Separating optical waveguides are expected to become important in integrated optics. They occur whenever a branching waveguide is used ( Fig. la) or whenever two parallel waveguides in close proximity are separated (Fig. lb). These situations will arise in virtually all applica tions of electrooptic switches and waveguide couplers that depend on evanescent coupling.Using quasi-normal modes and a step transition model we have studied theoretically the mode conversion behavior of separating planar dielec tric waveguides as a function of taper slope and mode synchronism, which is adjusted by branch asymm etry. ��en the taper slope is large in a near symmetric structure, considerable mode conversion will occur and the structure will act as a power divider (Fig. 2b). Incident power concentrated in the upper and lower parts of the structure will end up in the upper and lower arms of the branch, respectively. This behavior is usually required for conventional evanescent couplers. However , in a more asymmetric structure with smaller taper slope, mode conversion is negligible' and the structure will act as a mode splitter (Fig. 2a). Mode power is then trans ferred to one arm of the branch or the other.The transition boundary between a power divider and mode splitter (Fig. 3) is conveniently described approximately by:As/ey ; .43Here AS is the difference in mode propagation constant for large waveguide separation, e is the taper slope, and y is the decay constant of the field in the separating region. This result will allow quantitative design of mode splitters or power dividers, as desired, and should aid in the design of separating waveguides with minimum radiation losses. 81 A ----� B FIGURE I -Two types of separating optical wave guides. Propagation in both structures can be described in an equivalent manner if the parallel sections are sufficiently close in (b) for the normal modes to. have power in both guiding regions.
5.1
In previous papers we described a family of multidimensional filters to suppress random noise based on matrix-rank reduction of constant-frequency slices. Here we extend these filters to perform multidimensional trace interpolation. This requires rank reduction when some, perhaps most, of the matrix elements are unknown, a procedure called matrix completion or matrix imputation. We show how this new interpolator improves the spatial resolution of 3D data when applied prior to prestack migration.
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