Incoherently coupled two-color vector dark solitons in self-defocusing media

L. Wu; W. Chen; M. Shen

doi:10.1364/OE.26.032194

1. Introduction

During the past decades, solitons have been the subject of intense theoretical and experimental studies in many different fields, especially in the area of nonlinear optics [1]. Generally, solitons discussed are described by a single nonlinear Schrödinger (NLS) equation for a scalar field [2]. When these conditions are not satisfied, one must consider the interaction of several field components at different frequencies or polarizations and solve simultaneously a set of coupled NLS equations [2]. Because of its multicomponent nature, a shape-preserving solution of such coupled NLS equations is called a vector soliton [2]. In particular, an integrable vector solitons, first proposed by Manakov, consist of two orthogonally polarized components in a nonlinear Kerr media in which self-phase modulation is identical to cross-phase modulation [3]. The nonlinear action of a field component on itself equals the action of one component on the other [3]. The Manakov model can support different classes of vector soliton pairs [4–6]. Experimentally, Manakov solitons was observed in AlGaAs waveguide [7].

Spatial dark solitons, in the form of localized light intensity dip in a plane wave background with nontrivial phase profile [8], have drawn considerable attention in the past years [9,10]. In particular, the vector dark solitons was described as a bound state of mutual trapping of two dark solitons through cross-phase modulation [11,12]. Exact solutions of vector dark solitons [13–15] were sensitively investigated using the inverse scattering transform method [6] and Hirota method [4,16]. Vector dark solitons have been observed experimentally in the orthogonal axes of a highly birefringent optical fiber [17,18] and a photorefractive SBN crystal [19].

Another example of vector solitons is the two-color vector solitons which form when two beams of different colors (wavelengths or frequencies) propagate through a nonlinear media and couple to each other [20–22]. During the past years, two-color vector solitons have been extensively studied, such as two-color solitons in photonic band-gap fibers [23], bound-soliton bunch in a two-color mode-locked fiber laser [24], and two-colored temporal solitons in a segmented quasi-phase-matching structure [25]. Two-color vector solitons have drawn considerable attention in quadratic nonlinear media theoretically [26] and experimentally [27]. Quadratic nonlinear media possesses stable propagation of two-color solitons clusters [28], periodic-wave arrays [29], discrete solitons [30], surface solitons [31], vortex solitons [32], interface solitons [33], and walking Peregrine solitons [34]. In nonlocal media, two-color vector solitons have also been investigated [35,36], including two-color vector soliton interactions [37,38] and stability of vector vortex solitons [39,40]. Recent works also suggest the coexistence of two-color optical filaments at resonant frequencies [41,42] and collapse of two-color optical beams in a nonresonant regime in nonlinear self-focusing media [43].

In this paper, the stability of two-color vector dark and grey solitons in a nonresonant regime is investigated in a nonlinear self-defocusing media. The approximate solutions of the two-color vector dark and grey solitons are obtained using the variational method. The propagation dynamics of the two-color vector dark and grey solitons are numerically studied with split step Fourier transform method. Because of different wavelengths in its two components, the two-color vector dark and grey soliton are always unstable due to different diffraction coefficients.

2. Model and variational method

We consider the system of two coupled one-dimensional NLS equations describing the co-propagation of two incoherent dark beams in a Kerr medium with a self-defocusing nonlinearity. We also assume that the co-propagation of two dark beams are in a non-resonant regime [43]. Thus the envelopes of the dark beams E₁(x, z) and E₂(x, z) are governed by [43]

i \frac{\partial E_{1}}{\partial z} + \frac{γ}{2} \frac{\partial^{2} E_{1}}{\partial x^{2}} - \frac{1}{γ} ({| E_{1} |}^{2} + 2 {| E_{2} |}^{2}) E_{1} = 0,

i \frac{\partial E_{2}}{\partial z} + \frac{1}{2} \frac{\partial^{2} E_{2}}{\partial x^{2}} - (2 {| E_{1} |}^{2} + {| E_{2} |}^{2}) E_{2} = 0,

where x and z are the transverse and longitudinal coordinates, respectively. γ = ω₂/ω₁ is the nonresonant coefficient, which represents the ratio of the angular frequencies ω₁ and ω₂ of the dark beams [43]. Therefore, the corresponding propagation constants of the dark beams have the following relation k₂ = γk₁ [43]. In the limit of γ = 1, the coupled equations describe two incoherently coupled beams with same wavelengths (color). When γ ≠ 1, the two incoherent dark beams experience different phase velocities and diffraction naturally.

Since the system discussed above is not integrable when γ ≠ 1, it is hard to obtain the rigorously exact solutions. For the sake of simplicity and analytical tractability, in this paper, we will employ the approximately analytical Lagrangian approach [44] to analyze the coupled NLS equations. The accuracy of the Lagrangian approach depends on the functional choice of the variational solution, and in this paper we introduce a proper ansatz of the two-color dark solitons in the following form [44–48],

E_{1} = B_{1} tanh [D (x - x_{1})] + i A_{1},

E_{2} = B_{2} tanh [D (x - x_{2})] + i A_{2} .

Parameter A_1,2 and B_1,2 satisfies the normalization condition

A_{1, 2}^{2} + B_{1, 2}^{2} = 1

. Here parameter A_1,2, B_1,2, D and x_1,2 are all supposed to be functions of the propagation variable z. The modulation depths of the solitons can be given by

B_{1, 2}^{2}

[49]. When

B_{1, 2}^{2} = 1

, it represents fully “black” solitons, whereas, it represents “grey” solitons with transverse velocities when

B_{1, 2}^{2} < 1

[49]. The parameter D⁻¹ can be used to define the soliton widths and x_1,2 are the soliton centers [44].

The renormalized Lagrangian densities corresponding to Eqs. (1) and (2) are given in the following forms

{\tilde{L}}_{1} = \frac{i}{2} (E_{1}^{*} \frac{\partial E_{1}}{\partial z} - E_{1} \frac{\partial E_{1}^{*}}{\partial z}) (1 - \frac{1}{{| E_{1} |}^{2}}) - \frac{γ}{2} {| \frac{\partial E_{1}}{\partial x} |}^{2} - \frac{1}{2 γ} ({| E_{1} |}^{2} + 2 {| E_{2} |}^{2} - 3) ({| E_{1} |}^{2} - 1),

{\tilde{L}}_{2} = \frac{i}{2} (E_{2}^{*} \frac{\partial E_{2}}{\partial z} - E_{2} \frac{\partial E_{2}^{*}}{\partial z}) (1 - \frac{1}{{| E_{2} |}^{2}}) - \frac{1}{2} {| \frac{\partial E_{2}}{\partial x} |}^{2} - \frac{1}{2} (2 {| E_{1} |}^{2} + {| E_{2} |}^{2} - 3) ({| E_{2} |}^{2} - 1) .

Substituting Eqs. (3) and (4) into the Lagrangian density Eqs. (5) and (6), then integrating over x, $L_{1, 2} = \int_{- \infty}^{\infty} {\tilde{L}}_{1, 2} d x$ , we can obtain the averaged Lagrangian:

L_{1} = 2 \frac{d x_{1}}{d z} [- A_{1} B_{1} + {tan}^{- 1} (\frac{B_{1}}{A_{1}})] - \frac{2 γ}{3} B_{1}^{2} D - \frac{2 B_{1}^{2} (B_{1}^{2} + 2 B_{2}^{2})}{3 γ D},

L_{2} = 2 \frac{d x_{2}}{d z} [- A_{2} B_{2} + {tan}^{- 1} (\frac{B_{2}}{A_{2}})] - \frac{2}{3} B_{2}^{2} D - \frac{2 B_{2}^{2} (2 B_{1}^{2} + B_{2}^{2})}{3 D} .

The total averaged Lagrangian of the system is L = L₁ + L₂. Based on the Euler-Lagrange equations, we obtain that

D^{2} = \frac{B_{1}^{2} (B_{1}^{2} + 2 B_{2}^{2})}{γ^{2} B_{1}^{2} + γ B_{2}^{2}} + \frac{B_{2}^{2} (2 B_{1}^{2} + B_{2}^{2})}{γ B_{1}^{2} + B_{2}^{2}},

\frac{d x_{1}}{d z} = \frac{A_{1}}{3 B_{1}} [γ D + \frac{2 (B_{1}^{2} + B_{2}^{2})}{γ D} + \frac{2 B_{2}^{2}}{D}],

\frac{d x_{2}}{d z} = \frac{A_{2}}{3 B_{2}} [D + \frac{2 B_{1}^{2}}{γ D} + \frac{2 (B_{1}^{2} + B_{2}^{2})}{D}] .

The formulas of Eqs. (9)–(11) give the analytical relations among parameters of the two-color vector dark solitons. From Eq. (9), we can obtain that the width of the two-color vector dark solitons (proportional to D⁻¹) is related to the nonresonant coefficient γ. We show in Fig. (1) the width (∝ D⁻¹) of the two-color vector dark solitons versus the nonresonant coefficient γ with different values of B₁ and B₂. From Fig. 1, it is obviously that the width of the two-color vector dark soliton is a monotonic function of the nonresonant coefficient γ. For given values of B₁ and B₂, the width of the two-color dark solitons will increase when γ increases. When $B_{1}^{2} = B_{2}^{2}$ , i.e., the modulation depths of the two-color vector dark beams equal, the width of the vector solitons will increase when the modulation depth $B_{1}^{2}$ or $B_{2}^{2}$ decreases [solid and dash-dotted lines in Fig. (1)]. From Fig. (1), at γ = 1 (same color), we also find that the widths equal to a same value (point A) when the two solitons exchange their modulation depths [dashed and dotted lines in Fig. (1)].

Fig. 1 Width (∝ D⁻¹) of the two-color vector dark solitons versus the nonresonant coefficient γ with different values of B₁ and B₂.

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Substituting Eq. (9) into Eqs. (10) and (11), we can obtain that the transverse velocities v₁ = dx₁/dz and v₂ = dx₂/dz of the soliton components E₁ and E₂ are also related to the nonresonant coefficient γ. In Fig. (2), we show the transverse velocities v₁ and v₂ of the two-color vector dark solitons when they have the same modulation depths $B_{1}^{2} = B_{2}^{2}$ . When B₁ = B₂ = 1, i.e., it is obviously the transverse velocities of the vector dark solitons are v₁ = v₂ = 0. Although they have the same initial modulation depths when B₁ = B₂ = 0.8, the transverse velocities of the vector dark solitons are same (point A) only for γ = 1 (same color), otherwise they are different for γ ≠ 1. Both v₁ and v₂ will increase when the modulation depths of the dark solitons decrease, e.g., B₁ = B₂ = 0.5 (v₁ = v₂ at point B for γ = 1).

Fig. 2 Transverse velocities of the two-color vector dark solitons versus the nonresonant coefficient γ when B₁ = B₂. The dashed and solid lines represent the transverse velocities v₁ and v₂ of two soliton components E₁ and E₂, respectively.

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We also show in Fig. (3) the transverse velocities v₁ and v₂ of the vector dark solitons with different modulation depths $B_{1}^{2} \neq B_{2}^{2}$ . For example, when we set B₁ = 0.8 and B₂ = 0.5, the transverse velocities v₁ and v₂ are displayed as lines 5 and 2 in Fig. (3), respectively. If we keep B₁ = 0.8 unchanged, and B₂ decreases to B₂ = 0.3, the transverse velocity v₁ (line 6) will decrease, whereas, the transverse velocity v₂ will increase (line 1). When we set B₁ = 0.5 and B₂ = 0.5, the transverse velocities v₁ and v₂ are displayed as lines 3 and 4 in Fig. (3), respectively. And they have the same value with γ = 1 [black point in Fig. (3)]. If we keep B₂ = 0.5 unchanged, and decrease B₁ from B₁ = 0.8 to B₁ = 0.5, the transverse velocity v₁ will increase (lines 5 and 3), whereas, the transverse velocity v₂ will decrease (lines 2 and 4). In summary, when the modulation depth of one of the components decrease, the transverse velocity of this component will increase, however, the transverse velocity of another component will decrease.

Fig. 3 Transverse velocities of the two-color vector dark solitons versus the nonresonant coefficient γ when B₁ ≠ B₂. The dashed and solid lines represent the transverse velocities v₁ and v₂ of two soliton components E₁ and E₂, respectively.

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The variational approach discussed above can only give us approximate solutions of the vector dark solitons. Generally, there are some methods to study the stability of these dark solitons. For instance, the stability of dark solitons can be determined by the Vakhitov-Kolokolov criterion [50]. With some special forms of the dark beams, the stable dark solitons can be obtained by the inequality ∂P/∂v < 0, here v is the solitons velocity, and P is the renormalized momentum [51]. However, in this paper, we will seek the numerical solutions of Eqs. (1) and (2) to investigate the stability of the two-color vector dark solitons. In general, the forms of dark solitons E₁ and E₂ can be written as $E_{1} = u_{0} ψ (x) exp (- i u_{0}^{2} z)$ and $E_{2} = v_{0} ϕ (x) exp (- i v_{0}^{2} z)$ [2, 44], $u_{0}^{2}$ and $v_{0}^{2}$ are the constant intensity backgrounds of dark solitons E₁ and E₂ [2, 44], respectively. Thus the numerical solutions of Eqs. (1) and (2) can be represented as E₁ = ψ(x) exp (−ikz) and E₂ = ϕ(x) exp (−ikz), so that

\frac{γ}{2} \frac{\partial^{2} ψ}{\partial x^{2}} + k ψ - \frac{1}{γ} ({| ψ |}^{2} + 2 {| ϕ |}^{2}) ψ = 0,

\frac{1}{2} \frac{\partial^{2} ϕ}{\partial x^{2}} + k ϕ - (2 {| ψ |}^{2} + {| ϕ |}^{2}) ϕ = 0,

with

k = u_{0}^{2} = v_{0}^{2} = 1

. Equations (12) and (13) can be solved using a Newton iteration method [52]. The variation result at γ = 1 is used as the initial guess, i.e., ψ(x) = ϕ(x) = tanh(Dx) for dark solitons with D = 1.735. The numerical solutions of the two-color dark solitons are shown in Fig. (4). From Fig. 4(a), it is obvious that the numerical solutions coincide with the initial guess when γ = 1, which indicates the vector dark solitons with same color are stable. However, when γ ≠ 1, as shown in Fig. 4(b), the intensity background of ψ(x) decreases, whereas, the intensity background of ϕ(x) increases. Thus the vector dark solitons with different wavelengths are unstable. These results of the numerical solutions will be confirmed with full numerical simulations in the next section.

Fig. 4 Numerical solutions ψ(x) and ϕ(x) of the two-color vector dark solitons with different nonresonant coefficients (a) γ = 1 and (b) γ = 1.02, respectively.

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3. Dynamics of two-color vector dark solitons in nonlinear self-defocusing media

In this section, we mainly study the propagation dynamics and stability of the two-color vector dark solitons in nonlinear self-defocusing media. We employ the split-step Fourier method to integrate the NLS Eqs. (1) and (2) numerically. The analytically variational results obtained above are used as the initial input in the numerical simulations.

Firstly, we consider the case of vector dark solitons with the soliton parameters B₁ = B₂ = 1 (v₁ = v₂ = 0). In Fig. (5), we show the evolution behaviors of the vector dark solitons under the particular case γ = 1 (same color). It is clear that the width of the vector solitons [Fig. 5(a)] and its two components E₁ [Fig. 5(c)] and E₂ [Fig. 5(d)] nearly remain unchanged. Both the vector solitons and its two components propagate stably. This conclusion can also be obtained from Fig. 5(b), which describe the intensity profile of the vector solitons at the beginning (z = 0) and the end of the propagation distance (z = 10), respectively. It is obvious that the profile at z = 10 (dash red lines) is almost coincide with the initial profile at z = 0 (solid black lines). This result is totally agree with the numerical solutions of the two-color vector dark solitons with γ = 1 [Fig. 4(a)].

Fig. 5 Dynamics of two-color vector dark solitons (a) and its two components E₁ (c), E₂ (d) in nonlinear media. The intensity profiles of the vector solitons (b) at the beginning (z = 0) and the end (z = 10) of the propagation distance. The initial parameters are B₁ = B₂ = 1 and γ = 1.

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Next, we investigate the propagation dynamics of vector dark solitons when γ ≠ 1 (two color). In Fig. (6), the evolution behaviors of the vector dark solitons are displayed with γ = 1.02. We find that that both the vector solitons [Fig. 6(a)] and its two components E₁ [Fig. 6(c)] and E₂ [Fig. 6(d)] can only propagate stably within a short propagation distances, and then decay into strong dispersive waves (radiation modes). This result is also shown as dashed line in Fig. 6(b), which describe the intensity profile of the vector solitons at the end of the propagation distance (z = 10). It is obviously the vector dark solitons is stable within a short propagation distance z = 4 and then it is unstable at the propagation distance z = 10. The result confirms the numerical solutions in Fig. 4(b) that the vector dark solitons are always unstable when γ ≠ 1.

Fig. 6 Dynamics of two-color vector dark solitons (a) and its two components E₁ (c), E₂ (d) in nonlinear media. The intensity profiles of the vector solitons (b) at a particular propagation distance z = 4 and the end of the propagation distance z = 10. The initial parameters are B₁ = B₂ = 1 and γ = 1.02.

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We also numerically check that the instability of the two-color vector dark solitons depends on the value of |γ − 1| (not shown). Larger values of |γ − 1|, worse stability of the vector dark solitons. This instability of the two-color vector dark solitons is induced by the difference in the diffraction coefficients (wavelengths) and occurs via incoherent interaction process. When γ ≠ 1, the components of the vector solitons suffer from different degrees of diffraction. For example, when γ > 1, the larger diffraction coefficient γ/2 of the component E₁ makes it initially diffract more than the component E₂. Thus the width of E₁ will become larger than that of E₂ initially. Through the incoherent interaction, the widening of the component E₁ induces the component E₂ to deformation, which will inversely induce the component E₁ to deformation [53].

4. Dynamics of two-color vector “grey” solitons in nonlinear self-defocusing media

Now, we will investigate the dynamics of vector grey solitons when the components E₁ and E₂ have the same transverse velocities [B₁ = B₂ < 1 (v₁ = v₂ ≠ 0)]. We show in Fig. (7) the evolution behaviors of the vector grey solitons with the initial parameters B₁ = B₂ = 0.8 and γ = 1 (same color). Similar with Fig. (5), both the vector solitons [Fig. 7(a)] and its two components E₁ [Fig. 7(c)] and E₂ [Fig. 7(d)] can propagate stably with their widths keep unchanged. The intensity profiles of the vector solitons at the beginning (z = 0) and the end of the propagation distance (z = 10) are also displayed in Fig. 7(b). The intensity profiles at z = 10 (dash red line) is almost same with the initial intensity profile at z = 0 (solid black lines) but with some transverse displacement of the center location of the intensity profile. This displacement depends on the modulation depths $B_{1}^{2} = B_{2}^{2}$ for that the initial transverse velocities of the vector grey solitons are related to the amplitudes of the solitons v_1,2 = dx_1,2/dz = A_1,2 [44–48]. The displacement will become larger for that they have larger initial transverse velocities A_1,2 when the modulation depths $B_{1}^{2} = B_{2}^{2}$ decrease. Compare Figs. 7(a), 7(c), and 7(d) with Figs. 5(a), 5(c), and 5(d), we also find that the width of the vector dark solitons also become larger when $B_{1}^{2} = B_{2}^{2}$ decrease, and this result agrees with the variational result [solid and dash-dotted lines in Fig. (1)].

Fig. 7 Dynamics of two-color vector grey solitons (a) and its two components E₁ (c), E₂ (d) in nonlinear media. The intensity profiles of the vector solitons at the beginning (z = 0) and the end (z = 10) of the propagation distance. The initial parameters are B₁ = B₂ = 0.8 and γ = 1.

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For the case of γ ≠ 1 (two color), in Fig. (8), the evolution behaviors of the vector grey solitons with same initial modulation depths B₁ = B₂ = 0.8 (same initial transverse velocities) are displayed when γ = 1.02. Again, similar with Fig. (6), both the vector solitons [Fig. 8(a)] and its two components E₁ [Fig. 8(c)] and E₂ [Fig. 8(d)] can only propagate stably within a short propagation distances, and then decay into strong dispersive waves [also shown as dashed line in Fig. 8(b)], due to the difference in the diffraction coefficients.

Fig. 8 Dynamics of two-color vector grey solitons (a) and its two components E₁ (c), E₂ (d) in nonlinear media. The intensity profiles of the vector solitons (b) at a particular propagation distance z = 4 and the end of the propagation distance z = 10. The initial parameters are B₁ = B₂ = 0.8 and γ = 1.02.

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5. Dynamics OF one-color vector grey solitons with different transverse velocities

Finally, we show in Fig. (9) the propagation dynamics of vector grey solitons with different greyness or transverse velocities (B₁ ≠ B₂) when γ = 1 (one color). We can see that both the vector solitons [Fig. 9(a)] and its two components E₁ [Fig. 9(c)] and E₂ [Fig. 9(d)] can only propagate stably within a short propagation distances, and then decay into strong dispersive waves [also shown as dashed line in Fig. 9(b)], due to the difference in the transverse velocities. The instability of the one-color vector grey solitons depends on the values of |B₁ − B₂|. Larger values of |B₁ − B₂|, worse stability of the vector grey solitons (not shown).

Fig. 9 Dynamics of one-color (γ = 1) vector grey solitons (a) and its two components E₁ (c), E₂ (d) in nonlinear media. The intensity profiles of the vector solitons (b) at a particular propagation distance z = 4 and the end of the propagation distance z = 10. The initial parameters are B₁ = 0.8, B₂ = 0.796 and γ = 1.

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6. Conclusion

In conclusion, we investigated the stability of two-color vector dark solitons in a nonlinear self-defocusing media. We obtained the approximate solutions of the two-color vector dark solitons using the variational method. Propagation dynamics of the two-color vector dark solitons were numerically demonstrated with split step Fourier transform method. Numerical results showed that, when γ = 1, the vector dark/grey solitons will be stable with their beam widths remain unchanged during the propagation. Otherwise, the two-color dark/grey solitons are always unstable due to different diffraction coefficients when γ ≠ 1.

Funding

National Natural Science Foundation of China (11504226).

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Incoherently coupled two-color vector dark solitons in self-defocusing media

Abstract

1. Introduction

2. Model and variational method

3. Dynamics of two-color vector dark solitons in nonlinear self-defocusing media

4. Dynamics of two-color vector “grey” solitons in nonlinear self-defocusing media

5. Dynamics OF one-color vector grey solitons with different transverse velocities

6. Conclusion

Funding

References

Cited By

Figures (9)

Equations (13)

Optics Express