SPATIAL AND TEMPORAL INSTABILITIES IN PASSIVE OPTICAL SYSTEMS

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SPATIAL AND TEMPORAL INSTABILITIES IN PASSIVE OPTICAL SYSTEMS

L. Lugiato, C. Oldano, Kaige Wang, L. Santirana, L. Narducci, F. Prati, M.

Brambilla

To cite this version:

L. Lugiato, C. Oldano, Kaige Wang, L. Santirana, L. Narducci, et al.. SPATIAL AND TEMPORAL

INSTABILITIES IN PASSIVE OPTICAL SYSTEMS. Journal de Physique Colloques, 1988, 49 (C2),

pp.C2-343-C2-348. �10.1051/jphyscol:1988281�. �jpa-00227698�

JOURNAL DE PHYSIQUE

Colloque C2, Suppl6ment au n06, Tome 49, juin 1988

SPATIAL AND TEMPORAL INSTABILITIES IN PASSIVE OPTICAL SYSTEMS

L.A. LUGIATO, C. OLDANO, Kaige W A N G ( ~ ) , L. SANTIRANA, L .M. NARDUCCI* , F. PRATI" * and M. BRAMBILLA* *

Dipartimento di Fisica del Politecnico, Corso Duca degli Abruzzi 24, I-10129 Torino. Italy

'~epartrnent of Physics and Atmospheric Sciences, Drexel University, Philadelphia, PA 19104, U.S.A.

* * Dipartimento di Fisica dell'Universith, Via Celoria 16, I-10133 Mflano, Italy

Abstract We focus on some o f t h e main i n s t a b i l i t i e s which a r i s e i n t h e framework o f t h e f u l l set o f Maxwell-Bloch equations. We review some t h e o r e t i c a l and experimental r e s u l t s which concern t h e spontaneous onset o f s e l f - o s c i l l a t o r y behaviour, o r o f s t a t i o n a r y s p a t i a l patterns, i n passive o p t i c a l systems.

1

- INTRODUCTION

The l a s t decade has witnessed an impressive development i n t h e f i e l d o f i n s t a b i l i t i e s i n passive o p t i c a l systems 11-41. B i s t a b i l i t y i s a l s o a t y p i c a l phenomenon displayed by these systems, b u t i s u s u a l l y n o t necessary f o r t h e r i s e o f i n s t a b i l i t i e s . We focus on t h e i n s t a b i l i t i e s which emerge i n t h e framework o f t h e c l a s s i c s e t o f !.laxwell-Bloch equations f o r a u n i d i r e c t i o n a l r i n g c a v i t y f i l l e d w i t h a homogeneously broadened system o f two-level atoms. We w i l l discuss i n order two classes o f i n s t a b i l i t i e s : temporal i n s t a b i l i t i e s t h a t l e a d t o t h e r i s e o f spontaneous o s c i l l a t i o n s o r p u l s a t i o n s , breaking t h e t i m e t r a n s 1 a t i o n a l symmetry o f t h e s t a t i o n a r y s t a t e , and s p a t i a l i n s t a b i 1 it i e s t h a t l e a d t o t h e onset o f s t e a d y - s t a t e s p a t i a l p a t t e r n s , breaking t h e space t r a n s l a t i o n a l symmetry o f t h e homogeneous s t a t i o n a r y s t a t e .

emane anent

address : Department of Physics, Normal University, Beijing. China

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988281

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2. TEMPORAL INSTABILITIES

When t h e atomic l i n e w i d t h i s much l a r g e r than t h e f r e e s p e c t r a l range one can e l i m i n a t e a d i a b a t i c a l l y t h e atomic p o l a r i z a t i o n , and t h i s step leads t o t h e r a t e equation approximation o f t h e Maxwell -Bloch equations. I keda demonstrated / 5 / t h a t t h e r a t e equations w i t h t h e a p p r o p r i a t e boundary c o n d i t i o n s can be rephrased as a s e t o f d i f f e r e n c e - d i f f e r e n t i a l equations, and t h i s f o r m u l a t i o n was t h e basis o f h i s discovery o f chaos i n passive o p t i c a l systems.

Here we consider t h e i n s t a b i l i t i e s t h a t r e q u i r e t h e f u l l s e t o f Maxwell-Bloch equations, because they a r i s e from t h e coherent dynamics o f t h e a t o m - f i e l d i n t e r a c t i o n . I t i s customary t o d i v i d e these i n s t a b i l i t i e s i n t o singlemode and multimode /Q/. This d i s t i n c t i o n i s based on t h e n o t i o n o f resonant mode, which i s t h e l o n g i t u d i a n l mode o f t h e c a v i t y nearest t o t h e frequency o f t h e i n p u t f i e l d . I n t h e s t a t i o n a r y s t a t e t h e f i e l d o s c i l l a t e s w i t h t h e d r i v i n g i n p u t frequency. I n t h e singlemode i n s t a b i l i t y t h e resonant mode becomes unstable and t h e frequency o f t h e spontaneous o s c i l l a t i o n s i n t h e output i n t e n s i t y i s b a s i c a l l y t h e d i f f e r e n c e

I f c - f o l

between t h e frequency o f t h e resonant mode and t h e i n p u t frequency. I r i t h e multimode i n s t a b i l i t y a sidemode o f t h e resonant mode becomes u n s t a b l e and t h e frequency o f t h e spontaneous osci 11 a t i o n s i s e s s e n t i a l l y t h e d i f f e r e n c e

I f s - f o l

between t h e frequency o f t h e sidemode and t h e i n p u t frequency.

I n order t o ensure singlemode operation, i t i s s u f f i c i e n t t o s a t i s f y two c o n d i t i o n s : 1 ) t h e m i r r o r t r a n s m i s s i v i t y must be small, t o guarantee t h a t t h e l o n g i t u d i n a l modes are narrow and w e l l separated; 2) t h e power-broadened atomic l i n e must n o t cover l o n g i t u d i n a l modes d i f f e r e n t from t h e resonant mode, -to exlude i n s t a b i l i t i e s i n t h e sidemodes. The singlemode i n s t a b i l i t y o f o p t i c a l b i s t a b i l i t y was p r e d i c t e d i n / 7 / from t h e plane wave model and i n /8/ from a model which assumes f o r t h e output f i e l d t h e same Gaussian r a d i a l p r o f i l e o f t h e i n p u t f i e l d . This i n s t a b i l i t y was observed e x p e r i m e n t a l l y by Orozco, Rosenberger and Kimble /9-10/ using a r i n g c a v i t y crossed a t r i g h t angle by t e n atomic beams o f sodium, i n a c o n f i g u r a t i o n t h a t s a t i s f i e s t h e singlenode c o n d i t i o n s . The observed osci 1 ladons are s i n u s o i d a l

,

and t h e i r frequency i s c l o s e t o

Ifc-fol

i n accord w i t h t h e singlemode character o f t h e i n s t a b i l i t y . The Gaussian model t u r n s o u t t o be i n good q u a l i t a t i v e , and most times a l s o q u a n t i t a t i v e agreement w i t h t h e experimental d a t a concerning t h e unstable domains i n t h e parameter space, t h e frequency and t h e shape o f t h e o s c i 1 la t i o n s /1 I/, The plane wave model p r e d i c t s period-doubling and c h a o t i c phenomena which are n o t observed i n t h e experiment.

The multimode i n s t a b i l i t y o f o p t i c a l b i s t a b i l i t y was f i r s t p r e d i c t e d i n /12/ (see a l s o /2/ and references quoted t h e r e i n ) . I t r e q u i r e s t h a t t h e power-broadened l i n e w i d t h i s on t h e order o r l a r g e r than t h e f r e e s p e c t r a l range, and i s c h a r a c t e r i z e d by t h e f a c t o f a r i s i n g i n c o n d i t i o n s o f quasi-resonance between t h e Rabi frequency o f t h e i n t r a c a v i t y f i e l d and t h e frequency d i f f e r e n c e

I f s - f o l .

F i g . 1 i l l u s t r a t e s t h e two r u l e s t h a t govern t h i s i n s t a b i l i t y /13/. I t compares t h e frequency o f t h e spontaneous o s c i l l a t i o n s i n t h e o u t p u t i n t e n s i t y , obtained by s o l v i n g n u m e r i c a l l y t h e plane-wave Maxwell-Bloch equations, w i t h t h e frequency d i f f e r e n c e

I f

-f

I

and

S 0 w i t h t h e Rabi frequency o f t h e i n t e r n a l c a v i t y f i e l d .

The multimode i n s t a b i l i t y has been r e c e n t l y observed e x p e r i m e n t a l l y by Segard and Macke u s i n g a microwave Fabry-Perot c a v i t y f i l l e d w i t h H C ' ~ N /14/. Remarkable are t h e c a v i t y length(182 n) and t h e power broadening on t h e order' o f f o r t y ; both elements are necessary t o push a l o n g i t u d i n a l sidemode under t h e atomic l i n e . The s t r o n g c o r r e l a t i o n between t h e frequency o f t h e sinusoidal o s c i 1 la t i o n s , t h e frequency d i f f e r e n c e

I f

s-fo

1

and t h e Rabi frequency o f t h e i n t r a c a v i t y f i e l d i d e n t i f i e s c l e a r l y t h e nature o f t h e i n s t a b i l i t y observed i n t h i s experiment

/15/. The plane-wave Maxwell-Bloch equations a r e c e r t a i n l y a c r u d e model f o r t h e experimental s i t u a t i o n , y e t t h e y reproduce s u r p r i s i n g l y we1 1 most q u a l i t , a t i v e f e a t u r e s o f t h e experimental r e s u l t s /13/.

0.0 100. 200. 30i). LOO.

SOURCE DETUNlPXj (kHz)

Fig. 1

-

The frequency o f t h e o u t p u t i n t e n s i t y o s c i l l a t i o n s ( c i r c l e s ) i s compared w i t h t h e frequency d i f f e r e n c e ( f

-fol

(broken l i n e ) and w i t h t h e Rabi frequency o f t h e i n t r a c a v i t y f i e l d (square$)

Both t h e s i n g l e and t h e multimode i n s t a b i l i t i e s o f o p t i c a l b i s t a b i l i t y a r i s e f r o m fundamental mechanisms o f t h e a t o m - f i e l d i n t e r a c t i o n , as t h e Rabi n u t a t i o n . Furthermore, t h e y a r e t h e p a s s i v e c o u n t e r p a r t s o f t h e c l a s s i c l a s e r i n s t a b i l i t i e s d i s c o v e r e d i n t h e s i x t i e s by Haken and h i s School /16-19/, which have n o t y e t r e c e i v e d on unambiguous experimental v e r i f i c a t i o n .

3

-

SPATIAL INSTABILITIES

The work o f FlcLaughlin, Moloney and Newel1 /20,21/ has c l a r i f i e d t h a t t h e p l a n e wave c o n f i g u r a t i o n can become u n s t a b l e w i t h r e s p e c t t o t r a n s v e r s a l l y nonuniform p e r t u r b a t i o n s . They analyzed a c l a s s o f m o d u l a t i o n a l i n s t a b i l i t i e s which l e a d t o t h e onset o f s p a t i o - t e m p o r a l s t r u c t u r e s .

We proposed r e c e n t l y a model /22-24/ which has t h e f o l l o w i n g p r o p e r t i e s : 1 ) i t demonstrates e x p l i c i t l y t h e p o s s i b i l i t y of i n s t a b i l i t i e s which l e a d t o t h e r i s e o f s t a t i o n a r y s p a t i a l s t r u c t u r e s , 2) i t a l l o w s f o r an a n a l y t i c a l t r e a t m e n t o f t h e phenomenon. We c o n s i d e r a Fabry-Perot c a v i t y which has n o t o n l y t h e usual m i r r o r s , orthogonal t o t h e l o n g i t u d i n a l d i r e c t i o n z , b u t a l s o

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two l a t e r a l m i r r o r s orthogonal t o t h e transverse d i r e c t i o n x. Assuming t h a t t h e magnetic f i e l d i s p a r a l l e l t o t h e t r a n s v e r s e d i r e c t i o n y i n which t h e c a v i t y i s open, t h e e.m. f i e l d i s independent o f y . For a c a v i t y w i t h p e r f e c t l y conducting w a l l s , t h e modes have s t r u c t u r e c o s ( x n n x / b ) s i n ( n zn,/L), where b ( L ) .is t h e t r a n s v e r s e ( l o n g i t u d i n a l ) l e n g t h and n,, n,=0,1

,...

The c a v i t y i s f i l l e d w i t h a Kerr medium o r a system of two-level atoms. The model i s singlemode w i t h respect t o t h e l o n g i t u d i n a l v a r i a b l e , b u t i n c l u d e s an i n f i n i t e number of transverse modes (nx=O, 1

,. . ^.I..

^It

admits a t r a n s v e r s a l l y homogeneous s t a t i o n a r y s o l u t i o n (nx=O) which c o i n c i d e s w i t h t h a t described i n t h e c l a s s i c paper by Gibbs, E.lcCal1 and Venkatesan /25/. On i n c r e a s i n g t h e i n p u t i n t e n s i t y , t h i s s t a t i o n a r y s o l u t i o n becomes unstable against t h e growth o f transverse modes w i t h nx&. Hence t h e system approaches a new s t a t i o n a r y s t a t e which d i s p l a y s a s p a t i a l s t r u c t u r e i n t h e transverse x d i r e c t i o n . Using t h e b i f u r c a t i o n theory, t h i s s t r u c t u r e can be c a l c u l a t e d a n a l y t i c a l l y i n t h e neighbourhood o f t h e c r i t i c a l p o i n t . Fig. 2 shows t h e n u m e r i c a l l y c a l c u l a t e d e v o l u t i o n o f t h e i n t e n s i t i e s o f t h e various transverse modes, as t h e i n p u t i n t e n s i t y i s increased w e l l beyond t h e c r i t i c a l p o i n t , and d i s p l a y s two t r a n s i t i o n p o i n t s i n a d d i t i o n t o t h e c r i t i c a l p o i n t .

F i g , 2

-

The i n t e n s i t i e s o f t h e modes w i t h n =0,1,

...,

6 are graphed as a f u n c t i o n o f t h e value o f t h e normalized o u t p u t i n t e n s i t y i n t h e t r a n s v e r s a l l y homogenous s t a t i o n a r y s o l u t i o n .

An appropriate name f o r t h i s i n s t a b i l i t y i s o p t i c a l Turing i n s t a b i l i t y /26/, because i t i s s i m i l a r t o t h e d i f f u s i o n a l i n s t a b i l i t i e s t h a t are f a m i l i a r i n n o n l i n e a r chemical r e a c t i o n s and i n b i o l o g i c a l systems /27,28/. Here, however, t h e i n s t a b i l i t y a r i s e s from d i f f r a c t i o n i n s t e a d o f d i f f u s i o n . I t i s again a mu1 timode i n s t a b i l i t y , b u t does n o t produce any o s c i 1 la t i o n because t h e i n p u t f i e l d imposes i t s fequency t o a l l t h e modes. A necessary c o n d i t i o n f o r t h e occurrence o f t h i s i n s t a b i l i t y i s t h a t t h e frequency difference between t h e t r a n s v e r s a l l y homogeneous mode and t h e nearest transverse modes i s on t h e order of t h e c a v i t y Ilinewjdth,

S i m i l a r i n s t a b i 1 i t i e s have been discovered a l s o i n c a v i t y c o n f i g u r a t i o n s w i t h o u t l a t e r a l m i r r o r s , as f a r as example c a v i t i e s w i t h s p h e r i c a l m i r r o r s i n which t h e modes have a Gauss- Laguerre shape /29/, and i n a c t i v e systems as detuned l a s e r s /30/. I n t h e l a t t e r case, t h e s p a t i a l p a t t e r n f o r m a t i o n i s accompanied by a new phenomenon t h a t we c a l l e d cooperative frequency l o c k i n g , because t h e s t a t i o n a r y character o f these multimode c o n f i g u r a t i o n s emerges from t h e f a c t t h a t t h e transverse modes o f t h e resonator l o c k onto a common frequency d u r i n g t h e nonlinear t r a n s i e n t .

ACKNOYLEDGEWENTS

Work supported by a NATO C o l l a b o r a t i v e Research Grant and by t h e EEC t w i n n i n g p r o j e c t on Dynamics o f Nonlinear O p t i c a l Systems. One o f us (Wang Kaige) thanks f o r a g r a n t o f t h e Government o f t h e Popular Republic o f China.

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