Ray transfer matrix analysis: Difference between revisions

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[[File:RayTransferMatrixDefinitions.svg|thumb|300px|In ray transfer (ABCD) matrix analysis, an optical element (here, a thick lens) gives a transformation between {{math|(''x''{{sub|1}}, ''θ''{{sub|1}})}} at the input plane and {{math|(''x''{{sub|2}}, ''θ''{{sub|2}})}} 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 {{mvar|x}} and {{mvar|y}} (below we only consider the {{mvar|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 {{math|''x''{{sub|1}}}} from the optical axis, traveling in a direction that makes an angle {{math|''θ''{{sub|1}}}} with the optical axis. After propagation to the output plane that ray is found at a distance {{math|''x''{{sub|2}}}} from the optical axis and at an angle {{math|''θ''{{sub|2}}}} with respect to it. {{math|''n''{{sub|1}}}} and {{math|''n''{{sub|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

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<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, {{mvar|λ{{sup|N}}}} must not grow without limit. Suppose {{nowrap|<math> g^2 > 1</math>.}} Then both eigenvalues are real. Since {{nowrap|<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, {{nowrap|<math> g^2 \leq 1</math>,}} and the eigenvalues can be represented by complex numbers:

<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'')}}.

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After {{mvar|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>

which represents a [[periodic function]].

== Gaussian beams ==

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== References ==

{{refbegin|30em|indent=yes}}

* {{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 |journal= Journal of the Optical Society of America A |volume= 24 |issue= 4 |pages= 1053–1062 |doi= 10.1364/josaa.24.001053 |pmid= 17361291 |bibcode= 2007JOSAA..24.1053B |url= https://eprints.ucm.es/id/eprint/27620/1/AlievaT31libre.pdf }}

* {{cite book |last= Brouwer |first= W. |date= 1964 |title= Matrix Methods in Optical Instrument Design |publisher= Benjamin |___location= New York |bibcode= 1964mmoi.book.....B }}

* {{cite book |last= Duarte |first= F. J. |date= 2003 |title= Tunable Laser Optics |publisher= Elsevier-Academic |___location= New York |author-link= F. J. Duarte }}

* {{cite book |last1= Gerrard |first1= A. |last2= Burch |first2= J. M. |date= 1994 |title= Introduction to Matrix Methods in Optics |edition= Dover |orig-year= 1975 |publisher= Dover Publications |url= https://archive.org/details/introductiontoma0000gerr_u8i1/ |url-access= registration |isbn= 0-486-68044-4 }}

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* {{cite journal |last1= Nazarathy |first1= Moshe |last2= Shamir |first2= Joseph |date= 1982-03-01 |title= First-order optics—a canonical operator representation: lossless systems |journal= Journal of the Optical Society of America |volume= 72 |issue= 3 |pages= 356 |doi= 10.1364/josa.72.000356 }}

* {{cite conference |last= Nussbaum |first= Allen |date= 1 March 1992 |title= Modernizing the Teaching of Advanced Geometric Optics |publisher= [[SPIE]] |conference= Education in Optics, 1991 |book-title= Proc. SPIE 1603 |___location= Leningrad, Russian Federation |pages= 389–400 |url= http://spie.org/ETOP/1991/389_1.pdf }}

* {{cite journal |last1= Rashidian Vaziri |first1= M. R. |last2= Hajiesmaeilbaigi |first2= F. |last3= Maleki |first3= |date= 2013 |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 |url= https://iopscience.iop.org/article/10.1088/2040-8978/15/3/035202/pdf |url-access= subscription }}

* {{cite book |last= Siegman |first= Anthony E. |date= 1986 |title= Lasers |publisher= University Science Books |___location= Mill Valley, California |author-link= Anthony E. Siegman }}

* {{cite book |last= Wollnik |first= H. |date= 1987 |title= Optics of Charged Particles |publisher= Academic |___location= New York }}