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Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers
Optics Communications 218 (2003) 167–172 www.elsevier.com/locate/optcom
Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers A.C. Peacock *, N.G.R. Broderick, T.M. Monro Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, UK Received 31 October 2002; accepted 9 January 2003
Abstract Numerical simulations are used to demonstrate parabolic pulse generation in a highly nonlinear, normally dispersive microstructured fibre Raman amplifier. The dependence of the parabolic pulse solution on the parameters of the microstructured fibre is discussed. In particular, it is shown that the effects of pump depletion can be reduced with the appropriate sign of the third order dispersion. Additional simulations are conducted to establish the input pulse parameters which allow for the most efficient convergence to the parabolic pulse regime. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 42.65; 42.81; 42.65.Dr; 42.65.Re
1. Introduction With the rapid advancements in the technology of optical fibre amplifiers to generate high power pulses, the main limitation to further increasing the power available for use in optical systems are the pulse distortions resulting from the interplay between the nonlinear and dispersive effects. In the normal dispersion regime these distortions lead to the phenomenon of optical wavebreaking. Optical wavebreaking occurs when the different frequency components generated via the Kerr effect in the leading and trailing regions of the pulse interfere resulting in temporal oscillations near the edges of
*
Corresponding author. Fax: +44-23-8059-3149. E-mail address: [email protected] (A.C. Peacock).
the pulse. For this reason, since their theoretical prediction [1] and later, their experimental verification [2,3], self-similar parabolic pulses have generated considerable interest due to their ability to propagate in highly nonlinear media avoiding the effects of wavebreaking and maintaining their linear chirp. Parabolic pulses thus offer themselves to a wide range of applications in many areas of optical technology and particularly, as their linear chirp facilitates efficient pulse compression, to the field of high-power short pulse generation. Initial studies of parabolic pulse generation has considered their formation in rare-earth-doped optical fibre amplifiers with a non-frequency dependent gain profile and normal dispersion [2–4], and recently this work has been extended to examine the effects of the finite width of the gain spectrum [5]. The results have shown that the most efficient
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01129-5
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A.C. Peacock et al. / Optics Communications 218 (2003) 167–172
parabolic pulse generation occurs when the nonlinear effects are large enough so that the nonlinear propagation dominates over the dispersive propagation which requires a gain medium with a large gain bandwidth to support their growing spectral width. In this paper we revisit the problem of parabolic pulse generation but now consider amplification in a fibre based on the Raman effect. Such an amplifier would enable us to exploit the broad band Raman amplification where bandwidths of more than 100 nm have been demonstrated [6]. In addition, as the Raman effect is not confined to any particular wavelength, this technology can be used to provide gain where no conventional amplifiers are possible and more importantly, can form the basis of a tunable high power pulse source. Such pulse sources will find wide application in all optical processes such as wavelength division multiplexing (WDM), terahertz optical asymmetric demultiplexing (TOAD), and optical logic. Commercial Raman amplifiers based on standard fibre are available and are typically pumped via continuous wave sources operating at several Watt power levels. In such a regime we expect that propagation lengths of the order of kilometers would be required for the pulse to become parabolic due to the small gain. As a result, it is questionable whether the nonlinear effects will be sufficiently large, and the dispersive effects sufficiently small, to generate parabolic pulses efficiently. To over come this problem here we consider using a high power pulsed pump source and, to further enhance the nonlinear effects, small-core microstructured fibres with large effective nonlinearities [7]. Importantly, unlike high nonlinearity fibres with an enhanced n2 which can have a reduced Raman gain bandwidth [8], the bandwidth of the single-material microstructured fibres considered here are simply that of pure silica. In addition, the dispersion properties of a microstructured fibre can be tailored such that they operate in the normal dispersion regime, necessary for parabolic pulse propagation, over a wide range of wavelengths extending up to and beyond the optical communications operating wavelength of 1:5 lm. The results of our numerical simulations presented here show that
parabolic pulse generation is indeed possible in these fibres via Raman amplification and that these pulses can be efficiently compressed to the sub picosecond regime.
2. Numerical model and simulations We consider a range of small-core pure silica microstructured fibre designs with the aim of determining whether they are suitable hosts for parabolic pulse generation. Predictions for the effective mode area Aeff and the propagation constant b are shown in Fig. 1 for three typical microstructured fibres. These predictions were made using a full-vector orthogonal function model, as described in [9]. The transverse refractive index profile for a fibre with d=K ¼ 0:8 is shown in the inset of Fig. 1(b) where K is the hole-to-hole spacing and d is the hole diameter. Specifically we consider fibres with K in the range 0:9–1:2 lm and ratios d=K of 0.8–0.85, where the numerical predictions indicate normal dispersion at wavelengths in the range 1:5–1:7 lm. As small (sub-k) values of K lead to small core sizes and large values of d=K give large N =A, the combined effects result in tight modal confinement thus offering large effective
Fig. 1. (a) Effective areas Aeff and (b) propagation constants b, as functions of wavelength, for fibre A (solid curves), fibre B (dashed curves) and fibre C (dotted curves). Inset: refractive index profile of a fibre with d=K ¼ 0:8.
A.C. Peacock et al. / Optics Communications 218 (2003) 167–172
nonlinearities. We note that Aeff is related to the effective fibre nonlinearity via c ¼ 2pn2 =kAeff and the propagation constant b is related to the group velocity dispersion (GVD) and the third order dispersion parameters via bm ¼ dm b=dxm jx0 (m ¼ 2; 3), respectively [10]. For reference, the nonlinear and dispersion parameters are given explicitly in Table 1. Such fibres are similar to that used previously as a Raman amplifier [7]. Given the modal properties of the microstructured fibre, pulse propagation can be described by the standard nonlinear Schr€ odinger equation (NLSE). Including the effects of Raman amplification the evolution of the pulses in our system can then be described by a modified form of the NLSE [11]: oW b2 o2 W b 3 o3 W i o i ¼ þ i c 1 þ oz x0 oT 2 oT 2 6 oT 3 Z 1 0 0 2 0 Wðz; T Þ RðT ÞjWðz; T T Þj dT : 0
ð1Þ We write the nonlinear response function as RðT Þ ¼ ð1 fR ÞdðT Þ þ fR hR ðT Þ, assuming that the electronic contribution dðT Þ is instantaneous and that the relative size of the vibrational (Raman) contribution hR ðT Þ is determined by fR [10]. The field W, in comoving coordinates, can be expressed in terms of the amplitude Aj and the phase Uj (j ¼ p; s) of the pump and the signal beams as: Wðz; T Þ ¼ Ap ðz; T Þ exp½iUp ðz; T Þ þ As ðz; T Þ exp½iUs ðz; T Þ ;
ð2Þ
where the signal field is downshifted in frequency by 13.2 THz from the pump beam corresponding to the peak of the Raman gain spectrum. An important feature of Eq. (1) is the inclusion of the time-derivative operator in the nonlinear term which is necessary to ensure that the photon
169
number is conserved, and not the optical energy, so that the Raman interaction is described correctly. Our numerical simulations are based on a Raman interaction pumped with a 20 W, 1 ns (FWHM) super-Gaussian pulse at 1:536 lm typical of the output from a diode pumped high power Er:doped fibre amplifier chain. The Gaussian signal pulses of 5 W, 1 ps duration (FWHM) are injected at 1:647 lm into the three fibres with lengths of 16 m so that the total pulse gains are of the order of 30 dB. We choose the shape of the Raman response function to be that calculated from experimental data obtained for silica fibres with the fractional contribution of the delayed Raman response fR ¼ 0:18 [12]. The output signal pulses (top curves) and spectra (bottom curves) resulting from amplification in the three fibres are plotted in Fig. 2, corresponding to total pulse gains of (A) 29 dB, (B) 25 dB and (C) 29 dB. These results illustrate the significance of the fibre parameters on the total pulse gain and the final pulse shape. Despite these differences, it is clear that in all cases the pulse displays the characteristic features of a pulse entering the parabolic regime including both the low intensity exponentially decaying wings and the oscillations on the edges of the spectrum [13]. In order to illustrate the level to which each of the pulses have developed into the parabolic regime, the top curves of Fig. 2 also include parabolic and linear fits (obtained via a minimisation scheme based on the Nelder–Mead Simplex method) to the intensity profile and the chirp, respectively (circles). These fits indicate that the pulse which has evolved to be the most parabolic is that corresponding to fibre B. This is to be expected as this fibre has the smallest GVD parameter relative to the fibre nonlinearity so that the nonlinear effects are more pronounced. However,
Table 1 Dispersion and nonlinearity parameters for the fibres corresponding to the profiles in Fig. 1 calculated at k ¼ 1:647 lm
A B C
K (lm)
d=K
b2 (ps2 m1 )
b3 (ps3 m1 )
c (W1 km1 )
0.9 1.2 1.0
0.8 0.8 0.85
0.4980 0.0929 0.3412
)0.0013 )0.0010 )0.0018
51.7 52.4 56.4
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Fig. 2. Top, intensity (left axis – logarithmic scale) and chirp (right axis – linear scale) of the fibre outputs obtained from simulated propagation in fibres A, B and C (solid lines) together with the corresponding parabolic and linear fits (circles). Bottom, the output spectra.
it is also apparent that in all cases the pulse has developed asymmetrically. Although some of the asymmetry in the output pulse can be attributed to the shape of the gain spectrum [12] it is, in fact, primarily due to pump depletion where the leading edge of the pulse experiences more gain than the trailing edge. This effect is significant because the signal intensity eventually exceeds that of the pump intensity [10] and, due to the large gains necessary to amplify a pulse to the parabolic regime, it is a difficult problem to avoid. Indeed, the peak powers of the output pulses are (A) 102 W, (B) 96 W and (C) 125 W, so that they are all much greater than the initial pump power of 20 W. Furthermore, on comparing the output pulses in Fig. 2, we see that the relative sizes of the peak powers are in agreement with the size of the induced asymmetry in the pulses with the pulse generated in fibre C displaying the largest asymmetry. In this context, we also note that given the high peak powers reached by the signal pulse it is possible for it to then pump a second Raman pulse out of the amplifier noise, downshifted from itself by 13.2 THz, thus creating a cascaded Raman effect. However, our observations indicate that over
the relatively short propagation lengths we are considering the gain that the signal provides to the second Raman pulse is not sufficient to degrade the quality of the generated parabolic pulse. An important consequence of the effects of the pump depletion is that the formation of a parabolic pulse is highly dependent on the sign of the third order dispersion. We have found that when b3 < 0 then the asymmetry induced by the third order dispersion acts in the opposite direction to that induced by the pump depletion and thus can actually improve the quality of the output pulse. However, when b3 > 0 the effects of the asymmetries combine, destroying the linearity of the chirp, and can lead to the pulse developing oscillations on the long sloping training edge. Such effects can be seen in Fig. 3 where (a) shows the intensity profile and chirp (plotted on linear scales) and (b) shows the spectrum of the output pulse generated under the same conditions as that in Fig. 2(c) but this time with b3 ¼ 0:0018 ps3 m1 . Although in standard single-mode fibres b3 is typically positive over most wavelengths, both positive and negative values of b3 are possible in small-core microstructured fibres. Indeed, from the slope of the
A.C. Peacock et al. / Optics Communications 218 (2003) 167–172
Fig. 3. (a) Intensity (left axis) and chirp (right axis) of the output obtained from simulated propagation in fibre C but with b3 ¼ 0:0018 ps3 m1 . (b) The corresponding spectrum.
dispersion profiles in Fig. 1(b), clearly in all cases considered here b3 < 0. From the results presented in Figs. 2 and 3 it is clear that the evolution of the signal pulse to the parabolic pulse regime depends strongly on the parameters of the microstructured fibre. Further design work could be performed to identify microstructed fibres that are optimised for parabolic pulse generation. However, previous investigations into parabolic evolution have shown that certain input pulse parameters in conjunction with the amplifier parameters allow for more efficient convergence to the parabolic regime [13]. Based on these results, we have conducted additional simulations to establish the input signal and pump pulse parameters which optimise the evolution to the parabolic pulse solution with the hope that a faster convergence to the parabolic regime will make the specific fibre design less critical. As a solution to Eq. (1) describing an analytic form for a Raman amplified parabolic pulse has yet to be determined, as a means of quantifying the level to which a pulse has become parabolic we introduce an rms intensity error between the numerically simulated pulse INUM ðT Þ ¼ jWðL; T Þj2 , where L is the fibre length, and the fitted parabolic pulse IFIT ðT Þ as [14]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 PN 2 u j¼0 ½INUM ðTj Þ IFIT ðTj Þ e I ¼ tN : ð3Þ P N 1 2 j¼0 INUM ðTj Þ N The results of the simulations have shown that the important pulse parameters are the input pulse width and the ratio of the peak powers of the pump and signal pulses. To illustrate this, Fig. 4(a) shows a plot of the rms intensity error as a function of the input pulse width (FWHM) where in all
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cases the input pulse energy is 5.3 pJ (as used for the input pulses in Fig. 2) and the fibre parameters correspond to fibre A. This clearly shows that for a fixed input pulse energy there is an optimum input pulse width in agreement with the earlier observations of [13]. The remaining curves of Fig. 4 show a comparison between parabolic pulses generated with the same input pulse and fibre as used to generate Fig. 2(a) (i.e. a signal peak power of 5 W) but with pump powers of (b) 40 W and (c) 100 W. As larger pump powers offer larger gains per unit length, the length of propagation was chosen to be 8 and 3 m for (b) and (c), respectively, so that in both cases the signal pulse experienced a gain of 28 dB. Although the intensity misfits for these output pulses are similar, (b) eI ¼ 0:17 and (c) eI ¼ 0:15, the steep sloping edges on the pulse in Fig. 4(c) (characteristic of a parabolic pulse
Fig. 4. (a) RMS intensity error as a function of input pulse duration (FWHM) for fixed signal pulse energy. (b, c) Top, intensity (left axis) and chirp (right axis), bottom, corresponding spectra, of the fibre outputs obtained from simulations with 40 and 100 W pump pulses, respectively. In all cases the fibre parameters correspond to fibre A.
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Fig. 5. Intensity profile of the output pulse from Fig. 2(b) after linear compression.
plotted on a logarithmic scale) indicate that this pulse is more parabolic [13]. Thus these results suggest that a larger gain per unit length can enhance the rate of evolution to the parabolic regime. Finally, to demonstrate the potential use of Raman amplified parabolic pulses for high-powered short pulse generation we considered the compression of the pulse in Fig. 2(b) via a simple linear grating pair. The resultant compressed pulse can be seen in Fig. 5 where the long pedestal on the trailing edge can be attributed to the asymmetry of the uncompressed pulse. Nonetheless, this still yields a pulse with a duration of 700 fs (FWHM) and a peak power of 1.4 kW and we expect that these results can be improved with the inclusion of higher order dispersion compensation in the compression stage.
3. Conclusions In conclusion, we have used numerical simulations to demonstrate parabolic pulse formation in a microstructured fibre Raman amplifier. The success of our results can be attributed to the large gain bandwidth offered by the pure silica host and the ability to tailor the dispersion profile of a microstructured fibre to provide normal dispersion over a wide range of wavelengths. The results have shown that the final shape of the parabolic pulse
depends on the fibre parameters. Importantly, the effects of pump depletion can be reduced with the appropriate sign of the third order dispersion and such values of b3 are currently available in microstructured fibres. In addition, we have also established the importance of the input signal and pump pulse characteristics to the evolution map of the parabolic pulse. The ease with which these pulses can be compressed suggests that Raman amplified parabolic pulses offer an efficient source of high-powered short pulses unrestricted by wavelength. We expect that they will find wide application in many areas of optical technology and further research to verify these results experimentally is currently under way.
References [1] D. Anderson, M. Desaix, M. Karlsson, M. Lisak, M.L. Quiroga-Teixeiro, J. Opt. Soc. Am. B 10 (1993) 1185. [2] K. Tamura, M. Nakazawa, Opt. Lett. 21 (1996) 68. [3] M.E. Fermann, V.I. Kruglov, B.C. Thomsen, J.M. Dudley, J.D. Harvey, Phys. Rev. Lett. 84 (2000) 6010. [4] V.I. Kruglov, A.C. Peacock, J.M. Dudley, J.D. Harvey, Opt. Lett. 25 (2000) 1753. [5] A.C. Peacock, R.J. Kruhlak, J.D. Harvey, J.M. Dudley, Opt. Commun. 206 (2002) 171. [6] S. Namiki, Y. Emori, J. Sel. Top. Quantum Electron. 7 (2001) 3. [7] Z. Yusoff, J.H. Lee, W. Belardi, T.M. Monro, P.C. Teh, D.J. Richarson, Opt. Lett. 27 (2002) 424. [8] M. Ho, K. Uesaka, M. Marhic, Y. Akasaka, L.G. Kazovsky, J. Lightwave Technol. 19 (2001) 977. [9] T.M. Monro, N.G.R. Broderick, D.J. Richardson, Nato Summer School on Nanoscale Linear and Nonlinear Optics, Erice Sicily, July, 2000. [10] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2001. [11] K.J. Blow, D. Wood, J. Quantum Electron. 25 (1989) 2665. [12] R.H. Stolen, J.P. Gordon, W.J. Tomlinson, H.A. Haus, J. Opt. Soc. Am. B 6 (1989) 1159. [13] V.I. Kruglov, A.C. Peacock, J.D. Harvey, J.M. Dudley, J. Opt. Soc. Am. B 19 (2002) 461. [14] D.N. Fittinghoff, K.W. DeLong, R. Trebino, C.L. Ladera, J. Opt. Soc. Am. B 12 (1995) 1955.
Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers A.C. Peacock *, N.G.R. Broderick, T.M. Monro Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, UK Received 31 October 2002; accepted 9 January 2003
Abstract Numerical simulations are used to demonstrate parabolic pulse generation in a highly nonlinear, normally dispersive microstructured fibre Raman amplifier. The dependence of the parabolic pulse solution on the parameters of the microstructured fibre is discussed. In particular, it is shown that the effects of pump depletion can be reduced with the appropriate sign of the third order dispersion. Additional simulations are conducted to establish the input pulse parameters which allow for the most efficient convergence to the parabolic pulse regime. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 42.65; 42.81; 42.65.Dr; 42.65.Re
1. Introduction With the rapid advancements in the technology of optical fibre amplifiers to generate high power pulses, the main limitation to further increasing the power available for use in optical systems are the pulse distortions resulting from the interplay between the nonlinear and dispersive effects. In the normal dispersion regime these distortions lead to the phenomenon of optical wavebreaking. Optical wavebreaking occurs when the different frequency components generated via the Kerr effect in the leading and trailing regions of the pulse interfere resulting in temporal oscillations near the edges of
*
Corresponding author. Fax: +44-23-8059-3149. E-mail address: [email protected] (A.C. Peacock).
the pulse. For this reason, since their theoretical prediction [1] and later, their experimental verification [2,3], self-similar parabolic pulses have generated considerable interest due to their ability to propagate in highly nonlinear media avoiding the effects of wavebreaking and maintaining their linear chirp. Parabolic pulses thus offer themselves to a wide range of applications in many areas of optical technology and particularly, as their linear chirp facilitates efficient pulse compression, to the field of high-power short pulse generation. Initial studies of parabolic pulse generation has considered their formation in rare-earth-doped optical fibre amplifiers with a non-frequency dependent gain profile and normal dispersion [2–4], and recently this work has been extended to examine the effects of the finite width of the gain spectrum [5]. The results have shown that the most efficient
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01129-5
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parabolic pulse generation occurs when the nonlinear effects are large enough so that the nonlinear propagation dominates over the dispersive propagation which requires a gain medium with a large gain bandwidth to support their growing spectral width. In this paper we revisit the problem of parabolic pulse generation but now consider amplification in a fibre based on the Raman effect. Such an amplifier would enable us to exploit the broad band Raman amplification where bandwidths of more than 100 nm have been demonstrated [6]. In addition, as the Raman effect is not confined to any particular wavelength, this technology can be used to provide gain where no conventional amplifiers are possible and more importantly, can form the basis of a tunable high power pulse source. Such pulse sources will find wide application in all optical processes such as wavelength division multiplexing (WDM), terahertz optical asymmetric demultiplexing (TOAD), and optical logic. Commercial Raman amplifiers based on standard fibre are available and are typically pumped via continuous wave sources operating at several Watt power levels. In such a regime we expect that propagation lengths of the order of kilometers would be required for the pulse to become parabolic due to the small gain. As a result, it is questionable whether the nonlinear effects will be sufficiently large, and the dispersive effects sufficiently small, to generate parabolic pulses efficiently. To over come this problem here we consider using a high power pulsed pump source and, to further enhance the nonlinear effects, small-core microstructured fibres with large effective nonlinearities [7]. Importantly, unlike high nonlinearity fibres with an enhanced n2 which can have a reduced Raman gain bandwidth [8], the bandwidth of the single-material microstructured fibres considered here are simply that of pure silica. In addition, the dispersion properties of a microstructured fibre can be tailored such that they operate in the normal dispersion regime, necessary for parabolic pulse propagation, over a wide range of wavelengths extending up to and beyond the optical communications operating wavelength of 1:5 lm. The results of our numerical simulations presented here show that
parabolic pulse generation is indeed possible in these fibres via Raman amplification and that these pulses can be efficiently compressed to the sub picosecond regime.
2. Numerical model and simulations We consider a range of small-core pure silica microstructured fibre designs with the aim of determining whether they are suitable hosts for parabolic pulse generation. Predictions for the effective mode area Aeff and the propagation constant b are shown in Fig. 1 for three typical microstructured fibres. These predictions were made using a full-vector orthogonal function model, as described in [9]. The transverse refractive index profile for a fibre with d=K ¼ 0:8 is shown in the inset of Fig. 1(b) where K is the hole-to-hole spacing and d is the hole diameter. Specifically we consider fibres with K in the range 0:9–1:2 lm and ratios d=K of 0.8–0.85, where the numerical predictions indicate normal dispersion at wavelengths in the range 1:5–1:7 lm. As small (sub-k) values of K lead to small core sizes and large values of d=K give large N =A, the combined effects result in tight modal confinement thus offering large effective
Fig. 1. (a) Effective areas Aeff and (b) propagation constants b, as functions of wavelength, for fibre A (solid curves), fibre B (dashed curves) and fibre C (dotted curves). Inset: refractive index profile of a fibre with d=K ¼ 0:8.
A.C. Peacock et al. / Optics Communications 218 (2003) 167–172
nonlinearities. We note that Aeff is related to the effective fibre nonlinearity via c ¼ 2pn2 =kAeff and the propagation constant b is related to the group velocity dispersion (GVD) and the third order dispersion parameters via bm ¼ dm b=dxm jx0 (m ¼ 2; 3), respectively [10]. For reference, the nonlinear and dispersion parameters are given explicitly in Table 1. Such fibres are similar to that used previously as a Raman amplifier [7]. Given the modal properties of the microstructured fibre, pulse propagation can be described by the standard nonlinear Schr€ odinger equation (NLSE). Including the effects of Raman amplification the evolution of the pulses in our system can then be described by a modified form of the NLSE [11]: oW b2 o2 W b 3 o3 W i o i ¼ þ i c 1 þ oz x0 oT 2 oT 2 6 oT 3 Z 1 0 0 2 0 Wðz; T Þ RðT ÞjWðz; T T Þj dT : 0
ð1Þ We write the nonlinear response function as RðT Þ ¼ ð1 fR ÞdðT Þ þ fR hR ðT Þ, assuming that the electronic contribution dðT Þ is instantaneous and that the relative size of the vibrational (Raman) contribution hR ðT Þ is determined by fR [10]. The field W, in comoving coordinates, can be expressed in terms of the amplitude Aj and the phase Uj (j ¼ p; s) of the pump and the signal beams as: Wðz; T Þ ¼ Ap ðz; T Þ exp½iUp ðz; T Þ þ As ðz; T Þ exp½iUs ðz; T Þ ;
ð2Þ
where the signal field is downshifted in frequency by 13.2 THz from the pump beam corresponding to the peak of the Raman gain spectrum. An important feature of Eq. (1) is the inclusion of the time-derivative operator in the nonlinear term which is necessary to ensure that the photon
169
number is conserved, and not the optical energy, so that the Raman interaction is described correctly. Our numerical simulations are based on a Raman interaction pumped with a 20 W, 1 ns (FWHM) super-Gaussian pulse at 1:536 lm typical of the output from a diode pumped high power Er:doped fibre amplifier chain. The Gaussian signal pulses of 5 W, 1 ps duration (FWHM) are injected at 1:647 lm into the three fibres with lengths of 16 m so that the total pulse gains are of the order of 30 dB. We choose the shape of the Raman response function to be that calculated from experimental data obtained for silica fibres with the fractional contribution of the delayed Raman response fR ¼ 0:18 [12]. The output signal pulses (top curves) and spectra (bottom curves) resulting from amplification in the three fibres are plotted in Fig. 2, corresponding to total pulse gains of (A) 29 dB, (B) 25 dB and (C) 29 dB. These results illustrate the significance of the fibre parameters on the total pulse gain and the final pulse shape. Despite these differences, it is clear that in all cases the pulse displays the characteristic features of a pulse entering the parabolic regime including both the low intensity exponentially decaying wings and the oscillations on the edges of the spectrum [13]. In order to illustrate the level to which each of the pulses have developed into the parabolic regime, the top curves of Fig. 2 also include parabolic and linear fits (obtained via a minimisation scheme based on the Nelder–Mead Simplex method) to the intensity profile and the chirp, respectively (circles). These fits indicate that the pulse which has evolved to be the most parabolic is that corresponding to fibre B. This is to be expected as this fibre has the smallest GVD parameter relative to the fibre nonlinearity so that the nonlinear effects are more pronounced. However,
Table 1 Dispersion and nonlinearity parameters for the fibres corresponding to the profiles in Fig. 1 calculated at k ¼ 1:647 lm
A B C
K (lm)
d=K
b2 (ps2 m1 )
b3 (ps3 m1 )
c (W1 km1 )
0.9 1.2 1.0
0.8 0.8 0.85
0.4980 0.0929 0.3412
)0.0013 )0.0010 )0.0018
51.7 52.4 56.4
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Fig. 2. Top, intensity (left axis – logarithmic scale) and chirp (right axis – linear scale) of the fibre outputs obtained from simulated propagation in fibres A, B and C (solid lines) together with the corresponding parabolic and linear fits (circles). Bottom, the output spectra.
it is also apparent that in all cases the pulse has developed asymmetrically. Although some of the asymmetry in the output pulse can be attributed to the shape of the gain spectrum [12] it is, in fact, primarily due to pump depletion where the leading edge of the pulse experiences more gain than the trailing edge. This effect is significant because the signal intensity eventually exceeds that of the pump intensity [10] and, due to the large gains necessary to amplify a pulse to the parabolic regime, it is a difficult problem to avoid. Indeed, the peak powers of the output pulses are (A) 102 W, (B) 96 W and (C) 125 W, so that they are all much greater than the initial pump power of 20 W. Furthermore, on comparing the output pulses in Fig. 2, we see that the relative sizes of the peak powers are in agreement with the size of the induced asymmetry in the pulses with the pulse generated in fibre C displaying the largest asymmetry. In this context, we also note that given the high peak powers reached by the signal pulse it is possible for it to then pump a second Raman pulse out of the amplifier noise, downshifted from itself by 13.2 THz, thus creating a cascaded Raman effect. However, our observations indicate that over
the relatively short propagation lengths we are considering the gain that the signal provides to the second Raman pulse is not sufficient to degrade the quality of the generated parabolic pulse. An important consequence of the effects of the pump depletion is that the formation of a parabolic pulse is highly dependent on the sign of the third order dispersion. We have found that when b3 < 0 then the asymmetry induced by the third order dispersion acts in the opposite direction to that induced by the pump depletion and thus can actually improve the quality of the output pulse. However, when b3 > 0 the effects of the asymmetries combine, destroying the linearity of the chirp, and can lead to the pulse developing oscillations on the long sloping training edge. Such effects can be seen in Fig. 3 where (a) shows the intensity profile and chirp (plotted on linear scales) and (b) shows the spectrum of the output pulse generated under the same conditions as that in Fig. 2(c) but this time with b3 ¼ 0:0018 ps3 m1 . Although in standard single-mode fibres b3 is typically positive over most wavelengths, both positive and negative values of b3 are possible in small-core microstructured fibres. Indeed, from the slope of the
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Fig. 3. (a) Intensity (left axis) and chirp (right axis) of the output obtained from simulated propagation in fibre C but with b3 ¼ 0:0018 ps3 m1 . (b) The corresponding spectrum.
dispersion profiles in Fig. 1(b), clearly in all cases considered here b3 < 0. From the results presented in Figs. 2 and 3 it is clear that the evolution of the signal pulse to the parabolic pulse regime depends strongly on the parameters of the microstructured fibre. Further design work could be performed to identify microstructed fibres that are optimised for parabolic pulse generation. However, previous investigations into parabolic evolution have shown that certain input pulse parameters in conjunction with the amplifier parameters allow for more efficient convergence to the parabolic regime [13]. Based on these results, we have conducted additional simulations to establish the input signal and pump pulse parameters which optimise the evolution to the parabolic pulse solution with the hope that a faster convergence to the parabolic regime will make the specific fibre design less critical. As a solution to Eq. (1) describing an analytic form for a Raman amplified parabolic pulse has yet to be determined, as a means of quantifying the level to which a pulse has become parabolic we introduce an rms intensity error between the numerically simulated pulse INUM ðT Þ ¼ jWðL; T Þj2 , where L is the fibre length, and the fitted parabolic pulse IFIT ðT Þ as [14]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 PN 2 u j¼0 ½INUM ðTj Þ IFIT ðTj Þ e I ¼ tN : ð3Þ P N 1 2 j¼0 INUM ðTj Þ N The results of the simulations have shown that the important pulse parameters are the input pulse width and the ratio of the peak powers of the pump and signal pulses. To illustrate this, Fig. 4(a) shows a plot of the rms intensity error as a function of the input pulse width (FWHM) where in all
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cases the input pulse energy is 5.3 pJ (as used for the input pulses in Fig. 2) and the fibre parameters correspond to fibre A. This clearly shows that for a fixed input pulse energy there is an optimum input pulse width in agreement with the earlier observations of [13]. The remaining curves of Fig. 4 show a comparison between parabolic pulses generated with the same input pulse and fibre as used to generate Fig. 2(a) (i.e. a signal peak power of 5 W) but with pump powers of (b) 40 W and (c) 100 W. As larger pump powers offer larger gains per unit length, the length of propagation was chosen to be 8 and 3 m for (b) and (c), respectively, so that in both cases the signal pulse experienced a gain of 28 dB. Although the intensity misfits for these output pulses are similar, (b) eI ¼ 0:17 and (c) eI ¼ 0:15, the steep sloping edges on the pulse in Fig. 4(c) (characteristic of a parabolic pulse
Fig. 4. (a) RMS intensity error as a function of input pulse duration (FWHM) for fixed signal pulse energy. (b, c) Top, intensity (left axis) and chirp (right axis), bottom, corresponding spectra, of the fibre outputs obtained from simulations with 40 and 100 W pump pulses, respectively. In all cases the fibre parameters correspond to fibre A.
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Fig. 5. Intensity profile of the output pulse from Fig. 2(b) after linear compression.
plotted on a logarithmic scale) indicate that this pulse is more parabolic [13]. Thus these results suggest that a larger gain per unit length can enhance the rate of evolution to the parabolic regime. Finally, to demonstrate the potential use of Raman amplified parabolic pulses for high-powered short pulse generation we considered the compression of the pulse in Fig. 2(b) via a simple linear grating pair. The resultant compressed pulse can be seen in Fig. 5 where the long pedestal on the trailing edge can be attributed to the asymmetry of the uncompressed pulse. Nonetheless, this still yields a pulse with a duration of 700 fs (FWHM) and a peak power of 1.4 kW and we expect that these results can be improved with the inclusion of higher order dispersion compensation in the compression stage.
3. Conclusions In conclusion, we have used numerical simulations to demonstrate parabolic pulse formation in a microstructured fibre Raman amplifier. The success of our results can be attributed to the large gain bandwidth offered by the pure silica host and the ability to tailor the dispersion profile of a microstructured fibre to provide normal dispersion over a wide range of wavelengths. The results have shown that the final shape of the parabolic pulse
depends on the fibre parameters. Importantly, the effects of pump depletion can be reduced with the appropriate sign of the third order dispersion and such values of b3 are currently available in microstructured fibres. In addition, we have also established the importance of the input signal and pump pulse characteristics to the evolution map of the parabolic pulse. The ease with which these pulses can be compressed suggests that Raman amplified parabolic pulses offer an efficient source of high-powered short pulses unrestricted by wavelength. We expect that they will find wide application in many areas of optical technology and further research to verify these results experimentally is currently under way.
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