Download 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 :
~
E(r,t )A1ei (k1r1t ) A2ei (k 2r2t ) c.c. ki n0i c
I
moving
grating
n0 c ~ 2
E
4





n0 c
A1A1*  A 2 A*2  A1A*2ei (k1 k 2 )r i (1 2 )t )  c.c
2
nc
q  k 1  k 2 : grating wave vector
 0 A1A1*  A 2 A*2  A1A*2 ei (qr t )  c.c
where,
2
  1  2 : frequency difference
I

Nonlinear Optics Lab.
Hanyang Univ.
Special case (q=180 degree)
q2k 2
I


n0 c
A1A1* A 2 A*2  A1A*2 ei ( 2 k z t ) c.c
2

 0 
 0 
Phase velocity : v| |/2k
Nonlinear Optics Lab.
Hanyang Univ.
Theoretical treatment
Nonlinear refractive index considering the dynamic response (Debye relaxation equation) :

dnNL
 nNL  n2 I
dt
Solution :
nNL  
n2
t


I(t )e (t t )  dt 
Ex) I(t ')e
 it 

t
e

it  ( t t )
e
dt e
t 
t
e

(  i 1) t 
e it
dt 
i 1 
n0 n2c
A1A*2ei (qrt ) A1*A 2e i (qrt )
*
*
 nNL 
A1A1 A 2 A 2 

2
1i
1i


Wave equation :
2
2~
n

E
~
2E  2 2  0
c t
nNL n0
where, n  n0  nNL
and
n 2 n02 2n0 nNL
Nonlinear Optics Lab.
Hanyang Univ.


2
n02 2
n02 n2 22
n02 n212 A1 A 2
d 2A2
dA 2 2
2
2

2ik 2
 k 2 A 2  2 A 2 
A1  A 2 A 2 
dz 2
dz
c
c
c 1i
stationary index
time-varying index


2
n n
n n  A1 A 2
dA 2
2
2
 i 0 2 A1  A 2 A 2  i 0 2
dz
2
2 1  i
d I 2 n0 c  * dA 2
dA*2 


  A 2
A 2
dz 2 
dz
dz 
Ii 

where,  1 2
2n2 
II
2 2 1 2
c 1 
: when >0 (1<1) I2 increases with z
n0 c
A i A*i
2
Maximum gain ;
dI 2

 n2 I1I 2
dz
c
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
~
~
Optical pulse : E
( z, t )  A( z, t )ei ( k0 z 0t )  c.c.
Refractive index of 3rd order nonlinear medium : n(t )n0 n2 I (t ), I (t )
2
n0 c ~
A( z,t )
2
Phase change by nonlinear refractive index :
NL (t )n2 I (t )0 L c
Frequency change :
d
dt
(t )  NL (t )  
n2 0 L dI (t )
c
dt
Nonlinear Optics Lab.
Hanyang Univ.
Example
Pulse shape :
I (t ) I 0 sech 2 (t  0 )
Nonlinear phase shift :
 NL (t )n2  0 c LI 0 sech 2 (t  0 )
Frequency shift :
d
(t )  NL (t )
dt
2n2  0 c 0 LI 0 sech 2 (t  0 )tan h (t  0 )
# Maximum frequency shift :
max 
max
 NL
0
max
,  NL
 n2
0
c
I0L
: Whenever max exceeds the spectral width of the
max
 2 ,
incident pulse (~2/0), that is  NL
the spectral broadening due to self-phase
modulation will be important.
Nonlinear Optics Lab.
Hanyang Univ.
Pulse Propagation Equation
Optical pulse :
~
~
E( z, t )  A( z, t )ei ( k0 z 0t )  c.c.
where, k0  nlin (0 ) 0 c
Wave equation :
~
~
 2E 1  2D
 2 2  0 (6.5.11)
2
z
c t
~
~
Let’s introduce Fourier transform of E( z, t ) and D( z, t ) ;

d
~
E( z,t )   E( z, )e it

2

d
~
D( z, )   ( )E( z, )
D( z,t )   D( z, )e it

2
(6.5.11) 
 2 E(z,  )
2
  ( ) 2 E(z,  )  0
2
z
c
(6.5.14)
Nonlinear Optics Lab.
Hanyang Univ.
Fourier transform of amplitude is given by
 ~
A( z,)   A( z,t )eit dt

The amplitude is related with the Fourier amplitude as
E( z, )A( z, 0 )eik0 z A* ( z, 0 )e ik0 z
A( z, 0 )eik0 z
(6.5.14), slow varying approximation 
2ik 0
A
[k 2 ( )k02 ]A0
z
where, k ( )  ( ) c
k() ~ k0  k 2  k02  2k0 (k  k0 )
A( z,ω,ω 0 )
 i (k  k0 )A( z,ω,ω 0 )  0
z
(6.5.19)
Nonlinear Optics Lab.
Hanyang Univ.
Power series expansion of k() :
1
k k0 k NL k1 (  0 ) k 2 (  0 ) 2
2
where, k NL nNL
0
c
 dk 
k1  

 d  0
n2 I
0
(6.5.20)
2
~
, I nlin ( 0 )c 2 A( z ,t )
c
dn ( ) 
1
1
 nlin ( )   lin

c
d   0 vg (0 )
 1 dvg 
 d 2k 
d  1 



k2   2 

  2




 d  0 d  vg ( )   0
 v g d  0
(6.5.19) and (6.5.20) 
A
1
 ik NL A  ik1 (ω-ω0 )A  ik 2 (ω-ω0 ) 2 A  0
z
2
~
~
~
A A 1  2 A
~

k1  ik 2 2 ik NL A0 (6.5.26)
z
t 2 t
Nonlinear Optics Lab.
AA(z, )
~ ~
AA(z, t )
Hanyang Univ.
The equation can be simplified by means of a coordinate transformation ;
 t
z
 t  k1 z : retarded time
vg
~
~
~
~
~
A s
A A s A s τ A s



 k1
z
z
τ t
z
τ
~
~
~
~
A A s z A s τ A s



t
z t
τ t
τ
~
~
 2A  2A s

t 2
τ 2
~ ~
As As (z, )
~
~
A s 1
 2A s
~
 ik 2

i

k
A
(6.5.26) 
NL
s 0
z 2
 2
0
n0 n20 ~ 2
~
I
As   As
If we express the nonlinear contribution to the propagation constant as k NL  n2
c
2
~
~
A s 1
 2A s
~ 2~
 ik 2
 i A s A s
2
z 2

group velocity dispersion
: nonlinear schrodinger equation
self-phase modulation
Nonlinear Optics Lab.
Hanyang Univ.
2
Optical Solitons
~
~
A s 1
 2A s
~ 2~
 ik 2

i

A
s As
2
z 2

~
As an example, a pulse whose amplitude is expressed by A s ( z,t )A0s sech(  0 )ei z
k2
k 2 c
0 2
 k2 and  n2 must have opposite sign
If A s  2 
2
 0 2n2 0
Report and
 k 2 2 02 , 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  4
dx 
dE 4

dx 
# 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
1
n n 3 reff E (reff 0)
2
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
( sI   )( N D0  N D ) ne N D
t
ne N D 1

 ( j )
t
t e
(10.5.1)
(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 ne emE eDne  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(ne  N A  N D )
(10.5.4)
Change in dielectric constant :
  eff |E|
(10.5.5)
Wave equation for the optical field :
1 2
~
~
 Eopt  2 2 (  ) Eopt 0
c t
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 :
~
ik r
Eopt (r,t )[ Ap e p  As eik s r ]e it c.c.
Intensity distribution of light within the crystal :
I
n0 c ~ 2
E opt  I 0 ( I1eiqx c.c.)
4
(10.6.2)
n0 c
(| Ap |2 | As |2 )
2
nc
I1  0 ( Ap As* )(eˆ p eˆs )
2
q  qxˆ  k p k s : grating wave vector
where, I 0 
Nonlinear Optics Lab.
Hanyang Univ.
Intensity distribution of light within the crystal can also be described by
I I 0 [1mcos(qx )]
where, m2|I1|/ I 0 : modulation index
 tan 1 (ImI1 /Re I1 )
Approximate steady-state solution (|I1|<<I0)
Put, E E0 ( E1eiqx c.c.)
ne ne0 (ne1eiqx c.c.)
j  j0 ( j1eiqx c.c.)
N D  N D 0 ( N D1eiqx 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 D0  N D 0 ) ne 0 N D 0
Report
j0 ne 0 emE0  j ph.0
j0 constant

D0
N ne 0  N A
Nonlinear Optics Lab.
(10.6.5)
Hanyang Univ.
19
3
16
3
13
3
In most realistic case, N D (~10 cm )  N A (~10 cm )  ne 0 (~10 cm )
and N D1 ne1
 N D 0  N A
( sI 0   )( N D0  N A )
ne 0 
 NA
2) From eiqx dependent term (assume E0=0),
Report
sI1 ( N D0  N D 0 )( sI 0   ) N D1  ne0 N D1  ne1 N A
j1 0
ne0eE1 iqk BTne1
iq dc E1 4e(ne1  N D1 )
eDk BTm : Einstein relation
 sI1  ED 


 E1 i


 sI 0    1 ED / Eq 
qk BT
: diffusion field strength
e
4e
Eq 
N eff : maximum space charge field
 dc q
Neff N A ( N D0 N A )/ N D0
where, ED 
Nonlinear Optics Lab.
Hanyang Univ.
 sI1  ED 


E1 i


 sI 0    1 ED / Eq 
(10.6.8)
i) Quarter period shift of the index grating with respective to the intensity distribution
E1  sI1 /( sI 0   )  I1
iii) E1  fn( E D and Eq ) : depends also on grating vector q
ii)
Defining the optimum value of q maximizing the second factor as qopt,
2(q/qopt )
 sI1 
 Eopt
E1 i
2
 sI 0    1(q/qopt )
where,
 4N eff e 2 
,
qopt 
 k T 
 B dc 
can be adjusted
q2n( /c)sinq
 N eff k BT 

Eopt 

dc


1/ 2
Nonlinear Optics Lab.
Hanyang Univ.
Spatial growth rate
1) Steady state
(10.6.2) and (10.6.8) 
 Ap As* 
E
E1 i 2
 | A | | A |2  m
p 
 s
ED
1 ED / Eq
where, Em 
 iqr

ik r
e c.c.( As e iks r  Ap e p )
 4

Nonlinear polarization : P NL 
Dielectric constant change :
  2 eff E1
2
2

i


E
|
A
|
As iks r


*
eff
m
p
iks r
NL
 Ps 
Ap e 
e
2
2
4
4
| Ap | | As |

ik p r i  eff Em | As | Ap
ik p r
P  As e 
e
4
4 | Ap |2 | As |2
2
2
(10.6.16)
NL
p
Nonlinear Optics Lab.
Hanyang Univ.
Wave equation (slow varying approx.) :
dAs iks r
 2 NL
2ik
e 4 2 Ps
dz s
c
| Ap |2 As
dAs  3

 n  eff Em
dzs 2c
| Ap |2 | As |2
Is 

Similarly,
nc 2
| As |
2
IsI p
dI s

dz s I s  I p
dI p
dz p


where,  n 3 eff Em
c
IsI p
I s I p
: when >0, Is is amplified and Ip is attenuated
Nonlinear Optics Lab.
Hanyang Univ.
2) Transient two-beam coupling
Assume, ne  N D , N D  N D0 ,  sI 0
Ap As*
E1

 E1 iEm
t
| Ap |2 | As |2
where,
  D
1 ED / EM
1 ED / Eq
D
 dc
4emne 0
EM 
N A
qm
Wave equations :
 Ap i

 eff As E1

 x p 2n p c

 As  i  A E *
 xs 2ns c eff p 1
Nonlinear Optics Lab.
Hanyang Univ.