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We study vectorial modulation instability occurring inside a birefringent graded-index (GRIN) fiber when the two polarization components of the optical field are coupled nonlinearly through cross-phase modulation. In the scalar case in which only modes of one polarization are excited, the geometric parametric instability is known to produce an infinite number of sidebands around the wavelength of the input optical beam. We show that the birefringence of a GRIN fiber splits each of these sidebands into a triplet, whose frequency spacing depends on the differential group delay between the orthogonally polarized components. We verify the predictions of the linear stability analysis numerically by solving two coupled nonlinear Schrödinger equations that include spatial self-imaging effects through an effective nonlinear parameter. We present results for both continuous and pulsed optical beams experiencing normal or anomalous group-velocity dispersion inside a GRIN fiber.

We present a general framework capable of describing the nonlinear propagation of pulsed optical beams of arbitrary shapes and phase fronts inside a graded-index (GRIN) fiber. The main assumption made is that the spatial self-imaging features of the beam are not affected by the temporal evolution of optical pulses. A propagation kernel known from the work done in the 1970s is used to obtain a distance-dependent nonlinear coefficient that captures all spatial effects within an effective nonlinear Schrödinger equation. We consider three specific beam shapes (Gaussian, circular, and square) to study the impact of the shape, position, and curvature of optical beams on the complex spatiotemporal dynamics specific to GRIN fibers. In particular, we focus on the impact of an input beam’s shape on the modulation-instability sidebands and the generation of multiple dispersive waves when higher-order solitons form inside a GRIN fiber. The results of our numerical analysis indicate that for beam widths chosen to yield the same value of the effective mode area at the input end of the fiber, the nonlinear effects are pronounced considerably when a Gaussian beam is launched into the fiber. We also found that even though the self-imaging period is doubled when an off-centered Gaussian beam is launched into a GRIN fiber, it does not affect the nonlinear evolution because the effective beam area still maintains the same periodicity, as long as the shift in the beam’s center is not so large that it does not remain confined to the fiber’s core.