Operator symbols play a pivotal role in both the exact, well-posed, one-way reformulation of solving the (elliptic) Helmholtz equation and the construction of the generalized Bremmer coupling series. The inverse square-root and square-root Helmholtz operator symbols are the initial quantities of interest in both formulations, in addition to providing the theoretical framework for the development and implementation of the 'parabolic equation' (PE) method in wave propagation modeling. Exact, standard (left) and Weyl symbol constructions are presented for both the inverse square-root and square-root Helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension, extending (and, ultimately, unifying) the previously published corresponding results for the defocusing quadratic case [J. Math. Phys. 33 (5), 1887-1914 (1992)]. Both (i) spectral (modal) summation representations and (ii) contour-integral representations, exploiting the underlying periodicity of the associated, quantum mechanical, harmonic oscillator problem, are derived, and, ultimately, related through the propagating and nonpropagating contributions to the operator symbol. High-and low-frequency, asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the PE method. Moreover, while the focusing quadratic profile is, in some respects, nonphysical, the corresponding Helmholtz operator symbols, nevertheless, establish canonical symbol features for more general profiles containing locally-quadratic wells.
The Bremmer series solution of the wave equation in generally inhomogeneous media, requires the introduction of pseudodifferential operators. In this paper, sparse matrix representations of these pseudodifferential operators are derived. The authors focus on designing sparse matrices, keeping the accuracy high at the cost of ignoring any critical scattering-angle phenomena. Such matrix representations follow from rational approximations of the vertical slowness and the transverse Laplace operator symbols, and of the vertical derivative, as they appear in the parabolic equation method. Sparse matrix representations lead to a fast algorithm. An optimization procedure is followed to minimize the errors, in the high-frequency limit, for a given discretization rate. The Bremmer series solver consists of three steps: directional decomposition into up-and downgoing waves, one-way propagation, and interaction of the counterpropagating constituents. Each of these steps is represented by a sparse matrix equation. The resulting algorithm provides an improvement of the parabolic equation method, in particular for transient wave phenomena, and extends the latter method, systematically, for backscattered waves.
Absh-nct-We report on the reflection properties of multimode interference (MMI) devices: we distinguish between reflection back into the input waveguides and internal resonance modes due to the occurrence of simultaneous self-images. Because of self-imaging, reflection CBn be extremely efficient, even in the case of MMI devices with optimized transmission. This conclusion is confirmed by the observed spectral behavior of InP-based ring lasers incorporating MMI 3dB couplers and MMI power splitters. Several techniques are proposed to minimize the influence of these reflections.
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