EFFECTS OF SPATIAL DISTRIBUTION IN OPTICAL BISTABILITY AND AT PHASE TRANSITIONS OF THE FIRST KIND

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EFFECTS OF SPATIAL DISTRIBUTION IN OPTICAL BISTABILITY AND AT PHASE

TRANSITIONS OF THE FIRST KIND

N. Rosanov

To cite this version:

N. Rosanov. EFFECTS OF SPATIAL DISTRIBUTION IN OPTICAL BISTABILITY AND AT PHASE TRANSITIONS OF THE FIRST KIND. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-429-C2-434. �10.1051/jphyscol:19882102�. �jpa-00227612�

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EFFECTS OF SPATIAL DISTRIBUTION IN OPTICAL BISTABILITY AND AT PHASE TRANSITIONS OF THE FIRST KIND

N.N. ROSANOV

S.I. Vavilov State Optical Institute, Leningrad 199034, USSR

RBsumd

-

L e s r t k u l t a t s d * a n a l y s e e t de comparaison s o n t donnes des ef'fets a m t r i b u t i o n s p a t i a l e (principalement t r a n s v e r s a l e l dans l e s systhmes

~ p t i q u e s b a s t a b l e s (auec une augmentation d ' a b s o r p t i o n des rayonnements, d e s r 6 s o n a t e u r s non-lin4aires e t une r e f l e x i o n n o n - l i n b a i r e ) e t dans l e s sys-tbmes a v e c une t r a n s i t i o n d e s phases ( l i q u i d e

-

^vageur).

A b s t r a c t

-

R e s u l t s of anal s i s and comparison a r e g i v e n of t h e e f f e c t s of spatial (mainly t r a n s r e r s a l ) d i s t r i b u t i o n i n o p t i c a l b i s t a b l e systems

( i a c r e a s i a b s o r p t i o n b i a t a b i - l i t y , n o n l i n e a r r e s o n a t o r s and n o n l i n e a r r e n e e t i e 3 and i n a system w i t h phase t r a n s i t i o n ( l i q u i d

-

^vapour).

L

-

IDITRQDUGTLW

Spatial dis-tsibutivity: i a an i m p o ~ t a n t f e a t u r e of a p t i c a l b i s t a b l e systems (OBS ).

TBu& f errtare ia og p r a c t i c a l importanae, f o r example, i n p a r a l l e l o p t i c a l pro- cessing an& it makes t h e n a t u r e o f h y s t e r e s i s (and s t o c h a s t i c ) phenomena m d i - c . a l y d i f f e r a t frm that i n t h e previously i a v e s t i g a t e d n o n l i n e a r mechanical, elcsctricaS and o t h e r pointwise (lumped) a x s t e m . It seems that i n t h e f i e l d of o p t i c a l b i s t a b i l i t y t h e q u e s t i o n a r o s e , f o r t h e first time, 10 y e a r s ago, of

rir2cipa4, p o s s i b i l i t g of b s t e r e s i s . i n the s p a t i a l l y distribute& systems /I , 2 / ,

k ~ t h o u g b -3 i d e a s c o n r e r n l n g this phenomenon were put forward a l r e a d y i n 1980 /3,4/, these n o n t r i v i a l and v a r i o u s q u e s t i o n s a t t r a c t t h e a t t e n t i o n o f theo- rists and e x p e r i m e n t a l i s t s u n t i l 1 now. I n t h e p r e s e n t a p e r t h e conceptions of h y s t e r e t i c phenomena i n t h e d i s t r i b u t e d (ride-apertureP OES formed by now a r e r e p r e s e n t e d and analysed. We s t a r t from t h e s i m p l e s t b a s i c model ( i n c r e a s i n g a b s o r p t i o n b i s t a b i l i t y and t h e n a d d i t i o n a l f a c t o r s and e f f e c t s

,

a r i s i n g under d i f f e r e n t c o n d i t i ~ n s and w i t h d i f f e r e n t types of t h e OBS ( n o n l i n e a r r e s o n a t o r s and n o n l i n e a r r e f l e c t i o n ) , a r e considered. S p e c i a l a t t e n t i o n is paid t o t h e comparison of t h e e f f e c t s of s p a t i a l d i s t r i b u t i o n i n tXe OBS and i n systems w i t h phase t r a n s i t i o n of t2Le filrst kind.

To make a =re complete p i c t u r e some d a t a of t h e reviews /5,6/ a r e sunnuarased.

l o r e a t t e n t i o n is p a i d t o t h e r e s u l t s which were ob-bied a f t e r the publica- t i o n o f /5-7/.

2

-

E4SXC W E L

R a d i a t i o n w i t h i n t t e w i t y I i s - i n c i d e n t on a l a t e r a l f a g e o f t h e rod of t h e t h i - c k n e s s d; t h e rod a x i s is x. he rod m a t e r i a l is c h a r a c t e r i z e d by light-absorp- ti- c a e f a c i e n t i n c r e a s i n g w i t h temperature. The b i s t a t i i l i t y o f a s i m i l a r kind was p r e d i c t e d by E p s t e i n /8/ and i t w a s demonstated experimentally i n / 9 / . The t r a n s v e r s e d i s t r i b u t i o n e f f e c t s and b i s t a b i l i t y f o r t h e r a d i a t i a n beams were considered f o r t h e first time in ^/4,10/. It should be noted t h a t n o n l i n e a r i t y is o f a thepmal c h a r a c t e r and a feedback (and t s a n c v e r s e coupling as w e l l ) r e - q u i r e d f o r b i s t a b i l i t y is due t o a thermal d i f f u s i o n . The t r a n s v e r s e d i s t r i b u - t i o n e f f e c t s m a n i f e s t themselves in this model most c l e a r l y ,

For s u f f i c i e n t l y s m a l l r o d s i z e s i n y-

,

^z- d i r e c t i o n s , t h e temperature T , aue- raged over t h e rod s e c t i o n , obeys t h e equation

a T aRT

- I = ' , (I 1

where

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882102

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C2-430 JOURNAL DE PHYSIQUE

Here

$I ,

c and

h

a r e t h e d e n s i t y , h e a t c a p a c i t y and h e a t c o n d u c t i v i t y coef- f i c i e n t s , hvis t h e h e a t exchange c o e f f i c i e n t , T is t h e ambient temperature, R i s t h e l i g h t r e f l e c t i o n c o e f f i c i e n t a t t h e l a t e r a l rod face.

2.1

-

I n c i d e n t plane wave (I = c o n s t ) . Under c e r t a i n c o n d i t i o n s t h e equation F(T) = 0 f o r t h e i n t e n s i t y range

( 2 )

has 3 s t a t i o n a r y s p a t i a l l y uniform s o l u t i o n s T =

@

one o f them being an u n s t a b l e i n t e r m e d i a t e branch

@

(Fig.1). S i n c e t h & s % ~ ? & n s i t y of t r a n s m i t t e d r a d i a t i o n i s determined by t h e t%mperature, t h e thermal b i s t a b i l i t y comes along w i t h t h e o p t i e a l b i s t a b i l i t y . Under c o n d i t i o n s ⁽² besides s t a t e s

@

t h e r e a r e t h e f o l l o w i n g s p a t i a l l y non-uniform s o l u t i o n s o f equation (I) /5,8~?1/:

a) S t a b l e s w i t c h i n waves (S 1 , c ~ s r e s p o u d i n g t o t h e t r a n s i t i o n i n space between t h e s t a b l e s t a t e s

6

^{I and}

h5.

The v e l o c i t y of t h e movement of t h e S W f r o n t c h a q e s t h e s i g n over t h e i n t e n s i t y range (2). The v e l o c i t y t u r n s t o z e r o f o r t h e Xaxwell value" of i n t e n s i t y T o defined by t h e c o n d i t i o n

63"

@&

The SW i s analogous t o t h e wave of phase t r a n s i t i o n c h a r a c t e r i z e d by a smooth i n t e r p h a s e l a y e r , and the c o n d i t i o n ( 3 ) i s analogous t o t h e Maxwell r u l e of c o e x i s t e n c e o f two phases.

bf F o r I ^jk I thexe a r e s t a t i o n a r y s o l u t i o n s w i t h incomplete s w i t c h i n g ( t h e c r i - t i c a l nuclei?. Fo them t h e Bitstribut ^'on T ( x ) is symmetrical w i t h asymptotic v a l u e T ( L W ) =

6

^{f o r}^{I 2}^I ^and

&

^{f o r}^I

^<

^I

.

The extreme1 value T depends on i n t e n s f i t s a s is o h o h on ~ i g ? ~ . These d f s t r i b u t i o n s a r e s i m i l a @ t o t h e c r i t i c a l n u c l e i i n t h e theory o f phase t r a n s i t i o n s . Although u n s t a b l e , t h e former a r e important f o r s w i t c h i n g k i n e t i c determination.

c ) F o r t h e same r e a s o t h e e x ' s t e n c e of u n s t a b l e SW (between u n s t a b l e

@

and one of t h e s t a b l e

- 6

o r

6 -

branches) should be noted. The v e l o c i t $ of u n s t a b l e SW doesns t t u r h t o z e r a , i t s f r o n t always moves toward t h e r e g i o n of t h e u n s t a b l e s t a t e ( t h e r e g i o n occupied by i t i s reduced).

The e x i s t e n c e o f the s t a b l e S W r e s u l t s i n t h e d i v i s i o n of t h e s t a b l e ( f o r t h e s m a l l p e r t u r b a t i o n s ) s t a t i o n a r y s t a t e s o f d i s t r i b u t e d OBS i n t o s t a b l e ( g l o b a l l y ) and m e t a s t a b l e ones /12/. Due t o an expansion ( i n t h e form of SW running i n o p p o s i t e d i r e c t i o n s ) of f l u c t u a t i o n a l thermal outshoots t h e r e g i o n s of t h e lower branch. ( f o r I > I o ) and o f t h e upper one ( f o r I ( I o ) correspond t o t h e m e t a s t a b l e s t a t e s (Fig.1).

The t ernperature o u t s h o o t s ( a g a i n s t a metas t a b l e s t a t e background) a r e d i v i d e d i n t o t h o s e s w i t c h i n g t h e 0% i n t o s t a b l e s t a t e ( " o v e r c r i t i c a l outshoots") and d i s s i p a t i n g ones /II/. The k i n e t i c s of outshoots c h a r a c t e r i z e d by a w i d t h . R and a n amplitude ( a n extremal value) T is i l l u s t r a t e d i n Fig.2. Besides t h e above-mentioned elements, some feaVffiHes of t h e SB formation should be noted aa . w e l l as nomonotonic k i n e t i c s R ( t ) and T ( t ) . The p o s s i b i l i t y of an experimen- t a l determination of u n s t a b l e s t a t e s c h a y 8 8 t e r i s t i c s ( i n t e r m e d i a t e branch )

by means of t h e k i n e t i c s of a r t i f i c i a l l y formed l a r g e o u t s h o o t s i n v e s t i g a t i o n g i s a l s o t o be pointed out.

S w i t c h i n initiat*. Generally speaking even when I t 1 t h e f l u c t u a t i o n s c a n r e s u l t 1:: t h e m e t a s t a b l e s t a t e switching. But t a k i n g i n t o account t h e s m a l l p r o b a b i l i t y of t h e l a r g e f l u c t u a t i o n s and t h e s t a b i l i t y of m e t a s t a b l e s t a t e s w i t h r e s p e c t t o s m a l l p e r t u r b a t i o n s t h e s e s t a t e s have a very l o n g l i f e - t i m e

( e x c e p t i n t h e c l o s e v i c i n i t y of t h e brahch edge), Thus, t h e wide-aperture 0 s s w i t c h i n g occurs r a t h e r by a n o t h e r mechanism, corresponding t o a heterogeneous

( n o t a homogeneous) n u c l e a t i o n i n t h e f i e l d of h a s e t r a n s i t i o n s . I n o u r c a s e a l o c a l i n c r e a s e of t h e a b s o r p t i o n c o e f f i c i e n t i n t h e a r e a s w i t h a h i g h e r c o n c e n t r a t i o n of a b s o r b i n i n t r u s i o n s i s i m l i e d . One dangerous non-uniformity is enough t o s w i t c h t h e O& i n t h e m e t a s t a b f e s t a t e s t i l l when I = Ic<

.

For ^{I <}I

.

a s i n g l e non-uniformity does n o t p r a c t i c a l l y d i s t u r b t h e i n t e g r a l c h a r a c t e s i s t i c s o f t h e wide-aperture 0 s .

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t h e i n i t i a t h g switching non-uniformity wldth. 2 1 is shown

i q

^Fig.^J ^&der

conditions c h a r a c t e r u s t i c of Ge f o r 1=d=0,2 cm, LC= 20 w/om

.

For I < 1, t h e non-uniformity deforms t h e temperature p r a f i l e in t h e region sf it8 l ~ c a t i o n only, But f o r I) I a s t a t i o n a r y temperature d i s t r i b u t i o n with m e "me- t m t a b l e N a s y q t o t i c s no longer e x i s t s . Therefore, t h e i a t e n s i t sfowly r i s i n g up t o t h e level+= th% Switching t o t h e hi&-temperature branch

f~

^not

l i m i t e d by t h e non-unifo&tY location. A s they a r e formed and spread, t h e Sw w i l l gradualLy sw&tch ax1 t h e system t o t h e high-temperature s t a t e ,

2.2

-

kWsteresis of p r o f i l e s ( S p a t i a l h y s t e ~ e a i s ) , , Above-mentioned elements of t h e plane-waves theory allow us t o d i s c r i b e t h e b i s t a b i l i t y and h y s t e r e s i s ghe- nomena f o r t h e c-e of r a d i a t i o n wide (with r e s p e c t t o t h e SW f r o n t width) beam inckdent on t h e d i s t r i b u t e d OBS /4 ,lo/. The diagram of t h e temperature p r o f i l e s determination is given by Fig.4. Note t h a t f o r t h e beams w i t h unimodal i n t e n s i t y p r o f i l e and t h e maximum i n t e n s i t y &=I/,0 t h e b i s t a b i l i t y is possible only when

I , : I,< I + .

(4 1

For t h e beams w i t h t h e l e s s e r i n t e n s i t y (I

<

I ) only the smooth p r o f i l e s of t h e I-st type (Fig.41, a r e r e a l i z e d , and f o r I

3

l,rOonly p r a f i l e s of t h e 2-nd type ex- ist with spatial s n i t c h i n g between t h e brffnches i n the v i c i n i t y of I=lo. Inten- s i t y Imslmly ~ i s i n g from t h e small values up t o I

,

t h e temperature p r o f i l e is a m o t k , of type I. Then, f o r I g I

,

i n the c e n t r a l p a r t of t h e beam a sharp tem- p e r a t u r e outshoot appears wich l a t e r , even i f t h e i n t e n s i t y I is s t a b i l i z e d , broadens in t h e form of oppositely d i r e c t e d SW, The SW v e l o ~ i ? ~ decreases and t u r n s t o zero when SW f r o n t approaches t h e region with t h e r a d i a t i o n l o c a l inten- sit3 I

.

S o , t h e switching on occurs n o t simultaneously over t h e whole beam sec- tion, But only i n i t s narrow zone

-

t h e moving SMr front. The switching time of t h e wide-aperture OBS i s determined by t h e SK velocity.

The witehaing off w i t h a subsequent i n t e n s i t y I decrease occurs i n a d i f f e r e n t ray. W l t b 5, 9pproacheing E the s i z e of t h e sw9tched on O E S reqion gradually decreases w t z l i t disappe&e enterely. F o r I,> I due t o m e t a s t a b i l i t y o f the above-mentioned lower branch p a r t , t h e I-st type $0 f i l e is metastable.

3

-

P D D I T I Q U L FrtCTQRS

The Lsportant f e a t u r e s of t h e examined simple d i s t r i b u t e d BBS a r e t h e following:

-

a smoothness and s u f f i c i e n t width of r a d i a t i o n i n t e n s i t y p r o f i l e (5.1) ;

-

ssstem u n l i a t n e s s ( i n the d i r e c t i o n x , ^{3 - 2 1};

- -

one-d&mensicun geometry the existence of only aae ( J -3 ; component ( 3 . 4 ) .

The violsttion of these r e s t r i c t i o n s leach t o i n t e r e s t i n g physics whose appropri- a t e discussion is beyond the scope of ^thisreport. W e s h a l l c e n t e r our a t t e n t i o n on t b conditions under which t h e main r e s u l t s of t h e basic model theory a r e maintained.

3.1. An asymptotical c h a r a c t e r of t h e s p a t i a l h y s t e r e s i s d e s c r i p t i o n f o r t h e be- ams w k s e width w))l i s i n the f i r s t place manifested i n t h e switching k i n e t i c s . The i d e a s of non-int8racting SW a r e v a l i d i f t h e i r f r o n t s a r e removed from one an o t b e r (with r e s p e c t t o t h e f r o n t width 1 1. Therefore f o r r a t h e r narrow beams t h e switching k i n e t i c s d e s c r i p t i o n by means0of t h e SW w i l l not be accurate. The b i s t a b i l i t y condition ( 4 ) is l e s s s e n s i t i v e t o a beam width decrease. The width m u s t however be g r e a t e r than some i n t e n s i t y dependent c r i t i c a l value ~1~~ otherwise bf s t a b i l i t y disappears /6/.

'3.2. The boundary conditions manifest themselves n o t only i n t h e rod edges v i c i - ni tar. Like t h e above discussed case of non-unif ormi t y

,

t h e boundary conditions can i n i t i a t e a progressive switching of t h e whole wide-aperture OBS i n t h e form of th SW. L e t , f o r example, a semi-infinite rod ( x 0 ) be heated by t h e radia- t i o n with i n t e n s i t y I=const within t h e r a n e a f tfre lower branch m e t a s t a b i l i t y condition ( 4 ) . The edge temperature (T

1 - f

r i s i n g up t o t h e value g r e a t e r than T ( t h e c r i t i c a l nucleus extremal ternpekeure), t h e SW ( @ ^-t@ 2 become exci-

$gd. The S W w i l l prapagate along x , w i t h v e l o c i t y e s t a b l i s h i n g a3 its f r o n t nio- ves awas from t h e rQd edge.

The b i a t a b i l i t y of t h e bounded rod with the l e t h ²¹i s o s s i b l e . 0 ~ 1 ~ i f 17%

For an i d e a l boundary thermoineulatlon we have? 4, b x s t a b l l ~ t g - ^LS

t a i n e & f o r an3 small rod s i z e . The c r i t i c a l s i z e m g p a n d s on t h e l n t e n s i t g and On

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C2-432 JOURNAL DE PHYSIQUE

t h e degree Q9 OBS t b e m a l connection with its environment, decreasing as t h e connection weakens /6/.

3.3, The l o n g i t u d i n a l d i s t r i b u t i o n becomes important f o r t h e l a r g e rod thickness d. I n this c a s e t h e i n s t a b i l i t i e s of t h e s t a t i o n a r y d i s t r i b u t i o n s and autamoda- l a t i o n a l regimes may a r i s e /14/, Basic model is s t i l l Valid f o r s u f f i c i e n t l y s W l rod thickness, when the temperature changes s l i g h t l y .

The two-dimensional geametry ( t h e s a p l e i s a p l a t e ) would become pronounced for t h e narrow beams w.hose width a r e comparable with 1

.

Otherwise t h e d i f f e r e n c e from the l o c a l l y one-dimensional d e s c r i p t i o n i s n o t l a r g e both f o r two- and three-dimensional geometry /II/.

3.4. The medium s t a t e may be characterized n o t only by one macroscopic parameter

-

temperature, but a l s o by Concentration of p a r t i c l e s , c o n s t i t u i n g t h e medium, and by the d i f f e r e n t temperatures of d i f f e r e n t p a r t i c l e s kinds, f o r example, ele- c t r o n s and ions f o r t h e plasma high-frequency heating. Under c e r t a i n conditions t h e s e a d d i t i o n a l parameters a r e expressed i n terms of t h e e l e c t r o n i c temperature onlx, and we come -to s l i g t l y modified one-component d e s c r i p t i o n /I5/.

The s i t u a t i o n is dkfferent if t h e i n t r o d u c t i o n of t h e new components r e s u t s i n t h e i n s t a b i l i t i e s growth I n a sense t h e nonlinear r e s o n a t o r s and o t h e r purely o p t i c a l schemes art:, a s a minimum, two-componental ( t h e f i e l d is determined bg its amplitude and phase),

4

-

OTHEfi OBS

4-1. g o n l i n e a r r e s o n a t o r s (NR)

,

excited by an e x t e r n a l r a d i a t i o n , a r e r a d i c a l l r d i f f e r e n t depending upon nonlinear medium and resonator c h a r a c t e r i s t i c s . The ca- s e of Low-quality ~ ~ e s o n a t o r and of s t r o n g d i f f u s i o n ( i n the medium i t s e l f /lo/ a r of h e a t i n t h e NR with thermooptical n o n l i n e a r i t y /16,I7/) is t h e simplest t o anaXyze. I n this case the d i f f r a c t i o n c o n t r i b u t i o n t o t r a n s v e r s a l coupling of l i g h t p i x e l l s is n o t l a r g e , and we come again t o t h e equation of tgge ( I ) . An a d d i t i o n a l NR f e a t u r e is t h e existence of m u l t i s t a b i l i t y t h a t r e s u l t s i n t h e gre- a t e r v a r i e t y of SW types i n this case. The p r o f i l e h y s t e r e s i s was experimentally demonstrated i n /18/, and experiments on t h e SW were described i n /IT-19/, The i n t e r e s t i n g autowave e f f e c t s a r i s e wheg t h e i n s t a b i l i t i e s grow i n d i s t r i b u t e d

"

two-componental" system with two competing n o n l i n e a r i t y mechanisms /20/.

If t h e r a d i a t i o n d i f f r a c t i o n is a dominant mechanism of t r a n s v e r s a l coupling, m e n t h e t ~ a n s v e r s a l d i s t r i b u t i ~ n leads t o t h e growth of t h e i n s t a b i l i t i e s i n a number of t h e wide-aperture DIR /? ,21,22/. To suppress them t h e s p a t i a l filtrati- a n of the r a d i a t i o n w i t k i n the NR should be made u s e of, I n t h i s way the s p a t i a l frrysteresis, Sk a d switching k i n e t i c s analogous t o above-mentioned ones were ob-

t a i n e d f o r NR w i t h a d i f f r a o t i o n a l t r a n s v e r s a l coupling / ~ , I I , 21-2~/, One more WR f e a t u r e i s t o be noted. The SW v e l o c i t y depends on t h e angle between t h e axes of t h e beam and ~ R & , ~ h u s , t h e v e l o c i t i e s and "Kaq$ 1 va?u_~s"(py i n @ p s i t d i f f e r f o r the SW ~ n n i n g i n t h e d i r e c t i o n s x and -x (

IP b b I,=

^#¹ ^1. ^{h e de-}

pendence v ( ⁾ is e s p e c i a l l y sharp f o r t h e i n e r t i o n e s s nonligear mgdia, f o r wuch the b i s t a b i P i t y is maintained f o r a r b i t r a r y wide beams only f o r s u f f i c i e n t - l y small

8

^/3,22/. ^Ifnontfjre

r i t x

i s i n e r t i a l , then ^theinfluence of

0

i s n o t s o great. The dependence I,

(8

⁾ ^is^@o ^{n on}F ~ g . 5 f o r t h e NR w i t h thermooptical no-inearity /13/. The knowledge of Ia

(3

⁾ allows ^US t o define t h e b i s t a b i l i t g conditions m d the s p a t i a l h y s t e r e s i s c h a r a c t e r f o r the oblique incidebce of t h e r a d i a t i o n an t h e N l i /22/.

4-2. Nonlinear beams r e f l e c t i o n , The normal incidence was studied of a c y l i n d r i - c a l s-polar zed beam with amplitude p r o f i l e E . ( x ) on t h e boundary of t h e medium w i t h e1eetr:cl permettivity € = E p

+ ^EL ( E l *

= / 2 4 / . T O avoid t h e beam break-up t b e case was chosen when E < 0. Flg.6 shows t h e existence of two d i f f e r e n t s t a - t i o n = ~ p r o f i l e s of r e f l e c t g d beam i n t e n s i t y f o r t h e same p r o f i l e o f i n c i d e n t beam i n t e n s i t y .

5

-

^PHASETWSITION "LIQUID

-

VAPOUR"

A simple c o n t i n u a l model f o r a homogeneous system i s considered where two phases ( " l i q u i d n and wvatp~ur") d i f f e r only i n t h e d e n s i t 3 value

,

temperature T being p o t e n t i a l constant* The ^U^, c0nsta:acy: phase equilibrium is achieved f o r the btaxwe

P

1 condition of chemical

a.

For V f i U. t h e mass and impulse c u r r e n t s ap- pear whose d e s c r i p t i o n i n terms of '*quasithermodynamical method" /25/ i s

(6)

/LC = / l l o ( T )

^f

T . F(9) + 6 ~ 9 -

⁽⁶⁾

X t s n o n l m a l i t y (b#U) r e f l e c t s t h e presence o f s u r f a c e tension. Nonlinear f u n c t i - o;n F( ) corresponcfs t o Van-der-Waals s t a t e equation. F o r T c T t h e r e a r e 3 spa- tia- uniform. s o l n t i a n s of (5 ) corresponding t o van-der-Waalsc branches ( i n t e r - mediate branch being unstable),. One can a l s o c o n s t r u c t t h e c r i t i c a l nucleus and tke waves of phase t r a n s i t i o n s (SW) betueen d i f f e r e n t phases w i t h smooth d e n s i t y p r o f i l e s ,

The s g a t i a f h x s t e r e s i s c o n s i s t s i n t h e occurence of two s t a t i o n a r y d e n s i t y d i s t - r i b u t i o n s ( x ) in t h e system w i t h a f i x e d smooth s p a t i a l d i s t r i b u t i o n of t e - mperature !J?(IY? The ^pr o f i l e s (XI a r e o b t a i n e d by t h e c o n s t r u c t i o n analogous t o that i n Fig.4. The r e w o n $ T d t h e following. The Maxwell c o n d i t i o n d i s t i n g - u i s h e s t h e r e g i o n s i n t h e system w i t h a smooth temperature non-uniformity where t h e S W f r o n t s rnay stop. The I-st (smooth) d e n s i t y p r o f i l e does n o t i n c l u d e any i n t e r p h a s e t r a n s i t i o n , i t is f u l l y d e f i n e d by one Van-der-Waals branch, Its cen- tral p a r t c ~ x r e s p o n d s , under c e r t a i n c o n d i t i o n s , t o m e t a s t a b l e s t a t e s . The 2-nd d e n s i t y p r o f i l e e x i s t i n g under t h e same c o n d i t i o n s c o n s i s t s of l i q u i d and vapour s t a b l e s t a t e s . I t has a s h a r p jump i n t h e r e g i o n of i n t e r f a s e l a y e r , where t h e Maxwell r u l e is obeyed, The s p a t i a l h y s t e r e s i s k i n e t i c s is analogous t o t h a t

mentione& f a r t h e OBS.

TPLe d e n s i t y o u t s h o o t s may be c r e a t e d a r t i f i c i a l l y , f o r example, by an introduc- t i o n of a c a p s u l e c o n t a i n i n g a medium i n an i n t e r m e d i a t e ⁽with r e s p e c t t o environment) s t a t e i n t o t h e volune w i t h a medium i n t h e m e t a s t a b l e s t a t e . A f t e r t h e f a s t opening of t h e c a p s u l e t h e r e i s a p o s s i b i l i t y t o observe t h e development of S u b c r i t i c a l and o v e r c r i t i c a l l a r g e outshoots. S o , t h e i n t e r m e d i a t e ( u n s t a b l e ) Van-der-Waals branch d e t e r m i n a t i o n becomes p o s s i b l e by k i n e t i c s i n v e s t i g a t i o n s . Such experiments could r a d i c a l l y e n l a r g e t h e i n f o r m a t i o n on t h e phase t r a n s i -

t i o n s i n d i f f e r e n t media.

I ₀

_R

₀

Fig. I

-

The s t a t i o n a r y s t a t e s c h a r a c t e r i s t i c s . The curve 1

-ol,

²

- 0 2 ,

³ - g 3 , 4

-

^Tm.

1,

Fig. ²

-

The outshoots a g a i n s t t h e metastable s t a t e @ ( a ) and t h e i r k i n e t i c s ( b ) . The arrows i n & i c a t e time d i r e c t i o n .

Pig. 5

0

²⁰ ⁰ 40

0 ,

-

"Maxwell i n t e n s i t y v a l u e a M f J v s

(7)

JOURNAL DE PHYSIQUE

Fig. 4

-

Temperatu:re p r o f i l e s T ( x ) ( q u a d r a n t 1 1 ) . Q u a d r a n t I X

-

^;-shaped ^depen-

d e n c e @ ( I ) . C;uad:rant I Y

-

i n t e n s i t y p r o f i l e s I ( x ) : a ) I,>i*, b) I o < I m < I ~ .

F i g . 6

-

The i n t e n s i t y p r o f i l e s of Fig. 7

-

The f l u i d d e n s i t y o u t e h o o t s i n c i d e n t ( 0 ) and r e f l e c t e d (I , 2 ) beams. k i n e t i c s , T/Tc= 0.5, ~ O O / ~ , , , a ~ 9.C:.

Ro i s m o l e c u l a r s i z e .

n u s w c s

/ X / X o l o k o l o ~ , &.A. and S U ~ O V , A.I.

,

1zv.Vys~h.Uch.Zav. Radiof.

5f

(1978) U 5 9 .

/ 2 / Hosanov, N.K., Q p t - S p e k t r . 4 7 (1979) 606 /Opt.Spectr. (UaSR) 4 (1979) 3'35.l-

/ 3 / &osanov, N.N. and Semenov, V.E.

,

0 p t . S p e c t r o s c . (U~;E) 4 8 (1980

3

^59.

/4/ itosanov, N.N., P i s ' m a v ZhTF 6_ (1980) 7 7 8 / S o v . ~ e c h . ~ h x . i e t t . 6 (1980) '5'35/.

/5/ aosanov, N.N.

,

Izv. Akbd. Hauh S S B , ser.phys. 46 (19823 1886.

/6/ liosanov, N,N., Trudy Gos. Opt, I n s t .

22

NI93 (1985) 3,

/7/ G i b b s , H.it~. 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 L i n L i g h t w i t h L i g h t , Orlando,l9E5.

/ 8 / E p s t e i n , E ; . i . , Zh.lekh.Fiz. s ( 1 9 7 8 ) 1799%ov.Tech.Phys. & ( I 9 7 8 1 981:.

/9/ G o l i k , L.Z. e t a l , P i s ' m a v Zh. Tekh. F i z . (1981) 118.

/IO/ Rosanov, N.N.

,

Zh.Eksp. Teor.Fiz. 80 (1981) 96 /Sov,Phys.JETP /II/ Rosanov, N.N. and Khodova, G.V.

,

Kvantovaya E l e k t r o n .

/X2/ liosanov, N.N.

,

Opt. S p e k t r o s k .

a

(1987,) 211.

/IT/ Bobxlkov, D.B. e t a l , A b s t r a c t s , 5 t h All-Union Conf. on L a s e r O p t i c s (1986).

/I4/ S t a d r i i k , V.A., P i s ' m a v Zh. Eksp. Teor. F i z .

fi

(1987) 142.

/ I 5 / Hosanov, N.N., Zh. Tekh. F i e . (1984) 1634.

/16/ Karpushko, F.V. and S i n i t s y n , G.V., Zh. P r i k l . S p e k t r . (1978) 870.

/17/ G r i g o r i a n t s , A.V. e t a l , Kvant. L l e k t r o n . (1983) 17x4.

/I8/ k p a n a s e v i c h , *S.P. e t a l , Kvant. h l e k t r o n . I2 (1985) 387.

/ I 9 / G r i g o r i a n t s , B.V. e t a l , Kvant. E l e k t r o n . 11 (1984) 1060.

/20/ B a l k a r e i , Xu. _{I. e t}a l , Kvant. E l e k t r o n . 1 4 7 1 9 8 7 ) 127.

/21/ I i o s m o v , N.N. and Zemenov, V.L., Opt. C o a u n . 38 (1981) 475.

/22/ Rosanov, N.N.

,

^S^emenov, V. 6. and Khodova, G.V.

,

Kvant.Elektron. 2(1982)35-?.

/23/ Eosanov, N.N.

,

^Semenov,^V.L. and Khodova, G.V.

,

Kvant. L l e k t r o n . Y(1982)36t, /24/ Rosanov, N.N. and Khodova, G.V., Q p t . S p e k t r o s k . 6 1 (1986) 198.

/%5/ Ono, 3. and Kondo, S. i v ~ o l e c u l a r Theory of S u r f a c e T e n s i o n , Iloscow, 1967.

EFFECTS OF SPATIAL DISTRIBUTION IN OPTICAL BISTABILITY AND AT PHASE TRANSITIONS OF THE FIRST KIND

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