Abstract

Generalized ray tracing is an algorithm for calculating the geometrical parameters of a wave front in the neighborhood of a traced ray. These calculations are applied, surface by surface, for each traced ray, to an optical system being designed. These calculations determine the two points of contact of each traced ray with the two sheets of a caustic surface. The caustic surfaces are, in fact, aberrated three-dimensional images of object points and therefore contain all information on the geometrical aberrations of the subject lens. Generalized bending is a procedure in which the curvature of a pair of adjacent spherical refracting surfaces, their separation, and the distance to the next succeeding or next preceding surface may be changed so that any paraxial ray is left invariant except at the two affected surfaces. In this study we show that the displacement of a caustic point caused by a generalized bending is in essentially a straight line, that the direction of the displacement is determined by which refracting surfaces are selected, and that the magnitude of the displacement is proportional to the logarithm of the bending parameter. This suggests that caustic surfaces can be used as a merit function in the optical design process, that the merit functions can be calculated by means of generalized ray tracing, and that generalized bending provides an effective means of optimizing the design when included in a feedback loop.

© 1980 Optical Society of America

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  1. H. Coddington, A System of Optics (Simkin and Marshal, London, 1829, 1830), Parts I and II.
  2. A. Gullstrand, "Die Reelle Optische Abbildung," Sven. Vetensk. Handl. 41, 1–119 (1906).
  3. See, for example, J. J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969).
  4. C. S. Hastings, New Methods in Geometrical Optics (MacMillan, New York, 1927).
  5. O. Altrichter and G. Schäfer, "Herleitung der Gullstrandschen Grundgleischungen für Schiefe Strahlenbuschel aus den Hauptkrummungen der Wellenflasche," Optik 13, 241–253 (1956).
  6. J. A. Kneisley, III, "Local Curvatures of Wave Fronts in an Optical System," J. Opt. Soc. Am. 54, 229–235 (1964).
  7. S. C. Parker, "Properties and Applications of Generalized Ray Tracing," Thesis, University of Arizona, 1971; Optical Sciences Center Technical Report 71, University of Arizona, Tucson, Arizona, November 1971 (unpublished).
  8. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York and London, 1972), pp. 313.
  9. O. N. Stavroudis, "A simpler derivation of the formulas for generalized ray tracing," J. Opt. Soc. Am. 66, 1330–1333 (1976).
  10. Dirk J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Addison-Wesley, Reading, Mass., 1961).
  11. Michel Cagnet, Maurice Francis, and Jean Claude Thrierr, Atlas of Optical Phenomena (Pretice-Hall, Englewood Cliffs, N.J., 1962), plate 1.
  12. O. N. Stavroudis and R. C. Fronczek, "Generalized Ray Tracing and the Caustic Surface," Opt. Laser Technol. 10, 185–19l (1978).
  13. David L. Shealy and Donald G. Burkhard, "Analytical Illuminance Calculation in a Multi-Interface Optical System," Opt. Acta 22, 485–501 (1975).
  14. David L. Shealy, "Analytical illuminance and caustic surface calculations in geometrical optics," Appl. Opt. 15; 2588–2596 (1976).
  15. David L. Shealy, "Caustic surface and the Coddington equations," J. Opt. Soc. Am. 66, 76–77 (1976).
  16. L. E. Sutton, "A method for localized variation of the paths of two paraxial rays," Appl. Opt. 2, 1275–1280 (1963).
  17. John Edward Campbell, Continuous Groups (Chelsea, New York, 1966), Chaps. 1 and 2.
  18. J. H. Darnauer, "Properties of Generalized Bending," Thesis, University of Arizona, 1970; Optical Sciences Center Technical Report No. 64, University of Arizona, Tucson, Arizona, 28 February 1971 (unpublished).
  19. R. S. Chang and O. N. Stavroudis, "Third-order approximation of the displacement of a caustic point due to a generalized bending," J. Opt. Soc. Am. 70, 535–538 (1980).

1980

1978

O. N. Stavroudis and R. C. Fronczek, "Generalized Ray Tracing and the Caustic Surface," Opt. Laser Technol. 10, 185–19l (1978).

1976

1975

David L. Shealy and Donald G. Burkhard, "Analytical Illuminance Calculation in a Multi-Interface Optical System," Opt. Acta 22, 485–501 (1975).

1964

1963

1956

O. Altrichter and G. Schäfer, "Herleitung der Gullstrandschen Grundgleischungen für Schiefe Strahlenbuschel aus den Hauptkrummungen der Wellenflasche," Optik 13, 241–253 (1956).

1906

A. Gullstrand, "Die Reelle Optische Abbildung," Sven. Vetensk. Handl. 41, 1–119 (1906).

Altrichter, O.

O. Altrichter and G. Schäfer, "Herleitung der Gullstrandschen Grundgleischungen für Schiefe Strahlenbuschel aus den Hauptkrummungen der Wellenflasche," Optik 13, 241–253 (1956).

Burkhard, Donald G.

David L. Shealy and Donald G. Burkhard, "Analytical Illuminance Calculation in a Multi-Interface Optical System," Opt. Acta 22, 485–501 (1975).

Cagnet, Michel

Michel Cagnet, Maurice Francis, and Jean Claude Thrierr, Atlas of Optical Phenomena (Pretice-Hall, Englewood Cliffs, N.J., 1962), plate 1.

Campbell, John Edward

John Edward Campbell, Continuous Groups (Chelsea, New York, 1966), Chaps. 1 and 2.

Chang, R. S.

Coddington, H.

H. Coddington, A System of Optics (Simkin and Marshal, London, 1829, 1830), Parts I and II.

Darnauer, J. H.

J. H. Darnauer, "Properties of Generalized Bending," Thesis, University of Arizona, 1970; Optical Sciences Center Technical Report No. 64, University of Arizona, Tucson, Arizona, 28 February 1971 (unpublished).

Francis, Maurice

Michel Cagnet, Maurice Francis, and Jean Claude Thrierr, Atlas of Optical Phenomena (Pretice-Hall, Englewood Cliffs, N.J., 1962), plate 1.

Fronczek, R. C.

O. N. Stavroudis and R. C. Fronczek, "Generalized Ray Tracing and the Caustic Surface," Opt. Laser Technol. 10, 185–19l (1978).

Gullstrand, A.

A. Gullstrand, "Die Reelle Optische Abbildung," Sven. Vetensk. Handl. 41, 1–119 (1906).

Hastings, C. S.

C. S. Hastings, New Methods in Geometrical Optics (MacMillan, New York, 1927).

Kneisley, III, J. A.

Parker, S. C.

S. C. Parker, "Properties and Applications of Generalized Ray Tracing," Thesis, University of Arizona, 1971; Optical Sciences Center Technical Report 71, University of Arizona, Tucson, Arizona, November 1971 (unpublished).

Schäfer, G.

O. Altrichter and G. Schäfer, "Herleitung der Gullstrandschen Grundgleischungen für Schiefe Strahlenbuschel aus den Hauptkrummungen der Wellenflasche," Optik 13, 241–253 (1956).

Shealy, David L.

Stavroudis, O. N.

R. S. Chang and O. N. Stavroudis, "Third-order approximation of the displacement of a caustic point due to a generalized bending," J. Opt. Soc. Am. 70, 535–538 (1980).

O. N. Stavroudis and R. C. Fronczek, "Generalized Ray Tracing and the Caustic Surface," Opt. Laser Technol. 10, 185–19l (1978).

O. N. Stavroudis, "A simpler derivation of the formulas for generalized ray tracing," J. Opt. Soc. Am. 66, 1330–1333 (1976).

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York and London, 1972), pp. 313.

Stoker, J. J.

See, for example, J. J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969).

Struik, Dirk J.

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

Sutton, L. E.

Thrierr, Jean Claude

Michel Cagnet, Maurice Francis, and Jean Claude Thrierr, Atlas of Optical Phenomena (Pretice-Hall, Englewood Cliffs, N.J., 1962), plate 1.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Acta

David L. Shealy and Donald G. Burkhard, "Analytical Illuminance Calculation in a Multi-Interface Optical System," Opt. Acta 22, 485–501 (1975).

Opt. Laser Technol.

O. N. Stavroudis and R. C. Fronczek, "Generalized Ray Tracing and the Caustic Surface," Opt. Laser Technol. 10, 185–19l (1978).

Optik

O. Altrichter and G. Schäfer, "Herleitung der Gullstrandschen Grundgleischungen für Schiefe Strahlenbuschel aus den Hauptkrummungen der Wellenflasche," Optik 13, 241–253 (1956).

Sven. Vetensk. Handl.

A. Gullstrand, "Die Reelle Optische Abbildung," Sven. Vetensk. Handl. 41, 1–119 (1906).

Other

See, for example, J. J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969).

C. S. Hastings, New Methods in Geometrical Optics (MacMillan, New York, 1927).

H. Coddington, A System of Optics (Simkin and Marshal, London, 1829, 1830), Parts I and II.

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

Michel Cagnet, Maurice Francis, and Jean Claude Thrierr, Atlas of Optical Phenomena (Pretice-Hall, Englewood Cliffs, N.J., 1962), plate 1.

S. C. Parker, "Properties and Applications of Generalized Ray Tracing," Thesis, University of Arizona, 1971; Optical Sciences Center Technical Report 71, University of Arizona, Tucson, Arizona, November 1971 (unpublished).

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York and London, 1972), pp. 313.

John Edward Campbell, Continuous Groups (Chelsea, New York, 1966), Chaps. 1 and 2.

J. H. Darnauer, "Properties of Generalized Bending," Thesis, University of Arizona, 1970; Optical Sciences Center Technical Report No. 64, University of Arizona, Tucson, Arizona, 28 February 1971 (unpublished).

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