Transfer-matrix method (optics)

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 electric field E\, and magnetic field (its normalized derivative) 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 (E(z),H(z))\,, where :H(z) = \frac{1}{Z_c} E_r e^{ikz} - \frac{1}{Z_c} E_l e^{-ikz}\,. Since there are two equations relating E\, and H\, to E_r\, and E_l\,, these two representations are equivalent. In the new representation, propagation over a distance L\, into the positive direction of z\, is described by the matrix belonging to the special linear group :M = \left( \begin{array}{cc} \cos kL & i Z_c \sin kL \\ \frac{i}{Z_c} \sin kL & \cos kL \end{array} \right), and :\left(\begin{array}{c} E(z+L) \\ H(z+L) \end{array} \right) = M\cdot \left(\begin{array}{c} E(z) \\ H(z) \end{array} \right) 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},\qquad z 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},\qquad z>L', where t\, is the amplitude transmittance, k_R\, is the wave number in the rightmost medium, and L' is the total thickness. If H_L = \frac{1}{ik} Z_c \frac{dE_L}{dz}\, and H_R = \frac{1}{ik} Z_c \frac{dE_R}{dz}\,, then one can solve :\left(\begin{array}{c} E(z_R) \\ H(z_R) \end{array} \right) = M\cdot \left(\begin{array}{c} E(0) \\ H(0) \end{array} \right) in terms of the matrix elements M_{mn}\, of the system matrix M_s\, and obtain :t = 2 i k_L e^{-i k_R L}\left[\frac{1}{-M_{21} + k_L k_R M_{12} + i(k_R M_{11} + k_L M_{22})}\right] and :r = \left[\frac{ (M_{21} + k_L k_R M_{12}) + i(k_L M_{22} - k_R M_{11})}{(-M_{21} + k_L k_R M_{12}) + i(k_L M_{22} + k_R M_{11})}\right]. The transmittance and reflectance (i.e., the fractions of the incident intensity \left|E_0\right|^2 transmitted and reflected by the layer) are often of more practical use and are given by T=\frac{k_R}{k_L}|t|^2\, and R=|r|^2\,, 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 k'=nk\,. The transfer matrix is :M=\left(\begin{array}{cc}\cos k'd & \sin(k'd)/k' \\ -k' \sin k'd & \cos k'd \end{array}\right). The amplitude reflection coefficient can be simplified to :r = \frac{(1/n - n) \sin(k'd)}{(n+1/n)\sin(k'd) + 2 i \cos(k'd)}. This configuration effectively describes a Fabry–Pérot interferometer or etalon: for k'd=0, \pi, 2\pi, \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 H, the displacement u and the stress \sigma=C du/dz, where C 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, : :Q_z=\frac{4\pi}{\lambda}\sin\theta=2k_z where is the angle of incidence/reflection of the incident radiation and is the wavelength of the radiation. The measured reflectivity depends on the variation in the scattering length density () profile, , 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 (), scattering length density () and roughness () 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 layers. Since the incident neutron beam is refracted by each of the layers the wavevector , in layer , is given by: :k_n=\sqrt{{k_z}^2-4\pi({\rho}_n-{\rho}_0)} The Fresnel reflection coefficient between layer and is then given by: : r_{n,n+1} = \frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}} 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, :r_{n,n+1} = \frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}}\exp(-2k_{n}k_{n+1}{\sigma_{n,n+1}}^2) . A phase factor, , is introduced, which accounts for the thickness of each layer. :\beta_{0} = 0 :\beta_{n} = i k_{n}d_{n} where . A characteristic matrix, is then calculated for each layer. :c_{n}=\left[\begin{array}{cc} \exp\left(\beta_{n}\right) & r_{n,n+1}\exp\left(\beta_{n}\right)\\ r_{n,n+1}\exp\left(-\beta_{n}\right) & \exp\left(-\beta_{n}\right)\end{array}\right] The resultant matrix is defined as the ordered product of these characteristic matrices :M=\prod_{n}c_{n} from which the reflectivity is calculated as: :R=\left|\frac{M_{10}}{M_{00}}\right|^{2} == See also ==