DYNAMICS OF THE SYSTEM

We have used an iterative evolving grat ing approach'8010 " to model the dynamics of this system, the equations for which are shown below;

Interaction Region

M i r r o r w i t h R e f l c c t i ' i t ) r . f o r \>. a n d Refl ec ti vi ty r * f o r \ z *

Fig. 4. Diagram of the interaction showing the geometry of the beams involved.

4 J. Opt. Soc. A m n/Vol. 10, No. 5/May 1993 J a a tin e n ct at. c\An dz d A iin dz d A 2, dz d A dz d A1« ~d7 cLA, dz g m 2"< fy ' A Ct A j, t g,„ A r - a A 2,„ , g , A p - a A2 ,, g,n A r + a A 3m , g , A r + a / \ •),,

- gm A3,,, ~ £,/Vi. + a Ar, Tc\g„, y — -— = —[ Ar A 2,„ + A rA3,„) - g,„, at / „ ^ = } ( ApA 2, + / L A 3>) - g. . at In (la) ( l b ) (lc) (ld) (lc) (If) d g ) ( l h )

As me nt i one d above, there is no dir ect coupling bet ween the mul t i mode and the single-mode fields; however, the two ar e l i nked by the means of the total int ensi t y / 0. The boundar y condi t i ons for this system ar e

/y O ) = 1, / \ 2in(0) = ( J , ( 0 ) , Ar,(0) = e,A„(0),

Arid) = 0, A3„,(d) = rmA 2r,,id),

Ai , ( d) = r \ A 2' i d) + p(r, - r „ ) A2m( d ) . (1 i)

The last b o u n d a r y condition describes boot st rappi ng of the si ngl e- mode r e f l e c t i v i t y to the mu l t i m o d e r ef lectivi ty. Thi s boot s t r a ppi ng arises because the mul t i mode gr at i ng r equir es all of its cons t i t ue nt modes to be r ef l ec t e d by a mi r r or of r ef l ec t i vi t y r,„ to sust ain itself. However, the si n g l e - mo d e c o m p o n e n t of the m u l t i m o d e r a di a t i on is u n a t t e n u a t e d a nd is reflected by a mi r r or of r eflectivity /•,, thereby c r e a t i n g a residual field. 'Phis residual field t he n e n h a n c e s t h e si ngl e- mode r e f l e c t e d be a m. T h e qua nt i t y p des cr i bes the fraction of the mul t i mode r a d i a ­ tion t h a t is single mode.

'Phis s yst em of equations is solved iteratively to yield the evolution of the single-mode and the mul t i mode fields m the oscillator. Figures 3 a - 3 d show the evolution of these fields for var ious pa r a me t e r values. An e s t i ma t e for the p a r a m e t e r p was obtained by me a s ur i ng the fraction of radiation t h a t is t r a n s m i t t e d thr ough the t r a n s p a r en c y (see Fig. 1) t h a t also passes t hr ough the pin hole before oscillations begi n in the SLPCM. However, in order to obtain b e t t e r a g r e e m e n t bet ween e x p e r i me n t and model we used a val ue approximately twice this es t i mat e. The model, like t h e exper i ment s, d e mons t r a t e s the t r end t ha t increasi ng t h e di fferential loss causes a mor e rapid t ur n on of the si ngl e- mode radiation. As well as pr edi c t i ng the t r an s i en t behavi or of the system, the model also gives r e a ­ sonable a g r e e m e n t with the observed steady- state behav­ ior, p a r t i cu l a r l y in showing the tr end t h a t increasi ng the differ ent i al loss increases the fraction of the power in the singl e-mode component in the steady state.

More i m p o r t a n t , the model agr e es with the observed behavior in t h e case where the di f fer ent i al loss is zero

(i.e., when the r e is no mode selector). For this case the model predicts t ha t all modes receive an equal sha r e of the di f f r a ct e d pump power, and as t he single mode is only one of a large numbe r of modes t h a t compose the mult imode radiation, it will receive a negligible a m o u n t of the total power. Anot her i m p o r t a n t a r e a of a g r e e m e n t bet ween the model and the observed behavi or occur s for the case when the differential loss was r emoved a f t e r the steady s t a t e had been r eached (Fig. 3d) T h i s r esult deserves p a r t i cu l a r mention because as a n a l t e r na t i ve model t ha t is based on the interact ion of one p u m p be a m with two s e p a ­ r ated g r a t i ng s it could be used to expl ain the switching of power bet we en two physically s e p a r at e d S LP CMs . " In t h a t p a r t i cu l a r model, once s wi t c h i n g occurred, there was no way to swi t ch the power b a c k again. However, as Fig. 3d indicates, in the syst em descr i bed in this paper the power can be swi t ched at will b e t we e n the mult imode and the single-mode S LP C Ms simply by addi ng or by r e ­ moving the mode selector. T h i s reversi bility is a c on­ s e que nc e of the fact t hat, in t h e p r e s e n t case, the two g r a t i n g s exist in one region; i.e., they ar e not physically se par at e d.

T h e q u a n t i t y r in Eqs. (Ig) a n d (lh) is an intensity- de p e n d e n t ti me co n s t a n t c ha r a c t er i s t i c of the photorefrac- tive r esponse of the m a t e r i a l . 12 For all the cases shown in Figs. 3 a - 3 d we used the sa me i nput power, so it was appr o­ pr iat e to use the s a me t i me c o n s t a n t in the various s i m u ­ lations. Thi s gave s a t i sf ac t or y r esult s when compar i ng s i m u l a t i o n s of the g r o wt h of s i ngl e- mode power in the pr esence of di f fer ent di f fer ent i al loss (Tugs. 3a-3c). How­ ever, to obtain good a g r e e m e n t be t we e n the theoretical and t he observed behavior for the case where the t r a n s ­ par e ncy was removed a f t e r st eady st at e had been reached (Fig. 3d), it was nec es sar y to d e c r eas e the time cons t ant by a factor of 2 below' its pr evi ous value.

T h e effect of boot st r a ppi ng t he singl e-mode mi r r or r e ­ f lectivit y to the mul t i mode r ef l ec t i vi t y is not critical to the pr edi c t ed mode-selecti ng behavior. Si mi l a r behavior occur s wi t hout bootst rapping, except when the t u r n on of the si ngl e- mode field is del ayed and the t i me t a k e n to reach t he steady st a t e is i ncreased. F i gur e 5 shows a t y p i ­ cal behavi or of the syst em wi t h a n d wi t hout bootstrapping.

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1. 0

TIME (arb. units)

Fig 5. Theoretically predicted evolution of the multimode and the single-mode components with (solid lines) and without (dashed lines) bootstrapping. The parameters used in this model were

yd = 3.75; a = 0.0; em2 = 20 X lCTfi, e,2 = 1 X 10'*, r,2 = 0.7; r,„2= 0.6

J a a t i n c n et nl Vol. 10, No. 5/May 1993/J. Opt. Soc. Am 13 5

Steady-st ate solutions of Eqs. (la) — (Ih) ar e needed to d e ­ t e r m i n e how much single-mode power can be obtained as a f unct i on of differential loss. Unf or t una t e l y the complex­ ity of the differential equations ma ke s it unlikely t ha t e x ­ act solutions can be found directly. However, by linki ng FVVM and TWM processes and ma k i n g use of the solutions of an equi valent TWM system, it has proved possible to find appr oxi mat e solutions for the system described here.

FOUR-WAVE MIXING: A TWO-WAVE MIXING INTERPRETATION

Du r i n g a wave mixing process a photorefract ive material, such as b a r i u m tit anate, has no way of telling how many be a ms ar e involved or the way in which they combine. All the ma t er i a l knows is the r e s u l t an t gr a t i ng formed by the beams. Therefore, if a g r a t i ng produced by a FWM pr o­ cess has the s a me form as a g r at i n g produced by a T WM process, t hen from the g r a t i ng alone it is impossible to d e t e r mi n e which process pr oduced the gr ati ng. Co n s e ­ quently, for any FWM process t he r e will exist a TWM pr o­ cess t h a t produces exactly the s a me gr at i ng. We shall show t ha t this is indeed the case by considering the e q u a ­ tions for the st a ndar d t r a n s mi s s i o n geomet r y four-wave m i x e r 12 for the idealized s i t uat i on in which

abs orpt i on and the coupling c o n s t a n t is real, si t uat i on we can write

t here is no In such a d A 1 y — = - — (A 1 A 2 + A 3 A 4) A 2 , dz In (2a) d A > y , , — = — (A 1A 2 + A 3 A A A1, dz In (2b) d A3 y — - = “ (A , A 2 + A3 A4) A4, dz In (2c) — = — — (A (A 2 9" A 3 A 4) A 3, dz In (2d)

wher e A 1, A2, A3, and A4 r e p r e s e n t the field ampl i t ude s of the four b ea ms involved in this F WM process and all o t her symbol s have t heir previous significance. The s e e qua t i ons yield four const ant s of i nt e gr at i on:

A,2 + A22 = K x , (3a) A32 + A42 = k2 , (3b) /\ 1A 4 + A2 A3 = E1, /\ 1A3 — A 2 A 4 = /?2 • (3c) (3d) Wh e n we use these constant s, it is possible to expr ess the field ampl i t udes At and A< as a linear combi nat i on of the ot he r two field amplitudes, A i and A 2:

(E \ A2 + E2 Ai ), (4a)

— ( E \ A \ — E2A 2).

A, (4b) Cons equent l y the gr ati ng s t r e n g t h can then be expr essed

solely in t e r m s of A, and A 2. ~ { A \ A2 + I o A 3 A ) y /o / G 2 x [ A , A 2( £ , 2 - E22 + K x2) + E XE2( AX2 - A22) ] . (5)

However, the field a mpl i t ude s IIx and II2 given by Eqs ((hi)

and (6b),

B\ — ( Q \ A2 + Q2A)).

E1 (6a)

B2 — — ( Q2 A2 — Qi A ,),

E1 (6b)

produce exactly the s a me g r a t i ng if ( ^ and Q2 ar c given by

Eqs. (7a) and (7b):

Q2 - Q2 = E 2 - E f + A.V, (7a)

Q\ Q2 = — E \ E 2 . (7 b)

By simple s ubst i t ut i on it can be seen t ha t B x

the following dif fer ential equations:

and B2 satisfy d B x r -T-- = — BxB2B 2, dz K2 (8a) d ß2 r o o n j ~ „ B x B2 B\ , dz K3 (8b) wher e K3 - ß , 2 + B22 = —- ( Q, 2 + Q22) , A i r K3 (8c) (8d) T h e s e are, however, t h e well-known expressions for TWM in t h e t r a n s m i s s i o n g e o m e t r y . 13 Simple m a t h e m a t i c s shows t h a t for any F W M process (that is any K x, K'2, a n d

E\) there exist real values for Q, and Q2, which me a n s t h a t the equations can be solved uniquely. Therefore we can expr ess each of the F W M field variables as a l inear combi ­ nat ion of the two T W M field variables.

A, (2) = D(1£ , ( z ) + Di2 B 2(z), i E {1,4}, (9) where U,x and Di2 ar e cons t ant s . Although the t r e a t m e n t

given above considered the case of a purely real coupling c o n s t a n t y, the anal ysi s is, in fact, also valid in the more general sit uati on wher e the coupling cons t ant is complex F u r t h e r m o r e , when we use a si mi l ar approach, solutions to the F WM equat i ons in the reflection geometr y in t e r ms of T W M solutions in t h e ref l ect i on geometry can be found also T h u s in the ideal case wher e there is no absorpti on it is possible to e xpr e ss any F WM process in t e r m s of an equi valent TWM process.

6 T Opt. Soc Am. B/Vol. 10, No. 5/May 1993 Jaatinen ct at.

We not use this duality bet ween TWM and FWM pr o­ cesses to clarify a complicated F WM behavior by relating it to a T W M situation. In par t i cul a r , the relationship be ­ t ween the various field ampl i t udes involved in the FWM process discussed in the first p a r t of this paper is far from obvious, an d no clear solution pr e s e nt s itself. However, the s i m i l a r TWM situation in which one pump beam ( Ar )

supplies ener gy to two signal be a ms ( , and A2) in one int e r act i on volume is, by comparison, relatively s t r a i g h t ­ f orwar d to solve and easy to u n de r s t a nd.

T h e differential equations for such a TWM process ar e given below: d A r y . dz = ~ —A p { A\ A \/n + A 2 A 2), (10a) d A i y dz = A r A i A r , (10b) d At y — = — Ap A 2 A n . (10c) dz /„

In t hi s si t uat i on it is obvious from Eqs. ( 10a) — ( 10c), t h a t the r at i o of the two signal field ampl i t ude s is a c ons t ant t h r o u g h o u t the interaction volume. Thi s must be the case, since t he rate of change with r es pe ct to di st ance for both s i gna l s depends on the s a me var iable, the p ump bea m intensity, which means t ha t the rati o of the two gr a t i ng s t r e n g t h s is also cons t ant t hr o u g h o u t the interact ion vol­ ume. T h u s in the TWM case the int er act i on of one pump beam wi t h two noni nt er act i ng signal b e a ms leads to the f or ma t i on of two gr at i ngs with ampl i t ude s tha t ar e pr opor ­