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NON CRITICAL SLOWING DOWN IN OPTICAL BISTABILITY
B. Segard, B. Macke
To cite this version:
B. Segard, B. Macke. NON CRITICAL SLOWING DOWN IN OPTICAL BISTABILITY. Journal de
Physique Colloques, 1988, 49 (C2), pp.C2-115-C2-118. �10.1051/jphyscol:1988226�. �jpa-00227643�
NON CRITICAL SLOWING DOWN IN OPTICAL BISTABILITY
B. SEGARD and B. MACKE
L a b o r a t o i r e de Spectroscopie H e r t z i e n n e , CNRS-UA 249, U n i v e r s i t e de L i l l e I , P-59655 V i l l e n e u v e - d ' A s c q Cedex, France
Rdsum& : Nous prGsentonc, u,:e kcude e x p e r i i n e n t a l c r?t a n a l y t i q u e 2u comportement ciyr~dul~quc.
d ' u n s y s t h m e b i s t a b l e p r & p a r & a u v o i s i n a g e d ' u n p o i n t i n s t a b l e d e l a b r a n c h e i n t e r m e d i a i r e d e s o n c y c l e d l h y s t C r & s i s .
A b s t r a c t : We h a v e e x p e r i m e n t a l l y and a n a l y t i c a l l y s t u d i e d t h e d y n a m i c s o f a b i s t a b l e d e v i c e p r e p a r e d i n t h e v i c i n i t y o f a n u n s t a b l e s t a t e o f t h e m i d d l e b r a n c h o f t h e h y s t e r e s i s c y c l e .
We r e p o r t i n t h i s c o m m u n i c a t i o n on t h e s w i t c h i n g d y n a m i c s o f a n o p t i c a l b i s t a b l e d e v i c e i n t h e c a s e when a h o l d i n g power m a i n t a i n s t h e s y s t e m i n a s t a b l e s t a t e o f t h e l o w e r b r a n c h o f t h e h y s t e r e s i s c y c l e ( p o i n t S 1 o f F i g . 1) a n d a p u l s e a d d r e s s o f f i n i t e d u r a t i o n (Tp) b r i n g s i t c l o s e t o t h e u n s t a b l e s t a t e o f t h e m i d d l e b r a n c h ( p o i n t U). F o r a c r i t i c a l p u l s e d u r a t i o n (T,), t h e p u l s e e x a c t l y b r i n g s t h e d e v i c e i n t h e u n s t a b l e s t a t e where i t would i n d e f i n i t e l y r e m a i n i n t h e a b s e n c e o f l l u c t u a t i o n s . F o r p u l s e d u r a t i o n c l o s e t o Tc t h e d e v i c e s w i t c h e s up t o t h e s t a t e S 2 of t h e u p p e r b r a n c h (T
>
Tc) o r r e l a x e s t o i t s i n i t i a l s t a t e (Tp<
Tc) a f t e r a l e t h a r g y t i m e w h i c h d i v e r g e s when1 Fp
- Tc1
+ 0, f o l l o w i n g a i o g a r i t h m i c law / 1 , 2 / .A
B
,
I'\ \ I I
'\ \ I I '\\ I
y,
I '\1
'\,I N P U T POWER
>
P 0 P I
v
F i g . 1
-
S-shaped c u r v e of a n o p t i c a l b i s t a b l e s y s t e m . The m i d d l e b r a n c h AB ( d a s h e d l i n e ) i s u n s t a b l e . Bottom : S q u a r e wave p u l s e a d d r e s s b r i n g i n g t h e s y s t e m c l o s e t o t h e u n s t a b l e s t a t e U . The e x p e r i m e n t s w e r e a c h i e v e d a t m i l l i m e t r e w a v e l e n g t h ( h 2 3,5 mm) where cw s o u r c e s ( k l y s t r o n ) p r e s e n t i n g a v e r y low a m p l i t u d e n o i s e a r e a v a i l a b l e . The b i s t a b l e d e v i c e , d e s c r i b e d e l s e w h e r e 1 2 1 , i s a 1 8 2 m-long F a b r y - P e r o t r e s o n a t o r f i l l e d w i t h H C ~ ~ M a t low p r e s s u r e (0.5-1.5 mTorr) a s a s a t u r a b l e a b s o r b e r . The r e s o n a t o r i s c h a r a c t e r i z e d by a c a v i t y r o u n d t r i p t i m e TR 1.2 u s ( f r e e s p e c t r a l r a n g e 1 8 3 0 kHz) and a p h o t o n l i f e t i m e ~ ~ h 2 3 . 3 (IS (modewidth 2 96 kHz FWmi), l e a d i n g t o a f i n e s s e F-
8.5. The r e s o n a t o r was t u n e d and e x t e r n a l l y d r i v e n a t t h e e x a c t f r e q u e n c y o f t h e 0-1 r o t a t i o r l a 1 l i n e o f H C ~ ~ N ( p u r e l y a b s o r p t i v e b i s t a b i l i t y ) . Owing t o t h e l a r g e d i p o l e moment o f H C ~ ~ N , t h i s l i n e is e a s i l y s a t u r a t e d a t m o d e r a t e power. F i n a l l y t h e l a r g e power a b s o r p t i o n c o e f f i c i e n t o f t h e l i n e ( a 2 1 m-l i n t h e c o l l i s i o n a l l i m i t ) l e a d s t o a p a r a m e t e r o f c o o p e r a t i - v i t y C 1 250 (C = a c ~ ~ ~ 1 4 ) . The i n p u t power s u p p l i e d by a p h a s e - l o c k e d k l y s t r o n , was c o n t r o l l e dArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988226
C2-116 JOURNAL
DE
PHYSIQUEby a P.I.N. diode modulator itself driven by a home made generator programmed to obtain the input power modulation shown on figure 1. A programmable external clock allowed us to adjust the duration of the address pulse. The transmitted wave was detected by a Schottky diode mixer
(quadratic detector) and the corresponding signal (proportionnal to the output power) was sent on a digital oscilloscope (Lecroy 9400). The power modulation rate, the frequency of the external clock and the data acquisition were controlled by a microcomputer. With this experimental arran- gement it was possible to store the signals corresponding to 62 different values of the pulse address duration in a single shot experiment, the total duration of which did not exceed30 s. This procedure allowed us to overcome the difficulties related to the unavoidable slow drifts of the experimental parameters. Figure 2 shows a set of traces which record the output power versus time for different pulse durations.
T I M E
< m i c r o s e c o n d s>
Fig. 2
-
Typical set of output power versus time. The duration (Tp) of the input pulse (insert) increases from 5 (Tp = 17.301 VS) toh
(Tc = 18.939 ps).This result is in good agreement with the previous description, in particular a stabilisation of the output power level corresponding to an unstable point is observed. Moreover all the traces have nearly the same shape both for the relaxation back process (Fig. 2 a-c) and for the up switching process (Fig. 2 d-h). This indicates that the system may be described by a simple model involving a single dynamical variable. In such a model / 2 / the time evolution of the system is quite generally described by the equation
where x is the dynamical variable and p is the control parameter (here the input power).
For a given control parameter
uo
inside the bistable domain, there are three possible steady states, corresponding to the points (Fig. 1) Sl(x=xl), U(x=x,) and S2(x=x2) such that :After the end of the pulse address (t=O) the dynamical variable x exponentially increase (up switching) or decrease (relaxation back) and reach a given value X (close to xu) in a time t given by :
where f: is the partial derivative af/ax in (xu,po) and p1 is the value of the control parameter during the pulse address. This shows that the switching delay is proportionnal to Ln
I
T-TcI
withthe same slope for the up switching and relaxation back processes. The experimental results (Fig. 3) are in fairly good agreement with this prediction.
0
RELRXRTION TO THE LOWER BRRNCH
.r(
E 6 0 -
V .
40
-
9;
-
3-
2-
1 0 1L n l T - T c l Fig. 3
-
Switching delay versus L ~ ~ T ~ - T ~I.
C
Finally the shape of signals corresponding to the upswitching process may be reproduced with the parabolic model introduced by Mandel 1 3 1 . In this model, valid when the unstable point U and the upper stable point S2 are in the vicinity of the turning point B (see figure l), the function f(x,p) is replaced by an hyperbola and the equation which rules the bistable device evolution is :
+ SUITCHING TO THE UPPER BRANCH
where JJ (the input power), x (the output power)and t' (t1=t/r) are suitably scaled.
After the pulse address the solution of (4) is :
where x2 = p,
+
~&1 and xu = pO- l & d .
xo, the value of x(t) at the end of the pulse (t=O) is given by :
The switching delay (At) measured at half way between xu and x2 is then At = a Log (
- -
1 )T,-Tc
The measurement of the ratio x2/x, and the determination of the parametersa and Ballow us to calculate the output power x(t). The figure 4A shows a set of experimental records obtained for different values of the pulse duration in a situation in which the parabolic approximation is valid. These curves are well reproduced by their corresponding calculated traces (Fig. 4B), the pulse duration being adjusted inside the error range in order to obtain the best fit between experimental and calculated results.
This work was supported by the European Economic Community under contract ST2J-0187 F and by the Rdgion Nord
-
Pas-de-Calais. We gratefully thank Paul Mandel who suggested this study for helpful discussions.C2-118 JOURNAL
DE
PHYSIQUE0
F
20 $0 6 0 80 1 0 0 1 2 0 I t 0 1 6 0 1 8 0 2 0 0 0 20 t o 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0T I M E
C m i c r o s e c o n d s> T I M E
( m i c r o s e c o n d s ) Fig. 4-
Output power versus time. A ) experimental curves, B) theoretical curves obtained in the parabolic approximation model. The two set of traces are drawn at the same scale. The pulse duration decreases from left to right (15,19 ps<
T p < 15,60 ps ; Tc = 15,230 ps).References
/ 1 / Bigot J.Y., Daunois A . and Mandel P., Phys. Lett. A
123
(1987) 123./ 2 / Sdgard B., Zemmouri J. and Macke B., Optics Comm.
63
(1987) 339./ 3 / Mandel P., Optics Comm.
