Semidiscrete optical vortex droplets in quasi-phase-matched photonic crystals

A new scheme for producing semidiscrete self-trapped vortices (“swirling photon droplets”) in photonic crystals with competing quadratic ( χ (2) ) and self-defocusing cubic ( χ (3) ) nonlinearities is proposed. The photonic crystal is designed with a striped structure, in the form of spatially periodic modulation of the χ (2) susceptibility, which is imposed by the quasi-phase-matching technique. Unlike previous realizations of semidiscrete optical modes in composite media, built as combinations of continuous and arrayed discrete waveguides, the semidiscrete vortex “droplets” are produced here in the fully continuous medium. This work reveals that the system supports two types of semidiscrete vortex droplets, viz ., onsite-and intersite-centered ones, which feature, respectively, odd and even numbers of stripes, N . Stability areas for the states with diﬀerent values of N are identiﬁed in the system’s parameter space. Some stability areas overlap with each other, giving rise to the multistability of states with diﬀerent N . The coexisting states are mutually degenerate, featuring equal values of the Hamiltonian and propagation constant. An experimental scheme to realize the droplets is outlined, suggesting new possibilities for the long-distance transmission of nontrivial vortex beams in nonlinear media.


I. INTRODUCTION
Semidiscrete vortex quantum droplets, a new type of vortices, were initially predicted in binary Bose-Einstein condensates trapped in an array of tunnel-coupled quasi-1D potential wells [1].Unlike vortex modes in fully continuous or fully discrete systems [2][3][4][5][6][7], these are stripe-shaped localized states, which are continuous in one direction and discrete in the perpendicular one, and do not exhibit rotational symmetry.It is well known that the stability of self-trapped vortex modes in two-dimensional (2D) and three-dimensional (3D) geometries is a challenging problem because the self-attractive nonlinearity gives rise to strong splitting instability of vortex rings and tori, even if the collapse instability that affects fundamental (zero-vorticity) solitons in the same media may be suppressed [8][9][10][11][12][13].Due to the competition between the mean-field (MF) and beyond-MF effects in the bosonic condensate [14][15][16][17][18][19][20][21][22][23], semidiscrete vortex quantum droplets may maintain stability in this setting against the azimuthal (splitting) perturbations.In the field of nonlinear optics, somewhat similar objects in the form of "photon droplets" were experimentally demonstrated in optical media with nonlocal (thermal) nonlinearity [24,25].Actually, the competition between different nonlinear terms is a common effect [26][27][28][29][30][31][32], which occurs in the propagation of high-power laser beams in various media [32,33].Optical semidiscrete vortex droplets can be maintained by the balance between the competing nonlinearities.In particular, stable self-bound semidiscrete vortex modes in the spatial domain were predicted in coupled planar waveguides with the cubic-quintic nonlinearity [34].Similarly, self-bound spatiotemporal vortex modes can be predicted in coupled arrays of nonlinear fibers [35].
Recently, patterned quasi-phase matched (QPM) nonlinear photonic crystals in the 3D space have been produced by means of the thermoelectric field polarization [36], laser erasing [37], and femtosecond laser poling technique [38], which provides more possibilities for the creation of vortex states.The QPM technique has developed to a well-known method for achieving accurate phase matching in χ (2) crystals for the nonlinear frequency conversion [39][40][41][42][43][44][45][46][47][48][49] and nonlinear beam shaping [50][51][52][53][54][55][56][57] in different dimensions.Very recently, stable vortex solitons were predicted in 3D QPM photonic crystals [58].The structure of the vortex solitons can be engineered by fixing different phase-matching conditions in different cells of the photonic crystals, thus inducing effective discreteness in this 3D optical medium.This technique offers a possibility of building semidiscrete vortex modes in the QPM-structured bulk photonic crystals.It is relevant to mention that effective 2D (but not 3D) discrete waveguiding structures for optical beams with the extraordinary polarization can be induced by means of a different technique in photorefractive materials illuminated by interfering beams with the ordinary polarization [59] .Such virtual photonic lattices were used for the creation of quasi-discrete 2D vortex solitons [60][61][62].
In this paper, we propose a scenario for the creation of semidiscrete vortex optical droplets in 3D photonic crystals with quadratic (χ (2) ) and defocusing cubic (χ (3) ) nonlinearities and a striped QPM-induced spatial structure, as shown in Fig. 1(a,b,c).The χ (3) nonlinearity is able to compete with the χ (2) interaction if intensities of the light fields reach the level of several GW/cm 2 [32], making it possible to create self-bound photon droplets.We demonstrate that the phase-matching condition, adjusted to the striped structure, induces effective discreteness between adjacent stripes, which is necessary for the design of semidiscrete states.The balance of the competition between quadratic and cubic nonlinearity allows the self-bound vortex modes to maintain their stability.
The subsequent material is arranged as follows.The model is introduced in Section 2. Numerical results and estimates for the experimental setup are presented in Section 3. The paper is concluded by Section 4.

II. THE MODEL
The paraxial propagation of light beams through the 3D QPM photonic crystals with the competing χ (2) and χ (3) nonlinearities is governed by coupled equations for the slowly varying fundamental frequency (FF) and second harmonic (SH) amplitudes, A 1 and A 2 : where Y is the paraxial-diffraction operator, c is the speed of light in vacuum, while n 1,2 , ω 1,2 (ω 2 = 2ω 1 ), and k 1,2 are, respectively, the refractive indices, carrier frequencies, and wavenumbers of the FF and SH components, and ∆k 0 = 2k 1 − k 2 is the phase-velocity mismatch.χ (3) > 0 is the third-order susceptibility, which accounts for the cubic self-defocusing.The local modulation of the second-order susceptibility χ (2) is determined by where σ(X) is the transverse striped OC (onsite-centered) or IC (intersite-centered) modulation pattern: ℓ being the width of a stripe.The OC and IC patterns correspond to the pivot of the vortex beam located, respectively, at the center of a stripe or at the border between two stripes, see Fig. 1(a,b).Further, the factor accounting in Eq.
(2) for the modulation in the Z direction, with amplitude d 0 and period Λ, is [63][64][65] see Fig. 1(c).Actually, only the terms with m = ±1 are kept in Eq. ( 4), as they play the dominant role in the QPM effect.Thus, m = 1 and −1 relate to the FF and SH components, respectively.By means of rescaling [66,67] Eqs. (1), which keep, as said above, the terms with m = ±1 in Eq. ( 4), are simplified to the form of where Neglecting the slight difference in the FF and SH refractive indices, i.e., setting n 1 = n 2 , results in coefficients g 11 = 3π √ 3/8, g 22 = g 12 = 4g 11 and η = 2 in Eqs. ( 6) and (7).Equations ( 6) and ( 7) conserve two dynamical invariants, viz., the Hamiltonian and total power (alias the Manley-Rowe invariant [68][69][70]): where The power sharing between the FF and SH components is defined as the ratio r = P 1 /P 2 .Control parameters for the subsequent analysis are P , ℓ and Ω (the total power, the stripe's width, and the scaled detuning).
The stability of the stationary vortex solitons was tested by direct real-z simulations of the perturbed evolution in the framework of Eqs. ( 6) and ( 7) up to z = 10000.Unstable solutions readily exhibit splitting in the course of the simulations.As the stripe modulation acts solely in the x-direction, the vortex solutions feature similar modulation in the same direction, and can be characterized by the number of stripes, N , in the localized solution According to Eq. ( 3), the solutions are also categorized into the OC and IC types.For the OC-and IC-type solutions, with the pivot located at the center of a stripe or at the border between adjacent stripes, numbers N are, respectively, odd or even.Typical examples for the OC-type vortex solution with N = 3, 5, 7, 9 and IC-type ones with N = 2, 4, 6, 8, all carrying the winding number S = 1 in their FF component, are shown, respectively, in Figs. 2 and 3.All these states are stable, as confirmed by direct simulations up to z = 10000.Because the modulation is applied only along the x-direction, the vortex solutions feature a typical semidiscrete configuration similar to that reported in previous works [1,34].However, unlike those works, the stable vortex modes are elaborated here in the bulk crystals with the spatially modulated local χ (2) susceptibility.The vortex phase patterns exhibited in Figs. 2 and 3 of the SH component are striped to obey the matching condition of Eq. ( 14), hence, the effective angular coordinate of this component is defined by Eq. ( 13), showing a complicated striped-mixed vorticity pattern of this component.
Stability areas for the vortex solutions of the OC and IC types, with different values of N are shown, in the (P, ℓ) plane with Ω = 0 and S = 1, in Fig. 4(a,b).These plots demonstrate that the vortex solutions with larger values of P and smaller values of ℓ produce overlaps between different stability areas [see the cyan and yellow areas in Fig. 4(a), and the cyan area in Fig. 4(b)].The overlaps indicate the presence of multistability in the system.To further illustrate this feature, we select ℓ = 10 and examine solutions with different values of N , up to P = 200.In this region, we find that four different values of N stably coexist at P = 200 for both OC and IC types of the solutions.Note also that the solutions with larger values of N require larger values of P to support the stability.Multistable semidiscrete vortex solutions are characterized by values of H, β and r, which are displayed, as functions of P , in Fig. 5, while keeping ℓ and Ω fixed.Notably, curves H(P ) and β(P ) for these solutions overlap almost completely, indicating degenerate solutions.The nearly flat dependences, with dβ/dP ≈ 0, indicate the existence of broad states, which may be considered as effectively liquid ones, cf.Ref. [14].A nearly constant value r(P ) ≈ 2.7 in Fig. 4(e,f) implies the domination of the FF component in the system.
Finally, in Fig. 6 we present the range of the effective detuning for a fixed total power, P = 100.The results indicate that stable semidiscrete vortex solitons exist for Ω = 0.
Exploring vortex states with S > 1 is a challenging problem, as they may be stable only for sufficiently large values of P .Two typical examples of stable vortex solutions of the OC and IC types with S = 2, and N = 5 and 6, are shown in Fig. 7.The stable solution with S = 2 exists at P > 280.This threshold is much higher than its counterpart for S = 1, in which case the stable solutions are found at P > 40.

B. An outline of the experimental setup
The fabrication of 3D photonic crystal by means of femtosecond laser pulses is a mature technology [71,72].To estimate parameters of the setting under the consideration, we consider the nonlinear photonic crystal implemented in LiNbO 3 , which has the second-order nonlinearity coefficient d 0 = d 22 = 2.1 pm/V [73], and the third-order one χ (3) = 6.6 × 10 −22 m 2 /V 2 [74].The wavelengths of the FF and SH components are selected as 1064 nm and 532 nm, respectively.The relations between the scaled units, in which Eqs. ( 6) and ( 7) are written, and their physical counterparts can be established by means of Eq. ( 5), as summarizes in Table I.According to results of the simulations (see Figs. 2 and 3), the peak droplet's intensity in the FF and SH components are I FF ≈ 4.32 GW/cm 2 and I SH ≈ 0.432 GW/cm 2 , respectively.If we select the pulse width as 200 ps, the energy densities for the FF and SH components are 0.86 J/cm 2 and 0.086 J/cm 2 , which are lower than the damage threshold (∼1.5 J/cm 2 [75,76]) of the PPLN crystals.The characteristic propagation distance, which is z = 10000, amounts to 2.8 m, which is several times the underlying diffraction length.Therefore, the stability of the solitons, predicted by the simulations, is a reliable prediction.
The sketch for the experimental observation of these droplets is shown in Fig. 1(d): A high-power sub-nanosecond laser (e.g., one with the power exceeding 0.6 mJ per pulse, 200 ps pulse width, and 400 Hz repetition rate) may be a suggested light source in the proposed experiment.The input patterns on the front surface of the PPLN can be generated by a spatial light modulator (SLM) and a Lens.Input FF and SF components can be selected with energies 0.37 mJ and 0.16 mJ per pulse, which are closed to the FF/SF power ratio r ≈ 2.6 for stationary semidiscrete vortex droplets in Figs.6(e) and (f).If the input's power ratio is essentially different from this value, it naturally gives rise to strong oscillations between the FF and SH components.The power and phase patterns of the FF component of the input, which are shown in the left inset of Fig. 1(d), can be produced by a properly designed SLM1 and L1, and the input SH patterns with 0.14 mJ per pulse can be produced by SLM2 and L2, respectively.The FF and SH beams are coupled by the dichroic mirror, which sends them onto the PPLN coaxially.The necessary PPLN crystal may be 4 cm long along the axial direction.Finally, the beams transmitted through the PPLN are imaged on to a camera by Lens L3.The simulated output patterns at the back surface of the crystals are shown in the right inset in Fig. 1(d).

IV. CONCLUSION
We have proposed 3D photonic crystals with the striped structure and the combination of the χ (2) and defocusing χ (3) nonlinearities.The results of the analysis predict the creation of two types of stable semidiscrete vortex solutions, OC and IC (onsite-and intersite-centered ones), which exhibit an odd or even number of stripes in their structure, respectively.The smallest number of the stripes is N OC = 3 or N IC = 2.The setting admits multistability, viz., the coexistence of stable solutions with different numbers of stripes for the same parameters.Unlike the multistability in systems with the competing cubic-quintic nonlinear the Hamiltonian and propagation constant of these coexisting states are equal, thus featuring degeneracy.The range of the multistability has been found.The stable solutions for semidiscrete vortices exist within a certain range of positive or negative values of the phase mismatch of the χ (2) interaction in the medium.
The scheme proposed in this paper can be developed in other settings, such as photonic crystals with ring-or fan-shaped structures.Those settings can be used, in particular, for implementation of various scenarios of beam shaping.

FIG. 1 :
FIG. 1: (Color online) (a,b) The structures corresponding to OC and IC modulations, which are defined as per Eq.(3), ℓ and O representing the stripe's width and center of the modulation pattern, respectively.The black and gray blocks represent σ = −1 and +1, respectively.(c) The periodic modulation along the Z axis defined as per Eq.(4), Λ being the period of the longitudinal modulation.(d) A schematic of the experimental setup for the creation of the semidiscrete vortex droplets: L1, L2, L3 -lenses, SLM1, SLM2 -spatial light modulators, PPLN -the periodically polarized lithium niobate crystal, FF -the fundamental frequency, SH -the second harmonic.Two bottom plots display the input and output intensity and phase patterns of the FF and SH beams.

FIG. 3 :
FIG. 3: (Color online) A example of a stable IC-type optical droplet with (P, Ω) = (80, 0) and S = 1.The first and second rows display the intensity and phase distributions of the FF component, while the third and fourth rows exhibit the same for the SH component.The stripe's widths in panels (a-d) are ℓ = 22, 13, 9, and 7, respectively, corresponding to points "A-D" in Fig. 4(b).

FIG. 4 :
FIG. 4: (Color online) Stability areas of semidiscrete vortex optical droplets of the OC (a) and IC (b) types, in the (P, ℓ) plane with Ω = 0 and S = 1.Digits in colored stability areas indicate the number of stripes in the droplets, which are odd in (a) and even in (b).Note that the presence of bistability in the cyan and yellow areas.Stability ranges of multistable states are shown in (c) and (d) for the droplets of the OC and IC types, respectively, with (ℓ, Ω) = (10, 0) and varying values of the total power, P .

FIG. 7 :
FIG. 7: (Color online) (a1)-(a4) A example of a stable semidiscrete vortex droplet of the OC type, with vorticity S = 2 of the FF component.Other parameters are (P, ℓ, Ω) = (300, 15, 0).(b1)-(b4) An example of a stable semidiscrete vortex droplet of the IC type, with vorticity S = 2 of the FF component, for the same parameters as in (a1)-(a4).The four columns of panels from left to right display the intensity and phase patterns of the FF and SH components.These solitons with S = 2 remain stable for the propagation distance z = 10000, up to which the simulations were running.

TABLE I :
Relations between scaled and physical units of the coordinates, stripe's width, total power, and intensity.