The transfer-matrix method is a method used in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
and
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
to analyze the propagation of
electromagnetic
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
or
acoustic waves through a
stratified medium. This is for example relevant for the design of
anti-reflective coating
An antireflective, antiglare or anti-reflection (AR) coating is a type of optical coating applied to the surface of lenses, other optical elements, and photovoltaic cells to reduce reflection. In typical imaging systems, this improves the effic ...
s and
dielectric mirrors.
The
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
of
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
from a single interface between two
media
Media may refer to:
Communication
* Media (communication), tools used to deliver information or data
** Advertising media, various media, content, buying and placement for advertising
** Broadcast media, communications delivered over mass el ...
is described by the
Fresnel equations
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresne ...
. However, when there are multiple
interfaces
Interface or interfacing may refer to:
Academic journals
* Interface (journal), ''Interface'' (journal), by the Electrochemical Society
* ''Interface, Journal of Applied Linguistics'', now merged with ''ITL International Journal of Applied Lin ...
, 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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
, there are simple continuity conditions for the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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
The transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A transmission coefficient describes the amplitude, intensity, or total power of a transmitte ...
s.
Formalism for electromagnetic waves
Below is described how the transfer matrix is applied to
electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
(for example light) of a given
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
propagating through a stack of layers at
normal incidence The angle of incidence, in geometric optics, is the angle between a ray incident on a surface and the line perpendicular (at 90 degree angle) to the surface at the point of incidence, called the normal. The ray can be formed by any waves, such as o ...
. It can be generalized to deal with incidence at an angle,
absorbing media, and media with
magnetic properties
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
. We assume that the stack layers are normal to the
axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with
wave number
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
,
:
.
Because it follows from
Maxwell's equation
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
that electric field
and magnetic field (its normalized derivative)
must be continuous across a boundary, it is convenient to represent the field as the vector
, where
:
.
Since there are two equations relating
and
to
and
, these two representations are equivalent. In the new representation, propagation over a distance
into the positive direction of
is described by the matrix belonging to the
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...
:
and
:
Such a matrix can represent propagation through a layer if
is the wave number in the medium and
the thickness of the layer:
For a system with
layers, each layer
has a transfer matrix
, where
increases towards higher
values. The system transfer matrix is then
:
Typically, one would like to know the
reflectance
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...
and
transmittance
Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is t ...
of the layer structure. If the layer stack starts at
, then for negative
, the field is described as
:
where
is the amplitude of the incoming wave,
the wave number in the left medium, and
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
:
where
is the amplitude transmittance,
is the wave number in the rightmost medium, and
is the total thickness. If
and
, then one can solve
:
in terms of the matrix elements
of the system matrix
and obtain
: