SPATIOTEMPORAL INSTABILITIES IN A SATURABLE HOMOGENEOUSLY BROADENED RING CAVITY

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SPATIOTEMPORAL INSTABILITIES IN A SATURABLE HOMOGENEOUSLY BROADENED

RING CAVITY

J. Zondy, M. Le Berre, E. Ressayre, A. Tallet

To cite this version:

J. Zondy, M. Le Berre, E. Ressayre, A. Tallet. SPATIOTEMPORAL INSTABILITIES IN A SAT-

URABLE HOMOGENEOUSLY BROADENED RING CAVITY. Journal de Physique Colloques, 1988,

49 (C2), pp.C2-487-C2-490. �10.1051/jphyscol:19882114�. �jpa-00227625�

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SPATIOTEMPORAL INSTABILITIES IN A SATURABLE HOMOGENEOUSLY BROADENED RING CAVITY

J.J. ZONDY, M. LE BERRE, E. RESSAYRE and A. TALLET

Laboratoire de Photophysique MolBculaire, Bdt. 213, universite Paris-Sud. F-91405 Orsay Cedex, France

Resume : L ' i n f l u e n c e des e f f e t s transverses dus a l a d i f f r a c t i o n l i b r e s u r l a dynamique n o n l i n e a i r e d'une c a v i t e haute finesse e s t etudiee dans d i f f e r e n t e s c o n f i g u r a t i o n s optiques. La nature des deux premieres b i f u r c a t i o n s semble e t r e t r e s robuste mais l e s e u i l d ' a p p a r i t i o n du chaos e s t fortement diminue.

A b s t r a c t : The i n f l u e n c e o f transverse e f f e c t s caused by free-space propagation on t h e n o n l i n e a r dynamics o f a h i g h f i n e s s e r i n g c a v i t y i s i n v e s t i g a t e d f o r d i f f e r e n t o p t i c a l c o n f i g u r a t i o n s . The nature o f t h e f i r s t two Hopf b i f u r c a t i o n s i s found t o be r o b u s t a g a i n s t s i g n i f i c a t i v e transverse e f f e c t s b u t t h e t h r e s h o l d f o r chaos i s lowered w i t h r e s p e c t t o t h e plane wave case.

1 - INTRODUCTION

Theoretical s t u d i e s /1/ o f the n o n l i n e a r dynamics o f passive e x t e r n a l l y pumped r i n g c a v i t y ignore transverse e f f e c t s caused by t h e free-space propagation o u t s i d e t h e n o n l i n e a r medium.

These t r a n s v e r s e e f f e c t s can be escaped i n a confocal geometry w i t h lenses o f l a r g e f o c a l l e n g t h placed a t a p p r o p r i a t e s i t e s on the t r a v e l o f t h e beam i n o r d e r t o counteract t h e divergence o f t h e beam and t o focus t h e beam back t o t h e entrance o f t h e c e l l ( F i g . 1 ) :

Fis.l

: Schematic o f t h e r i n g c a v i t y . The i n p u t and output m i r o r s have r e f l e c t i v i t y R

6

1. An example o f confocal geometry c o u l d be p i = p2 = 2 f and Z = 4 f where f i s t h e f o c a l l e n g t h o f the lenses. With l a r g e f t h e r e t u r n e d e l e c t r i c f i e l d Eback(F) a t the medium entrance has the same shape than t h e o u t p u t f i e l d , Eback (F) = E ~ ~ ~ ~ ~ ~ ( - ? ) .

But s l i g h t changes o f t h e lenses s i t e s may cause s i g n i f i c a t i v e transverse e f f e c t s when the n o n l i n e a r i n t e r a c t i o n between t h e l i g h t and medium m o d i f i e s t h e f i e l d phases p r o f i l e s . Free d i f f r a c t i o n on such modified beams leads t o t h e formations o f r i n g s /2/ i f t h e c a v i t y l e n g t h L

= p i

+

p2

+

Z i s s u f f i c i e n t l y l o n g w i t h respect t o the Rayleigh l e n g t h zd o f the i n p u t beam.

Since t h e c a v i t y r o u n d - t r i p t i m e r R = L/C i s o f t h e order o f o r g r e a t e r than t h e medium r e l a x a t i o n time 6-1 f o r t h e r i n g c a v i t y t o e x h i b i t i n s t a b i l i t y , the c o n d i t i o n L >> Zd can be e a s i l y v e r i f i e d . Then t h e r e t u r n e d beam has no more t h e i n j e c t e d beam shape. This note presents r e s u l t s o f numerical s i m u l a t i o n s i n c l u d i n g free-space propagation o f t h e dynamic o f

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882114

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C2-488 JOURNAL

DE

PHYSIQUE

a high-finess ring cavity in a case where a transition to chaos via a quasi-periodic motion occurs in the plane-wave limit. The main observations are a) weak transverse effects are sufficient to ensure a whole beam dynamic motion and b) the nature of the first two bifurcations appears to be surprisingly robust against significative transverse modifi

-

cations of the intracavity field.

This study does not deal with transverse effects arising from the diffraction inside the medium which have been extensively studied by Moloney et a1 .

^/3/.

The length of the two-level medium will be assumed small ltzd in order to neglect diffr-action as a first approximation. In the adiabatic approximation the equations of the ring cavity are for the intracavity electric field,

oll i~

- (1-i~) Q(r9,t

-

7

)

E(r,t)

=

~ ~ ( r )

^tR

dr' H(r,r9) e ' E(r,t - -

^{? R}_C )

and for the atomic energy s

The calculations are made for primarily dispersive medium with an off-resonance absorption coefficient a1

=

0.1. a is the detuning of the laser from the resonance in units of the

-. -.

resonance half-width. H(r, r') accounts for free-space propagation. The transmitted input -. ₂

beam is a Gaussian sO(r)

=

EO e-r /uO2 with its waist located at the medium entrance.

^i~

is equal to 8-1.

In the plane wave limit, with these set of parameters the lower branch of the stationary hysterisis loop is unstable, moreover a transition to chaos via a quasi -periodic motion occurs for a very large range of detuning . The critical value Eth for the first Hopf bifurcation and the period

T

at threshold of the periodic solution are very sensitive to the

1

inear refractive index

4 = alb /

2. The table

1

resumes the main characteristics for the first two Hopf bifurcations for +

⁼

7n/2 and 4n.

Table

:

Plane wave threshold values of the input field amp1 itude for the 1 imit cycle (CL) and quasi-periodicity (lip), T label the periods.

-

o =

7n/2

LC ELC = 0.26 TLC

r

3

^i~

QP Eqp

¹

0.32

T l r i ~ T2

= 2.5

* R

chaos ECH

a

0.42

4 = 4n

ELC

c

0.07 TLC

^c1 1 i~

Epp

E

0.11

T l = i ~

Tp =

7 * R

ECH

o

0.35

i

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( c o n f i g u r a t i o n A) w i t h f = 27 zd, p l = f

-

2.25 zd, p2 = f

+

2.25 zd and z = 4 f ( c f F i g . 1). The o t h e r c o n f i g u r a t i o n ( c o n f i g u r a t i o n B)

,

a1 so n e a r l y confocal w i t h f = 9 zd, p i = p2 = f

+

0.225 zd and z = 198 zd corresponds t o s l i g h t l y s t r o n g e r transverse e f f e c t s .

The case 4 = 4n has been simulated w i t h d i f f e r e n t o p t i c s : p e r f e c t a1 ignement between t h e i n p u t f i e l d and t h e lenses, s l i g h t misalignement and f i n a l l y a Gaussian i n p u t f i e l d w i t h a weak i r r e g u l a r i t y on t h e edge. It was always observed two successive Hopf b i f u r c a t i o n s b e f o r e t h e c h a o t i c regime. For t h e c o n f i g u r a t i o n A, the r o l e o f t h e weak transverse e f f e c t s i s m a i n l y t o a l l o w whole beam o s c i l l a t i o n s ( F i g . 2 ) .

Fiq. 2 : Quasiconfocal o p t i c s ( 4 = 4n) : t h e l i m i t c y c l e a t Eo = 0.07 has a p e r i o d P 11 i~

I n the case B, t h e d e s t a b i l i z a t i o n o f t h e f i x e d p o i n t (Fig. 3 ) occurs f o r Eo

--

0.155 f o r a l l t h e geometries t r i e d . As i n t h e plane wave case t h e p e r i o d i c signal i s slow i t corresponds t o an o s c i l l a t i o n between a c e n t r a l spot and a doughnut ( F i g . 4 ) .

f i q .

3

: C o n f i g u r a t i o n B (o = 4n) : a) I n t e n s i t y p r o f i l e a f t e r an i n t e g r a t i o n over one f R _; b) o f t h e s t a t i o n a r y s t a t e a t

E,

= 0.15

F i a . 4 : C o n f i g u r a t i o n B, L i m i t c y c l e f o r c y l i n d r i c a l geometry (Eo=0.155, 4 = 4n). a) I n t e n s i t y t i m e t r a c e f o r t h e r a d i u s = 0 ; b) f o r t h e r a d i u s i n d i c a t e d by t h e arrow i n Fig. 3. c ) ,

d ) ,

^e) i n t e n s i t y p r o f i l e s a t o , p, 8 r e s p e c t i v e l y .

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C2-490 JOURNAL DE PHYSIQUE

The q u a s i p e r i o d i c regime appears a t Eo n 0.162 and chaos a t Eo

-

0.2. The main e f f e c t o f t h e m i s a l ignement i s t o g i v e r i s e t o a frequency l o c k i n g , w i t h a t r a n s v e r s e r i n g d i v i d e d i n t o few s t r o n g peaks ( F i g . 5 ) .

-

In

(t) ( T V , Y )

-. ^{r _}

F i q . 5 : C o n f i g u r a t i o n B, a) q u a s i - p e r i o d i c s i g n a l a t Eo = 0.162; 4 = 4n w i t h TI = 12 T R T2 =

1.6 T R i n t h e c y l i n c l r i c a l symmetry ; b) frequency l o c k i n g i n t h e case o f s l i g h t

-

misalignement ; c ) i n t e n s i t y p r o f i l e a t 460 rR. An arrow i n d i c a t e s t h e l o c a t i o n Ro o f t h e

-

gaussian i n p u t . The c e n t r a l s p o t i s i n a1 ignement w i t h t h e l e n s e s . O p p o s i t e t o Ro, t h e r i n g e x h i b i t s t h r e e s p i k e s .

The f i x e d p o i n t f o r 4 = 7n/2 i n c y l i n d r i c a l symmetry has a doughnut s p a t i a l p r o f i l e . T h i s shape seems t o be l e s s s t a b l e t h a n t h e f i x e d p o i n t i n F i g . 3, s i n c e l i m i t c y c l e and chaos appear f o r s m a l l i n p u t (ELC = 0.06, Echaos

=

0.08). The 1 im i t c y c l e has a p e r i o d T ⁿ3 f R o f t h e o r d e r o f t h e p l a n e wave p e r i o d .

REFERENCES

1 ) K . Ikeda, Opt. Comm.

30,

257 (1979) ; see H.M. Gibbs, O p t i c a l B i s t a b i l i t y C o n t r o l 1 i n g L i g h t w i t h L i g h t (Academic, New York, 1986) and r e f e r e n c e s t h e r e i n ; M. Le B e r r e e t a1

. ,

Phys. Rev.

L e t t .

56,

278 (1986).

2) M. Le Berre, E. Ressa,yre, A. T a l l e t , K. T a i , H.M. Gibbs, H.C. R u s h f o r d and N. Peyghambarian, JOSA B,

1,

591 (1984).

3) J. Moloney, H. Adachihara, D.W. MacLaughin and A.C. Newell, i n Chaos, N o i s e and F r a c t a l s , Ed. E.R. P i k e and L.A. L u g i a t o , Adam H i l g e r , B r i s t o : , 1987 and r e f e r e n c e s t h e r e i n .

The a u t h o r s acknow'ledge NATO g r a n t n' 8 5 4 7 3 4 .