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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:

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.

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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.

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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.

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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.

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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.

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