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Self-induced transparency effect on the two-soliton interaction
Optics Communications 356 (2015) 426–430
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Self-induced transparency effect on the two-soliton interaction C.C.D. da Silva, R.V. Moreira, S.A. Garcia, D.P. Caetano n Departamento de Ciências Exatas, Escola de Engenharia Industrial Metalúrgica de Volta Redonda, Universidade Federal Fluminense, Volta Redonda, RJ 27255125, Brazil
art ic l e i nf o
a b s t r a c t
Article history: Received 13 July 2015 Received in revised form 10 August 2015 Accepted 12 August 2015
We report on the numerical investigation of the interaction between two solitons in a doped nonlinear dispersive medium. The dopant is modeled as a two-level atomic system coherently driven to support simultaneous nonlinear Schrödinger equation and self-induced transparency solitons. We investigate the influence of the self-Induced transparency in the case where the two-solitons are in phase. It is found that the periodical soliton collision effect disappears due to the presence of the atomic system. This new phenomenon is understood considering the detuning parameter on the atomic system description. This result can be exploited in multiple pulse propagation where the soliton interactions play an important role. & 2015 Elsevier B.V. All rights reserved.
Keywords: Optical solitons Self-induced transparency Soliton interaction
1. Introduction Solitons are one of the most fascinating phenomena in the nonlinear science and can be found in different research themes, for example, fluid mechanics, superconductivity, Bose–Einstein condensates, plasm physics, and biology. In the context of the interaction between light and matter, solitons can be found in the study of the propagation of light pulses through a nonlinear dispersive medium [1,2] or an ensemble of atomic system [3], as well as in a combination of these two propagation media [4–6]. An interesting scenario to study the dynamics of the solitons propagation is a doped optical fiber, due to the possibility to investigate how well-known effects associated only to the nonlinear dispersive medium or the atomic system can be modified giving rise to new effects. In the coherent propagation regime, where the time duration of the pulses is smaller than the relaxation times of the atomic system, self-induced transparency solitons (SIT solitons) are observed when light pulses resonantly interact with a two-level atomic system [7–9]. To observe this effect, the pulse area should be an integer multiple of 2π. On the other hand, in the nonlinear dispersive regime, the interplay between the nonlinear and dispersive effects allows us to obtain stable solutions of the nonlinear Schrödinger equation, which describes the pulse propagation through an optical fiber, so-called nonlinear Schrödinger equation solitons (NLSE solitons) [10,11]. To observe this effect, a perfect match between the pulse and optical fiber parameters should be n
Corresponding author. E-mail address: [email protected] (D.P. Caetano).
http://dx.doi.org/10.1016/j.optcom.2015.08.036 0030-4018/& 2015 Elsevier B.V. All rights reserved.
achieved. Coexistence solitons (SIT–NLSE solitons) [4–6] arrive from the combination of the above-mentioned effects when light pulses coherently propagate through a doped optical fiber, where the dopant is modeled as a two-level atomic system. In this case, both power and area of the pulse should satisfy the requirement to observe simultaneous SIT and NLSE soliton. Recently, it has been demonstrated that this requirement can be relaxed [12]. After its demonstration, SIT–NLSE solitons have been exploited to investigate generalizations of coherent propagation effects in the nonlinear dispersive regime, for example soliton cloning [13] and matched pulses in three-level atomic systems [14,15]. In the soliton cloning context, besides the observation of the SIT–NLSE soliton cloning [16], it was observed the effect of pulse break-up [17,18]. In the generalization of matched pulses, it was demonstrated a condition to observe the formation of coupled solitons [19]. It was also demonstrated that the SIT–NLSE soliton cloning effect can be applied to suppress the frequency chirp along optical pulses [20]. Recently, self-induced transparency effect has been observed in surface plasmon polariton together with nonlinear and dispersive effects, giving rise to plasmon solitons [21]. For fundamental NLSE solitons, which correspond to the firstorder solutions of the NLSE, a remarkable feature is that the pulse shape remains unchangeable along the propagation. Another feature related to solitons is that the pulse shape also remains unchangeable after collision. These features give to solitons particlelike behavior. In addition, concerning the interaction of two NLSE solitons it has been observed that attraction or repulsion can be observed depending on the relative phase between the pulses [22,23]. This last effect is relevant in the design of optical communications systems [24]. In this work, we investigate the
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interaction of two SIT–NLSE solitons. The idea is to observe how the presence of the atomic system affects the NLSE soliton dynamics, in particular, the fact that solitons collide periodically along the propagation when they possess a zero phase difference. Our results show that the solitons does not collide, inhibiting the soliton collapse effect, and the propagation dynamics in the asymptotic limit is basically dictated by the atomic system. To confirm the role of the atomic system we studied the effect of the detuning. As a result, we observe that by increasing the detuning parameter the propagation dynamics dictated by the nonlinear dispersive medium is recovered. These results add new perspectives for the study about the propagation of SIT–NLSE solitons and its applications as well as new dynamics effects which can be pursued in other nonlinear phenomena.
2. Basic theory Next, we present the basic equations concerning the propagation of two SIT–NLSE solitons through a doped optical fiber and the numerical procedure used to solve the equations. Let A(z, t ) be the optical field describing a stream of two pulses. This optical field coherently interacts with the atomic system as illustrated in Fig. 1. The extend nonlinear Schrödinger equation (NLSE) given by
β ∂ 2A iωna ∂A μ c2⁎c1 = − i 2 2 + iγ |A|2 A + 2 ∂t 2ε0c 12 ∂z
(1)
describes the propagation of the pulse by taking into account the self-phase modulation, group velocity dispersion (GVD), and selfinduced transparency effects. β2 is the GVD coefficient, γ is the Kerr nonlinearity coefficient, μ12 is the atomic dipole momentum associated with transition |2〉 → |1〉, na is the density of dopant atoms, ε0 is the electric permittivity of the vacuum, and c is the velocity of light in vacuum. c1 and c2 are the probability amplitudes for the level |1〉 and |2〉, respectively. The Bloch equations
dc1 i = c2μ12 A = dt
(2)
dc2 i ⁎ = c1μ12 A − iΔc2 = dt
(3)
describe the time evolution of the probability amplitudes, where Δ is the detuning parameter. It is worth to mention that this set of equations are derived considering the slowly varying envelope and rotating-wave approximation. Furthermore, it governs the propagation dynamics in the regime where the atomic system is coherently driven in the sense that the pulse duration is smaller than the dephasing times of the atomic dipole. To numerically solve Eqs. (1)–(3) we implement an algorithm based on the split-step Fourier and Runge–Kutta method in
Fig. 2. Soliton interaction for two in-phase solitons with q0 ¼4.0; (a) NLSE soliton interaction; (b) SIT soliton interaction; (c) NLSE–SIT soliton interaction.
MATLAB. The initial optical field corresponds to two in-phase NLSE–SIT solitons given by
A(z, τ ) = A 0 [sech(τ − q0) + sech(τ + q0)], Fig. 1. Optical field A interacting with a two-level atoms. The resonant interaction is achieved when the detuning parameter Δ is equal to zero.
(4)
where q0 defines the pulse separation and τ is the normalized time with respect to the initial pulse width T0, τ = t /T0 . A0 is the optical
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Fig. 3. 2D plot of the pulse shape after propagating a distance equivalent to the collision length Lcol. Black solid line corresponds to NLSE solitons and gray solid line corresponds to NLSE–SIT solitons. Dotted line represents the initial pulse. q0 ¼4.0.
Fig. 4. Behavior of the collision length Lcol as a function of the pulse separation q0. Solid line corresponds to the NLSE interaction and the squared points correspond to the SIT–NLSE soliton interaction.
field amplitude which is adjusted to the SIT–NLSE soliton condition together with the atomic system and nonlinear dispersive medium parameters. In the simulations T0 = 200 fs, γ = 4.41 × 10−4 m−1 W−1, and β2 = − 10 ps2 /km . For completeness, in this work we also simulate the case where only the nonlinear dispersive or the atomic system effects are present. The idea is to compare our results for the SIT–NLSE soliton interaction with the results which are known for SIT soliton and NLSE soliton.
3. Results and discussion Initially, we set detuning parameter to zero (Δ ¼0). In this case, it is well-known that the interaction between NLSE solitons gives rise to an attraction between the solitons. The solitons periodically collide with a well defined collision length, which can be analytically predicted [1]. On the other hand, when we consider the propagation of two identical co-propagating SIT solitons, they do not interact. In Fig. 2(a) we show the typical result for the NLSE soliton interaction, μ12 = 0 in Eq. (1). Throughout the presentation
Fig. 5. Soliton interaction for two in-phase solitons in the asymptotic regime with q0 ¼ 4.0; (a) NLSE soliton interaction; (b) SIT soliton interaction; (c) NLSE–SIT soliton interaction.
of our results, the solitons propagation dynamics will be illustrated in a density plot, where the intensity of the optical field is given in terms of the initial normalized intensity of one of the solitons and the propagation distance ξ is the normalized distance with respect to the dispersive length LD, ξ = z /LD . In Fig. 2(b) we show the
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Fig. 6. Detuning effect in the NLSE–SIT soliton interaction with q0 ¼ 3.5; (a) Δ ¼0.005; (b) Δ ¼0.008; (c) Δ ¼0.02; (d) Δ ¼ 0.03.
typical result for the SIT soliton interaction, β2 = 0 and γ = 0 in Eq. (1). When both nonlinear dispersive and atomic system effect are turned on, the SIT–NLSE solitons interact in a new fashion as showed in Fig. 2(c). The dynamics initially tends to be equal to the NLSE solitons, but the solitons do not collide. In order to understand what is happening in this situation, we plot the intensity distribution for the optical field at the propagation distance equivalent to the collision length. The results are presented in Fig. 3. It can be noted that initially the two solitons overlap their fields due to the nonlinear dispersive effects, but the atomic system reshapes the fields in such a way that instead of the soliton fusion (black solid line) a different shape is observed, resembling a broken-up 4π SIT soliton (gray solid line). To test this effect, we have changed the pulse separation q0 for different values. In all cases, the same behavior is observed for the same collision length predicted by the NLSE soliton interaction [1], which is given by π Lcol = 2 LD exp(q0). It is interesting to note that the propagation distance where the broken-up 4π SIT soliton is resembled coincides with the collision length. This result is summarized in Fig. 4, where we plot the theoretical prediction for the collision length as a function the pulse separation (solid line) and the propagation distance where the broken-up 4π SIT soliton is resembled (squared points). This is an indication that the nonlinear dispersive effects
are present in the initial evolution of the SIT–NLSE soliton interaction. To conclude our analysis, we simulate an asymptotic regime by increasing the propagation distance. The results are shown in Fig. 5. As can be seen, in the SIT–NLSE soliton interaction no collision is observed and the propagation dynamics tends to be equal to the SIT interaction. We understand this behavior as a consequence of the constant absorption and re-emission of the optical fields by the atomic system, suppressing the nonlinear dispersive effects, as also observed in the SIT–NLSE soliton cloning [20]. To confirm this statement, we investigate the role of the presence of the atomic system by increasing the value of the detuning parameter Δ. To demonstrate that the atomic system plays an important role in the new propagation dynamics in the soliton interaction, we include the effect of the detuning on the simulations for different values of the detuning parameter, namely, Δ equal to 0.005, 0.008, 0.02, and 0.03. The results are shown in Fig. 6. As we can see, when the detuning parameter is increased the propagation dynamics changes. For the smaller value of Δ, the result looks like the result showed in Fig. 2c. However, by increasing the value of Δ, the role of the atomic system is less relevant. In particular, when Δ ¼0.03, we observe that the result is quite similar to the result showed in Fig. 2a, indicating that the atomic system is not playing any role.
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From these results, we can attribute to the presence of the atomic system the observation of the SIT–NLSE soliton interaction. Therefore, a suppression of the soliton collision effect is observed.
4. Conclusion In summary, we have numerically investigated the interaction of two in-phase SIT–NLSE solitons propagating in a nonlinear dispersive medium in the presence of a dopant. Our results show that the propagation dynamics is quite different for the wellknown NLSE solitons interaction. However, it was possible to identify the presence of the nonlinear dispersive and atomic system effects governing the dynamics. In particular, it is observed that the soliton collision effect disappears and the dynamics is similar to two SIT solitons in the asymptotic limit. We understand that our results give new perspectives on the soliton propagation dynamics. In particular, we have extended the observation of the effects of soliton interaction to the coherent propagation regime. It is worth to mention that distinct nonlinear dispersive as well coherent propagation effect have been studied using SIT–NLSE solitons, for instance soliton cloning effect [16] and formation of matched pulses [19]. Our results also find application on the development of optical communication systems where the soliton attraction effect plays an important role on the development of networks exploiting optical solitons as information carriers.
Acknowledgments The authors thank the financial support from Brazilian Founding Agency FAPERJ (Fundação Carlos Chagas Filho de Apoio à Pesquisa do Estado do Rio de Janeiro).
References [1] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2012. [2] Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, New York, 2003.
[3] L. Allen, J.H. Eberly, Optical Resonance and Two-Level Atoms, Dover Publications, New York, 1987. [4] A.I. Maı˘mistov, E.A. Manykin, Propagation of ultrashort optical pulses in resonant non-linear light guides, Zh. Eksp. Teor. Fiz. 85 (1983) 1177–1181 [Sov. Phys. JETP 58 (1983) 685–687]. [5] M. Nakazawa, E. Yamada, H. Kubota, Coexistence of self-induced transparency soliton and nonlinear Schrödinger soliton, Phys. Rev. Lett. 66 (1991) 2625–2628. [6] M. Nakazawa, E. Yamada, H. Kubota, Coexistence of a self-induced-transparency soliton and a nonlinear Schrödinger soliton in an erbium-doped fiber, Phys. Rev. A 44 (1991) 5973–5987. [7] S.L. McCall, E.L. Hahn, Self-induced transparency by pulsed coherent light, Phys. Rev. Lett. 18 (1967) 908–911. [8] S.L. McCall, E.L. Hahn, Self-induced transparency, Phys. Rev. 183 (1969) 457–485. [9] H.M. Gibbs, R.E. Slusher, Peak amplification and breakup of a coherent optical pulse in a simple atomic absorber, Phys. Rev. Lett. 24 (1970) 638–641. [10] A. Hasegawa, F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett. 23 (1973) 142–144. [11] L.F. Mollenauer, R.H. Stolen, J.P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett. 45 (1980) 1095–1098. [12] D.P. Caetano, S.B. Cavalcanti, J.M. Hickmann, R.A. Kraenkel, A.M. Kamchatnov, E.A. Makarova, Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach, Phys. Rev. E 67 (2003) 046615. [13] G. Vemuri, G.S. Agarwal, K.V. Vasavada, Cloning, dragging, and parametric amplification of solitons in a coherently driven, nonabsorbing system, Phys. Rev. Lett. 79 (1997) 3889–3892. [14] J.H. Eberly, Transmission of dressed fields in three-level media, Quantum Semiclass. Opt. 7 (1995) 373–384. [15] A. Rahman, J.H. Eberly, Theory of shape-preserving short pulses in inhomogeneously broadened three-level media, Phys. Rev. A 58 (1998) 805–808. [16] D.P. Caetano, S.B. Cavalcanti, J.M. Hickmann, Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium, Phys. Rev. E 65 (2002) 036617. [17] E.J.S. Fonseca, S.B. Cavalcanti, J.M. Hickmann, Soliton interaction in a nonlinear waveguide in the presence of resonances, Phys. Rev. E 64 (2001) 016610. [18] E.J.S. Fonseca, S.B. Cavalcanti, J.M. Hickmann, Control of soliton interaction in a coherently excited three-level system embedded in a nonlinear waveguide, J. Opt. Soc. Am. B 19 (2002) 492–497. [19] T.N. Dey, S. Dutta Gupta, G.S. Agarwal, Coupled solitons in rare-earth doped two-mode fiber, Opt. Express 16 (2008) 17441–17450. [20] D.P. Caetano, J.M. Hickmann, Suppressing the frequency chirp in optical solitons using coherent effects, J. Opt. Soc. Am. B 23 (2006) 1655–1659. [21] A. Marini, F. Biancalana, Ultrashort self-induced transparency plasmon solitons, Phys. Rev. Lett. 110 (2013) 243901. [22] J.P. Gordon, Interaction forces among solitons in optical fibers, Opt. Lett. 8 (1983) 596–598. [23] F.M. Mitschke, L.F. Mollenauer, Experimental observation of interaction forces between solitons in optical fibers, Opt. Lett. 12 (1987) 355–357. [24] G.P. Agrawal, Fiber-Optic Communications Systems, John Wiley & Sons, New York, 2002.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Self-induced transparency effect on the two-soliton interaction C.C.D. da Silva, R.V. Moreira, S.A. Garcia, D.P. Caetano n Departamento de Ciências Exatas, Escola de Engenharia Industrial Metalúrgica de Volta Redonda, Universidade Federal Fluminense, Volta Redonda, RJ 27255125, Brazil
art ic l e i nf o
a b s t r a c t
Article history: Received 13 July 2015 Received in revised form 10 August 2015 Accepted 12 August 2015
We report on the numerical investigation of the interaction between two solitons in a doped nonlinear dispersive medium. The dopant is modeled as a two-level atomic system coherently driven to support simultaneous nonlinear Schrödinger equation and self-induced transparency solitons. We investigate the influence of the self-Induced transparency in the case where the two-solitons are in phase. It is found that the periodical soliton collision effect disappears due to the presence of the atomic system. This new phenomenon is understood considering the detuning parameter on the atomic system description. This result can be exploited in multiple pulse propagation where the soliton interactions play an important role. & 2015 Elsevier B.V. All rights reserved.
Keywords: Optical solitons Self-induced transparency Soliton interaction
1. Introduction Solitons are one of the most fascinating phenomena in the nonlinear science and can be found in different research themes, for example, fluid mechanics, superconductivity, Bose–Einstein condensates, plasm physics, and biology. In the context of the interaction between light and matter, solitons can be found in the study of the propagation of light pulses through a nonlinear dispersive medium [1,2] or an ensemble of atomic system [3], as well as in a combination of these two propagation media [4–6]. An interesting scenario to study the dynamics of the solitons propagation is a doped optical fiber, due to the possibility to investigate how well-known effects associated only to the nonlinear dispersive medium or the atomic system can be modified giving rise to new effects. In the coherent propagation regime, where the time duration of the pulses is smaller than the relaxation times of the atomic system, self-induced transparency solitons (SIT solitons) are observed when light pulses resonantly interact with a two-level atomic system [7–9]. To observe this effect, the pulse area should be an integer multiple of 2π. On the other hand, in the nonlinear dispersive regime, the interplay between the nonlinear and dispersive effects allows us to obtain stable solutions of the nonlinear Schrödinger equation, which describes the pulse propagation through an optical fiber, so-called nonlinear Schrödinger equation solitons (NLSE solitons) [10,11]. To observe this effect, a perfect match between the pulse and optical fiber parameters should be n
Corresponding author. E-mail address: [email protected] (D.P. Caetano).
http://dx.doi.org/10.1016/j.optcom.2015.08.036 0030-4018/& 2015 Elsevier B.V. All rights reserved.
achieved. Coexistence solitons (SIT–NLSE solitons) [4–6] arrive from the combination of the above-mentioned effects when light pulses coherently propagate through a doped optical fiber, where the dopant is modeled as a two-level atomic system. In this case, both power and area of the pulse should satisfy the requirement to observe simultaneous SIT and NLSE soliton. Recently, it has been demonstrated that this requirement can be relaxed [12]. After its demonstration, SIT–NLSE solitons have been exploited to investigate generalizations of coherent propagation effects in the nonlinear dispersive regime, for example soliton cloning [13] and matched pulses in three-level atomic systems [14,15]. In the soliton cloning context, besides the observation of the SIT–NLSE soliton cloning [16], it was observed the effect of pulse break-up [17,18]. In the generalization of matched pulses, it was demonstrated a condition to observe the formation of coupled solitons [19]. It was also demonstrated that the SIT–NLSE soliton cloning effect can be applied to suppress the frequency chirp along optical pulses [20]. Recently, self-induced transparency effect has been observed in surface plasmon polariton together with nonlinear and dispersive effects, giving rise to plasmon solitons [21]. For fundamental NLSE solitons, which correspond to the firstorder solutions of the NLSE, a remarkable feature is that the pulse shape remains unchangeable along the propagation. Another feature related to solitons is that the pulse shape also remains unchangeable after collision. These features give to solitons particlelike behavior. In addition, concerning the interaction of two NLSE solitons it has been observed that attraction or repulsion can be observed depending on the relative phase between the pulses [22,23]. This last effect is relevant in the design of optical communications systems [24]. In this work, we investigate the
C.C.D. da Silva et al. / Optics Communications 356 (2015) 426–430
427
interaction of two SIT–NLSE solitons. The idea is to observe how the presence of the atomic system affects the NLSE soliton dynamics, in particular, the fact that solitons collide periodically along the propagation when they possess a zero phase difference. Our results show that the solitons does not collide, inhibiting the soliton collapse effect, and the propagation dynamics in the asymptotic limit is basically dictated by the atomic system. To confirm the role of the atomic system we studied the effect of the detuning. As a result, we observe that by increasing the detuning parameter the propagation dynamics dictated by the nonlinear dispersive medium is recovered. These results add new perspectives for the study about the propagation of SIT–NLSE solitons and its applications as well as new dynamics effects which can be pursued in other nonlinear phenomena.
2. Basic theory Next, we present the basic equations concerning the propagation of two SIT–NLSE solitons through a doped optical fiber and the numerical procedure used to solve the equations. Let A(z, t ) be the optical field describing a stream of two pulses. This optical field coherently interacts with the atomic system as illustrated in Fig. 1. The extend nonlinear Schrödinger equation (NLSE) given by
β ∂ 2A iωna ∂A μ c2⁎c1 = − i 2 2 + iγ |A|2 A + 2 ∂t 2ε0c 12 ∂z
(1)
describes the propagation of the pulse by taking into account the self-phase modulation, group velocity dispersion (GVD), and selfinduced transparency effects. β2 is the GVD coefficient, γ is the Kerr nonlinearity coefficient, μ12 is the atomic dipole momentum associated with transition |2〉 → |1〉, na is the density of dopant atoms, ε0 is the electric permittivity of the vacuum, and c is the velocity of light in vacuum. c1 and c2 are the probability amplitudes for the level |1〉 and |2〉, respectively. The Bloch equations
dc1 i = c2μ12 A = dt
(2)
dc2 i ⁎ = c1μ12 A − iΔc2 = dt
(3)
describe the time evolution of the probability amplitudes, where Δ is the detuning parameter. It is worth to mention that this set of equations are derived considering the slowly varying envelope and rotating-wave approximation. Furthermore, it governs the propagation dynamics in the regime where the atomic system is coherently driven in the sense that the pulse duration is smaller than the dephasing times of the atomic dipole. To numerically solve Eqs. (1)–(3) we implement an algorithm based on the split-step Fourier and Runge–Kutta method in
Fig. 2. Soliton interaction for two in-phase solitons with q0 ¼4.0; (a) NLSE soliton interaction; (b) SIT soliton interaction; (c) NLSE–SIT soliton interaction.
MATLAB. The initial optical field corresponds to two in-phase NLSE–SIT solitons given by
A(z, τ ) = A 0 [sech(τ − q0) + sech(τ + q0)], Fig. 1. Optical field A interacting with a two-level atoms. The resonant interaction is achieved when the detuning parameter Δ is equal to zero.
(4)
where q0 defines the pulse separation and τ is the normalized time with respect to the initial pulse width T0, τ = t /T0 . A0 is the optical
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Fig. 3. 2D plot of the pulse shape after propagating a distance equivalent to the collision length Lcol. Black solid line corresponds to NLSE solitons and gray solid line corresponds to NLSE–SIT solitons. Dotted line represents the initial pulse. q0 ¼4.0.
Fig. 4. Behavior of the collision length Lcol as a function of the pulse separation q0. Solid line corresponds to the NLSE interaction and the squared points correspond to the SIT–NLSE soliton interaction.
field amplitude which is adjusted to the SIT–NLSE soliton condition together with the atomic system and nonlinear dispersive medium parameters. In the simulations T0 = 200 fs, γ = 4.41 × 10−4 m−1 W−1, and β2 = − 10 ps2 /km . For completeness, in this work we also simulate the case where only the nonlinear dispersive or the atomic system effects are present. The idea is to compare our results for the SIT–NLSE soliton interaction with the results which are known for SIT soliton and NLSE soliton.
3. Results and discussion Initially, we set detuning parameter to zero (Δ ¼0). In this case, it is well-known that the interaction between NLSE solitons gives rise to an attraction between the solitons. The solitons periodically collide with a well defined collision length, which can be analytically predicted [1]. On the other hand, when we consider the propagation of two identical co-propagating SIT solitons, they do not interact. In Fig. 2(a) we show the typical result for the NLSE soliton interaction, μ12 = 0 in Eq. (1). Throughout the presentation
Fig. 5. Soliton interaction for two in-phase solitons in the asymptotic regime with q0 ¼ 4.0; (a) NLSE soliton interaction; (b) SIT soliton interaction; (c) NLSE–SIT soliton interaction.
of our results, the solitons propagation dynamics will be illustrated in a density plot, where the intensity of the optical field is given in terms of the initial normalized intensity of one of the solitons and the propagation distance ξ is the normalized distance with respect to the dispersive length LD, ξ = z /LD . In Fig. 2(b) we show the
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Fig. 6. Detuning effect in the NLSE–SIT soliton interaction with q0 ¼ 3.5; (a) Δ ¼0.005; (b) Δ ¼0.008; (c) Δ ¼0.02; (d) Δ ¼ 0.03.
typical result for the SIT soliton interaction, β2 = 0 and γ = 0 in Eq. (1). When both nonlinear dispersive and atomic system effect are turned on, the SIT–NLSE solitons interact in a new fashion as showed in Fig. 2(c). The dynamics initially tends to be equal to the NLSE solitons, but the solitons do not collide. In order to understand what is happening in this situation, we plot the intensity distribution for the optical field at the propagation distance equivalent to the collision length. The results are presented in Fig. 3. It can be noted that initially the two solitons overlap their fields due to the nonlinear dispersive effects, but the atomic system reshapes the fields in such a way that instead of the soliton fusion (black solid line) a different shape is observed, resembling a broken-up 4π SIT soliton (gray solid line). To test this effect, we have changed the pulse separation q0 for different values. In all cases, the same behavior is observed for the same collision length predicted by the NLSE soliton interaction [1], which is given by π Lcol = 2 LD exp(q0). It is interesting to note that the propagation distance where the broken-up 4π SIT soliton is resembled coincides with the collision length. This result is summarized in Fig. 4, where we plot the theoretical prediction for the collision length as a function the pulse separation (solid line) and the propagation distance where the broken-up 4π SIT soliton is resembled (squared points). This is an indication that the nonlinear dispersive effects
are present in the initial evolution of the SIT–NLSE soliton interaction. To conclude our analysis, we simulate an asymptotic regime by increasing the propagation distance. The results are shown in Fig. 5. As can be seen, in the SIT–NLSE soliton interaction no collision is observed and the propagation dynamics tends to be equal to the SIT interaction. We understand this behavior as a consequence of the constant absorption and re-emission of the optical fields by the atomic system, suppressing the nonlinear dispersive effects, as also observed in the SIT–NLSE soliton cloning [20]. To confirm this statement, we investigate the role of the presence of the atomic system by increasing the value of the detuning parameter Δ. To demonstrate that the atomic system plays an important role in the new propagation dynamics in the soliton interaction, we include the effect of the detuning on the simulations for different values of the detuning parameter, namely, Δ equal to 0.005, 0.008, 0.02, and 0.03. The results are shown in Fig. 6. As we can see, when the detuning parameter is increased the propagation dynamics changes. For the smaller value of Δ, the result looks like the result showed in Fig. 2c. However, by increasing the value of Δ, the role of the atomic system is less relevant. In particular, when Δ ¼0.03, we observe that the result is quite similar to the result showed in Fig. 2a, indicating that the atomic system is not playing any role.
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From these results, we can attribute to the presence of the atomic system the observation of the SIT–NLSE soliton interaction. Therefore, a suppression of the soliton collision effect is observed.
4. Conclusion In summary, we have numerically investigated the interaction of two in-phase SIT–NLSE solitons propagating in a nonlinear dispersive medium in the presence of a dopant. Our results show that the propagation dynamics is quite different for the wellknown NLSE solitons interaction. However, it was possible to identify the presence of the nonlinear dispersive and atomic system effects governing the dynamics. In particular, it is observed that the soliton collision effect disappears and the dynamics is similar to two SIT solitons in the asymptotic limit. We understand that our results give new perspectives on the soliton propagation dynamics. In particular, we have extended the observation of the effects of soliton interaction to the coherent propagation regime. It is worth to mention that distinct nonlinear dispersive as well coherent propagation effect have been studied using SIT–NLSE solitons, for instance soliton cloning effect [16] and formation of matched pulses [19]. Our results also find application on the development of optical communication systems where the soliton attraction effect plays an important role on the development of networks exploiting optical solitons as information carriers.
Acknowledgments The authors thank the financial support from Brazilian Founding Agency FAPERJ (Fundação Carlos Chagas Filho de Apoio à Pesquisa do Estado do Rio de Janeiro).
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