Abstract

The matrix method of Gaussian optical ray trace analysis has been extended to facilitate the analysis of skew rays in mixed systems of spherical and orthogonal cylindrical lenses. The position and angle of the ray are specified by a Cartesian-altazimuth representation. Lenses and lens systems are analyzed to demonstrate the method.

© 1984 Optical Society of America

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Errata

Alfred E. Attard, "Matrix optical analysis of skew rays in mixed systems of spherical and orthogonal cylindrical lenses: errata," Appl. Opt. 23, 3740_2-3740 (1984)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-23-21-3740_2

References

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  1. A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976).
  2. W. Brouwer, Matrix Methods in Optical Instrument Design (BenjaminNew York, 1964).
  3. M. V. Klein, Optics (Wiley, New York, 1970).
  4. J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice-Hall, Englewood Cliffs, N.J., 1972).
  5. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1975).
  6. G. W. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975).
  7. W. T. Rhodes, “Acousto-Optic Signal Processing: Convolution and Correlation,” Proc. IEEE, 69, 65 (1981).
    [CrossRef]
  8. W. G. Peck, “Automated Lens Design,” in Applied Optics and Optical Engineering, Vol. 8, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1980).
    [CrossRef]

1981 (1)

W. T. Rhodes, “Acousto-Optic Signal Processing: Convolution and Correlation,” Proc. IEEE, 69, 65 (1981).
[CrossRef]

Brouwer, W.

W. Brouwer, Matrix Methods in Optical Instrument Design (BenjaminNew York, 1964).

Fowles, G. W.

G. W. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975).

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970).

Meyer-Arendt, J. R.

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Nussbaum, A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Peck, W. G.

W. G. Peck, “Automated Lens Design,” in Applied Optics and Optical Engineering, Vol. 8, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1980).
[CrossRef]

Phillips, R. A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Rhodes, W. T.

W. T. Rhodes, “Acousto-Optic Signal Processing: Convolution and Correlation,” Proc. IEEE, 69, 65 (1981).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1975).

Proc. IEEE (1)

W. T. Rhodes, “Acousto-Optic Signal Processing: Convolution and Correlation,” Proc. IEEE, 69, 65 (1981).
[CrossRef]

Other (7)

W. G. Peck, “Automated Lens Design,” in Applied Optics and Optical Engineering, Vol. 8, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1980).
[CrossRef]

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976).

W. Brouwer, Matrix Methods in Optical Instrument Design (BenjaminNew York, 1964).

M. V. Klein, Optics (Wiley, New York, 1970).

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice-Hall, Englewood Cliffs, N.J., 1972).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1975).

G. W. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975).

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Figures (2)

Fig. 1
Fig. 1

System of two orthogonal cylindrical lenses is shown. O is the object at distance T from the first lens whose power is P. The distance between the first lens and the second lens, whose power is P′, is L. The distance from the second lens to the image I is K.

Fig. 2
Fig. 2

System of a cylindrical lens and a spherical lens. The arrangement is similar to that in Fig. 1, except that the second cylindrical lens is replaced by a spherical lens.

Equations (22)

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n 1 n ^ × i ^ = n 2 n ^ × r ^ ,
i ¯ = J ^ 1 sin a + J ^ 2 cos a sin b + J ^ 3 cos a cos b , r ¯ = J ^ 1 sin a + J ^ 2 cos a sin b + J ^ 3 cos a cos b , ns ¯ = J ^ 1 ( x / R ) + J ^ 2 ( y / R ) - J ¯ 3 1 - ( x 2 + y 2 ) / R 2 , ncx ¯ = J ^ 2 ( y / R ) - J ^ 3 1 - y 2 / R 2 . ncy ¯ = J ^ 1 ( x / R ) - J ^ 3 1 - x 2 / R 2 .
i ¯ = J ^ 1 a + J ^ 2 b + J ^ 3 . r ¯ = J ^ 1 a + J ^ 2 b + J ^ 3 , ns ¯ = J ^ 1 ( x / R ) + J ^ 2 ( y / R ) - J ^ 3 , ncx = J ^ 2 ( y / R ) - J ^ 3 , ncy ¯ = J ^ 1 ( x / R ) - J ^ 3 .
| X Y a b | = | A 1 A 2 A 3 A 4 B 1 B 2 B 3 B 4 C 1 C 2 C 3 C 4 D 1 D 2 D 3 D 4 | · | X Y a b |
T = | 1 0 t 0 0 1 0 t 0 0 1 0 0 0 0 1 | .
R ( s ) = | 1 0 0 0 0 1 0 0 - P / n 2 0 n 1 / n 2 0 0 - P / n 2 0 n 1 / n 2 | ,
R ( c x ) = | 1 0 0 0 0 1 0 0 0 0 n 1 / n 2 0 0 - P / n 2 0 n 1 / n 2 | ,
R ( c y ) = | 1 0 0 0 0 1 0 0 - P / n 2 0 n 1 / n 2 0 0 0 0 n 1 / n 2 | ,
R ( s ) = | 1 - s P 1 / n 0 s / n 0 0 1 - s P 1 / n 0 s / n - P 0 1 - s P 2 / n 0 0 - P 0 1 - s P 2 / n | ,
R ( c x ) = | 1 0 s / n 0 0 1 - s P 1 / n 0 s / n 0 0 1 0 0 - P 0 1 - s p 2 / n | .
R ( c y ) = | 1 - s P 1 / n 0 s / n 0 0 1 0 s / n - P 0 1 - s P 2 / n 0 0 0 0 0 | .
A 3 = A 4 = B 3 = B 4 = 0 , A 1 = B 2 = 1.
H 1 = - s P 2 / n P , H 2 = s P 1 / n P .
R ( s ) = | 1 0 0 0 0 1 0 0 - P 0 1 0 0 - P 0 1 | .
R ( c x ) = | 1 0 0 0 0 1 0 0 0 0 1 0 0 - P 0 1 | .
R ( c y ) = | 1 0 0 0 0 1 0 0 - P 0 1 0 0 0 0 1 | .
X = X ( 1 - P K ) + a ( T + L + K - P K T - P K L ) , Y = Y ( 1 - P L - P K ) + b ( T + L + K - P L T - P K T ) , a = X ( - P ) + a ( 1 - P L - P T ) , b = Y ( - P ) + b ( 1 - P T )
M x / M y = ( P K 1 - 1 ) / ( P K 2 + P L - 1 ) .
X = X ( 1 - P K ) + a [ T + L + K ( 1 - P T - P L ) ] , Y = Y ( 1 - P L - P K - P K + P P K L ) + b [ T + L - P L T + K ( 1 - P T - P L - P T + P P L T ) ] , a = X ( - P ) + a ( 1 - P T - P L ) , b = Y ( - P - P + P P L ) + b [ - P T + ( 1 - P T ) ( 1 - P L ) ] .
n 1 ns ¯ × i = n 2 ns ¯ × r ¯ = | J 1 ¯ J 2 ¯ J ¯ 3 x / R y / R - 1 a b 1 J ¯ 1 J ¯ 2 J ¯ 3 x / R y / R - 1 a b 1 | .
A 1 = B 2 = 1 , A 2 = A 3 = A 4 = B 1 = B 3 = B 4 = 0 , C 1 = D 2 = - ( n 2 - n 1 ) / R n 2 , C 3 = D 4 = n 1 / n 2 , C 2 = C 4 = D 1 = D 3 = 0.
R ( 1 ) = T ( s ) = R ( 2 ) = | 1 0 0 0 0 1 0 0 - P 1 / n 0 1 / n 0 0 - P 1 / n 0 1 / n 1 0 s 0 0 1 0 s 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 - P 2 0 n 0 0 - P 2 0 n | ,

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