Metric - Mode analysis procedures

Remarks about the Metric mode solver

This concerns lossless and isotropic multilayer slab waveguides with 1-D cross sections and piecewise constant refractive index profiles. Modal solutions are sought on infinite, half-infinite, or finite computational intervals that are enclosed by Dirichlet or Neumann boundary conditions. Standard transfer-matrix techniques allow to identify the eigenmodes in a quite efficient and reasonably robust way. The solutions of the symmetric eigenvalue problems consist of real eigenvalues (the squared propagation constants) and of the corresponding eigenfunctions (the mode profiles, with real transverse field components).

The mode solver engine relies on a search algorithm along the real axis of squared propagation constants, driven by the nodal properties of the basis modes. Where applicable, symmetry properties are exploited to resolve mode degeneracies. In view of the rather nonstandard refractive index profiles that occur along with the spectral discretization for the Helmholtz solvers, a few additional comments seem to be appropriate:

In some instances, permittivity profiles become relevant that consist of individual regions with high refractive index, separated by relatively large low index gaps. For the lowest order modes with exponential profile shapes in the intermediate low index regions, the high index parts are numerically decoupled. The mode solver treats these cases accordingly: Up to a suitable limit of effective index, the lowest order fields are determined separately for the decoupled regions. The mode set then contains some entries with modified refractive index profiles (perhaps degenerate, in case of a properly symmetric structure), whose field profiles are numerically vanishing in the regions where the waveguide has been altered. When computing a spectral discretization, these special modes are joined with higher order fields computed on the entire structure to form the full expansion basis.
The mode solver can only take advantage of symmetry properties, if the full problem, structure and boundary conditions, is symmetric. This should be observed when selecting the computational window for a spectral discretization.

Still cases may occur where the mode solver fails, indicated by some cryptic error message (like "slams: findmodes: missing mode.") at the unexpected end of the program execution. Then before starting to examine the mode solver source code, a few issues should be considered:

Is the structure under investigation correctly and reasonably defined? See the remarks on the construction of Waveguide objects. Perhaps inspect the log output of the mode solver, or try a plot of the refractive index profile.
A slight modification of the problem can sometimes help to avoid singular failures in the eigenvalue search procedure. This concerns in particular the positions of boundaries when computing a spectral discretization (see below), or the placement of "artificial" interfaces in an homogeneous region (related to the constraint nx, nz >= 1).
During the mode spectrum computations (field discretization for BEP / QUEP), occasionally configurations are encountered, where a mode close to the transition from propagating to evanescent type (with a "mode angle" next to the transverse direction, and an almost vanishing propagation constant) occurs. This can cause the linear systems of equations in the BEP / QUEP solvers to become numerically very close so singular, i.e. can lead to highly erroneous results. The mode solver issues a warning message, a (!)-symbol in quiet mode. As a workaround, one should try a slight change of the width of the computational window.

Apart from the direct arguments of the analysis procedures, a few internal options influence the behaviour of the mode solver:

`SLAMS_BQTOL`	Tolerance for fixing the propagation constants; relative or absolute, depending on the absolute value of the squared propagation constants that are involved.
`SLAMS_BQSEP`	Minimum distance at which propagation constants are considered different; relative or absolute, depending on the absolute value of the squared propagation constants that are involved.
`SLAMS_DISCERRTOL`	Tolerance for the maximum relative error in the continuity of the basic field component for accepting a field as a valid mode.
`SLAMS_MODEANGLELIM`	The mode solver issues a warning message, if the absolute value of a mode propagation constant happens to be smaller than `k0` times the level of this option times the maximum refractive index of the respective `Waveguide` (see the comment above).
`DECOUPLED`	A normalized parameter that specifies whether localized fields related to pieces of a waveguide are used to establish approximate modal basis sets (cf. the comment above).

These options are declared in slams.h with generic values defined in slams.cpp (for the value DECOUPLED: structure.h and structure.cpp). Assigning new values to these variables in a driver file affects the operation of the mode solver in subsequent calls to the procedures below.

Nevertheless, in most cases the mode analysis procedures should operate completely autonomously, without any necessity to change the internal options.

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

Nevertheless, in most cases the mode analysis procedures should operate completely autonomously, without any necessity to change the internal options.

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

General mode analysis procedures

A mode analysis problem is specified in terms of the refractive index distribution, the polarization and vacuum wavelength, and possibly accompanied by a computational interval and types of boundary conditions. The Metric collection provides two more generic analysis procedures, defined in slams.h, slams.cpp:

`findmodes`	Determine all modes with squared propagation constants within a specified range, for a general problem given by an arbitrary refractive index profile, with possibly different types of boundary conditions at the upper and lower end of a (possibly infinite or half-infinite) computational window.
`freespacemodes`	For a constant refractive index profile that represents a homogeneous medium: Determine all modes up to a certain order, for a problem with either Dirichlet or Neumann boundary conditions on a finite interval.

Simpler "wrapper"-procedures exist for two typical situations, the identification of guided modes, and the computation of full spectral discretizations. These are described in two sections below.

The procedures operate on the following arguments (naming according to the declaration in slams.h):

wg The multilayer structure that is to be analyzed, including the wavelength, a Waveguide object.

pol The polarization of the 2-D optical waves, either TE or TM, a parameter of type Polarization as defined in gengwed.h.

bdtb, bdtt
or bdt

The type of boundary conditions at the lower and upper end of the computational interval (bdtb, bdtt for findmodes, or bdt for freespacemodes), parameters of type Boundary_type as defined in gengwed.h. The allowed values are

`OPEN`	No boundary conditions, the computational interval extends towards `+`infinity (upper boundary) or `-`infinity (lower boundary). The boundary condition is replaced by the constraint of a square integrable mode profile.
`LIMD`	A homogeneous Dirichlet boundary condition (vanishing field) is imposed on the principal mode profile component at the respective end of the computational interval.
`LIMN`	A homogeneous Neumann boundary condition (vanishing transverse (x-) derivative) is imposed on the principal mode profile component at the respective end of the computational interval.

bpb, bpt The x-positions of the bottom and top endpoints of the computational interval. In case of an OPEN boundary condition, the position parameter of the respective end becomes irrelevant.

ma (output) A ModeArray object that contains the found modes upon completion of the procedure.

verb An integer value that controls the level of log output (stderr): 2: full log information, 1: progress only, 0: suppress all log output.

Besides these explicit arguments, the mode solver procedure can be influenced by a few internal options. The ModeArray parameter ma is passed as a reference argument, and filled with the found modes. See the section on modes and mode arrays for a description of the corresponding objects and for possibilities for their further evaluation.

Procedures: findmodes, freespacemodes.
Relevant source files: slams.h, slams.cpp.

Guided mode analysis

Given a refractive index profile, a vacuum wavelength, and a polarization state, the procedure modeanalysis as defined in slams.h and slams.cpp tries to find all guided modes that are supported by the structure. The open problem is considered, with the constraint of square integrable mode profiles. Internally, the modeanalysis procedure is based on a call to the generic findmodes routine; cf. also the general remarks on the Metric mode solvers. Here only a few most intuitive arguments are required: (naming according to the declaration in slams.h):

`wg`	The `Waveguide` under consideration, including the wavelength.
`pol`	The `Polarization` of the 2-D light waves, either `TE` or `TM` as defined in gengwed.h.
`ma`	(output) A `ModeArray` object that is cleared first, then filled with the found modes. These are ordered by their squared propagation constants, highest values first. Hence, with the exception of cases with numerical degeneracies, the array index corresponds to the mode order; index 0 identifies the fundamental mode.
`quiet`	(optional) An integer parameter that suppresses the log output (stderr), if `quiet == 1`.

The function returns the number of found modes as an integer value. For further details on how to access and process the mode objects stored in the ModeArray argument see the section on modes and mode arrays. The driver file tlwg.cpp provides an example for the usage of the mode analysis routine and for the handling of the mode objects. Modal dispersion curves, here the dependence of effective indices on a core layer thickness, can be calculated according to the example in disp.cpp.

Procedure: modeanalysis.
Relevant source files: slams.h, slams.cpp.

Spectral discretization

Enclosing an internal refractive index profile by suitable boundary conditions leads to a discretization of the mode spectrum on the respective computational interval. Within the Metric collection, finite discrete sets of these modes are employed as an expansion basis for the BEP and QUEP Helmholtz solvers.

Given the refractive index profile, a vacuum wavelength, polarization state, the computational interval, and the type of boundary conditions, the procedure modespectrum as defined in slams.h, slams.cpp computes all eigenmodes of the problem with orders up to a specified number. Internally, the modespectrum procedure is based on a call to the generic findmodes routine; cf. also the general remarks on the Metric mode solvers. The following arguments are required (names according to the declaration in slams.h):

wg The Waveguide under consideration, including the wavelength.

pol The Polarization of the 2-D light waves, either TE or TM as defined in gengwed.h.

cw The computational window, an Interval as declared in structure.h. The endpoints are meant as x-positions on the cross section axis of wg.

bdt

(optional) The type of boundary conditions at the ends of the computational interval, parameters of type Boundary_type as defined in gengwed.h. Here the allowed values are

`LIMD`	Homogeneous Dirichlet boundary conditions (vanishing field) are imposed on the principal mode profile component.
`LIMN`	Homogeneous Neumann boundary conditions (vanishing transverse (x-) derivative) are imposed on the principal mode profile component.

If the bdt parameter is missing, the procedure selects Dirichlet boundary conditions (LIMD).

mnm Modes from order 0 up to order mnm-1 are sought. Hence the integer value mnm specifies the number of modes to be identified.

ma (output) A ModeArray object that is cleared first, then filled with the found modes. These are ordered by their squared propagation constants, highest values first. Hence, with the exception of cases with numerical degeneracies, the array index corresponds to the mode order.

quiet (optional) An integer parameter that suppresses the log output (stderr), if quiet == 1.

The function returns the number of found modes as an integer value (i.e. the value mnm upon successful completion). For further details on how to access and process the mode objects stored in the ModeArray argument see the section on modes and mode arrays. An example for the usage of the routine can be found in the driver file spec.cpp. Note that the spectral discretization procedure is invoked automatically by some of the constructors of the field containers BepField and QuepField of the Helmholtz solvers.

Procedure: modespectrum.
Relevant source files: slams.h, slams.cpp.