Characteristic matrix method (transfer matrix for thin films)

A single interface obeys the Fresnel equations; a coated surface has several interfaces whose reflections interfere, and the arithmetic of adding them up interface by interface becomes unmanageable beyond two layers. The characteristic matrix method, due to Abelès, replaces that bookkeeping with matrix multiplication. It is exact for homogeneous, isotropic, plane-parallel layers illuminated by a plane wave, and it is what every coating-design program computes.

The recipe

Work at wavelength $\lambda$ and angle of incidence $\theta_0$ in a lossless incident medium of index $n_0$. For each medium of index $n$ (complex if absorbing), Snell's law fixes $n\cos\theta$, and the optical admittance that the wave sees depends on polarization:

$$(n\cos\theta)^2 = n^2 - n_0^2\sin^2\theta_0$$ $$\eta_s = n\cos\theta, \qquad \eta_p = \frac{n}{\cos\theta}$$

A layer of physical thickness $d$ has phase thickness $\delta = 2\pi n d \cos\theta/\lambda$ and characteristic matrix

$$M = \begin{pmatrix} \cos\delta & i\sin\delta/\eta \\ i\eta\sin\delta & \cos\delta \end{pmatrix}$$

Multiply the matrices in order from the incident side to the substrate and apply the product to the substrate admittance $\eta_{sub}$:

$$\begin{pmatrix} B \\ C \end{pmatrix} = M_1 M_2 \cdots M_N \begin{pmatrix} 1 \\ \eta_{sub} \end{pmatrix}$$ $$r = \frac{\eta_0 B - C}{\eta_0 B + C}, \qquad R = |r|^2$$ $$T = \frac{4\eta_0\,\mathrm{Re}(\eta_{sub})}{|\eta_0 B + C|^2}$$

The ratio $Y = C/B$ is the admittance of the whole stack, and $r$ is the Fresnel coefficient of an equivalent single interface between $\eta_0$ and $Y$. Absorbing layers enter with complex $n$ and lose nothing in the formalism; $R + T$ then falls below one by the absorbed fraction.

What the method makes obvious

Two special cases carry most of coating design. A quarter-wave layer ($\delta = \pi/2$) has $M = \begin{pmatrix} 0 & i/\eta \\ i\eta & 0 \end{pmatrix}$, which transforms the admittance beneath it into $\eta^2/Y$: a quarter-wave of index $n_1$ on a substrate $n_s$ presents $n_1^2/n_s$ to the incident medium, so the reflectance at normal incidence is $\left[(n_0 n_s - n_1^2)/(n_0 n_s + n_1^2)\right]^2$ and vanishes when $n_1 = \sqrt{n_0 n_s}$, the rule behind every single-layer anti-reflection coating. A half-wave layer ($\delta = \pi$) has $M = -I$ and is invisible at its design wavelength, the absentee layer. Stacking quarter-waves of alternating high and low index multiplies the admittance by $(n_H/n_L)^2$ per pair, which is why a Bragg mirror of $N$ pairs reaches $R = \left[(1 - Y)/(1 + Y)\right]^2$ with $Y = (n_H/n_L)^{2N} n_s$, and why ten pairs of Ta₂O₅ and SiO₂ give 99.8 % at 1550 nm.

Off the design wavelength the phase thicknesses drift and the algebra has to be done numerically, which is the tool's job. The Thin-Film Reflectance Calculator implements the recipe above for any stack, either polarization, and any angle, plots the spectrum and the angular response, and checks itself on every load against the closed forms in this entry, the Fresnel equations, the Airy formula for an etalon, the Brewster and critical angles, energy conservation, and reciprocity. The design reasoning for the single-layer case is worked in Anti-reflection coating design.

References: H. A. Macleod, Thin-Film Optical Filters, 5th ed. (CRC Press, 2018), chapter 2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), section 1.6. F. Abelès, "Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés," Annales de Physique 12, 596 and 706 (1950).