2.1. Quasi-optical approach

The way to derive the equation describing the interplay between dispersion, diffraction, inhomogeneity, and nonlinearity was described in detail in [3] for Kerr nonlinearity. There the authors started from Maxwell’s equations supplemented by the dependence of the refractive index on carrier frequency and transversal coordinates. Then a standard procedure was fulfilled, the paraxial and the slowly varying envelope approximations were applied. It is important to underline that the equation for a slowly varying envelope obtained in that paper is similar to the standard multidimensional nonlinear Schrodinger equation (NLSE) except the only one additional term due to nonlinear medium inhomogeneity.

In the present work for the consideration of the SHG in a quadratically nonlinear waveguide we applied about the same procedure.

To describe waveguide geometry we represent the linear frequency susceptibility χ_ω(r_⊥) in the form Here r_⊥ is the transverse radius-vector perpendicular to the central axis of the waveguide, is the linear susceptibility of the medium at the center of the waveguide cross section; the dimensionless function f_ω(r_⊥) characterizes the transverse inhomogeneity of the susceptibility and satisfies the condition f_ω(0) = 0.

Similar to [3], the paraxial approximation is also applied. We assume that both harmonics propagate along the z-axis, and at that, we suppose the linear group velocities and (envelope carrier frequencies ω and 2ω correspondingly) at the center of the waveguide (r_⊥ = 0) are related by the following inequality

Provided inhomogeneity, nonlinearity, dispersion, and diffraction are weak, we use the slowly varying envelope approximation [22]. The resulting system of equations for the envelopes of the fundamental Φ₁ and second Φ₂ harmonics looks as follows: (1.1) In (1.1) , t is time, (1.2) f_ω(r_⊥) and f_2ω(r_⊥) are the dimensionless functions satisfying the condition f_ω(0) = f_2ω(0) = 0, is the mismatch of group velocities, k₁ = k(ω) and k₂ = k(2ω) are the wave numbers, corresponding to the fundamental frequency ω and second harmonics 2ω, respectively. β_ω and β_2ω are the parameters of the dispersion of group velocities (DGV), and are the refractive indexes at the center of the waveguide cross section of the fundamental and second harmonics, respectively, c is the speed of light in vacuum, Δ_⊥ is the transversal Laplacian, , . χ⁽²⁾(2ω,−ω) and χ⁽²⁾(ω,ω) are the second order nonlinear optical susceptibilities at the waveguide center.

The first and second terms in the right-hand sides of Eq (1.1) describe the effect of the waveguide (transverse inhomogeneity) on the phase and group velocities of the harmonics, respectively. Spatial inhomogeneity of the linear refractive indices for both harmonics in (1.1) is taken into account in the same way as it was done in [3, 4] when studying the waveguide propagation mode of the soliton of the NLSE.

Putting in (1.1) g_1,2 = 0, we come to the well-known system of equations for the pulsed mode of SHG in a homogeneous medium [23,24].

Dependences of the refractive indexes n_ω,2ω on the transverse coordinates r_⊥ of the waveguide are expressed through the functions f_ω,2ω(r_⊥) as follows: (1.3)

If we deal with a focusing waveguide, then the functions f_ω(r_⊥) and f_2ω(r_⊥) decrease from the center to the periphery. In the opposite case, the waveguide is defocusing. Below we consider planar waveguides, i.e. Δ_⊥ = ∂²/∂x² and f_ω,2ω = f_ω,2ω(x), the profiles of the waveguide functions are conveniently chosen, for example, in the form of parabolic profile with saturation: (1.4) Here x is the distance from the centre of the waveguide to a current point in transversal direction, ε_w = 1 for the defocusing waveguide and ε_w = −1 for the focusing waveguide.

In addition, we can consider profiles in the forms (1.5) or (1.6)

Note that in the case of focusing waveguides (ε_w = −1) the transverse susceptibility profiles (see (1.3)) for the dependences (1.4), (1.5) and (1.6) have the Lorentzian, exponential and Gaussian modes, respectively: (1.7) (1.8) (1.9)

In all these cases, the refractive indexes n_ω,2ω(x) decrease from the maximum value at the center of the waveguide (at x = 0) to unity at its periphery (at x→∞).

2.2. Dimensionless equations and integrals of motion

We still consider a planar (Δ_⊥ = ∂²/∂x²) waveguide and use dimensionless parameters related to the physical parameters in the following way: Φ_1,2 = A_1,2A_in, , , , , Δk = 2k₁−k₂, l_nl = (α_ωA_in)⁻¹, . Here A_in is the input peak amplitude of the fundamental harmonic, R_in and τ_in are initial pulse spatial and temporal widths, respectively. We introduce also the following propagation and waveguide characteristics: , , , , , , γ = α_2ω/α_w, a_w and a_2ω are the characteristic lengths of waveguide transversal inhomogeneity. Then, we suppose the group velocity matching conditions are satisfied (). Finally, we get the following system of the dimensionless equations which we use as a base for our numerical simulation: (2.1) Boundary and initial conditions are as follows.

(2.2) In (2.1)–(2.2) L_z is the dimensionless length of the nonlinear medium, L_τ is the dimensionless time interval during which laser pulse interaction with the medium is analyzed. L_x is the dimensionless length of the transversal domain. We choose finite lengths of the transversal coordinate and time with zero conditions at the boundaries of these coordinates from the following considerations. Since we deal with a finite pulse, it is naturally to take a sufficiently long time interval whose boundaries are not influenced by the pulse. Besides, the transversal size of the studied bullet is also finite due to narrow laser radiation. Thus, we can choose a wide transversal domain with boundaries which are not affected by radiation. Obviously, this choice assumes the need to monitor the fulfillment of zero boundary conditions, and, if necessary, a widening of the computational domain. It may cause specific computational difficulties, which nevertheless, can be eliminated by imposing artificial boundary conditions (see [25] and references therein).

In (2.1) p_1,2 are dimensionless functions describing waveguide inhomogeneity and corresponding to (1.4)–(1.6). Below we discuss simulation with (2.3) (2.4) (2.5) (2.6) Here is the distance from the center of the waveguide to the current point. In general, a_ω and a_2ω are not equal to each other (). Form (2.3) describes a waveguide with parabolic profile, (2.4)–(2.5) correspond to profiles with Lorentzian (2.4), tangential (2.5), and exponential (2.6) saturation. Below we will omit bars in the notations of dimensionless variables.

The system (2.1)–(2.2) possesses the motion integrals (2.7) (2.8)

In (2.8) φ_1,2 are the wave phases. To derive (2.7) we multiply the equations of (2.1) by correspondingly, then integrate both parts of these transformed equations with respect to x,τ, sum them up and take real parts of all components. Finally we get the equality (2.7) expressing a law of energy conservation. At the next step we proceed, multiplying the equations of (2.1) by correspondingly. Then we repeat the successive steps used when receiving (2.7), but at the last step we take imaginary parts of all components. So, we obtain (2.8) concerning the evolution of the phases of harmonics.

3.1. Approximation of the equations

We introduce uniform grids in the domain Γ and in the z domain: (3.1) A numerical approximation to the exact solution of the problem (2.1)–(2.2) we consider on the grid ω_z×ω_Γ and denote it by . To approximate first- and second-order derivatives with respect to x and τ we use the standard expressions: and are used for and . We also introduce the notation for the half-sums: . Then we write down the following nonlinear symmetric finite difference scheme in the case of phase matching (Δk = 0): (3.2) Initial and boundary conditions are approximated exactly.

(3.3)

This scheme is known to be of the second order of approximation with respect to all coordinates [12–14]. It is easily generalized to the case of phase mismatch (Δk≠0).

3.2. Two-step iterative process

Nonlinear scheme (3.2)–(3.3) can be resolved with the help of an iteration process [12]. But the computational complexity of the direct matrix inversion after linearization makes such an approach practically useless [12, 15]. FFT technique [15, 19] is not straightforward in the case under consideration due to the x-dependent coefficients p_1,2 in (3.2). Therefore, we develop the approach proposed in [20, 21] and write down the following two-step iteration process for the implementation of (3.2)- (3.3).

At the first step we seek for the iteration (s+1) of the difference functions : (3.4) (3.5)

At the second step we complete the procedure at the current iteration cycle finding : (3.6) (3.7) We see that at each step of the iteration process we deal with one-dimensional problem. Thus, matrix inversion in (3.4)–(3.7) can be made with the help of the tridiagonal matrix algorithm, and in general, the proposed method is time-saving. The iteration procedure stops when the criterion where ε is a constant determining computation accuracy.

Theorem 3.1. Provided that h_z≤C(h_xh_τ)² the unique solution of the difference scheme (3.2)-(3.3) exists and the two-step iteration process (3.4)-(3.7) converges to it as a geometric progression with the denominator q≈h_z/(h_xh_τ)².

Proof. The proof is made with the help of the contraction mapping theorem. We have to demonstrate that (3.4)–(3.7) is a contraction, i.e that all iterations are uniformly limited and that in a certain difference norm ‖∙‖_(h) (3.8)

Passing to the difference norm C we use at each step of the iteration process the Cauchy–Bunyakovsky–Schwarz inequality, estimate the inverse norm of the difference operators and [12]. This procedure allows us firstly, to show that the iterations are uniformly limited if h_z≤C(h_xh_τ)². Then, considering consecutively the iteration differences , , and using similar estimations, we get the inequality (3.8).

3.3. Conservativeness

Theorem 3.2. The scheme (3.1)-(3.2) is conservative: it preserves the following difference analogues of the integrals (2.7)-(2.8) in the norm (3.9) (3.10)

In (3.10) , .

Proof. We follow the procedure described in [13]. Multiplying both parts of the first equation of (3.1) by and both parts of the second equation by and summing them up with respect to x and τ, we take the real parts of the resulting expressions. Then we apply the difference analogue of the integration by parts formula [12] and finally, get (3.8).

We use a similar procedure for the integral (2.3). But to derive (3.9) we multiply both equations of (3.1) by , , sum them up with respect to x, τ and take the imagine parts of the obtained equations.

4.1. Initial pulse and absorbing boundary conditions

In this Section we discuss the outcome of a direct numerical simulation of the system (2.1) on the base of the finite-difference scheme (3.2). In computations we launch the initial pulse at both frequencies (4.1) or at the fundamental frequency only (4.2) Implementing calculations we usually choose L_x and L_τ so that the amplitudes of both harmonics decay to zero at the boundaries of this domain. In particular cases, when L_x and L_τ are too large, the efficiency of the method can be improved by using absorbing boundary conditions along the coordinates x and τ. To this end we embed an artificial absorption in the system (2.1) (4.3) (4.4) defining the function σ(x,τ) = σ_min(σ_x(x)+σ_τ(τ)), where

Fig 1 illustrates our choice of the function σ and the values of the constants included in it as well. We demonstrate the selection of parameters for σ_x(x). Our considerations concerning σ_τ(τ) are similar. In our computations we take the values x_ab≈(0.1÷0.2)L_x, τ_ab≈(0.1÷0.2)L_τ and the constants σ_min≈0.01÷0.1, ε_xτ≈3÷5. Analyzing Fig 1(A), which represents linear diffraction of the pulse, one can see that changing the value of σ_min at fixed values of x_ab, L_x, and ε_xτ, we observe the best coincidence with the reference curve obtained with the larger computational transversal domain (black dotted line) at σ_min = 0.1 (green line). Fig 1(B) demonstrates the choice of optimal ε_xτ = 5 in a similar way. Here we fix x_ab, L_x, and σ_min and vary ε_xτ. Parameter x_ab allows us to improve absorption but reduces the useful part of the computational domain. Values of τ_ab are taken from the considerations of computational efficiency and minimization of the tails’ influence on the bullet formation process. Thus, we can reduce the computational domain along the transverse coordinates. It is obvious that in such calculations the integrals (2.7)–(2.8) may not be conserved. Typically we take the domains L_x = 20, L_τ = 30 which are longer than those presented in Fig 1. It seems these dimensions are rather redundant. In fact, in the process of bullet trapping the pulse-beam spreads in time and x and parts of it which are close to the boundaries of absorption domains may be of intensities comparable with the peak ones. Chosen dimensions allow us to minimize this effect. At the same time they are shorter by an order of magnitude than those that should be taken in the absence of absorption layers.

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Fig 1. Linear diffraction of the Gaussian beam of the initial unit radius at the distance z = 10. x_ab = 0.2, L_x = 2.

Solid black lines in figures (a,b) represent diffraction with zero boundary conditions at the points x = −5 and x = 5, at that, σ_min = 0. Dotted black lines correspond to computations with transversal calculation area which is much larger than that shown in the figures (a,b), σ_min = 0 (reference curves). Beam profile dependence on σ_min for ε_xτ = 5 (a). σ_min = 0.01 (red line), σ_min = 0.1 (green line), σ_min = 1 (blue line). Beam profile dependence on ε_xτ for σ_min = 0.01 (b). ε_xτ = 2 (red line), ε_xτ = 5 (green line), ε_xτ = 7 (blue line).

https://doi.org/10.1371/journal.pone.0220840.g001

Our computations are carried out on the basis of the difference scheme described in the Sec. 3. We choose also the following computational parameters: the step size along the propagation coordinate is h_z = 0.001, while the step sizes along space and time coordinates are h_x = 0.05, h_τ = 0.05. Typical size of the computational domain along z axis in our simulation is L_z = 500.

4.2. Simulation of light bullet trapping in different regimes

Since anomalous dispersion supports light bullets in 2D+1 media even without waveguide [6, 8], our principal interest is to investigate waveguide at normal group velocity dispersion. The simulation and observation are conducted up to z = 500l_nl. Such distance is an optimal one to investigate effects appearing due to the interference of diffraction, dispersion, nonlinearity and waveguide geometry.

In the first series of numerical experiments, we use a pulse of the Gaussian form at both frequencies (4.1) as a two-component input radiation. This case implies the phase-matching conditions k₂ = 2k₁ or Δk = 0. We have demonstrated that the regime of robust soliton propagation is established with the termination of energy exchange approximately at 200l_nl. This propagation is accompanied by regular in-phase oscillations of peak intensities at both harmonics. The generalized phase Φ = 2φ₁−φ₂−Δkz oscillates near optimal value equal to −π. It also confirms the formation of a stable parametrically coupled pair of solitons and means that we observe the reactive regime of the classic parametric solitons when there is no energy exchange between the fundamental and second harmonics [5]. We have examined the temporal and spatial profiles at z = 500l_nl as well. Due to waveguide influence the spatial profile is narrower than the temporal one. The calculated profiles have been compared with those of the Gaussian form having the same amplitudes, widths, and durations. This comparison demonstrates quite a good match between them. Thus, the soliton solution in this case has the form close to the Gaussian one. In total, in this computational series we have showed the trapping of a two-component “breathing” light bullet and its stability. One should underline that it forms at normal dispersion due to waveguide geometry only.

Consecutive processes of SHG and two-component light bullet formation are in the focus of the second series of our numerical simulation. In this experiment we launch a pulse in the Gaussian form (4.1) at the fundamental frequency only. Fig 2(A) demonstrates the dependence of the peak intensities of both harmonics on the propagation coordinate z. Firstly, energy transfer and second harmonic generation are observed. Approximately at a distance of 100l_nl we see that both waves are trapped in a coupled pair and the regime of robust two-component soliton propagation is established. As in the previous case wave propagation is accompanied by regular in-phase oscillations of peak intensities at both harmonics (Fig 2(A)). Peak amplitude maxima are lower than in the previous case due to the lower initial total energy. Fig 2(B) shows the generalized phase oscillations near optimal value equal to π. In contrast to the previous case, the optimal generalized phase changes a sign. Note that the generalized phase is calculated as arccos function in the following way. If at a current point it is close to 2π, then at the next point the value 2πm is added to the calculated value in order to avoid a jump. Thus, at long distances the phase will be approximately 2πm (m>1 if it increases, or m<−1, if it decreases). Then, when a soliton is trapped, the phase will be close to π+2πm. The dimensionless temporal (Fig 2(C)) and spatial (Fig 2(D)) widths of the harmonics oscillates in-phase to each other and in-anti-phase to the intensity. Due to the influence of waveguide geometry spatial profile (Fig 2(F)) is narrower than a temporal one (Fig 2(E)). Fig 2(G) and Fig 2(H) show the evolution of temporal and spatial intensity profiles, correspondingly, at the fundamental frequency along the longitudinal coordinate z.

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Fig 2. Light bullet formation in the process of SHG at normal dispersion in a focusing waveguide.

Input pulse of the Gaussian form (4.2) is launched at the fundamental frequency only. Peak intensities of the fundamental (red line) and second (blue line) harmonics vs the propagation distance (a). Generalized phase vs the propagation distance (b). Temporal width of the fundamental (red line) and second (blue line) harmonics vs the propagation distance (c). Spatial width of the fundamental (red line) and second (blue line) harmonics vs the propagation distance (d). Transversal profiles of both harmonics calculated at the distance z = 500 (black solid lines) (e), (f), the approximations of the Gaussian form (4.1) having the same amplitudes, durations (e) and spatial widths (f) (red and blue dashed lines for the fundamental and second harmonics, correspondingly). Temporal and spatial profile evolution for the fundamental harmonic (g), (h). Waveguide with parabolic profile (2.4). Parameters: D_q1 = −0.5, D_q2 = −1.0, D_x1 = 0.1, D_x2 = 0.05, D_τ1 = 0.1, D_τ2 = 0.2, Δk = 0, γ = 0.5.

https://doi.org/10.1371/journal.pone.0220840.g002

Fig 3 illustrates the consecutive second harmonic generation and two-component light bullet trapping provided a small phase-mismatching. We see that the process of light bullet propagation is rather stable at Δk = 0.1 (Fig 3(A)). If we increase the value of the phase velocity detuning up to 0.5, it distorts the picture of the formation of a bullet due to a sufficient decay of the resulting pulse intensity. Even at Δk = 0.15 we observe a soliton close to the stability limit (Fig 3(B)). High values of phase mismatching prevent SHG. Fig 3(C) represents maximum amplitudes of both harmonics in time and transversal coordinate, which are averaged over the distance from z = 100 to z = 200. It can be seen that, starting from Δk = 0.2, the averaged maximum amplitudes drastically fall and soliton formation does not occur.

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Fig 3.

Light bullet formation in the process of SHG at the phase mismatch Δk = 0.1 (a) and the Gaussian pulse evolution at Δk = 0.15 (b). In (a,b) red lines depict the fundamental harmonic intensities, the blue lines correspond to those of the second harmonic. Dependence of averaged amplitudes of the first (red line) and second (blue line) harmonics on phase mismatch. Maximum amplitudes (in time and transversal coordinate) of the first (red line) and second (blue line) harmonics, averaged over the distance from z = 100 to z = 200 (c). Input pulse of the Gaussian form (4.2) is launched at the fundamental frequency only. Other parameters are the same as in Fig 2.

https://doi.org/10.1371/journal.pone.0220840.g003

As it was discussed in the Sec. 2.1 the transverse susceptibility profiles may have different form. The next series of experiments deals with the Lorentzian (Fig 4), tangential (Fig 5) and Gaussian (Fig 6) modes, respectively. The consecutive SHG and two-component light bullet formation is demonstrated in all three cases. The most evident confirmation of this fact can be received when we analyze the behavior of generalized phase–at the distance of 50l_nl it begins to oscillate near optimal value equal to π. Since the waveguide in these cases effectively influences only on pulse-beams of a finite width, the soliton tails can gradually move away from the pulse center, a part of the pulse energy gradually goes to the periphery. This results in forming a quasi-stable bullet little by little losing its energy. In Fig 4(C) and Fig 5(C) it can be seen that the time profile also strongly deviates from the Gaussian one. At the same time, the generalized phase experiences significant oscillations, which indicate an obvious deviation of this regime from the soliton one. For the Gaussian profile (Fig 6), the deviations from the basic test case (Fig 2) are less noticeable. It may be explained by a good match between initial beam size and waveguide size.

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Fig 4.

Peak intensities of the fundamental (red line) and second (blue line) harmonics vs the propagation distance (a). Generalized phase vs the propagation distance (b). Transversal profiles of both harmonics calculated at the distance z = 500 (black solid lines) (c), (d), the approximations of the form (4.1) with the maximum intensity and temporal (c) and spatial (d) widths of the calculated first and second harmonics (red blue dashed lines). The Gaussian input pulse at the fundamental frequency (4.2) is launched into the waveguide with the Lorentzian profile (2.4). Parameters: D_q1 = −0.5, D_q2 = −1.0, D_x1 = 0.1, D_x2 = 0.05, D_τ1 = 0.1, D_τ2 = 0.2, Δk = 0, γ = 0.5.

https://doi.org/10.1371/journal.pone.0220840.g004

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Fig 5. The same as in Fig 4 but waveguide has a tangential profile (2.5).

https://doi.org/10.1371/journal.pone.0220840.g005

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Fig 6. The same as in Fig 4 but waveguide has the Gaussian profile (2.6).

https://doi.org/10.1371/journal.pone.0220840.g006

Fig 7 illustrates a remarkable property of the investigated system to capture both pulses in a waveguide and to trap a two-component light bullet while initially the waveguide presents at one frequency only. In this experiment we launch input pulses of the Gaussian form at both frequencies (4.1), provided waveguide with parabolic profile (2.3) is just at the fundamental frequency. We see that the bullet energy gradually decreases down to bullet disappearing. However, one should note that the distance of total bullet destruction is not so small–it is equal approximately to 30 dispersion lengths. It is worthwhile to underline once again that provided normal dispersion, trapping of a two-component light bullet is impossible in the absence of a waveguide at least at the fundamental frequency.

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Fig 7.

Peak intensities of the fundamental (red line) and second (blue line) harmonics vs the propagation distance (a). Generalized phase vs the propagation distance (b). Temporal width of the fundamental (red line) and second (blue line) harmonics vs the propagation distance (c). Spatial width of the fundamental (red line) and second (blue line) harmonics vs the propagation distance (d). Input pulses of the Gaussian form at both frequencies (4.1), waveguide with parabolic profile (2.3) is at the fundamental frequency only. Parameters: D_q1 = −0.5, D_q2 = 0, D_x1 = 0.025, D_x2 = 0.0125, D_τ1 = 0.1, D_τ2 = 0.2, Δk = 0, γ = 0.5.

https://doi.org/10.1371/journal.pone.0220840.g007