Abstract

The concept of using the atmospheric backscatter from a pulsed laser as an artificial guide star (AGS) for an adaptive-optics system in an astronomical telescope is analyzed. The extent to which such an AGS can be used to provide the information that is needed for adaptive-optics-system compensation of a wave front that is distorted by propagation through the atmosphere is studied. Attention is directed to the effect of focus anisoplanatism, the measurement error that is introduced when the probe light from the AGS at a finite range travels a different path than does the light from an astronomical object at a larger (infinite) range. Because of focus anisoplanatism there is a residual wave-front error that the wave-front-distortion sensor, relying on the AGS, is unable to sense and for which the adaptive-optics system is therefore unable to compensate. This residual wave-front error has a mean-square value that takes the form σφFA2 = (D/d0)5/3, where d0 is an aperture-diameter-sized quantity that measures the magnitude of the effect of focus anisoplanatism. The value of d0 depends on the vertical distribution of the optical strength of turbulence, the optical wavelength of the imaging system, the zenith angle, and the backscatter altitude. An expression is given for d0, and sample results for d0 are presented. Analytic results are also developed for the effect of focus anisoplanatism on the Strehl definition (or normalized antenna gain) of the imaging system. Numerical results are presented for the normalized antenna gain for a wide variety of backscatter altitudes and for several vertical distributions of the optical strength of turbulence. These results indicate that the diameter dependence of the normalized antenna gain is fully expressed as a function of D/d0. The peak achievable antenna gain is approximately equal to 40% of that of a diffraction-limited system with an aperture diameter that is equal to d0, and the peak is achieved with an actual aperture diameter in the range of (7/6)d0 to (9/6)d0.

© 1994 Optical Society of America

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References

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  1. D. L. Fried, ed., topical issue on adaptive optics, J. Opt. Soc. Am.67, 269–422 (1977).
    [Crossref]
  2. F. Roddier, ed., Active Telescope Systems, Proc. Soc. Photo-Opt. Instrum. Eng.1114(1989).
    [Crossref]
  3. D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
    [Crossref]
  4. R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).
  5. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).
  6. C. A. Primmerman, “Adaptive optics experiments using synthetic beacons,” Bull. Am. Astron. Soc. 23, 898–899 (1991).
  7. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
    [Crossref]
  8. E. D. Kibblewhite, U. of Chicago, Chicago, Ill. (personal communication, 1989).
  9. T. R. O’Meara, Hughes Research Laboratory, Malibu, Calif., is responsible for the introduction of the term focus anisoplanatism.
  10. C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
    [Crossref]
  11. There is an apparent relationship between ordinary anisoplanatism (discussed in Ref. 3) and the geometry of focus anisoplanatism (presented in Fig. 1). However, because the former does not allow for the neglect of tilt error (discussed in Section 2 of the text), because it does not (easily) allow for the accommodation of the variety of angular offsets that are involved, and because it does not allow for the weakened sensing of even low spatial-frequency turbulence at altitudes approaching the backscatter altitude (and the complete insensitivity to turbulence at altitudes above the backscatter altitude), the estimation of the magnitude of focus-anisoplanatism effects from ordinary anisoplanatism theory must be considered questionable.
  12. B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [Crossref]
  13. J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefronts,” (the Optical Sciences Company, Placentia, Calif., 1984).
  14. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,”J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [Crossref]
  15. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  16. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  17. D. L. Fried, “Diffusion analysis for the propagation of mutual coherence,”J. Opt. Soc. Am. 58, 961–969 (1968).
    [Crossref]

1991 (4)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

C. A. Primmerman, “Adaptive optics experiments using synthetic beacons,” Bull. Am. Astron. Soc. 23, 898–899 (1991).

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[Crossref]

1990 (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

1985 (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

1982 (1)

1976 (1)

1968 (1)

1965 (1)

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Belsher, J. F.

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefronts,” (the Optical Sciences Company, Placentia, Calif., 1984).

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Foy, R.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Fried, D. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
[Crossref]

D. L. Fried, “Diffusion analysis for the propagation of mutual coherence,”J. Opt. Soc. Am. 58, 961–969 (1968).
[Crossref]

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,”J. Opt. Soc. Am. 55, 1427–1435 (1965).
[Crossref]

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefronts,” (the Optical Sciences Company, Placentia, Calif., 1984).

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Gardner, C. S.

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[Crossref]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

Kibblewhite, E. D.

E. D. Kibblewhite, U. of Chicago, Chicago, Ill. (personal communication, 1989).

Labeyrie, A.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Noll, R. J.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Primmerman, C. A.

C. A. Primmerman, “Adaptive optics experiments using synthetic beacons,” Bull. Am. Astron. Soc. 23, 898–899 (1991).

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

Tyler, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Welsh, B. M.

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[Crossref]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Astron. Astrophys. (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Bull. Am. Astron. Soc. (2)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Real time atmospheric compensation using adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

C. A. Primmerman, “Adaptive optics experiments using synthetic beacons,” Bull. Am. Astron. Soc. 23, 898–899 (1991).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (1)

Nature (London) (1)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Proc. IEEE (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

Other (7)

There is an apparent relationship between ordinary anisoplanatism (discussed in Ref. 3) and the geometry of focus anisoplanatism (presented in Fig. 1). However, because the former does not allow for the neglect of tilt error (discussed in Section 2 of the text), because it does not (easily) allow for the accommodation of the variety of angular offsets that are involved, and because it does not allow for the weakened sensing of even low spatial-frequency turbulence at altitudes approaching the backscatter altitude (and the complete insensitivity to turbulence at altitudes above the backscatter altitude), the estimation of the magnitude of focus-anisoplanatism effects from ordinary anisoplanatism theory must be considered questionable.

D. L. Fried, ed., topical issue on adaptive optics, J. Opt. Soc. Am.67, 269–422 (1977).
[Crossref]

F. Roddier, ed., Active Telescope Systems, Proc. Soc. Photo-Opt. Instrum. Eng.1114(1989).
[Crossref]

E. D. Kibblewhite, U. of Chicago, Chicago, Ill. (personal communication, 1989).

T. R. O’Meara, Hughes Research Laboratory, Malibu, Calif., is responsible for the introduction of the term focus anisoplanatism.

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefronts,” (the Optical Sciences Company, Placentia, Calif., 1984).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

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Figures (6)

Fig. 1
Fig. 1

Propagation geometry. We study viewing in a direction that is not necessarily normal to the ground. We indicate the telescope aperture plane and the bounds of an aperture of diameter D. We consider the center of this region to be our origin and define the s axis as starting from this origin and as being perpendicular to the aperture plane. The backscatter region is at a distance S along the s axis. The vector r, with the same origin, denotes the position on the aperture plane. The dotted lines indicate the propagation paths to the position r in the aperture plane from (1) the backscatter region and (2) an astronomical object of interest, i.e., an object that is located in a direction associated with the s axis. The difference between the effects of turbulence along these two propagation paths is what gives rise to focus anisoplanatism.

Fig. 2
Fig. 2

Vertical distribution of the optical strength of turbulence CN2(h). The value of the refractive-index structure constant CN2(h) as a function of altitude h is shown for three turbulence models. We refer to these models as SLC-Day (denoted by the solid curve), SLC-Night (denoted by the long-dashed curve), and Hufnagel–Valley 27 (denoted by the short-dashed curve).

Fig. 3
Fig. 3

Dependence of d0 on the backscatter altitude H. The quantity d0 is defined by Eq. (53), and its numerical value is calculated with Eq. (54). The results that are shown here were calculated for the three strengths of turbulence models that are shown in Fig. 2, with the same relationship of line type to turbulence model as in Fig. 2. Results were calculated for an imaging wavelength of λ = 0.5 μm and a zenith angle of ψ = 0°. The physical significance of d0 may be inferred from consideration of Eq. (56), according to which the mean-square residual wave-front error σφFA2, which is induced by the effect of focus anisoplanatism in an AGS/AOS, is proportional to the aperture diameter D to the five-thirds power, with the quantity d0−5/3 serving as the constant of proportionality. It can be seen that, if we assume that a mean-square residual wave-front error σφFA2 of 1 rad squared corresponds to the transition from a well-compensated (σφFA2 < 1) to a poorly compensated (σφFA2 > 1) AOS, d0 represents the value of the largest aperture diameter for which the AGS/AOS will provide good compensation.

Fig. 4
Fig. 4

Normalized antenna gain 〈GFA〉/GDL or Strehl definition as a function of aperture diameter D. The quantity 〈GFA〉 corresponds to the average antenna gain or the peak power density in the focal plane when imaging a point source with the deleterious effects of focus anisoplanatism present, whereas GDL corresponds to the diffraction-limited antenna gain or the peak power density in the focal plane for diffraction-limited imaging of the point source. Using Monte Carlo methods, we performed the integration in Eq. (57) to obtain the results that are shown here. The calculations were performed for a zenith angle of ψ = 0° and an imaging wavelength of λ = 0.5 μm. (a) shows the results that were obtained with the SLC-Day turbulence model that is shown in Fig. 2. (b) and (c) correspond to the SLC-Night and the Hufnagel–Valley 27 turbulence models that are shown in Fig. 2. In each of these three figures we show ten curves, corresponding to AGS backscatter altitudes of H = 5, 10, 20, 30, 40, 50, 60, 70, 80, and 90 km, with the smaller value of H corresponding to the curve that is closer to the lower left-hand corner of each figure.

Fig. 5
Fig. 5

Normalized antenna gain 〈GFA〉/GDL as a function of the normalized aperture diameter D/d0. Here all the curves in Fig. 4 are replotted to show the dependence on D/d0 rather than on D only. It is quite obvious that, except for some small discrepancies when the normalized antenna gain is less than 0.1, which we are inclined to attribute to the limited accuracy of the Monte Carlo integrations, all the results appear to reduce to a single curve. We interpret this fact (i.e., that the normalized antenna gain depends on the aperture diameter only through a dependence on D/d0) as an indication that the quantity d0 is the proper measure of the magnitude of the effect of focus anisoplanatism on the performance of an AGS/AOS, not only in terms of the mean-square residual wave-front error but also, more significantly, in terms of the ability of the system to perform in a diffraction-limited manner, i.e., in terms of the Strehl definition that the system can achieve.

Fig. 6
Fig. 6

Alternative normalization of antenna gain. Here we show the same results as are presented in Fig. 5, except that here, instead of normalizing the average antenna gain 〈GFA〉 by dividing by the antenna gain of a diffraction-limited aperture of the same diameter [i.e., dividing by GDL = (1/4)π(D/λ)2], we have normalized by dividing by the diffraction-limited antenna gain of an aperture of diameter d0 [i.e., dividing by (1/4)π(d0/λ)2]. In interpreting these results it is important to note, based on a consideration of Eqs. (14)(16), that we have defined the antenna gain in terms of unit power that is collected by the full aperture of the imaging system. This is why the curve peaks and then falls off as the aperture diameter is increased, since the increase of the aperture diameter represents some of this unit power’s being collected in the outer regions of the aperture, in which the focus-anisoplanatism-induced residual wave-front error is so large that this collected power hardly contributes to the peak focal-plane power density, with less of the unit power in the central (good part) of the aperture contributing to the peak power density. Note that the peak antenna gain that is achieved is only ~40% of that of a diffraction-limited aperture of diameter do and is achieved with an aperture diameter of D = (7/6)d0 ↔ (9/6)d0.

Equations (62)

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ϕ Δ ( r ) = ϕ ( r ) - ϕ AGS ( r ) .
φ FA ( r ) = ϕ Δ ( r ) - ϕ ¯ Δ - ϕ Δ T ( r ) ,
W ( x ) = { 1 if x 1 2 0 if x > 1 2 ,
ϕ ¯ Δ = ( 1 4 π D 2 ) - 1 d r W ( r / D ) ϕ Δ ( r ) .
ϕ Δ T ( r ) = k θ Δ · r ,
θ Δ = ( 1 64 π D 4 k ) - 1 d r W ( r / D ) ϕ Δ ( r ) r ,
k = 2 π / λ .
ϕ Δ T ( r ) = ( 1 4 π D 2 ) - 1 d ρ W ( ρ / D ) ϕ Δ ( ρ ) [ ( 4 / D ) 2 r · ρ ] .
ϕ Δ ( r ) = ( 1 4 π D 2 ) - 1 d ρ W ( ρ / D ) ϕ Δ ( r )
φ FA ( r ) = ( 1 4 π D 2 ) - 1 d ρ W ( ρ / D ) × [ ϕ Δ ( r ) - ϕ Δ ( ρ ) - ( 4 / D ) 2 ϕ Δ ( ρ ) ρ · r ] .
d ρ W ( ρ / D ) [ ( 4 / D ) 2 ϕ Δ ( r ) ] ρ · r = 0 ,
φ FA ( r ) = ( 1 4 π D 2 ) - 1 d ρ W ( ρ / D ) [ 1 + ( 4 / D ) 2 r · ρ ] × [ ϕ Δ ( r ) - ϕ Δ ( ρ ) ] .
σ φ FA 2 = ( 1 4 π D 2 ) - 1 d r W ( r / D ) [ φ FA ( r ) ] 2 ,
P Max = P Coll G DL ,
G DL = 1 4 π ( D / λ ) 2 F - 2 .
G FA = G DL | ( 1 4 π D 2 ) - 1 d r W ( r / D ) exp [ - i φ FA ( r ) ] | 2 .
σ φ FA 2 = 16 π 2 D 4 d r 1 d r 2 W ( r 1 / D ) W ( r 2 / D ) × { 1 2 [ φ FA ( r 1 ) ] 2 - [ φ FA ( r 1 ) φ FA ( r 2 ) ] + 1 2 [ φ FA ( r 2 ) ] 2 } .
D φ FA ( r 1 , r 2 ) = [ φ FA ( r 1 ) - φ FA ( r 2 ) ] 2 ,
σ φ FA 2 = 8 π 2 D 4 d r 1 d r 2 W ( r 1 / D ) W ( r 2 / D ) D φ FA ( r 1 , r 2 ) .
exp ( c ν ) = exp ( 1 2 c 2 ( ν 2 ) ) .
exp { i [ φ FA ( r 1 ) - φ FA ( r 2 ) ] } = exp [ - 1 2 D φ FA ( r 1 , r 2 ) ] .
G FA / G DL = 16 π 2 D 4 d r 1 d r 2 W ( r 1 / D ) W ( r 2 / D ) × exp [ - 1 2 D φ FA ( r 1 , r 2 ) ] .
ϕ ( r ) = k 0 d s n ( r , s ) ,
ϕ AGS ( r ) = k 0 d s H ( S - s ) n [ χ ( s ) r , s ] ,
χ ( s ) = 1 - s / S ,
H ( x ) = { 1 if x > 0 0 if x 0 .
ϕ Δ ( r ) = k 0 d s { n ( r , s ) - H ( S - s ) n [ χ ( s ) r , s ] } .
φ FA ( r ) = 4 k π D 2 d ρ W ( ρ / D ) 0 d s [ 1 + ( 4 / D ) 2 r · ρ ] × ( [ n ( r , s ) - n ( ρ , s ) ] - H ( S - s ) × { n [ χ ( s ) r , s ] - n [ χ ( s ) ρ , s ] } ) .
φ FA ( r 1 ) - φ FA ( r 2 ) = 4 k π D 4 d ρ W ( ρ / D ) 0 d s × [ [ 1 + ( 4 / D ) 2 r 1 · ρ ] ( [ n ( r 1 , s ) - n ( ρ , s ) ] - H ( S - s ) { n [ χ ( s ) r 1 , s ] - n [ χ ( s ) ρ , s ] } ) - [ 1 + ( 4 / D ) 2 r 2 · ρ ] ( [ n ( r 2 , s ) - n ( ρ , s ) ] - H ( S - s ) { n [ χ ( s ) r 2 , s ] - n [ χ ( s ) ρ , s ] } ) ] .
D φ FA ( r 1 , r 2 ) = ( 4 k π D ) 2 d ρ d ρ W ( ρ / D ) W ( ρ / D ) 0 0 d s d s × [ [ 1 + ( 4 / D ) 2 r 1 · ρ ] ( [ n ( r 1 , s ) - n ( ρ , s ) ] - H ( S - s ) { n [ χ ( s ) r 1 , s ] - n [ χ ( s ) ρ , s ] } ) - [ 1 + ( 4 / D ) 2 r 2 · ρ ] ( [ n ( r 2 , s ) - n ( ρ , s ) ] - H ( S - s ) { n [ χ ( s ) r 2 , s ] - n [ χ ( s ) ρ , s ] } ) ] × [ [ 1 + ( 4 / D ) 2 r 1 · ρ ] ( [ n ( r 1 , s ) - n ( ρ , s ) ] - H ( S - s ) { n [ χ ( s ) r 1 , s ] - n [ χ ( s ) ρ , s ] } ) - [ 1 + ( 4 / D ) 2 r 2 · ρ ] ( [ n ( r 2 , s ) - n ( ρ , s ) ] - H ( S - s ) { n [ χ ( s ) r 2 , s ] - n [ χ ( s ) ρ , s ] } ) ] .
0 A 0 B d s d s { n [ α ( s ) , s ] - n [ β ( s ) , s ] } { n [ γ ( s ) , s ] - n [ δ ( s ) , s ] } = - 1 2 0 A 0 B d s d s { D n [ α ( s ) , γ ( s ) , s , s ] - D n [ α ( s ) , δ ( s ) , s , s ] - D n [ β ( s ) , γ ( s ) , s , s ] + D n [ β ( s ) , δ ( s ) , s , s ] } = - 2.914 2 0 min ( A , B ) d s C N 2 ( s ) [ α ( s ) - γ ( s ) 5 / 3 - α ( s ) - δ ( s ) 5 / 3 - β ( s ) - γ ( s ) 5 / 3 + β ( s ) - δ ( s ) 5 / 3 ] ,
min ( A , B ) = { A if A < B B if A > B .
D φ FA ( r 1 , r 2 ) = - 2.914 2 ( 4 k π D 2 ) 2 0 d s C N 2 ( s ) d ρ d ρ W ( ρ / D ) W ( ρ / D ) { [ 1 + ( 4 / D ) 2 r 1 · ρ ] [ 1 + ( 4 / D ) 2 r 1 · ρ ] × [ r 1 - r 1 5 / 3 - r 1 - ρ 5 / 3 - ρ - r 1 5 / 3 + ρ - ρ 5 / 3 - H ( S - s ) × ( r 1 - χ r 1 5 / 3 - r 1 - χ ρ 5 / 3 - ρ - χ r 1 5 / 3 + ρ - χ ρ 5 / 3 ) - H ( S - s ) × ( χ r 1 - r 1 5 / 3 - χ r 1 - ρ 5 / 3 - χ ρ - r 1 5 / 3 + χ ρ - ρ 5 / 3 ) + H ( S - s ) × ( χ r 1 - χ r 1 5 / 3 - χ r 1 - χ ρ 5 / 3 - χ ρ - χ r 1 5 / 3 + χ ρ - χ ρ 5 / 3 ) ] - [ 1 + ( r / D ) 2 r 1 · ρ ] × [ 1 + ( 4 / D ) 2 r 2 · ρ ] [ r 1 - r 2 5 / 3 - r 1 - ρ 5 / 3 - ρ - r 2 5 / 3 + ρ - ρ 5 / 3 - H ( S - s ) × ( r 1 - χ r 2 5 / 3 - r 1 - χ ρ 5 / 3 - ρ - χ r 2 5 / 3 + ρ - χ ρ 5 / 3 ) - H ( S - s ) × ( χ r 1 - r 2 5 / 3 - χ r 1 - ρ 5 / 3 - χ ρ - r 2 5 / 3 + χ ρ - ρ 5 / 3 ) + H ( S - s ) × ( χ r 1 - χ r 2 5 / 3 - χ r 1 - χ ρ 5 / 3 - χ ρ - χ r 2 5 / 3 + χ ρ - χ ρ 5 / 3 ) ] - [ 1 + ( 4 / D ) 2 r 2 · ρ ] × [ 1 + ( 4 / D ) 2 r 1 · ρ ] [ r 2 - r 1 5 / 3 - r 2 - ρ 5 / 3 - ρ - r 1 5 / 3 + ρ - ρ 5 / 3 - H ( S - s ) × ( r 2 - χ r 1 5 / 3 - r 2 - χ ρ 5 / 3 - ρ - χ r 1 5 / 3 + ρ - χ ρ 5 / 3 ) - H ( S - s ) × ( χ r 2 - r 1 5 / 3 - χ r 2 - ρ 5 / 3 - χ ρ - r 1 5 / 3 + χ ρ - ρ 5 / 3 + H ( S - s ) × ( χ r 2 - χ r 1 5 / 3 - χ r 2 - χ ρ 5 / 3 - χ ρ - χ r 1 5 / 3 + χ ρ - χ ρ 5 / 3 ) ] + [ 1 + ( 4 / D ) 2 r 2 · ρ ] × [ 1 + ( 4 / D ) 2 r 2 · ρ ] [ r 2 - r 2 5 / 3 - r 2 - ρ 5 / 3 - ρ - r 2 5 / 3 + ρ - ρ 5 / 3 - H ( S - s ) × ( r 2 - χ r 2 5 / 3 - r 2 - χ ρ 5 / 3 - ρ - χ r 2 5 / 3 + ρ - χ ρ 5 / 3 ) - H ( S - s ) × ( χ r 2 - r 2 5 / 3 - χ r 2 - ρ 5 / 3 - χ ρ - r 2 5 / 3 + χ ρ - ρ 5 / 3 ) + H ( S - s ) × ( χ r 2 - χ r 2 5 / 3 - χ r 2 - χ ρ 5 / 3 - χ ρ - χ r 2 5 / 3 + χ ρ - χ ρ 5 / 3 ) ] } .
d ρ W ( ρ / D ) r a · ρ Q r b - ρ 5 / 3 = 0             if r a r b ,
d ρ d ρ W ( ρ / D ) W ( ρ / D ) ( r a · ρ ) ( r b · ρ ) Q ρ - ρ 5 / 3 = 1 2 ( r a · r b ) d ρ d ρ W ( ρ / D ) W ( ρ / D ) ρ · ρ Q ρ - ρ 5 / 3 .
D φ FA ( r 1 - r 2 ) = - 2.914 k 2 0 d s C N 2 ( s ) { [ - r 1 - r 2 5 / 3 + H ( S - s ) ( r 1 - χ r 2 5 / 3 + χ r 1 - r 2 5 / 3 - χ r 1 - χ r 2 5 / 3 - r 1 - χ r 1 5 / 3 - r 2 - χ r 2 5 / 3 ) ] + 64 π D 4 ( r 1 - r 2 ) × d ρ W ( ρ / D ) ρ [ - r 1 - ρ 5 / 3 + r 2 - ρ 5 / 3 + H ( S - s ) ( r 1 - χ ρ 5 / 3 - r 2 - χ ρ 5 / 3 + χ r 1 - ρ 5 / 3 - χ r 2 - ρ 5 / 3 - χ r 1 - χ ρ 5 / 3 + χ r 2 - χ ρ 5 / 3 ) ] + 1 4 ( 64 π D 4 ) 2 r 1 - r 2 2 × d ρ d ρ W ( ρ / D ) W ( ρ / D ) ρ · ρ [ ρ - ρ 5 / 3 - H ( S - s ) ( ρ - χ ρ 5 / 3 + χ ρ - ρ 5 / 3 - χ ρ - χ ρ 5 / 3 ) ] } .
F 2 ( u ) = - 64 π d x W ( x ) u - x 5 / 3 x · 1 u ,
F 3 ( u ) = - ( 64 2 π ) 2 d x d x W ( x ) W ( x ) ( x · x ) u x - x 5 / 3 .
F 3 ( u ) = 32 0 1 / 2 d x x 2 F 2 ( u x ) .
0 2 π d ν cos ( ν ) [ a 2 - 2 a b cos ( ν ) + b 2 ] 5 / 6 = 2 π ( a b ) 5 / 6 { I ( a / b ) if a b I ( b / a ) if a > b ,
I ( t ) = t 1 / 6 k = 0 Γ ( k - 5 / 6 ) Γ ( k + 1 / 6 ) Γ ( k + 1 ) Γ ( k + 2 ) [ Γ ( - 5 / 6 ) ] 2 t 2 k .
ξ ( x ) = x - 1 0 2 π d ν cos ( ν ) [ 1 - 2 x cos ( ν ) + x 2 ] 5 / 6 ,
ξ ( x ) = 2 π x - 1 / 6 { I ( x ) if x 1 I ( x - 1 ) if x > 1 .
F 2 ( u ) = - 64 π u 0 1 / 2 d x x 8 / 3 ξ ( u / x ) .
D φ FA ( r 1 , r 2 ) = 2.914 k 2 0 d s C N 2 ( s ) [ r 1 - r 2 5 / 3 × [ 1 + H ( S - s ) χ 5 / 3 ] - H ( S - s ) × [ r 1 - χ r 2 5 / 3 + χ r 1 - r 2 5 / 3 - ( r 1 5 / 3 + r 2 5 / 3 ) ( 1 - χ ) 5 / 3 ] - D 5 / 3 ( r 1 - r 2 D ) · ( { F 2 ( r 1 D ) - H ( S - s ) × [ χ 5 / 3 F 2 ( r 1 χ D ) + F 2 ( χ r 1 D ) - χ 5 / 3 F 2 ( r 1 D ) ] } 1 r 1 - { F 2 ( r 2 D ) - H ( S - s ) × [ χ 5 / 3 F 2 ( r 2 χ D ) + F 2 ( χ r 2 D ) - χ 5 / 3 F 2 ( r 2 D ) ] } 1 r 2 ) - D 5 / 3 ( r 1 - r 2 2 D 2 ) × { F 3 ( 1 ) [ 1 + H ( S - s ) χ 5 / 3 ] - H ( S - s ) × [ χ 5 / 3 F 3 ( χ - 1 ) + F 3 ( χ ) ] } ] .
h = s cos ( ψ ) ,             H = S cos ( ψ ) .
χ = 1 - h / H ,             C N 2 ( s ) C N 2 ( h ) ,
D φ FA ( r 1 , r 2 ) = 2.914 k 2 sec ( ψ ) D 5 / 3 [ | r 1 - r 2 D | 5 / 3 0 d s C N 2 ( h ) [ 1 + H ( H - h ) χ 5 / 3 ] - 0 H d h C N 2 ( h ) [ r 1 - χ r 2 ) / D 5 / 3 + ( χ r 1 - r 2 ) / D 5 / 3 ] + [ ( r 1 D ) 5 / 3 + ( r 2 D ) 5 / 3 ] 0 H d h C N 2 ( h ) ( 1 - χ ) 5 / 3 - ( r 1 - r 2 D ) · ( 1 r 1 { F 2 ( r 1 D ) × 0 d h C N 2 ( h ) [ 1 + H ( H - h ) χ 5 / 3 ] - 0 H d h C N 2 ( h ) [ χ 5 / 3 F 2 ( r 1 χ D ) + F 2 ( χ r 1 D ) ] } - 1 r 2 { F 2 ( r 2 D ) × 0 d h C N 2 ( h ) [ 1 + H ( H - h ) χ 5 / 3 ] - 0 H d h C N 2 ( h ) [ χ 5 / 3 F 2 ( r 2 χ D ) + F 2 ( χ r 2 D ) ] } ) - r 1 - r 2 2 D 2 × { F 3 ( 1 ) 0 d h C N 2 ( h ) [ 1 + H ( H - h ) χ 5 / 3 ] - 0 H d h C N 2 ( h ) [ χ 5 / 3 F 3 ( χ - 1 ) + F 3 ( χ ) ] } ] .
S 0 ( Q , μ , H ) = ( Q - 2 μ + Q - 1 ) 5 / 6 0 d h C N 2 ( h ) × [ 1 + H ( H - h ) χ 5 / 3 ] + ( Q 5 / 6 + Q - 5 / 6 ) × 0 H d h C N 2 ( h ) ( h / H ) 5 / 3 - 0 H d h C N 2 ( h ) × [ ( Q - 2 χ μ - χ 2 Q - 1 ) 5 / 6 + ( χ 2 Q - 2 χ μ + Q - 1 ) 5 / 6 ] ,
S m ( ρ , H ) = { [ F 2 ( ρ ) - p F 3 ( 1 ) ] × 0 d h C N 2 ( h ) [ 1 + H ( H - h ) χ 5 / 3 ] - 0 H d h C N 2 ( h ) [ χ 5 / 3 F 2 ( ρ / χ ) - ρ χ 5 / 3 F 3 ( χ - 1 ) + F 2 ( χ ρ ) - F 3 ( χ ) ] } ,
D φ FA ( r 1 , r 2 ) = 2.914 k 2 sec ( ψ ) D 5 / 3 × { ( r 1 r 2 D 2 ) 5 / 6 S 0 ( r 1 r 2 , 1 r 1 · 1 r 2 , H ) - ( r 1 - r 2 D ) × [ 1 r 1 S m ( r 1 D , H ) - 1 r 2 S m ( r 2 D , H ) ] } . }
σ φ FA 2 = D 5 / 3 { ( 8 π 2 ) 2.914 k 2 sec ( ψ ) d x d x W ( x ) W ( x ) × [ ( x x ) 5 / 6 S 0 ( x / x , 1 x · 1 x , H ) - 2 ( x - x ) · 1 x × S m ( x , H ) ] } .
d 0 = { 2.914 ( 8 π 2 ) k 2 sec ( ψ ) d x d x W ( x ) W ( x ) × [ ( x x ) 5 / 6 S 0 ( x / x , 1 x · 1 x , H ) - 2 ( x - x ) · 1 x × S m ( x , H ) ] } - 3 / 5 .
d 0 = { ( 2.914 ) 32 k 2 sec ( ψ ) 0 1 / 2 0 1 / 2 x x d x d x × [ ( x x ) 5 / 6 S ¯ 0 ( x / x , H ) - 2 x S m ( x , H ) ] } - 3 / 5 ,
S ¯ 0 ( y , H ) = ( 2 π ) - 1 0 2 π d ν S 0 [ t , cos ( ν ) , H ] .
σ φ FA 2 = ( D / d 0 ) 5 / 3 .
G FA / G DL = 16 π 2 d x d x W ( x ) W ( x ) × exp { - [ ( 2.914 / 2 ) k 2 sec ( ψ ) D 5 / 3 ] × [ ( x x ) 5 / 6 S 0 ( x / x , 1 x · 1 x , H ) - 2 ( x - x ) · 1 x S m ( x , H ) ] } .
G FA / G DL exp [ - ( D / d 0 ) 5 / 3 ]             if D / d 0 < 1.
G FA / G DL 1.15 ( D / d 0 ) - 4             if D / d 0 > 2.
f low ( x ) = [ 1 + ( 0.667 x ) 10 ] - 1 ,
f high ( x ) = ( 0.667 x ) 10 [ 1 + ( 0.667 x ) 10 ] - 1
G FA / G DL = exp [ - ( D / d 0 ) 5 / 3 ] f low ( D / d 0 ) + 1.15 ( D / d 0 ) - 4 f high ( D / d 0 ) = exp [ - ( D / d 0 ) 5 / 3 ] + 0.0199 ( D / d 0 ) 6 1.00173 ( D / d 0 ) 10 .

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