Abstract

Spatial-frequency domain techniques have traditionally been applied to obtain estimates for the independent effects of a variety of individual error sources in adaptive optics (AO). Overall system performance is sometimes estimated by introducing the approximation that these individual error terms are statistically independent, so that their magnitudes may be summed in quadrature. More accurate evaluation methods that account for the correlations between the individual error sources have required Monte Carlo simulations or large matrix calculations that can take much longer to compute, particularly as the order of the AO system increases beyond a few hundred degrees of freedom. We describe an approach to evaluating AO system performance in the spatial-frequency domain that is relatively computationally efficient but still accounts for many of the interactions between the fundamental error sources in AO. We exploit the fact that (in the limits of an infinite aperture and geometrical optics) all the basic wave-front propagation, sensing, and correction processes that describe the behavior of an AO system are spatial-filtering operations in the Fourier domain. Essentially all classical wave-front control algorithms and evaluation formulas are expressed in terms of these filters and may therefore be evaluated one spatial-frequency component at a time. Performance estimates for very-high-order AO systems may be obtained in 1 to 2 orders of magnitude less time than needed when detailed simulations or analytical models in the spatial domain are used, with a relative discrepancy of 5% to 10% for typical sample problems.

© 2005 Optical Society of America

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References

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  1. P. L. Wizinowich, D. Bonaccini, eds., Adaptive Optical System Technologies II, Proc. SPIE4839 (2003).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  7. C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. L. A. Poyneer, B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004).
    [CrossRef]
  12. A. Tokovinin, M. Le Louarn, “Isoplanatism in a multiconjguate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
    [CrossRef]
  13. A. Tokovinin, E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. A 18, 873–8827 (2001).
    [CrossRef]
  14. F. Rigaut, J.-P. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
    [CrossRef]
  15. L. Jollisaint, J.-P. Veran, “Fast computation and morphologic interpretation of the adaptive optics point spread function,” in Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002).
  16. E. P. Wallner, “Optimal wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  17. 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]
  18. D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
    [CrossRef]
  19. B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [CrossRef]
  20. T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, V. Michau, “Optimal wave-front reconstruction strategies for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
    [CrossRef]
  21. W. H. Press, B. P. Flanner, S. A. Teukolosky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).
  22. C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high-resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 943–957 (1992).
  23. B. L. Ellerbroek, “Optimizing closed-loop adaptive-optics performance with use of multiple control bandwidths,” J. Opt. Soc. Am. A 11, 2871–2886 (1994).
    [CrossRef]
  24. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, Publications, New York, 1972), Eq. 4.3.133, p. 78.
  25. R. Hudgin, “Wave-front compensation error due to finite corrector-element size,” J. Opt. Soc. Am. 67, 393–396 (1977).
    [CrossRef]
  26. F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
    [CrossRef]

2004 (1)

2003 (2)

2001 (2)

2000 (1)

1994 (3)

1993 (1)

1991 (1)

1984 (1)

1983 (1)

1982 (1)

1977 (2)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, Publications, New York, 1972), Eq. 4.3.133, p. 78.

Ahmadia, A. J.

A. J. Ahmadia, B. L. Ellerbroek, “Parallelized simulation code for multiconjugate adaptive optics,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 218–227 (2003).
[CrossRef]

Arcidiacono, C.

C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
[CrossRef]

Baruffolo, A.

C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
[CrossRef]

Boyer, C.

C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high-resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 943–957 (1992).

Brindisi, A.

C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
[CrossRef]

Conan, J. M.

Diolaiti, E.

C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
[CrossRef]

Ellerbroek, B. L.

Farinato, J.

C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
[CrossRef]

Flanner, B. P.

W. H. Press, B. P. Flanner, S. A. Teukolosky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

Fried, D. L.

Fusco, T.

Gardner, C. S.

Gendron, E.

C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high-resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 943–957 (1992).

Gilles, L.

Graves, J. E.

Greenwood, D. P.

Hudgin, R.

Johnston, D. C.

Jollisaint, L.

L. Jollisaint, J.-P. Veran, “Fast computation and morphologic interpretation of the adaptive optics point spread function,” in Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002).

Lai, O.

F. Rigaut, J.-P. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Le Louarn, M.

Macintosh, B.

Madec, P. Y.

C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high-resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 943–957 (1992).

McKenna, D. L.

Michau, V.

Mugnier, L. M.

Northcott, M. J.

Poyneer, L. A.

Press, W. H.

W. H. Press, B. P. Flanner, S. A. Teukolosky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

Ragazzoni, R.

C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
[CrossRef]

Rigaut, F.

F. Rigaut, J.-P. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Roddier, D.

Roddier, F.

Rousset, G.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, Publications, New York, 1972), Eq. 4.3.133, p. 78.

Teukolosky, S. A.

W. H. Press, B. P. Flanner, S. A. Teukolosky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

Tokovinin, A.

Tyler, G. A.

Veran, J.-P.

F. Rigaut, J.-P. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

L. Jollisaint, J.-P. Veran, “Fast computation and morphologic interpretation of the adaptive optics point spread function,” in Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002).

Vernet-Viard, E.

C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flanner, S. A. Teukolosky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

Viard, E.

Vogel, C. R.

Wallner, E. P.

Welsh, B. M.

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (10)

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
[CrossRef]

B. L. Ellerbroek, “Optimizing closed-loop adaptive-optics performance with use of multiple control bandwidths,” J. Opt. Soc. Am. A 11, 2871–2886 (1994).
[CrossRef]

G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
[CrossRef]

L. A. Poyneer, B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004).
[CrossRef]

A. Tokovinin, M. Le Louarn, “Isoplanatism in a multiconjguate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
[CrossRef]

A. Tokovinin, E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. A 18, 873–8827 (2001).
[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]

D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
[CrossRef]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, V. Michau, “Optimal wave-front reconstruction strategies for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
[CrossRef]

Other (10)

W. H. Press, B. P. Flanner, S. A. Teukolosky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high-resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 943–957 (1992).

B. L. Ellerbroek, C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 206–217 (2003).
[CrossRef]

A. J. Ahmadia, B. L. Ellerbroek, “Parallelized simulation code for multiconjugate adaptive optics,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 218–227 (2003).
[CrossRef]

C. Arcidiacono, E. Diolaiti, R. Ragazzoni, A. Baruffolo, A. Brindisi, J. Farinato, E. Vernet-Viard, “Sky coverage and Strehl ratio uniformity in layer-oriented MCAO systems,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 169–180 (2003).
[CrossRef]

F. Rigaut, J.-P. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

L. Jollisaint, J.-P. Veran, “Fast computation and morphologic interpretation of the adaptive optics point spread function,” in Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, Publications, New York, 1972), Eq. 4.3.133, p. 78.

P. L. Wizinowich, D. Bonaccini, eds., Adaptive Optical System Technologies II, Proc. SPIE4839 (2003).

T. Andersen, A. Ardeberg, eds., Extremely Large Telescopes II, Proc. SPIE (to be published).

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Figures (5)

Fig. 1
Fig. 1

AO linear systems model. This figure is a block diagram illustrating the abstract linear model of an AO system that is the starting point for this work. Here x is the phase disturbance introduced by atmospheric turbulence, s is the WFS measurement vector, n is the noise component of this measurement, a is the DM actuator command vector, ϕ is the residual phase profile after compensation by the AO system, and σ2 is the mean square, piston-removed value of ϕ. The operators Gx and Ga are linear transformations describing the relationship between x, a, and s; Hx and Ha similarly relate x and a to the value of ϕ. The matrix R is the reconstruction algorithm, and f(τ) is a temporal filter used to condition the output of the reconstructor before it is applied to the DM. The symmetric, semi-positive-definite operator Wϕ defines a norm on the space of phase profiles and is used to compute σ2 from ϕ.

Fig. 2
Fig. 2

Fitting error and WFS spatial aliasing error for NGS AO. This figure plots the normalized mean square phase errors due to DM fitting [Eq. (50)] and WFS spatial aliasing [Eq. (51)] as a function of the one-dimensional order D/Δ of the AO system. A conventional least-squares wave-front reconstruction algorithm is assumed, and there is no WFS measurement noise, servo lag, or anisoplanatism.

Fig. 3
Fig. 3

Wave-front reconstruction noise gain for NGS AO. This figure plots the noise gain for a least-squares wave-front reconstruction algorithm [Eq. (52)] as a function of the order of the AO system.

Fig. 4
Fig. 4

Wave-front correction error due to anisoplanatism. This figure plots the mean square phase error due to anisoplanatism, normalized according to Eqs. (53) and (54), for the case of a NGS AO system with no DM fitting error, WFS spatial aliasing, WFS measurement noise, or servo lag. The Cn2(h) profile consists of a single layer displaced from the telescope aperture plane. The parameter Q is defined as (D/r0)/(θ/θ0).

Fig. 5
Fig. 5

Wave-front correction error due to servo lag. This figure plots the mean square phase error due to servo lag, normalized according to Eqs. (55) and (56), for the case of a NGS AO system with no DM fitting error, WFS spatial aliasing, WFS measurement noise, or anisoplanatism. The atmospheric profile consists of a single layer with a known wind speed and an unknown, uniformly distributed wind velocity. The parameter Q is defined as (D/r0)/(fg/f).

Tables (1)

Tables Icon

Table 1 Comparison of Frequency-Domain Performance Estimates versus the Results of Spatial Domain Simulations for a Sample NGS MCAO Systema

Equations (56)

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ϕ=ϕ(a)=Hxx-Haa,
σ2(a)=ϕT(a)Wϕϕ(a),
a*arg minaσ2(a)=Wa-1Caxx,
σ2(a*)=xT[Wx-CaxTWa-1Cax]x,
σ2(a)=σ2(a*)+(a*-a)TWa(a*-a),
σ2(a*)=tr[WxxxT]-tr[Wa-1CaxxxTCaxT],
σ2(a)=σ2(a*)+tr[Wa(a*-a)(a*-a)T].
s0=Gxx+n,
a=Rs0,
(a*-Rs0)(a*-Rs0)T=a*a*T-Rs0a*T-a*s0TRT+Rs0s0TRT,
s0s0T=GxxxTGxT+nnT,
a*s0T=Wa-1CaxxxTGxT,
a*a*T=Wa-1CaxxxTCaxTWa-1.
R*arg minRσ2(Rs0)=a*s0Ts0s0T-1.
s(t)=s0(t)-Gaa(t),
a(t)=0dτf(τ)[Rs(t-τ)].
aˆ(ν)=[I+fˆ(ν)RGa]-1fˆ(ν)Rsˆ0(ν),
aˆ(ν)=R{sˆ0(ν)fˆ(ν)/[1+fˆ(ν)]}.
a(t)=Rsf(t)=R[Gxxf(t)+nf(t)],
Rcarg minR{σ2(Rsf):RGa=I}=R*-(I-R*Ga)(GaTsfsfT-1Ga)-1GaTsfsfT-1.
Wa=diag[(Wa)1,,(Wa)N],
Wa=i PiTWaPi,
σ2(a)=σ2(a*)+i tr[Wa(Pia*-Pia)(Pia*-Pia)T].
Piaˆ(ν)=PiR{sˆ0(ν)fˆi(ν)/[1+fˆi(ν)]},
Pia(t)=PiRsfi(t)=PiR[Gxxfi(t)+nfi(t)].
M=a*a*T-(a*-Rsf)(a*-Rsf)T.
Dij=1ifΛij>00otherwise
R=Wa-1/2UDUTWa1/2R.
tr[Waa*a*T]-[Wa(a*-Rsf)(a*-Rsf)T]=tr(DΛ),
ϕ(r; θ)=j x(r+hjθ; j),
ϕˆ(κ, θ)=jexp(2πihjθκ)xˆ(κ; j).
α(r; k)=m a(Δkm; k)uk(r-Δkm),
αˆ(κ; k)=Δk-2aˆ(κ; k)uˆk(κ).
αˆ(κ; k)=aˆ(κ; k)ifmax{|κx|, |κy|}<1/2Δk0otherwise,
s(Δm; l)=Δ-2 d2rφ(r; l)w(r-Δm)+n(Δm; l),
sˆ(κ, l)=2πim (κ+Δ-1m)φˆ(κ+Δ-1m; l)×wˆ*(κ+Δ-1m)+nˆ(κ; l)=2πiκφˆ(κ; l)wˆ*(κ)+2πim0 (κ+Δ-1m)φˆ(κ+Δ-1m; l)×wˆ*(κ+Δ-1m)+nˆ(κ; l).
ϕTWϕψ= dθWΩ(θ) dκ[1-|WˆA(κ)|2]ϕˆ*(κ; θ)ψˆ(κ; θ).
WˆA(κ)=2J1(πDκ)/(πDκ),
ϕTWϕϕ= dθWΩ(θ) drWA(r)ϕ(r; θ)- drWA(r)ϕ(r; θ)2.
Wx(κ; (k, l))=WˆΩ[(hk-hl)κ][1-|WˆA(κ)|2].
|xˆ(κ; j)|2=cxr05/3Ψ(κ)cj,
cj=hj-hj+dhCn2(h)  0dhCn2(h),
θ0=r06.88j hj5/3cj-3/5,
fg=0.241j |vj|5/3cj3/5  r0,
|nˆx(κ; l)|2=|nˆy(κ; l)|2=σn2(l).
x(r; j, t)=x(r-tvj; j, 0),
xˆf(κ; j, t)= dr exp(-2πirκ)×0dτx[r-(t-τ)vj; j, 0]fIR(τ)=0dτ exp[-2πi(τ-t)vjκ]fIR(τ)×xˆ(κ; j, 0)=exp(2πitvjκ)fˆ(vjκ)1+fˆ(vjκ)×xˆ(κ; j, 0),
|xˆf(κ; j)|2=xˆ(κ; j)xˆf*(κ; j)=|xˆ(κ; j)|2(1+|vjκ/fc|2)1/2.
|nˆf,x(κ; l)|2=|nˆf,y(κ; l)|2=σ2(l) dtfIR2(t),
σ2(a*)(Δ/r0)5/3=cxmax{κx,κy}1/2dκκ11/31-2J1(πκD/Δ)(πκD/Δ)2,
σ2(RLSs0)-σ2(a*)(Δ/r0)5/3=cx-1/21/2-1/21/2dκ1-2J1(πκD/Δ)(πκD/Δ)2×m0|κ+m|-11/3κxκyκT(κ+m)(κx+mx)(κy+my)κTκ2.
σ2(RLSs0)σn2=14π2-1/21/2-1/21/2dκκ2×1-2J1(πκD/Δ)(πκD/Δ)2×1[sinc(πκx)sinc(πκy)]2.
σ2(RLSs0)(θ/θ0)5/3=0.581πcx0dκκ8/3[1-J0(2πκ)]×1-2J1(3.18πκQ)(3.18πκQ)2,
σ2(RLSs0)(D/r0)5/3=0.581πcx0dκκ8/3[1-J0(2πκ/Q)]×1-2J1(3.18πκ)(3.18πκ)2,
σ2(RLSsf)(fg/f)5/3=8.30πcx0dκκ8/3[1-(1+κ2)-1/2]×1-2J1(0.407πκQ)0.407πκQ2,
σ2(RLSsf)(D/r0)5/3=8.30πcx0dκκ8/3×{1-[1+(κ/Q)2]-1/2}×1-2J1(0.407πκ)0.407πκ2,

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