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; a stack of thin films.
[1][2] 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.
Depending on the exact path length, these reflections can interfere destructively or constructively.
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.
The final step of the method involves converting the system matrix back into reflection and transmission coefficients.
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.
We assume that the stack layers are normal to the
axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with wave number
, Because it follows from Maxwell's equation that electric field
and magnetic field (its normalized derivative)
must be continuous across a boundary, it is convenient to represent the field as the vector
is described by the matrix belonging to the special linear group SL(2, C) and Such a matrix can represent propagation through a layer if
The system transfer matrix is then Typically, one would like to know the reflectance and transmittance of the layer structure.
the wave number in the left medium, and
is the wave number in the rightmost medium, and
, then one can solve in terms of the matrix elements
and obtain and The transmittance and reflectance (i.e., the fractions of the incident intensity
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).
The transfer matrix is The amplitude reflection coefficient can be simplified to This configuration effectively describes a Fabry–Pérot interferometer or etalon: for
It is possible to apply the transfer-matrix method to sound waves.
Instead of the electric field E and its derivative H, the displacement u and the stress
The Abeles matrix method[3][4][5] 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.
Since the incident neutron beam is refracted by each of the layers the wavevector k, in layer n, is given by: The Fresnel reflection coefficient between layer n and n+1 is then given by: Because 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,[6] A phase factor, β, is introduced, which accounts for the thickness of each layer.
A characteristic matrix, cn is then calculated for each layer.
The resultant matrix is defined as the ordered product of these characteristic matrices from which the reflectivity is calculated as: There are a number of computer programs that implement this calculation:
