Abstract

A hybrid method for image formation through inhomogeneous media is presented. This method embodies principles of superresolution by incoherent-to-coherent conversion and one-way phase conjugation. A general equation is derived for an arbitrary configuration embodying a synthesis of the two methods. A variety of specific configurations are evaluated. The effect of a double-pass aberration-correction method is achieved even though the object wave has made only a single pass through the inhomogeneity.

© 1989 Optical Society of America

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  1. E. N. Leith and J. Upatnieks, “Reconstructed wave fronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [CrossRef]
  2. E. N. Leith and J. Upatnieks, “Holograms: their properties and uses,” SPIE J. 4, 3–6 (1965).
  3. H. Kogelnik, “Holographic image projection through inhomogeneous media,” Bell Syst. Tech. J. 44, 2451–2455 (1965).
    [CrossRef]
  4. J. P. Huignard and J. P. Herriau, “Real-time coherent object edge reconstruction with Bi12SiO20crystals,” Appl. Opt. 17, 2671–2672 (1978).
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  5. J. P. Huignard, J. P. Herriau, P. Aubourg, and E. Spitz, “Phase-conjugate wave-front generation via real-time holography in Bi12SiO20crystals,” Opt. Lett. 4, 21–23 (1979).
    [CrossRef]
  6. J. O. White and A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
    [CrossRef]
  7. J. Upatnieks, A. Vander Lugt, and E. N. Leith, “Correction of lens aberrations by means of holograms,” Appl. Opt. 5, 589–593 (1966).
    [CrossRef] [PubMed]
  8. H. Kogelnik and K. S. Pennington, “Holographic imaging through a random medium,” J. Opt. Soc. Am. 58, 273–274 (1968).
    [CrossRef]
  9. A. Yariv and T. L. Koch, “One-way coherent imaging through a distorting medium using four-wave mixing,” Opt. Lett. 7, 113–115 (1982).
    [CrossRef] [PubMed]
  10. B. Fisher, M. Cronin-Golomb, J. O. White, and A. Yariv, “Real-time phase conjugate window for one-way optical field imaging through a distortion,” Appl. Phys. Lett. 41, 141–143 (1982).
    [CrossRef]
  11. K. R. MacDonald, W. R. Tompkin, and R. W. Boyd, “Passive one-way aberration correction using four-wave mixing,” Opt. Lett. 13, 485–487 (1988).
    [CrossRef] [PubMed]
  12. A. Cunha and E. N. Leith, “One-way phase conjugation with partially coherent light and superresolution,” Opt. Lett. 13, 1105–1107 (1988).
    [CrossRef] [PubMed]
  13. G. Hohberg, “Holographische Pupillenfortsetgung mit partiell kohörenter Beleuchtung,” Optik 28, 288–293 (1968).
  14. M. Ueda, T. T. Sato, and M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
    [CrossRef]
  15. D. Gorlitz and F. Lanzl, “Methods of zero-order noncoherent filtering,” Opt. Commun. 20, 68–72 (1977).
    [CrossRef]
  16. A. Lohmann, “Incoherent optical processing of complex data,” Appl. Opt. 16, 261–263 (1977).
    [CrossRef] [PubMed]
  17. W. T. Rhodes, “Bipolar point-spread function synthesis by phase switching,” Appl. Opt. 16, 265–267 (1977).
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  18. A. W. Lohmann and W. T. Rhodes, “Two-pupil synthesis of optical transfer function,” Appl. Opt. 17, 1141–1151 (1978).
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  19. W. Stoner, “Incoherent optical processing via spatially offset pupil masks,” Appl. Opt. 17, 2454–2467 (1978).
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  20. E. N. Leith and C.-P. Kuei, “Interferometric method for imaging through inhomogeneities,” Opt. Lett. 12, 149–151 (1987).
    [CrossRef] [PubMed]
  21. E. N. Leith, D. K. Angell, and C.-P. Kuei, “Superresolution by incoherent-to-coherent conversion,” J. Opt. Soc. Am. A 4, 1050–1054 (1987).
    [CrossRef]
  22. E. N. Leith and D. K. Angell, “Generalization of some incoherent spatial filtering techniques,” Appl. Opt. 25, 499–502 (1986).
    [CrossRef] [PubMed]
  23. A. Cunha and E. N. Leith, “Generalized phase-conjugation system using partially coherent light,” IEEE J. Quantum Electron. QE-25, 351–359 (1989).
    [CrossRef]

1989 (1)

A. Cunha and E. N. Leith, “Generalized phase-conjugation system using partially coherent light,” IEEE J. Quantum Electron. QE-25, 351–359 (1989).
[CrossRef]

1988 (2)

1987 (2)

1986 (1)

1982 (2)

A. Yariv and T. L. Koch, “One-way coherent imaging through a distorting medium using four-wave mixing,” Opt. Lett. 7, 113–115 (1982).
[CrossRef] [PubMed]

B. Fisher, M. Cronin-Golomb, J. O. White, and A. Yariv, “Real-time phase conjugate window for one-way optical field imaging through a distortion,” Appl. Phys. Lett. 41, 141–143 (1982).
[CrossRef]

1980 (1)

J. O. White and A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

1979 (1)

1978 (3)

1977 (3)

1973 (1)

M. Ueda, T. T. Sato, and M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

1968 (2)

G. Hohberg, “Holographische Pupillenfortsetgung mit partiell kohörenter Beleuchtung,” Optik 28, 288–293 (1968).

H. Kogelnik and K. S. Pennington, “Holographic imaging through a random medium,” J. Opt. Soc. Am. 58, 273–274 (1968).
[CrossRef]

1966 (1)

1965 (2)

E. N. Leith and J. Upatnieks, “Holograms: their properties and uses,” SPIE J. 4, 3–6 (1965).

H. Kogelnik, “Holographic image projection through inhomogeneous media,” Bell Syst. Tech. J. 44, 2451–2455 (1965).
[CrossRef]

1962 (1)

Angell, D. K.

Aubourg, P.

Boyd, R. W.

Cronin-Golomb, M.

B. Fisher, M. Cronin-Golomb, J. O. White, and A. Yariv, “Real-time phase conjugate window for one-way optical field imaging through a distortion,” Appl. Phys. Lett. 41, 141–143 (1982).
[CrossRef]

Cunha, A.

A. Cunha and E. N. Leith, “Generalized phase-conjugation system using partially coherent light,” IEEE J. Quantum Electron. QE-25, 351–359 (1989).
[CrossRef]

A. Cunha and E. N. Leith, “One-way phase conjugation with partially coherent light and superresolution,” Opt. Lett. 13, 1105–1107 (1988).
[CrossRef] [PubMed]

Fisher, B.

B. Fisher, M. Cronin-Golomb, J. O. White, and A. Yariv, “Real-time phase conjugate window for one-way optical field imaging through a distortion,” Appl. Phys. Lett. 41, 141–143 (1982).
[CrossRef]

Gorlitz, D.

D. Gorlitz and F. Lanzl, “Methods of zero-order noncoherent filtering,” Opt. Commun. 20, 68–72 (1977).
[CrossRef]

Herriau, J. P.

Hohberg, G.

G. Hohberg, “Holographische Pupillenfortsetgung mit partiell kohörenter Beleuchtung,” Optik 28, 288–293 (1968).

Huignard, J. P.

Koch, T. L.

Kogelnik, H.

H. Kogelnik and K. S. Pennington, “Holographic imaging through a random medium,” J. Opt. Soc. Am. 58, 273–274 (1968).
[CrossRef]

H. Kogelnik, “Holographic image projection through inhomogeneous media,” Bell Syst. Tech. J. 44, 2451–2455 (1965).
[CrossRef]

Kondo, M.

M. Ueda, T. T. Sato, and M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

Kuei, C.-P.

Lanzl, F.

D. Gorlitz and F. Lanzl, “Methods of zero-order noncoherent filtering,” Opt. Commun. 20, 68–72 (1977).
[CrossRef]

Leith, E. N.

Lohmann, A.

Lohmann, A. W.

MacDonald, K. R.

Pennington, K. S.

Rhodes, W. T.

Sato, T. T.

M. Ueda, T. T. Sato, and M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

Spitz, E.

Stoner, W.

Tompkin, W. R.

Ueda, M.

M. Ueda, T. T. Sato, and M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

Upatnieks, J.

Vander Lugt, A.

White, J. O.

B. Fisher, M. Cronin-Golomb, J. O. White, and A. Yariv, “Real-time phase conjugate window for one-way optical field imaging through a distortion,” Appl. Phys. Lett. 41, 141–143 (1982).
[CrossRef]

J. O. White and A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Yariv, A.

B. Fisher, M. Cronin-Golomb, J. O. White, and A. Yariv, “Real-time phase conjugate window for one-way optical field imaging through a distortion,” Appl. Phys. Lett. 41, 141–143 (1982).
[CrossRef]

A. Yariv and T. L. Koch, “One-way coherent imaging through a distorting medium using four-wave mixing,” Opt. Lett. 7, 113–115 (1982).
[CrossRef] [PubMed]

J. O. White and A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. Lett. (2)

B. Fisher, M. Cronin-Golomb, J. O. White, and A. Yariv, “Real-time phase conjugate window for one-way optical field imaging through a distortion,” Appl. Phys. Lett. 41, 141–143 (1982).
[CrossRef]

J. O. White and A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Holographic image projection through inhomogeneous media,” Bell Syst. Tech. J. 44, 2451–2455 (1965).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Cunha and E. N. Leith, “Generalized phase-conjugation system using partially coherent light,” IEEE J. Quantum Electron. QE-25, 351–359 (1989).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Acta (1)

M. Ueda, T. T. Sato, and M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

Opt. Commun. (1)

D. Gorlitz and F. Lanzl, “Methods of zero-order noncoherent filtering,” Opt. Commun. 20, 68–72 (1977).
[CrossRef]

Opt. Lett. (5)

Optik (1)

G. Hohberg, “Holographische Pupillenfortsetgung mit partiell kohörenter Beleuchtung,” Optik 28, 288–293 (1968).

SPIE J. (1)

E. N. Leith and J. Upatnieks, “Holograms: their properties and uses,” SPIE J. 4, 3–6 (1965).

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Figures (6)

Fig. 1
Fig. 1

Superresolution technique. H1 can be a vanishingly small aperture, without loss of resolution.

Fig. 2
Fig. 2

One-way phase-conjugation and superresolution combined technique. This figure serves all five cases described in Section 4. The inhomogeneity is either s2 or H2, and s1 is the object. For the superresolution mode, H1 becomes a δ function.

Fig. 3
Fig. 3

Experimental setup for the hybrid system. Object s1 is a resolution chart; H1, a horizontal slit; s2, a piece of frosted glass. uobs is the plane of observation, located just to the left of s2.

Fig. 4
Fig. 4

Image obtained by using a conventional imaging system.

Fig. 5
Fig. 5

Corrected image obtained with the hybrid system operating in the pure one-way phase-conjugation mode.

Fig. 6
Fig. 6

Corrected image obtained with the hybrid system operating in the one-way phase-conjugation and superresolution combined mode.

Tables (2)

Tables Icon

Table 1 Possible Configurations of the Superresolution One-Way Phase-Conjugation Hybrid Systema

Tables Icon

Table 2 Possible Configurations of the Superresolution Two-Way Phase Conjugation Hybrid Systema

Equations (30)

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α = 2 π sin θ λ 2 π θ λ .
exp ( j { ( a x + b x 2 + c x 3 + d x 4 ) [ a ( x x 1 ) + b ( x x 1 ) 2 + c ( x x 1 ) 3 + d ( x x 1 ) 4 ] } ) = exp { j [ a x 1 + b ( 2 x x 1 x 1 2 ) + c ( 3 x 2 x 1 3 x x 1 2 + x 1 3 ) + d ( 4 x 3 x 1 6 x 2 x 1 2 + 3 x x 1 3 x 1 4 ) ] } ,
P DFWM ( x ) = χ ( 3 ) u 1 ( x ) u 2 * ( x ) u 3 ( x ) .
u obs ( x ) = κ χ ( 3 ) A 3 s 2 ( x ) H 2 ( f x ) h 1 ( ξ ) × s 1 ( x 1 ξ ) exp [ j π λ d 1 ( x x 1 ) 2 ] h 2 * ( η ) s 2 * ( x 2 η ) × exp [ j π λ d 2 ( x 2 2 2 x 2 x + 2 x 3 x x 3 2 ) ] × s ( x 1 x 2 + η ξ ) exp ( j 2 π f x x 3 ) d x 3 d ξ d η d x 1 × d x 2 d x exp ( j 2 π f x x ) d f x .
u obs ( x ) = κ χ ( 3 ) A 3 s 2 ( x ) s 2 * ( x ) h 1 ( ξ ) s 1 ( x 1 ξ ) × s ( x 1 ξ x ) exp [ j π λ d 1 ( x x 1 ) 2 ] d ξ d x 1 .
u obs ( x ) = κ χ ( 3 ) A 3 h 1 ( ξ ) s 1 ( x 1 ξ ) s ( x 1 ξ x ) × exp [ j π λ d 1 ( x x 1 ) 2 ] d ξ d x 1 .
u obs pc ( x ) = κ χ ( 3 ) A 3 s 1 ( x 1 ) exp [ j π λ d 1 ( x x 1 ) 2 ] d x 1 .
u obs ( x ) = κ χ ( 3 ) A 3 I s 1 ,
I = exp ( j π λ d 1 ξ 2 ) d ξ = ( 1 + j ) ( λ d 1 / 2 ) 1 / 2 ,
u obs ( x ) = κ χ ( 3 ) A 3 s 2 ( x ) exp ( j π λ d 2 x 2 ) s 2 * ( η ) × h 1 ( ξ ) s 1 ( x ξ ) s ( x ξ η ) × exp [ j π λ d 2 ( η 2 2 x η + 2 x x ) ] d ξ d η d x .
u obs ( x ) = κ χ ( 3 ) A 3 s 2 ( x ) exp ( j π λ d 2 x 2 ) s 2 * ( η ) s 1 ( x ) × exp [ j π λ d 2 ( η 2 x η + 2 x x ) ] d η d x .
u obs ( x ) = κ χ ( 3 ) A 3 s 2 ( x ) exp ( j π λ d 2 x 2 ) s 2 * ( η ) s 1 ( x ξ ) × δ ( x ξ η ) exp [ j π λ d 2 ( η 2 2 x η + 2 x x ) ] d ξ d η d x .
u obs ( x ) = κ χ ( 3 ) A 3 s 2 ( x ) s 2 * ( x ) s 1 ( x ) ,
u obs ( x ) = κ χ ( 3 ) A 3 s 1 ( x ) ;
u obs ( x ) = κ χ ( 3 ) A 3 H 2 ( f x ) H 2 * ( f x ) S 1 ( f x ) exp ( j 2 π f x x ) d f x .
S 1 ( f x ) = s 1 ( x ) exp ( j 2 π f x x ) d x ,
u obs ( x ) = κ χ ( 3 ) A 3 s 1 ( x ) .
u out ( x ) = κ χ ( 3 ) u 1 ( x ) u 2 * ( x ) u 3 ( x ) d f 0 .
S 1 im ( x 1 ) = S ( f 0 ) H 1 ( f x ) s 1 ( x ) exp [ j 2 π ( f x f 0 ) x ] d x × exp ( j 2 π f x x 1 ) d f x , S 2 im ( x 2 ) = S ( f 0 ) H 2 ( f x ) s 2 ( x ) exp [ j 2 π ( f x f 0 ) x ] d x × exp ( j 2 π f x x 2 ) d f x .
S 1 im ( x 1 ) = S ( f 0 ) h 1 ( ξ ) s 1 ( x 1 ξ ) exp [ j 2 π f 0 ( x 1 ξ ) ] d ξ , S 2 im ( x 2 ) = S ( f 0 ) h 2 ( η ) s 2 ( x 2 η ) exp [ j 2 π f 0 ( x 2 η ) ] d η .
u 1 ( x ) = S ( f 0 ) { h 1 ( ξ ) s 1 ( x 1 ξ ) exp [ j 2 π f 0 ( x 1 ξ ) ] d ξ } × exp [ j π λ d 1 ( x x 1 ) 2 ] d x 1 , u 2 ( x ) = S ( f 0 ) { h 2 ( η ) s 2 ( x 2 η ) exp [ j 2 π f 0 ( x 2 η ) ] d η } × exp [ j π λ d 2 ( x x 2 ) 2 ] d x 2 .
u out ( x ) = κ χ ( 3 ) A 3 | S ( f 0 ) | 2 { h 1 ( ξ ) s 1 ( x 1 ξ ) × exp [ j 2 π f 0 ( x 1 ξ ) ] d ξ exp [ j π λ d 1 ( x x 1 ) 2 ] d x 1 } × { h 2 ( η ) s 2 ( x 2 η ) exp [ j 2 π f 0 ( x 2 η ) ] d η × exp [ j π λ d 2 ( x x 2 ) 2 ] d x 2 } * d f 0 .
u out ( x ) = κ χ ( 3 ) A 3 h 1 ( ξ ) s 1 ( x 1 ξ ) exp [ j π λ d 1 ( x x 1 ) 2 ] × h 2 ( η ) s 2 ( x 2 η ) exp [ j π λ d 2 ( x x 2 ) 2 ] × { | S ( f 0 ) | 2 exp [ j 2 π f 0 ( x 1 x 2 + η ξ ) ] d f 0 } × d ξ d η d x 1 d x 2 .
s ( x ) = | S ( f 0 ) | 2 exp ( j 2 π f 0 x ) d f 0 .
u out ( x ) = κ χ ( 3 ) A 3 h 1 ( ξ ) s 1 ( x 1 ξ ) exp [ j π λ d 1 ( x x 1 ) 2 ] × h 2 ( η ) s 2 ( x 2 η ) exp [ j π λ d 2 ( x x 2 ) 2 ] × s ( x 1 x 2 + η ξ ) d ξ d η d x 1 d x 2 .
u out ( 2 ) ( x 3 ) = u out ( x ) exp [ j π λ d 2 ( x 3 x ) 2 ] d x .
u out ( 2 ) ( x 3 ) = κ χ ( 3 ) A 3 h 1 ( ξ ) s 1 ( x 1 ξ ) × exp [ j π λ d 1 ( x x 1 ) 2 ] h 2 * ( η ) s 2 * ( x 2 η ) × exp { j π λ d 2 [ ( x x 2 ) 2 ( x 3 x ) 2 ] } × s ( x 1 x 2 + η ξ ) d ξ d η d x 1 d x 2 d x .
U freq ( f x ) = κ χ ( 3 ) A 3 h 1 ( ξ ) s 1 ( x 1 ξ ) × exp [ j π λ d 1 ( x x 1 ) 2 ] h 2 * ( η ) s 2 * ( x 2 η ) × exp [ j π λ d 2 ( x 2 2 2 x 2 x + 2 x 3 x x 3 2 ) ] × s ( x 1 x 2 + η ξ ) exp ( j 2 π f x x 3 ) × d x 3 d ξ d η d x 1 d x 2 d x .
u obs ( x ) = s 2 ( x ) 1 { H 2 ( f x ) U freq ( f x ) } ,
u obs ( x ) = κ χ ( 3 ) A 3 s 2 ( x ) H 2 ( f x ) h 1 ( ξ ) × s 1 ( x 1 ξ ) exp [ j π λ d 1 ( x x 1 ) 2 ] h 2 * ( η ) s 2 * ( x 2 η ) × exp [ j π λ d 2 ( x 2 2 2 x 2 x + 2 x 3 x x 3 2 ) ] × s ( x 1 x 2 + η ξ ) exp ( j 2 π f x x 3 ) d x 3 d ξ d η × d x 1 d x 2 d x exp ( j 2 π f x x ) d f x .

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