Abstract

The problem of high-resolution imaging through long horizontal-path ground-level turbulence has gone unsolved since it was first addressed many decades ago. In this paper I describe a method that shows promise for diffraction-limited imaging through ground-level turbulence with large (meters) apertures and at large (kilometers) distances. The key lies in collecting image data in the spatial frequency domain via the method of Fourier telescopy and taking suitable time averages of the magnitude and phase of the Fourier telescopy signal. The method requires active illumination of the target with laser light, and the time averages required will likely be over many tens of seconds if not tens of minutes or more. The scheme will thus not be suitable for time-varying scenes. The basic scheme is described, and principle challenges briefly discussed.

© 2012 Optical Society of America

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References

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  1. R. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010).
  2. J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).
  3. G. W. Carhart and M. A. Vorontsov, “Synthetic imaging: nonadaptive anisoplanatic image correction in atmospheric turbulence,” Opt. Lett. 23, 745–747 (1998).
    [CrossRef]
  4. M. A. Vorontsov and G. W. Carhart, “Anisoplanatic imaging through turbulent media: image recovery by local information fusion from a set of short-exposure images,” J. Opt. Soc. Am. A 18, 1312–1324 (2001).
    [CrossRef]
  5. C. D. Mackay, Institute of Astronomy, University of Cambridge, Madingly Road, Cambridge CB3 0HA, UK (private communication, 2011). See also the paper by S. Zhang, F. F. Suess, and C. D. Mackay, “Anisoplanatic lucky imaging for surveillance,” https://www.ast.cam.ac.uk/sites/default/files/OSApaper_211106-2.pdf .
  6. A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University, 2006), Chap. 5.
  7. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
  8. W. T. Rhodes, “Optical imaging through horizontal-path turbulence: a new solution to a difficult problem?” in Imaging Systems Applications, OSA Technical Digest (CD) (Optical Society of America, 2011, paper IW82. A related presentation, “High-resolution imaging through horizontal path turbulence,” was given at the SPIE Symposium A Tribute to Joseph W. Goodman, San Diego, 21 August 2011.
  9. R. B. Holmes, S. Ma, A. Bhowmik, and C. Greninger, “Analysis and simulation of a synthetic-aperture technique for imaging through a turbulent medium,” J. Opt. Soc. Am. A 13, 351–364 (1996).
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    [CrossRef]
  12. Light bucket is a colloquial expression for a flux collecting detector designed solely to collect radiation with no attempt made to form an image.
  13. S. R. Silva and W. T. Rhodes are preparing a manuscript to be called “Sinusoidal motion of Young’s fringes for Fourier telescopy tests.”
  14. A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
    [CrossRef]
  15. See, e.g., J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef]
  16. N. L. Swanson, C. N. Pham, and D. H. VanWinkle, “Fringe visibility of multimode laser light scattered through turbid water,” Appl. Opt. 36, 9509–9514 (1997).
    [CrossRef]
  17. J. R. Fienup, Institute of Optics, University of Rochester, New York 14627, USA (private communication, 2011).
  18. D. Feldkhun and K. H. Wagner, “Doppler encoded excitation pattern tomographic optical microscopy,” Appl. Opt. 49, H47–H63 (2010).
    [CrossRef]
  19. D. Pava and W. T. Rhodes are preparing a manuscript to be called “Spatio-temporal non-redundant arrays for Fourier telescopy applications.”

2010 (1)

2001 (1)

1998 (1)

1997 (1)

1996 (1)

1993 (2)

1981 (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Bhowmik, A.

Carhart, G. W.

Feldkhun, D.

Fienup, J. R.

J. R. Fienup, Institute of Optics, University of Rochester, New York 14627, USA (private communication, 2011).

Greninger, C.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

Holmes, R. B.

Huntley, J. M.

Labeyrie, A.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University, 2006), Chap. 5.

Lim, J. S.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Lipson, S. G.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University, 2006), Chap. 5.

Ma, S.

Mandrosov, V. I.

V. I. Mandrosov, Coherent Fields and Images in Remote Sensing (SPIE, 2003).

Nisenson, P.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University, 2006), Chap. 5.

Oppenheim, A. V.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Pava, D.

D. Pava and W. T. Rhodes are preparing a manuscript to be called “Spatio-temporal non-redundant arrays for Fourier telescopy applications.”

Pham, C. N.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Rhodes, W. T.

S. R. Silva and W. T. Rhodes are preparing a manuscript to be called “Sinusoidal motion of Young’s fringes for Fourier telescopy tests.”

D. Pava and W. T. Rhodes are preparing a manuscript to be called “Spatio-temporal non-redundant arrays for Fourier telescopy applications.”

W. T. Rhodes, “Optical imaging through horizontal-path turbulence: a new solution to a difficult problem?” in Imaging Systems Applications, OSA Technical Digest (CD) (Optical Society of America, 2011, paper IW82. A related presentation, “High-resolution imaging through horizontal path turbulence,” was given at the SPIE Symposium A Tribute to Joseph W. Goodman, San Diego, 21 August 2011.

Saldner, H.

Sica, L.

Silva, S. R.

S. R. Silva and W. T. Rhodes are preparing a manuscript to be called “Sinusoidal motion of Young’s fringes for Fourier telescopy tests.”

Swanson, N. L.

Tyson, R.

R. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010).

VanWinkle, D. H.

Vorontsov, M. A.

Wagner, K. H.

Appl. Opt. (3)

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Proc. IEEE (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Other (11)

V. I. Mandrosov, Coherent Fields and Images in Remote Sensing (SPIE, 2003).

J. R. Fienup, Institute of Optics, University of Rochester, New York 14627, USA (private communication, 2011).

D. Pava and W. T. Rhodes are preparing a manuscript to be called “Spatio-temporal non-redundant arrays for Fourier telescopy applications.”

R. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

C. D. Mackay, Institute of Astronomy, University of Cambridge, Madingly Road, Cambridge CB3 0HA, UK (private communication, 2011). See also the paper by S. Zhang, F. F. Suess, and C. D. Mackay, “Anisoplanatic lucky imaging for surveillance,” https://www.ast.cam.ac.uk/sites/default/files/OSApaper_211106-2.pdf .

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University, 2006), Chap. 5.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

W. T. Rhodes, “Optical imaging through horizontal-path turbulence: a new solution to a difficult problem?” in Imaging Systems Applications, OSA Technical Digest (CD) (Optical Society of America, 2011, paper IW82. A related presentation, “High-resolution imaging through horizontal path turbulence,” was given at the SPIE Symposium A Tribute to Joseph W. Goodman, San Diego, 21 August 2011.

Light bucket is a colloquial expression for a flux collecting detector designed solely to collect radiation with no attempt made to form an image.

S. R. Silva and W. T. Rhodes are preparing a manuscript to be called “Sinusoidal motion of Young’s fringes for Fourier telescopy tests.”

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Figures (6)

Fig. 1.
Fig. 1.

Sampling (a) in the space domain and (b) in the spatial frequency domain.

Fig. 2.
Fig. 2.

Space-domain imaging and the effect of turbulence and time averaging on the PSF. All high spatial frequency content is lost to the long-exposure PSF and, therefore, to the image.

Fig. 3.
Fig. 3.

Fourier domain imaging illustrating the effect of time averaging on spatial frequency components: (a) sinusoidal test pattern, (b) snapshots of the test pattern as viewed through turbulence in different states, (c) result of time averaging the blurred and distorted imagery. Although its contrast is reduced, perhaps greatly, the time-average pattern is unblurred and undistorted, and the phase of the sinusoid is the same as that of the test target. It is important to note that the contrast-reducing time average illustrated in this figure is not what is actually performed in the proposed time-average Fourier telescopy scheme.

Fig. 4.
Fig. 4.

Production of moving Young’s fringes through the interference of light from two spatially separated but mutually coherent point sources of light that are offset in temporal frequency. The spatial frequency of the fringes is given by u=S/λd. In a practical system two mutually coherent laser beams are interfered. With turbulence, optical path lengths l1(t) and l2(t) vary with time, introducing position- and time-varying perturbations in the phase of the fringe pattern.

Fig. 5.
Fig. 5.

Temporal frequency spectrum (left) and phasor diagram (right) for the bandpass output signal from the Fourier telescopy detector. (a) In the absence of turbulence, the complex amplitude of the detected sinusoidal signal has the desired magnitude and phase, |F(u,v)| and ψ(u,v). (b) In the presence of turbulence, the complex amplitude s˜(t;u,v) of the signal wanders in time with a probability cloud suggested by the shading. Under reasonably assumed conditions, the time-varying phasor has average magnitude |s˜(t;u,v)| proportional to |F(u,v)| and average phase θ(t;u,v) equal to ψ(u,v).

Fig. 6.
Fig. 6.

The phase θ(t) of the Fourier telescopy signal in the presence of turbulence must be sampled (dots) at a rate sufficiently high that it allows tracking through 2π phase jumps. To the right of the signal appears a representative probability density function fθ(θ). For ground-level turbulence over path lengths of hundreds of meters, this function may easily have a standard deviation σθ that is significantly larger than 2π radians.

Equations (9)

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Iill(x,y)=1+cos[ωt2π(ux+vy)],
i(t)=f(x,y){1+cos[ωt2π(ux+vy)]}dxdy.
i(t)=B+s(t;u,v),
s(t;u,v)=f(x,y)cos[ωt2π(ux+vy)]dxdy=Re{f(x,y)exp{i[ωt2π(ux+vy)]}dxdy}=Re{eiωtf(x,y)ei2π(ux+vy)dxdy}=Re{eiωtF(u,v)}=Re{eiωt|F(u,v)|eiψ(u,v)}=|F(u,v)|cos[ωt+ψ(u,v)],
F(u,v)=|F(u,v)|ejψ(u,v),
Iill(x,y;t)=b(x,y,t)+a(x,y,t)cos[ωt2π(ux+vy)+ϕ(x,y,t)].
s(t;u,v)=f(x,y)a(x,y,t)cos[ωt2π(ux+vy)+ϕ(x,y,t)]dxdy,
s(t;u,v)=|s˜(t;u,v)|cos[ωt+arg{s˜(t;u,v)}],
s˜(t;u,v)=F{f(x,y)a(x,y,t)ejϕ(x,y,t)},

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