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Symmetry-breaking instability in gap soliton
Optics Communications 227 (2003) 295–299 www.elsevier.com/locate/optcom
Symmetry-breaking instability in gap soliton K. Senthilnathan a, K. Porsezian b
b,*
a Department of Physics, Anna University, Chennai 600 025, India Raman School of Physics, Pondicherry University, Pondicherry 605 014, India
Received 16 July 2003; received in revised form 18 September 2003; accepted 18 September 2003
Abstract We derive the existence of bright gap soliton solution for nonlinear-coupled mode equations, which governs the pulse propagation in fiber Bragg grating. We find the occurrence of symmetry-breaking instability in gap soliton for arbitrary values of detuning parameter. We confirm this instability by linear stability analysis, potential well plots and phase-plane diagrams. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Fiber Bragg grating; Gap soliton; Instability
Nonlinear effects in fiber Bragg grating (FBG) continue to attract much attention of many researchers in recent times. Bragg gratings in optical fibers are excellent devices for studying nonlinear phenomena particularly based on the Kerr-nonlinearity. This Kerr-nonlinearity in FBG is combated with a strong effective dispersion induced by resonant reflection of light on the Bragg grating that gives rise to an interesting family of solitons called, in general, Bragg Grating Solitons. They are often referred to as Gap Solitons if their spectrum lies within the photonic band gap. Many research groups [1–5] theoretically predicted the existence of gap soliton in FBG and still the investigation on
*
Corresponding author. Tel.: +91-413-2655991494; fax: 91413-2655255. E-mail address: [email protected] (K. Porsezian).
this exciting phenomenon is going on in multidimensions. These solitons in FBG were experimentally verified and successfully demonstrated, independently, by Eggleton et al. [6] and Taverner et al. [7]. Recent developments in theory as well as experimental investigation of these solitons challenges very interesting applications and devices in the optical fiber communication field. The observation of soliton in FBG also paves the way for many potential applications such as all opticallogic gates [7], pulse compression [8], all-optical switching [9], and limiting [10]. As far as the soliton communication is concerned, the stability of the soliton pulse is very important. In this context, it is also interesting to see the stability of gap soliton in FBG. Based on this fact, for the first time, Sterke [11] investigated the stability of the gap soliton in a nonlinear periodic media. Using linear stability analysis (LSA),
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.09.049
296
K. Senthilnathan, K. Porsezian / Optics Communications 227 (2003) 295–299
he predicted both the stable and unstable regions as a function of intensity of the incoming radiation and the frequency respectively. Recently, Rossi et al. [12] also investigated the stability of optical gap soliton in the frame of generalized Massive Thirring Model (MTM). From the phase-plane diagram they showed the existence of stable and unstable branches of gap soliton. In this letter, we report the occurrence of symmetry-breaking instability (SBI) of gap soliton in FBG. For this purpose, using LSA we obtain the condition for the occurrence of instability in gap soliton. Then, we explain the unequal and quasiperiodic energy transfer in forward and backward propagating modes in FBG in terms of potential well plots and phase-plane diagrams. The forward and backward propagational modes of an optical pulse in a uniform grating are governed by the following nonlinear-coupled mode (NLCM) equations: df ^ i þ df þ jb þ Cs j f j2 þ 2Cx jbj2 f ¼ 0; dz ð1Þ db i þ d^b þ jf þ Cs jbj2 þ 2Cx j f j2 b ¼ 0; dz where f and b are the slowly varying amplitudes of forward and backward propagating modes, respectively. d^ ¼ n0 =cðx0 xB Þ is the detuning parameter, it is a measure of detuning from the Bragg resonance condition. Here, xB ¼
pc 2pc ¼ n0 ^ kB
is the Bragg frequency, where kB is the Bragg freespace wavelength, n is the average refractive index, Cs and Cx are self-phase and cross-phase modulation coefficients. To investigate SBI of gap soliton, we choose the Stokes parameters because they provide useful information about the total energy and energy difference among the forward and backward propagating modes. Using these parameters, we obtain a new set of Stokes parameters from the NLCM equations. The Stokes parameters are defined by [13] A0 ¼ j f j2 þ jbj2 ; A2 ¼ iðfb f bÞ;
A1 ¼ fb þ f b; A3 ¼ j f j2 jbj2 ;
with the constraint A20 ¼ A21 þ A22 þ A23 . We note that these Stokes parameters, for the first time, have been successfully applied to nonlinear directional coupler (NLDC) by Diano, Gregory and Wabnitz [14] and derived the stationary solution in terms of Jacobi elliptic function. Then, de Sterke and Sipe [4] applied the same Stokes parameter formalism to the NLCM equations and also obtained the solution in terms of elliptic function. However, in this letter, we are interested in deriving the soliton like solution from the NLCM equation by applying the same Stokes parameter formalism. To obtain the analytical gap soliton solution from Eq. (1), we convert Eq. (1) in terms of following first-order differential equations dA0 ¼ 2jA2 ; dz dA1 ¼ 2d^A2 þ 3CA0 A2 ; dz ð2Þ dA2 ¼ 2d^A1 2jA0 3CA0 A1 ; dz dA3 ¼ 0: dz Now we turn to investigate the existence of optical soliton of Eq. (1) in nonlinear periodic media under consideration. For this purpose, we construct the anharmonic oscillator equation from Eq. (2) as carried out by Daino et al. [14]. Before embarking into the gap soliton investigation, we discuss the construction of anharmonic oscillator type equation. To derive this equation, first we find the conserved quantity in the form d^A0 þ 34CA20 þ jA1 ¼ C, where C is the constant of integration. With the help of conserved quantity in Eq. (2), we obtain the anharmonic oscillator equation d2 A0 þ aA0 þ bA20 þ 2cA30 ¼ 4d^C; ð3Þ dz2 where a ¼ 2½2d^2 2j2 3CC, b ¼ 9Cd^ and c ¼ 94 C2 : Eq. (3) contains all the physical parameters of the NLCM equations under consideration. On solving the Eq. (3), we obtain 3a A0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : pffiffiffi 2 ð Þz b þ 9ac cosh a þ 1
ð4Þ
The above solution represents the bright solitary wave for the forward and backward propagating
K. Senthilnathan, K. Porsezian / Optics Communications 227 (2003) 295–299
modes in FBG under the influence of Kerr-nonlinearity. At this juncture, it is interesting to note that the bright soliton solution given in Eq. (4) is different from the results reported in the literature. It is of great interest and practical importance to analyze the stability of the above predicted stationary gap soliton. We test the stability of the gap soliton by means of LSA to Eq. (2). There is only one singular point in our dynamical system and it is found to be (0, 0, 0). After perturbing the system, we find the following eigenvalues from the n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio stability matrix, 0; 2 j2 d2 ; 2 j2 d2 . If j > d, then these roots have real values with equal and opposite sign. Hence, we anticipate unstable branch of gap soliton. Similarly, if j < d; in this case the roots have imaginary values. So we expect the stable branch of gap soliton. We also confirm these stability and instability phenomenon through phase-plane diagrams. Now we proceed to investigate the SBI of Eq. (1) from the numerical point of view. To do this, instead of conventional direct integration of Eq. (1), we integrate the anharmonic oscillator equation constructed from Eq. (2). The reason is we have performed the LSA of Eq. (1) by converting it into a set of first order differential equations, to check our analytical predictions of LSA.
297
As has been discussed in the previous paragraph, to predict the SBI in the NLCM equations we integrate the Eq. (3). From Figs. 1 and 2, it is evident that the qualitative aspect of the potential well drastically changes as we increase the nonlinearity parameter and there by it also increases the detuning of the system. In addition to 2D plots, we also provide 3D plots because they clearly illustrate how the shape of the potential changes from single potential to double-well potential. In this connection, we intend to analyze how the energy is shared and transferred between the forward and backward propagating modes in terms of potential well. To investigate the impact of nonlinearity in FBG through the potential well plots, first we discuss only the linear case. At low input power, it is well known that the entire incident light gets reflected in FBG. Hence, the energy is now completely transferred to the backward wave through the process of Bragg reflection. Under this condition, the system obeys the harmonic oscillator equation. Now, by applying the stability condition obtained from LSA to Eq. (3), we find that the system possesses a single well potential. This is clearly seen in Figs. 1(a) and 2(a) where the energy transfer is periodic and complete. From the shape of the potential, we expect a stable behavior.
Fig. 1. Potential well plots: (a) Single well potential when a ¼ 0:47, b ¼ 0 and c ¼ 0. (b) Double well potential when a ¼ 0:47, b ¼ 0:01 and c ¼ 0:1. (c) Asymmetric double well potential when a ¼ 0:5, b ¼ 0:19 and c ¼ 0:35. (d) Asymmetric doule well potential when a ¼ 0.5, b ¼ 0.33 and c ¼ 0.42.
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K. Senthilnathan, K. Porsezian / Optics Communications 227 (2003) 295–299
Fig. 2. 3D Potential well plots: (a) Single well potential when a ¼ 0:47, b ¼ 0 and c ¼ 0. (b) Double well potential when a ¼ 0:47, b ¼ 0:01 and c ¼ 0:1. (c) Asymmetric double well potential when a ¼ 0:5, b ¼ 0:19 and c ¼ 0:35.
Further, at low intensities of the incident light, the particle is initially at rest at the bottom of the quasi-harmonic potential well. When we introduce the small amount of nonlinearity into the system, we find that the system possesses the double well potential as shown in Figs. 1(b) and 2(b). If we increase the nonlinearity, the anharmonicity becomes dominant and it plays an indispensable role in the potential well plots. Due to high intensity, refractive index of the system increases which in turn shifts the central frequency to the lower frequency side i.e., to the upper branch of the dispersion curve. This nonlinearity
induces and increases the detuning value of the system and therefore some of the spectral components of the propagating pulse do not obey the Bragg resonance condition. Hence the light i.e., photons migrate from lower branch to the upper branch of the dispersion curve and causes the transmission of light in FBG. As a result, we have both the transmitted light as well as reflected light. Consequently, under this condition, the energy is shared between the forward and backward propagating modes. This kind of energy transfer is explained in Figs. 1(c) and 2(c). This type of energy transfer is mainly because of the anharmo-
Fig. 3. Phase plane diagrams: (a) For single well potential when a ¼ 1, b ¼ 0 and c ¼ 0. (b) For double well potential when a ¼ 1, b ¼ 0 and c ¼ 0:06. (c) For asymmetric double well potential when a ¼ 1, b ¼ 0:1 and c ¼ 0:12. (d) For asymmetric double well potential when a ¼ 1, b ¼ 0:51 and c ¼ 0:14.
K. Senthilnathan, K. Porsezian / Optics Communications 227 (2003) 295–299
nicity, which breaks the symmetry property of the potential well. More over, the migration of photons from one potential well at A0 ¼ þ1 to the other at A0 ¼ 1 also confirms the transfer of photons from lower branch of the dispersion curve to the upper branch of the dispersion curve in FBG. At this juncture, we point out that Anderson et al. [15] also reported this type of stability and instability in the nonlinear directional coupler (NLDC). Further increase in nonlinearity and detuning parameters, there will be more unequal sharing of energy between the forward and backward propagating modes. This is shown in Fig. 1(d), where the energy is unequally transferred from one potential well to the other. This means that more photons migrate from one potential side to the other. Here the photons takes the longer time to reach the position at A0 ¼ 1, which means that the period is increased. From Fig. 1(d) it should be noted that the photons available in a well where the potential function has minimum at A0 ¼ 1 are more stable when compared to the other well. So the migration of photons from one potential to the other makes more asymmetry in the potential well diagram. From this process, SBI can easily be anticipated. This kind of unequal sharing of energy between the modes can be thought of as switching mechanism. This finds application in nonlinear grating structures as optical switches as pointed out by Winful et al. [16]. In addition to the above potential well discussion, we also plot the phase-plane diagram, which also provides clear information about the energy transfer between the forward and backward propagating modes in FBG. The elliptic and hyperbolic points in the phaseplane diagrams clearly represent the stable and unstable branches of the gap soliton. Fig. 3(d) shows the existence of SBI of gap soliton in FBG. It is also clear that there exist two types of motions like wobbling and decay as shown from the closed and open orbits of the phase-plane diagrams [12]. In conclusion, using the Stokes parameter formalism, we have derived the bright gap soliton solution for the NLCM equations in FBG. Using LSA we obtained the stable and unstable regions
299
of the stationary gap soliton. Further, we explained the energy transfer/sharing between the forward and backward propagating modes in FBG in terms of potential well plots and phase-plane diagrams. From this process, we predicted the existence of SBI of gap soliton in FBG. This approach can also be applied to other interesting physical system like quadratic and quintic nonlinearity, which is unavoidable at high intensity of the pulse. The work is in progress.
Acknowledgements K.P. expresses his thanks to DST, BRNS (DAE) and CSIR Government of India for financial support through research grant. KSN wishes to thank CSIR for providing him SRF.
References [1] W. Chen, D.L. Mills, Phys. Rev. Lett. 58 (1987) 160. [2] D.N. Christodoulides, R.I. Joseph, Phys. Rev. Lett. 62 (1989) 1746. [3] A.B. Aceves, S. Wabnitz, Phys. Lett. A 141 (1989) 37. [4] C.M. de Sterke, J.E. Sipe, in: E. Wolf (Ed.), Progress in Optics XXXIII, Elsevier, Amsterdam, 1994 (Chapter III). [5] K. Senthilnathan, K. Porsezian, P. Malathi, J. Opt. Soc. Am. B 20 (2003) 366. [6] B.J. Eggleton, R.E. Slusher, C.M. de Sterke, P.A. Krug, J.E. Sipe, Phys. Rev. Lett. 76 (1996) 1627. [7] D. Taverner, N.G.R. Broderick, D.J. Richardson, M. Ibsen, R.I. Laming, Opt. Lett. 23 (1998) 259, 328. [8] N.G.R. Broderick, D. Taverner, D.J. Richardson, M. Ibsen, R.I. Laming, Opt. Lett. 22 (1997) 1837. [9] A. Melloni, M. Chinello, M. Martinelli, IEEE Photon. Technol. Lett. 12 (2000) 42. [10] D. Pelinovsky, J. Sears, L. Brzozowski, E.H. Sargent, J. Opt. Soc. Am. B 19 (2002) 43. [11] C.M. de Sterke, Phys. Rev. A 45 (1992) 8252. [12] A.D. Rossi, C. Conti, S. Trillo, Phys. Rev. Lett. 81 (1998) 85. [13] E. Collett, Polarized Light – Concepts and Application, Marcel Decker Inc., New York, 1989. [14] B. Daino, G. Gregori, S. Wabnitz, Appl. Phys. 58 (1985) 4512. [15] D. Anderson, M. Lisak, Phys. Rev. A 32 (1985) 2270. [16] H.G. Winful, J.H. Marburger, E. Gamire, Appl. Phys. Lett. 35 (1979) 379.
Symmetry-breaking instability in gap soliton K. Senthilnathan a, K. Porsezian b
b,*
a Department of Physics, Anna University, Chennai 600 025, India Raman School of Physics, Pondicherry University, Pondicherry 605 014, India
Received 16 July 2003; received in revised form 18 September 2003; accepted 18 September 2003
Abstract We derive the existence of bright gap soliton solution for nonlinear-coupled mode equations, which governs the pulse propagation in fiber Bragg grating. We find the occurrence of symmetry-breaking instability in gap soliton for arbitrary values of detuning parameter. We confirm this instability by linear stability analysis, potential well plots and phase-plane diagrams. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Fiber Bragg grating; Gap soliton; Instability
Nonlinear effects in fiber Bragg grating (FBG) continue to attract much attention of many researchers in recent times. Bragg gratings in optical fibers are excellent devices for studying nonlinear phenomena particularly based on the Kerr-nonlinearity. This Kerr-nonlinearity in FBG is combated with a strong effective dispersion induced by resonant reflection of light on the Bragg grating that gives rise to an interesting family of solitons called, in general, Bragg Grating Solitons. They are often referred to as Gap Solitons if their spectrum lies within the photonic band gap. Many research groups [1–5] theoretically predicted the existence of gap soliton in FBG and still the investigation on
*
Corresponding author. Tel.: +91-413-2655991494; fax: 91413-2655255. E-mail address: [email protected] (K. Porsezian).
this exciting phenomenon is going on in multidimensions. These solitons in FBG were experimentally verified and successfully demonstrated, independently, by Eggleton et al. [6] and Taverner et al. [7]. Recent developments in theory as well as experimental investigation of these solitons challenges very interesting applications and devices in the optical fiber communication field. The observation of soliton in FBG also paves the way for many potential applications such as all opticallogic gates [7], pulse compression [8], all-optical switching [9], and limiting [10]. As far as the soliton communication is concerned, the stability of the soliton pulse is very important. In this context, it is also interesting to see the stability of gap soliton in FBG. Based on this fact, for the first time, Sterke [11] investigated the stability of the gap soliton in a nonlinear periodic media. Using linear stability analysis (LSA),
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.09.049
296
K. Senthilnathan, K. Porsezian / Optics Communications 227 (2003) 295–299
he predicted both the stable and unstable regions as a function of intensity of the incoming radiation and the frequency respectively. Recently, Rossi et al. [12] also investigated the stability of optical gap soliton in the frame of generalized Massive Thirring Model (MTM). From the phase-plane diagram they showed the existence of stable and unstable branches of gap soliton. In this letter, we report the occurrence of symmetry-breaking instability (SBI) of gap soliton in FBG. For this purpose, using LSA we obtain the condition for the occurrence of instability in gap soliton. Then, we explain the unequal and quasiperiodic energy transfer in forward and backward propagating modes in FBG in terms of potential well plots and phase-plane diagrams. The forward and backward propagational modes of an optical pulse in a uniform grating are governed by the following nonlinear-coupled mode (NLCM) equations: df ^ i þ df þ jb þ Cs j f j2 þ 2Cx jbj2 f ¼ 0; dz ð1Þ db i þ d^b þ jf þ Cs jbj2 þ 2Cx j f j2 b ¼ 0; dz where f and b are the slowly varying amplitudes of forward and backward propagating modes, respectively. d^ ¼ n0 =cðx0 xB Þ is the detuning parameter, it is a measure of detuning from the Bragg resonance condition. Here, xB ¼
pc 2pc ¼ n0 ^ kB
is the Bragg frequency, where kB is the Bragg freespace wavelength, n is the average refractive index, Cs and Cx are self-phase and cross-phase modulation coefficients. To investigate SBI of gap soliton, we choose the Stokes parameters because they provide useful information about the total energy and energy difference among the forward and backward propagating modes. Using these parameters, we obtain a new set of Stokes parameters from the NLCM equations. The Stokes parameters are defined by [13] A0 ¼ j f j2 þ jbj2 ; A2 ¼ iðfb f bÞ;
A1 ¼ fb þ f b; A3 ¼ j f j2 jbj2 ;
with the constraint A20 ¼ A21 þ A22 þ A23 . We note that these Stokes parameters, for the first time, have been successfully applied to nonlinear directional coupler (NLDC) by Diano, Gregory and Wabnitz [14] and derived the stationary solution in terms of Jacobi elliptic function. Then, de Sterke and Sipe [4] applied the same Stokes parameter formalism to the NLCM equations and also obtained the solution in terms of elliptic function. However, in this letter, we are interested in deriving the soliton like solution from the NLCM equation by applying the same Stokes parameter formalism. To obtain the analytical gap soliton solution from Eq. (1), we convert Eq. (1) in terms of following first-order differential equations dA0 ¼ 2jA2 ; dz dA1 ¼ 2d^A2 þ 3CA0 A2 ; dz ð2Þ dA2 ¼ 2d^A1 2jA0 3CA0 A1 ; dz dA3 ¼ 0: dz Now we turn to investigate the existence of optical soliton of Eq. (1) in nonlinear periodic media under consideration. For this purpose, we construct the anharmonic oscillator equation from Eq. (2) as carried out by Daino et al. [14]. Before embarking into the gap soliton investigation, we discuss the construction of anharmonic oscillator type equation. To derive this equation, first we find the conserved quantity in the form d^A0 þ 34CA20 þ jA1 ¼ C, where C is the constant of integration. With the help of conserved quantity in Eq. (2), we obtain the anharmonic oscillator equation d2 A0 þ aA0 þ bA20 þ 2cA30 ¼ 4d^C; ð3Þ dz2 where a ¼ 2½2d^2 2j2 3CC, b ¼ 9Cd^ and c ¼ 94 C2 : Eq. (3) contains all the physical parameters of the NLCM equations under consideration. On solving the Eq. (3), we obtain 3a A0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : pffiffiffi 2 ð Þz b þ 9ac cosh a þ 1
ð4Þ
The above solution represents the bright solitary wave for the forward and backward propagating
K. Senthilnathan, K. Porsezian / Optics Communications 227 (2003) 295–299
modes in FBG under the influence of Kerr-nonlinearity. At this juncture, it is interesting to note that the bright soliton solution given in Eq. (4) is different from the results reported in the literature. It is of great interest and practical importance to analyze the stability of the above predicted stationary gap soliton. We test the stability of the gap soliton by means of LSA to Eq. (2). There is only one singular point in our dynamical system and it is found to be (0, 0, 0). After perturbing the system, we find the following eigenvalues from the n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio stability matrix, 0; 2 j2 d2 ; 2 j2 d2 . If j > d, then these roots have real values with equal and opposite sign. Hence, we anticipate unstable branch of gap soliton. Similarly, if j < d; in this case the roots have imaginary values. So we expect the stable branch of gap soliton. We also confirm these stability and instability phenomenon through phase-plane diagrams. Now we proceed to investigate the SBI of Eq. (1) from the numerical point of view. To do this, instead of conventional direct integration of Eq. (1), we integrate the anharmonic oscillator equation constructed from Eq. (2). The reason is we have performed the LSA of Eq. (1) by converting it into a set of first order differential equations, to check our analytical predictions of LSA.
297
As has been discussed in the previous paragraph, to predict the SBI in the NLCM equations we integrate the Eq. (3). From Figs. 1 and 2, it is evident that the qualitative aspect of the potential well drastically changes as we increase the nonlinearity parameter and there by it also increases the detuning of the system. In addition to 2D plots, we also provide 3D plots because they clearly illustrate how the shape of the potential changes from single potential to double-well potential. In this connection, we intend to analyze how the energy is shared and transferred between the forward and backward propagating modes in terms of potential well. To investigate the impact of nonlinearity in FBG through the potential well plots, first we discuss only the linear case. At low input power, it is well known that the entire incident light gets reflected in FBG. Hence, the energy is now completely transferred to the backward wave through the process of Bragg reflection. Under this condition, the system obeys the harmonic oscillator equation. Now, by applying the stability condition obtained from LSA to Eq. (3), we find that the system possesses a single well potential. This is clearly seen in Figs. 1(a) and 2(a) where the energy transfer is periodic and complete. From the shape of the potential, we expect a stable behavior.
Fig. 1. Potential well plots: (a) Single well potential when a ¼ 0:47, b ¼ 0 and c ¼ 0. (b) Double well potential when a ¼ 0:47, b ¼ 0:01 and c ¼ 0:1. (c) Asymmetric double well potential when a ¼ 0:5, b ¼ 0:19 and c ¼ 0:35. (d) Asymmetric doule well potential when a ¼ 0.5, b ¼ 0.33 and c ¼ 0.42.
298
K. Senthilnathan, K. Porsezian / Optics Communications 227 (2003) 295–299
Fig. 2. 3D Potential well plots: (a) Single well potential when a ¼ 0:47, b ¼ 0 and c ¼ 0. (b) Double well potential when a ¼ 0:47, b ¼ 0:01 and c ¼ 0:1. (c) Asymmetric double well potential when a ¼ 0:5, b ¼ 0:19 and c ¼ 0:35.
Further, at low intensities of the incident light, the particle is initially at rest at the bottom of the quasi-harmonic potential well. When we introduce the small amount of nonlinearity into the system, we find that the system possesses the double well potential as shown in Figs. 1(b) and 2(b). If we increase the nonlinearity, the anharmonicity becomes dominant and it plays an indispensable role in the potential well plots. Due to high intensity, refractive index of the system increases which in turn shifts the central frequency to the lower frequency side i.e., to the upper branch of the dispersion curve. This nonlinearity
induces and increases the detuning value of the system and therefore some of the spectral components of the propagating pulse do not obey the Bragg resonance condition. Hence the light i.e., photons migrate from lower branch to the upper branch of the dispersion curve and causes the transmission of light in FBG. As a result, we have both the transmitted light as well as reflected light. Consequently, under this condition, the energy is shared between the forward and backward propagating modes. This kind of energy transfer is explained in Figs. 1(c) and 2(c). This type of energy transfer is mainly because of the anharmo-
Fig. 3. Phase plane diagrams: (a) For single well potential when a ¼ 1, b ¼ 0 and c ¼ 0. (b) For double well potential when a ¼ 1, b ¼ 0 and c ¼ 0:06. (c) For asymmetric double well potential when a ¼ 1, b ¼ 0:1 and c ¼ 0:12. (d) For asymmetric double well potential when a ¼ 1, b ¼ 0:51 and c ¼ 0:14.
K. Senthilnathan, K. Porsezian / Optics Communications 227 (2003) 295–299
nicity, which breaks the symmetry property of the potential well. More over, the migration of photons from one potential well at A0 ¼ þ1 to the other at A0 ¼ 1 also confirms the transfer of photons from lower branch of the dispersion curve to the upper branch of the dispersion curve in FBG. At this juncture, we point out that Anderson et al. [15] also reported this type of stability and instability in the nonlinear directional coupler (NLDC). Further increase in nonlinearity and detuning parameters, there will be more unequal sharing of energy between the forward and backward propagating modes. This is shown in Fig. 1(d), where the energy is unequally transferred from one potential well to the other. This means that more photons migrate from one potential side to the other. Here the photons takes the longer time to reach the position at A0 ¼ 1, which means that the period is increased. From Fig. 1(d) it should be noted that the photons available in a well where the potential function has minimum at A0 ¼ 1 are more stable when compared to the other well. So the migration of photons from one potential to the other makes more asymmetry in the potential well diagram. From this process, SBI can easily be anticipated. This kind of unequal sharing of energy between the modes can be thought of as switching mechanism. This finds application in nonlinear grating structures as optical switches as pointed out by Winful et al. [16]. In addition to the above potential well discussion, we also plot the phase-plane diagram, which also provides clear information about the energy transfer between the forward and backward propagating modes in FBG. The elliptic and hyperbolic points in the phaseplane diagrams clearly represent the stable and unstable branches of the gap soliton. Fig. 3(d) shows the existence of SBI of gap soliton in FBG. It is also clear that there exist two types of motions like wobbling and decay as shown from the closed and open orbits of the phase-plane diagrams [12]. In conclusion, using the Stokes parameter formalism, we have derived the bright gap soliton solution for the NLCM equations in FBG. Using LSA we obtained the stable and unstable regions
299
of the stationary gap soliton. Further, we explained the energy transfer/sharing between the forward and backward propagating modes in FBG in terms of potential well plots and phase-plane diagrams. From this process, we predicted the existence of SBI of gap soliton in FBG. This approach can also be applied to other interesting physical system like quadratic and quintic nonlinearity, which is unavoidable at high intensity of the pulse. The work is in progress.
Acknowledgements K.P. expresses his thanks to DST, BRNS (DAE) and CSIR Government of India for financial support through research grant. KSN wishes to thank CSIR for providing him SRF.
References [1] W. Chen, D.L. Mills, Phys. Rev. Lett. 58 (1987) 160. [2] D.N. Christodoulides, R.I. Joseph, Phys. Rev. Lett. 62 (1989) 1746. [3] A.B. Aceves, S. Wabnitz, Phys. Lett. A 141 (1989) 37. [4] C.M. de Sterke, J.E. Sipe, in: E. Wolf (Ed.), Progress in Optics XXXIII, Elsevier, Amsterdam, 1994 (Chapter III). [5] K. Senthilnathan, K. Porsezian, P. Malathi, J. Opt. Soc. Am. B 20 (2003) 366. [6] B.J. Eggleton, R.E. Slusher, C.M. de Sterke, P.A. Krug, J.E. Sipe, Phys. Rev. Lett. 76 (1996) 1627. [7] D. Taverner, N.G.R. Broderick, D.J. Richardson, M. Ibsen, R.I. Laming, Opt. Lett. 23 (1998) 259, 328. [8] N.G.R. Broderick, D. Taverner, D.J. Richardson, M. Ibsen, R.I. Laming, Opt. Lett. 22 (1997) 1837. [9] A. Melloni, M. Chinello, M. Martinelli, IEEE Photon. Technol. Lett. 12 (2000) 42. [10] D. Pelinovsky, J. Sears, L. Brzozowski, E.H. Sargent, J. Opt. Soc. Am. B 19 (2002) 43. [11] C.M. de Sterke, Phys. Rev. A 45 (1992) 8252. [12] A.D. Rossi, C. Conti, S. Trillo, Phys. Rev. Lett. 81 (1998) 85. [13] E. Collett, Polarized Light – Concepts and Application, Marcel Decker Inc., New York, 1989. [14] B. Daino, G. Gregori, S. Wabnitz, Appl. Phys. 58 (1985) 4512. [15] D. Anderson, M. Lisak, Phys. Rev. A 32 (1985) 2270. [16] H.G. Winful, J.H. Marburger, E. Gamire, Appl. Phys. Lett. 35 (1979) 379.