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The similar dynamics is embedded in Gross-Pitaevskii equation for macroscopic wavefunction Ψ of Bose- E +1 E n ,m − n,m = f(E )E + Einstein condensates [24]: ∆t n,m n,m En,m+1 + En,m−1 2En,m χ − . (1) 2 2 2 ∂Ψ(z, ~r, t) ~ 4π~ aS 2 ∆z i~ = ∆⊥Ψ+ U(~r, t)Ψ + Ψ Ψ. ∂t −2m m | | Behavior of such discrete (DDS) (5) ranges from exactly solvable diffusion for purely real χ where U(~r, t) is confining potential of arbitrary com- and diffraction for purely imaginary χ towards spatiotem- plexity [25], aS is scattering length, m is mass of boson. poral instabilities [14], localized structures[15] and tur- In a presence of gain G and losses γ the NLS-GLE may bulence. The role of nonlinear term f(En,m) in pattern have a form of NLS with Frantz-Nodvik resonant gain formation is crucial [1, 16]. In many cases the numeri- term [26], relevant to amplification with stimulated emis- cal schemes (1) are replaced for much more sophisticated sion cross-section σ of light pulse of duration T2 <τ

∂E(z, ~r, t) n ∂E i 2 ∞ + + ∆⊥E = ikn2 E E+ , , 3 , ∂z c ∂t 2k En+1(~r)= K(~r ~r )f(En(~r ))d ~r | | − ≈ t ′ Z−∞ 2 σNo(z, ~r)E exp[ 2σ E dt ] γE, (6) K(~r ~r )f(E (~r ))S(mx,my,mz), (2) − −∞ | | − − m n m Z mx,my,mz X We will discuss discrete iterative maps (2) for numeri- where S(mx,my,mz) is a finite volume element in- cal modeling of chaotic and regular spatiotemporal prop- stead of infinitesimal one d3~r, kernel K(~r ~r,) is a Green agation inherent to equations (3, 4, 5, 6). The article function of linear version of (1, 2), i.e. response− for delta- is organized as follows: section II is devoted to overview function δ(~r ~r,) in right part of scalar diffusion equation, of regular and chaotic iterative dynamics of real 1D [27] reads as − and complex point maps [28], section III outlines the di- K(~r ~r,) (χ/∆t)−1/2exp( √χ ~r ~r, 2/∆t) rect link between evolution PDEs (3-4) and maps (1-2), or for− scalar≈ parabolic diffraction− | equation− | this reads as section IV contains examples of localized solutions ob- K(~r ~r,) (χ/∆t)−1/2exp( i√χ ~r ~r, 2/∆t) [21]. tained with nonlocal maps being equivalent to equations Such− discrete≈ numerical approach− proved| − | to be extremely (3-4) , the spatially periodic and chaotic lattices are ob- effective for modeling of nonlinear dynamics inherent tained numerically in section V and nonlinear dynamics to evolution partial differential equations of so-called in the presence of multimode fluctuations is presented in parabolic type [22]. Apart from nonlocal dispersive section VI with conclusive discussion in section VII. laplacian there exists a nonlinear term f(En) which de- pends on square modulus of field E as in Kolmogorov- Petrovskii-Piskounov (KPP) equation [21]: II. ITERATIVE MAPS WITH UNIVERSAL BEHAVIOR.

∂E(z, ~r, t) 1 ∂E ∂2E Feigenbaum demonstrated the universality in itera- + + χ 2 = f(E)E. (3) ∂z V ∂t ∂t tions of real mapping with parabolic maximum of the Here the diffusion term χ = F 2/k2D2 may be respon- unit interval into itself alike [29]: sible for spatial filtering of high transverse harmonics in laser cavity by intracavity diaphragm and iteratively E +1 = λ E (1 E ), (7) repeated nonlinear transformation of the scalar field E n F n − n in ring laser with intracavity second harmonic genera- where the sole control parameter λF completely de- tion or Raman scattering [23]. More realistic models are termines the discrete time evolution of this simplest dy- based upon nonlinear Shrodinger equation, known also namical system. In particular he shown that two univer- as Ginzburg-Landau equation (NLS - GLE) which cap- sal irrational numbers, namely δF = 4.6692.. scales the tures the interplay of phase-amplitude modulation during separation of the values of λF = λ1, λ2, λ3...λM−1.λM ... propagation of complex field which is the source of mod- where period-doubling bifurcations occur(fig.1): ulational instability [22], solitons [2] and collapse [1]. For propagation of light pulse in Kerr dielectric NLS - GLE λM λM−1 reads as: δF = lim − 4.6692, (8) M→∞ λ −1 λ −2 → M − M ∂E(z, ~r, t) n ∂E i 2 and αF =2.5... scales the location of limit cycle points + + ∆⊥E = ikn2 E E. (4) ∂z c ∂t 2k | | in phase-space. 3

FIG. 2: a) Layout of unidirectional single transverse mode ring laser with nonlinear losses . Envelope of the laser pulse is modulated consecutively from one passage to another by hy- perbolic tangent chaotic map. b) Layout of single transverse mode unidirectional ring cavity described by Ikeda map. The phase lag between entrance field E0 and intracavity field En 2 is proportional to light intensity |En| . c)Layout of confocal cavity of length Lc = 2F with saturable gain G(E) at right mirror and saturable absorber α(E) at the opposite one. The fields on opposite mirrors E˜n(~rbot) and En(~rbot) are linked via Fourier transform. Spatial soliton is formed by transverse modelocking. b) Layout of diode-pumped solid-state laser with slightly focusing output mirror where vortex-antivortex lattices appear due to transverse modelocking at high Fresnel 2 3 numbers Nf ∼ 10 − 10 .

are described by following maps:

E +1 = g(E ) 1 tanh[g(E )] , (9) n n { − n }

g(En) En+1 = 2 2 2 , (10) 1+ χ3 L g(E ) | n | Both maps have parabolic maxima and their bifurca- tion points are condensed to different values G = λchaos (fig.1) of laser system gain with the same universal speed FIG. 1: a) of ring laser shows distribution δF = 4.6692.... At these bifurcation points the deter- of electric field amplitudes En at gradually increased gain G = ministic dynamical system generates chaotic time series. G1,2...chaos, and histograms representing chaotic probability densities P (E) for gain G = 10.595 (b) and G = 39 (c) after Thus in a model of single transversal mode solid-state 5000 iterations. laser with nonlinear losses [27] this dynamical regime cor- responds to generation of spatially coherent but tempo- rally chaotic radiation. There exists a variety of nonlinear optical systems Apart from universality the above 1D dynamical sys- whose dynamics might be approximated by iterates of tems might be considered as deterministic source of ran- maps with parabolic maxima. The intuitively attrac- dom numbers. In contrast to logistic maps whose range tive example is a toy model of ring laser with intracavity of chaotic oscillation amplitudes is limited within inter- nonlinear losses [27] (fig.2a). Radiation of electric field val λ [0, 1), the time series generated by chaotic optical ∈ amplitude En circulates between confining mirrors along cavities are produced by mapping of semi-infinite interval closed trajectory and it passes repeatedly through gain on itself [21, 23, 27]. element, diaphragms and nonlinear elements. The suc- Randomization of time series is so strong at chaotic cessive passages of field through fast amplifying medium accumulation points λchaos that probability density func- with gain E +1 = g(E ) GE at small E and nonlin- tions (PDF) for amplitudes E at a certain values of bi- n n → n n n ear medium with quadratic χ2 or cubic χ3 susceptibilities furcation parameter λ, G are very close to experimentally 4 obtained histograms for interference of statistically inde- pendent Stockes pulses reflected from independent phase- conjugating Brillouin mirrors [30]. Noteworthy phases of these Stockes pulses are random because stimulated Brillouin scattering originates from thermal acoustic fluc- tuations. As a result the recorded interference pattern I ∼= 1 + cos(∆φ)/2 of the two beams with phase dif- ference distributed uniformly at interval ∆φ [0, π] is also random though light intensity I has exact∈ theoreti- cal probability density P (I) [31] (fig.3c):

P (I)dI ∼= 1 + cos(∆φ) d(∆φ) , d(∆φ) h i1 P (I)dI = , P (I)= . (11) π π I(1 I) − Both histograms (b, c) at fig.3 arep perfectly fitted with P (I)=1/[π I(1 I)] exact probability densities. For this particular case− there exists a remarkable coincidence of dynamicalp chaos PDF and interference pattern PDF. In a more realistic models the nonlinear self-phase modulation inherent to the semiconductor lasers the Kerr cubic nonlinearity [16] may be directly introduced into point maps (9). For this purpose Ikeda constructed a map for complex envelopes of electric field of ultrashort pulses circulating in single transverse mode ring cavity [28] (fig.2b):

2 E +1 = RE exp ikn L + ikn2 E L + T E0, (12) n n · o c | n| nl h i where E0 is external optical pump, R is reflectivity of entrance mirror, T is transmission of entrance mirror, n0 is linear index of refraction, n2 is Kerr component of nonlinear index of refraction, Lnl is width of Kerr slice, Lc is length of cavity. For more general model of laser cavity with nonstationary gain and population inversion FIG. 3: a) Comparison of logistic map (1) x → λx(1 − x) and lifetime T1 the generalized Ikeda map had been intro- duced in [32] and subsequently generalized for wide area hyperbolic tangent map x → Gx[1 − tanh(x)] (2, 3) for laser with nonlinear losses at (G = 6, 9), b) histogram for full chaos laser [9] : after period 3, representing chaotic probability density P (E) obtained by 5000 iterations of logistic maps with λ = 3.9, c) 2 identical probability density P (I) ∼ 1/pI(1 − I) for interfer- E +1 = T E0 + RE exp ikn L + σN L ikn2 E L n n · o c n nl | n| nl ence pattern for Michelson interferometer with independent Nn+1 Nn h N0 Nn 2 i phase-conjugating mirrors obtained by averaging over ensem- − =+ − σNn En , (13)ble of 300 000 counts. ∆t T1 − | | where Nn is population inversion at a given bounce of pulse En in cavity, σ is stimulated emission cross-section, sufficiently long operator products as is shown in [21]. N0 is the pump rate of amplifying medium, ∆t = Lcn/c Indeed each equation (9,10,12,13) in medium of length is discrete time step of map. L admits the decomposition for sequence of thin slices of linear dispersive elements and nonlinear nondispersive el- ements. One may expect that in the limit of infinite num- III. ITERATIONS OF DISCRETE MAPS AND ber n of an infinitely thin slices with ∆L = L/n PARABOLIC PARTIAL DIFFERENTIAL such an→ artificial ∞ medium will be equivalent to perfect EQUATIONS. continuous medium. Within each slice the propagation of pulse En is exactly integrable so that the following The spatiotemporal evolution of pulse envelops En de- product of maps (2) is evident for passage through one scribed by equations (9,10,12,13) may be represented by slice: 5

∞ E +1(~r)= DˆFrfˆ (E (~r)), (14) 1 2 n n E(z =0, ~r⊥)= exp[i~κ ~r⊥]E˜(z =0, ~κ)d ~κ, (23) 2π −∞ · where Z The substitution in (22) leads to Fourier components ξ∂2 ∂2k Dˆ =1+ , ξ = ∆z , (15) after first iterate, i.e. after propagation distance ∆z: 2∂t2 ∂ω2 is dispersion operator, idz~κ2 E(∆z, ~κ)= 1 E(0, ~κ), (24) i∆z − 2k Fˆ r =1+ ∆⊥, (16) n o 2k The next iterates are represented as follows: is diffraction operator. In this picture the propagation of 2 pulse En(~r, t) through m slices is modeled as a product idz~κ 2 E(2∆z, ~κ)= 1 E(0, ~κ), (25) of operators: − 2k n o ˆ ˆ ˆ ˆ ˆ ˆ En+m(~r)= DF r...f DFrf DF r(f[En(~r))] , (17) idz~κ2 m E(m∆z, ~κ)= 1 E(0, ~κ). (26) h h ii − 2k for continuous time variable t = m∆t this product be- n o comes: In order to obtain solution at finite distance L let us use limit m under apparent constraint m∆z = L : → ∞

En+m(~r, t) = lim DˆFˆ r...f DˆFrfˆ DˆFˆ r(f[En(~r))] , m→∞ iL~κ2 m h h ii(18) E(L, ~κ) = lim 1 E(0, ~κ). (27) m→∞ − m2k Consider infinitesimal slice ∆z = L/n and use the n o map (13) for calculation of pulse envelope after passage After rearrangement of this formula we have: through it:

2 iL~κ 1 2k 2km ˆ ˆ E(L, ~κ) = lim 1 ξ E(0, ~κ), ξ = . (28) E(z + ∆z, ~r⊥,t)= DFrf(E(z, ~r⊥,t)). (19) ξ→∞ { − ξ } iL~κ2 h i ˆ ˆ 1 ξ Substitution of operators D and F r in this product Next because lim →∞(1 ) = e is known as Euler ξ − ξ gives: number, we have for Fourier components after propaga- tion at finite distance L:

∂E 2 E(z + ∆z, ~r⊥,t)= E + ∆z = ik ∆z E n0 + n2 E ∂z | | iL~κ2 2 2 ∂ k ∂ E i∆hz∆⊥E i E(L, ~κ) = exp E(0, ~κ), (29) +E ∆z + , (20) 2k ∂ω2 ∂t2 k h i next after return to coordinate space we have: where the second identity leads immediately to NLS- GLE equation: ∞ 2 1 i~κ L 2 E(L, ~r⊥)= exp i~κ ~r⊥ + E˜(0, ~κ) d ~κ, 2π · 2k 2 2 Z−∞ ∂E(z, ~r, t) ∂ k ∂ E i 2 h i (30) + 2 2 + ∆⊥E +ik n0 +n2 E E =0. ∂z ∂ω ∂t 2k | | after substitution of initial Fourier spectrum: h i (21) On the other hand the infinite chain of operators ∞ 1 2 may be used for construction of exact solution of linear E(0, ~κ)= exp[ i~κ ~r⊥] E(0, ~r) d ~r⊥, (31) 2π −∞ − · Shrodinger equation (NLS with n2 = 0) at finite prop- Z agation distance L = m∆z. For this purpose consider we obtain exact solution as Fresnel-Kirchoff integral [21]: pulse propagation in free space using the following map:

ik exp[ikL] idz E(L, ~r⊥)= E(z + ∆z, ~r⊥)= 1+ ∆⊥ E(z, ~r⊥), (22) 2πL × 2k ∞ , 2 n o ik ~r⊥ ~r⊥ ˜ , 2 , exp | − | E(0, ~r⊥) d ~r⊥, (32) Let us decompose envelope E in Fourier integral: −∞ 2L Z h i 6

IV. LOCALIZED WAVETRAINS AS FIXED eigenmodes with identical frequencies greatly facilitates POINTS OF NONLOCAL MAPS the phase-locking of eigenmodes and formation of stable nonlinear localized wavetrains. Noteworthy the excita- Bifurcation diagrams of discrete maps show the loca- tion of threshold solitons due to saturable absorption and tion of fixed points (fig.1a). One may use constructive gain [13, 36] had been realized experimentally in confocal analogy between fixed points of finite-dimensional maps resonator(fig.2c) [37]. and stationary self-similar solutions of evolution PDEs The other possibility is the excitation of thresholdless [33]. Original idea was to construct eigenfunctions of spatial solitons due to gain saturation or nonlinear para- nonlinear resonators with Kerr medium from soliton so- metric processes alike second harmonic generation [21]. lutions of NLS-GLE using boundary conditions [34]. The The cavity is again the confocal Fabry-Perot resonator complications of this technique arise from the fact, that with two spherical mirrors M1, M2 both have the same exact soliton solutions of NLS-GLE are asymptotic ob- focal length F . jects on the whole propagation axis z. As a result such a For detailed numerical modeling it is worthwhile con- generalization of conventional theory of solitons to finite sider the saturable gain medium is described by complex space interval bounded by cavity mirrors is not trivial. maps similar to Ikeda equations [28]: The alternative approach is to use Fox-Lee method when diffraction is taken into account by calculation of Fresnel- 2 E +1 = RE exp ikn F + ikn2 E L + σ N L Kirchoff integrals at each round-trip [21]. For the sim- n n · o | n| am am n am plest Fabry-Perot resonator the mapping of field at n-th Nn+1h Nn N0 Nn 2 i passage into field at n + 1-th passage is as follows: − =+ − σamNn En . (34) ∆t T1am − | |

∞ , 2 The saturable absorbing medium described by analo- ik exp[ikL] ik ~r ~r gous set of Ikeda-like maps: En+1(~r)= exp | − | 2πL −∞ 2L Z h i , , 2 , ˆ f[En(~r )]D(~r )d ~r , En+1(~r)= Frf[En(~r)], (33) 2 E˜ +1 = RE˜ exp ikn F + ikn2 E˜ L σ N˜ L , n n · o | n| ab − ab n ab where D(~r) may be a smooth, say hypergaussian or even ˜ h ˜ ˜ ˜ i Nn+1 Nn N0 Nn 2 step-like Heaviside function θ(d ~r ) , where d is di- − =+ − σabN˜n E˜n , (35) aphragm width, L is distance between− | | mirrors. This ∆t T1ab − | | product of convolution integral operator Fˆ r and point where Lam,Lab << Lr are the thicknesses of ampli- map f(En(~r)) had been named as nonlocal nonlinear map fying medium and absorber media correspondingly, rel- [21]. evant to experimental situation, σam, σab are the stimu- The basic properties of nonlocal maps are visible lated emission cross-sections, T1am,T1ab are longitudi- clearly when different spatial scales are taken into ac- nal relaxation times for amplifier and absorber corre- count. Here the key parameter is Fresnel number Nf = spondingly placed near opposite confocal mirrors or de- 2 kD /L of resonator. For the simplest plane-parallel posited upon their surfaces. The propagation of fields Fabry-Perot cavity Nf is the number of Fresnel zones En, E˜n between mirrors is described by already de- on a given mirror M1 visible from opposite one M2 [35]. fined Fresnel-Kirchoff integral nonlocal maps (33) with There are two limits each have clear physical meaning. parabolic phase-modulation exp[ i2k ~r 2/F ] induced by The first one is the limit corresponding to geometrical op- mirrors: − | | tics i.e. λ . In this case evolution of spatial struc- → ∞ ture En(~r) follows to point transformations [23]. The ∞ , 2 , 2 other case is a single spatial mode limit Nf 1 when ik exp[ikL] ik ~r ~r i2k ~r E +1(~r)= exp | − | | | spatial filtering during each passage through∼ the cavity n 2πL 2L − F Z−∞ is strong enough so En(~r) has a predefined single mode  , 2  , ~r 2 , shape, say TEMoo whose amplitude evolve in time as f[En(~r )] exp | | d ~r , (36) − 2d2 (8-9) [27] or as (11-12) [9] in a presence of self-phase h i modulation. The most interesting case is the intermedi- The standard numerical evaluation with FFT to equi- ate one Nf > 1 when mode interactions are mediated by librium stationary eigenmodes for dicrete time step δt = nonlinearities, diffraction and dispersion. 2Lrn/c is achieved for a different levels of accuracy via 10 Consider first the formation of solitary waves as a re- - 150 iterates [20]. The 2D spatial solitons (fig.4) were sult of phase-locking of transverse modes. Such a local- obtained for the cavity (fig.2c). The key assumptions ized wavetrain is expected to be the eigenfunction of non- for generation of spatial solitons were threshold excita- local map En+1(~r)= Frfˆ [En(~r)]=ΥEn(~r) , where Υ is tion guaranteed via smallness in linear regime of gain eigenvalue. To get exact solutions for localized transverse σampNnLamp compared to absorption σabN˜nLab, faster structures one may consider a specially configured non- saturation of absorption compared to gain, Kerr self- linear cavities. In particular the confocal cavities (fig.2c) focusing in gain slice and higher saturable self-defocusing [13, 21] which are known to have the set of degenerate in absorber slice. Under above restrictions the threshold 7

amplifier plane (fig.2c) is Fourier transform of E(x):

ik ∞ ikxx, E˜(x)= E(x,)exp dx,, E˜(x)= 2πF −∞ F r Z h i π2G 2πx, G = Aam sech , (40) s2σamT1am(1 αG) d 1 αG − h r − i Noteworthy both solutions in Fourier conjugated planes are hyperbolic secants [13] as it follows from exact result for the sech spectrum [38]. The inherent solitonic rela- tion [33] between amplitude and soliton width known as ”area theorem” is embedded here in explicit form. Linear stability analysis with respect to small ampli- tude spatial harmonics had been realized in [13] following the standard perturbative technique [26] with linearized master equation (37): FIG. 4: Spatial soliton En(~r) excited in a given area of [512x512] computational mesh fitted for confocal cavity fig.2c E +1(x)= E (x)+ iζψ , ζψ = αGψ of length 2F with saturable gain G(E) at left mirror and n n ζ ζ − ζ · 2 2 saturable absorber α(E) at opposite mirror. The oscillatory 2 4F ∂ ψζ 1 3σ T1 sech(νx) + G , (41) rings around central part of soliton subjected to self-phase − ab ab kd ∂x2 modulation occur due to interference with background. h i   where ζ is instability increment, ψζ are linear exci- tation modes in effective potential produced by solitons excitations in the form of bright spatial soliton [2] emerge (38), (40). The spectrum of these infinitesimal excita- after a light δ-like click in a given section of computa- tions ψζ consists of the two sets [13]. Noteworthy the tional mesh. above eigenvalue problem is isomorphic to quantum me- chanical problem of scattering a particle of mass m on In order to get analytical solutions the following de- 2 composition in Taylor series to the third order proved to sech (νx) potential well [40]: be useful: 2 2 ~ ∂ ψζ 2 [iζ U0 sech (νx)]ψ =0, 2m ∂x2 − − ζ 4F 2 kd 1 αG E˜n+1(~r)= Gf[En(~r)] + G ∆⊥En(~r),Gα < 1 ν = − , (42) kd 4F r G 2  f(E ) = (α 1)E 1 β E + E , β = σ T1 , (37) n − n − | n| n ab ab The first set of excitations consists of unbounded running   plane waves ψζ ∼= exp(ζt+ipx) with continuum spectrum where G = exp[σamN0Lam]. The exact analytical solu- ζ. All ζ in this set are negative, thus all unbounded tion E(x)= En+1(x)= En(x) in absorber plane (fig.2c) plane wave excitations ψζ quench exponentially in time. in 1D case has the following form: The other set of bounded excitations ψζ , contains both negative and positive energies ζ being equal to positive and negative instability increments correspondingly:

2 xkd 1 αG 2 E(x)= Aab sech − , (38) 1 αG 24αG σabT1ab 4F G ζ = αG + − 1 2n + 1+ . (43) r h r i − 4 − − 1 αG h r − i This solution is generalization of the conventional One may get exact formulas for boundaries of zero ζ Rayleigh formula [35] for focal spot size known as ∆x ∼= which separate areas of stable increments from unstable λF/d: ones as is shown at fig.5. The vital consequence from this linear stability analysis is a necessity of specially λF 4 G arranged filtering of spatial harmonics ψ , belonging to ∆x = λF/d, ∆x = . (39) ζ ∼ the bounded set of excitations with negative spectrum of d2π r1 αG − energies. Filtering of these positive instability increments The effective width is not standard Rayleigh one WRal ∼= ζ will stop the growth of excitations and the stability of λF/d. Indeed in our case the dissipative soliton width is solitons (38), (40) is guaranteed in this case. modified in the presence of gain and losses as: Wsol1 ∼= The other confocal cavity configuration is ring Sagnac- λF √G/(d√1 αG). As a result the effective width of like cavity with thin-slice nonlinear gain medium and spa- spatial soliton− diverges when gain approaches to lasing tial filter in Fourier conjugated planes [21] where exact threshold G α−1. The stationary solution E˜(x) in the solutions for localized solitonic excitations do exist. The → 8

a various nonlinear phase-locking regimes from spatially periodic lattices to fully chaotic states which are speckle fields characterized by randomly spaced optical vortices collocated with zeros of complex field amplitudes En(~r) [4]. The simplest case of periodic structure formation is possible in a plane-parallel Fabry-Perot cavity with a thin slice gain medium having periodic gain distribu- tion in transverse directions G(~r) = G(~r + ~p) , where ~p = ~xpx+~ypy , and ~x, ~y, are unit orts in Cartesian coordi- nates. The self-imaging or Talbot effect is inherent to pe- riodic field distributions. Because Fox-Lee nonlocal maps (22-24) derived above are exact solutions of Maxwell- Bloch equations in paraxial approximation direct sub- stitution of spatially periodic field En(~r) = En(~r + ~p) in (22- 24) immediately proves the Talbot identity of so- called self-imaging of spatially periodic fields at propa- FIG. 5: Location of negative instability increments (hatched 2 area) for spatial soliton in confocal cavity of length 2F with gation distances zT = 2mp /λ, m is integer in a set of saturable gain G(E) at left mirror and saturable absorber cases of commensurability of px,py. The corrections due α(E) at opposite mirror. The vertical axis is for number n of to finite asymmetric aperture having widths dx, dy are spatial harmonic of excitation ψζ . given by:

xnx xnx E(x,y,zT )= Enx,ny exp i2π + px px × exact solution for spatial solitons had been obtained by nx,ny X h  i searching eigenfunctions of similar nonlocal maps which 2 2 (2pxnx x) (2pyny y) also include Fox-Lee convolution integral: exp − 2 2 − 2 2 −(1 + izT /kdx)2px − (1 + izT /kdy)2py × h −1/2 i √ izT izT 2G 1 xkd G 1 1+ 1+ exp[ikz ] . (46) E(x)= Aab2 − sech − . (44) kd2 kd2 · T s σT1ab 4F x y h i h  i The stability analysis in this case is almost identical to The generalization of Talbot theorem which is exact re- the previous one summarized above in eqs. (38, 40, 43, sult of conventional diffraction theory to nonlinear res- 44) and fig.5. In both cavities with gain and losses con- onators is as follows [41]. It had been shown that for sidered above the 1D spatial solitons have generic link the thin gain slice of arbitrary optical nonlinearity the of width and amplitudes in accordance with area theo- spatially periodic field is self-imaged from one mirror upon another provided the distance between mirrors is rem. The interesting common feature of solitonic widths 2 in both cases: a multiple of a Talbot one Lc = mzT = m2p /λ , m is positive integer. The issue of stability of such periodic 1 4F √G 4F configurations requires a special attention. Nevertheless Wsol1 = = ; Wsol2 = ; ν kd√1 αG kd√G 1 the numerical investigations with Fox-Lee nonlocal maps − − λF (33, 34, 35) implemented with the aid of standard fast WRal ∼= 1.22 ; (45) Fourier transform (FFT) routines upon [128x128] and d [1024x1024] computational meshes have shown the stable is that their widths Wsol1 and Wsol2 are basically the eigenmodes composed of 5x5 and 8x8 phase-locked peri- generalization of Rayleigh formula for the width of the odically spaced filaments [19, 20] with field distributions focal spot of a thin parabolic lens WRal illuminated by a En+1(~r) ∼= En(~r) close to in-phase wavefronts and almost plane monochromatic wave of wavelength λ and aperture single-lobe far field Fourier spectra with suppressed side d [35]. lobes. The self-organized vortex lattices in laser output be- came known since 2001 [7] for microchip laser oscillators V. PERIODIC TRANSVERSE STRUCTURES IN composed from diode-pumped thin Nd:YAG gain slice LASER CAVITIES OBTAINED WITH in stable Fabry-Perot cavity with long focus F output NONLOCAL MAP NUMERICAL SCHEMES mirror. The qualitative agreement with predicted peri- odic vortex lattices [8] had been found. The separation Nonlocal maps (33, 34, 35) contain a rich self- of longitudinal modes ∆ωc = πc/(nL) was large enough organization dynamics. Apart from spatial solitons to facilitate the interaction of a dense set of transverse 2 3 whose shape fit the exact solutions (38, 40) the eigen- modes. Fresnel number was in the range 10

VI. MODELING OF THERMAL NOISE MEDIATED PATTERNS

The inclusion of noise in dynamical equations (33, 34, 35) for multimode laser dynamics is natural, because each computational building block of nonlocal maps has a clear physical meaning [16]. In particular the spon- taneous emission in cavity means the addition to En(~r) the multimode speckle field composed via superposition of randomly directed plane waves [43]:

δEn(~r)= Aδ exp[iknx x + ikny y], (47) n ,n Xx y 2 2 2 2 where Aδ is normalization constant, knx +kny +kz = k . The noisy additions to population inversion δNn(~r) and density of carriers δN˜n(~r) are generated via the identical procedure. With these noise sources the master equa- tions are transformed to somewhat more complicated one without decrease of computational speed:

∞ , 2 ikz exp[ikzL] ikz ~r ~r E +1(~r)= exp | − | n 2πL 2L Z−∞ , , h2 , i f[En(~r )]D(~r )d ~r + δEn(~r), (48) The equations for thin gain slice are also modified FIG. 6: Vortex lattices with topological charge ℓ = ±2 per straightforwardly as follows: each singularity in plane-parallel microchip laser with high 2 3 Fresnel number 10 < Nf < 10 obtained by virtue of numer- 2 E +1 = RE exp ik n F + ik n2 E L + σ N L ical modeling with nonlocal maps (33, 34, 35). Apart from n n z o z | n| am am n am [7x7] regular lattice around the center of mesh the chaotic Nn+1 Nn hN0 Nn + δNn(~r) 2 i background of randomly spaced vortex-antivortex pairs is − =+ − σamNn En . (49) seen in bottom ”Phase” plot. ∆t T1am − | | The same holds for the thin saturable absorber slice:

2 (fig.6). For accurate numerical simulation the computa- E˜n+1 = RE˜n exp ikznoF + ikzn2 E˜n Lab + σabN˜nLab tional mesh of a moderate resolution [512,512] proved | | ˜ ˜ h˜ ˜ ˜ i to be sufficient to get the stationary vortex-antivortex Nn+1 Nn N0 Nn + δNn(~r) 2 − =+ − σabN˜n E˜n . (50) state. The detailed pattern of phase dislocations, wave- ∆t T1ab − | | fronts and effective fields of velocities of such lattices [42] For low Fresnel number 5 < Nf < 40 solid state demonstrated the vortex pairing (fig.6). The effective laser cavity the off-axis alignment of optical pump may ~ magnetic fields B˜ which may be realized with such vor- lead to emission of stable topologically charged vortices tex configurations [53, 54] associated with the each phase- (fig.7) [46–48]. Such output patterns are highly desirable locked vortex are counter-directed for adjacent vortices. for metrological [49, 50] and secure free space applica- This happens in contrast to conventional Abrikosov vor- tions [51]. The temporal spectrum of such laser oscil- tex lattices [5] where all vortices produce co-rotating cur- lator is broadened by a set of factors including relax- rents around magnetic field lines. ation oscillations [7]. The fig.8 demonstrates the typical Numerical results on melting of this vortex-antivortex behavior of laser output in the presence of noise. The lattices [39] proved to be in a close similarity with stepwise switching on the population inversion leads to Berezinskii- Kosterlitz- Thauless scenario [44, 45]. The relaxation oscillation [52] with characteristic time scale τ τ T1 because of different time scales of T1 melting of vortex-antivortex lattices ignited by increase rel ∼= c am am of optical gain σ N0L was envisioned as unbound- and photon lifetime in a cavity τc(fig.8a). As expected in am am p ing the vortices and subsequent loss of the long-range a steady-state regime the power spectrum of microchip order. The observed dynamics and proliferation of vor- laser oscillator has a well visible relaxation peak due to tices to random locations fits with conventional model of perpetual disturbances of intracavity field δEn(~r), popu- speckle fields generated by superposition of plane waves lation inversion δNn(~r) and density of carriers in absorber with random phases and directions of propagation [4]. δN˜n(~r) (fig.8b). 10

requires the accurate calculation of eigenvalues and eigen- functions of perturbations of solitons, vortices and their lattices. The easiness of numerical implementation of nonlocal maps especially with the aid of FFT routines promise new results in this field. In particular the clear physical meaning of the Fox-Lee method [55], the natural inclusion of boundary conditions in numerical and ana- lytical schemes and well elaborated numerical procedures for each computational block of nonlocal maps gives the

FIG. 7: Vortex with topological charge ℓ = 1 in plane- parallel microchip laser with low Fresnel number 5 < Nf < 40 generated via nonlocal Fox-Lee map on [512x512] mesh in the 2 presence of multimode noise. a),b) are intensity |En(~r⊥)| plots, c) is phase distribution arg[En(~r⊥)] (enlarged).

VII. CONCLUSIONS

The toy models of 1D map for unidirectional ring laser and 2D Ikeda map for standing wave lasers are shown to be easily modified into much more realistic models via simple Fresnel-Kirchoff convolution integral transform to mediate interaction of spatially distributed point maps. The exact and numerical solutions for spatial solitons were obtained and their stability analysis had been per- formed. For high Fresnel number the iterations of nonlo- cal maps have shown the fast convergence rate to stable square vortex lattices known from table-top experiments FIG. 8: a)Nonstationary relaxation dynamics of output in- 2 [7]. The inclusion of noise leads to realistic relaxation tensity |En| and b) power spectrum in a model of plane- oscillations power spectra during sufficiently short itera- parallel microchip laser with obtained by virtue of numerical tions time intervals. modeling with nonlocal maps (48, 49, 50) shows relaxation Diversity of dynamical regimes of self-organization in oscillation hump. spatially distributed nonlinear systems and computation- ally fast generation of stable spatial structures via nonlo- cal maps had been demonstrated in this work. The close links with conventional evolution equations [17, 22, 52] firm guaranties for avoiding numerical artifacts, better had been established. The stability issues remain still a convergence rates and accurate comparison with exact subject of a very complicated analytical studies, as this results and alternative numerical approaches.

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