Ray transfer matrix analysis: Difference between revisions

Revision as of 14:50, 19 October 2023

Revision as of 17:50, 1 November 2023

Line 74:

# <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^{ij\theta}</math> and <math>e^{-ij\theta}</math>. This case is similar to a separable [[Fractional Fourier transform|Fractional Fourier Transform]].

== Matrices for simple optical components ==

Line 248:

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

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

Line 261:

== 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{ij\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) + ij 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}}: