Abstract

The use of Hamilton’s mixed and angle characteristic functions in wave and diffraction aberration calculations is theoretically examined. The relation of Hamilton's mixed and angle characteristic functions to a new wave-aberration function is shown. This aberration function is to be used in the Luneburg-Debye diffraction integrals. The mixed and angle characteristic functions as utilized in diffraction theory via the Luneburg-Debye integrals are examined. The mathematical and physical approximations are discussed. The use of the Luneburg-Debye diffraction integrals for image evaluation is examined and some difficulties are pointed out. It is concluded that the above methods should not be used for microwave and radio-frequency imaging systems; they are of limited validity for optical imaging systems.

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  1. W. Hamilton, Mathemotical Papers, A. Conway and J. Synge, Eds. (Cambridge University Press, London, 1931), Vol. 1.
  2. R. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964).
  3. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, N. Y., 1959).
  4. J. Synge, Geometrical Optics (Cambridge University Press, London, 1937).
  5. J. Synge, J. Opt. Soc. Am. 27, 75, 138 (1937).
  6. M. Herzberger, J. Opt. Soc. Am. 27, 133 (1937).
  7. A. Sommerfeld and J. Runge, Ann. Phys. (Leipzig) 35, 277 (1911).
  8. N. Arley, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 22, No. 8, 1 (1945).
  9. M. Kline, in Proceedings of a Symposiuml on Electromagnetic Waves, R. Langer, Ed. (University of Wisconsin Press, Madison, Wis., 1962), p. 3.
  10. M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, Inc. and John Wiley & Sons, Inc., New York, 1965).
  11. F. Zernike, Pieter Zeeman (Martinus Nijhoff, Hague, The Netherlands, 1935), p. 323.
  12. H. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
  13. Lord Rayleigh, Collected Scientific Papers (Cambridge University Press, London, 1912), Vol. 5, (1902–1910), p. 456.
  14. M. Herzberger, J. Opt. Soc. Am. 53, 661 (1963).
  15. M. Herzberger and D. Wilder, J. Opt. Soc. Soc. Am. 54, 773 (1964).
  16. M. Born, Optik (Springer-Verlag, Berlin, 1933).
  17. P. Debye, Ann. Physik 30, 775 (1909).
  18. J. Picht, Ann. Physik 77, 685 (1925).
  19. J. Focke, Opt. Acta 3, 110 (1956).
  20. E. Wolf, Proc. Roy. Soc. (London) 253A, 349 (1959).
  21. (a) B. Richards, in Astrontomtcal Optics and Related Subjects, Ed. Z. Kopal (North-Holland Publishing Co., Amsterdam 1956); (b) B. Richards and E. Wolf, Proc. Roy. Soc. (London) 253A, 285 (1959); (c) H. Osterberg and J. Wilkins, J. Opt. Soc. Am. 39, 553 (1949); (d) H. Osterberg, in Phase Microscopy by A. H. Bennett et al. (John Wiley & Sons, New York, N. Y., 1951); (e) H. Osterberg and R. McDonald, in Optical Imzage Evaluation (Natl. Bur. Std. Circ. 526, 1954).
  22. F. Kottler, Ann. Physik 71, 457 (1923).
  23. (a) J. Stratton and L. Chu, Phys. Rev. 56, 99 (1939); (b) J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, New York, N. Y., 1941).
  24. H. Severin, Z. Physik 129, 426 (1951) and Nuovo Cimento Ser. 9, Suppl. Al. 9, 381 (1952).
  25. J. Vasseur, Ann. Phys. (Paris) 1, 506 (1952).
  26. C. Bouwkamp, Rept. Prog. Phys. 17, 35 (1954).
  27. (a) A. Sommerfeld, Lectures in Theoretical Physics (Academic Press Inc., New York, 1954), Vol. IV; (b) D. Jones, Proc. Camb. Phil. Soc. 48, 733 (1952).
  28. H. Weyl, Ann. Phys. (Leipzig) 60, 481 (1919).
  29. For no aberration or phase retardation due to apodization, etc., Φ = 0. It does not follow that W = 0 or is a constant, as is often assumed. The function W is not an aberration function but a characteristic function. As previously defined W = V - (αX1 + βY1 + γZ1,) where X1, Y1, Z1 define a point such as P1, Fig. 2. For perfect imaging where the origin of coordinates is not the perfect image point P0, it can be seen that X1, Y1, Z1, and V are constants (Fermat’s principle); but the ray components α, β and γ are variables and will be different for each ray; i.e., ∂W/∂α = -X1, etc. The exponent of the integrand of Eqs. (8) and (9) can be written as W (Q1′) + αX0 + βY0 + γZ0 + α(X1′ - X0) + β(Y1′ - Y0) + γ(Z1′ - Z0). What is constant in the above equation for perfect imaging is the sum of the first four terms. They can be identified as V(P0). Hence, in Eqs. (8) and (9), when there is perfect imaging, the last three terms of the preceding equation above must remain. If the perfect imaging is on axis and only the image plane Z0 = O is considered, then W(Q1′) can be suppressed since X0 = Y0 = Z0 = 0.
  30. P. Clemmowv, Proc. Roy. Soc. (London) 205A, 286 (1951)
  31. R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).
  32. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1142 (1965).
  33. W. Brouwer, E. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).
  34. W. Charman, J. Opt. Soc. Am. 53, 410, 415 (1963).
  35. B. Watrasiewicz, Optica Acta 12, 167 (1965).
  36. H. Osterberg and G. Pride, J. Opt. Soc. Am. 40, 14 (1950); H. Osterberg and L. Smith, J. Opt. Soc. Am. 50, 362 (1960); L. Smith, ibid., p. 369; W. Welford, Optics in Metrology (Pergamon Press, London, 1960).
  37. J. Focke, Opt. Acta 4, 124 (1957).
  38. R. Luneburg, Lecture Notes (New York University, 1947, 1948).
  39. G. Lansraux, Can. J. Phys. 40, 1101 (1962).
  40. R. Tremblay and A. Boivin, Appl. Opt. 5, 251 (1966).
  41. M. Bachynski and G. Bekefi, J. Opt. Soc. Am. 47, 428 (1957).
  42. (a) G. Farnell, J. Opt. Soc. Am. 48, 643 (1958); (b) Can. J. Phys. 35, 777 (1957); (c) Can. J. Phys. 36, 935 (1958).
  43. The Rayleigh equation evaluated is from H. Osterberg and L. Smith, J. Opt. Soc. Am. 51, 1050 (1961); the Maggi-Rubinowvicz-Kirchhoff equation is from Ref. 42 (a).
  44. A. Schell, IEEE Trans. Antennas Propagation AP-11, 428 (1963).
  45. M. Gravel and A. Boivin, J. Opt. Soc. Am. 56, 1438A (1966) and private communication.

Arley, N.

N. Arley, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 22, No. 8, 1 (1945).

Bachynski, M.

M. Bachynski and G. Bekefi, J. Opt. Soc. Am. 47, 428 (1957).

Barakat, R.

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

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1142 (1965).

Bekefi, G.

M. Bachynski and G. Bekefi, J. Opt. Soc. Am. 47, 428 (1957).

Beutler, H.

H. Beutler, J. Opt. Soc. Am. 35, 311 (1945).

Boivin, A.

R. Tremblay and A. Boivin, Appl. Opt. 5, 251 (1966).

M. Gravel and A. Boivin, J. Opt. Soc. Am. 56, 1438A (1966) and private communication.

Born, M.

M. Born, Optik (Springer-Verlag, Berlin, 1933).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, N. Y., 1959).

Bouwkamp, C.

C. Bouwkamp, Rept. Prog. Phys. 17, 35 (1954).

Brouwer, W.

W. Brouwer, E. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).

Charman, W.

W. Charman, J. Opt. Soc. Am. 53, 410, 415 (1963).

Chu, L.

(a) J. Stratton and L. Chu, Phys. Rev. 56, 99 (1939); (b) J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, New York, N. Y., 1941).

Clemmowv, P.

P. Clemmowv, Proc. Roy. Soc. (London) 205A, 286 (1951)

Debye, P.

P. Debye, Ann. Physik 30, 775 (1909).

Farnell, G.

(a) G. Farnell, J. Opt. Soc. Am. 48, 643 (1958); (b) Can. J. Phys. 35, 777 (1957); (c) Can. J. Phys. 36, 935 (1958).

Focke, J.

J. Focke, Opt. Acta 3, 110 (1956).

J. Focke, Opt. Acta 4, 124 (1957).

Gravel, M.

M. Gravel and A. Boivin, J. Opt. Soc. Am. 56, 1438A (1966) and private communication.

Hamilton, W.

W. Hamilton, Mathemotical Papers, A. Conway and J. Synge, Eds. (Cambridge University Press, London, 1931), Vol. 1.

Herzberger, M.

M. Herzberger, J. Opt. Soc. Am. 27, 133 (1937).

M. Herzberger, J. Opt. Soc. Am. 53, 661 (1963).

M. Herzberger and D. Wilder, J. Opt. Soc. Soc. Am. 54, 773 (1964).

Houston, A.

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1142 (1965).

Kay, I.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, Inc. and John Wiley & Sons, Inc., New York, 1965).

Kline, M.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, Inc. and John Wiley & Sons, Inc., New York, 1965).

M. Kline, in Proceedings of a Symposiuml on Electromagnetic Waves, R. Langer, Ed. (University of Wisconsin Press, Madison, Wis., 1962), p. 3.

Kottler, F.

F. Kottler, Ann. Physik 71, 457 (1923).

Lansraux, G.

G. Lansraux, Can. J. Phys. 40, 1101 (1962).

Luneburg, R.

R. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964).

R. Luneburg, Lecture Notes (New York University, 1947, 1948).

O’Neill, E.

W. Brouwer, E. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).

Osterberg, H.

H. Osterberg and G. Pride, J. Opt. Soc. Am. 40, 14 (1950); H. Osterberg and L. Smith, J. Opt. Soc. Am. 50, 362 (1960); L. Smith, ibid., p. 369; W. Welford, Optics in Metrology (Pergamon Press, London, 1960).

The Rayleigh equation evaluated is from H. Osterberg and L. Smith, J. Opt. Soc. Am. 51, 1050 (1961); the Maggi-Rubinowvicz-Kirchhoff equation is from Ref. 42 (a).

Picht, J.

J. Picht, Ann. Physik 77, 685 (1925).

Pride, G.

H. Osterberg and G. Pride, J. Opt. Soc. Am. 40, 14 (1950); H. Osterberg and L. Smith, J. Opt. Soc. Am. 50, 362 (1960); L. Smith, ibid., p. 369; W. Welford, Optics in Metrology (Pergamon Press, London, 1960).

Rayleigh, Lord

Lord Rayleigh, Collected Scientific Papers (Cambridge University Press, London, 1912), Vol. 5, (1902–1910), p. 456.

Richards, B.

(a) B. Richards, in Astrontomtcal Optics and Related Subjects, Ed. Z. Kopal (North-Holland Publishing Co., Amsterdam 1956); (b) B. Richards and E. Wolf, Proc. Roy. Soc. (London) 253A, 285 (1959); (c) H. Osterberg and J. Wilkins, J. Opt. Soc. Am. 39, 553 (1949); (d) H. Osterberg, in Phase Microscopy by A. H. Bennett et al. (John Wiley & Sons, New York, N. Y., 1951); (e) H. Osterberg and R. McDonald, in Optical Imzage Evaluation (Natl. Bur. Std. Circ. 526, 1954).

Runge, J.

A. Sommerfeld and J. Runge, Ann. Phys. (Leipzig) 35, 277 (1911).

Schell, A.

A. Schell, IEEE Trans. Antennas Propagation AP-11, 428 (1963).

Severin, H.

H. Severin, Z. Physik 129, 426 (1951) and Nuovo Cimento Ser. 9, Suppl. Al. 9, 381 (1952).

Smith, L.

The Rayleigh equation evaluated is from H. Osterberg and L. Smith, J. Opt. Soc. Am. 51, 1050 (1961); the Maggi-Rubinowvicz-Kirchhoff equation is from Ref. 42 (a).

Sommerfeld, A.

A. Sommerfeld and J. Runge, Ann. Phys. (Leipzig) 35, 277 (1911).

(a) A. Sommerfeld, Lectures in Theoretical Physics (Academic Press Inc., New York, 1954), Vol. IV; (b) D. Jones, Proc. Camb. Phil. Soc. 48, 733 (1952).

Stratton, J.

(a) J. Stratton and L. Chu, Phys. Rev. 56, 99 (1939); (b) J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, New York, N. Y., 1941).

Synge, J.

J. Synge, Geometrical Optics (Cambridge University Press, London, 1937).

J. Synge, J. Opt. Soc. Am. 27, 75, 138 (1937).

Tremblay, R.

R. Tremblay and A. Boivin, Appl. Opt. 5, 251 (1966).

Vasseur, J.

J. Vasseur, Ann. Phys. (Paris) 1, 506 (1952).

Walther, A.

W. Brouwer, E. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).

Watrasiewicz, B.

B. Watrasiewicz, Optica Acta 12, 167 (1965).

Weyl, H.

H. Weyl, Ann. Phys. (Leipzig) 60, 481 (1919).

Wilder, D.

M. Herzberger and D. Wilder, J. Opt. Soc. Soc. Am. 54, 773 (1964).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, N. Y., 1959).

E. Wolf, Proc. Roy. Soc. (London) 253A, 349 (1959).

Zernike, F.

F. Zernike, Pieter Zeeman (Martinus Nijhoff, Hague, The Netherlands, 1935), p. 323.

Other (45)

W. Hamilton, Mathemotical Papers, A. Conway and J. Synge, Eds. (Cambridge University Press, London, 1931), Vol. 1.

R. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, N. Y., 1959).

J. Synge, Geometrical Optics (Cambridge University Press, London, 1937).

J. Synge, J. Opt. Soc. Am. 27, 75, 138 (1937).

M. Herzberger, J. Opt. Soc. Am. 27, 133 (1937).

A. Sommerfeld and J. Runge, Ann. Phys. (Leipzig) 35, 277 (1911).

N. Arley, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 22, No. 8, 1 (1945).

M. Kline, in Proceedings of a Symposiuml on Electromagnetic Waves, R. Langer, Ed. (University of Wisconsin Press, Madison, Wis., 1962), p. 3.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, Inc. and John Wiley & Sons, Inc., New York, 1965).

F. Zernike, Pieter Zeeman (Martinus Nijhoff, Hague, The Netherlands, 1935), p. 323.

H. Beutler, J. Opt. Soc. Am. 35, 311 (1945).

Lord Rayleigh, Collected Scientific Papers (Cambridge University Press, London, 1912), Vol. 5, (1902–1910), p. 456.

M. Herzberger, J. Opt. Soc. Am. 53, 661 (1963).

M. Herzberger and D. Wilder, J. Opt. Soc. Soc. Am. 54, 773 (1964).

M. Born, Optik (Springer-Verlag, Berlin, 1933).

P. Debye, Ann. Physik 30, 775 (1909).

J. Picht, Ann. Physik 77, 685 (1925).

J. Focke, Opt. Acta 3, 110 (1956).

E. Wolf, Proc. Roy. Soc. (London) 253A, 349 (1959).

(a) B. Richards, in Astrontomtcal Optics and Related Subjects, Ed. Z. Kopal (North-Holland Publishing Co., Amsterdam 1956); (b) B. Richards and E. Wolf, Proc. Roy. Soc. (London) 253A, 285 (1959); (c) H. Osterberg and J. Wilkins, J. Opt. Soc. Am. 39, 553 (1949); (d) H. Osterberg, in Phase Microscopy by A. H. Bennett et al. (John Wiley & Sons, New York, N. Y., 1951); (e) H. Osterberg and R. McDonald, in Optical Imzage Evaluation (Natl. Bur. Std. Circ. 526, 1954).

F. Kottler, Ann. Physik 71, 457 (1923).

(a) J. Stratton and L. Chu, Phys. Rev. 56, 99 (1939); (b) J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, New York, N. Y., 1941).

H. Severin, Z. Physik 129, 426 (1951) and Nuovo Cimento Ser. 9, Suppl. Al. 9, 381 (1952).

J. Vasseur, Ann. Phys. (Paris) 1, 506 (1952).

C. Bouwkamp, Rept. Prog. Phys. 17, 35 (1954).

(a) A. Sommerfeld, Lectures in Theoretical Physics (Academic Press Inc., New York, 1954), Vol. IV; (b) D. Jones, Proc. Camb. Phil. Soc. 48, 733 (1952).

H. Weyl, Ann. Phys. (Leipzig) 60, 481 (1919).

For no aberration or phase retardation due to apodization, etc., Φ = 0. It does not follow that W = 0 or is a constant, as is often assumed. The function W is not an aberration function but a characteristic function. As previously defined W = V - (αX1 + βY1 + γZ1,) where X1, Y1, Z1 define a point such as P1, Fig. 2. For perfect imaging where the origin of coordinates is not the perfect image point P0, it can be seen that X1, Y1, Z1, and V are constants (Fermat’s principle); but the ray components α, β and γ are variables and will be different for each ray; i.e., ∂W/∂α = -X1, etc. The exponent of the integrand of Eqs. (8) and (9) can be written as W (Q1′) + αX0 + βY0 + γZ0 + α(X1′ - X0) + β(Y1′ - Y0) + γ(Z1′ - Z0). What is constant in the above equation for perfect imaging is the sum of the first four terms. They can be identified as V(P0). Hence, in Eqs. (8) and (9), when there is perfect imaging, the last three terms of the preceding equation above must remain. If the perfect imaging is on axis and only the image plane Z0 = O is considered, then W(Q1′) can be suppressed since X0 = Y0 = Z0 = 0.

P. Clemmowv, Proc. Roy. Soc. (London) 205A, 286 (1951)

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

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1142 (1965).

W. Brouwer, E. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).

W. Charman, J. Opt. Soc. Am. 53, 410, 415 (1963).

B. Watrasiewicz, Optica Acta 12, 167 (1965).

H. Osterberg and G. Pride, J. Opt. Soc. Am. 40, 14 (1950); H. Osterberg and L. Smith, J. Opt. Soc. Am. 50, 362 (1960); L. Smith, ibid., p. 369; W. Welford, Optics in Metrology (Pergamon Press, London, 1960).

J. Focke, Opt. Acta 4, 124 (1957).

R. Luneburg, Lecture Notes (New York University, 1947, 1948).

G. Lansraux, Can. J. Phys. 40, 1101 (1962).

R. Tremblay and A. Boivin, Appl. Opt. 5, 251 (1966).

M. Bachynski and G. Bekefi, J. Opt. Soc. Am. 47, 428 (1957).

(a) G. Farnell, J. Opt. Soc. Am. 48, 643 (1958); (b) Can. J. Phys. 35, 777 (1957); (c) Can. J. Phys. 36, 935 (1958).

The Rayleigh equation evaluated is from H. Osterberg and L. Smith, J. Opt. Soc. Am. 51, 1050 (1961); the Maggi-Rubinowvicz-Kirchhoff equation is from Ref. 42 (a).

A. Schell, IEEE Trans. Antennas Propagation AP-11, 428 (1963).

M. Gravel and A. Boivin, J. Opt. Soc. Am. 56, 1438A (1966) and private communication.

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