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.

© 1967 Optical Society of America

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References

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  1. W. Hamilton, Mathematical 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).
    [Crossref]
  6. M. Herzberger, J. Opt. Soc. Am. 27, 133 (1937).
    [Crossref]
  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 Symposium 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).
    [Crossref]
  13. 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).
    [Crossref]
  15. M. Herzberger and D. Wilder, J. Opt. Soc. Soc. Am. 54, 773 (1964).
    [Crossref]
  16. M. Born, Optik (Springer–Verlag, Berlin, 1933).
    [Crossref]
  17. P. Debye, Ann. Physik 30, 775 (1909).
  18. J. Picht, Ann. Physik 77, 685 (1925).
  19. J. Focke, Opt. Acta 3, 110 (1956).
    [Crossref]
  20. E. Wolf, Proc. Roy. Soc. (London) 253A, 349 (1959).
  21. (a)B. Richards, in Astronomical 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 and et al. (John Wiley & Sons, New York, N. Y., 1951); (e)H. Osterberg and R. McDonald, in Optical Image Evaluation (Natl. Bur. Std. Circ. 526,1954).
  22. F. Kottler, Ann. Physik 71, 457 (1923).
    [Crossref]
  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. A1. 9, 381 (1952).
    [Crossref]
  25. J. Vasseur, Ann. Phys. (Paris) 1, 506 (1952).
  26. C. Bouwkamp, Rept. Prog. Phys. 17, 35 (1954).
    [Crossref]
  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 = 0 is considered, then W(Q1′) can be suppressed since X0 = Y0 = Z0 = 0.
  30. P. Clemmow, Proc. Roy. Soc. (London) 205A, 286 (1951)
  31. R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).
    [Crossref]
  32. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1142 (1965).
    [Crossref]
  33. W. Brouwer, E. O’Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).
    [Crossref]
  34. W. Charman, J. Opt. Soc. Am. 53, 410, 415 (1963).
    [Crossref]
  35. B. Watrasiewicz, Optica Acta 12, 167 (1965).
    [Crossref]
  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).
    [Crossref]
  37. J. Focke, Opt. Acta 4, 124 (1957).
    [Crossref]
  38. R. Luneburg, Lecture Notes (New York University, 1947, 1948).
  39. G. Lansraux, Can. J. Phys. 40, 1101 (1962).
    [Crossref]
  40. R. Tremblay and A. Boivin, Appl. Opt. 5, 251 (1966).
    [Crossref]
  41. M. Bachynski and G. Bekefi, J. Opt. Soc. Am. 47, 428 (1957).
    [Crossref]
  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–Rubinowicz–Kirchhoff equation is from Ref. 42 (a).
    [Crossref]
  44. A. Schell, IEEE Trans. Antennas Propagation AP-11, 428 (1963).
    [Crossref]
  45. M. Gravel and A. Boivin, J. Opt. Soc. Am.56, 1438A (1966) and private communication.

1966 (1)

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

1965 (2)

1964 (1)

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

1963 (4)

1962 (2)

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

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

1961 (1)

1959 (1)

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

1958 (1)

1957 (2)

1956 (1)

J. Focke, Opt. Acta 3, 110 (1956).
[Crossref]

1954 (1)

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

1952 (1)

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

1951 (2)

H. Severin, Z. Physik 129, 426 (1951) and Nuovo Cimento Ser. 9, Suppl. A1. 9, 381 (1952).
[Crossref]

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

1950 (1)

1945 (2)

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

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

1939 (1)

(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).

1937 (2)

1925 (1)

J. Picht, Ann. Physik 77, 685 (1925).

1923 (1)

F. Kottler, Ann. Physik 71, 457 (1923).
[Crossref]

1919 (1)

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

1911 (1)

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

1909 (1)

P. Debye, Ann. Physik 30, 775 (1909).

Arley, N.

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

Bachynski, M.

Barakat, R.

Bekefi, G.

Beutler, H.

Boivin, A.

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

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

Born, M.

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

M. Born, Optik (Springer–Verlag, Berlin, 1933).
[Crossref]

Bouwkamp, C.

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

Brouwer, W.

Charman, W.

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).

Clemmow, P.

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

Debye, P.

P. Debye, Ann. Physik 30, 775 (1909).

Farnell, G.

Focke, J.

J. Focke, Opt. Acta 4, 124 (1957).
[Crossref]

J. Focke, Opt. Acta 3, 110 (1956).
[Crossref]

Gravel, M.

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

Hamilton, W.

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

Herzberger, M.

Houston, A.

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, in Proceedings of a Symposium 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).

Kottler, F.

F. Kottler, Ann. Physik 71, 457 (1923).
[Crossref]

Lansraux, G.

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

Luneburg, R.

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

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

O’Neill, E.

Osterberg, H.

Picht, J.

J. Picht, Ann. Physik 77, 685 (1925).

Pride, G.

Rayleigh,

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

Richards, B.

(a)B. Richards, in Astronomical 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 and et al. (John Wiley & Sons, New York, N. Y., 1951); (e)H. Osterberg and R. McDonald, in Optical Image 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).
[Crossref]

Severin, H.

H. Severin, Z. Physik 129, 426 (1951) and Nuovo Cimento Ser. 9, Suppl. A1. 9, 381 (1952).
[Crossref]

Smith, L.

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, J. Opt. Soc. Am. 27, 75, 138 (1937).
[Crossref]

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

Tremblay, R.

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

Vasseur, J.

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

Walther, A.

Watrasiewicz, B.

B. Watrasiewicz, Optica Acta 12, 167 (1965).
[Crossref]

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).
[Crossref]

Wolf, E.

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

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

Zernike, F.

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

Ann. Phys. (Leipzig) (2)

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

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

Ann. Phys. (Paris) (1)

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

Ann. Physik (3)

F. Kottler, Ann. Physik 71, 457 (1923).
[Crossref]

P. Debye, Ann. Physik 30, 775 (1909).

J. Picht, Ann. Physik 77, 685 (1925).

Appl. Opt. (2)

Can. J. Phys. (1)

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

IEEE Trans. Antennas Propagation (1)

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

J. Opt. Soc. Am. (11)

J. Opt. Soc. Soc. Am. (1)

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

Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. (1)

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

Opt. Acta (2)

J. Focke, Opt. Acta 3, 110 (1956).
[Crossref]

J. Focke, Opt. Acta 4, 124 (1957).
[Crossref]

Optica Acta (1)

B. Watrasiewicz, Optica Acta 12, 167 (1965).
[Crossref]

Phys. Rev. (1)

(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).

Proc. Roy. Soc. (London) (2)

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

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

Rept. Prog. Phys. (1)

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

Z. Physik (1)

H. Severin, Z. Physik 129, 426 (1951) and Nuovo Cimento Ser. 9, Suppl. A1. 9, 381 (1952).
[Crossref]

Other (14)

M. Born, Optik (Springer–Verlag, Berlin, 1933).
[Crossref]

(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).

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 = 0 is considered, then W(Q1′) can be suppressed since X0 = Y0 = Z0 = 0.

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

(a)B. Richards, in Astronomical 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 and et al. (John Wiley & Sons, New York, N. Y., 1951); (e)H. Osterberg and R. McDonald, in Optical Image Evaluation (Natl. Bur. Std. Circ. 526,1954).

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

M. Kline, in Proceedings of a Symposium 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.

W. Hamilton, Mathematical 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).

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

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Figures (4)

Fig. 1
Fig. 1

The geometry illustrates a coordinate system centered at 0 with an incident wave ψ and a reference sphere ∑ of radius R = QS = 0′S. The point S is a defined image point on the image plane XY. The ray 0′P0 is a reference ray; it can be the principal ray. The ray along Q ¯Q1′ is a general ray and the wave aberration is Φ = Q ¯Q. The ray 0′P0 is the ray through the point 0′ which is the point of intersection or tangency of ψ and ∑. The length R′ is QP1 and R″ is 0′P0. The point Q0′ is a foot for the perpendicular 0Q0′ to R″; Q0 and Q1 are feet for the perpendiculars SQ0 and SQ1 drawn to R″ and R′, respectively.

Fig. 2
Fig. 2

This drawing is similar to Fig. 1. The defined image point is now P0, the intersection of the principal ray, or some other ray used as a reference ray, with the image plane XY. The lengths S = Q1P1′ (shown twice) illustrate the variation of Q1P1′ with S. Note that Q1P1′ is perpendicular to P0Q1 and hence parallel to the general ray QP1. The point P1′ is the point where the complex amplitude is to be determined.

Fig. 3
Fig. 3

(a) Compares the axial intensity from (1) the Luneburg–Debye integral, (2) the Rayleigh integral, and (3) the Maggi–Rubinowicz–Kirchhoff integral. The geometric focal point is at Z = 0 and the aperture at Z = − 105 (not shown). The wavelength is λ = 3.2 cm, and the aperture diameter is 50 cm. In (a) and (b) the ordinate is log102|U(P)|2] and the abscissa is the axial distance in centimeters measured from the geometrical focus. (b) is the same as (a) except that the aperture is now at Z = −95 cm, and the solid angle in image space remains constant.

Fig. 4
Fig. 4

This is the same as Fig. 3 except that the wavelength has been shortened to λ = 1.0 cm.

Equations (68)

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Φ = Q ¯ Q = V ( Q ) - V ( 0 ) = V ( P 1 ) - R - [ V ( P 0 ) - R ] ,
Φ = W ( Q 1 ) + Q 1 P 1 - R - [ W ( Q 0 ) + Q 0 P 0 - R ] ,
Q 0 P 0 - R ~ - R             and             Q 1 P 1 - R ~ - R .
or             Φ = W ( Q 1 ) - W ( Q 0 ) , Φ = T ( Q 1 ) - T ( Q 0 ) .
Φ = W ( Q 1 ) - W ( Q 0 ) + ( α X s + β Y s ) - ( α 0 X s + β 0 Y s ) ,
Φ = W ( Q 1 ) - W ( Q 0 ) + ( α X 0 + β Y 0 ) - ( α 0 X 0 + β 0 Y 0 ) .
Φ = Φ + Q Q 1 - 0 Q 0 .
Φ = Φ + Q Q 1 - R
Φ = Φ + R - R + P 1 Q 1 .
R 2 = X Q 2 + Y Q 2 + Z Q 2 ,
R 2 = ( X Q - X 1 ) 2 + ( Y Q - Y 1 ) 2 + Z Q 2 .
R - R = [ ( X Q X 1 + Y Q Y 1 ) / R ] - [ ( X 1 2 + Y 1 2 ) / 2 R ] - { [ 2 ( X Q X 1 + Y Q Y 1 ) - ( X 1 2 + Y 1 2 ) ] 2 } / 8 R 3 - .
P 1 Q 1 = Y 1 sin w = Y 1 ( Y Q - Y 1 ) / R ,
Φ - Φ - 0.001 μ ;
Φ - Φ - 0.1 μ ;
Φ - Φ - 2.5 μ .
W = V - Σ i α i x i ,
W = V - Σ α i x i = W ( 0 ) + W ( 1 ) + W ( 2 ) + W ( 3 ) + + W ( n ) + .
Φ / α = - X 1 + X S             and             Φ / β = - Y 1 + Y S ,
Φ / α = - X 1 + X 0             and             Φ / β = - Y 1 + Y 0 .
E ( P 1 ) = - i λ Ω a ( α , β ) × exp { i k [ W ( α , β ) + α X 1 + β Y 1 + γ Z 1 ] } d α d β / γ
H ( P 1 ) = - i λ Ω b ( α , β ) × exp { i k [ W ( α , β ) + α X 1 + β Y 1 + γ Z 1 ] } d α d β / γ .
U ( P 1 ) = - i λ Ω A ( α , β ) × exp { i k [ W ( α , β ) + α X 1 + β Y 1 + γ Z 1 ] } d α d β / γ .
E ( P 1 ) = × 1 4 π ( n × E ) G d S - × × 1 4 π i k ( n × H ) G d S ,
E ( P 1 ) = 1 2 π ( E G n - G E n ) d S - 1 4 π G ( E × s ) d l + 1 4 π i k ( H · s ) G d l ,
E ( P 1 ) = - 1 4 π [ i k ( n × H ) G + ( n × E ) × G + ( E · n ) G ] d S - 1 4 π i k ( H · s ) G d l .
E ( P ) = 1 2 π [ i k ( n × H ) G + ( n · H ) G ] d S + ( H · s ) G d l ,
E ( P ) = 1 2 π ( n × E ) × G d S .
G = ( 1 / r - i k ) G r ,
d l = D m sin θ m d φ .
E = E ( V ) e i k V             and             H = H ( V ) e i k V ,
exp [ i k ( W + α X 1 + β Y 1 + γ Z 1 ) ] ,
U ( P 1 ) = - i k 2 π - U ( α , β ) × exp ( α X 1 + β Y 1 + γ Z 1 ) d α d β / γ ,
U ( P 1 ) = - i k 2 π α 2 + β 2 1 U ( α , β ) × exp ( α X 1 + β Y 1 + γ Z 1 ) d α d β / γ ,
k [ V ( Q ) + S ] .
V ( Q ) + S = V ( P 1 ) - R + S = W ( Q 1 ) + ( α X 1 + β Y 1 ) - R + S .
lim R , S ( - R + S ) = - ( α X 1 + β Y 1 ) ,
V ( Q ) + S W ( Q 1 ) + α X 1 + β Y 1 + γ Z 1 .
S 2 = ( X Q - X 1 ) 2 + ( Y Q - Y 1 ) 2 + ( Z Q - Z 1 ) 2
R 2 = ( X Q - X 0 ) 2 + ( Y Q - Y 0 ) 2 + ( Z Q - Z 0 ) 2 .
S - R = - ( 1 / R ) [ X Q ( X 1 - X 0 ) + Y Q ( Y 1 - Y 0 ) + Z Q ( Z 1 - Z 0 ) ] - ( 1 / 2 R ) [ ( X 1 2 - X 0 2 ) + ( Y 1 - Y 0 2 ) + ( Z 1 2 - Z 0 2 ) ] - 0 ( 1 / R 3 ) .
- X Q / R ~ α ;             Y Q / R ~ β ;             and             - Z Q / R ~ γ .
lim R , S ( S - R ) = α ( X 1 - X 0 ) + β ( Y 1 - Y 0 ) + γ ( Z 1 - Z 0 ) .
Φ + S - R ~ Φ + Σ α ( X 1 - X 0 ) .
Φ + S - R ~ Φ + Σ α ( X 1 - X s ) .
V ( Q ) + S ;
W ( Q 1 ) + α X 1 + β Y 1 + γ Z 1 ,
V ( Q ) + R + α ( X 1 - X 1 ) + β ( Y 1 - Y 1 ) + γ ( Z 1 - Z 1 ) ;
V ( Q ) + R - ( 1 / R ) [ X Q ( X 1 - X 0 ) + Y Q ( Y 1 - Y 0 ) + Z Q ( Z 1 - Z 0 ) ] .
x q = X Q / X Q m             and             y q = Y Q / Y Q m .
x 1 = X 1 ( X Q m / R λ )             and             y 1 = Y 1 ( Y Q m / R λ ) ;
s = F x R / X Q m             and             t = F y R / Y Q m .
and that             X Q = D sin θ sin φ = X 1 + R cos u Y Q = D sin θ cos φ = Y 1 + R cos v ,
α = cos u ,             β = cos v ,             and             γ = cos w .
sin φ = ( X 1 / D sin θ ) + R cos u / D sin θ
cos φ = ( Y 1 / D sin θ ) + R cos v / D sin θ .
sin φ = cos u / s i n θ
cos φ = cos v / sin θ ,
and             x q ~ cos u / cos u e = α / α e = α y q ~ cos v / cos v e = β / β e = β .
x 1 ~ X 1 cos u e / λ = X 1 α e / λ
y 1 ~ Y 1 cos v e / λ = Y 1 β e / λ ;
s ~ F α / α e             and             t ~ F β / β e .
Φ = ω 1 λ ( x q 2 + y q 2 ) + ω 2 y q 2 Φ = ω 1 λ ( α 2 + β 2 ) + ω 2 β 2
Φ = ω 1 λ ( x q 2 + y q 2 ) + ω 2 λ ( x q 2 + y q 2 ) 2 + ω 3 λ ( x q 2 + y q 2 ) 3 + Φ = ω 1 λ ( α 2 + β 2 ) + ω 2 λ ( α 2 + β 2 ) 2 + ω 3 λ ( α 2 + β 2 ) 3 +
Φ = ω 1 λ ( x q 2 + y q 2 ) y q + ω 2 λ ( x q 2 + y q 2 ) 2 y q + Φ = ω 1 λ ( α 2 + β 2 ) β + ω 2 λ ( α 2 + β 2 ) 2 β + .
U ( α , β ) = - B ( α , β ) exp ( α x 1 + β y 1 ) d α d β ,
B ( α , β ) = A ( α , β ) e i k W / ( 1 - α 2 - β 2 ) 1 2
B ( α , β ) = { 0 ,             α 2 + β 2 > 1 Eq . ( 25 ) ,             α 2 + β 2 1