Learn: the theory behind the calculator

The equations every page of this site rests on: Maxwell's equations in a piecewise homogeneous medium, the time convention, the continuity of tangential E and H at an interface, and how separation of variables turns the vector problem for a film, a cylinder or a sphere into one scalar second-order equation per channel.

The one idea behind this site's engine: a second-order wave equation has two independent solutions, so the two boundary conditions at an interface always give an invertible 2×2 matrix. Multiplying those matrices across the interfaces solves a film, a coated cylinder or a multilayered sphere with any number of layers, for plane waves and dipoles alike, and the same construction shows where the numerics bite and how PyStratify keeps them stable.

Reflection and transmission of a plane wave at a single planar interface, derived from the two boundary conditions: Snell's law, the Fresnel coefficients for s and p polarization and their phases, the power coefficients and why T is not |t|², the Brewster angle, total internal reflection and the evanescent wave, absorbing media and the branch of the complex angle, and the ellipsometric ratio ρ = r_p / r_s.

Abelès' characteristic matrix of a layer, derived from the two boundary conditions; how a stack multiplies them; the admittance that gives R and T and Macleod's admittance locus; the amplitude form with interface and propagation matrices that also gives the field inside; quarter-wave and half-wave layers and Herpin's equivalent layer; and the numerics: growing exponentials, the scattering-matrix alternative, and what PyStratify does.

Why a thin film draws interference fringes and a millimetre-thick substrate does not: the coherence length and the fringe spacing, the incoherent sum of intensities derived as a phase average, the intensity transfer matrix that joins coherent sub-stacks through incoherent layers, the asymmetry R ≠ R′ of an absorbing film, partial coherence, and what a spectrometer actually records.

A quarter-wave stack of two materials reflects almost perfectly over a band of wavelengths: the peak reflectance from the admittance, Bloch's theorem and the dispersion relation of the one-dimensional crystal, the stop band width from the index contrast, the N-period reflectance through Chebyshev polynomials, penetration depth, higher-order bands and the ones that vanish, oblique incidence and the omnidirectional mirror.

A transparent spacer between two partial mirrors transmits a comb of narrow peaks: the Airy function derived from the multiple reflections, the resonance condition with the mirrors' phases, free spectral range, finesse and resolving power, what absorption in the mirrors costs, angle tuning and the Jacquinot advantage, the photon lifetime and Q of the cavity, and the Gires–Tournois phase.

A half-wave spacer between two quarter-wave mirrors passes one narrow band inside a broad stop band: the design (HL)^p 2L (LH)^p and why it transmits, Macleod's half-width formula and what each parameter does, the Lorentzian single-cavity line and the square-topped multi-cavity filter, the angle shift and the converging-beam limit, blocking, absorption and the narrowest filters, and the notch and rugate filters that do the opposite.

A few nanometres of silver divide a beam into reflected and transmitted halves: the Drude metal and the skin depth, why a metal film is partly transparent, the absorption that no design removes and Macleod's potential transmittance that bounds it, the induced-transmission trick of dielectric matching layers, the polarizing and phase-shifting behaviour at 45°, and what dielectric splitters trade for their losslessness.

An ultrathin silver film between two dielectric layers lets daylight through and reflects heat: the Drude metal below its plasma frequency, the induced-transmission design that antireflects the silver in the visible, Kirchhoff's law and the Hagen–Rubens emissivity of a conductor, the sheet-resistance rule of thumb, the glazing figures of merit, and why double- and triple-silver stacks steepen the edge.

A thin gold film on a prism couples p-polarized light to the surface plasmon at the gold/analyte boundary: the plasmon as a pole of the Fresnel coefficient, its dispersion relation and decay lengths, prism coupling in the Kretschmann and Otto geometries, the Lorentzian dip and critical coupling, why only p light couples, the sensitivity and figure of merit of an SPR sensor, and the limits of the model.

A sub-wavelength taper removes the index step instead of cancelling its reflection: the effective-medium picture and where it holds, Maxwell Garnett and Bruggeman mixing rules, the WKB view of a graded profile as the Fourier transform of its index gradient, why height matters more than shape and why smoothness matters at all, Southwell's quintic, and what the moth's eye adds that a graded film cannot.

Upright pyramids on silicon give reflected light a second chance: the ray geometry of a facet, the product of two reflectances and what a coating on the facets adds, the oblique path inside the wafer, the Lambertian limit of 4n² from étendue and the Yablonovitch argument, how pyramids and random textures approach it, where the ray picture stops (feature size against wavelength), and why this needs ray tracing beside the transfer matrix.

A periodic surface couples light into discrete orders: the grating equation from the conservation of in-plane momentum modulo 2π/Λ, propagating and evanescent orders and the cutoff, Rayleigh–Wood anomalies, the effective film a sub-wavelength grating becomes (Rytov's zeroth- and second-order indices and form birefringence), and how rigorous coupled-wave analysis solves the rest, with the Fourier factorization rules that make it converge.

Total absorption is not what a solar cell needs: the absorbed power density from the field, the layer-resolved absorptance from the Poynting flux and its conservation check, the standing wave in a thin stack and how it moves with thickness, the photocurrent from the AM1.5 spectrum, parasitic absorption in contacts and transport layers, and the Lambertian ceiling that textures aim at.

A surface that reflects sunlight and emits in the 8–13 µm atmospheric window can cool below ambient: the power balance of a radiative cooler and the sky's own emission, Kirchhoff's law and emissivity from a film calculation, the phonon bands of silica and silicon nitride as Lorentz oscillators and the reststrahlen reflection they cause, the silver mirror underneath, directional emissivity, and what the ideal selective emitter looks like.

How a molecule, a quantum dot or an OLED emitter radiates when it sits in a stratified medium: the plane-wave expansion of a dipole field, the Chance–Prock–Silbey integrals for its decay rate, the channels its power goes to (escaping light, substrate modes, guided modes, absorption), quenching near a metal and the microcavity effect that OLED design is built on.

The plane-wave solution for an infinitely long circular cylinder: cylindrical waves, the two decoupled polarizations at normal incidence and their coefficients, efficiencies per unit length, the thin-wire limit with its ε = −1 resonance, coated cylinders by the same recursion, oblique incidence, and the single-channel bound that superscattering beats.

A point dipole next to a nanowire or an optical fibre: the continuous spectrum of axial wavenumbers, the guided modes as poles, and how the radiated, guided and absorbed powers are separated. The thin metal wire's plasmon at the local limit. And a fibre under an infrared microscope's Cassegrain: why a fibre that absorbs nothing shows a spectrum.

How a plane wave scatters off a homogeneous sphere: the plane wave as a sum of multipoles, the coefficients a_ℓ and b_ℓ from two boundary conditions per order, the efficiencies and the amplitude functions of the far field, the Rayleigh and geometric limits and the extinction paradox, van de Hulst's anomalous diffraction, where the series stops, and the numerics of the Riccati–Bessel functions that every Mie code lives or dies by.

Reading a spectrum by its electric and magnetic multipoles: each order's share of the scattering and the single-channel bound, the small-sphere expansions of a_1 and b_1, how a magnetic dipole arises in a dielectric and where it sits, the forward and backward amplitudes, the first and second Kerker conditions and their generalisation to quadrupoles, Huygens metasurfaces, and the anapole as a zero of a_1.

Mie's solution extended to any number of concentric shells: the fields of each shell as regular and outgoing multipoles, the 2×2 transfer matrix of one interface written in Riccati–Bessel functions, raising and lowering through the shells, the two conditions that close the system, and the numerical discipline (logarithmic amplitudes, ratios of Bessel functions) that makes thousands of orders and nanometre shells safe. Plasmon hybridization in a nanoshell as the first application.

A molecule or quantum dot beside, or inside, a sphere: how its field is expanded in multipoles, how the sphere's transfer matrices return it, the radiative and non-radiative decay rates that result, the quantum yield, and the competition between enhancement and quenching that fixes the best distance for enhanced fluorescence.