A broad class of exact self-similar solutions to the nonlinear Schrödinger equation (NLSE) with distributed dispersion, nonlinearity, and gain or loss has been found. Appropriate solitary wave solutions applying to propagation in optical fibers and optical fiber amplifiers with these distributed parameters have also been studied. These solutions exist for physically realistic dispersion and nonlinearity profiles in a fiber with anomalous group velocity dispersion. They correspond either to compressing or spreading solitary pulses which maintain a linear chirp or to chirped oscillatory solutions. The stability of these solutions has been confirmed by numerical simulations of the NLSE.
With continued steep growth in the volume of data transmitted over optical networks there is a widely recognized need for more sophisticated photonics technologies to forestall a 'capacity crunch' 1 . A promising solution is to open new spectral regions at wavelengths near 2 μm and to exploit the long-wavelength transmission and amplification capabilities of hollowcore photonic-bandgap fibres 2,3 and the recently available thulium-doped fibre amplifiers 4 . To date, photodetector devices for this window have largely relied on III-V materials 5 or, where the benefits of integration with silicon photonics are sought, GeSn alloys, which have been demonstrated thus far with only limited utility 6-9 . Here, we describe a silicon photodiode operating at 20 Gbit s -1 in this wavelength region. The detector is compatible with standard silicon processing and is integrated directly with silicon-on-insulator waveguides, which suggests future utility in silicon-based mid-infrared integrated optics for applications in communications.The advantages of silicon photonics, which have been well documented for traditional communication wavelengths around 1.3 and 1.5 µm (refs 8,9), extend to operation in the mid-infrared (MIR) region 10 . Silicon photonic components are fabricated using complementary metal-oxide semiconductor (CMOS)-compatible technologies, with the potential for integration with electronic control. Recently, groups have demonstrated several silicon-based components operating in the MIR wavelength range of 2-20 μm, including low-loss waveguides, couplers, splitters and multiplexers 11 , as well as some with hybrid active functionality 12,13 . However, photodetectors that are compatible with silicon waveguides, are capable of detection beyond 2 μm, and operate at the bandwidths required by future optical communication networks remain elusive. The sig-
Glass fibres with silicon cores have emerged as a versatile platform for all-optical processing, sensing and microscale optoelectronic devices. Using SiGe in the core extends the accessible wavelength range and potential optical functionality because the bandgap and optical properties can be tuned by changing the composition. However, silicon and germanium segregate unevenly during non-equilibrium solidification, presenting new fabrication challenges, and requiring detailed studies of the alloy crystallization dynamics in the fibre geometry. We report the fabrication of SiGe-core optical fibres, and the use of CO 2 laser irradiation to heat the glass cladding and recrystallize the core, improving optical transmission. We observe the ramifications of the classic models of solidification at the microscale, and demonstrate suppression of constitutional undercooling at high solidification velocities. Tailoring the recrystallization conditions allows formation of long single crystals with uniform composition, as well as fabrication of compositional microstructures, such as gratings, within the fibre core.
Pulse propagation in high-gain optical fiber amplifiers with normal group-velocity dispersion has been studied by self-similarity analysis of the nonlinear Schrödinger equation with gain. For an amplifier with a constant distributed gain, an exact asymptotic solution has been found that corresponds to a linearly chirped parabolic pulse that propagates self-similarly in the amplifier, subject to simple scaling rules. The evolution of an arbitrary input pulse to an asymptotic solution is associated with the development of low-amplitude wings on the parabolic pulse whose functional form has also been found by means of self-similarity analysis. These theoretical results have been confirmed with numerical simulations. A series of guidelines for the practical design of fiber amplifiers to operate in the asymptotic parabolic pulse regime has also been developed.
Self-similarity techniques are used to study pulse propagation in a normal-dispersion optical fiber amplifier with an arbitrary longitudinal gain profile. Analysis of the nonlinear Schrödinger equation that describes such an amplifier leads to an exact solution in the high-power limit that corresponds to a linearly chirped parabolic pulse. The self-similar scaling of the propagating pulse in the amplifier is found to be determined by the functional form of the gain profile, and the solution is confirmed by numerical simulations. The implications for achieving chirp-free pulses after compression of the amplifier output are discussed.
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