Abstract

A solution is obtained to the inverse diffraction problem for a monochromatic scalar wave field propagated into the half space z>0. It is shown how to determine the field distribution throughout the region 0≤z<z<sub>1</sub> from the knowledge of the field distribution in the plane z=z<sub>1</sub>>0. The solution takes a particularly simple form when the spatial-frequency spectrum of the distribution in the plane z=z<sub>1</sub> (or in any other plane z= const>0) is bandlimited to a circle whose radius is equal to the wavenumber of the field. In this case, the solution to the inverse diffraction problem may be expressed in a form strictly similar to that for the direct-propagation problem (exterior boundary-value problem), given by Rayleigh’s diffraction formula of the first kind. A comparison of these two solutions leads to the formulation of a new reciprocity theorem, valid for a wide class of wave fields.

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  1. Our solution may also be regarded as a solution to what can be called the interior reconstruction problem (cf. Ref. 3, p. 622) for a domain bounded by mutually parallel planes, namely the determination of the field in the domain 0 ≤ z <z1, from the knowledge of the field distribution on the plane z=z1>0, if we assume that the field obeys the Sommerfeld radiation condition at infinity in the half space z>0.
  2. G. C. Sherman, J. Opt. Soc. Am. 57, 1490 (1967).
  3. R. Mittra and R. L. Ranson in Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967, J. Wiley & Sons, Inc., New York, distr.), p. 633.
  4. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Clarendon Press, Oxford, 1960), 2nd ed., p. 25.
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford and New York, 1965), 3rd ed., §8.3.3
  6. D. Gabor, Proc. Roy. Soc. (London) 197A, 462 (1949).
  7. L. Mertz, Transformations in Optics (J. Wiley & Sons, Inc., New York, 1965), Ch. 3.
  8. J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 588 (1966).
  9. While the present paper was being written, a paper by G. C. Sherman (Ref. 2) appeared, which deals with some aspects of this problem. Sherman’s point-convergence function corresponds to our inverse-propagation kernel K01.
  10. C. J. Bouwkamp, in Rep. Progr. Phys. 17, 41 (1954).
  11. E. Wolf, Proc. Phys. Soc. (London) 74, 269 (1959), Appendix.
  12. E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).
  13. This formula follows by a straightforward modification of a formula due to H. Weyl, Ann. Physik 60, 481 (1919). For a direct derivation of this formula, see, for example, A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space [Pergamon Press (Ltd.), Oxford, 1966], Eq. (2.19).
  14. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1966), p. 311 et seq.
  15. E. Lalor, J. Math. Phys. 9, 2001 (1968); J. Opt. Soc. Am. 58, 720A (1968).
  16. A referee has pointed out to us that although the reciprocity theorem is not stated there explicitly, the result appears in §4 of the paper by Mittra and Ranson (Ref. 3). We are obliged to the referee for bringing this fact to our attention.
  17. G. C. Sherman [Phys. Rev. Letters 21, 761 (1968), Theorem II] recently showed that a field which obeys these conditions can be continued into the whole half-space z<0. With the help of this result the reciprocity theorem may be formulated in a more general form, in which the restrictions z0≥0 and |d|≤z1 are removed.
  18. Footnote added in proof. Since this paper was written an article by W. Lukosz, J. Opt. Soc. Am. 58, 1084 (1968) appeared, in which this reciprocity theorem is also given [Lukosz's equation (3.12)]. However, Lukosz does not explicitly assume that evanescent waves are absent, i.e., that the field in any plane z=const>0 is bandlimited to the domain u2+v2k2. As is evident from our analysis, the theorem applies only to fields that are bandlimited in this wav.

Ranson, R. L.

R. Mittra and R. L. Ranson in Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967, J. Wiley & Sons, Inc., New York, distr.), p. 633.

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Clarendon Press, Oxford, 1960), 2nd ed., p. 25.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford and New York, 1965), 3rd ed., §8.3.3

Bouwkamp, C. J.

C. J. Bouwkamp, in Rep. Progr. Phys. 17, 41 (1954).

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Clarendon Press, Oxford, 1960), 2nd ed., p. 25.

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) 197A, 462 (1949).

Lalor, E.

E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).

E. Lalor, J. Math. Phys. 9, 2001 (1968); J. Opt. Soc. Am. 58, 720A (1968).

Lukosz, W.

Footnote added in proof. Since this paper was written an article by W. Lukosz, J. Opt. Soc. Am. 58, 1084 (1968) appeared, in which this reciprocity theorem is also given [Lukosz's equation (3.12)]. However, Lukosz does not explicitly assume that evanescent waves are absent, i.e., that the field in any plane z=const>0 is bandlimited to the domain u2+v2k2. As is evident from our analysis, the theorem applies only to fields that are bandlimited in this wav.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1966), p. 311 et seq.

Mertz, L.

L. Mertz, Transformations in Optics (J. Wiley & Sons, Inc., New York, 1965), Ch. 3.

Mittra, R.

R. Mittra and R. L. Ranson in Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967, J. Wiley & Sons, Inc., New York, distr.), p. 633.

Sherman, G. C.

G. C. Sherman, J. Opt. Soc. Am. 57, 1490 (1967).

While the present paper was being written, a paper by G. C. Sherman (Ref. 2) appeared, which deals with some aspects of this problem. Sherman’s point-convergence function corresponds to our inverse-propagation kernel K01.

G. C. Sherman [Phys. Rev. Letters 21, 761 (1968), Theorem II] recently showed that a field which obeys these conditions can be continued into the whole half-space z<0. With the help of this result the reciprocity theorem may be formulated in a more general form, in which the restrictions z0≥0 and |d|≤z1 are removed.

Weyl, H.

This formula follows by a straightforward modification of a formula due to H. Weyl, Ann. Physik 60, 481 (1919). For a direct derivation of this formula, see, for example, A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space [Pergamon Press (Ltd.), Oxford, 1966], Eq. (2.19).

Winthrop, J. T.

J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 588 (1966).

Wolf, E.

E. Wolf, Proc. Phys. Soc. (London) 74, 269 (1959), Appendix.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford and New York, 1965), 3rd ed., §8.3.3

Worthington, C. R.

J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 588 (1966).

Other (18)

Our solution may also be regarded as a solution to what can be called the interior reconstruction problem (cf. Ref. 3, p. 622) for a domain bounded by mutually parallel planes, namely the determination of the field in the domain 0 ≤ z <z1, from the knowledge of the field distribution on the plane z=z1>0, if we assume that the field obeys the Sommerfeld radiation condition at infinity in the half space z>0.

G. C. Sherman, J. Opt. Soc. Am. 57, 1490 (1967).

R. Mittra and R. L. Ranson in Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967, J. Wiley & Sons, Inc., New York, distr.), p. 633.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Clarendon Press, Oxford, 1960), 2nd ed., p. 25.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford and New York, 1965), 3rd ed., §8.3.3

D. Gabor, Proc. Roy. Soc. (London) 197A, 462 (1949).

L. Mertz, Transformations in Optics (J. Wiley & Sons, Inc., New York, 1965), Ch. 3.

J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 588 (1966).

While the present paper was being written, a paper by G. C. Sherman (Ref. 2) appeared, which deals with some aspects of this problem. Sherman’s point-convergence function corresponds to our inverse-propagation kernel K01.

C. J. Bouwkamp, in Rep. Progr. Phys. 17, 41 (1954).

E. Wolf, Proc. Phys. Soc. (London) 74, 269 (1959), Appendix.

E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).

This formula follows by a straightforward modification of a formula due to H. Weyl, Ann. Physik 60, 481 (1919). For a direct derivation of this formula, see, for example, A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space [Pergamon Press (Ltd.), Oxford, 1966], Eq. (2.19).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1966), p. 311 et seq.

E. Lalor, J. Math. Phys. 9, 2001 (1968); J. Opt. Soc. Am. 58, 720A (1968).

A referee has pointed out to us that although the reciprocity theorem is not stated there explicitly, the result appears in §4 of the paper by Mittra and Ranson (Ref. 3). We are obliged to the referee for bringing this fact to our attention.

G. C. Sherman [Phys. Rev. Letters 21, 761 (1968), Theorem II] recently showed that a field which obeys these conditions can be continued into the whole half-space z<0. With the help of this result the reciprocity theorem may be formulated in a more general form, in which the restrictions z0≥0 and |d|≤z1 are removed.

Footnote added in proof. Since this paper was written an article by W. Lukosz, J. Opt. Soc. Am. 58, 1084 (1968) appeared, in which this reciprocity theorem is also given [Lukosz's equation (3.12)]. However, Lukosz does not explicitly assume that evanescent waves are absent, i.e., that the field in any plane z=const>0 is bandlimited to the domain u2+v2k2. As is evident from our analysis, the theorem applies only to fields that are bandlimited in this wav.

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