Dispersion managed self-similar parabolic pulses

Transcription

1 Disersion managed self-similar arabolic ulses Christohe Finot To cite this version: Christohe Finot. Disersion managed self-similar arabolic ulses. Journal of Otics A: Pure and Alied Otics, IOP Publishing, 008, 10 (8), HAL Id: hal htts://hal.archives-ouvertes.fr/hal Submitted on 15 Ar 010 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 Disersion managed self-similar arabolic ulses C FINOT Institut Carnot de Bourgogne, UMR 509 CNRS-Université de Bourgogne, 9 Av. A. Savary, Dijon, France christohe.finot@u-bourgogne.fr Abstract. We describe the roagation of a arabolic self-similar ulse in an anomalous disersive nonlinear fibre. Given the caacity of a linearly chired arabolic ulse to maintain unaffected its tyical shae over a short roagation distance, we introduce the concet of disersion-managed self-similar ulses and outline otential benefits in terms of sectral broadening enhancement. Keywords. Parabolic ulses, nonlinear roagation, nonlinear fibre 1. Introduction Since their first exerimental demonstration in 000 in normally disersive fibre amlifiers [1], the unique roerties of otical arabolic ulses have generated a great deal of attention. Indeed, irresective to its initial shae, any ulse rogressively evolves towards a arabolic intensity rofile combined with a strictly linear chir []. This tyical shae exeriences then a self-similar evolution, with exonential sectral and temoral broadenings and remarkably resists to the deleterious effects of the otical wave-breaking [3, 4]. Several methods have been successfully tested to generate arabolic ulses : rare-earth doed fibres (ytterbium [1, 5] or erbium [6, 7] ), Raman amlifiers [8] as well as a wide set of attractive assive means such as suerstructured fibre Bragg gratings [4], a set of two carefully chosen fibres [9], comblike fibre [10] or disersion decreasing fibre [11]. The exerimental interest has strongly been driven by the ossibility to fully cancel the linear chir and thus to generate high-ower ultrashort ulses with noticeably low substructures [1, 4-6, 1, 13]. Other alications in the telecommunication field such as multiwavelength sources [4] have also taken advantage of the broadened smooth sectrum of arabolic ulses []. Previous works have reorted the role layed by the disersive roerties of the fibre and arrangements of fibres with various normal disersion values have been suggested to efficiently generate arabolic ulses [9-11]. The urose of the resent article is to extend the concet of arabolic similaritons to a combination of fibres with alternating negative and ositive disersions. Based on disersion management, a concet that has found imressive alications in high bit-rate otical transmissions [14-16], we demonstrate that a arabolic ulse can maintain its unique features even in resence of anomalous disersion and nonlinearity. This article is thus organized as follows. We first describe the model we use and the arameters chosen to characterize the ulse evolution. We then focus on the roagation of arabolic ulses in the

3 anomalous regime, with or without nonlinearity. Evolution in disersion managed fibres is discussed and arabolic tyical behaviour is comared with other ulse shaes. Finally, we investigate the influence of the average disersion.. Model The evolution of the comlex electrical field ψ(z,t) of a ulse roagating in a assive otical fibre in resence of second order disersion β and Kerr nonlinearity γ can be described by the well-known scalar nonlinear Schrödinger equation (NLSE) [17] : ψ β ψ i = γ ψ ψ. (1) z t with z the roagation distance and T the time in a coroagating frame. For the sake of simlicity, we do not include in the resent discussion the effects of higher-order linear and nonlinear effects such as third order disersion, linear attenuation or Raman scattering [17]. We study the evolution of an initial arabolic ulse ψ having the following characteristic rofile : t C ψ ( t) = P 1 ex i t if t T T () ψ ( t) = 0 if t > T with T, P and C the temoral width, the eak ower and the temoral chir coefficient of the arabolic ulse resectively [1, ]. As long as the chir coefficient C is high, given the stationary hase theorem, the sectral shae ψ ( ω ) reflects the temoral rofile and is then arabolic and linearly chired [18] : π P ω π C ψ ( ω) 1 ex i ω + (3) C ω 4 with ω and C the sectral width and chir coefficient resectively given by : ω = C T and C = 1 C (4) Equation (1) will be numerically integrated by using the standard slit-ste Fourier method [17]. In order to avoid some numerical artefacts due to the unhysical comact nature of the arabolic ulse, we have used a slight temoral aodization window based on a 5 th -order Gaussian shae [4] with a full width at half maximum (FWHM) of 1.6 T. We have carefully checked that this aodization window does not influence our results. During the roagation in the fibre, the ulse will be affected by the combined effects of non-linearity and disersion. To qualitatively illustrate the ulse dynamics and similarly to the slit-ste Fourier algorithm, let us consider non-linearity and disersion acting searately on an infinitesimal length δz. On the one hand, second order disersion alone induces in the sectral domain an additional chir given by δ C = β δ z. Given the relationshi (3), the arabolic shae is maintained and the disersion leads in the temoral domain to new chir coefficient C, eak ower P and temoral width T : P C ' ' ' P =, T = (1 + β C δ z) T and C =. (5) 1 + β C δ z 1+ β C δ z On the second hand, self-hase modulation (SPM) alone as defined in this model does not affect the temoral intensity rofile but results in an additional temoral linear chir given by [1, 17]: ' ' δ C = γ δ z P T (6)

4 We can then see that the evolution highly deends on the disersion regime. For normal disersion, nonlinearity and disersion act conjointly to increase the chir coefficient C + δc and the temoral width of the ulse increases. On the contrary, it is clear from (5) that the ulse will shorten in the anomalous disersion regime. In order to more rigorously redict the evolution of the ulse arameters, the following set of couled equations can be used [3, 1] : T γ β = T 0 P 0 z T 1 T C = (7) β T z T0 P 0 = T P with T 0, P 0 and C 0 the initial ulse roerties. Equations (7) have been derived from NLSE and imlicitly assume that the arabolic tyical rofile is maintained over the roagation. In order to check this assumtion, we characterize the ulse shae thanks to the misfit factor between the ulse intensity and the aodized arabolic ulse ψ [1, 18] : 4 M = ψ ψ _ fit dt ψ dt _ fit (8) To illustrate this article, we study an initial arabolic ulse obtained for examle in an erbium doed similariton amlifier at telecommunication wavelengths [6] and having the following roerties : T 0 = 5 s, P 0 = 500 W and C 0 = 1.7 THz.s -1. We consider the evolution in assive fibres with a nonlinearity γ = W -1.m -1. Both disersion regimes will be investigated with fibres having disersion values of β 1 = s.m -1 (fibre 1) and β = s.m -1 (fibre ). Such fibre arameters are tyical of non-zero disersion-shifted fibres [19] with reduced disersion sloe. Let us add that the following results could also be demonstrated at other wavelengths such as 1.06 µm where high ower arabolic ulse generation has been successfully reorted [1, 5]. Indeed, taking advantage of the rogresses in microstructured design and fabrication techniques, using hotonic crystal fibres [0] is conceivable with the additional otential benefit of a reduced third-order disersion. 3. Parabolic ulse roagating in a single assive fibre We first examine the behaviour of a arabolic ulse in a single 15-m fibre. Let us recall that in the case of a single normally disersive fibre (fibre ), the arabolic ulse undergoes a temoral broadening illustrated in figure 1(a1). We can also make out that the increase of the temoral width (figure 1(b), solid black line) is accurately described by (7) and that the arabolic shae is ket unaffected as highlighted by the low misfit factor seen in figure 1(c) [3, 9]. A negative initial chir arameter would have lead to a different dynamics, with a noticeable sectral comression stage [1, 1]. In an anomalous medium, the evolution is rather different. Let us consider the evolution in absence of nonlinearity, as it can haen when using a air of linear diffraction gratings [1, 5], a hollow core hotonic bandga fibre [6, 0] or a strongly attenuated ulse [4, 13]. As it can be seen in figure 1(a) and figure 1(b) (results obtained with fibre 1, dotted line), the ulse undergoes a temoral comression leading to a transformed limited ulse at a distance z c given by : 1 z = (9) c C β P0 A with β A the second order disersion of the anomalous medium. At the otimum comression stage, the shae is not arabolic anymore but becomes a Bessel function of the first kind [1, 13, 18], which exlains the significant increase of the misfit factor. Following this comression stage, the ulse undergoes temoral broadening. In order to better outline this

5 broadening, we have also lotted in figure 1(b) the evolution of a ulse having a higher initial chir coefficient C 0 of.56 THz.s -1 : according to (9), the comression occurs faster and we can note that after the comression stage, the ulse broadens again and regains its tyical arabolic intensity rofile. Indeed, as in a urely disersive media the ulse sectrum does not change, the relationshi between equations () and (3) is still valid in the limit of a highly anomalous chir. Figure 1 : (a) Evolution of the intensity rofile with resect to the roagation distance in the case of a arabolic ulse roagating in a normally disersive nonlinear fibre (fibre ) (a1), in an anomalous disersive fibre (fibre 1) without (a) and with (a3) nonlinearity. Evolution of the FWHM temoral duration of the ulse (b) and of the misfit arameter M (c) of a ulse roagating in a normal disersive nonlinear fibre (solid line), in an anomalous disersive fibre without (dotted line) and with (dashed line) nonlinearity. Results obtained for an initial chir coefficient of C 0 = 1.7 THz.s -1 and C 0 =.56 THz.s -1 are lotted in black and grey resectively. Results redicted by (7) are lotted with circles. Otimal comression distance as redicted by (9) is indicated by full diamonds. Let us now evaluate the imact of Kerr nonlinearity (with the chir arameter C 0 = 1.7 THz.s -1 ). On the first stage (below 3 m), the ulse evolution is similar to the comression obtained without nonlinearity : the arabolic ulse shortens and maintains its shae as assessed by the low misfit factor. A direct consequence of the temoral comression is a eak ower enhancement so that nonlinear effects have an increasing role. This results in a rather different evolution as it can be visualized in figure 1(a3). Indeed, the dynamics of the ulse is severely imaired by soliton comression and ulse break-u [17, ] : the temoral comression of the ulse is faster than in the case of a urely disersive roagation and more remarkably, the ulse slits into four ulses evolving with various velocities. Parabolic ulse aroximation becomes obviously comletely out of context and contrary to a urely disersive fibre, the arabolic shae is not regained after further roagation. One can also notice that the best recomressed ulse is achieved in a nonlinearity free fibre, whereas in resence of nonlinearity, much higher sidelobes are resent [18, 3].

6 4. Parabolic ulse roagation in a disersion-managed fibre In the receding section, we have ointed out that arabolic ulses evolving in an anomalous medium in resence of nonlinearity were able to maintain their arabolic features only uon a reduced distance. We now study the evolution of the ulse in a fibre with anomalous disersion (fibre 1) followed by a normally disersive one (fibre ). Both fibres have the same length (3 meters) so that the average disersion of this ma (ma 1) is null. Temoral and sectral evolutions are lotted in figure (a,b) resectively. After the temoral comression stage described in art 3, the ulse undergoes both temoral and sectral broadening in the second segment. Note that the ulse maintains a arabolic intensity rofile as well as a high linear chir during the whole roagation. The initial and final rofiles are comared in figure (c-d). Whereas the temoral intensity rofile does not exhibit significant changes, the most striking oint is that due to SPM, the ulse sectrum has noticeably broadened. This confirms that, as the initial and final roerties are not identical, arabolic ulses are intrinsically different from analytical stationary solutions such as disersion managed solitons [16] or chired solitons observed in an all-normal disersion fibre lasers [4]. Figure : Longitudinal evolution of the temoral (a) and sectral (b) rofiles of a arabolic ulse in a set of two fibres (ma1). Initial (solid line) and final (dashed line) temoral (c) and sectral (d) rofiles are comared. Let us now investigate the evolution in a concatenation of the receding ma. The roagation over four mas is dislayed in figure 3(a). The evolution is arabolic over the first three mas, but the rogressive growth of the misfit factor (figure 3(b)) reveals a degradation of the intensity rofile. The arabolic ulse finally collases. To better understand the origin of this dramatic ulse change, let us consider the sectra recorded after, 3 and 4 mas (figure 3(c)). After three mas, the growth of unwanted bands of amlified regions becomes aarent. This can be interreted as a manifestation of the scalar stimulated modulation instability (MI) which occurs in the anomalous segment [17] and which selectively amlifies bands of

7 the arabolic broad sectrum. As sectral and temoral rofiles are in the case of highly linear chired ulses closely linked, the change of the sectral rofile will directly translate into rohibitive temoral oscillations. Figure 3 : (a) Longitudinal evolution of the temoral intensity over 4 mas made of fibres 1 and (b) Longitudinal evolution of the misfit arameter for ma 1 (dashed grey line) and for ma (solid black line) (c) Sectra after, 3 and 4 mas (solid line, dotted line and dashed grey line resectively). (d) Longitudinal evolution of the temoral intensity over 10 mas made of fibres and 3. Longitudinal evolution of the eak ower (e-left) and of the -0 db sectral width (eright) of a ulse evolving in a single normally disersive fibre (fibre, grey dashed line) and in 10 mas made of fibres and 3 (solid black line). Results derived from (7) are lotted with grey circles. Even if MI can not be comletely suressed and will ultimately limit the roagation length, different solutions can be undertaken to minimize this detrimental issue. A first ossibility could be to use a fibre with lower nonlinearity γ, which would lead to a lower MI gain. One can also use a fibre with higher disersion (fibre 3, second order disersion β 3 = 4 x β 1 = s.m -1 ). According to (9), this aroach leads to a faster comression and then reduces the integrated MI gain. In order to maintain the same integrated disersion, we have then used an anomalous disersive segment with a fibre length reduced by a factor four (ma ) and we have tested this aroach in figure 3(d). The roagation over 10 mas (and over u to 1 mas according to additional simulations not shown here) does not exhibit MI degradation. The arabolic shae is reserved, as it is confirmed in figure 3(b) with a misfit factor remaining as low as 0.05 after 10 mas.

8 Also shown in figure 3(e) is the evolution of the eak ower and sectral width. Equations (7) have been found excellent in redicting this evolution. The ulse undergoes a rogressive nonlinear sectrum exansion, as it can also be seen in figure 4(a1). In a single normally disersive fibre, due to the ulse broadening, the eak ower of the ulse raidly decreases (figure 3(e) bottom, dashed grey line), inducing less and less SPM, so that the sectral width saturates (figure 3(e) to, dashed grey line). In contrast, in the roosed disersion-managed system, as the temoral broadening if eriodically cancelled, the ulse kees a high eak ower at the inut of the normally disersive segment and self-hase modulation is thus cumulatively efficient. An increase of the sectral broadening by more than a factor comared to a single normally disersive fibre can be demonstrated (figure 3(e) to). Let us however note that in exerimental situations, sectral broadening will be ultimately limited not by MI but by the imact of higher order effects such as slicing losses, third order disersion (if the fibre disersion is not flattened) or Raman scattering. 5. Influence of the initial ulse shae and of the ma average disersion Let us now highlight the influence of the initial ulse rofile. We comare in figure 4 the evolutions of initial arabolic and hyerbolic secant (sech) ulses. In both cases, the ulses undergo sectral broadening. But, whereas the arabolic ulse maintains its shae, the sech ulse develos a highly eaked sectrum with shar edges. Details of the sectrum after 10 mas (figure 4(b)) illustrate the major distortions undergone by the sectral sech shae. This change also affects the temoral rofile (figure 4(c)) with the develoment of a flattened ulse shae. Figure 4 : (a) Longitudinal sectral evolution over ten mas made of fibres and 3 for an initial arabolic ulse (a1) and an initial sech ulse (a). Comarison of the outut sectral (b) and temoral rofiles (c). Sech and arabolic ulses are lotted with solid black lines and dotted grey line resectively.

9 Finally, we have studied the imact of the integrated disersion L 3 β 3 + L β of a ma. We have ket the length L 3 of the anomalous fibre (fibre 3) constant and we have varied the length L of the normally disersive segment (fibre ). Evolution of the temoral width across the roagation in six mas is lotted in figure 5(a). For negative average disersions, after each ma, the ulse is further comressed, which is beneficial for sectral width and eak-ower enhancement. However, after a given number of mas, the temoral width of the ulse becomes so short that roblems mentioned in section 3 of the aer such as soliton comression will negatively act : the ulse will loose its characteristic arabolic features, as outlined by the high misfit factor observed in figure 5(a). Let us now focus on the zero average disersion case. We can notice that the breathing of the arabolic ulse is more and more ronounced after each ma. This is mainly due to the continuous sectral broadening exerienced by the ulse. Indeed, as the sectral width increases (as well as the chir coefficient C ), the comression stage in the anomalous fibre (equation (9)) is faster and faster. As a consequence, the eak ower after the anomalous segment is higher and higher, as it can be seen on figure 3(e). With enhanced eak ower and sectrum, the temoral broadening in the normally disersive fibre will also be enhanced. If one targets outut arabolic ulse with temoral width identical to the inut, it is then desirable to use a slightly negative net disersion. Regarding ositive net disersions, the ulse temoral broadening is higher, which seems beneficial for a low misfit factor M (figure 5(c)). From the curves simulated after 6, 8 and 10 mas (figures 5(b,c)), it is clear that higher the number of mas is, more ronounced will be the tendancies reviously described. Figure 5 : Imact of the integrated disersion of the ma. Longitudinal evolution of the temoral width (a-bottom) and misfit factor (a-to) of the ulse for three values of integrated disersion of the ma ( , 0 and s, dotted grey line, solid black line and dashed line resectively). Evolution of the outut FWHM temoral width and sectral width (-0 db width) (b) and of the misfit arameter (c) after 6, 8 and 10 mas (black solid, dashed and dotted grey lines resectively).

10 6. Conclusion In the resent article, we have outlined the ability of a highly chired arabolic ulse to maintain its shae through an arrangement of normally and anomalous disersive fibres desite the resence of nonlinearity. By avoiding soliton comression and limiting MI deleterious imact, it is then ossible to take advantage of the arabolic ulse ability to maintain its shae in normally disersive fibres. By maintaining high eak-ower, enhanced sectral broadening is then demonstrated. We exect this work to oen new ersectives in the fields of coherent continuum generation as well as in the comrehension of the dynamics of ulses in self-similar fibre lasers [4]. Controlling the temoral sreading of a arabolic ulse will also be of ractical interest in the context of telecommunication alications where ulse-to-ulse interactions can severely damage arabolic self-similar dynamics [5]. Acknowledgements We would like to thank Bertrand Kibler, Stéhane Pitois and Guy Millot for fruitful discussions. This research was suorted by the Conseil Régional de Bourgogne and by the Agence Nationale de la Recherche (SUPERCODE roject - ANR-06-BLAN-0401 as well as SOFICARS roject - ANR-07- RIB ). References [1] Fermann M E, Kruglov V I, Thomsen B C, Dudley J M and Harvey J D 000 Self-similar roagation and amlification of arabolic ulses in otical fibers Phys. Rev. Lett [] Dudley J M, Finot C, Millot G and Richardson D J 007 Self-similarity in ultrafast nonlinear otics Nat. Phys [3] Anderson D, Desaix M, Karlson M, Lisak M and Quiroga-Teixeiro M L 1993 Wave-breakingfree ulses in nonlinear otical fibers J. Ot. Soc. Am. B [4] Parmigiani F, Finot C, Mukasa K, Ibsen M, Roelens M A F, Petrooulos P and Richardson D J 006 Ultra-flat SPM-broadened sectra in a highly nonlinear fiber using arabolic ulses formed in a fiber Bragg grating Ot. Exress [5] Duriez P, Finot C, Malinowski A, Sahu J K, Nilsson J, Richardson D J, Wilcox K G, Foreman H D and Troer A C 006 High-ower, high reetition rate icosecond and femtosecond sources based on Yb-doed fiber amlification of VECSELS Ot. Exress [6] Billet C, Dudley J M, Joly N and Knight J C 005 Intermediate asymtotic evolution and hotonic bandga fiber comression of otical similaritons around 1550 nm Ot. Exress [7] Nicholson J W, Yablon A, Westbrook P S, Feder K S and Yan M F 004 High ower, single mode, all-fiber source of femtosecond ulses at 1550 nm and its use in suercontinuum generation Ot. Exress [8] Finot C, Millot G, Pitois S, Billet C and Dudley J M 004 Numerical and exerimental study of arabolic ulses generated via Raman amlification in standard otical fibers IEEE J. Sel. To. Quantum Electron [9] Finot C, Provost L, Petrooulos P and Richardson D J 007 Parabolic ulse generation through assive nonlinear ulse reshaing in a normally disersive two segment fiber device Ot. Exress [10] Kibler B, Billet C, Lacourt P A, Ferrière R, Larger L and Dudley J M 006 Parabolic ulse generation in comb-like rofiled disersion decreasing fibre Electron. Lett

11 [11] Finot C, Barviau B, Millot G, Guryanov A, Sysoliatin A and Wabnitz S 007 Parabolic ulse generation with active or assive disersion decreasing otical fibers Ot. Exress [1] Finot C, Parmigiani F, Petrooulos P and Richardson D J 006 Parabolic ulse evolution in normally disersive fiber amlifiers receding the similariton formation regime Ot. Exress [13] Finot C, Fatome J, Pitois S and Millot G 007 All-Fibered High-Quality Low Duty-Cycle 0- GHz and 40-GHz Picosecond Pulse Sources IEEE Photon. Technol. Lett [14] Mollenauer L F, Mamyshev P V, Gri J, Neubelt M J, Mamysheva N, Grüner-Nielsen L and Veng T 000 Demonstration of massive wavelength-division multilexing over transoceanic distances by use of disersion-managed solitons Ot. Lett [15] Fatome J, Pitois S, Tchofo-Dinda P and Millot G 003 Exerimental demonstration of 160- GHz densely disersion-managed soliton transmission in a single channel over 896 km of commercial fibers Ot. Exress [16] Agrawal G P 001 Alications of nonlinear fiber otics (San Francisco, CA: Academic Press) [17] Agrawal G P 001 Nonlinear Fiber Otics, Third Edition (San Francisco, CA: Academic Press) [18] Peacock A C 001 Self-similar amlification and roagation of arabolic ulses in otical fibers. Master thesis at Auckland University, New-Zealand) [19] International Telecommunication Union 006 G.655 : Characteristics of a non-zero disersion-shifted single-mode otical fibre and cable. [0] Russell P S 003 Photonic crystal fibers Science [1] Finot C, Guenot A and Duriez P 007 Sectral comression of otical arabolic similaritons Ann. Phys. Fr [] Mollenauer L F, Stolen R H, Gordon J P and Tomlinson W J 1983 Extreme icosecond ulse narrowing by means of soliton effect in single-mode otical fibers Ot. Lett [3] Tomlinson W J, Stolen R H and Shank C V 1984 Comression of otical ulses chired by self-hase modulation in fibers J. Ot. Soc. Amer. B [4] Chong A, Renninger W and Wise F W 007 Proerties of normal-disersion femtosecond fiber lasers J. Ot. Soc. Am. B [5] Finot C and Millot G 005 Interactions of otical similaritons Ot. Exress