The scalar polarized 2-D Helmholtz equations are addressed, on a rectangular computational domain with transparent boundary conditions that permit guided wave in- and outflux. Given a problem specification in terms of interface positions, a matrix of refractive index values, polarization and wavelength parameters, and guided wave input, the script determines modal output amplitudes (elements of the scattering matrices), and the power levels associated with guided and nonguided directional outgoing waves (transmittances / reflectances, power balance). Facilities for detailed inspection of the optical electromagnetic field are provided, including animations of the harmonic oscillations in time, with options for exporting figures and data.
Input
Cartesian coordinates x and z span the 2-D plane of interest. For convenience, we refer to these directions as vertical (x) and horizontal (z). The terms "top", "right", "bottom", and "left" are used to identify the boundaries of the computational domain. For a rectangular circuit with N_{l} inner layers and N_{s} inner slices, the input mask receives the vacuum wavelength λ, a specification of the polarization, a matrix of refractive index values n_{l,s}, values t_{l} for the thicknesses of the inner layers, and values w_{s} for the widths of the inner slices. All dimensions are meant in micrometers. At least one inner layer and one inner slice is required. The figure illustrates the relevant geometry, where blue elements relate to the boundaries of the computational domain:
The optical electromagnetic field is assumed to be constant along the y-direction. One then distinguishes waves that are polarized perpendicular to the x-z-plane (TE polarization), and waves that are polarized parallel to that plane (TM polarization).
In certain cases the default-fill-functionality of the script can ease the data entry. Select the appropriate number of inner layers and slices, and Clear the input mask. Enter refractive indices and width for one slice. Then Fill will repeat the values of refractive indices and width row-wise over all slices. Likewise, if only a single refractive index value, width, or thickness is provided, the Fill procedure enters these values into all other fields.
Guided wave excitation
Initialize starts a pre-analysis of the structure. Guided modes supported by the slab waveguides that cross the boundaries of the computational domain are identified. Initial amplitudes can be specified for each of these modes.
Computational parameters
The solver works on a cross-shaped computational domain, with an inner rectangular computational window (x, z) ∈ [x_{b}, x_{t}] × [z_{l}, z_{r}] (cf. the figure above). That window is specified by the distances Δx_{t}, Δz_{r}, Δx_{b}, Δz_{l} between the outermost interface positions, and the boundary locations.
The electromagnetic field is expanded into the sets of normal modes (1-D slab modes with Dirichlet boundary conditions) supported by the refractive index profiles associated with the layers and slices that make up the circuit. Inside the inner computational window, quadridirectional expansions into modes that propagate in the ±x- and ±z-directions apply. Bidirectional expansions in the half-infinite exterior regions ensure that the boundaries of the inner window become transparent for outgoing (scattered) waves from the interior. The numbers M_{x}, M_{z} of terms taken into account for these expansions are specified through the number of terms spend for each (vacuum) wavelength-unit in the respective x-, z-extension of the inner window.
Remarks:
Output
Upon completion of a simulation, the solver lists the complex amplitudes of guided modes of different order (if any), that cross the boundary lines in the four directions, and the levels of outgoing guided and total optical power, per boundary line and for the entire domain. Depending on the specific excitation setting, the values can be directly interpreted as elements of the scattering matrix of the circuit, or as reflectance and transmittance levels, respectively.
The solver then offers a panel with controls for detailed inspection of the optical field solution. Referring to the coordinate system as introduced above, this concerns electric fields E and magnetic fields H that depend on the spatial coordinates x, z and on time t, with angular frequency ω, as
All fields are constant along the y-direction.
The frequency-domain electric and magnetic fields E, H depend on the in plane coordinates x and z only. For TE polarization, these fields are of the form E = (0, E_{y}, 0), H = (H_{x}, 0, H_{z}). Likewise, TM-polarized fields are of the form E = (E_{x}, 0, E_{z}), H = (0, H_{y}, 0). The plot functionality refers to the complex components of these field functions. Further, the x- and z-components S_{x}, S_{z} of the time-averaged Poynting vector and the time averaged energy density w can be inspected. Note that the y-component of the Poynting vector vanishes for the present y-constant fields. Single components ψ ∈ {E_{x}, . . . , H_{z}} of the complex-valued frequency domain electromagnetic field relate to time-varying physical fields Ψ(x, z, t) = Re ψ(x, z) exp(i ω t). The global phase of the solution can be adjusted; the animations show the respective component at recurring points in time, equally distributed over the period of 2π/ω.
Being solutions of linear differential equations, the field profiles scale linearly with the incident waves. No units are shown for the electric or magnetic field components. Still the given values correspond to a normalization to unit incident power per unit length (y-direction), of the incoming modes (W/µm). 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).
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. The boundary positions of the mapped rectangle can be adjusted; defaults are averages between the outermost interfaces and the computational window boundaries. The plot rectangle can extend beyond the inner computational window, e.g. in order to visualize corner effects. Click in the figure for a precise evaluation of field levels. A click outside the actual axis closes the coordinate display. 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 contour is removed by a click in the colorbar area outside the axes.
Example
The following list of parameters reproduces the field solution that corresponds to the resonant state of a square 2-D resonator with perpendicular bus waveguides, as reported in section 3.4 of Ref. [1] (different: position of the coordinate origin; size of the computational window, precise spectral density, precise plot region). Quantities are as introduced in the preceding paragraphs of this page.
n_{4,0} = 1.0, | n_{4,1} = 1.0, | n_{4,2} = 1.0, | n_{4,3} = 3.4, | n_{4,4} = 1.0, |
n_{3,0} = 1.0, | n_{3,1} = 3.4, | n_{3,2} = 1.0, | n_{3,3} = 3.4, | n_{3,4} = 1.0, |
n_{2,0} = 1.0, | n_{2,1} = 1.0, | n_{2,2} = 1.0, | n_{2,3} = 1.0, | n_{2,4} = 1.0, |
n_{1,0} = 3.4, | n_{1,1} = 3.4, | n_{1,2} = 3.4, | n_{1,3} = 3.4, | n_{1,4} = 3.4, |
n_{0,0} = 1.0, | n_{0,1} = 1.0, | n_{0,2} = 1.0, | n_{0,3} = 1.0, | n_{0,4} = 1.0, |
The solver predicts a guided reflectance of 22% (output to the left port), and transmittances of 22% (right port) and 46% (top port).
Reference
[1] M. Hammer
Quadridirectional eigenmode expansion scheme for 2-D modeling of wave propagation in integrated optics
Optics Communications 235 (4-6), 285-303 (2004)