Solution Of Coupled-mode Equations Research Articles

Exact scaling solution of the mode coupling equations for non-linear fluctuating hydrodynamics in one dimension

We obtain the exact solution of the one-loop mode-coupling equations for the dynamical structure function in the framework of non-linear fluctuating hydrodynamics in one space dimension for the strictly hyperbolic case where all characteristic velocities are different. All solutions are characterized by dynamical exponents which are Kepler ratios of consecutive Fibonacci numbers, which includes the golden mean as a limiting case. The scaling form of all higher Fibonacci modes are asymmetric Lévy-distributions. Thus a hierarchy of new dynamical universality classes is established. We also compute the precise numerical value of the Prähofer–Spohn scaling constant to which scaling functions obtained from mode coupling theory are sensitive.

Analytical solutions of coupled-mode equations for microring resonators

We present a study on analytical solutions of coupled-mode equations for microring resonators with an emphasis on occurrence of all-optical EIT phenomenon, obtained by using a cofactor. As concrete examples, analytical solutions for a 3×3 linearly distributed coupler and a circularly distributed coupler are obtained. The former corresponds to a non-degenerate eigenvalue problem and the latter corresponds to a degenerate eigenvalue problem. For comparison and without loss of generality, analytical solution for a 4×4 linearly distributed coupler is also obtained. This paper may be of interest to optical physics and integrated photonics communities.

Coupled mode equations and gap solitons for the 2D Gross–Pitaevskii equation with a non-separable periodic potential

Gap solitons near a band edge of a spatially periodic nonlinear PDE can be formally approximated by solutions of Coupled Mode Equations (CMEs). Here we study this approximation for the case of the 2D Periodic Nonlinear Schrödinger/Gross–Pitaevskii Equation with a non-separable potential of finite contrast. We show that unlike in the case of separable potentials [T. Dohnal, D. Pelinovsky, G. Schneider, Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, J. Nonlinear Sci. 19 (2009) 95–131] the CME derivation has to be carried out in Bloch rather than physical coordinates. Using the Lyapunov–Schmidt reduction we then give a rigorous justification of the CMEs as an asymptotic model for reversible non-degenerate gap solitons and provide H s estimates for this approximation. The results are confirmed by numerical examples, including some new families of CMEs and gap solitons absent for separable potentials.

Analytical solutions of coupled-mode equations for multiwaveguide systems, obtained by use of Chebyshev and generalized Chebyshev polynomials.

A novel approach is proposed for obtaining the analytical solutions of the coupled-mode equations (CMEs); the method is applicable for an arbitrary number of coupled waveguides. The mathematical aspects of the CMEs and their solution by use of Chebyshev polynomials are discussed. When mode coupling between only adjacent waveguides is considered (denoted weak coupling), the first and second kinds of the usual Chebyshev polynomials are appropriate for evaluating the CMEs for linearly distributed and circularly distributed multiwaveguide systems, respectively. However, when one is considering the coupling effects between nonadjacent waveguides also (denoted strong coupling), it is necessary to use redefined generalized Chebyshev polynomials to express general solutions in a form similar to those for the weak-coupling case. As concrete examples, analytical solutions for 2 x 2, 3 x 3, and 4 x 4 linearly distributed directional couplers are obtained by the proposed approach, which treats the calculation as a nondegenerate eigenvalue problem. In addition, for the 3 x 3 circularly distributed directional coupler, which gives rise to a degenerate eigenvalue problem, an analytical solution is obtained in an improved way. Also, for comparison and without loss of generality, to clarify the difference between the two coupling cases, analytical solutions for a 5 x 5 circularly distributed directional coupler are obtained by use of the usual and the redefined generalized Chebyshev polynomials.

Effect of Spontaneous and Carrier Noise on Mode-Locked Hybrid Soliton Pulse Source

Noise characteristics of mode-locked hybrid soliton pulse source (HSPS) where fiber Bragg grating is used as external cavity is presented. Model is based on time-domain solution of coupled-mode equations including spontaneous noise. Relative intensity noise (RIN) is calculated by using numerical solution of these equations. It is shown that inclusion of carrier density noise in the rate equations is necessary for the accurate analysis of noise in mode-locked lasers. It is also found that transform-limited pulses are not generated over a wide tuning range around the fundamental mode-locking frequency with spontaneous and carrier density noise sources if noise level is high. Therefore mode-locking range where transform limited pulses are obtained reduces.

Electro-optic modulation and frequency translation of light in a gyrotropic medium

The solution of coupled-mode equations is obtained for a strong coupling and a short interaction length. It is used in an analysis of electro-optic modulation in a gyrotropic or optically active medium. The gyrotropy and electro-optic interaction make it possible to transform phase modulation into intermode coupling and also to translate the phase matching from low to high frequencies and thus achieve efficient hf mode conversion. This can be used in wideband electro-optic modulators of the state of polarisation and in optical frequency translators operating in the gigahertz range.

Solutions of coupled-mode equations with a large dimension in underwater acoustics

A new fast and efficient acoustic propagation model is introduced which solves the coupled-mode differential equations in underwater acoustics for range-dependent, fluid-layered, ocean environments. The coupled range equations are first modified and then solved using the Lanczos method, an approach recently introduced in nuclear theory. This method has been successfully applied to the description of the nuclear response of heavy atomic nuclei in the giant resonance region of the energy excitation spectrum [T. Udagawa and B. T. Kim, Phys. Rev. C 40, 2271–2275 (1989)], as well as, other physical processes in low to medium energy in nuclear theory. The efficiency of this method permits the solution of coupled mode equations with large dimensions. The eigenvalues and depth-dependent mode functions are quickly obtained using an Airy function approach, which allows the range-dependent mode-coupling matrices to be evaluated analytically. The validity of the model is demonstrated, and example calculations are presented. [This work is supported by the Office of Naval Research (ONR 4530) Project RO35K01.]

Spatial instabilities, all-optical limiting, and thresholding in nonlinear distributed-feedback devices

Solutions of coupled-mode equations describing intensity-induced Bragg coupling of counterpropagating modes in a nonlinear distributed-feedback structure may become spatially unstable. Instability leads to nonpassing plateaus in the intensity transmission characteristics. This induces strong optical limiting action in the transmission mode and lower thresholding in the reflection mode.

Coupling from Multimode to Single-Mode Linear Waveguides Using Horn-Shaped Structures

Coupling from a multimode to a single-mode linear waveguide using horn-shaped structures is investigated. The approximate coupling efficiency is found by numerical solution of coupled-mode equations that apply to the reciprocal problem, i.e., to the problem of propagation in an expanding horn. A coupling efficiency in excess of 90 percent is calculated when coupling is from the principal mode of a sample 50-mu m-wide multimode waveguide to a 3-mu m-wide single-mode guide (lambda = 0.63mu m). This efficiency results from a uniformly tapered horn whose length is on the order of 2 mm. The length can be decreased by using a shaped coupling region. One such region is found to result in a coupling length of approximately 1.6 mm.

Unbounded Solutions of Coupled Mode Equations

A number of sufficient conditions for the existence of unbounded solutions of two and three coupled mode equations are obtained when some modes are linearly unstable and all initial amplitudes are arbitrarily small. The difficulty of obtaining sufficient conditions for boundedness of all solutions is discussed, and only two such conditions are obtained. In certain cases it is proved that the unbounded solutions are not more rapid than exponential, whereas they can be shown to be singular (``explosive'') in other cases.