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.