Abstract

<p>Kirchhoff’s diffraction theory, which is often criticized because of an apparent internal inconsistency, is shown to be a rigorous solution to a certain boundary-value problem that has a clear physical meaning. This new interpretation of Kirchhoff’s theory is a direct consequence of the Rubinowicz theory of the boundary diffraction wave.</p><p>We argue that these true boundary conditions of Kirchhoff’s theory are physically reasonable for diffraction at an aperture in a <i>black</i> screen whose linear dimensions are large compared with the wavelength. The boundary values which Kirchhoff’s solution takes in the plane of the aperture and in the near zone on the axis for the case of a normally incident plane wave diffracted at a circular aperture are compared with previously published results of experiments with microwaves, and reasonable agreement is found. (Strict agreement cannot be expected since the screens used in the microwave experiments were not black.) Moreover, Kirchhoff’s solution is found to be in closer agreement with the experimental results than the “manifestly consistent” Rayleigh-Sommerfeld theory.</p><p>We also suggest an extension of the Kirchhoff theory, which might provide a physically reasonable approximation to the solution of the problems of diffraction at an aperture in a black screen whose linear dimensions are of the order of magnitude of or smaller than the wavelength of the light.</p>

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed.
  2. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.
  3. H. Poincaré, Théorie Mathématique de la Lumnière (Georges Carré, Paris, 1892), Pt. II, pp. 187–188.
  4. G. Toraldo di Francia, Atti Fond. Giorgio Ronchi 11, 503 (1956).
  5. M. Boron, Optik (Julius Springer-Verlag, Berlin, 1933; reprinted (1965), p. 152.
  6. W. Franz, Z. Physik 125, 563 (1949).
  7. S. A. Schelkunoff, Commun. Pure Appl. Math. 4, 43 (1951).
  8. C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
  9. Some of the modified theories are reviewed in Ref. 8, Sec. 4.
  10. F. Kottler, (a) Ann. Physik (Leipzig) 70, 405 (1923). (b) in Progress in Optics IV, E. Wolf, Ed. (North-Holland Publ. Co. Amsterdam and John Wiley & Sons Inc., New York, 1965), p. 281.
  11. A. Rubinowicz, Ann. Physik (Leipzig) 53, 257 (1917). For other references and an account of the theory, see Ref. 1, Sec. 8.9.
  12. The experimental evidence is tentative in the sense that it uses data obtained from diffraction experiments with screens that are not black.
  13. When the incident wave is a plane wave, propagated in a direction specified by the unit vector p, (2.6) must be replaced by [Equation], where ŝ= QP/|QP|. A similar change must then be made in the expression for W(i), given by Eq. (3.1). In (2.6), and in several other places, an adjacent pair of capital letters indicates a vector. Also, ŝ represents a vector, even though not printed in boldface type.
  14. A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).
  15. K. Miyamoto and E. Wolf, (a) J. Opt. Soc. Am. 52, 615 (1962); (b) 52, 626 (1962).
  16. J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
  17. M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young's ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young's ideas, Andrews's views are essentially in qualitative agreement witb those of the present paper.
  18. Some aspects of the Rayleigh-Sommerfeld theory are discussed in the papers by E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 52, 761 (1962); 54, 587 (1964). In the first of these two papers, the theory was associated with the names of Rayleigh and Kirchhoff, but it seems more appropriate to associate it with the names of Rayleigh and Sommerfeld.
  19. N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).
  20. We have shown elsewhere [J. Opt. Soc. Am. 54, 587 (1964)] that, if the linear dimensions of the aperture are large compared with the wavelength, the difference in the behavior of the Rayleigh-Sommerfeld solutions and of the Kirchhoff solution in the plane of the aperture does not lead to significantly different results for the predicted field in the far zone, in the neighborhood of the forward direction.
  21. See also R. W. Dunham, J. Opt. Soc. Am. 54, 1102 (1964).
  22. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, Calif., 1964).
  23. It is clear from an examination of the left-hand side of (B4) that, if QP′≠0, the expression could only vanish if cos(SQ′, QP′) = 1, i.e., if the directions SQ′ and QP′ were parallel. This is impossible, since S is situated off the plane of the aperture, whereas both Q′ and P′ are in this plane.
  24. O. D. Kellogg, Foundations of Potential Theory (Dover Publications, Inc., New York, 1953), p. 144.

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed.

Boron, M.

M. Boron, Optik (Julius Springer-Verlag, Berlin, 1933; reprinted (1965), p. 152.

Bouwkamp, C. J.

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.

di Francia, G. Toraldo

G. Toraldo di Francia, Atti Fond. Giorgio Ronchi 11, 503 (1956).

Dunham, R. W.

See also R. W. Dunham, J. Opt. Soc. Am. 54, 1102 (1964).

Ehrlich, M. J.

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young's ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young's ideas, Andrews's views are essentially in qualitative agreement witb those of the present paper.

Franz, W.

W. Franz, Z. Physik 125, 563 (1949).

Held, G.

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young's ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young's ideas, Andrews's views are essentially in qualitative agreement witb those of the present paper.

Keller, J. B.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).

Kellogg, O. D.

O. D. Kellogg, Foundations of Potential Theory (Dover Publications, Inc., New York, 1953), p. 144.

Kottler, F.

F. Kottler, (a) Ann. Physik (Leipzig) 70, 405 (1923). (b) in Progress in Optics IV, E. Wolf, Ed. (North-Holland Publ. Co. Amsterdam and John Wiley & Sons Inc., New York, 1965), p. 281.

Lewis, R. M.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).

Luneburg, R. K.

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

Marchand, E. W.

Some aspects of the Rayleigh-Sommerfeld theory are discussed in the papers by E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 52, 761 (1962); 54, 587 (1964). In the first of these two papers, the theory was associated with the names of Rayleigh and Kirchhoff, but it seems more appropriate to associate it with the names of Rayleigh and Sommerfeld.

Miyamoto, K.

K. Miyamoto and E. Wolf, (a) J. Opt. Soc. Am. 52, 615 (1962); (b) 52, 626 (1962).

Mukunda, N.

N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumnière (Georges Carré, Paris, 1892), Pt. II, pp. 187–188.

Rubinowicz, A.

A. Rubinowicz, Ann. Physik (Leipzig) 53, 257 (1917). For other references and an account of the theory, see Ref. 1, Sec. 8.9.

A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).

Schelkunoff, S. A.

S. A. Schelkunoff, Commun. Pure Appl. Math. 4, 43 (1951).

Seckler, B. D.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).

Silver, S.

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young's ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young's ideas, Andrews's views are essentially in qualitative agreement witb those of the present paper.

Wolf, E.

K. Miyamoto and E. Wolf, (a) J. Opt. Soc. Am. 52, 615 (1962); (b) 52, 626 (1962).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed.

Some aspects of the Rayleigh-Sommerfeld theory are discussed in the papers by E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 52, 761 (1962); 54, 587 (1964). In the first of these two papers, the theory was associated with the names of Rayleigh and Kirchhoff, but it seems more appropriate to associate it with the names of Rayleigh and Sommerfeld.

Other (24)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.

H. Poincaré, Théorie Mathématique de la Lumnière (Georges Carré, Paris, 1892), Pt. II, pp. 187–188.

G. Toraldo di Francia, Atti Fond. Giorgio Ronchi 11, 503 (1956).

M. Boron, Optik (Julius Springer-Verlag, Berlin, 1933; reprinted (1965), p. 152.

W. Franz, Z. Physik 125, 563 (1949).

S. A. Schelkunoff, Commun. Pure Appl. Math. 4, 43 (1951).

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).

Some of the modified theories are reviewed in Ref. 8, Sec. 4.

F. Kottler, (a) Ann. Physik (Leipzig) 70, 405 (1923). (b) in Progress in Optics IV, E. Wolf, Ed. (North-Holland Publ. Co. Amsterdam and John Wiley & Sons Inc., New York, 1965), p. 281.

A. Rubinowicz, Ann. Physik (Leipzig) 53, 257 (1917). For other references and an account of the theory, see Ref. 1, Sec. 8.9.

The experimental evidence is tentative in the sense that it uses data obtained from diffraction experiments with screens that are not black.

When the incident wave is a plane wave, propagated in a direction specified by the unit vector p, (2.6) must be replaced by [Equation], where ŝ= QP/|QP|. A similar change must then be made in the expression for W(i), given by Eq. (3.1). In (2.6), and in several other places, an adjacent pair of capital letters indicates a vector. Also, ŝ represents a vector, even though not printed in boldface type.

A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).

K. Miyamoto and E. Wolf, (a) J. Opt. Soc. Am. 52, 615 (1962); (b) 52, 626 (1962).

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young's ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young's ideas, Andrews's views are essentially in qualitative agreement witb those of the present paper.

Some aspects of the Rayleigh-Sommerfeld theory are discussed in the papers by E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 52, 761 (1962); 54, 587 (1964). In the first of these two papers, the theory was associated with the names of Rayleigh and Kirchhoff, but it seems more appropriate to associate it with the names of Rayleigh and Sommerfeld.

N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).

We have shown elsewhere [J. Opt. Soc. Am. 54, 587 (1964)] that, if the linear dimensions of the aperture are large compared with the wavelength, the difference in the behavior of the Rayleigh-Sommerfeld solutions and of the Kirchhoff solution in the plane of the aperture does not lead to significantly different results for the predicted field in the far zone, in the neighborhood of the forward direction.

See also R. W. Dunham, J. Opt. Soc. Am. 54, 1102 (1964).

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

It is clear from an examination of the left-hand side of (B4) that, if QP′≠0, the expression could only vanish if cos(SQ′, QP′) = 1, i.e., if the directions SQ′ and QP′ were parallel. This is impossible, since S is situated off the plane of the aperture, whereas both Q′ and P′ are in this plane.

O. D. Kellogg, Foundations of Potential Theory (Dover Publications, Inc., New York, 1953), p. 144.

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