Abstract

A consistent and conceptually simple derivation is given for the basic integral equation of the scattered field in the presence of a phase-conjugate mirror characterized by a constant amplitude-reflection coefficient. This integral equation is exact, except for the fact that the incident field is assumed to contain no evanescent components and the effects of the evanescent waves are neglected at the phase-conjugate mirror.

© 1983 Optical Society of America

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  1. See, for example, the review article by D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 156–183 (1982), and references therein.
    [CrossRef]
  2. The fact that the complex amplitude reflectivity μ of the PCM is assumed to be a constant represents, of course, an idealization. In practice μ depends, in general, both on the position across the PCM and on the direction of propagation of the wave incident onto the PCM. Although these generalizations could be readily incorporated into the analysis, we choose to exclude them here so as not to obscure the main ideas of the paper.
  3. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
    [CrossRef] [PubMed]
  4. 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]
  5. G. S. Agarwal, A. T. Friberg, and E. Wolf, “Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. (to be published).
  6. From now on we omit the harmonic, time-dependent factor e−iwt of the various fields.
  7. See, for example, P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.
  8. In this connection, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).
    [CrossRef]
  9. See, for example, A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966), Eq. (2.19).
  10. E. Wolf, “Phase conjugacy and symmetries in spatially band-limited wavefields containing no evanescent components,” J. Opt. Soc. Am. 70, 1311–1319 (1980).
    [CrossRef]
  11. We emphasize that the effects of the evanescent waves are neglected only in the generation of the conjugate field Uc(r) by the PCM in the plane z= z1[cf. Eq. (15)]. The evanescent waves are included in the scattered field U(s)(r) given by Eq. (17), and the effects of these waves are therefore taken into account, for example, in the half-space R− [compare with Eq. (12)] and in all multiple-scattering processes inside the scattering medium.
  12. G. S. Agarwal, “Dipole radiation in the presence of a phase-conjugate mirror,” Opt. Commun. 42, 205–207 (1982).
    [CrossRef]

1982 (4)

See, for example, the review article by D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 156–183 (1982), and references therein.
[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]

In this connection, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).
[CrossRef]

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

1980 (1)

1977 (1)

Agarwal, G. S.

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]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. (to be published).

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.

In this connection, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).
[CrossRef]

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, “Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. (to be published).

Pepper, D. M.

See, for example, the review article by D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 156–183 (1982), and references therein.
[CrossRef]

A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
[CrossRef] [PubMed]

Roman, P.

See, for example, P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.

Wolf, E.

In this connection, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (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]

E. Wolf, “Phase conjugacy and symmetries in spatially band-limited wavefields containing no evanescent components,” J. Opt. Soc. Am. 70, 1311–1319 (1980).
[CrossRef]

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. (to be published).

Yariv, A.

J. Opt. Soc. Am. (1)

Opt. Commun. (3)

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

In this connection, see E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).
[CrossRef]

Opt. Eng. (1)

See, for example, the review article by D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 156–183 (1982), and references therein.
[CrossRef]

Opt. Lett. (1)

Other (6)

We emphasize that the effects of the evanescent waves are neglected only in the generation of the conjugate field Uc(r) by the PCM in the plane z= z1[cf. Eq. (15)]. The evanescent waves are included in the scattered field U(s)(r) given by Eq. (17), and the effects of these waves are therefore taken into account, for example, in the half-space R− [compare with Eq. (12)] and in all multiple-scattering processes inside the scattering medium.

The fact that the complex amplitude reflectivity μ of the PCM is assumed to be a constant represents, of course, an idealization. In practice μ depends, in general, both on the position across the PCM and on the direction of propagation of the wave incident onto the PCM. Although these generalizations could be readily incorporated into the analysis, we choose to exclude them here so as not to obscure the main ideas of the paper.

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

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. (to be published).

From now on we omit the harmonic, time-dependent factor e−iwt of the various fields.

See, for example, P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.

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Figures (1)

Fig. 1
Fig. 1

Illustration of the notation relating to the interaction of an incident field U(i) with a scattering medium ( V ) and with a PCM.

Equations (23)

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U ( i ) ( r , t ) = U ( i ) ( r ) e i ω t ,
U ( r ) = U ( i ) ( r ) + U ( s ) ( r ) ,
U ( s ) ( 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 ]
G ( r , r ) = G ( H ) ( r , r ) + G ( I ) ( r , r ) ,
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 if | κ | k ,
= + i ( κ 2 k 2 ) 1 / 2 if | κ | > k .
U > ( s , H ) ( r ) = 1 4 π V G > ( H ) ( r , r ) F ( r ) U ( r ) d 3 r ,
G > ( H ) ( r , r ) i 2 π | κ | k 1 w × exp { i [ κ ( ρ ρ ) + w | z z | ] } d 2 κ .
U < ( s , H ) ( r ) = 1 4 π V G < ( H ) ( r , r ) F ( r ) U ( r ) d 3 r ,
G < ( H ) ( r , r ) i 2 π | κ | k 1 w × exp { i [ κ ( ρ ρ ) w ( z z ) ] d 2 κ .
U > ( H ) ( r ) = U ( i ) ( r ) + U > ( s , H ) ( r ) ,
U c ( r ) = μ [ U ( i ) ( r ) + U > ( s , H ) ( r ) ] * ,
U ( r ) = U ext ( r ) + U ( s ) ( r ) ,
U ( s ) ( r ) = 1 4 π V G ( r , r ) F ( r ) U ( r ) d 3 r ,
U ext ( r ) = U ( i ) ( r ) + U c ( r ) ,
Ĉ U ( r ) U * ( r ) .
U ( r ) = U ( 0 ) ( r ) 1 4 π V Ĝ c ( r , r ) F ( r ) U ( r ) d 3 r ,
U ( 0 ) ( r ) = ( 1 + μ Ĉ ) U ( i ) ( r ) ,
Ĝ c ( r , r ) = G ( r , r ) + μ G > ( H ) * ( r , r ) Ĉ .