Abstract

A new technique of high-resolution imaging through atmospheric turbulence is described. As in speckle interferometry, short-exposure images are recorded, but in addition the associated wave fronts are measured by a Hartmann–Shack wave-front sensor. The wave front is used to calculate the point-spread function. The object is then estimated from the correlation of images and point-spread functions by a deconvolution process. An experimental setup is described, and the first laboratory results, which prove the capabilities of the method, are presented. A signal-to-noise-ratio calculation, permitting a first comparison with the speckle interferometry, is also presented.

© 1990 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 283–368.
    [CrossRef]
  2. A. Labeyrie, “High resolution techniques in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 14, pp. 47–87.
    [CrossRef]
  3. J. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
    [CrossRef]
  4. G. P. Weigelt, “Modified astronomical speckle interferometry. Speckle masking,” Opt. Commun. 21, 55–59 (1977).
    [CrossRef]
  5. J. Primot, G. Rousset, J. C. Fontanella, “Image deconvolution from wavefront sensing: atmospheric turbulence simulation cell results,” in Proceedings of the ESO Conference on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 683–692; G. Rousset, J. Primot, J. C. Fontanella, “Turbulent wavefront sensing and image processing,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 311–318 (1988).
    [CrossRef]
  6. F. Roddier, “Passive versus active methods in optical interferometry,” in Proceedings of the ESO/NOAO Conference on High Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 683–692.
  7. G. Rousset, J. Primot, J. C. Fontanella, “Visible wavefront development,” presented at the Workshop on Adaptive Optics in Solar Observations, Freiburg, Federal Republic of Germany, September 8–9, 1987.
  8. J. C. Fontanella, “Analyse de surface d’onde, déconvolution et optique adaptative,” J. Opt. (Paris) 16, 257–268 (1985).
    [CrossRef]
  9. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  10. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  11. C. L. Lawson, R. J. Hanson, Solving Least Square Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  12. S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).
  13. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  15. J. W. Goodman, Statistical Optics (Wiley, New York, 1984).
  16. F. Roddier, “Atmospheric limitations to high angular resolution imaging,” in Proceedings of the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjär, eds. (European Southern Observatory, Garching, Federal Republic of Germany, 1981), pp. 5–23.
  17. J. Primot, “Application des techniques d’analyse de surface d’onde à la restauration d’images dégradées par la turbulence atmosphérique,” Ph.D. dissertation (Université de Paris-Sud, Orsay, France, 1989).
  18. M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” in Proceedings of the 4th International Symposium on Gas Flow and Chemical Lasers, M. Onorato, ed. (Plenum, New York, 1982), pp. 525–532.
  19. D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
    [CrossRef]

1985 (1)

J. C. Fontanella, “Analyse de surface d’onde, déconvolution et optique adaptative,” J. Opt. (Paris) 16, 257–268 (1985).
[CrossRef]

1980 (1)

1978 (1)

1977 (2)

1976 (1)

1974 (2)

J. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
[CrossRef]

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Ameer, G. A.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Bezdid’ko, S. N.

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Billard, M.

M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” in Proceedings of the 4th International Symposium on Gas Flow and Chemical Lasers, M. Onorato, ed. (Plenum, New York, 1982), pp. 525–532.

Brown, S. L.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Cochran, G. C.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Dueck, R.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Fertin, G.

M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” in Proceedings of the 4th International Symposium on Gas Flow and Chemical Lasers, M. Onorato, ed. (Plenum, New York, 1982), pp. 525–532.

Fontanella, J. C.

J. C. Fontanella, “Analyse de surface d’onde, déconvolution et optique adaptative,” J. Opt. (Paris) 16, 257–268 (1985).
[CrossRef]

M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” in Proceedings of the 4th International Symposium on Gas Flow and Chemical Lasers, M. Onorato, ed. (Plenum, New York, 1982), pp. 525–532.

J. Primot, G. Rousset, J. C. Fontanella, “Image deconvolution from wavefront sensing: atmospheric turbulence simulation cell results,” in Proceedings of the ESO Conference on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 683–692; G. Rousset, J. Primot, J. C. Fontanella, “Turbulent wavefront sensing and image processing,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 311–318 (1988).
[CrossRef]

G. Rousset, J. Primot, J. C. Fontanella, “Visible wavefront development,” presented at the Workshop on Adaptive Optics in Solar Observations, Freiburg, Federal Republic of Germany, September 8–9, 1987.

Fried, D. L.

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
[CrossRef]

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1984).

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Square Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Knox, J. T.

J. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “High resolution techniques in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 14, pp. 47–87.
[CrossRef]

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Square Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Lussier, D. M.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Markey, J. K.

Moretti, W.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Noll, R. J.

Primot, J.

J. Primot, “Application des techniques d’analyse de surface d’onde à la restauration d’images dégradées par la turbulence atmosphérique,” Ph.D. dissertation (Université de Paris-Sud, Orsay, France, 1989).

J. Primot, G. Rousset, J. C. Fontanella, “Image deconvolution from wavefront sensing: atmospheric turbulence simulation cell results,” in Proceedings of the ESO Conference on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 683–692; G. Rousset, J. Primot, J. C. Fontanella, “Turbulent wavefront sensing and image processing,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 311–318 (1988).
[CrossRef]

G. Rousset, J. Primot, J. C. Fontanella, “Visible wavefront development,” presented at the Workshop on Adaptive Optics in Solar Observations, Freiburg, Federal Republic of Germany, September 8–9, 1987.

Roberts, P. H.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 283–368.
[CrossRef]

F. Roddier, “Passive versus active methods in optical interferometry,” in Proceedings of the ESO/NOAO Conference on High Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 683–692.

F. Roddier, “Atmospheric limitations to high angular resolution imaging,” in Proceedings of the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjär, eds. (European Southern Observatory, Garching, Federal Republic of Germany, 1981), pp. 5–23.

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, “Image deconvolution from wavefront sensing: atmospheric turbulence simulation cell results,” in Proceedings of the ESO Conference on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 683–692; G. Rousset, J. Primot, J. C. Fontanella, “Turbulent wavefront sensing and image processing,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 311–318 (1988).
[CrossRef]

G. Rousset, J. Primot, J. C. Fontanella, “Visible wavefront development,” presented at the Workshop on Adaptive Optics in Solar Observations, Freiburg, Federal Republic of Germany, September 8–9, 1987.

Southwell, W. H.

Steinhoss, K. E.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Thompson, B. J.

J. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
[CrossRef]

Tyler, G. A.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Wang, J. Y.

Weigelt, G. P.

G. P. Weigelt, “Modified astronomical speckle interferometry. Speckle masking,” Opt. Commun. 21, 55–59 (1977).
[CrossRef]

Winker, D. M.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Astron. J. (1)

J. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
[CrossRef]

J. Opt. (Paris) (1)

J. C. Fontanella, “Analyse de surface d’onde, déconvolution et optique adaptative,” J. Opt. (Paris) 16, 257–268 (1985).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Commun. (1)

G. P. Weigelt, “Modified astronomical speckle interferometry. Speckle masking,” Opt. Commun. 21, 55–59 (1977).
[CrossRef]

Sov. J. Opt. Technol. (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Other (11)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 283–368.
[CrossRef]

A. Labeyrie, “High resolution techniques in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 14, pp. 47–87.
[CrossRef]

J. Primot, G. Rousset, J. C. Fontanella, “Image deconvolution from wavefront sensing: atmospheric turbulence simulation cell results,” in Proceedings of the ESO Conference on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 683–692; G. Rousset, J. Primot, J. C. Fontanella, “Turbulent wavefront sensing and image processing,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 311–318 (1988).
[CrossRef]

F. Roddier, “Passive versus active methods in optical interferometry,” in Proceedings of the ESO/NOAO Conference on High Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 683–692.

G. Rousset, J. Primot, J. C. Fontanella, “Visible wavefront development,” presented at the Workshop on Adaptive Optics in Solar Observations, Freiburg, Federal Republic of Germany, September 8–9, 1987.

J. W. Goodman, Statistical Optics (Wiley, New York, 1984).

F. Roddier, “Atmospheric limitations to high angular resolution imaging,” in Proceedings of the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjär, eds. (European Southern Observatory, Garching, Federal Republic of Germany, 1981), pp. 5–23.

J. Primot, “Application des techniques d’analyse de surface d’onde à la restauration d’images dégradées par la turbulence atmosphérique,” Ph.D. dissertation (Université de Paris-Sud, Orsay, France, 1989).

M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” in Proceedings of the 4th International Symposium on Gas Flow and Chemical Lasers, M. Onorato, ed. (Plenum, New York, 1982), pp. 525–532.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoss, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds., Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

C. L. Lawson, R. J. Hanson, Solving Least Square Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

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Figures (10)

Fig. 1
Fig. 1

Block diagram of the processing algorithm from the generation of the turbulence-degraded images to the object restoration. A function symbol marked with a tilde denotes the Fourier transform of the function.

Fig. 2
Fig. 2

Principle of a Hartmann–Shack wave-front sensor.

Fig. 3
Fig. 3

Image pattern generated by a Hartmann–Shack wave-front sensor for 13 × 13 subapertures in the case of a circular pupil.

Fig. 4
Fig. 4

Experimental setup used to obtain simultaneous records of turbulence-degraded images and wave fronts.

Fig. 5
Fig. 5

Left, autocorrelations of 20 first Zernike polynomial coefficients as a function of the number of the polynomials; right, cross correlations between a2a8, a3a7, a4a11, a5a13, a6a12, a7a17, a8a16, a9a19, and a10a18. Crosses, theoretical values; filled circles, experimental values from 100 records.

Fig. 6
Fig. 6

Squared speckle transfer function calculated from a set of 100 measured turbulent wave fronts.

Fig. 7
Fig. 7

Improvement in the reconstruction process versus the number of Zernike polynomials considered: a, long-exposure image; b, with tilt correction; c, correction with 11 Zernike polynomials; d, correction with 22 Zernike polynomials; e, correction with 55 Zernike polynomials.

Fig. 8
Fig. 8

Encircled normalized energy as a function of the image field x. x = 1 corresponds to the first dark ring of a diffraction-limited point source. 1, unperturbed point source; 2, long-exposure image for D/r0 = 13; 3, tilt correction; 4, correction with 11 Zernike polynomials; 5, correction with 22 Zernike polynomials; 6, correction with 55 Zernike polynomials.

Fig. 9
Fig. 9

Laboratory results of the image and wave-front processing technique obtained with images generated in a turbulence simulation cell. a, The unperturbed image; b, the associated point-spread function; c, one typical turbulence-degraded image for D/r0 = 13; d, the corresponding point-spread function calculated from the measured wave front (88 Zernike polynomials); e, the direct deconvolution of this image by its corresponding point-spread function; and f, the best estimate of the object from 100 records. © The Walt Disney Company; reproduced by permission.

Fig. 10
Fig. 10

Optical scheme for a particular lenslet of the Hartmann–Shack wave-front sensor.

Equations (80)

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

I i = 0 * S i .
I ˜ i = O ˜ S ˜ i ,
O ˜ e = I ˜ i S ˜ i * S ˜ i S ˜ i * ,
| S ˜ i | 2 = S ˜ i 2 + 0.342 ( r 0 / D ) 2 T 0 ,
E = | O ˜ e S ˜ i I ˜ i | 2 ¯ ,
S ˜ i = e i Φ * e i Φ .
x c = x i s i s i ,
x c = λ f 2 π A sa sa Φ u d u d υ ,
σ s = 2.6 2 π ( d / r 0 ) 5 / 6 ( waves ) ,
σ M = π 2 n a δ a d λ f ( waves ) ,
σ M = 4 3 σ n G s N s 3 ( waves ) ,
Φ = i = 1 L a i Z i .
Φ / u j = i = 2 L a i Z i / u j , Φ / υ j = i = 2 L a i Z i / υ j , j = 1 , , K .
g = B · a ,
a = ( B t · B ) 1 · t B · g ,
Z even i = ( n + 1 ) 1 / 2 R n m ( r ) 2 cos m θ ( m 0 ) , Z odd i = ( n + 1 ) 1 / 2 R n m ( r ) 2 sin m θ ( m 0 ) , Z i = ( n + 1 ) 1 / 2 R n 0 ( r ) ( m = 0 ) ,
R n m = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! r n 2 s s ! [ ( n + m ) / 2 s ] ! [ ( n m ) / 2 s ] ! ,
( 1 / π ) aperture Z i ( r , θ ) Z i ( r , θ ) r d r d θ = δ i i ,
δ i i = { 1 if i = i 0 if i i .
σ t = 1 2 π ( a i 2 ) 1 / 2 ( waves ) .
σ p = σ M { Tr [ ( B · B ) 1 ] } 1 / 2 ( waves ) ,
σ m = 0.54 L 3 / 4 ( D / r 0 ) 5 / 6 ( waves ) ,
σ Φ 2 = σ p 2 + σ m 2 ( waves squared ) .
O ˜ e ( f ) = i = 1 M I ˜ i ( f ) S ˜ i c * ( f ) i = 1 M | S ˜ i c ( f ) | 2 ,
S ˜ i ( f ) = T 0 ( f ) / N ( f ) n = 1 N ( f ) exp ( i D Φ n ) , f > r 0 / λ ,
S ˜ i c ( f ) = T 0 ( f ) / N ( f ) n = 1 N ( f ) exp [ i ( i D Φ n + δ Φ n ) ] ,
σ δ Φ n 2 = 8 π 2 σ Φ 2 ( rad 2 ) ,
S ˜ i c ( f ) = S ˜ i ( f ) + δ S ˜ i ( f )
δ S ˜ i ( f ) = i T 0 ( f ) / N ( f ) n = 1 N ( f ) δ Φ n exp ( i D Φ n ) .
| S ˜ i c ( f ) | 2 = | S ˜ i ( f ) | 2 + | δ S ˜ i ( f ) | 2 .
| δ S ˜ i ( f ) | 2 = 8 π 2 σ Φ 2 | S ˜ i ( f ) | 2 ;
| S ˜ i c ( f ) | 2 = ( 1 + 8 π 2 σ Φ 2 ) | S ˜ i ( f ) | 2 .
Num ( O ˜ e ) = i = 1 M I ˜ i ( f ) S ˜ i c * ( f ) ,
I ˜ i ( f ) = O ˜ ( f ) S ˜ i ( f ) + δ I ˜ i ( f )
O ˜ ( f ) = N p W 1 / 2 ( f ) .
| δ I ˜ i ( f ) | 2 = N p .
i = 1 M | S ˜ i c ( f ) | 2 = M ( 1 + 8 π 2 σ Φ 2 ) | S ˜ i ( f ) | 2 .
O ˜ e ( f ) = O ˜ ( 1 + 8 π 2 σ Φ 2 ) + 1 M i = 1 M α i exp ( i Φ α i ) + 1 M i = 1 M β i exp ( i Φ β i ) ,
α i = O ˜ | S ˜ i ( f ) | | δ S ˜ i ( f ) | M ( 1 + 8 π 2 σ Φ 2 ) | S ˜ i ( f ) | 2
= N 2 p W 1 / 2 ( f ) | S ˜ i ( f ) | | δ S ˜ i ( f ) | M ( 1 + 8 π 2 σ Φ 2 ) T 0 ( f )
β i = O ˜ | δ I ˜ i ( f ) | | S ˜ i ( f ) | M ( 1 + 8 π 2 σ Φ 2 ) | S ˜ i ( f ) | 2
= N 2 | δ I ˜ i ( f ) | | S ˜ i ( f ) | M ( 1 + 8 π 2 σ Φ 2 ) T 0 ( f ) .
| S ˜ i ( f ) | 2 T 0 ( f ) / N .
σ 0 e 2 = 2 N 2 p 2 W ( f ) M ( 1 + 8 π 2 σ Φ 2 ) [ 4 π 2 σ Φ 2 + 1 2 p T 0 ( f ) W ( f ) ]
σ Φ 0 e 2 = 1 M [ 4 π 2 σ Φ 2 + 1 2 p T 0 ( f ) W ( f ) ] .
SNRdec = 1 / σ Φ 0 e = M [ 2 p T 0 ( f ) W ( f ) 1 + 8 π 2 σ Φ 2 p T 0 ( f ) W ( f ) ] 1 / 2 .
SNRsi = M [ p T 0 ( f ) W ( f ) 1 + p T 0 ( f ) ( f ) ] .
SNRdec = M / 2 π σ Φ .
σ 88 = ± λ / 10 rms .
σ 3 = ± λ / 2 rms ,
σ p = ± λ / 80 rms .
( 1 + 8 π 2 σ Φ 2 ) 4 ,
σ Φ 0.7 rad .
σ Φ = 0.6 rad .
Φ s a = j = 1 a j Z j .
δ Φ ¯ / δ u = 2 / d δ Φ / δ u
= 2 / d j = 2 a j δ Z j ¯ / δ u ,
δ Z j / δ u = k γ k j u Z j , k < j .
δ Φ ¯ / δ u = j = 2 a j k γ k j u Z k ¯ .
Z k ¯ = 0 if k 1 Z 1 ¯ = 1 ( piston mode ) .
δ Φ ¯ / δ u = n = 1 n odd a 1 , n [ 2 ( n + 1 ) ] 1 / 2 .
( δ Φ ¯ / δ u ) 2 = n = 1 n , n odd n = 1 2 ( n + 1 ) ( n + 1 ) a 1 , n a 1 , n ,
( δ Φ ¯ / δ u ) 2 = 1.685 ( d / r 0 ) 5 / 3 ,
σ s 2 = d 2 / 4 π 2 ( δ Φ ¯ / δ u ) 2 = 1.685 / π 2 ( d / r 0 ) 5 / 3 ( waves squared ) ,
α 2 = ( λ / π d ) 2 ( δ Φ ¯ / δ u ) 2 .
α 2 = 0.98 × 6.88 ( λ / 2 π ) 2 d 1 / 3 r 0 5 / 3 .
δ N = δ λ f l d .
σ M = l d f σ δ N .
s ( N x , N y ) = G s exp [ 4 ( N x 2 + N y 2 ) / N t 2 ] ,
G s = 4 n a π N t 2 ,
δ N = s ( N x , N y ) N x n a .
σ δ N 2 = N s N x 2 σ n 2 n a 2 ,
σ M 2 = l 2 d 2 σ n 2 12 n a 2 f 2 N s 4 ( waves squared ) .
N t = λ f d l .
= N t / N s ,
σ M = 4 σ n 3 G s N s 3 ( wave ) .
σ δ N 2 = 1 / n a 2 ( σ c 2 ( N x , N y ) N x ) ,
σ δ N 2 = 4 n a π N t 2 exp [ 4 ( N x 2 + N y 2 ) / N t 2 ] N x 2 .
σ δ N 2 = N t 2 8 n a .
σ M = π 2 n a δ a d λ f ( waves ) .

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