Ray transfer matrix analysis (also known as ABCD matrix analysis) is a mathematical form for performing
ray tracing calculations in sufficiently simple problems which can be solved considering only paraxial rays. Each optical element (surface, interface, mirror, or beam travel) is described by a 2×2 ''ray transfer
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 ...
'' which operates on a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
describing an incoming
light ray
In optics a ray is an idealized geometrical model of light, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy flow. Rays are used to model the propagation o ...
to calculate the outgoing ray. Multiplication of the successive matrices thus yields a concise ray transfer matrix describing the entire optical system. The same mathematics is also used in
accelerator physics
Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams ...
to track particles through the magnet installations of a
particle accelerator
A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams.
Large accelerators are used for fundamental research in particle ...
, see
electron optics
Electron optics is a mathematical framework for the calculation of electron trajectories along electromagnetic fields. The term ''optics'' is used because magnetic and electrostatic lenses act upon a charged particle beam similarly to optical le ...
.
This technique, as described below, is derived using the ''
paraxial approximation
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens).
A paraxial ray is a ray which makes a small angle (''θ'') to the optica ...
'', which requires that all ray directions (directions normal to the wavefronts) are at small angles ''θ'' relative to the
optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight.
The optical axis is an imaginary line that defines the path along which light propagat ...
of the system, such that the approximation
remains valid. A small θ further implies that the transverse extent of the ray bundles (''x'' and ''y'') is small compared to the length of the optical system (thus "paraxial"). Since a decent imaging system where this is ''not'' the case for all rays must still focus the paraxial rays correctly, this matrix method will properly describe the positions of focal planes and magnifications, however
aberrations still need to be evaluated using full
ray-tracing techniques.
Definition of the ray transfer matrix
The ray tracing technique is based on two reference planes, called the ''input'' and ''output'' planes, each perpendicular to the optical axis of the system. At any point along the optical train an optical axis is defined corresponding to a central ray; that central ray is propagated to define the optical axis further in the optical train which need not be in the same physical direction (such as when bent by a prism or mirror). The transverse directions ''x'' and ''y'' (below we only consider the ''x'' direction) are then defined to be orthogonal to the optical axes applying. A light ray enters a component crossing its input plane at a distance ''x''
1 from the optical axis, traveling in a direction that makes an angle ''θ''
1 with the optical axis. After propagation to the output plane that ray is found at a distance ''x''
2 from the optical axis and at an angle ''θ''
2 with respect to it. ''n''
1 and ''n''
2 are the
indices of refraction of the media in the input and output plane, respectively.
The ABCD matrix representing a component or system relates the output ray to the input according to
:
where the values of the 4 matrix elements are thus given by
:
and
:
This relates the ''ray vectors'' at the input and output planes by the ''ray transfer matrix'' (RTM) M, which represents the optical component or system present between the two reference planes. A
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
argument based on the
blackbody
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
radiation can be used to show that the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of a RTM is the ratio of the indices of refraction:
:
As a result, if the input and output planes are located within the same medium, or within two different media which happen to have identical indices of refraction, then the determinant of M is simply equal to 1.
A different convention for the ray vectors can be employed. Instead of using ''θ''≈sin ''θ'', the second element of the ray vector is ''n'' sin ''θ'', which is proportional not to the ray angle ''per se'' but to the transverse component of the
wave vector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
.
This alters the ABCD matrices given in the table below where refraction at an interface is involved.
The use of transfer matrices in this manner parallels the 2×2 matrices describing electronic
two-port networks, particularly various so-called ABCD matrices which can similarly be multiplied to solve for cascaded systems.
Some examples
* For example, if there is free space between the two planes, the ray transfer matrix is given by:
where ''d'' is the separation distance (measured along the optical axis) between the two reference planes. The ray transfer equation thus becomes:
and this relates the parameters of the two rays as:
* Another simple example is that of a
thin lens
In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
. Its RTM is given by:
where ''f'' is the
focal length
The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foca ...
of the lens. To describe combinations of optical components, ray transfer matrices may be multiplied together to obtain an overall RTM for the compound optical system. For the example of free space of length ''d'' followed by a lens of focal length ''f'':
Note that, since the multiplication of matrices is non-
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
, this is not the same RTM as that for a lens followed by free space:
:
Thus the matrices must be ordered appropriately, with the last matrix premultiplying the second last, and so on until the first matrix is premultiplied by the second. Other matrices can be constructed to represent interfaces with media of different
refractive indices
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
, reflection from
mirror
A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
s, etc.
Eigenvalues of Ray Transfer Matrix
A ray transfer matrix can be regarded as a
linear canonical transformation In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the ac ...
. According to the eigenvalues of the optical system, the system can be classified into several classes. Assume the ABCD matrix representing a system relates the output ray to the input according to
.
We compute the eigenvalues of the matrix
that satisfy eigenequation
,
by calculating the determinant
.
Let
, and we have eigenvalues
.
According to the values of
and
, there are several possible cases. For example:
# A pair of real eigenvalues:
and
, where
. This case represents a magnifier
#
or
. This case represents unity matrix (or with an additional coordinate reverter)
.
#
. This case occurs if but not only if the system is either a unity operator, a section of free space, or a lens
# A pair of two unimodular, complex conjugated eigenvalues
and
. This case is similar to a separable
Fractional Fourier Transformer.
Table of ray transfer matrices
for simple optical components
Relation between geometrical ray optics and wave optics
The theory of
Linear canonical transformation In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the ac ...
implies the relation between ray transfermatrix (
geometrical optics
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of ''rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...
) and wave optics.
Common Decomposition of Ray Transfer Matrix
There exist infinite ways to decompose a ray transfer matrix
into a concatenation of multiple transfer matrix. For example:
#
.
#
#
#
Resonator stability
RTM analysis is particularly useful when modeling the behavior of light in
optical resonator An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that forms a cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and provi ...
s, such as those used in lasers. At its simplest, an optical resonator consists of two identical facing mirrors of 100%
reflectivity
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 radius of
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
''R'', separated by some distance ''d''. For the purposes of ray tracing, this is equivalent to a series of identical thin lenses of focal length ''f''=''R''/2, each separated from the next by length ''d''. This construction is known as a ''lens equivalent duct'' or ''lens equivalent
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
''. The RTM of each section of the waveguide is, as above,
:
.
RTM analysis can now be used to determine the ''stability'' of the waveguide (and equivalently, the resonator). That is, it can be determined under what conditions light traveling down the waveguide will be periodically refocused and stay within the waveguide. To do so, we can find all the "eigenrays" of the system: the input ray vector at each of the mentioned sections of the waveguide times a real or complex factor ''λ'' is equal to the output one. This gives:
:
which is an
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
equation:
:
where I is the 2×2
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
.
We proceed to calculate the eigenvalues of the transfer matrix:
:
leading to the
characteristic equation
:
where
:
is the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the RTM, and
:
is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the RTM. After one common substitution we have:
:
where
:
is the ''stability parameter''. The eigenvalues are the solutions of the characteristic equation. From the
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, gr ...
we find
:
Now, consider a ray after ''N'' passes through the system:
:
If the waveguide is stable, no ray should stray arbitrarily far from the main axis, that is, ''λ''
''N'' must not grow without limit. Suppose
. Then both eigenvalues are real. Since
, one of them has to be bigger than 1 (in absolute value), which implies that the ray which corresponds to this eigenvector would not converge. Therefore, in a stable waveguide,
≤ 1, and the eigenvalues can be represented by complex numbers:
:
with the substitution ''g'' = cos(''ϕ'').
For
let
and
be the eigenvectors with respect to the eigenvalues
and
respectively, which span all the vector space because they are orthogonal, the latter due to
. The input vector can therefore be written as
:
for some constants
and
.
After ''N'' waveguide sectors, the output reads
:
which represents a periodic function.
Ray transfer matrices for Gaussian beams
The same matrices can also be used to calculate the evolution of
Gaussian beam
In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This ...
s. propagating through optical components described by the same transmission matrices. If we have a Gaussian beam of wavelength
, radius of curvature ''R'' (positive for diverging, negative for converging), beam spot size ''w'' and refractive index ''n'', it is possible to define a
complex beam parameter In optics, the complex beam parameter is a complex number that specifies the properties of a Gaussian beam at a particular point z along the axis of the beam. It is usually denoted by ''q''. It can be calculated from the beam's vacuum wavelength λ0 ...
''q'' by:
[
:
(''R'', ''w'', and ''q'' are functions of position.) If the beam axis is in the ''z'' direction, with waist at and ]Rayleigh range
In optics and especially laser science, the Rayleigh length or Rayleigh range, z_\mathrm, is the distance along the propagation direction of a light beam, beam from the beam waist, waist to the place where the area of the Cross_section_(geometry ...
, this can be equivalently written as[ especiall]
Chapter 5
/ref>
:
This beam can be propagated through an optical system with a given ray transfer matrix by using the equation:
:
where ''k'' is a normalization constant chosen to keep the second component of the ray vector equal to 1. Using matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
, this equation expands as
:
and
:
Dividing the first equation by the second eliminates the normalization constant:
:
It is often convenient to express this last equation in reciprocal form:
:
Example: Free space
Consider a beam traveling a distance ''d'' through free space, the ray transfer matrix is
:
and so
:
consistent with the expression above for ordinary Gaussian beam propagation, i.e. . As the beam propagates, both the radius and waist change.
Example: Thin lens
Consider a beam traveling through a thin lens with focal length ''f''. The ray transfer matrix is
: .
and so
:
:
Only the real part of 1/''q'' is affected: the wavefront curvature 1/''R'' is reduced by the power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
of the lens 1/''f'', while the lateral beam size ''w'' remains unchanged upon exiting the thin lens.
Higher rank matrices
Methods using transfer matrices of higher dimensionality, that is 3×3, 4×4, and 6×6, are also used in optical analysis.[H. Wollnik, ''Optics of Charged Particles'' (Academic, New York, 1987).] In particular, 4×4 propagation matrices are used in the design and analysis of prism sequences for pulse compression Pulse compression is a signal processing technique commonly used by radar, sonar and echography to increase the range resolution as well as the signal to noise ratio. This is achieved by modulating the transmitted pulse and then correlating th ...
in femtosecond lasers.
See also
* Transfer-matrix method (optics)
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. This is for example relevant for the design of anti-reflective coatings and diele ...
* Linear canonical transformation In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the ac ...
References
Further reading
*{{cite book , title = Fundamentals of Photonics , author = Bahaa E. A. Saleh and Malvin Carl Teich , publisher = John Wiley & Sons , location = New York , year = 1991 Section 1.4, pp. 26 – 36.
External links
Thick lenses (Matrix methods)
ABCD Matrices Tutorial
Provides an example for a system matrix of an entire system.
ABCD Calculator
An interactive calculator to help solve ABCD matrices.
Simple Optical Designer (Android App)
An application to explore optical systems using the ABCD matrix method.
Geometrical optics
Accelerator physics