Soliton-effect compression and dispersive radiation

Transcription

1 1 March 000 Optics Communications Soliton-effect compression and dispersive radiation Noel F. Smyth ) Department of Mathematics and Statistics, UniÕersity of Edinburgh, The King s Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland, UK Received November 1999; accepted 0 December 1999 Abstract Soliton-effect pulse compression is investigated using two approximate methods which do not include the dispersive radiation shed as the soliton evolves and by one approximate method which does. Results derived from the approximate equations are compared with full numerical solutions of the governing equation. It is found that while the methods without radiation can be used to predict the position of optimal compression, to also have accurate prediction of the pulse amplitude requires inclusion of the dispersive radiation. A careful analysis of the solutions of the various approximate equations shows that care must be taken in using length predictions derived from approximate methods. This is because there are phase differences between the numerical and approximate solutions. Such phase differences are not predicted by the approximate methods. While approximate methods do exist to determine phase evolution, these methods do not predict the initial phase, which is of importance when distance predictions are made. q 000 Elsevier Science B.V. All rights reserved. PACS: 4.50.Rh; k; i; 4.81.Dp; Kf Keywords: Solitons; Compression; Optical solitons 1. Introduction The equation governing the propagation of a pulse down a monomode, polarisation preserving, nonlinear optical fibre in the anomalous group velocity dispersion regime is the nonlinear Schrodinger Ž NLS. equation, which can be written in non-dimensional form as wx 1 E u 1 E u i q q< u< us0. Ž 1. E z E t ) noel@maths.ed.ac.uk Here u is the complex-valued envelope of the pulse, z is the distance down the fibre and t is the reduced time Žtime measured in a frame which moves with the group velocity.. While the NLS equation can in principle be solved exactly using the inverse scatterwx, in practice it is difficult to determine ing method the transient evolution of a pulse using inverse scattering. This is because this transient evolution involves an interaction between the pulse and the dispersive radiation shed as it evolves. Inverse scattering shows that this dispersive radiation is determined by an integral equation. However this integral equation is difficult to solve. As an alternative to inverse scattering, various approximate techniques have been developed to determine the transient evolution of a pulse in a nonlin r00r$ - see front matter q 000 Elsevier Science B.V. All rights reserved. PII: S

2 470 N.F. SmythrOptics Communications ear optical fibre. One successful method has been the chirp method of Anderson w3,4 x, which is based on using an appropriate trial function in a Lagrangian for the NLS equation. One drawback of this method is that it does not include the dispersive radiation shed by the pulse as it evolves. Hence solutions of the chirp equations cannot evolve to a steady state. To overcome this drawback, Kath and Smyth wx 5 developed an alternative approximate method based on an averaged Lagrangian and conservation equations which did include this shed dispersive radiation. This new method was found to give solutions in excellent agreement with full numerical solutions of the NLS equation wx 5. In subsequent work, the method of Kath and Smyth was extended to model nonlinear twin-core fibres wx 6, non-uniform fibres wx 7 and fibre compressors wx 8. In all cases, good agreement was found with full numerical solutions of the relevant governing equations. Afanasjev et al. wx 9 used the chirp method of Anderson wx 4, and an extension of this method using a new trial function, to study so-called soliton-effect compression wx 1. They found that for low initial pulse amplitudes the standard chirp method gave good predictions for the point at which the pulse has its first amplitude maximum, the so-called optimal compression length. However for high initial amplitudes, the modified chirp method gave better predictions. In the present work the method of Kath and Smyth wx 5 is used to study soliton-effect compression. Similar to the modified chirp equations, the approximate equations derived using this method give good predictions of the optimal compression length for high initial pulse amplitudes, but not for low initial amplitudes. However for all initial amplitudes the method of Kath and Smyth gives good predictions of the pulse amplitude along the fibre, while both of the chirp methods do not. This is because the method of Kath and Smyth includes the effect of the dispersive radiation shed by the pulse as it evolves. The question arises as to why there is this varying agreement between the approximate and numerical solutions. It is shown that the varying agreement is due to initial phase differences between the approximate and numerical solutions which are not calculated by the approximate methods. This shows approximate methods must be used with caution when used to predict lengths to certain events.. Approximate equations Ž. The NLS equation 1 possesses the Lagrangian ) ) Lsi u u yuu y< u < q< u < 4, Ž. z z t where the superscript ) denotes the complex conjugate. Approximate solutions of the NLS equation Ž. 1 are now obtained by substituting various trial functions for u into this Lagrangian. Afanasjev et al. wx 9 used two related trial functions usa sechž trw. e i sqi bt, Ž 3. usa sech trw e i sqi btanh Žtr w.. Ž 4. The first of these is the chirped trial function of Anderson wx 4 and the second is an extension of this trial function. The extension was introduced by Afanasjev et al. wx 9 to try to improve agreement with full numerical solutions of the NLS equation. The new trial function has phase saturation as t "`, which is in better accord with full numerical solutions than the quadratic phase variation of Ž. 3. Ordinary differential equations governing the amplitude a, width w and chirp b of the trial functions are obtained by substituting the trial functions Ž. 3 and Ž. 4 into the averaged Lagrangian H ` Ls L dt Ž 5. y` and taking variations with respect to the parameters a, w and b. In this manner the equations governing the chirp trial function Ž. 3 can be found to be d dw Ž a w. s0, swb, d z d z db syb q Ž 1ya w. Ž 6 4. d z p w and those governing the extended trial function Ž. 4 are d dw 3b Ž a w. s0, s, d z d z 35w db 3b 1 s q Ž 1ya w.. Ž 7. d z 35w w

3 N.F. SmythrOptics Communications It is to be noted that the trial functions Ž. 3 and Ž. 4 do not account for the effect of the dispersive radiation shed as a pulse evolves. In particular this means that the solutions of the variational equations Ž. 6 and Ž. 7 cannot evolve to a steady state. To try to include the effect of the dispersive radiation shed as a pulse evolves, Kath and Smyth wx 5 used a different approach for obtaining a suitable trial function. The trial function used by these authors was usa sechž trw. e is qi ge i s. Ž 8. The amplitude a, width w, phase u and g are functions of distance z. This trial function consists of two parts. The first term is a varying soliton, similar to that of Anderson wx 4 and Afanasjev et al. wx 9. The second term is new and represents the low frequency dispersive radiation which stays in the vicinity of the pulse. In this regard it is noted that the group velocity for the linearised NLS equation is cg syk, so that long wavelength dispersive radiation has low velocity. The dispersive radiation in the vicinity of the pulse is then flat, which is why the second term in the trial function Ž. 8 has no t dependence. Kath and Smyth wx 5 noted that numerical solutions showed that the shed dispersive radiation was of small amplitude compared to the main pulse, so that < g < <a. The radiation cannot remain independent of t, soitis assumed that the radiation term in Ž. 8 contributes to the Lagrangian in a region ylr- t- lr about the pulse. Away from the pulse the extended trial function Ž. 4 takes the form usa sechž trw.ž 1qib. e i s Ž 9. for small b, which is similar to the trial function Ž. 8. The envelope a sechž trw. of the ib term serves to ensure that this term makes a finite contribution to the averaged Lagrangian. In the trial function Ž. 8 this was ensured by taking the radiation flat for ylr- t-lr. As for the previous trial functions Ž. 3 and Ž. 4, the trial function Ž. 8 is now used to calculate the averaged Lagrangian Ž. 5. After some manipulation the variational equations for this averaged Lagrangian can be found to be d l 1 y Ž aw. s gž a y w., Ž 10. d z p d g y sy až a yw., Ž 11. d z 3p ž / d a 4 y aw s0. Ž 1. d z w These variational equations have the fixed point 1r3 a y1 4 ksasw s awy. 13 w The parameter l was determined by Kath and Smyth wx 5 by the requirement that the frequency of oscillation of the solution of the variational equations near the fixed point is equal to the steady soliton oscillation frequency k r. This requirement gave that 3p ls. Ž 14. 8k The variational equations 10 to 1 hold in the vicinity of the pulse and so do not include the dispersive radiation shed as the pulse evolves. Since the amplitude of the radiation is small, away from the pulse this radiation is governed by the linearised NLS equation E u 1 E u i q s0. E z E t Ž 15. This linearised equation was solved by Kath and Smyth wx 5 using Laplace transforms. It was then shown using conservation equations that loss to dispersive radiation results in Eq. Ž 11. for g being modified to d g y sy až a yw. ya g, d z 3p Ž 16. where 3k 1 d z r Ž j. as H dj, Ž r d z 0 (p Ž zyj. ž / 3k 3p r s a wykq g. Ž k

4 47 N.F. SmythrOptics Communications For boundary value problems for which u is speciwx 5 fied at zs0, it was shown by Kath and Smyth that the integral in Ž 17. could be approximated. The loss parameter a was then approximated by 3k r as. Ž r Ž 0. ' p z The final approximate equations, including loss to dispersive radiation, governing the evolution of the pulse are then Ž 10., Ž 1. and Ž 16.. It was shown by Kath and Smyth wx 5 that these approximate equations gave solutions in good agreement with full numerical solutions of the NLS equation Ž. 1 and that the agreement with numerical solutions was better than the agreement of solutions of the chirp equations of Anderson wx Comparison with numerical solutions Solutions of the three sets of approximate equations of the previous section will now be compared with full numerical solutions of the NLS equation Ž. 1. Particular attention will be paid to the optimal compression length for soliton-effect compression. The systems of approximate equations Ž. 6, Ž. 7 and Ž 10., Ž 1., Ž 16. were solved numerically using a fourth order Runge Kutta scheme. The full NLS equation Ž. 1 was solved numerically using a pseudo-spectral method based on that of Fornberg and Whitham w10 x. To determine the optimal compression length for a soliton-effect compressor, the boundary condition už 0,t. sa sech t Ž 0. was used. Fig. 1 shows the optimal compression length L as a function of the initial amplitude A as given by the full numerical solution of the NLS equation Ž. 1 and by the three sets of approximate equations of the previous section. As found by Afanasjev et al. wx 9 the chirp equations of Anderson wx 4 give better agreement with the numerical solution for low initial amplitudes A, while the modified chirp equations Ž. 7 give better agreement for high initial amplitudes. The shelf equations Ž 10., Ž 1. and Ž 16. are seen to give results similar to those of the modified chirp equations. This is not unexpected given the relationship between these approximate Fig. 1. Optimal compression length L against initial amplitude A. Full numerical solution: solid line; solution of approximate equations Ž 10., Ž 1., Ž 16.: dashed line; solution of chirp equations Ž 6.: dotted line; solution of modified chirp equations Ž. 7 : dash dotted line. equations noted in the previous section. For large initial amplitudes the shelf equations give a length in better agreement with the numerical length than do the modified chirp equations. However for low initial amplitudes the chirp equations give the best agreement with the numerical solution. The reason for this varying agreement of the various approximate solutions with numerical solutions can be found from a detailed examination of the evolution of the pulse amplitude along the fibre. Fig. shows the evolution of the amplitude a of the pulse along the length z of the fibre for different final lengths for a low initial amplitude A s 1.5. Fig. a shows the comparison until just past the optimal compression length, Fig. b shows the comparison for a longer length, while Fig. c shows the comparison for the chirp and shelf equations only. The optimal compression lengths as predicted by the different approximate equations can be seen from Fig. a. It can also be noted from this figure that the shelf equations give a much better prediction of the amplitude of the pulse than either of the chirp equations. The reason for this is that the shelf equations include loss to dispersive radiation, as noted by Kath and Smyth wx 5. The reason why the optimal length as given by the chirp equations is in better agreement than the length as given by the shelf equations can be seen from Fig. b and Fig. c. Particularly from Fig. c, it is apparent that the wavelength of the

5 N.F. SmythrOptics Communications is a constant phase difference between the shelf equation and numerical amplitude oscillations. This phase difference is the reason for the disagreement between the optimal compression lengths. The derivation of the shelf equations does not include this phase, which is a higher order effect. While methods exist for determining equations for this higher order phase Žsee for example w11,1 x., these methods do not determine the initial phase, which is of importance here. In the light of this, it can now be seen that the good agreement, for low initial amplitude, between the optimal compression lengths as given by the full numerical solution and by the chirp equations is due to two effects canceling out. Fig. a shows that the wavelength of the chirp solution is significantly less than the numerical wavelength. The phase difference between the chirp and numerical solutions then cancels out with this shorter wavelength to result in the position of the first amplitude maximum agreeing with the numerical position. It is noted from Fig. b that the position of only the first maximum agrees with the numerical position. Fig. 3 shows the evolution of the amplitude of the pulse down the fibre as given by the full numerical solution and by the approximate equations for the high initial amplitude A s 3.0. The optimal compression length as given by the solution of the shelf equations now agrees well with that given by the full Fig.. Amplitude a of pulse as a function of z. Initial condition as1.5 and ws1.0. Full numerical solution: solid line; solution of approximate equations Ž 10., Ž 1., Ž 16.: dashed line; solution of chirp equations Ž. 6 : dotted line; solution of modified chirp equations Ž. 7 : dash dotted line. Ž. a Comparison to zs5. Ž. b Comparison to zs10. Ž. c Comparison to zs50. amplitude oscillations as given by the shelf equations agrees with the numerical wavelength. However there Fig. 3. Amplitude a of pulse as a function of z. Initial condition as3.0 and ws1.0. Full numerical solution: solid line; solution of approximate equations Ž 10., Ž 1., Ž 16.: dashed line; solution of chirp equations Ž. 6 : dotted line; solution of modified chirp equations Ž. 7 : dash dotted line.

6 474 N.F. SmythrOptics Communications numerical solution. This is because the phase difference between the two solutions is now not as great as for A s 1.5. The optimal compression length as given by the modified chirp equations is also in good agreement with the numerical length, as noted by Afanasjev et al. wx 9. However the amplitude of the pulse as given by these modified chirp equations is not in good agreement with the numerical amplitude, while the amplitude as given by the shelf equations is. This is because the shelf equations include radiative loss, which the chirp equations do not. Both the optimal compression length and pulse amplitude as given by the standard chirp equations are not in good agreement with the corresponding numerical values, again as noted by Afanasjev et al. wx 9. It can further be seen from Fig. 3 that after the first amplitude maximum none of the approximate solutions are in agreement with the numerical solution. This is because for A)1.5 the initial pulse breaks up into two or more solitons w13 x. As all the approximate methods assume a trial function consisting of a single soliton, they cannot be expected to give equations whose solutions are in agreement with numerical solutions for large distances for A) 1.5. The agreement between the numerical solution and the solution of the shelf equations is good up until the first amplitude maximum as the second soliton develops after this first maximum. Fig. 4 compares the pulse profiles < u< as given by the full numerical solution and the solutions of the approximate equations for the high initial condition As3.0. Fig. 4a shows this comparison at zs0.37, which is the position of the point of optimal compression for the numerical solution. It can be seen that the solution of the shelf equations is in fair agreement with the numerical solution. The amplitude as given by the shelf equations is lower than the numerical amplitude, as expected from Fig. 3, while the widths of the two solutions are in good agreement. The figure also shows that the shape of the pulse as given by the modified chirp equations is not in good agreement with the numerical shape. Part of the difference in shape between the numerical and approximate solutions is due to phase differences. To subtract out these differences, the pulse profiles have been compared at the position of their respective first maximums in Fig. 4b. It can now be seen that there is excellent agreement between the solution of the Fig. 4. Comparison of pulse profiles. Full numerical solution: solid line; solution of approximate equations Ž 10., Ž 1., Ž 16.: dashed line; solution of modified chirp equations Ž. 7 : dotted line. Ž. a Profiles at zs0.37. Ž. b Numerical profile at zs0.37. Approximate profiles at position of their respective amplitude maximums. shelf equations and the full numerical solution. However there is still not good agreement between the solution of the modified shelf equations and the numerical solution since the amplitude of the modified shelf solution is too large. 4. Conclusions Predictions of the optimal compression length for a soliton-effect compressor using various approximate methods have been compared with the optimal length as given by full numerical solutions. The first of the approximate methods was based on the chirp method of Anderson w3,4 x, while the second was an

7 N.F. SmythrOptics Communications extension of this method by Afanasjev et al. wx 9. The third approximate method was the shelf method of Kath and Smyth wx 5. This final method includes the effect of the dispersive radiation shed as a pulse evolves. It was found that the optimal compression length was well predicted by the chirp method for low initial pulse amplitudes, but not for high initial amplitudes. For high initial pulse amplitudes the modified chirp and shelf methods both gave good predictions for the optimal length. The amplitude and shape of the pulse as it evolved down the fibre was well predicted by the shelf equations, but not by the chirp equations. This is because the shelf equations were the only approximate equations to include the effect of shed dispersive radiation. As well as comparing predictions for the optimal compression length, the reasons for the differences between the predictions of the approximate equations and the full numerical solution were discussed. It was shown that a major source for these differences was the phase difference between solutions of the approximate equations and the numerical solution. It was found that if this phase difference was factored out, then the shelf equations gave an optimal compression length in agreement with the numerical value, even for low initial amplitudes. None of the approximate methods determined this phase of the amplitude oscillations, which is a higher order effect. It was further shown that for low initial amplitudes, the wavelength of the pulse amplitude oscillations as given by the chirp equations was less than the numerical wavelength. This low wavelength then cancelled out with the phase difference between the oscillations to yield an optimal compression length in agreement with the numerical value. This shows that lengths as predicted by the approximate methods should be treated with caution. To have confidence in the lengths predicted by approximate methods, these methods will need to be extended to determine the initial phase and equations for the phase. References wx 1 G.P. Agrawal, Nonlinear Fibre Optics, Academic Press, New York, wx A.C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, wx 3 D. Anderson, Phys. Rev. A 7 Ž wx 4 D. Anderson, M. Lisak, T. Reichel, J. Opt. Soc. Am. B 5 Ž wx 5 W.L. Kath, N.F. Smyth, Phys. Rev. E 51 Ž wx 6 N.F. Smyth, A.L. Worthy, J. Opt. Soc. Am. B 14 Ž wx 7 N.F. Smyth, A.H. Pincombe, Phys. Rev. E 57 Ž wx 8 K.I.M. McKinnon, N.F. Smyth, A.L. Worthy, J. Opt. Soc. Am. B 16 Ž wx 9 V.V. Afanasjev, B.A. Malomed, P.L. Chu, M.K. Islam, Opt. Commun. 147 Ž w10x B. Fornberg, G.B. Whitham, Phil. Trans. Roy. Soc. London A 89 Ž w11x R. Haberman, Stud. Appl. Math. 78 Ž w1x R. Haberman, Stud. Appl. Math. 84 Ž w13x J. Satsuma, N. Yajima, Prog. Theor. Phys. Suppl. 55 Ž