Abstract

A theory is developed for the resolution of an optical synthetic-aperture imaging system viewing an object through an inhomogeneous refractive medium. The inhomogeneities of the propagation medium create errors in the phase history data with resultant space-variant image effects, including geometric distortions and broadening of the impulse response or point-spread function. I relate the intensity-impulse response to the usual wave structure function. I determine the modulation transfer function for synthetic apertures of any size and exposure time, valid whenever the optical bandwidth is small compared with the carrier frequency, and derive the resolution for monostatic and bistatic synthetic apertures, valid whenever the real sampling aperture is small compared with the medium’s coherence length. The results take the same form as the well-known turbulence-limited resolution of incoherent, real-aperture imaging with short exposure. Turbulence-limited synthetic-aperture resolution is somewhat better than incoherent real-aperture resolution under the same conditions. Autofocus processing improves synthetic-aperture resolution beyond this limit, and adaptive correction of higher-order phase history errors would improve it further.

© 2003 Optical Society of America

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  1. D. Park, J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” in Laser Radar III, R. J. Becherer, ed., Proc. SPIE999, 100–116 (1989).
    [CrossRef]
  2. S. Marcus, B. D. Colella, T. J. Green, “Solid-state laser synthetic-aperture radar,” Appl. Opt. 33, 960–964 (1994).
    [CrossRef] [PubMed]
  3. T. J. Green, S. Marcus, B. D. Colella, “Synthetic-aperture-radar imaging with a solid-state laser,” Appl. Opt. 34, 6941–6949 (1995).
    [CrossRef] [PubMed]
  4. S. Yoshikado, T. Aruga, “Short-range verification experiment of a trial one-dimensional synthetic aperture infrared laser radar operated in the 10-μm band,” Appl. Opt. 39, 1421–1425 (2000).
    [CrossRef]
  5. C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
    [CrossRef]
  6. T. G. Kyle, “High resolution laser imaging system,” Appl. Opt. 28, 2651–2656 (1989).
    [CrossRef] [PubMed]
  7. T. S. Lewis, H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781–1782 (1970).
    [CrossRef]
  8. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985 (2002).
    [CrossRef]
  9. C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  10. R. L. Lucke, L. J. Rickard, “Photon-limited synthetic-aperture imaging for planet surface studies,” Appl. Opt. 41, 5084–5095 (2002).
    [CrossRef] [PubMed]
  11. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  12. J. L. Walker, “Range-doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst. AES-16, 23–52 (1980).
    [CrossRef]
  13. W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995).
  14. If the real sampling aperture is moved over a curved surface, then the PHD represents the three-dimensional spatial Fourier spectrum of the target object.
  15. J. Way, E. A. Smith, “The evolution of synthetic aperture radar systems and their progression to the EOS SAR,” IEEE Trans. Geosci. Remote Sens. 29, 962–985 (1991).
    [CrossRef]
  16. R. L. Jordan, B. L. Huneycutt, M. Werner, “The SIR-C/X-SAR synthetic aperture radar system,” Proc. IEEE 79, 827–838 (1991).
    [CrossRef]
  17. The projection is done along the arc of a circle with center at the center of the synthetic aperture.
  18. R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  19. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
    [CrossRef]
  20. R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  21. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  22. S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Heidelberg, 1978), pp. 9–43.
  23. R. R. Beland, “Propagation through atmospheric turbulence,” in The Infrared and Electro-Optical Systems Handbook, Vol. 2, Fred G. Smith, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 157–232.
  24. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 361–464.
  25. G. R. Heibreider, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Antennas Propag. AP-15, 90–98 (1967).
    [CrossRef]
  26. A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2.
  27. A. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds mumbers,” in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley, New York, 1961), pp. 151–155.
  28. We have chosen to chirp the local oscillator signal. This measurement process is known as dechirp-on-receive or stretch processing. The same result can be obtained by interfering sE(t)with a monochromatic optical wave and then dechirping the electronic beat signal from the optical detector either by mixing with a chirped electrical LO signal or by digital filtering.
  29. For a finite detector the interference signal is the convolution of the complex video signal with the detector impulse response backpropagated to the receiver entrance pupil.
  30. The circumflex always denotes a quantity defined for the sample as a whole, rather than at points in space.
  31. Walter G. Carrara, Ron S. Goodman, Ronald M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995), pp. 501–506.
  32. We use the untapered domain in Kspace throughout this paper because it produces the narrowest possible IPR in image space (κ0≈0.886).Other tapering may be used with only minor changes in the analysis. In scan mode the impulse response usually is tapered by the reciprocal of the two-way antenna gain G(p,θ).
  33. The slant plane coordinates pof the scattering point are translated from its focus plane coordinates p0by the layover, p=p0+δlpwhere the layover is δlp=h(p0)×(tanηe^x+tanψ e^r)for height h(p0)and tilt angle η. After removal of the carrier frequency, the scattering amplitude spcontains the height-dependent phase factor exp[i2k0(p⋅e^r+h(p0)tan η)].
  34. sinc(x)≡sin(πx)/πx.
  35. For any random variable α, the variance is Δα≡〈(α- 〈α〉)2〉.
  36. R. E. Hufnagel, N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,” J. Opt. Soc. Am. 54, 52–61 (1964).
    [CrossRef]
  37. D∝|r|2for |r|≤l0.
  38. D. L. Fried, “Limiting resolution looking down through the atmosphere,” J. Opt. Soc. Am. 56, 1380–1384 (1966).
    [CrossRef]
  39. Robert A. Schmeltzer, “Means, variances, and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).
  40. This sometimes is called the down-looking r0because it was introduced in the context of looking down at a point scatterer through the atmosphere from a space platform. But the definition, Eq. (2.25) in Ref. 38, is valid for any geometry (looking down, up, or horizontally) viewing a point scatterer. It differs from the coherence length of an infinite plane wave by the (ζ/R)5/3factor in the integrand.
  41. rect(x)≡1for |x| ≤1/2,≡0 for |x| >1/2.
  42. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]

2002 (2)

2000 (1)

1995 (1)

1994 (1)

1991 (3)

J. Way, E. A. Smith, “The evolution of synthetic aperture radar systems and their progression to the EOS SAR,” IEEE Trans. Geosci. Remote Sens. 29, 962–985 (1991).
[CrossRef]

R. L. Jordan, B. L. Huneycutt, M. Werner, “The SIR-C/X-SAR synthetic aperture radar system,” Proc. IEEE 79, 827–838 (1991).
[CrossRef]

R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[CrossRef] [PubMed]

1989 (1)

1980 (1)

J. L. Walker, “Range-doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst. AES-16, 23–52 (1980).
[CrossRef]

1979 (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

1971 (1)

1970 (1)

T. S. Lewis, H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781–1782 (1970).
[CrossRef]

1967 (2)

G. R. Heibreider, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Antennas Propag. AP-15, 90–98 (1967).
[CrossRef]

Robert A. Schmeltzer, “Means, variances, and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

1966 (2)

1965 (1)

1964 (1)

Abshier, J. O.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

Accetta, J. S.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

Aleksoff, C. C.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

Aruga, T.

Bashkansky, M.

Beland, R. R.

R. R. Beland, “Propagation through atmospheric turbulence,” in The Infrared and Electro-Optical Systems Handbook, Vol. 2, Fred G. Smith, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 157–232.

Carrara, W. G.

W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995).

Carrara, Walter G.

Walter G. Carrara, Ron S. Goodman, Ronald M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995), pp. 501–506.

Clifford, S. F.

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Heidelberg, 1978), pp. 9–43.

Colella, B. D.

Eichel, P. H.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Fee, M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

Frehlich, R. G.

Fried, D. L.

Funk, E.

Ghiglia, D. C.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 361–464.

Goodman, R. S.

W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995).

Goodman, Ron S.

Walter G. Carrara, Ron S. Goodman, Ronald M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995), pp. 501–506.

Green, T. J.

Heibreider, G. R.

G. R. Heibreider, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Antennas Propag. AP-15, 90–98 (1967).
[CrossRef]

Hufnagel, R. E.

Huneycutt, B. L.

R. L. Jordan, B. L. Huneycutt, M. Werner, “The SIR-C/X-SAR synthetic aperture radar system,” Proc. IEEE 79, 827–838 (1991).
[CrossRef]

Hutchins, H. S.

T. S. Lewis, H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781–1782 (1970).
[CrossRef]

Jakowatz, C. V.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Jordan, R. L.

R. L. Jordan, B. L. Huneycutt, M. Werner, “The SIR-C/X-SAR synthetic aperture radar system,” Proc. IEEE 79, 827–838 (1991).
[CrossRef]

Kavaya, M. J.

Klooster, A.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

Kolmogorov, A.

A. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds mumbers,” in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley, New York, 1961), pp. 151–155.

Kyle, T. G.

Lewis, T. S.

T. S. Lewis, H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781–1782 (1970).
[CrossRef]

Lucke, R. L.

Lutomirski, R. F.

Majewski, R. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995).

Majewski, Ronald M.

Walter G. Carrara, Ron S. Goodman, Ronald M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995), pp. 501–506.

Marcus, S.

Monin, A. S.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2.

Park, D.

D. Park, J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” in Laser Radar III, R. J. Becherer, ed., Proc. SPIE999, 100–116 (1989).
[CrossRef]

Peterson, L. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

Reintjes, J.

Rickard, L. J.

Schmeltzer, Robert A.

Robert A. Schmeltzer, “Means, variances, and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

Schroeder, K. S.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

Shapiro, J. H.

D. Park, J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” in Laser Radar III, R. J. Becherer, ed., Proc. SPIE999, 100–116 (1989).
[CrossRef]

Smith, E. A.

J. Way, E. A. Smith, “The evolution of synthetic aperture radar systems and their progression to the EOS SAR,” IEEE Trans. Geosci. Remote Sens. 29, 962–985 (1991).
[CrossRef]

Stanley, N. R.

Tai, A. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Thompson, P. A.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Wahl, D. E.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Walker, J. L.

J. L. Walker, “Range-doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst. AES-16, 23–52 (1980).
[CrossRef]

Way, J.

J. Way, E. A. Smith, “The evolution of synthetic aperture radar systems and their progression to the EOS SAR,” IEEE Trans. Geosci. Remote Sens. 29, 962–985 (1991).
[CrossRef]

Werner, M.

R. L. Jordan, B. L. Huneycutt, M. Werner, “The SIR-C/X-SAR synthetic aperture radar system,” Proc. IEEE 79, 827–838 (1991).
[CrossRef]

Yaglom, A. M.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2.

Yoshikado, S.

Yura, H. T.

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
[CrossRef] [PubMed]

Appl. Opt. (7)

IEEE Trans. Aerosp. Electron. Syst. (1)

J. L. Walker, “Range-doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst. AES-16, 23–52 (1980).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

G. R. Heibreider, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Antennas Propag. AP-15, 90–98 (1967).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

J. Way, E. A. Smith, “The evolution of synthetic aperture radar systems and their progression to the EOS SAR,” IEEE Trans. Geosci. Remote Sens. 29, 962–985 (1991).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Acta (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (2)

T. S. Lewis, H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781–1782 (1970).
[CrossRef]

R. L. Jordan, B. L. Huneycutt, M. Werner, “The SIR-C/X-SAR synthetic aperture radar system,” Proc. IEEE 79, 827–838 (1991).
[CrossRef]

Q. Appl. Math. (1)

Robert A. Schmeltzer, “Means, variances, and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

Other (23)

This sometimes is called the down-looking r0because it was introduced in the context of looking down at a point scatterer through the atmosphere from a space platform. But the definition, Eq. (2.25) in Ref. 38, is valid for any geometry (looking down, up, or horizontally) viewing a point scatterer. It differs from the coherence length of an infinite plane wave by the (ζ/R)5/3factor in the integrand.

rect(x)≡1for |x| ≤1/2,≡0 for |x| >1/2.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Heidelberg, 1978), pp. 9–43.

R. R. Beland, “Propagation through atmospheric turbulence,” in The Infrared and Electro-Optical Systems Handbook, Vol. 2, Fred G. Smith, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 157–232.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 361–464.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2.

A. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds mumbers,” in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley, New York, 1961), pp. 151–155.

We have chosen to chirp the local oscillator signal. This measurement process is known as dechirp-on-receive or stretch processing. The same result can be obtained by interfering sE(t)with a monochromatic optical wave and then dechirping the electronic beat signal from the optical detector either by mixing with a chirped electrical LO signal or by digital filtering.

For a finite detector the interference signal is the convolution of the complex video signal with the detector impulse response backpropagated to the receiver entrance pupil.

The circumflex always denotes a quantity defined for the sample as a whole, rather than at points in space.

Walter G. Carrara, Ron S. Goodman, Ronald M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995), pp. 501–506.

We use the untapered domain in Kspace throughout this paper because it produces the narrowest possible IPR in image space (κ0≈0.886).Other tapering may be used with only minor changes in the analysis. In scan mode the impulse response usually is tapered by the reciprocal of the two-way antenna gain G(p,θ).

The slant plane coordinates pof the scattering point are translated from its focus plane coordinates p0by the layover, p=p0+δlpwhere the layover is δlp=h(p0)×(tanηe^x+tanψ e^r)for height h(p0)and tilt angle η. After removal of the carrier frequency, the scattering amplitude spcontains the height-dependent phase factor exp[i2k0(p⋅e^r+h(p0)tan η)].

sinc(x)≡sin(πx)/πx.

For any random variable α, the variance is Δα≡〈(α- 〈α〉)2〉.

The projection is done along the arc of a circle with center at the center of the synthetic aperture.

W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, Norwood, Mass., 1995).

If the real sampling aperture is moved over a curved surface, then the PHD represents the three-dimensional spatial Fourier spectrum of the target object.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majewski, J. O. Abshier, M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser ,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 29–40 (1987).
[CrossRef]

D. Park, J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” in Laser Radar III, R. J. Becherer, ed., Proc. SPIE999, 100–116 (1989).
[CrossRef]

D∝|r|2for |r|≤l0.

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Figures (5)

Fig. 1
Fig. 1

The synthetic-aperture, image-collection process.

Fig. 2
Fig. 2

Coordinate system for synthetic-aperture phase history data collection.

Fig. 3
Fig. 3

Atmospheric coherence length r˜0 of a monostatic SAL actively sampling a point source, as a function of wavelength for several grazing angles. All curves are scaled by Eqs. (7.3) and (7.5) from a one-way coherence length r0=30 cm at λ=0.5 μm looking at nadir down to the ground (ψ=90°) from an altitude of 15 km. (The lines for ψ = 65° and ψ = 90° are nearly indistinguishable from each other.)

Fig. 4
Fig. 4

The SA resolution integral I [Eq. (8.7)], for the near field (α=1) and the far field (α=0.5). The SA limit is the resolution that would be achieved in the absence of a randomly inhomogeneous medium.

Fig. 5
Fig. 5

Comparison of the across-range or azimuth point-spread function FWHM (the IPR 3-dB width) of an optical, passive, real-aperture, image-formation system Eq. (9.1) with an optical SA system [Eq. (9.2)] looking down from an altitude of 15 km, as a function of grazing angle, for several wavelengths. The SA curves are scaled by Eqs. (7.3) and (7.5) to a passive real-aperture system with r0=30 cm at λ=0.5 μm looking at nadir down to the ground (ψ=90°) from an altitude of 15 km.

Tables (3)

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Table 1 Conditions and Assumptions

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Table 2 Effect of Propagation Medium on Optical Synthetic-Aperture Images

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Table 3 Resolution and Coherence Length of Real and Synthetic Apertures in Turbulence

Equations (149)

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ρ0r=cκr2Δf,
ρ0a=λ0Rcκa2L.
|θ|  θmax=L/2Rc.
d=θRc.
2πΔf/ω01.
Dr0,
l0LL0.
sT(x, t)=Re{sT(x)WTθ(x)exp[iϕT(t)]},
ϕT(t)=-(ω0t+πf˙t2)for|t|<τc/2.
G(p, x; ω, tθ)=G0(p, x; ω)exp[χ(x, p; ω, tθ)+iφ(x, p; ω, tθ)],
G(x, p; ω, t)=G(p, x; ω, t).
φbr(x, p; ω, tθ)φ(x, p; ω, tθ)-k0Δr¯.
sE(x, t)=RespWRθ(x)Gbr(p, x)dxGbr(p, x)×WTθ(x)sT(x, t-2R/c),
=RespWRθ(x)G0(p, x)exp[χ(x)+iφbr(x)]dxWTθ(x)G0(p, x)sT(x)×expiϕT(t-2R/c)×exp[χ(x)+iφbr(x)].
slo(x, t)=Re{slo(x)WRθ(x)exp[iϕlo(t)]},
ϕlo(t)=-[ω0(t-2R0/c)+πf˙(t-2R0/c)2].
sV(t)=s^V(p, θ)exp[iϕV(t; p)],
ϕV(t; p, θ)ϕT(t-2R/c)-ϕlo(t)=K(t)(Rp+Δr¯-R0)-(4πf˙/c2)(Rp+Δr¯-R0)2,
K(t)2ω0/c+(4πf˙/c)(t-2R0/c),
s^V(p, θ)spdxdxWRθ(x)×WTθ(x)slo*(x)sT(x)G0(p, x; θ)×G0(p, x; θ)exp[χ(x, p; tθ)+χ(x, p; tθ)+iφbr(x, p; tθ)+iφbr(x, p, tθ)].
s^V0(p, θ)=spG(p, θ)/R02(θ)
=spdxdxWRθ(x)×WTθ(x)slo*(x)sT(x)G0(p, x; θ)×G0(p, x; θ),
sV(t)=s^V(p, θ)exp[iK(t)(Rp+Δr¯-R0)]
for|t-2R0/c| τc/2.
P(K)Ke^θsV[t(K)].
ΔK=4πΔfc,
Δθ=2 arctanL2Rc.
ΔKa4k0|θ|max=4πLλ0Rc=2πκaρ0a,
ΔKr4πΔfc=2πκrρ0r.
exp(l^θ+iϕ^brθ)
spdxdxWRθ(x)WTθ(x)slo*(x)sT(x)G0(p, x)G0(p, x)×exp[χ(x; tθ)+χ(x; tθ)+iφbr(x; tθ)+iφbr(x; tθ)]s^V0.
s^V=s^V0(p, θ)exp[l^θ+iϕ^brθ].
δKaθmaxΔKa=ΔKaΔKr2k0=L2Rc4πΔfc,
2k0δθk0DRc,
δKa2k0δθ2πΔfω0LD1.
K(t)e^θK(t)e^r+2k0θe^a,
witherrorofOδKa24k0.
d=θRc=KaRc2k0.
Δa¯ : uu=Δa¯dKa[ϕ^br(Ka)-uKa]2=0,
ϕ^btrϕ^br-Δa¯Ka.
H0(K)1forKdomainofEq.(3.13),0elsewhere,
H(K)H0(K)exp[lˆ(Ka)+iϕ^btr(Ka)].
S0(K)s^V0exp[iK(t)(Rp-R0)]forKdomainofEq.(3.13)0elsewhere,
S(K)S0(K)exp[i(KaΔa¯+KrΔr¯)].
P(K)S(K)H(K).
h(u)ξdKH(K)exp(iuK),
h0(u)ξdKH0(K)exp(iuK)=sinc(aΔKa/2π)sinc(rΔKr/2π).
s=s0(p+δgp-Δa¯e^a-Δr¯e^r)h,
M(J)=μduh(u)h*(u)exp(-iuJ),
M(J)=ξ2μdKH(K)H*(K-J)
=ξ2μdKH0(K)H0(K-J)exp{lˆ(Ka)+lˆ(Ka-Ja)+i[ϕ^btr(Ka)-ϕ^btr(Ka-Ja)]},
s^Vs^V0exp[χR(x; tθ)+χT(x; tθ)+i(φbrR(x; tθ)+φbrT(x; tθ))],
lˆ(Ka)χR(x; tK)+χT(x; tK),
ϕ^br(Ka)φbrR(x; tK)+φbrT(x; tK).
φtr(x)φ(x)-xe^aaφ¯,
aφ¯: uu=aφ¯dx[φ(xe^a; tx)-ux]2=0.
ΔaR¯Rc2k0aφR¯,
ΔaT¯Rc2k0aφT¯,
Δa¯=ΔaR¯+ΔaT¯,
H(K)H0(K)exp{χT(x)+χT(x)+i[φtrR(x)+φtrT(x)]}.
M(J)ξ2μdKH0(K)H0(K-j)×exp(χR(x; tK)+χT(x; tK)+χR(y; tK-J)+χT(y; tK-J)+i{φR(x; tK)+φT(x; tK)-φR(y; tK-J)-φT(y; tK-J)-e^a[xaφR¯(x)+xaφT¯(x)-yaφR¯(y)-yaφT¯(y)]}),
(x+x)e^ak0Rc=Ka,
(y+y)e^ak0Rc=Ka-Ja,
12 (x-y+x-y)e^a=JaRc/2k0.
[χ(x)+χ(y)][φ(x)-φ(y)]=0,
[φ(x)-φ(y)]=0.
[χ(x)+χ(y)][φ(x)-φ(y)-(x-y)e^aaφ¯]=0,
exp[uα+vβ]
=exp[12 (u2Δα+v2Δβ)+uα+vβ],
exp(2χ)=1.
χ=-Δχ,
exp[-4χ]=I2/I2,
M(J)=ξ2μdKH0(K)H0(K-J)exp{χR(x)+χT(x)+χR(y)+χT(y)+i[φR(x)+φT(x)-φR(y)-φT(y)-e^x(xaφR¯+xaφT¯-yaφR¯-yaφT¯)]}.
exp{χR(x)+χT(x)+χR(y)+χT(y)+i[φR(x)+φT(x)-φR(y)-φT(y)-e^x(xaφR¯+xaφT¯-yaφR¯-yaφT¯)]}=exp{12 [χR(x)+χT(x)+χR(y)+χT(y)-2χR-2χT]2+2χR+2χT]-12 [φR(x)+φT(x)-φR(y)-φT(y)]2+12 [e^x(xaφR¯+xaφT¯-yaφR¯-yaφT¯)]2}
=exp{-12 [χR(x)+χT(x)-χR(y)-χT(y)]2-[χR(x)-χT(x)]2-[χR(y)-χT(y)]2-2χR-2χT-12 [φR(x)+φT(x)-φR(y)-φT(y)]2+12 [e^x(xxφR¯+xaφT¯-yaφR¯-yaφT¯)]2}.
|(x-y)e^a|=JaRc/2k0.
M(J)=ξ2μ exp(-4χ)dKH0(K)H0(K-J)×exp(-12D(Ja)+12 Δa¯2Ja2),
D(Ja)({2[χ(x; 0)-χ(y; tJ)]}2+[2{φ(x;0)-φ(y; tJ)]}2),
Δa¯2=4ΔaR¯2=4Rc2k02aφ¯2.
M(J)=M0(J)MA(J),
M0(J)ξ2μdKH0(K)H0(K-J),
MA(J)exp-12 D(Ja)+12 Δa¯2Ja2.
M(0)=1;
ξ2μ exp(-4χ)=1ΔKaΔKr.
M(J)=ξ2μdKH0(K)H0(K-J)exp[-12DR(JR)-12DT(JT)+12 ΔaR¯2JR2+12 ΔaT¯2JT2],
DR(JR){[χR(x; 0)-χR(y; tJR)]2+[φR(x; 0)-φR(y; tJR)]2},
JR2k0|(x-y)e^a|/Rc,
JT2k0|(x-y)e^a|/Rc,
12 (JR+JT)=Ja.
ξ2μ=1/ΔKaΔKr.
D(r)[χ(0)-χ(r)]2+[φ(0)-φ(r)]2=b(|r|/r0)5/3,
b2[245Γ(65)]5/66.884,
r010π227Γ(43)sin(π6)28/3sin(5π6)Γ(116)2bk02×0RdζCn2(ζ)(ζ/R)5/3-3/5,
r0(sin ψ)3/5λ06/5.
D(Ja)4D(JaRc/2k0).
r˜0(monostatic)r0/26/5,
D(J)=bJRc2k0r˜05/3.
D(J)=4DJRc2k0r˜0LL+VAT
r˜0=r026/5L+VATL.
R1(2π)2dJM(J).
R=1ΔKaΔKr(2π)2dJdKH0(K)H0(K-J)×exp[-12D(Ja)+12 Δa¯2Ja2].
zJaRc2k0r˜0,
|z|zmΔKaRc2k0r˜0=Lr˜0.
R=ΔKr2πΔKa2π|z|zmdzzm1-|z|zm×exp-12 bz5/3+12 Δa¯2ΔKa2zzm2.
R=R1I(L/r˜0),
R1=ΔKr2π2r˜0λ0Rc ,
I(zm)2zm01du(1-u)×exp{-12 bzm5/3[u5/3-min(αβ1, 1)u2]}.
R0limr0R=ΔKrΔKa(2π)2=κ0ρ0rκ0ρ0a.
R1κ0ρ0rκ0ρ1a,
ρ1aλ0Rcκ02r˜0
ρa=ρ1aI(L/r˜0)=λ0Rcκ02r˜01I(L/r˜0).
ρa_min0.591(λ0Rc/2r˜0).
RlimL/r0R0.851R1,
ρa_κ00.851λ0Rc2r˜01.04 λ0Rc2r˜0.
JR=2Ja,
DR(JR)=D2JaRc2k0,
DT(JT)=Dτ(tJR)D2JaRc2k0VAV,
ΔaR¯2=Rc2k02aφ¯2,
ΔaT¯2=0.
r˜0(bistatic-1)r02=21/5r˜0(monostatic).
JR=JT=Ja.
DR(JR)=DT(JT)=DJaRc2k0,
ΔaR¯2=ΔaT¯2=Rc2k02aφ¯2.
r˜0(bistatic2)r023/5=23/5r˜0(monostatic).
ρminra=2λRπr0.
ρasa0.591 λ0Rc2r˜0,
ρasa=0.591π26/54 ρminra0.602ρminra.
Δa¯2=(Rc/k0)2aφ¯2.
aφ¯2L2=β1Dφ(L),
β1144-132+344-134=4053741.083.
Dφ(r)=αD(r),α=1inthenearfield,
=½inthefarfield.
Δa¯2=1ΔKa2min(αβ1, 1)b(L/r˜0)5/3.
ΔSφ=12ΔKa2dJdJrectJ+12 JΔKa×rectJ-12 JΔKaDφ(J).
=988 αb(L/r˜0)5/30.7036α(L/r˜0)5/3.
ΔSφtr=988-β112αb(L/r˜0)5/30.08283α(L/r˜0)5/3.
L1(φ)1.234α-3/5r˜0,
L1(φtr)4.458α-3/5r˜0.
q ¯: uu=q¯dKa[ϕˆ(Ka)-Δa¯Ka -u(Ka2-ΔKa2/12)]2=0
φtfrφ-Δa¯Ka-q¯(Ka2-ΔKa2/12).
q¯=180ΔKa5dKrectKΔKaK2-ΔKa212φ(K).
q¯2=180/ΔKa4(ΔSφtr-ΔSφtfr),
q¯2=-18022ΔKa10dJdJrectJ+12 JΔKa×rectJ-12 JΔKa×(J+12 J)2-ΔKa212 (J-12 J)2-ΔKa212Dφ(J).
q¯2=(1/ΔKx4)αβ2b(L/r˜0)5/3,
β2-1802323-180+124-116+320×116+116-324+317-18+172+324+124+31418-124-124-124+311×-116-1144+124+38180-172+11441.177.
ΔSφtfr=ΔSφtr-q¯2(ΔKa4/180)0.03781α(L/r˜0)5/3,
L1(ϕ^tfr)7.135α-3/5r˜0.
Δx¯q¯=1ΔKa3 αγ12b(L/r˜0)5/3
γ12-(12)(180)320132-124+31418-16-148+38×132-1483.616.

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