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Whispering gallery mode solver

An online solver for the whispering gallery resonances of microrings, microdisks, or more general 2-D circular dielectric multilayer cavities in integrated optics / photonics. Following the structure definition in terms of cavity radius, refractive indices, and layer thicknesses, if applicable, and the specification of a target vacuum wavelength or of an interval of potential resonance wavelengths, the script calculates the complex angular eigenfrequencies of the (leaky) whispering gallery modes (WGMs) supported by the cavity, and their resonance wavelengths and linewidths, resonance frequencies, cavity lifetimes, and quality (Q-) factors, respectively. Facilities for detailed inspection of the corresponding mode profiles and resonant field patterns, and for exporting data and figures, are provided.

Input

For a circular cavity with N intermediate layers, the input mask receives
the radius R of the *outer rim* of the cavity,
a specification of the polarization,
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 uniform circular dielectric disk.
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 cavity is located at the origin of both coordinate systems. 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 tries to identify whispering gallery modes (WGMs) with resonance wavelengths that are either close to a target vacuum wavelength, or fall within a specific interval of vacuum wavelengths (Interval-checkbox). A somewhat heuristic procedure is applied for generating initial guesses for the roots of the transverse resonance condition in the complex plane. A complex secant method converges these 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 influences the process to determine the integer angular mode orders that relate to the resonances of interest. If the numbers of trials is not directly specified (Auto-checkbox), the solver applies a heuristic based on bend mode solutions for an equivalent "tilted" effective refractive index profile for a straight graded-index slab. Alternatively, a number of initial guesses for regularly spaced angular mode numbers and attenuation levels can be specified explicitly.

The range in the complex plane examined (initial guesses) for angular mode orders can be restricted to values that corresponded to angular light propagation with effective indices
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^{-2}
are considered.

While the found solutions should be valid whispering gallery modes, one must not rely too much on the solver being able to find all existing WGMs, nor on the correct classification of the modes. In many cases, however, these heuristics have been observed to work adequately. The solver should reproduce those results of Ref. [1] that concern resonances of separate, uncoupled rings or disks 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 whispering gallery mode, the complex WGM eigenvalue in the form of either

- the complex angular eigenfrequency ω
_{c}= ω + i α, for (real) angular resonance frequency ω and attenuation constant α, - the time period T and the cavity lifetime τ = 1/Δω, for a linewidth (full width at half maximum, FWHM) Δω of the resonance concerning angular frequency,
- the resonance wavelength λ and the linewitdh (full width at half maximum, FWHM) Δλ of the resonance on the vacuum wavelength axis,
- the resonance frequency f and the quality factor Q associated with the resonance.

Relations τ = 1/(2 α), Q = ω/(2 α) = λ/(Δλ) , λ = 2 π c/ω, Δω = 2 α = ω/Q = 1/τ, T = 2 π/ω, and f = 1/T = ω/(2 π) apply, where c is the vacuum speed of light.

The WGM identifiers comprise two indices. A first index indicates the radial order of the mode, determined as 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). The second index gives the angular order m of the WGM, the number of full exponential cycles in the principal fields on an angular interval of 2π.

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 polarized WGMs 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 polarized WGMs can be written as
**E**(r) = (E_{r}, 0, E_{θ})(r), and
**H**(r) = (0, H_{y}, 0)(r).

Being solutions of eigenvalue problems, the whispering gallery modes are determined up to some
complex constant only. No units are shown for their electric or
magnetic fields. The given values correspond to a level of one in the principal field component
at the rim r = R of the cavity.
Units of V/µm can be assumed for all electric fields, 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_{c} t)
at θ = 0. The animations show the respective component
at recurring points in time, including the decay effected by the imaginary part of ω_{c}.

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

Referring to the coordinates as introduced in the figure, the WGMs as calculated by the solver, with positive angular mode order m, represent fields that propagate *clockwise* in postive θ-direction. For each mode there exists an eigenfunction with the same complex eigenfrequency and negative angular mode order -m, that corresponds to an *anticlockwise* propagating resonant wave. The solver permits to inspect the respective fields, if the Rev-checkbox is selected. Note that the profiles of the reversed modes differ from the profiles of the mode with positive angular orders in the signs of the radial components. Superpositions of those two degenerate WGMs, with arbitrary complex coefficients, constitute further valid solutions for that particular resonance frequency. Options for inspecting the fields of those superimposed states are available for the propagation plots.

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]** E.F. Franchimon, K.R. Hiremath, R. Stoffer, M. Hammer

*Interaction of whispering gallery modes in integrated optical micro-ring or -disk circuits: Hybrid CMT model*

Journal of the Optical Society of America B **30** (4), 1048-1057 (2013)