Tansmittance and reflectance of dielectric multilayer configurations

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.

Input

For a multilayer structure with N inner layers, the input mask receives the
vacuum wavelength λ, the angle of incidence θ_{I} (in degrees), a specification of the polarization, refractive index values
n_{I} (region I, domain of the incoming & reflected waves),
n_{1}, ... ,
n_{N} (inner layers 1 to N),
n_{II} (region II, domain of the transmitted wave),
and thicknesses
t_{1}, ... ,
t_{N} 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:

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 (θ_{I} ≠ 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
**k**_{I},
**k**_{R}, and
**k**_{T} 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.

Output

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

- the angle of incidence θ
_{I}, as specified, - the reflectance R,
- for R≠0: the complex amplitude reflection coefficient ρ
_{E}(s/TE polarization) related to the principal electric field component, or ρ_{H}(s/TM polarization) related to the principal magnetic component, - for R≠0: the angle of reflection θ
_{I}, - the transmittance T,
- for T≠0: the complex amplitude transmission coefficient τ
_{E}(s/TE polarization) related to the principal electric field component, or τ_{H}(s/TM polarization) related to the principal magnetic component, - for T≠0: the angle of refraction θ
_{II}.

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
n_{I} sinθ_{I} = n_{II} sinθ_{II}.
For
n_{I}>n_{II}, this requires the angle of incidence (absolute value) to be lower than the critical angle θ_{c} for total internal reflection, given by
sinθ_{c} = n_{II}/n_{I}. 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 S_{x} 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

All fields are constant along the y-direction. The z-wavenumber k_{z} is given by the angle of incidence as
k_{z} = k n_{I} sinθ_{I}, 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, E_{y}, 0),
**H** = (H_{x}, 0, H_{z}). Likewise,
the profiles of s/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 profile functions. Further, the time-averaged z-component S_{z} 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 (S_{x} = 1-R = T, for unit incident intensity).
Single components ψ ∈ {E_{x}, . . . , H_{z}}
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 = E_{y}; for p/TM polarization,
the solver considers the principal magnetic field component ϕ = H_{y}. In regions I and II, respectively, the principal field is of the form

ϕ(x) = u_{I} exp(-i k n_{I} cosθ_{I} x) + r_{I} exp(i k n_{I} cosθ_{I} x), if x<0,
ϕ(x) = u_{II} exp(-i k_{x,II} x), if x>∑_{j} t_{j},

where
k_{x,II} = k n_{II} cosθ_{II}, if T≠0.
The complex valued amplitude reflection- and transmission coefficients relate to this principal component. These are defined as
ρ_{E} = r_{I}/u_{I},
τ_{E} = u_{II}/u_{I},
for s/TE polarization,
and
ρ_{H} = r_{I}/u_{I},
τ_{H} = u_{II}/u_{I},
for p/TM polarization.

Taking into account the angles of propagation and the local refractive index, the intensities
I_{I}, I_{R}, and I_{T} of the incident, reflected, and transmitted waves
can be expressed in terms of the amplitudes u_{I}, r_{I}, and u_{II} of the principal field components as
I_{I} = k n_{I} cosθ_{I}|u_{I}|^{2}/(2ωμ_{0}),
I_{R} = k n_{I} cosθ_{I}|r_{I}|^{2}/(2ωμ_{0}),
I_{T} = k n_{II} cosθ_{II}|u_{II}|^{2}/(2ωμ_{0}),
for s/TE polarization, and as
I_{I} = k n_{I} cosθ_{I}|u_{I}|^{2}/(2ωε_{0}n_{I}^{2}),
I_{R} = k n_{I} cosθ_{I}|r_{I}|^{2}/(2ωε_{0}n_{I}^{2}),
I_{T} = k n_{II} cosθ_{II}|u_{II}|^{2}/(2ωε_{0}n_{II}^{2}),
for p/TM polarization, where, for both polarizations, reflectance and transmittance are defined
as the ratios
R = I_{R}/I_{I} and
T = I_{T}/I_{I}
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 = |ρ_{E}|^{2}, or
R = |ρ_{H}|^{2},
while the amplitude transmission coefficients are related to the transmittances
by
T = (n_{II} cosθ_{II})|τ_{E}|^{2}/(n_{I} cosθ_{I})
for s/TE polarization, and by
T = (n_{I} cosθ_{II})|τ_{H}|^{2}/(n_{II} cosθ_{I})
for p/TM polarized fields. Note that the values of the amplitude reflection coefficients are thus limited to |ρ_{E,H}|^{2} ≤ 1, while the
amplitude transmission coefficients can well reach absolute levels |τ_{E,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 u_{I} of the (polarization dependent) principal component ϕ has been adjusted such that
I_{I} = 1 W/µm^{2}. 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). 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 θ_{I}, the vacuum wavelength λ, the refractive indices
n_{I}, n_{II} of region I & region II, 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), 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
ρ_{E,H} and τ_{E,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).