Structure definition Coordinate system, polarization, units Waveguide objects Sequences of waveguide segments Rectangular circuits

Convention for coordinate system, polarization, units

All Metric routines rely on a representation of the 2-D problems in a Cartesian coordinate system with axes x and z. As suggested by the figure below, occasionally these directions are referred to as "horizontal" (z) and "vertical" (x). The structure under investigation and all optical fields are assumed to be constant along the direction y perpendicular to the computational x-z-plane. For the description of 2-D waveguides and the associated modes, the z-axis takes the role of the direction of light propagation, the "longitudinal" direction, while the x-axis is oriented along the "transverse" direction. In the context of the scattering problems, the distinction between the roles of the axes becomes irrelevant.

The propagation of continuous optical electromagnetic waves in linear isotropic dielectric structures is considered. The vacuum wavelength lambda = 2 pi / k₀ = 2 pi c / omega specifies the fixed angular frequency omega, corresponding to a time dependence  ~exp(i omega t) of all fields, for vacuum wavenumber k₀ and vacuum speed of light c. Conventionally, optical waves are described in terms of the electric field E and the magnetic field H. Under the above assumptions, the Maxwell equations for the optical electromagnetic field split into two independent subsets of equations, that apply to solutions with different polarization states:

• TE fields combine the electric component E_y and the components H_x and H_z of the magnetic field. Usually E_y serves as the principal component for a scalar description of TE fields.
• TM fields involve the magnetic component H_y and the electric components E_x and E_z. Usually one uses H_y as the principal component for a scalar description of TM fields.

Relevant source files: gengwed.h, gengwed.cpp.

Multilayer slab waveguide objects

Waveguide objects represent a 1-D piecewise constant refractive index profile. The waveguide cross section, the x-axis, is to be divided into an array of intervals with constant refractive index, where the bottom and top intervals are unbounded. Since the specification of permittivity or refractive index levels is always related to optical oscillations at a definite frequency, the value of the vacuum wavelength is made part of the waveguide container.

The information on the number of layers, the layer boundaries, the refractive index values, and the vacuum wavelength constitute a Waveguide object as defined in structure.h, structure.cpp. See matrix.h and matrix.cpp for the definition of Dvectors used for the boundary locations and refractive indices.

Among the members of a Waveguide-object are:

nx		The number of inner layers in the division of the cross section axis.
hx		The x-positions of the layer boundaries, a Dvector(nx+1).
lambda		The vacuum wavelength of the light propagating in this waveguide.
n		The array of refractive index values, a Dvector(nx+2).
eps		Functions which return permittivity values, for positions specified by a layer index or by an x-coordinate.
layer		Returns the layer boundaries (an Interval object, also declared in structure.h) for a given x-coordinate or a layer index.
layeridx		Returns the index of a layer that corresponds to a given x-coordinate.
plot		Generate a .m-script that produces a plot of the refractive index profile when executed in MATLAB; for purposes of inspecting a waveguide specification. See the section on visualization of structural information.

Additionally, objects of class Waveguide can be copied and assigned, they know how to write and read themselves to/from FILEs, they can translate themselves to new coordinates, they have some knowledge about the bounds (defaultepseffmin, defaultepseffmax) on prospective effective indices (permittivities) of guided modes, they can compare (equal) themselves to another Waveguide, and they supply a number of methods for testing (consistency, bdmatch), information (constmed, checksymmetry, decoupepseff, lbdecepseff, ubdecepseff), and transformation (optimize, split, expand, mirror) that are used by the routines for mode analysis, and by the Helmholtz solvers. Further details can be found in the source files structure.h, structure.cpp directly.

Note the following points:

• All length parameters are given in micrometers, if not otherwise stated.
• The Waveguide member nx denotes the number of inner layers in the decomposition. Accordingly, for a Waveguide(nx) one has to specify the locations hx(0), ..., hx(nx) of the layer boundaries, and refractive index values n(0), ..., n(nx+1). Consequently, hx is a Dvector(nx+1), and n is a Dvector(nx+2).
• The number nx of inner layers must be greater or equal than one. Disregarding this constraint is likely to end in a core dump. A homogeneous piece of material can still be specified as a Waveguide(1) with two arbitrary layer positions and three equal refractive index values. In case of a structure with only one dielectric interface one should split one of the layers into two with equal permittivity.
• The boundary locations must be in ascending order, without any equal entries: hx(l) < hx(l+1). Disregarding this constraint is likely to end in a core dump.

The Waveguide constructors take one of the sets {}, {nx}, or {nx, hx, lambda, n} as arguments, where the remaining information must be directly assigned. See one of the example files (e.g. tlwg.cpp) for an actual waveguide definition.

Class: Waveguide.
Relevant source files: structure.h, structure.cpp.

Sequences of waveguide segments

The BEP and QUEP Helmholtz solvers operate on sequences of multilayer slab waveguides as containers for the geometrical and material specification of the scattering problems. Sequences of z-homogeneous waveguide segments are combined into SegWgStruct objects, as defined in structure.h and structure.cpp. The ingredients are an array of 1-D (x-dependent) refractive index profiles, i.e. Waveguides, and the accompanying z-positions of the vertical interfaces between the segments.

The SegWgStruct objects provide the following methods / members:

nz		The number of inner segments in the sequence.
hz		The z-positions of the segment interfaces, a Dvector(nz+1).
()		Integer subscript operators, range 0, ..., nz+1: Access to the individual Waveguides in the sequence.
init		Initialize all segments with the same Waveguide profile.
segidx		The segment index corresponding to a given z-coordinate.
segment		The boundaries of a segment corresponding to a given z-coordinate or segment index.
n		A function that returns the local refractive index, given a pair of x-z-coordinates.
lambda		The vacuum wavelength, assuming that it is unique in all Waveguides.
defaultwindow		Determines a default window around the "inner" structure, e.g. for plotting purposes.
plot		Generate a .m-script that produces a plot of the refractive index profile when executed in MATLAB; for purposes of inspecting a structure specification. See the section on visualization of structural information.

Additionally, SegWgStruct-objects can be copied and assigned, they can write and read themselves to / from files, and they know how to shift all coordinate data along the x and z directions (ztranslate, xtranslate). The QUEP routines access methods that convert the horizontal sequence to a vertical one (rotate), procedures that add and remove additional outer segments (expand, shrink), or a function that eliminates unnecessary boundaries between identical segments (optimize). Finally, a SegWgStruct can convert itself into a Circuit (see below).

Note that, in contrast to the waveguide container, here numbers of inner segments of less than 1 are allowed: SegWgStruct(-1) defines a z-homogeneous waveguide (with hz not to be accessed, used only as a means to enable the plot routines for mode arrays). SegWgStruct(0) is a structure with a single vertical interface, e.g. a waveguide facet.

Two SegWgStruct-constructors take one of the sets {}, {nz} as arguments; the waveguides and z-positions have to be assigned individually. As an example, in the driver files sgrat.cpp and defcav.cpp the structure definition as an array of pieces of waveguides is quite intuitive. A third SegWgStruct-constructor converts a a Circuit object (see below) into a SegWgStruct.

Class: SegWgStruct.
Relevant source files: structure.h, structure.cpp.

Rectangular optical circuits

In some cases the arrangement of the structural data for a scattering problem in the particular form of sequences of waveguides appears to be rather cumbersome. Where appropriate, Circuit objects can be used for a specification of the material and geometrical information. A Circuit as defined in structure.h, structure.cpp combines a matrix of refractive index values, two arrays of positions of horizontal and vertical interfaces, and the vacuum wavelength.

Since the Circuits are - so far - only meant to provide an alternative way of constructing the SegWgStructs, only basic members and methods are provided:

nx		The number of inner layers in the rectangular division.
hx		The x-positions of the horizontal layer interfaces, a Dvector(nx+1).
nz		The number of inner slices in the rectangular division.
hz		The z-positions of the vertical slice interfaces, a Dvector(nz+1).
lambda		The vacuum wavelength for the scattering problem.
n		Refractive index values for the rectangular patches, a Dmatrix(nx+2, nz+2).
plot		Generate a .m-script that produces a plot of the refractive index profile when executed in MATLAB; for purposes of inspecting a structure specification. See the section on visualization of structural information.

Copying and assignment of Circuits is automatically implemented via the respective methods of the member objects. To avoid core dumps caused by access to illegal index positions, note the following issues:

• nx and nz are the numbers of inner layers and slices in the decomposition. Consequently, hx is to de declared as a Dvector(nx+1), hz is to de declared as a Dvector(nz+1), and n is a Dmatrix(nx+2, nz+2).
• Both nx and nz must be greater or equal than 1. To define even simpler structures, one needs to provide additional interfaces between regions with equal refractive indices. Examples are the definitions of a waveguide facet in facet.cpp, or the simulations of beams in a homogeneous medium in bcross.cpp, bspread.cpp
• The interface positions in hx and hz must be in strictly ascending order, without any equal entries.

Class: Circuit.
Relevant source files: structure.h, structure.cpp.