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Raman perturbation and surface core soliton in hollow photonic crystal fiber
Physics Letters A 372 (2008) 2391–2399 www.elsevier.com/locate/pla
Raman perturbation and surface core soliton in hollow photonic crystal fiber Mousumi Ballav, A. Roy Chowdhury ∗ High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata 700032, India Received 11 July 2007; received in revised form 18 November 2007; accepted 26 November 2007 Available online 5 December 2007 Communicated by A.R. Bishop
Abstract Effect of Raman scattering and higher order dispersion are investigated in case of surface core or gap solitons, known to exist in hollow core photonic crystal fibers. The necessary conditions for the existence of the soliton are analyzed in the parameter plane. It is observed that these conditions are compatible with the famous Vakhitov–Kolokolov (VK) criterion. In the later part the system of partial differential equations are solved by ETDRK method from which the shapes of the pulses are investigated as it travels along the z direction. Both the terms, intrapulse Raman scattering and higher order dispersion have important effect on the propagation characteristics © 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Lc; 42.55.-f; 42.60.-v; 42.65.Sf Keywords: Gap soliton; Hollow photonic crystal fiber; Raman perturbation
1. Introduction Optical soliton [1] have received considerable attention for a decade in the context of optical data transmission in long distances. The formation of optical soliton is a mechanism of the balance between group velocity dispersion due to the frequency dependent refractive index of the fiber and self-phase modulation due to nonlinear Kerr effect. In 1989, Aceves et al. [2] obtained soliton solution which describes Bragg-resonant wave propagation in nonlinear periodic medium. In this Letter, it is observed that mean wavelength of the soliton lies in the centre of the forbidden gap. Now a days it has been shown that optical gap solitons exist in a nonlinear Bragg grating [3,4] fiber and a nonlinear photonic crystal fiber (PCF) [5,6] where the void has a crystal like structure running parallel to the fiber’s axis. PCF is extremely useful for the delivery of powerful light signals with an energy up to 100 pJ over a distance of 200 m. Such PCF structures share the propagation properties of photonic crystals, based on the existence of the frequency gap with the transmission, as well as the properties of conventional optical fibers, due to the presence of a defect in the structure acting as PCF core. Light confinement is usually restricted to the core of a PCF. This can be attributed to nonlinear effects such as self trapping which leads to the formation of optical solitons. These periodically modulated nonlinear systems can also support self-trapped localized pulses or beams in the form of gap solitons [7]. A novel property of gap solitons is that unlike ordinary solitons which requires self-focusing nonlinearity, they can exist in both self-focusing and self-defocussing media. The recent surge of interest in theoretical and experimental studies of optical parametric process in PCFs is related to their high nonlinearities, achieved by reduction in core size [8] and to the control of dispersion by suitable design of the fiber core and photonic crystal cladding [9]. In Ref. [10] authors studied systems of equations for gap solitons in one-dimensional and quasi-onedimensional nonlinear photorefractive crystal. * Corresponding author. Tel.: +91 033 2416 3708; fax: +91 033 2413 7121, +91 033 2414 6414.
E-mail address: [email protected] (A.R. Chowdhury). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.11.052
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One of the most recent findings in the domain of optical soliton is the observation of gap like soliton in hollow photonic crystal fibers [11,12], i.e. photonic crystal fibers with air-filled core. In hollow photonic optical fiber one can make various choices of gases to fill the core and also have different geometrical shapes to control dispersion and nonlinear properties, thereby enhancing the propagation characteristics [13]. In Ref. [14], author studied nonlinear regimes of pulse propagation and proximity of the avoided crossing without considering Raman scattering and higher order dispersion. But higher order nonlinear effects (intrapulse Raman scattering, third order dispersion) have to be considered in the case of femtosecond pulses and the pulse evolution is no longer periodic. In Ref. [15] experimentally it was observe that within a hollow core photonic band gap fiber large third order dispersion limits soliton propagation. The simplest model that is now in vougue is a linear coupling between the wave propagating in the surface and that in the hollow core. Here in this Letter we introduce the equations for the surface core solitons to take into account of the intrapulse Raman scattering and investigate the existence of gap like soliton and its stability. Lastly these set of equations are numerically integrated with the help of ETDRK, so estimate its parameters as it propagates along the axis. 2. Formulation To formulate the coupled mode system, we designate the soliton propagating in the surface as As and that in the core as Ac . The equation is ∂|As |2 ∂ 3 As i ∂ ∂As iβ2 ∂ 2 As ∂As 2 2 − T + γ − iA = iΓ |A | A + | A A −v − |A (1) c s s s s R s ∂z ∂t 2 ∂t 2 ω0 ∂t ∂t ∂t 3 for the surface wave, while that one propagating in the core is governed by ∂Ac ∂Ac +v − iAs = 0. ∂z ∂t Here t and z are the reduced time and propagating distance. Stationary fundamental surface core solitons of Eqs. (1) and (2) are looked for in the form As = (UR + iUI )eikz Ac = (VR + iVI )eikz
(2)
(3)
whence we get
2 d 3 UR dUI dUR β2 dUI 1 2 2 2 dUR 2 dUR γ −v − kUI + VI + Γ UR + UI UI + + + 2UR + 2UR UI UR + UI ∂t 2 dt 2 ω0 dt dt dt ∂t 3
dUR dUI − 2TR UI UR (4) + UI = 0, dt dt dUI dUR dUI dUI 1 2 β2 d 2 UR d 3 UI − v + kUR − VR − Γ UR2 + UI2 UR − − + 2UR UI + 2UI2 UR + UI2 γ 3 2 dt 2 dt ω0 dt dt dt dt
dUR dUI − 2TR UR UR (5) + UI = 0, dt dt dVR − k VI + UI = 0, v (6) dt dVI + kVR − UR = 0. v (7) dt Eqs. (4) to (6) are solved in the form of boundary value problem. Before that it is interesting to note that the dispersion relation obtained from the linearized version of Eqs. (1) and (2) can be written as
β2 2 β2 2 2 3 3 ω + γ ω − 1 + vω vω + ω + γ ω = 0. k +k (8) 2 2 The name surface core soliton or gap soliton can only be justified if one can show that the energy associated with the solution of Eqs. (4) to (7) lies between the two branches of the curve given by Eq. (8). To compute the corresponding energy a well-posed boundary value problem associated to Eqs. (4) to (7) are to be solved. The analysis of the associated linearized problem gives an idea about the stability zones of the soliton. The complex third order and coupled linear ode in the set of Eqs. (4) to (7) can be recast as coupled set of eight-dimensional system in the following variables T UR , UR1 = dt UR , UR2 = dtt UR , UI , U= (9) UI 3 = dt UI , UI 4 = dtt UI , VR , VI
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where T denotes the transposed. To this end Eqs. (4) to (7) are linearized around a stationary wave solution U0 . Substituting the ansatz U = U0 + δU and on linearization with respect to δU yields an eigenvalue problem for λ LδU = λδU where L stands for ⎛ 0 1 0 ⎜ 0 ⎜ ⎜ L31 L32 ⎜ 0 ⎜ 0 L=⎜ 0 ⎜ 0 ⎜ ⎜ L61 L62 ⎝ 0 0 L81 0
(10)
0 1 0 0 0 L63 0 0
0 0 L34 0 0 L64 L74 0
0 0 L35 1 0 L65 0 0
0 0 L36 0 1 0 0 0
0 0 0 0 0 L67 0 L87
⎞ 0 0 ⎟ ⎟ L38 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎠ L78 0
(11)
with 2Γ c Γ 1 2 3UR0 + UI20 − 2TR UR0 UI 0 ; L32 = − UR0 UI 0 ; γ γ γ ω0
k 2Γ UR0 UI 0 β2 Γ 2 2 2 L34 = − L35 = − − TR UI 0 ; U + 3UI 0 ; L36 = − ; γ γ R0 γ ω0 2γ
1 2Γ UR0 UI 0 β2 2 L38 = − ; + TR UR0 L62 = − ; ; L63 = γ γ ω0 2γ 2Γ c Γ 1 2 L64 = UR0 + 3UI20 + 2TR UR0 UI 0 ; L65 = − UR0 UI 0 ; γ γ γ ω0 1 k k 1 1 L74 = ; L81 = − ; L78 = − ; L87 = . L67 = ; γ c c c c L31 = −
On simplification the eigenvalue equation is written as λ8 + b6 λ6 + b5 λ5 + b4 λ4 + b3 λ3 + b2 λ2 + b1 λ + b0 = 0
(12)
where the coefficients bi (i = 1, 2, . . . , 6) are complicated combinations of Lij . For example b6 = −(L32 + L78 L87 + L63 L36 + L65 ), b5 = −(L63 L35 + L62 L36 + L64 + L31 ) and so on. Stability corresponds to purely imaginary value of λ and complex value of λ with negative real part. Instability is detected by the opposite events. A necessary and sufficient condition for the eigenvalue λ to be with negative real part is supplied by the Routh–Hurwitz criterion which reads Δ7 , Δ5 , Δ3 > 0,
b 3 , b7 , b0 > 0
where Δk (k = 3, 5, 7) are the Hurwitz determinant. The actual numerical computation proceeds by evolution of U0 by the help of boundary value solution and then the determinants are evaluated for various parameter values and the common region are sought for. In Fig. 1 we summarize the output of the extensive numerical calculation using Eq. (12) aimed to identify the domain of the existence and stability of the fundamental surface core soliton in the plane (k, v). It is observed that the stability domain is asymmetric with respect to the group velocity between two modes. Assuming the parameters values β2 = 1.0 and ω0 = 1.0, it is observed that the stability zone and also the zone of existence of the soliton decrease with increase of TR . However the zone of no soliton is greatly influenced with different values of γ . The whole scenario is explicitly exhibited in Figs. 1(a) and (b). It becomes evident that the zone of no soliton increases with the increase of γ or the third order dispersion. Since we have not taken into account any dispersion management, we have normalized β2 and ω0 to one. It is seen that with the change of these values the region gets altered, but the physical essence remains the same. Bright solitary waves are now calculated as stationary solution with appropriate boundary condition. We have found that families of surface core solitons exist at different values of material parameters β2 , Γ , TR with different wave amplitudes and group velocity asymmetry. In Fig. 2, we have plotted the amplitude and the real and imaginary part of stationary solutions for two representative
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(a)
(b)
Fig. 1. (a) Numerically found domains of the existence and stability of surface core soliton for TR = 1.0 and γ = 0.1. The values of other parameters are β2 = 1.0, ω0 = 1.0. (b) Domain of the existence and stability of surface core soliton for TR = 1.0 and γ = 0.4, i.e. 3rd order dispersion increases no soliton region and stability region. Values of β2 , ω0 are the same as in (a).
Fig. 2. Amplitudes of core and surface solitons as a function of transverse coordinate when there is no intrapulse Raman scattering term (TR = 0) in the material property of hollow photonic crystal fiber. The parameters are v = 1.0, γ = 0.3, β2 = 1.0 and Γ = 1.0.
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Fig. 3. Amplitudes of core and surface solitons as a function of transverse coordinate when TR = 0.1. There is a temporal shifting of core and surface solitons. Values of other parameters are same as in Fig. 2.
solitons with TR = 0. If we include the effect of intrapulse Raman scattering with finite values of TR , it is observed that local amplitude gets shifted on either side of t = 0. These are displayed in Figs. 3 and 4 for two different set of values of third order dispersion and TR . To confirm the robustness of the solitons, we have verified that surface core solitons with different group velocity exists where the initial inputs are slightly changed. The shape of the surface core solitons depends strongly on group velocity. We now try to confirm that these type solitons can actually be called as gap solitons. A conventional way to represent the solution is to exhibit its energy dependence on the wave number k at a fixed value of the group velocity v. Now the total energy is ∞ E=
|As |2 + |Ac |2 dt.
−∞
In Fig. 5 we display the curve E = E(k) for the family of soliton solutions obtained by the boundary value problem. In this figure, the solid curves are two branches of the dispersion relation given in Eq. (8) for v = 1.779, TR = 0.1, β2 = 1.0 and γ = 0.2. The bold dashed curves show the dependence E = E(k) for the soliton family. We observe that this curve lies in between the two branches defined by the dispersion relation. So that it can be called a gap soliton or a surface core soliton. From the dependence E = E(k), for the family of the stationary solitons one can predict their stability on the basis of the Vakhitov–Kolokolov (VK) [16,17] criterion dE dk > 0, which is a necessary stability condition, securing the absence of instability eigenmodes with real eigenvalues. It being the most useful criterion for stability analysis of optical solitons. Fig. 5 illustrates that band gap is filled by the family of soliton and it always satisfies the VK criterion, i.e. the family of soliton is stable in direct simulation. In the next phase of our computation we have solved the original coupled pde (Eqs. (1) and (2)) with the help of ETDRK4 scheme. Exponential time differencing fourth order Runge–Kutta (ETDRK) is a numerical scheme designed for solving partial differential equations, where it is possible to split the problem into its linear and a nonlinear part. In the present situation the system for surface core solitons can be written as y˙ = My + N(y, ξ )
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Fig. 4. Amplitudes of core and surface solitons as a function of transverse coordinate for different values of TR = 0.12 and group velocity v = 1.79. Values of parameters are β2 = 1.0, γ = 0.22.
Fig. 5. Dispersion curves for cubic nonlinear media and there is a band gap between the curves. Dotted line represents nonlinear wave number versus energy of surface core solitons for group velocity v = 1.779 and TR = 0.1.
with y(ξn−1 ) = yn−1 . Integrating above equation formally over a single step length h, it is found h yn+1 = e
Mh
yn + e
Mh
e−Mζ N y(ξn + ζ ), ξn + ζ dζ.
0
There are different ETDRK schemes for approximating the integral on the right-hand side. We have used framework of fourth order ETD of Runge–Kutta type proposed by Cox and Matthews [18] to solve the equations for surface core solitons in Matlab.
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Fig. 6. Intensity of the surface and core soliton showing stable propagation up to a certain distance perturbed by intrapulse Raman scattering.
Fig. 7. Evolution of the surface and core solitons.
Figs. 6 and 7 display a typical stable propagation of surface core soliton supported by intrapulse Raman scattering. From these figures it is observed that not only surface soliton changes its temporal position (i.e. linear shift of central frequency) due to intrapulse Raman scattering, but also mutual effect shifts the pulse position of core soliton during propagation. To characterize the output pulse we have calculated the following parameters associated with it: (a) The intensity distribution: |As,c (z, t)|2 ; 2 −∞ |As,c (z, t)| dt
IN = ∞
(b) The root mean square temporal width: 1/2 σ = x 2 − x2 where
x
p
∞
∞ = −∞
x p |As,c (z, t)|2 dt
2 −∞ |As,c (z, t)| dt
.
The corresponding results for the width and intensity of the core and surface soliton is shown in Fig. 8. Due to the effect of intrapulse Raman scattering, the width of surface and core solitons changes with the propagation distance. 3. Conclusion The concept of a nonlinear periodic structure in which refractive index depends on the local intensity of the electromagnetic wave has been introduced in 1979 [19]. This periodic structure supports gap solitons. The stationary gap soliton would be promising for applications such as optical buffers and delay lines. Winful et al. [20] showed that an intense pump pulse, detuned far from Bragg resonance of a nonlinear periodic structure of Raman active medium can excite a gap soliton. This Raman gap soliton is stable and remains trapped inside the grating.
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Fig. 8. The local soliton pulse width calculated as function of pulse propagation length. (a) and (c) display an example of pulse compression of surface and core solitons. (b) and (d) display temporal intensity envelope of surface and core soliton respectively.
In our above analysis we have tried to visualize the effect of Raman scattering and higher order dispersion on the stability of surface core (gap) solitons—now a days very important in the light of photonic cystal fiber (nonlinear photonic band gap structure). It is observed that the stability region is decreased due to the intrapulse Raman scattering term TR . The time evolution of a single solitary pulse as obtained with the help of ETDRK shows clearly the Raman shift of central frequency taking place both for the surface and core soliton. Finally we have extracted the pulse parameters from the time series data, which shows that it gets compressed as it propagates. The dispersion along with the self-phase modulation that arises from the nonlinear index can result in pulse compression. It may be added that one should take into account a suitable dispersion map instead of constant dispersion to control the pulse. Also it will be interesting to investigate the effect of periodic pumping as in case of usual solitons. We will be coming back to these problems in near future. Acknowledgement One of the authors (M.B.) is grateful to CSIR (Government of India) for J.R.F. in a project which was run in collaboration with CGCRI Kolkata. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23 (1973) 142. A.B. Aceves, S. Wabnitz, Phys. Lett. A 141 (1989) 37. C.M. de Sterke, B.J. Eggleton, Phys. Rev. E 59 (1999) 1267. A.B. Aceves, G. Fibich, B. Ilan, Physica D 189 (2004) 277. A. Ferrando, M. Zacarés, P.F. de Cordoba, D. Binosi, J.A. Monsoriu, Opt. Exp. 11 (2003) 452. J.R. Salgueiro, Y.S. Kivshar, D.E. Pelinovsky, V. Simon, H. Michinl, nlin.PS/050133. Yu.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego, 2003, p. 591. V. Finazzi, T.M. Monro, D.J. Richardson, J. Opt. Soc. Am. B 20 (2003) 1427. W.H. Reeves, D.V. Skryabin, F. Biancalana, J.C. Knight, P.St.J. Russell, F.G. Omentto, A. Efimov, A.J. Taylor, Nature 424 (2003) 511. B.A. Malomed, T. Mayteevarunyoo, E.A. Ostrovskaya, Y.S. Kivshar, Phys. Rev. E 71 (2005) 056616.
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[11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
D.G. Ouzounov, F.R. Ahmad, D. Müler, N. Venkataraman, M.T. Gallagher, M.G. Thomas, J. Silcox, K.W. Koch, A.L. Gaeta, Science 301 (2003) 1702. A.D. Bessonov, A.M. Zheltikov, Phys. Rev. E 73 (2006) 066618. F. Luan, J. Knight, P. Russell, S. Campbell, D. Xiao, D. Reid, B. Mangan, D. Williams, P. Roberts, Opt. Exp. 12 (2005) 835. D.V. Skryabin, Opt. Exp. 12 (2004) 4841. D.G. Ouzounov, A.L. Gaeta, N. Venkataraman, M.T. Gallagher, K.W. Koch, Conference on lasers and electro-optics, 2005. N.G. Vakhitov, A.A. Kolokolov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16 (1973) 10120 (in Russian). N.G. Vakhitov, A.A. Kolokolov, Radiophys. Quantum Electron. 16 (1973) 783 (English translation). S.M. Cox, P.C. Matthews, J. Comput. Phys. 176 (2002) 430. H.G. Winful, J.M. Marburger, E. Garmire, Appl. Phys. Lett. 35 (1979) 376. H.G. Winful, V. Perlin, Phys. Rev. Lett. 84 (2000) 3586.
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Raman perturbation and surface core soliton in hollow photonic crystal fiber Mousumi Ballav, A. Roy Chowdhury ∗ High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata 700032, India Received 11 July 2007; received in revised form 18 November 2007; accepted 26 November 2007 Available online 5 December 2007 Communicated by A.R. Bishop
Abstract Effect of Raman scattering and higher order dispersion are investigated in case of surface core or gap solitons, known to exist in hollow core photonic crystal fibers. The necessary conditions for the existence of the soliton are analyzed in the parameter plane. It is observed that these conditions are compatible with the famous Vakhitov–Kolokolov (VK) criterion. In the later part the system of partial differential equations are solved by ETDRK method from which the shapes of the pulses are investigated as it travels along the z direction. Both the terms, intrapulse Raman scattering and higher order dispersion have important effect on the propagation characteristics © 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Lc; 42.55.-f; 42.60.-v; 42.65.Sf Keywords: Gap soliton; Hollow photonic crystal fiber; Raman perturbation
1. Introduction Optical soliton [1] have received considerable attention for a decade in the context of optical data transmission in long distances. The formation of optical soliton is a mechanism of the balance between group velocity dispersion due to the frequency dependent refractive index of the fiber and self-phase modulation due to nonlinear Kerr effect. In 1989, Aceves et al. [2] obtained soliton solution which describes Bragg-resonant wave propagation in nonlinear periodic medium. In this Letter, it is observed that mean wavelength of the soliton lies in the centre of the forbidden gap. Now a days it has been shown that optical gap solitons exist in a nonlinear Bragg grating [3,4] fiber and a nonlinear photonic crystal fiber (PCF) [5,6] where the void has a crystal like structure running parallel to the fiber’s axis. PCF is extremely useful for the delivery of powerful light signals with an energy up to 100 pJ over a distance of 200 m. Such PCF structures share the propagation properties of photonic crystals, based on the existence of the frequency gap with the transmission, as well as the properties of conventional optical fibers, due to the presence of a defect in the structure acting as PCF core. Light confinement is usually restricted to the core of a PCF. This can be attributed to nonlinear effects such as self trapping which leads to the formation of optical solitons. These periodically modulated nonlinear systems can also support self-trapped localized pulses or beams in the form of gap solitons [7]. A novel property of gap solitons is that unlike ordinary solitons which requires self-focusing nonlinearity, they can exist in both self-focusing and self-defocussing media. The recent surge of interest in theoretical and experimental studies of optical parametric process in PCFs is related to their high nonlinearities, achieved by reduction in core size [8] and to the control of dispersion by suitable design of the fiber core and photonic crystal cladding [9]. In Ref. [10] authors studied systems of equations for gap solitons in one-dimensional and quasi-onedimensional nonlinear photorefractive crystal. * Corresponding author. Tel.: +91 033 2416 3708; fax: +91 033 2413 7121, +91 033 2414 6414.
E-mail address: [email protected] (A.R. Chowdhury). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.11.052
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One of the most recent findings in the domain of optical soliton is the observation of gap like soliton in hollow photonic crystal fibers [11,12], i.e. photonic crystal fibers with air-filled core. In hollow photonic optical fiber one can make various choices of gases to fill the core and also have different geometrical shapes to control dispersion and nonlinear properties, thereby enhancing the propagation characteristics [13]. In Ref. [14], author studied nonlinear regimes of pulse propagation and proximity of the avoided crossing without considering Raman scattering and higher order dispersion. But higher order nonlinear effects (intrapulse Raman scattering, third order dispersion) have to be considered in the case of femtosecond pulses and the pulse evolution is no longer periodic. In Ref. [15] experimentally it was observe that within a hollow core photonic band gap fiber large third order dispersion limits soliton propagation. The simplest model that is now in vougue is a linear coupling between the wave propagating in the surface and that in the hollow core. Here in this Letter we introduce the equations for the surface core solitons to take into account of the intrapulse Raman scattering and investigate the existence of gap like soliton and its stability. Lastly these set of equations are numerically integrated with the help of ETDRK, so estimate its parameters as it propagates along the axis. 2. Formulation To formulate the coupled mode system, we designate the soliton propagating in the surface as As and that in the core as Ac . The equation is ∂|As |2 ∂ 3 As i ∂ ∂As iβ2 ∂ 2 As ∂As 2 2 − T + γ − iA = iΓ |A | A + | A A −v − |A (1) c s s s s R s ∂z ∂t 2 ∂t 2 ω0 ∂t ∂t ∂t 3 for the surface wave, while that one propagating in the core is governed by ∂Ac ∂Ac +v − iAs = 0. ∂z ∂t Here t and z are the reduced time and propagating distance. Stationary fundamental surface core solitons of Eqs. (1) and (2) are looked for in the form As = (UR + iUI )eikz Ac = (VR + iVI )eikz
(2)
(3)
whence we get
2 d 3 UR dUI dUR β2 dUI 1 2 2 2 dUR 2 dUR γ −v − kUI + VI + Γ UR + UI UI + + + 2UR + 2UR UI UR + UI ∂t 2 dt 2 ω0 dt dt dt ∂t 3
dUR dUI − 2TR UI UR (4) + UI = 0, dt dt dUI dUR dUI dUI 1 2 β2 d 2 UR d 3 UI − v + kUR − VR − Γ UR2 + UI2 UR − − + 2UR UI + 2UI2 UR + UI2 γ 3 2 dt 2 dt ω0 dt dt dt dt
dUR dUI − 2TR UR UR (5) + UI = 0, dt dt dVR − k VI + UI = 0, v (6) dt dVI + kVR − UR = 0. v (7) dt Eqs. (4) to (6) are solved in the form of boundary value problem. Before that it is interesting to note that the dispersion relation obtained from the linearized version of Eqs. (1) and (2) can be written as
β2 2 β2 2 2 3 3 ω + γ ω − 1 + vω vω + ω + γ ω = 0. k +k (8) 2 2 The name surface core soliton or gap soliton can only be justified if one can show that the energy associated with the solution of Eqs. (4) to (7) lies between the two branches of the curve given by Eq. (8). To compute the corresponding energy a well-posed boundary value problem associated to Eqs. (4) to (7) are to be solved. The analysis of the associated linearized problem gives an idea about the stability zones of the soliton. The complex third order and coupled linear ode in the set of Eqs. (4) to (7) can be recast as coupled set of eight-dimensional system in the following variables T UR , UR1 = dt UR , UR2 = dtt UR , UI , U= (9) UI 3 = dt UI , UI 4 = dtt UI , VR , VI
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where T denotes the transposed. To this end Eqs. (4) to (7) are linearized around a stationary wave solution U0 . Substituting the ansatz U = U0 + δU and on linearization with respect to δU yields an eigenvalue problem for λ LδU = λδU where L stands for ⎛ 0 1 0 ⎜ 0 ⎜ ⎜ L31 L32 ⎜ 0 ⎜ 0 L=⎜ 0 ⎜ 0 ⎜ ⎜ L61 L62 ⎝ 0 0 L81 0
(10)
0 1 0 0 0 L63 0 0
0 0 L34 0 0 L64 L74 0
0 0 L35 1 0 L65 0 0
0 0 L36 0 1 0 0 0
0 0 0 0 0 L67 0 L87
⎞ 0 0 ⎟ ⎟ L38 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎠ L78 0
(11)
with 2Γ c Γ 1 2 3UR0 + UI20 − 2TR UR0 UI 0 ; L32 = − UR0 UI 0 ; γ γ γ ω0
k 2Γ UR0 UI 0 β2 Γ 2 2 2 L34 = − L35 = − − TR UI 0 ; U + 3UI 0 ; L36 = − ; γ γ R0 γ ω0 2γ
1 2Γ UR0 UI 0 β2 2 L38 = − ; + TR UR0 L62 = − ; ; L63 = γ γ ω0 2γ 2Γ c Γ 1 2 L64 = UR0 + 3UI20 + 2TR UR0 UI 0 ; L65 = − UR0 UI 0 ; γ γ γ ω0 1 k k 1 1 L74 = ; L81 = − ; L78 = − ; L87 = . L67 = ; γ c c c c L31 = −
On simplification the eigenvalue equation is written as λ8 + b6 λ6 + b5 λ5 + b4 λ4 + b3 λ3 + b2 λ2 + b1 λ + b0 = 0
(12)
where the coefficients bi (i = 1, 2, . . . , 6) are complicated combinations of Lij . For example b6 = −(L32 + L78 L87 + L63 L36 + L65 ), b5 = −(L63 L35 + L62 L36 + L64 + L31 ) and so on. Stability corresponds to purely imaginary value of λ and complex value of λ with negative real part. Instability is detected by the opposite events. A necessary and sufficient condition for the eigenvalue λ to be with negative real part is supplied by the Routh–Hurwitz criterion which reads Δ7 , Δ5 , Δ3 > 0,
b 3 , b7 , b0 > 0
where Δk (k = 3, 5, 7) are the Hurwitz determinant. The actual numerical computation proceeds by evolution of U0 by the help of boundary value solution and then the determinants are evaluated for various parameter values and the common region are sought for. In Fig. 1 we summarize the output of the extensive numerical calculation using Eq. (12) aimed to identify the domain of the existence and stability of the fundamental surface core soliton in the plane (k, v). It is observed that the stability domain is asymmetric with respect to the group velocity between two modes. Assuming the parameters values β2 = 1.0 and ω0 = 1.0, it is observed that the stability zone and also the zone of existence of the soliton decrease with increase of TR . However the zone of no soliton is greatly influenced with different values of γ . The whole scenario is explicitly exhibited in Figs. 1(a) and (b). It becomes evident that the zone of no soliton increases with the increase of γ or the third order dispersion. Since we have not taken into account any dispersion management, we have normalized β2 and ω0 to one. It is seen that with the change of these values the region gets altered, but the physical essence remains the same. Bright solitary waves are now calculated as stationary solution with appropriate boundary condition. We have found that families of surface core solitons exist at different values of material parameters β2 , Γ , TR with different wave amplitudes and group velocity asymmetry. In Fig. 2, we have plotted the amplitude and the real and imaginary part of stationary solutions for two representative
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(a)
(b)
Fig. 1. (a) Numerically found domains of the existence and stability of surface core soliton for TR = 1.0 and γ = 0.1. The values of other parameters are β2 = 1.0, ω0 = 1.0. (b) Domain of the existence and stability of surface core soliton for TR = 1.0 and γ = 0.4, i.e. 3rd order dispersion increases no soliton region and stability region. Values of β2 , ω0 are the same as in (a).
Fig. 2. Amplitudes of core and surface solitons as a function of transverse coordinate when there is no intrapulse Raman scattering term (TR = 0) in the material property of hollow photonic crystal fiber. The parameters are v = 1.0, γ = 0.3, β2 = 1.0 and Γ = 1.0.
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Fig. 3. Amplitudes of core and surface solitons as a function of transverse coordinate when TR = 0.1. There is a temporal shifting of core and surface solitons. Values of other parameters are same as in Fig. 2.
solitons with TR = 0. If we include the effect of intrapulse Raman scattering with finite values of TR , it is observed that local amplitude gets shifted on either side of t = 0. These are displayed in Figs. 3 and 4 for two different set of values of third order dispersion and TR . To confirm the robustness of the solitons, we have verified that surface core solitons with different group velocity exists where the initial inputs are slightly changed. The shape of the surface core solitons depends strongly on group velocity. We now try to confirm that these type solitons can actually be called as gap solitons. A conventional way to represent the solution is to exhibit its energy dependence on the wave number k at a fixed value of the group velocity v. Now the total energy is ∞ E=
|As |2 + |Ac |2 dt.
−∞
In Fig. 5 we display the curve E = E(k) for the family of soliton solutions obtained by the boundary value problem. In this figure, the solid curves are two branches of the dispersion relation given in Eq. (8) for v = 1.779, TR = 0.1, β2 = 1.0 and γ = 0.2. The bold dashed curves show the dependence E = E(k) for the soliton family. We observe that this curve lies in between the two branches defined by the dispersion relation. So that it can be called a gap soliton or a surface core soliton. From the dependence E = E(k), for the family of the stationary solitons one can predict their stability on the basis of the Vakhitov–Kolokolov (VK) [16,17] criterion dE dk > 0, which is a necessary stability condition, securing the absence of instability eigenmodes with real eigenvalues. It being the most useful criterion for stability analysis of optical solitons. Fig. 5 illustrates that band gap is filled by the family of soliton and it always satisfies the VK criterion, i.e. the family of soliton is stable in direct simulation. In the next phase of our computation we have solved the original coupled pde (Eqs. (1) and (2)) with the help of ETDRK4 scheme. Exponential time differencing fourth order Runge–Kutta (ETDRK) is a numerical scheme designed for solving partial differential equations, where it is possible to split the problem into its linear and a nonlinear part. In the present situation the system for surface core solitons can be written as y˙ = My + N(y, ξ )
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Fig. 4. Amplitudes of core and surface solitons as a function of transverse coordinate for different values of TR = 0.12 and group velocity v = 1.79. Values of parameters are β2 = 1.0, γ = 0.22.
Fig. 5. Dispersion curves for cubic nonlinear media and there is a band gap between the curves. Dotted line represents nonlinear wave number versus energy of surface core solitons for group velocity v = 1.779 and TR = 0.1.
with y(ξn−1 ) = yn−1 . Integrating above equation formally over a single step length h, it is found h yn+1 = e
Mh
yn + e
Mh
e−Mζ N y(ξn + ζ ), ξn + ζ dζ.
0
There are different ETDRK schemes for approximating the integral on the right-hand side. We have used framework of fourth order ETD of Runge–Kutta type proposed by Cox and Matthews [18] to solve the equations for surface core solitons in Matlab.
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Fig. 6. Intensity of the surface and core soliton showing stable propagation up to a certain distance perturbed by intrapulse Raman scattering.
Fig. 7. Evolution of the surface and core solitons.
Figs. 6 and 7 display a typical stable propagation of surface core soliton supported by intrapulse Raman scattering. From these figures it is observed that not only surface soliton changes its temporal position (i.e. linear shift of central frequency) due to intrapulse Raman scattering, but also mutual effect shifts the pulse position of core soliton during propagation. To characterize the output pulse we have calculated the following parameters associated with it: (a) The intensity distribution: |As,c (z, t)|2 ; 2 −∞ |As,c (z, t)| dt
IN = ∞
(b) The root mean square temporal width: 1/2 σ = x 2 − x2 where
x
p
∞
∞ = −∞
x p |As,c (z, t)|2 dt
2 −∞ |As,c (z, t)| dt
.
The corresponding results for the width and intensity of the core and surface soliton is shown in Fig. 8. Due to the effect of intrapulse Raman scattering, the width of surface and core solitons changes with the propagation distance. 3. Conclusion The concept of a nonlinear periodic structure in which refractive index depends on the local intensity of the electromagnetic wave has been introduced in 1979 [19]. This periodic structure supports gap solitons. The stationary gap soliton would be promising for applications such as optical buffers and delay lines. Winful et al. [20] showed that an intense pump pulse, detuned far from Bragg resonance of a nonlinear periodic structure of Raman active medium can excite a gap soliton. This Raman gap soliton is stable and remains trapped inside the grating.
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Fig. 8. The local soliton pulse width calculated as function of pulse propagation length. (a) and (c) display an example of pulse compression of surface and core solitons. (b) and (d) display temporal intensity envelope of surface and core soliton respectively.
In our above analysis we have tried to visualize the effect of Raman scattering and higher order dispersion on the stability of surface core (gap) solitons—now a days very important in the light of photonic cystal fiber (nonlinear photonic band gap structure). It is observed that the stability region is decreased due to the intrapulse Raman scattering term TR . The time evolution of a single solitary pulse as obtained with the help of ETDRK shows clearly the Raman shift of central frequency taking place both for the surface and core soliton. Finally we have extracted the pulse parameters from the time series data, which shows that it gets compressed as it propagates. The dispersion along with the self-phase modulation that arises from the nonlinear index can result in pulse compression. It may be added that one should take into account a suitable dispersion map instead of constant dispersion to control the pulse. Also it will be interesting to investigate the effect of periodic pumping as in case of usual solitons. We will be coming back to these problems in near future. Acknowledgement One of the authors (M.B.) is grateful to CSIR (Government of India) for J.R.F. in a project which was run in collaboration with CGCRI Kolkata. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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