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 medium. 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 , such as in the figure, the reflections themselves are also partially transmitted and then 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.
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 formula_1 axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with wave number formula_2,
Because it follows from Maxwell's equation that formula_4 and formula_5 must be continuous across a boundary, it is convenient to represent the field as the vector formula_6, where
Since there are two equations relating formula_4 and formula_9 to formula_10 and formula_11, these two representations are equivalent. In the new representation, propagation over a distance formula_12 into the positive formula_1 direction is described by the unimodular matrix
and
Such a matrix can represent propagation through a layer if formula_2 is the wave number in the medium and formula_12 the thickness of the layer:
For a system with formula_18 layers, each layer formula_19 has a transfer matrix formula_20, where formula_19 increases towards higher formula_1 values. The system transfer matrix is then
Typically, one would like to know the reflectance and transmittance of the layer structure. If the layer stack starts at formula_24, then for negative formula_1, the field is described as
where formula_27 is the amplitude of the incoming wave, formula_28 the wave number in the left medium, and formula_29 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
where formula_31 is the amplitude transmittance, formula_32 is the wave number in the rightmost medium, and formula_33 is the total thickness. If formula_34 and formula_35, then we can solve
in terms of the matrix elements formula_37 of the system matrix formula_38 and obtain
and
The transmittance and reflectance (i.e., the fractions of the incident intensity formula_41 transmitted and reflected by the layer) are often of more practical use and are given by formula_42 and formula_43, respectively (at normal incidence).
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 formula_44. The transfer matrix is
The amplitude reflection coefficient can be simplified to 
This configuration effectively describes a Fabry–Pérot interferometer or etalon: for formula_47, 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 formula_48, where formula_49 is the p-wave modulus, should be used.
Abeles matrix formalism.
The Abeles matrix method is a computationally fast and easy way to calculate the specular reflectivity from a stratified interface, as a function of the perpendicular momentum transfer, "Qz":
where "θ" is the angle of incidence/reflection of the incident radiation and "λ" is the wavelength of the radiation.
The measured reflectivity depends on the variation in the scattering length density (SLD)
profile, "ρ"("z"), perpendicular to the interface. Although the scattering length density profile
is normally a continuously varying function, the interfacial structure can often be well approximated 
by a slab model in which layers of thickness ("dn"), scattering length density ("ρn") and roughness (σn,n+1) are sandwiched between the super- and sub-phases. One then uses a refinement procedure to minimise the differences between the theoretical and measured reflectivity curves, by changing the parameters that describe each layer.
In this description the interface is split into "n" layers. Since the incident neutron beam
is refracted by each of the layers the wavevector, "k", in layer "n", is given by:
Since the interface between each layer is unlikely to be perfectly smooth the roughness/diffuseness of each interface modifies the Fresnel coefficient and is accounted for by an error function, as described by Nevot and Croce (1980).
A phase factor, "β", is introduced, which accounts for the thickness of each layer.
where formula_55.
A characteristic matrix, cn is then calculated for each layer.
The resultant matrix is defined as the product of these characteristic matrices
from which the reflectivity is calculated as:
External links.
There are a number of computer programs that implement this calculation: