Transcript Creation of Colloidal Periodic Structure

Chapter 6. Processes Resulting from
the Intensity-Dependent Refractive Index
- Optical phase conjugation
- Self-focusing
- Optical bistability
- Two-beam coupling
- Optical solitons
- Photorefractive effect (Chapter 10)
: cannot be described by a nonlinear susceptibility c(n) for any value of n
Reference :
R.W. Boyd, “Nonlinear Optics”, Academic Press, INC.
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 :
~
i ( k r  t )
i ( k r  t )
E ( r ,t )  A 1 e 1 1  A 2 e 2 2  c.c . k i  n 0 i c
I
I
n0c

n0c
2
2
moving
grating
n0c ~ 2
E
4

*

*
A A
*
1
 A 2 A 2  A 1A 2 e
A A
*
1
 A 2 A 2  A 1A 2 e
1
1
*
*
i ( k 1  k 2 ) r  i (  1   2 ) t )
i ( q r   t )
 c .c

 c .c

where,
q  k 1  k 2 : grating wave vector
   1   2 : frequency difference
Nonlinear Optics Lab.
Hanyang Univ.
Special case (q=180 degree)
q   2k 2
I
n0c
2
A

A 1  A 2 A 2  A 1A 2 e
*
1
*
*
i (  2 k z  t )
 c .c

 0 
 0 
Phase velocity :
v  | |/ 2 k
Nonlinear Optics Lab.
Hanyang Univ.
Theoretical treatment
Nonlinear refractive index considering the dynamic response (Debye relaxation equation) :

dn NL
 n NL  n 2 I
dt
Solution :
n NL 
t
n2



I( t  ) e
Ex) I ( t ')  e
 i t 
 n NL 
( t  t ) 
dt
t

e
n0 n2 c
A A
2
i t 

1
*
1
e
( t  t )
d t  e

A 2A 
*
2
t 
t
e
(  i  1 ) t 
*
i ( q r  t )

A 1A 2 e
2
2~
n

E
~
2
 E 2
0
2
c t
 i t
 i  1 
*

1 i 
Wave equation :
d t 
e
A 1A 2e
 i ( q r  t )
1 i 
n NL  n 0
where, n  n 0  n NL
and
n  n 0  2 n 0 n NL
2
2
Nonlinear Optics Lab.
Hanyang Univ.
2

d A2
dz
2
 2 ik 2
dA 2
dz
n0 
2
k A 2 
2
2
c
2
2
A 2 
2
n 0 n 2 2
2
c
A
2
1
 A2
stationary index
dA 2
i
dz

n 0 n 2
2
A
2
1
 A2
n c  * dA 2
dA 2
 0  A 2
A 2
dz 2 
dz
dz
*
d I2
Ii 
n0 c
2
2
A




2
A
2
2

2
2
n 0 n 2 1 A 1 A 2
c
1 i 
time-varying index
2
2

i
n 0 n 2 A 1 A 2
2
1  i 
2 n 2

c
1  
2
2
where,    1   2
I 1 I 2 : when >0 (1<1) I2 increases with z
*
A iA i
Maximum gain ;
dI2
dz
 n2

c
I1 I 2
when  1
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
~
~
i(k
Optical pulse : E
( z , t )  A ( z , t )e
0 z  0t )
 c.c.
Refractive index of 3rd order nonlinear medium : n ( t )  n 0  n 2 I ( t ),
I (t ) 
2
n0 c ~
A ( z ,t )
2
Phase change by nonlinear refractive index :
 NL ( t )   n 2 I ( t ) 0 L c
Frequency change :
 ( t ) 
d
dt
 NL ( t )  
n 2 0 L dI ( t )
c
dt
Nonlinear Optics Lab.
Hanyang Univ.
Example
I ( t )  I 0 sec h ( t  0 )
Pulse shape :
2
Nonlinear phase shift :
 NL ( t )   n 2  0 c LI 0 sec h ( t  0 )
2
Frequency shift :
 ( t ) 
d
dt
 NL ( t )
 2 n 2  0 c  0  LI 0 sec h ( t  0 )tan h ( t  0 )
2
# Maximum frequency shift :
max
  NL

max
 max 
,   NL
 n2 0 I 0 L
c
0
: Whenever max exceeds the spectral width of the
incident pulse (~2/0), that is   NLmax  2 ,
the spectral broadening due to self-phase
modulation will be important.
Nonlinear Optics Lab.
Hanyang Univ.
Pulse Propagation Equation
Optical pulse :
~
~
i ( k z  t )
E ( z , t )  A ( z , t ) e 0 0  c.c.
where, k 0  n lin ( 0 )  0 c
Wave equation :
2~
 E
z
2

2~
1  D
c
2
t
2
 0 (6.5.11)
~
~
Let’s introduce Fourier transform of E ( z , t ) and D ( z , t ) ;
~
E ( z,t ) 
~
D ( z,t ) 






E( z,  ) e
 i t
D( z,  ) e
d
2
 i t
d
2
D( z,  )   ( ) E( z,  )
(6.5.11) 
 E(z,  )
2
z
2
  ( )

2
c
2
E(z,  )  0
(6.5.14)
Nonlinear Optics Lab.
Hanyang Univ.
Fourier transform of amplitude is given by
A( z,   ) 



~
i  t
A ( z,t ) e dt
The amplitude is related with the Fourier amplitude as
E( z,  )  A( z,   0 ) e
ik 0 z
 A( z,   0 ) e
ik 0 z
 A ( z,   0 ) e
*
 ik 0 z
(6.5.14), slow varying approximation 
2 ik 0
A
z
 [ k ( )  k 0 ] A  0
2
2
where, k ( )   ( )  c
k() ~ k0  k 2  k 02  2 k 0 ( k  k 0 )
 A( z, ω,ω 0 )
z
 i ( k  k 0 ) A( z, ω,ω 0 )  0
(6.5.19)
Nonlinear Optics Lab.
Hanyang Univ.
Power series expansion of k() :
1
2
k  k 0   k NL  k 1 (  0 )  k 2 (  0 )
2
where,  k NL   n NL
0
c
 n2 I
1
 dk 
k1  


c
 d     0
 d 2k
k 2  
2
 d
0
,
c
(6.5.20)
2
~
I  n lin ( 0 ) c 2  A ( z ,t )
dn lin ( ) 
1

n
(

)



lin

d      0
v g ( 0 )

 1 dv g 

d  1 



  2




d   v g ( ) 
d 
   0
 vg
   0
  0
(6.5.19) and (6.5.20) 
A

z
~
A
z
 i  k NL A  ik 1 ( ω-ω 0 ) A 
 k1
~
A
1
2~
 A
1
2
ik 2 ( ω-ω 0 ) A  0
~
 ik 2 2  i  k NL A  0
t 2
t
2
(6.5.26)
Nonlinear Optics Lab.
A  A(z,  )
~ ~
A  A (z, t )
Hanyang Univ.
The equation can be simplified by means of a coordinate transformation ;
z
 t
 t  k 1 z : retarded time
vg
~
A
z
~
A

~
A s

~
A s τ

~
A s
 k1
z
τ t
z
~
~
~
A s z A s τ
A s



t
z t
τ t
τ
~
~
2
2
 As
 A

2
2
t
τ
~
A s
(6.5.26) 
z

1
2
ik 2
2~
 As

2
~
A s
~ ~
A s  A s (z,  )
τ
~
 i  k NL A s  0
If we express the nonlinear contribution to the propagation constant as  k NL  n 2
~
A s
z

1
2
ik 2
2~
 As

group velocity dispersion
2
~ 2~
 i A s A s
0
n n  ~
I  0 2 0 As
c
2
: nonlinear schrodinger equation
self-phase modulation
Nonlinear Optics Lab.
Hanyang Univ.
2
~
  As
2
Optical Solitons
~
A s
z

1
2
2~
 As
ik 2

2
~ 2~
 i A s A s
~
As an example, a pulse whose amplitude is expressed by A s ( z , t )  A 0s sec h (  0 ) e i  z
If A
Report
0
s
2

k2

2
 0
k 2c
2 n 2
2
0

k2 and  n2 must have opposite sign
and
2
   k 2 2 0 , the pulse can propagate with an invariant shape : Optical soliton
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 ;
  D  4

dE
dx
dE
dx

4

1 3
 n   n reff E
2

4

# 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
( reff  0 )
Nonlinear Optics Lab.
Hanyang Univ.
10.5 Photorefractive Equations of Kukhtarev et al.
Assume that the crystal contains NA acceptors and ND0 donors per unit volume, with NA<<ND0
Rate equations :

N D
t
ne
t


 ( sI   )( N D  N D )   n e N D (10.5.1)
0


N D
1
 (  j )
t
e
(10.5.2)
where, s : photoionization cross section of a donor
 : thermal generation rate (thermal ionization)
 : recombination coefficient
j : electrical current density
Nonlinear Optics Lab.
Hanyang Univ.
Electrical current density :
j  n e e m E  eD  n e  j ph
(10.5.3)
where, m : electron mobility
D : Diffusion constant
jph : photovoltaic contribution to the current
Local field within the crystal :

 dc   E   4 e ( n e  N A  N D )
(10.5.4)
Change in dielectric constant :
     eff |E |
(10.5.5)
Wave equation for the optical field :
1 
~
~
 E opt  2 2 (     ) E opt  0
c t
2
2
(10.5.6)
: Cannot easily be solved exactly
Nonlinear Optics Lab.
Hanyang Univ.
10.6 Two-Beam Coupling in Photorefractive Materials
Optical field within the crystal :
~
i k r
i k r
 i t
E opt ( r , t ) [ Ap e p  As e s ]e
 c.c .
Intensity distribution of light within the crystal :
I
n0c ~ 2
iqx
E opt  I 0  ( I 1 e  c .c .)
4
(10.6.2)
where, I 0 
n0c
I1 
n0c
2
(| A p |  | A s | )
2
2
( A p A s )( eˆ p eˆ s )
*
2
q  q xˆ  k p  k s : grating wave vector
Nonlinear Optics Lab.
Hanyang Univ.
Intensity distribution of light within the crystal can also be described by
I  I 0 [1 m cos ( qx  )]
where, m  2| I 1 |/ I 0 : modulation index
  tan
1
(Im I 1 / Re I 1 )
Approximate steady-state solution (|I1|<<I0)
Put, E  E 0  ( E 1 e iqx  c .c .)
n e  n e 0  ( n e1 e
iqx
j  j 0  ( j1 e

 c .c .)
iqx
 c .c .)


N D  N D 0  ( N D 1e
iqx
 c .c .)
(10.5.1)~(10.5.6)  (Assume E1, j1, ne1, ND1 are small that the product of any of them can be neglect)
1) From x independent term,


( sI 0   )( N D  N D 0 )   n e 0 N D 0
0
Report
j 0  n e 0 e m E 0  j ph .0
j 0  constant
N

D0
 ne0  N A
Nonlinear Optics Lab.
(10.6.5)
Hanyang Univ.
19
In most realistic case, N D (~ 10 cm
3
3
)  N A (~ 10 cm
16
)  n e 0 (~ 10 cm
13
3
)

N D 1  n e1
and

 N D0  N A
( sI 0   )( N D  N A )
0
ne0 
 NA
2) From eiqx dependent term (assume E0=0),


j1  0
 n e 0 eE 1  iqk B Tn e1
iq  dc E 1   4 e ( n e1  N D 1 )
0
Report

sI 1 ( N D  N D 0 )  ( sI 0   ) N D 1   n e 0 N D 1   n e1 N A
eD  k B T m
 sI 1
 E 1   i 
 sI 0  

: Einstein relation

ED

  1 E / E
D
q





where, E D 
Eq 
qk B T
e
4 e
 dc q
: diffusion field strength
N eff : maximum space charge field
N eff  N A ( N D  N A )/ N D
0
Nonlinear Optics Lab.
0
Hanyang Univ.
 sI 1
E 1   i 
 sI 0  

ED

  1 E / E
D
q





(10.6.8)
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
Defining the optimum value of q maximizing the second factor as qopt,
 sI 1
E 1   i 
 sI 0  
2 ( q / q opt )

 E opt
2

1 ( q / q opt )

q  2 n ( / c ) sin q
where,
 4 N eff e 2
q opt  
 k T
 B dc

,


  N eff k B T
E opt  
 dc

can be adjusted




1/ 2
Nonlinear Optics Lab.
Hanyang Univ.
Spatial growth rate
1) Steady state
(10.6.2) and (10.6.8) 
*

A p As
E1   i 
 | A |2 | A |2
p
 s

E
 m

where, E m 
Nonlinear polarization : P NL  

 4
e
iq r
ED
1 E D / E q

ik r
ik r
 c .c .  ( A s e s  A p e p )

Dielectric constant change :       eff E 1
2
 Ps
P
NL
NL
p


 *
4

4
 i   eff E m
| A p | As
4
| A p |  | As |
2
Ape
ik s r

As e

2
i   eff E m
| As | A p
4
| A p |  | As |
2
ik p r
2
e
2
ik s r
(10.6.16)
2
2
e
2
ik p r
Nonlinear Optics Lab.
Hanyang Univ.
Wave equation (slow varying approx.) :
2 ik
dA s
e
ik s r
  4
dz s

dA s
dz s


dI s
dz s
Similarly,
dI
p
dz p
c
2
n  eff E m
3
2c

2
Ps
NL
2

Is

nc
2
| As |
| A p | | As |
2
2
2
IsI p
IsI p

| A p | As
where,  

c
n  eff E m
3
IsI p
IsI p
: when >0, Is is amplified and Ip is attenuated
Nonlinear Optics Lab.
Hanyang Univ.
2) Transient two-beam coupling
Assume, n e  N D , N D  N D0 ,   sI 0

 E1
t
*
 E 1   iE m
A p As
| A p |  | As |
2
2
where,   D
1 E D / E M
1 E D / E q
D
 dc
4 e m n e 0
EM 
N A
qm
Wave equations :
  A p  i

 eff A s E 1

 x p 2 n p c

  As   i  A E *
  x 2 n c eff p 1
s
 s
Nonlinear Optics Lab.
Hanyang Univ.