Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih(...
-
date post
22-Dec-2015 -
Category
Documents
-
view
222 -
download
4
Transcript of Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih(...
![Page 1: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/1.jpg)
Pattern Formation Induced by Modulation Instability in Nonlinear
Optical System
Ming-Feng Shih(石明豐 )
Department of Physics
National Taiwan University
![Page 2: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/2.jpg)
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
![Page 3: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/3.jpg)
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
![Page 4: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/4.jpg)
Phenomenon well known: Modulation instability
Light
![Page 5: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/5.jpg)
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
![Page 6: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/6.jpg)
22 2
0
(| | )2 2
A n Aik A k A
z n
![Page 7: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/7.jpg)
![Page 8: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/8.jpg)
![Page 9: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/9.jpg)
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
![Page 10: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/10.jpg)
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
![Page 11: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/11.jpg)
“Slow” materials
![Page 12: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/12.jpg)
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()
![Page 13: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/13.jpg)
![Page 14: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/14.jpg)
Threshold coherence exists for a fixed nonlinear strength.
![Page 15: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/15.jpg)
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
![Page 16: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/16.jpg)
• Vortex light beam carrying orbital angular momentum
l=0 l=1
Plane of constant phase
Intensity
![Page 17: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/17.jpg)
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
![Page 18: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/18.jpg)
Experimental Observationsa. Single-charge vortices : l = 1
Coherent
Partially incoherent
Speckle pattern Input Output(2.5kV)
![Page 19: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/19.jpg)
coherence: High Low
Simulation
Experiment
PRL v. 92, 043904 (2004)
![Page 20: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/20.jpg)
Interaction between Optical Spatial Solitons
In-phase
out-of-phase
![Page 21: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/21.jpg)
Meng et al. OPTICS LETTERS / Vol. 22, No. 7 / April 1, 1997
Input Diffraction indivisual in-phase
out-of-phase
![Page 22: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/22.jpg)
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
![Page 23: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/23.jpg)
[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.
![Page 24: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/24.jpg)
In Phase
Threshold at 0.0028
![Page 25: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/25.jpg)
out-of-phase
Threshold at 0=0.0022
![Page 26: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/26.jpg)
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
![Page 27: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/27.jpg)
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.
![Page 28: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/28.jpg)
In-phase
Coherent
Partially
incoherent
![Page 29: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/29.jpg)
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)
![Page 30: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/30.jpg)
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
![Page 31: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/31.jpg)
Time=t1 t2 t3 ........
In instantaneous nonlinear self-focusing media
![Page 32: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/32.jpg)
Time=t1 t2 t3 ........
In noninstantaneous nonlinear self-focusing media
![Page 33: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/33.jpg)
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
![Page 34: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/34.jpg)
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.
![Page 35: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/35.jpg)
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
![Page 36: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/36.jpg)
![Page 37: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/37.jpg)
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
![Page 38: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/38.jpg)
![Page 39: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/39.jpg)
In self-defocusing medium, MI cannot happen.
Pro
paga
tion
However, MI still can happen for moving pattern.
![Page 40: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/40.jpg)
Output face of the crystal: self-defocusing nonlinearity is on
Input faceof the crystal
![Page 41: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/41.jpg)
![Page 42: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/42.jpg)
L
1slope= /k h
Feedback
![Page 43: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/43.jpg)
0% 9% 16% 28%
With feedback percentage equal to
![Page 44: Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d7f5503460f94a62b0c/html5/thumbnails/44.jpg)
)}][exp(Re{ ,, zihhirkitiaa 210
2121
And of course, when the MI forms, it moves with time.