1. Introduction

Over the past 20 years, the theory of spatial solitons in Kerr media has been well established [1–13]. The evolution of spatial solitary waves with Kerr χ ⁽³⁾ nonlinearity is described by the nonlinear Schrödinger equation (NLSE). The dark soliton solution was first discovered by Zakharov and Shabat using the inverse scattering method [14]. Experimentally, Kerr solitons have been observed in CS₂, glass, semiconductors, and polymer waveguides [15]. Recently the issue of evolution and interaction between optical spatial solitons has been investigated [16,17]. It was reported that certain parameters, e.g., phase shift [18], center of the solitons [19], can be steerable with various methods, which can be a potential application in all-optical devices.

As we know, quadratic nonlinearity can effectively mimic cubic properties. The existence of nonlinear phase shift in fundamental waves during SHG was first discussed by Ostrovskii in 1967 [20], and the existence of solitons was predicted in 1974 by V. E. Zakharov and S. V. Manakov [21]. In the past 12 years, it was already proved experimentally that bright solitary waves can exist with cascaded second-order nonlinearities (χ⁽²⁾:χ⁽²⁾) in quadratic media [22–25]. An advanced study on the cascading effect and the induced Kerr effect was conducted by Bang. Theoretically, Bang and his colleagues showed that quadratic solitons are equivalent to solitons of a nonlocal Kerr medium [26]. Assanto and Stegeman used the concepts of cascading phase shift to interpret the formation and properties of quadratic solitons [27]. In quasi-phase-matched (QPM) cascaded SHG processes, both positive and negative values of $n_{\mathit{\text{cascad}}}^2$ are observed and measured [28]. Bright spatial solitons have been observed in periodically poled lithium niobate and KTP crystals during second-harmonic generation [29,30]. Dark spatial solitons by χ ⁽²⁾:χ ⁽²⁾ nonlinearity in QPM configuration can also be expected.

Since a cascading second-order nonlinearity is much larger than a third-order nonlinearity, lower power is needed to achieve similar effects in cascaded second-order quadratic media. It has been reported that the optical branching effect was experimentally observed in Kerr media [4]. The beam propagating in the media splits into several beams. These beams propagate at different angles on each side of the beam. Theoretical calculation and experimental measurement of the transverse velocity of dark solitons in Kerr media are reported by Anderson et al. [31]. By the wavefront modulation method, dark spatial solitons and soliton pairs can be easily generated, and they can be steerable by controlling the blocking width or the phase difference at the wavefront of the beam. The purpose of this paper is to investigate the evolution characters of dark solitons in cascaded media and to compare them with those of solitons in Kerr media. We numerically simulate this effect in quadratic media to study its concrete properties and the potential influences on it. In addition, we investigate the generation of a hollow in a beam with a uniform background and the method to control the location of the generated hollow.

2. Theory

Under the condition of slowly varying envelope approximation, the wave coupling equations in a QPM quadratic nonlinearity media in a reduced normalized form is

where a ₁,a ₂ are the normalized amplitudes of the fundamental and harmonic waves, respectively, and α = k ₁/k ₂, k ₁, k ₂ are the wavenumbers at the two frequencies. The parameter β ( β = k ₁ η ²Δk) is proportional to the phase mismatch Δk ( Δk = 2k ₁-k ₂ + 2π/Λ ; Λ is the one-order QMP grating period), and η is the characteristic beam transverse width. ξ is the propagation distance in the unit of k ₁ η ². δ accounts for the Poynting vector walk-off when propagation is not along the crystal optical axes. We can set δ = 0 because Poynting vector walk-off is absent in typical QPM geometries. The function d(ξ) stands for the effective nonlinear coefficients involved in QPM. ∇_⊥ represents ∂/∂x. in the situation of one dimension; here X is the normalized transverse coordinate in units of η.

Based on the above wave coupling equation, the generation and evolution of dark solitons under various conditions can be numerically simulated. In our calculation, we integrate the wave coupling equation numerically with a split-step approach. The linear part (${\nabla }_{\perp }^2$) is integrated in the Fourier space, and the nonlinear part is integrated by a fourth Runge-Kutta algorithm. We divide the propagation process into many steps. In every step, the diffraction effect is first considered exclusively, and then only the nonlinear process is calculated.

3. Results

3.1 Optical branching effect in cascaded quadratic media

In Kerr media, it has been observed experimentally that beam splitting occurs after a certain value of the input beam intensity has been reached. The dark solitons are evolved and generated in pairs [4]. The nth pair from the center of the beam is called the nth-order pair. These phenomena result from the inherent repulsive character of their interaction: it is energetically favorable to break a soliton into two pieces traveling away from each other. Naturally, a problem is raised: does the optical branching effect occur in the cascaded quadratic media?

A box-like dark spatial pulse as an initial input condition is suitable for analyzing the branching effect. An ideal box-like pulse is given by

If above ideal box-like pulse is employed as the initial condition, the simulation results are singular due to abrupt discontinuity at the points x = ±a. To avoid this issue, an input beam described by the function below is introduced to simulate the box-like dark solitons:

In the calculation, we found that the soliton pairs do not appear simultaneously when the beam enters the media. Along with the evolution of dark pulses in the media, soliton pairs appear one after another symmetrically beside the center of the hollow and evolve to the edges. For fixed intensity and as time increases, more branches appear until a steady state is reached where the number of branches is fixed. The above behaviors are the same as those in Kerr media [4]. To observe the steady state, we studied the evolution of box-like dark solitons after propagating various diffraction lengths, and we find that after a certain number of diffraction lengths, the soliton pair number does not increase.

Figure 1 shows the generation and evolution process of dark soliton pairs at different diffraction lengths. At 15 diffraction lengths [Fig. 1(b)], only the first-order pair is completely produced; at 20 diffraction lengths [Fig. 1(c)], the second-order pair is found; at 35 diffraction lengths [Fig. 1(d)], the third-order pair is produced. After 35 diffraction lengths, no higher-order pair appears and the interval between the existing three pairs become large. In addition, it is shown that the low-order pairs are darker than the high-order ones: the intensity of bottoms of high-order pairs is no longer zero, and it increases with the order of the pairs.

 

Fig. 1. (a) Input box-like pulse (partial) (A=6); evolution of box-like dark soliton in cascaded media (A=6 β=-25 ) at diffraction length of n =15(b), 20(c), and 35(d). After about 30 diffraction length, the number of soliton pairs (3 pairs for A=6) does not increase further.

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The steady soliton pair number is governed by the parameters of the input beam and the cascading effect. As the input intensity increases, the generation of multiple solitons is favored. We investigated the steady branch numbers for input beams with different intensities. In this process, since the required diffraction length for stable branch number varies for different intensity, and since much longer evolution length will arouse reflections from the beam edges, repeated calculations are conducted to ensure that the branch number for each input beam is what it can finally engender. The results are shown in Fig. 2. When the input amplitude is 2, the steady state is reached at about 100 diffraction lengths and only the first-order pair is produced. When the amplitude is 4, 80 diffraction lengths are needed and three pairs are produced. While the amplitude is 7, only 60 diffraction lengths are needed and four pairs are produced. These results confirm the prediction that the higher the intensity of the input beam, the more soliton pairs are generated.

 

Fig. 2. Steady state of optical branching effect with input normalized amplitude of 2(a), 4(b), and 7(c). The required diffraction lengths are 100(a), 80(b), and 60(c), respectively, and numbers of soliton pairs are 1(a), 3(b), and 4(c).

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The number of branches also differs when the initial width of the input hollow is changed. With the increase of initial width, the branch number increases notably. We changed ω [the width of the input hollow described by function (2)] to 1/2ω, 2ω, 4ω and simulated the evolution process. The results are shown in Fig. 3. It can be found that, when the initial width is large, more pairs are produced and they are closer to each other than the pairs produced with a narrow input hollow. In addition, due to the conservation of energy, when a number of dark soliton pairs are produced, the intensity of the background increases obviously: in Fig. 3(d), the normalized intensity of the background reaches 20, while its initial value is 16.

 

Fig. 3. Steady state of optical branching effect with initial width of ω/2 (a), ω(b), 2ω(c), and 4ω(d).

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From the inverse scattering theory, in Kerr media the eigenvalue of a box-like dark soliton pair is determined by

where v is the transverse velocity measured experimentally and E ₀ represents the amplitude of the input beam. This shows that for a given pair, the transverse velocity is directly proportional to the amplitude of the input beam. As in cascaded media, it is relatively difficult to obtain the analytic solution from the wave coupling equation. Through our numerical calculation, we proved that the soliton pairs in cascaded media have similar performance to those in Kerr media. With different fundamental wave amplitudes, we recorded the transverse displacements of the first-order pair at same diffraction length to analyze the relation between the amplitude and its transverse velocity. The results shown in Fig. 4 demonstrate exactly the linear relation between the displacement (indicating the transverse velocity) and the input beam amplitude, which is just as predicted in Kerr media.

 

Fig. 4. Relation between the y and the displacements of the first-order pairs at 100 diffraction lengths.

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From the above discussions we can conclude that the optical branching effect in cascaded quadratic media proves to have analog performance compared with that in Kerr media: the soliton pairs are evolved and generated gradually through the propagation in the media; there is a steady state where the number of soliton pairs does not increase; the steady number of soliton pairs increases with the intensity of the input beam and the initial width of the hollow; the transverse velocity of a given pair is directly proportional to the input beam amplitude.

3.2 Generation of dark solitons in uniform background

In nonlinear-optical fibers, the generation of dark solitons in uniform background with unequal phase at t = ±∞ has been theoretically studied [32]. Dark solitons can be generated by an input pulse with a phase difference at its edges that equals π. We demonstrate that, in cascaded quadratic media, spatial dark solitons can also be generated in a uniform background. In our calculation, the uniform background is given by

where A represents the amplitude of input beam. A factor of e ^(iδ) is introduced in the calculation to introduce different phases. As we know, spatial dark solitons can be used to form steerable optical waveguides to be applied in all-optical apparatuses [33–34]. In cascaded media, the steerablity of dark solitons by changing the phase difference at the center of the uniform background is investigated. The simulation results are shown in Fig. 5; dark solitons in a uniform background are generated with different transverse velocity, depending on the phase difference δ.

 

Fig. 5. Generation of dark solitons in a uniform background. (a) Input uniform background, (b) hollow generated in the center of background with a phase difference of π at two edges of the uniform background, (c) hollow generated at left side of the background when the left side of the hollow has a phase of +π/2 ahead of the right side, (d) hollow generated at the right side of the background when the left side of the hollow has a phase of +3π/2 ahead of the right side.

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In Fig. 5(b), when the two sides of the uniform background have a phase difference of π, a dark hollow in the center of the uniform background is generated. When the left side of the hollow has a phase of +π/2 ahead of the right side, the soliton hollow is generated at the left side of the center as shown in Fig. 5(c). In this case, the soliton hollow sinks less than the hollow generated in the center of background, which is called the gray soliton. In Fig. 5(d), the left side of the hollow has a phase of +3π/2 ahead of the right side and the hollow is generated right to the center.

4. Conclusion

In summary, the results of the numerical simulation show that optical branching phenomena also occur in cascaded media and the characteristics of the effect are similar to those in Kerr media. In addition, we demonstrate that hollows in a uniform background can be generated in cascaded media and the position of the hollow are steerable by means of changing the phase difference of two sides of the input beam. The dark solitons caused by the cascading effect may have potential application in all-optical apparatuses because lower power is required.