Abstract

A generalized ray-transfer matrix for describing the action of an optical element having an arbitrary wavefront aberration is obtained. In this generalized ray-transfer matrix, the action of the aberrated optical element is represented by the product of radial ray-transfer matrices and tangential ray-transfer matrices. The refraction angle of an incident ray is calculated from the gradient of the wavefront aberration at the point of incidence, and the radial and tangential ray-transfer matrices directly use the gradient as a matrix component. To show the validity of the generalized ray-transfer matrix, intercept heights from a spot diagram are calculated with the generalized ray-transfer matrix and compared with those calculated with commercial ray-tracing software.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 1991), p. 26.
  2. A. Tovar and L. Casperson, J. Opt. Soc. Am. A 14, 882 (1997).
    [CrossRef]
  3. J. Bourderionnet, A. Brignon, J. Huignard, and R. Frey, Opt. Commun. 204, 299 (2002).
    [CrossRef]
  4. J. Blows, G. Forbes, and J. Dawes, Opt. Commun. 186, 111 (2000).
    [CrossRef]
  5. A. Guirao, D. Williams, and I. Cox, J. Opt. Soc. Am. A 18, 1003 (2001).
    [CrossRef]
  6. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), p. 98.

2002

J. Bourderionnet, A. Brignon, J. Huignard, and R. Frey, Opt. Commun. 204, 299 (2002).
[CrossRef]

2001

2000

J. Blows, G. Forbes, and J. Dawes, Opt. Commun. 186, 111 (2000).
[CrossRef]

1997

Blows, J.

J. Blows, G. Forbes, and J. Dawes, Opt. Commun. 186, 111 (2000).
[CrossRef]

Bourderionnet, J.

J. Bourderionnet, A. Brignon, J. Huignard, and R. Frey, Opt. Commun. 204, 299 (2002).
[CrossRef]

Brignon, A.

J. Bourderionnet, A. Brignon, J. Huignard, and R. Frey, Opt. Commun. 204, 299 (2002).
[CrossRef]

Casperson, L.

Cox, I.

Dawes, J.

J. Blows, G. Forbes, and J. Dawes, Opt. Commun. 186, 111 (2000).
[CrossRef]

Forbes, G.

J. Blows, G. Forbes, and J. Dawes, Opt. Commun. 186, 111 (2000).
[CrossRef]

Frey, R.

J. Bourderionnet, A. Brignon, J. Huignard, and R. Frey, Opt. Commun. 204, 299 (2002).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), p. 98.

Guirao, A.

Huignard, J.

J. Bourderionnet, A. Brignon, J. Huignard, and R. Frey, Opt. Commun. 204, 299 (2002).
[CrossRef]

Saleh, B.

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 1991), p. 26.

Teich, M.

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 1991), p. 26.

Tovar, A.

Williams, D.

J. Opt. Soc. Am. A

Opt. Commun.

J. Bourderionnet, A. Brignon, J. Huignard, and R. Frey, Opt. Commun. 204, 299 (2002).
[CrossRef]

J. Blows, G. Forbes, and J. Dawes, Opt. Commun. 186, 111 (2000).
[CrossRef]

Other

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), p. 98.

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 1991), p. 26.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Beam propagation before and after an optical element having an arbitrary wavefront aberration. S i ( i = 1 , 2 ) are the propagation vectors before and after an optical element.

Fig. 2
Fig. 2

Intercept heights calculated with the generalized ray-transfer matrix (open squares) and the ray-tracing software (open diamonds).

Tables (1)

Tables Icon

Table 1 Radial ( R ) and Tangential ( T ) Ray-Transfer Matrices for Optical Elements Having Primary Wavefront Aberrations

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

S ( r ) = z ̂ r ̂ W ( r ) r ϕ ̂ 1 r W ( r ) ϕ
r 2 = r 1 , α = θ 2 θ 1 = tan 1 ( W 2 ( r ) r r = r 1 ) tan 1 ( W 1 ( r ) r r = r 1 ) ,
ϕ 2 = ϕ 1 , β = ϑ 2 ϑ 1 = tan 1 ( W 2 ( r ) r ϕ r = r 1 ) tan 1 ( W 1 ( r ) r ϕ r = r 1 ) .
[ r 2 θ 2 ] = [ 1 0 1 r 1 W ( r ) r r = r 1 1 ] [ r 1 θ 1 ] = M radial [ r 1 θ 1 ] ,
[ ϕ 2 ϑ 2 ] = [ 1 0 1 ϕ 1 W ( r ) r ϕ r = r 1 1 ] [ ϕ 1 ϑ 1 ] = M tang [ ϕ 1 ϑ 1 ] .
W ( r , ϕ ) = n , m a n m Z n m ( r , ϕ ) ,
n = 1 , 2 , , and m = n , n 2 , , n .
W ( r , ϕ ) r = n , m a n m Z n m ( r , ϕ ) r ,
W ( r , ϕ ) ϕ = n , m a n m Z n m ( r , ϕ ) ϕ .
M radial = n , m [ 1 0 a n m 1 r 1 Z n m ( r , ϕ ) r r = r 1 1 ] ,
M tang = n , m [ 1 0 a n m 1 ϕ 1 Z n m ( r , ϕ ) r r r = r 1 1 ] .
M radial = [ 1 0 1 f 1 ] , M tang = [ 1 0 0 1 ] = I .

Metrics