Ray transfer matrix analysis: Difference between revisions

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'''Ray transfer matrix analysis''' (also known as '''ABCD matrix analysis''') is a mathematical form for performing [[Ray tracing (physics)|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 (math)|matrix]]'' which operates on a [[vector space|vector]] describing an incoming [[ray (optics)|light ray]] 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]] to track particles through the magnet installations of a [[particle accelerator]], see [[electron optics]].

This technique, as described below, is derived using the ''[[paraxial approximation]]'', which requires that all ray directions (directions normal to the wavefronts) are at small angles ''θ'' relative to the [[optical axis]] of the system, such that the approximation <math>\sin \theta \approx \theta</math> 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 [[Optical aberration|aberrations]] still need to be evaluated using full [[Ray tracing (physics)#Optical design|ray-tracing]] techniques.<ref>Extension of matrix methods to tracing (non-paraxial) meridional rays is included [http://spie.org/Documents/ETOP/1991/389_1.pdf here].</ref>

== Definition of the ray transfer matrix ==

[[File:RayTransferMatrixDefinitions.svg|thumb|300px|In ray transfer (ABCD) matrix analysis, an optical element (here, a thick lens) gives a transformation between <math>(x_1, \theta_1)</math> at the input plane and <math>(x_2, \theta_2)</math> when the ray arrives at the output plane.]]

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 [[index of refraction|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

:<math> \begin{bmatrix}x_2 \choose\ \theta_2\end{bmatrix} = \begin{pmatrixbmatrix} A & B \\ C & D \end{pmatrixbmatrix} \begin{bmatrix}x_1 \choose\ \theta_1\end{bmatrix}, </math>

where the values of the 4 matrix elements are thus given by

:<math>A = \left.\frac{x_2 \over }{x_1 } \biggright|_{\theta_1 = 0} \qquad B = \left.\frac{x_2 \over }{\theta_1 } \biggright|_{x_1 = 0},</math>

and

:<math>C = \left.\frac{\theta_2 \over}{ x_1 } \biggright|_{\theta_1 = 0} \qquad D = \left.\frac{\theta_2 \over }{\theta_1 } \biggright|_{x_1 = 0}.</math>

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]] argument based on the [[blackbody]] radiation can be used to show that the [[determinant]] of a RTM is the ratio of the indices of refraction:

:<math>\det(\mathbf{M}) = AD - BC = \frac{ n_1 \over }{n_2 }. </math>

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.

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|isbn= 9780486680446

|url=https://books.google.com/books?id=naUSNojPwOgC

}}</ref> for the ray vectors can be employed. Instead of using ''θ''≈sin 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]].

This alters the ABCD matrices given in the table below where refraction at an interface is involved.

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== Some examples ==

* For example, if there is free space between the two planes, the ray transfer matrix is given by: <math display="block"> \mathbf{S} = \begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix} , </math> where ''d'' is the separation distance (measured along the optical axis) between the two reference planes. The ray transfer equation thus becomes: <math display="block"> \begin{bmatrix} x_2 \\ \theta_2 \end{bmatrix} = \mathbf{S} \begin{bmatrix} x_1 \\ \theta_1\end{bmatrix} , </math> and this relates the parameters of the two rays as: <math display="block"> \begin{matrix}

* For example, if there is free space between the two planes, the ray transfer matrix is given by:

:<math> \begin{matrix} x_2 & = & x_1 + d\theta_1 \\

\theta_2 & = & \theta_1 \end{matrix} </math>

:<math> \mathbf{S} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} </math>,

\end{matrix} </math>

* Another simple example is that of a [[thin lens]]. Its RTM is given by: <math display="block"> \mathbf{L} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{bmatrix} , </math> where ''f'' is the [[focal length]] 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'': <math display="block">\mathbf{L}\mathbf{S} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{f} & 1\end{bmatrix}

where ''d'' is the separation distance (measured along the optical axis) between the two reference planes. The ray transfer equation thus becomes:

:<math> \mathbf{S} = \begin{pmatrixbmatrix} 1 & d \\ 0 & 1 \end{pmatrixbmatrix} </math>,

:<math>= \begin{x_2bmatrix} 1 & d \\choose -\theta_2frac{1}{f} =& 1-\mathbffrac{Sd}{x_1f} \choose \theta_1end{bmatrix} . </math>,

and this relates the parameters of the two rays as:

:<math> \begin{matrix} x_2 & = & x_1 + d\theta_1 \\

\theta_2 & = & \theta_1 \end{matrix} </math>

* Another simple example is that of a [[thin lens]]. Its RTM is given by:

:<math> \mathbf{L} = \begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix} </math>,

where ''f'' is the [[focal length]] 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'':

:<math>\mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1\end{pmatrix}

\begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}

= \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} </math>.

Note that, since the multiplication of matrices is non-[[commutative]], this is not the same RTM as that for a lens followed by free space:

:<math> \mathbf{SL} =

\begin{pmatrixbmatrix} 1 & d \\ 0 & 1 \end{pmatrixbmatrix}

\begin{pmatrixbmatrix} 1 & 0 \\ -\frac{-1}{f} & 1 \end{pmatrixbmatrix}

= \begin{pmatrixbmatrix} 1-\frac{d}{f} & d \\ -\frac{-1}{f} & 1 \end{pmatrixbmatrix} . </math>.

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 index|refractive indices]], reflection from [[mirror]]s, etc.

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| Refraction at a curved interface

| align="center" | <math>\begin{pmatrix} 1 & 0 \\ \frac{n_1-n_2}{R \cdot n_2} & \frac{n_1}{n_2} \end{pmatrix} </math>

| ''R'' = radius of curvature, ''R'' > 0 for convex (centrecenter of curvature after interface)<br/>

''n''₁ = initial refractive index<br/>''n''₂ = final refractive index.

''n''₂ = final refractive index.

|-

| Reflection from a flat mirror

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| Single prism

| align="center" | <math> \begin{pmatrix} k & \frac{d}{nk} \\ 0 & \frac{1}{k} \end{pmatrix} </math>

| ''<math>k'' = (\cos<math>\psi</math> / \cos<math>\phi)</math>) is the [[beam expander|beam expansion]] factor, where <math>\phi</math> is the angle of incidence, <math>\psi</math> is the angle of refraction, ''d'' = prism path length, ''n'' = refractive index of the prism material. This matrix applies for orthogonal beam exit.<ref name=TLO>{{cite book | title = Tunable Laser Optics | author = [[F. J. Duarte]] | publisher = Elsevier-Academic | ___location = New York | year = 2003 }} Chapter 6.</ref>

|-

| Multiple prism beam expander using ''r'' prisms

| align="center" | <math> \begin{pmatrix} M & B \\ 0 & \frac{1}{M} \end{pmatrix} </math>

| ''M'' is the total beam magnification given by <math>M = k_1 k_2 k_3... \cdots k_r</math>, where ''k'' is defined in the previous entry and ''B'' is the total optical propagation distance{{clarify|date=July 2019}} of the multiple prism expander.<ref name=TLO />

|}

== Resonator stability ==

RTM analysis is particularly useful when modeling the behaviourbehavior of light in [[optical resonator]]s, such as those used in lasers. At its simplest, an optical resonator consists of two identical facing mirrors of 100% [[reflectivity]] and radius of [[curvature]] ''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]]''. The RTM of each section of the waveguide is, as above,

:<math>\mathbf{M} =\mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} </math>.

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 travellingtraveling down the waveguide will be periodically refocussedrefocused 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:

:<math> \mathbf{M} \begin{bmatrix}x_1 \choose\ \theta_1\end{bmatrix} = \begin{bmatrix}x_2 \choose\ \theta_2\end{bmatrix} = \lambda \begin{bmatrix}x_1 \choose\ \theta_1\end{bmatrix} . </math>.

which is an [[eigenvalue]] equation:

:<math> \left[ \mathbf{M} - \lambda\mathbf{I} \right] \begin{bmatrix}x_1 \choose\ \theta_1\end{bmatrix} = 0 , </math>,

where '''I''' is the 2x22×2 [[identity matrix]].

We proceed to calculate the eigenvalues of the transfer matrix:

:<math>\operatorname{det} \left[ \mathbf{M} - \lambda\mathbf{I} \right] = 0 , </math>,

leading to the [[Characteristic polynomial#Characteristic equation|characteristic equation]]

:<math> \lambda^2 - \operatorname{tr}(\mathbf{M}) \lambda + \operatorname{det}( \mathbf{M}) = 0 , </math>,

where

:<math> \operatorname{tr} ( \mathbf{M} ) = A + D = 2 - \frac{ d \over }{f } </math>

is the [[trace (linear algebra)|trace]] of the RTM, and

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is the [[determinant]] of the RTM. After one common substitution we have:

:<math> \lambda^2 - 2g \lambda + 1 = 0 , </math>,

where

:<math> g \ \stackreloverset{\mathrm{def}}{{}={}}\ \frac{ \operatorname{tr}(\mathbf{M}) \over}{ 2 } = 1 - \frac{ d \over}{ 2 f } </math>

is the ''stability parameter''. The eigenvalues are the solutions of the characteristic equation. From the [[Quadratic equation#Quadratic formula and its derivation|quadratic formula]] we find

:<math> \lambda_{\pm} = g \pm \sqrt{g^2 - 1} \,. </math>

Now, consider a ray after ''N'' passes through the system:

:<math> \begin{bmatrix}x_N \choose\ \theta_N \end{bmatrix} = \lambda^N \begin{bmatrix}x_1 \choose\ \theta_1\end{bmatrix}. </math>.

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 <math> g^2 > 1</math>. Then both eigenvalues are real. Since <math> \lambda_+ \lambda_- = 1</math>, 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, <math> g^2 </math> ≤ 1, and the eigenvalues can be represented by complex numbers:

:<math> \lambda_{\pm} = g \pm i \sqrt{1 - g^2} = \cos(\phi) \pm i \sin(\phi) = e^{\pm i \phi} , </math>,

with the substitution ''g'' = cos(''ϕ'').

For <math> g^2 < 1 </math> let <math> r_+ </math> and <math> r_- </math> be the eigenvectors with respect to the eigenvalues <math> \lambda_+ </math> and <math> \lambda_- </math> respectively, which span all the vector space because they are orthogonal, the latter due to <math>\lambda_+</math> ≠\neq <math>\lambda_- </math>. The input vector can therefore be written as

:<math> c_+ r_+ + c_- r_- , </math>,

for some constants <math> c_+ </math> and <math> c_- </math>.

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After ''N'' waveguide sectors, the output reads

:<math> \mathbf{M}^N (c_+ r_+ + c_- r_-) = \lambda_+^N c_+ r_+ + \lambda_-^N c_- r_- = e^{i N \phi} c_+ r_+ + e^{- i N \phi} c_- r_- , </math>,

which represents a periodic function.

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The same matrices can also be used to calculate the evolution of [[Gaussian beam]]s.<ref>{{cite journal|last1=Rashidian vaziri|first1=M R|title=New ducting model for analyzing the Gaussian beam propagation in nonlinear Kerr media and its application to spatial self-phase modulations|journal=Journal of Optics|volume=15|issue=3|pages=035202|doi=10.1088/2040-8978/15/3/035202|bibcode=2013JOpt...15c5202R|year=2013}}</ref> propagating through optical components described by the same transmission matrices. If we have a Gaussian beam of wavelength <math>\lambda_0</math>, 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]] ''q'' by:<ref name=Lei/>

:<math> \frac{1}{q} = \frac{1}{R} - \frac{i\lambda_0}{\pi n w^2} . </math>.

(''R'', ''w'', and ''q'' are functions of position.) If the beam axis is in the ''z'' direction, with waist at <math>z_0</math> and [[Rayleigh range]] <math>z_R</math>, this can be equivalently written as<ref name=Lei>{{cite web|url=http://www.colorado.edu/physics/phys4510/phys4510_fa05/ |author=C. Tim Lei |title=Physics 4510 Optics webpage}} especially [http://www.colorado.edu/physics/phys4510/phys4510_fa05/Chapter5.pdf Chapter 5]</ref>

:<math> q = (z - z_0) + i z_R .</math>.

This beam can be propagated through an optical system with a given ray transfer matrix by using the equation{{explain|date=July 2019}}:

:<math> \begin{bmatrix} q_2 \choose\ 1 \end{bmatrix} = k \begin{pmatrixbmatrix} A & B \\ C & D \end{pmatrixbmatrix} \begin{bmatrix}q_1 \choose\ 1 \end{bmatrix} , </math>,

where ''k'' is a normalisationnormalization constant chosen to keep the second component of the ray vector equal to 1. Using [[matrix multiplication]], this equation expands as

:<math> q_2 = k(Aq_1 + B) \,</math>

and

:<math> 1 = k(Cq_1 + D) \, </math>

Dividing the first equation by the second eliminates the normalisationnormalization constant:

:<math> q_2 =\frac{Aq_1+B}{Cq_1+D} ,</math>,

It is often convenient to express this last equation in reciprocal form:

:<math> \frac{ 1 \over}{ q_2 } = \frac{ C + D/q_1 \over }{ A + B/q_1 } . </math>

=== Example: Free space ===

Consider a beam traveling a distance ''d'' through free space, the ray transfer matrix is

: <math>\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix} .</math>.

and so

: <math>q_2 = \frac{Aq_1A q_1+B}{Cq_1C q_1+D} = \frac{q_1+d}{1} = q_1+d</math>

consistent with the expression above for ordinary Gaussian beam propagation, i.e. <math> q = (z-z_0) + i z_R</math>. As the beam propagates, both the radius and waist change.

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and so

: <math>q_2 =\frac{Aq_1+B}{Cq_1+D} = \frac{q_1}{-\frac{q_1}{f}+1} </math>

: <math>\frac{1}{q_2} = \frac{-\frac{q_1}{f} + 1}{q_1} = \frac{1}{q_1} - \frac{1}{f} .</math>.

Only the real part of 1/''q'' is affected: the wavefront curvature 1/''R'' is reduced by the [[Optical power|power]] of the lens 1/''f'', while the lateral beam size ''w'' remains unchanged upon exiting the thin lens.