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).

[Crossref]

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.

[Crossref]

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).

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.

[Crossref]

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).

[Crossref]

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.

[Crossref]

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

[Crossref]

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

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

[Crossref]

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.

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.

[Crossref]

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.

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).

[Crossref]

E. Collett and E. Wolf, “Symmetry Properties of Focused Fields,” Opt. Lett., 5, 264–266 (1980).

[Crossref]
[PubMed]

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.

[Crossref]

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.

[Crossref]

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

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).

[Crossref]

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

[Crossref]

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).

[Crossref]

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.

[Crossref]

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.

[Crossref]

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).

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.

[Crossref]

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).

[Crossref]

An example is provided by a Gaussian laser beam. If the spot size of the beam is w0, the angular spread of the beam is well known to be given by Θ = λ/πw0, where λ is the wavelength. With w0 ≫ 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/πw0)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,υ), namely, p = u/k, q = υ/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+υ2=k2(λ/πw0)2≪k2.Typically, the ratio λ/πw0 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+υ2⩽k2 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+υ2⩽k2 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.