Chapter 6. Processes Resulting from the Intensity-Dependent Refractive Index

Nonlinear Optics Lab . Hanyang Univ.

Chapter 6. Processes Resulting from

the Intensity-Dependent Refractive Index

- Optical phase conjugation - Self-focusing

- Optical bistability - Two-beam coupling - Optical solitons

Reference :

R.W. Boyd, “Nonlinear Optics”, Academic Press, INC.

- Photorefractive effect (Chapter 10)

:

cannot be described by a nonlinear susceptibility c⁽ⁿ⁾for any value of n

Nonlinear Optics Lab . Hanyang Univ.

6.4 Two-Beam Coupling

: Under certain condition, energy is transferred from one beam to the other

 Refractive index experienced by either wave is modified by the intensity of the other wave

Total optical field :

c n

k e

e

t) A₁ ^{i( t)} A₂ ^{i(  t)} c.c. _{i 0}



_i ,

~(r  ^k₁^r ₁  ^k₂^r ₂   E

0 E~2

I 4

 c

 n

 

 

 



^{e cc}



c n

c c c e

n

t i

t i

i

. A

A A

A A

2 A

. A

A A

A A

2 A I

) (

* 2 1

* 2 2

* 1 1 0

) ) ( )

* ( 2 1

* 2 2

* 1 1

0 1 2 1 2







































r q

r k k

2 1

2 1





  



k k

where, q : grating wave vector : frequency difference

moving grating

Nonlinear Optics Lab . Hanyang Univ.

Special case (q=180 degree)

2k2

q

 

 ^{e c c} 

c

n

_{i kz t}

. A

A A

A A 2 A

I 

⁰ _{1 1}^*



₂ ^*₂



₁ ^*₂ ^{(2  )}











 0 0





Phase velocity : v||/2k

Nonlinear Optics Lab . Hanyang Univ.

Theoretical treatment

Nonlinear refractive index considering the dynamic response (Debye relaxation equation) :

2I n dt n

dn

NL

NL  



Solution :

t d e

n t

n_NL 



^t  ^{t t} 

 



) 2 I( ) (





 







1

) 1 ( )

(





 

  ^





























^{te i te t t dt e t te i t dt} _i^{e i t}

 











i e i

e c

n n n

t i t

i

NL  

 





 ^{    }

1 A A 1

A A A

A A 2 A

) ( 2

* 1 )

(

* 2

* 1 2 2

* 1 1 2

0

r q r

q

Wave equation : E~ 0 E~

2 2 2 2

2 



 

 c t

n where, n  n₀ n_{NL and}

n

NL

n n

n

²



₀²

 2

₀

n0

n_NL 

Ex)

I ( t ' )  e

^it

Nonlinear Optics Lab . Hanyang Univ.

 

_ ^ _







i c

n n c

n n c

k n dz ik d dz

d

 













 1

A A A

A A

A A A

A 2 ₀² _{2 1}² ₁ ² ₂

2 2 2 2 1 2 2 2 2 0 2 2

2 2 0 2 2 2 2 2 2

2 2

stationary index time-varying index

 

_^ _





i n

i n n

i n dz d

 



 1

A A A 2

A 2 A

A _{0 2 1} ² ₂

2 2 2 2

1 2

0

2 where, ₁₂



 



 



 dz

d dz

d c

n dz

d ^*₂

2

* 2 2 0

2 A

A A 2 A

I



0 A A*

I_i n2 c _{i i}

 

2 2 1 2

2

I I

1 2









  c

n

: when >0 (₁<₁) I₂ increases with z

Maximum gain ;

2 1 2

2

I I

I

n c dz

d _ 

 1



when

Nonlinear Optics Lab . Hanyang Univ.

# There is no energy coupling if 0

i) 0 (nonlinearity has a fast response)

ii) ₁₂0 (input waves are at the same frequency)



Two-beam coupling can occur in certain photorefractive crystal even between beams of the same frequency.

In such case, energy transfer occurs as a result of a spatial phase shift between the nonlinear index grating and the optical intensity distribution.

Nonlinear Optics Lab . Hanyang Univ.

6.5 Pulse Propagation and Optical Solitons

Optical solitons : Under certain condition, an exact cancellation of group velocity dispersion can occur by a nonlinear optical process so called self-phase modulation.

Self-Phase Modulation

Optical pulse :

A ~ ( , ) c.c.

) , (

E ~ z t  z t e

ⁱ₍^k₀^z₀^t₎



Refractive index of 3^rd order nonlinear medium : ⁰ A~( , )² ) 2

( n c zt t

I ^ 

), ( )

(t n₀ n₂I t

n  

Phase change by nonlinear refractive index : c

L t

I n

NL(t) 2 ( )0

  Frequency change :

dt t dI c

L t n

dt

t d _NL ( )

) ( )

(



²



⁰



  

Nonlinear Optics Lab . Hanyang Univ.

Example

) (

h sec )

(t I₀ ² t



₀ I 

Pulse shape :

Nonlinear phase shift :

 

sech ( ) )

( ₂ _{0 0} ² ₀

_NL t n c LI t Frequency shift :

 

^{sech ( )tanh( )}

2

) ( )

(

0 0

2 0

0 0

2    





t t

LI c

n dt t t d _NL





# Maximum frequency shift :

,

0 max

max 

 ^^NL _{I L}

n c

NL 0

0 2

max 

 



: Whenever _max exceeds the spectral width of the incident pulse (~2/₀), that is ,

the spectral broadening due to self-phase modulation will be important.



^max 2

 _NL

Nonlinear Optics Lab . Hanyang Univ.

Pulse Propagation Equation

c.c.

) , ( A~ ) , (

E~ z t  z t eⁱ₍^k₀^z₀^t₎  Optical pulse :

where, k₀  n_lin(



₀)



₀ c Wave equation :

D~ 0 1

E~

2 2 2 2

2 



 





t c z

Let’s introduce Fourier transform of D~( , ); )

, ( E~

and z t t

z



^

 





^



) 2 D(

) (

D~ d

e z,

z,t ^{i t}



^

 





^



) 2 E(

) (

E~ d

e z,

z,t ^{i t}

) E(

) ( )

D(z,   z,

(6.5.11)

(6.5.11) 

0 ) E(z, )

) ( E(z,

2 2 2

2  





    

c z

(6.5.14)

Nonlinear Optics Lab . Hanyang Univ.

dt e z,t z,



^ ^{i t}

 

 ^

 A~( ) )

A(

Fourier transform of amplitude is given by

The amplitude is related with the Fourier amplitude as

z ik

z ik z

ik

e z,

e z,

e z,

z,

0

0 0

) A(

) (

A )

A(

) E(

0

0

* 0



























 ^

(6.5.14), slow varying approximation 

0 A ] )

( A [

2

₀



²



₀²





 k k

ik z 

^{where, k() () c}

k() ~ k₀ 

k

²

 k

₀²

 2 k

₀

( k  k

₀

)

0 ) A(

) ) (

A(

0 0

0

  



 i k k z,ω,ω

z z,ω,ω

(6.5.19)

Nonlinear Optics Lab . Hanyang Univ.

Power series expansion of k() :

2 0 2

0 1

0

( )

2 ) 1

(       







 k k k k

k

_NL

where, ⁰ ₂ ^{0, I}



^{n (} ₀^{)c 2}



^A~(z,t)2

I c c n

n

k_{NL NL}  _lin  











0 0 0

0 0

2 2

2 2

0 1

1 )

( 1

) (

1 )

) ( 1 (



 

 























 

 

 



















 









 



 





 

 

 

 



 





d dv v v

d d d

k k d

v d

n dn c d

k dk

g g g

g lin

lin

(6.5.20)

(6.5.19) and (6.5.20) 

0 A ) 2 (

A 1 ) (

A A

₂

0 2

0

1

 





 

 i k ik ω-ω ik ω-ω

z

^NL

0 A ~ A ~

2 1 A ~ A ~

2 2 2

1

  



 



 



  i k

_NL

ik t k t

z

) A(z,

A



) (z, A ~ A ~

 t

(6.5.26)

Nonlinear Optics Lab . Hanyang Univ.

The equation can be simplified by means of a coordinate transformation ; z

k v t

t z

g

 1







 : retarded time

2 2 2 2

1

A~ A~

A~ A~

A~ A~

A~ A~

A~ A~

A~

τ t

τ t

τ τ t z z t

k τ z t

τ τ z

z

s

s s

s

s s

s s



 







 















 







 



 







 



 



 A~ (z, )

A~

s

s 

(6.5.26)  A~ A~ 0

2 1 A~

2 2

2   



 





s NL s

s ik i k

z 

If we express the nonlinear contribution to the propagation constant as ₂ ^{0 0 2 0 2} A~ ² A~

I 2 _{s s}

NL

n n n c

k 





 _{ }





s s

s

s

ik i

z A ~

A ~ A ~

2 1 A ~

2 2

2

2



 ^



 





: nonlinear schrodinger equation

group velocity dispersion self-phase modulation

Nonlinear Optics Lab . Hanyang Univ.

Optical Solitons

s s s

s ik i

z A~

A~ A~

2 1 A~

2 2

2

2 

 ^



 





As an example, a pulse whose amplitude is expressed by

A ~

_s

( z , t ) A

_s

sec h (   ) e

^iz

0



0

 





 ₂

0 2

2 2

0 2 2

0

A 2



 n c k k

s

, 2 ₀²

2 

k and

k₂ and  n₂ must have opposite sign If

the pulse can propagate with an invariant shape : Optical soliton Report

Ex) Fused silica optical fiber

i) n₂ > 0 (electronic polarization)

ii) Group velocity dispersion parameter k₂ : k₂ > 0 for visible region

k₂ < 0 for l > 1.3mm

#

Nonlinear Optics Lab . Hanyang Univ.

10.4 Introduction to the Photorefractive Effect

: The change in refractive index resulted from the optically induced redistribution of electrons and holes.

# Photorefractive effect gives rise to a strong optical nonlinearity, however, the effect tends to be rather slow with response time of 0.1 s being typical.

Origin of photorefractive effect

Maxwell equation ;



 ⁴

4  





 dx

D dE





4 dx dE

) 0 2 (

1 ₃







n n r_eff E r_eff

# Refractive index distribution is shifted

by 90 degree with respect to the intensity distribution

 Leads to the transfer of energy between the two incident beams

Nonlinear Optics Lab . Hanyang Univ.

10.5 Photorefractive Equations of Kukhtarev et al.

Assume that the crystal contains N_Aacceptors and N_D⁰ donors per unit volume, with N_A<<N_D⁰

Rate equations :

) 1 (

) )(

(

⁰

e j t

N t

n

N n N

N t sI

N

D e

D e D

D D





 

 











 







 





where, s : photoionization cross section of a donor

 : thermal generation rate (thermal ionization)

 : recombination coefficient j : electrical current density (10.5.1)

(10.5.2)

Nonlinear Optics Lab . Hanyang Univ.

ph e

e

e E eD n j

n

j  m   

Electrical current density :

where, m : electron mobility D : Diffusion constant

j_ph : photovoltaic contribution to the current

Local field within the crystal :

) (

4  

^









_{e A D}

dc

E  e n N N



Change in dielectric constant :

|

eff

|E



  



Wave equation for the optical field :

~ 0 ) 1 (

~

2 2 2

2

  



 



_opt

E

_opt

t

E c  

: Cannot easily be solved exactly

(10.5.3)

(10.5.4)

(10.5.5)

(10.5.6)

Nonlinear Optics Lab . Hanyang Univ.

10.6 Two-Beam Coupling in Photorefractive Materials

. c.c ]

[ ) ,

~ ( _{p s}

s

p  

 ^{i  i  i t}

opt r t A e^{k r} Ae^{k r} e ^ E

Optical field within the crystal :

Intensity distribution of light within the crystal :

.) . (

E ~

4

^{0 1}

2 opt

0

c I I e c c

I  n  

^iqx





^where,

s p

s p s p

s p

x q q

e e A c A I n

A c A

I n

k ˆ k

ˆ ) )(ˆ 2 (

)

|

|

| 2 (|

0 * 1

2 0 2

0



















: grating wave vector (10.6.2)

Nonlinear Optics Lab . Hanyang Univ.

Intensity distribution of light within the crystal can also be described by

)]

( cos 1

0

[   

 I m qx

I

^where,

) Re / (Im tan

/

|

| 2

1 1

1 0 1

I I

I I m

 





: modulation index

Approximate steady-state solution (|I

₁

|<<I

₀

)

Put,

.) . (

.) . (

.) . (

.) . (

1 0

1 0

1 0 1

0

c c e

N N

N c

c e

n n

n

c c e

j j

j c

c e

E E

E

iqx D D

D iqx

e e

e

iqx iqx































(10.5.1)~(10.5.6)  (Assume E₁, j₁, ne1, N_D1are small that the product of any of them can be neglect)

1) From x independent term,

A e

D ph

e

D e D

D

N n

N j

E e n j

j N

n N

N sI























0 0

0 . 0

0 0

0 0

0 0

0

0

)( ) constant

(

m





(10.6.5)

Report

Nonlinear Optics Lab . Hanyang Univ.

In most realistic case, N_D(~10¹⁹cm^3)N_A(~10¹⁶cm^3)n_e0(~10¹³cm^3) and N_D^₁n_e1

A

A D

e

A D

N

N N

n sI

N N





)( )

( ₀ ⁰

0 0



 



 ^

2) From e^iqx dependent term (assume E₀=0),

) (

4 0

) (

) (

1 1

1 1

1 0

1 1

1 0 1

0 0

0 1































D e

dc e

B e

A e D

e D

D D

N n

e E

iq Tn

iqk eE

n

j N

n N

n N

sI N

N sI











 





 





 



 





 





q D

D

E E

E sI

i sI

E

₀

1 /

1

1



where,

e T

E_Dqk^B : diffusion field strength

eff dc

q N

q E e





4 : maximum space charge field

0

0 )/

( _{D A D}

A

eff N N N N

N  

Report

Tm k

eD _B : Einstein relation

Nonlinear Optics Lab . Hanyang Univ.

 





 





 



 





 



q D

D

E E

E sI

i sI

E

₀

1 /

1

1



i) Quarter period shift of the index grating with respective to the intensity distribution ii)

E

₁

 sI

₁

/( sI

₀

  )  I

₁

iii)

E

₁

 fn ( E

_D and

E

_q

)

: depends also on grating vector q (10.6.8)

Defining the optimum value of q maximizing the second factor as q_opt,

2 0

1

1

1 ( / )

) /

( 2

opt opt

opt

q q

q E q

sI i sI

E   

 





 

 

where,

2 / 2 1

4 ,



 

















dc B eff opt

dc B

eff opt

T k E N

T k

e q N









q

/ )sin (

2n c q

can be adjusted

Nonlinear Optics Lab . Hanyang Univ.

Spatial growth rate

(10.6.2) and (10.6.8) 

m p

s

s

p E

A A

A i A

E 











 

 _{2 2}

*

1 | | | | ^where, D q

D

m E E

E E

/ 1



Nonlinear polarization :

. . ( )

4

r ik p r ik s r

iq

NL

e c c A e

s

A e

p

P

^



^



^



 



  

 



Dielectric constant change :

    

²



_eff

E

₁

r ik s

p

p s

m r eff

ik s NL

p

r ik s

p

s p m

r eff ik p NL

s

p p

s s

A e A

A A E

e i A P

A e A

A A

E e i

A P









 

 



 

 



2 2

2 2

2 2

2 2

|

|

|

|

|

| 4

4

|

|

|

|

|

| 4

4

*





















(10.6.16)

1) Steady state

Nonlinear Optics Lab . Hanyang Univ.

Wave equation (slow varying approx.) :

NL s r

ik s

s P

e c dz

ikdA ^s ₂

2

4

2 ^ 

 

2 2

2 3

|

|

|

|

|

|

2

_{p s}

s p m

eff s

s

A A

A E A

c n dz

dA

 

  

|2

2 | ^s

s nc A

I ^ 

p s

p s s

s

I I

I I dz

dI

 





where, n _eff E_m

c 

 ₃





Similarly,

p s

p s p

p

I I

I I dz

dI

 





: when >0, I_s is amplified and I_p is attenuated

Nonlinear Optics Lab . Hanyang Univ.

2) Transient two-beam coupling

0 0,

, N N sI

N

n_e _D^ _D^ _D





Assume,

2 2

* 1

1

|

|

|

|

_{p s}

s p

m

A A

A iE A

t E E

 



 

 

^where,

q D

M D

D E E

E E

/ 1

/ 1



 



4 _e0 dc D emn

  

m

 q E_M  N^A

Wave equations :

 



 





 





 





* 1

1

2 2

E c A

n i x

A

E c A

n i x

A

p eff s

s s

s eff p

p p

 

 