Ray transfer matrix analysis: Difference between revisions

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The ABCD matrix representing a component or system relates the output ray to the input according to

<math display="block"> \begin{bmatrix}x_2 \\ \theta_2\end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix}, </math>

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

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

<math display="block">A = \left.\frac{x_2}{x_1} \right|_{\theta_1 = 0} \qquad B = \left.\frac{x_2}{\theta_1} \right|_{x_1 = 0},</math>

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

and

<math display="block">C = \left.\frac{\theta_2}{ x_1 } \right|_{\theta_1 = 0} \qquad D = \left.\frac{\theta_2}{\theta_1 } \right|_{x_1 = 0}.</math>

:<math>C = \left.\frac{\theta_2}{ x_1 } \right|_{\theta_1 = 0} \qquad D = \left.\frac{\theta_2}{\theta_1 } \right|_{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 {{Citation needed|date=August 2023}} can be used to show that the [[determinant]] of a RTM is the ratio of the indices of refraction:

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

:<math>\det(\mathbf{M}) = AD - BC = \frac{n_1}{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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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 display="block"> \mathbf{SL} =

:<math> \mathbf{SL} =

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

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

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== Eigenvalues ==

A ray transfer matrix can be regarded as a [[linear canonical transformation]]. According to the eigenvalues of the optical system, the system can be classified into several classes.<ref>{{Cite journal|last1=Bastiaans|first1=Martin J.|last2=Alieva|first2=Tatiana|date=2007-03-14|title=Classification of lossless first-order optical systems and the linear canonical transformation|url=http://dx.doi.org/10.1364/josaa.24.001053|journal=Journal of the Optical Society of America A|volume=24|issue=4|pages=1053–1062|doi=10.1364/josaa.24.001053|pmid=17361291 |issn=1084-7529}}</ref> Assume the ABCD matrix representing a system relates the output ray to the input according to

<math display="block"> \begin{bmatrix}x_2 \\ \theta_2\end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix}

=\mathbf{T}\mathbf{v} .</math>

<math> \begin{bmatrix}x_2 \\ \theta_2\end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix}

=\mathbf{T}\mathbf{v} </math>.

We compute the eigenvalues of the matrix <math> \mathbf{T} </math> that satisfy eigenequation

<math display="block"> [\boldsymbol{T}-\lambda I] \mathbf{v} = \begin{bmatrix}

<math> [\boldsymbol{T}-\lambda I] \mathbf{v}=\left[\begin{array}{cc}

A-\lambda & B \\

C & D-\lambda

\end{arraybmatrix}\right] \mathbf{v} = 0 ,</math>,

by calculating the determinant

<math display="block"> \begin{vmatrix}

<math> \left|\begin{array}{cc}

A-\lambda & B \\

C & D-\lambda

\end{arrayvmatrix}\right| = \lambda^{2} - (A+D) \lambda + 1 = 0 .</math>.

Let <math>m = \frac{(A+D)}{2}</math>, and we have eigenvalues <math>\lambda_{1}, \lambda_{2}=m \pm \sqrt{m^{2}-1}</math>.

According to the values of <math>\lambda_{1}</math> and <math>\lambda_{2}</math>, there are several possible cases. For example:

# A pair of real eigenvalues: <math>r</math> and <math>r^{-1}</math>, where <math>r\neq1</math>. This case represents a magnifier <math> \begin{bmatrix} r & 0 \\ 0 & r^{-1} \end{bmatrix} </math>

# <math>\lambda_{1}=\lambda_{2}=1</math> or <math>\lambda_{1}=\lambda_{2}=-1</math>. This case represents unity matrix (or with an additional coordinate reverter) <math> \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} </math>.

</math>

# <math>\lambda_{1}=\lambda_{2}=1</math> or <math>\lambda_{1}=\lambda_{2}=-1</math>. This case represents unity matrix (or with an additional coordinate reverter) <math> \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}

</math>.

# <math>\lambda_{1}, \lambda_{2}=\pm1</math>. 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 <math>e^{i\theta}</math> and <math>e^{-i\theta}</math>. This case is similar to a separable [[Fractional Fourier transform|Fractional Fourier Transform]].

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| Reflection from a curved mirror

| align="center" | <math> \begin{pmatrix} 1 & 0 \\ -\frac{2}{R_e} & 1 \end{pmatrix} </math>

| <math>R_e = R\cos\theta</math> effective radius of curvature in tangential plane (horizontal direction) <br/>

<math>R_e = R/\cos\theta</math> effective radius of curvature in the sagittal plane (vertical direction)<br/>

''R'' = radius of curvature, R > 0 for concave, valid in the paraxial approximation<br/>

<math>\theta</math> is the mirror angle of incidence in the horizontal plane.

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== Relation between geometrical ray optics and wave optics ==

The theory of [[Linear canonical transformation]] implies the relation between ray transfer matrix ([[geometrical optics]]) and wave optics.<ref>{{Cite journal|last1=Nazarathy|first1=Moshe|last2=Shamir|first2=Joseph|date=1982-03-01|title=First-order optics—a canonical operator representation: lossless systems|url=http://dx.doi.org/10.1364/josa.72.000356|journal=Journal of the Optical Society of America|volume=72|issue=3|pages=356|doi=10.1364/josa.72.000356|issn=0030-3941}}</ref>

{| border="1" cellspacing="0" cellpadding="4" class=wikitable

!Element

!Matrix in geometrical optics

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== Resonator stability ==

RTM analysis is particularly useful when modeling the behavior 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 display="block">\mathbf{M} =\mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} </math>.

:<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 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:

<math display="block"> \mathbf{M} \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix} = \begin{bmatrix}x_2 \\ \theta_2\end{bmatrix} = \lambda \begin{bmatrix} x_1 \\ \theta_1 \end{bmatrix} . </math>

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

which is an [[eigenvalue]] equation:

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

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

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

We proceed to calculate the eigenvalues of the transfer matrix:

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

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

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

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

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

where

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

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

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

<math display="block">\det(\mathbf{M}) = AD - BC = 1 </math>

:<math>\operatorname{det}(\mathbf{M}) = AD - BC = 1 </math>

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

<math display="block"> \lambda^2 - 2g \lambda + 1 = 0 , </math>

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

where

<math display="block"> g \overset{\mathrm{def}}{{}={}} \frac{ \operatorname{tr}(\mathbf{M}) }{ 2 } = 1 - \frac{ d }{ 2 f } </math>

:<math> g \overset{\mathrm{def}}{{}={}} \frac{ \operatorname{tr}(\mathbf{M}) }{ 2 } = 1 - \frac{ d }{ 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 display="block"> \lambda_{\pm} = g \pm \sqrt{g^2 - 1} . </math>

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

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

<math display="block"> \begin{bmatrix}x_N \\ \theta_N \end{bmatrix} = \lambda^N \begin{bmatrix}x_1 \\ \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 \leq 1</math>, and the eigenvalues can be represented by complex numbers:

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

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

with the substitution {{math|1=''g'' = cos(''ϕ'')}}.

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_+ \neq \lambda_-</math>. The input vector can therefore be written as

<math display="block"> c_+ r_+ + c_- r_- , </math>

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

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

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

<math display="block"> \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>

:<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.

== Gaussian beams ==

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| s2cid=123550261 }}</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 display="block"> \frac{1}{q} = \frac{1}{R} - \frac{i\lambda_0}{\pi n w^2} . </math>

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

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

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

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

<math display="block"> q_2 = k (A q_1 + B)</math>

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

and

<math display="block"> 1 = k (C q_1 + D) </math>

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

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

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

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

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

<math display="block"> \frac{ 1 }{ q_2 } = \frac{ C + D/q_1 }{ A + B/q_1 } . </math>

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

=== Example: Free space ===

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

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

and so

: <math display="block">q_2 = \frac{A q_1+B}{C 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.

=== Example: Thin lens ===

Consider a beam traveling through a thin lens with focal length ''f''. The ray transfer matrix is

: <math display="block">\begin{bmatrix}A&B\\C&D\end{bmatrix}=\begin{bmatrix}1&0\\-1/f&1\end{bmatrix}</math>.

and so

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

: <math display="block">\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.