Transfer-matrix method (optics) explained

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. 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.

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 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)=Ereikz+Ele-ikz

.Because it follows from Maxwell's equation that electric field

E

and magnetic field (its normalized derivative) H=\frac Z_c \frac\, must be continuous across a boundary, it is convenient to represent the field as the vector (E(z),H(z))\,, where

M=\left(\begin{array}{cc}\coskL&iZc\sinkL\

i
Zc

\sinkL&\coskL\end{array}\right),

and

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

Mj

, where

j

increases towards higher

z

values. The system transfer matrix is then

Ms=MN\ldotsM2M1.

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

EL(z)=E0

ikLz
e

+rE0

-ikLz
e

,    z<0,

where

E0

is the amplitude of the incoming wave,

kL

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

ER(z)=tE0

ikRz
e

,    z>L',

where

t

is the amplitude transmittance,

kR

is the wave number in the rightmost medium, and

L'

is the total thickness. If H_L = \frac Z_c \frac\, and H_R = \frac Z_c \frac\,, then one can solve

\left(\begin{array}{c}E(zR)\H(zR)\end{array}\right)= M\left(\begin{array}{c}E(0)\H(0)\end{array}\right)

in terms of the matrix elements

Mmn

of the system matrix

Ms

and obtain

t=2ikL

-ikRL
e\left[
1
-M21+kLkRM12+i(kRM11+kLM22)

\right]

and

r=\left[

(M21+kLkRM12)+i(kLM22-kRM11)
(-M21+kLkRM12)+i(kLM22+kRM11)

\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|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}\cosk'd&\sin(k'd)/k'\ -k'\sink'd&\cosk'd\end{array}\right)

.The amplitude reflection coefficient can be simplified to

r=

(1/n-n)\sin(k'd)
(n+1/n)\sin(k'd)+2i\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

\sigma=Cdu/dz

, where

C

is the p-wave modulus, should be used.

Abeles matrix formalism

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, :

Q
z=4\pi
λ

\sin\theta=2kz

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 profileis 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 beamis refracted by each of the layers the wavevector, in layer, is given by:

kn=\sqrt{{k

2-4\pi({\rho}
n-{\rho}

0)}

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

rn,n+1=

kn-kn+1
kn+kn+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,

rn,n+1=

kn-kn+1
kn+kn+1

\exp(-2knkn+1{\sigman,n+1

}^2) .A phase factor,, is introduced, which accounts for the thickness of each layer.

\beta0=0

\betan=ikndn

where .A characteristic matrix, is then calculated for each layer.

cn=\left[\begin{array}{cc} \exp\left(\betan\right)&rn,n+1\exp\left(\betan\right)\\ rn,n+1\exp\left(-\betan\right)&\exp\left(-\betan\right)\end{array}\right]

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

M=\prodncn

from which the reflectivity is calculated as:
R=\left|M10
M00

\right|2

See also

References

  1. Born, M.; Wolf, E., Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Oxford, Pergamon Press, 1964.
  2. Mackay, T. G.; Lakhtakia, A., The Transfer-Matrix Method in Electromagnetics and Optics. San Rafael, CA, Morgan and Claypool, 2020.
  3. O. S. Heavens. Optical Properties of Thin Films. Butterworth, London (1955).
  4. Névot . L. . Croce . P. . Caractérisation des surfaces par réflexion rasante de rayons X. Application à l'étude du polissage de quelques verres silicates . Revue de Physique Appliquée . EDP Sciences . 15 . 3 . 1980 . 0035-1687 . 10.1051/rphysap:01980001503076100 . 761–779. 128834171 . fr.
  5. Abelès . Florin . Florin Abelès. La théorie générale des couches minces . The generalized theory of thin films. Journal de Physique et le Radium . EDP Sciences . 11 . 7 . 1950 . 0368-3842 . 10.1051/jphysrad:01950001107030700 . 307–309. fr.

Further reading

first-principles derivation of the transmission and reflection probabilities from a multilayer with complex indices of refraction.

External links

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