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978-1-4799-0683-3/13/$31.00 ©2013 IEEE 1

Passive Reshaping of Ultrashort Pulses in Optical Fibers Igor A. Sukhoivanov1, Oleksiy V. Shulika1, Sergii O. Iakushev2,

Jose A. Andrade Lucio1, Gabriel Ramos Ortiz3 1 University of Guanajuato, Mexico, e-mail: a.shulika@ieee.org

2 Kharkiv National University of Radio Electronics, Ukraine 3 Centro de Investigaciónes en Óptica, México

ABSTRACT The passive nonlinear reshaping in normally dispersive optical fibers in the steady-state regime is studied numerically. It is found that normal dispersion and self-phase modulation are able to provide pulse reshaping towards a parabolic pulse profile at the distances exceeding the optical wave breaking length. By matching properly initial pulse parameters and fiber parameters it is possible to change the pulse shape in the steady-state regime. We have found conditions for producing of parabolic pulses from initial secant or Gaussian pulses. The influence of initial pulse shape, initial chirp, third-order dispersion and loss on the parabolic pulse formation is studied consistently, and estimation of practical conditions which are needed for parabolic pulses formation in a passive fiber is given. Keywords: Nonlinear pulse reshaping, optical wave breaking, parabolic pulses.

1. INTRODUCTION The propagation of ultrashort laser pulses in the optical fibers is connected with a plenty of interesting and practically important phenomena. Unique dispersive and nonlinear properties of the optical fibers lead to various scenarios of the pulse evolution which are resulted in particular changes of the pulse shape, spectrum and chirp. The modern age of the optical fibers starts from the 1960s with the appearance of the first lasers. The investigations of the nonlinear phenomena in the optical fibers have been continuously gained by the decreasing loss. Loss reduction in the fibers made possible the observation of such nonlinear processes which required longer propagation path length at the available power levels in the 1970s. Stimulated Raman Scattering (SRS) and Brillouin scattering (SBS) were studied first [1]. Optical Kerr-effect [2], parametric four-wave mixing (FWM) [3] and self-phase modulation (SPM) [4] were observed later. The theoretical prediction of the optical solitons as an interplay of the fiber dispersion and the fiber nonlinearity was done as early as 1973 [5] and the soliton propagation was demonstrated seven years later in a single mode optical fiber [6].

Discovery of the optical solitons have revolutionized the field of the optical fiber communications. Nowadays solitons are used as the information carrying ‘‘bits’’ in optical fibers [7]. This is resulted from unique properties of the solitons which are when GVD is anomalous, and also, the pulse energy and duration have to meet a particular condition in order to provide the formation of a stable soliton called the fundamental soliton. Fundamental solitons preserve their temporal and spectral shape even over long propagation distances. Owing to this stability the solitons play important role not only in optical fiber communications but also in ultrashort lasers as well. Soliton mode locking is a frequently used technique for producing high-quality ultrashort pulses in the bulk and fiber lasers [8, 9].

In spite of unique properties of the optical solitons the difficulties arise with generation of the high power pulses and their delivering through the optical fibers due to distortions and break-up effects. Particularly, the fundamental soliton exists at only one particular power level and propagation of higher-power pulses excites the higher-order solitons which are sensitive to perturbation and break-up through the soliton fission [10, 11]. However, in the normal dispersion regime the pulse propagation is subject to instability through the appearance of the optical wave breaking (OWB) [12, 13]. It was shown that optical intensity shock formation precedes the OWB appearance [14]. The shock occurs when the more intense parts of the pulse, owing to the nonlinearity, are travelling at the speeds which are different from those of the weaker parts. This reshaping evolves to a breaking singularity, when the top of the shock actually overtakes its bottom, resulting in a region of multivalued solutions.

Break-up effects in general restrict pulse propagation in an optical fiber, especially concerning the high-power pulses. However, in 1990s it was theoretically predicted that pulses with parabolic intensity profile and a linear frequency chirp can propagate without wave breaking in the normal dispersive optical fiber [15]. Then it was shown numerically that ultrashort pulses injected into a normal dispersion optical fiber amplifier evolve towards the parabolic pulse profile with amplification and, moreover, retained their parabolic shape even as they continued to be amplified to higher powers [16].

This novel class of the optical pulses exhibits several fundamental features. Parabolic pulses propagate in a self-similar manner, holding certain relations (scaling) between the pulse power, width, and chirp parameter. The scaling of the amplitude and width of both the temporal amplitude and spectrum depends only on the amplifier parameters and the input pulse energy, and is completely independent of the input pulse shape.

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Self-similarity is a fundamental property of many physical systems and has been studied extensively in diverse areas of physics such as hydrodynamics, mechanics, solid-state physics and theory of elasticity [17]. Discovery of the self-similar pulse propagation in the optical fibers attracts much interest not only from theoretical standpoint but due to possible practical applications as well [18]. By analogy with the well-known stable dynamics of the solitary waves – solitons, these self-similar parabolic pulses have been termed as similaritons. One has to note that solitons themselves can also be interpreted as an example of self-similarity [19]. In contrast to the optical solitons however, the similaritons in optical fibers can tolerate strong nonlinearity without wave breaking. The normal GVD effectively linearizes the accumulated phase of the pulse allowing for the spectral bandwidth to increase without destabilizing the pulse. Solitons maintain their shape, width, and amplitude, whereas similaritons maintain their shape but not their width or amplitude; instead of that the certain relations between pulse power, width, and chirp parameter are hold.

Application of similaritons provided a significant progress in the field of pulse generation and amplification in fiber systems. The remarkable properties of the parabolic pulses have been applied to the development of a new generation of high-power optical fiber amplifier [20]. Self-similar propagation regime allows avoiding the catastrophic pulse break up due to excessive nonlinearity. However, in contrast to the well-known chirped pulse amplification method, where the aim is to avoid nonlinearity by dispersive pre-stretching before amplification, a self-similar amplifier actively exploits the nonlinearity. This allows obtaining output pulses that are actually shorter than after recompression of the initial input pulse. Similariton approach has been applied also to the development of high-power fiber lasers (“similariton laser”) allowing overcome existing power limitations of soliton mode-locking [21]. Parabolic pulses are of great interest for a number of applications including, amongst others, the highly coherent continuum sources development and optical signal processing [22].

2. PASSIVE RESHAPING OF ULTRASHORT PULSES Parabolic pulses are generated usually in the active fiber systems such as amplifiers or lasers. However, last time it is also of interest to study alternative methods of generating parabolic pulses, especially in the context of non-amplification usage, such as optical telecommunications. Some approaches were proposed for the generation of parabolic pulses in the passive fiber systems, such as dispersion decreasing fibers [23] and fiber Bragg gratings [24]. Then it was found that nonlinearity and normal dispersion in the simple passive fiber can provide pulse reshaping towards the parabolic pulse at the propagation distance preceding the optical wave breaking [25]. But with longer propagation distance the pulse shape doesn’t remain parabolic. However, recently the possibility to obtain nonlinear-dispersive similaritons in the passive fibers at longer propagation distances has been demonstrated [26]. However, the shape of the pulses obtained in passive fiber is different from the fully parabolic one. Under some special initial conditions even triangular pulses can be obtained [27]. A number of research have been made in this area by the author of this proposal, and it was shown that due to the unique properties of parabolic pulses they are attractive for such implementations as pulse amplification, pulse compression, pulse synthesis [28-39].

In the strong nonlinear case (soliton order is larger than 10) optical wave breaking becomes significant. Previously, it was shown that oscillations induced by OWB are vanishing at longer distances and pulse transforms to the chirped trapezium-shaped pulse [43]. However, further pulse evolution has not been considered yet. We will present numerical results concerning the pulse transformations at much longer distance, well above of the distance for OWB onset. We have found that OWB is a transient stage of pulse propagation, after which steady-stage regime is achieved. In this regime the pulse shape becomes nearly smooth with a linear chirp and no shape transformations occur, hence the term “steady-state propagation regime”.

Some practical conditions for achieving parabolic pulses in a passive fiber have been made [42]. The super-Gaussian pulse (2-nd order) is the best choice to obtain parabolic pulses in a passive optical fiber, but the state of the art ultrashort lasers generate usually the secant or the Gaussian waveforms. In this case the initial Gaussian pulse is more preferable than secant, because the resulted pulse shape can be closer to the parabolic one at the shorter propagation distance. Next point is that the selected fiber should provide suitable dispersion properties at the operation wavelength. Dispersion should be normal and flat enough in order to reduce the impact of TOD. The higher amount of second-order dispersion allows usage of shorter pieces of fiber due to the smaller dispersion length DL and, moreover, could provide higher amount of DD LL /′ . For example, realistic dispersion parameters of conventional single-mode fibers are: /kmps 18 2

2 =β , /kmps 04.0 33 =β . For the initial pulse with

duration fs 1000 =T we obtain m 5.0=DL , 45/ =′ DD LL . By varying the peak power of the initial pulse and/or selecting fiber with appropriate nonlinear coefficient one can achieve operation within the desired range

5.25.1 << N , where the pulse shape in the steady-state regime is the closest to the parabolic one. Thus, we can obtain parabolic pulse from the initial Gaussian pulse or the secant pulse applying a few meters of such a fiber. Because the required fiber length is quite short the influence of attenuation on the pulse reshaping process is expected to be negligible. The application of initially chirped pulses will allow to obtain pulses with better parabolic profile at the shorter propagation distance. However, we have to note that 45/ =′ DD LL might be too

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small for operation at the higher soliton order (such as 10=N ). In this case one can use longer initial pulses or a fiber with lower amount of TOD. The application of photonic crystal fibers seems to be attractive in this case due to their flexibility in tailoring its dispersion properties by varying geometrical fiber parameters.

3. CONCLUSIONS The passive nonlinear pulse reshaping in normal dispersion optical fibers at the propagation distances substantially exceeding the OWB length OWBξ where a steady-state regime is achieved is discussed. Pulses with parabolic intensity profile, parabolic spectrum and linear chirp can be obtained in the steady-state regime from various initial pulse shapes due to the combined action of normal dispersion and self-phase modulation. The resulted pulse shape in the steady-state regime strongly depends on the initial pulse parameters and fiber parameters. The range of soliton order 5.25.1 << N is the most preferable one for producing parabolic pulses. If this condition is satisfied, one can obtain pulses with nearly parabolic profile even from initially unchirped pulses. However, the application of chirped pulses allows obtaining better parabolic profile from initial Gaussian and secant pulses. With the increasing of soliton order the pulse shape in the steady-state regime moves away from the parabolic profile. In spite of that it is possible to obtain parabolic pulses even at the high soliton order applying stronger initial chirp. Although third-order dispersion is able to distort the pulse shape in the steady-state regime due to the optical shock-type instabilities, one can avoid pulse distortions by choosing the proper ratio of dispersion lengths. The presence of the gain in the fiber enhances pulse reshaping towards the parabolic profile, whereas the loss enhances pulse reshaping opposite to the parabolic profile.

Parabolic pulses obtained in the steady-state regime possesses the following advantages: 1) both the pulse temporal intensity shape and the spectrum are parabolic; 2) pulse temporal intensity shape and spectrum remain parabolic during further pulse propagation in the fiber; 3) the chirp is perfectly linear even with increasing of the soliton order; 4) the spectral width approaches its maximal asymptotic value providing a maximal compression factor. However, these features achieved at the longer propagation distance and therefore the pulse can undergo the stronger dispersion broadening as compared to the parabolic pulse formation at the short propagation distance preceding the OWB length.

ACKNOWLEDGEMENTS This work was supported by University of Guanajuato in the frame of DAIP grant 59/12 and by CONACyT within the sabbatical grant 175573/12.

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