CRITERIA FOR OPTICAL BISTABILITY IN A LOSSY SATURATING CAVITY

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CRITERIA FOR OPTICAL BISTABILITY IN A LOSSY SATURATING CAVITY

E. Garmire

To cite this version:

E. Garmire. CRITERIA FOR OPTICAL BISTABILITY IN A LOSSY SATURATING CAVITY.

Journal de Physique Colloques, 1988, 49 (C2), pp.C2-165-C2-167. �10.1051/jphyscol:1988239�. �jpa-

00227656�

JOURNAL DE PHYSIQUE

Colloque C2, Suppl6ment au n06, Tome 49,

j u i n 1988

CRITERIA FOR OPTICAL BISTABILITY IN A LOSSY SATURATING CAVITY E. GARMIRE

Center for Laser Studies, University of Southern California, Los Angeles, CA 90089-1112, U.S.A.

Abstract

-

We d i s c u s s the conditions for multiple quantum well and nipi materials to exhibit bistability i n a F a b r y - P e r o t or a distributed feedback structure.

1

-

INTRODUCTION

L a r g e optical n o n l i n e a r i t i e s h a v e been s e e n in a variety of multiple quantum well and nipi materials. In all c a s e s the nonlinearities occur in the presence of a saturating nonlinearity and a residual a b s o r p t i o n . We d i s c u s s here the conditions for s u c h materials to exhibit bistability in a Fabry-Perot or a distributed

feedback s t r u c t u r e . In t h e F a b r y - P e r o t w e s h o w that there is a n a b s o l u t e

requirement on the ratio of the optically induced index c h a n g e a t s a t u r a t i o n ,

A",,

t o the loss per unit l e n g t h , a: A n /a = l/k. W e furthermore find that there is a trade-off in cavity d e s i g n between sinintizing the switching intensity and

minimizing the required nonlinear phase c h a n g e . T h e analysis is performed f o r reflecting c a v i t i e s , which s h o w optimum system performance. F o r distributed f e e d b a c k , h o w e v e r , t h i s requirement i s n o t a b s o l u t e , but can be reduced with nonlinear quarter-wave s t a c k s .

2 - RESULTS

In t h e c a s e of a s a t u r a t i n g n o n l i n e a r i t y , w e n a y w r i t e the optically induced refractive index a s

A",I

'

~n =

-.

^where

1+I'

Invert to express the nonlinear index (and induced phase change) in terms of the cavity intensity. T h e equation for bistability becomes:

where A d s = A n s k L s

T h i s a l l o w s the input intensity to be calculated in terms of t h e internal half-round-trip phase.

S w i t c h i n g between t w o s t a b l e points is calculated by dI /dd = 0 . An e x a c t

a n a l y s i s y i e l d s a c u b i c equation. We a r e interested i n i f w o regimes: switch-up a n d holding. Approximations h a v e been m a d e t o o b t a i n analytic estimations by a s s u m i n g that t h e nonlinear phase c h a n g e a t switch-up i s s m a l l , w h i l e a t h o l d i n g it i s l a r g e and may be near s a t u r a t i o n , yielding quadratic equations.

We o b t a i n f o r t h e up-switch:

A d U p = d o

-

(1/3)

,/ ⁽⁶⁰⁾²

^{- 3/F}

^'.

F o r l a r g e d , the holding phase c h a n g e is:

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988239

C2-166 JOURNAL

DE

PHYSIQUE

We have compared these estimations to the exact results and found that they are adequate

.

T h e observation of bistability requires that there be sufficient nonlinear phase c h a n g e . n o t only to a c h i e v e the s w i t c h - u p c o n d i t i o n , but to a c h i e v e the

nonlinearity necessary for holding. We c o n s i d e r here operation of the cavity parameters to a c h i e v e bistability with the minimum total phase c h a n g e . Minimizing A d H . this occurs for do - > ~ 6 T h i s means that the minimum holding phase is ~ .

T o find the absolute minimum holding nonlinear phase c h a n g e . we need to minimize do. However. w e see from switch-up that

d o

'm.

^{T h u s Ads,} = d o L - m .

When this condition is f u l f i l l e d , both up-switching and holding in the up-switched state will occur. T h i s puts a n absolute limit on the ability of a medlum to switch which depends on A n l a . T h i s is shown by inserting numbers f o r F to obtain

When the loss i s small and the reflectivities h i g h , write

where p = [l-m )/aL. describes ratio of reflection loss to ahsorption loss The lowest possible limit i s when p = 0 , which g i v e s an absolute criterion to the observation of optical bistability. Bistability cannot be seen unless

bns/a > ( l / k ) ( F / 2 )

-

^l/k (k i s the f r e e - s p a c e wave v e c t o r )

T h e limit p = 0 occurs when aL > > 1 -

T.

That is. the m i r r o r s have very low loss compared the the absorption loss in t e d e v i c e .

T h e optimum u s e of a lossy nonlinear Fabry-Perot is in reflection. rather than in transmission. T h i s is because the loss is largest when the cavity is on resonance.

However. the reflected power i s a m i n i m u m , s o this loss d o e s n o t matter. By

c o n t r a s t , the loss i s smallest when the c a v i t y is off resonance. which is when the reflected signal is in the o n - s t a t e , the reverse of the transmission case.

In an optimized l o s s y , saturating nonlinear F a b r y - P e r o t . the reflectivity o n resonance g o e s t o z e r o , a s long a s R F = R e . a n "impedance-matched" c a v i t y . The reflection bistable d e v i c e requires

T h i s is two t i m e s the mininun

(p

= 0) and s h o w s that there is a trade-off in cavity d e s i g n between minimizing Ans/a and minimizing the incident intensity.

We may choose impedance m a t c h e d , with A n /a a factor of t w o larger than the m i n i m u m , or w e may o p e r a t e with p = 0 , t$ a c h i e v e the lowest possible Ans/a. T h e requireaent p = 0 neans that the incident i n t e n s i t y , I S W - > m . T h u s w e c a n never.

in p r i n c i p a l , reach p = 0 . T h e trade-off for cavity d e s i g n i s between k e e p i n g I

reasonable and Ans/a sufficiently small. SW

T h e reason f o r this s t u d y i s the d e s i r e t o be a b l e to test materials prior t o d e v i c e f a b r i c a t i o n t o predict performance. W e h a v e inserted n u m b e r s f o r a variety of multiple quantum well a n d hetero-nipi materials w e h a v e measured. In a l l cases.

o n r e s o n a n c e of the e x c i t o n f e a t u r e t h e l o s s i s sufficiently large t h a t the criterion A n / a > l/k i s n o t met. H o w e v e r , o n the long-wavelength s i d e the nonlinearity a e c r e a s e s m o r e slowly t h a n the absorption and eventually t h i s criterion m a y be met.

With importance of u s i n g Fabry-Perots operated in r e f l e c t f o n and t h e concept o f integrated mirrors g r o w n on the o p a q u e s u b s t r a t e s , c o m e s a re-thinking of the u s e of distributed feedback. Modelling f o r t h e l o s s y , saturating nonlinear Bragg

reflector indicates a decided a d v a n t a g e over the Fabry-Perot. T h e fractional c h a n g e in coupling coefficient (KL) c a n be much larger than the fractional c h a n g e In refractive index s i n c e

w h e r e A n is the built-in index c h a n g e per quarter-wavelength layer. S i n c e K =

AnH*. an! loss limits t h e length to 221.. w e obtain a condition o n switching:

where d(KL) is the c h a n g e in coupling coefficient necessary t o a c h i e v e bistability.

From numerical m o d e l l i n g , w e find a heuristic relation:

T h u s the condition t o s e e bistability becomes

T h i s is smaller than the requirement f o r a Pabry-Perot a s long a s K L i s

sufficiently large (KL > 6). S i n c e K L i s limited by the naximua reflectance c h a n g e per facet ( 4 n H < 0 . 5 ) , and loss (length L t 2 / a ) , the condition f o r bistability becomes :

Unlike the nonlinear F a b r y - P e r o t , there is n o absolute condition on A n /a. F o r sufficiently l o w loss and large index discont'nuities, t h i s can alwaysSbe

f

s a t i s f i e d . A s a n e x a m p l e , consider a = 1 pm and A n H = 0 . 5 . T h i s g i v e s a

coupling coefficient KL = 7.6, and the conditions to s e e bistability c a n be n e t in a distributed feedback s t r u c t u r e .