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Transverse nonlinear optics: Introduction and review
Chaos. Soliton~ & Fractals Vol. 4. Nos 8/'9, pp. 1251-1258. 1994 Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-t)779/9457,00 + .~10
Pergamon
0960-0779(94)E0090-C
Transverse Nonlinear Optics: Introduction and Review L. A. L U G I A T O Dipartimento di Fisica dell" Universit~ di Milano, via Celoria 16, Milano 20133, Italy
A b s t r a c t - - T h e characteristic features of the field of Transverse Nonlinear Optics are briefly illustrated, as an introduction to this special issue of Chaos, Solitons & Fractals.
1. I N T R O D U C T I O N
The analysis of unstable phenomena in nonlinear optical systems has been the object of intensive studies, both theoretical and experimental, over the last 15 years (see, for example [1-7]). More recently, the main focus in this field shifted gradually from purely temporal effects to spatial and spatio-temporal phenomena, especially spontaneous pattern formation in the structure of the electromagnetic field in the planes orthogonal with respect to the direction of propagation. Even though Transverse Nonlinear Optics is basically a newly born field, the existence of such effects is well known since the earliest days of laser physics. As a matter of fact, in order to obtain the simple Gaussian T E M ~ transverse structure that is desired for most applications, one introduces apertures or other means into the laser cavity. Otherwise, the system generates spontaneously more or less complex patterns. These phenomena were, however, mostly considered as undesirable or difficult to be understood and controlled, even if there were pioneering works which analysed several interesting aspects of these effects [8-22]. In the last decade, and especially in the last few years, the interest in these phenomena emerged with evidence and activated systematic investigations in this field. It is by now clear that the spatial and spatio-temporal effects in the transverse structure are similar to the phenomena of pattern formation, turbulence, etc., that are familiar in other fields such as, for example, hydrodynamics, nonlinear chemical reactions, or biology. These phenomena represent the general object of study of Haken's synergetics [23] and of Prigogine's theory of dissipative structures [24] and can now be studied in optics, in systems whose dynamics is governed by the fundamental laws of radiation-matter interaction. In the case of the laser, the analogy with hydrodynamics can be substantiated also formally [16, 25, 26]. G i n z b u r g - L a n d a u [26-32], or Kuramoto-Sivashinsky [33, 341, or Newell-Whitehead [32] type equations have been derived and used to describe spatio-temporal phenomena in optics. As a matter of fact, optics addressed the issue of pattern formation and transformation much later than the more traditional fields of nonlinear chemical reactions and, especially, hydrodynamics. This fact is due to the circumstance that, until recently, it was not 'natural' to realize optical systems with a large aspect ratio; this is, however, no longer true now, for example in the case of high-power lasers or surface-emitting lasers. On the other hand, optics presents at least two special features that are interesting and stimulating. First, 1251
1252
1,. \. 1.~,,1\1,,
optical systems are very fast and havc a large frequency bandwidth: hcncc t h e \ lend themselves naturally for application, for instance to t e l e c o m m u n i c a t i o n s and to informati~m technology. O n e relevant example of the useful application of optical structures is alrcad~ provided hy the soliton transmission in optical fibres. The invcstigations of t|-ans~crsc nonlinear optics offer, in principle, the possibility of an approach to parallel optical information processing, bv encoding information in the transverse Stltlcturu e l lhc electromagnetic field. The researches in this direction arc at a rather initial slag~. (suc 135-4fl1) but therc is a definite evolution, and some i)f Ihc papcrs in this issuc prmid< ~ interesting hints in thai direction. The second special featurc is that ~ptical s\,stems ha~c the tendencv to display interesting q u a n t u m effects at roonl t c m p e r a t u r c . Recent \ c a r , have witncssed the start of investigations on q u a n t u m effects in trans'~erse optical SlltlCttllt.,,, (sec [41-451; these researches ure relevant from the viewpoints ~1 l:undamcntal quanlttln mechanics, of q u a n t u m noise reduction {squeezing) and, hol)efully. ~,l applications, l~,J example to noiseless transmission of images. The mechanism which gives rise to pattern formation is. as usual, Ihc combination ,~i nonlinearity and of coupling bctween the different spatial poinls. In the case ~d optical systems, this coupling ix provided b \ diffraction, which plays the same role as diflusioJ~ in nonlinear chcmical reactions and in biolog}. In lhc paraxial approximation ~hich p, currently uscd in these studies, diffraction is described by the transverse l,aplaeian, lh;fl i,. the sum of the second derivatives ~ith respect to the t,ao transverse spatial variable,, Both stationary and dwmmical patterns can be formed in this way. The kind of pattcln d e p e n d s crucially on the type of nonlinearity which may he, for cxample, quadratic in file case of parametric d o w n - c o n v e r s i o n and second harmonic generation, cubic in Z' ~' media (four-wave mixing), or the saturable nonlinearity which governs the singlemodc gain in lasers. The nonlinearity determines the c o m b i n a t i o n of wave xectors in the transverse plane, which in turn produces the structures that c m c r g c . General reviews of the field of transverse nonlinear optics can he found in rcfs ]4~ 51 ] The investigations have covered both theoretical and experimental aspects, and haxc c o n c e r n e d the case of passive (i.e. without population inversion) systems [3{L 31. 35. 3i~. 39. 52-92], the case of active systems such as lasers [25 2g. 3 2 - 3 4 , 37, 38, 4(l, g3 12L)t and svstems with gain which does not arise from population inversion {e.g. p h o t o r e l r a c l i \ e s , optical parametric oscillators) [ 13()- 137]. Thc sxstcms that have been studied can I~c. ~crv roughly speaking, dividcd inI~ l\~,~ classes. The first class is c o m p o s e d of systems which display translaticmat svnlnlctr\, t | c n c u , ii thcrc are mirrors, they must be plane mirrors: if there arc driving fields, they must bu plane waves. Thesc systems allox~, the study of the onset of t~atlerns from tt honlogcilcotl~, state, that is. the s p o n t a n e o u s breaking of the translational svmnletr\. This p h c n o m c m m arises w h e n e v e r the h o m o g e n e o u s staiional-\ slatc beconles tillstablc against the grov~lh ~)t small i n h o m o g e n e o u s perturbations, for example small-amplitude waves with a nltidulaiud sinusoidal configuration. It may happen that, when the wavelength of the modulation lies in an appropriate range, Ihe perturbation grm\.s instead of dying away. In this casc !hr, svstcm forms s p o n t a n e o u s l y a pattern which, according to the nature of the instahilit\, max be stationary or dwmmical. C o m m o n examples of patterns which may a p p e a r arc sn-ipc~ (also called rolls due to the similarity with the R a y l e i g h - B e n a r d instability in fluids) :lnd hexagons. Each of these configurations may display defects [2{} 22], for example dish,c> tions in roll patterns [137] or p e n t a - h e p t a defects in hexagonal patterns [7S]. Also, lhL' h o m o g e n e o u s configurations can show defects called optical phase singularities or optical \ortices [28. It2, 134. 13S, 13g] centred at points of zero intensit\ el the electric lichl in the transverse plane.
Introduction and review
1253
The second class of systems which have been extensively investigated is characterized by the inclusion of spherical mirrors; this is the standard situation for optical systems contained in resonant cavities. In this case the system has no translational but, at most, rotational symmetry in the transverse plane. The spherical mirrors confine the radiation field in the region which surrounds the axis of the system, so that the field configuration is boundary-dependent. The dynamics of the system is governed by the nonlinear interaction and competition among the different discrete modes (of G a u s s - H e r m i t e or G a u s s Laguerre type) of the resonator. When the Fresnel number of the cavity is at most of order unity, instead of the periodic (in space) patterns met in the first class of systems, one finds localized structures composed by a limited number of peaks and, possibly, phase singularities. These structures arise from the contribution of a reduced set of cavity modes. The variety of different configurations, both of stationary and of dynamical nature, that can emerge in this way is spectacular. On the other hand, when the Fresnel number is very large the behaviour of the system becomes more and more boundary-free and similar to the case of the first class. The field configurations arise from the contributions of a very large number of cavity modes, so that the modal composition itself becomes basically irrelevant. Space-time chaos is one of the common scenarios in this limit [130]. Of course, there are cases of systems which are intermediate between the two classes, for example systems with plane mirrors driven by external field with a Gaussian or a flat-top profile [90]; and there are systems, the behaviour of which is governed by propagation, in combination with diffraction, without any infuence from feedback such as that provided, for example, by a cavity resonator (e.g. see [140-142]). Laser arrays have also been the object of extensive investigations (e.g. see [143, 144]). Even if the list of references in this article is long, it is in no way complete and, certainly, the selection of papers is influenced by my personal taste and limits of information. Hence I apologize for all omissions. This special issue collects contributions from most leading groups active in the field of transverse nonlinear optics; the quality of the papers and the coverage of subjects ensure that this volume provides a rather faithful description of the state-of-the-art in this domain. Undeniably, the field of optical patterns developed mainly in Europe. This was in part the cause and in part the consequence of the illuminated support provided by the Programs of the Commission of the European Communities to the researches on Optical Instabilities, Opti.cal Chaos and Optical Patterns. An especially important role was, and is, played by the E S P R I T Basic Research Project 3260 TOPP (Transverse Optical Patterns) and 7118 T O N I C S (Transverse Nonlinear Optics). A significant proportion of the papers in this issue arise from research activities in the framework of TONICS.
Acknowledgements'--I am grateful to Professor M. S. E1 Naschie and Dr A. Boehm for their kind invitation to organize this issue. I thank the support of the TONICS Project and, especially, Drs Ingemar Hussla, Manuela Pinheiro and Brendan Hawdon of the ESPRIT Basic Research staff, for their wise and continuous assistance and guidance provided to the partners of the projects TOPP and TONICS. Thanks are also due to Drs Massimo Brambilla, Fabrizio Castelli, Alessandra Gatti, Irene Marzoli, Marco Pinna, Roberta Pirovano, Franco Prati, Alice Sinatra and Martino Travagnin for their continuous help in the preparation of this issue.
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74. A . A . s\[{lll~l~," t.'\. B. z\. 8{1111~,t)11and P,. "~'akit¢, l.a~cr I'hv~. 1. 3c~cJ t lqCJl I. 75. A . Potrossiail. M. Pinard. A . M a i t r e . . 1 . "f. ('ourloi', and (;. (h-vnb¢rg. Irall~,\c_'r,,c-pilttt.'i'll lcUll/all~qt hq ¢Oti llt¢l-l]ropagatillg laser b¢alll~; in rtibiditllll rapt)ill, EU#ol)l(v~. /.,'st. 18, 138t) ( ICJ92i. 7(~..I. "st. Courtois mid (-i. ( ] r y n b ¢ r g , Spatial patL,2rll f o r m a t i o l ] 1or ct)tilllC'lprol-~at!Cltilig beN|ill,, ill ci k ( r l mCdlliiii a simpl¢ modol. Opl. ( o m m . 87. ISf~ (lCJ92). 77. W. J. Firth. A . J. Scroggi¢, (i. S. M c D o n a l d {illd I.. ,~t. Lugiato, ]lcxa~onal paltClip, ill ()ptical hiM|lib|hi\ Phys. Rcl'. A46, R3(~()9 (1992). 78. I~. ('h~.1117, ~V. J FiFlh, R. lndik. ,I. V. Molonmv and [{. M. \ ~ l i g h l , ['hiec-dimeilsional <,lllltli;lli()llX ,)1 dcgt_'n¢ral0 t_'Otilllc'rl)rol~agalillg bt.'ai11 instabilities in tl ncmlim'ar in(ditilil. ()[)t. ('<)him. 88. lfi7 (lcJc)2) 7cj G. P. D'Alc,~sandro anti "~'..I. Firth. Hexagonal stxilial pattern foi ;i K e r r kli¢¢ \~itll a lt_'cdha(k Iiiii1 H IJhv~ 17('1'. A46, 537 (lt)CJ2). ~1i ~. R. Lit| and (i. lndchmlou\~, Periodic ;llld cha<>tic ~l)~lliOl¢lllpt)r;l[ ,,ICll(n i[1 ~t ]'~hci,q:-¢oili/lgalc lltf%till;{{l,i usiIIg, pholorefracti',c B a T i O ; [;htl~,t2IeotljHg{lIC i l l i r r o r . . / ()/W..S'oc..'t/H. Bg. 15i17 (lt. ( ' o m m . 91. 321 33(l(IC)92}. <<,2..1. Nalik. 1.. M. Hoflci-. (}. 1_ Lippi. ('h, 'v'oigmi'd ~lnd ~\'. I.anoc. Yltillk~.¢i$¢ ()lViual bistabilitv cmd t()illl;ll!<,i! cif ti-any, vt21"<~¢ MrtietUl-O.'4 hi a ~odium-filicd P a b r y - P e r o l rt_'<,Oilalt)r. ]Jhl,'~,. R('v. ,\-15. R4237 (1c)c#2l St M. t [ a ¢ i l ( r m a n , Period-doubling bifuraelions and nlodulutional in~tabilit\ in the nonlinca) ring c';i\ll,, :ill allCllvtiea] study. OpL Lct~. 17, 7c}2 (19c,i21. ~4. M Branlbil]~i, (i, Bloggi and F. Prclti. S p a t i o i ( n l p o l a l pallc'ln 1orill |it t)~i~si~c ~2). ~(). ~'1. ~tiLler {llld F. K{li~,t+'l. 8p~lLia] ~;ylnll]7[ly brcakillg all(I c.'ot_'M~lt.'llCt_' ot ;illl~lC.'[t)l <-, ill {I IlC)lliillc'{ll I111~ c;)\H\ .lpph'. Phys. 11~5, 13,<'4 14-3 (19'-,12). ,',;7 E. V. |)cgtyarc'x and M. A . \/orontso~<. Spatial dvilallliCy; ot two t.'Olllpoilcill optical ',\,M¢in %~it]1 lai~c. ,<';~i~" int¢iaction. Mo/. (. )'vw l.kl. ('rvvr. <;ci 7"cchno/. 3, 295-31(i 119c~23. S~. S. It. I.iu ;lilt] (l. fndt_'b¢louw. ~l'~aliotC, ml'loral piillOrll anti \t)lticcx llit){iOll ~, ill l~lla'~c-ct)tliugcil¢ rc~,t)lltlltli~ Opt. ( ' o n l m . f i l l . 442 (IC~C)3/. ~t) ~|. }|;IcllCl'lllall. fkeda instability and {l{lllN\Cl~.k ' clI¢ct~, ill vl()illillc~ir rillg l¢~tiilalois, ()lr)[. ( <)~#11tl 11111. ;Xc! ( i tjcJ3 ). '-lit. P. Papofl. (}. 1)" Alc'>,~,alldl'(). (l. f.. ()pp,, aild ~ , \ . . 1 f:irih. Local ~illcl gh)bal elloots ()1 bottndaric,, ,~l optical-patteri~ ft)rmation in K c r l media, PJlv~. i~t,l'. A48, (~34 (Ic7931. c)l M. T a m b t i r r h l i , M. B o m l v i l a , S. V~'abllil/ ~iilcl ~. ~Cilltalllato, Hcxagt)nal]} pcillc'lil(d be'am fil~/lllCnt')83}. cJ4. W. J. Hahis, S. N. l.in and N. B. A b r a h a m , Route to n/ode-locking in a three-mr)de 3.39 um las¢l h~clticilng chaos iI1 the sccondar}, beal fr¢ql.lei1cy, P/ivy,. t~t,l'. A28, 2tJl5 2920 (1c)833. t)5. P. H o l l i n g c r and ('. Jtlilg, Single Ionl~itudinal-nlodc ]Hsur {1<~ '<1 discrclc d \ m i m i c a ] s}stc111..1. ()[H S J( tiM, B2, 218-225 (feSS). c)¢~. M. f.. Shih. P. VV. M i l o n n i :lnd .I. F,. : \ c k ( r h a h , M o d c l m g Icls(r instabililics cind chaos, .1. ()pt. 5,o( li~z 112. 13(i ( 1985i. c-i7. D, .l. Biswas alld R. (i, Harrison, E x p e r i m e n t a l ( x i d c n c c o I lhrcc-nlodc quasipcriodi¢ity ttlld chtlo., 111 i single [Ollgittldiiltll multi-trallSvc'rse-nlodc" cw ( ' O - las,._'r, ])/tvv. I¢cl. :4132. 3835 3837 (1985). c38. L. A. Lugiato, F. Prati. [_. M. Narducci. P. Ru, J. R. Trcdiccc aiid D. K. I-~aildv, P,ole o l irans\crsc c.ltc'ct~ in hiscr instabilities, t'hvs. Rot'. A37. 384% 38()(~ (It>lSV;). cjc-i. L. A. L u g i a t o . (i. L. O p p o . M. A. Pernigo. ,1. R. Trod|coo. L M. Nardu¢¢i and D. K. Bciild), .%pOllianc<>tls ~,palial pcitlcrn formatit)n in lasers and coopcralivc frcclUCi~cy locking. Opt. ( ' o m m . 68, f~3 f~£ (10NS) 1(10. (.'. T[IITlIll, Prc'qUcncy locking in i w o tral/Svc'rs,2 optical i11c)(.t~_',,of a laser. Phys. R,'v A38. 5tJ6() 5t)o3 i [cJ,'-;xJ [ ( l [ L. ,.\. Lugiato. F. Prati. L. M. Nurducci alld (}. I . O p p o . Spt)I/lall(t)tlS b r e a k i n g of the cylindrical S~lllll/c, ll'. in lascis, Opt. ('omnt. 09. 387 392 (19893. 1(32. J. R. Trcdiccc, E. J. Q u c l . A . M. G h a z z a ~ i . (_'. ( h ' c ( n . M. ,.\. Pcl'nigt). 1.. ]kl. Naiducci and I ,-\. l.u~i~ih, Spalial and t e m p o r a l i n s t a b i l i l i c s in a ('02 laser, Phys. R(,v. Len. 62. 1274 1277 (198c,i). 11 ~,. V. K[ische. C. O. Weiss and B. Wellcgehauscii, 8pciiiolemporal chm)s trom ;i ¢olltinuous "N;i l~L,c'I Phvv. Rev. A39, c)19 922 (198'4). I!)4. 1_,. A . l.ugiato. (i. L. O p p o , J. R. Trcdiccc. 1.. M. Nalducci CiIld M. A. Pcrnigo. lnqllbiliti¢~ and sp;m~i c o m p l e x i t y in a laser. J. O[~t. 5,o('. ,,tm. 7. 1(119-1033 (It,it~l)). 1115. P. T. Are¢¢hi, R. MeLlcci alld [.. Pczzali. (_'haos in ;i ( O - wa\'cguid¢ Idsc'r chic it~ lltlll,,\t_'l~C lilt,tic' c o m p e t i t i o n . Phys. Rt>l'. A42, 5 7 9 1 - 5 7 9 3 (it)c)()). L(I(~. ('. Green. G. B. Mindlin. E. J. 1)'Angelo. H. (i. Solar| anti .1. R. Trcdiccc, Sl)ontan¢otis Syllll]]CIi% I'qt_'HkillE in laser: the cxperinleiHal sick', Phys. Rev. Let|. 65, 3124-3127 (l'~CJ0). ll37. L. i \ . M e l n i k o v . . S . T a r l a k o \ ' a allCi ('. N. T:.iru.tkov. N o n l i n e a r dynamics o f bee|ill [~aranl(t(ry, aild lillc'il,qu, ~II ~ill unidirectional rmg lay,er with a flOlIiOgl211cous]y broadened line..J. Opt..';or. Am. B7. 128(~ 1292 (Itjt-i(li 1138. ('. T a m m alld (" (7). V~/ciy,s. ,~,flonUtn¢ous brc:.lking of cylindrical >,vnlillcLr~ in all opiically pumped ta-
Introduction and review
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Bortolotti and E. R. Pike, pp. 133-144. Institute of Physics, Bristol (1991). li5. P. Colet, M. San Miguel, M. Brambilla and L. A. Lugiato, Fluctuations in transverse laser patterns, Phys. Rev. A43, 3862-3876 (1991). 116. L. Gil. K. Emilsson, and G. L. Oppo, Dynamics of spiral waves in a spatially inhomogeneous Hopf bifurcation, Phys. Rev. A45, R567-R570 (1992). 117. P. K. Jakobsen, J. V. Moloney, A. C. Newell and R. Indik, Space-time dynamics of wide-gain-section lasers, Phys. Rev. A45, 8129-8137 (1992). 118. E. J. D'Angelo, E. Izaguirre, G. B. Mindlin, G. Huyet, L. Gil and J. R. Tredicce, Spatiotemporal dynamics of lasers in the presence of an imperfect 0(2) symmetry, Phys. Rev. Lett. 68, 3702-3705 (1992). 119. E. J. D'Angelo, C, Green, J. R. Tredicce, N. B. Abraham, S. Balle, Z. Chert and G.-L. Oppo, Symmetry breaking, dynamical pulsations, and turbulence in the transverse intensity patterns of a laser: the role played by defects, Physica D61, 6-24 (1992). 120. D. Hennequin, C. 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Pergamon
0960-0779(94)E0090-C
Transverse Nonlinear Optics: Introduction and Review L. A. L U G I A T O Dipartimento di Fisica dell" Universit~ di Milano, via Celoria 16, Milano 20133, Italy
A b s t r a c t - - T h e characteristic features of the field of Transverse Nonlinear Optics are briefly illustrated, as an introduction to this special issue of Chaos, Solitons & Fractals.
1. I N T R O D U C T I O N
The analysis of unstable phenomena in nonlinear optical systems has been the object of intensive studies, both theoretical and experimental, over the last 15 years (see, for example [1-7]). More recently, the main focus in this field shifted gradually from purely temporal effects to spatial and spatio-temporal phenomena, especially spontaneous pattern formation in the structure of the electromagnetic field in the planes orthogonal with respect to the direction of propagation. Even though Transverse Nonlinear Optics is basically a newly born field, the existence of such effects is well known since the earliest days of laser physics. As a matter of fact, in order to obtain the simple Gaussian T E M ~ transverse structure that is desired for most applications, one introduces apertures or other means into the laser cavity. Otherwise, the system generates spontaneously more or less complex patterns. These phenomena were, however, mostly considered as undesirable or difficult to be understood and controlled, even if there were pioneering works which analysed several interesting aspects of these effects [8-22]. In the last decade, and especially in the last few years, the interest in these phenomena emerged with evidence and activated systematic investigations in this field. It is by now clear that the spatial and spatio-temporal effects in the transverse structure are similar to the phenomena of pattern formation, turbulence, etc., that are familiar in other fields such as, for example, hydrodynamics, nonlinear chemical reactions, or biology. These phenomena represent the general object of study of Haken's synergetics [23] and of Prigogine's theory of dissipative structures [24] and can now be studied in optics, in systems whose dynamics is governed by the fundamental laws of radiation-matter interaction. In the case of the laser, the analogy with hydrodynamics can be substantiated also formally [16, 25, 26]. G i n z b u r g - L a n d a u [26-32], or Kuramoto-Sivashinsky [33, 341, or Newell-Whitehead [32] type equations have been derived and used to describe spatio-temporal phenomena in optics. As a matter of fact, optics addressed the issue of pattern formation and transformation much later than the more traditional fields of nonlinear chemical reactions and, especially, hydrodynamics. This fact is due to the circumstance that, until recently, it was not 'natural' to realize optical systems with a large aspect ratio; this is, however, no longer true now, for example in the case of high-power lasers or surface-emitting lasers. On the other hand, optics presents at least two special features that are interesting and stimulating. First, 1251
1252
1,. \. 1.~,,1\1,,
optical systems are very fast and havc a large frequency bandwidth: hcncc t h e \ lend themselves naturally for application, for instance to t e l e c o m m u n i c a t i o n s and to informati~m technology. O n e relevant example of the useful application of optical structures is alrcad~ provided hy the soliton transmission in optical fibres. The invcstigations of t|-ans~crsc nonlinear optics offer, in principle, the possibility of an approach to parallel optical information processing, bv encoding information in the transverse Stltlcturu e l lhc electromagnetic field. The researches in this direction arc at a rather initial slag~. (suc 135-4fl1) but therc is a definite evolution, and some i)f Ihc papcrs in this issuc prmid< ~ interesting hints in thai direction. The second special featurc is that ~ptical s\,stems ha~c the tendencv to display interesting q u a n t u m effects at roonl t c m p e r a t u r c . Recent \ c a r , have witncssed the start of investigations on q u a n t u m effects in trans'~erse optical SlltlCttllt.,,, (sec [41-451; these researches ure relevant from the viewpoints ~1 l:undamcntal quanlttln mechanics, of q u a n t u m noise reduction {squeezing) and, hol)efully. ~,l applications, l~,J example to noiseless transmission of images. The mechanism which gives rise to pattern formation is. as usual, Ihc combination ,~i nonlinearity and of coupling bctween the different spatial poinls. In the case ~d optical systems, this coupling ix provided b \ diffraction, which plays the same role as diflusioJ~ in nonlinear chcmical reactions and in biolog}. In lhc paraxial approximation ~hich p, currently uscd in these studies, diffraction is described by the transverse l,aplaeian, lh;fl i,. the sum of the second derivatives ~ith respect to the t,ao transverse spatial variable,, Both stationary and dwmmical patterns can be formed in this way. The kind of pattcln d e p e n d s crucially on the type of nonlinearity which may he, for cxample, quadratic in file case of parametric d o w n - c o n v e r s i o n and second harmonic generation, cubic in Z' ~' media (four-wave mixing), or the saturable nonlinearity which governs the singlemodc gain in lasers. The nonlinearity determines the c o m b i n a t i o n of wave xectors in the transverse plane, which in turn produces the structures that c m c r g c . General reviews of the field of transverse nonlinear optics can he found in rcfs ]4~ 51 ] The investigations have covered both theoretical and experimental aspects, and haxc c o n c e r n e d the case of passive (i.e. without population inversion) systems [3{L 31. 35. 3i~. 39. 52-92], the case of active systems such as lasers [25 2g. 3 2 - 3 4 , 37, 38, 4(l, g3 12L)t and svstems with gain which does not arise from population inversion {e.g. p h o t o r e l r a c l i \ e s , optical parametric oscillators) [ 13()- 137]. Thc sxstcms that have been studied can I~c. ~crv roughly speaking, dividcd inI~ l\~,~ classes. The first class is c o m p o s e d of systems which display translaticmat svnlnlctr\, t | c n c u , ii thcrc are mirrors, they must be plane mirrors: if there arc driving fields, they must bu plane waves. Thesc systems allox~, the study of the onset of t~atlerns from tt honlogcilcotl~, state, that is. the s p o n t a n e o u s breaking of the translational svmnletr\. This p h c n o m c m m arises w h e n e v e r the h o m o g e n e o u s staiional-\ slatc beconles tillstablc against the grov~lh ~)t small i n h o m o g e n e o u s perturbations, for example small-amplitude waves with a nltidulaiud sinusoidal configuration. It may happen that, when the wavelength of the modulation lies in an appropriate range, Ihe perturbation grm\.s instead of dying away. In this casc !hr, svstcm forms s p o n t a n e o u s l y a pattern which, according to the nature of the instahilit\, max be stationary or dwmmical. C o m m o n examples of patterns which may a p p e a r arc sn-ipc~ (also called rolls due to the similarity with the R a y l e i g h - B e n a r d instability in fluids) :lnd hexagons. Each of these configurations may display defects [2{} 22], for example dish,c> tions in roll patterns [137] or p e n t a - h e p t a defects in hexagonal patterns [7S]. Also, lhL' h o m o g e n e o u s configurations can show defects called optical phase singularities or optical \ortices [28. It2, 134. 13S, 13g] centred at points of zero intensit\ el the electric lichl in the transverse plane.
Introduction and review
1253
The second class of systems which have been extensively investigated is characterized by the inclusion of spherical mirrors; this is the standard situation for optical systems contained in resonant cavities. In this case the system has no translational but, at most, rotational symmetry in the transverse plane. The spherical mirrors confine the radiation field in the region which surrounds the axis of the system, so that the field configuration is boundary-dependent. The dynamics of the system is governed by the nonlinear interaction and competition among the different discrete modes (of G a u s s - H e r m i t e or G a u s s Laguerre type) of the resonator. When the Fresnel number of the cavity is at most of order unity, instead of the periodic (in space) patterns met in the first class of systems, one finds localized structures composed by a limited number of peaks and, possibly, phase singularities. These structures arise from the contribution of a reduced set of cavity modes. The variety of different configurations, both of stationary and of dynamical nature, that can emerge in this way is spectacular. On the other hand, when the Fresnel number is very large the behaviour of the system becomes more and more boundary-free and similar to the case of the first class. The field configurations arise from the contributions of a very large number of cavity modes, so that the modal composition itself becomes basically irrelevant. Space-time chaos is one of the common scenarios in this limit [130]. Of course, there are cases of systems which are intermediate between the two classes, for example systems with plane mirrors driven by external field with a Gaussian or a flat-top profile [90]; and there are systems, the behaviour of which is governed by propagation, in combination with diffraction, without any infuence from feedback such as that provided, for example, by a cavity resonator (e.g. see [140-142]). Laser arrays have also been the object of extensive investigations (e.g. see [143, 144]). Even if the list of references in this article is long, it is in no way complete and, certainly, the selection of papers is influenced by my personal taste and limits of information. Hence I apologize for all omissions. This special issue collects contributions from most leading groups active in the field of transverse nonlinear optics; the quality of the papers and the coverage of subjects ensure that this volume provides a rather faithful description of the state-of-the-art in this domain. Undeniably, the field of optical patterns developed mainly in Europe. This was in part the cause and in part the consequence of the illuminated support provided by the Programs of the Commission of the European Communities to the researches on Optical Instabilities, Opti.cal Chaos and Optical Patterns. An especially important role was, and is, played by the E S P R I T Basic Research Project 3260 TOPP (Transverse Optical Patterns) and 7118 T O N I C S (Transverse Nonlinear Optics). A significant proportion of the papers in this issue arise from research activities in the framework of TONICS.
Acknowledgements'--I am grateful to Professor M. S. E1 Naschie and Dr A. Boehm for their kind invitation to organize this issue. I thank the support of the TONICS Project and, especially, Drs Ingemar Hussla, Manuela Pinheiro and Brendan Hawdon of the ESPRIT Basic Research staff, for their wise and continuous assistance and guidance provided to the partners of the projects TOPP and TONICS. Thanks are also due to Drs Massimo Brambilla, Fabrizio Castelli, Alessandra Gatti, Irene Marzoli, Marco Pinna, Roberta Pirovano, Franco Prati, Alice Sinatra and Martino Travagnin for their continuous help in the preparation of this issue.
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