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. 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 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.

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 ${\displaystyle z\,}$ axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with wave number ${\displaystyle k\,}$,

Because it follows from Maxwell's equation that electric field ${\displaystyle E\,}$ and magnetic field (its normalized derivative) ${\textstyle H={\frac {1}{ik}}Z_{c}{\frac {dE}{dz}}\,}$ must be continuous across a boundary, it is convenient to represent the field as the vector ${\textstyle (E(z),H(z))\,}$, where

Since there are two equations relating ${\displaystyle E\,}$ and ${\displaystyle H\,}$ to ${\displaystyle E_{r}\,}$ and ${\displaystyle E_{l}\,}$, these two representations are equivalent. In the new representation, propagation over a distance ${\displaystyle L\,}$ into the positive direction of ${\displaystyle z\,}$ 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 ${\displaystyle k\,}$ is the wave number in the medium and ${\displaystyle L\,}$ the thickness of the layer: For a system with ${\displaystyle N\,}$ layers, each layer ${\displaystyle j\,}$ has a transfer matrix ${\displaystyle M_{j}\,}$, where ${\displaystyle j\,}$ increases towards higher ${\displaystyle z\,}$ values. The system transfer matrix is then