Chaotic Explosions Eduardo G. Altmann,1 Jefferson S. E. Portela,1 and Tam´as T´el2 1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 2Institute for Theoretical Physics - MTA-ELTE Theoretical Physics Research Group, E¨otvo¨s University, Budapest, H–1117, Hungary Weinvestigatechaoticdynamicalsystemsforwhichtheintensityoftrajectoriesmightgrowunlim- itedintime. Weshowthat(i)theintensitygrowsexponentiallyintimeandisdistributedspatially accordingtoafractalmeasurewithaninformationdimensionsmallerthanthatofthephasespace, (ii) such exploding cases can be described by an operator formalism similar to the one applied to chaoticsystemswithabsorption(decayingintensities),but(iii)theinvariantquantitiescharacteriz- ingexplosionandabsorptionaretypicallynotdirectlyrelatedtoeachother,e.g.,thedecayrateand fractal dimensions of absorbing maps typically differ from the ones computed in the corresponding inverse (exploding) maps. We illustrate our general results through numerical simulation in the 5 cardioid billiard mimicking a lasing optical cavity, and through analytical calculations in the baker 1 map. 0 2 PACSnumbers: 05.45.-a,05.45.Df,42.55.Sa n a J I. INTRODUCTION overthephasespace)approachesafractaldensityρ . We c 2 call this phenomenon a chaotic explosion. 2 Fractality is a signature of chaos appearing in strange In this Letter we show that chaotic systems with gain attractors and in the invariant sets of open dynamical can be treated with the same formalism of systems with ] D systems[1–3]. Hereweareinterestedinsystemsinwhich absorption but that the properties of these two classes C trajectories have associated to them a time-varying in- of systems cannot be trivially related to each other. We tensity. In a recent work [4] we showed that fractality in obtain general results for the fractality and for the in- . n chaotic systems in which the intensity of trajectories de- verse map of systems with gain/absorption, which are il- li cays due to absorption can be described by an operator lustrated analytically in the baker map and numerically n formalism and that absorption leads to a multi-fractal in the cardioid billiard. For optical microcavities, our [ spectrum of the decaying state. In this work we inves- results show how gain can be introduced in the ray de- 1 tigate the dynamical and fractal properties of systems scriptionandhowitaffectsthefar-fieldemission,demon- v containing gain (i.e., negative absorption). In systems strating also in the ray description that lasing is not de- 3 withgaintheenergyorintensityoftrajectoriesincreases termined by the shape of the (passive) cavity alone. 4 in time, e.g., the intensity of a ray grows while it is in a 4 gain medium or is multiplied by a factor larger than one 5 0 when reflected on a wall. II. TRUE-TIME MAPS WITH GAIN . Optical microcavities provide a representative physi- 1 cal system of the general dynamical-systems picture de- We consider an extended map which includes, besides 0 5 scribed above. The formalism of open chaotic systems a usual map f, the true physical time tn and the ray 1 hasbeenextensivelyusedtodescribelasingpropertiesof intensity J at the n-th intersection (cid:126)x with a Poincar´e n n : two-dimensionalopticalcavities[5,6]. Thesuccessofthis section as [10, 11] v approach relies on the use of long-living ray trajectories i X to describe the lasing modes. This is justified because (cid:126)x =f((cid:126)x ),  n+1 n r lasing modes are induced by the gain medium present in f : t =t +τ((cid:126)x ), (1) a ext n+1 n n optical cavities and only long-living trajectories are able J =J R((cid:126)x ), n+1 n n toprofitfromthisgain. Therelevanceofgainledtospe- cificinvestigationsofitsroleinexperiments[7]andwave where the return time τ((cid:126)x) 0, chosen as the time be- simulations [8], but we are not aware of ray simulations tween intersections (cid:126)x = (cid:126)x ≥and (cid:126)x = (cid:126)x(cid:48) f((cid:126)x) (for n n+1 ≡ which have explicitly included gain. This is a crucial is- billiards, τ is the collision time between two consecu- sueespeciallywhenthegainisnotuniformlydistributed tive bounces with the wall), and the reflection coeffi- in the cavity, as in the experiments of Ref. [7]. cient R((cid:126)x) are known functions of the coordinate (cid:126)x on We consider chaotic billiards with gain as models of the Poincar´e section. Cases in which gain occurs con- optical microcavities. In Fig. 1 we show simulations of tinuously in time (not only at the intersection with the trajectoriesthatbounceelastically,butwhoseintensities Poincar´e section) correspond to a reflection coefficient increase exponentially in time with rate g while passing R((cid:126)x) = egτg((cid:126)x), where g is the gain rate and τg is the throughagainregion(thegraydiscinFig.1a). Forlong time spent in the gain region (τ = τ if gain is uniform g times,thetotalintensitygrowsexponentiallyintimeand in the billiard table). Explosion occurs if R((cid:126)x)>1 for a the spatial distribution (obtained for any t normalizing sufficiently large region of (cid:126)x. 2 FIG. 1: Explosion in a fully chaotic billiard. (a) Cardioid billiard, whose boundary in polar coordinates is r(φ) = 1+cos(φ) withφ∈[−π,π][9]. Thegainregion(gray,markedbyg)isadiscofradius0.5inthemiddleofthebilliard. (b)Collisiontime distribution τ((cid:126)x) in the cardioid billiard (velocity modulus is unity). Birkhoff coordinates (cid:126)x=(s,p=sinθ) are used, where s isthearclengthalongtheboundaryandθ isthecollisionangle. (c)Reflectioncoefficientdistribution: R((cid:126)x)=egτg withg=1 and τ given by the length of the intersection of the ray with the gain region. (d) Time-independent density ρ in the phase g c space. (e) Time-dependence of the intensity integrated over the phase space. The explosion rate κ≈0.215 is the slope of the curve (note the log-scale). Inset shows the non-exponential behavior for short times. We are interested in the density ρ((cid:126)x,t) (i.e., the col- Comparingwithourpreviousresults[4,6],weseethat lective intensity of an ensemble of trajectories) at time t (3) is an extension to cases with R>1 of the Frobenius- in (cid:126)x. Here we consider the class of (ergodic and chaotic) Perron operator considered previously only in systems mapsf((cid:126)x )forwhichweshowthatforanysmoothinitial with absorption (R 1 for any (cid:126)x). The explosion rate κ n ≤ ρ((cid:126)x,t=0) one observes for long times playstheroleofanegativeescaperateandtheattracting limit distribution ρ is the conditionally invariant den- c ρ((cid:126)x,t) ρ ((cid:126)x)eκt, (2) sity [6] (also known as the steady probability distribu- c ∼ tion in the optics community [12, 13]). By integrating, where κ is the temporal rate of the total energy change forn , (3)overall(cid:126)x(cid:48), thelefthandsideisunitydue →∞ (an explosion rate for κ > 0) and is independent of tonormalizationofρ ((cid:126)x),whiletherighthandsideisan n ρ((cid:126)x,t=0). average taken with respect to ρ . This yields (see also c sitWy eofcaannexitpeercattitohnatscρhce((cid:126)xm)eoffo(r2)ρi(s(cid:126)xt)heofattthreacteixntgenddeend- Rtheaft.[t4y]p)i(cid:104)cea−llκyτRR(cid:105)>c =1,1b.uFtoritcshraeodtuicceedxpvlaolsuioenes−tκhτiRsmtaekaenns map (1). With a compensation factor e−κτ((cid:126)x) per itera- with the proper explosion rate averages out to unity if tion,thisschemeevolvesadensityρn((cid:126)x)atdiscretetime the average is taken with the ρc associated to κ. nintoρ ((cid:126)x)atthenextintersectionwiththePoincar´e n+1 surface of section as III. GENERAL FEATURES ρ ((cid:126)x(cid:48))= (cid:88) e−κτ((cid:126)x)R((cid:126)x)ρn((cid:126)x), (3) n+1 f((cid:126)x) It is remarkable that the formalism of transient (cid:126)x∈f−1((cid:126)x(cid:48)) |D | chaos [2, 3] can be applied with slight modifications to where representstheJacobianofthePoincar´emapf. describechaoticexplosions. Belowweusethisformalism f IntheDspecialcaseofinvertiblearea-preservingdynamics to explore the most interesting effects of gain (R>1). (as in the billiard of Fig. 1), ((cid:126)x) = 1 and there is no f D sum in (3) (map f has a single preimage). In any extended map, there exists one κ – the one ap- A. Fractality pearinginEq.(2)–forwhichρ ((cid:126)x)arisesasthelimiting c (n ) distribution of ρ ((cid:126)x) iterated by scheme (3). We first derive general relations between the fractality n → ∞ The right-hand-side of Eq. (3), with the proper κ, is an of ρ – the eigenfunctions of Eq. (3) – and the distribu- c operator with largest eigenvalue unity and ρ as the cor- tions (τ((cid:126)x),R((cid:126)x)) characterizing the extended map f . c ext responding eigenfunction. The second-largest eigenvalue Letusconsiderthemapf in(1)tobeinvertibleandtwo- controls the convergence of a smooth initial density ρ dimensional. We are interested in the dynamics within a 0 to ρ . In agreement with the physical picture, we see region of interest Γ only, which has a size of order unity c that for extended maps f there are three different fac- in appropriately chosen units. Considering the n-th im- ext tors contributing to the density ρ: reflectivity R, return age and preimage of Γ taken with the map f, we find for times τ, and stretching of phase space volume . The n 1 a set of narrow “columns” in the unstable direc- f D (cid:29) rate κ follows from the constraint that the compensated tion and narrow “strips” along the stable directions, as intensity neither increases nor decreases for large n [3]. illustrated in Fig. 2. They are good approximants of the 3 accumulating on column i is µ(n) =e−κτj(n)n+nlnRj(n)ε(n) =e(cid:16)lnRj(n)−κτj(n)−λj(n)(cid:17)n. i j (4) Theexistenceofatime-independentmeasureimpliesthat with the proper value of κ we have (cid:80) µ(n) =1. i i Inthespiritofdynamicalsystemstheory,weassociate with strip j the measure that its points represent after n steps. Therefore the measure µ(n) of strip j is j µ(n) =µ(n). (5) j i Therelationexpressesthatf mapsthemeasuresofthe FIG. 2: Schematic diagram of the phase-space partitioning ext around a hyperbolic chaotic set obtained taking the n (cid:29) 1 stable and unstable directions into each other. fold image and preimage of the region of interest Γ with re- We now focus on fractal properties of this measure. spect to the (usual) map f. These define narrow strips and Systems described by closed maps f with gain have a columns overlapping with branches of the stable (s) and un- trivial fractal dimension D equal to the phase space di- 0 stable (u) manifold, respectively. The emphasized strip and mension. Their fractality requires thus the computation column belong to a n-cycle point (•) on the chaotic set. of the generalized dimensions [1, 3] 1 ln(cid:80) µq D = lim k k (6) q 1 q ε→0 ln1/ε − unstableandstablefoliationofthechaoticset(achaotic where µk is the measure of the k-th box in a coverage sea or an attractor in closed systems, or a chaotic sad- with a uniform grid of box size ε, and the sum is over dleinopenones)underlyingthedynamics. Eachofthem non-empty boxes. containsanelementofann-cycle[1,3],i.e.,apointwhich A general result can be obtained for the informa- is mapped by f onto itself after n iterations. tion dimension D1. For a general one-dimensional dis- tribution containing measures µ in intervals of dif- α Letusfocusonsuchann-cyclepointattheintersection ferent sizes ε we have for small ε [1, 3]: D = α α 1 (cid:80) (cid:80) ofstripj andcolumni(drawnwithboldlinesinFig.2). µ lnµ / µ lnε . Now, take a section of strip α α α α α α Duetothepermanentcontractioninthestabledirection, j alongtheunstablefoliation(u),andassociatethemea- the width of column i is approximately ε(cid:48)i(n) = eλ(cid:48)i(n)n sureofthestripµ(jn) totheintervalsizeε(jn). Inthelimit where λ(cid:48)(n) < 0 is the contracting Lyapunov exponent n 1thesizesaresmall,andsubstitutingEq.(4),(5)in i (cid:29) around the cycle point over discrete time n. Similarly, the D1 formula above, we obtain the partial information theheightε(jn) ofstripj isε(jn) =e−λ(jn)n,whereλ(jn) >0 dimension D1(1) along the unstable direction as denotes the corresponding positive Lyapunov exponent. κτ¯ lnR Byconstruction,pointsstartinginstripjspendthedom- D(1) =1+ − . (7) 1 λ¯ inant part of their lifetime n in the close vicinity of the hyperboliccycleinquestion(whichbelongstothechaotic The averages denoted by overbars are taken over the set). Therefore, for large n nearly all points in the strip chaotic set (e.g., λ¯ is the positive Lyapunov exponent are subjected to an average stretching factor eλj(n)n (in on the saddle). This is so because, as argued above, the trajectoryeffectivelyexperiencescollisiontimesτ,reflec- the unstable direction). In an extended map f , ob- ext tion coefficients R, and local Lyapunov exponents λ (of tained from f through Eq. (1), these points experience an average collision time τ(n) =(cid:80)n τ /n and an aver- the usual map f) close to an unstable cycle (which be- j k=1 k longs to the chaotic set). Quantities characterizing the age gain/reflection coefficient Rj(n) =(Πnk=1Rk)1/n while map(λ¯),thecollisiontimes(τ¯),andthegain(lnR)deter- being around the n-cycle, where τk and Rk denote the mine a fractal property (D1) via the simple relation (7). collision time and reflection coefficient, respectively, be- Fractality is nontrivial if D(1) < 1, implying κτ¯ < lnR longing to element k of the n-cycle. (for a positive rate κ), i.e.,1R > 1 in a sufficiently ex- tended region. In the spirit of operator (3) valid for f , the density ext Repeating the calculation along a section parallel to ontheimagesofstripj isthedensityofstripj multiplied the stable foliation (s) we find D(2), the partial infor- (ineachiteration)byafactorRke−κτk. Startingwithan mation dimension along the stable1direction. Since the initial unit density, the area ε(n) 1 of strip j should j × measures are the same (see (5)), the difference follows be multiplied in fext by a factor e−κτj(n)n+nlnRj(n) by the from the sizes which are ε(cid:48)i(n) now, and we find end of n iterations. Since the n-th image of strip j with respect to f is column i, by construction, the measure D(2) λ¯(cid:48) =D(1)λ¯. (8) 1 | | 1 4 Theoverallinformationdimensionofthechaoticsetwith Many systems of interest are not restricted to a the measure of f is D =D(1)+D(2). The derivation bounded phase space Ω, as considered above. For ext 1 1 1 above holds for any value of R and therefore generalizes instance, in scattering systems one usually defines a theresultsofRef.[4],whereweconjecturedEq.(7)(with bounded region Γ of interest (which contains all peri- a negative κ) for strictly absorbing cases (R 1). The odic orbits and the chaotic saddle), but the dynamics is ≤ case of usual maps (τ = 1,R = 1) follows as a limit: defined in the full phase space. In this case, the inverse if the map is closed (no loss in any sense), κ = 0, and mapisdefinedasaboveinthewholespace,withthesame D = 1+λ¯/ λ¯(cid:48) , i.e. we recover the Kaplan-Yorke region of interest Γ since the chaotic saddle is invariant. 1 | | formula[1](D =2forareapreservingmaps);ifthemap Another case of interest is the one of total absorption in 1 isopen(trajectoriesescape), κ<0, andEqs.(7)and(8) a localized region, i.e., R((cid:126)x)=0 for some(cid:126)x Ω . In this ∈ go over into the Kantz-Grassberger formulas [2, 3]. case, the inverse map can be defined only as the limiting case of R 0 . + → B. Inverse map C. Who wins? Besides the physical motivation to study gain, a natu- ral question that can only be answered considering both In a general system both gain and absorption coexist R>1andR<1istheoneoftheinverseoftheextended and a natural question is whether decay or explosion is map defined in Eq. (1), which we denote by finv and de- observed globally . This is a priori unclear because of ext fine implicitly by f finv =I, where I is the identity. the non-trivial nature of the asymptotic density ρ ((cid:126)x). ext◦ ext c If f has R>1, then finv should compensate this with When half of the phase space has R and the other half ext ext L R<1. R as reflection coefficients one would intuitively expect R For finv to exist, the usual map f has to be invertible for an ergodic system that a steady state is found with ext (i.e., f must have a single pre-image). Consider the case R R = 1. When taking the return times τ into ac- L R i inwhichthedynamicsisdefinedinaboundedregion,i.e., count, intuition says that for R R = 1 the behavior L R f isclosed. Inthiscasetheprocedurefordefiningfinv is that dominates is the one associated with the half space ext totakef−1oftheusualmapf andattributetox(cid:48) =f(x) characterized by smaller collision times (with more col- the negative of the same return time τinv(x(cid:48))= τ(x)< lisions per time unit). As shown below, these two intu- 0andtheinversereflectioncoefficientRinv(x(cid:48))=−1/R(x). itions are not generally correct. We can therefore write: A particularly interesting case is the one of energetic  steady states when the total energy neither grows nor (cid:126)x =finv((cid:126)x ) =f−1((cid:126)x ), finv :tn+1 =t +τninv((cid:126)x ) =t τn(f−1((cid:126)x )), decreases in time (κ = 0), in spite of local gains and ext Jnn++11 =JnnRinv((cid:126)xn)n =Jnn/−R(f−1((cid:126)xnn)). ltoimsseess.aSnidtuaarteionthsuwsitehasκy≈to0orbesmeravien.uInnchtahnisgecdasfeorthloenrge (9) is no need for any extraction of intensity and the cor- The iteration corresponding to (3) of the inverted ex- responding iteration scheme is given by Eq. (3) without tended map in cases when f describes a closed map is the term e−κτ((cid:126)x), i.e., the usual Frobenius-Perron equa- Rinv((cid:126)x) tion for closed systems (even if f is open). The density ρ ((cid:126)x(cid:48) f−1((cid:126)x)) =eκinvτinv((cid:126)x) ρ ((cid:126)x) n+1 ≡ inv((cid:126)x) n ρc is invariant (not conditionally invariant), does not de- |(D1/fR((cid:126)x(cid:48))|) pendexplicitlyonthereturntimesτ,andD1 isgivenby =e−κinvτ((cid:126)x(cid:48)) 1/ ((cid:126)x(cid:48)) ρn((cid:126)x). (7) with κ=0. As shown below, this configuration leads | Df | to a fractal invariant density even for volume-preserving (10) dynamics. This equation is different from the operator of f in ext Eq. (3). Therefore, for extended maps the chaotic prop- erties (κ, fractality, etc.) of maps f and finv typically ext ext IV. EXAMPLES differ. In particular, the rate κinv of the inverse map is not κ (nor any simple function of κ)[14] − In this section we use specific systems to explore the Thedifferenceintheasymptoticratesκinv = κseems (cid:54) − general results reported above. First, we consider the to contradict the fact that, by definition, maps f and ext analytically treatable closed area-preserving baker map finv cancel each other, i.e., the gain and loss resulting ext definedinFig.3[1,3]. Initiallyρ 1. Inthenextstep, fbryomn anpapplipcalitciaotniosnosfothfeonoethmerapmaispe(xfoarctalyrbcitormarpielynslaatregde ρ0 is multiplied by Rke−κτk, k =01≡or 2, leading to two vertical columns of width 1/2 lying along the x=0 and n). The solution of this apparent paradox is that the asymptotic densities of the forward ρ and inverse ρinv x=1 lines. They carry measures P1 and P2 where c c maps differ. Taking ρc (or ρicnv) as initial condition ρ0, Pk =Rke−κτk/2, k =1,2. (11) κinv = κ. However, these are atypical ρ . Generic ones 0 − convergeexponentiallyfasttotheρ ofthecorresponding The construction goes on in a self-similar way, and the c map. condition that the measure after appropriate compen- 5 R 1015 2 τ2 f f ρ(t=0)=1 { 1010 Forward Map Rτ11{ yx ΣJ(t)/Nii 110005 (I‡nppv624e5−−r1s1·e)t− /Mτt/τap FIG.3: Actionoftheclosedarea-preservingbakermapf with )= 10-5 gain on the unit square. The map f is defined as (x,y)(cid:48) = (t ρ (x/2,2y) for y < 0.5 and (x,y)(cid:48) = (x/2+0.5,2y −1) for 10-10 y >0.5 and extended – as in Eq. (1) – by assigning R =R 1 (R = R ) and τ = τ (τ = τ ) to trajectories in the lower 10-15 2 1 2 50 25 0 25 50 (upper) half. t/τ sation remains bounded (ρ is reached) corresponds to c FIG. 4: Explosion and escape rates κ in the forward and (P +P )n =1, even after n 1 steps. We thus find as 1 2 (cid:29) inverse(extended)bakermap. Pointsaredistributedatt=0 an equation specifying the explosion rate κ: uniformly in the phase space with J = 1 and then iterated i using Eq. (1) with f defined in Fig. 3, τ = 2τ ≡ 2τ, and 2 1 P +P =1. (12) 1 2 R = 4 = 2R . The energy density ρ(t) is computed as the 1 2 average intensity of all trajectories ρ = (cid:80)N J /N at time i=1 i t. The forward map shows a growing energy in time (t > 0, blue line) while the corresponding inverse map (t < 0, red A. Fractality line)showsadecayingenergy. Dashedlinesaretheanalytical calculations. The fractality of the baker map can be calculated ex- plicitly. The generalized dimensions D(2) of ρ along q c the stable (horizontal) direction is derived from Eqs. (6) In the example discussed above this corresponds thus to and(11)asDq(2) =ln(P1q+P2q)/[(1−q)ln2].Inthelimit τx2 =e−2κττ1 ==(−√26τ5,R11)=/4.1/T4h,isRi2s l=arg1e/r2t,hwanhic1h, imleapdlysintgo of q →1 the D1 is obtained as a n≡egative κinv −0.568/τ, i.e. an escape rate, a global ≈− D(2) =1 [ κτ¯+lnR]/ln2 decay of intensity towards zero. In Fig. 4 we confirm 1 − − that these analytical calculations agree with simulations κ(τ P +τ P )+P lnR +P lnR =1 − 1 1 2 2 1 1 2 2, of individual trajectories. The different slopes confirm − ln2 (13) the general result κ = κinv . In the previous section | |(cid:54) | | which is an equivalent derivation of the general for- we related this apparent paradox to the difference be- mula (7) obtained by identifying Pi with the measure tween ρc and ρicnv, which can be quantified by D1. From on the chaotic set. Eq. (13), D(2)inv 0.783 = 0.661 D(2). Non-trivial 1 ≈ (cid:54) ≈ 1 As a particular case, consider τ = 2τ 2τ and fractality is preserved despite the change of sign in κ. 2 1 ≡ R = 4,R = 2. From Eq. (12) a quadratic equa- 1 2 tion is obtained for x e−κτ > 0, leading to x = ≡ √2 1 and therefore a positive (explosion) rate κ = C. Who wins? − ln(√2 1)/τ 0.881/τ . Parameters P and P 1 2 − − ≈ are then 2x and x2, respectively. The average return We allow for competition between gain and absorp- time is τ¯ = τ2x + 2τx2 = 2τ(2 √2) while the av- − tion in the particular example defined above by writing erage of lnR is found to be (2√2 1)ln2. The nu- − Ri(α) = αRi, with 0 < α < 1. Because of (12), this merator of the fraction in the parenthesis of Eq. (13) leads with R = 4 = 2R to an α-dependent explosion is 2(2 √2)ln(√2 1)+(2√2 1)ln2 0.235. This rate κ(α) giv1en by e−κ(α2)τ = (cid:112)1+1/α 1. Decreas- − − − ≈ is positive, rendering D(2) = 1 0.235/ln2 0.616 < ing α from unity, this quantity is less th−an unity but D(2) =1, a clear sign of1the mul−tifractality of≈ρ . increases with decreasing α. At α (cid:47) 1/2 absorption 0 c and gain coexist (R (α) < 1) but explosion still domi- 2 nates (κ > 0). For α < α = 1/3 absorption dominates c B. Inverse map (κ < 0). At the critical value α = αc, κ = 0, which is the steady state condition[15], R R (α=α )=4/3, 1c 1 c ≡ The inverse of the baker map discussed above is com- R2c R2(α = αc) = 2/3, P1c = 2/3, P2c = 1/3, the ≡ puted using the general relation (9). Symmetries and average of lnR is 5/3ln2 ln3, and thus Eq. (7) yields − = 1 make the inverse map to be equivalent (after again fractality D(2) = 1 lnR/ln2 = 0.919 for any Df 1 − a transformation x y,y x) to the forward map τ1,τ2. (cid:55)→ (cid:55)→ after replacing R by 1/R and τ by τ (see Fig. 3). Even though the return times τ are irrelevant at the i i i − 6 dex η). Figure 5 reports results for transverse-electric polarized light in such a configuration. Whether explo- sionoccurs,dependsonthegainparameterg (κdepends smoothly on g, see panel a). At the critical value g =g ∗ gain and loss cancel each other (κ=0) and an energetic steady state sets in. The density ρ is fractal for any g, c also at g = g∗ (see lower inset of panel a). The trans- mitted rays can be detected outside the billiard as an emissioninagivendirection(representedbythefar-field angle Φ, see upper inset in Fig. 5a). We obtain that the (observable) far-field emission (as a function of angle Φ) is modified by the gain (compare panels b-d). We thus conclude that (non-uniform[16]) gain has to be included in ray simulations. V. CONCLUSIONS Weconsideredchaoticsystemsinwhichtheintensities of trajectories may grow in time (e.g., due to gain or a reflection coefficient R > 1). We extended the formal- ism of systems with absorption (R<1) to show how the theoryof (open) chaoticsystemscanbe usedinthis new class of systems. For instance, we derived a formula – Eq. (7) – that relates the invariant properties (e.g., frac- FIG. 5: Ray emission from an optical cardioid billiard with taldimensionsandtheexplosion/escaperates)ofsystems gainandtransmission. (a)Asymptoticrateκasafunctionof with gain/absorption, a generalization of important re- thegainstrengthg. Thesteadystateκ=0isachievedatg= g∗ ≈1.0367,thelowerinsetshowstheinvariantdensityρ for sults in the theory of open systems (in which R is ei- c thiscase. (b-d)Long-time(t(cid:29)1)emissionE(Φ)ofintensity ther 0 or 1) [2, 3]. Despite the unifying formalism, our J in the far-field angle Φ (normalized as (cid:82) E(Φ)dΦ = 1) for results reveal an intricate relationship between systems (b) g =0, no gain and decaying intensity: κ<0; (c) g =g∗, with gain and with absorption. For instance, the inverse steadystate,κ=0;(d)g=2,explosion,κ>0. Theintensity of a system with gain is a system with absorption, but – of rays grows with rate g at the gain region (gray circle, see in contrast to usual dynamical systems – their invariant also Fig. 1) and is split between reflected and transmitted properties are not trivially related to each other. raysforsmallcollisionanglesθ. Thisleadstoapartialoptical For applications in optical microcavities, our results reflection R (θ) = [sin(θ −θ)/sin(θ +θ)]2 < 1 for | p |≡| o T T show how gain can be incorporated in the usual ray sim- sinθ |<1/η, where we used η =3 and the transmitted angle ulations. Wheneverthegainisnotuniforminthecavity, θ is given by Snell’s law as ηsinθ=sinθ [12, 13]. T T wefindthattheemissionismodifiedandcanbedescribed through our formalism of chaotic explosions. These re- sultscanbetestedexperimentallybyconstructingoptical steady state, the steady state is not achieved at R R = 1 2 1. In our example, R (α)R (α) = 8α2 = 1 for α = cavitieswithdifferentlocalizedgainregions[7]andcom- 1 2 paring the emission to ray simulations with and without 1/√8 = 0.354 > α = 1/3 (explosion). This remains c gain. valid in maps (τ =1) for any R ,R : introducing R = i 1 2 1 1/R in Eq. (12) leads to κτ = ln((R +1/R )/2) 0. 2 1 1 ≥ The stead-state condition is thus not R R = 1, but 1 2 rather R +R =2. VI. ACKNOWLEDGMENTS 1 2 As a final illustration of the significance of our general results,weperformnumericalsimulationofraysinanop- WethankM.Sch¨onwetterforthecarefulreadingofthe tical cavity with gain and absorption. Imagine that the manuscript. This work has been supported by OTKA cardioid billiard with localized gain introduced in Fig. 1 grant No. NK100296 and by the Alexander von Hum- is composed of a dielectric material (with refraction in- boldt Foundation. [1] Ott E, Chaos in dynamical systems, Cambrdige Univ. Cambridge Univ. Press, 2005. Press, 1993. [3] LaiY-CandT´elT,TransientChaos: ComplexDynamics [2] Gaspard P, Chaos, Scattering and Statistical Mechanics, in Finite Time Scales, Springer, 2011. 7 [4] Altmann E. G., Portela J. S. E., and T´el, T, Phys. Rev. [12] LeeS-Y,RimS,RyuJ-W,Kwon,T-Y,ChoiM,andKim Lett. 111, 144101 (2013). C-M, Phys. Rev. Lett. 93 13 (2004). 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