Abstract

The practice of using complex-valued notation in classical electromagnetic theory is shown to have a more fundamental basis than that of simplifying mathematical manipulations. In the algebraic formulation, the positive and negative time-frequency components of the real-valued field quantities are propagated by two different diffraction operators, each of which is the dual (adjoint) of the other. In addition to the basic distinguishability of these positive and negative time-frequency components, other features distinguish them in general, owing to the constraint of Maxwell’s equations. In the monochromatic case, there are boundary fields for which these other features vanish. For some of these fields, a pseudoscopic image can be propagated. The appearance and location of the pseudoscopic image in holography is shown to be a direct consequence of the unitary property of the diffraction operator in the absence of evanescent waves.

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  1. W. Heitler, The Quantum Theory of Radiation (Clarendon Press, Oxford, 1954), pp. 3–5.
  2. M. H. Stone, Linear Transformations in Hilbert Space (American Mathematical Society, New York, 1932), p. 23.
  3. W. D. Montgomery, J. Opt. Soc. Am. 58, 1112 (1968).
  4. The emphasis in the present paper is on the vector potential A rather than on E, as in (I). The reason for this is that the potential enters directly into the interaction hamiltonian between radiation and electron, as well as the fact that the total electromagnetic energy is simply written in terms of the vector potential. See P.A.M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1958), pp. 239–241.
  5. R. K. Luneburg, Mathematical Theory of Optics (Univ. of Calif. Press, Berkeley, 1964), pp. 311–319.
  6. N. Dunford and J. Schwartz, Linear Operators Part I: General Theory (Wiley-Interscience, Inc., New York, 1967), pp. 119–121, 241.
  7. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon Press Ltd., London, 1966), p. 33.
  8. This equivalence has been proven earlier by C. J. Bouwkamp, Rept. Progr. Phys. 17, 39 (1954) and G. C. Sherman, J. Opt. Soc. Am. 57, 546 (1967) in mathematical contexts having less algebraic structure. Part of the equivalence proof in (I) was later repeated by E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).
  9. We will term the behavior of Dz on b as that of a homogeneous operator, because it acts on each component separately.
  10. Reference 2, pp. 29–32.
  11. Note that Dz*, as a homogeneous operator in H, is still the adjoint of Dz in H.
  12. Equation (16) is implicit in Eq. 2.107 of Clemmow (Ref. 7), p. 33.
  13. A subset m of the Hilbert space H is a subspace if it is both a linear manifold and closed. A linear manifold M is one for which f, g∈M implies αf +βg∈M for all complex numbers α, β. It is closed (in the norm) if for every sequence {fn}⊆ M such that ‖fn — f‖ → 0 as n → ∞ for some f∈H we must have f∈M.
  14. P. R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1957), p. 22.
  15. R. Mittra and P. Ransom in Proceedings of the Symposium on Modern Optics, Jerome Fox, Ed. (Polytechnic Press, New York, 1967), pp. 619–647.
  16. G. C. Sherman, J. Opt. Soc. Am. 57, 1490 (1967).
  17. E. Wolf and J. Shewell, Phys. Letters 25A, 417 (1967).
  18. E. Lalor, J. Math. Phys. 9, 12, 2001 (1968).
  19. J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968).
  20. W. D. Montgomery, J. Opt. Soc. Am. 59, 136 (1969).
  21. W. D. Montgomery, J. Opt. Soc. Am. 58, 720A (1968).
  22. G. C. Sherman, Phys. Rev. Letters 21, 761 (1968).
  23. W. D. Montgomery, Nuovo Cimento LIX, 137 (1969).
  24. Reference 2, pp. 104, 105.
  25. Reference 14, pp. 24, 25.
  26. Reference 6, p. 190.

Bouwkamp, C. J.

This equivalence has been proven earlier by C. J. Bouwkamp, Rept. Progr. Phys. 17, 39 (1954) and G. C. Sherman, J. Opt. Soc. Am. 57, 546 (1967) in mathematical contexts having less algebraic structure. Part of the equivalence proof in (I) was later repeated by E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon Press Ltd., London, 1966), p. 33.

Dirac, P.A.M.

The emphasis in the present paper is on the vector potential A rather than on E, as in (I). The reason for this is that the potential enters directly into the interaction hamiltonian between radiation and electron, as well as the fact that the total electromagnetic energy is simply written in terms of the vector potential. See P.A.M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1958), pp. 239–241.

Dunford, N.

N. Dunford and J. Schwartz, Linear Operators Part I: General Theory (Wiley-Interscience, Inc., New York, 1967), pp. 119–121, 241.

Halmos, P. R.

P. R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1957), p. 22.

Heitler, W.

W. Heitler, The Quantum Theory of Radiation (Clarendon Press, Oxford, 1954), pp. 3–5.

Lalor, E.

E. Lalor, J. Math. Phys. 9, 12, 2001 (1968).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Univ. of Calif. Press, Berkeley, 1964), pp. 311–319.

Mittra, R.

R. Mittra and P. Ransom in Proceedings of the Symposium on Modern Optics, Jerome Fox, Ed. (Polytechnic Press, New York, 1967), pp. 619–647.

Montgomery, W. D.

W. D. Montgomery, J. Opt. Soc. Am. 59, 136 (1969).

W. D. Montgomery, J. Opt. Soc. Am. 58, 720A (1968).

W. D. Montgomery, J. Opt. Soc. Am. 58, 1112 (1968).

W. D. Montgomery, Nuovo Cimento LIX, 137 (1969).

Ransom, P.

R. Mittra and P. Ransom in Proceedings of the Symposium on Modern Optics, Jerome Fox, Ed. (Polytechnic Press, New York, 1967), pp. 619–647.

Schwartz, J.

N. Dunford and J. Schwartz, Linear Operators Part I: General Theory (Wiley-Interscience, Inc., New York, 1967), pp. 119–121, 241.

Sherman, G. C.

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

G. C. Sherman, Phys. Rev. Letters 21, 761 (1968).

Shewell, J.

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

Shewell, J. R.

J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968).

Stone, M. H.

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

Wolf, E.

J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968).

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

Other (26)

W. Heitler, The Quantum Theory of Radiation (Clarendon Press, Oxford, 1954), pp. 3–5.

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

W. D. Montgomery, J. Opt. Soc. Am. 58, 1112 (1968).

The emphasis in the present paper is on the vector potential A rather than on E, as in (I). The reason for this is that the potential enters directly into the interaction hamiltonian between radiation and electron, as well as the fact that the total electromagnetic energy is simply written in terms of the vector potential. See P.A.M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1958), pp. 239–241.

R. K. Luneburg, Mathematical Theory of Optics (Univ. of Calif. Press, Berkeley, 1964), pp. 311–319.

N. Dunford and J. Schwartz, Linear Operators Part I: General Theory (Wiley-Interscience, Inc., New York, 1967), pp. 119–121, 241.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon Press Ltd., London, 1966), p. 33.

This equivalence has been proven earlier by C. J. Bouwkamp, Rept. Progr. Phys. 17, 39 (1954) and G. C. Sherman, J. Opt. Soc. Am. 57, 546 (1967) in mathematical contexts having less algebraic structure. Part of the equivalence proof in (I) was later repeated by E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).

We will term the behavior of Dz on b as that of a homogeneous operator, because it acts on each component separately.

Reference 2, pp. 29–32.

Note that Dz*, as a homogeneous operator in H, is still the adjoint of Dz in H.

Equation (16) is implicit in Eq. 2.107 of Clemmow (Ref. 7), p. 33.

A subset m of the Hilbert space H is a subspace if it is both a linear manifold and closed. A linear manifold M is one for which f, g∈M implies αf +βg∈M for all complex numbers α, β. It is closed (in the norm) if for every sequence {fn}⊆ M such that ‖fn — f‖ → 0 as n → ∞ for some f∈H we must have f∈M.

P. R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1957), p. 22.

R. Mittra and P. Ransom in Proceedings of the Symposium on Modern Optics, Jerome Fox, Ed. (Polytechnic Press, New York, 1967), pp. 619–647.

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

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

E. Lalor, J. Math. Phys. 9, 12, 2001 (1968).

J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968).

W. D. Montgomery, J. Opt. Soc. Am. 59, 136 (1969).

W. D. Montgomery, J. Opt. Soc. Am. 58, 720A (1968).

G. C. Sherman, Phys. Rev. Letters 21, 761 (1968).

W. D. Montgomery, Nuovo Cimento LIX, 137 (1969).

Reference 2, pp. 104, 105.

Reference 14, pp. 24, 25.

Reference 6, p. 190.

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