Abstract

Several theorems are formulated, regarding symmetry relations between two monochromatic fields that propagate either into the same half-space (<i>z</i> > 0) or into two complementary half-spaces (<i>z</i> > 0 and <i>z</i> < 0) and that satisfy one of two simple phase-conjugacy conditions in a cross sectional plane <i>z</i> = constant. The theorems are rigorously valid for fields whose two-dimensional spatial-frequency spectrum in the cross sectional plane is bandlimited to a circle of radius equal to the wave number of the field. One of the theorems elucidates some recently predicted symmetry properties of focused fields.

© 1980 Optical Society of America

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  1. For reviews of this subject see, for example, A. Yariv, "Phase Conjugate Optics and Real-Time, Holography," IEEE J. Quantum Electron. QE-14, 650–660 (1978) or J. AuYeung and A. Yariv, "Phase-Conjugate Optics," Opt. News, 5, 13–17 (1979).
  2. E. Collett and E. Wolf, "Symmetry Properties of Focused Fields," Opt. Lett., 5, 264–266 (1980).
  3. Sufficiency conditions are discussed in papers by (a) W. D. Montgomery, "Algebraic Formulation of Diffraction Applied to Self Imaging," J. Opt. Soc. Am. 58, 1112–1124 (1968); (b) E. Lalor, "Conditions for the Validity of the Angular Spectrum of Plane Waves," J. Opt. Soc. Am. 58, 1235–1237 (1968); (c) G. C. Sherman, "Diffracted Wavefields Expressible by Plane-Wave Expansions Containing Only Homogeneous Waves," J. Opt. Soc. Am. 59, 697–711 (1969).
  4. Although not explicity stated, this theorem is implicit in the analysis of J. R. Shewell and E. Wolf, "Inverse Diffraction and a New Reciprocity Theorem," J. Opt. Soc. Am. 58, 1596–1603 (1968), Sec. III.
  5. Ref. 3(c) above. See also G. C. Sherman, "Diffracted Wavefields Expressible by Plane-Wave Expansions Containing Only Homogeneous Waves," Phys. Rev. Lett., 21, 761–764 (1968); ibid. 21, 1220(E) (1968).
  6. An example is provided by a Gaussian laser beam. If the spot size of the beam is ω0, the angular spread of the beam is well known to be given by θ = λ/πω0, where λ is the wavelength. With ω0>>λ, as is always the case in practice, the effective values of p and q are then restricted to the domainp2+q2≤sin2θ≈θ2=(λ/πω0)2≪1,as may readily be established with the help of the asymptotic formula (1.15a). Now, as is clear from (1.8), there is a one to one correspondence between the directional parameters (p,q) and the spatial frequencies (u,v), namely, p = u/k, q = v/k. Hence the above inequality implies that in any cross section perpendicular to the beam axis, the beam field is effectively bandlimited to the spatial frequency domainu2+v2=k2(λ/πω0)2k2.Typically, the ratio λ/πω0 may be of the order of 10-4.
  7. Theorem II, p. 701 of Ref. 3(c) above.
  8. It is not difficult to show that if the bandlimited wavefield U(x,y,z) is defined throughout the whole space, it is bandlimited to the spatial-frequency domain u2 + v2k2 in any planar cross section, whether or not the cross section is perpendicular to the z direction.
  9. K. Miyamoto and E. Wolf, "Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave—Part I," J. Opt. Soc. Am. 52, 615–625 (1962), Appendix.
  10. M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.
  11. A strictly analogous result to that stated in Footnote 8 then applies: The "extended" wavefield V(x,y,z) may be shown to be bandlimited to the spatial-frequency domain u2 + v2k2 in any planar cross section.
  12. W. Lukosz, "Equivalent-Lens Theory of Holographic Imaging," J. Opt. Soc. Am. 58, 1084–1091 (1968), Sec. III. Our method of proof is similar to that given by Lukosz but it is free of an inaccuracy contained in his derivation. Lukosz did not impose any restriction regarding bandlimitation; however, it is not difficult to see that the theorem then no longer holds. In fact, the angular spectrum integral then diverges in the half-space z < 0.
  13. R. Mittra and P. L. Ransom, "Imaging with Coherent Fields," in Modern Optics, edited by J. Fox (Polytechnic, Brooklyn, 1967; Wiley, New York, distr.), pp. 619–647.
  14. P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30, 755–776 (1909).

1980 (1)

1978 (1)

For reviews of this subject see, for example, A. Yariv, "Phase Conjugate Optics and Real-Time, Holography," IEEE J. Quantum Electron. QE-14, 650–660 (1978) or J. AuYeung and A. Yariv, "Phase-Conjugate Optics," Opt. News, 5, 13–17 (1979).

1968 (4)

1962 (1)

1909 (1)

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30, 755–776 (1909).

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.

Collett, E.

Debye, P.

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30, 755–776 (1909).

Lukosz, W.

Mittra, R.

R. Mittra and P. L. Ransom, "Imaging with Coherent Fields," in Modern Optics, edited by J. Fox (Polytechnic, Brooklyn, 1967; Wiley, New York, distr.), pp. 619–647.

Miyamoto, K.

Montgomery, W. D.

Ransom, P. L.

R. Mittra and P. L. Ransom, "Imaging with Coherent Fields," in Modern Optics, edited by J. Fox (Polytechnic, Brooklyn, 1967; Wiley, New York, distr.), pp. 619–647.

Sherman, G. C.

Ref. 3(c) above. See also G. C. Sherman, "Diffracted Wavefields Expressible by Plane-Wave Expansions Containing Only Homogeneous Waves," Phys. Rev. Lett., 21, 761–764 (1968); ibid. 21, 1220(E) (1968).

Shewell, J. R.

Wolf, E.

Yariv, A.

For reviews of this subject see, for example, A. Yariv, "Phase Conjugate Optics and Real-Time, Holography," IEEE J. Quantum Electron. QE-14, 650–660 (1978) or J. AuYeung and A. Yariv, "Phase-Conjugate Optics," Opt. News, 5, 13–17 (1979).

Ann. Phys. (1)

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30, 755–776 (1909).

IEEE J. Quantum Electron. (1)

For reviews of this subject see, for example, A. Yariv, "Phase Conjugate Optics and Real-Time, Holography," IEEE J. Quantum Electron. QE-14, 650–660 (1978) or J. AuYeung and A. Yariv, "Phase-Conjugate Optics," Opt. News, 5, 13–17 (1979).

J. Opt. Soc. Am. (4)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

Ref. 3(c) above. See also G. C. Sherman, "Diffracted Wavefields Expressible by Plane-Wave Expansions Containing Only Homogeneous Waves," Phys. Rev. Lett., 21, 761–764 (1968); ibid. 21, 1220(E) (1968).

Other (6)

An example is provided by a Gaussian laser beam. If the spot size of the beam is ω0, the angular spread of the beam is well known to be given by θ = λ/πω0, where λ is the wavelength. With ω0>>λ, as is always the case in practice, the effective values of p and q are then restricted to the domainp2+q2≤sin2θ≈θ2=(λ/πω0)2≪1,as may readily be established with the help of the asymptotic formula (1.15a). Now, as is clear from (1.8), there is a one to one correspondence between the directional parameters (p,q) and the spatial frequencies (u,v), namely, p = u/k, q = v/k. Hence the above inequality implies that in any cross section perpendicular to the beam axis, the beam field is effectively bandlimited to the spatial frequency domainu2+v2=k2(λ/πω0)2k2.Typically, the ratio λ/πω0 may be of the order of 10-4.

Theorem II, p. 701 of Ref. 3(c) above.

It is not difficult to show that if the bandlimited wavefield U(x,y,z) is defined throughout the whole space, it is bandlimited to the spatial-frequency domain u2 + v2k2 in any planar cross section, whether or not the cross section is perpendicular to the z direction.

M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.

A strictly analogous result to that stated in Footnote 8 then applies: The "extended" wavefield V(x,y,z) may be shown to be bandlimited to the spatial-frequency domain u2 + v2k2 in any planar cross section.

R. Mittra and P. L. Ransom, "Imaging with Coherent Fields," in Modern Optics, edited by J. Fox (Polytechnic, Brooklyn, 1967; Wiley, New York, distr.), pp. 619–647.

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