Characteristic matrix method (transfer matrix for thin films)

The characteristic matrix method, introduced by Abelès in 1950 and also called the Abelès matrix or transfer matrix method, computes the reflectance and transmittance of a stack of thin films. Each layer is represented by a 2×2 matrix built from its phase thickness and its optical admittance; the matrices are multiplied in order from the incident medium to the substrate, and the product gives the reflection coefficient of the whole stack for either polarization at any angle of incidence. A quarter-wave layer of MgF₂ (index 1.38) on glass (index 1.52), for example, lowers the normal-incidence reflectance at its design wavelength from 4.26% to 1.26%, and ten quarter-wave pairs of Ta₂O₅ and SiO₂ on fused silica reflect 99.83% at 1550 nm. The method is exact for homogeneous, isotropic, plane-parallel layers illuminated by a plane wave, and it is the standard calculation of thin-film coating design.

A single interface obeys the Fresnel equations; a coated surface has several interfaces whose reflections interfere, and adding them up interface by interface becomes unmanageable beyond two layers. The matrix method replaces that bookkeeping with one matrix per layer, whatever the number of layers.

Characteristic matrix of a thin film

The calculation is made at a wavelength $\lambda$ and an angle of incidence ${\theta }_0$ in a lossless incident medium of index $n_0$. In each medium of index $n$, Snell's law fixes $n\cos \theta$, and the optical admittance that the wave sees, in units of the admittance of free space, depends on polarization:

$$(n\cos \theta {)}^2=n^2-n_0^2{\sin }^2{\theta }_0$$ $${\eta }_s=n\cos \theta ,{\eta }_p=\frac{n}{\cos \theta }$$

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

$$M=\left(\begin{array}{cc}\cos \delta & i\sin \delta \mathrm{/}\eta \\ i\eta \sin \delta & \cos \delta \end{array}\right)$$

The matrix connects the tangential components of the electric and magnetic fields at the front surface of the layer to those at its back surface. Since the tangential components are continuous across every interface, the matrix of a whole stack is the product of the matrices of its layers. The determinant of each matrix is 1.

Transfer matrix for a multilayer stack

The matrices are multiplied in order from the incident side to the substrate, and the product is applied to the substrate admittance ${\eta }_{\text{sub}}$:

$$\left(\begin{array}{c}B\\ C\end{array}\right)=M_1{M}_2\cdots M_N\left(\begin{array}{c}1\\ {\eta }_{\text{sub}}\end{array}\right)$$

The ratio $Y=C\mathrm{/}B$ is the admittance of the layers and substrate together, as seen from the incident medium, so a stack of any complexity reflects like a single interface between the incident medium and a medium of admittance $Y$.

The name transfer matrix is also used for a different bookkeeping of the same physics, in which 2×2 matrices act on the amplitudes of the forward and backward waves in each layer, one matrix for each interface and one for the propagation across each layer (Yeh); the two formulations give identical reflectance and transmittance. The ray-transfer ABCD matrix of paraxial optics is a separate formalism, which acts on ray heights and angles.

Reflectance from the characteristic matrix

The reflection coefficient, the reflectance and the transmittance of the stack follow from $B$ and $C$:

$$r=\frac{{\eta }_{0}B-C}{{\eta }_{0}B+C},R=\mathrm{\mid }r{\mathrm{\mid }}^2$$ $$T=\frac{4{\eta }_0\mathrm{R}\mathrm{e}({\eta }_{\text{sub}})}{\mathrm{\mid }{\eta }_{0}B+C{\mathrm{\mid }}^2}$$

Here ${\eta }_0$ is the admittance of the incident medium for the same polarization, $n_0\cos {\theta }_0$ or $n_0\mathrm{/}\cos {\theta }_0$. With lossless layers $R+T=1$, and the phase of $r$ is the phase change on reflection, which matters for mirrors inside resonators. At normal incidence the s and p results coincide. For p polarization, $r$ is the ratio of the tangential field components and has the opposite sign to the $r_p$ written in the Fresnel equations entry; $R$ is the same.

As an example, a quarter-wave layer of MgF₂, 99.6 nm thick, on glass at 550 nm and normal incidence has $\delta =\pi \mathrm{/}2$, and

$$\left(\begin{array}{c}B\\ C\end{array}\right)=\left(\begin{array}{cc}0& i\mathrm{/}1.38\\ 1.38i& 0\end{array}\right)\left(\begin{array}{c}1\\ 1.52\end{array}\right)=\left(\begin{array}{c}1.1014i\\ 1.38i\end{array}\right),$$

so $Y=1.2529$, $r=(1-1.2529)\mathrm{/}(1+1.2529)=-0.1123$, $R$ = 1.26% and $T$ = 98.74%. Bare glass of index 1.52 reflects 4.26%.

Quarter-wave and half-wave layers

Two special cases carry most of coating design. A quarter-wave layer ($\delta =\pi \mathrm{/}2$) has $M=\left(\begin{array}{cc}0& i\mathrm{/}\eta \\ i\eta & 0\end{array}\right)$, which transforms the admittance $Y$ beneath it into ${\eta }^2\mathrm{/}Y$. A quarter-wave of index $n_1$ on a substrate $n_s$ therefore presents $n_1^2\mathrm{/}{n}_s$ to the incident medium, and the reflectance at normal incidence is

$$R={\left(\frac{n_0{n}_s-n_1^2}{n_0{n}_s+n_1^2}\right)}^2,$$

which vanishes when $n_1=\sqrt{n_0{n}_s}$, the rule behind every single-layer anti-reflection coating. On glass of index 1.52 in air the rule asks for an index of 1.233; MgF₂, the standard single layer, leaves 1.26%. Two quarter-waves, of index $n_1$ on the incident side and $n_2$ beneath it, present $Y=(n_1\mathrm{/}{n}_2{)}^2{n}_s$, which equals $n_0$ when $n_2\mathrm{/}{n}_1=\sqrt{n_s\mathrm{/}{n}_0}$: MgF₂ over a quarter-wave of index 1.70 on the same glass gives $Y=1.0016$ and $R=6.6\times {10}^{-7}$ at the design wavelength, the two-layer V-coat.

A half-wave layer ($\delta =\pi$) has $M=-I$ and leaves the admittance unchanged. At its design wavelength it is an absentee layer, invisible to the reflectance, and it alters the spectrum only away from that wavelength.

Quarter-wave stacks and Bragg mirrors

Alternating quarter-waves of high and low index multiply the admittance by $(n_H\mathrm{/}{n}_L{)}^2$ per pair. For $N$ pairs with a high-index layer facing the incident medium and a low-index layer on the substrate, $Y=(n_H\mathrm{/}{n}_L{)}^{2N}{n}_s$ and

$$R={\left(\frac{n_0-Y}{n_0+Y}\right)}^2,$$

the peak reflectance of a Bragg mirror. Ten pairs of Ta₂O₅ ($n_H=2.10$) and SiO₂ ($n_L=1.45$) on fused silica ($n_s=1.444$), 184.5 nm and 267.2 nm thick for 1550 nm, give $Y=2380$ and $R$ = 99.83% in air. An extra high-index layer between the stack and the substrate multiplies $Y$ by $(n_H\mathrm{/}{n}_s{)}^2$, to 5034, and raises $R$ to 99.92%. Once $R$ is high, each added pair divides the transmission by $(n_H\mathrm{/}{n}_L{)}^2$, 2.10 for this pair of materials. The reflectance stays high over a stop band, 1387 to 1756 nm for this stack, whose width the Bragg mirror entry derives.

Absorbing layers and sign conventions

Absorbing layers and metals enter through a complex refractive index, with no other change to the method; $R+T$ then falls below one by the absorbed fraction. The sign of the imaginary part has to match the form of the matrix. The form above, Macleod's, belongs to fields varying as $e^{i\omega t}$ and takes the index as $n-ik$. With the $n+i\kappa$ of the complex refractive index entry, which belongs to $e^{-i\omega t}$, the off-diagonal elements take $-i$ in place of $+i$, as in Born and Wolf. Mixing the two conventions turns the decay in an absorbing layer into growth: a 2 µm layer of index $2.0+0.5i$ on glass at 550 nm, entered in the form above, returns a reflectance of 7.4 instead of the 13.5% of the bulk material.

Spectra away from the design wavelength

Off the design wavelength the phase thicknesses change in proportion to $1\mathrm{/}\lambda$, and at oblique incidence they shrink with the cosine of the angle in each layer, which moves the spectrum toward shorter wavelengths; the products are then evaluated numerically. The single MgF₂ layer above, with its indices taken as constant, reflects 2.21% at 400 nm and 1.94% at 800 nm. The Thin-Film Reflectance Calculator implements the method 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.

Limits of the method

The layers are taken as homogeneous and isotropic, bounded by flat, parallel, smooth interfaces, and the light as a plane wave. Interference inside the layers is treated as fully coherent, which is correct for films but not for a substrate millimeters thick under a broadband source: the reflection from the back of a real substrate, 4.26% for uncoated glass of index 1.52, is added incoherently. A graded-index layer is approximated by many thin homogeneous sublayers, and anisotropic layers, such as birefringent films and liquid crystals, need 4×4 matrices (Berreman).

Common questions

What is the characteristic matrix method?

A way of computing the reflectance and transmittance of a stack of thin films. Each layer is a 2×2 matrix built from its phase thickness and optical admittance, the matrices are multiplied from the incident side to the substrate, and the product gives the reflection coefficient of the stack. It is exact for flat, homogeneous, isotropic layers under plane-wave illumination.

Is the characteristic matrix the same as the transfer matrix?

In thin-film optics the two names are used for the same calculation. Strictly, the characteristic matrix acts on the tangential electric and magnetic fields at the layer boundaries, while some texts build a transfer matrix that acts on the amplitudes of the forward and backward waves; both give the same reflectance and transmittance. The ABCD ray-transfer matrix of paraxial optics is unrelated.

How is the reflectance of a multilayer coating calculated?

The characteristic matrices of the layers are multiplied in order and applied to the vector $(1,{\eta }_{\text{sub}})$ to give $B$ and $C$, and the reflectance is $R=\mathrm{\mid }({\eta }_{0}B-C)\mathrm{/}({\eta }_{0}B+C){\mathrm{\mid }}^2$. For a quarter-wave MgF₂ layer on glass of index 1.52 at normal incidence this gives $Y=C\mathrm{/}B=1.253$ and $R$ = 1.26%.

Does the method work for absorbing layers and metals?

Yes. Absorbing layers enter with a complex refractive index, and $R+T$ then falls short of one by the absorbed fraction. The sign of the imaginary part must match the form of the matrix: $n-ik$ with the $+i$ form used here (Macleod), $n+i\kappa$ with the $-i$ form (Born and Wolf).

Why does a quarter-wave layer reduce reflection?

A quarter-wave layer transforms the admittance beneath it into ${\eta }^2\mathrm{/}Y$, so a layer of index $n_1$ on a substrate of index $n_s$ presents $n_1^2\mathrm{/}{n}_s$ to the incident medium. The reflectance vanishes when $n_1=\sqrt{n_0{n}_s}$, an index of 1.233 for glass of index 1.52 in air; MgF₂, at 1.38, lowers the reflectance from 4.26% to 1.26%.

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, 12th series, 5, 596 and 706 (1950). P. Yeh, Optical Waves in Layered Media (Wiley, 1988). D. W. Berreman, "Optics in stratified and anisotropic media: 4×4-matrix formulation," Journal of the Optical Society of America 62, 502 (1972).