Abstract

The correction of wave distortions by the technique of optical phase conjugation is examined first on the basis of a newly derived integral equation for scattering of monochromatic scalar waves in the presence of a phase-conjugate mirror. The solution is developed in an iterative series, and the first- and second-order terms are analyzed and illustrated diagrammatically. A generalization of the integral equation is then presented, which takes into account the electromagnetic nature of light. It is also shown that, if the conjugate wave is generated without losses or gains and with a complete reversal of polarization, a total elimination of distortions may be achieved by this technique under circumstances that frequently occur in practice.

© 1983 Optical Society of America

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  1. For reviews of the technique of optical phase conjugation see, for example, A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978); D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 156–183 (1982).
    [Crossref]
  2. See, for example, 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); 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); and V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
    [Crossref] [PubMed]
  3. G. S. Agarwal and E. Wolf, “Theory of phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 321–326 (1982).
    [Crossref]
  4. G. S. Agarwal, A. T. Friberg, and E. Wolf, “Effect of backscattering in phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 861–863 (1982).
    [Crossref]
  5. G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
    [Crossref]
  6. Several schemes, such as those based on three-wave mixing [A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976)], four-wave mixing [R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977)], stimulated Brillouin scattering (Zel’dovich et al.2), etc. have been proposed and used for the generation of the conjugated field. The polarization properties of the conjugated field will, in general, be different from those of the incident (probe) field. A complete reversal of polarization can be achieved by suitably arranging the experimental geometry and by choosing the polarization properties of the various interacting waves appropriately [see, for example, Ref. 16 below and J. F. Lam, D. G. Steel, R. A. McFarlane, and R. C. Lind, “Atomic coherence effects in resonant degenerate four-wave mixing,” Appl. Phys. Lett. 38, 977–979 (1981)].
    [Crossref]
  7. This integral equation is derived, in the context of time-independent quantum-mechanical potential scattering, for example in P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.
  8. The concept of a PCM is a convenient idealization that describes the effect of a true physical device located beyond the plane z= z1, by means of which a field distribution U is replaced by a new field distribution μU* in that plane. The transformation U→ μU* is usually achieved by nonlinear optical interactions, such as stimulated scattering processes or optical parametric interactions [see Ref. 1].
  9. The term “conjugate” (or “conjugated”) field is somewhat ambigous and must be interpreted with caution. The field Uc(r) is generated as a result of the interaction of the incident field U(i)(r) with the scattering medium in the presence of the PCM. In general, it will include, in addition to U(i)(r), also contributions (usually ignored) arising from backscattering of the incident field and from successive conjugations of waves backscattered onto the PCM [see, for example, Figs. 3(a), 3(d), 4(a), and 4(h)].
  10. For a discussion of some of the complications that arise when the evanescent waves are taken into account, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982). The approximation resulting from the neglection of the evanescent waves in the scattered field is examined, within the accuracy of the first Born approximation, in App. A of Ref. 3.
    [Crossref]
  11. A. T. Friberg, “Integral equation of the scattered field in the presence of a phase-conjugate mirror” J. Opt. Soc. Am. (to be published.)
  12. Throughout this paper, a caret above a symbol denotes an operator.
  13. See, for example, A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966), Eq. (2.19).
  14. We will not discuss here the conditions under which series (3.1) will converge, a subject that would require a separate investigation.
  15. 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 with 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]
  16. B. Ya. Zel’dovich and V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979); B. Ya. Zel’dovich and T. V. Yakovleva, “Spatial-polarization wavefront reversal in stimulated scattering of the Rayleigh line wing,” Sov. J. Quantum Electron. 10, 501–505 (1980).
    [Crossref]
  17. Alternatively, by using the Maxwell equation ∇ · D= 0 (where D≡ E+ 4πP is the electric displacement vector), the constitutive relation D= n2E and the vector identity ∇ · (n2E) = n2∇ · E+ E· ∇n2, Eq. (7.2) may be rewritten in the familiar form ∇2E(r)+k2n2(r)E(r)+∇{E(r)·∇ log[n2(r)]}=0.However, form (7.2) is more convenient for the present purposes.
  18. See, for example, C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971), Sec. 14.
  19. G. S. Agarwal, “Dipole radiation in the presence of a phase conjugate mirror,” Opt. Commun. 42, 205–207 (1982).
    [Crossref]
  20. In this connection, see V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971); E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]

1982 (5)

G. S. Agarwal and E. Wolf, “Theory of phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 321–326 (1982).
[Crossref]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Effect of backscattering in phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 861–863 (1982).
[Crossref]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[Crossref]

For a discussion of some of the complications that arise when the evanescent waves are taken into account, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982). The approximation resulting from the neglection of the evanescent waves in the scattered field is examined, within the accuracy of the first Born approximation, in App. A of Ref. 3.
[Crossref]

G. S. Agarwal, “Dipole radiation in the presence of a phase conjugate mirror,” Opt. Commun. 42, 205–207 (1982).
[Crossref]

1980 (1)

1979 (1)

B. Ya. Zel’dovich and V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979); B. Ya. Zel’dovich and T. V. Yakovleva, “Spatial-polarization wavefront reversal in stimulated scattering of the Rayleigh line wing,” Sov. J. Quantum Electron. 10, 501–505 (1980).
[Crossref]

1978 (1)

For reviews of the technique of optical phase conjugation see, for example, A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978); D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 156–183 (1982).
[Crossref]

1976 (1)

Several schemes, such as those based on three-wave mixing [A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976)], four-wave mixing [R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977)], stimulated Brillouin scattering (Zel’dovich et al.2), etc. have been proposed and used for the generation of the conjugated field. The polarization properties of the conjugated field will, in general, be different from those of the incident (probe) field. A complete reversal of polarization can be achieved by suitably arranging the experimental geometry and by choosing the polarization properties of the various interacting waves appropriately [see, for example, Ref. 16 below and J. F. Lam, D. G. Steel, R. A. McFarlane, and R. C. Lind, “Atomic coherence effects in resonant degenerate four-wave mixing,” Appl. Phys. Lett. 38, 977–979 (1981)].
[Crossref]

1972 (1)

See, for example, 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); 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); and V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
[Crossref] [PubMed]

Agarwal, G. S.

G. S. Agarwal and E. Wolf, “Theory of phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 321–326 (1982).
[Crossref]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Effect of backscattering in phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 861–863 (1982).
[Crossref]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[Crossref]

G. S. Agarwal, “Dipole radiation in the presence of a phase conjugate mirror,” Opt. Commun. 42, 205–207 (1982).
[Crossref]

Baños, A.

See, for example, A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966), Eq. (2.19).

Carter, W. H.

For a discussion of some of the complications that arise when the evanescent waves are taken into account, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982). The approximation resulting from the neglection of the evanescent waves in the scattered field is examined, within the accuracy of the first Born approximation, in App. A of Ref. 3.
[Crossref]

Faizullov, F. S.

See, for example, 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); 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); and V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
[Crossref] [PubMed]

Friberg, A. T.

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[Crossref]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Effect of backscattering in phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 861–863 (1982).
[Crossref]

A. T. Friberg, “Integral equation of the scattered field in the presence of a phase-conjugate mirror” J. Opt. Soc. Am. (to be published.)

Popovichev, V. I.

See, for example, 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); 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); and V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
[Crossref] [PubMed]

Ragul’skii, V. V.

See, for example, 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); 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); and V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
[Crossref] [PubMed]

Roman, P.

This integral equation is derived, in the context of time-independent quantum-mechanical potential scattering, for example in P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.

Shkunov, V. V.

B. Ya. Zel’dovich and V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979); B. Ya. Zel’dovich and T. V. Yakovleva, “Spatial-polarization wavefront reversal in stimulated scattering of the Rayleigh line wing,” Sov. J. Quantum Electron. 10, 501–505 (1980).
[Crossref]

Tai, C.-T.

See, for example, C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971), Sec. 14.

Tatarskii, V. I.

In this connection, see V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971); E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

Wolf, E.

For a discussion of some of the complications that arise when the evanescent waves are taken into account, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982). The approximation resulting from the neglection of the evanescent waves in the scattered field is examined, within the accuracy of the first Born approximation, in App. A of Ref. 3.
[Crossref]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[Crossref]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Effect of backscattering in phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 861–863 (1982).
[Crossref]

G. S. Agarwal and E. Wolf, “Theory of phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 321–326 (1982).
[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 with 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]

Yariv, A.

For reviews of the technique of optical phase conjugation see, for example, A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978); D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 156–183 (1982).
[Crossref]

Several schemes, such as those based on three-wave mixing [A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976)], four-wave mixing [R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977)], stimulated Brillouin scattering (Zel’dovich et al.2), etc. have been proposed and used for the generation of the conjugated field. The polarization properties of the conjugated field will, in general, be different from those of the incident (probe) field. A complete reversal of polarization can be achieved by suitably arranging the experimental geometry and by choosing the polarization properties of the various interacting waves appropriately [see, for example, Ref. 16 below and J. F. Lam, D. G. Steel, R. A. McFarlane, and R. C. Lind, “Atomic coherence effects in resonant degenerate four-wave mixing,” Appl. Phys. Lett. 38, 977–979 (1981)].
[Crossref]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich and V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979); B. Ya. Zel’dovich and T. V. Yakovleva, “Spatial-polarization wavefront reversal in stimulated scattering of the Rayleigh line wing,” Sov. J. Quantum Electron. 10, 501–505 (1980).
[Crossref]

See, for example, 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); 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); and V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

Several schemes, such as those based on three-wave mixing [A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976)], four-wave mixing [R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977)], stimulated Brillouin scattering (Zel’dovich et al.2), etc. have been proposed and used for the generation of the conjugated field. The polarization properties of the conjugated field will, in general, be different from those of the incident (probe) field. A complete reversal of polarization can be achieved by suitably arranging the experimental geometry and by choosing the polarization properties of the various interacting waves appropriately [see, for example, Ref. 16 below and J. F. Lam, D. G. Steel, R. A. McFarlane, and R. C. Lind, “Atomic coherence effects in resonant degenerate four-wave mixing,” Appl. Phys. Lett. 38, 977–979 (1981)].
[Crossref]

IEEE J. Quantum Electron. (1)

For reviews of the technique of optical phase conjugation see, for example, A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978); D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 156–183 (1982).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Commun. (3)

For a discussion of some of the complications that arise when the evanescent waves are taken into account, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982). The approximation resulting from the neglection of the evanescent waves in the scattered field is examined, within the accuracy of the first Born approximation, in App. A of Ref. 3.
[Crossref]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[Crossref]

G. S. Agarwal, “Dipole radiation in the presence of a phase conjugate mirror,” Opt. Commun. 42, 205–207 (1982).
[Crossref]

Sov. J. Quantum Electron. (1)

B. Ya. Zel’dovich and V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979); B. Ya. Zel’dovich and T. V. Yakovleva, “Spatial-polarization wavefront reversal in stimulated scattering of the Rayleigh line wing,” Sov. J. Quantum Electron. 10, 501–505 (1980).
[Crossref]

Sov. Phys. JETP (1)

See, for example, 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); 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); and V. Wang and C. R. Guiliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
[Crossref] [PubMed]

Other (10)

This integral equation is derived, in the context of time-independent quantum-mechanical potential scattering, for example in P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.

The concept of a PCM is a convenient idealization that describes the effect of a true physical device located beyond the plane z= z1, by means of which a field distribution U is replaced by a new field distribution μU* in that plane. The transformation U→ μU* is usually achieved by nonlinear optical interactions, such as stimulated scattering processes or optical parametric interactions [see Ref. 1].

The term “conjugate” (or “conjugated”) field is somewhat ambigous and must be interpreted with caution. The field Uc(r) is generated as a result of the interaction of the incident field U(i)(r) with the scattering medium in the presence of the PCM. In general, it will include, in addition to U(i)(r), also contributions (usually ignored) arising from backscattering of the incident field and from successive conjugations of waves backscattered onto the PCM [see, for example, Figs. 3(a), 3(d), 4(a), and 4(h)].

Alternatively, by using the Maxwell equation ∇ · D= 0 (where D≡ E+ 4πP is the electric displacement vector), the constitutive relation D= n2E and the vector identity ∇ · (n2E) = n2∇ · E+ E· ∇n2, Eq. (7.2) may be rewritten in the familiar form ∇2E(r)+k2n2(r)E(r)+∇{E(r)·∇ log[n2(r)]}=0.However, form (7.2) is more convenient for the present purposes.

See, for example, C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971), Sec. 14.

A. T. Friberg, “Integral equation of the scattered field in the presence of a phase-conjugate mirror” J. Opt. Soc. Am. (to be published.)

Throughout this paper, a caret above a symbol denotes an operator.

See, for example, A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966), Eq. (2.19).

We will not discuss here the conditions under which series (3.1) will converge, a subject that would require a separate investigation.

In this connection, see V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971); E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram illustrating the notation. (a) In the absence of the PCM, the total field U is given by the sum of the incident field U(i) and the scattered field U(s). (b) In the presence of the PCM in the plane z = z1, the total field throughout the domain z < z1 is denoted by Uc.

Fig. 2
Fig. 2

Diagrammatic illustration of the zeroth-order term Uc(0) in iterative expansion (3.1) for the conjugate field Uc. U(i) denotes the incident wave, and μU(i)* denotes the wave that is generated by the PCM in the absence of the scatterer.

Fig. 3
Fig. 3

Diagrammatic illustrations of the four contributions [(a)–(d)], given in Eq. (4.4), to the first-order term Uc(1)(r<) of iterative expansion (3.1).

Fig. 4
Fig. 4

Diagrammatic illustrations of the eight contributions [(a)–(h)], given in Eq. (5.4), to the second-order term Uc(2)(r<) of iterative expansion (3.1).

Equations (73)

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U ( i ) ( r , t ) = U ( i ) ( r ) e - i ω t
U ( r ) = U ( i ) ( r ) + U ( s ) ( r )
U ( r ) = U ( i ) ( r ) - 1 4 π V G ( r , r ) F ( r ) U ( r ) d 3 r ,
G ( r , r ) = exp ( i k r - r ) r - r
F ( r ) = - k 2 [ n 2 ( r ) - 1 ]
U c ( r ) = U c ( 0 ) ( r ) - 1 4 π V Ĝ c ( r , r ) F ( r ) U c ( r ) d 3 r ,
U c ( 0 ) ( r ) = U ( i ) ( r ) + μ U ( i ) * ( r )
Ĉ f ( r ) f * ( r ) ,
G ( r , r ) = Θ ( z - z ) G > ( H ) ( r , r ) + Θ ( z - z ) G < ( H ) ( r , r ) + G ( I ) ( r , r ) ,
Θ ( ζ ) = 1 if ζ > 0 , = 0 if ζ < 0.
G > ( H ) ( r , r ) = i 2 π κ k 1 w exp { i [ κ · ( ρ - ρ ) + w ( z - z ) ] } d 2 κ ,
G < ( H ) ( r , r ) = i 2 π κ k 1 w exp { i [ κ · ( ρ - ρ ) - w ( z - z ) ] } d 2 κ ,
G ( I ) ( r , r ) = i 2 π κ > k 1 w exp { i [ κ · ( ρ - ρ ) + w z - z ] } d 2 κ ,
w = + ( k 2 - κ 2 ) 1 / 2             when κ k ,
= + i ( κ 2 - k 2 ) 1 / 2             when κ > k .
Ĝ c ( r , r ) = G ( r , r ) + μ G > ( H ) * ( r , r ) Ĉ .
G > ( H ) * ( r , r ) = - G < ( H ) ( r , r )
Ĝ c ( r , r ) = Θ ( z - z ) ( 1 + μ Ĉ ) G > ( H ) ( r , r ) + Θ ( z - z ) G < ( H ) ( r , r ) ( 1 - μ Ĉ ) + G ( I ) ( r , r ) .
Ĝ c ( r , r ) = G ( r , r )             ( μ = 0 ) ,
U c ( 0 ) ( r ) = U ( i ) ( r )             ( μ = 0 ) .
U c ( r ) = n = 0 U c ( n ) ( r ) ,
U c ( n ) ( r ) = - 1 4 π V Ĝ c ( r , r ) F ( r ) U c ( n - 1 ) ( r ) d 3 r             ( n = 1 , 2 , 3 , ) .
U c ( r ) U c ( 0 ) ( r ) = U ( i ) ( r ) + μ U ( i ) * ( r ) = ( 1 + μ Ĉ ) U ( i ) ( r ) ,
Ĝ c ( H ) ( r < , r ) = G < ( H ) ( r < , r ) + μ G > ( H ) * ( r < , r ) Ĉ .
U c ( n ) ( r < ) = - 1 4 π V Ĝ c ( H ) ( r < , r ) F ( r ) U c ( n - 1 ) ( r ) d 3 r ,
U c ( r < ) U c ( I ) ( r < ) = U c ( 0 ) ( r < ) + U c ( 1 ) ( r < ) ,
U c ( 1 ) ( r < ) = - 1 4 π V Ĝ c ( H ) ( r < , r ) F ( r ) U c ( 0 ) ( r ) d 3 r .
U c ( 1 ) ( r < ) = - 1 4 π V [ G < ( H ) ( r < , r ) + μ G > ( H ) * ( r < , r ) Ĉ ] F ( r ) ( 1 + μ Ĉ ) U ( i ) ( r ) d 3 r .
U c ( 1 ) ( r < ) = - 1 4 π V G < ( H ) ( r < , r ) F ( r ) U ( i ) ( r ) d 3 r ( a ) - μ 4 π V G < ( H ) ( r < , r ) F ( r ) U ( i ) * ( r ) d 2 r ( b ) - μ 4 π V G > ( H ) * ( r < , r ) F ( r ) U ( i ) * ( r ) d 3 r ( c ) - μ 2 4 π V G > ( H ) * ( r < , r ) F ( r ) U ( i ) ( r ) d 3 r . ( d )
U c ( 1 ) ( r < ) = ( 1 - μ 2 ) U bs ( 1 ) ( r < ) ,
U bs ( 1 ) ( r < ) = - 1 4 π V G < ( H ) ( r < , r ) F ( r ) U ( i ) ( r ) d 3 r
U c ( I ) ( r < ) U c ( 0 ) ( r < ) + U bs ( 1 ) ( r < )             ( μ 1 ) .
U c ( r < ) U c ( II ) ( r < ) = U c ( 0 ) ( r < ) + U c ( 1 ) ( r < ) + U c ( 2 ) ( r < ) .
U c ( 2 ) ( r < ) = ( - 1 4 π ) 2 V V Ĝ c ( H ) ( r < , r ) × F ( r ) Ĝ c ( r , r ) F ( r ) U c ( 0 ) ( r ) d 3 r d 3 r .
U c ( 2 ) ( r < ) = ( 1 4 π ) 2 V V [ G < ( H ) ( r < , r ) + μ G > ( H ) * ( r < , r ) Ĉ ] F ( r ) × [ G ( r , r ) + μ G > ( H ) * ( r , r ) Ĉ ] F ( r ) × ( 1 + μ Ĉ ) U ( i ) ( r ) d 3 r d 3 r .
U c ( 2 ) ( r < ) = + ( 1 4 π ) 2 V V G < ( H ) ( r < , r ) F ( r ) G ( r , r ) F ( r ) U ( i ) ( r ) d 3 r d 3 r ( a ) + μ ( 1 4 π ) 2 V V G < ( H ) ( r < , r ) F ( r ) G ( r , r ) F ( r ) U ( i ) * ( r ) d 3 r d 3 r ( b ) + μ ( 1 4 π ) 2 V V G < ( H ) ( r < , r ) F ( r ) G > ( H ) * ( r , r ) F ( r ) U ( i ) * ( r ) d 3 r d 3 r ( c ) + μ ( 1 4 π ) 2 V V G > ( H ) * ( r < , r ) F ( r ) G * ( r , r ) F ( r ) U ( i ) * ( r ) d 3 r d 3 r ( d ) + μ 2 ( 1 4 π ) 2 V V G < ( H ) ( r < , r ) F ( r ) G > ( H ) * ( r , r ) F ( r ) U ( i ) ( r ) d 3 r d 3 r ( e ) + μ 2 ( 1 4 π ) 2 V V G > ( H ) * ( r < , r ) F ( r ) G * ( r , r ) F ( r ) U ( i ) ( r ) d 3 r d 3 r ( f ) + μ 2 ( 1 4 π ) 2 V V G > ( H ) * ( r < , r ) F ( r ) G > ( H ) ( r , r ) F ( r ) U ( i ) ( r ) d 3 r d 3 r ( g ) + μ μ 2 ( 1 4 π ) 2 V V G > ( H ) * ( r < , r ) F ( r ) G > ( H ) ( r , r ) F ( r ) U ( i ) * ( r ) d 3 r d 3 r . ( h )
G ( I ) * ( r , r ) = G ( I ) ( r , r )
U c ( 2 ) ( r < ) = ( 1 - μ 2 ) [ U bs ( 2 ) ( r < ) + U ( 2 ) ( r < ) ] ,
U bs ( 2 ) ( r < ) = ( 1 4 π ) 2 V V G < ( H ) ( r < , r ) F ( r ) × G ( r , r ) F ( r ) U ( i ) ( r ) d 3 r d 3 r
U ( 2 ) ( r < ) = - μ ( 1 4 π ) 2 V V G > ( H ) * ( r < , r ) F ( r ) × G > ( H ) ( r , r ) F ( r ) U ( i ) * ( r ) d 3 r d 3 r
U c ( 2 ) ( r < ) U bs ( 2 ) ( r < ) + U ( 2 ) ( r < )             ( μ 1 ) ,
U c ( n ) ( r < ) = ( - 1 4 π ) n V d 3 r 1 F ( r 1 ) V d 3 r n F ( r n ) × G < ( H ) ( r < , r 1 ) ( 1 - μ Ĉ ) P ˆ 1 , 2 P ˆ n - 1 , n ( 1 + μ Ĉ ) U ( i ) ( r n ) ,
P ˆ j , j + 1 Ĝ c ( r j , r j + 1 ) = ( 1 + μ Ĉ ) A j , j + 1 + B j , j + 1 ( 1 - μ Ĉ ) + G ( I ) ( r j , r j + 1 ) ,
A j , k = Θ ( z j - z k ) G > ( H ) ( r j , r k )
B j , k = Θ ( z k - z j ) G < ( H ) ( r j , r k ) .
( 1 - μ Ĉ ) P ˆ 1 , 2 P ˆ 2 , 3 P ˆ n - 1 , n = Q ˆ 1 , 2 Q ˆ 2 , 3 Q ˆ n - 1 , n ( 1 - μ Ĉ ) ,
Q ˆ j , j + 1 = ( 1 - μ Ĉ ) B j , j + 1 + G ( I ) ( r j , r j + 1 ) .
( 1 - μ Ĉ ) ( 1 + μ Ĉ ) = ( 1 - μ 2 ) ,
U c ( n ) ( r < ) = ( - 1 4 π ) n V d 3 r 1 F ( r 1 ) V d 3 r n F ( r n ) × G < ( H ) ( r < , r 1 ) Q ˆ 1 , 2 Q ˆ n - 1 , n × ( 1 - μ Ĉ ) ( 1 + μ Ĉ ) U ( i ) ( r n )             ( n = 1 , 2 , 3 , ) .
U c ( n ) ( r < ) = 0 ,             n = 1 , 2 , 3 , .
U c ( r < ) = U ( i ) ( r < ) + e i ϕ U ( i ) * ( r < )             ( μ = 1 ) ,
E ( i ) ( r ) z = z 1 μ E ( i ) * ( r ) z = z 1 ,
× × E ( r ) - k 2 n 2 ( r ) E ( r ) = 0 ,
n 2 ( r ) = 1 + 4 π χ ( r )
P ( r ) = χ ( r ) E ( r ) ,
× × E ( r ) - k 2 E ( r ) = - F ( r ) E ( r ) ,
F ( r ) = - k 2 [ n 2 ( r ) - 1 ] = - 4 π k 2 χ ( r )
× × G ( r , r ) - k 2 G ( r , r ) = 4 π δ ( 3 ) ( r - r ) I ,
G ( r , r ) = ( I + 1 k 2 ) G ( r , r ) .
E ( r ) = E ( i ) ( r ) - 1 4 π V G ( r , r ) · F ( r ) E ( r ) d 3 r .
E c ( r ) = E c ( 0 ) ( r ) - 1 4 π V G c ( r , r ) · F ( r ) E c ( r ) d 3 r ,
E c ( 0 ) ( r ) = E ( i ) ( r ) + μ E ( i ) * ( r ) = ( 1 + μ Ĉ ) E ( i ) ( r ) ,
G c ( r , r ) = Θ ( z - z ) ( 1 + μ Ĉ ) G > ( H ) ( r , r ) + Θ ( z - z ) G < ( H ) ( r , r ) ( 1 - μ Ĉ ) + G ( I ) ( r , r ) .
G > ( H ) ( r , r ) = ( I + 1 k 2 ) G > ( H ) ( r , r ) ,
G < ( H ) ( r , r ) = ( I + 1 k 2 ) G < ( H ) ( r , r ) ,
G ( I ) ( r , r ) = ( I + 1 k 2 ) G ( I ) ( r , r ) ,
E c ( r ) = n = 0 E c ( n ) ( r ) ,
E c ( n ) ( r ) = - 1 4 π V G c ( r , r ) · F ( r ) E c ( n - 1 ) ( r ) d 3 r             ( n = 1 , 2 , 3 , ) .
E c ( n ) ( r < ) = 0             for n = 1 , 2 , 3 ,             ( μ = 1 ) .
E c ( r < ) = E ( i ) ( r < ) + e i ϕ E ( i ) * ( r < )             ( μ = 1 ) ,
E c ( r < ) E c ( 0 ) ( r < ) + E c ( 1 ) ( r < ) ,
E c ( 1 ) ( r < ) = - ( 1 - μ 2 ) 1 4 π ( I + 1 k 2 ) G ( H ) ( r < , r ) · F ( r ) E ( i ) ( r ) d 3 r .
2E(r)+k2n2(r)E(r)+{E(r)·log[n2(r)]}=0.