Abstract

One of Rayleigh’s diffraction integrals is generalized as to the nature of the functions on which it operates. These functions comprise the well-known Hilbert space £<sub>2</sub> of square-integrable functions. The algebraic properties are apparent in the existence of a left inverse for the diffraction operator, and in other places where the ordering of a sequence of operations is important. The conditions for self imaging are completely characterized by a certain part of the mathematical spectrum of the diffraction operator. The contribution of the evanescent waves to this spectrum is clearly shown. Both the infinite- and finite-aperture cases of diffraction are treated. In the infinite-aperture case we can employ a vector theory in which we obtain a complete characterization of self imaging. This self imaging is of an approximate nature. For the finiteaperture case we employ a scalar theory and obtain some partial results. However, in contrast to the infinite aperture we prove, for the case of an arbitrary finite aperture in a black screen, that there exist fields which image themselves exactly through the diffraction process. These fields are the eigenfunctions of a diffraction operator which corresponds to the sequential operations of inserting a screen into the incident field, discarding the evanescent-wave contribution, and diffracting through a distance <i>z</i>. The operator which corresponds to just the first two processes of inserting the screen and discarding the evanescent waves is of the type which generates the so-called generalized prolate spheroidal functions. These functions correspond to fields which have the interesting optical property of giving the same diffraction pattern whether the screen is applied or not (i.e., to within a constant positive factor). A conjecture is put forth concerning the evanescent-wave contribution to measurements that have been taken near the aperture. If the conjecture proves correct, it may be possible to show a consistency between the first Rayleigh diffraction integral and near-aperture measurements.

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).
  2. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, 1962), 4th ed.
  3. F. Riesz and B. Nagy, Functional Analysis (Frederick Ungar Publishing Co., New York, 1965).
  4. A. Zaanen, Linear Analysis (North-Holland Publishing Co., Amsterdam, 1956).
  5. N. Dunford and J. Schwartz, Linear Operators (Interscience Publishers, John Wiley & Sons, Inc., New York, 1957, 1963), Parts I, II.
  6. L. Schwartz, Théorie des Distributions (Hermann, Paris, 1957-1959) Vols. I, II.
  7. I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vols. 1-3 (Academic Press Inc., New York, 1964); I. M. Gel'fand and N. Vilenkin, Generalized Functions, Vol. 4 (Academic Press Inc., New York, 1964); I. M. Gel'fand, M. I. Graev, and N. Vilenkin, Generalized Functions, Vol. 5 (Academic Press Inc.. New York, 1966).
  8. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, 1962).
  9. A. H. Zemanian, Distribution Theory and Transformn Analysis (McGraw-Hill Book Co., New York, 1965).
  10. J. Arsac, Fourier Transforms and the Theory of Distributions (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1966).
  11. W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).
  12. G. Sherman, J. Opt. Soc. Am. 57, 1490 (1967).
  13. E. Wolf and J. R. Shewell, Phys. Letters 25A, 417 (1967).
  14. M. Born and E. Wolf, Principles of Optics (The Macmillan Co., New York, 1959), p. 492.
  15. See Ref. 1, p. 6.
  16. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
  17. M. H. Stone, Linear Transformations in Hilbert Space (American Mathematical Society, New York, 1932), p. 23.
  18. In vector notation, the electric vector E(r,t) = φ(r)e-iωt+φ* (r)eiωt where r= (x,y,z) and the components of φe-iωt are the analytic signals of the Cartesian components of E. From this we have φ(x,z). φ*(x,z)dx = ½(E2)tdx, where φ.φ* denotes the dot product between φ and its conjugate and 〈〉t is the time average over a period. Thus the integral | φ (x) | 2dx in vector notation denotes one half of the time-average energy in the electric field per unit length in the z direction. We will henceforth refer to this integral as simply the energy in φ. The z parameter (for z≥0) will frequently be supressed in such functions as φ, f, g, h etc.
  19. See Ref. 1, pp. 311-319.
  20. See Ref. 14, p. 582.
  21. See Ref. 3, p. 145.
  22. E. C. Titchmarch, Introduction to the Theory of Fourier Integrals (Oxford University Press, 1948) pp. 75, 76.
  23. See Ref. 17, pp. 104, 105.
  24. See Ref. 4, p. 134.
  25. See Ref. 7, Vol. 4, p. 105.
  26. For a scalar theory we may take all of £2 for the domain of definition of Dz. For the vector theory (with none of the source in the z = 0 plane), it can readily be shown that all of Maxwell's equations are satisfied for z > 0 if and only if ϱ.Ff = 0 for all ¾, where the vector ϱ(ξ) = [ξ, η, (λ-2 - ξ2)½] and f(x) = φ(x,0) as given in Ref. 18. All complex-boundary fields f satisfying this relation constitute a proper subspace Γ of the vector £ functions. Therefore, in the vector theory, the domain of Dz is a proper subspace of £2.
  27. See Ref. 11, for a partial list.
  28. See Ref. 7, Vol. 4, p. 103.
  29. See Ref. 9, p. 103.
  30. See Ref. 4, p. 301.
  31. P. R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1957), 2nd ed., p. 51.
  32. See Ref. 4, p. 305.
  33. See Ref. 31, p. 51.
  34. See Ref. 14, pp. 558-560.
  35. H. M. Barlow and J. Brown, Radio Surface Waves (Oxford University Press, 1962).
  36. P. R. Halmos, Finite-Dimensional Vector Spaces (D. Van Nostrand Co., Inc., Princeton, N. J., 1958), 2nd ed., pp. 71, 72.
  37. See Ref. 4, p. 306.
  38. In the vector theory we must deal with the subspace ß∩M (see Ref. 26) in place of ß. (ß∩M denotes the space of functions common to ß∩M is invariant under Dz.
  39. See Ref. 3, p. 380.
  40. See Ref. 14, p. 553.
  41. E. Wolf in, Progress in Optics IV (North-Holland Publishing Company, Amsterdam, 1965), p. 201.
  42. See Ref. 41, p. 283.
  43. This result follows from Appendix D regardless of the size of the screen, as long as it has positive area.
  44. J. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 342 (1955).
  45. Apart from the extensive literature which shows that the evanescent waves are surface waves, existing only in close proximity (of the order of a wavelength λ) to matter, we can see from Eq. (11) that they obey a dispersive relation. That is, the phase velocity ν/|ξ| is a function of the wavenumber. These things tell tell us that we should not expect to measure contributions from such waves in the aperture unless the aperture is small (diameter of the order of λ).
  46. D. Slepian, Bell System Tech. J. 43, 3009 (1964).
  47. D. Slepian and H. Pollak, Bell System Tech. J. 40, 43 (1961).
  48. See Ref. 5, Part II, p. 1010.
  49. See Ref. 5, p. 1009.
  50. See Ref. 4, p. 338.
  51. See Ref. 4, p. 330.
  52. P. R. Halmos, Measure Theory (D. Van Nostrand Co., Inc., New York, 1950), 3rd ed., p. 62.
  53. See Ref. 31, p. 52.
  54. See Ref. 17, p. 104.
  55. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Graylock Press, Rochester, N. Y., 1957) Vol. I, p. 63.
  56. Ref. 52, p. 110.
  57. Ref. 52, p. 147.
  58. K. Knopp, Theory of Functions (Dover Publications, Inc., New York, 1945) Vol. 1, p. 87.
  59. See Ref. 31, p. 16.
  60. See Ref. 4, p. 311.
  61. See Ref. 3, p. 231-234.
  62. See Ref. 31, p. 27.
  63. See Ref. 5, p. 1012.
  64. See Ref. 31, p. 27.
  65. See Ref. 5, p. 1025.
  66. See Ref. 5, p. 1032.
  67. See Ref. 31, p. 40.
  68. See Ref. 58, p. 73.
  69. See Ref. 2, p. 18.
  70. See Ref. 58, p. 74.
  71. See Ref. 58, p. 87.

Arsac, J.

J. Arsac, Fourier Transforms and the Theory of Distributions (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1966).

Barlow, H. M.

H. M. Barlow and J. Brown, Radio Surface Waves (Oxford University Press, 1962).

Born, M.

M. Born and E. Wolf, Principles of Optics (The Macmillan Co., New York, 1959), p. 492.

Brown, J.

H. M. Barlow and J. Brown, Radio Surface Waves (Oxford University Press, 1962).

Dunford, N.

N. Dunford and J. Schwartz, Linear Operators (Interscience Publishers, John Wiley & Sons, Inc., New York, 1957, 1963), Parts I, II.

Ehrlich, J. J.

J. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 342 (1955).

Fomin, S. V.

A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Graylock Press, Rochester, N. Y., 1957) Vol. I, p. 63.

Gel’fand, I. M.

I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vols. 1-3 (Academic Press Inc., New York, 1964); I. M. Gel'fand and N. Vilenkin, Generalized Functions, Vol. 4 (Academic Press Inc., New York, 1964); I. M. Gel'fand, M. I. Graev, and N. Vilenkin, Generalized Functions, Vol. 5 (Academic Press Inc.. New York, 1966).

Glauber, R. J.

R. J. Glauber, Phys. Rev. 130, 2529 (1963).

Halmos, P. R.

P. R. Halmos, Finite-Dimensional Vector Spaces (D. Van Nostrand Co., Inc., Princeton, N. J., 1958), 2nd ed., pp. 71, 72.

P. R. Halmos, Measure Theory (D. Van Nostrand Co., Inc., New York, 1950), 3rd ed., p. 62.

P. R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1957), 2nd ed., p. 51.

Held, G.

J. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 342 (1955).

Knopp, K.

K. Knopp, Theory of Functions (Dover Publications, Inc., New York, 1945) Vol. 1, p. 87.

Kolmogorov, A. N.

A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Graylock Press, Rochester, N. Y., 1957) Vol. I, p. 63.

Lighthill, M. J.

M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, 1962).

Luneburg, R. K.

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

Montgomery, W. D.

W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).

Nagy, B.

F. Riesz and B. Nagy, Functional Analysis (Frederick Ungar Publishing Co., New York, 1965).

Pollak, H.

D. Slepian and H. Pollak, Bell System Tech. J. 40, 43 (1961).

Riesz, F.

F. Riesz and B. Nagy, Functional Analysis (Frederick Ungar Publishing Co., New York, 1965).

Schwartz, J.

N. Dunford and J. Schwartz, Linear Operators (Interscience Publishers, John Wiley & Sons, Inc., New York, 1957, 1963), Parts I, II.

Schwartz, L.

L. Schwartz, Théorie des Distributions (Hermann, Paris, 1957-1959) Vols. I, II.

Sherman, G.

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

Shewell, J. R.

E. Wolf and J. R. Shewell, Phys. Letters 25A, 417 (1967).

Shilov, G. E.

I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vols. 1-3 (Academic Press Inc., New York, 1964); I. M. Gel'fand and N. Vilenkin, Generalized Functions, Vol. 4 (Academic Press Inc., New York, 1964); I. M. Gel'fand, M. I. Graev, and N. Vilenkin, Generalized Functions, Vol. 5 (Academic Press Inc.. New York, 1966).

Silver, S.

J. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 342 (1955).

Slepian, D.

D. Slepian and H. Pollak, Bell System Tech. J. 40, 43 (1961).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).

Stone, M. H.

M. H. Stone, Linear Transformations in Hilbert Space (American Mathematical Society, New York, 1932), p. 23.

Titchmarch, E. C.

E. C. Titchmarch, Introduction to the Theory of Fourier Integrals (Oxford University Press, 1948) pp. 75, 76.

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, 1962), 4th ed.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, 1962), 4th ed.

Wolf, E.

E. Wolf and J. R. Shewell, Phys. Letters 25A, 417 (1967).

E. Wolf in, Progress in Optics IV (North-Holland Publishing Company, Amsterdam, 1965), p. 201.

M. Born and E. Wolf, Principles of Optics (The Macmillan Co., New York, 1959), p. 492.

Zaanen, A.

A. Zaanen, Linear Analysis (North-Holland Publishing Co., Amsterdam, 1956).

Zemanian, A. H.

A. H. Zemanian, Distribution Theory and Transformn Analysis (McGraw-Hill Book Co., New York, 1965).

Other

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

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, 1962), 4th ed.

F. Riesz and B. Nagy, Functional Analysis (Frederick Ungar Publishing Co., New York, 1965).

A. Zaanen, Linear Analysis (North-Holland Publishing Co., Amsterdam, 1956).

N. Dunford and J. Schwartz, Linear Operators (Interscience Publishers, John Wiley & Sons, Inc., New York, 1957, 1963), Parts I, II.

L. Schwartz, Théorie des Distributions (Hermann, Paris, 1957-1959) Vols. I, II.

I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vols. 1-3 (Academic Press Inc., New York, 1964); I. M. Gel'fand and N. Vilenkin, Generalized Functions, Vol. 4 (Academic Press Inc., New York, 1964); I. M. Gel'fand, M. I. Graev, and N. Vilenkin, Generalized Functions, Vol. 5 (Academic Press Inc.. New York, 1966).

M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, 1962).

A. H. Zemanian, Distribution Theory and Transformn Analysis (McGraw-Hill Book Co., New York, 1965).

J. Arsac, Fourier Transforms and the Theory of Distributions (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1966).

W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).

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

E. Wolf and J. R. Shewell, Phys. Letters 25A, 417 (1967).

M. Born and E. Wolf, Principles of Optics (The Macmillan Co., New York, 1959), p. 492.

See Ref. 1, p. 6.

R. J. Glauber, Phys. Rev. 130, 2529 (1963).

M. H. Stone, Linear Transformations in Hilbert Space (American Mathematical Society, New York, 1932), p. 23.

In vector notation, the electric vector E(r,t) = φ(r)e-iωt+φ* (r)eiωt where r= (x,y,z) and the components of φe-iωt are the analytic signals of the Cartesian components of E. From this we have φ(x,z). φ*(x,z)dx = ½(E2)tdx, where φ.φ* denotes the dot product between φ and its conjugate and 〈〉t is the time average over a period. Thus the integral | φ (x) | 2dx in vector notation denotes one half of the time-average energy in the electric field per unit length in the z direction. We will henceforth refer to this integral as simply the energy in φ. The z parameter (for z≥0) will frequently be supressed in such functions as φ, f, g, h etc.

See Ref. 1, pp. 311-319.

See Ref. 14, p. 582.

See Ref. 3, p. 145.

E. C. Titchmarch, Introduction to the Theory of Fourier Integrals (Oxford University Press, 1948) pp. 75, 76.

See Ref. 17, pp. 104, 105.

See Ref. 4, p. 134.

See Ref. 7, Vol. 4, p. 105.

For a scalar theory we may take all of £2 for the domain of definition of Dz. For the vector theory (with none of the source in the z = 0 plane), it can readily be shown that all of Maxwell's equations are satisfied for z > 0 if and only if ϱ.Ff = 0 for all ¾, where the vector ϱ(ξ) = [ξ, η, (λ-2 - ξ2)½] and f(x) = φ(x,0) as given in Ref. 18. All complex-boundary fields f satisfying this relation constitute a proper subspace Γ of the vector £ functions. Therefore, in the vector theory, the domain of Dz is a proper subspace of £2.

See Ref. 11, for a partial list.

See Ref. 7, Vol. 4, p. 103.

See Ref. 9, p. 103.

See Ref. 4, p. 301.

P. R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1957), 2nd ed., p. 51.

See Ref. 4, p. 305.

See Ref. 31, p. 51.

See Ref. 14, pp. 558-560.

H. M. Barlow and J. Brown, Radio Surface Waves (Oxford University Press, 1962).

P. R. Halmos, Finite-Dimensional Vector Spaces (D. Van Nostrand Co., Inc., Princeton, N. J., 1958), 2nd ed., pp. 71, 72.

See Ref. 4, p. 306.

In the vector theory we must deal with the subspace ß∩M (see Ref. 26) in place of ß. (ß∩M denotes the space of functions common to ß∩M is invariant under Dz.

See Ref. 3, p. 380.

See Ref. 14, p. 553.

E. Wolf in, Progress in Optics IV (North-Holland Publishing Company, Amsterdam, 1965), p. 201.

See Ref. 41, p. 283.

This result follows from Appendix D regardless of the size of the screen, as long as it has positive area.

J. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 342 (1955).

Apart from the extensive literature which shows that the evanescent waves are surface waves, existing only in close proximity (of the order of a wavelength λ) to matter, we can see from Eq. (11) that they obey a dispersive relation. That is, the phase velocity ν/|ξ| is a function of the wavenumber. These things tell tell us that we should not expect to measure contributions from such waves in the aperture unless the aperture is small (diameter of the order of λ).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).

D. Slepian and H. Pollak, Bell System Tech. J. 40, 43 (1961).

See Ref. 5, Part II, p. 1010.

See Ref. 5, p. 1009.

See Ref. 4, p. 338.

See Ref. 4, p. 330.

P. R. Halmos, Measure Theory (D. Van Nostrand Co., Inc., New York, 1950), 3rd ed., p. 62.

See Ref. 31, p. 52.

See Ref. 17, p. 104.

A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Graylock Press, Rochester, N. Y., 1957) Vol. I, p. 63.

Ref. 52, p. 110.

Ref. 52, p. 147.

K. Knopp, Theory of Functions (Dover Publications, Inc., New York, 1945) Vol. 1, p. 87.

See Ref. 31, p. 16.

See Ref. 4, p. 311.

See Ref. 3, p. 231-234.

See Ref. 31, p. 27.

See Ref. 5, p. 1012.

See Ref. 31, p. 27.

See Ref. 5, p. 1025.

See Ref. 5, p. 1032.

See Ref. 31, p. 40.

See Ref. 58, p. 73.

See Ref. 2, p. 18.

See Ref. 58, p. 74.

See Ref. 58, p. 87.

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