Abstract

The concept of slope discrepancy developed in the mid-1980’s to assess measurement noise in a wave-front sensor system is shown to have additional contributions that are due to fitting error and branch points. This understanding is facilitated by the development of a new formulation that employs Fourier techniques to decompose the measured gradient field (i.e., wave-front sensor measurements) into two components, one that is expressed as the gradient of a scalar potential and the other that is expressed as the curl of a vector potential. A key feature of the theory presented here is the fact that both components of the phase (one corresponding to each component of the gradient field) are easily reconstructable from the measured gradients. In addition, the scalar and vector potentials are both easily expressible in terms of the measured gradient field. The work concludes with a wave optics simulation example that illustrates the ease with which both components of the phase can be obtained. The results obtained illustrate that branch point effects are not significant until the Rytov number is greater than 0.2. In addition, the branch point contribution to the phase not only is reconstructed from the gradient data but is used to illustrate the significant performance improvement that results when this contribution is included in the correction applied by an adaptive optics system.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. Moretti, G. M. Cochran, K. E. Steinhoff, G. A. Tyler, “SOR-3 data reduction,” (the Optical Sciences Company, Anaheim, Calif., 1988).
  2. D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
    [CrossRef]
  3. D. M. Lussier, “Slope discrepancy due to four point phase approximation in the SOR-3 experiment,” (the Optical Sciences Company, Anaheim, Calif., 1987).
  4. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  5. G. A. Tyler, T. J. Brennan, W. Moretti, J. L. Vaughn, R. H. Dueck, “Recent analytical and experimental results,” (the Optical Sciences Company, Anaheim, Calif., 1992).
  6. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  7. J. D. Jackson, Classical Electrodynamics, 3rd ed. (J Wiley, New York, 1998).
  8. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Sec. 4.1.
  9. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (9.1.21).
  10. Ref. 9, Eq. (11.4.17).
  11. S. M. Brown, J. W. Hardy, R. Hutchin, P. J. Mailhot, M. B. Michalik, S. Paley, “Reconstruction development study,” (Itek Optical Systems, Lexington, Mass.1987).
  12. P. S. Idell, “Image synthesis for nonimaged laser-speckle patterns,” Opt. Lett. 12, 858–860 (1987).
    [CrossRef] [PubMed]
  13. P. S. Idell, J. D. Gonglewski, “Image synthesis from wave-front sensor measurements of a coherent diffraction field,” Opt. Lett. 15, 1309–1311 (1990).
    [CrossRef] [PubMed]
  14. D. G. Voelz, J. D. Gonglewski, P. S. Idell, “Image synthesis from nonimaged laser-speckle patterns: comparison of theory, computer simulation, and laboratory results,” Appl. Opt. 30, 3333–3344 (1991).
    [CrossRef] [PubMed]
  15. J. D. Gonglewski, P. S. Idell, D. G. Voelz, D. C. Dayton, B. K. Spielbusch, R. E. Pierson, “Coherent image synthesis from wave-front sensor measurements of a nonimaged laser speckle field: a laboratory demonstration,” Opt. Lett. 16, 1893–1895 (1991).
    [CrossRef] [PubMed]
  16. R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging ,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
    [CrossRef]
  17. D. G. Voelz, J. D. Gonglewski, P. S. Idell, “SCIP computer simulation and laboratory verification ,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 169–176 (1993).
    [CrossRef]
  18. P. H. Roberts, “A wave optics propagation code,” (the Optical Sciences Company, Anaheim, Calif., 1986).
  19. G. A. Tyler, D. L. Fried, “A wave optics propagation algorithm,” (the Optical Sciences Company, Anaheim, Calif., 1982).
  20. M. C. Roggemann, A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. A 17, 53–62 (2000).
    [CrossRef]

2000 (1)

1998 (1)

1992 (1)

1991 (2)

1990 (1)

1987 (1)

Ameer, G. A.

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

Brennan, T. J.

G. A. Tyler, T. J. Brennan, W. Moretti, J. L. Vaughn, R. H. Dueck, “Recent analytical and experimental results,” (the Optical Sciences Company, Anaheim, Calif., 1992).

Brown, S. M.

S. M. Brown, J. W. Hardy, R. Hutchin, P. J. Mailhot, M. B. Michalik, S. Paley, “Reconstruction development study,” (Itek Optical Systems, Lexington, Mass.1987).

Browne, S. L.

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

Cochran, G. C.

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

Cochran, G. M.

W. Moretti, G. M. Cochran, K. E. Steinhoff, G. A. Tyler, “SOR-3 data reduction,” (the Optical Sciences Company, Anaheim, Calif., 1988).

Dayton, D. C.

Dueck, R.

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

Dueck, R. H.

G. A. Tyler, T. J. Brennan, W. Moretti, J. L. Vaughn, R. H. Dueck, “Recent analytical and experimental results,” (the Optical Sciences Company, Anaheim, Calif., 1992).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Sec. 4.1.

Fried, D. L.

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

G. A. Tyler, D. L. Fried, “A wave optics propagation algorithm,” (the Optical Sciences Company, Anaheim, Calif., 1982).

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

Gonglewski, J. D.

Hardy, J. W.

S. M. Brown, J. W. Hardy, R. Hutchin, P. J. Mailhot, M. B. Michalik, S. Paley, “Reconstruction development study,” (Itek Optical Systems, Lexington, Mass.1987).

Hutchin, R.

S. M. Brown, J. W. Hardy, R. Hutchin, P. J. Mailhot, M. B. Michalik, S. Paley, “Reconstruction development study,” (Itek Optical Systems, Lexington, Mass.1987).

Hutchin, R. A.

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging ,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
[CrossRef]

Idell, P. S.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (J Wiley, New York, 1998).

Koivunen, A. C.

Lussier, D. M.

D. M. Lussier, “Slope discrepancy due to four point phase approximation in the SOR-3 experiment,” (the Optical Sciences Company, Anaheim, Calif., 1987).

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

Mailhot, P. J.

S. M. Brown, J. W. Hardy, R. Hutchin, P. J. Mailhot, M. B. Michalik, S. Paley, “Reconstruction development study,” (Itek Optical Systems, Lexington, Mass.1987).

Michalik, M. B.

S. M. Brown, J. W. Hardy, R. Hutchin, P. J. Mailhot, M. B. Michalik, S. Paley, “Reconstruction development study,” (Itek Optical Systems, Lexington, Mass.1987).

Moretti, W.

W. Moretti, G. M. Cochran, K. E. Steinhoff, G. A. Tyler, “SOR-3 data reduction,” (the Optical Sciences Company, Anaheim, Calif., 1988).

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

G. A. Tyler, T. J. Brennan, W. Moretti, J. L. Vaughn, R. H. Dueck, “Recent analytical and experimental results,” (the Optical Sciences Company, Anaheim, Calif., 1992).

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Sec. 4.1.

Paley, S.

S. M. Brown, J. W. Hardy, R. Hutchin, P. J. Mailhot, M. B. Michalik, S. Paley, “Reconstruction development study,” (Itek Optical Systems, Lexington, Mass.1987).

Pierson, R. E.

Roberts, P. H.

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

P. H. Roberts, “A wave optics propagation code,” (the Optical Sciences Company, Anaheim, Calif., 1986).

Roggemann, M. C.

Spielbusch, B. K.

Steinhoff, K. E.

W. Moretti, G. M. Cochran, K. E. Steinhoff, G. A. Tyler, “SOR-3 data reduction,” (the Optical Sciences Company, Anaheim, Calif., 1988).

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

Tyler, G. A.

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

G. A. Tyler, D. L. Fried, “A wave optics propagation algorithm,” (the Optical Sciences Company, Anaheim, Calif., 1982).

W. Moretti, G. M. Cochran, K. E. Steinhoff, G. A. Tyler, “SOR-3 data reduction,” (the Optical Sciences Company, Anaheim, Calif., 1988).

G. A. Tyler, T. J. Brennan, W. Moretti, J. L. Vaughn, R. H. Dueck, “Recent analytical and experimental results,” (the Optical Sciences Company, Anaheim, Calif., 1992).

Vaughn, J. L.

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

G. A. Tyler, T. J. Brennan, W. Moretti, J. L. Vaughn, R. H. Dueck, “Recent analytical and experimental results,” (the Optical Sciences Company, Anaheim, Calif., 1992).

Voelz, D. G.

Winker, D. M.

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. A (2)

Opt. Lett. (3)

Other (13)

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging ,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
[CrossRef]

D. G. Voelz, J. D. Gonglewski, P. S. Idell, “SCIP computer simulation and laboratory verification ,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 169–176 (1993).
[CrossRef]

P. H. Roberts, “A wave optics propagation code,” (the Optical Sciences Company, Anaheim, Calif., 1986).

G. A. Tyler, D. L. Fried, “A wave optics propagation algorithm,” (the Optical Sciences Company, Anaheim, Calif., 1982).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (J Wiley, New York, 1998).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Sec. 4.1.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (9.1.21).

Ref. 9, Eq. (11.4.17).

S. M. Brown, J. W. Hardy, R. Hutchin, P. J. Mailhot, M. B. Michalik, S. Paley, “Reconstruction development study,” (Itek Optical Systems, Lexington, Mass.1987).

G. A. Tyler, T. J. Brennan, W. Moretti, J. L. Vaughn, R. H. Dueck, “Recent analytical and experimental results,” (the Optical Sciences Company, Anaheim, Calif., 1992).

W. Moretti, G. M. Cochran, K. E. Steinhoff, G. A. Tyler, “SOR-3 data reduction,” (the Optical Sciences Company, Anaheim, Calif., 1988).

D. M. Winker, G. A. Ameer, S. L. Browne, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, 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. SPIE926, 360–366 (1988).
[CrossRef]

D. M. Lussier, “Slope discrepancy due to four point phase approximation in the SOR-3 experiment,” (the Optical Sciences Company, Anaheim, Calif., 1987).

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 (4)

Fig. 1
Fig. 1

Pupil plane irradiance profile. Four pupil plane irradiance profiles computed in a wave optics simulation of the four turbulence levels are illustrated. The diameter of the receiving aperture is 0.70 m, the transmitter wavelength is 1 μm and the spherical wave propagation path length is 50 km. (a)–(d) correspond to different strengths of turbulence. The theoretical log amplitude variances (Rytov numbers) for these turbulence levels are (a) 0.11; (b) 0.22, (c) 0.44, and (d) 0.88.

Fig. 2
Fig. 2

Logarithm of the slope discrepancy Strehl ratio versus Rytov number. Equation (108) was used to compute -log(SSD) as a function of Rytov number for the four cases presented in Fig. 1. Since the cases studied were designed to minimize the strength of other effects, it is believed that the effects of branch points dominate the results presented here. For levels of the Rytov numbers below 0.2 the effects of branch points are negligible. Beyond this level, the quantity -log(SSD), which is reminiscent of a wave-front variance, is monotonic and is nearly linearly related to the Rytov number.

Fig. 3
Fig. 3

Components of the phase and gradient field for σl2=0.8. The results are presented for the strongest turbulence case considered (σl2=0.8) and correspond to the same frame as illustrated in Fig. 1(d). (a), (b), and (c) illustrate the components of the wave-front gradient field and correspond to the least-squares component, the slope discrepancy component, and the total, respectively. Equations (78) and (80) were applied to the wave optics simulation data to obtain these results. The reconstructed phase associated with each of the gradient components is represented in (d)–(f). These wave fronts were reconstructed from the corresponding components of the gradient field, as discussed in the text. An ordinary least-squares reconstructor was applied to gLS to obtain ϕLS, and the branch cut reconstructor used in Ref. 4 was applied to gSD to obtain ϕSD.

Fig. 4
Fig. 4

Strehl ratio performance of an adaptive optics system. In this figure the Strehl ratio is presented as a function of the Rytov number. Each curve corresponds to a different component of the wave-front distortion. Curve BC is the adaptive optics Strehl ratio obtained when both components of the phase are used in the adaptive optics correction. The Strehl ratio is not unity in this case, primarily because of scintillation, but the fitting error associated with the finite subapertures’ size does play a secondary role. Curve LS is the adaptive optics Strehl ratio obtained when only the least-squares component of the reconstructed phase is used to compensate the wave front. Curve SD is the Strehl ratio loss associated with not using the slope discrepancy phase [computed from Eq. (107)]. Curve BC*SD, is the product of the full adaptive optics Strehl and the Strehl degradation that is due to the slope discrepancy, which is almost entirely due to the effects of branch points in this example.

Equations (108)

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

  A(r, z)=[  A(r, z)]-××A(r, z),
A(r, z)=dαA˜(κ, s)exp[2πi(α  x)]
A˜(κ, s)=dxA(r, z)exp[-2πi(α  x)],
x=(r, z)
α=(κ, s).
A(r, z)=-4π2dαααA˜(κ, s)×exp[2πi(αx)],
A(r, z)=-4π2dαααA˜(κ, s)×exp[2πi(αx)],
-××A(r, z)=4π2dαα×α×A˜(κ, s)×exp[2πi(αx)].
-4π2ααA˜(α)=-4π2ααA˜(α)+4π2α×α×A˜(α).
A˜(α)=2πiααA˜(α)2πiα2+2πiα×α×A˜(α)-2πiα2.
A(x)=ϕ(x)+×V(x),
ϕ(x)=dααA˜(α)2πiα2exp[2πi(αx)],
V(x)=dαα×A˜(α)-2πiα2exp[2πi(αx)].
g(r)=gx(r)ux+gy(r)uy,
A(x)=g(r)
A˜(α)=g˜x(κ)δ(s)ux+g˜y(κ)δ(s)uy,
g˜x(κ)=drgx(r)exp(-2πiκr),
g˜y(κ)=drgy(r)exp(-2πiκr).
g(r)=ϕ(r)+uzV(r),
ϕ(r)=dκκxg˜x(κ)+κyg˜y(κ)2πiκ2exp(2πiκr),
V(r)=dκκyg˜x(κ)-κxg˜y(κ)2πiκ2exp(2πiκr),
V(r)=V(r)uz.
g(r)=gLS(r)+gSD(r),
gLS(r)=ϕ(r),
gSD(r)=×uzV(r).
ϕtotal(r)=ϕLS(r)+ϕSD(r).
ϕtotal(r)=g(r),
ϕLS(r)=ϕ(r),
ϕSD(r)=×uzV(r).
ϕLS(r)x=ϕ(r)x,
ϕLS(r)y=ϕ(r)y.
ϕSD(r)x=V(r)y,
ϕSD(r)y=-V(r)x.
ϕLS(r)=ϕ(r).
ϕtotal=tan-1(y/x),
g(r)=ϕtotal(r).
gx(r)=-yx2+y2,
gy(r)=xx2+y2.
ddztan-1(z)=11+z2.
g˜x(κ)=-dryx2+y2exp(-2πiκ  r).
x=r cos ϕ,
y=r sin ϕ,
g˜x(κ)=--ππdϕ0dr sin ϕ exp[-2πiκr cos(ϕ-θ)],
φ=ϕ-θ,
r=-r.
g˜x(κ)=2 sin θ0-dr0πdφ cos φ exp(2πiκr cos φ)+cos θ0-dr-ππdφ sin φ exp(2πiκr cos φ).
0πdφ cos φ exp(iz cos φ)=iπJ1(z),
g˜x(κ)=2πi sin θ 0-drJ1(2πκr).
t=-2πκr
κx=κ cos θ,
κy=κ sin θ
g˜x(κ)=i κxκ20dtJ1(t),
0dtJ1(t)=1.
g˜x(κ)=i κyκ2.
g˜y(κ)=-i κxκ2.
ϕ(r)=0.
V(r)=12πdκκ-2exp(2πiκr),
gLS(r)=0.
gSD(r)=×uz12πdκκ-2exp(2πiκr).
gSD(r)=ux2πydκκ-2exp(2πiκr)-uy2πxdκκ-2exp(2πiκr).
gSD(r)=iux0dκ-ππdθ sin θ exp[2πiκr cos(θ-ϕ)]-iuy0dκ-ππdθ cos θ×exp[2πiκr cos(θ-ϕ)].
φ=θ-ϕ,
gSD(r)=2i sin ϕux0dκ0πdφ cos φ exp(2πiκr cos φ)-2i cos ϕuy0dκ0πdφ cos φ×exp(2πiκr cos φ).
gSD(r)=-2π sin ϕux0dκJ1(2πκr)+2π cos ϕuy0dκJ1(2πκr).
gSD(r)=-yx2+y2ux+xx2+y2uy.
ϕLS(r)=0,
ϕSD(r)=0ydγ xx2+γ2.
ϕSD(r)=tan-1(y/x).
s=Γp,
pˆ=(ΓTΓ)-1ΓTs,
sˆ=Γpˆ.
sˆ=Γ(ΓTΓ)-1ΓTs.
s-sˆ=[I-Γ(ΓTΓ)-1ΓT]s,
ΓGradient,
ΓT Divergence,
ΓTΓLaplacian.
ϕ=(ΓTΓ)-1ΓTg,
gLS=Γ(ΓTΓ)-1ΓTg.
gSD=g-gLS.
gSD=[I-Γ(ΓTΓ)-1ΓT]g.
g=n,
nnT=Insσn2,
n=0,
gLS=Γ(ΓTΓ)-1ΓTn,
gSD=[Ins-Γ(ΓTΓ)-1ΓT]n,
σLS2=1nsTrgLS gLST,
σLS2=1nsTr[Γ(ΓTΓ)-1ΓTnnTΓ(ΓTΓ)-1ΓT].
σLS2=σn2nsTr[Γ(ΓTΓ)-1ΓT].
σLS2=σn2nsTr[(ΓTΓ)-1ΓTΓ].
Tr[(ΓTΓ)-1ΓTΓ]=np-nu.
σLS2=np-nuns σn2.
σSD2=1nsTrgSDgSDT.
σSD2=1nsTr{[Ins-Γ(ΓTΓ)-1ΓT]nnT×[Ins-Γ(ΓTΓ)-1ΓT]}.
σSD2=σn2nsTr[Ins-Γ(ΓTΓ)-1ΓT].
σSD2=ns-np+nuns σn2.
σLS2+σSD2=σn2.
=Γp-c.
gSD=[I-Γ(ΓTΓ)-1ΓT].
gSD=[I-Γ(ΓTΓ)-1ΓT]Γp-[I-Γ(ΓTΓ)-1ΓT]c.
[I-Γ(ΓTΓ)-1ΓT]Γp=0.
gSD=[I-Γ(ΓTΓ)-1ΓT]c.
σSD2=1nsTr{[I-Γ(ΓTΓ)-1ΓT]ccT[I-Γ(ΓTΓ)-1ΓT]},
σSD2=1nsTr{[1-Γ(ΓTΓ)-1ΓT]2ccT}.
[I-Γ(ΓTΓ)-1ΓT]2=[I-Γ(ΓTΓ)-1ΓT],
σSD2=1nsTr{[I-Γ(ΓTΓ)-1ΓT]ccT}.
σl2=1.056λ-7/6L11/6Cn2.
SSD=1Nn=1Nexp[iϕSD(rn)]2.
-log(SSD)=-2 log1Nn=1Nexp[iϕSD(rn)].

Metrics