Chapter 8. Second-Harmonic Generation and Parametric Oscillation

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Chapter 8. Second-Harmonic Generation

and Parametric Oscillation

8.0 Introduction

Second-Harmonic generation : Parametric Oscillation :





   2

) (

_{1 2 3}

2 1

3

    

    

Reference :

R.W. Boyd, Nonlinear Optics,

Chapter 1. The nonlinear Optical Susceptibility

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The Nonlinear Optical Susceptibility

General form of induced polarization :









 ( ) ( ) ( )

)

(t ⁽¹⁾E t ⁽²⁾E² t ⁽³⁾E³ t

P

  









P⁽¹⁾(t) P⁽²⁾(t) P⁽³⁾(t)

: Linear susceptibility where,



⁽¹⁾

: 2



^nd(2 order nonlinear susceptibility⁾ : 3



^rd(3 order nonlinear susceptibility⁾

) 2

P(

: 2^nd order nonlinear polarization

) 2

P(

: 3^rd order nonlinear polarization

Maxwell’s wave equation :

2 2 2

2 2 2 2

t P t

E c

E n







 



Source term : drives (new) wave

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Second order nonlinear effect

) ( )

( ^{(2) 2}

) 2

( t E t

P 



Let’s us consider the optical field consisted of two distinct frequency components ; c.c.

)

(t E₁e^{i 1t}E₂e^{i 2t}

E ^{ }

] [

2

] c.c.

2 2

[ )

(

* 2 2

* 1 1 ) 2 (

) (

* 2 1 )

( 2 1 2

2 2 2

2 1 ) 2 ( )

2

( _{1 2 1 2 1 2}

E E E E

e E E e

E E e

E e

E t

P

^{i t i t i t i t}

















^{     }





^{     }

(OR) )

( 2

) 0 (

) DFG (

2 ) (

) SFG (

2 ) (

) SHG (

) 2 (

) SHG (

) 2 (

* 2 2

* 1 1 ) 2 (

* 2 1 ) 2 ( 2

1

2 1 ) 2 ( 2

1

2 2 ) 2 ( 2

2 1 ) 2 ( 1

E E E E P

E E P

E E P

E P

E P







































: Second-harmonic generation

: Sum frequency generation

: Difference frequency generation : Optical rectification

# Typically, no more than one of these frequency component will be generated  Phase matching !

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Nonlinear Susceptibility and Polarization

1) Centrosymmetric media (inversion symmetric) : V (  x )  V ( x )

Potential energy for the electric dipole can be described as

4 ...

) 2

( 

₀^{2 2}

 m Bx

⁴

 m x

x

V 

Restoring force :

3

...

2

0

 



 

 

 m x mBx

x

F V 

Equation of motion :

m t eE Bx

x x

x _  2  _  

₀²



³

  ( )/

Damping force

Restoring force

Coulomb force

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Purtubation expansion method :

c.c.

)

(t E₁e^{i 1t}E₂e^{i 2t}

E ^{ }

Assume,

) ( )

(t E t E 











 x^{(1) (2)}x^{(2) (3)}x⁽³⁾

x   

Each term proportional to ⁿ should satisfy the equation separately

m t eE x

x

x_⁽¹⁾2



_⁽¹⁾



₀^{2 (1)} ( )/

0 2 ⁽²⁾ ₀^{2 (2)}

) 2

(  x  x 

x_



_



0 2 ⁽³⁾ ₀^{2 (3) 3(1)}

) 3

(  x  x Bx 

x_



_





: Damped oscillator

 x 

⁽²⁾

 0

Second order nonlinear effect in centrosymmetric media can not occur !

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2) Noncentrosymmetric media (inversion anti-symmetric) : V (  x )  V ( x )

Potential energy for the electric dipole can be described as

3 ...

) 2

( 

₀^{2 2}

 m Dx

³

 m x

x

V 

Restoring force :

2

...

2

0

 



 

 

 m x mDx

x

F V 

Equation of motion :

m t eE Dx

x x

x _  2  _  

₀²



²

  ( )/

Damping force

Restoring force

Coulomb force

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Similarly,

c.c.

)

(t E₁e^{i 1t}E₂e^{i 2t}

E ^{ }

Assume,

) ( )

(t E t E 











 x^{(1) (2)}x^{(2) (3)}x⁽³⁾

x   

Each term proportional to ⁿ should satisfy the equation separately

m t eE x

x

x_⁽¹⁾2



_⁽¹⁾



₀^{2 (1)} ( )/

0 ] [

2 ⁽²⁾ ₀^{2 (2) (1) 2}

) 2

(  x  x D x 

x_



_



0 2

2 ⁽³⁾ ₀^{2 (3) (1) (2)}

) 3

(  x  x  DBx x 

x_



_





Solution :

c c e

x e

x t

x_⁽¹⁾( ) ⁽¹⁾(



₁) ^i1t ⁽¹⁾(



₂) ^i2t .









 

j j

j j

j

j i

E m

e L

E m x e

2 )

) (

( _{2 2}

0 )

1 (



 





 _{: Report}

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Example) Solution for SHG

) ( ) / 2 (

1 2

2 1 2

2 )

2 ( 2 0 ) 2 ( )

2

( ¹

 



^

L

E e

m e x D

x x

t

 i





  



Put general solution as x⁽²⁾(t)x⁽²⁾(2



₁)e^2i1t

) ( ) 2 (

) / ) (

2 (

1 2 1

2 1 2 1

) 2 (



 

L L

E m e

x D





: Report Similarly,

) ( ) 2 (

) / ) (

2 (

2 2 2

2 2 2 2

) 2 (



 

L L

E m e x  D

) ( ) ( ) (

) / ( ) 2

(

2 1

2 1

2 1 2 2

1 ) 2 (







 

 L L L

E E m e x D



 



) ( ) ( ) (

) / ( ) 2

(

2 1

2 1

* 2 1 2 2

1 ) 2 (







 

  

 

 L L L

E E m e x D

) (

) ( ) 0 (

) / ( 2 )

( ) ( ) 0 (

) / ( ) 2

0 (

2 2

* 2 2 2

1 1

* 1 1 ) 2

2 (







 

 



 

L L

L

E E m e D L

L L

E E m e x D

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Susceptibility

) ( )

(j Nex j

P

 















^{( ) ( ) ( )}

)

(t P^{( ) (1)}E t ⁽²⁾E² t ⁽³⁾E³ t P

j

j

  

Polarization :



( )

) / ) (

(

2 )

1 (

j

j L

m e N

 

  : linear susceptibility

2 )

1 ( )

1 ( 3 2 2

2 ) 3

2

( (2 )[ ( )]

) ( ) 2 (

) / ) (

, , 2

( _{j j}

j j

j j

j N e

mD L

L

a m e

N    



 





   : SHG

) ( ) ( ) (

) / ) (

, , (

2 1

2 1

2 3 2

1 2 1 ) 2 (







 







 L L L

D m e N

 

 ^ _{2 3}⁽¹⁾(₁^₂)⁽¹⁾(₁)⁽¹⁾(₂) e

N mD

) ( ) ( ) (

) / ) (

, , (

2 1

2 1

2 3 2

1 2 1 ) 2 (







 







     

L L

L

D m e N

: SFG

: DFG

: OR

) ( ) ( )

( _{1 2} ⁽¹⁾ ₁ ⁽¹⁾ ₂

) 1 ( 3

2       

N e mD

) (

) ( ) 0 (

) / ) (

, , 0 (

2 ) 3

2 (

j j

j

j L L L

D m e N



 



    _{2 3} ⁽¹⁾(0) ⁽¹⁾( _j) ⁽¹⁾( _j)

e N

mD     



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<Miller’s rule>

- empirical rule, 1964

) ( ) ( ) (

) , , (

2 ) 1 ( 1 ) 1 ( 2 1 ) 1 (

2 1 2 1 ) 2 (





























3 2e N

 mDis nearly constant for all noncentrosymmetric crystals.

# N ~ 10²³ cm^-3 for all condensed matter

# Linear and nonlinear contribution to the restoring force would be comparable when the displacement is approximately equal to the size of the atom (~order of lattice constant d) :

m₀²d=mDd  D=w₀²/d : roughly the same for all noncentrosymmetric solids.

4 4 0 2

3 )

2 (

d m

e

  (non-resonant case) : used in rough estimation of nonlinear coefficient.

2 0 2

2

0 2

)

(_j  _j  i_j

L N1 d/ ³ D₀²/d



6

0

2 0 2 3 3

2 1

2 1

2 3 2

1 2 1 ) 2

( (1/ )( / )( / )

) ( ) ( ) (

) / ) (

, ,

( 









 







 ^{d e m d}

L L

L

D m e

N 

 

 310^8esu

: good agreement with the measured values

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Qualitative understanding of Second order nonlinear effect

in a noncentrosymmetric media

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2 component

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General expression of nonlinear polarization and

Nonlinear susceptibility tensor

General expression of 2^nd order nonlinear polarization :

t i

m n

i t i

m n

i

i t P e ^{n m} P e ^{n m}

P(r, ) (







) ^{ (  )}  (







) ^{(  )}

), (

) ( ) , , (

) (

) (

) 2 (

m k n j m n m n jk nm

ijk m

n

i E E

P   



      

where,

2^nd order nonlinear susceptibility tensor

# Full matrix form of P_i(_n_m)

) ( ) ( ) , , (

) ( ) ( ) , , (

) ( ) ( ) , , (

) ( ) ( ) , , (

) (

2 2

2 2 2 2 ) 2 (

1 2

1 2 1 2 ) 2 (

2 1

2 1 2 1 ) 2 (

1 1

1 1 1 1 ) 2 (





























































k j

jk ijk

k j

jk ijk

k j

jk ijk

k j

jk ijk m

n i

E E

E E

E E

E E

P



























2 , 1 , m n

: SHG

: SHG : SFG : SFG

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Example 1. SHG





































































1 2

2 1

1 3

3 1

2 3

3 2

3 3

2 2

1 1

321 312

331 313

332 323

333 322

311

221 212

231 213

232 223

233 222

211

121 112

131 113

132 123

133 122

111

) 2 (

) 2 (

) 2 (

E E

E E

E E

E E

E E

E E

E E

E E

E E

P P P

n z

n y

n x





























































Example 2. SFG





























































































.

) ( ) (

. .

. .

. ) , , (

.

. .

.

.

) ( ) (

. .

. .

. ) , , (

.

. .

. ) (

) (

) (

n k m j n

m m n ijk

m k n j m

n m n ijk m

n z

m n y

m n x

E E

E E

P P P









































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Properties of the nonlinear susceptibility tensor

1) Reality of the fields

) , r ( ), , r

( t E t

P_i are real measurable quantities : )*

( )

( _{n m i n m}

i P

P     

*

*

) ( )

(

) ( )

(

m k m

k

n j n

j

E E

E E

















 

_ijk⁽²⁾

(  

_n

 

_m

,  

_n

,  

_m

)  

_ijk⁽²⁾

( 

_n

 

_m

, 

_n

, 

_m

)

^*

2) Intrinsic permutation symmetry

) , , (

) , , (

)

( _{n m ijk}⁽²⁾ _{n m n m ijk}⁽²⁾ _{n m m n} Pi









 



  



 



  

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4) Kleinman symmetry (nonresonant,  is frequency independent)

) (

) (

) (

) (

) (

) (

2 1 3 ) 2 ( 2

1 3 ) 2 ( 2

1 3 ) 2 (

2 1 3 ) 2 ( 2

1 3 ) 2 ( 2

1 3 ) 2 (



















































































kji jik

ikj

kij jki

ijk

intrinsic

3) Full permutation symmetry (lossless media :  is real)

) (

* ) (

) (

) (

3 2 1

) 2 (

3 2 1

) 2 ( 3

2 1 )

2 ( 2

1 3 ) 2 (





























































jki

jki jki

ijk

: Indices can be freely permuted !

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Define, 2

^nd

order nonlinear tensor, d

_ijk



₂¹



_ijk⁽²⁾

) ( ) ( 2

) (

) (

m k n jk nm

j ijk m

n

i

d E E

P  ^  ^    

## If the Kleinman’s symmetry condition is valid, the last two indices can be simplified to one index as follows ;

6 5

4 3

2 1 :

21 , 2 1 13 , 31 32 , 23 33 22 11 : l

jk

and,

 







 









36 35 34 33 32 31

26 25 24 23 22 21

16 15 14 13 12 11

d d d d d d

d d d d d d

d d d d d d

d_il : 18 elements

d

can be represented as the 3x6 matrix ;ijk

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Again, by Kleinman symmetry (Indices can be freely permuted),

 







 









14 13

23 33

24 15

12 14

24 23

22 16

16 15

14 13

12 11

d d

d d

d d

d d

d d

d d

d d

d d

d d

d

_il ^{: Report}

d_il has only 10 independent elements :

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

Example 1. SHG

























































) ( ) ( 2

) ( ) ( 2

) ( ) ( 2

) (

) (

) (

2 ) 2 (

) 2 (

) 2

( ₂

2 2

36 35

34 33

32 31

26 25

24 23

22 21

16 15

14 13

12 11

























y x

z x

z y

z y x

z y x

E E

E E

E E

E E E

d d

d d

d d

d d

d d

d d

d d

d d

d d

P P P

Example 2. SFG





























 































) ( ) ( ) ( ) (

) ( ) ( ) ( ) (

) ( ) ( ) ( ) (

) ( ) (

) ( ) (

) ( ) (

4 ) (

) (

) (

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

36 35

34 33

32 31

26 25

24 23

22 21

16 15

14 13

12 11

3 3 3











































x y

y x

x z

z x

y z

z y

z z

y y

x x

z y x

E E

E E

E E

E E

E E

E E

E E

E E

E E

d d

d d

d d

d d

d d

d d

d d

d d

d d

P P P

: Report

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

8.2 Formalism of Wave Propagation in Nonlinear Media

Maxwell equation

t







 d

i

h t

 





 h

e



d



₀eP _{i σ e} Polarization :

P  

₀



_e

e  P

_NL

Assume, the nonlinear polarization is parallel to the electric field, then

2 NL 2 2

2

e

2

e P (r , )

e

t

t t

t



 



 



 

   

Total electric field propagating along the z-direction :

.]

. )

( 2[ ) 1 , ( e

.]

. )

( 2[ ) 1 , ( e

.]

. )

( 2[ ) 1 , ( e

) (

3 )

(

) (

2 )

(

) (

1 )

(

3 3 3

2 2 2

1 1 1

c c e

z E t

z

c c e

z E t

z

c c e

z E t

z

z k t i

z k t i

z k t i































) , ( e ) , ( e ) , ( e

e ^(1) z t  ^(2) z t  ^(2) z t

where,

2 1

3

 



 

and

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.



₁

term

 

 



 



 



 



 



^{  }

. .

2

) ( ) ( e

e e

^{[( ) ( )}

* 2 3

2 2 2

) 2 ( 1 )

( 1 )

(

2 ₁ ^{1 1} E z E z e _{3 2 3 2} cc

d t t

t

z k k t i  









 

 



 

  



 



 



 



 

^

( )

^

( )

^

. .

) 2 ( 2

1

_{( )}

1 2 1 ) 1 (

1 )

( 2

1 2

1 1 1

1 1

1

e k E z e c c

z z ik E

z e z

E

_{i t k z i t k z i t k z}

. ) .

2 ( ) 2 (

1

_{1 ( )}

1 1

2

1

e

^{1 1}

c c

dz z ik dE

z E

k

^{i t k z}



 

  





^{ }

2 1 2 1 1

) ( )

(

dz z E d dz

z

k dE  (slow varying approximation)

...

Text

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

z k k k

e

i

E E i d

dz E

dE

_{* ( )}

3 1 2

* 2 2 2 2

*

2 1 3 2

2 2











 











z k k k

e

i

E E i d

dz E

dE

( )

2 1 3

3 3

3 3

3 1 2 3

2 2











 











z k k k

e

i

E E i d

dz E

dE

_{* ( )}

2 3 1

1 1 1 1

1 3 2 1

2 2













 











Similarly,