Underlying conservation and stability laws in nonlinear propagation of axicon-generated Bessel beams
aa r X i v : . [ phy s i c s . op ti c s ] A ug Underlying conservation and stability laws in nonlinear propagation ofaxicon-generated Bessel beams
Miguel A. Porras
Grupo de Sistemas Complejos, Universidad Polit´ecnica de Madrid,ETSI de Minas y Energ´ıa, Rios Rosas 21, 28003 Madrid, Spain
Carlos Ruiz-Jim´enez and Juan Carlos Losada
Grupo de Sistemas Complejos, Universidad Polit´ecnica de Madrid,ETSI Agr´onomos, Ciudad Universitaria s/n, 28040 Madrid, Spain
In light filamentation induced by axicon-generated, powerful Bessel beams, the spatial propagationdynamics in the nonlinear medium determines the geometry of the filament channel and hence itspotential applications. We show that the observed steady and unsteady Bessel beam propagationregimes can be understood in a unified way from the existence of an attractor and its stabilityproperties. The attractor is identified as the nonlinear unbalanced Bessel beam (NL-UBB) whoseinward H¨ankel beam amplitude equals the amplitude of the linear Bessel beam that the axicon wouldgenerate in linear propagation. A simple analytical formula that determines de NL-UBB attractoris given. Steady or unsteady propagation depends on whether the attracting NL-UBB has a small,exponentially growing, unstable mode. In case of unsteady propagation, periodic, quasi-periodic orchaotic dynamics after the axicon reproduces similar dynamics after the development of the smallunstable mode into the large perturbation regime.
I. INTRODUCTION
The nonlinear propagation of very intense pulsedBessel beams (BBs) has attracted a lot of attention inrecent years, specially because of the ability of BBs ofcreating filamentary ionized channels that may be longerand more spatially controllable [1–3] that the filamentscreated by focusing standard, Gaussian-like light pulses[4, 5]. The versatility of Bessel beams for filamentationhas been dramatically demonstrated very recently withthe generation of tubular plasma channels when the seed-ing Bessel beam carries an optical vortex [2, 3]. Theseachievements have opened new perspectives in ultrafastlaser material processing in transparent dielectrics, suchas waveguide writing and micro- or nanomachining [6, 7],or in long-range filamentation in gases with application,for instance, in microwave guiding by filaments in theatmosphere [8].The first studies on BB propagation in nonlinear mediadate from the beginning of the past decade [9, 10]. Non-linear Bessel beams as stationary (non-diffracting) solu-tions to the nonlinear Schr¨odinger equation (NLSE) withKerr nonlinearity were first introduced in [11]. Nonlin-ear unbalanced Bessel beams (NL-UBBs) [12] were laterfound as stationary solutions of the NLSE in media withKerr nonlinearity and nonlinear losses (NLLs), just thetwo key nonlinearities determining the spatial dynamicsin BB filamentation. These NL-UBBs have indeed beenproven to play a prominent role in the filamentation withaxicon focused BBs [1, 13, 14], acting as attractors of thedynamics. Matter waves of this kind can also exist inBose-Einstein condensates [15]. More recently NL-UBBscarrying vortices have been also described [16], and havesimilarly found to act as attractors in the filamentationseeded by axicon focused vortex BBs [2, 3]. In the experimental and numerical studies on nonlin-ear BB propagation, two different initial conditions forthe light entering the medium are usually considered. Ina first arrangement, BBs are launched into the mediumwhen they are already formed [9, 10, 14, 16–19], e. g.,an ideal BB at any transversal plane, or an apodizedBB at the focus of an axicon. Except if NLLs domi-nate initially the dynamics [16–18], Kerr nonlinearity in-duces in this case large temporal and spatial instabilities[14, 19]. In most of filamentation experiments with BBs[1–3, 13, 14] and related numerical studies [20], the radi-ation exiting from the BB generator enters the nonlinearmedium prior to the formation of the BB, in a state ofwidespread energy at low intensity levels, so that the lin-ear BB is never formed. With an axicon, for example, themedium is placed in contact with it, or simply fills thespace surrounding it, as in filamentation in gases. This“soft” input condition has been proven useful to preventthe onset of large temporal instabilities in the nonlinearmedium [14]. With this arrangement, two different Besselbeam propagation regimes have been observed [1, 13].In a steady Bessel propagation regime, the input radia-tion undergoes a transformation into a quasi-stationarystate within the Bessel zone that has been identified asa NL-UBB. In a unsteady regime, the light intensity andfluence feature periodic, quasi-periodic, even disorderedspikes in the Bessel zone (and azimuthal breaking in thecase of vortex BBs [3]). This regime has been associatedwith sufficiently small cone angles and relatively low in-put powers so that self-focusing is the dominant nonlin-earity.In this paper we aim at providing a unified understand-ing of these two regimes of BB propagation under softinput conditions. We show that these two regimes aredifferent manifestations of the same underlying dynam-ics. Either steady or unsteady, the spatial dynamics isdominated by the existence of an attractor in the form ofa specific NL-UBB. We identify the attracting NL-UBBand derive an approximate analytical expression specify-ing it in terms of the properties of the nonlinear mediumand the light beam illuminating the axicon. However, anattractor is not necessarily a stable attractor; its insta-bility may lead to a richer dynamics around it, includ-ing periodic, quasi-periodic and chaotic behavior. Theunsteady or steady regimes are seen to be determinedby the existence or not of a small, exponentially grow-ing unstable mode of the attracting NL-UBB. In the un-steady regime, the unstable dynamics in the Bessel zoneof the axicon is triggered by the unstable mode and isseen to reproduce its characteristic oscillation frequency,its development into large periodic or quasi-periodic an-harmonic oscillations, or its development into chaoticoscillations, depending on the gain of the small unsta-ble mode. Although the unsteady Bessel filamentationregime has been previously suggested to be associatedwith NL-UBB instability [13, 14], it is only the identifi-cation of the attracting NL-UBB that have allowed us toanalyze its stability properties, and hence to verify thathypothesis, putting it in quantitative terms.For simplicity we focus on BBs generated by axicons inmost of the numerical simulations, but the same resultsare seen to hold for other soft input conditions that wouldgenerated BBs in linear propagation. We illustrate theresults in air at 800 nm, in which case the characteristicangles separating the different regimes are quite small,but we have verified that the same results hold at largerangles (but still paraxial) in condensed media.
II. NONLINEAR UNBALANCED BESSELBEAMS
We consider diffraction, Kerr nonlinearity and NLLsas the key effects determining the propagation of thelight beam coming from the BB generator. In the parax-ial approximation, the envelope A of the light beam E = A exp( − iωt + ikz ) of frequency ω and propagationconstant k , is then suitably described by the NLSE ∂ z A = i k ∆ ⊥ A + ikn n | A | A − β ( M ) | A | M − A , (1)where n , n and β ( M ) are, respectively, the linear andnonlinear refractive indexes and the M -photon absorp-tion coefficient. For the initial conditions of interest, andaccording to [14], temporal effects are assumed to play asecondary role.In order to properly understand the propagation, it is im-portant to review the properties of NL-UBBs, stressingtheir asymptotic properties. NL-UBBs were introducedin [12] as non-diffracting and non-attenuating solutionsof (1) of the form A = a ( r ) exp[ iφ ( r )] exp( iδz ), where . I (TW/cm ) ( deg r ee s ) no NL-UBBr (mm) a m p li t ude W / / c m . (a) (b) | b i n , ou t | ( T W / c m ) I ( TW/cm ) |b in | |b out | =1 o =0.15 o FIG. 1. (a) In air at 800 nm ( n ≃ n = 3 . × − cm /W, M = 8, β (8) = 1 . × − cm /W ), region in pa-rameters space ( I , θ ) of existence of NL-UBBs. The dashedblack curve is the approximate analytical curve ( I , max , θ )given in the text. To the right of the gray dotted curve,NL-UBB are said to be NLL-dominated in the sense that L MPA < L dif and L MPA < L
Kerr , where the characteristiclengths are L MPA = 2 /β ( M ) I M − for multiphoton absortion, L dif = 1 / | δ | (Rayleigh range of central lobe of Bessel function)for diffraction, and L Kerr = n/kn I for Kerr nonlinearity.Insets 1 and 2: Amplitude radial profiles of NL-UBBs with θ = 0 .
15 deg, and I = 12 and 28 TW/cm (solid curves),and of linear BBs of the same intensity (dashed curves). Thesquares 1 and 2 indicate the location of those NL-UBBs in pa-rameters space. (b) Values of | b in | and | b out | for NL-UBBswith θ = 0 .
15 and 1 degrees, extracted from the numericallyevaluated radial profiles (gray dotted curves), and approxi-mately evaluated from Eq. (9) (solid curves). δ = − kθ / θ . The real amplitude a ( r ) > φ ( r ) satisfy a ′′ + a ′ r + k θ a − ( φ ′ ) a + 2 k n n a = 0 , (2) − F r ≡ − k πrφ ′ a = β ( M ) π Z r drra M ≡ N r , (3)(prime signs stand for d/dr ) with boundary conditions a (0) = √ I , a ′ (0) = 0, φ ′ (0) = 0, where I is thepeak intensity of the NL-UBB. Equation (3) is the re-filling condition for stationarity with nonlinear absorp-tion, stating that the nonlinear power losses N r in eachcircle of radius r are compensated by an inward radialflux − F r through its circumference. For each cone an-gle θ , NL-UBB exist up to a maximum value I , max of the peak intensity whose value depends on the op-tical properties of the medium at ω . Figure 1(a) showsthe region of existence in the parameters space ( I , θ )of NL-UBBs in air at 800 nm, and the insets two typ-ical radial amplitude profiles. An approximate for-mula relating I , max to θ is θ = σ M β ( M ) I M − , max /k − n I , max /n [dashed curve in Fig. 1(a)], where σ M = 0 . , . , . , . , . , . . . . for M =3 , , . . . [12]. At large radius r , NL-UBBs behave asymp-totically as BBs, but with unbalanced amplitudes of itsoutward and inward H¨ankel components: a ( r ) e iφ ( r ) ≃ h b out H (1)0 ( kθr ) + b in H (2)0 ( kθr ) i , (4)with | b in | ≥ | b out | , and only b in = b out = √ I B for a lin-ear BB of intensity I B . The amplitudes | b in | and | b out | can be easily extracted from the radial intensity profile,known numerically or experimentally: Using the asymp-totic forms of H¨ankel functions for large argument onegets, from Eq. (4), a ≃ ( a m /r )[1 + C cos(2 kθr + ϕ )],with C = 2 | b out || b in | / ( | b out | + | b in | ) and a m = ( | b out | + | b in | ) / (2 π ), meaning that the asymptotic radial intensityprofile consists of oscillations of contrast C ≤ a m /r . From these features, the H¨ankel am-plitudes are obtained to be | b in , out | = πa m (1 ±√ − C ).Examples of their values extracted from the numerical in-tensity profiles are depicted in Fig. 1(b) for increasingNL-UBB intensities I at two cone angles. III. ON THE DYNAMICS OF REAL BESSELBEAMS IN NONLINEAR MEDIA
In Ref. [18], it was demonstrated experimentally thata BB A ( r,
0) = √ I B J ( kθr ) launched in a nonlinearmedium in a regime where NLLs are significant trans-forms spontaneously into a NL-UBB that preserves thecone angle. In the ideal case of BBs carrying infinitepower (and for the broader class of vortex NL-UBBs) thespecific attracting NL-UBB has been identified as thatwhose inward H¨ankel amplitude equal the amplitude ofthe launched BB, that is, | b in | = √ I B [16]. Since for theinput BB b in = b out = √ I B , the amplitude of the inwardH¨ankel component can be said to be a preserved quantityin the nonlinear dynamics.In actual settings the input power is finite, and themedium is placed close to or in contact with an axicon(or other BB generators), or simply fills the space sur-rounding the axicon, so that the BB is not formed whenthe radiation enters the medium and propagates linearlyinitially. With an axicon, for instance, the field enteringthe medium at z = 0 is usually modelled by A ( r,
0) = p I G exp( − r /w ) exp( − ikθr ) , (5)where w and I G are the width and the intensity of theGaussian beam illuminating the axicon. In linear prop-agation, this would produce an apodized BB of inten-sity I B = πkwθI G / √ e at a distance z B = w/ θ , whichis one-half the length w/θ of the so-called Bessel zone.Under these soft input conditions, the unsteady Besselfilamentation regime, associated with small cone anglesand relatively low intensities, results in periodic or quasi-periodic field oscillations [1, 13], though it can also resultin chaos (see below). The steady filamentation regime,associated with large cone angles or higher intensities,has been explained in terms of the formation of a NL-UBB [1, 13, 14].As pointed out in the introduction, these regimes areshown here to be different manifestations of the same un-derlying dynamics. Either steady or unsteady the spa-tial dynamics in BB filamentation is dominated by the existence of an attractor in the form of a specific NL-UBB, unsteady or steady regimes being determined bythe existence or not of a small, exponentially growingunstable mode of the attracting NL-UBB. In Section IVwe identify the attractor and obtain approximate analyt-ical formulas determining it. In Section V we perform alinearized stability analysis of ideal NL-UBB and note abiunivocal relation between NLUBB stability/insabilityunder small perturbations and steady/unsteady propa-gation after the real BB generator. The unsteady Besselregime appears then to be triggered by the existence ofa small unstable mode in the attracting NL-UBB. De-pending on the gain, signatures of this mode, or of its de-velopment into large periodic, quasi-periodic or chaoticperturbation regimes are indeed observed in the Besselzone of the axicon. IV. THE ATTRACTING NONLINEARUNBALANCED BESSEL BEAM
We identify the attracting NL-UBB as that whose in-ward H¨ankel amplitude coincides with the amplitude ofthe BB that the BB generator would create in linear prop-agation, i. e., the NL-UBB with | b in | = √ I B . Since b in = b out = √ I B for BBs, the amplitude of the inwardH¨ankel component is not affected by nonlinearities, andin this sense can be said to be conserved. This conclu-sion is extracted from extensive numerical simulations,of which only a few examples are shown. Conceptually,it is not difficult to understand that the inward H¨ankelcomponent created by the axicon, even if of finite power,and supplying power conically inwards is not affected bynonlinear absorption at the beam center in the Besselzone.Figure 2 illustrates this law for an axicon imprint-ing a cone angle θ = 0 .
15 deg illuminated by Gaussianbeams of width w = 1 . I G = 0 . I G = 0 . and propagating in air at λ = 800 nm. In linear propagation these Gaussianbeams would create linear BBs [dashed curves in Figs.2(a) and (c)] of intensities I B = 39 .
17 TW/cm and I B = 10 .
22 TW/cm , respectively [dashed horizontallines I B in Figs. 2(a) and (c)]. In the steady regime ofFig. 2(a), the on-axis intensity [solid curve in Fig. 2(a)]in the Bessel zone stabilizes in the numerically evaluatedintensity I = 28 TW/cm (solid horizontal line I ) ofthe NL-UBB having | b in | = √ I B = 39 .
17 [see Fig. 1(b)].The whole beam transforms in fact into the attractingNL-UBB, as seen in Fig. 2(b) showing intensity profilesat increasing distances up to the center of the Bessel zone z B = 286 cm. In the unsteady regime of Fig. 2(c), the on-axis intensity [solid curve in Fig. 2(c)] also approaches, z (cm) on - a x i s i n t en s i t y ( T W / c m ) I B I (a) on - a x i s i n t en s i t y ( T W / c m ) (c) z (cm) I I B z=74.3 cm I n t en s i t y ( T W / c m ) z=0 (b) z=149 cm z=286 cm radius r (cm) z=0 z=286 cmz=149 cmz=74.3 cm radius r (cm)(d) FIG. 2. (a) For a Gaussian beam of with w = 1 . I G = 0 . illuminating an axiconthat forms a BB of cone angle θ = 0 .
150 deg and peak inten-sity I B = 39 .
17 TW/cm at the center z B ≃
286 cm of theBessel zone in linear propagation (dashed curve and horizon-tal dashed line), on-axis intensity in nonlinear propagationin air at 800 nm (solid curve), and numerically evaluated in-tensity I = 28 TW/cm2 of the NL-UBB with | b in | = I B (horizontal solid line). (b) Radial intensity profile at increas-ing propagation distances up to the center of the Bessel zone(solid curves), and radial intensity profile of the attractingNL-UBB with | b in | = I B (gray dashed curve). (c) and (d)The same as in (a) and (b) but with I G = 0 . forthe input Gaussian beam, I B = 10 .
22 TW/cm for the linearBB, and I = 12 TW/cm for the NL-UBB with | b in | = I B . but now oscillates about the numerically evaluated in-tensity I = 12 TW/cm (solid horizontal line I ) of theNL-UBB with | b in | = √ I B = 10 .
22 TW/cm [see Fig.1(b)], and the same happens to the whole radial profileat increasing distances. Oscillations may be much morepronounced and disordered, but clear signatures of theattracting NL-UBB, its dominant small unstable mode,and its development into a large perturbation regime, arealways observable, as shown below (see Fig. 7).The law | b in | = √ I B holds for other finite-power ver-sions of BBs, as the Bessel-Gauss beam A ( r, z ) = p I B v v + izk J v v + izk kθr ! × exp − r + z β k v + izk ! , (6) -600 -400 -200 0 200 400 600051015 I B I -600 -400 -200 0 200 400 600010203040 (b) I B I on - a x i s i n t en s i t y ( T W / c m ) z (cm) (a) FIG. 3. The same as in Fig. 2 (a) and (c) but the beamentering into the medium is the Bessel-Gauss beam of Eq.(6) with v = 2 cm at z = −
929 cm from the waist at z = 0.In (a), I B = 39 .
17 TW/cm leading to I = 28 TW/cm , andin (b), I B = 10 .
22 TW/cm , leading to I = 12 TW/cm . producing at z = 0 the Gaussian-apodized BB √ I B J ( kθr ) e − r /v . For a soft input into the medium,the entrance plane is at z = z in ≪ v = 2 cm and z in = −
929 cm, propagationin air at the same wave length and the same linear BBpeak intensities I B = 39 .
17 and I B = 10 .
22 TW/cm asin Fig. 2, the attracting NL-UBBs are seen in Fig. 3 tohave the same intensities I = 28 and I = 12 TW/cm as with the axicon.Thus, given the intensity I B that a BB generator wouldcreate, it is possible to foresee the attracting NL-UBB. Inpractice, this requires to extract the values of | b in | fromthe numerical radial profiles of NL-UBBs of different in-tensities I with the given cone angle in the particularmedium, as explained above [dotted curve | b in | in Fig.1(b)] and in Ref. [16], and to pick up the particular NL-UBB with | b in | = I B . This long numerical procedurewould be greatly simplified if we had analytical expres-sions for | b in | as functions of the NL-UBB parameters( θ , I ) and the optical properties of the medium.An approximate expression can be obtained as fol-lows. We first note that Eq. (4) implies − F ∞ =( | b in | − | b out | ) /k = N ∞ [12], meaning that the un-balance of the H¨ankel amplitudes sets a net constantinward radial power flux coming from a reservoir atlarge radial distances to refill the total NLL N ∞ dur-ing the propagation. At the same time, most of NLLtake place in the beam center, where the NL-UBB canbe approached, if the NL-UBBs is not well within theNLL-dominated region (see caption of Fig. 1), by A ≃√ I J ( √ k θ + 2 kk NL r ) exp( iδz ), with k NL = kn I /n [12]. Evaluation of N ∞ with this profile yields the fol-lowing approximate expression for the NLL of a NL-UBB, and hence an approximate relation between | b in | and | b out | : kN ∞ = | b in | − | b out | ≃ β ( M ) I M kθ (1 + 2 n I /nθ ) γ ( M ) (7)where γ ( M ) ≡ π R ∞ J M ( x ) xdx is a number.Second, numerical evaluation of | b in , out | reveals that,except for NLL-dominated NL-UUBs (see caption of Fig.1), their average value ( | b out | + | b in | ) / | b in | = | b out | ≡ | b Kerr | of the asymp-totic form of the solutions of Eqs. (2) and (3) in theabsorption-less case (i. e., with β ( M ) = 0). Without ab-sorption, φ ( r ) = 0 and the amplitude of nonlinear BBsbehaves as a ( r ) ≃ [ b Kerr H (1)0 ( kθr ) + b ⋆ Kerr H (2)0 ( kθr )] atlarge r . The scaling ρ = kθr and ˜ a = a/ √ I in Eqs. (2),leads to the one-parameter problem ˜ a ′′ + ˜ a ′ /ρ + η ˜ a =0, with initial conditions ˜ a (0) = 1, ˜ a ′ (0) = 0, andwhere η = 2 n I /nθ . From the numerical solutionof this problem with different values of η , we find thevalue of | ˜ b Kerr | of the scaled asymptotic form ˜ a ( ρ ) ≃ [˜ b Kerr H (1)0 ( ρ ) + ˜ b ⋆ Kerr H (2)0 ( ρ )] as a function of η , whichis found to fit accurately to the function | ˜ b Kerr | = f ( η ) =(1 + cη ) / (1 + dη ) with c = 0 .
63 and d = 0 .
76. Com-ing back to real variables, | b Kerr | = √ I (1 + cη ) / (1 + dη )provides semi-analytical solution to the asymptotic formof nonlinear BBs in transparent media. Finally, since( | b out | + | b in | ) / ≃ | b Kerr | in the nonlinearly lossy medium,we obtain | b out | + | b in | = 2 1 + c (2 n I /nθ )1 + d (2 n I /nθ ) p I . (8)The two relations (7) and (8) lead to the approximateformulas | b in , out | ≃ f ( η ) p I ± γ ( M ) β ( M ) I M kθ (1 + η ) f ( η ) √ I , (9)for the amplitudes of the inward and outward Hankelcomponents of NL-UBBs as functions of their cone angle θ and peak intensity I and the medium properties [seeFig. 1(b), solid curves].If we now set | b in | = √ I B in Eq. (9), we obtain theapproximate equation p I B ≃ f ( η ) p I + γ ( M ) β ( M ) I M kθ (1 + η ) f ( η ) √ I (10)( η = 2 n I /nθ ) relating the intensity I of the attractingNL-UBB to the cone angle θ and the intensity I B of thelinear BB that the Bessel-beam generator would create.As an example, the values of I provided by Eq. (10)for NL-UBBs in air at 800 nm, two cone angles θ andincreasing I B are plotted in Fig. 4, and are seen to matchquite accurately the numerically obtained values for NL-UBBs of intensities I below the NLL-dominated case(horizontal dotted lines). Equation (10) is seen to give areasonably good estimate of I even at huge intensities of I B = 100 TW/cm well within the NLL-dominated case. I ( T W / c m ) I B ( TW/cm )=0.15 o =1 o FIG. 4. In air at 800 nm and for the indicated cone angles,intensity I of the attracting NL-UBB as a function of the in-tensity I B of the linear BB that the BB generator would cre-ate in linear propagation, numerically evaluated (dotted graycurves), and obtained from Eq. (10) (solid curves). Abovethe horizontal dotted lines NL-UBBs are NLL-dominated. =10 I =14 TW/cm I =20 θ (deg) ga i n − I m κ ( c m − ) (a) (b)(c) (d) =10I =14 TW/cm I =20 θ (deg) R e κ ( c m − ) θ =0.05 deg θ =0.075 θ =0.1 θ =0.15 θ =0.2 I (TW/cm ) ga i n − I m κ ( c m − ) θ =0.05 θ =0.075 θ =0.1 θ =0.15 θ =0.2 deg I (TW/cm ) R e κ ( c m − ) FIG. 5. (a) Gain − Im κ and (b) oscillation frequency Re κ ofthe most unstable mode mode of NL-UBBs in air at 800 nmas functions of their cone angle for a few values of their peakintensity I . (c) and (d) The same as in (a) and (b) but asfunctions of the peak intensity I for a few values of the coneangle θ . V. STEADY AND UNSTEADY REGIMESVERSUS NL-UBB STABILITY
Once the attractor is specified, our numerical simu-lations indicate that there is a bi-univocal relation be-tween steady/unsteady propagation regime and stabil-ity/instability of the attracting NL-UBB against smallradial perturbations. propagation distance z (cm) on - a x i s i n t en s i t y I ( T W / c m ) I = 14 TW/cm , = 0.075 deg (a) (c) de r i v a t i v e d I/ d z ( T W / c m ) on-axis intensity I (TW/cm )z=400 - 3000 cm = 0.075 degI = 14 TW/cm de r i v a t i v e d I/ d z ( T W / c m ) on-axis intensity I (TW/cm )I = 14 TW/cm = 0.075 deg z=0 - 12000 cm(e) I = 14 TW/cm , = 0.05 deg on - a x i s i n t en s i t y I ( T W / c m ) propagation distance z (cm) (d) de r i v a t i v e d I/ d z ( T W / c m ) z=400 - 3000 cmon-axis intensity I (TW/cm ) = 0.05 degI = 14 TW/cm (f) z=0 - 12000 cm = 0.05 degI = 14 TW/cm on-axis intensity I (TW/cm ) de r i v a t i v e d I/ d z ( T W / c m ) FIG. 6. In air at 800 nm, on-axis intensity I versus propagation distance z of perturbed, ideal NL-UBBs of peak intensitiesintensities I = 14 TW/cm and cone angles (a) θ = 0 .
075 deg and (b) θ = 0 .
05 deg. (c) and (d) Corresponding phase spaces I – dI/dz in [400 , , For fundamental (vortex-less) NL-UBBs, radial insta-bility appears to be the dominant instability, since no az-imuthal breaking has been observed in experiments andsimulations [1, 12, 13, 18, 19], particularly under soft in-put conditions [14]. Linearized stability analysis of NL-UBB against radial perturbations has been performednumerically for typical values of NL-UBB parameters inRef. [12] and [14], where all details of the procedure areexplained. In short, supposing a solution to the NLSE(1) of the form A = a ( r ) e iφ ( r ) e iδz + ǫ [ u ( r ) e iκz + v ⋆ ( r ) e − iκ ⋆ z ] e iδz , (11) that is, a NL-UBB plus a small ( ǫ →
0) mode ( u, v ) thatgrow exponentially in case that Im κ < κ , a differen-tial eigenvalue problem is obtained for the eigenvalues κ and eigenmodes modes ( u, v ), which has to be solvednumerically. As pointed out in [14], the difficulty withthis analysis for NL-UBBs compared to that for stan-dard solitons lies in the weak localization of NL-UBBs.Truncation of the NL-UBB in any finite radial box im-posed by the numerical procedure sets a lower bound tothe reliable values of | Im κ | [14]. Figure 5 shows exam- I propagation distance z (cm) on - a x i s i n t en s i t y I ( T W / c m ) (a) I B = 0.075 deg = 0.075 deg(c) de r i v a t i v e d I/ d z ( T W / c m ) on-axis intensity I (TW/cm ) = 0.050 deg I B I (b) on - a x i s i n t en s i t y I ( T W / c m ) propagation distance z (cm) (d) = 0.05 degon-axis intensity I (TW/cm ) de r i v a t i v e d I/ d z ( T W / c m ) FIG. 7. (a) On axis intensity I (solid curve) in air at 800 nm after and axicon imprinting a cone angle θ = 0 .
075 deg illuminatedby a Gaussian beam of width w = 2 . I G = 0 . . In linear propagation (dashed curve) theintensity of the BB would be I B = 10 .
508 TW/cm (horizontal dashed line) at z B = 955 cm, so that the attracting NL-UBB isdefined by I = 14 TW/cm (horizontal solid line) and θ = 0 .
075 deg. (b) The same except that θ = 0 .
05 deg and I G = 0 . . In linear propagation the BB of intensity I B = 10 .
165 TW/cm would be formed at z B = 1432 cm, so that theattracting NL-UBB is defined by I = 14 TW/cm and θ = 0 .
05 deg. (c) Phase space I – dI/dz about the focal region of theaxicon in case (a). (d) Phase space in the whole propagation in case (b). The dashed lines in (c) and (d) locate the attractor. ples of the exponential gain − Im κ and the oscillationfrequency Re κ of the most unstable mode of NL-UBBsin air at 800 nm as functions of the cone angle and fixedvalues of the peak intensity [Figs. 5(a) and (b)], and asfunctions of intensity and fixed values of the cone angle[Figs. 5(c) and (d)]. As noted in [1], NL-UBBs tend tostabilize as the cone angle increases. According to ouranalysis, no signs of instability are present for θ abovea certain threshold angle (about θ ≃ .
23 deg in Fig.5), but a definitive response to the question of the ab-solute stabilization of NL-UBBs cannot be given. Thetrend of − Im κ with increasing cone angle suggests anexponential decay. With increasing intensity, Kerr non-linearity renders NL-UBBs increasingly unstable at first,but the increasing NLLs has an opposite stabilizing effectat higher intensities. Stabilization by NLL appears to becomplete above a certain cone angle (about θ ≃ .
15 degin Fig. 5). It is interesting that below this angle, oncethe Kerr-induced unstable mode disappears by the ac-tion of NLLs (at about I = 18 TW/cm ), an underlyingunstable mode with a different oscillation frequency Re κ becomes dominant.In connection with Figs. 2(a) and 3(a), the steadypropagation regime after the axicon can be seen to beassociated with the absence of unstable modes of the at-tracting NL-UBB with θ = 0 .
15 and I = 28 TW/cm that tends to be formed at the center of the Bessel zone.Even if repeated self-focusing cycles are observed beforethe focus of the axicon for these small cone angles (cy- cles that may be much more pronounced), and beforethe waist of the Bessel-Gauss beam, the input radiationis pushed stably towards the NL-UBB about the center ofthe Bessel zone. In Fig. 2(c) and Fig. 3(b), instead, the(weak) unstable regime appears to reflect the instabilityof the attracting NL-UBB with θ = 0 .
15 and I = 12TW/cm . The gain is indeed low [Fig. 5(c)] (comparedto next situations below), and the oscillation frequencyabout the focus or waist is seen to coincide with the os-cillation frequency Re κ of the dominant unstable modeof the attracting NL-UBB [Fig. 5(d)].The connection between unsteady Bessel propaga-tion regime and instability of the attracting NL-UBB isclearer in situations of higher gain. For two NL-UBBswith increasing gain, Fig. 6(a) and (b) shows the growthof the respective dominant unstable modes with propa-gation distance. These small modes develop into large,periodic (but no longer harmonic) perturbation regimes,that gradually turn into quasi-periodic, and eventuallyinto chaos. In all cases we have studied, this process isfound to be faster as the gain − Im κ triggering this pro-cess is higher. Also, the oscillation frequency in the large,periodic perturbation regime is close to but slightly lowerthan Re κ . In simulations as those of Fig. 6, NL-UBBsare directly launched into the medium and the dominant,unstable modes of each NL-UBB are found to emergespontaneously from numerical noise with the gain and os-cillation frequency predicted by the linearized instabilityanalysis [left part in Figs. 6(a) and (b)]. To simulate thepropagation of ideal, non-truncated NL-UBBs, we usethe procedure of replacing the propagated field at eachaxial numerical step of propagation with the initial NL-UBB in a narrow annulus touching the end of the (quitelarge) numerical radial grid. This procedure is justifiedsince no dynamics is expected to take place in the linearasymptotic tails. In Figs. 6(c-f), phase spaces I – dI/dz ( I on-axis intensity) in relevant propagation intervals areshown. In the case of lower gain, the large perturbationregime remains periodic for a considerable propagationdistance [Fig. 6(c)], whereas in the case of higher gain, itbecomes quasi-periodic from the beginning of the largeperturbation regime [Fig. 6(d)] and enters sooner intochaos. The phase spaces up to the longest propagationdistance [Figs. 6(e) and (f)] evidence that NL-UBBs areactually chaotic attractors, whose morphology dependson the specific NL-UBB.On the other hand, Figs. 7(a) and (b) show the on-axisintensities after an axicon illuminated with two Gaussianbeams of width and intensities such that the attractorsare the two NL-UBBs analyzed above. In the case of Fig.7(a) with lower gain, the oscillation frequency about thefocus of the axicon coincides with that of the large, pe-riodic, perturbation regime. Indeed the structure of theoscillations in the phase space of Fig. 7(c) about the fo-cus of the axicon mimics the structure of the anharmonicoscillations in the periodic perturbation regime of the at-tracting NL-UBB in Fig. 6(c). Small differences orig-inate from the slow decay of intensity along the Besselzone of the finite-power BB. The structure of the phasespace in the whole Bessel zone (not shown) does not re-produce the morphology of the chaotic attractor in Fig.6(e). In the case of Fig. 7(b) with higher gain, the on-axis intensity in a considerable part of the Bessel zoneexhibits a highly disordered dynamics. Comparison ofthe morphology of the phase space in Fig. 7(d) for thewhole Bessel zone with that of the attracting NL-UBBin Fig. 6(f) evidences that the dynamics after the axi-con is reproducing the chaotic dynamics about the idealNL-UBB chaotic attractor. VI. CONCLUSIONS
From a series of diagnostic numerical simulations, wehave extracted the underlying laws governing the spatialdynamics of the light beam emerging from an axicon,and entering a medium where self-focusing Kerr effectand multiphoton absorption are relevant. If as pointed out, temporal and plasma effects play a secondary role indetermining the spatial dynamics in filamentation withBBs, these laws provide an unified understanding of thedifferent Bessel filamentation regimes described previ-ously. In a few words, the nonlinear propagation is deter-mined by an attracting NL-UBB and its stability proper-ties under small perturbations. The attracting NL-UBBis that whose inward H¨ankel amplitude equals the am-plitude of the BB that the BB generator would createat the center of the Bessel zone in linear propagation.We have derived an approximate analytical expressionthat determines the attracting NL-UBB given the opti-cal properties of the medium, the cone angle, and theintensity of the linear BB (or equivalently, the axiconbase angle and the input Gaussian width and intensity).Steady/unsteady propagation regimes are shown to cor-respond to stability/instability of the attracting NL-UBBunder small radial perturbations, i. e., to the existence ofa small unstable radial mode that tends to grow exponen-tially. We have performed an extensive stability analysisunder small radial perturbations that put in quantitativeterms the stabilization effect of increasing the cone an-gle and the intensity. In case of instability under smallperturbations, NL-UBBs are seen to develop a large per-turbation and chaotic regimes with increasing propaga-tion distances. In the Bessel zone after the axicon, anddepending on how large the gain of the small unstablemode of the attracting NL-UBB is, the unsteady dynam-ics reproduces the dynamics of the small perturbation,large, or chaotic perturbation regimes of the attractingNL-UBB. Though a direct relation with increasing gainis obvious, further research would be needed to specifyin more quantitative terms the particular perturbationregime (small, large or chaotic) of the attracting NL-UBBthat is observed in the Bessel zone. We have restrictedourselves to vortex-less NL-UBBs, but the generality ofthese ideas suggests a relatively simple generalization toaxicon-generated vortex NL-UBB, in which case not onlyradial instability but also azimuthal instability should betaken into account.
ACKNOWLEDGMENTS
M.A.P. acknowledges support from Projects of theSpanish Ministerio de Econom´ıa y Competitividad No.MTM2012-39101-C02-01 and No. FIS2013-41709-P.J.C.L. acknowledges support from Project of the Span-ish Ministerio de Econom´ıa y Competitividad MTM2012-39101-C02-01. [1] P. Polesana, M. Franco, A. Couairon, D. Faccio, and P.Di Trapani, “Filamentation in Kerr media from pulsedBessel beams,” Phys. Rev. A [3] V. Jukna, Mili´an, C. Xie, T. Itina, J. Dudley, F. Cour-voisier, and A. Couairon, “Filamentation with nonlinearBessel vortices,” Opt. Express15,