A mode solver for integrated optical dielectric multilayer slab waveguides with 1-D cross sections. Following the waveguide definition in terms of refractive indices, layer thicknesses, and the vacuum wavelength, the script 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.

Input

For an N-layer structure, the input mask receives the
vacuum wavelength, a specification of the polarization, refractive index values
n_{s} (substrate),
n_{1}, ... ,
n_{N} (inner layers 1 to N),
n_{c} (cover),
and thicknesses
t_{1}, ... ,
t_{N} of the inner layers.
All dimensions are meant in micrometers.
The figure illustrates the relevant geometry:

Light propagates along the z-direction, with the refractive index profile and all fields assumed to be constant along the y-axis. The x-direction is perpendicular to the film plane.

Output

A table shows, for each guided mode,
the propagation constant
β (in µm^{-1}) and the effective index
N_{eff} = β/k, where k = 2π/λ
is the vacuum wavenumber associated with the specified vacuum wavelength
λ.
The mode identifier (order) indicates the number of nodes in the principal electric
component
E_{y} of TE modes, and in the principal magnetic component
H_{y} of TM modes.
B is a normalized effective permittivity, the
ratio
B = (N_{neff}^{2}-n_{min}^{2})/(
n_{max}^{2}-n_{min}^{2}), with
the maximum refractive index n_{max} of all layers.
n_{min} denotes the larger one of the substrate and cover
refractive indices.

In a waveguide with one inner layer, a mode angle θ
(output in degrees) can be associated with a guided mode, defined by
cos(θ) = β/kn_{1} =
N_{eff}/n_{1}; this refers to the common ray picture
for confined wave propagation in a single core waveguide.
For a structure that supports more than one guided mode, the program calculates
the coupling lengths or half beat lengths
L_{c} = π/|β_{0} - β_{1}|
that correspond to the interference pattern of each pair of modes with
different propagation constants β_{0} and β_{1}.

If the interior stack contains (thick) layers with, when compared to other guiding regions, low refractive index, it can happen that the multilayer structure supports modes that are, for all practical purposes, restricted to some part of the structure only. A typical example would be a directional coupler (here 1-D), consisting of two high index cores with a thick buffer layer of low index material in between. With increasing distance between the cores, the "supermodes" of the full structure become more and more degenerate, i.e. "numerically decoupled". The solver detects these scenarios, and replaces the true "supermodes" by modes of the individual cores, with (near) zero amplitudes in the respective other core. Modes determined in this way show up with an additional entry (x) in the list of effective indices and propagation constants. Note that these fields are true eigenfunctions (e.g. orthogonal to each other, and to any other modes), for all numerical and practical purposes.

Referring to the coordinate system as introduced above, the profiles of guided TE modes supported by the
present lossless dielectric multilayer slab waveguides are of the form
**E**(x) = (0, E_{y}, 0)(x), and
**H**(x) = (H_{x}, 0, i H_{z})(x), where
**E** and **H** are the electric and magnetic parts of the 1-D mode profile, respectively, depending
on the out-of-plane coordinate x only (note the imaginary unit).
Likewise, the profiles of guided TM modes can be written as
**E**(x) = (E_{x}, 0, i E_{z})(x), and
**H**(x) = (0, H_{y}, 0)(x). For both TE and TM modes, the overall phase of the
mode can be adjusted such that the component functions
E_{x}, E_{y}, E_{z}, H_{x}, H_{y}, and H_{z}
are real. Consequently, after selection of
"E_{z}" or "H_{z}", the plot window shows the *imaginary* part of the
longitudinal electric or magnetic component of the mode profile, while for all other components
the *real* part is displayed.

Being solutions of eigenvalue problems, the mode profiles are determined up to some
complex constant only. No units are shown for their electric or
magnetic fields. Still the given values correspond to a normalization of
the modes to unit power flow: The integral along the x-axis of the longitudinal component S_{z}
of the Poynting vector evaluates to 1 W/µm (power per lateral (y) unit length).
Correspondingly, all electric fields are given in units of V/µm, magnetic fields
are measured in A/µm, the components of the Poynting vector **S** have units of
V·A/µm^{2} = W/µm^{2}, and the electromagnetic energy density
w is measured in W·fs/µm^{3}. In this context the vacuum permittivity and
permeability, respectively, are
ε_{0} = 8.85·10^{-3} A·fs/(V·µm) and
µ_{0} = 1.25·10^{3} V·fs/(A·µm).

Integrals of the mode profiles can serve for purposes of normalization,
for an assessment of the confinement, or for the evaluation of
perturbational expressions, where standard ratios concern the influence of material loss or gain, or phase shifts
caused by small changes in refractive index. The script evaluates respective integrals of
the squared principal components
|E_{y}|^{2},
|H_{y}|^{2}
of TE and TM modes, integrals of the squared absolute values
|**E**|^{2},
|**H**|^{2}
of the vectorial electric and magnetic fields, and integrals of the longitudinal component S_{z} of the Poynting vector.
Absolute and relative expressions (confinement factors, Γ) are evalutated piecewise for each layer with constant refractive index.
In case of a local perturbation in only part of a layer, you might wish to split the respective region by introducing additional
layers of proper location and thickness with equal refractive index.

Mode profile plots show the field (real or imaginary, as discussed above), its absolute value, or the squared field. The background shading indicates the dielectric structure, where darker shading means higher refractive index. After selecting "Plot", the extent of the vertical axis is being adjusted such that it covers the maximum values, determined separately for the electric field strength, magnetic field strength, Poynting vector, and the energy density, over all modes (and all their field components) that have been identified by the solver. This is to make the plots comparable. Select the button labeled "↕" to adjust the vertical plot range to the functions that are actually displayed.

Parameter scans

The *Scan* facilities concern variations of all quantities that define the multilayer waveguide configuration, for a given number of inner layers and fixed polarization.
The list of options covers the vacuum wavelength λ, the refractive indices
n_{s}, n_{c} of the substrate and cover regions, and the refractive indices n_{j} and thicknesses t_{j} of the interior layers.
Choose the parameter interval of interest (tests for physical plausibility apply), and specify a number of samples. The solver then determines effective index levels of guided modes,
for the specified range of the parameter. *Note that curves far varying vacuum wavelength neglect material dispersion, i.e. all refractive
index values are assumed to be constant for the respective wavelength/frequency interval.*

Similar to the facilities for field inspection, *Plot* controls are offered that permit to enlarge (+) and to reduce (-) the size of the figure, to *Export* the curve data, to export
the figure in *SVG* format, and to *Detach* the figure into a separate browser window. Click in the figure for a precise evaluation of curve levels. *Accept* a specific
parameter for inspecting the fields of the modified configuration (this closes all scan-related controls).