E. Wolf, “Phase conjugacy and symmetries in spatially bandlimited wavefields containing no evanescent components,” J. Opt. Soc. Am. 70, 1311–1319 (1980). Equation (2.1) of this reference contains a misprint. U(2)(x, y, z)eiωt should be replaced by U(2)(x, y, z)e−iωt. Also, Eq. (1.8) should read A(u/k, v/k) = k2Ũ(u, v; z)e−iwz. These corrections do not affect any other equations or conclusions of that paper.

[CrossRef]

R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977); A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four wave mixing,” Opt. Lett. 1, 16–18 (1977).

[CrossRef]

See, for example, Ref. 1 and D. N. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977); V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).

[CrossRef]
[PubMed]

See, for example, D. N. Pattanayak and E. Wolf, “Scattering states and bound states as solutions of the Schrödinger equation with nonlocal boundary conditions,” Phys. Rev. D 13, 913–923 (1976), Sec. II.

[CrossRef]

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976); “On transmission and recovery of three-dimensional image information in optical waveguides,” J. Opt. Soc. Am. 66, 301–306 (1976).

[CrossRef]

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972); O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Sov. Phys. JETP 16, 435–438 (1972).

This formula can be obtained by a straightforward modification of a well-known representation that is due to H. Weyl, derived in his paper, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919). See also A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966), Eq. (2.19).

See, for example, Ref. 1 and D. N. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977); V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).

[CrossRef]
[PubMed]

See, for example, Ref. 1 and D. N. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977); V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).

[CrossRef]
[PubMed]

In this connection see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” (to be published). Opt. Commun.

This result is, essentially, a three-dimensional analog of a well-known theorem that the Fourier transform of a continuous function that vanishes outside a finite interval is a boundary value of an entire analytic function. This theorem follows at once from a well-known result in the theory of analytic functions defined by definite integrals [cf. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford U. Press, London, 1935), Sec. 5.5]. The multidimensional form of the theorem is the well-known Plancherel–Pólya theorem [cf. B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, R.I., 1963), pp. 352et seq.].

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972); O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Sov. Phys. JETP 16, 435–438 (1972).

In more picturesque language, condition (3.10) is equivalent to the neglection of spatial Fourier components of the scattering potential for which the representative K vectors lie outside Ewald’s limiting sphere. [See, for example, R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1948), pp. 14–15, or H. Lipson and C. A. Taylor, “X-ray crystal-structure determination as a branch of physical optics,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 287–350, Sec. 4.

[CrossRef]

See, for example, D. N. Pattanayak and E. Wolf, “Scattering states and bound states as solutions of the Schrödinger equation with nonlocal boundary conditions,” Phys. Rev. D 13, 913–923 (1976), Sec. II.

[CrossRef]

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972); O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Sov. Phys. JETP 16, 435–438 (1972).

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972); O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Sov. Phys. JETP 16, 435–438 (1972).

This formula can be obtained by a straightforward modification of a well-known representation that is due to H. Weyl, derived in his paper, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919). See also A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966), Eq. (2.19).

E. Wolf, “Phase conjugacy and symmetries in spatially bandlimited wavefields containing no evanescent components,” J. Opt. Soc. Am. 70, 1311–1319 (1980). Equation (2.1) of this reference contains a misprint. U(2)(x, y, z)eiωt should be replaced by U(2)(x, y, z)e−iωt. Also, Eq. (1.8) should read A(u/k, v/k) = k2Ũ(u, v; z)e−iwz. These corrections do not affect any other equations or conclusions of that paper.

[CrossRef]

See, for example, D. N. Pattanayak and E. Wolf, “Scattering states and bound states as solutions of the Schrödinger equation with nonlocal boundary conditions,” Phys. Rev. D 13, 913–923 (1976), Sec. II.

[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).

[CrossRef]

In this connection see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” (to be published). Opt. Commun.

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976); “On transmission and recovery of three-dimensional image information in optical waveguides,” J. Opt. Soc. Am. 66, 301–306 (1976).

[CrossRef]

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972); O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Sov. Phys. JETP 16, 435–438 (1972).

This formula can be obtained by a straightforward modification of a well-known representation that is due to H. Weyl, derived in his paper, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919). See also A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966), Eq. (2.19).

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976); “On transmission and recovery of three-dimensional image information in optical waveguides,” J. Opt. Soc. Am. 66, 301–306 (1976).

[CrossRef]

See, for example, Ref. 1 and D. N. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977); V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).

[CrossRef]
[PubMed]

Sufficiency conditions are discussed by W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968); E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968); G. C. Sherman, “Diffracted wavefields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).

[CrossRef]

G. C. Sherman, “Diffracted wavefields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969), Theorem II, p. 701.

[CrossRef]

R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977); A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four wave mixing,” Opt. Lett. 1, 16–18 (1977).

[CrossRef]

E. Wolf, “Phase conjugacy and symmetries in spatially bandlimited wavefields containing no evanescent components,” J. Opt. Soc. Am. 70, 1311–1319 (1980). Equation (2.1) of this reference contains a misprint. U(2)(x, y, z)eiωt should be replaced by U(2)(x, y, z)e−iωt. Also, Eq. (1.8) should read A(u/k, v/k) = k2Ũ(u, v; z)e−iwz. These corrections do not affect any other equations or conclusions of that paper.

[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).

[CrossRef]

See, for example, D. N. Pattanayak and E. Wolf, “Scattering states and bound states as solutions of the Schrödinger equation with nonlocal boundary conditions,” Phys. Rev. D 13, 913–923 (1976), Sec. II.

[CrossRef]

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972); O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Sov. Phys. JETP 16, 435–438 (1972).

In this connection see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” (to be published). Opt. Commun.

It is important to appreciate that when zL the integral on the right-hand side of formula (3.12) does not represent, even approximately, the scattered field that arises from the interaction of the incident field U(i) with the scatterer. This can be most easily seen by examining the asymptotic behavior of the integral in formula (3.12), in the half-space z 0, at large distances from the origin. One finds that the integral represents an incoming wave in that half-space, whereas the scattered field must clearly behave there as an outgoing wave.

This result is, essentially, a three-dimensional analog of a well-known theorem that the Fourier transform of a continuous function that vanishes outside a finite interval is a boundary value of an entire analytic function. This theorem follows at once from a well-known result in the theory of analytic functions defined by definite integrals [cf. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford U. Press, London, 1935), Sec. 5.5]. The multidimensional form of the theorem is the well-known Plancherel–Pólya theorem [cf. B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, R.I., 1963), pp. 352et seq.].

In more picturesque language, condition (3.10) is equivalent to the neglection of spatial Fourier components of the scattering potential for which the representative K vectors lie outside Ewald’s limiting sphere. [See, for example, R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1948), pp. 14–15, or H. Lipson and C. A. Taylor, “X-ray crystal-structure determination as a branch of physical optics,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 287–350, Sec. 4.

[CrossRef]

From now on we omit the time-dependent factor e−iωt, and the term “field” or “wave” refers to their space-dependent parts.