Abstract

A matrix equation for the refraction of a thin pencil of rays by a surface of arbitrary shape is derived. The equivalence of this equation and previous nonmatrix equations derived for the same purpose is established. Potential applications of this matrix equation are in the field of ophthalmic lens design. The design of progressive-addition lenses, or just the thorough evaluation of spherotoric lenses, is an example of a task that requires the ability to propagate a thin pencil of rays under very general conditions. The matrix version of the generalized Coddington equations proposed here is a fitting tool for this end.

© 1996 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Kingslake, “Who discovered Coddington equations?” Opt. Photonics News 5, 20–23 (1994).
    [CrossRef]
  2. J. C. Sturm, “Mémoire sur l’optique,”J. Math. Pures Appl. 3, 357 (1838).
  3. A. Gullstrand, “Die reelle optische abbildung,”K. Sven. Ventenskapsakad. Akad. Handl. XLI, 1–139 (1906).
  4. J. B. Keller, H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,”J. Opt. Soc. Am. 40, 48–52 (1950).
    [CrossRef]
  5. J. A. Kneisly, “Local curvature of wavefronts in an optical system,”J. Opt. Soc. Am. 54, 229–235 (1964).
    [CrossRef]
  6. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972), Chaps. IX and X.
  7. O. N. Stavroudis, “A simpler derivation of the formulas of generalized ray tracing,”J. Opt. Soc. Am. 66, 1330–1333 (1976).
    [CrossRef]
  8. D. G. Burkhard, D. L. Shealy, “Simplified formula for the illuminance of an optical system,” Appl. Opt. 20, 897–909 (1981).
    [CrossRef] [PubMed]
  9. O. N. Stavroudis, “Caustic as an expression of the image errors of a lens,” in International Optical Design, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 87–91.
  10. G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]
  11. A long list of references on the work of M. P. Keating, W. F. Harris is given in W. F. Long, “Lens power matrices and the sum of equivalent spheres,” Optom. Vis. Sci. 68, 821–822 (1991). The main ideas behind matrix methods in optometry can be found in M. P. Keating, Geometrical, Physical and Visual Optics (Butterworths, Boston, 1988), Chap. 16.
    [CrossRef]
  12. W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
  13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 15.
  14. H. H. Arsenault, “The rotation of light fans by cylindrical lenses,” Opt. Commun. 31, 275–278 (1979).
    [CrossRef]
  15. S. W. Lee, “Electromagnetic reflection from a conducting surface: geometrical optics solution,”IEEE Trans. Antennas Propag. AP-23, 184–191 (1975).
  16. J. K. Davis, “Geometric optics in ophthalmic lens design,” in Applications of Geometrical Optics II, W. J. Smith, ed., Proc. SPIE39, 65–100 (1973).
  17. A. E. Murray, “Skew astigmatism at toric surfaces, with special reference to spectacle lenses,”J. Opt. Soc. Am. 47, 599–602 (1957).
    [CrossRef]
  18. G. M. Fuerter, “Spline surfaces as means for optical design,” in 1985 International Lens Design Conference, D. T. Moore, W. H. Taylor, eds., Proc. SPIE554, 118–127 (1985).
  19. As is customary, the direction of the normal to the surface is taken in the direction of the incident ray.
  20. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1986), Chap. 4.
  21. Formally, in a one-dimensional expansion o(hn) means terms such that limh→ 0o(hn)/hn= 0.
  22. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), Chap. 4.
  23. At normal incidence we cannot define a plane of incidence, and the interpretation of the Euler angles that we have chosen must be reconsidered. When β= I= 0, one of the two other rotations is evidently redundant. Since we have shown that α1= α, for consistency we must set α′= α= 0 when β′= β= 0. This means that in this case P1= T1(γ) (an orthogonal matrix), where γ is now the angle from the O¯usaxis to the O¯uaxis. Clearly P1′=T1(γ′), and in general γ′≠ γ. But none of these considerations affects the subsequent arguments of our proof; they are included here only for the purpose of giving a clear interpretation to the results of Section 3 in the special case of normal incidence.
  24. See Ref. 20, Chap. 9.
  25. N. V. Yefimov, Quadratic Forms and Matrices (Academic, New York, 1964), Chap. III.
  26. The other diagonal element of matrix U, 1−a31′2=a32′2+a33′2, cannot be zero unless a33′=0.
  27. D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Addison-Wesley, Reading, Mass., 1961). Chap. 2.
  28. We can verify this condition in Eq. (54) with the equation tr Z= z11+ z22= λ1+ λ2. We have already sketched a proof in Section 4 to show that z12= z21.

1994 (1)

R. Kingslake, “Who discovered Coddington equations?” Opt. Photonics News 5, 20–23 (1994).
[CrossRef]

1991 (1)

A long list of references on the work of M. P. Keating, W. F. Harris is given in W. F. Long, “Lens power matrices and the sum of equivalent spheres,” Optom. Vis. Sci. 68, 821–822 (1991). The main ideas behind matrix methods in optometry can be found in M. P. Keating, Geometrical, Physical and Visual Optics (Butterworths, Boston, 1988), Chap. 16.
[CrossRef]

A long list of references on the work of M. P. Keating, W. F. Harris is given in W. F. Long, “Lens power matrices and the sum of equivalent spheres,” Optom. Vis. Sci. 68, 821–822 (1991). The main ideas behind matrix methods in optometry can be found in M. P. Keating, Geometrical, Physical and Visual Optics (Butterworths, Boston, 1988), Chap. 16.
[CrossRef]

1981 (1)

1979 (1)

H. H. Arsenault, “The rotation of light fans by cylindrical lenses,” Opt. Commun. 31, 275–278 (1979).
[CrossRef]

1976 (1)

1975 (1)

S. W. Lee, “Electromagnetic reflection from a conducting surface: geometrical optics solution,”IEEE Trans. Antennas Propag. AP-23, 184–191 (1975).

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

1964 (1)

1957 (1)

1950 (1)

1906 (1)

A. Gullstrand, “Die reelle optische abbildung,”K. Sven. Ventenskapsakad. Akad. Handl. XLI, 1–139 (1906).

1838 (1)

J. C. Sturm, “Mémoire sur l’optique,”J. Math. Pures Appl. 3, 357 (1838).

Arsenault, H. H.

H. H. Arsenault, “The rotation of light fans by cylindrical lenses,” Opt. Commun. 31, 275–278 (1979).
[CrossRef]

Brouwer, W.

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

Burkhard, D. G.

Davis, J. K.

J. K. Davis, “Geometric optics in ophthalmic lens design,” in Applications of Geometrical Optics II, W. J. Smith, ed., Proc. SPIE39, 65–100 (1973).

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Fuerter, G. M.

G. M. Fuerter, “Spline surfaces as means for optical design,” in 1985 International Lens Design Conference, D. T. Moore, W. H. Taylor, eds., Proc. SPIE554, 118–127 (1985).

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), Chap. 4.

Gullstrand, A.

A. Gullstrand, “Die reelle optische abbildung,”K. Sven. Ventenskapsakad. Akad. Handl. XLI, 1–139 (1906).

Harris, W. F.

A long list of references on the work of M. P. Keating, W. F. Harris is given in W. F. Long, “Lens power matrices and the sum of equivalent spheres,” Optom. Vis. Sci. 68, 821–822 (1991). The main ideas behind matrix methods in optometry can be found in M. P. Keating, Geometrical, Physical and Visual Optics (Butterworths, Boston, 1988), Chap. 16.
[CrossRef]

Keating, M. P.

A long list of references on the work of M. P. Keating, W. F. Harris is given in W. F. Long, “Lens power matrices and the sum of equivalent spheres,” Optom. Vis. Sci. 68, 821–822 (1991). The main ideas behind matrix methods in optometry can be found in M. P. Keating, Geometrical, Physical and Visual Optics (Butterworths, Boston, 1988), Chap. 16.
[CrossRef]

Keller, H. B.

Keller, J. B.

Kingslake, R.

R. Kingslake, “Who discovered Coddington equations?” Opt. Photonics News 5, 20–23 (1994).
[CrossRef]

Kneisly, J. A.

Lee, S. W.

S. W. Lee, “Electromagnetic reflection from a conducting surface: geometrical optics solution,”IEEE Trans. Antennas Propag. AP-23, 184–191 (1975).

Long, W. F.

A long list of references on the work of M. P. Keating, W. F. Harris is given in W. F. Long, “Lens power matrices and the sum of equivalent spheres,” Optom. Vis. Sci. 68, 821–822 (1991). The main ideas behind matrix methods in optometry can be found in M. P. Keating, Geometrical, Physical and Visual Optics (Butterworths, Boston, 1988), Chap. 16.
[CrossRef]

Murray, A. E.

Shealy, D. L.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 15.

Stavroudis, O. N.

O. N. Stavroudis, “A simpler derivation of the formulas of generalized ray tracing,”J. Opt. Soc. Am. 66, 1330–1333 (1976).
[CrossRef]

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972), Chaps. IX and X.

O. N. Stavroudis, “Caustic as an expression of the image errors of a lens,” in International Optical Design, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 87–91.

Struik, D. J.

D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Addison-Wesley, Reading, Mass., 1961). Chap. 2.

Sturm, J. C.

J. C. Sturm, “Mémoire sur l’optique,”J. Math. Pures Appl. 3, 357 (1838).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1986), Chap. 4.

Yefimov, N. V.

N. V. Yefimov, Quadratic Forms and Matrices (Academic, New York, 1964), Chap. III.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

S. W. Lee, “Electromagnetic reflection from a conducting surface: geometrical optics solution,”IEEE Trans. Antennas Propag. AP-23, 184–191 (1975).

J. Math. Pures Appl. (1)

J. C. Sturm, “Mémoire sur l’optique,”J. Math. Pures Appl. 3, 357 (1838).

J. Opt. Soc. Am. (4)

K. Sven. Ventenskapsakad. Akad. Handl. (1)

A. Gullstrand, “Die reelle optische abbildung,”K. Sven. Ventenskapsakad. Akad. Handl. XLI, 1–139 (1906).

Opt. Commun. (1)

H. H. Arsenault, “The rotation of light fans by cylindrical lenses,” Opt. Commun. 31, 275–278 (1979).
[CrossRef]

Opt. Photonics News (1)

R. Kingslake, “Who discovered Coddington equations?” Opt. Photonics News 5, 20–23 (1994).
[CrossRef]

Optom. Vis. Sci. (1)

A long list of references on the work of M. P. Keating, W. F. Harris is given in W. F. Long, “Lens power matrices and the sum of equivalent spheres,” Optom. Vis. Sci. 68, 821–822 (1991). The main ideas behind matrix methods in optometry can be found in M. P. Keating, Geometrical, Physical and Visual Optics (Butterworths, Boston, 1988), Chap. 16.
[CrossRef]

Proc. IEEE (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Other (16)

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

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 15.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972), Chaps. IX and X.

O. N. Stavroudis, “Caustic as an expression of the image errors of a lens,” in International Optical Design, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 87–91.

J. K. Davis, “Geometric optics in ophthalmic lens design,” in Applications of Geometrical Optics II, W. J. Smith, ed., Proc. SPIE39, 65–100 (1973).

G. M. Fuerter, “Spline surfaces as means for optical design,” in 1985 International Lens Design Conference, D. T. Moore, W. H. Taylor, eds., Proc. SPIE554, 118–127 (1985).

As is customary, the direction of the normal to the surface is taken in the direction of the incident ray.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1986), Chap. 4.

Formally, in a one-dimensional expansion o(hn) means terms such that limh→ 0o(hn)/hn= 0.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), Chap. 4.

At normal incidence we cannot define a plane of incidence, and the interpretation of the Euler angles that we have chosen must be reconsidered. When β= I= 0, one of the two other rotations is evidently redundant. Since we have shown that α1= α, for consistency we must set α′= α= 0 when β′= β= 0. This means that in this case P1= T1(γ) (an orthogonal matrix), where γ is now the angle from the O¯usaxis to the O¯uaxis. Clearly P1′=T1(γ′), and in general γ′≠ γ. But none of these considerations affects the subsequent arguments of our proof; they are included here only for the purpose of giving a clear interpretation to the results of Section 3 in the special case of normal incidence.

See Ref. 20, Chap. 9.

N. V. Yefimov, Quadratic Forms and Matrices (Academic, New York, 1964), Chap. III.

The other diagonal element of matrix U, 1−a31′2=a32′2+a33′2, cannot be zero unless a33′=0.

D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Addison-Wesley, Reading, Mass., 1961). Chap. 2.

We can verify this condition in Eq. (54) with the equation tr Z= z11+ z22= λ1+ λ2. We have already sketched a proof in Section 4 to show that z12= z21.

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.


Metrics