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```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.
6.1 Optical Phase Conjugation
: Generation of a time-reversed wavefront
~
Signal wave : Es (r,t )Es ei t  c.c.
~
Phase conjugate wave : Ec (r,t )rE*sei t  c.c.
where, r : amplitude reflection coefficient
E s eˆs A s e ik s r
 * * * ik s r
E s eˆs A s e
Nonlinear Optics Lab.
Hanyang Univ.
Properties of phase conjugate wave : E*s eˆs*A*se ik s r
1)
eˆs*
: The polarization state of circular polarized light does not change in reflection from PCM
Ex)
ês e0 i e0 ei/2 j
i) In reflection from metallic mirror [-phase shift]
ii) In reflection from PCM [-phase shift &
y component : i/2  -i/2 (- : delayed)]
2)
A *s
: The wavefront is reversed
A s as ei ( rt )  A*s as e i ( rt )
3)
e ik s r : The incident wave is reflected back into its direction of incidence
k s  k s
Nonlinear Optics Lab.
Hanyang Univ.
Aberration correction by optical phase conjugation
Wave equation :
~
~  (r )  2 E
 E 2
0
2
c t
2
~
Solution : E (r,t ) A(r ) ei ( kz t )  c.c.
 2 (r ) 2 
A
With slow varying approximation,  A 2 k  A2ik 0
z
 c

2
T
Since the equation is generally valid, so is its complex conjugate, which is given explicitly by
 2 (r ) 2  *
A*
 A  2 k  A 2ik
0
z
 c

2
T
*
~
Solution : Ec (r,t ) A* (r) ei (  kzt )  c.c.
: A wave propagating in the –z direction whose complex amplitude
is everywhere the complex of the forward-going wave
Nonlinear Optics Lab.
Hanyang Univ.
Nonlinear Optics Lab.
Hanyang Univ.
Phase Conjugation by Degenerated Four-Wave Mixing
New wave (k4) source term !
1) Qualitative understanding
~
Four interacting waves : Ei (r,t )Ei (r) eit  c.c. Ai (r) ei ( ki rt )  c.c. (i 1,2,3,4)
Nonlinear polarization :
P NL 6 c (3) E1 E2 E3* 6 c (3) A1 A2 A3*ei ( k1 k2 k3 )r 6 c (3) A1 A2 A3*e ik3r
k1 k2 0 ( Counter-propagating waves)
So, A4 is proportional to A3* (complex conjugate of A3)
and its propagation direction is –k3(in the case of perfect phase matching)
Nonlinear Optics Lab.
Hanyang Univ.
2) Rigorous treatment
Total field amplitude : E E1  E2  E3  E4
Nonlinear polarization : P NL 3c (3) (    ) E 2 E *
 P1 3c (3)
P2 3c (3)
P3 3c (3)
P4 3c (3)
E E 2E E E 2E E E 2E E E 2E E E 
E E 2E E E 2E E E 2E E E 2E E E 
E E 2E E E 2E E E 2E E E 2E E E 
E E 2E E E 2E E E 2E E E 2E E E 
2
1
2
2
2
3
2
4
*
1
*
2
*
3
*
4
1
2
3
4
*
2
*
1 1
*
1 1
*
1 1
2
1
3
2
3
3
2
4
2
*
3
*
3
*
2
*
2
1
2
3
4
*
4
*
4 4
*
4 4
*
3 3
4
*
2
*
3 4 1
*
1 2 4
*
1 2 3
3
4
E1 , E2 E3 , E4  Neglect the 2nd order terms of E3 and E4


E E 2E E E 
2E E E 2E E E 2E E E 
2E E E 2E E E 2E E E 
P1 3c (3) E12 E1* 2E1E2 E2*
P2 3c (3)
P3 3c (3)
P4 3c (3)
2
2
*
2
3
4
2
1
*
1
1
*
1
*
1
1
3
4
2
*
2
2
*
2
1
1
2
*
4
2
*
3
Nonlinear Optics Lab.
Hanyang Univ.
Wave equation :
~
  2 Ei 4  2 ~
2~
 Ei  2 2  2 2 Pi
c t
c t
~
where, Ei (r,t )Ei (r) eit  c.c. Ai (r) ei ( ki rt )  c.c.
(1) Pump waves, A1 and A2 (slow varying approximation)

dA1 6i (3)

c | A1|2 2| A2 |2 A1 i 1 A1
dz
nc
dA2 6i (3)

c | A2 |2 2| A1|2 A2 i 2 A2
dz
nc




# Each wave shifts the phase of the other wave
by twice as much as it shift its own phase
# Since only the phase of the pump waves are
affected by nonlinear coupling, the quantities
|A1|2 and |A2|2 are spatially invariant, and hence
the k1 and k2 are in fact constant
Solution :
A2 ( z ) A2 (0)ei 2 z
A1 ( z ) A1 (0)ei1z
(6.1.15)
P4  A1 A2 A3*  A1 (0) A2 (0)ei (1  2 ) z A3* : Nonlinear polarization responsible for producing the phase
conjugate wave varies spatially.
Therefore, two pump beams should have equal intensities :
A1 (0) A2 (0) 1  2 A1 ( z) A2 ( z) A1 (0) A2 (0)
Nonlinear Optics Lab.
Hanyang Univ.
(2) Signal (A3) and conjugate waves (A4)
dA3 12i (3)

c (| A1|2 2| A2 |2 ) A3  A1 A2 A4* i 3 A3 iA4*
dz
nc
dA4 12i (3)

c (| A1|2 2| A2 |2 ) A4  A1 A2 A3* i 3 A4 iA3*
dz
nc
12i (3)
where,  3 
c (| A1|2 2| A2 |2 )
nc
12i (3)

c A1 A2
nc
put, A3  A3' ei 3 z
'
A4  A4' ei 4 z  dA3 iA'
4
dAi'
dz
2 '


|

|
Ai 0 (i 3,4)
2
'
dz
dA4
iA3'
dz




Nonlinear Optics Lab.
Hanyang Univ.
Solution :
i| |sin | |z '
cos[| |( z  L)] '*
A4 ( L)
A3 (0)
 cos| |L
cos| |L
cos| |z '
i sin[|  |( z  L)] '*
A4' ( z )
A4 ( L)
A3 (0)
cos| |L
 | cos| |L
A3'* ( z )
A4' ( L)0 (  conjugate wave at z=L is zero)
A3'* ( L) A3'* (0)/cos| |L
A4' (0)
i
(tan| |L) A3'* (0)
| |
i) A3'* ( L) A3'* (0)
: amplification
ii) A4' (0)0~
: depends on | |L (can exceed 100%  pump wave energy)
Nonlinear Optics Lab.
Hanyang Univ.
Processes of degenerated four-wave mixing
: One photon from each of the pump waves is annihilated
and one photon is added to each of the signal and conjugate waves
 Amplification of A3 and over 100% conversion of A4/A3 are possible
one photon transition
two photon transition
wave-vectors
Nonlinear Optics Lab.
Hanyang Univ.
Experimental set-ups
(k1 k 2 )0
(k 3 k 4 )0
A3  A4
A2
A1
A4
A3
Nonlinear Optics Lab.
Hanyang Univ.
17.5 Optical Resonator with Phase Conjugate Reflectors (A. Yariv)
R
l (m,n)

2  1  R
3  2  l (m, n)  1  R  l (m, n)
4  3    (1  R  l (m, n))  
5  4  l (m, n)  1  R  
6  5  R  1  
7  6  l (m, n)  1    l (m, n)
8  7    1  l (m, n)
9  8  l (m, n)  1
# The self-consistence condition is satisfied
automatically every two round trips.
 The phase conjugate resonator is stable
regardless of the radius of curvature R
of the mirror and the spacing l.
Nonlinear Optics Lab.
Hanyang Univ.
17.6 The ABCD Formalism of Phase Conjugate Optical Resonator
The wave incident upon the PCM :


kr2 r 2 
kr2 
Ei εi (r)exp i(t kz ) 2 εi (r)exp i(t kz )
2 w 
2qi 


where,
1 1 i
 - 2
qi   w
Reflected conjugate wave :


kr2 r 2  *
kr2 
E r ε (r)exp i(t kz
) 2 εi (r)exp i(t kz
)
2

w
2
q


r 

*
i

1
1 i
1
  2  *
qr  w
qi
By comparing the ABCD law for ordinary optical elements,

Aqi* B
qr  * ,
Cqi D
 A B 1 0 


M
 C D   0 -1
Ray transfer matrix for the PCM mirror
Nonlinear Optics Lab.
Hanyang Univ.
ABCD law at any plane following the PCM :
qout
A T qi*  BT

CT qi*  DT
Example)
Matrix after one round trip :
A
M1   1
 C1
0  A B  1 0  A B 
B1   1

   - 2



D1   R 1  C D  0  1 C D 
0  1 0 
 1


 - 2

1  0  1
 R

Matrix after two round trip :
M 2 M1 
2
0  1 0  1
0  1 0   1 0 
 1







I
 -2

1  0 1 -2
1  0 1  0 1 
 R

 R

: Self-consistence condition is satisfied
automatically every two round trips
Nonlinear Optics Lab.
Hanyang Univ.
17.7 Dynamic Distortion Correction within a Laser Resonator
Phase conjugate resonator
Distortion corrected beam
Nonlinear Optics Lab.
Hanyang Univ.
17.8 Holographic Analogs of Phase Conjugate Optics
1) Holography
recording
Nonlinear Optics Lab.
Hanyang Univ.
2) Phase conjugate optics
Holography by phase conjugation
A3  A4
- Real time processing (no developing process)
- Distortion free image transmission
Nonlinear Optics Lab.
Hanyang Univ.
17.9 Imaging through a Distorted Medium
Distortion free
transmission (A2)
Nonlinear Optics Lab.
Hanyang Univ.
6.2 Self-Focusing of Light
nn0 n2 I
Gaussian beam : I(r )  e r
2
w2
n2 0 : self -focusing

n2 0 : self -defocusing
Nonlinear Optics Lab.
Hanyang Univ.
Self-Trapping
: Beam spread due to diffraction is precisely compensated by the contraction due to self-focusing
Simple model for self-trapping
Critical angle for total internal reflection : cos  0 
 0 0
1
n
 1  02 1
2
n0
  0  2n 
n0 

Nonlinear Optics Lab.
1
2
Hanyang Univ.
n0
n0  n
A laser beam of diameter d will contain rays within a cone whose maximum angular extent
is of the order of magnitude of diffraction angle ;
2

d 
 0.6
n0 d
n0 d
So, the condition for self-trapping :
d  0
1
2
1  0.6 

  2n  0.6
 n n0 
n0 

n0 d
2  dn0 

nn2 I  d 0.6 2n0 n2 I 
1
2
2
Critical laser power :
Pcr 

4
d 2I 
 0.6 2
8n0 n2
Ex) CS2, n2=2.6x10-14 cm2/W, n0=1.7, =1mm
 Pcr = 33 kW
# Independent of the beam diameter
Nonlinear Optics Lab.
Hanyang Univ.
Simple model of self-focusing
1
n0  n
2
n0 n
2w0
zf
1
1
z f (n0  n)  ( z 2f  w02 ) 2 (n0  n)
2
z f (n0  n)  z f n0 
 z f  w0 

 2
n0
w0
0

n 
w
1
1
z f n  z f n0  0
z
2
2
 f




2
1
2
nn2 I
1
n 
 n 
 z f w0  0  w02  0 
 n2 I 
 2n2 P 
2
1
2
1
2
where, P  w02 I : total power
1
2
where,  0   2 n  : critical angle
n0 

2n0 w02
1
zf 
0.6  P Pcr 12
Nonlinear Optics Lab.
Hanyang Univ.
6.3 Optical Bistability
: Two different output intensities for a given input intensity
 Switch in optical computing and in optical computing
Bistability in a nonlinear medium inside of a Fabry-Perot resonator
A2  A 2e2ikl l
A2  A1  A2
A1
 A2 
1  2e 2ikll
Intensity reflectance and transmittance :
  R,  T
2
where,
2
 ,  : amplitude reflectance and transmittance
Nonlinear Optics Lab.
Hanyang Univ.
(6.3.3)
1) Absorptive Bistability
In the case when only the absorption coefficient depends nonlinearly on the field intensity,
at the resonance condition,
Assume,
A2 
 2e 2ikl R
 l 1
A1
1  R(1  l )
I i nc 2  A i
2

I2 
TI1
1  R(1   l )
2
Introducing the dimensionless parameter C, C 
I2 
1
I1
T (1  C ) 2
R l
1 R
(6.3.7)
Nonlinear Optics Lab.
Hanyang Univ.
Assume the absorption coefficient obeys the relation valid for a two-level saturable absorber ;

0
1  I IS
C0
R l
, where,C0  0
(1 R)
12I 2 I S
Intracavity intensity : I 2 I '2  2I 2  C 
(6.3.7) 

C0 

I1 TI 2 1
 12I 2 I S 
2
I3  TI 2
Nonlinear Optics Lab.
Hanyang Univ.
2) Dispersive Bistability
In the case when only the refractive index depends nonlinearly on the field intensity,
  0, n  f (I)
(6.3.3)  A 2 
A1
A1

1   2 e 2ikl 1  R eiδ
where,  2  R ei
  0  2
 0    2n0

c
l : linear phase shift

 2 2n2 I l : nonlinear phase shift
Similarly as before,
c
T I1
T I1

(1 Reiδ )(1 Re iδ ) (1 R 2 2 Rcos )
I1 T

1 4 R T 2 sin 2  2
I2 


Nonlinear Optics Lab.
Hanyang Univ.
I2
1T

I1 1  4 R T 2 sin 2  2




 
   0   4n2
l  I2
C 
I3  TI 2
Nonlinear Optics Lab.
Hanyang Univ.
```
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