A solver for problems of plane wave reflection from dielectric multilayer systems at oblique incidence. Given the stack definition in terms of refractive indices, layer thicknesses, vacuum wavelength, and the angle of incidence, the script calculates reflectance and transmittance properties for s- and p-polarized waves, and allows to inspect the corresponding optical fields. Facilities for evaluating parameter scans / spectra are available. The script can serve as a basic tool for the design of dielectric multilayer coatings / reflectors / filters, in particular for purposes of demonstration.


For a multilayer structure with N inner layers, the input mask receives the vacuum wavelength λ, the angle of incidence θI (in degrees), a specification of the polarization, refractive index values nI (region I, domain of the incoming & reflected waves), n1, ... , nN (inner layers 1 to N), nII (region II, domain of the transmitted wave), and thicknesses t1, ... , tN of the inner layers. All dimensions are meant in micrometers. The inner layers can be absent; the setting N=0 refers to a single interface plane between regions I and II. The figure illustrates the relevant geometry:

Planar multilayer geometry

The x-direction of the Cartesian coordinate system is perpendicular to the film plane, with constant refractive index profile along the y- and z-directions. For oblique incidence (θI ≠ 0), the x- and z-axes span the plane of incidence. All fields are assumed to be constant along the y-direction. One then distinguishes incoming waves that are polarized perpendicular to the plane of incidence (s/TE polarization), and waves that are polarized parallel to that plane (p/TM polarization). Wave vectors kI, kR, and kT are associated with the incident, the reflected, and the transmitted wave, respectively.

Frequently, periodic or partly periodic layer configurations are relevant, where in certain cases the default-fill-functionality of the script can ease the data entry. Select the appropriate number of inner layers, and Clear the input mask. Enter refractive indices and thicknesses for one period into the fields of the lowermost inner layers. Then Fill will repeat that sequence of values upwards towards the uppermost layer.


A table lists the properties of the incident, reflected, and transmitted waves:

The law of refraction applies; the angle of reflection equals the angle of incidence. In case of nonzero transmittance, Snell's law relates the angles of incidence and of refraction as nI sinθI = nII sinθII. For nI>nII, this requires the angle of incidence (absolute value) to be lower than the critical angle θc for total internal reflection, given by sinθc = nII/nI. These statements hold irrespectively of the properties of the intermediate layer stack.

Reflectance and transmittance correspond to ratios of the intensities associated with the reflected and transmitted waves, respectively, relative to the unit intensity of the incident wave. Intensity levels are evaluated with respect to the stack surface, the y-z-plane, i.e. the levels correspond to the (absolute value of) the x-component Sx of the Poynting vector associated with the individual plane waves. Note that one also finds the terms "reflection coefficient" / "transmission coefficient" used for the real intensity ratios R and T.

Referring to the coordinate system as introduced above, this concerns optical electric fields E and magnetic fields H that depend on the spatial coordinates x, z (these span the plane of incidence), and on time t, with angular frequency ω, as

E(x, z, t) = Re{E(x) exp(i ω t-i kz z)},   H(x, z, t) = Re{H(x) exp(i ω t-i kz z)}.

All fields are constant along the y-direction. The z-wavenumber kz is given by the angle of incidence as kz = k nI sinθI, where k is the vacuum wavenumber k = 2π/λ.

The electric and magnetic field profiles E, H depend on the normal coordinate x only. For s/TE polarization, these vector fields are of the form E = (0, Ey, 0), H = (Hx, 0, Hz). Likewise, the profiles of p/TM-polarized fields are of the form E = (Ex, 0, Ez), H = (0, Hy, 0). The plot functionality refers to the complex components of these profile functions. Further, the time-averaged z-component Sz of the Poynting vector and the time averaged energy density w can be inspected. Note that the y-component of the Poynting vector vanishes for present the y-constant fields, and that the x-component is constant (Sx = 1-R = T, for unit incident intensity). Single components ψ ∈ {Ex, . . . , Hz} of the complex-valued frequency domain electromagnetic profile relate to time-varying physical fields Ψ(x, t) = Re ψ(x) exp(i ω t) at z = 0. The animations show the respective component at recurring points in time, equally distributed over the period of 2π/ω.

The solver relies on a transfer-matrix method for the underlying 1-D scalar Helmholtz-problem for a principal field component ϕ; all other electromagnetic components are derived from this principal field. For s/TE polarization, this is the electric component ϕ = Ey; for p/TM polarization, the solver considers the principal magnetic field component ϕ = Hy. In regions I and II, respectively, the principal field is of the form

ϕ(x) = uI exp(-i k nI cosθI x) + rI exp(i k nI cosθI x), if x<0,       ϕ(x) = uII exp(-i kx,II x), if x>∑j tj,

where kx,II = k nII cosθII, if T≠0. The complex valued amplitude reflection- and transmission coefficients relate to this principal component. These are defined as ρE = rI/uI, τE = uII/uI, for s/TE polarization, and ρH = rI/uI, τH = uII/uI, for p/TM polarization.

Taking into account the angles of propagation and the local refractive index, the intensities II, IR, and IT of the incident, reflected, and transmitted waves can be expressed in terms of the amplitudes uI, rI, and uII of the principal field components as II = k nI cosθI|uI|2/(2ωμ0), IR = k nI cosθI|rI|2/(2ωμ0), IT = k nII cosθII|uII|2/(2ωμ0), for s/TE polarization, and as II = k nI cosθI|uI|2/(2ωε0nI2), IR = k nI cosθI|rI|2/(2ωε0nI2), IT = k nII cosθII|uII|2/(2ωε0nII2), for p/TM polarization, where, for both polarizations, reflectance and transmittance are defined as the ratios R = IR/II and T = IT/II of reflected or transmitted intensity, respectively, and incident intensity. Consequently, for both polarizations, the reflectances are the absolute square of the amplitude reflection coefficients R = |ρE|2, or R = |ρH|2, while the amplitude transmission coefficients are related to the transmittances by T = (nII cosθII)|τE|2/(nI cosθI) for s/TE polarization, and by T = (nI cosθII)|τH|2/(nII cosθI) for p/TM polarized fields. Note that the values of the amplitude reflection coefficients are thus limited to |ρE,H|2 ≤ 1, while the amplitude transmission coefficients can well reach absolute levels |τE,H|2 above one.

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: The coefficient uI of the (polarization dependent) principal component ϕ has been adjusted such that II = 1 W/µm2. 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/µm2 = W/µm2, and the electromagnetic energy density w is measured in W·fs/µm3. In this context the vacuum permittivity and permeability, respectively, are ε0 = 8.85·10-3 A·fs/(V·µm) and µ0 = 1.25·103 V·fs/(A·µm). Note that this normalization leads to comparatively large numbers for field strengths in case of near-grazing incidence.

Parameter scans

The Scan facilities concern variations of all quantities that define the multilayer configuration, for a given number of inner layers and fixed polarization. The list of options covers the angle of incidence θI, the vacuum wavelength λ, the refractive indices nI, nII of region I & region II, and the refractive indices nj and thicknesses tj of the interior layers. Choose the parameter interval of interest (tests for physical plausibility apply), specify a number of samples, and select one of the options for plotting levels of reflectance and/or transmittance and/or amplitude reflection and transmission coefficients. Depending on polarization, the definitions of ρE,H and τE,H apply, as given above. 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 respective modified configuration (this closes all scan-related controls).