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
~
n  n0  n2 E 2 t   ......
where, n0 : weak-field refractive index
n2 : 2nd order index of refraction
~
Et   E()eit  c.c.

nn0  2n2 E  
2

~
2
E 2 t  2 E ( )E ( )* 2 E ( )
: optical Kerr effect (3rd order nonlinear effect)
Nonlinear Optics Lab.
Hanyang Univ.
Polarization :
P( )   (1) E    3 ( 3) E   E     eff E  
2
In Gaussian units, n2  1  4eff

n 2n E   14
2 2
0
2
(1)
12 (3) E  
2
n02  4n0 n2 E    4n22 E    1  4 (1)  12 ( 3) E  
2

 n0  14 (1)
4
2

12
n2  3 (3) n0
Nonlinear Optics Lab.
Hanyang Univ.
Appendix A. Systems of Units in Nonlinear Optics (Gaussian units and MKS units)
1) Gaussian units ;
~
~
~
~
P(t )   (1) Et    (2) E 2 t   (3) E 3 t   ....
  
~ ~ statvolt statcoulomb
PE

cm
cm2
#  (1) dimensionless
 
cm
1
#  ( 2 )  ~  
 E  statvolt
 
# 
( 3)
2
 1  cm
 ~ 2  
2
 E  statvolt

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 ;
P~   mC
E~   Vm
2


~
~
~
~
P(t )   0  (1) Et    ( 2) E 2 t   (3) E 3 t   .... : MKS 1
~
~
~
~
P(t )   0  (1) Et    (2) E 2 t   (3) E 3 t   ....
: MKS 2
#  0 8.851012 F/m, [F][C/V]
In MKS 1,
 dimensionless
In MKS 2,
 (1) dimensionless
(1)
 
1 m
   ~ 
E  V
 (2)   VC2
( 2)
Nonlinear Optics Lab.
 
m2
  2
V
 (3)   Cm
V3
( 3)
Hanyang Univ.
3) Conversion among the systems
~
~
(1) 1 statvolt  300V  E (MKS)  3104 E (Gaussian)


~ ~
~ ~
(Gaussian)
(2) D E 4PE 14 (1)
(1)
(1)


(MKS)

4

(Gaussian)
~
~ ~
~
(1)
(MKS)
D   0 E  P   0 E 1 

(3)

( 2)

4
(MKS 1) 
 ( 2) (Gaussian)
4
310
 (3) (MKS1) 
4
( 3)

(Gaussian)
4 2
(310 )
4 0 ( 2)
 (Gaussian)
4
310
 (3) (MKS2) 
4 0
( 3)

(Gaussian)
4 2
(310 )
 ( 2 ) (MKS 2) 
Nonlinear Optics Lab.
Hanyang Univ.
Alternative way of defining the intensity-dependent refractive index
n  n0  n2 I
I
n0 c
2
E ( )
2
n  n0  2n2 E ( )
2
(4.1.4)
4
4 3 (3) 12 2 (3)
 n2  n2 
 2 
n0c
n0c n0
n0 c
n2   (3) ?
 cm2  12 2 7 (3)
0.0395 (3)
 2 10  esu 
n2 
 esu 
2
W
n
c
n

 0
0
Example) CS2 ( (3) :1.91012esu), n0 1.58, I1MW /cm2 , n ?
n2 
0.0395 (3)
0.0395
12



esu

1
.
9

10
n02
1.582
31014 cm2 /W
n  n2 I  1106  31014  3108
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 3rd Susceptibility
Centrosymmetric media
~
Fres  m02~
r  mb(~
r ~
r )~
r
Equation of motion :
~
~
r  2~
r   2~
r  b(~
r ~
r )~
r  eE(t ) / m
0
Solution :
i t
~
E(t )  E1ei1t  E2ei2t  E3ei3t  c.c   E(n )e
n
n
Perturbation expansion method ;
~
r (t )  ~
r (1) (t )  2~
r ( 2) (t )  3~
r (3) (t )  ...
~
 ~
r(1)  2~
r (1)  02~
r (1)  eE(t ) / m
~
r(2)  2~
r ( 2)   2~
r ( 2)  0
0


~
r(3)  2~
r (3)  02~
r (3)  b ~
r (1)  ~
r (1) ~
r (1)  0
Nonlinear Optics Lab.
Hanyang Univ.
3rd order polarization :
P (q )Ner (q )
(3)
(3)
4th-rank tensor : 81 elements
( 3)
 Pi(3) (q ) ijkl
(q ,m ,n , p ) E j m Ek n El  p 
jkl ( mnp )
( 3)
D ijkl
(q ,m ,n , p ) E j m Ek n El  p 
jkl
Element with
even number
of index
where, D : Degeneracy factor
(The number of distinct permutations of the frequency m, n, p)
Let’s consider the 3rd order susceptibility for the case of an isotropic material.
1111   2222  3333
21 nonzero 1122  1133   2211  2233  3311  3322
elements :
1212  1313   2323  2121  3131  3232
(Report)
1221  1331  2112  2332  3113  3223
and,
1111 1122 1212  1221
(Report)
Nonlinear Optics Lab.
Hanyang Univ.
Expression for the nonlinear susceptibility in the compact form :
ijkl  1122ij kl  1212ik jl  1221il jk
Example) Third-harmonic generation : ijkl (3    )
1122 1212 1221
 ijkl (3    )1122 (3    )( ij kl  ik jl  il jk )
Example) Intensity-dependent refractive index : ijkl (    )
1122 1212 1221
 ijkl (3    )1122 (    )( ij kl  ik jl )1221 (    ) il jk
Nonlinear Optics Lab.
Hanyang Univ.
Nonlinear polarization for Intensity-dependent refractive index
Pi ()  3 ijkl (      ) E j  Ek  El  
jkl


 Pi ( )61122 Ei EE 31221 Ei EE


 P61122 EE E31221 EEE in vector form
Defining the coefficients, A and B as
A61122 , B61221 (Maker and Terhune’s notation)


P  A E  E E 
1
BE  E E
2
Nonlinear Optics Lab.
Hanyang Univ.
In some purpose, it is useful to describe the nonlinear polarization by in terms of an
effective linear susceptibility,  ij as
Pi   ij E j
j




1
  ij  A EE  ij  B Ei E j  Ei E j
2
1
where, A  A  B  6 1122  31221
2
B  B  61221
Physical mechanisms ;
B/ A6, B'/ A3 : molecularorientation
B/A1, B'/A'2 : nonresonant electronic response
B/A0, B'/A'0 : electrostriction
Nonlinear Optics Lab.
Hanyang Univ.
4.3 Nonresonant Electronic Nonlinearities
# The most fast response :
 2a0 /v 1016 s
[a0(Bohr radius)~0.5x10-8cm, v(electron velocity)~c/137]
Classical, Anharmonic Oscillator Model of Electronic Nonlinearities
Approximated Potential : U r   m02 r  mb r
1
2
 (q , m , n , p ) 
( 3)
ijkl

1
4
4
Nbe4  ij kl  ik jl  il  jk

(1.4.52)
3m3 D( q ) D( m ) D( n ) D( p )
  (   ) 
( 3)
ijkl
2

Nbe4  ij kl  ik jl  il  jk
3 m3 D3  D 

where, D   02   2  2i
According to the notation of Maker and Terhune,
2 Nbe4
A B 3 3
m D  D  
Nonlinear Optics Lab.
Hanyang Univ.
Far off-resonant case, 0  D( )  02 , b02 /d 2

( 3)
Ne 4
 3 6 2
m 0 d
( 3)
Typical value of 
N 41022 cm3 , d 3108 cm, e4.81010 esu
 0 71015 rad/s, m9.11028 g
  (3) ~ 21014 esu
Nonlinear Optics Lab.
Hanyang Univ.
4.4 Nonlinearities due to Molecular Orientation
The torque exerted on the molecule when an electric field is applied :
τ  PE
induced dipole moment
Nonlinear Optics Lab.
Hanyang Univ.
Second order index of refraction
Change of potential energy :
dU   p  dE   p3 dE3  p1 dE1 3 E3 dE3 1E1 dE1



1
1
 3 E32  1 E12    3 E 2 cos 2   1 E 2 sin 2 
2
2
1
1
  1 E 2   3  1 E 2 cos 2 
2
2
U 

Optical field (orientational relaxation time ~ ps order) : E 2 E 2 t   E 2
~
~
1) With no local-field correction
n2  1  4N 
Mean polarization :
   3 cos 2   1 sin 2   1  ( 3  1 ) cos 2 
Nonlinear Optics Lab.
Hanyang Univ.
d cos  exp U   / kT 


 d exp U   / kT 
2
cos 
2
Defining intensity parameter, J 
cos 
2



0


cos2  exp J cos2  sin d


0

2



0
0

exp J cos2  sin d
i) J  0
cos 
1
 3  1 E~ 2 / kT
2
cos2  sin d


0
sin d

1
3
1
2
  0   3  1
3
3
2 
1
 n02 14N   3  1  : linear refractive index
3 
3
Nonlinear Optics Lab.
Hanyang Univ.
ii) J  0

n 2  1  4N 1   3  1  cos 2 

1 
1
 n 2 n02 4N  1 3 1  cos2   3 
3 
3
1

 4N  3  1  cos2   
3

n2  n02  n02
  n nn0 
2N
3 1  cos2 1 
n0
3

Nonlinear Optics Lab.
Hanyang Univ.
cos 
2



0


cos2  exp J cos2  sin d


0


exp J cos2  sin d
1 4 J 8J 2
  
....
3 45 945
~2
4N
4 J 4 N
2E
 3 1  
 3 1 
 n
2n0
45 45 n0
kT
~
 n2 E 2
Second-order index of refraction :
4N  3  1 
n2 
45n0
kT
2
Nonlinear Optics Lab.
Hanyang Univ.
2) With local-field correction
~
~ 4 ~
Eloc  E  P
3
~
 ~ 4 ~ 
 P N  E P 
3 

N
N
~
~
(1)
(1) ~



 P
E  E
 4

 4

1 N 
1 N 
 3

 3

~
~
PNp and pEloc
 14
(1)
n 2 1 4
 (1) 1 4
 (1)  N or 2  N 
n 2 3
 2 3
(1)
: Lorentz-Lorenz law
4N  n02 2   3 1 


 n2 
45n0  3 
kT
4
2
Nonlinear Optics Lab.
Hanyang Univ.
8.2 Electrostriction
: Tendency of materials to become compressed in the presence of an electric field
Molecular potential energy :
density change
E
E
1
1
U    p  dE    E  dE   E  E   E 2
0
0
2
2
Force acting on the molecule :
1
F   U  E 2 
2
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
V

E2
E 2   
   w p st
u 
 
 pst
8
8   
V

So, electrostrictive pressure :
 E 2
E2
p st  
 e
 8
8

: electrostrictive constant
where,  e  

Nonlinear Optics Lab.
Hanyang Univ.
Density change :
 1  
 Pst   CPst
    


P


E2
   C e
8
where, C  1  : compressibility
 P
For optical field,   C e
 
~
E2
8
 1   
  

e  
4 4   

~
E 2    1
1
~2
 
 C e

C

E
e
8    4
32 2
Nonlinear Optics Lab.
Hanyang Univ.
~
Et   Eeit  c.c.

~
E 2 2EE
1
2

  
C

E

E
e
16 2
<Classification according to the Maker and Terhune’s notation>
Nonlinear polarization : P  E
1
2
2
P
C

E
E
e
16 2
 3 (3) (    ) E E
  (3) (    ) 
2
1
C e2
2
48
CT  e2
That is, A 
, B0
2
16
Nonlinear Optics Lab.
Hanyang Univ.
e  


 e  n2 1
: For dilute gas
 e  n2 1n2  2/ 3 : 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.