Abstract

The aberrations of a slit aperture are classified in terms of Legendre polynomials instead of the usual power series. An expression is obtained for the Strehl criterion in the presence of all aberrations, provided they are small. Power-series expansions are developed for the point spread function for a single aberration. Finally, the Maréchal balancing theory is studied and shown to lead to the same aberration polynomials.

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  1. B. R. A. Nijhoer, "The Diffraction Theory of Aberrations," Doctoral thesis, University of Delft, 1942. Convenient summaries may be found in: E. H. Linfoot, Recent Advances in Optics (Oxford University Press, Oxford, 1955), or M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959).
  2. G. C. Steward, The Symmetrical Optical System (Cambridge University Press, Cambridge, England, 1928).
  3. A formal proof of the arguments presented below can be found in Appendix A.
  4. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1952), 4th ed., Chap. 15.
  5. The Sn polynomials possess the minimum property: "they have the smallest distance in the mean from zero of all polynomials of the nth degree with the leading coefficient unity." R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience Publishers, Inc., New York, 1953), p. 86.
  6. See Ref. 1.
  7. The integrand could be made more general by including an additional defocusing term; however, as the integral now stands, we are computing a (v) in the plane in which the maximum Strehi criterion occurs.
  8. P. Morse and H. Feshbach, Methods of Theoretical Physics Part II (McGraw-Hill Book Co., Inc., New York, 1953), p. 1575.
  9. A. Maréchal, Rev. Opt. 26, 257 (1947).
  10. See Ref. 1.
  11. F. Zernike and H. C. Brinkman, Proc. Akad. Sci. Amsterdam 38, 161 (1935).

Brinkman, H. C.

F. Zernike and H. C. Brinkman, Proc. Akad. Sci. Amsterdam 38, 161 (1935).

Courant, R.

The Sn polynomials possess the minimum property: "they have the smallest distance in the mean from zero of all polynomials of the nth degree with the leading coefficient unity." R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience Publishers, Inc., New York, 1953), p. 86.

Feshbach, H.

P. Morse and H. Feshbach, Methods of Theoretical Physics Part II (McGraw-Hill Book Co., Inc., New York, 1953), p. 1575.

Hilbert, D.

The Sn polynomials possess the minimum property: "they have the smallest distance in the mean from zero of all polynomials of the nth degree with the leading coefficient unity." R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience Publishers, Inc., New York, 1953), p. 86.

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

Morse, P.

P. Morse and H. Feshbach, Methods of Theoretical Physics Part II (McGraw-Hill Book Co., Inc., New York, 1953), p. 1575.

Nijhoer, B. R. A.

B. R. A. Nijhoer, "The Diffraction Theory of Aberrations," Doctoral thesis, University of Delft, 1942. Convenient summaries may be found in: E. H. Linfoot, Recent Advances in Optics (Oxford University Press, Oxford, 1955), or M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959).

Steward, G. C.

G. C. Steward, The Symmetrical Optical System (Cambridge University Press, Cambridge, England, 1928).

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1952), 4th ed., Chap. 15.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1952), 4th ed., Chap. 15.

Zernike, F.

F. Zernike and H. C. Brinkman, Proc. Akad. Sci. Amsterdam 38, 161 (1935).

Other

B. R. A. Nijhoer, "The Diffraction Theory of Aberrations," Doctoral thesis, University of Delft, 1942. Convenient summaries may be found in: E. H. Linfoot, Recent Advances in Optics (Oxford University Press, Oxford, 1955), or M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959).

G. C. Steward, The Symmetrical Optical System (Cambridge University Press, Cambridge, England, 1928).

A formal proof of the arguments presented below can be found in Appendix A.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1952), 4th ed., Chap. 15.

The Sn polynomials possess the minimum property: "they have the smallest distance in the mean from zero of all polynomials of the nth degree with the leading coefficient unity." R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience Publishers, Inc., New York, 1953), p. 86.

See Ref. 1.

The integrand could be made more general by including an additional defocusing term; however, as the integral now stands, we are computing a (v) in the plane in which the maximum Strehi criterion occurs.

P. Morse and H. Feshbach, Methods of Theoretical Physics Part II (McGraw-Hill Book Co., Inc., New York, 1953), p. 1575.

A. Maréchal, Rev. Opt. 26, 257 (1947).

See Ref. 1.

F. Zernike and H. C. Brinkman, Proc. Akad. Sci. Amsterdam 38, 161 (1935).

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