Abstract

A liquid-crystal adaptive optics system using all-optical feedback interferometry is applied to partially coherent imaging through a phase disturbance. A theoretical analysis based on the propagation of the cross-spectral density shows that the blurred image due to the phase disturbance can be restored, in principle, irrespective of the state of coherence of the light illuminating the object. Experimental verification of the theory has been performed for two cases when the object to be imaged is illuminated by spatially coherent light originating from a He–Ne laser and by spatially incoherent white light from a halogen lamp. We observed in both cases that images blurred by the phase disturbance were successfully restored, in agreement with the theory, immediately after the adaptive optics system was activated. The origin of the deviation of the experimental results from the theory, together with the effect of the feedback misalignment inherent in our optical arrangement, is also discussed.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. E. Pearson, ed., Selected Papers on Adaptive Optics for Atmospheric Compensation, MS 92 (SPIE Press, Bellingham, Wash., 1994).
  2. R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic Press, New York, 1998).
  3. D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 287–310.
  4. D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology, U. Efron, ed. (Marcel Dekker, New York, 1994), pp. 585–654.
  5. G. D. Love, “Liquid crystal adaptive optics,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 273–285.
  6. T. Shirai, T. H. Barnes, T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25, 773–775 (2000).
    [CrossRef]
  7. T. Shirai, T. H. Barnes, T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive-optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
    [CrossRef]
  8. A. D. Fisher, C. Warde, “Simple closed-loop system for real-time optical phase measurement,” Opt. Lett. 4, 131–133 (1979).
    [CrossRef] [PubMed]
  9. A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 8, 353–355 (1983).
    [CrossRef] [PubMed]
  10. A. D. Fisher, “Self-referenced high-resolution adaptive wavefront estimation and compensation,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 102–112 (1985).
    [CrossRef]
  11. T. H. Barnes, T. Eiju, K. Matsuda, “High resolution adaptive optics using an Interference Phase Loop,” Opt. Commun. 132, 494–502 (1996).
    [CrossRef]
  12. M. A. Vorontsov, V. A. Katulin, A. F. Naumov, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
    [CrossRef]
  13. T. H. Barnes, T. Eiju, S. Kokaji, K. Matsuda, N. Yoshida, “Bistable optically writable image memory using phase-only spatial light modulator,” Optik (Stuttgart) 96, 107–114 (1994).
  14. T. H. Barnes, T. Eiju, K. Matsuda, “Direct, unambiguous display of phase-difference using optical feedback to a phase-only spatial light modulator,” Optik (Stuttgart) 101, 17–23 (1995).
  15. A. R. D. Somervell, T. H. Barnes, “Unambiguous measurement of surface profile using a Sagnac interferometer with phase feedback,” Opt. Commun. 150, 61–65 (1998).
    [CrossRef]
  16. T. Shirai, T. H. Barnes, T. G. Haskell, “Surface-profile measurement by means of a polarization Sagnac interferometer with parallel optical feedback,” Opt. Lett. 24, 297–299 (1999).
    [CrossRef]
  17. N. Mukohzaka, N. Yoshida, H. Toyoda, Y. Kobayashi, T. Hara, “Diffraction efficiency analysis of a parallel-aligned nematic-liquid-crystal spatial light modulator,” Appl. Opt. 33, 2804–2811 (1994).
    [CrossRef] [PubMed]
  18. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 4.4.
  19. For example, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Sec. 6.5.
  20. For phase modulation characteristics of the PAL-SLM used in our experiment, see Fig. 2 of Ref. 7.
  21. The focal length of each lens: 30 cm for L1, 20 cm for L2, 30 cm for L3, and 30 cm for L4.
  22. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 1.5.
  23. This hypothesis can, in principle, be justified experimentally by putting a color filter in front of the halogen lamp. However, the limited power of our halogen lamp (100 W) and/or the limited sensitivity of our CCD camera prevented us from obtaining meaningful data.
  24. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 7.1.3.

2001 (1)

T. Shirai, T. H. Barnes, T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive-optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

A. R. D. Somervell, T. H. Barnes, “Unambiguous measurement of surface profile using a Sagnac interferometer with phase feedback,” Opt. Commun. 150, 61–65 (1998).
[CrossRef]

1996 (1)

T. H. Barnes, T. Eiju, K. Matsuda, “High resolution adaptive optics using an Interference Phase Loop,” Opt. Commun. 132, 494–502 (1996).
[CrossRef]

1995 (1)

T. H. Barnes, T. Eiju, K. Matsuda, “Direct, unambiguous display of phase-difference using optical feedback to a phase-only spatial light modulator,” Optik (Stuttgart) 101, 17–23 (1995).

1994 (2)

T. H. Barnes, T. Eiju, S. Kokaji, K. Matsuda, N. Yoshida, “Bistable optically writable image memory using phase-only spatial light modulator,” Optik (Stuttgart) 96, 107–114 (1994).

N. Mukohzaka, N. Yoshida, H. Toyoda, Y. Kobayashi, T. Hara, “Diffraction efficiency analysis of a parallel-aligned nematic-liquid-crystal spatial light modulator,” Appl. Opt. 33, 2804–2811 (1994).
[CrossRef] [PubMed]

1989 (1)

M. A. Vorontsov, V. A. Katulin, A. F. Naumov, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

1983 (1)

1979 (1)

Barnes, T. H.

T. Shirai, T. H. Barnes, T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive-optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
[CrossRef]

T. Shirai, T. H. Barnes, T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25, 773–775 (2000).
[CrossRef]

T. Shirai, T. H. Barnes, T. G. Haskell, “Surface-profile measurement by means of a polarization Sagnac interferometer with parallel optical feedback,” Opt. Lett. 24, 297–299 (1999).
[CrossRef]

A. R. D. Somervell, T. H. Barnes, “Unambiguous measurement of surface profile using a Sagnac interferometer with phase feedback,” Opt. Commun. 150, 61–65 (1998).
[CrossRef]

T. H. Barnes, T. Eiju, K. Matsuda, “High resolution adaptive optics using an Interference Phase Loop,” Opt. Commun. 132, 494–502 (1996).
[CrossRef]

T. H. Barnes, T. Eiju, K. Matsuda, “Direct, unambiguous display of phase-difference using optical feedback to a phase-only spatial light modulator,” Optik (Stuttgart) 101, 17–23 (1995).

T. H. Barnes, T. Eiju, S. Kokaji, K. Matsuda, N. Yoshida, “Bistable optically writable image memory using phase-only spatial light modulator,” Optik (Stuttgart) 96, 107–114 (1994).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 1.5.

Eiju, T.

T. H. Barnes, T. Eiju, K. Matsuda, “High resolution adaptive optics using an Interference Phase Loop,” Opt. Commun. 132, 494–502 (1996).
[CrossRef]

T. H. Barnes, T. Eiju, K. Matsuda, “Direct, unambiguous display of phase-difference using optical feedback to a phase-only spatial light modulator,” Optik (Stuttgart) 101, 17–23 (1995).

T. H. Barnes, T. Eiju, S. Kokaji, K. Matsuda, N. Yoshida, “Bistable optically writable image memory using phase-only spatial light modulator,” Optik (Stuttgart) 96, 107–114 (1994).

Fisher, A. D.

Gaeta, C. J.

D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology, U. Efron, ed. (Marcel Dekker, New York, 1994), pp. 585–654.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 7.1.3.

For example, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Sec. 6.5.

Hara, T.

Haskell, T. G.

Katulin, V. A.

M. A. Vorontsov, V. A. Katulin, A. F. Naumov, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

Kobayashi, Y.

Kokaji, S.

T. H. Barnes, T. Eiju, S. Kokaji, K. Matsuda, N. Yoshida, “Bistable optically writable image memory using phase-only spatial light modulator,” Optik (Stuttgart) 96, 107–114 (1994).

Liang, J.

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 287–310.

Love, G. D.

G. D. Love, “Liquid crystal adaptive optics,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 273–285.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 4.4.

Matsuda, K.

T. H. Barnes, T. Eiju, K. Matsuda, “High resolution adaptive optics using an Interference Phase Loop,” Opt. Commun. 132, 494–502 (1996).
[CrossRef]

T. H. Barnes, T. Eiju, K. Matsuda, “Direct, unambiguous display of phase-difference using optical feedback to a phase-only spatial light modulator,” Optik (Stuttgart) 101, 17–23 (1995).

T. H. Barnes, T. Eiju, S. Kokaji, K. Matsuda, N. Yoshida, “Bistable optically writable image memory using phase-only spatial light modulator,” Optik (Stuttgart) 96, 107–114 (1994).

Miller, D. T.

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 287–310.

Mitchell, P. V.

D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology, U. Efron, ed. (Marcel Dekker, New York, 1994), pp. 585–654.

Mukohzaka, N.

Naumov, A. F.

M. A. Vorontsov, V. A. Katulin, A. F. Naumov, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

Pepper, D. M.

D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology, U. Efron, ed. (Marcel Dekker, New York, 1994), pp. 585–654.

Roorda, A.

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 287–310.

Shirai, T.

Somervell, A. R. D.

A. R. D. Somervell, T. H. Barnes, “Unambiguous measurement of surface profile using a Sagnac interferometer with phase feedback,” Opt. Commun. 150, 61–65 (1998).
[CrossRef]

Toyoda, H.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic Press, New York, 1998).

Vorontsov, M. A.

M. A. Vorontsov, V. A. Katulin, A. F. Naumov, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

Warde, C.

Williams, D. R.

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 287–310.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 1.5.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 4.4.

Yoshida, N.

N. Mukohzaka, N. Yoshida, H. Toyoda, Y. Kobayashi, T. Hara, “Diffraction efficiency analysis of a parallel-aligned nematic-liquid-crystal spatial light modulator,” Appl. Opt. 33, 2804–2811 (1994).
[CrossRef] [PubMed]

T. H. Barnes, T. Eiju, S. Kokaji, K. Matsuda, N. Yoshida, “Bistable optically writable image memory using phase-only spatial light modulator,” Optik (Stuttgart) 96, 107–114 (1994).

Appl. Opt. (1)

Opt. Commun. (4)

A. R. D. Somervell, T. H. Barnes, “Unambiguous measurement of surface profile using a Sagnac interferometer with phase feedback,” Opt. Commun. 150, 61–65 (1998).
[CrossRef]

T. H. Barnes, T. Eiju, K. Matsuda, “High resolution adaptive optics using an Interference Phase Loop,” Opt. Commun. 132, 494–502 (1996).
[CrossRef]

M. A. Vorontsov, V. A. Katulin, A. F. Naumov, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

T. Shirai, T. H. Barnes, T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive-optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
[CrossRef]

Opt. Lett. (4)

Optik (Stuttgart) (2)

T. H. Barnes, T. Eiju, S. Kokaji, K. Matsuda, N. Yoshida, “Bistable optically writable image memory using phase-only spatial light modulator,” Optik (Stuttgart) 96, 107–114 (1994).

T. H. Barnes, T. Eiju, K. Matsuda, “Direct, unambiguous display of phase-difference using optical feedback to a phase-only spatial light modulator,” Optik (Stuttgart) 101, 17–23 (1995).

Other (13)

J. E. Pearson, ed., Selected Papers on Adaptive Optics for Atmospheric Compensation, MS 92 (SPIE Press, Bellingham, Wash., 1994).

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic Press, New York, 1998).

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 287–310.

D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology, U. Efron, ed. (Marcel Dekker, New York, 1994), pp. 585–654.

G. D. Love, “Liquid crystal adaptive optics,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), pp. 273–285.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 4.4.

For example, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Sec. 6.5.

For phase modulation characteristics of the PAL-SLM used in our experiment, see Fig. 2 of Ref. 7.

The focal length of each lens: 30 cm for L1, 20 cm for L2, 30 cm for L3, and 30 cm for L4.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 1.5.

This hypothesis can, in principle, be justified experimentally by putting a color filter in front of the halogen lamp. However, the limited power of our halogen lamp (100 W) and/or the limited sensitivity of our CCD camera prevented us from obtaining meaningful data.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 7.1.3.

A. D. Fisher, “Self-referenced high-resolution adaptive wavefront estimation and compensation,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 102–112 (1985).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Experimental setup for adaptive optics based on all-optical feedback interferometry. The meanings of the abbreviations are as follows: BS1, 2, beam splitters; WP1, 2, wedge plates; L1–9, lenses; F1, 2, fictitious planes; PBS, polarizing beam splitter; CCD1, 2, CCD cameras; CA, circular aperture; λ/2, half-wavelength plate; IRP, image-rotating prism.

Fig. 2
Fig. 2

Experimental setup for imaging through a phase disturbance with adaptive optics. This setup is combined with the adaptive optics system illustrated in Fig. 1 by BS1 and the write light to the PAL-SLM.

Fig. 3
Fig. 3

Analyzing model for the imaging system illustrated in Fig. 2.

Fig. 4
Fig. 4

Images of a USAF test target illuminated by (a) spatially coherent light originating from a He–Ne laser and (b) spatially incoherent white light from a halogen lamp. These images were captured by CCD3 without the phase disturbance AP and the adaptive optics system.

Fig. 5
Fig. 5

Interference fringes captured by CCD1 (a) without and (b) with feedback. The actual diameter of the dashed circle is 4 mm at CCD1, but it corresponds to 6 mm at AP.

Fig. 6
Fig. 6

Images of a USAF test target (a) without and (b) with feedback in the case of coherent illumination.

Fig. 7
Fig. 7

Images of a USAF test target (a) without and (b) with feedback in the case of incoherent white-light illumination.

Fig. 8
Fig. 8

Effects of the misalignment in the feedback optics. (a) Interference fringe captured by CCD1 with misaligned feedback. Images of a USAF test target with misaligned feedback in the case of (b) coherent illumination and (c) incoherent white-light illumination.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

Iout(x, y)=A{1+V cos[ϕmod(x, y)+ϕin(x, y)]}.
cos[ϕmod(x, y)+ϕin(x, y)]=1Vϕmod(x, y)kAG-1,
ϕmod(x, y)=kGIout(x, y)
cos[ϕmod(x, y)+ϕin(x, y)]=-1,
ϕmod(x, y)+ϕin(x, y)=const.
WT(ρo1, ρo2, ω)=O*(ρo1, ω)O(ρo2, ω)Wo(ρo1, ρo2, ω),
So(ρo, ω)Wo(ρo, ρo, ω)=Io(ω).
Wi(ρi1, ρi2, ω)=k2πf4d2ρϕ1d2ρϕ2d2ρo1d2ρo2×WT(ρo1, ρo2, ω)a*(ρϕ1, ω)a(ρϕ2, ω)×exp-i kf [ρϕ2(ρo2+ρi2)-ρϕ1(ρo1+ρi1)]×exp-i Δf2 f2 k(-ρo12+ρo22+ρi12-ρi22),
a(ρϕ, ω)=exp[iϕ(ρϕ, ω)],
WI(ρI1, ρI2, ω)=k2πf4d2ρΦ1d2ρΦ2d2ρi1d2ρi2×Wi(ρi1, ρi2, ω)A*(ρΦ1, ω)A(ρΦ2, ω)×exp-i kf [ρΦ2(ρi2+ρI2)-ρΦ1(ρi1+ρI1)]×exp-i Δf2 f2 k(-ρi12+ρi22+ρI12-ρI22),
A(ρΦ, ω)=exp[iΦ(ρΦ, ω)],
WI(ρI1, ρI2, ω)
=k2πf4d2ρΦ1d2ρΦ2×d2ρo1d2ρo2×WT(ρo1, ρo2, ω)×a*(-ρΦ1, ω)a(-ρΦ2, ω) A*(ρΦ1,ω) A(ρΦ2,ω)×exp-i kf [ρΦ2  (ρI2-ρo2)-ρΦ1(ρI1-ρo1)]×exp-i Δf2 f2 k(-ρo12+ρo22+ρI12-ρI22).
A(ρ, ω)=a*(-ρ, ω),
Φ(ρ, ω)+ϕ(-ρ, ω)=const.
WI(ρI1, ρI2, ω)=WT(ρI1, ρI2, ω).
SI(ρI, ω)WI(ρI, ρI, ω)=|O(ρI, ω)|2Io(ω).
Wf(ρf1, ρf2, ω)
=k2πf2d2ρo1d2ρo2WT(ρo1, ρo2, ω)×exp-i kf (ρo2  ρf2-ρo1  ρf1).
Wϕ(ρϕ1, ρϕ2, ω)
=k2πΔf2d2ρf1d2ρf2Wf(ρf1, ρf2, ω)×exp-i k2Δf [(ρf1-ρϕ1)2-(ρf2-ρϕ2)2].
Wϕ(ρϕ1, ρϕ2, ω)=a*(ρϕ1, ω)a(ρϕ2, ω)Wϕ(ρϕ1, ρϕ2, ω),
Wl(ρl1, ρl2, ω)
=k2π(f-Δf )2d2ρϕ1d2ρϕ2Wϕ(ρϕ1, ρϕ2, ω)×exp-i k2(f-Δf ) [(ρϕ1-ρl1)2-(ρϕ2-ρl2)2].
Wl(ρl1, ρl2, ω)=exp-i k2 f (ρl22-ρl12)Wl(ρl1, ρl2, ω).
Wi(ρi1, ρi2, ω)
=k2πf2d2ρl1d2ρl2Wl(ρl1, ρl2, ω)×exp-i k2 f [(ρl1-ρi1)2-(ρl2-ρi2)2].
Wϕ(ρϕ1, ρϕ2, ω)
=k2πf2a*(ρϕ1, ω)a(ρϕ2, ω)d2ρo1d2ρo2
×WT(ρo1, ρo2, ω)×exp-i kf (ρo2  ρϕ2-ρo1  ρϕ1)×exp-i Δf2 f2 k(ρo22-ρo12).
Wl(ρl1, ρl2, ω)
=k2π41f2(f-Δf )2exp-i k2 f (ρl22-ρl12)×d2ρϕ1d2ρϕ2d2ρo1d2ρo2×WT(ρo1, ρo2, ω)a*(ρϕ1, ω)a(ρϕ2, ω)×exp-i kf (ρo2  ρϕ2-ρo1  ρϕ1)×exp-i Δf2 f2 k(ρo22-ρo12)×exp-ik2(f-Δf ) [(ρϕ1-ρl1)2-(ρϕ2-ρl2)2].
Wi(ρi1, ρi2, ω)
=k2πf4d2ρϕ1d2ρϕ2d2ρo1d2ρo2×WT(ρo1, ρo2, ω)a*(ρϕ1, ω)a(ρϕ2, ω)×exp-i kf [ρϕ2(ρo2+ρi2)-ρϕ1(ρo1+ρi1)]×exp-i Δf2 f2 k(-ρo12+ρo22+ρi12-ρi22).

Metrics