FIRST PASSAGE TIME DISTRIBUTION FOR DELAYED LASER THRESHOLD INSTABILITY

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FIRST PASSAGE TIME DISTRIBUTION FOR DELAYED LASER THRESHOLD INSTABILITY

M. San Miguel, M. Torrent

To cite this version:

M. San Miguel, M. Torrent. FIRST PASSAGE TIME DISTRIBUTION FOR DELAYED LASER THRESHOLD INSTABILITY. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-149-C2-151.

�10.1051/jphyscol:1988235�. �jpa-00227652�

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JOURNAL DE PHYSIQUE

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

FIRST PASSAGE TIME DISTRIBUTION FOR DELAYED LASER THRESHOLD INSTABILITY

M. SAN MIGUEL and M.C. TORRENT'

Departament d e Fisica, Universitat Illes Balears,

S P - 0 7 0 7 1

Palma de Mallorca, Spain

' ~ e p a r t a m e n t d'Estructura i Constituents de la Materia, Universitat d e Barcelona, Diagonal

647, S P - 0 8 0 2 8

Barcelona, Spain

A b s t r a c t - The d e l a y i c h a r a c t e r i z e d by a c a l c o r r e s p o n d i n g t o slow

n t h e t h r e s h o l d i n s t a b i l i t y induced by a sweep o f c a v i t y l o s s e s i s . c u l a t i o n o f a f i r s t p a s s a g e t i m e d i s t r i b u t i o n . Two d i f f e r e n t c a s e s

and f a s t sweeping r a t e s a r e c o n s i d e r e d .

1

-

INTRODUCTION

The d e l a y i n t h e o b s e r v a t i o n o f a b i f u r c a t i o n induced by a sweep o f t h e c o n t r o l p a r a m e t e r h a s been s t u d i e d i n a v a r i e t y o f o p t i c a l i n s t a b i l i t e s a s r e c e n t l y r e v i e w / I / . Experimental r e s u l t s a r e a v a i l a b l e f o r t h e t h r e s h o l d o f a n Argon l a s e r /2/ and f o r s e v e r a l s y s t e m s d i s p l a y i n g O p t i c a l B i s t a - b i l i t y /3/. Most t h e o r e t i c a l s t u d i e s g i v e a d e t e r m i n i s t i c d e s c r i p t i o n o f t h e problem ( f o r a n ex- c e p t i o n s e e / 4 / ) . We a d d r e s s h e r e t h i s problem t h r o u g h a s t o c h a s t i c dynamics c h a r a c t e r i z a t i o n . The b a s i c i d e a b e h i n d t h i s a n a l y s i s is t h a t t h e d e l a y is due t o a dynamical s t a b i l i z a t i o n , t h e s t a b i - l i t y b e i n d a s s o c i a t e d w i t h t h e l i f e t i m e o f a s t a t e and t h u s i s d e t e r m i n e d by f l u c t u a t i o n s . The l i f e t i m e i s d e f i n e d a s a Mean F i r s t P a s s a g e Time (MFPT) t o r e a c h a c e r t a i n t h r e s h o l d o b s e r v a b l e v a l u e . We w i l l c a l c u l a t e t h e d i s t r i b u t i o n o f t h e s e F i r s t P a s s a g e Times (FPT). Connection w i t h t h e e x p e r i m e n t i s made i d e n t i f y i n g t h e MFPT w i t h t h e t i m e a t which t h e i n s t a b i l i t y is o b s e r v e d . Such i d e n t i f i c a t i o n g i v e s a n o p e r a t i o n a l d e f i n i t i o n o f t h e dynamical b i f u r c a t i o n p o i n t and t h e o b s e r v e d d e l a y .

The c a l c u l a t i o n o f t h e FPT d i s t r i b u t i o n i s d i f f e r e n t when t h e r e l a x a t i o n i s from a n u n s t a b l e s t a t e o r a s t a t e o f m a r g i n a l s t a b i l i t y a s i n o p t i c a l b i s t a b i l i t y /3/. Our r e s u l t s below r e f e r t o t h e r e l a x a t i o n from a n u n s t a b l e s t a t e and s h o u l d b e compared w i t h c o r r e s p o n d i n g n u m e r i c a l r e s u l t s / 5 / , a n a l o g i c s i m u l a t i o n s /6/ o r e x p e r i m e n t s /2/. To d e s c r i b e t h e l a s e r t h r e s h o l d i n s t a b i l i t y we c o n s i - d e r t h e u s u a l model f o r a s i n g l e mode i n t h e good c a v i t y l i m i t

E i s t h e e l e c t r i c f i e l d complex a m p l i t u d e and a=P-l( where

r

^and

^X

a r e , r e s p e c t i v e l y , t h e g a i n and l o s s p a r a m e t e r s . The complex random term

$

^{( t ) =}^fl^{( t}) + i f 2 ( t ) model s p o n t a n e o u s e m i s s i o n f l u c - t u a t i o n s o f s t r e n g t h € . We a r e i n t e r e s t e d i n t h e s i t u a t i o n i n which t h e l a s e r i s c o n t i n u o u s l y d r i v e n from below t o above t h r e s h o l d a t a c e r t a i n sweeping r a t e N . We c o n s i d e r two c a s e s . I n a f i r s t c a s e o f i n t e r e s t t o s t u d y slow v a r i a t i o n s o f a we assume

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988235

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C2-150 JOURNAL

DE

PHYSIQUE

v a l i d f o r a l l t > O . A s e c o n d c a s e o f i n t e r e s t is t o s t u d y f a s t v a r i a t i o n s o f a when s w i t c h i n g on t h e l a s e r . I n t h i s c a s e , ( 2 ) i s v a l i d f o r 'I < t i t and a ( t ) = a f o r a l l t > t l . I n any o f t h e two

1

c a s e s , v a r i a t i o n s o f a c a n be induced by t i m e d e p e n d e n t c a v i K y i c s s e s , a s i n t h e s e t - u p o f Ref.

2 .

The p a s s a g e t i m e d i s t r i b u t i o n t o l e a v e t h e i n s t a b i l i t y i s e s s e n t i a l l y d e t e r m i n e d by l i n e a r dyna- m i c s / 7 / . The l i n e a r e q u a t i o n f o r t h e l a s e r i n t e n s i t y f o l l o w i n g from (1) c a n b e s o l v e d a s

2 2 2

where h =hl+h2, and

The key i d e a o f t h e c a l c u l a t i o n i s t o i n v e r t ( 3 ) t o have t a s a f u n c t i o n o f I . Time t i s t h e n ex- p r e s s e d a s a s t o c h a s t i c q u a n t i t y t o r e a c h a v a l u e I . Taking G a u s s i a n a v e r a g e s o v e r t h e d i s t r i b u - t i o n o f h i i t i s t h e n p o s s i b l e t o c a l c u l a t e t h e moments o f t h e p a s s a g e t i m e d i s t r i b u t i o n . R e s u l t s a r e g i v e n below a n d d e t a i l s w i l l be p u b l i s h e d e l s e w h e r e .

2 - SLOW SWEEPING CASE!

The s l o w sweeping case! c o r r e s p o n d s t o t h e f i r s t o n e mentioned above. The c a l c u l a t i o n c a n b e c a r r i e d o u t i n t h e l i m i t &'

<<

^\I;;T< < a . The f i r s t i n e q u a l i t y r e p r e s e n t s t h e s m a l l n o i s e l i m i t and t h e s e c o n d one c o r r e s p o n d s t o s l o w s w e e p i n g . We o b t a i n

la01

1

(

^(h;(-J/2 ⁺

$

^T-1/2

( t )

- -

= - l o g ---

o( E

2

E

-1

where T = l o g ( I / < h ( a ) > ) is a l a r g e q u a n t i t y d i v e r g i n g a s l o g

,

I i s t h e t h r e s h o l d o b s e r v a b l e v a l u e o f I , and

y

t h e digamma f u n c t i o n . S i n c e l a o

1

/ u( i s t h e t i m e ? s u c h t h a t a ( ? ) = 0 , t h e r i g h t hand s i d e o f ( 5 ) give:; t h e postponement o f d e l a y f o r t h e o b s e r v a t i o n o f t h e i n s t a b i l i t y . I t i s c l e a r i n t h i s s t o c h a s t ; i c framework t h a t s u c h d e l a y i s a q u a n t i t y d e t e r m i n e d by t h e n o i s e i n t e n s i t y

5 .

The dependences on

1

o f t h e MFPT < t > and i t s v a r i a n c e <( A t ) 2

>

g i v e n by ( 5 ) and ( 6 ) a r e e s s e n t i a l l y d i f f e r e n t from t h e o n e s found i n s t a n d a r d c a l c u l a t i o n s o f t r a n s i e n t l a s e r s t a t i s t i c s c o r r e s p o n d i n g t o d - 3 w

.

The dynamical b i f u r c a t i o n p o i n t c a n now b e d e f i n e d by t h e mean v a l u e o f a ( t ) needed t o r e a c h a n o b s e r v a b l e v a l u e I . Fyom ( 5 ) , t h i s g i v e s

The &'I2 dependence o f ( 7 ) c o i n c i d e s w i t h t h e o n e f o u n d by S c h a r p f e t a l . / 2 / . However, t h e i d e n - t i f i c a t i o n o f o u r MFP'Y w i t h t h e t i m e o f maximum r a t e o f g r o w t h o f t h e o u t p u t i n t e n s i t y m o n i t o r e d i n R e f . 2 is n o t compLetely o b v i o u s . T h a t t i m e c o u l d b e beyond t h e l i n e a r r e g i m e a n a l y z e d h e r e , a n d a more s t r i n g e n t t e s t o f t h e r e s u l t s ( 5 ) and ( 6 ) would b e g i v e n by a measurement of t h e va- r i a n c e o f a ( t ) , which from ( 6 ) g i v e s

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In any case, and in spite of the slightly different definitions of models and delays, our results (5)-(8) give a g o o d description of the main findings of earlier numerical studies /5/ and analogic simulations /6/.

3 - FAST SWEEPING CASE

The calculations can be carried out here for

i

<i a

$ E .

The first inequality is again a small noise intensity condition while the second one implies fast sweeping. This case includes the limit

J( - m

.

^{We obtain}

1 I (a+ja~l )2 1 c t > = - log ---- +

2 - --y(1)

2a < h ~ ) > 2a3. 2a

A meaningful delay in this case is with respect to instantaneous limitu ->- :

where

C

t>inst is the value of L t> for o( ->a

,

and

<

^h2 are, respectively,

<

hw)>calcu- ² lated in the limit ?( -& and for the present case. The first term in the right hand side of (11) is the corresponding delay obtained in a deterministic framework. The second term is the con- tribution of fluctuations which in the range of parameters considered is a negative quantity in- dependent of E

.

Financial support from Direcci6n General de Investigaci6n Cientifica y Tecnica (Spain), project PB 86-0534, is acknowledged. Stimulating discussions with P. Mandel are also acknowledged.

REFERENCES

/1/ MANDEL, P., in Instabilites and chaos in Quantum Optics, Eds. N. Abraham and E. Arimondo (Plenum Press, to be published).

/2/ SCHARPF, W., SQUICCIARINI, M., BROMLEY, D., GREEN, C., IREDUCCE, J.R., and NARDUCCI, L.M., Opt. Commun.

63

(1987) 344.

/3/ ARIhIONDO, E., DANGOISSE, D., GABBANINI, C., MENCHI, E., and PAPOFF, F., J. Opt. Soc. Am.

B4 (1987) 842.

-

t.lITSDIKE, F., DESERNO, R., LANGE, W., and MLYNEK, J., Phys. Rev. A33 (1986) 3219. - /4/ BROECK, C. VAN DEN, and MANDEL, P., Phys. Lett.

A122

(1987) 36.

/5/ BROGGI, G., COLOMBO, A., LUGIATO, L., and MANDEL, P., Phys. Rev. A33 (1986) 3635.

/6/ MANNELLA, R., MOSS, F., and MCCLINTOCK, P.V., Phys. Rev.

A35

(198r2560.

/7/ DE PASQUALE, F., SANCHO, J-lil., SAN MIGUEL, M., and TARTAGLIA, P., Phys. Rev. Lett.

2

⁽¹⁹⁸⁶⁾

2473.