P P 



 

  



 







 



 1

Density change :

C P



 



where, 1 : compressibility

 



 8

E2

C _e







For optical field,

 



 8

~₂ E C _e





 







 

 







 



 4

1 4







 



1 8

~₂



 







 E 

C _e ²

2

~ 32

1 C









 

 

 

e

Nonlinear Optics Lab . Hanyang Univ.

 

~ t Ee^it cc

E ^{ E~2 2EE}

 





 ₂ ²EE

16 1

C



 

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

Nonlinear polarization : P 



E E

P ₂ ^{2 2}

16 1

Ce

   3⁽³⁾()E²E

2 2 )

3 (

48 ) 1

( C



 











0 16 ,

2

is,

That

 C B 

A





Nonlinear Optics Lab . Hanyang Univ.



 

 

 

e

2 1

n

e







 n n

e

: 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