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Stability of Bragg grating solitons in a cubic–quintic nonlinear medium with dispersive reflectivity
Physics Letters A 375 (2010) 225–229
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Stability of Bragg grating solitons in a cubic–quintic nonlinear medium with dispersive reflectivity Sahan Dasanayaka, Javid Atai ∗ School of Electrical and Information Engineering, The University of Sydney, NSW, 2006, Australia
a r t i c l e
i n f o
Article history: Received 17 October 2010 Accepted 23 October 2010 Available online 26 October 2010 Communicated by V.M. Agranovich Keywords: Bragg grating solitons Cubic–quintic nonlinearity Dispersive reflectivity
a b s t r a c t We investigate the existence and stability of Bragg grating solitons in a cubic–quintic medium with dispersive reflectivity. It is found that the model supports two disjoint families of solitons. One family can be viewed as the generalization of the Bragg grating solitons in Kerr nonlinearity with dispersive reflectivity. On the other hand, the quintic nonlinearity is dominant in the other family. Stability regions are identified by means of systematic numerical stability analysis. In the case of the first family, the size of the stability region increases up to moderate values of dispersive reflectivity. However for the second family (i.e. region where quintic nonlinearity dominates), the size of the stability region increases even for strong dispersive reflectivity. For all values of m, there exists a subset of the unstable solitons belonging to the first family for which the instability development leads to deformation and subsequent splitting of the soliton into two moving solitons with different amplitudes and velocities. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Fiber Bragg gratings (FBGs) have been the subject of intensive research due to their applications in sensing, and optical communications [1]. FBGs are produced by periodically varying the refractive index along an optical fiber. The variation of the refractive index gives rise to a photonic bandgap in their spectrum inside which linear waves cannot propagate. The linear cross-coupling between the counterpropagating waves results in a large effective dispersion. The effective dispersion induced by the FBG is approximately six orders of magnitude greater than the underlying chromatic dispersion of the fiber [2,3]. At sufficiently high intensities the FBG-induced dispersion may be counterbalanced by nonlinearity resulting in the formation of a Bragg grating (BG) soliton. BG soliton have been investigated theoretically [3–6] and experimentally [7–10] during the past two decades. In the case of uniform gratings, exact analytical soliton solutions have been found and it has been shown that the solitons form a two-parameter family of solutions [4,6]. One of these parameters is related to soliton’s velocity, which can range between zero (quiescent) and the speed of light in the medium, and the other is dependent on the detuning frequency, peak power and soliton width. Stability of these solitons has been investigated by means of variational approximation and numerical techniques. It has been demonstrated that approximately half of the family is unstable against oscillatory perturbations [11–13]. Experimental efforts have been focused on
*
Corresponding author. E-mail address: [email protected] (J. Atai).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.10.043
generating zero-velocity BG solitons due to their potential applications in optical buffers and optical memory. To date, BG solitons with a velocity in excess of 23% of the speed of light in the medium have been observed in FBGs [14]. The existence and stability of BG solitons have been investigated in more sophisticated systems and nonlinearities. For example, it has been shown that solitons exist in a dual core fiber where one or both cores may be nonlinear [15,16]. BG solitons have also been predicted in photonic crystals [17] and in optical media with quadratic [18–20] and cubic–quintic nonlinearities [21,22]. More recently, a model has been proposed that takes into account the spatial dispersion of Bragg reflectivity [23,24]. The model is applicable to nonuniform gratings and slightly disordered ones. A key finding in Refs. [23,24] is that, for both quiescent and moving solitons, dispersion of reflectivity results in the expansion of the stability region compared to the standard model. In this Letter we investigate the existence and stability of quiescent BG solitons in a cubic–quintic medium equipped with a Bragg grating with dispersive reflectivity. In Section 2, the model is presented and the characteristics of the stationary solutions are discussed. In Section 3, stability of solitons is analyzed. In particular, the effect of dispersive reflectivity on the stability regions is discussed. A summary of the main results is provided in Section 4. 2. The model and soliton solutions Following a similar approach as the one described in Ref. [23], the model of Ref. [21] can be modified to arrive at the following system of normalized coupled equations:
226
S. Dasanayaka, J. Atai / Physics Letters A 375 (2010) 225–229
Fig. 1. Profiles of Type 1 (solid line) and Type 2 (dashed line) solitons. (a)
iut + iu x +
1 2
1 3 3 |u |2 + | v |2 u − ν |u |4 + |u |2 | v |2 + | v |4 u 4
2
4
+ v + mv xx = 0, 1 2 1 3 3 i vt − i v x + | v | + |u |2 v − ν | v |4 + | v |2 |u |2 + |u |4 v 2
4
2
+ u + mu xx = 0,
4
(1)
where u (x, t ) and v (x, t ) are the amplitudes of the forward- and backward-propagating waves. ν > 0 is a real-valued parameter that controls the strength of quintic nonlinearity. The BG-induced coupling coefficient is normalized to unity. The coefficients of self- and cross-phase modulation in front of the cubic terms have been normalized in the usual way, with their ratio being 1:2 corresponding to Kerr nonlinearity. The ratio of the coefficients for the quintic terms is 1:6:3 and has been derived in Ref. [25]. m > 0 is a real parameter which accounts for dispersive reflectivity. Substituting (u , v ) ∼ exp(ikx − i ωt ) into the system (1) and linearizing, the following dispersion relation is obtained:
ω2 = 1 − mk2
2
+ k2 .
(2)
Eq. (2) results in the following bandgap:
2
ω =
1 4m−1 4m2
if m 12 , if m > 12 .
Further analysis of the bandgap shows that for values of m 12 the bandgap is the same as the standard model (i.e. Bragg grating in cubic nonlinearity). However, for values of m > 12 the gap shrinks but it never closes. Since the values m >
1 2
may not be physically
realizable [24], we will confine our analysis to m 12 . In the absence of dispersive reflectivity, i.e. m = 0, it was shown that the model of Eqs. (1) admits an exact analytical solution [21]. In particular, it was shown that above a certain value of ν , there exist two disjoint families of soliton solutions in the bandgap (−1 < ω < +1). The border between these soliton families in the plane (ν , ω ) was found to be the curve 1 − ω = 27/(160ν ). Solitons corresponding to the region 1 − ω < 27/(160ν ) may be regarded as a generalization of the standard BG solitons in the cubic model. On the other hand, quintic nonlinearity dominates in the region 1 − ω > 27/(160ν ). In the case of m = 0, there are no analytical solutions and soliton solutions must be found numerically. We sought for stationary
ν = 0.6, m = 0.1 and (b) ν = 0.6, m = 0.5.
solutions to Eqs. (1) as
u (x, t ), v (x, t ) = U (x), V (x) e −i ωt .
(3)
Substituting (3) into Eqs. (1) we obtain the following system of ordinary differential equations:
mV xx + ω U + iU x + 3
1 2
1 3 |U |2 + | V |2 U − ν |U |4 + |U |2 | V |2 4
2
+ | V |4 U + V = 0 , 4
mU xx + ω V − i V x + 3
1 2
+ | U |4 V + U = 0 . 4
1 3 | V |2 + |U |2 V − ν | V |4 + | V |2 |U |2 4
2
(4)
We have numerically solved Eqs. (4) for various values of m up to 0.5 by means of the relaxation algorithm. Similar to the case of m = 0, it is found that, above a certain value of ν , there are two disjoint families of zero-velocity solitons that fill the entire bandgap. The border between the soliton families has been determined numerically. A very noteworthy and interesting feature of the solutions is that the border separating the two families is almost the same as the one for the case m = 0. More specifically, the maximum deviation from the curve 1 − ω = 27/(160ν ) occurs for m = 0.5 and is ≈ 0.03. The soliton families differ in their phase structure and parities. For solitons belonging to the region to the left and above the border (henceforth Type 1), (u (x)) and ( v (x)) are odd and (u (x)) and ( v (x)) are even functions of x. For solitons in the region to the right and below the border (henceforth Type 2), the parities of the real and imaginary parts of the solutions are opposite to those of Type 1 solitons. As was shown in Ref. [23], in the standard model (i.e. BG with cubic nonlinearity) with dispersive reflectivity, when m exceeds a certain value sidelobes appear in the profiles of solitons. The same holds true for the model of Eqs. (1), i.e. for m 0.26 (for ω = 0), sidelobes emerge in both Type 1 and Type 2 solitons. Examples of solitons with and without sidelobes are displayed in Fig. 1. 3. Stability analysis We have carried out systematic numerical stability analysis to test the stability of both families of solitons. To this end, Eqs. (1) were solved by means of the symmetrized split-step
S. Dasanayaka, J. Atai / Physics Letters A 375 (2010) 225–229
Fig. 2. Examples of Type 1 soliton evolution for m = 0.1. (a) A stable soliton with
ω = 0.20, ν = 0.05 and (b) an unstable soliton with ω = −0.60, ν = 0.05.
Fig. 3. Examples of Type 2 soliton evolution for m = 0.1. (a) A stable soliton with
ω = −0.60, ν = 0.50 and (b) an unstable soliton with ω = 0.20, ν = 0.50.
Fourier method. In all simulations, the stationary solution obtained from the relaxation algorithm was propagated to t = 2000. The intrinsic numerical noise was found to be sufficient to trigger the instability development. Examples of stable and unstable Type 1 and Type 2 solitons are shown in Figs. 2 and 3, respectively. To characterize the effect of dispersive reflectivity on the stability of solitons, we have investigated stability of soliton families for various values of m ranging from 0.1 to 0.5. The results of the stability analyses are summarized in the stability diagrams shown in Fig. 4. The presence of dispersive reflectivity drastically alters the stability of solitons. In the case of Type 1 solitons, dispersive reflectivity results in the initial expansion of the stability region. In particular, as shown Fig. 4(a), for m = 0.1, a V-shaped notch is formed which extends into the previously unstable region (see Ref. [21]). The notch attains its maximum size at m ≈ 0.2 (see Fig. 4(b)). A striking feature of Fig. 4(b) is that when ν = 0.08 the solitons corresponding to −0.93 < ω < +1 are stable. As m increases to 0.3, the notch transforms into a stable region within the unstable region which is separate from the upper stable region. Increasing m further results in the shrinkage of this region (see Fig. 4(d)) such that at m = 0.5 the stability region resembles that
227
of m = 0 (see Ref. [21]). It should be noted that the stability region for Type 1 solitons in Fig. 4(e) is marginally smaller than that of Ref. [21]. Comparing the stability region for Type 2 solitons shown in Fig. 4 with that corresponding to m = 0 reported in [21], it is readily seen that the inclusion of dispersive reflectivity significantly enlarges the stability region. As m increases, the lower edge of the stable region moves towards the lower edge of the bandgap. Also, increasing m leads to leftward expansion of the stable region. Similar to the case of m = 0, unstable solitons of both types may either radiate some energy and then evolve into a stable soliton belonging to the same family (e.g. Fig. 2(b)) or completely decay into radiation (e.g. Fig. 3(b)). However, when m = 0, some unstable Type 1 solitons exhibit a different behavior. As is shown in Fig. 5, upon propagation the unstable quiescent soliton is deformed and then splits into two moving solitons with unequal amplitudes and velocities. We have observed this symmetry breaking in the unstable region in the vicinity of the dashed curve in Fig. 4. To the best of our knowledge, such an instability development has not previously been reported for quiescent BG solitons. This result also demonstrates that moving solitons exist in the model of Eqs. (1).
228
S. Dasanayaka, J. Atai / Physics Letters A 375 (2010) 225–229
Fig. 4. Soliton stability diagrams for (a) m = 0.1, (b) m = 0.2, (c) m = 0.3, (d) m = 0.4 and (e) m = 0.5. Dashed curve is the border between the soliton families, see the text.
4. Conclusions
Type 2 solitons). The boundary separating the families in the plane
(ν , ω) is almost independent of the value of m. Stability regions for The existence and stability of quiescent Bragg grating solitons were numerically investigated in a Bragg grating in a cubic–quintic nonlinear medium with dispersive reflectivity. It is found that, similar to the cubic–quintic model with no dispersive reflectivity, the solitons occur in two disjoint families (referred to as Type 1 and
the soliton families are determined through systematic direct numerical simulations. In the case of Type 1 solitons, the stability region expands as m increases from 0 to 0.3. Increasing the value m beyond 0.3 results in the shrinking of the stability region. However, the stability region for Type 2 solitons increases as m varies
S. Dasanayaka, J. Atai / Physics Letters A 375 (2010) 225–229
Fig. 5. Examples of spontaneous pulse splitting for (a)
ω = −0.40, ν = 0.12, m = 0.1 and (b) ω = −0.58, ν = 0.12, m = 0.5.
from 0 to 0.5. For all values of m, there exists a small subset of unstable Type 1 solitons that are deformed as they propagate and subsequently split into two moving solitons with unequal amplitudes and velocities. References [1] [2] [3] [4] [5] [6] [7] [8]
229
R. Kashyap, Fiber Bragg Gratings, Academic Press, San Diego, 1999. P.St.J. Russell, J. Mod. Opt. 38 (1991) 1599. C.M. De Sterke, J.E. Sipe, Prog. Opt. 33 (1994) 203. A.B. Aceves, S. Wabnitz, Phys. Lett. A 141 (1989) 37. J.E. Sipe, H.G. Winful, Opt. Lett. 13 (1988) 132. D.N. Christadoulides, R.I. Joseph, Phys. Rev. Lett. 62 (1989) 1746. B.J. Eggleton, C.M. de Sterke, R.E. Slusher, J. Opt. Soc. Am. B 14 (1997) 2980. B.J. Eggleton, R.E. Slusher, C.M. de Sterke, P.A. Krug, J.E. Sipe, Phys. Rev. Lett. 76 (1996) 1627. [9] D. Taverner, N.G.R. Broderick, D.J. Richardson, R.I. Laming, M. Ibsen, Opt. Lett. 23 (1998) 328.
[10] C.M. de Sterke, B.J. Eggleton, P.A. Krug, J. Lightwave Technol. 15 (1997) 1494. [11] B.A. Malomed, R.S. Tasgal, Phys. Rev. E 49 (1994) 5787. [12] I.V. Barashenkov, D.E. Pelinovsky, E.V. Zemlyanaya, Phys. Rev. Lett. 80 (1998) 5117. [13] A. De Rossi, C. Conti, S. Trillo, Phys. Rev. Lett. 81 (1998) 85. [14] J.T. Mok, C.M. de Sterke, I.C.M. Littler, B.J. Eggleton, Nat. Phys. 2 (2006) 775. [15] W.C.K. Mak, P.L. Chu, B.A. Malomed, J. Opt. Soc. Am. B 15 (1998) 1685. [16] J. Atai, B.A. Malomed, Phys. Rev. E 62 (2000) 8713; J. Atai, B.A. Malomed, Phys. Rev. E 64 (2001) 066617. [17] F. Biancalana, A. Amann, E.P. O’Reilly, Phys. Rev. A 77 (2008) 011801(R). [18] C. Conti, S. Trillo, G. Assanto, Phys. Rev. Lett. 78 (1997) 2341. [19] H. He, P.D. Drummond, Phys. Rev. Lett. 78 (1997) 4311. [20] W.C.K. Mak, B.A. Malomed, P.L. Chu, Phys. Rev. E 58 (1998) 6708. [21] J. Atai, B.A. Malomed, Phys. Lett. A 284 (2001) 247. [22] J. Atai, J. Opt. B: Quantum Semiclass. Opt. 6 (2004) S177. [23] J. Atai, B.A. Malomed, Phys. Lett. A 342 (2005) 404. [24] D.R. Neill, J. Atai, B.A. Malomed, J. Opt. A: Pure Appl. Opt. 10 (2008) 085105. [25] A.I. Maimistov, B.A. Malomed, A. Desyatnikov, Phys. Lett. A 254 (1999) 179.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Stability of Bragg grating solitons in a cubic–quintic nonlinear medium with dispersive reflectivity Sahan Dasanayaka, Javid Atai ∗ School of Electrical and Information Engineering, The University of Sydney, NSW, 2006, Australia
a r t i c l e
i n f o
Article history: Received 17 October 2010 Accepted 23 October 2010 Available online 26 October 2010 Communicated by V.M. Agranovich Keywords: Bragg grating solitons Cubic–quintic nonlinearity Dispersive reflectivity
a b s t r a c t We investigate the existence and stability of Bragg grating solitons in a cubic–quintic medium with dispersive reflectivity. It is found that the model supports two disjoint families of solitons. One family can be viewed as the generalization of the Bragg grating solitons in Kerr nonlinearity with dispersive reflectivity. On the other hand, the quintic nonlinearity is dominant in the other family. Stability regions are identified by means of systematic numerical stability analysis. In the case of the first family, the size of the stability region increases up to moderate values of dispersive reflectivity. However for the second family (i.e. region where quintic nonlinearity dominates), the size of the stability region increases even for strong dispersive reflectivity. For all values of m, there exists a subset of the unstable solitons belonging to the first family for which the instability development leads to deformation and subsequent splitting of the soliton into two moving solitons with different amplitudes and velocities. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Fiber Bragg gratings (FBGs) have been the subject of intensive research due to their applications in sensing, and optical communications [1]. FBGs are produced by periodically varying the refractive index along an optical fiber. The variation of the refractive index gives rise to a photonic bandgap in their spectrum inside which linear waves cannot propagate. The linear cross-coupling between the counterpropagating waves results in a large effective dispersion. The effective dispersion induced by the FBG is approximately six orders of magnitude greater than the underlying chromatic dispersion of the fiber [2,3]. At sufficiently high intensities the FBG-induced dispersion may be counterbalanced by nonlinearity resulting in the formation of a Bragg grating (BG) soliton. BG soliton have been investigated theoretically [3–6] and experimentally [7–10] during the past two decades. In the case of uniform gratings, exact analytical soliton solutions have been found and it has been shown that the solitons form a two-parameter family of solutions [4,6]. One of these parameters is related to soliton’s velocity, which can range between zero (quiescent) and the speed of light in the medium, and the other is dependent on the detuning frequency, peak power and soliton width. Stability of these solitons has been investigated by means of variational approximation and numerical techniques. It has been demonstrated that approximately half of the family is unstable against oscillatory perturbations [11–13]. Experimental efforts have been focused on
*
Corresponding author. E-mail address: [email protected] (J. Atai).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.10.043
generating zero-velocity BG solitons due to their potential applications in optical buffers and optical memory. To date, BG solitons with a velocity in excess of 23% of the speed of light in the medium have been observed in FBGs [14]. The existence and stability of BG solitons have been investigated in more sophisticated systems and nonlinearities. For example, it has been shown that solitons exist in a dual core fiber where one or both cores may be nonlinear [15,16]. BG solitons have also been predicted in photonic crystals [17] and in optical media with quadratic [18–20] and cubic–quintic nonlinearities [21,22]. More recently, a model has been proposed that takes into account the spatial dispersion of Bragg reflectivity [23,24]. The model is applicable to nonuniform gratings and slightly disordered ones. A key finding in Refs. [23,24] is that, for both quiescent and moving solitons, dispersion of reflectivity results in the expansion of the stability region compared to the standard model. In this Letter we investigate the existence and stability of quiescent BG solitons in a cubic–quintic medium equipped with a Bragg grating with dispersive reflectivity. In Section 2, the model is presented and the characteristics of the stationary solutions are discussed. In Section 3, stability of solitons is analyzed. In particular, the effect of dispersive reflectivity on the stability regions is discussed. A summary of the main results is provided in Section 4. 2. The model and soliton solutions Following a similar approach as the one described in Ref. [23], the model of Ref. [21] can be modified to arrive at the following system of normalized coupled equations:
226
S. Dasanayaka, J. Atai / Physics Letters A 375 (2010) 225–229
Fig. 1. Profiles of Type 1 (solid line) and Type 2 (dashed line) solitons. (a)
iut + iu x +
1 2
1 3 3 |u |2 + | v |2 u − ν |u |4 + |u |2 | v |2 + | v |4 u 4
2
4
+ v + mv xx = 0, 1 2 1 3 3 i vt − i v x + | v | + |u |2 v − ν | v |4 + | v |2 |u |2 + |u |4 v 2
4
2
+ u + mu xx = 0,
4
(1)
where u (x, t ) and v (x, t ) are the amplitudes of the forward- and backward-propagating waves. ν > 0 is a real-valued parameter that controls the strength of quintic nonlinearity. The BG-induced coupling coefficient is normalized to unity. The coefficients of self- and cross-phase modulation in front of the cubic terms have been normalized in the usual way, with their ratio being 1:2 corresponding to Kerr nonlinearity. The ratio of the coefficients for the quintic terms is 1:6:3 and has been derived in Ref. [25]. m > 0 is a real parameter which accounts for dispersive reflectivity. Substituting (u , v ) ∼ exp(ikx − i ωt ) into the system (1) and linearizing, the following dispersion relation is obtained:
ω2 = 1 − mk2
2
+ k2 .
(2)
Eq. (2) results in the following bandgap:
2
ω =
1 4m−1 4m2
if m 12 , if m > 12 .
Further analysis of the bandgap shows that for values of m 12 the bandgap is the same as the standard model (i.e. Bragg grating in cubic nonlinearity). However, for values of m > 12 the gap shrinks but it never closes. Since the values m >
1 2
may not be physically
realizable [24], we will confine our analysis to m 12 . In the absence of dispersive reflectivity, i.e. m = 0, it was shown that the model of Eqs. (1) admits an exact analytical solution [21]. In particular, it was shown that above a certain value of ν , there exist two disjoint families of soliton solutions in the bandgap (−1 < ω < +1). The border between these soliton families in the plane (ν , ω ) was found to be the curve 1 − ω = 27/(160ν ). Solitons corresponding to the region 1 − ω < 27/(160ν ) may be regarded as a generalization of the standard BG solitons in the cubic model. On the other hand, quintic nonlinearity dominates in the region 1 − ω > 27/(160ν ). In the case of m = 0, there are no analytical solutions and soliton solutions must be found numerically. We sought for stationary
ν = 0.6, m = 0.1 and (b) ν = 0.6, m = 0.5.
solutions to Eqs. (1) as
u (x, t ), v (x, t ) = U (x), V (x) e −i ωt .
(3)
Substituting (3) into Eqs. (1) we obtain the following system of ordinary differential equations:
mV xx + ω U + iU x + 3
1 2
1 3 |U |2 + | V |2 U − ν |U |4 + |U |2 | V |2 4
2
+ | V |4 U + V = 0 , 4
mU xx + ω V − i V x + 3
1 2
+ | U |4 V + U = 0 . 4
1 3 | V |2 + |U |2 V − ν | V |4 + | V |2 |U |2 4
2
(4)
We have numerically solved Eqs. (4) for various values of m up to 0.5 by means of the relaxation algorithm. Similar to the case of m = 0, it is found that, above a certain value of ν , there are two disjoint families of zero-velocity solitons that fill the entire bandgap. The border between the soliton families has been determined numerically. A very noteworthy and interesting feature of the solutions is that the border separating the two families is almost the same as the one for the case m = 0. More specifically, the maximum deviation from the curve 1 − ω = 27/(160ν ) occurs for m = 0.5 and is ≈ 0.03. The soliton families differ in their phase structure and parities. For solitons belonging to the region to the left and above the border (henceforth Type 1), (u (x)) and ( v (x)) are odd and (u (x)) and ( v (x)) are even functions of x. For solitons in the region to the right and below the border (henceforth Type 2), the parities of the real and imaginary parts of the solutions are opposite to those of Type 1 solitons. As was shown in Ref. [23], in the standard model (i.e. BG with cubic nonlinearity) with dispersive reflectivity, when m exceeds a certain value sidelobes appear in the profiles of solitons. The same holds true for the model of Eqs. (1), i.e. for m 0.26 (for ω = 0), sidelobes emerge in both Type 1 and Type 2 solitons. Examples of solitons with and without sidelobes are displayed in Fig. 1. 3. Stability analysis We have carried out systematic numerical stability analysis to test the stability of both families of solitons. To this end, Eqs. (1) were solved by means of the symmetrized split-step
S. Dasanayaka, J. Atai / Physics Letters A 375 (2010) 225–229
Fig. 2. Examples of Type 1 soliton evolution for m = 0.1. (a) A stable soliton with
ω = 0.20, ν = 0.05 and (b) an unstable soliton with ω = −0.60, ν = 0.05.
Fig. 3. Examples of Type 2 soliton evolution for m = 0.1. (a) A stable soliton with
ω = −0.60, ν = 0.50 and (b) an unstable soliton with ω = 0.20, ν = 0.50.
Fourier method. In all simulations, the stationary solution obtained from the relaxation algorithm was propagated to t = 2000. The intrinsic numerical noise was found to be sufficient to trigger the instability development. Examples of stable and unstable Type 1 and Type 2 solitons are shown in Figs. 2 and 3, respectively. To characterize the effect of dispersive reflectivity on the stability of solitons, we have investigated stability of soliton families for various values of m ranging from 0.1 to 0.5. The results of the stability analyses are summarized in the stability diagrams shown in Fig. 4. The presence of dispersive reflectivity drastically alters the stability of solitons. In the case of Type 1 solitons, dispersive reflectivity results in the initial expansion of the stability region. In particular, as shown Fig. 4(a), for m = 0.1, a V-shaped notch is formed which extends into the previously unstable region (see Ref. [21]). The notch attains its maximum size at m ≈ 0.2 (see Fig. 4(b)). A striking feature of Fig. 4(b) is that when ν = 0.08 the solitons corresponding to −0.93 < ω < +1 are stable. As m increases to 0.3, the notch transforms into a stable region within the unstable region which is separate from the upper stable region. Increasing m further results in the shrinkage of this region (see Fig. 4(d)) such that at m = 0.5 the stability region resembles that
227
of m = 0 (see Ref. [21]). It should be noted that the stability region for Type 1 solitons in Fig. 4(e) is marginally smaller than that of Ref. [21]. Comparing the stability region for Type 2 solitons shown in Fig. 4 with that corresponding to m = 0 reported in [21], it is readily seen that the inclusion of dispersive reflectivity significantly enlarges the stability region. As m increases, the lower edge of the stable region moves towards the lower edge of the bandgap. Also, increasing m leads to leftward expansion of the stable region. Similar to the case of m = 0, unstable solitons of both types may either radiate some energy and then evolve into a stable soliton belonging to the same family (e.g. Fig. 2(b)) or completely decay into radiation (e.g. Fig. 3(b)). However, when m = 0, some unstable Type 1 solitons exhibit a different behavior. As is shown in Fig. 5, upon propagation the unstable quiescent soliton is deformed and then splits into two moving solitons with unequal amplitudes and velocities. We have observed this symmetry breaking in the unstable region in the vicinity of the dashed curve in Fig. 4. To the best of our knowledge, such an instability development has not previously been reported for quiescent BG solitons. This result also demonstrates that moving solitons exist in the model of Eqs. (1).
228
S. Dasanayaka, J. Atai / Physics Letters A 375 (2010) 225–229
Fig. 4. Soliton stability diagrams for (a) m = 0.1, (b) m = 0.2, (c) m = 0.3, (d) m = 0.4 and (e) m = 0.5. Dashed curve is the border between the soliton families, see the text.
4. Conclusions
Type 2 solitons). The boundary separating the families in the plane
(ν , ω) is almost independent of the value of m. Stability regions for The existence and stability of quiescent Bragg grating solitons were numerically investigated in a Bragg grating in a cubic–quintic nonlinear medium with dispersive reflectivity. It is found that, similar to the cubic–quintic model with no dispersive reflectivity, the solitons occur in two disjoint families (referred to as Type 1 and
the soliton families are determined through systematic direct numerical simulations. In the case of Type 1 solitons, the stability region expands as m increases from 0 to 0.3. Increasing the value m beyond 0.3 results in the shrinking of the stability region. However, the stability region for Type 2 solitons increases as m varies
S. Dasanayaka, J. Atai / Physics Letters A 375 (2010) 225–229
Fig. 5. Examples of spontaneous pulse splitting for (a)
ω = −0.40, ν = 0.12, m = 0.1 and (b) ω = −0.58, ν = 0.12, m = 0.5.
from 0 to 0.5. For all values of m, there exists a small subset of unstable Type 1 solitons that are deformed as they propagate and subsequently split into two moving solitons with unequal amplitudes and velocities. References [1] [2] [3] [4] [5] [6] [7] [8]
229
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