Observation of Quadratic Optical Vortex Solitons - APS Link Manager

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Apr 24, 2000 - 2Department of Quantum Electronics, Vilnius University, Building 3 Sauletekio avenue 9, 2040 Vilnius, Lithuania. (Received 2 November 1999).
VOLUME 84, NUMBER 17

PHYSICAL REVIEW LETTERS

24 APRIL 2000

Observation of Quadratic Optical Vortex Solitons Paolo Di Trapani,1, * Walter Chinaglia,1 Stefano Minardi,1 Algis Piskarskas,2 and Gintaras Valiulis2 1

INFM and Department of Chemical, Physical and Mathematical Sciences, University of Insubria at Como, 22100 Como, Italy 2 Department of Quantum Electronics, Vilnius University, Building 3 Sauletekio avenue 9, 2040 Vilnius, Lithuania (Received 2 November 1999) We report on the generation of stable dark-vortex solitons in large-phase-mismatched second-harmonic generation of self-defocusing type, sustained by a combined effect of transverse walk-off and finite beam size. PACS numbers: 42.65.Tg, 05.45.Yv, 42.65.Ky

The intriguing physics of the wave-particle analogy, with its potential for all-optical, ultrafast, processing of digital signals, makes the light soliton one of the most fascinating objects in optics, especially when the sustaining mechanism is the x 共2兲 cascading nonlinearity [1–3]. In fact, the x 共2兲 provides fast response, relatively low thresholds, the possibility of generating new wavelengths or polarizations, and finally that of processing very weak signals by means of the achievable parametric gain. Unfortunately, the x 共2兲 has been shown to support only the lowest-order solitons, i.e., those made by a single, bright element of light, higer-order structures being typically unstable with respect to spontaneous breakup into single, lowest-order units. In this Letter, we report on achievement of the solitonpropagation regime for optical vortices in a x 共2兲 material. The present observation was made possible by getting rid of the modulation instability of the intense background (where the vortex is nested) via the combined effect of lateral walk-off and finite beam size, in regime of large phase mismatch (LPMM) second-harmonic generation (SHG). This result, which represents the first observation of a non-zero-order soliton in a quadratic material, remarkably enriches the light-beam nonlinear dynamics. In fact, optical vortices (screw-type dislocations of the wave front with a helical phase ramp around a phase singularity) show several unique features that distinguish them from conventional beams, like their (finite) angular momentum [4], which provides a new scenario for the investigation of the wave-particle analogy, and their “quantized” topological charge, which is suitable for carrying and processing units (“bit”) of information. Note that such a vortex charge follows algebraic rules when processed via the x 共2兲 nonlinearity [5], quite a relevant issue if the applications to optical computing are considered. For what concerns the solitary regime in non-x 共2兲 media we mention that vortex-soliton solutions of the 共2 1 1兲-dimensional cubic nonlinear Schrödinger equation were introduced and analyzed in the pioneering paper by Pitaevskii [6] as topological excitations in superfluids. In the context of nonlinear optics, they were theoretically suggested by Snyder et al. [7] and demonstrated experimentally in Kerr [8], photorefractive 0031-9007兾00兾84(17)兾3843(4)$15.00

[9], saturable-atomic [10], and photovoltaic [11] nonlinear media (see the review paper [12] for an overview of recent experimental and theoretical results on vortex dynamics). The type of vortex considered in the present work is the so-called “dark-vortex” beam, in which the core singularity is embedded in a large-beam background (in contrast to the case of the “bright” or “doughnut” vortex type [4], where the singularity is nested in a bright, focused, Gaussian-like beam). Note that the sign of the self-phase modulation required to obtain the dark-vortex soliton (DVS) regime is that of self-defocusing (SD). In fact, in this case, the dark core accumulates a phase delay (with respect to the intense, flattop background) that attracts energy from the surroundings, thus leading to a core-shrinking effect which eventually balances its diffraction. Relying upon the mentioned DVS achievements in Kerr media [8,12] and the analogy between the Kerr and the LPMM quadratic nonlinear interactions [2,3] one might expect DVS’s to be easily obtainable also in the regime of SD LPMM quadratic interaction. This is actually not the case since the SD-x 共2兲 interaction is affected by the parametric modulation instability (MI) of the plane wave [13], which is conversely stable in the SD-x 共3兲 case. Because of this MI the large-beam background embedding the vortex core is expected to break up into several bright-solitary spikes [14,15]. This fact is known to prevent the achievement of the DVS regime, at least if conventional settings are concerned. Recently a method for parametric MI elimination has been proposed that uses the contribution of a weak SD x 共3兲 nonlinearity [15]. We are not aware of any experiment performed in this direction. In our settings (see below), as in most of the commonly used bulk x 共2兲 crystals, the Kerr contribution is of the self-focusing (SF) type. Our numerical calculations and measurements (data non shown) indicate that, for the adopted crystal length, intensity, and phase-mismatch values, the only effect of this Kerr contribution is that of slightly speeding up the beam-breakup dynamics, which remains essentially controlled by the parametric MI process. We performed a first experiment in condition of LPMM SHG in the absence of walk-off with the purpose of verifying the impossibility of obtaining the DVS formation, © 2000 The American Physical Society

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FIG. 1. Dark-vortex dynamics in the absence of walk-off: experimental results. (a) FH input dark-vortex beam profile; beam diameter: 500 mm; core diameter: 50 mm; FH wavelength: l 苷 1055 nm. ( b) FH output profile for low input intensity. (c) FH output profile for high input intensity (I 苷 30 GW兾cm2 ) in self-defocusing SHG; nonlinear crystal: 30 mm type I LBO operated at u 苷 90± and f 苷 0±; phase mismatch: Dk 苷 kSH 2 2kFH 苷 20 cm21 ; temperature: 142 ±C; pulse duration: 1.5 ps. (d ) SH output profile for the same input as in (c); SHG conversion efficiency: ⯝2.5%. Picture size: 1.4 3 1.4 mm2 .

in this regime. The experiment was realized by launching a dark-vortex beam at the 1055 nm laser wavelength into a 30-mm-long lithium triborate crystal (LBO), cut

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for noncritical SHG. The high power (5 mJ in 1.5 ps) of our Nd:glass laser system allowed us to use a simple, low-diffraction-efficiency (⯝3%), fork hologram for the vortex generation. The obtained dark-vortex intensity profile at the SHG-crystal input face is shown in Fig. 1a. The desired phase mismatch was reached by properly adjusting the crystal temperature (see the caption for further details). The impact of diffraction on the linear propagation of the vortex through the SHG crystal is evident from the result in Fig. 1b, where the dark-core enlargement in the reported low-intensity output is evident. On increasing the intensity we always observed the expected MIinduced beam breakup and spike formation, without any appreciable reduction in the core size (the largest core shrinkage was less than 20%). Typical output profiles at the FH and second-harmonic (SH) wavelengths are depicted in Figs. 1c and 1d, respectively, taken just above the threshold of detectable MI-induced breakup. The conversion efficiency was ⯝2.5%, thus confirming that we are operating in the LPMM regime. To the best of our knowledge, this is the first experimental report of the nonlinear dynamics of dark-vortex beams in quadratic nonlinear media. These results can be compared with the previous measurements done by Petrov et al. [16] in the different case of the bright-vortex beams. Note that also for the bright-vortices unstable soliton solutions were predicted for SF, SD, and phase-matched quadratic interaction [4,17]. As expected, beam breakup was observed in all three cases [16]. Because of the absence of the large-beam background, however, the breakup mechanism was not attributed, as in our case, to the MI of the plane wave, but to a modulation instability of azimuthal type (AMI) [4,17]. The idea that we are proposing using the lateral walk-off to prevent the MI-induced beam breakup can be intuitively

FIG. 2. The impact of the walk-off on the dark-vortex dynamics in self-defocusing SHG: numerical results. Top: calculated FH profiles at propagation length z 苷 30 mm (corresponding to 143 the confocal beam parameter, for the input core diameter), for different values of the walk-off angle (r). Bottom: FH profiles at z 苷 50 mm. Input beam (in both cases): dark vortex with core diameter d 苷 30 mm and beam diameter d 苷 400 mm; average intensity: 21 GW兾cm2 ; ⯝10% (peak-to-peak) noise level. Model parameters: phase mismatch Dk 苷 kSH 2 2kFH 苷 25 cm21 , quadratic nonlinearity deff 苷 0.82 pm兾V (LBO value), cubic nonlinearity n2 苷 15 3 10214 esu (BBO value, since we are not aware of the LBO one). The white arrow indicates the walk-off direction.

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explained as follows: in the LPMM regime, the undesired spikes should have a SH energy content much larger than the rest of the beam. By choosing a crystal in which the SH field walks away from that of the FH (i.e., operated in type I critical phase matching) we can force the spikes to propagate with a given transverse velocity, with respect to the FH background. If the beam is not too large the spikes will then soon move out of the beam itself, thus leading to a self-quenching of the instability process. The impact of walk-off on dark-vortex dynamics in SD LPMM SHG is well described by our numerical results in Fig. 2, reporting the calculated FH profiles for 2 propagation lengths and 4 different values of the walk-off angle r (see the caption for details). The data show that, for the chosen 0.4 mm beam size, a walk-off angle r ⯝ 0.5± should be sufficient for removing all spikes far from the core region. This allows the soliton regime to be established in the numerical experiment, as indicated by the achieved diffraction-free propagation of the dark core over more than 103 the confocal beam parameter (until whole-beam diffraction occurs). The second experiment in conditions of LPMM SHG in the presence of walk-off required a rather troublesome setup just for the generation of the suitable dark-vortex beam for DVS excitation. In fact, we had to select a FH wavelength slightly smaller (e.g., lFH ⯝ 970 nm) than that of the laser in order to obtain the desired walk-off and phase mismatch while keeping reasonable the crystal temperature. The schematic layout of the adopted setup is depicted in Fig. 3: a first portion of the frequency-doubled laser pulse pumps a three-pass parametric generator/ amplifier, based on a type II b-barium metaborate crystal (BBO), whose output signal beam at ⯝970 nm passes through the fork hologram for the vortex generation. In order to compensate for the low diffraction efficiency of the hologram, and boost up the vortex energy to the desired level, the vortex is launched as a seeding beam in a further single-pass optical parametric amplifier (OPA), pumped by the residual SH laser pulse. This OPA stage is made by a walk-off-free type II LBO (tuning range limited to 960–980 nm), the walk-off absence being needed for amplification of a vortex beam with a sufficiently small core (in the non-soliton regime) and the type II operation to narrow the parametric gain bandwidth, and so prevent undesired cascading effects due to close-to-degeneracy opFork hologram

Filters

Filter LBO-I SHG

LBO-II OPA

λ=970nm

φ = 0 (ρ = 0 ) o

o

Glan pol.

φ = 15o (ρ = 0.5o)

3-PASS BBO-II OPA

Delay line 1.5 ps, 6mJ Nd-glass LASER

SHG

BS

FIG. 3.

λ=527nm

Experimental setup.

CCD

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eration. The OPA is operated in gain-saturation regime, in order to obtain a sufficiently large and flattop beam shape. A low-pass filter and polarizer stop the residual pump and the undesired idler waves, respectively. A two-lens imaging telescope delivers the vortex to the SHG crystal. Figure 4 reports our experimental results on the achievement of the DVS regime. The experiment was performed by directing the FH vortex shown in Fig. 4a into a 30 mm LBO crystal cut for SHG with a walk-off angle r ⯝ 0.5± (see the caption for further details). Note the rather noisy profile of the input vortex, due to the fact that the beam was passed through a saturated OPA. The achievement of the DVS regime in SD SHG is evident from comparison of the FH profiles at the output-face of the SHG crystal in the case of low (Fig. 4b) and high (Fig. 4c) input intensity, respectively, where the last exhibits a diffraction-free core propagation over ⯝10 confocal lengths. For a more quantitative description, see also the transverse intensity profiles along

FIG. 4. The dark-vortex dynamics in the presence of walk-off: experimental results. (a) FH input dark-vortex beam profile. ( b) FH output profile for low input intensity. (c) FH output profile for high input intensity (I 苷 30 GW兾cm2 ) in selfdefocusing SHG; pulse duration: 0.8 ps; nonlinear crystal: 30 mm type I LBO, cut at u 苷 90± and f 苷 15±; FH wavelength: l 苷 970 nm; Dk 苷 20 cm21 ; r 苷 0.47± (crystal rotated to get f 苷 13.7±); temperature: T 苷 130 ±C. (d ) The SH wave corresponding to case (c); conversion #5%. (e) FH output profile for: FH wavelength: l 苷 962 nm; Dk 苷 10 cm21 , r 苷 0.57± (f 苷 17±); T 苷 60 ±C; (f ) FH output profile for: FH wavelength: l 苷 970 nm; Dk 苷 220 cm21 (selffocusing); input intensity half than in case (d ); r 苷 0.5± (f 苷 15±); T 苷 130 ±C. The white arrow indicates the walk-off direction. Single picture width: 1.2 mm.

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stable way by suitably playing with the walk-off and the finite beam size in phase mismatch SHG, an operating condition that was never considered before. These darkvortex solitons represent the first achievement of a stable non-zero-order solitary structure obtained via x 共2兲 :x 共2兲 interaction. We would also like to mention that, to the best of our knowledge, the general understanding of the impact of the walk-off on the gain shape of the modulation instability is still an open issue in the literature, especially if the presence of topological charge and of a nonconstant transverse intensity profile is considered. We hope that the reported data will serve for a deeper understanding of this process. This work was partially supported by MURST (Ministero dell’Università, Ricerca Scientifica e Tecnologica) Project No. 97022268683-001 and by Unesco UVO-ROSTE under Contract No. 875.670.9. Paolo Di Trapani would like to thank Hao He for charming discussions on vortex solitons. All authors are thankful to Yuri Kivshar for many useful suggestions.

FIG. 5. X and Y transverse coordinates profiles of beams in Fig. 4a (dotted line; core diameter: dx 苷 34 mm, dy 苷 31 mm; beam diameter: dx 苷 473 mm, dy 苷 460 mm), Fig. 4b (dashed line; core diameter: dx 苷 104 mm, dy 苷 110 mm), and Fig. 4c (full line; core diameter: dx 苷 52 mm, dy 苷 52 mm).

the X and Y axis in Fig. 5, with the corresponding core diameters listed in the caption. Figure 4d shows the SH output beam, whose dark core is placed at the same position as the FH one (note that the beam is shifted in the figure, due to slightly wedged adopted color filters). According to our numerical results, this SH phase dislocation is expected to have topological charge 苷2. Work is in progress to measure it. Note also the two bright stripes leaving the core in the SH beam, and the corresponding gray depletion line barely visible in the FH profile (Fig. 4c), representing as a sort of shadow of the core singularity in the walk-off direction. These bright SH stripes define the region of the major instability in the beam, as is evident from the calculated FH profiles in Fig. 2. In order to make the spike-formation dynamics in our experiment clearer we operated the crystal closer to phase matching, to increase instability, and at the maximum achievable walk-off (by rotating the crystal), to send the spikes to the very edge of the beam. A typical FH profile is shown in Fig. 4e, with an intense single spike leaving the beam and the dark solitonlike core still evident in the center. Further reduction in the mismatch detuning leads to a complete beam breakup, as also occurs on the SF side of the mismatch detuning (see Fig. 4f). In conclusion, our measurements of vortex dynamics sustained by cascading quadratic interaction have demonstrated how dark vortex solitons can be obtained in a 3846

*Email address: [email protected] [1] W. E. Torruellas et al., Phys. Rev. Lett. 74, 5036 (1995). [2] G. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996). [3] P. Di Trapani et al., Phys. Rev. Lett. 81, 570 (1998). [4] W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997). [5] A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, Opt. Commun. 140, 273 (1997). [6] L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646 (1961) [Sov. Phys. JETP 13, 451 (1961)]. [7] A. W. Snyder, L. Poladian, and J. D. Mitchell, Opt. Lett. 17, 789 (1992). [8] G. A. Swartzlander, Jr., and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992). [9] G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995). [10] B. Luther-Davies, R. Powels, and V. Tikhonenko, Opt. Lett. 19, 1816 (1994). [11] Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, Phys. Rev. Lett. 78, 2948 (1997). [12] Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998). [13] A. V. Buryak and Yu. S. Kivshar, Phys. Rev. A 51, R41 (1995); A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 20, 834 (1995); S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995). [14] T. Alexander, Yu. S. Kivshar, A. V. Buryak, and R. Sammut, Phys. Rev. E 61, 2042 (2000). [15] T. J. Alexander, A. V. Buryak, and Yu. S. Kivshar, Opt. Lett. 23, 670 (1998). [16] D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, Opt. Lett. 23, 1 (1998). [17] D. V. Petrov and L. Torner, Electron. Lett. 33, 608 (1997); L. Torner and D. V. Petrov, J. Opt. Soc. Am. B 14, 2017 (1997); J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, J. Opt. Soc. Am. B 15, 625 (1998).