Cross-phase Modulation Terms Research Articles

Discrete vector light bullets in coupled [formula omitted] nonlinear cavities

We consider coupled weakly birefringent cavities filled-in with nonlinear Kerr material and subject to linearly polarized optical injection. Light propagation in such a system is described by a system of discrete Lugiato–Lefever-type equations for each linear polarization component of the electric field into each cavity, coupled by the cross-phase modulation terms and the neighboring waveguides field overlap integrals. We demonstrate that this system supports stable three-dimensional vector localized structures often called discrete vector light bullets. We consider both anomalous and normal dispersion and show that it results in the generation of, respectively, bright and dark discrete vector light bullets. Due to the polarization multistability of the system, we demonstrated coexisting light bullets with polarization at the light bullets peaks as different as predominantly linear to predominantly circular. We have shown that chaotic spatio-temporal dynamics can be realized even for such an injection strengths for which the light bullets distribution in the system is stationary by increasing the coupling strength C between the cavities.

Generalized theory of optical resonator and waveguide modes and their linear and Kerr nonlinear coupling

We derive a general theory of linear coupling and Kerr nonlinear coupling between modes of dielectric optical resonators from first principles. The treatment is not specific to a particular geometry or choice of mode basis, and can therefore be used as a foundation for describing any phenomenon resulting from any combination of linear coupling, scattering, and Kerr nonlinearity, such as bending and surface roughness losses, geometric backscattering, self- and cross-phase modulation, four-wave mixing, third-harmonic generation, and Kerr frequency comb generation. The theory is then applied to a translationally symmetric waveguide in order to calculate the evanescent coupling strength to the modes of a microresonator placed nearby, as well as the Kerr self- and cross-phase modulation terms between the modes of the resonator. This is then used to derive a dimensionless equation describing the symmetry-breaking dynamics of two counterpropagating modes of a loop resonator and prove that cross-phase modulation is exactly twice as strong as self-phase modulation only in the case that the two counterpropagating modes are otherwise identical.

Figure of merit for optimizing nonlinear transmission in fiber ring lasers

Passive mode-locking based on nonlinear polarization evolution as a saturable absorber is a powerful technique which has been performed in many optical configurations. The saturable absorption effect results from the intensity discrimination due to the interplay of the optical Kerr nonlinearity of the fiber, wave plates, and polarizers. This paper highlights the nonlinear transmission of a single-mode birefringent fiber exhibiting an optical Kerr nonlinearity placed between two polarizers and a set of several phase plates. Optical Kerr effect involves self-phase and cross-phase modulation terms and eventually four-wave mixing term. Different optical configurations are investigated involving various number of phase plates at both fiber ends. A figure of merit is defined which provides a universal way of comparison between different optical configurations. It demonstrates that the nonlinear transmission has a minimal sensitivity to both the exact optical configuration and the exact considered nonlinear terms. Consequently, the simplest model based on a single-mode birefringent fiber, exhibiting self-phase and cross-phase modulation, placed between two half-wave plates can be used without loss of generality.

Modulation instability analysis and optical solitons in birefringent fibers to RKL equation without four wave mixing

In this present work, we retrieve the optical solitons in birefringent fibers modeled by Radhakrishnan-Kundu-Lakshmanan (RKL) equation in coupled vector form. The system is composed of self-phase, cross-phase modulation terms, self-steepening terms and the effect of four-wave mixing is discarded. In this article, by employing innovative integration norms known as the extended rational sine-cosine/sinh-cosh techniques different forms of optical solitons like dark, singular, combined and complex soliton solutions are recovered. Furthermore, singular periodic wave solutions are also observed. The constraint conditions which ensure the formation of soliton solutions are also specified. By using the suitable values of the parameters involved, the 2-, 3-dimensional and the contour graphs of some of the reported solutions are plotted. The stability of the given model is studied by exercising the modulation instability analysis which confirms that the model is stable and guarantees that all extracted solutions are stable and exact.

Theoretical investigation of pulse-to-pulse instabilities in a synchronously pumped femtosecond optical parametric oscillator

A singly resonant synchronously pumped optical parametric oscillator (SPOPO) based on the experimental setup of J. Opt. Soc. Am. B 36, 131 (2019)JOBPDE0740-322410.1364/JOSAB.36.000131 is studied theoretically. Stable, oscillatory, and chaotic operation modes of the SPOPO are investigated. The need of the self- and cross-phase modulation terms in the theoretical model for the explanation of the instabilities is demonstrated. The theoretical values of the wavelengths of the signal spectrum maxima are found. The evidence of chaos by the calculation of the Lyapunov exponent is provided. The possibilities to avoid the instabilities are discussed.

Effects of self- and cross-phase modulation on the spontaneous symmetry breaking of light in ring resonators

We describe spontaneous symmetry breaking in the powers of two optical modes coupled into a ring resonator, using a pair of coupled Lorentzian equations, featuring tunable self- and cross-phase modulation terms. We investigate a wide variety of nonlinear materials by changing the ratio of the self- and cross-phase interaction coefficients. Static and dynamic effects range from the number and stability of stationary states to the onset and nature of oscillations. Minimal conditions to observe symmetry breaking are provided in terms of the ratio of the self-/cross-phase coefficients, detuning, and input power. Different ratios of the nonlinear coefficients also influence the dynamical regime, where they can induce or suppress bifurcations and oscillations. A generalised description on this kind is useful for the development of all-optical components, such as isolators and oscillators, constructed from a wide variety of optical media in ring resonators.

Rogue waves for a coupled nonlinear Schrödinger system in a multi-mode fibre

In this paper, we investigate the rogue waves for an integrable coupled nonlinear Schrödinger (CNLS) system with the self-phase modulation, cross-phase modulation and four-wave mixing term, which can describe the propagation of optical waves in a multi-mode fibre. We construct a generalized Darboux transformation (GDT) for the CNLS system and find a gauge transformation which converts the Lax pair into the constant-coefficient differential equations. Solving those equations, we can obtain the vector solutions of the Lax pair. Using the GDT, we derive an iterative formula for the nth-order rogue-wave solutions for the CNLS system. We derive the first- and second-order rogue-wave solutions for the CNLS system and analyse the profiles for the rogue waves with respect to the self-phase modulation term a, cross-phase modulation term c and four-wave mixing term b, respectively. The rogue waves become thinner with the increase in the value for the real part of b and that the effect of a or c on the rogue waves is the same as the one of the real part of b.

Mid-IR femtosecond frequency conversion by soliton-probe collision in phase-mismatched quadratic nonlinear crystals.

We show numerically that ultrashort self-defocusing temporal solitons colliding with a weak pulsed probe in the near-IR can convert the probe to the mid-IR. A near-perfect conversion efficiency is possible for a high effective soliton order. The near-IR self-defocusing soliton can form in a quadratic nonlinear crystal (beta-barium borate) in the normal dispersion regime due to cascaded (phase-mismatched) second-harmonic generation, and the mid-IR converted wave is formed in the anomalous dispersion regime between λ=2.2-2.4 μm as a resonant dispersive wave. This process relies on nondegenerate four-wave mixing mediated by an effective negative cross-phase modulation term caused by cascaded soliton-probe sum-frequency generation.

Integrability and soliton solutions for an [formula omitted]-coupled nonlinear Schrödinger system in optical fibers

In this paper, an N-coupled nonlinear Schrödinger system in optical fibers with energy-exchanging and different self-phase modulation and cross-phase modulation terms is investigated. The system is found to be integrable in the sense of passing the Painlevé analysis. The Lax pair for the system is given in the form of the block matrices through generalizing that for a 2-coupled nonlinear Schrödinger system with the Ablowitz–Kaup–Newell–Segur scheme. Through the Hirota bilinear method with an auxiliary function, one- and two-soliton solutions are derived via symbolic computation. One-soliton solutions can present the profile of one peak or two peaks in different components. With the asymptotic analysis, two-soliton solutions are found to admit the elastic collision, and the propagation and collision of solitons are analyzed.

Analysis of the effects and causes of numerical error in four-wave mixing simulations

We present an investigation of accumulated numerical error in the simulation of four-wave mixing, which is frequently used to model a variety of nonlinear optical devices. The Dormand-Prince method (commonly used by commercial solving software such as MATLAB) has been found to be susceptible to numerical error, which manifests itself in a 3.8% increase in the total power over 200 m of nonlinear interaction length. This numerical error can lead to qualitatively mistaken physical interpretations of simulation results, which are similar to those found in previously published materials. We use a home-built Adams-Bashforth solver to simulate four-wave mixing, which produces results that do not lead to unphysical results, even for simulation over a large nonlinear interaction length. By comparing the results of these two methods we were able to illustrate the qualitative effects of the accumulated numerical error in the former. The source of this cumulative power error is traced to the solutions provided by the Dormand-Prince method for the self- and cross-phase modulation terms of the coupled mode equations; this error increases for larger nonlinearities or if step size increases. Even when this power accumulates from infinitesimal per-step errors, significant changes occur that could lead to qualitative differences in generated power values and conversion efficiencies. This reveals the potential danger of applying commonly used numerical solvers in simulating nonlinear optical processes.

Nonlinear Propagation in Multimode and Multicore Fibers: Generalization of the Manakov Equations

We investigate theoretically nonlinear transmission in space-division multiplexed (SDM) systems using multimode fibers exhibiting rapidly varying birefringence. A primary objective is to generalize the Manakov equations, well known in the case of single-mode fibers. We first investigate the case where linear coupling among spatial modes of the fiber is weak and derive new Manakov equations after averaging over random birefringence fluctuations. Such an averaging reduces the number of intermodal nonlinear terms drastically since all four-wave-mixing terms vanish. Cross-phase modulation terms still affect multimode transmission but their effectiveness is reduced. We verify the accuracy of new Manakov equations by simulating the transmission of multiple 114-Gb/s bit streams in the PDM-QPSK format over different modes of a multimode fiber and comparing the numerical results with those obtained by solving the full stochastic equations. The agreement is excellent in all cases studied. A major benefit of the new Manakov equations is that they typically reduce the computation time by more than a factor of 10. Our results show that birefringence fluctuations improve system performance by reducing the impact of fiber nonlinearities. The extent of improvement depends on the fiber design and how many spatial modes are used for SDM transmission. We also consider the case where all spatial modes experience strong random linear coupling modeled using a random matrix. We derive new Manakov equations in this regime and show that the impact of some nonlinear effects can be reduced relatively to single-modes fibers. Finally, we extend our analysis to multicore fibers and show that the Manakov equations obtained in the strong- and weak-coupling regimes can still be used depending on the extent of coupling among fiber cores.

Phase interaction of short vector solitons

An interaction of vector solitons in the frame of coupled third-order nonlinear Schrödinger equations taking into account third-order linear dispersion, nonlinear dispersion, and cross-phase modulation terms is considered. Phase nature of the solitonsʼ interaction is shown. In particular, dependence of solitonsʼ trajectories on initial distance between solitons is shown. Conditions of reflection and propagation of solitons through each other are obtained.

Stationary pulses in two cubic–quintic Ginzburg–Landau equations coupled with a cross phase modulation term

Existence of stationary pulses are investigated in system modelled by two cubic–quintic Ginzburg–Landau equations coupled with a cross phase modulation term. Only for the symmetric case, that is, when the two components are equal, stationary pulses are obtained and several solutions coexist simultaneously. Direct numerical simulations were run in order to study the stability and the dynamics of the stationary pulses.

Modulational instability in two cubic–quintic Ginzburg–Landau equations coupled with a cross phase modulation term

Modulation instability is investigated in two cubic–quintic Ginzburg–Landau equations coupled with a cross phase modulation type term. After carrying out a stability analysis an expression for gain is obtained. Some direct simulations to see the evolution of different continuous wave states are reported. These show the formation of modulation instability pulses as well as transitions from lower amplitude continuous wave states to higher amplitude continuous wave states.

Fundamental modes of a trapped probe photon in optical fibers conveying periodic pulse trains

Wave modes induced by cross-phase reshaping of a probe photon in the guiding structure of a periodic train of temporal pulses are investigated theoretically with emphasis on exact solutions to the wave equation for the probe. The study has direct connection with recent advances on the issue of light control by light, the focus being on the trapping of a low-power probe by a temporal sequence of periodically matched high-power pulses of a dispersion-managed optical fiber. The problem is formulated in terms of the nonlinear optical fiber equation with averaged dispersion, coupled to a linear equation for the probe including a cross-phase modulation term. Shape-preserving modes which are robust against the dispersion are shown to be induced in the probe, they form a family of mutually orthogonal solitons the characteristic features of which are determined by the competition between the self-phase and cross-phase effects. Considering a specific context of this competition, the theory predicts two degenerate modes representing a train of bright signals and one mode which describes a train of dark signals. When the walk-off between the pump and probe is taken into consideration, these modes have finite-momentum envelopes and none of them is totally transparent vis-\`a-vis the optical pump soliton.

Parametric light bullets supported by quasi-phase-matched quadratically nonlinear crystals

We present a comprehensive analysis of the dynamics of three-dimensional spatiotemporal nonspinning and spinning solitons in quasi-phased-matched (QPM) gratings. By employing an averaging approach based on perturbation theory, we show that the soliton's stability is strongly affected by the QPM-induced third-order nonlinearity (which is always of a mixed type, with opposite signs in front of the corresponding self-phase and cross-phase modulation terms). We study the dependence of the stability of the spatiotemporal soliton (STS) on its energy, spin, the wave-vector mismatch between the fundamental and second harmonics, and the relative strength of the intrinsic quadratic and QPM-induced cubic nonlinearities. In particular, all the spinning solitons are unstable against fragmentation, while zero-spin STS's have their stability regions on the system's parameter space.

Loss and non-linearity saturation effects on the bistability of non-linear distributed feedback structures

We investigate theoretically the effects of optical absorption and saturation of the non-linear index of refraction on the bistability of non-linear distributed feedback structures (NLDFBs). By assuming that the Kerr non-linearity saturates in an exponential fashion, we obtain for the first time closed-form expressions for the so-called self-phase and cross-phase modulation terms. Our investigation shows that both absorption and, in particular, saturation significantly affect the bistable properties of this structure and in many cases eliminate this response completely. In some cases, however, saturation alters the NLDFB's transfer characteristics in a potentially useful manner. We find that weak levels of saturation may increase the contrast ratio between the intensities of the high and low bistable states. At increasing levels of saturation, where bistability is no longer observed, we find regimes where the NLDFB structure could possibly be used for optical amplification.

Bright spatial soliton dynamics in a symmetric optical planar waveguide structure.

The behavior of spatial solitons in coupled planar optical waveguides is investigated. A full derivation of the appropriate coupled nonlinear Schr\"odinger equations is given that includes all the nonlinear cross-phase modulation terms. The dynamics of the spatial solitons are developed using the average variational method of Whitham $[---Linear and Nonlinear Waves (Wiley, New York, 1974)]. The main problems addressed concern spatial soliton beam displacement, beam switching, and a few stability problems. Significant mathematical progress has been made, which is also verified by accurate numerical simulations.

Experiments and numerical simulations on beam bending and transverse beam encoding by spatial self- and cross-phase modulation in ruby

Experiments and numerical simulations are performed on the nonlinear beam evolution of simultaneously propagating cw argon-ion and HeNe laser beams through a ruby rod. The HeNe probe beam exhibits a deflection of nearly 12 mrad when a 1.5 W, 514.5 nm laser beam is injected into the ruby rod at an angle to the probe. The copropagating geometry is also studied where the azimuthally symmetric transverse beam encoding leads to complex ring structures in both the HeNe and argon-ion laser beams. The numerical simulations are compared with the experimental data to obtain the cross-phase modulation term. Using previously reported values for the saturable nonlinear index of ruby and a nonlinear index for cross-phase modulation n c = 1.35Re( n 2), the nonlinear wave equation in two transverse dimensions is numerically solved to predict the bending angles and emerging beam profiles. These findings are discussed in light of previous experiments in ruby as well as the possibility of using these effects for producing nanosecond optical switching systems.