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Eigenvalues of Ray Transfer Matrix

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A ray transfer matrix can been ragarded as Linear canonical transformation. According to the eigenvalues of the optical system, the system can be classified into several classes[1]. Assume the the ABCD matrix representing a system relates the output ray to the input according to

.

We compute the eigenvlaues 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:

  1. A pair of real eigenvalues: and , where . This case represents a magnifier
  2. or . This case represents unity matrix (or with an additional coordinate reverter) .
  3. . This case occurs if but not only if the system is either a unity operator, a section of free space, or a lens
  4. A pair of two unimodular, complex conjugated eigenvalues and . This case is similar to a separable Fractional Fourier Transformer.

Relation between geometrical ray optics and wave optics

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The theory of Linear canonical transformation implies the relation between ray transfermatrix (geometrical optics) and wave optics[2].

Element Matrix in geometrical optics Operator in wave optics Remarks
Scaling
Quadratic phase factor : wave number
Fresnel free-space-propagation operator : coordinate of the source

: coordinate of the goal

Normalized Fourier-transform operator

Common Decomposition of Ray Transfer Matrix

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There exist infinite ways to decompose a ray transfer matrix into a concatenation of multiple transfer matrix. For example:

  1. .
  1. ^ Bastiaans, Martin J.; Alieva, Tatiana (2007-03-14). "Classification of lossless first-order optical systems and the linear canonical transformation". Journal of the Optical Society of America A. 24 (4): 1053. doi:10.1364/josaa.24.001053. ISSN 1084-7529.
  2. ^ Nazarathy, Moshe; Shamir, Joseph (1982-03-01). "First-order optics—a canonical operator representation: lossless systems". Journal of the Optical Society of America. 72 (3): 356. doi:10.1364/josa.72.000356. ISSN 0030-3941.