Multiple branching of vectorial spatial solitary waves in quadratic media

15 January 1998

Optics Communications 146 Ž1998. 356–362

Full length article

Multiple branchg of vectorial spatial solitary waves in quadratic media Giuseppe Leo, Gaetano Assanto

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Department of Electronic Engineering, Terza UniÕersity of Rome, Via della Vasca NaÕale 84, 00146 Rome, Italy Received 20 May 1997; revised 27 August 1997; accepted 3 September 1997

Abstract Branching of spatial solitary waves due to quadratic parametric interactions in the vectorial frequency-degenerate case is discussed. The phenomenon, favoured by balanced inputs at the fundamental frequency in the absence of a second-harmonic seed, does not rely on the presence of birefringent walk-off. Beam breakup into several solitary beams propagating along different directions is interpreted in terms of amplitude and phase modulation of the wave transverse distributions. q 1998 Elsevier Science B.V.

After the first demonstrations of spatial solitary waves ŽSSWs. in bulk KTP w1x and in LiNbO 3 waveguides w2x, the investigation of solitary waves in quadratic media has evolved from the strictly mathematical approach w3–5x to more physical grounds w6x. Accounting for an increasing number of realistic issues, stability analyses of scalar SSWs have been performed for waveguides w7x and bulk media w8x, initially neglecting walk-off. More recently, two-parameter families of quadratic solitons have been discovered, which are associated to Type I second harmonic generation ŽSHG. in the presence of walk-off Ž‘‘walking solitons’’. w9,10x, or three-wave mixing without w11x, and with walk-off w12x. Among the potential applications based on self-trapping and dragging of SSWs, novel forms of beam conditioning have been proposed and demonstrated w13x, and SSW collisional interactions have been addressed in both the scalar w14,15x and the vectorial cases w16x. In this Communication, with reference to Type II SHG interactions between intense beams propagating in a planar waveguide containing a x Ž2. material, we discuss multiple branching of SSWs. This phenomenon, addressed in the context of eigenvalue switching by Torner et al. for the scalar Type I case w17x, consists of the beam breakup into

1

Corresponding author. E-mail: [email protected].

self-confined channels departing from each other. Analysing a pure SHG upconversion scheme, we show that the transition between a single SSW and a multi-branched self-confined field preferably occurs for nearly balanced inputs at the fundamental frequency. Moreover, this transition is strongly dependent on the sign of the phase-mismatch. Beam branching is not due to birefringent walk-off, which is instead responsible for asymmetric propagation with respect to the launch direction into the crystal. In the framework of the slowly-varying envelope and Fresnel ` approximations with Kleinman symmetry holding, the cw Ž1 q 1.D evolution of the three electric fields interacting via ‘‘eoe’’ SHG is described by the set of differential equations:

Ez Ev ,o q

Ex x Ev ,o 2i k v ,o

s i k E2 v ,e Ev),e exp Ž yiD kz . ,

Ž Ez y rv Ex . Ev ,e q

Ex x Ev ,e 2i k v ,e

Ž Ez y r 2 v Ex . E2 v ,e q

s i k E2 v ,e Ev),o exp Ž yiD kz . ,

Ex x E2 v ,e 2i k 2 v ,e

s 2i k Ev ,o Ev ,e exp Ž iD kz . ,

Ž1. where the propagation coordinate z defines with x the waveguide plane, and diffraction occurs in the latter trans-

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 5 0 7 - 5

G. Leo, G. Assantor Optics Communications 146 (1998) 356–362

357

verse dimension only. The subscripts ‘ore’ and ‘ vr2 v ’ ŽFFrSH. indicate the ordinaryrextraordinary polarizations and the fundamentalrharmonic frequencies, respectively; r i Ž i s v ,2 v . are the walk-off angles of the ‘e’ components, and the normalizations are such that intensities in Wrm are given by the square moduli of the fields. The nonlinear coupling constant is defined as

(

k ' v d eff

2 c ´ 0 nv ,o nv ,e n 2 v ,e h eff

,

with refractive indices n defining the phase velocity of each field, d eff the effective susceptibility, and h eff the effective waveguide depth, accounting for the overlap between the eigenmodal profiles. The mismatch between the wavevectors is D k ' k v ,o q k v ,e y k 2 v ,e . Referring to experimental conditions in a KTP slab guide, we chose h eff s 2 mm, l s 1064 nm, beam waist w 0 s 20 mm in x, diffraction length L D ' p w 02 nv ,orl f 2 mm, rv s 3.1 = 10y3, r 2 v s 4.9 = 10y3, and k f 0.2 Wy1r2 my1r2 . Hereafter, the peak intensities of the three field components will be indicated by Iv ,o , Iv ,e , I2 v , distances along x and z expressed in units of w 0 and L D , respectively Ž z ' z L D , x ' j w 0 ., and the outputs taken at z s 30. Eqs. Ž1. form a non-integrable set, and we employed a numerical split-step propagator to explore the space of experimentally relevant parameters Žmismatch, input power, input polarization.. Typical laser-emitted Gaussian Žalong x . beams were launched at intensities comparable to those actually injected in KTP experiments, from pure upconversion with either balanced or unbalanced inputs at v w13x to seed-controlled downconversion w18x. In particular, we spanned a large region of wavevector mismatches, encompassing both the Kerr limit investigated in Ref. w17x and the inherently quadratic phenomenology described by Eqs. Ž1. for large conversion efficiencies w19x, i.e. far from the nonlinear-Schrodinger-equation limit. ¨ Here we treat the standard upconversion SHG process, by initially neglecting walk-off Ž rv s r 2 v s 0.. In wide ranges of mismatch Ž< D k < F 500p my1 . and total input power Ž Pv R 1 kW., we verified the existence of unbalanced SSWs, with ratio between polarization components at v dictated by the corresponding input imbalance. One typical case is shown in Fig. 1, where the outputs Iv ,o , Iv ,e , I2 v are plotted versus the input imbalance d s wŽ Iv ,o y Iv ,e .rŽ Iv ,o q Iv ,e .x zs0 , for D k s 10p my1 and total input power Pv s 10 kW. Due to high excitation, self-trapping occurs for < d < Q 0.95, i.e. except in the proximity of the purely diffractive linear cases Ž d s "1.. In Fig. 1a, where the whole imbalance range is shown, it can be noticed that the curve relative to the SH field lies between the other two, and that an abrupt polarization switching occurs near d s 0. Zooming into the low-imbalance region, a local prevalence of the SH over the FF components is apparent ŽFig. 1b.. Moreover, for small focusing mismatches Ž0 F D k F 50p my1 . in the above excitation

Fig. 1. Ža. Output peak intensities Žat z s 30. of the SSW field components versus the FF input imbalance d , with D k s10p my1 and Pv s10 kW. Žb. Detail of the polarization switch, near the balanced-input condition. Solid lines refer to Iv ,o , dashed lines to Iv ,e , and dotted lines to I2 v . Intensities are normalized to the total input peak intensity.

range, this feature is always verified and is consistently associated to branching of the SSW. An example of these imbalance-controlled transitions between single and multiple SSWs is given in Fig. 2, where contour plots show the SSW propagation ŽFF field. for the same conditions as in Fig. 1, with imbalances d s y0.02 ŽFig. 2a., d s 0 ŽFig. 2b., d s 0.02 ŽFig. 2c.: these changes from a single channel to a ‘‘fork’’ junction or viceversa are sharp and subject to a purely incoherent control. These are appealing features for an all-optical reconfigurable multiplexer, also in view of the fact that the SSW total power is well distributed between the three channels of Fig. 2b, with the lateral channels carrying 2.64 kW each and the central channel 2.85 kW. Since device cascadability demands a single carrier frequency, it is noteworthy that the FF power fraction is approximately 64% in each split branch, and

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G. Leo, G. Assantor Optics Communications 146 (1998) 356–362

Fig. 2. Intensity propagation along z of the total FF field Žsimilar evolution for the SH., with imbalances d s y0.02 Ža., d s 0 Žb., d s 0.02 Žc., and other parameters as in Fig. 1.

about 69% in the single channel cases. Radiation losses amount to 19% in Fig. 2b and 38% in Figs. 2a, 2c and can be taken as typical in the range 5 kW F Pv F 20 kW. Finally, the total self-trapped power in the one-channel cases is approximately 23% less than the total self-trapped power in the ‘‘fork-junction’’ case. This observation qualitatively agrees with the theoretical statement by Buryak et al. w20x, i.e. that the threshold energy above which stable solitons can exist is the highest in the polarization-balanced case Ž d s 0.. It is worth emphasizing that, in order to obtain the ‘‘fork-junction’’, it is not necessary to have exactly d s 0: as long as the condition I2 v ) Iv ,o , Iv ,e is fulfilled Žsee Fig. 1b., different values of d simply alter the power sharing between the three channels. This sharing results optimally equalized for an exactly balanced input, as in the reported case. Insight into the SSW branching phenomenon can be gained by monitoring the evolution of the three fields in the early stages of propagation in the case of splitting, such as the one in Fig. 2b. Defining the interaction length Lint as the distance required for a one-way intensity transfer between FF and SH at j s 0 Žon axis., the transverse intensity profiles are plotted in Fig. 3 after L int ŽFig. 3a., 2 L int ŽFig. 3b., and 3 Lint ŽFig. 3c.. In Fig. 3a, a central dip in the intensity profiles at v and a corresponding peak at 2 v occur. With depletion being generally limited by the

weakest FF input, the greater the total input intensity the deeper and broader the dip becomes. In the successive down-conversion cycle ŽFig. 3b., since parametric gain is higher where the field overlap is larger, the profile at 2 v is no longer Gaussian. The formation of multi-humped profiles develops through cycles of up- and down-conversions and eventually leads to breakup and generation of several solitary waves ŽFig. 3c.. Corresponding to the amplitude profiles, the phase fronts of each wave undergo distortion. As the diffractive regime is altered by the parametric interaction, the convex wavefronts become locally modulated in a step-wise fashion due to the nonlinearly-induced phase shift w21x, corresponding to selftrapped lobes in the intensity profile. Associated to each branch of the split SSW shown in Fig. 2b, flattened portions appear in the phase profiles. The angular separation between the central and the lateral SSW channels can therefore be related to the angle between the average wavenormals to each flattened wavefront portion. At lower input levels than those considered above, the sidelobes do not carry enough power to ‘‘turn on’’ further phase steps besides the central one: therefore they radiate away even in the case of FF balanced inputs, giving rise to the central SSW only. At higher input levels Ž Pv R 30 kW., due to transverse modulation of parametric gain, more than three channels can be generated. Eventually, for even stronger excitations, each SSW channel can in turn

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Fig. 3. Intensity profiles along x of the SSW field components of Fig. 2b, at z s 0.45 Ža., z s 1.65 Žb., z s 2.1 Žc.. Solid lines refer to Iv ,o , dashed lines to Iv ,e , dotted lines to I2 v r10 Ža. and I2 v Žb, c.. Intensities are normalized as in Fig. 1.

become a source of ‘‘secondary’’ branches, until the splitting patterns become quite intricated. This occurrence, although intriguing in the framework of nonlinear fractal growth, is an issue only at powers G 40 kW and close to phase matching. In addition to the input intensities, the phase mismatch D k plays a key role in the transition between single and multiple SSWs. Restricting to the case d f 0, which is relevant in terms of branching, we studied SSW propagation for several values of D k, in a wide interval of input powers ranging from SSW formation threshold Ž Pv R 1 kW. up to high excitations Ž Pv F 35 kW.. Compared to the case D k R 0 examined so far, the presence of moderately large positive mismatches Žup to f 200p my1 . increases the branching threshold of factors as large as five. Furthermore, larger mismatches inhibit the branching, because ‘‘cascading’’ becomes inefficient in the beam tails during the first interaction lengthŽs.. Conversely, negative mismatches tend to counteract the branching for any given input. For small < D k < and excitations, therefore, the sign of the mismatch is crucial in determining the occurrence of branching. This feature is in agreement with the notion that Type II solitons have lower formation thresholds for D k ) 0 w1,2x. For large < D k <, on the other hand, radiation losses increase and branching no longer occurs, irrespective of the mismatch sign. One such example is shown in Fig. 4, where Pv s 15 kW, d s 0,

and the cases D k s "500p my1 ŽFigs. 4a, 4c. and D k s 10p my1 ŽFig. 4b. are presented. Regardless of branching, however, the SSW propagation depends in all cases on the sign and magnitude of D k, as shown in Fig. 5 with reference to the central channels of Fig. 4. For D k s 500p my1 Žfocusing., long-lasting in-phase peak-intensity oscillations of the three mutually trapped fields occur along z. The oscillation amplitude increases with the total input power, and the SH component is weaker than the other two Žthe higher Pv , the smaller such intensity difference.. For D k s y500p my1 Ždefocusing., highly unbalanced SSWs propagate, which are less intense and exhibit higher thresholds and radiation losses than for a positive mismatch. Above SSW threshold, breathing oscillations are negligible, and the SH peak intensity is bounded by the FF ones. A further example of the combined influence of mismatch and input intensity is given in Fig. 6, where both the peak intensity Žsolid lines. and the symmetric displacements of the lateral channels Ždashed lines. are reported versus Pv , for different D k. For D k s 10p my1 Žsame case of Fig. 2b., an increase of the total input power reduces the angle between lateral and central channels ŽFig. 6a.. For D k s "100p my1, the lateral channels are only weakly confined, far less intense than in the previous case, and tend to depart from the central SSW as the input increases ŽFig. 6b, where only the case of positive mis-

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G. Leo, G. Assantor Optics Communications 146 (1998) 356–362

Fig. 4. Intensity propagation along z of the total FF field Žfor the SH similar evolution., with Pv s 15 kW, d s 0, and nonlinear mismatch: Ža. D k s 500p my1 , Žb. D k s 10p my1 , Žc. D k s y500p my1 .

Fig. 5. Peak intensity evolutions along z of the three SSW components within the central channel, in the same conditions as in Fig. 4. Solid lines refer to Iv ,o , dashed lines to Iv ,e , and dotted lines to I2 v . Intensities are normalized as in Fig. 1.

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Fig. 6. Output peak intensity Žsolid lines, left scale. and lateral displacement Ždotted lines, right scale. of either lateral SSW Žat z s 30., versus total input power Pv , with d s 0 and phase mismatch: Ža. D k s 10p my1 , Žb. D k s 100p my1 . Intensities are normalized as in Fig. 1.

Fig. 7. Intensity propagation along z of the total FF field Žsimilar evolution for the SH. in the presence of walk-off, with imbalances d s y0.2 Ža., d s 0 Žb., d s 0.2 Žc., and other parameters as in Fig. 1.

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G. Leo, G. Assantor Optics Communications 146 (1998) 356–362

match is shown.. Notice that, due to the small distance between adjacent channels, interactions between them can arise for very intense excitations and small mismatches. Evidence of the branching phenomenology, which occurs regardless of walk-off, was observed by Torruellas and coworkers during experiments in bulk KTP w1,13x. At very high input intensities, two output SSW spots were observed, symmetrically placed with respect to the walk-off plane w22x. When walk-off is introduced in the Ž1 q 1.D simulations, both single and multiple SSWs simply undergo symmetry breaking, as suggested by Fig. 7. In conclusion, we have investigated the branching phenomenology intrinsic to the propagation of SHG vectorial spatial solitary waves. The effects discussed here depend neither on asymmetries nor on multi-peaked input profiles, as those recently addressed in Refs. w23,24x. We have numerically demonstrated that abrupt transitions between single and multiple self-guided channels can be incoherently controlled through the input imbalance, and are associated to digital polarization switching. The role of wavevector mismatch on the occurrence of branching and the effects of birefringent walk-off have been addressed in conditions suitable for experiments in KTP waveguides.

Acknowledgements We thank W.E. Torruellas ŽWSU, Pullman. and G.I. Stegeman ŽCREOL-UCF, Orlando. for enlightening discussions, Ministero dell’Universita` e della Ricerca Scientifica e Tecnologica Ž40%, Photonic Technologies. and Consiglio Nazionale delle Ricerche Ž96.001844.CT11, 96.02238.CT07. for financial support.

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