Nonlinear Susceptibilities: Quantum Mechanical Treatment
The nonlinear harmonic oscillator model used earlier for calculating (2) did not capture
the essential physics of the nonlinear interaction of radiation with molecules. It was useful
because knowledge of the sign of (2) is not usually important and because normally
experimentally measured nonlinear susceptibilities are used in calculations. BUT, there is
( 2)
no reliable way to evaluate the required nonlinear force constant kijk
.
In contrast to the nonlinear harmonic oscillator model, the quantum treatment uses first order
perturbation theory for allowed electric dipole transitions to derive formulas for the second
and third order nonlinear susceptibilities of a single isolated molecule with a given set of
energy levels. The results, called the “some over states (SOS)”, will be expressed in terms
of the energy separations between the excited state energy levels m and the ground state g,
mg , between excited states m and n,  nm , the photon energy of the incident light 1 and


the transition electric dipole moments  mg and  nm between the states. The average electron
lifetime in the excited state is  mg . All of these parameters can either be calculated from first
principles or can be obtained from linear and nonlinear spectroscopy.
The electrons are assumed to be initially in the ground
state. This theory can be extended to electrons already
in excited states when the optical field is incident. This
the density matrix approach which deals with state
populations in addition to the parameters stated above.
Perturbation Theory of Field Interaction with Molecules
 2 
ˆ (r, t )

is the electron wave function and ̂ (r , t ) dr is the probability of finding an electron in
2 

   ˆ 
volumedr  dxdydz at time t with the normalization
 (r , t ) dr  1. The stationary
  
discrete states are solutions of Schrödinger’s equation i ˆ / t  H 0 ˆ . The wave function for
 iˆ t

the m’th eigenstate is written as ˆ m  um (r )e m where um (r ) is the spatial distribution of the
wave function and ˆ mg  mg  i /  mg is a complex quantity with usually mg mg  1 which
reduces to ˆ g   g for the ground state which does not decay. The eigenstates are
   * 

“orthogonal” in the sense that    um (r )un (r )dxdydz   mn . The ground state wave
function is Ψˆ ( s  0) (r, t )  aˆ0u g (r )e i g t. The superscript s =0 identifies the case that no
interaction has yet occurred and s>0 identifies the number of interactions between the electron
 
and an electromagnetic field.
E (r , t )
 
An incident field Eloc (r , t ) distorts the molecular (atomic)
electron cloud and mixes the states via the induced
electric dipole interaction for the duration of the field.
The probability of the 2electron in the m’th excited state
is proportional to aˆm(1) . The total wavefunction becomes



Ψˆ (r , t )  Ψˆ (0) (r , t )  Ψˆ (1) (r , t )


ˆ
Ψˆ (1) (r , t )  m aˆ m(1)um (r )e imt  c.c.
with aˆ m( 1 ) (t )  0 for some m.
loc
A second and third interaction with the same or different electromagnetic fields lead to





Ψˆ (r , t )  Ψˆ (0) (r , t )  Ψˆ (1) (r , t )  Ψˆ ( 2) (r , t )  Ψˆ (3) (r , t )


ˆ
Ψˆ ( 2) (r , t )   aˆ n( 2)un (r )e int  c.c.
with aˆ n( 2 ) (t )  0 for some n.


ˆ
Ψˆ (3) (r , t )   aˆv(3)uv (r )e ivt  c.c.
with aˆv( 3 ) (t )  0 for some v respectively.
n
v
For example
Interactions in quantum mechanics are governed by the interaction potentials V(t)


i   ( H 0  V (r , t ))  ,   0  interaction off;   1  interaction on
t
 
  


  
Electric dipole : V (r , t )   (r )  Eloc (r , t )  e r  Eloc (r , t ) Eloc (r , t ) - " local field" at molecule
 
in which  (r ) is the induced or permanent dipole moment.
Thus the total wave function can be written in terms of the number of interactions as
ˆ  ˆ (0)  ˆ (1)   2 ˆ (2)   3ˆ (3)  ...
In quantum mechanics the expectation value of a physical parameter is an average
of that parameter over the electron distribution

 p(t)   induced molecular polarization, averaged over the electron distribution,
 ˆ
ˆ (r, t )dxdydz
ˆ * (r, t )r
ˆ (r, t )dr  e    
ˆ * (r, t )r
ˆ
  |  |    -e  
  





 p(t )    p (0) (t )     p (1) )t )    2  p ( 2) (t )    3  p (3) (t )   ...
Permanent
dipole moment
Linear polarizability
 ij   ij(1)
First hyperpolarizability
( 2)
( 2)
 ijk
  ijk
Second hyperpolarizability
(3)
(3)
 ijk



ijk
Susceptibilities are calculated via successive applications of first order perturbation theory
i

 

Equating terms with the same power of  gives
 ˆ ( 0)
ˆ ( 0)
  H0
t
 ˆ ( 2)
ˆ ( 2)  V (t ) 
ˆ (1)
i 
 H 0
t
i

 ˆ ( 0)
ˆ (1)   2 
ˆ ( 2)   3 
ˆ (3)  ..  H 
ˆ ( 0)  
ˆ (1)   2 
ˆ ( 2)   3 
ˆ (3)  ..
  
0
t
ˆ (0)  
ˆ (1)   2 
ˆ (3)  .. ,
 V (t ) 
 ˆ (1)
ˆ (1)  V (t ) 
ˆ ( 0)
  H0
t
 ˆ (3)
ˆ (3)  V (t ) 
ˆ ( 2) .
i 
 H0
t
i

,

After N interactions  i[aˆ( N ) (t )  iˆ  aˆ( N ) (t )]u (r )e it
ˆ

ˆ


ˆ
 ˆ  aˆ( N ) (t )u (r )e it   aˆ(N 1) (t )V (r , t )u (r )e it .

Multiplying by u* ( r )ei̂t , integrating over all space and applying the orthogonality relations

ˆ ˆ


 
1
 aˆ( N ) (t )   aˆ(N 1) (t )V (t )ei (  )t with V (t )   u* (r )V (r , t )u (r )dr .

i
Defining ˆ  ˆ  ˆ and integrating from t=- to t,
t
ˆ 
1
General quantum mechanics result  aˆ( N ) (t )    V (t )aˆ(N 1) (t )eit dt .

i


The total electromagnetic field present at the site of a molecule, Eloc (r , t ) is written as
 

*
1
i pt
i t
Eloc (r , t )   p [E loc ( p )e
 E loc ( p )e p ] where  p is over all fields present
2
*

*

i t
i pt
Aside: E loc ( p )  E loc ( p ) and that in nonlinear optics, E loc ( p )e
and E loc ( p )e p
can be considered to be separate input modes for operational purposes.
 *    

 
For the electric dipole interaction : V (t )   u (r )  (r )  Eloc (r , t )u (r )dr


 
*   
u (r )  (r )u (r )dr .

  


 1 
i t
 i t
  p   {E loc ( p )e p  E*loc ( p )e p },
2
.
Interaction of the Molecules With the Field

*


Integrating the first
mg  Eloc ( p ) i (ˆ mg  p )t mg  Eloc ( p ) i (ˆ mg  p )t
1
(1)
ˆ
e

e
}.
interaction from t’=  am (t )   p{
ˆ mg   p
ˆ mg   p
2



- to t
Redefine the summation over pʹ to a summation over p with p going from -p max  pmax where
pmax is the total number of fields present, and for negative p,  p   p and Eloc ( p )  E*loc ( p )..




E
1
mg
loc ( p ) i (ˆ mg  p )t
 aˆ m(1) (t )   p
e
.
ˆ
2
mg   p
ˆ 
1
aˆ n( 2) (t )  m t  Vnm (t )aˆ m(1) (t )einmt dt 
i


1
i (ˆ nm q )t 
t ˆ (1)

a
(
t
'
)


E
(

)
e
dt 


nm
loc q
2 m q  m
4i




[  nm  E loc (q )][  mg  E loc ( p )] i (ˆ ng q  p )t
1
 aˆ n( 2) (t )  2 q  p m
e
.
ˆ
ˆ
(     )(  
4
Second Interaction:
ng
q
p
mg
p)
Third Interaction:


 


[


E
(

)][


E
(

)][


E
nm
loc q
mg
loc ( p )] i (ˆg r q  p )t
ˆa(3) (t )  1 n,m r ,q , p n loc r
e
.
3
ˆ
ˆ
ˆ
(







)(





)(



)
8
g
r
q
p
ng
q
p
mg
p
The summations over n and m are both over all the states. Also summations over p, q and r are
each over all of the fields present. Note that states m and n can be the same state, m and  can
be same state etc. Finally, note that there appears to be a time sequence for the interactions with
fields which is p, q, r. However, since each of p, q, r is over the total field, all the possible
permutations of p, q, r approximate an “instantaneous interaction”. For example, assume there
are 2 optical fields present, . Therefore for a(2), p and q each run from -2 to +2, excluding 0,
and there are 4x4=16 different contributing field combinations, each defining a time sequence!
For each field combination, there are multiple possible “intermediate” states (pathways to
state v), denoted by “m” and “n” which can be identical, different etc. For example if there is the
ground state “g” and 3 excited states, one of which is the state “v=2”, then the “pathways” to
“v=2” could be g 2 1 2, g 3 1 2, g 2 g 2 etc. The probability for each step
in the pathway,
 2 for example state ”n” to state “m” is given by the transition dipole matrix
element | μmn | . Normally, there are only a few states linked by strong transition moments in a
given molecule which simplifies the “sum over states, SOS” calculation. The probability of
exciting state “m” also depends, via the resonant denominators, on how close the energy
difference is between the ground state (initial electronic state before any interaction) and the
state “m”, i.e. whether it matches the energy obtained from the EM fields in reaching state “m”
via state “n” and the other states in that particular pathway.
Optical Susceptibilities
 

ˆ
ˆ
 
* 
*
ˆ
Recall:  p (t )    a (t )u (r )  (r ) aˆ (t )u (r )dr e i (  )t  c.c.
 
( )
ˆ
|  (r ) | ˆ ( ) 
    

ˆ
ˆ
ˆ
( 0)  ˆ ( 0)
( 0)  ˆ (1)
(1)  ˆ ( 0)
 p (t )  [  | μ |    {  | μ |      | μ |  }
ˆ
ˆ
ˆ
2
( 0)  ˆ ( 2)
(1)  ˆ (1)
( 2)  ˆ ( 0)
  {  | μ |      | μ |      | μ |  }
ˆ (0) | μ | 
ˆ (3)    
ˆ (1) | μ | 
ˆ ( 2)    
ˆ ( 2) | μ | 
ˆ (1) (   
ˆ (3) | μ | 
ˆ (0) }],
  3{ 

ˆ (0) | μ | 
ˆ (1)    
ˆ (1) | μ | 
ˆ (0)    (1) .
Linear Susceptibility  1   p (1)  
*  *

[
μ
 
1
mg  E loc ( p )] i g t i ( p  g )t
ˆ
ˆ
(1)
( 0)
*   
  | μ |  
e
u
(
r
)
μ
(
r
)
u
(
r
 m  p 0

m
g ) dr  c.c. ( p  0)
*
2
ˆ mg
p
* 
[
μ
 
1
mg  E loc ( p )] i g t i ( p  g )t
*   

e
u
(
r
)
μ
(
r
)
u
(
r
 m  p 0

m
g ) dr  c.c. ( p  0)
*
2
ˆ mg
p


*  *
* 
[ μmg  E loc ( p )] mg i pt
[ μmg  E loc ( p )] mg i pt
1
 {m  p 0
e

e
 c.c.,


m p 0
*
*
ˆ
ˆ
2
 
 
mg
p
mg
p
ˆ
*( 0)  ˆ (1) 
Similarly :   μ dr

*




μ gm [ μmg  E loc ( p )] i pt
( μmg  E loc ( p )  gm i pt
1
 [  m  p 0
e
  m  p 0
e
 c.c.].
ˆ
ˆ
2
 
 
mg
p
mg
p
*
Since  gm


 


 (1)
μ
[
μ

E
(

)]
μ
[
μ

E
1
gm mg
loc
p
mg gm
loc ( p )] i pt
  mg ,  p (t ) 

}e
 c.c.
  { ˆ  
*
ˆ
2 m p
ng   p
mg
p

The two denominator terms are referred
 (1) to as “resonant” and “anti-resonant”. The former has
1
the form [mg  i /  mg   p ] and  p  is enhanced when  mg   p , hence the name “resonant”.
For the term [mg  i /  mg   p ]1 , the denominator always remains large and hence the name
“anti-resonant” is appropriate. Note that although the resonant contribution is dominant when the
photon energy is comparable to mg , in the zero frequency limit (mg   p ) the two terms
are comparable.
Perhaps a more physical interpretation can be given in terms of the time that the field interacts
with the molecule as interpreted by the uncertainty principle. When an EM field interacts with
the electron cloud, there can be energy exchange between molecule and field. The uncertainty
principle can interpreted in terms of E being the allowed “uncertainty” in energy and t as the
maximum time over which it can occur. Within this constraint, a photon can be absorbed and
re-emitted, OR emitted and then re-absorbed.
Adding in the approximate local field correction term from lecture 1, and writing
ˆ (1)

 (1) 
 (1) 
1
i t
(1)
P (r , t )  Nf  p (r , t )   0  p  ( p ;  p )  E ( p )e p  c.c.
2
 
 


ˆ (1)
μmg μ gm
μ gm μmg
N (1)


f m 
  ( p ;  p ) 
1 
1
 0
mg   p  i mg mg   p  i mg 
ˆ (1)

f (1) 
 r ( )  2
3
1
2
2


 2
p
 2i mg
  p2   mg
mg
N (1)
f m |  mg | 2mg 
( p ;  p ) 
,
2

2
2

2
 0
[(mg   p )   mg ][(mg   p )   mg ] 
which is almost identical to the SHO result, with physical quantities for the oscillator strength.




ˆ ( 0) | μ | 
ˆ (2)    
ˆ (1) | μ | 
ˆ (1)    
ˆ ( 2) | μ | 
ˆ (0)  .
Second Order Susceptibility  p (2) (t )  
Sum
frequency
 p  q


 

 (2)
μ gn [ μnm  E loc (q )][ μmg  E loc ( p )]
1
 p (t )  2 n, m q , p {
(ˆ ng  q   p )(ˆ mg   p )
4





 



[ μ gn  E loc (q )] μnm [ μmg  E loc ( p )] [ μnm  E loc ( p )][ μ gn  E loc (q )] μmg i ( p q )t


}e
 c.c.
*
*
*
ˆ
ˆ
ˆ
ˆ
(ng  q )(mg   p )
(mg  q   p )(ng  q )
Difference
frequency
 p  q
*

 

 (2)
μ
[
μ

E
(

)][
μ

E
1
gn nm
loc q
mg
loc ( p )]
 p (t )  2 n,m q , p {
(ˆ ng  q   p )(ˆ mg   p )
4
*


*






[ μ gn  E loc (q )] μnm [ μmg  E loc ( p )] [ μnm  E loc ( p )][ μ gn  E loc (q )] μmg i ( p q )t


}e
 c.c.
(ˆ *   )(ˆ   )
(ˆ *   )(ˆ *     )
ng
q
mg
p
ng
q
mg
p
q
Local Field Corrections in Nonlinear Optics (not just
 r ( p  2)  r (q  2)
for  p  q !)
3
3
A Maxwell polarization exists throughout the medium at the nonlinearly generated frequency
ʹ=pq
Maxwell field


1 
Eloc ( ' )  E( ' ) 
P( ' )
(spatial average)
3 0 
 
 (2)
1 
P( ' )]  p ( ' ) .
The total dipole moment induced at the molecule is p ( ' )    [E( ) 
3 0
Maxwell polarization


Nonlinear polarization at
Since P( ' )  N p ( ' )
(induced on walls of
molecule due to mixing
 (2)
 r ( ' )  2  (2)
spherical
cavity)
 P ( ' )  N [
]p ( ' )
of fields


 

 (2)

(



)

2
μ
[
μ

E
(


)][
μ

E
N
r
p
q
gn nm
loc
q
mg
loc ( p )]
 P ( p  q )  2 [
]n,m q , p {
3
(ˆ ng  q   p )(ˆ mg   p )
2





 



[ μ gn  E loc (q )] μnm [ μmg  E loc ( p )] [[ μ gn  E loc (q )][ μnm  E loc ( p )] μmg


}
*
*
*
ˆ
ˆ
ˆ
ˆ
(ng  q )(mg   p )
(mg  q   p )(ng  q )




 ( 2)

1
  0 q , p [  ([ p  q ];  p ,q )   ( 2) ([ p  q ];q ,  p )]E( p )E(q )
2
3
Extra term
N  r ( p  q )  2  r ( p )  2  r (q )  2
( 2)
 ˆ ijk ([ p  q ];  p ,q )  2


n ,m
3
3
3
 0
{
(ˆ ng
μ gn,i μnm,k μmg , j
μ gn,k μnm,i μmg , j
μnm, j μ gn,k μmg ,i
 *
 *
}.
*
ˆ
ˆ
ˆ
ˆ
ˆ
 q   p )(mg   p ) (ng  q )(mg   p ) (mg  q   p )(ng  q )
Examples of Second Order Processes
e.g. Type 2 Sum Frequency Generation [ E x (1 ) and E z (2 ) input; Px( 2) (1  2 ) generated
1
( 2)
( 2)
Px( 2) (1  2 )   0 [ ˆ xxz
((1  2 ); 1 , 2 )  ˆ xzx
((1  2 ); 2 , 1 )]E x (1 )E z (2 )
2
Note that order of polarization subscripts must match order of frequencies in susceptibility!
e.g. nonlinear DC field generation by mixing of E x ( ) and E*z ( )
1
( 2)
( 2)
Px( 2) (0)   0 [ ˆ xxz
(0;  , )  ˆ xzx
(0; ,  )] : E x ( )E z ( )
2
( 2)
( 2)
ˆ ijk
(0;  , )  ˆ ikj
(0; ,  )
Since the summations are over all states, n and m include the ground state which produces
divergences as marked by red circles – unphysical divergences!
These divergences can be removed, see B. J. Orr and J. F. Ward, “Perturbation Theory of the
Nonlinear Optical Polarization of an Isolated System”, Molecular Physics 20, (3), 513-26 (1971).
( 2)
 ˆ ijk
([ p  q ];  p , q ) 
N
 0
2

( 2)

'
n ,m
μ gn,i ( μnm,k  μ gg ,k ) μmg,j
{
(ˆ ng   p  q )(ˆ mg   p )
μ gn,k ( μnm,j  μ gg , j ) μmg,i
μ gn,k ( μnm,i  μ gg ,i ) μmg,j
 *

}
*
*
ˆ
ˆ
ˆ
ˆ
(ng  q )(mg   p  q ) (ng  q )(mg   p )
The prime in the ground state is excluded from the summation over the states, i.e. the
summation is taken over only the excited states. Note that the summation includes contributions
from permanent dipole moments in the ground state and excited states (case n=m).
Non-resonant Limit (ω0)  ( 2)  [  r (0)  2 ][  r (0)  2 ]2
3
3
N ( 2) '
1
( 2)
~ijk
(0;0,0) 

{ μ gn,i ( μnm,k  μ gg ,k ) μmg,j

2
n ,m  
 0
ng mg
 μ gn,k ( μnm,j  μ gg , j ) μmg,i  μ gn,k ( μnm,i  μ gg ,i ) μmg,j }
 0,Type 1 1
( 2)
Pi(2) (2 )     0 ~ijj
(0;0,0)E 2j ;
2
1
 0,Type 2
( 2)
( 2)
( 2)
Pi(2) (2 )     0{~ijk
(0;0,0)  ~ikj
(0;0,0)}E j E k   0 ~ijk
(0;0,0)E j E k etc.
2
The same susceptibility is obtained for SHG, sum frequency and difference frequency generation,
as expected for Kleinman symmetry.
Third Order Susceptibility (Corrected for Divergences)
(3)
ˆ ijkl
([ p  q  r ];  p , q , r ) 
x{
N
 0 3
 (3) v,n,m
'
μ gv,i ( μn,l  μ gg ,l )( μnm,k  μ gg ,k ) μmg , j
μ gv, j ( μvn,k  μ gg ,k )( μnm,i  μ gg ,i ) μmg ,l

*
(ˆg   p  q  r )(ˆ ng  q   p )(ˆ mg   p ) (ˆ*g   p )(ˆ ng
 q   p )(ˆ mg  r )
μ gv,l ( μvn,i  μ gg ,i )( μnm,k  μ gg ,k ) μmg , j
μ gv, j ( μn,k  μ gg ,k )( μnm,l  μ gg ,l ) μmg ,i


}
*
*
(ˆ*g  r )(ˆ ng  q   p )(ˆ mg   p ) (ˆ*g   p )(ˆ ng
 q   p )(ˆ mg
  p  q  r )


N
 0 3
*
(ωˆ ng

(3)

'
n ,m
{
(ˆ ng
μ gn,i μng,l μ gm,k μmg,j
μ gn,i μng,l μ gm,k μmg,j
 *
  p  q  r )(ˆ ng  r )(ˆ mg   p ) (ˆ mg
 q )(ˆ ng  r )(ˆ mg   p )
μ gn,l μng,i μ gm,j μmg,k
μ gn,l μng,i μ gm,j μmg,k
 *
},
*
*
*
 ωr )(ωˆ mg
 ω p )(ωˆ mg  ωq ) (ˆ ng
 r )(ˆ mg
  p )(ˆ ng
  p  q  r )

(3)
[
 r ( p )  2  r ( q )  2  r ( r )  2  r ( p   q   r )  2
3
][
3
In general for   0 (Kleinman limit)
][
 (3)  [
][
3
3
]
 r ( 0)  2  r ( 0)  2  r ( 0)  2  r ( 0)  2
3
][
3
][
3
][
(3)
(3)
(3)
~ijkl
(3 ;  ,  ,  )  ~ijkl
( ;  ,  , )  ~ijkl
([2a  b ]; a , a ,b ) etc.
~ ' s are equal!
In the limit   0, all the third order 
ijk
(3)
3
][
 r ( 0)  2
3
]3
Symmetry Properties of ˆ ijkl : Isotropic Media
(3)
Isotropic media: simplest case of relationships between elements
In an isotropic medium, all co-ordinate systems are equivalent, i.e. any rotation of axes
must yield the same results!
(3)
(3)
xxxx  yyyy  zzzz; in general for ˆ ijkl,ij jk  k  ˆ1111
(3)
(3)
yyzz  yyxx  xxzz  xxyy  zzxx zzyy; in general for ˆ ijkl ij k  ˆ1122
(3)
(3)
 i jk  ˆ1221
xyyx  xzzx  yxxy  yzzy  zxxz  zyyz; in general for ˆ ijkl
(3)
(3)
xyxy  xzxz  yxyx  yzyz  zxzx  zyzy. in general for ˆ ijkl ik  j  ˆ1212
Assume the general case of three, parallel, co-polarized (along, for example, the x-axis) input
fields E1(1), E 2 (2 ) and E 3 (3 ) with arbitrary frequencies 1 ,  2 , 3.
y
y

1
x
(3)
 Px(3) (4 )   0 ˆ xxxx
(4 ; 3 , 2 , 1 )E1(1 )E 2 (2 )E 3 (3 ).
4
x
The axis system (x', y') is rotated 450 from the original x-axis in the x-y plane.
1
1
1
1
1
1

E1x ' 
E1 ; E 2 x ' 
E 2 ; E3x' 
E 3 E1y ' 
E1 ; E 2 y ' 
E 2 ; E3 y' 
E3.
2
2
2
2
2
2
arbitrary choice of axes 
(3)
ˆ x(3' x)' x ' x ' (4 ; 3 , 2 , 1 )  ˆ xxxx
(4 ; 3 , 2 , 1 ),
(3)
ˆ xxyy
(4 ; 3 , 2 , 1 )  ˆ x(3' x)' y ' y ' (4 ; 3 , 2 , 1 ) etc.
1
(3)
(3)
Px(3' ) (4 )   0 [ ˆ xxxx
(4 ; 3 , 2 , 1 )E1x' E 2 x 'E 3 x'  ˆ xxyy
(4 ; 3 , 2 , 1 )E1y ' E 2 y 'E 3 x'
4
(3)
(3)
 ˆ xyyx
(4 ; 3 , 2 , 1 )E1x' E 2 y 'E 3 y'  ˆ xyxy
(4 ; 3 , 2 , 1 )E1y ' E 2 x 'E 3 y' ]  c.c.
1
1
(3)
(3)
 Px(3' ) (4 )   0
[ ˆ xxxx
(4 ; 3 , 2 , 1 )  ˆ xxyy
(4 ; 3 , 2 , 1 )
4 2 2
(3)
(3)
 ˆ xyyx
(4 ; 3 , 2 , 1 )  ˆ xyxy
(4 ; 3 , 2 , 1 )]E1 E 2 E 3  c.c.
Also : PxNL
' ( 4 ) 
1
1 (3)
0
ˆ xxxx (4 , 3 , 2 , 1 )E1E 2 E 3  c.c.
4
2
y
y
Px(3)
x
x
Px(3)
Valid for any arbitrary set of frequencies 1 ,  2 , 3
(3)
(3)
(3)
(3)
 ˆ xxxx
(4 ; 3 , 2 , 1 )  ˆ xxyy
(4 ; 3 , 2 , 1 )  ˆ xyyx
(4 ; 3 , 2 , 1 )  ˆ xyxy
(4 ; 3 , 2 , 1 ).
Kleinman (0) limit
1 (3)
(3)
(3)
(3)
 ~1111
( ;  p , q , r )  ~1122
( ; 3 , 2 , 1 )  ~1221
( ; 3 , 2 , 1 )  ~1212
( ; 3 , 2 , 1 ).
3
There is a maximum of 34=81 terms in the ˆ ijkl tensor. The symmetry properties of the
medium reduce this number and the number of independent terms for different symmetry
classes was given in lecture 4. The inter-relationships between the non-zero terms are given
in the Appendix. All materials have some non-zero elements.
(3)
Appendix: Symmetry Properties For Different Crystal Classes
Triclinic For both classes (1 and 1 ) there are 81 independent non-zero elements.
Monoclinic For all three classes (2, m and 2/m) there are 41 independent non-zero elements:
3 elements with suffixes all equal,
18 elements with suffixes equal in pairs,
12 elements with suffixes having two y’s, one x and one z,
4 elements with suffixes having three x’s and one z,
4 elements with suffixes having three z’s and one x.
Orthorhombic For all three classes (222, mm2 and mmm) there are 21 independent nonzero
elements,
3 elements with all suffixes equal,
18 elements with suffixes equal in pairs
Tetragonal For the three classes 4, and 4/m, there are 41 nonzero elements of which only 21 are
independent. They are:
xxxx=yyyy
zzzz
zzxx=zzyy
xyzz=-yxzz
xxyy=yyxx
xxxy=-yyyx
xxzz=yyzz
zzxy=-zzyx
xyxy=yxyx
xxyx=-yyxy
zxzx=zyzy
xzyz=-yzxz
xyyx=yxxy
xyxx=-yxyy
xzxz=yzyz
zxzy=-zyzx
yxxx=-xyyy
zxxz=zyyz
zxyz=-zyxz
xzzx=yzzy
xzzy=-yzzx
For the four classes 422, 4mm, 4/mmm and 2m, there are 21 nonzero elements of which
only 11 are independent. They are:
xxxx=yyyy zzzz
yyzz=xxzz yzzy=xzzx
xxyy=yyxx
zzyy=zzxx
yzyz=xzzx xyxy=yxyx
zyyz=zxxz zyzy=zxzx xyyx=yxxy
Cubic For the two classes 23 and m3, there are 21 nonzero elements of which only 7 are
independent. They are:
xxxx=yyyy=zzzz
yyzz=zzxx=xxyy
zzyy=xxzz=yyxx
yzyz=zxzx=xyxy
zyzy=xzxz=yxyx
yzzy=zxxz=xyyx
zyyz=xzzx=yxxy
For the three classes 432, 3m and m3m, there are 21 nonzero elements of which only
4 are independent. They are:
xxxx=yyyy=zzzz
yyzz=zzxx=xxyy=zzyy=xxzz=yyxx
yzyz=zxzx=xyxy=zyzy=xzxz=yxyx
yzzy=zxxz=xyyx=zyyz=xzzx=yxxy
Trigonal For the two classes 3 and , there are 73 nonzero elements of which only 27 are
independent. They are:
zzzz
xxxx=yyyy=xxyy+xyyx+xyxy xxyy=yyxx
xyyx=yxxy
xyxy=yxyx
yyzz=zzxx xyzz=-yxzz
zzyy=zzxx
zzxy=-zzyx
zyyz=zxxz
zxyz=-zyxz
yzzy=xzzx
xzzy=-yzzx
xxyy=-yyyx=yyxy+yxyy+xyyyyyxy=-xxyx
yxyy=-xyxx
xyyy=-yxxx
yyyz=-yxxz=-xyxz=-xxyz
yyzy=-yxzx=-xyxz=-xxzy
yzyy=-yzxx=-zxyx=-xxzy
zyyy=-zyxx=-zxyx=-zxxy
xxxz=-xyyz=-yxyz=-zzxz
xxzx=-xyzy=-xyzy=-yyzx
xzxx=-yzxy=-yzyx=-xzyy
zxxx=-zxyy=-zyxy=-zyyx
For the three classes 3m, m and 3,2 there are 37 nonzero elements of which only 14 are
independent. They are:
zzzz
xxxx=yyyy=xxyy+xyyx+xyxy xxyy=yyxx
xyyx=yxxy
xyxy=yxyx
yyzz=xxzz zzyy=zzxx
zyyz=zxxz yzzy=xzzx yzyz=xzxz
zyzy=zxzx
xxxz=-xyyz=-yxyz=-yyxz
xxzx=-xyzy=-yxzy=-yyzx
zxxx=-zxyy=-zyxy=-zyyx
Hexagonal For the three classes 6, and 6/m there are 41 non-zero elements of which only 19
are independent. They are:
zzzz
xxxx=yyyy=xxyy+xyyx+xyxy
xxyy=yyxx
xyyx=yxxy
xyxy=yxyx
yyzz=zzxx
xyzz=-yxzz
zzyy=zzxx
zzxy=-zzyx
zyyz=zxxz
zxyz=-zyxz
yzzy=xzzx
xzzy=-yzzx
yzyz=xzxz
xzyz=-yzxz
zyzy=zxzx
zxzy=-zyzx
xxyy=-yyyx=yyxy+yxyy+xyyy
yyxy=-xxyx
yxyy=-xyxx
xyyy=-yxxx
For the four classes 622, 6mm, 6/mmm and m2, there are 21 nonzero elements of
which only 10 are independent. They are:
zzzz
xxxx=yyyy=xxyy+xyyx+xyxy
xxyy=yyxx xyyx=yxxy xyxy=yxyx
yyzz=xxzz zzyy=zzxx zyyz=zxxz yzzy=xzzx yzyz=xzxz zyzy=zxzx
Common Third Order Nonlinear Phenomena
Most general expression for the nonlinear polarization in the frequency domain is
(3)
Pi(3) ( )   0     ijkl
( ;  p , q , r ) E j ( p )Ek (q ) E (r )  (   p  q  r )d p dq dr
Each is the total field!
Consider just isotropic media, more complicated but same physics for anisotropic media
Single Incident Beam
Third Harmonic Generation
 a
3a

1
(3)
PxNL (r , t )   0 ˆ xxxx
(3a ; a , a , a )E x3 (a )e3i ( ka at )  c.c.
8
Intensity-Dependent Refraction and Absorption

PxNL (r , t )
 a
1
(3)
(3)
  0 [ ˆ xxxx
(a ;a , a , a )  ˆ xxxx
(a ; a ,a , a )
8
 
1
E (r , t )  E( )e-it  c.c.
2
1 
 E ( )ei[ kz t ]  c.c
2
n||
 a
(3)
 ˆ xxxx
(a ; a , a ,a )] | E x (a ) |2 E x (a )ei ( ka z at )  c.c.
NL 
 Px (r , t )
 0
3 ~ (3)
  0  xxxx (a ; a ,a , a ) | E x (a ) |2 E x (a )ei ( ka z at )  c.c.
8
Two Coherent Input Beams
Case I Equal Frequencies, Orthogonal Polarization
Third Harmonic Generation

1
(3)
Px(3) (r , t )   0 [ ˆ xxyy
(3a ; a , a , a )
8
ê x
3

ê y
(3)
(3)
 ˆ xyyx
(3a ; a , a , a )  ˆ xyxy
(3a ; a , a , a )]E x (a )E y2 (a )e
i ( 2 k y  k x ) z 3iat
 c.c.

1
3)
3)
Py(3) (r , t )   0 [ ˆ (yxxy
(3a ; a , a , a )  ˆ (yxyx
(3a ; a , a , a )
8
3)
 ˆ (yyxx
(3a ; a , a , a )]E x2 (a )E y (a )e
(3) 
 Px, y (r , t )
 0
i ( 2 k x  k y ) z 3iat
 c.c.
3 ~ (3)
  0  xxxx (3a ; a , a , a )E y2, x (a )E x, y (a )e3i ( kz at )  c.c.
8
Cross Intensity-Dependent Refraction and Absorption (also known as cross-phase modulation)
for example


(3)
P(3) (r ,t )  0 | E|| (a ) |2 E  (a )[{ˆ xxyy
(a ; a ,a , a )
8
(3)
(3)
 ˆ xxyy
(a ; a , a ,a )}  {ˆ xyyx
(a ; a ,a , a )
ê||

ê
n||
n
(3)
(3)
(3)
  xyyx
(a ;a , a , a )}  {ˆ xyxy
(a ;a , a , a )  ˆ xyxy
(a ; a , a ,a )}]ei ( ka z at )  c.c.

2 ~ (3)
 0
 P(3) (r ,t )  0 { xxxx
(a ; a ,a , a ) | E|| (a ) |2 E  (a )ei ( ka z at )  c.c.
8
4-Wave-Mixing
ê||

ê


(3)
P(3) (r , t )  0 E||2 (a )E * (a ){ˆ xxyy
(a ;a , a , a )
8
(3)
(3)
 ˆ xyyx
(a ; a , a ,a )  ˆ xyxy
(a ; a ,a , a )}e
i ( 2 k||  k ) z i ( 2||  ) iat
 c.c.

 ~ (3)
i ( 2 k  k ) z i ( 2||  ) iat
 0
 P(3) (r , t )  0  xxxx
(a ; a ,a , a )E||2 (a )E * (a ) e || 
 c.c.
8
Case II Unequal Frequencies, Parallel Polarization
Cross Intensity-Dependent Refraction and Absorption (also known as cross-phase modulation)
Most common is effect of strong beam on a weak beam
(3) 
Px (r , t )
1
(3)
  0{ˆ xxxx
(a ; a , b ,b )
8
(3)
(3)
(3)
 ˆ xxxx
(a ; a ,b , b )  ˆ xxxx
(a ; b , a ,b )  ˆ xxxx
(a ;b , a , b )
(3)
(3)
 ˆ xxxx
(a ;b , b , a )  ˆ xxxx
(a ; b ,b , a )} | E x (b ) |2 E x (a )ei[ k x (a ) z at ]  c.c.

6 ~ (3)
 0
 Px(3) (r , t )   0  xxxx
(a ; b ,b , a )} | E x (b ) |2 E x (a )ei[ k x (a ) z at ]  c.c.
8

6 ~ (3)
 0
 Px(3) (r , t )   0  xxxx
(a ; b ,b , a )} | E x (b ) |2 E x (a )ei[ k x (a ) z at ]  c.c.
8

3 ~ (3)
 0
 PxNL (r , t )   0  xxxx
(a ; a ,a , a ) | E x (a ) |2 E x (a )ei ( ka z at )  c.c.
8
Coherent Anti-Stokes Raman Scattering CARS) 2a-b, a > b

1
(3)
Px(3) (r , t )   0 [  xxxx
([2a  b ]; a , a ,b )
8
(3)
 ˆ xxxx
([2a  b ]; a ,b , a )
(3)
 ˆ xxxx
([2a  b ];b , a , a )] E x2 (a )E x* (b )ei[{2k (a )k (b )}z ( 2a b )t ]  c.c.
Case III Incoherent Beams
Cross Intensity-Dependent Refraction and Absorption
(also known as cross-phase modulation)
Most common is effect of strong beam on a weak beam

1
(3)
Px(3) (r , t )   0{ˆ xxxx
(a ; a , b ,b )
8
(3)
(3)
(3)
 ˆ xxxx
(a ; a ,b , b )  ˆ xxxx
(a ; b , a ,b )  ˆ xxxx
(a ;b , a , b )
(3)
(3)
 ˆ xxxx
(a ;b , b , a )  ˆ xxxx
(a ; b ,b , a )} | E x (b ) |2 E x (a )ei[ k x (a ) z at ]  c.c.

6 ~ (3)
 0
 Px(3) (r , t )   0  xxxx
(a ; b ,b , a )} | E x (b ) |2 E x (a )ei[ k x (a ) z at ]  c.c.
8