Abstract

Several theorems are known concerning symmetry relations between monochromatic wave fields that propagate either into the same half-space (z > 0) or into complementary half-spaces (z > 0 and z < 0) and that are complex conjugates of each other in some cross-sectional plane z = constant. The theorems derived up to now apply only to wave fields that do not contain inhomogeneous (evanescent) components. In the present paper two of the main theorems are generalized to a wider class of fields. It is found that homogeneous and inhomogeneous components of a wave field have quite different symmetry properties under phase conjugation. The results are illustrated by a discussion of the behavior of plane waves, both homogeneous and evanescent ones, which undergo phase conjugation followed by transmission or by reflection.

© 1985 Optical Society of America

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  1. See, for example, the following review articles: A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978);J. Au Yeung, A. Yariv, “Phase-conjugate optics,” Opt. News 5 (2), 13–17 (1979).
    [CrossRef]
  2. G. S. Agarwal, A. T. Friberg, E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982);“Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (1982).
    [CrossRef]
  3. E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugate waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.
  4. See also R. Mittra, T. M. Habashy, “On the theory of wave-front distortion by phase conjugation,” J. Opt. Soc. Am. A 1, 1103–1109 (1984).
    [CrossRef]
  5. These predictions were illustrated by explicit calculations relating to a modified Fabry–Perot interferometer consisting of an ordinary partially reflecting dielectric mirror and a phase-conjugate mirror in the following papers: M. Nazarathy, “A Fabry–Perot etalon with one phase-conjugate mirror,” Opt. Commun. 45, 117–121 (1983);A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983);P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
    [CrossRef]
  6. E. Wolf, “Phase conjugacy and symmetries in spatially band-limited 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, υ/k) = k2Ũ(u, υ; z)e−iωz. These corrections do not affect any other equations or conclusions of that paper.
    [CrossRef]
  7. See also W. D. Montgomery, “Unitary operators in the homogeneous wavefield,” Opt. Lett. 6, 314–315 (1981);M. Nazarathy, J. Shamir, “Phase conjugacy and symmetries in general optical systems,” J. Opt. Soc. Am. 73, 910–915 (1983).
    [CrossRef] [PubMed]
  8. E. Wolf, W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).See also Ref. 3.
    [CrossRef]
  9. By saying that a (nonplanar) wave field propagates into a particular half-space, we mean, of course, that at points in that half-space, which is sufficiently far away from the origin, the field behaves as an outgoing spherical wave. In the expressions (2.4) and (2.11) below, the terms UI and VI do not contribute to the far-zone behavior, because of the exponential decay of their integrands as k|z| → ∞. Hence the far-zone behavior [more precisely, the asymptotic behavior as kr= k(x2+ y2+ z2) → ∞ in a fixed direction] of U and V is the same as that of UH and VH, respectively.That UH and VH represent fields that propagate into the half-spaces z> 0 and z< 0, respectively, is shown in K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave, Part I,” J. Opt. Soc. Am. 52, 615–625 (1962), App.
    [CrossRef]
  10. Sufficiency conditions are discussed, for example, in papers by W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968) andE. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
    [CrossRef]
  11. For fields UH this result was established by G. C. Sherman in “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969),Theorems II and IV. The corresponding result for fields VH can be derived in a strictly similar manner.
    [CrossRef]
  12. Phase conjugation of evanescent waves has been discussed, for example, in connection with four-wave mixing involving surface plasmons [see, for example, K. Ujihara, “Four-wave mixing and two-dimensional conjugation of surface plasmons,” Opt. Commun. 42, 1–4 (1982);“Phase-conjugation of surface plasmon waves by the third order nonlinearity of a free electron gas,” Opt. Commun. 43, 225–228 (1982)].
    [CrossRef]

1984 (1)

1983 (1)

These predictions were illustrated by explicit calculations relating to a modified Fabry–Perot interferometer consisting of an ordinary partially reflecting dielectric mirror and a phase-conjugate mirror in the following papers: M. Nazarathy, “A Fabry–Perot etalon with one phase-conjugate mirror,” Opt. Commun. 45, 117–121 (1983);A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983);P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
[CrossRef]

1982 (3)

G. S. Agarwal, A. T. Friberg, E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982);“Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (1982).
[CrossRef]

E. Wolf, W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).See also Ref. 3.
[CrossRef]

Phase conjugation of evanescent waves has been discussed, for example, in connection with four-wave mixing involving surface plasmons [see, for example, K. Ujihara, “Four-wave mixing and two-dimensional conjugation of surface plasmons,” Opt. Commun. 42, 1–4 (1982);“Phase-conjugation of surface plasmon waves by the third order nonlinearity of a free electron gas,” Opt. Commun. 43, 225–228 (1982)].
[CrossRef]

1981 (1)

1980 (1)

1978 (1)

See, for example, the following review articles: A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978);J. Au Yeung, A. Yariv, “Phase-conjugate optics,” Opt. News 5 (2), 13–17 (1979).
[CrossRef]

1969 (1)

1968 (1)

1962 (1)

Agarwal, G. S.

G. S. Agarwal, A. T. Friberg, E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982);“Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (1982).
[CrossRef]

E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugate waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.

Carter, W. H.

E. Wolf, W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).See also Ref. 3.
[CrossRef]

Friberg, A. T.

G. S. Agarwal, A. T. Friberg, E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982);“Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (1982).
[CrossRef]

E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugate waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.

Habashy, T. M.

Mittra, R.

Miyamoto, K.

Montgomery, W. D.

Nazarathy, M.

These predictions were illustrated by explicit calculations relating to a modified Fabry–Perot interferometer consisting of an ordinary partially reflecting dielectric mirror and a phase-conjugate mirror in the following papers: M. Nazarathy, “A Fabry–Perot etalon with one phase-conjugate mirror,” Opt. Commun. 45, 117–121 (1983);A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983);P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
[CrossRef]

Sherman, G. C.

Ujihara, K.

Phase conjugation of evanescent waves has been discussed, for example, in connection with four-wave mixing involving surface plasmons [see, for example, K. Ujihara, “Four-wave mixing and two-dimensional conjugation of surface plasmons,” Opt. Commun. 42, 1–4 (1982);“Phase-conjugation of surface plasmon waves by the third order nonlinearity of a free electron gas,” Opt. Commun. 43, 225–228 (1982)].
[CrossRef]

Wolf, E.

E. Wolf, W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).See also Ref. 3.
[CrossRef]

G. S. Agarwal, A. T. Friberg, E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982);“Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (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).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, υ/k) = k2Ũ(u, υ; z)e−iωz. These corrections do not affect any other equations or conclusions of that paper.
[CrossRef]

By saying that a (nonplanar) wave field propagates into a particular half-space, we mean, of course, that at points in that half-space, which is sufficiently far away from the origin, the field behaves as an outgoing spherical wave. In the expressions (2.4) and (2.11) below, the terms UI and VI do not contribute to the far-zone behavior, because of the exponential decay of their integrands as k|z| → ∞. Hence the far-zone behavior [more precisely, the asymptotic behavior as kr= k(x2+ y2+ z2) → ∞ in a fixed direction] of U and V is the same as that of UH and VH, respectively.That UH and VH represent fields that propagate into the half-spaces z> 0 and z< 0, respectively, is shown in K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave, Part I,” J. Opt. Soc. Am. 52, 615–625 (1962), App.
[CrossRef]

E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugate waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.

Yariv, A.

See, for example, the following review articles: A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978);J. Au Yeung, A. Yariv, “Phase-conjugate optics,” Opt. News 5 (2), 13–17 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

See, for example, the following review articles: A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978);J. Au Yeung, A. Yariv, “Phase-conjugate optics,” Opt. News 5 (2), 13–17 (1979).
[CrossRef]

J. Opt. Soc. Am. (4)

By saying that a (nonplanar) wave field propagates into a particular half-space, we mean, of course, that at points in that half-space, which is sufficiently far away from the origin, the field behaves as an outgoing spherical wave. In the expressions (2.4) and (2.11) below, the terms UI and VI do not contribute to the far-zone behavior, because of the exponential decay of their integrands as k|z| → ∞. Hence the far-zone behavior [more precisely, the asymptotic behavior as kr= k(x2+ y2+ z2) → ∞ in a fixed direction] of U and V is the same as that of UH and VH, respectively.That UH and VH represent fields that propagate into the half-spaces z> 0 and z< 0, respectively, is shown in K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave, Part I,” J. Opt. Soc. Am. 52, 615–625 (1962), App.
[CrossRef]

Sufficiency conditions are discussed, for example, in papers by W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968) andE. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
[CrossRef]

For fields UH this result was established by G. C. Sherman in “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969),Theorems II and IV. The corresponding result for fields VH can be derived in a strictly similar manner.
[CrossRef]

E. Wolf, “Phase conjugacy and symmetries in spatially band-limited 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, υ/k) = k2Ũ(u, υ; z)e−iωz. These corrections do not affect any other equations or conclusions of that paper.
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (4)

These predictions were illustrated by explicit calculations relating to a modified Fabry–Perot interferometer consisting of an ordinary partially reflecting dielectric mirror and a phase-conjugate mirror in the following papers: M. Nazarathy, “A Fabry–Perot etalon with one phase-conjugate mirror,” Opt. Commun. 45, 117–121 (1983);A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983);P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
[CrossRef]

G. S. Agarwal, A. T. Friberg, E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982);“Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (1982).
[CrossRef]

Phase conjugation of evanescent waves has been discussed, for example, in connection with four-wave mixing involving surface plasmons [see, for example, K. Ujihara, “Four-wave mixing and two-dimensional conjugation of surface plasmons,” Opt. Commun. 42, 1–4 (1982);“Phase-conjugation of surface plasmon waves by the third order nonlinearity of a free electron gas,” Opt. Commun. 43, 225–228 (1982)].
[CrossRef]

E. Wolf, W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).See also Ref. 3.
[CrossRef]

Opt. Lett. (1)

Other (1)

E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugate waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.

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Figures (4)

Fig. 1
Fig. 1

Illustrating Theorem I: If U(2) = [U(1)]* at every point in the plane z = 0, then U H ( 2 ) ( C + ) = [ U H ( 1 ) ( C ) ] * , U I ( 2 ) ( D ) = [ U I ( 1 ) ( D ) ] * ,where C+ and C are any two points that possess reflection symmetry with respect to the plane z = 0 and D is any point in the half-space z ≥ 0.

Fig. 2
Fig. 2

Illustrating Theorem II: If V = U* at every point in the plane z = 0 then V H ( E ) = [ U H ( E ) ] * , V I ( F ) = [ U I ( F + ) ] * ,where E is any point, F+ is any point in the half-space z > 0, and F is the mirror image of F+ in the plane z = 0.

Fig. 3
Fig. 3

Illustrating phase conjugation of a plane wave at a plane z = 0, followed by transmission. Upper figures: homogeneous wave; lower figures: evanescent wave.

Fig. 4
Fig. 4

Illustrating phase conjugation of a plane wave at a plane z = 0, followed by reflection. Upper figures: homogeneous wave; lower figures: evanescent wave.

Equations (51)

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U ( x , y , z , t ) = U ( x , y , z ) e i ω t ,
2 U + k 2 U = 0 ,
k = ω / c ,
U ( x , y , z ) = U H ( x , y , z ) + U I ( x , y , z ) ,
U H ( x , y , z ) = p 2 + q 2 1 A ( p , q ) exp [ i k ( p x + q y + m z ) ] d p d q ,
U I ( x , y , z ) = p 2 + q 2 > 1 A ( p , q ) exp [ i k ( p x + q y + m z ) ] d p d q .
m = + ( 1 p 2 q 2 ) 1 / 2 when p 2 + q 2 1 ,
m = + i ( p 2 + q 2 1 ) 1 / 2 when p 2 + q 2 > 1 .
V ( x , y , z , t ) = V ( x , y , z ) e i ω t ,
2 V + k 2 V = 0 ,
V ( x , y , z ) = V H ( x , y , z ) + V I ( x , y , z ) ,
V H ( x , y , z ) = p 2 + q 2 1 B ( p , q ) exp [ i k ( p x + q y m z ) ] d p d q ,
V I ( x , y , z ) = p 2 + q 2 > 1 B ( p , q ) exp [ i k ( p x + q y m z ) ] d p d q .
U ( 1 ) ( x , y , z ) i ω t , U ( 2 ) ( x , y , z ) e i ω t ( z 0 )
U ( 2 ) ( x , y , 0 ) = [ U ( 1 ) ( x , y , 0 ) ] *
U H ( 2 ) ( x , y , z ) = [ U H ( 1 ) ( x , y , z ) ] * with all z ,
U I ( 2 ) ( x , y , z ) = [ U I ( 1 ) ( x , y , z ) ] * with all z 0 ,
U ( j ) ( x , y , z ) = A ( j ) ( p , q ) exp [ i k ( p x + q y + m z ) d p d q ( z 0 )
A ( 2 ) ( p , q ) exp [ i k ( p x + q y ] d p d q = [ A ( 1 ) ( p , q ) ] * exp [ i k ( p x + q y ) ] d p d q .
A ( 2 ) ( p , q ) exp [ i k ( p x + q y ) ] d p d q = [ A ( 1 ) ( p , q ) ] * exp [ i k ( p x + q y ) ] d p d q .
A ( 2 ) ( p , q ) = [ A ( 1 ) ( p , q ) ] * .
U H ( 2 ) ( x , y , z ) = p 2 + q 2 1 [ A ( 1 ) ( p , q ) ] * exp [ i k ( p x + q y + l m z ) ] d p d q ,
U I ( 2 ) ( x , y , z ) = p 2 + q 2 > 1 [ A ( 1 ) ( p , q ) ] * exp [ i k ( p x + q y + m z ) ] d p d q ,
U H ( 2 ) ( x , y , z ) = [ p 2 + q 2 1 A ( 1 ) ( p , q ) exp [ i k ( p x + q y m z ) ] d p d q ] *
U I ( 2 ) ( x , y , z ) = [ p 2 + q 2 > 1 A ( 1 ) ( p , q ) exp [ i k ( p x + q y + m z ) ] d p d q ] * .
U ( x , y , z ) e i ω t ( z 0 ) ,
V ( x , y , z ) e i ω t ( z 0 )
V ( x , y , 0 ) = [ U ( x , y , 0 ) ] *
V H ( x , y , z ) = [ U H ( x , y , z ) ] * with all z ,
V I ( x , y , z ) = [ U I ( x , y , z ) ] * with all z 0 ,
U ( x , y , z ) = A ( p , q ) exp [ i k ( p x + q y + m z ) ] d p d q ( z 0 ) ,
V ( x , y , z ) = B ( p , q ) exp [ i k ( p x + q y m z ) ] d p d q ( z 0 ) ,
B ( p , q ) exp [ i k ( p x + q y ) ] d p d q = [ A ( p , q ) ] * exp [ i k ( p x + q y ) ] d p d q .
B ( p , q ) = [ A ( p , q ) ] * .
V H ( x , y , z ) = p 2 + q 2 1 [ A ( p , q ) ] * exp [ i k ( p x + q y m z ) ] d p d q ( z 0 )
V I ( x , y , z ) = p 2 + q 2 > 1 [ A ( p , q ) ] * exp [ i k ( p x + q y m z ) ] d p d q ( z 0 ) .
V H ( x , y , z ) = [ p 2 + q 2 1 A ( p , q ) exp [ i k ( p x + q y + m z ) ] d p d q ] * ( z 0 ) ,
V I ( x , y , z ) = [ p 2 + q 2 > 1 A ( p , q ) exp [ i k ( p x + q y m z ) ] d p d q ] * ( z 0 ) .
V H ( x , y , z ) = [ U H ( x , y , z ) ] * ( z 0 ) .
V I ( x , y , z ) = [ p 2 + q 2 > 1 A ( p , q ) exp [ i k ( p x + q y + m z ) ] d p d q ] * ( z 0 ) .
V I ( x , y , z ) = [ U I ( x , y , z ) ] * ( z 0 ) ,
U H ( 1 ) ( x , y , z ) = A exp [ i k ( p x + q y + m z ) ] ( z 0 ) ,
U H ( 2 ) ( x , y , z ) = A * exp [ i k ( p x q y + m z ) ] ( z 0 ) .
U I ( 1 ) ( x , y , z ) = A exp [ i k ( p k + q y ) ] exp ( k | m | z ) ( z 0 ) ,
U I ( 2 ) ( x , y , z ) = A * exp [ i k ( p x + q y ) ] exp ( k | m | z ) ( z 0 ) .
U H ( x , y , z ) = A exp [ i k ( p x + q y + m z ) ] ( z 0 ) .
V H ( x , y , z ) = A * exp [ i k ( p x + q y + m z ) ] ( z 0 ) .
U I ( x , y , z ) = A exp [ i k ( p x + q y ) ] exp ( k | m | z ) ( z 0 )
V I ( x , y , z ) = A * exp [ i k ( p x + q y ) ] exp ( k | m | | z | ) ( z 0 ) .
U H ( 2 ) ( C + ) = [ U H ( 1 ) ( C ) ] * , U I ( 2 ) ( D ) = [ U I ( 1 ) ( D ) ] * ,
V H ( E ) = [ U H ( E ) ] * , V I ( F ) = [ U I ( F + ) ] * ,

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