Abstract

There has long been interest in reducing or circumventing the limitations imposed by atmospheric turbulence on optical imaging systems. Recent studies have shown that irradiance or speckle interferometry, or real-time atmospheric compensation, may be used to regain diffraction-limited performance. In this paper, the relationship between real-time phase compensation and the optimum channel-matched filter compensator is developed, with emphasis on the fundamental limits imposed by the propagation medium. In particular, the effects of uncompensated amplitude fluctuations and finite isoplanatic diameter are evaluated. It is shown that the former does not usually present any limitation on imaging performance, whereas the latter may severely limit the field of view over which diffraction-limited imaging can be realized. Results are presented for both coherently and incoherently illuminated objects.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  2. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  3. P. H. Deitz and F. P. Carlson, J. Opt. Soc. Am. 63, 274 (1973).
    [Crossref]
  4. P. H. Deitz, J. Opt. Soc. Am. 65, 279 (1975).
    [Crossref]
  5. M. Greenebaum, “The Residual Effects of Atmospheric Turbulence on a Class of Holographic Imaging and Correlography Systems,” presented at the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.
  6. A. Labeyrie, Astron. Astrophys,  6, 85 (1970).
  7. D. Y. Gezari, A. Labeyrie, and R. V. Stachnik, Astrophys. J. 173, L1 (1972).
    [Crossref]
  8. D. Korff, J. Opt. Soc. Am. 63, 971 (1973).
    [Crossref]
  9. Recent studies indicate that with more complicated processing the object phase information that is lacking in the correlation function may also be retrieved, c.f., K. T. Knox and B. J. Thompson, Astrophys. J. 193, L45 (1974); P. H. Deitz and F. P. Carlson, J. Opt. Soc. Am. 64, 11 (1974).
    [Crossref]
  10. R. A. Muller and A. Buffington, J. Opt. Soc. Am. 64, 1200 (1974).
    [Crossref]
  11. J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Phase Correction of Optical Imaging Systems,” presented at the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.
  12. W. T. Cathey, C. L. Hayes, W. C. Davis, and V. F. Pizzuro, Appl. Opt. 9, 701 (1970).
    [Crossref] [PubMed]
  13. J. H. Shapiro, IEEE Trans. Commun. Tech.,  COM-19, 410 (1971).
    [Crossref]
  14. G. Q. McDowell, “Pre-Distortion of Local-Oscillator Wave-front for Improved Optical Heterodyne Detection through a Turbulent Atmosphere,” Sc. D. thesis (Dept. of Electrical Engineering, M. I. T., Cambridge, Mass., April1971).
  15. W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
    [Crossref] [PubMed]
  16. V. N. Mahajan, J. Opt. Soc. Am. 65, 271 (1975).
    [Crossref]
  17. F. J. Dyson, J. Opt. Soc. Am. 65, 551 (1975).
    [Crossref]
  18. S. R. Robinson, “Spatial Phase Compensation Receivers for Optical Communication,” Ph. D. thesis (Dept. of Electrical Engineering, M.I.T., Cambridge, Mass., May1975).
  19. J. C. Wyant, “Active Wavefront Compensation for Astronomical Applications,” presented at the Topical Meeting on Imaging in Astronomy, Cambridge, Mass., June 1975.
  20. J. H. Shapiro, Appl. Opt. 13, 2609 (1974).
    [Crossref] [PubMed]
  21. In other words, we shall assume a uniform turbulence-strength distribution along the propagation path.
  22. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  23. R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [Crossref] [PubMed]
  24. H. T. Yura, Appl. Opt. 11, 1399 (1972).
    [Crossref] [PubMed]
  25. H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph. D. thesis (Case Western Reserve University, Cleveland, Ohio, August1973).
  26. J. H. Shapiro, Appl. Opt. 13, 2614 (1974).
    [Crossref] [PubMed]
  27. J. H. Shapiro, J. Opt. Soc. Am. 65, 65 (1975).
    [Crossref]
  28. R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
    [Crossref]
  29. R. L. Fante and J. L. Poirier, Appl. Opt. 12, 2247 (1973).
    [Crossref] [PubMed]
  30. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [Crossref]
  31. R. L. Fante, Proc. IEEE 63, 1669 (1975).
    [Crossref]
  32. We assume the atmosphere is ergodic, i.e., time averages equal ensemble averages.
  33. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.
  34. J. H. Shapiro, “Diffraction-limited atmospheric imaging of extended objects,” J. Opt. Soc. Am. 66, 469 (1976).
    [Crossref]
  35. This result also applies to nonuniform turbulence within the region of validity of Rytov’s method.
  36. J. H. Shapiro, J. Opt. Soc. Am. 64, 540A (1974).
  37. J. H. Shapiro, J. Opt. Soc. Am. 61, 492 (1971).
    [Crossref]
  38. J. D. Gaskill, J. Opt. Soc. Am. 58, 600 (1968).
    [Crossref]
  39. J. L. Bufton, P. O. Minott, M. W. Fitzmaurice, and P. J. Titterton, J. Opt. Soc. Am. 62, 1068 (1972).
    [Crossref]
  40. L. S. Taylor and C. J. Infosino, J. Opt. Soc. Am. 65, 78 (1975).
    [Crossref]
  41. By quasimonochromatic we mean that Δλ/λ is small enough to ensure that the atmospheric log-amplitude and phase perturbations are completely correlated across the wavelength band of the source, c.f., D. L. Fried, Appl. Opt. 10, 721 (1971).
    [Crossref] [PubMed]

1976 (1)

1975 (6)

1974 (7)

J. H. Shapiro, Appl. Opt. 13, 2609 (1974).
[Crossref] [PubMed]

W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
[Crossref] [PubMed]

Recent studies indicate that with more complicated processing the object phase information that is lacking in the correlation function may also be retrieved, c.f., K. T. Knox and B. J. Thompson, Astrophys. J. 193, L45 (1974); P. H. Deitz and F. P. Carlson, J. Opt. Soc. Am. 64, 11 (1974).
[Crossref]

R. A. Muller and A. Buffington, J. Opt. Soc. Am. 64, 1200 (1974).
[Crossref]

J. H. Shapiro, Appl. Opt. 13, 2614 (1974).
[Crossref] [PubMed]

R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
[Crossref]

J. H. Shapiro, J. Opt. Soc. Am. 64, 540A (1974).

1973 (3)

1972 (3)

1971 (4)

1970 (2)

1968 (1)

1966 (1)

1965 (1)

1964 (1)

Bridges, W. B.

Brown, W. P.

Brunner, P. T.

Buffington, A.

Bufton, J. L.

Carlson, F. P.

Cathey, W. T.

Davis, W. C.

Deitz, P. H.

Dyson, F. J.

Fante, R. L.

Feinleib, J.

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Phase Correction of Optical Imaging Systems,” presented at the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Fitzmaurice, M. W.

Fried, D. L.

Gaskill, J. D.

Gezari, D. Y.

D. Y. Gezari, A. Labeyrie, and R. V. Stachnik, Astrophys. J. 173, L1 (1972).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

Greenebaum, M.

M. Greenebaum, “The Residual Effects of Atmospheric Turbulence on a Class of Holographic Imaging and Correlography Systems,” presented at the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Hardy, J. W.

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Phase Correction of Optical Imaging Systems,” presented at the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Hayes, C. L.

Hufnagel, R. E.

Infosino, C. J.

Knox, K. T.

Recent studies indicate that with more complicated processing the object phase information that is lacking in the correlation function may also be retrieved, c.f., K. T. Knox and B. J. Thompson, Astrophys. J. 193, L45 (1974); P. H. Deitz and F. P. Carlson, J. Opt. Soc. Am. 64, 11 (1974).
[Crossref]

Korff, D.

Labeyrie, A.

D. Y. Gezari, A. Labeyrie, and R. V. Stachnik, Astrophys. J. 173, L1 (1972).
[Crossref]

A. Labeyrie, Astron. Astrophys,  6, 85 (1970).

Lazzara, S. P.

Lin, H. S.

H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph. D. thesis (Case Western Reserve University, Cleveland, Ohio, August1973).

Lutomirski, R. F.

Mahajan, V. N.

McDowell, G. Q.

G. Q. McDowell, “Pre-Distortion of Local-Oscillator Wave-front for Improved Optical Heterodyne Detection through a Turbulent Atmosphere,” Sc. D. thesis (Dept. of Electrical Engineering, M. I. T., Cambridge, Mass., April1971).

Minott, P. O.

Muller, R. A.

Nussmeier, T. A.

O’Meara, T. R.

Pizzuro, V. F.

Poirier, J. L.

Robinson, S. R.

S. R. Robinson, “Spatial Phase Compensation Receivers for Optical Communication,” Ph. D. thesis (Dept. of Electrical Engineering, M.I.T., Cambridge, Mass., May1975).

Sanguinet, J. A.

Shapiro, J. H.

Stachnik, R. V.

D. Y. Gezari, A. Labeyrie, and R. V. Stachnik, Astrophys. J. 173, L1 (1972).
[Crossref]

Stanley, N. R.

Tatarski, V. I.

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

Taylor, L. S.

Thompson, B. J.

Recent studies indicate that with more complicated processing the object phase information that is lacking in the correlation function may also be retrieved, c.f., K. T. Knox and B. J. Thompson, Astrophys. J. 193, L45 (1974); P. H. Deitz and F. P. Carlson, J. Opt. Soc. Am. 64, 11 (1974).
[Crossref]

Titterton, P. J.

Wyant, J. C.

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Phase Correction of Optical Imaging Systems,” presented at the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

J. C. Wyant, “Active Wavefront Compensation for Astronomical Applications,” presented at the Topical Meeting on Imaging in Astronomy, Cambridge, Mass., June 1975.

Yura, H. T.

Appl. Opt. (8)

Astron. Astrophys (1)

A. Labeyrie, Astron. Astrophys,  6, 85 (1970).

Astrophys. J. (2)

D. Y. Gezari, A. Labeyrie, and R. V. Stachnik, Astrophys. J. 173, L1 (1972).
[Crossref]

Recent studies indicate that with more complicated processing the object phase information that is lacking in the correlation function may also be retrieved, c.f., K. T. Knox and B. J. Thompson, Astrophys. J. 193, L45 (1974); P. H. Deitz and F. P. Carlson, J. Opt. Soc. Am. 64, 11 (1974).
[Crossref]

IEEE Trans. Commun. Tech. (1)

J. H. Shapiro, IEEE Trans. Commun. Tech.,  COM-19, 410 (1971).
[Crossref]

J. Opt. Soc. Am. (17)

Proc. IEEE (1)

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[Crossref]

Other (11)

We assume the atmosphere is ergodic, i.e., time averages equal ensemble averages.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph. D. thesis (Case Western Reserve University, Cleveland, Ohio, August1973).

This result also applies to nonuniform turbulence within the region of validity of Rytov’s method.

M. Greenebaum, “The Residual Effects of Atmospheric Turbulence on a Class of Holographic Imaging and Correlography Systems,” presented at the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Phase Correction of Optical Imaging Systems,” presented at the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

S. R. Robinson, “Spatial Phase Compensation Receivers for Optical Communication,” Ph. D. thesis (Dept. of Electrical Engineering, M.I.T., Cambridge, Mass., May1975).

J. C. Wyant, “Active Wavefront Compensation for Astronomical Applications,” presented at the Topical Meeting on Imaging in Astronomy, Cambridge, Mass., June 1975.

G. Q. McDowell, “Pre-Distortion of Local-Oscillator Wave-front for Improved Optical Heterodyne Detection through a Turbulent Atmosphere,” Sc. D. thesis (Dept. of Electrical Engineering, M. I. T., Cambridge, Mass., April1971).

In other words, we shall assume a uniform turbulence-strength distribution along the propagation path.

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

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 (6)

FIG. 1
FIG. 1

Propagation geometry.

FIG. 2
FIG. 2

Block diagram of imaging system.

FIG. 3
FIG. 3

Geometry used for mean-square error calculation.

FIG. 4
FIG. 4

( η ¯/2)2 vs d1/ρ0 for various values cf Df0.

FIG. 5
FIG. 5

( η ¯/2)2 vs d2ρ0z for various values of Df0.

FIG. 6
FIG. 6

Geometry for enhanced amplitude effects in space-to-earth imaging.

Equations (42)

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

E 0 ( r ¯ ) = R 1 d r ¯ E i ( r ¯ ) h 21 ( r ¯ , r ¯ ) ,             r ¯ R 2 .
h 21 ( r ¯ , r ¯ ) = { exp [ i 2 π ( z + r ¯ - r ¯ 2 / 2 z ) / λ ] / i λ z } × exp [ χ ( r ¯ , r ¯ ) + i ϕ ( r ¯ , r ¯ ) ] ,
exp { χ ( r ¯ + ρ ¯ , r ¯ + ρ ¯ ) + χ ( r ¯ , r ¯ ) + i [ ϕ ( r ¯ + ρ ¯ , r ¯ + ρ ¯ ) - ϕ ( r ¯ , r ¯ ) ] } = exp ( - D ( ρ ¯ , ρ ¯ ) / 2 ) ,
D ( ρ ¯ , ρ ¯ ) = 2.91 k 2 C n 2 z 0 1 d s ρ ¯ s + ρ ¯ ( 1 - s ) 5 / 3
D ( ρ ¯ , ρ ¯ ) = ( ρ ¯ 2 + ρ ¯ · ρ ¯ + ρ ¯ 2 ) / ρ 0 2 ,
d r ¯ 1 d r ¯ 2 f ( r ¯ 1 ) exp ( - r ¯ 1 - r ¯ 2 5 / 3 / 2 ρ 0 5 / 3 ) f * ( r ¯ 2 )
d r ¯ 1 d r ¯ 2 f ( r ¯ 1 ) exp ( - r ¯ 1 - r ¯ 2 2 / 2 ρ 0 2 ) f * ( r ¯ 2 ) ,
exp ( i π r ¯ 2 2 / λ z ) Ê i L ( r ¯ 2 ) = R 2 d r ¯ R 1 d r ¯ 1 [ exp ( i π r ¯ 1 2 / λ z ) E i ( r ¯ 1 ) / ( λ z ) 2 ] × exp [ χ ( r ¯ , r ¯ 1 ) + i ϕ ( r ¯ , r ¯ 1 ) + i 2 π r ¯ · ( r ¯ 2 - r ¯ 1 ) / λ z ] .
Ê i L ( r ¯ ) 2 = R 2 d r ¯ 1 R 2 d r ¯ 2 R 1 d r ¯ 1 R 1 d r ¯ 2 { exp [ i π ( r ¯ 1 2 - r ¯ 2 2 ) / λ z ] E i ( r ¯ 1 ) E i * ( r ¯ 2 ) / ( λ z ) 4 } × exp { i 2 π [ r ¯ 1 · ( r ¯ - r ¯ 1 ) - r ¯ 2 · ( r ¯ - r ¯ 2 ) ] / λ z - D ( r ¯ 1 - r ¯ 2 , r ¯ 1 - r ¯ 2 ) / 2 } .
h 21 ( r ¯ , r ¯ ) h 21 * ( r ¯ - r ¯ , 0 ¯ ) 2 / h 21 ( r ¯ , r ¯ ) 2 × h 21 ( r ¯ - r ¯ , 0 ¯ ) 2 = exp [ - D ( r ¯ , r ¯ ) ] .
Ê i L ( r ¯ ) 2 = d f ¯ 1 d f ¯ 2 circ ( 2 λ z f ¯ 1 / d 2 ) circ ( 2 λ z f ¯ 2 / d 2 ) E ˜ i ( f ¯ 1 ) E ˜ i * ( f ¯ 2 ) × exp [ i 2 π r ¯ · ( f ¯ 1 - f ¯ 2 ) - D ( λ z ( f ¯ 1 - f ¯ 2 ) , 0 ¯ ) / 2 ] ,
E ˜ i L ( f ¯ ) = E ˜ i ( f ¯ ) circ ( 2 λ z f ¯ / d 2 ) ,
exp ( i π r ¯ 2 / λ z ) E i ( r ¯ ) = A cos ( 2 π f ¯ 0 · r ¯ ) ,
Ê i L ( r ¯ ) 2 = ( A 2 / 2 ) { 1 + cos ( 4 π f ¯ 0 · r ¯ ) exp [ - D ( 2 λ z f ¯ 0 , 0 ¯ ) / 2 ] } ;
Ê i CF ( r ¯ ) = R 2 d r ¯ E 0 ( r ¯ ) h 21 * ( r ¯ , r ¯ ) ,
E ˜ i CF ( f ¯ ) = E ˜ i ( f ¯ ) H CF ( f ¯ ) ,
H CF ( f ¯ ) = 0 d x 2 π J 1 ( 2 π x ) J 0 ( 4 π λ z f ¯ x / d 2 ) × exp [ - D ( 0 ¯ , 2 λ z x / d 2 ) / 2 ]
Var ( v ) = v 2 - v 2 0 ,
Ê i CF ( r ¯ ) 2 = Var [ Ê i CF ( r ¯ ) ] + Ê i CF ( r ¯ ) 2 .
Ê i CF ( r ¯ ) 2 = R 2 d r ¯ 1 R 2 d r ¯ 2 R 1 d r ¯ 1 R 1 d r ¯ 2 { exp [ i π ( r ¯ 1 2 - r ¯ 2 2 ) / λ z ] E i ( r ¯ 1 ) E i * ( r ¯ 2 ) / ( λ z ) 4 } × exp { i 2 π [ r ¯ 1 · ( r ¯ - r ¯ 1 ) - r ¯ 2 · ( r ¯ - r ¯ 2 ) ] / λ z - [ D ( 0 ¯ , r ¯ 1 - r ¯ ) + D ( 0 ¯ , r ¯ 2 - r ¯ ) + D ( r ¯ 1 - r ¯ 2 , r ¯ 1 - r ¯ 2 ) + D ( r ¯ 1 - r ¯ 2 , 0 ¯ ) - D ( r ¯ 1 - r ¯ 2 , r ¯ 1 - r ¯ ) - D ( r ¯ 1 - r ¯ 2 , r ¯ - r ¯ 2 ) ] / 2 + 2 [ C χ ( r ¯ 1 - r ¯ 2 , r ¯ 1 - r ¯ ) + i C χ , ϕ ( r ¯ 1 - r ¯ 2 , r ¯ 1 - r ¯ ) + C χ ( r ¯ 1 - r ¯ 2 , r ¯ - r ¯ 2 ) - i C χ , ϕ ( r ¯ 1 - r ¯ 2 , r ¯ - r ¯ 2 ) ] } ,
Ê i PM ( r ¯ ) = - R 2 d r ¯ E 0 ( r ¯ ) exp { - i [ π r ¯ - r ¯ 2 / λ z + ϕ ( r ¯ , r ¯ ) ] } / i λ z ,
E ˜ i PM ( f ¯ ) = E ˜ i ( f ¯ ) H PM ( f ¯ ) ,
Ê i PM ( r ¯ ) 2 Ê i PM ( r ¯ ) 2 ,
H PM ( f ¯ ) = 0 d x 2 π J 1 ( 2 π x ) J 0 ( 4 π λ z f ¯ x / d 2 ) × exp { - [ D ( 0 ¯ , 2 λ z x / d 2 ) / 2 + C χ ( 0 ¯ , 2 λ z x / d 2 ) ] + C χ ( 0 ¯ , 0 ¯ ) / 2 } .
circ ( 2 λ z f ¯ / d 2 ) exp ( C χ ( 0 ¯ , 0 ¯ ) / 2 )
Ê i TR ( r ¯ ) = R 2 d r ¯ E 0 ( r ¯ ) h 21 * ( r ¯ - r ¯ + r ¯ 0 , r ¯ 0 ) .
= R 1 d r ¯ Ê i TR ( r ¯ ) - Ê i CF ( r ¯ ) 2 / R 1 d r ¯ E i ( r ¯ ) 2 .
K ( r ¯ 1 , r ¯ 2 ) = R 2 d r ¯ [ h 21 * ( r ¯ , r ¯ 1 ) - h 21 * ( r ¯ - r ¯ 1 , 0 ¯ ) ] [ h 21 ( r ¯ , r ¯ 2 ) - h 21 ( r ¯ - r ¯ 2 , 0 ¯ ) ] = [ 2 d 2 2 / ( λ z ) 2 ] exp [ i π ( r ¯ 2 2 - r ¯ 1 2 ) / λ z ] sinc [ d 2 ( r 1 x - r 2 x ) / λ z ] sinc [ d 2 ( r 1 y - r 2 y ) / λ z ] × { exp [ - D ( 0 ¯ , r ¯ 1 - r ¯ 2 ) / 2 ] - exp [ - D ( r ¯ 1 , r ¯ 2 ) / 2 ] } .
0 ( 3 d 1 2 / 4 ρ 0 2 ) 2 ,
0 [ 2 π ( d 2 ρ 0 / λ z ) 2 ] 2
η ¯ = [ R 1 d r ¯ 1 R 1 d r ¯ 2 K ( r ¯ 1 , r ¯ 2 ) 2 ] / R 1 d r ¯ K ( r ¯ , r ¯ ) .
Ê i TR ( r ¯ 2 ) = | R 1 d r ¯ 1 [ exp ( i π r ¯ 1 2 / λ z ) E i ( r ¯ 1 ) ] exp [ - D ( r ¯ 2 , r ¯ 1 ) / 2 ] × d 2 J 1 ( π r ¯ 1 - r ¯ 2 d 2 / λ z ) / 2 λ z r ¯ 1 - r ¯ 2 | Ê i CF ( r ¯ 2 ) exp ( - D ( r ¯ 2 , r ¯ 2 ) / 2 ) .
Ê i TR ( r ¯ ) 2 = R 2 d r ¯ 1 R 2 d r ¯ 2 R 1 d r ¯ 1 R 1 d r ¯ 2 { exp [ i π ( r ¯ 1 2 - r ¯ 2 2 ) / λ z ] E i ( r ¯ 1 ) E i * ( r ¯ 2 ) / ( λ z ) 4 } × exp { i 2 π [ r ¯ 1 · ( r ¯ - r ¯ 1 ) - r ¯ 2 · ( r ¯ - r ¯ 2 ) ] / λ z - [ D ( r ¯ , r ¯ 1 ) + D ( r ¯ , r ¯ 2 ) + D ( r ¯ 1 - r ¯ 2 , 0 ¯ ) + D ( r ¯ 1 - r ¯ 2 , r ¯ 1 - r ¯ 2 ) - D ( r ¯ 1 - r ¯ 2 + r ¯ , r ¯ 1 ) - D ( r ¯ 1 - r ¯ 2 - r ¯ , - r ¯ 2 ) ] / 2 + 2 [ C χ ( r ¯ 1 - r ¯ 2 + r ¯ , r ¯ 1 ) + i C χ , ϕ ( r ¯ 1 - r ¯ 2 + r ¯ , r ¯ 1 ) + C χ ( r ¯ 1 - r ¯ 2 - r ¯ , - r ¯ 2 ) - i C χ , ϕ ( r ¯ 1 - r ¯ 2 - r ¯ , - r ¯ 2 ) ] } .
Ê i TR ( r ¯ ) 2 = Var [ Ê i TR ( r ¯ ) ] + Ê i TR ( r ¯ ) 2 .
D ( ρ ¯ , ρ ¯ ) = 2.91 k 2 z 0 1 d s C n 2 ( s z ) ρ ¯ s + ρ ¯ ( 1 - s ) 5 / 3 ,
d 1 < ρ 0 = ( 1.09 k 2 z 0 1 d s C n 2 ( s z ) ) - 3 / 5
E i ( r ¯ 1 ) E i * ( r ¯ 2 ) s = I i ( r ¯ 1 ) δ ( r ¯ 1 - r ¯ 2 ) ,
I ˆ i L ( r ¯ ) = Ê i L ( r ¯ ) 2 s = d f ¯ Ĩ I ( f ¯ ) H L ( f ¯ ) exp ( i 2 π f ¯ · r ¯ ) ,
H L ( f ¯ ) = [ π d 2 2 / 4 ( λ z ) 2 ] exp [ - D ( λ z f ¯ , 0 ¯ ) / 2 ] { cos - 1 ( λ z f ¯ / d 2 ) - ( λ z f ¯ / d 2 ) [ 1 - ( λ z f ¯ / d 2 ) 2 ] 1 / 2 } 2 / π .
I ˆ i CF ( r ¯ ) = Ê i CF ( r ¯ ) 2 s d f ¯ Ĩ i ( f ¯ ) ( d f ¯ H CF ( f ¯ + f ¯ ) H CF * ( f ¯ ) )
I ˆ i PM ( r ¯ ) = Ê i PM ( r ¯ ) 2 s d f ¯ Ĩ i ( f ¯ ) ( d f ¯ H PM ( f ¯ + f ¯ ) H PM * ( f ¯ ) ) .
I ˆ i TR ( r ¯ ) = Ê i TR ( r ¯ ) 2 s = Var [ Ê i TR ( r ¯ ) ] s + I ˆ i CF ( r ¯ ) exp [ - D ( r ¯ , r ¯ ) ] .