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Lett. 92, 213902 (2004). 10. A. Ruehl, O. Prochnow, D. Wandt, D. Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, "Dynamics of ...... through a Marie-Curie grant.
Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device Christophe Finot, Lionel Provost, Periklis Petropoulos and David J. Richardson Optoelectronics Research Centre, University of Southampton, Southampton SO 17 1BJ, United-Kingdom [email protected]

Abstract: We numerically and experimentally demonstrate that pulses with a parabolic intensity profile can be formed by passive reshaping of more conventional laser pulses using nonlinear propagation in a length of normally dispersive nonlinear fibre. Moreover, we show that the parabolic shape can be stabilised by launching these pulses into a second length of fiber with suitably different nonlinear and dispersive characteristics relative to the initial reshaping fiber. ©2007 Optical Society of America OCIS codes: (060.4370) Nonlinear optics, fibers; (060.7140) Ultrafast processes in fibers.

(060.4510) Optical communications;

References and Links 1. 2. 3. 4. 5.

6. 7.

8. 9. 10. 11. 12. 13. 14. 15.

D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M. L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Am. B 10, 1185-1190 (1993). W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, "Optical wave-breaking of pulses in nonlinear optical fibers," Opt. Lett. 10, 457-459 (1985). D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, "Wave-breaking in nonlinear optical fibers," J. Opt. Soc. Am. B 9, 1358-1361 (1992). M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000). P. Dupriez, C. Finot, A. Malinowski, J. K. Sahu, J. Nilsson, D. J. Richardson, K. G. Wilcox, H. D. Foreman, and A. C. Tropper, "High-power, high repetition rate picosecond and femtosecond sources based on Ybdoped fiber amplification of VECSELs," Opt. Express 14, 9611-9616 (2006). C. Billet, J. M. Dudley, N. Joly, and J. C. Knight, "Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm," Opt. Express 13, 3236-3241 (2005). Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004). C. Finot, G. Millot, C. Billet, and J. M. Dudley, "Experimental generation of parabolic pulses via Raman amplification in optical fiber," Opt. Express 11, 1547-1552 (2003). F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004). A. Ruehl, O. Prochnow, D. Wandt, D. Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, "Dynamics of parabolic pulses in an ultrafast fiber laser," Opt. Lett. 31, 2734-2736 (2006). C. Finot, F. Parmigiani, P. Petropoulos, and D. J. Richardson, "Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime," Opt. Express 14, 3161-3170 (2006). T. Hirooka, and M. Nakazawa, "Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion," Opt. Lett. 29, 498-500 (2004). A. Latkin, S. K. Turitsyn, and A. Sysoliatin, "On the theory of parabolic pulse generation in tapered fibre," Opt. Lett. (to be published). S. V. Chernikov, and P. V. Mamyshev, "Femtosecond soliton propagation in fibers with slowly decreasing dispersion," J. Opt. Soc. Am. B 8, 1633-1641 (1991). B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42, 965-966 (2006).

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16. F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, "Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating," Opt. Express 14, 7617-7622 (2006). 17. S. Pitois, C. Finot, J. Fatome, and G. Millot, "Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers," Opt. Commun. 260, 301-306 (2006). 18. G. P. Agrawal, Nonlinear Fiber Optics, Third Edition (San Francisco, CA : Academic Press, 2001). 19. H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910-913 (1981). 20. C. Jirauschek, F. Ö. Ilday, and F. X. Kärtner, "A Semi-Analytic Theory of the Self-Similar Laser Oscillator," in Non Linear Guided Waves and their Applications (NLGW)(Dresden, 2005). 21. C. Finot, G. Millot, and J. M. Dudley, "Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers," Opt. Lett. 29, 2533-2535 (2004). 22. R. Trebino, Frequency-Resolved Optical Gating : the measurement of ultrashort laser pulses (Norwell, MA : Kluwer Academic Publishers, 2000). 23. F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, "Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating," IEEE Photon. Technol. Lett. 18, 829-831 (2006). 24. A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).

1. Introduction The generation and applications of optical pulses with a parabolic intensity profile has developed into an area of acute research activity over recent years. These pulses can propagate in a normally dispersive optical fiber in the presence of nonlinearity without loss of their characteristic parabolic profile [1], and thus do not suffer from the deleterious effects of optical wave-breaking [2, 3]. Moreover, the combination of normal dispersion and nonlinearity results in the development of a highly linear temporal chirp. As a consequence parabolic pulses are of great interest for a number of applications including amongst others high power pulse generation, and all optical signal processing. Since 2000, various methods have been suggested to generate such pulses. For example it has been demonstrated that parabolic pulses represent an asymptotic solution of the Nonlinear Schrödinger Equation (NLSE) with a gain term and thus that such pulses can be formed under appropriate conditions during pulse amplification in a normally dispersive fiber amplifier [4]. Indeed, parabolic pulses have now been experimentally demonstrated using rare-earth doped fibers (based on Ytterbium [4, 5] or Erbium [6, 7] dopants), as well as using Raman amplification [8]. The self-similar behaviour of these pulses has led to the term “parabolic similariton” to describe parabolic pulses formed as a result of the interplay of gain, dispersion and nonlinearity. The use of optical amplifiers or similariton lasers [9, 10] for the generation of high quality, high-intensity ultrashort pulses [4-6, 11] is of great scientific and technological interest. However, for many applications, the requirement for amplification is undesireable since this requires a high power pump source and adds cost and complexity to the system. Hence parabolic pulse generation schemes based on passive fiber components are of great interest and a variety of possibilities have been proposed. The use of dispersion decreasing fibers to reshape discrete optical pulses has been theoretically suggested [12, 13], the decreasing dispersion acting in a similar way to gain in terms of pulse reshaping [14]. The feasibility of such an approach has recently been demonstrated by using a dispersion decreasing comb-like fiber [15]. A completely different approach relies on simple and direct linear pulse shaping using a superstructured fiber Bragg grating [16]. Another proposed solution is to convert a sinusoidal beat signal into a train of parabolic pulses during nonlinear propagation in a normally dispersive fiber [17]. In the present article, we propose a new and simple passive method of generating parabolic pulses that relies upon nonlinear propagation in discrete sections of commercially available normally dispersive fibers. We first show how an initial Gaussian pulse evolves during propagation in a normally dispersive fiber and reshapes at a particular point into a pulse with a parabolic temporal intensity profile with a nearly linear chirp. We then discuss the influence of the initial pulse energy as well as the strong influence of the initial pulse shape. We next #76338 - $15.00 USD

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present experimental results based on an accurate characterization both in intensity and phase which confirm our theoretical conclusions. Finally, we show numerically and experimentally that the resulting parabolic pulse can maintain its profile during propagation in a second length of highly nonlinear fiber of suitably chosen dispersion and nonlinearity. 2. Progressive nonlinear reshaping of a Gaussian pulse in a normally dispersive fiber 2.1 Modelling We consider the longitudinal evolution of the complex electric field ψ (z,T) which can be modelled by the standard NLSE [18] :

β ∂ 2ψ ∂ψ 2 = 2 − γ ψ ψ , 2 ∂T 2 ∂z

i

(1)

with β2 the second order dispersion, γ the non-linear coefficient, T the time in a copropagating time-frame and z the propagation distance. In the normally dispersive case, Eq. (1) can be transformed into the following normalized form [18] : i

∂u 1 ∂ 2u 2 = − u u , 2 ∂τ 2 ∂ξ

(2)

with u, τ and ξ normalized parameters defined by : U (ξ , τ ) =

u ( ξ ,τ ) = N U ,

ψ

τ =

,

PC

T , T0

ξ =

z . LD

(3)

T0 and PC are respectively the characteristic temporal width and the peak power of the initial pulse. LD, LNL and N are respectively the dispersion length, the nonlinear length and the “soliton” number defined as : LD =

T02 , β2

LNL =

1

γ PC

,

N =

LD . LNL

(4)

In Fig. 1, we present the evolution of an initial Gaussian pulse U(0, τ ) = exp(-τ 2 / 2) with N = 4 in a normally dispersive fiber. Figure 1(a) illustrates the interplay between dispersion and nonlinearity which leads to a progressive reshaping of the pulse [19]. At ξ = 0.2 (Fig. 1(a3)), the pulse exhibits a parabolic intensity profile, as confirmed by the good agreement between the intensity profile and a parabolic fit which shows only a small discrepancy in the wings of the pulse. Beyond this distance, the intensity profile begins to deviate more strongly from the parabolic shape with a flattening of the top of the pulse (Fig. 1(a4)). In order to quantify the evolution towards a parabolic shape, different approaches are possible. Ruehl et al. have used the kurtosis parameter [10] whereas Jirauschek et al. based their analysis on the use of an ansazt [20]. We prefer to compute the evolution of the misfit parameter M between the pulse intensity profile |u (τ)|2 and a parabolic fit |p(τ)|2 of the same energy [11] : M2 =

⎡ ∫⎣u

2

− p

2

⎤ ⎦

2



u dτ 4



(5)

with p(τ) corresponding to the normalized intensity profile |p(τ)|2 = 1 – (τ / τP)2 if τ ≤ τP and |p (τ)|2 = 0 otherwise. The trends shown in Fig. 1(a) are confirmed in Fig. 1(b) : M(ξ) reaches a minimum Mopt after a distance ξopt(N = 4) = 0.2 where the pulse has an intensity profile close to the desired parabolic shape. #76338 - $15.00 USD

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Fig. 1. (a) Evolution of the temporal intensity profile of an initial Gaussian pulse in a normally dispersive fiber for N = 4 and for different propagation distance ξ : 0, 0.1, 0.2 and 0.6 (Fig. a1, a2, a3 and a4 respectively). Results from numerical integration of Eq. (2) (solid lines) are compared to parabolic fits (blue circles). The discrepancy between the two is highlighted by the yellow shaded areas. (b) Evolution of the misfit parameter M versus the normalized propagation distance ξ for N = 4.

2.2 Influence of N We next discuss the influence of the initial power, or equivalently the value of N on the pulse reshaping process. We have plotted on the map presented in Fig. 2(a) the evolution of M versus N (ranging from 1 to 12) and ξ (between 0 and 1). We can see that ξopt depends strongly on N, as does the behaviour of the pulse after ξopt. When N is increased, ξopt decreases. We can also compare the values of ξopt for which Mopt is reached with the analytical condition proposed by Anderson et al. to predict the distance ξwb where the wave-breaking occurs [3] (Fig. 2(a), white solid line) : 1 ξ wb = . (6) 4 exp(−3 / 2) N 2 − 1 It is clear that the distance for which the pulse becomes parabolic precedes the onset of wave-breaking (let us recall that Eq. (6) is only accurate for high N or small ξwb [3], i.e. ξopt 4, the extrema lie within the central part of the pulse, τext being less than the FWHM of the parabolic pulse. From this non-monotonic chirp profile, we can anticipate that the pulse will be subject to wave-breaking deformations on further propagation [3]. Again, this constitutes a major difference compared to the asymptotic generation of a parabolic similariton which leads to a perfectly linear chirp [4].

Fig. 3. (a) Temporal intensity and chirp profiles at ξopt for different N values (N = 2, 2.6, 3 and 4, curves A, B, C, D, mixed, solid, dotted and dashed lines respectively). The initial Gaussian pulse is presented in blue. (b) Evolution of τext (solid line) versus N compared to the temporal widths of the pulse at various intensity points (mixed lines)

We also studied the evolution of Mopt versus N (Fig. 4, blue line). We can see that N influences Mopt and that an optimal N exists: for N = 2.6, the misfit parameter is as low as 0.033. For this Nopt value, according to Fig. 3(b), τext remains below the -7 dB temporal width so that we can consider the chirp extrema to lie within in the wings of the pulse and not in its central part.

Fig. 4. Evolution of Mopt versus N for different initial pulse shapes: Gaussian, sech, truncatedcosine or supergaussian (blue, red , green and purple solid lines respectively). Results are obtained at the optimum distance ξopt adapted for each initial condition.

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3. Influence of the initial pulse shape We highlighted in the previous section the strong influence of N on pulse evolution for an initial chirp-free Gaussian pulse. We now describe to what extent the initial pulse shape can affect the nonlinear evolution towards a parabolic intensity profile in a normally dispersive fiber. Specifically, we consider the evolution of hyperbolic-secant pulses ψ(0,Τ) ∝ sech(Τ /T0), truncated-cosine pulses ψ(0,Τ) ∝ cos(T/T0) rect(Τ /π T0) (with rect(Τ /π T0) the unity rectangular gate function of total width π T0) and supergaussian pulses ψ (0, T ) ∝ exp(− T 4 / 2T04 ) . Fig. 5(a) compares the intensity profiles of these different initial pulse shapes which are in each case specified to have the same FWHM temporal width and energy.

Fig 5. (a) Comparison of intensity profiles of different pulse shape with the same FWHM and the same energy (parabolic pulse are presented in black solid line, other pulses are plotted with the same convention as Fig. 4). (b) Evolution of the misfit parameter M versus N and ξ for a sech pulse.

In Fig. 5(b), we have plotted a map similar to that presented in Fig. 2(a) but using now a sech pulse as the input. We observe that pulse reshaping is again obtained and that just as for the Gaussian case an operational band exist for which the output intensity profile approaches the parabolic shape. We should however point out some major differences. Firstly, the band corresponding to M < 0.01 is much narrower than for a Gaussian input pulse. Secondly, the optimal values ξopt = 1 and Nopt = 1.4 also differ significantly from the values obtained for a Gaussian pulse. We also present similar maps for truncated cosine pulses (Fig. 6(a)). For this type of pulse, two bands where M is low exist. The results obtained here are consistent with those demonstrated by Pitois et al. [17] who have used the progressive reshaping of a sinusoidal beat signal in order to obtain a parabolic pulse profile before compressing them to generate short pulses. The corresponding experimental conditions used in that case are denoted by the cross E. Note that the slight difference with respect to the theoretical optimum of the first band can be accounted for by considering the influence of fiber loss within these experiments. It is also to be appreciated that in this work the constraint of having a continuous train of cosine pulses restricted operation to this first band in order to avoid temporal broadening and hence overlap of the pulses. However, as will be shown in more detail for the case of supergaussian pulses, the choice of band 2 seems more suitable if the target is to generate parabolic pulses with a nearly linear chirp profile. The optimum Mopt obtained for this band is ξopt = 0.27 and Nopt = 7.38.

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Fig. 6. Evolution of the misfit parameter M versus N and ξ for a truncated cosine pulse (a) and for a supergaussian pulse (b).

We next study the evolution of a supergaussian initial pulse, the corresponding map is presented in Fig. 6(b). Three different bands can be distinguished. Details of the evolution of M for N = 16.5 is presented in Fig. 7(a). The temporal intensity and chirp profiles corresponding to the three minima of M are presented in Fig. 7(b). We can see that for the first minimum, the chirp profile of the pulse differs significantly from the behaviour obtained in the case of a Gaussian pulse as presented Fig. 3(a): the chirp at the centre of the pulse is not linearly increasing (which is also the case for sech and cosine pulses) but on the contrary is nearly flat. This can simply be explained by noting that whereas cosine, Gaussian or sech pulses can be approximated by a parabola at the pulse center, this is not the case for a supergaussian. As a result, it is clear that the pulses obtained in the first band are bound to be heavily affected by wave-breaking. It is thus preferable to focus our attention on the second band, where we can see that the chirp is far more linear. In the third band (Fig. 7(b2)), the linearity of the chirp is very high, and the pulse shape very close to parabolic with an Mopt as low as 9 10-3 for Nopt = 16.5 and ξopt = 0.36. However, within this regime, the pulse undergoes very significant temporal broadening, which can be a major drawback for many practical applications.

Fig. 7 : (a) Evolution of M for N = 16.5 in the case of a supergaussian pulse. (b) Temporal intensity and chirp profiles obtained at the different propagation distances ξ (positions 1, 2 and 3 on Fig a). Initial supergaussian pulse is plotted in purple solid line.

In conclusion, in this section we have shown (through maps Fig. 2(a), 5(b), 6 and Fig. 4) that the initial pulse shape has a strong influence on the passive evolution towards a parabolic profile in a normally dispersive fiber. It is to be re-iterated that this represents a major difference compared to asymptotic similariton based techniques where the resulting parabolic pulse generated is independent of the initial pulse shape [4, 21]. #76338 - $15.00 USD

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According to the current approach then one should carefully adapt the length of fiber ξopt used and the initial power Nopt for a given input pulse profile in order to optimize the convergence to a parabolic shape. In order to illustrate how significantly the parameters can differ for particular input pulse shapes, we present in Table 1 the length of fiber L and the initial peak power PC needed for optimal conditions. (For the purpose of these calculations we have considered the propagation of a 7 ps FWHM initial pulse in a dispersion compensating fiber with β2 = 0.146 ps2.m-1 and γ = 8.7 10-3 W-1.m-1.) Finally, one can also see that the higher Mini (i.e . the greater the initial pulse deviates from a parabolic pulse), the longer the fiber and the lower the initial power required. Table 1. Comparison of optimal parameters for different initial pulse shapes

ψ (T) exp( - T² / 2T0² ) Gaussian Hyperbolic sech( T / T0 ) secant Trunc. cosine Cos (T/To) x (second band)

rect(T/ πTo)

Supergaussian exp( - T4 / 2T04 ) (second band)

Mini

ξopt

Nopt

Lopt

PCopt

m

W

Energy pJ

0.131

0.42

2.6

51

6.4

47

0.176

1

1.4

108

2.1

17

0.075

0.27

7.38

36

46

320

0.053

< 0.065

> 15

239

> 1440

4 Experimental results In order to validate our theoretical results, we have carried out the following experimental verification. The experimental set-up is presented in Fig. 8(a). We used Gaussian pulses generated by a gain-switched DFB laser operating at a repetition rate of 10 GHz at 1550 nm. The initial chirp of the pulses at the output of the laser is nearly fully compensated by using 200 m of DCF fiber, so that the pulses are nearly transformed-limited (time-bandwidth product of 0.49, see the inset of Fig. 8(a)) with a FWHM of 7.0 ps. The pulses were then amplified using a high-power erbium fiber amplifier. Note that in order to reach higher pulse energies we had the option to use a modulator in order to decrease the repetition rate down to 5 GHz or 2.5 GHz. The passive reshaping into parabolic pulses is obtained by propagating the pulses in 50 m of dispersion compensating fiber (all critical parameters of the fiber are presented in the previous section). The input and output pulses were characterized in intensity and phase using a secondharmonic generation frequency-resolved optical gating device (SHG-FROG) [22]. The quality of the retrieval was checked by usual means (comparison of the results with independentlymeasured autocorrelation and spectrum). We highlight the good agreement obtained in Fig. 8(b) between the experimental results (blue circles) and those of numerical simulations (solid lines) based on the numerical solution of the NLSE. We incorporate in our model the effect of third order dispersion β3 = 1.3 10-3 ps3.m-1 and take as an initial condition ψ(0,T) the input pulse form as experimentally characterized by the FROG technique (the experimental FROG results are presented in the inset of Fig. 8(a)). We can clearly see that for low energy (11 pJ, corresponding to N = 1.1) the output pulses are still very close to Gaussian in profile. The reshaping is obvious for higher power (47 pJ, N = 2.6) where the retrieved intensity profile is in very good agreement with a parabolic fit. The small discrepancy in the falling edge of the pulse can be attributed to slight deviations of the initial pulse shape from the idealised Gaussian form. The temporal chirp is linear over most of the centre of the pulse and well into the wings as expected. The quality of the fit to a parabolic profile decreases for even higher powers (111 pJ, N = 4) and flattening of the top of the pulse is observed as predicted. These experimental results confirm our simulations and theoretical approach, and demonstrate the possibility to generate parabolic intensity pulses with a nearly linear chirp. #76338 - $15.00 USD

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Fig. 8. (a) Experimental set-up used to passively generate parabolic pulses. The initial experimental pulses characterized by FROG are plotted in the inset (circles) and compared with a Gaussian fit (solid black line). (b) Temporal chirp (Fig. b1) and intensity (Fig. b2) profiles. The results are for initial pulse energies of 11, 47 and 115 pJ (repetition rate of 10, 5 and 2.5 GHz respectively). Experimental results obtained by the FROG technique (blue circles) are compared to numerical simulations (solid black line) and to Gaussian or parabolic fits (green dashed and red mixed lines respectively). The misfit area is shaded in yellow.

3. Two-stage device We now show how, under certain conditions, a parabolic intensity profile generated through the preceding approach can be maintained during subsequent propagation in a further piece of fiber. To do this, we use a double stage device. This would enable us to work with a high soliton number in the second fiber, and then to use the parabolic pulse for signal processing [23] or spectral broadening for recompression [16]. 3.1 Principle As we pointed out in the first part of this paper it is not possible to generate a stable parabolic pulse with a high N in a single piece of fiber. To overcome this limitation, we propose to first generate a parabolic pulse in a length of fiber under optimal conditions as discussed previously (for a Gaussian pulse, Nopt = 2.6, which leads to a close to linear chirp across the entire pulse). We then stabilise the parabolic pulse shape by abruptly increasing N (to a value N’) by launching the pulses into a second fiber with different characteristics. We define N’ in the second fiber by : T 2 γ ' G PC N '2 = 0 . (7) β '2 where β2’ and γ’ are the parameters of second fiber, and T0 and PC relate to the pulse parameters at the input of the first fiber. G is an optional gain (or loss) between the two fibers. Under these conditions pulse propagation in the second fiber segment will be dominated by the non-linearity ensuring that the linearly chirped parabolic pulses maintain their shape even in the presence of dispersion [1]. Two approaches can be used to increase N’ relative to N. Firstly, it is possible to amplify the pulses before onward propagation. Secondly, it is possible to choose a different fiber such that the ratio of γ’ / β2’ of this fiber is higher than γ / β2 of the first fiber. The second approach has the advantage of being completely passive and avoids any potential perturbation of the pulse due to amplification itself. We have plotted in Fig. 9(a) a map similar to Fig. 2(a). The white solid line corresponds to the points where M is optimal, and at which physical position we have performed the abrupt increase in N to N’. To illustrate the method, we have chosen here to use N’ = 8. Within Fig. 9 we can readily distinguish a band of operating conditions for which the misfit parameter M #76338 - $15.00 USD

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remains very low. In agreement with our previous observations this band corresponds to N values between 2 and 2.7, for which the pulse before the second segment is parabolic and has a linear chirp for most of its duration.

Fig. 9. (a) Evolution of the misfit parameter M versus N and ξ for an initial Gaussian pulse. The change from N to N’=8 is done when Mopt is reached in the first segment. (b) Evolution of M in the second stage for a parabolic pulse generated in the second stage (solid black line). The results are compared with the evolution a Gaussian (blue line) or sech (red line) pulse of the same FWHM temporal width, same linear chirp and same energy launched in the second fiber. Inset: intensity profile at the output of the second fiber.

To outline the benefits of using pulse shaping in the first fiber section we compare in Fig. 9(b) the evolution of the M parameter for parabolic pulses generated in this fashion in the second fiber segment with the evolution of Gaussian or sech pulses of the same linear chirp, temporal width and energy. The parabolic pulses are the only ones able to maintain their shape during onward propagation whereas the other pulses exhibit large changes in M parameter. We can also see in the inset of Fig. 9(b) that the parabolic pulses at the output of the second segment do not exhibit the characteristic signs of wave-breaking effects in the wings of the pulse (as observed for both the Gaussian and sech pulses) [2]. 3.2 Spectral properties We next study the spectral reshaping of the pulse. Fig 10(a) shows the spectral intensity profile of the initial pulse (dotted line), after the first stage (mixed line) and after the second stage (solid line, for N’ = 8 and ξ = 4). We can see that during the propagation in the first stage the spectrum has only broadened slightly, indicating relatively moderate nonlinear effects. At this stage, the spectrum does not have the parabolic spectral intensity profile characteristic of a highly chirped parabolic pulse [1]. The result is very different at the output of the second stage, where a large degree of spectral broadening and reshaping is observed which can be fitted well with a parabolic shape. The parabolic pulse is thus seen to undergo large SPM-based spectral broadening without evidence of any oscillatory structure [16], or the large wings characteristic of wave-breaking [2]. We have also plotted the evolution of the FWHM spectral width of the pulse. The parabolic spectral width increases rapidly in the second segment before saturating. The evolution here is consistent with the analytical formulation given by Anderson et al. describing the evolution of a parabolic pulse in a non-linear medium [1, 11]. This saturation of the spectral width is different from the behavior expected in a self-similar amplifier where the spectral width continuously increases [11], the only limitation being the amplification bandwidth of the fiber amplifier [24].

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Fig 10. (a) Spectral intensity profiles of the initial Gaussian pulse (grey dotted line) and of the parabolic pulse obtained by numerical simulations at the output of the first segment (N = 2.6, mixed line). Results at the output of the second segment (N’ = 8, ξ = 4) are compared to a parabolic fit. (b) Longitudinal evolution of the FWHM spectral width of the pulse. Numerical results (solid line) are compared with analytical results (blue diamonds).

3.3 Stability of the stage In the last section we have chosen N’ = 8. We show here that our qualitative results do not depend on a specific value of N’, as long as N’ is sufficiently higher than N. Fig. 11(a) shows the evolution of the misfit function Mout at the output of the second segment (ξ = 4). We can see that even with N’ = 5, Mout remains quite low, illustrating the fact that the parabolic intensity profile is preserved. The higher N’ the lower the achievable value of Mout. We can also see that increasing N’ leads to an increase of the temporal width of the output pulse, thereby providing a simple means to tune the temporal width of the output parabolic pulse.

Fig 11. (a) Influence of the N’ value on the evolution of the misfit parameter Mout at the output of the second segment and on the evolution of the output temporal FWHM. (b) Map similar to the one presented Fig 9(a) but for a sech initial pulse.

We can also check from Fig. 11(b) that our conclusions are still valid with an initial sech pulse, as long as we use parameters for the first segment close to the optimum parameters defined in Section 3. The same conclusions are also applicable to parabolic pulses generated through reshaping of truncated cosine or supergaussian pulses (with pulses in the second or third band), as long as the N’ value is always higher than the optimum N value.

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Received 24 October 2006; revised 7 December 2006; accepted 7 December 2006

5 February 2007 / Vol. 15, No. 3 / OPTICS EXPRESS 862

3.4 Experimental results To validate our numerical approach, we decided to launch the parabolic pulses generated in our first stage experiment into a second stage based on a 1 km highly non-linear fiber. The parameters of this second stage were β'2 = 2.16 10-3 ps2.m-1 and γ’ = 18.7 10-3 W-1.m-1. Since we wished to conduct this demonstration at a repetition rate of 10GHz then the power limits of our amplifier meant that the maximum initial energy available at the input of the first fiber was limited to 31 pJ, corresponding to N = 2.1. Including the losses between the two segments of fiber (due to non-optimal splicing between DCF and HNLF fiber, G = -2.5 dB) then, according to the definition of Eq. (7), we obtain an estimate of N’ = 8.8N. Although the initial N value was somewhat below the optimal value of N=2.6, we can see from the maps in Fig. 2(b) and Fig. 9(a) that the parameters used are still quite close to the optimum region so that we would not expect any great deterioration of pulse quality at the first stage output. We confirmed this using FROG measurements and, as can be seen in Fig. 12(a), the pulses generated at the output of the first stage (pink diamonds) remain reasonably close to the desired parabolic pulse form (red mixed line).

Fig. 12. Temporal chirp (top) and intensity (bottom) profiles of the pulses characterized by FROG after the first stage (pink diamonds) and after the second stage (blue circles). Experimental results are compared with parabolic fits (red mixed lines) and with numerical simulations (solid black line).

The temporal intensity and chirp profiles after propagation in the second stage HNLF are shown with blue circles and we can see the good agreement between the numerical simulations based on the NLSE which includes also the effects of third order dispersion β'3 = 34 10-6 ps3.m-1 and the linear losses α. The output pulses are seen to exhibit an intensity profile in good agreement with a parabolic fit, and a highly linear chirp. These experimental results confirm our proposed approach and the possibility to generate parabolic pulses at high repetition rates in a completely passive way. These pulses are then able to maintain their shape during further propagation in fiber. Compared to previous demonstrations at high-repetition rate, this approach relaxes the need to use an amplifying medium (and thus the need for a pumping source) [5, 7], or the need for additional custom devices [16]. Note also that this method has the advantage of being highly tuneable in terms of wavelength, and can be straightforwardly implemented with widely available commercial fibers. #76338 - $15.00 USD

(C) 2007 OSA

Received 24 October 2006; revised 7 December 2006; accepted 7 December 2006

5 February 2007 / Vol. 15, No. 3 / OPTICS EXPRESS 863

4. Conclusion We have demonstrated the possibility of generating linearly chirped parabolic pulses in a completely passive manner simply by using progressive pulse reshaping in a normally dispersive fiber. By carefully choosing the initial power and the length of the fiber according to the nature of the pulses, one can expect to have a parabolic pulse at the system output. This offers thus a very promising and attractive way to passively generate parabolic pulses. Such pulses have a range of potential uses including ultrashort pulse generation and compression, multi-wavelength pulse generation [7, 23], and signal processing [23]. Our theoretical results are confirmed experimentally. Furthermore, we anticipate that these results might be of great interest in the field of the parabolic lasers where the propagation of a pulse in a normally dispersive fiber leads to a high degree of pulse reshaping [10]. Acknowledgments Christophe Finot gratefully acknowledges the financial support of the European Union through a Marie-Curie grant. We acknowledge John Dudley and Kazunori Mukasa for fruitful discussions and Furukawa Electric for the loan of the highly non-linear fiber used in these experiments.

#76338 - $15.00 USD

(C) 2007 OSA

Received 24 October 2006; revised 7 December 2006; accepted 7 December 2006

5 February 2007 / Vol. 15, No. 3 / OPTICS EXPRESS 864