Abstract

The transfer function of an optical system in the presence of off-axis aberrations is evaluated via Gauss quadrature. The transfer function in the presence of third-order coma is studied and the phase of the transfer function is shown to be a smooth function. The variance of the wavefront is obtained for all third- and fifth-order aberrations and its use is illustrated for an actual system along with the transfer function.

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  1. R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).
  2. R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).
  3. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).
  4. R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).
  5. R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).
  6. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538, (1965).
  7. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 881 (1965).
  8. 8 R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964) Chap. 6. This is a corrected reprint of Luenburg's original notes published by Brown University Press, 1944.
  9. The senior author wishes to thank M. Herzberger for a stimulating series of conversations in 1960 on the implications of Luneburg's work with regard to the mixed characteristic and the usual treatment of wavefront aberrations.
  10. B. R. A. Nijboer, "The diffraction theory of aberrations" doctoral thesis, Groningen, 1942.
  11. 11 H. H. Hopkins, Wavs Theory of Aberrations (Oxford University Press, London, 1950).
  12. A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.
  13. See Ref. 8, p. 103.
  14. See Ref. 8, p. 270, as well as M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958) Chaps. 26, 27.
  15. The generalization to nonrotationally symmetric wavefronts has been treated by us and preliminary results communicated at the 1964 Fall Meeting of the Optical Society of America, J. Opt. Soc. Am. 54, 1407A (1964). We hope to publish the final results shortly.
  16. There are a number of ways of obtaining W for an actual optical system. One common method is to trace rays and then curve fit via least squares: E. Marchand and R. Phillips, Appl. Opt. 2, 359 (1963); W. Brouwer, E. L. O'Neill, and A. Walther, Appl. Opt, 2, 1239 (1963). Another procedure developed by us and as yet unpublished utilizes certain interpolating properties of the Zernike polynomials. The calculations in this paper for an actual system (see Sec. 6) employed the Buchdahl aberration coefficients. Our lens-design computing facilities directly calculate all the third- and fifth-order aberrations as well as seventh-order spherical aberration as a matter of routine using an expanded version of Buchdahl's work developed by M. Rimmer of our design department. It was a simple matter to adapt these coefficients to our requirements.
  17. Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).
  18. V. I. Krylov, Approximate Calculation of Integrals (The Macmillan Co., New York, 1962).
  19. A. S. Marathay, Proc. Phys. Soc. (London) 74, 721 (1959).
  20. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).
  21. M. De and B. K. Nath, Optik 15, 739 (1958).
  22. 22 E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).
  23. The senior author wishes to thank E. L. O'Neill for a series of discussions concerning De and Nath's work shortly after its publication. It is largely due to O'Neill that this problem has come to be re-examined by us and independently by DeVelis. DeVelis has kindly informed us (private communication) that he has located the error in their calculations.
  24. A. Maréchal, Rev. Opt. 26, 257 (1947).
  25. We are presently investigating the use of this merit factor in lens design.

Barakat, R.

R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).

R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).

R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).

R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538, (1965).

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 881 (1965).

Bennett, A. H.

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

De, M.

M. De and B. K. Nath, Optik 15, 739 (1958).

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

11 H. H. Hopkins, Wavs Theory of Aberrations (Oxford University Press, London, 1950).

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

Houston, A.

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 881 (1965).

R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538, (1965).

R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).

Jupnik, J.

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

Kopal, Z.

Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

Krylov, V. I.

V. I. Krylov, Approximate Calculation of Integrals (The Macmillan Co., New York, 1962).

Luneburg, R. K.

8 R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964) Chap. 6. This is a corrected reprint of Luenburg's original notes published by Brown University Press, 1944.

Marathay, A. S.

A. S. Marathay, Proc. Phys. Soc. (London) 74, 721 (1959).

Maréchal, A.

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

Morello, M. V.

R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).

Nath, B. K.

M. De and B. K. Nath, Optik 15, 739 (1958).

Nijboer, B. R. A.

B. R. A. Nijboer, "The diffraction theory of aberrations" doctoral thesis, Groningen, 1942.

O'Neill, E. L.

22 E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).

Osterberg, H.

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

Richards, O. W.

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

Other (25)

R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).

R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).

R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538, (1965).

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 881 (1965).

8 R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964) Chap. 6. This is a corrected reprint of Luenburg's original notes published by Brown University Press, 1944.

The senior author wishes to thank M. Herzberger for a stimulating series of conversations in 1960 on the implications of Luneburg's work with regard to the mixed characteristic and the usual treatment of wavefront aberrations.

B. R. A. Nijboer, "The diffraction theory of aberrations" doctoral thesis, Groningen, 1942.

11 H. H. Hopkins, Wavs Theory of Aberrations (Oxford University Press, London, 1950).

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

See Ref. 8, p. 103.

See Ref. 8, p. 270, as well as M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958) Chaps. 26, 27.

The generalization to nonrotationally symmetric wavefronts has been treated by us and preliminary results communicated at the 1964 Fall Meeting of the Optical Society of America, J. Opt. Soc. Am. 54, 1407A (1964). We hope to publish the final results shortly.

There are a number of ways of obtaining W for an actual optical system. One common method is to trace rays and then curve fit via least squares: E. Marchand and R. Phillips, Appl. Opt. 2, 359 (1963); W. Brouwer, E. L. O'Neill, and A. Walther, Appl. Opt, 2, 1239 (1963). Another procedure developed by us and as yet unpublished utilizes certain interpolating properties of the Zernike polynomials. The calculations in this paper for an actual system (see Sec. 6) employed the Buchdahl aberration coefficients. Our lens-design computing facilities directly calculate all the third- and fifth-order aberrations as well as seventh-order spherical aberration as a matter of routine using an expanded version of Buchdahl's work developed by M. Rimmer of our design department. It was a simple matter to adapt these coefficients to our requirements.

Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

V. I. Krylov, Approximate Calculation of Integrals (The Macmillan Co., New York, 1962).

A. S. Marathay, Proc. Phys. Soc. (London) 74, 721 (1959).

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

M. De and B. K. Nath, Optik 15, 739 (1958).

22 E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).

The senior author wishes to thank E. L. O'Neill for a series of discussions concerning De and Nath's work shortly after its publication. It is largely due to O'Neill that this problem has come to be re-examined by us and independently by DeVelis. DeVelis has kindly informed us (private communication) that he has located the error in their calculations.

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

We are presently investigating the use of this merit factor in lens design.

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