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A mode solver for bent integrated optical dielectric multilayer slab waveguides and curved dielectric interfaces with 1-D cross sections. Following the structure definition in terms of bend radius, refractive indices, layer thicknesses, if applicable, and the vacuum wavelength, the script calculates the complex effective indices of the (leaky) modes supported by the bend, or their phase propagation and attenuation constants, respectively, 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 a curved slab waveguide with N intermediate layers, the input mask receives the
vacuum wavelength, a specification of the polarization, the radius R of the *outer rim* of the bend, further
refractive index values
n_{i} (interior region),
n_{1}, ... ,
n_{N} (inner layers 1 to N),
n_{e} (exterior region),
and thicknesses
t_{1}, ... ,
t_{N} of the intermediate layers.
Select N=0 to specify a curved dielectric interface only.
All dimensions are meant in micrometers.
The figure illustrates the relevant geometry:

Polar coordinates r, θ span the Cartesian x-z-plane; the center of the bend is located at the origin of both coordinate systems. Light propagates along the angular coordinate θ. The refractive index profile is independent of θ and piecewise constant in the radial direction r. All electromagnetic fields and the refractive index distribution are assumed to be constant along the y-axis (perpendicular to the x-z- and r-θ-plane, not shown).

Computational parameters

The solver relies on a heuristic procedure for generating initial guesses for the roots of the transverse resonance condition in the complex plane. A complex secant method then converges the initial guesses numerically to actual roots. Further heuristics are applied for the classification and ordering of these roots.

The solver accepts a few parameters that influence this search for valid mode wavenumbers in the complex plane. If the numbers of trials is not directly specified (Auto-checkbox), the solver applies a heuristic based on modal solutions for an equivalent "tilted" effective refractive index profile for a straight graded-index slab. Alternatively, a number of initial guesses on a regular rectangular grid in the complex plane can be specified explicitly.

The range in the complex plane examined (initial guesses) for effective mode indices can be restricted to real parts larger than the
smallest refractive index in the structure. The solver then responds with the modes of lowest radial order only (checkbox). Alternatively, an
interval from close-to-zero up to the largest refractive index in the structure is examined. Then also modes of higher radial
order can be identified. In all cases, attenuation constants in a range between 10^{-14} and 10^{-1}
are considered.

While the found solutions should be valid bend modes, one must not rely too much on the solver being able to find all existing bend modes, nor on the correct classification of the modes. In many cases, however, these heuristics have been observed to work adequately. The solver should reproduce the results of Ref. [1] with (almost) no changes to the default computational parameters. You might wish to try this out as a test of the solver, and to become familiar with the interface.

Output

A table shows, for each bend mode, the complex bend mode eigenvalue in the form of either

- the effective index N
_{eff}, - the propagation constant γ = β - i α, for phase propagation constant β and attenuation constant α,
- the angular wavenumber ν.

Relations
γ = k N_{eff} and
ν = γ R apply,
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 region r < R, in the real part of the principal electric component
E_{y} of the radial profile of TE modes, and in the principal magnetic component
H_{y} of the radial profile of TM modes (where the phase of the mode profile has been adjusted such that the principal components are real at
r = R).

One should be aware that the definition of the bend radius R is, in some respect, arbitrary. For a curved slab with one intermediate layer of thickness t_{1}, a frequent choice is to describe the bend with a radius R' = R - t_{1}/2 at the center of the slab (cf. the figure), rather than at the outer rim, as in case of the present solver. This choice should lead to the same solutions, i.e. to the same angular dependence ∼exp(-i ν θ) of the bend modes. These are then characterized by *different* values
γ' = γ R/R'
and
N_{eff}' = N_{eff} R/R'
of propagation constant and effective mode index, respectively.

Referring to the polar coordinate system as introduced above, this concerns optical electric fields
** E** and magnetic fields

All fields are constant along the y-direction. The profiles of TE bend modes are of the form
**E**(r) = (0, E_{y}, 0)(r), and
**H**(r) = (H_{r}, 0, H_{θ})(r), where
**E** and **H** are the electric and magnetic parts of the mode profile, respectively, depending
on the radial coordinate r only.
Likewise, the profiles of TM bend modes can be written as
**E**(r) = (E_{r}, 0, E_{θ})(r), and
**H**(r) = (0, H_{y}, 0)(r).

Primarily, the solver determines *forward* bend modes, with the former expressions for the optical electromagnetic field, with positive propagation constant β, positive attenuation constant α, and corresponding signs of the real and imaginary parts of γ. Referring to the above figure, these relate to clockwise wave propagation in positive θ-direction. For each of these solutions a *backward* bend mode can be constructed, with a functional form

Note that, while the backward modes can thus be characterized by the same values of β, α and γ,
the profiles **E**_{-}, **H**_{-} *differ* from the profiles **E**_{-}, **H**_{-} of the forward modes in the signs of the radial components. The backward modes relate to anticlockwise wave propagation in negative θ-direction. The solver permits to inspect the respective fields, if the Rev-checkbox is selected.

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 (rough) normalization of
the modes to unit angular power flow: The radial integral of the angular component S_{θ}
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).

Mode profile plots show the real and imaginary parts of the complex field profile, its absolute value, or the absolute squared profile. 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, on a default radial range. This is to make the plots comparable. Select the button labeled "↕" to adjust the vertical plot range to the functions that are actually displayed.

Single components ψ ∈ {E_{r}, …, H_{θ}}
of the complex-valued electromagnetic mode profile relate to time-varying physical fields
*Ψ*(r, t) = Re ψ(r) exp(i ω t)
at θ = 0. The animations show the respective component
at recurring points in time, equally distributed over the period of 2π/ω.

Alternatively, propagation plots can be displayed. These show a component of the total physical
electromagnetic field ** E**,

Select points on the planes of the plots to inspect precise local field levels. Clicks outside the actual axes close the coordinate displays. In case of a propagation plot, the "▯"-button toggles a colorbar. Select a field level on the colorbar to superimpose the field plot with a pseudo-contour at that level. Also here the contours are removed by a click in the colorbar area outside the axes.

Reference

**[1]** K.R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, J. Ctyroky

*Analytical approach to dielectric optical bent slab waveguides*

Optical and Quantum Electronics
**37** (1-3), 37-61 (2005)