Transfer-matrix method (optics)

The transfer-matrix method is a method used in optics and acoustics to analyze the propagation of electromagnetic or acoustic waves through a stratified (layered) medium. [Born, M.; Wolf, E., "Principles of optics: electromagnetic theory of propagation, interference and diffraction of light". Oxford, Pergamon Press, 1964.] This is for example relevant for the design of anti-reflective coatings and dielectric mirrors.

The reflection of light from a single interface between two media is described by the Fresnel equations. However, when there are multiple interfaces, such as in the figure, the reflections themselves are also partially reflected. Depending on the exact path length, these reflections can interfere destructively or constructively. The overall reflection of a layer structure is the sum of an infinite number of reflections, which is cumbersome to calculate.

The transfer-matrix method is based on the fact that, according to Maxwell's equations, there are simple continuity conditions for the electric field across boundaries from one medium to the next. If the field is known at the beginning of a layer, the field at the end of the layer can be derived from a simple matrix operation. A stack of layers can then be represented as a system matrix, which is the product of the individual layer matrices. The final step of the method involves converting the system matrix back into reflection and transmission coefficients.

Formalism for electromagnetic waves

Below is described how the transfer matrix is applied to electromagnetic waves (for example light) of a given frequency propagating through a stack of layers at normal incidence. It can be generalized to deal with incidence at an angle, absorbing media, and media with magnetic properties. We assume that the stack layers are normal to the $z$ axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with wave number $k$,:$E(z)\; =\; E\_r\; e^\{ikz\}\; +\; E\_l\; e^\{-ikz\}.$Because it follows from Maxwell's equation that $E$ and $F=dE/dz$ must be continuous across a boundary, it is convenient to represent the field as the vector $(E(z),F(z))$, where:$F(z)\; =\; ik\; E\_r\; e^\{ikz\}\; -\; ik\; E\_l\; e^\{-ikz\}.$Since there are two equations relating $E$ and $F$ to $E\_r$ and $E\_l$, these two representations are equivalent. In the new representation, propagation over a distance $L$ into the positive $z$ direction is described by the matrix:and:Such a matrix can represent propagation through a layer if $k$ is the wave number in the medium and $L$ the thickness of the layer:For a system with $N$ layers, each layer $j$ has a transfer matrix $M\_j$, where $j$ increases towards higher $z$ values. The system transfer matrix is then:$M\_s\; =\; M\_N\; cdot\; ldots\; cdot\; M\_2\; cdot\; M\_1.$

Typically, one would like to know the reflectance and transmittance of the layer structure. If the layer stack starts at $z=0$, then for negative $z$, the field is described as:$E\_L(z)\; =\; E\_0\; e^\{ik\_Lz\}\; +\; r\; E\_0\; e^\{-ik\_Lz\},$where $E\_0$ is the amplitude of the incoming wave, $k\_L$ the wave number in the left medium, and $r$ is the amplitude (not intensity!) reflectance coefficient of the layer structure. On the other side of the layer structure, the field consists of a right-propagating transmitted field:$E\_R(z)\; =\; t\; E\_0\; e^\{ik\_R\; z\},$where $t$ is the amplitude transmittance and $k\_R$ is the wave number in the rightmost medium. If $F\_L\; =\; dE\_L/dz$ and $F\_R\; =\; dE\_R/dz$, then we can solve:in terms of the matrix elements $M\_\{mn\}$ of the system matrix $M\_s$ and obtain:$r\; =\; frac\{-i\; k\_R\; M\_\{11\}\; +\; k\_L\; k\_R\; M\_\{12\}\; +\; M\_\{21\}\; +\; ik\_L\; M\_\{22\{-M\_\{21\}\; +\; ik\_L\; M\_\{22\}\; +\; i\; k\_R\; M\_\{11\}\; +\; k\_L\; k\_R\; M\_\{12,$and:$t\; =\; frac\{(-M\_\{21\}\; +\; i\; k\_L\; M\_\{22\})(M\_\{11\}\; +\; i\; k\_L\; M\_\{12\})\; +\; (M\_\{11\}\; -\; i\; k\_L\; M\_\{12\})(M\_\{21\}\; +\; i\; k\_L\; M\_\{22\})\}\{-M\_\{21\}\; +\; ik\_L\; M\_\{22\}\; +\; i\; k\_R\; M\_\{11\}\; +\; k\_L\; k\_R\; M\_\{12.$The intensity transmittance and reflectance, which are often of more practical use, are $T=|t|^2$ and $R=|r|^2$, respectively.

Example

As an illustration, consider a single layer of glass with a refractive index "n" and thickness "d" suspended in air at a wave number "k" (in air). In glass, the wave number is $k\text{'}=nk$. The transfer matrix is:The amplitude reflection coefficient can be simplified to :$r\; =\; frac\{(1/n\; -\; n)\; sin\; k\text{'}d\}\{(n+1/n)sin\; k\text{'}d\; +\; 2\; i\; cos(k\text{'}d)\}.$This configuration effectively describes a Fabry-Pérot interferometer or etalon: for $k\text{'}d=0,\; pi,\; 2pi,\; cdots$, the reflection vanishes.

Acoustic waves

It is possible to apply the transfer-matrix method to sound waves. Instead of the electric field "E" and its derivative "F", the displacement "u" and the stress $sigma=C\; du/dz$, where $C$ is Young's modulus should be used.

References