Abstract

The phase structure function and the phase-slope structure function have been used to characterize atmospheric-turbulence parameters—parameters that are important in the performance evaluation and ultimately in the design of adaptive optics systems for astronomical imaging applications and for laser beam propagations. We have extended the method to account for the effects of strong beam-path-induced aberrations and additive measurement noise. A stable noise-suppressant algorithm is devised in which the estimation is done sequentially: first the atmospheric-turbulence parameters are estimated independently, and then the noise-variance parameter is estimated. The developed theory is applied to the Shack–Hartmann wave-front-sensor data collected in the recently completed Airborne Laser Extended Atmospheric Characterization Experiment for very-long-horizontal-path (between 60 and 100 km) laser beam propagations. The estimated r0 values are in general agreement with the values obtained from the high-bandwidth scintillometer analysis on data taken in the same experiment. In addition, if the atmospheric turbulence spectrum can be described by a simple power law, then the data reduction with use of the developed estimation equations, which allow arbitrary atmospheric-turbulence power-law values, indicates that deviation from Kolmogorov turbulence is negligible.

© 1998 Optical Society of America

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References

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  1. T. F. Elbert. Estimation and Control of Systems (Van Nostrand Reinhold, New York, 1984).
  2. R. K. Szeto, G. A. Tyler, “ABLE-ACE wavefront sensor: optical wavefront reconstruction in terms of zernike modes,” (The Optical Sciences Company, Anaheim, Calif., 1994).
  3. R. K. Szeto, M. S. Lodin, “Effects of correlated noise on ABLE-ACE wavefront sensor analysis,” (The Optical Sciences Company, Anaheim, Calif., 1995).
  4. D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.
  5. R. R. Butts, “ABLE ACE wavefront sensor,” Proc. SPIE3065, (to be published).
  6. E. E. Silbaugh, B. M. Welsh, M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wave-front slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460 (1996).
    [CrossRef]
  7. A. S. Gurvich, B. M. Koprov, L. R. Tsvang, A. M. Yaglom, “Data on the small scale structure of atmospheric turbulence,” in Atmospheric Turbulence and Radio Wave Propagation, A. M. Yaglom, V. I. Tatarskii, eds. (Nauka, Moscow, 1967), pp. 30–52.
  8. D. Dayton, B. Pierson, B. Spielbusch, J. Gonglewski, “Atmospheric structure function measurements with a Shack–Hartmann wave-front sensor,” Opt. Lett. 17, 1737–1739 (1992).
    [CrossRef] [PubMed]
  9. T. W. Nicholls, G. D. Boreman, J. C. Dainty, “Use of a Shack–Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum,” Opt. Lett. 20, 2460–2462 (1995).
    [CrossRef]
  10. A. N. Kolomogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk. Sssr. 30, 299–303 (1941).
  11. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
    [CrossRef]
  12. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

1996

1995

1992

1967

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

1941

A. N. Kolomogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk. Sssr. 30, 299–303 (1941).

Benton, D. W.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Boreman, G. D.

Brennan, T. T.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Brown, W. P.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Butts, R. R.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

R. R. Butts, “ABLE ACE wavefront sensor,” Proc. SPIE3065, (to be published).

Coy, S. C.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Dainty, J. C.

Dayton, D.

Dueck, R. H.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Elbert, T. F.

T. F. Elbert. Estimation and Control of Systems (Van Nostrand Reinhold, New York, 1984).

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Gonglewski, J.

Gurvich, A. S.

A. S. Gurvich, B. M. Koprov, L. R. Tsvang, A. M. Yaglom, “Data on the small scale structure of atmospheric turbulence,” in Atmospheric Turbulence and Radio Wave Propagation, A. M. Yaglom, V. I. Tatarskii, eds. (Nauka, Moscow, 1967), pp. 30–52.

Koenig, K. W.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Kolomogorov, A. N.

A. N. Kolomogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk. Sssr. 30, 299–303 (1941).

Koprov, B. M.

A. S. Gurvich, B. M. Koprov, L. R. Tsvang, A. M. Yaglom, “Data on the small scale structure of atmospheric turbulence,” in Atmospheric Turbulence and Radio Wave Propagation, A. M. Yaglom, V. I. Tatarskii, eds. (Nauka, Moscow, 1967), pp. 30–52.

Lodin, M. S.

R. K. Szeto, M. S. Lodin, “Effects of correlated noise on ABLE-ACE wavefront sensor analysis,” (The Optical Sciences Company, Anaheim, Calif., 1995).

Masson, B. S.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Merritt, P. H.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Nicholls, T. W.

O’Keefe, S. O.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Peterson, D. H.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Pierson, B.

Praus, R. W.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Roggemann, M. C.

Silbaugh, E. E.

Spielbusch, B.

Szeto, R. K.

R. K. Szeto, G. A. Tyler, “ABLE-ACE wavefront sensor: optical wavefront reconstruction in terms of zernike modes,” (The Optical Sciences Company, Anaheim, Calif., 1994).

R. K. Szeto, M. S. Lodin, “Effects of correlated noise on ABLE-ACE wavefront sensor analysis,” (The Optical Sciences Company, Anaheim, Calif., 1995).

Tatarski, V. I.

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

Tsvang, L. R.

A. S. Gurvich, B. M. Koprov, L. R. Tsvang, A. M. Yaglom, “Data on the small scale structure of atmospheric turbulence,” in Atmospheric Turbulence and Radio Wave Propagation, A. M. Yaglom, V. I. Tatarskii, eds. (Nauka, Moscow, 1967), pp. 30–52.

Tyler, G. A.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

R. K. Szeto, G. A. Tyler, “ABLE-ACE wavefront sensor: optical wavefront reconstruction in terms of zernike modes,” (The Optical Sciences Company, Anaheim, Calif., 1994).

Venet, B. P.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Washburn, D. C.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Weaver, L. O.

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

Welsh, B. M.

Yaglom, A. M.

A. S. Gurvich, B. M. Koprov, L. R. Tsvang, A. M. Yaglom, “Data on the small scale structure of atmospheric turbulence,” in Atmospheric Turbulence and Radio Wave Propagation, A. M. Yaglom, V. I. Tatarskii, eds. (Nauka, Moscow, 1967), pp. 30–52.

Dokl. Akad. Nauk. Sssr.

A. N. Kolomogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk. Sssr. 30, 299–303 (1941).

J. Opt. Soc. Am. A

Opt. Lett.

Proc. IEEE

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Other

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

T. F. Elbert. Estimation and Control of Systems (Van Nostrand Reinhold, New York, 1984).

R. K. Szeto, G. A. Tyler, “ABLE-ACE wavefront sensor: optical wavefront reconstruction in terms of zernike modes,” (The Optical Sciences Company, Anaheim, Calif., 1994).

R. K. Szeto, M. S. Lodin, “Effects of correlated noise on ABLE-ACE wavefront sensor analysis,” (The Optical Sciences Company, Anaheim, Calif., 1995).

D. C. Washburn, D. W. Benton, W. P. Brown, R. R. Butts, K. W. Koenig, B. S. Masson, P. H. Merritt, B. P. Venet, S. O. O’Keefe, D. H. Peterson, L. O. Weaver, S. C. Coy, R. W. Praus, T. T. Brennan, R. H. Dueck, G. A. Tyler, “Airborne laser extended atmospheric characterization experiment, final report,” May1996.

R. R. Butts, “ABLE ACE wavefront sensor,” Proc. SPIE3065, (to be published).

A. S. Gurvich, B. M. Koprov, L. R. Tsvang, A. M. Yaglom, “Data on the small scale structure of atmospheric turbulence,” in Atmospheric Turbulence and Radio Wave Propagation, A. M. Yaglom, V. I. Tatarskii, eds. (Nauka, Moscow, 1967), pp. 30–52.

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Figures (14)

Fig. 1
Fig. 1

Condition number of the associated estimation matrix for (d/r0)p+1 and σθ2 plotted as a function of the number of separations for a square Shack–Hartmann sensor array. Nsep denotes the number of subapertures in either the x direction or the y direction of a square aperture placed parallel to the axes. This illustrates the difficulty in the simultaneous estimation of (d/r0)p+1 and σθ2 for a given turbulence-spectrum power-law parameter p.

Fig. 2
Fig. 2

Effects of point-subaperture approximation. There is very little difference between the point-subaperture approximation and the finite-subaperture result obtained in Ref. 6. The top two curves correspond to β=0, the middle two curves correspond to α=β, and the bottom two curves correspond to α=0. For large separations we see the asymptotic behavior of (separation/d)-1/3 as is shown in Eq. (27).

Fig. 3
Fig. 3

δΔϕx(αd,βd)-σΔϕx(αd,βd) is shown for a data frame in Science 6.

Fig. 4
Fig. 4

Normalized form of the negative of the kernel function in Eq. (20) plotted for p=2/3. There is some resemblance between Fig. 3 and the theory. However, the effect of measurement noise in Fig. 3 is quite evident.

Fig. 5
Fig. 5

Frame-to-frame estimated r0 values for a typical Science 6 flight data sequence, the propagation distance for this sequence is approximately 60 km. The solid line is the estimated average value for the data sequence. There are substantial variations from frame to frame.

Fig. 6
Fig. 6

Frame-to-frame estimated σθ values are shown for the same data sequence as in Fig. 5. The solid line is the estimated average value for the data sequence. The relative frame-to-frame variation is quite small compared with that in Fig. 5.

Fig. 7
Fig. 7

Estimated values of r0 plotted as a function of the turbulence-spectrum power law parameter p. We can see that the dependence of r0 on p is quite weak.

Fig. 8
Fig. 8

Estimated σθ values plotted as a function of the turbulence-spectrum power law parameter p for the same data sequence as in Fig. 8. The dependence on p is quite weak here, too.

Fig. 9
Fig. 9

Quadratic functional Φ(r0, p;12) plotted as a function of p for a data sequence in the Science 6 flight. The minimum occurs at p=15/24 (p=2/3 is Kolmogorov).

Fig. 10
Fig. 10

Estimated r0 values obtained with the phase-slope structure-function approach plotted against the estimated r0 values from the HBWS experiment. The points that lie on the straight line imply exact agreement between the two approaches.

Fig. 11
Fig. 11

Estimated r0 values: Zernike reconstruction approach versus HBWS. The figure is reproduced with permission from the ABLE ACE final report.4,5

Fig. 12
Fig. 12

Histogram of the estimated σθ for all data sequences in Science 6–10. The graph shows results when both the x phase and the y-slope measurements are used. The average value is 3.02 µrad.

Fig. 13
Fig. 13

Histogram of turbulence-spectrum parameter p for all data sequences in Science 6–10. We can deduce that atmospheric turbulence is (almost) Kolmogorov.

Fig. 14
Fig. 14

Comparison of estimated r0 values with the assumption of delta-correlated noise and with the assumption of fully correlated noise. Results shown indicate that the delta-correlated-noise assumption is quite good for the data analyzed.

Equations (67)

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sx(xi, yj)=1kd[ϕ(xi, yj)-ϕ(xi-1, yj)].
mx(xi, yj)=sx(xi, yj)+nx(xi, yj),
nx(xi, yj)nx(xi, yj)=σθ,x2if(i, j)=(i, j)0otherwise.
DΔϕx(αx, βy)=[mx(x, y)-mx(x-αx, y-βy)]2,
SΔϕx(αx, βy)=[mx(x, y)+mx(x-αx, y-βy)]2.
DΔϕx(αd, βd)=[sx(x, y)+nx(x, y)-sx(x-αd, y-βd)-nx(x-αd, y-βd)]2.
sx(x, y)nx(x, y)=0;
DΔϕx(αd, βd)=[sx(x, y)-sx(x-αd, y-βd)]2+[nx(x, y)-nx(x-αd, y-βd)]2.
[nx(x, y)-nx(x-αd, y-βd)]2=2σθ,x2.
(a-b)(c-d)-12[(α-c)2-(a-d)2-(b-c)2+(b-d)2]
[sx(x, y)-sx(x-αd, y-βd)]2=(kd)-2  {[ϕ(x, y)-ϕ(x-d, y)]-[ϕ(x-αd, y-βd)-ϕ(x-αd-d, y-βd)]}2=(kd)-2{[ϕ(x, y)-ϕ(x-d, y)]2+[ϕ(x-αd, y-βd)-ϕ(x-αd-d, y-βd)]2+[ϕ(x, y)-ϕ(x-αd, y-βd)]2-[ϕ(x, y)-ϕ(x-αd-d, y-βd)]2-[ϕ(x-d, y)-ϕ(x-αd, y-βd)]2+[ϕ(x-d, y)-ϕ(x-αd-d, y-βd)]2}.
[ϕ(r1)-ϕ(r2)]2=γ(p)|r1-r2|r0p+1.
ψ(U)=(32U2/π)01udu12[cos-1(u)-u(1-u2)1/2]exp[-12γ(p)(Uu)p+1].
v=12γ(p)(Uu)p+1
Γ(z)=0dttz-1 exp(-t),
γ(p)=28Γ[2/(p+1)]p+1(p+1)/2.
[ϕ(r1)-ϕ(r2)]2=6.88|r1-r2|r05/3.
[sx(x, y)-sx(x-αd, y-βd)]2=(kd)-2γ(p)(d/r0)p+1×{1+1+(α2+β2)(p+1)/2-[(α+1)2+β2](p+1)/2-[(α-1)2+β2](p+1)/2+(α2+β2)(p+1)/2}.
DΔϕx(αd, βd)=2σθ,x2+(kd)-2γ(p)(d/r0))p+1×[2+K(α, β; p)],
K(α, β; p)=2(α2+β2)(p+1)/2-[(α+1)2+β2](p+1)/2-[(α-1)2+β2](p+1)/2.
Fx(r0, p, σθ, x2; α, β)=2σθ,x2+γ(p)(d/r0)p+1×(kd)-2[2+K(α, β; p)].
Fx(r0, p, σθ, x2; α, β)=δΔϕx(αd, βd),
Gx(r0, p, σθ, x2; α, β)=2σθ,x2+γ(p)(d/r0)p+1×(1/kd)2[2-K(α, β; p)].
Gx(r0, p, σθ, x2; α, β)=σΔϕx(αd, βd),
Q(r0, p, σθ, x2, σθ, y2)
=α,β[δΔϕx(αd, βd)-Fx(r0, p, σθ, x2; α, β)]2
+[σΔϕx(αd, βd)-Gx(r0, p, σθ, x2; α, β)]2+[δΔϕy(αd, βd)-Fy(r0, p, σθ, y2; α, β)]2+[σΔϕy(αd, βd)-Gy(r0, p, σθ, y2; α, β)]2.
β=μβα,μβ>0;
K(α, β; p)=α5/3{2[1+μβ2]5/6-(1+μβ2+2α-1+α-2)5/6-(1+μβ2-2α-1+α-2)5/6}-(56)(1+μβ2)-1/6α-1/3[2-13(1+μβ2)-1].
DΔϕx(αd, βd)-SΔϕx(αd, βd)
=2γ(p)(d/r0)p+1(kd)-2K(α, β; p).
DΔϕx(αd, βd)+SΔϕx(αd, βd)
=4σθ2+4γ(p)(d/r0)p+1(kd)-2.
2γ(p)(d/r0)p+1(kd)-2K(α, β; p)
=δΔϕx(αd, βd)-σΔϕx(αd, βd),
4σθ,x2+4γ(p)(d/r0)p+1(kd)-2
=δΔϕx(αd, βd)+σΔϕx(αd, βd).
2γ(p)(d/r0)p+1(kd)-2K(β, α; p)
=δΔϕy(αd, βd)-σΔϕy(αd, βd),
4σθ,y2+4γ(p)(d/r0)p+1(kd)-2
=δΔϕy(αd, βd)+σΔϕy(αd, βd).
Φ(r0, p; ω)=ωα,β[δΔϕx(αd, βd)-σΔϕx(αd, βd)-2γ(p)(d/r0)p+1×(kd)-2K(α, β; p)]2+(1-ω)α,β[δΔϕy(αd, βd)-σΔϕy(αd, βd)-2γ(p)(d/r0)p+1×(1/kd)2K(β, α; p)]2
dr0p+1=(kd)22γ(p)K(α, β; p)×[δΔϕx(αd, βd)-σΔϕx(αd, βd)],
Ψ(σθ, x2, σθ, y2; ω)
=ωα,β[δΔϕx(αd, βd)+σΔϕx(αd, βd)-4σθ,x2
-4γ(p)(d/r0)p+1(kd)-2]2+(1-ω)×α,β[δΔϕy(αd, βd)+σΔϕy(αd, βd)-4σθ,y2-4γ(p)(d/r0)p+1(kd)-2]2.
Kx(α, β)= dy{2(α2+y2)(p+1)/2-[(α+1)2+y2](p+1)/2-[(α-1)2+y2](p+1)/2}tri(y-β),
tri(x)=1-|x|if|x|1,0otherwise.
M(xc, yc, A)=dxdyW(x, y)[f(x-xc, y-yc; A)-m(x, y)]2,
W(x, y)=1if(x, y)subaperture0otherwise,
f(x, y; A)=A sinc2(14πx)sinc2(14πy).
[n(x, y)-n(x-αd, y-βd)]2
=2μη0,0-2μηα,β,
DΔϕx(αd, βd)-SΔϕx(αd, βd)
=-4μηα,β+48Γ[2/(p+1)]p+1(p+1)/2
×(d/r0)p+1(kd)-2K(α, β; p),
DΔϕx(αd, βd)+SΔϕx(αd, βd)
=4μη0,0+88Γ[2/(p+1)]p+1(p+1)/2
×(d/r0)p+1(kd)-2.
Φ˜(r0, p; ω)=ωα,β[mr0,px(α, β)-K˜(α, β; p)(d/r0)p+1]2+(1-ω)α,β[mr0,py(α, β)-K˜(β, α; p)(d/r0)p+1]2,
mr0,px(α, β)=η0,0[δΔϕx(αd, βd)-σΔϕx(αd, βd)]+ηα,β[δΔϕx(αd, βd)+σΔϕx(αd, βd)],
mr0,py(α, β)=η0,0[δΔϕy(αd, βd)-σΔϕy(αd, βd)]+ηα,β[δΔϕy(αd, βd)+σΔϕy(αd, βd)],
K˜(α, β; p)=48Γ[2/(p+1)]p+1(p+1)/2×1kd2[2ηα,β+η0,0K(α, β; p)].
Ψ˜(μ; ω)=ωα,β[mμx(α, β)-B˜(α, β; p)μ]2+(1-ω)α,β[mμy(α, β)-B˜(β, α; p)μ]2,
mμx(α, β)=2K(α, β; p)[δΔϕx(αd, βd)-σΔϕx(αd, βd)]-[δΔϕx(αd, βd)+σΔϕx(αd, βd)],
mμy(α, β)=2K(β, α; p)[δΔϕy(αd, βd)-σΔϕy(αd, βd)]-[δΔϕy(αd, βd)+σΔϕy(αd, βd)],
B˜(α, β; p)=-4{[2ηα,β(kd)-2K(α, β; p)-η0,0]}.

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