View source for Transfer-matrix method (optics)

{{dablink|For the transfer-matrix method in geometric optics, see [[Ray transfer matrix analysis]]. For the transfer-matrix method in statistical physics, see [[Transfer-matrix method]].}}

[[Image:Etalon-1-corr.svg|thumb|Propagation of a [[Ray (optics)|ray]] through a layer]]

The '''transfer-matrix method''' is a method used in [[optics]] and [[acoustics]] to analyze the propagation of [[electromagnetic wave|electromagnetic]] or [[acoustic wave]]s through a [[stratified medium]]; a stack of [[thin film]]s.<ref>Born, M.; Wolf, E., ''[[Principles of Optics|Principles of optics: electromagnetic theory of propagation, interference and diffraction of light]]''.  Oxford, Pergamon Press, 1964.</ref><ref> Mackay, T. G.; Lakhtakia, A., ''The Transfer-Matrix Method in Electromagnetics and Optics''.  San Rafael, CA, Morgan and Claypool, 2020. {{doi|10.2200/S00993ED1V01Y202002EMA001}}</ref> This is, for example, relevant for the design of [[anti-reflective coating]]s and [[dielectric mirror]]s.

The [[Reflection (physics)|reflection]] of [[light]] from a single interface between two [[medium (optics)|media]] is described by the [[Fresnel equations]]. However, when there are multiple [[Wiktionary:interface|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 [[Interference (wave propagation)|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 (mathematics)|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 coefficient]]s.

==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 [[surface normal|normal incidence]]. It can be generalized to deal with incidence at an angle, [[Absorption (electromagnetic radiation)|absorbing media]], and media with [[Permeability (electromagnetism)|magnetic properties]]. We assume that the stack layers are normal to the <math>z\,</math> axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with [[wave number]] <math>k\,</math>,
:<math>E(z) = E_r e^{ikz} + E_l e^{-ikz}\,</math>.
Because it follows from [[Maxwell's equation]] that electric field <math>E\,</math> and magnetic field (its normalized derivative) <math display=inline>H=\frac{1}{ik} Z_c \frac{dE}{dz}\,</math> must be continuous across a boundary, it is convenient to represent the field as the vector <math display=inline>(E(z),H(z))\,</math>, where
:<math>H(z) = \frac{1}{Z_c} E_r e^{ikz} - \frac{1}{Z_c} E_l e^{-ikz}\,</math>.
Since there are two equations relating <math>E\,</math> and <math>H\,</math> to <math>E_r\,</math> and <math>E_l\,</math>, these two representations are equivalent. In the new representation, propagation over a distance <math>L\,</math> into the positive direction of <math>z\,</math> is described by the matrix belonging to the [[special linear group]] {{nowrap|SL(''2'', '''C''')}}
:<math>M = \left( \begin{array}{cc} \cos kL & i Z_c \sin kL \\ \frac{i}{Z_c} \sin kL & \cos kL \end{array} \right),</math>
and
:<math>\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)</math>
Such a matrix can represent propagation through a layer if <math>k\,</math> is the wave number in the medium and <math>L\,</math> the thickness of the layer:
For a system with <math>N\,</math> layers, each layer <math>j\,</math> has a transfer matrix <math>M_j\,</math>, where <math>j\,</math> increases towards higher <math>z\,</math> values. The system transfer matrix is then
:<math>M_s = M_N \cdot \ldots \cdot M_2 \cdot M_1.</math>

Typically, one would like to know the [[reflectance]] and [[transmittance]] of the layer structure. If the layer stack starts at <math>z=0\,</math>, then for negative <math>z\,</math>, the field is described as
:<math>E_L(z) = E_0 e^{ik_Lz} + r E_0 e^{-ik_Lz},\qquad z<0,</math>
where <math>E_0\,</math> is the amplitude of the incoming wave, <math>k_L\,</math> the wave number in the left medium, and <math>r\,</math> 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
:<math>E_R(z) = t E_0 e^{ik_R z},\qquad z>L',</math>
where <math>t\,</math> is the amplitude transmittance, <math>k_R\,</math> is the wave number in the rightmost medium, and <math>L'</math> is the total thickness. If <math display=inline>H_L = \frac{1}{ik} Z_c \frac{dE_L}{dz}\,</math> and <math display=inline>H_R = \frac{1}{ik} Z_c \frac{dE_R}{dz}\,</math>, then one can solve
:<math>\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)</math>
in terms of the matrix elements <math>M_{mn}\,</math> of the system matrix <math>M_s\,</math> and obtain

:<math>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]</math>

and

:<math>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]</math>.

The transmittance and reflectance (i.e., the fractions of the incident intensity <math display=inline>\left|E_0\right|^2</math> transmitted and reflected by the layer) are often of more practical use and are given by <math display=inline>T=\frac{k_R}{k_L}|t|^2\,</math> and <math>R=|r|^2\,</math>, 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 <math>k'=nk\,</math>. The transfer matrix is
:<math>M=\left(\begin{array}{cc}\cos k'd & \sin(k'd)/k' \\ -k' \sin k'd & \cos k'd \end{array}\right)</math>.
The amplitude reflection coefficient can be simplified to 
:<math>r = \frac{(1/n - n) \sin(k'd)}{(n+1/n)\sin(k'd)  + 2 i \cos(k'd)}</math>.
This configuration effectively describes a [[Fabry–Pérot interferometer]] or etalon: for <math display=inline>k'd=0, \pi, 2\pi, \cdots\,</math>, 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 (physics)|stress]] <math>\sigma=C du/dz</math>, where <math>C</math> is the [[p-wave modulus]], should be used.

==Abeles matrix formalism==
[[Image:Stratifiedinterface.svg|thumb|400px|right|Reflection from a stratified interface]]
The '''Abeles matrix method'''<ref>O. S. Heavens. ''Optical Properties of Thin Films''. Butterworth, London (1955).</ref><ref>{{cite journal | last1=Névot | first1=L. | last2=Croce | first2=P. | title=Caractérisation des surfaces par réflexion rasante de rayons X. Application à l'étude du polissage de quelques verres silicates | journal=Revue de Physique Appliquée | publisher=EDP Sciences | volume=15 | issue=3 | year=1980 | issn=0035-1687 | doi=10.1051/rphysap:01980001503076100 | pages=761–779| s2cid=128834171 | url=https://hal.archives-ouvertes.fr/jpa-00244786/file/ajp-rphysap_1980_15_3_761_0.pdf |language=fr}}</ref><ref>{{cite journal | last=Abelès | first=Florin |author-link=Florin Abelès| title=La théorie générale des couches minces |trans-title=The generalized theory of thin films| journal=Journal de Physique et le Radium | publisher=EDP Sciences | volume=11 | issue=7 | year=1950 | issn=0368-3842 | doi=10.1051/jphysrad:01950001107030700 | pages=307–309|language=fr}}</ref> 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<sub>z</sub>'':
:<math>Q_z=\frac{4\pi}{\lambda}\sin\theta=2k_z</math>
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<sub>n</sub>''), scattering length density (''ρ<sub>n</sub>'') and roughness (σ<sub>n,n+1</sub>) 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:
:<math>k_n=\sqrt{{k_z}^2-4\pi({\rho}_n-{\rho}_0)}</math>
The [[Fresnel equations|Fresnel reflection]] coefficient between layer ''n'' and ''n+1'' is then given by:
:<math> r_{n,n+1} = \frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}} </math>
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 [[#Nevot1980|Nevot and Croce (1980)]].

:<math>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) </math>
A phase factor, ''β'', is introduced, which accounts for the thickness of each layer.
:<math>\beta_{0} = 0</math>
:<math>\beta_{n} = i k_{n}d_{n}</math>
where <math>i^2 = -1</math>.
A characteristic matrix, c<sub>n</sub> is then calculated for each layer.
:<math>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]</math>
The resultant matrix is defined as the ordered product of these characteristic matrices
:<math>M=\prod_{n}c_{n}</math>
from which the reflectivity is calculated as:
:<math>R=\left|\frac{M_{10}}{M_{00}}\right|^{2}</math>

== See also ==
*[[Neutron reflectometry]]
*[[Ellipsometry]]
*[[Jones calculus]]
*[[X-ray reflectivity]]
*[[Scattering-matrix method]]

==References==
<references/>

==Further reading==

*[http://www.jedsoft.org/physics/notes/multilayer.pdf Multilayer Reflectivity]: first-principles derivation of the transmission and reflection probabilities from a multilayer with complex indices of refraction.
*[https://ocw.mit.edu/courses/materials-science-and-engineering/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/lecture-notes/MIT3_024S13_2012lec23.pdf Layered Materials and Photonic Band Diagrams] (Lecture 23) in MIT Open Course [https://ocw.mit.edu/courses/materials-science-and-engineering/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/lecture-notes/ Electronic, Optical and Magnetic Properties of Materials].
*[https://ocw.mit.edu/courses/mechanical-engineering/2-57-nano-to-macro-transport-processes-spring-2012/video-lectures/lecture-13-em-wave-propagation-through-thin-films-multilayers/ EM Wave Propagation Through Thin Films & Multilayers] (Lecture 13) in MIT Open Course [https://ocw.mit.edu/courses/mechanical-engineering/2-57-nano-to-macro-transport-processes-spring-2012 Nano-to-Macro Transport Processes]. Includes short discussion acoustic waves.

==External links==
There are a number of computer programs that implement this calculation:
*[http://people.csail.mit.edu/jaffer/FreeSnell/ FreeSnell] is a stand-alone computer program that implements the transfer-matrix method, including more advanced aspects such as granular films.
*[http://thinfilm.hansteen.net/ Thinfilm] is a web interface that implements the transfer-matrix method, outputting reflection and transmission coefficients, and also [[ellipsometer|ellipsometric]] parameters Psi and Delta.
*[http://www.luxpop.com/#Thin_Film_Stack Luxpop.com] is another web interface that implements the transfer-matrix method.
*[http://sjbyrnes.com/?page_id=12 Transfer-matrix calculating programs in ''Python'' and in ''Mathematica''].
*[https://lbolla.github.io/EMpy/ EMPy ("Electromagnetic Python") software].
*[http://motofit.sourceforge.net/ motofit] is a program for analysing neutron and X-ray reflectometry data.
*[http://larfis.polymtl.ca/index.php/en/links/openfilters OpenFilters] is a program for designing optical filters.
*[https://github.com/gevero/py_matrix 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.
*[https://ncnr.nist.gov/instruments/magik/calculators/calcR_d3_dark.html In-browser calculator and fitter] Javascript interactive reflectivity calculator using matrix method and Nevot-Croce roughness approximation (calculation kernel converted from C via [[Emscripten]])

[[Category:Physical optics]]
[[Category:Neutron-related techniques]]
[[Category:Scattering theory]]
[[Category:Electromagnetism]]