pattern formation induced by modulation instability in nonlinear optical system ming-feng shih(...
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Pattern Formation Induced by Modulation Instability in Nonlinear
Optical System
Ming-Feng Shih(石明豐 )
Department of Physics
National Taiwan University
Alan Turing recognized that the formation of organized structures can arise from the interplay between reaction and diffusion. Such structures are universal and form in a variety of different systems.
1. Introduction
Nonlinear Optical Medium2 2
22 2 2 2
0
1 1E PE
c t c t
( , ) ( , ) ( , )L NLP r t P r t P r t 0NL NLP E
L NL 2
0 2 | |n n n E
2
2
0
0
n self-focusing
n self-defocusing
Phenomenon well known: Modulation instability
Light
The spatial period of the pattern is decided bythe balance between diffraction and self-focusing.
Diffraction
2| |u
n
Self-focusing
22 2
0
(| | )2 2
A n Aik A k A
z n
22 2
0
(| | )2 2
A n Aik A k A
z n
ikz)tAexp(iΕ 0An
Ank2
z
Aik2A
0
222
)|(|
with
)exp()( ziaAA 0 21 iaaa 0
1,2 1,2 1 2Re{ exp( [ ] )}a a ik r i h ih z
with
2 4 2 2 4 2 1/ 2 1/ 22 0
0
( 2 ( 4 4 ) )1
h kA P P P P Pn 2
)/2/( 00 kAnkP
,max
2
k
,maxk
No threshold
Phenomena to be explored: How about incoherent light?
(First proposed by M. Soljacic et al., PRL v.84, 467(2000))
Why?
1. Incoherent optical soliton. (1997)
2. Closely related to the BEC around the critical temperature.
Incoherent light different :
“fast” phase variation,
or phase cannot be defined “exactly”
which sets the material requirement.
2. Modulation instability with incoherent light
Correlation length
,
,
,
,
0
0
c coherent
c pure incoherent
c partially incoherent
“Slow” materials
Coherent Density Approach:
| |2jjI E
at | | ( )2
jE G I
Assuming partially incoherent light consists of many coherent but mutually incoherent light fields, each field propagates with angle with respective to z -direction.
Nonlinearity n is a function of I
+ +.......
G()
Threshold coherence exists for a fixed nonlinear strength.
D. Kip, et al, Science 290, 495 (2000)
Degree of coherence: Low High
The result is reasonable:
Incoherent light diffracts more than coherent light.
Therefore it requires higher nonlinearity to form MI pattern.
Threshold coherence
• Vortex light beam carrying orbital angular momentum
l=0 l=1
Plane of constant phase
Intensity
In self-focusing media, vortex ring is unstable due to azimuthal instability
V. Tikhonenko, et al, J. Opt. Soc. Am. B 12, 2046 (1995); Phys. Rev. Lett. 76, 2698 (1996).
D.V. Skryabin and W. J. Firth, Phys. Rev. Lett. 79, 2450 (1997); Phys. Rev. E 58, 3916 (1998).
Single-charge : l = 1
Double-charge : l = 2
Experimental Observationsa. Single-charge vortices : l = 1
Coherent
Partially incoherent
Speckle pattern Input Output(2.5kV)
coherence: High Low
Simulation
Experiment
PRL v. 92, 043904 (2004)
Interaction between Optical Spatial Solitons
In-phase
out-of-phase
Meng et al. OPTICS LETTERS / Vol. 22, No. 7 / April 1, 1997
Input Diffraction indivisual in-phase
out-of-phase
Coherent Density Approach:
( ) exp[ / ] /( )2 2
0 0G 2 2
Assuming partially incoherent light consists of many coherent but mutually incoherent light fields, each field propagates with angle with respective to z -direction.
Nonlinearity n is a function of I
[sech(( ) / )]s 0 0E E x d W
+ +.......
G()
| |2jjI E
at | | ( )2
jE G I
[sech(( ) / ) exp( )sech(( ) / )]s 0 0 0E E x d W i x d W Soliton interaction is controlled by the coherence
Note: two beams as a whole are made partially incoherent, but the relative phase between the two parts is fixed.
In Phase
Threshold at 0.0028
out-of-phase
Threshold at 0=0.0022
Why the coherence affects the interaction?We use the in-phase interaction as an example to illustrate.
+ + + .......
G()
Larger separation, 2d, means smaller threshold value of 0 !
smaller
larger
0,thθ 1/d
d
d=10 d=12th=0.0022 0,th=0.0018
0.0022 x 10 / 12 = 0.183
out-of-phase
Similar for the in-phase interaction.
In-phase
Coherent
Partially
incoherent
Partiallyincoherent
less
coh
eren
t
m
ore
cohe
rent
out-of-phase
Highlighted by Optics in 2005 by Optics and Photonics News (OSA magazine)
PRL v.94, 063904(2005)
0 sec 10 sec 20 sec 30 sec 40 sec
293 mμ 572 mW/cm2
0.75 kV
3. coherent MI with time-varying noise
Time=t1 t2 t3 ........
In instantaneous nonlinear self-focusing media
Time=t1 t2 t3 ........
In noninstantaneous nonlinear self-focusing media
ikz)tAexp(iΕ 0An
Ank2
z
Aik2A
0
222
)|(|
with
)exp()( ziaAA 0 21 iaaa )}][exp(Re{ ,, zihhirkitiaa 21
02121
with
assuming nonlinearity of relaxation type:
For a wave
21212
242
2
242
00
2 Q1
P44PP
Q1
P2PkA
nh // ))((
2
1
)/2/( 00 kAnkP and Q
We have
22( 1) | |n n A
t
1. When (noise is static), no difference for instantaneous and noninstantaneous media.
2 If is large enough, h2 is leveled off and contains no peak.
Increasing the material response time can arrest the MI:
PRL v88, 133902, Apr. 2002
0.7 kV 5mm
(a)
(b)
572 mW/cm2
143 mW/cm2
57.2 mW/cm2
371.8 mW/cm2
0.9 kV 5mm
57.2 mW/cm2
286 mW/cm2
143 mW/cm2
572 mW/cm2
smalllarge τ
293 mμ
293 mμ
5 7 2 m W / c m 22 8 . 6 m W / c m 2 5 7 . 2 m W / c m 2 2 8 6 m W / c m 21 4 . 3 m W / c m 28 5 . 8 m W / c m 2 1 4 3 m W / c m 2
M o d u la tio n In sta b ility O p tic a l Tu rb u le n c e
optical intensitysmall large
In self-defocusing medium, MI cannot happen.
Pro
paga
tion
However, MI still can happen for moving pattern.
Output face of the crystal: self-defocusing nonlinearity is on
Input faceof the crystal
L
1slope= /k h
Feedback
0% 9% 16% 28%
With feedback percentage equal to
)}][exp(Re{ ,, zihhirkitiaa 210
2121
And of course, when the MI forms, it moves with time.
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