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

${\displaystyle E(z)=E_{r}e^{ikz}+E_{l}e^{-ikz}\,}$.

Because it follows from Maxwell's equation that ${\displaystyle E\,}$ and ${\displaystyle H=1/ikZ_{c}dE/dz\,}$ must be continuous across a boundary, it is convenient to represent the field as the vector ${\displaystyle (E(z),H(z))\,}$, where

${\displaystyle H(z)=1/Z_{c}E_{r}e^{ikz}-1/Z_{c}E_{l}e^{-ikz}\,}$.

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 ${\displaystyle z\,}$ direction is described by the unimodular matrix

${\displaystyle M=\left({\begin{array}{cc}\cos kL&iZ_{c}\sin kL\\{\frac {i}{Z_{c}}}\sin kL&\cos kL\end{array}}\right),}$

and

${\displaystyle \left({\begin{array}{c}E(z+L)\\H(z+L)\end{array}}\right)=M\cdot \left({\begin{array}{c}E(z)\\H(z)\end{array}}\right)}$

Such a matrix can represent propagation through a layer if ${\displaystyle k\,}$ 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

${\displaystyle M_{s}=M_{N}\cdot \ldots \cdot M_{2}\cdot M_{1}.}$

Typically, one would like to know the reflectance and transmittance of the layer structure. If the layer stack starts at ${\displaystyle z=0\,}$, then for negative ${\displaystyle z\,}$, the field is described as

${\displaystyle E_{L}(z)=E_{0}e^{ik_{L}z}+rE_{0}e^{-ik_{L}z},\qquad z<0,}$

where ${\displaystyle E_{0}\,}$ is the amplitude of the incoming wave, ${\displaystyle k_{L}\,}$ the wave number in the left medium, and ${\displaystyle r\,}$ 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

${\displaystyle E_{R}(z)=tE_{0}e^{ik_{R}z},\qquad z>L',}$

where ${\displaystyle t\,}$ is the amplitude transmittance, ${\displaystyle k_{R}\,}$ is the wave number in the rightmost medium, and ${\displaystyle L'}$ is the total thickness. If ${\displaystyle H_{L}=1/ikZ_{c}dE_{L}/dz\,}$ and ${\displaystyle H_{R}=1/ikZ_{c}dE_{R}/dz\,}$, then we can solve

${\displaystyle \left({\begin{array}{c}E(z_{R})\\H(z_{R})\end{array}}\right)=M\cdot \left({\begin{array}{c}E(0)\\H(0)\end{array}}\right)}$

in terms of the matrix elements ${\displaystyle M_{mn}\,}$ of the system matrix ${\displaystyle M_{s}\,}$ and obtain

${\displaystyle t=2ik_{L}e^{-ik_{R}L}\left[{\frac {1}{-M_{21}+k_{L}k_{R}M_{12}+i(k_{R}M_{11}+k_{L}M_{22})}}\right]}$

and

${\displaystyle r=\left[{\frac {(M_{21}+k_{L}k_{R}M_{12})+i(k_{L}M_{22}-k_{R}M_{11})}{(-M_{21}+k_{L}k_{R}M_{12})+i(k_{L}M_{22}+k_{R}M_{11})}}\right]}$.

The transmittance and reflectance (i.e., the fractions of the incident intensity ${\displaystyle \left|E_{0}\right|^{2}}$ transmitted and reflected by the layer) are often of more practical use and are given by ${\displaystyle T={\frac {k_{R}}{k_{L}}}|t|^{2}\,}$ and ${\displaystyle R=|r|^{2}\,}$, respectively (at normal incidence).

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 ${\displaystyle \sigma =Cdu/dz}$, where ${\displaystyle C}$ is the p-wave modulus, should be used.

Reflection from a stratified interface

The Abeles matrix method[2][3][4] 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:

${\displaystyle Q_{z}={\frac {4\pi }{\lambda }}\sin \theta =2k_{z}}$

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 (d_n), scattering length density (ρ_n) and roughness (σ_n,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:

${\displaystyle k_{n}={\sqrt {{k_{z}}^{2}-4\pi ({\rho }_{n}-{\rho }_{0})}}}$

The Fresnel reflection coefficient between layer n and n+1 is then given by:

${\displaystyle r_{n,n+1}={\frac {k_{n}-k_{n+1}}{k_{n}+k_{n+1}}}}$

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).

${\displaystyle r_{n,n+1}={\frac {k_{n}-k_{n+1}}{k_{n}+k_{n+1}}}\exp(-2k_{n}k_{n+1}{\sigma _{n,n+1}}^{2})}$

A phase factor, β, is introduced, which accounts for the thickness of each layer.

${\displaystyle \beta _{0}=0}$

${\displaystyle \beta _{n}=ik_{n}d_{n}}$

where ${\displaystyle i^{2}=-1}$. A characteristic matrix, c_n is then calculated for each layer.

${\displaystyle c_{n}=\left[{\begin{array}{cc}\exp \left(\beta _{n}\right)&r_{n,n+1}\exp \left(\beta _{n}\right)\\r_{n,n+1}\exp \left(-\beta _{n}\right)&\exp \left(-\beta _{n}\right)\end{array}}\right]}$

The resultant matrix is defined as the product of these characteristic matrices

${\displaystyle M=\prod _{n}c_{n}}$

from which the reflectivity is calculated as:

${\displaystyle R=\left|{\frac {M_{10}}{M_{00}}}\right|^{2}}$

• Neutron reflectometry
• Ellipsometry
• Jones calculus
• X-ray reflectivity

• Multilayer Reflectivity: first-principles derivation of the transmission and reflection probabilities from a multilayer with complex indices of refraction.
• Layered Materials and Photonic Band Diagrams (Lecture 23) in MIT Open Course Electronic, Optical and Magnetic Properties of Materials.
• EM Wave Propagation Through Thin Films & Multilayers (Lecture 13) in MIT Open Course Nano-to-Macro Transport Processes. Includes short discussion acoustic waves.

There are a number of computer programs that implement this calculation:

• FreeSnell is a stand-alone computer program that implements the transfer-matrix method, including more advanced aspects such as granular films.
• Thinfilm is a web interface that implements the transfer-matrix method, outputting reflection and transmission coefficients, and also ellipsometric parameters Psi and Delta.
• Luxpop.com is another web interface that implements the transfer-matrix method.
• Transfer-matrix calculating programs in Python and in Mathematica.
• EMPy ("Electromagnetic Python") software.
• motofit is a program for analysing neutron and X-ray reflectometry data.
• OpenFilters is a program for designing optical filters.
• Py_matrix is an open source Python code that implements the transfer-matrix method for multilayers with arbitrary dielectric tensors. It was especially created for plasmonic and magnetoplasmonic calculations.
• In-browser calculator and fitter Javascript interactive reflectivity calculator using matrix method and Nevot-Croce roughness approximation (calculation kernel converted from C via Emscripten)