Abstract

A discrete ordinates code is developed with which to compute the beam spread function (BSF) without invoking the small-angle scattering approximation or performing Monte Carlo calculations. The computed BSF is used to predict the response of a detector versus its distance to the origin of a highly collimated beam, its angle with respect to the beam, and the two local angles that specify the detector orientation. Numerical results have been obtained for water models that simulate a clear ocean, a coastal ocean, and a turbid harbor. Six orders of magnitude or more change in the detector response caused by scattered photons can be predicted for different detector locations while simultaneously obtaining small changes for different detector orientations. This capability is useful for assessment of the sensitivity of the detector response to the interpretation of time-independent underwater imaging systems or visibility models.

© 2002 Optical Society of America

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References

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  1. K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. GPO, Washington, D.C., 1953).
  2. D. E. Kornreich, B. D. Ganapol, “The suite of analytical benchmarks for neutral particle transport in infinite isotropically scattering media,” Nucl. Sci. Eng. 125, 24–50 (1997).
  3. H. R. Gordon, “Equivalence of the point and beam spread functions of scattering media: a formal demonstration,” Appl. Opt. 33, 1120–1122 (1994).
    [CrossRef] [PubMed]
  4. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  5. W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea (Interface and In-Water Transmission and Imaging), P. Halley, ed., Lecture Series no. 61 (North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development, Neuilly-sur-Seine, France, 1973).
  6. K. J. Voss, A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
    [CrossRef] [PubMed]
  7. K. J. Voss, “Simple empirical model of the oceanic point spread function,” Appl. Opt. 30, 2647–2651 (1991).
    [CrossRef] [PubMed]
  8. R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994).
  9. N. L. Swanson, V. M. Gehman, B. D. Billard, T. L. Gennaro, “Limits of the small-angle approximation to the radiative transport equation,” J. Opt. Soc. Am. A 18, 385–391 (2001).
    [CrossRef]
  10. N. L. Swanson, B. D. Billard, V. M. Gehman, T. L. Gennaro, “Application of the small-angle approximation to ocean water types,” Appl. Opt. 40, 3608–3613 (2001).
    [CrossRef]
  11. C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
    [CrossRef]
  12. A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
    [CrossRef]
  13. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).
  14. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, LaGrange Park, Ill., 1993).
  15. K. D. Lathrop, F. W. Brinkley, “twotran sphere: a fortran program to solve the multigroup transport equation in two-dimensional spherical geometry,” Los Alamos Report LA-4567 (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1970).
  16. R. Sanchez, N. J. McCormick, “Discrete ordinate solutions for highly forward-peaked scattering,” Ann. Nucl. Energy, submitted for publication.
  17. R. Sanchez, “On the singular structure of the uncollided and first-collided components of the Green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
    [CrossRef]
  18. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).
  19. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  20. T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).
  21. J. W. McLean, J. D. Freeman, R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37, 4701–4711 (1998).
    [CrossRef]
  22. G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Methods Appl. Sci. 2, 21–36 (1992).
    [CrossRef]
  23. G. C. Pomraning, A. K. Prinja, J. W. VanDenburg, “An asymptotic model for the spreading of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).
  24. C. Börgers, E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

2002 (1)

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

2001 (2)

2000 (1)

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the Green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[CrossRef]

1998 (2)

C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
[CrossRef]

J. W. McLean, J. D. Freeman, R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37, 4701–4711 (1998).
[CrossRef]

1997 (1)

D. E. Kornreich, B. D. Ganapol, “The suite of analytical benchmarks for neutral particle transport in infinite isotropically scattering media,” Nucl. Sci. Eng. 125, 24–50 (1997).

1996 (1)

C. Börgers, E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

1994 (1)

1992 (3)

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Methods Appl. Sci. 2, 21–36 (1992).
[CrossRef]

G. C. Pomraning, A. K. Prinja, J. W. VanDenburg, “An asymptotic model for the spreading of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

1991 (1)

1990 (1)

Billard, B. D.

Billmers, R.

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

Börgers, C.

C. Börgers, E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

Brinkley, F. W.

K. D. Lathrop, F. W. Brinkley, “twotran sphere: a fortran program to solve the multigroup transport equation in two-dimensional spherical geometry,” Los Alamos Report LA-4567 (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1970).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. GPO, Washington, D.C., 1953).

Chapin, A. L.

Concannon, B.

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

Contarino, V.

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

Cota, G. F.

C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
[CrossRef]

Davis, J.

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

de Hoffmann, F.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. GPO, Washington, D.C., 1953).

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Freeman, J. D.

Ganapol, B. D.

D. E. Kornreich, B. D. Ganapol, “The suite of analytical benchmarks for neutral particle transport in infinite isotropically scattering media,” Nucl. Sci. Eng. 125, 24–50 (1997).

Gehman, V. M.

Gennaro, T. L.

Gordon, H. R.

Grenfell, T. C.

C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
[CrossRef]

Kornreich, D. E.

D. E. Kornreich, B. D. Ganapol, “The suite of analytical benchmarks for neutral particle transport in infinite isotropically scattering media,” Nucl. Sci. Eng. 125, 24–50 (1997).

Larsen, E. W.

C. Börgers, E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

Lathrop, K. D.

K. D. Lathrop, F. W. Brinkley, “twotran sphere: a fortran program to solve the multigroup transport equation in two-dimensional spherical geometry,” Los Alamos Report LA-4567 (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1970).

Laux, A.

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

Lewis, E. E.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, LaGrange Park, Ill., 1993).

Maffione, R. A.

C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
[CrossRef]

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

McCormick, N. J.

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

R. Sanchez, N. J. McCormick, “Discrete ordinate solutions for highly forward-peaked scattering,” Ann. Nucl. Energy, submitted for publication.

McLean, J. W.

Miller, W. F.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, LaGrange Park, Ill., 1993).

Mobley, C. D.

C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
[CrossRef]

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

Mullen, L.

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

Pegau, W. S.

C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
[CrossRef]

Perovich, D. K.

C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
[CrossRef]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

Placzek, G.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. GPO, Washington, D.C., 1953).

Pomraning, G. C.

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Methods Appl. Sci. 2, 21–36 (1992).
[CrossRef]

G. C. Pomraning, A. K. Prinja, J. W. VanDenburg, “An asymptotic model for the spreading of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

Prentice, J.

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

Prinja, A. K.

G. C. Pomraning, A. K. Prinja, J. W. VanDenburg, “An asymptotic model for the spreading of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

Sanchez, R.

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the Green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[CrossRef]

R. Sanchez, N. J. McCormick, “Discrete ordinate solutions for highly forward-peaked scattering,” Ann. Nucl. Energy, submitted for publication.

Swanson, N. L.

VanDenburg, J. W.

G. C. Pomraning, A. K. Prinja, J. W. VanDenburg, “An asymptotic model for the spreading of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

Voss, K. J.

Walker, R. E.

Wells, W. H.

W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea (Interface and In-Water Transmission and Imaging), P. Halley, ed., Lecture Series no. 61 (North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development, Neuilly-sur-Seine, France, 1973).

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Ann. Nucl. Energy (1)

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the Green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Geosci. Remote Sens. (1)

C. D. Mobley, G. F. Cota, T. C. Grenfell, R. A. Maffione, W. S. Pegau, D. K. Perovich, “Modeling light propagation in sea ice,” IEEE Trans. Geosci. Remote Sens. 36, 1743–1749 (1998).
[CrossRef]

J. Mod. Opt. (1)

A. Laux, R. Billmers, L. Mullen, B. Concannon, J. Davis, J. Prentice, V. Contarino, “The a, b, cs of oceanographic lidar predictions: a significant step toward closing the loop between theory and experiment,” J. Mod. Opt. 49, 439–451 (2002).
[CrossRef]

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

Limnol. Oceanogr. (1)

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

Math. Models Methods Appl. Sci. (1)

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Methods Appl. Sci. 2, 21–36 (1992).
[CrossRef]

Nucl. Sci. Eng. (3)

G. C. Pomraning, A. K. Prinja, J. W. VanDenburg, “An asymptotic model for the spreading of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

C. Börgers, E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

D. E. Kornreich, B. D. Ganapol, “The suite of analytical benchmarks for neutral particle transport in infinite isotropically scattering media,” Nucl. Sci. Eng. 125, 24–50 (1997).

Other (10)

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea (Interface and In-Water Transmission and Imaging), P. Halley, ed., Lecture Series no. 61 (North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development, Neuilly-sur-Seine, France, 1973).

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion (U.S. GPO, Washington, D.C., 1953).

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, LaGrange Park, Ill., 1993).

K. D. Lathrop, F. W. Brinkley, “twotran sphere: a fortran program to solve the multigroup transport equation in two-dimensional spherical geometry,” Los Alamos Report LA-4567 (Los Alamos Scientific Laboratory, Los Alamos, N. Mex., 1970).

R. Sanchez, N. J. McCormick, “Discrete ordinate solutions for highly forward-peaked scattering,” Ann. Nucl. Energy, submitted for publication.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

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Figures (11)

Fig. 1
Fig. 1

Coordinate systems for the BSF and the associated PSF problems. The figure shows when the detector’s axis points toward the beam’s emission point.

Fig. 2
Fig. 2

Different detector orientations. The detector in position 1 has θ d = 0°, the detector in position 2 has θ d = 30° and ϕ d = 0°, the detector in position 3 has θ d = 30° and ϕ d = 180°.

Fig. 3
Fig. 3

Uncollided photon beam in CLW, COW, and THW.

Fig. 4
Fig. 4

Collided photons in CLW for different polar elevations (θ = 0°, 1°, 3°, 9°, and 20°) for a detector looking straight at the emission point (θ d = 0°, ϕ d = 0°).

Fig. 5
Fig. 5

Collided photons in THW for different polar elevations (θ = 0°, 1°, 3°, 9°, and 20°) for a detector looking straight at the emission point (θ d = 0°, ϕ d = 0°).

Fig. 6
Fig. 6

Collided signal for a detector in CLW aimed at the origin of the beam (θ d = 0°, ϕ d = 0°) for angles from θ = 0° to 30°.

Fig. 7
Fig. 7

Collided signal for a detector in CLW aimed at the origin of the beam (θ d = 0°, ϕ d = 0°), a detector looking inward (θ d = 30°, ϕ d = 0°), and a detector looking outward (θ d = 30°, ϕ d = 180°) versus distance for θ = 0°, 3°, and 18°.

Fig. 8
Fig. 8

Collided signal for a detector in CLW at r = 0.8 m and θ = 2° for inward-looking detector (at left) and outward-looking detector (at right) showing variations of the response as a function of the detector polar orientations θ d .

Fig. 9
Fig. 9

Collided signal for a detector in CLW at r = 0.4 m and θ = 2° for polar detector orientation θ d for ϕ d = 0°, 10°, 20°, and 30° (detector looking inward) and 150°, 160°, 170°, and 180° (detector looking outward).

Fig. 10
Fig. 10

Same as Fig. 4 except for a 5-cm-diameter detector rather than a 1-cm-diameter detector.

Fig. 11
Fig. 11

Coordinate notation for a finite detector.

Tables (4)

Tables Icon

Table 1 Basic Properties for Model Watersa

Tables Icon

Table 2 Details of the Calculation Mesh

Tables Icon

Table 3 Position (r, θ) and Orientation (θ d , ϕ d ) of Detectors

Tables Icon

Table 4 Ratio of Scattered to Uncollided Signal versus Distance r for a Straight Detector (μ d = 0, ϕ d = 0) on Axis (θ = 0)

Equations (42)

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

Ω·+cLBSF=bHLBSF+δ3r-r0δ2Ω·Ω0, limrLBSFr, Ω<,
HLr, Ω=4πdΩβ˜Ω·ΩLr, Ω,
D= dr  dΩσdr, ΩLBSFr, Ω.
Gr0, Ω0r, Ω=Gr, -Ωr0, -Ω0,
D= dr  dΩσdr, -ΩGr, Ωr0, -Ω0=LPSFr0, -Ω0,
Ω·+cLPSF=bHLPSF+σdr, -Ω, limrLPSFr, Ω<.
σdr, Ω=δ3r-rpdgPSFΩ·nd,
gPSFΩ·nd=|Ω·nd|ΘΩ·nd,
Ω·+cL=bHL+δ3rgΩ·ez, limrLr, Ω<,
H1/2Lr, Ω=2πdΩβ˜1/2Ω·ΩLr, Ω,
β˜1/2Ω·Ω=β˜Ω·Ω+β˜szΩ·Ω
Lr, Ω=Luncr, Ω+Lcolr, Ω,
Luncr, Ω=exp-crr2δ2Ω·erger·ez=exp-crr2δ1-μ2πgcos θ,
Ω·+cLcol=bH1/2Lcol+exp-crr2β˜μgcos θ, limrLcolr, Ω<,
Ω·μr2rr2+1rμ1-μ2+ηr sin θθsin θ-cot θrϕ ξ.
Eout=0ρd ρdρ 02πdαDr, nd,
Dr, nd=2πdΩ|Ω·nd|LBSFr, Ω,
1=max1-LnewLold, 2=maxV1-LnewLold,3= VLnew-Lold2 VLold21/2,4= V1-Lnew/Lold2 V1/2,
ψunc=exp-cz, ψcol=exp-czexpωcz-1.
ψcol=α0gωcz2πα0gωcz2+s23/2exp-cz×expgωcz-1,
=O1-μ2  1,
B0μr2rr2+1rμ1-μ2+c-bH,B1-ηr sin θθsin θ+cot θrϕξ.
Lr, θ, μ, ϕ=n0Lnr, θ, μ, ϕ,
LRmax, θ, μ, ϕLasRmax, μ=σAνexp-cRmax/νφνμ,
φνμ=gνμ1ν-μ
σBνφνμp=1Nϕq=1q=NϕLRmax, θ, μp, ϕq, μp>0,
σBνJν+=μp>0q=1q=Nϕwpw¯qμpLRmax, θ, μp, ϕq,
Jν+=μp>0wpμpφνμp
LRmax, θ, μp, ϕq=σBνφνμp, μp<0.
Eoutrd; ρd, nd=0ρd ρdρ 02πdαEoutrpd; nd,
e1=1-μd2, ϕd+πd, e2=0, ϕd+π2d.
rs2=rd2+ρ2+2rdρ cos α1-μd2, cos θs=-nd·rs/rs=rdrsμd.
μs=rd cos θd+ρΩ0·e1 cos α+Ω0·e2 sin αrs, cos ϕs=-Ω0·nd+μs cos θs1-μs2sin θs.
Ω0·e1=1-μd2cos θd+μd cos ϕd sin θd, Ω0·e2=sin ϕd sin θd, Ω0·nd=μd cos θd-1-μd2cos ϕd sin θd.
Drd; nd=LPSFrs, Ωs,
rs=rd, θs=cos-1 μd, μs=cos θd, ϕs=ϕd.
rs2=rd2+ρ2, cos θs=rdrs, μs=rd cos θd+ρ sin θd cosα-ϕdrs, cos ϕs=-cos θd+μs cos θs1-μs2sin θs.
0<Ψ1<Ψ2<<ΨN=π.
0Ψ0<Ψ1<Ψ2<<ΨN=π,
0πβ˜Ψsin ΨdΨ=12π.
lnβ˜Ψ=1-αilnβ˜i+αilnβ˜i-1, αi=lnΨi/ΨlnΨi/Ψi-1, Ψ  Ψi-1, Ψi,
1-cos Ψ0β˜0+Ψ0Ψ1 β˜Ψsin ΨdΨ=12π-Ψ1πβ˜Ψsin ΨdΨ.

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