2-D effective index mode solver

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Effective index mode solver

An approximate, quasianalytic mode solver for dielectric integrated optical waveguides with two-dimensional light confinement and weak lateral guiding. Following the waveguide definition in terms of refractive indices, layer thicknesses, a rib width and etching depth, and the vacuum wavelength, the Java applet calculates the propagation constants / effective indices of guided modes and allows to inspect the corresponding optical field patterns. It is intended as a basic tool for integrated optics design, in particular for purposes of demonstration. Note the remarks on the effective-index approximation the mode solver relies on.

Input

For a N-layer rib waveguide, the Define frame receives the vacuum wavelength, the refractive index values ns (substrate), n1, ... , nN (layers 1 to N), nc (cover), the thicknesses t1, ... , tN of the inner layers, the etching depth h, and the width w of the rib. All dimensions are meant in micrometers. The figure illustrates the relevant geometry:

Rib waveguide geometry

Note the orientation of the transverse coordinate axes with the x-direction perpendicular to the film plane. Light propagates along the z-direction. The entire structure is meant to be symmetric with respect to the x-z-plane.

The text fields accept ordinary numbers as well as simple C-style expressions, as long as the result remains in a proper range. Try e.g. an input '2*pi' for the rib width.

Output

A window titled "Mode solver status" informs about propagation constants beta (in Ám-1) and effective mode indices neff. The effective index simulation starts with computing the one-dimensional modes of the central slice -w/2 < y < w/2 and of the two identical lateral slices y < -w/2 and w/2 < y. The single mode index indicates the number of nodes in the basic electric component Ey of the 1-D TE modes, and in the basic magnetic component Hy of the 1-D TM modes. Subsequently, the mode solver forms approximations to the two-dimensional modes by combining appropriate pairs of 1-D modes. An output line like '[TE 1|TE 1|TE 1] >>>' means that the first order TE modes of the central slice and of the lateral slices constitute the 2-D modes listed afterwards. An entry '****' indicates that no suitable lateral 1-D mode was available, hence the substrate refractive index has been substituted. The 2-D modes are identified by the polarization and by two indices, where the first one is directly inherited from the underlying 1-D mode of the central slice. The indices are the number of horizontal (first digit) and vertical (the second one) nodal lines in the mode profile.

The mode profile plots show the field or squared field (Intensity checkbox) of the basic electric component Ey for TE modes, and of the basic magnetic component Hy for TM modes. The 2-D profile checkbox selects an intensity figure. Blue scale levels represent the absolute value of the mode field, or the square of this quantity, plotted over the selected region of the waveguide cross section plane. Choose Horizontal profile or Vertical profiles for a more quantitative inspection of the profile shape. For these 1-D profile plots, the background shading (Geometry checkbox) indicates the dielectric structure. Darker shading means higher refractive index.

Note that the effective index approximation assumes factorizing mode fields. Hence the horizontal profile yields the correct y-dependence for all positions x. According to the initial decomposition in vertical slices, there are two different vertical profiles, one corresponding to the central region (plotted in black) and one for the lateral slices (the white curve). These curves are the constituting 1-D mode profiles. Multiplying them with the horizontal profile results in the 2-D mode fields.

Limitations

The effective index method is meant to be used for quasiplanar modes, assuming small y-derivatives of the permittivity. For structures produced by etching techniques, the approximation may be valid, if either the etching depth remains small when compared with the typical vertical guiding dimensions, or if the local field strength at the rib flanks is small for the mode under investigation. The plot of the two basic 1-D vertical profiles allows to check the validity of the approximation. A pronounced disagreement between the black and the white curve causes the non-physical discontinuities, which are visible in the 2-D images, indicating that the basic assumptions of the effective index approach are not satisfied for the actual problem. This applies in particular to a configuration, where the lateral film does not support a guided mode (the white curve disappears).

While the results may still provide a base impression regarding the real modal properties of such a geometry, one should be aware that the calculations are somewhat arbitrary. This concerns the choice of the basic 1-D modes, which are combined to form a 2-D field, and the amplitudes, which are applied to join the slices. (For a given 1-D mode of the central slice, the implemented algorithm selects that mode of the lateral slice with the nearest smaller propagation constant. The profiles are joined by minimizing the squared difference on the vertical slice boundaries.) For modes with a weak lateral confinement, however, this procedure is usually well defined.