HAL Id: jpa-00227708
https://hal.archives-ouvertes.fr/jpa-00227708
Submitted on 1 Jan 1988
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
OPTIMISED INSTABILITY THRESHOLDS IN A NONLINEAR RING CAVITY CONTAINING A KERR
MEDIUM
W. Firth, S. Sinclair
To cite this version:
W. Firth, S. Sinclair. OPTIMISED INSTABILITY THRESHOLDS IN A NONLINEAR RING CAV-
ITY CONTAINING A KERR MEDIUM. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-385-
C2-388. �10.1051/jphyscol:1988291�. �jpa-00227708�
JOURNAL DE PHYSIQUE
Colloque C2, Suppl6ment au n06, Tome 49, juin 1988
OPTIMISED INSTABILITY THRESHOLDS IN A NONLINEAR RING CAVITY CONTAINING A KERR MEDIUM
W. J. FIRTH and S
.
W. SINCLAIRDepartment of Physics and Applied Physics, University of Strathclyde, GB-Glasgow G4 ONG, Scotland, Great-Britain
Abstract Stability analysis of an arbitrary finesse ring cavity containing a Kerr medium reveals an absolute minimum threshold for the driving intensity necessary for realisation of self-pulsing instabilities. This threshold is minimised not only with respect to cavity mistuning, but also with respect to the medium response time. The instability threshold is typically six times that for optical bktability. We have also generalised this model to include the effects of transverse diffusion.
1
-
INTRODUCTION:Prediction of oscillatory instabilities and chaos in passive ring cavities containing a Kerr medium in bot.h the short [I] and long 1121 response time limits has already been made by Ikeda and co-workers. One may ask under what conditions is the instability threshold minimised? This question of minimum thresholds is the main, and important topic of our paper.
We have re--examined the linear stability analysis, appropriate to the problem, and have shown elsewhere 131, that a parametrir decomposition of the characteristic equation renders possible a unifying analytical scheme for instability thresholds, without the need for the approximations adopted by most previous authors (e.g. high/low cavity finesse, long/short medium response times) [4-81. Using this model, and given only the cavity finesse, we have demonstrate the rxistence of an absolutr minimum in driving, fic.1~1 intensity ncc.essary for instability. This minimum occurs at uniquely defined values of mistuning and ~R/T, both of which, along with oscillation frequency at threshold, and threshold intensity itself are given explicitly by our analysis[g]. It is then possible to compare thresholds for OR and instability. Both are equal at very low finese. At high fjnese, the minimum instability threshold is some six times that for OB. We find generally that lowest thresholds occur in this very high finesse/long response time regime. Finally, we propose a possible physical mechanism for these instabilities.
2 - BASIC MODEL
We assume that the ring cavity under consideration is governed by the following delay-differential system of equations [1,2]:
where: R s entrance mirror reflect,ivity.
E(t) E internal optical field amplitude.
Ei s pump field incident on the cavity
.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988291
JOURNAL DE PHYSIQUE
t~ = c a v i t y round t r i p time = / c where is t h e t o t a l o p t i c a l p a t h l e n g t h of t h e c a v i t y .
@ ( t )
=
n o n l i n e a r phase s h i f t of o p t i c a l f i e l d due t o Kerr e f f e c t i n medium3 c a v i t y mistuning.
T
=
response time of medium.Our "feedback f a c t o r " B is r e l a t e d t o t h e c a v i t y f i n e s s e , and t h e c a v i t y f i e l d decay r a t e is given by (-PnB/tR). We assume t h a t t h e dynamics o f @ ( t ) a r e f u l l y described by a Debye r e l a x a t i o n equation (2). Fixed p o i n t s of (1) a r e given by E ( t + t R ) = E ( t ) = Es, and from (2)
eS
= IEsI2;as
being t h e s t e a d y s t a t e n o n l i n e a r phase s h i f t . The s t a b i l i t y o f a f i x e d p o i n t is i n v e s t i g a t e d by l i n e a r s t a b i l i t y a n a l y s i s . A t t h o i n s t a b i l i t y t h r e s h o l d , a small p e r t u r b a t i o n o f frlequency Q w i l l c e a s e t o b e damped and w i l l i n i t i a l l y experience exponential growth. The c a s e Q=0 simply g i v e s t h e OB switching p o i n t s . Excluding t h i s c a s e , we have shown [3] t h a t a l l o t h e r v a r i a b l e s i n t h i s problem can b e expressed uniquely and e x p l i c i t l y i n terms o f , w&tR, Y = t R / r , and B which we r e - e x p r e s s a s B = ~ - P w i t h P p o s i t i v e . In p a r t i c u l a r , we f i n d :( i ) t h e t h r e s h o l d i n t e n s i t y A2(w) i s c o n s t r a i n e d t o l i e i n w e l l defined bands, t h e n t h band l y i n g i n t h e range ((n-l)n,nn),,
( i i ) A2(w) has a minirmun w i t h i n allowed bands, and i s divergent a t band edges.
This minimum, a t w = b ~ corresponds t o minimising A2 with r e s p e c t t o ao;
optimal (O0 then being d i r e c t l y c a l c u l a t e d .
I n t h e low finesse, l i m i t (B=O) minimum A2 is i n f i n i t e , and i n t h e high f i n e s s e l i m i t (B=l) minimum A2=0, i n d i c a t i n g t h a t no optimum f i n e s s e e x i s t s ( s i m i l a r l y f o r OB). However, t h e r e is an optimum response time T, t h e e x i s t e n c e and e x p l i c i t e v a l u a t i o n o f which is our main r e s u l t . This optimum 7: is c a l c u l a t e d by minimising A2 w i t h r e s p e c t t o b o t h w and y f o r f i x e d f i n e s s e (through B ) , o r indeed w i t h r e s p e c t t o any two independent v a r i a b l e s . For convenience we choose x, d e f i n e d by x = cos(6st@0) and w, where:
W2 = xi:
+
(cOsh2p-X2) ( 1 + (E c o t h ~ ) ' )s o t h a t Y
Our o b j e c t i v e now is t o minimise A2(w,x) s u b j e c t t o f i x e d P and t h e c o n s t r a i n t s -1<x<l, w>cosM. The l a t t e r c o n s t r a i n t follows immediately from (3): t h e c a s e w=coshP corresponds t o -0. If we s e t w=coshP and minimise (4) w i t h r e s p e c t t o x a l o n e we o b t a i n t h e tuning-optimised minimum t h r e s h o l d f o r t h e Ikeda i n s t a b i l i t y i n t h e mapping l i m i t , T=O [3].
When w=cosM, we f i n d
a ~ ~ / a r
is p o s i t i v e i n d i c a t i n g t h a t t h i s is a t l e a s t alocal
minimum t h r e s h o l d f o r T>O.I n t h e g e n e r a l c a s e we f i n d t h e c o n d i t i o n f o r a s t a t i o n a r y value o f A2(w,x) t o be:
which b e i n g c u b i c i n coshP, (11) can b e solved a n a l y t i c a l l y . However, it is si.mple and direct.
t o s o l v e numerically f o r x(cosh/3).
W e f i n d t h a t (5) h a s no p h y s i c a l l y a c c e p t a b l e r o o t s i n t h e range 0<B<Bc=0.3827. n2min given by T=O is t h e n a g l o b a l minimum i n t h e range
Bat.
For B>Bc, however,two
v a l i d r o o t sof (11) exist, corresponding to a minimum and a maximum of A2; these roots coalesce at Bc.
For B>-0.4 the minimum A2 is than A2min given by T=O, which is then only a minimum, the global minimum being at finite t - the maximum given by the second root is a "ridge in the B-T plane between the global and local minima. These extrema of A2 are shown in Figure 1.
The solid curve is for r=O, the dotted curves are obtained from ( 5 ) . For moderate and high finesse values, we see that the lowest possible instability threshold is obtained at finite T.
Furthermore, at high finesse the absolute minimum thresholds are much lower than those predicted from the Ikeda map. We have shown that in the high finesse limit, Bal, the optimised thresholds (correct to first order in the parameter 8) are [3]:
For OB in this limit ~2 = (8/3J3)~~, so that minimum thresholds for oscillating instabilities and OB scale in the same favourable way with increasing finesse, the OB threshold being lower by a factor of about six.
Ikeda and Akimoto [2] found A2-83 for small 8, while Silberberg and Bar-Joseph 1101 have also addressed this problem - choosing to study
as
& A2, which has no finite optimum value of 7. Somewhat surprisingly therefore, (6), as well as our generalisation to arbitrary finesse, seems to be new.3
-
PHYSICAL MECHANISMFigure 1, and our analysis, suggest that the long [2] and short [l] response time instabilities must have physically distinct mechanisms. Firth et a1 [11,12] and Silberberg and Bar-Joseph [lo] have given an interpretation of the short-t instability in terms of sideband amplification through four wave mixing. We now suggest that the long-7 instability should be interpreted as Rayleigh-wing scattering discussed in this context by Silberberg and Bar-Joseph [lo]. Normally this process requires material dispersion to break the symmetry between the Stokes and anti-Stokes waves. We suggest that the cavity transmission function acts here as a pseudo phase mismatch, and evidence the fact that the oscillation frequency of 1.6/7 exceeds the 117 which one expects from the pure scattering mechanism [lo] as support for this assertion. At this Stokes shift, the anti-Stokes frequency is displaced from the Stokes by several cavity linewidths, and thus strongly suppressed by the cavity. This will allow the Rayleigh mechanism to dominate the four wave mixing mechanism, and hence be primarily responsible for the long-t instability.
In summary, we have identified and calculated the absolute minimum instability threshold in a Kerr cavity and proposed that its physical mechanism is of Rayleigh type in the most favourable case.
Equations (1) and (2) may be extended to include the effects of transverse diffusion.
Preliminary calculations indicate a raising of the OB threshold above the plane wave case, along with the possible existence of modulation instabilities. Such instabilities have been studied in the case of diffraction [13.14], but without addressing the questipn of optimised thresholds. It will be interesting to investigate a formal connection between these two distinct cases.
Our preliminary analysis indicates that the characteristic with finite wave vector K, is of the same form (and hence has the same general threshold behaviour) as the plane wave case K=O. The only formal difference between the two cases is that the effective nonlinearity n2 and response time 7 are reduced for finite K in comparison with K=O. With diffusion we find that oscillations with finite K are favoured if t exceeds the optimim response time calculated from the plane wave model. We find the threshold condition for such instabilities is when the tangent to the curve A2(7) passes through the origin. This is shown in Figure 2 for a high finesse resonator.
References
[l] Ikeda, K., Opt. Comm.
30
(1979) 257.[2] Ikeda, K., Akimoto, O., Phys. Rev. Lett.
48
(1982) 617.[3] Firth, W.J., Sinclair, S.W., "to be published" Journal of Modern Optics.
[4] Ikeda, K., Daido, H., Akimoto, O., Phys. Rev. Lett.
45
(1980) 709.C2-388 JOURNAL DE PHYSIQUE
[5] Lugiato, L.A., Narducci, L.M. Bandy, D.K., Pennise, C.A., Opt.
Comm.
43
(1982) 281.[6] Bonifacio, R., Lugiato, L.A., L e t t . Nuovo C i m . ,
21
(1978) 510.[7] Gronchi, M., Benza, W., Lugiato, L.A., Meystre, P., Sargent 111, M., Phys. Rev. A
3
(1981) 1419.[8] Abraham, E., F i r t h , W . J . , Optica Acta
3
(1983) 1541.[9] F i r t h , W . J . , S i n c l a i r , S.W., " t o be published" Optics L e t t e r s . [ l o ] S i l b e r b e r g , U., Bar-Joseph, I., J. Opt. Soc. Am., B l ( 1 9 8 4 ) 662.
[ l l ] F i r t h , W . J . , Harrison, R.G., Al-Saidi, I . A . , Phys. Rev. A ( 1986), 24413.
[12] F i r t h , W . J . , Wright, E.M., Cummins, E.J.D., O p t i c a l B i s t a b i l i t y 11 e d i t e d by C.M. Bowden, H.M. Gibbs, S.L. McCall (Plenum, 1983) 111.
1131 McLaughlin, D.W., Moloney, J . V . , Newel, A.C., Phys. Rev. L e t t . ,
54
(1985) 681.[14j Lugiato, L.A., Lefever, R . , Phys. Rev. L e t t . , 58 (1987) 2209.
Acknowledgements
One of u s (SWE;) wishes t o thank BTRL and SERC f o r f i n a n c i a l support, and M r . G. S. Mc:Donal tl
f o r h e l p f u l discussion. This work was c a r r i e d out i n p a r t w i t h i n t h e Stimulation Action Program of t h e European Community.
Fig.1
-
Extrema oT (4) s o l i d curve corresponds t c r T = O , d o t t e d curve is for f i n i t P T .Fig. 2 - Tu~ling opt.im:iserl mirrimr-I a s a f u n c t i o n trf soalctl r.c:..por;r:r. ! imr. f o r :i l r ~ i c l - I F ~ I I c . : ~ : ; ~ .
