Abstract

A theory relating to correction of distortions that may be achieved by phase conjugation is developed on the basis of the first Born approximation. It is shown that, to good accuracy, the effect of a distorting medium on an incident wave is eliminated by phase conjugation if the following conditions are satisfied: the incident field contains no evanescent components, the transmitting medium is a weak, nonabsorbing scatterer, and backscattering of the incident and of the conjugate wave and also the effects of scattered evanescent waves are negligible.

© 1982 Optical Society of America

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  1. 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).
  2. 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).
  3. 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).
  4. 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).
  5. In this connection see E. Wolf and W. H. Carter, "Comments on the theory of phase-conjugated waves," (to be published). Opt. Commun.
  6. 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).
  7. From now on we omit the time-dependent factor e-iωt, and the term "field" or "wave" refers to their space-dependent parts.
  8. 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.
  9. 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.
  10. E. Wolf, "Three-dimensional structure determination of semitransparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).
  11. 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).
  12. 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 crystalstructure determination as a branch of physical optics," Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 287–350, Sec. 4.
  13. 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.
  14. It is important to appreciate that when z < L 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.
  15. This result is, essentially, a three-dimensional analog of a wellknown 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. 352 et seq.].

1980 (1)

1977 (2)

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).

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).

1976 (2)

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).

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.

1972 (1)

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).

1969 (2)

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.

E. Wolf, "Three-dimensional structure determination of semitransparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).

1968 (1)

Bjorklund, G. C.

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).

Bloom, D. N.

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).

Carter, W. H.

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

Faizullov, F. S.

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).

Hellwarth, R. W.

Montgomery, W. D.

Pattanayak, D. N.

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.

Popovichev, V. I.

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).

Ragul’skii, V. V.

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).

Sherman, G. C.

Wolf, E.

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.

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.

E. Wolf, "Three-dimensional structure determination of semitransparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).

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

Yariv, A.

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).

Zel’dovich, B. Ya.

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).

Appl. Phys. Lett. (2)

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).

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).

J. Opt. Soc. Am. (4)

Opt. Commun. (1)

E. Wolf, "Three-dimensional structure determination of semitransparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).

Phys. Rev. D (1)

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.

Sov. Phys. JETP (1)

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).

Other (6)

It is important to appreciate that when z < L 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 wellknown 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. 352 et seq.].

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

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).

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 crystalstructure determination as a branch of physical optics," Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 287–350, Sec. 4.

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

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