- Email: [email protected]
Vector soliton switching in a fiber nonlinear directional coupler
Optics Communications 284 (2011) 186–190
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Vector soliton switching in a fiber nonlinear directional coupler Amarendra K. Sarma Department of Physics, Indian Institute of Technology Guwahati 781039, India
a r t i c l e
i n f o
Article history: Received 28 October 2008 Received in revised form 1 September 2010 Accepted 1 September 2010 Keywords: Soliton switching Nonlinear directional coupler Vector soliton
a b s t r a c t This paper reports a detailed numerical study of soliton switching in a high as well as low birefringent nonlinear coupler. It is shown that by controlling the polarization angle one can have nearly 100% transmission with excellent switching characteristics. It is shown that soliton remains stable during its propagation inside the coupler. However it is observed that high birefringent coupler exhibits relatively better soliton stability. We show that the coupler could be used as a soliton switch even at an input peak power less than the critical power, the power at which 50–50 power sharing takes place between the two cores, just by a judicious choice of the polarization angle. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Optical couplers provide a means for controlling light by light. Intensity and amplitude behavior of linear couplers has been investigated by many authors [1]. Substantial progress in controlling light beams has been achieved after nonlinear waveguides with both linear and nonlinear coupling have been taken into account [2]. Nonlinear couplers provide excellent possibilities in constructing switching and memory elements for all-optical devices. In fact the nonlinear directional couplers (NLDC) are nowadays considered to be one of the vital building blocks of all-optical communication systems and signal processing devices. Optical fibers and certain organic polymers with high third order nonlinearities may be used for the construction of couplers based on Kerr effect. The nonlinear directional couplers are interesting as they exchange energy periodically between the guides like linear ones for low intensities while they trap the energy in the guide into which it has been launched initially for high intensities. Nonlinear directional couplers with cores identical in all respects have been studied extensively in the context of all-optical soliton switching after the pioneering work of Jensen [3] and Trillo, Wabnitz, Wright and Stegeman [4] owing to their use in multitude of fiber-optic devices which require splitting of an optical field into two coherent but physically separated parts [5–17]. Jensen showed that one can switch a continuous signal from one core to the other by varying the input power of the signal. The idea when applied to pulse switching led to pulse distortion and breakup, resulting in inefficient switching. Trillo, Wabnitz, Wright and Stegeman showed
that pulse break up could be avoided, if one used soliton pulse as a signal. Recently, nonlinear directional couplers with dissimilar cores have attracted a considerable attention as several new effects can occur in them [18–22].The study of NLDC is no longer confined to the conventional silica based optical fiber coupler, it has recently been extended to AlGaAs nanowire [16] and lead silicate based holey fiber coupler also [9]. Most of the works relating to soliton switching reported in literatures is carried on NLDC made of polarization maintaining fibers. Interesting situations occurs when a fiber coupler exhibits large or low birefringence. In this work we report a detailed numerical study of the effect of large and low birefringence on soliton self-switching in an NLDC. When birefringence of the optical fiber is taken into consideration it may be appropriate to call the soliton as vector soliton. It may be noted that T. I. Lakoba et al. [23] have already carried out a detailed theoretical study on solitons in nonlinear fiber couplers with two orthogonal polarizations, but not in the context of soliton self-switching. In this work we are trying to address exactly this issue. We show that the coupler could be used as a soliton switch even at an input peak power less than the critical power, the power at which 50–50 power sharing takes place between the two cores, just by a judicious choice of the polarization angle. This has the advantage of reducing the required switching energy in a coupler. It is worthwhile to mention that recently vector soliton switching in a nonlinear amplifying loop mirror have been reported [24].
2. The model
E-mail address: [email protected]. 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.09.001
We consider a symmetric nonlinear coupler with either very high or low birefringence. It requires four coupled nonlinear partial
A.K. Sarma / Optics Communications 284 (2011) 186–190
187
differential equations to describe the pulse dynamics in such a coupler [25]: 2 ∂A1 ∂A iβ ∂ A1 2 2 = iκA2 + iγ jA1 j + χjB1 j A1 + β1x 1 + 2 2 ∂T 2 ∂z ∂T ∂B1 ∂B iβ ∂2 B1 2 2 = iκB2 + iγ jB1 j + χjA1 j B1 + β1y 1 + 2 2 ∂T 2 ∂z ∂T ∂A2 ∂A iβ ∂2 A2 2 2 = iκA1 + iγ jA2 j + χjB2 j A2 + β1x 2 + 2 2 ∂T 2 ∂z ∂T ∂B2 ∂B iβ ∂2 B2 2 2 = iκB1 + iγ jB2 j + χjA2 j B2 + β1y 2 + 2 2 2 ∂T ∂z ∂T
ð1Þ
where Aj and Bj with j = 1, 2, refers to the slowly varying linearly polarized components of pulse envelopes in the j-th core. β 1x = 1/ vgx and β 1y = 1/ vg y, where vg x and vg y are group velocities for the two polarization components. β2 and γ are the well known group velocity dispersion (GVD) and nonlinear parameter respectively. In the anomalous GVD regime, β2 b 0 and this is our domain of interest in this work. κ is the zeroth order coupling coefficient between the two cores. χ is the cross-phase modulation parameter. For the high birefringence case χ = 2/3 while χ = 2 for the low birefringence one. Now considering a coordinate system moving with the ˜ where β ˜ = β1x + β1y = 2, average group velocity 1 = β, and introduc ˜ = T0 ; ing the normalized units as follows: ξ = jβ2 jz = T02 ; τ = T−z β pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi A1 = U1 P0 ; B1 = U2 P0 ; A2 = U3 P0 ; B2 = U4 P0 ;N 2 = γ P 0 T 20 / |β2| ; ui = NUi with i = 1,2, 3, 4, the set of Eq. (1) can now be expressed as a set of normalized coupled equations in the following: ∂u1 ∂u i ∂ u1 2 2 = iκ0 u3 + i ju1 j + χju2 j u1 +g 1− 2 2 ∂ξ ∂τ ∂τ ∂u2 ∂u2 i ∂2 u2 2 2 = iκ0 u4 + i ju2 j + χju1 j u1 −g − 2 2 ∂τ ∂ξ ∂τ ∂u3 ∂u i ∂2 u3 2 2 = iκ0 u1 + i ju3 j + χju4 j u3 +g 3− 2 ∂τ2 ∂ξ ∂τ ∂u4 ∂u i ∂2 u4 2 2 = iκ0 u2 + i ju4 j + χju3 j u4 −g 4 − 2 2 ∂τ ∂ξ ∂τ
Fig. 1. (color online) plot of the transmission coefficient as a function of the normalized peak power for θ = 0, 30 and 45 (in degrees) with χ = 2/3.
conditions correspond to the case when a soliton of a given pulse duration is launched into the first core while the second core is kept empty [7]. The transmission coefficient T1(2) representing the fractional output energy in core 1 (2) after one coupling length, is calculated numerically according to the formula in the following: h
∞
∫
2
T1ð2Þ =
∞
−∞
i 2 2 ju1ð3Þ j + ju2ð4Þ j dτ
∫ ju1 j2 + ju2 j2 + ju3 j2 + ju4 j2 dτ
ð4Þ
−∞
ð2Þ
where g = T0(β1x − β1y) /2|β2| is the normalized group velocity mismatch, let us call it as the ‘detuning parameter', and κ0 = κT20 / |β2| is the normalized coupling coefficient. For low birefringence coupler g = 0. The above system of coupled nonlinear Schrodinger equations is the basic system of equations in this work.
In this work, we adhere to the usual notion of switching in which a signal launched into the input core emerges from the same core at the output, after traversing one coupling length inside the coupler (i.e. the bar state). A parameter of importance which describes the soliton stability inside the coupler is the so called RMS pulse width σ defined as follows [25]: h i1 2 2 2 σ = 〈T 〉−〈T〉
ð5Þ
∞
∞
−∞
−∞
where 〈T m 〉 = ∫ T m juj2 dT = ∫ juj2 dT
3. Results and discussions The system of four coupled Eq. (2) are hard to solve analytically, so we solve them numerically by the fast Fourier transform method for the linear dispersive part and by the fourth-order Runge-Kutta method, with auto-control of the step size for a given accuracy of the results, for the nonlinear part. For a detailed discussion on numerical methods to solve the NLSE readers are referred to Ref. [25]. The initial conditions for our numerical integration are as follows: pffiffiffiffiffi P0 cos θ sechðτÞ pffiffiffiffiffi u2 = P0 sinθ sechðτÞ u1 =
ð3Þ
u3 = 0 u4 = 0 where P0 is the normalized input peak power of the soliton. The angle θ defines linear polarization of the incident pulse measured from the slow axis. Let us call it the ‘polarization angle’, which refers to the relative strengths of the partial pulses in each of the two polarizations. The initial
Fig. 2. (color online) plot of the transmission coefficient as a function of the polarization angle θ with χ = 2/3 for two different values of normalized peak powers.
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Fig. 3. (color online) Temporal profiles of |u1|2, |u2|2, |u3|2 and |u4|2 after one coupling length of the coupler with χ = 2/3.
We would firstly discuss the case of high birefringent coupler for which χ = 2/3. In order to see the effect of the input peak power on the transmission of the coupler, in Fig. 1 we plot the transmission coefficient T as function of the normalized peak power for three different polarization angles taking the detuning parameter g =0.15. In this work the angle is measured in degrees. It can be seen that the coupler exhibits better switching characteristics if the polarization angle deviates from zero. The role of polarization angle on the transmission of the coupler can be seen more clearly from Fig. 2 where we plot the transmission coefficient T as function of the polarization angle θ for two different normalized peak powers P0 = 1.5and 0.5 taking the detuning parameter g to be 0.15. We observe that when the input peak power is
above the critical power of switching, as in the case of P0 = 1.5, the transmission is not critically dependent on the polarization angle. However a slight increase in the transmission is observed with increase in θ. Interesting results occur for input peak power below the critical power of switching. As could be seen from Fig. 2, for P0 = 0.5, which is three times smaller than the case considered above, the transmission in the first core increases from a mere 30% to more than 90% with increase in the polarization angle. Hence by manipulating the polarization angle one may achieve soliton switching at a much reduced input peak power. Now, in order to have an idea about the spatio-temporal evolution of the pulses inside the coupler, in Fig. 3 we plot the temporal profiles of the pulses after the propagation of one coupling length inside the coupler for θ = 80, P0 = 1 and g = 0.15. It is quite evident from the scale of the plots that almost all the power is confined to the |u2|mode while a very small amount of power is leaking to |u3|and |u4|mode. The energy in the second core is equally distributed between the modes, where they take the form of small radiation, which may be attributed to the fact that in the second core both the modes see the same propagation constant
Fig. 4. (color online) RMS pulse width variation of the pulse in |u2| mode with normalized distance with χ = 2/3.
Fig. 5. Plot of the transmission coefficient in core 1 vs. the detuning parameter ‘g’ for P0 = 1 and θ = 80.
Our numerical study is presented for the following typical parameters: κ0 = 0.1, T0 = 10ps and β2 = − 20ps2 / km. The length of the coupler is assumed to be one coupling length, i.e. LC = π /2κ0 [6]. 3.1. Case I: high birefringence
A.K. Sarma / Optics Communications 284 (2011) 186–190
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Fig. 6. (color online) plot of the transmission coefficient as a function of the polarization angle θ with χ = 2 for P0 = 1.
Fig. 8. (color online) RMS pulse width variation of the pulse in |u1| (line with ‘*’) and |u3| (line with ‘triangle’) mode with normalized distance with χ = 2.
while it is not so in the first core due to the presence of perturbations. |u2| mode is supporting a stable soliton pulse inside the coupler. This is more evident from Fig. 4 where the RMS pulse width variation of |u2| with normalized distance is depicted. One could see that the variation in the RMS pulse width, fluctuating between (nearly) 0.91 and 0.98, is very small. Hence we may conclude that stable soliton propagation is supported in the given range of propagation distance. However it may be noted that the soliton is getting shifted slightly in the temporal domain along the negative time axis. This happens due to the presence of detuning. Finally, in order to see the role of the detuning parameter on the transmission characteristics of the highly birefringent coupler, in Fig. 5 we plot the variation of the transmission as a function of the detuning parameter ‘g’ with θ = 80 and P0 = 1. We find that the transmission drops only very slightly from the case without detuning (i.e. g = 0). Hence we may conclude that the switching characteristic of the coupler is not significantly affected by the presence of detuning. One obvious effect of detuning is to shift the soliton along the temporal axis during its propagation; higher the value of detuning, higher would be the shift.
3.2. Case II: low birefringence Low birefringence case for which χ = 2 and g = 0, can be studied in a similar way. We find that for a given polarization angle the coupler shows similar transmission characteristics with that of the high birefringent one with the variation of the input peak power. The only notable difference is that, the low birefringent coupler has a comparatively larger critical power of switching as compared to the high birefringent one. Fig. 6 depicts the variation of the transmission characteristics of the coupler with the polarization angle with P0 = 1. We observe that by a judicious choice of the polarization angle one could increase the transmission significantly from a mere 30% to more than 90% over a wide range of polarization angles. In Fig. 7 we plot the temporal profiles of the pulses at the output end of the low birefringence coupler for P0 = 1 and θ = 30. It can be seen clearly that maximum of the power is confined in |u1| mode and the solitonic character is maintained, which is again more evident by Fig. 8 where we plot the RMS pulse width variation of |u1| (the line with ‘*’) and |u3| (the line with ‘triangle’) mode withpropagation distance. It can be clearly seen that the pulse in the |u1| mode is a soliton while |u3| is
Fig. 7. (color online) temporal profiles of |u1|2, |u2|2, |u3|2 and |u4|2 after one coupling length of the coupler.
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a non-solitonic radiation. We observe almost identical RMS pulse width variation for |u2| mode with that of |u1| mode, confirming it's nearly solitonic nature. Our study reveals that the RMS pulse width variation of the soliton in |u2| mode in high birefringent coupler is negligibly small compared to its low birefringent counterpart. Hence we may conclude that high birefringent coupler exhibits better soliton stability than the low birefringent one during propagation inside the coupler. Moreover one interesting difference from that of the high birefringence one is that, here the soliton is not shifted in the temporal domain because of the absence of detuning. 4. Conclusions In this work we have carried out a detailed numerical study of soliton switching in high as well as low birefringent coupler. The transmission characteristics of the coupler could be manipulated by varying the polarization angle. The coupler could be used as a soliton switch even at an input peak power less than the critical power of switching, just by a judicious choice of the polarization angle. The soliton is relatively more stable during its propagation inside the high birefringent coupler. Acknowledgement The author would like to thank the reviewers for their constructive comments which helped in improving the presentation of the work.
This research work is supported by the Department of Science and Technology, Government of India by Grant no. 100/IFD/7523/ 2008-09. References [1] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, J. Wiley, New York, 1991. [2] N. Finlayson, W.C. Banyai, E.M. Wright, C.T. Seaton, G.I. Stegeman, T.J. Cullen, C.N. Ironside, Appl. Phys. Lett. 53 (1988) 1144. [3] S.M. Jensen, IEEE J. Quantum Electron. 18 (1982) 1580. [4] S. Trillo, S. Wabnitz, E.M. Wright, G.I. Stegeman, Opt. Lett. 13 (1988) 672. [5] S. Trillo, S. Wabnitz, Opt. Lett. 16 (1991) 1. [6] P.M. Ramos, C.R. Paiva, IEEE J. Quantum Electron. 35 (1999) 983. [7] A. Kumar, A.K. Sarma, Opt. Commun. 234 (2004) 427. [8] A.K. Sarma, A. Kumar, Opt. Eng. 46 (11) (2007) 115005-1. [9] A.K. Sarma, Jpn. J. Appl. Phys. 47 (Pt.1) (2008) 5493. [10] A.K. Sarma, A. Kumar, Pramana-J. Phys. 69 (2007) 575. [11] Y. Wang, W. Wang, J. Lightwave Technol. 24 (2006) 1041. [12] A. Kumar, A.K. Sarma, Jpn. J. Appl. Phys. 44 (2005) 8498. [13] I.M. Skinner, G.D. Peng, B.A. Malomed, P.L. Chu, Opt. Commun. 113 (1995) 493. [14] B.A. Umarov, F.Kh. Abdullaev, M.K.B. Wahiddin, Opt. Commun. 162 (1999) 340. [15] B.A. Malomed, I.M. Skinner, R.S. Tasgal, Opt. Commun. 139 (1997) 247. [16] A.K. Sarma, Appl. Opt. 48 (2009) 5067. [17] I.M. Uzunov, R. Muschall, M. Gölles, Y.S. Kivshar, B.A. Malomed, F. Lederer, Phys. Rev. E 51 (1995) 2527. [18] P. Shum, M. Liu, IEEE Photonics Technol. Lett. 14 (2002) 1106. [19] B.A. Malomed, Phys. Rev. E 51 (1995) R864. [20] B.A. Malomed, I.M. Skinner, P.L. Chu, G.D. Peng, Phys. Rev. E 53 (1996) 4084. [21] D.J. Kaup, T.I. Lakoba, B.A. Malomed, J. Opt. Soc. Am. B 14 (1997) 1199. [22] D.J. Kaup, B.A. Malomed, J. Opt. Soc. Am. B 15 (1998) 2838. [23] T.I. Lakoba, D.J. Kaup, B.A. Malomed, Phys. Rev. E 55 (1997) 6107. [24] R. Gomez-Alcala, A. Dengra, Opt. Lett. 31 (2006) 3137. [25] G.P. Agrawal, Nonlinear Fiber Optics, 4th ed., Academic Press, New Work, 2007.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Vector soliton switching in a fiber nonlinear directional coupler Amarendra K. Sarma Department of Physics, Indian Institute of Technology Guwahati 781039, India
a r t i c l e
i n f o
Article history: Received 28 October 2008 Received in revised form 1 September 2010 Accepted 1 September 2010 Keywords: Soliton switching Nonlinear directional coupler Vector soliton
a b s t r a c t This paper reports a detailed numerical study of soliton switching in a high as well as low birefringent nonlinear coupler. It is shown that by controlling the polarization angle one can have nearly 100% transmission with excellent switching characteristics. It is shown that soliton remains stable during its propagation inside the coupler. However it is observed that high birefringent coupler exhibits relatively better soliton stability. We show that the coupler could be used as a soliton switch even at an input peak power less than the critical power, the power at which 50–50 power sharing takes place between the two cores, just by a judicious choice of the polarization angle. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Optical couplers provide a means for controlling light by light. Intensity and amplitude behavior of linear couplers has been investigated by many authors [1]. Substantial progress in controlling light beams has been achieved after nonlinear waveguides with both linear and nonlinear coupling have been taken into account [2]. Nonlinear couplers provide excellent possibilities in constructing switching and memory elements for all-optical devices. In fact the nonlinear directional couplers (NLDC) are nowadays considered to be one of the vital building blocks of all-optical communication systems and signal processing devices. Optical fibers and certain organic polymers with high third order nonlinearities may be used for the construction of couplers based on Kerr effect. The nonlinear directional couplers are interesting as they exchange energy periodically between the guides like linear ones for low intensities while they trap the energy in the guide into which it has been launched initially for high intensities. Nonlinear directional couplers with cores identical in all respects have been studied extensively in the context of all-optical soliton switching after the pioneering work of Jensen [3] and Trillo, Wabnitz, Wright and Stegeman [4] owing to their use in multitude of fiber-optic devices which require splitting of an optical field into two coherent but physically separated parts [5–17]. Jensen showed that one can switch a continuous signal from one core to the other by varying the input power of the signal. The idea when applied to pulse switching led to pulse distortion and breakup, resulting in inefficient switching. Trillo, Wabnitz, Wright and Stegeman showed
that pulse break up could be avoided, if one used soliton pulse as a signal. Recently, nonlinear directional couplers with dissimilar cores have attracted a considerable attention as several new effects can occur in them [18–22].The study of NLDC is no longer confined to the conventional silica based optical fiber coupler, it has recently been extended to AlGaAs nanowire [16] and lead silicate based holey fiber coupler also [9]. Most of the works relating to soliton switching reported in literatures is carried on NLDC made of polarization maintaining fibers. Interesting situations occurs when a fiber coupler exhibits large or low birefringence. In this work we report a detailed numerical study of the effect of large and low birefringence on soliton self-switching in an NLDC. When birefringence of the optical fiber is taken into consideration it may be appropriate to call the soliton as vector soliton. It may be noted that T. I. Lakoba et al. [23] have already carried out a detailed theoretical study on solitons in nonlinear fiber couplers with two orthogonal polarizations, but not in the context of soliton self-switching. In this work we are trying to address exactly this issue. We show that the coupler could be used as a soliton switch even at an input peak power less than the critical power, the power at which 50–50 power sharing takes place between the two cores, just by a judicious choice of the polarization angle. This has the advantage of reducing the required switching energy in a coupler. It is worthwhile to mention that recently vector soliton switching in a nonlinear amplifying loop mirror have been reported [24].
2. The model
E-mail address: [email protected]. 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.09.001
We consider a symmetric nonlinear coupler with either very high or low birefringence. It requires four coupled nonlinear partial
A.K. Sarma / Optics Communications 284 (2011) 186–190
187
differential equations to describe the pulse dynamics in such a coupler [25]: 2 ∂A1 ∂A iβ ∂ A1 2 2 = iκA2 + iγ jA1 j + χjB1 j A1 + β1x 1 + 2 2 ∂T 2 ∂z ∂T ∂B1 ∂B iβ ∂2 B1 2 2 = iκB2 + iγ jB1 j + χjA1 j B1 + β1y 1 + 2 2 ∂T 2 ∂z ∂T ∂A2 ∂A iβ ∂2 A2 2 2 = iκA1 + iγ jA2 j + χjB2 j A2 + β1x 2 + 2 2 ∂T 2 ∂z ∂T ∂B2 ∂B iβ ∂2 B2 2 2 = iκB1 + iγ jB2 j + χjA2 j B2 + β1y 2 + 2 2 2 ∂T ∂z ∂T
ð1Þ
where Aj and Bj with j = 1, 2, refers to the slowly varying linearly polarized components of pulse envelopes in the j-th core. β 1x = 1/ vgx and β 1y = 1/ vg y, where vg x and vg y are group velocities for the two polarization components. β2 and γ are the well known group velocity dispersion (GVD) and nonlinear parameter respectively. In the anomalous GVD regime, β2 b 0 and this is our domain of interest in this work. κ is the zeroth order coupling coefficient between the two cores. χ is the cross-phase modulation parameter. For the high birefringence case χ = 2/3 while χ = 2 for the low birefringence one. Now considering a coordinate system moving with the ˜ where β ˜ = β1x + β1y = 2, average group velocity 1 = β, and introduc ˜ = T0 ; ing the normalized units as follows: ξ = jβ2 jz = T02 ; τ = T−z β pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi A1 = U1 P0 ; B1 = U2 P0 ; A2 = U3 P0 ; B2 = U4 P0 ;N 2 = γ P 0 T 20 / |β2| ; ui = NUi with i = 1,2, 3, 4, the set of Eq. (1) can now be expressed as a set of normalized coupled equations in the following: ∂u1 ∂u i ∂ u1 2 2 = iκ0 u3 + i ju1 j + χju2 j u1 +g 1− 2 2 ∂ξ ∂τ ∂τ ∂u2 ∂u2 i ∂2 u2 2 2 = iκ0 u4 + i ju2 j + χju1 j u1 −g − 2 2 ∂τ ∂ξ ∂τ ∂u3 ∂u i ∂2 u3 2 2 = iκ0 u1 + i ju3 j + χju4 j u3 +g 3− 2 ∂τ2 ∂ξ ∂τ ∂u4 ∂u i ∂2 u4 2 2 = iκ0 u2 + i ju4 j + χju3 j u4 −g 4 − 2 2 ∂τ ∂ξ ∂τ
Fig. 1. (color online) plot of the transmission coefficient as a function of the normalized peak power for θ = 0, 30 and 45 (in degrees) with χ = 2/3.
conditions correspond to the case when a soliton of a given pulse duration is launched into the first core while the second core is kept empty [7]. The transmission coefficient T1(2) representing the fractional output energy in core 1 (2) after one coupling length, is calculated numerically according to the formula in the following: h
∞
∫
2
T1ð2Þ =
∞
−∞
i 2 2 ju1ð3Þ j + ju2ð4Þ j dτ
∫ ju1 j2 + ju2 j2 + ju3 j2 + ju4 j2 dτ
ð4Þ
−∞
ð2Þ
where g = T0(β1x − β1y) /2|β2| is the normalized group velocity mismatch, let us call it as the ‘detuning parameter', and κ0 = κT20 / |β2| is the normalized coupling coefficient. For low birefringence coupler g = 0. The above system of coupled nonlinear Schrodinger equations is the basic system of equations in this work.
In this work, we adhere to the usual notion of switching in which a signal launched into the input core emerges from the same core at the output, after traversing one coupling length inside the coupler (i.e. the bar state). A parameter of importance which describes the soliton stability inside the coupler is the so called RMS pulse width σ defined as follows [25]: h i1 2 2 2 σ = 〈T 〉−〈T〉
ð5Þ
∞
∞
−∞
−∞
where 〈T m 〉 = ∫ T m juj2 dT = ∫ juj2 dT
3. Results and discussions The system of four coupled Eq. (2) are hard to solve analytically, so we solve them numerically by the fast Fourier transform method for the linear dispersive part and by the fourth-order Runge-Kutta method, with auto-control of the step size for a given accuracy of the results, for the nonlinear part. For a detailed discussion on numerical methods to solve the NLSE readers are referred to Ref. [25]. The initial conditions for our numerical integration are as follows: pffiffiffiffiffi P0 cos θ sechðτÞ pffiffiffiffiffi u2 = P0 sinθ sechðτÞ u1 =
ð3Þ
u3 = 0 u4 = 0 where P0 is the normalized input peak power of the soliton. The angle θ defines linear polarization of the incident pulse measured from the slow axis. Let us call it the ‘polarization angle’, which refers to the relative strengths of the partial pulses in each of the two polarizations. The initial
Fig. 2. (color online) plot of the transmission coefficient as a function of the polarization angle θ with χ = 2/3 for two different values of normalized peak powers.
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A.K. Sarma / Optics Communications 284 (2011) 186–190
Fig. 3. (color online) Temporal profiles of |u1|2, |u2|2, |u3|2 and |u4|2 after one coupling length of the coupler with χ = 2/3.
We would firstly discuss the case of high birefringent coupler for which χ = 2/3. In order to see the effect of the input peak power on the transmission of the coupler, in Fig. 1 we plot the transmission coefficient T as function of the normalized peak power for three different polarization angles taking the detuning parameter g =0.15. In this work the angle is measured in degrees. It can be seen that the coupler exhibits better switching characteristics if the polarization angle deviates from zero. The role of polarization angle on the transmission of the coupler can be seen more clearly from Fig. 2 where we plot the transmission coefficient T as function of the polarization angle θ for two different normalized peak powers P0 = 1.5and 0.5 taking the detuning parameter g to be 0.15. We observe that when the input peak power is
above the critical power of switching, as in the case of P0 = 1.5, the transmission is not critically dependent on the polarization angle. However a slight increase in the transmission is observed with increase in θ. Interesting results occur for input peak power below the critical power of switching. As could be seen from Fig. 2, for P0 = 0.5, which is three times smaller than the case considered above, the transmission in the first core increases from a mere 30% to more than 90% with increase in the polarization angle. Hence by manipulating the polarization angle one may achieve soliton switching at a much reduced input peak power. Now, in order to have an idea about the spatio-temporal evolution of the pulses inside the coupler, in Fig. 3 we plot the temporal profiles of the pulses after the propagation of one coupling length inside the coupler for θ = 80, P0 = 1 and g = 0.15. It is quite evident from the scale of the plots that almost all the power is confined to the |u2|mode while a very small amount of power is leaking to |u3|and |u4|mode. The energy in the second core is equally distributed between the modes, where they take the form of small radiation, which may be attributed to the fact that in the second core both the modes see the same propagation constant
Fig. 4. (color online) RMS pulse width variation of the pulse in |u2| mode with normalized distance with χ = 2/3.
Fig. 5. Plot of the transmission coefficient in core 1 vs. the detuning parameter ‘g’ for P0 = 1 and θ = 80.
Our numerical study is presented for the following typical parameters: κ0 = 0.1, T0 = 10ps and β2 = − 20ps2 / km. The length of the coupler is assumed to be one coupling length, i.e. LC = π /2κ0 [6]. 3.1. Case I: high birefringence
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Fig. 6. (color online) plot of the transmission coefficient as a function of the polarization angle θ with χ = 2 for P0 = 1.
Fig. 8. (color online) RMS pulse width variation of the pulse in |u1| (line with ‘*’) and |u3| (line with ‘triangle’) mode with normalized distance with χ = 2.
while it is not so in the first core due to the presence of perturbations. |u2| mode is supporting a stable soliton pulse inside the coupler. This is more evident from Fig. 4 where the RMS pulse width variation of |u2| with normalized distance is depicted. One could see that the variation in the RMS pulse width, fluctuating between (nearly) 0.91 and 0.98, is very small. Hence we may conclude that stable soliton propagation is supported in the given range of propagation distance. However it may be noted that the soliton is getting shifted slightly in the temporal domain along the negative time axis. This happens due to the presence of detuning. Finally, in order to see the role of the detuning parameter on the transmission characteristics of the highly birefringent coupler, in Fig. 5 we plot the variation of the transmission as a function of the detuning parameter ‘g’ with θ = 80 and P0 = 1. We find that the transmission drops only very slightly from the case without detuning (i.e. g = 0). Hence we may conclude that the switching characteristic of the coupler is not significantly affected by the presence of detuning. One obvious effect of detuning is to shift the soliton along the temporal axis during its propagation; higher the value of detuning, higher would be the shift.
3.2. Case II: low birefringence Low birefringence case for which χ = 2 and g = 0, can be studied in a similar way. We find that for a given polarization angle the coupler shows similar transmission characteristics with that of the high birefringent one with the variation of the input peak power. The only notable difference is that, the low birefringent coupler has a comparatively larger critical power of switching as compared to the high birefringent one. Fig. 6 depicts the variation of the transmission characteristics of the coupler with the polarization angle with P0 = 1. We observe that by a judicious choice of the polarization angle one could increase the transmission significantly from a mere 30% to more than 90% over a wide range of polarization angles. In Fig. 7 we plot the temporal profiles of the pulses at the output end of the low birefringence coupler for P0 = 1 and θ = 30. It can be seen clearly that maximum of the power is confined in |u1| mode and the solitonic character is maintained, which is again more evident by Fig. 8 where we plot the RMS pulse width variation of |u1| (the line with ‘*’) and |u3| (the line with ‘triangle’) mode withpropagation distance. It can be clearly seen that the pulse in the |u1| mode is a soliton while |u3| is
Fig. 7. (color online) temporal profiles of |u1|2, |u2|2, |u3|2 and |u4|2 after one coupling length of the coupler.
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a non-solitonic radiation. We observe almost identical RMS pulse width variation for |u2| mode with that of |u1| mode, confirming it's nearly solitonic nature. Our study reveals that the RMS pulse width variation of the soliton in |u2| mode in high birefringent coupler is negligibly small compared to its low birefringent counterpart. Hence we may conclude that high birefringent coupler exhibits better soliton stability than the low birefringent one during propagation inside the coupler. Moreover one interesting difference from that of the high birefringence one is that, here the soliton is not shifted in the temporal domain because of the absence of detuning. 4. Conclusions In this work we have carried out a detailed numerical study of soliton switching in high as well as low birefringent coupler. The transmission characteristics of the coupler could be manipulated by varying the polarization angle. The coupler could be used as a soliton switch even at an input peak power less than the critical power of switching, just by a judicious choice of the polarization angle. The soliton is relatively more stable during its propagation inside the high birefringent coupler. Acknowledgement The author would like to thank the reviewers for their constructive comments which helped in improving the presentation of the work.
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