Abstract

From the geometrical path statistics of rays in an anomalous-diffraction theory (ADT) [Opt. Lett. 28, 179 (2003)] closed-form expressions for the geometrical path distribution of rays and analytical formulas for the optical efficiencies of finite circular cylinders oriented in an arbitrary direction with respect to the incident light are derived. The characteristics of the shapes of the cylinders produce unique features in the geometrical path distributions of the cylinders compared with spheroids. Gaussian ray approximations, which depend only on the mean and the mean-squared geometrical paths of rays, of the optical efficiencies of finite circular cylinders and spheroids are compared with the exact optical efficiencies in ADT. The influence of the difference in shape between cylinders and spheroids on the optical efficiencies in ADT is illustrated by their respective geometrical path distributions of rays.

© 2003 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  2. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, San Diego, Calif., 1999).
  3. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [Crossref]
  4. B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [Crossref]
  5. M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
    [Crossref]
  6. S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
    [Crossref]
  7. W. A. Farone, M. J. I. Robinson, “The range of validity of the anomalous diffraction approximation to electromagnetic scattering by a sphere,” Appl. Opt. 7, 643–645 (1968).
    [Crossref] [PubMed]
  8. S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
    [Crossref] [PubMed]
  9. A. Maslowska, P. J. Flatau, G. L. Stephens, “On the validity of the anomalous diffraction theory to light scattering by cubes,” Opt. Commun. 107, 35–40 (1994).
    [Crossref]
  10. Y. Liu, W. P. Arnott, J. Hallett, “Anomalous diffraction theory for arbitrarily oriented finite circular cylinders and comparison with exact T-matrix results,” Appl. Opt. 37, 5019–5030 (1998).
    [Crossref]
  11. A. J. Baran, J. S. Foot, D. L. Mitchell, “Ice-crystal absorption: a comparison between theory and implications for remote sensing,” Appl. Opt. 37, 2207–2215 (1998).
    [Crossref]
  12. F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
    [Crossref]
  13. P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
    [Crossref]
  14. D. A. Cross, P. Latimer, “General solutions for the extinction and absorption efficiencies of arbitrarily oriented cylinder by anomalous-diffraction methods,” J. Opt. Soc. Am. 60, 904–907 (1970).
    [Crossref]
  15. P. Chýlek, J. D. Klett, “Extinction cross sections of nonspherical particles in the anomalous diffraction approximation,” J. Opt. Soc. Am. A 8, 274–281 (1991).
    [Crossref]
  16. M. Xu, M. Lax, R. R. Alfano, “Light anomalous diffraction using geometrical path statistics of rays and Gaussian ray approximation,” Opt. Lett. 28, 179–181 (2003).
    [Crossref] [PubMed]
  17. A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).
  18. A. A. Kokhanovsky, Optics of Light Scattering Media: Problems and Solutions (Wiley, New York, 1999).

2003 (1)

1998 (3)

1994 (2)

A. Maslowska, P. J. Flatau, G. L. Stephens, “On the validity of the anomalous diffraction theory to light scattering by cubes,” Opt. Commun. 107, 35–40 (1994).
[Crossref]

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[Crossref]

1991 (1)

1987 (1)

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[Crossref]

1980 (1)

1975 (1)

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[Crossref]

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

1970 (1)

1969 (1)

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[Crossref]

1968 (1)

Ackerman, S. A.

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[Crossref]

Alfano, R. R.

M. Xu, M. Lax, R. R. Alfano, “Light anomalous diffraction using geometrical path statistics of rays and Gaussian ray approximation,” Opt. Lett. 28, 179–181 (2003).
[Crossref] [PubMed]

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Alimova, A.

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Arnott, W. P.

Asano, S.

Baran, A. J.

Bryant, F. D.

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[Crossref]

Chýlek, P.

Cross, D. A.

Draine, B. T.

Farone, W. A.

Flatau, P. J.

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[Crossref]

A. Maslowska, P. J. Flatau, G. L. Stephens, “On the validity of the anomalous diffraction theory to light scattering by cubes,” Opt. Commun. 107, 35–40 (1994).
[Crossref]

Foot, J. S.

Hallett, J.

Katz, A.

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Klett, J. D.

Kokhanovsky, A. A.

A. A. Kokhanovsky, Optics of Light Scattering Media: Problems and Solutions (Wiley, New York, 1999).

Latimer, P.

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[Crossref]

D. A. Cross, P. Latimer, “General solutions for the extinction and absorption efficiencies of arbitrarily oriented cylinder by anomalous-diffraction methods,” J. Opt. Soc. Am. 60, 904–907 (1970).
[Crossref]

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[Crossref]

Lax, M.

Liu, Y.

Maslowska, A.

A. Maslowska, P. J. Flatau, G. L. Stephens, “On the validity of the anomalous diffraction theory to light scattering by cubes,” Opt. Commun. 107, 35–40 (1994).
[Crossref]

McCormick, S. A.

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[Crossref]

Mitchell, D. L.

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Robinson, M. J. I.

Rosen, R.

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Rudolph, E.

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Sato, M.

Savage, H.

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Shah, M.

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Stephens, G. L.

A. Maslowska, P. J. Flatau, G. L. Stephens, “On the validity of the anomalous diffraction theory to light scattering by cubes,” Opt. Commun. 107, 35–40 (1994).
[Crossref]

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[Crossref]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Xu, M.

M. Xu, M. Lax, R. R. Alfano, “Light anomalous diffraction using geometrical path statistics of rays and Gaussian ray approximation,” Opt. Lett. 28, 179–181 (2003).
[Crossref] [PubMed]

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

Appl. Opt. (4)

Astrophys. J. (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

J. Atmos. Sci. (1)

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[Crossref]

J. Colloid Interface Sci. (2)

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[Crossref]

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Quant. Spectrosc. Radiat. Transfer (1)

M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[Crossref]

Opt. Commun. (1)

A. Maslowska, P. J. Flatau, G. L. Stephens, “On the validity of the anomalous diffraction theory to light scattering by cubes,” Opt. Commun. 107, 35–40 (1994).
[Crossref]

Opt. Lett. (1)

Other (4)

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron.9 (to be published).

A. A. Kokhanovsky, Optics of Light Scattering Media: Problems and Solutions (Wiley, New York, 1999).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, San Diego, Calif., 1999).

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Figures (12)

Fig. 1
Fig. 1

Finite circular cylinder bounded by sides I, II, and III. The revolutional axis of the cylinder makes angle χ with the incident beam.

Fig. 2
Fig. 2

Projected area of a cylinder whose revolutional axis makes angle χ with the incident light: (a) β < 1, (b) β ≥ 1.

Fig. 3
Fig. 3

Ray distributions for a finite circular cylinder at a fixed orientation χ = π/4 (FX), randomly oriented (RN), polydisperse at a fixed orientation χ = π/4 (POL FX), and randomly oriented polydisperse (POL RN). Axial ratio of the cylinder: (a) ∊ = 0.5, (b) ∊ = 2. Height of the cylinder, L = 2 for the monosized cylinder. The log-normal size distribution n(x) of the half-height (L/2) of the polydisperse cylinder with a m = 1 and σ = 0.2 is also as insets. The height of the delta-function peak in (b) for the monosized cylinder at a fixed orientation is P = (4π - 33)/6(π + 2) ≃ 0.239.

Fig. 4
Fig. 4

Ray distributions for a spheroid at a fixed orientation χ = π/4 (FX), randomly oriented (RN), polydisperse at a fixed orientation (POL FX), and randomly oriented polydisperse (POL RN). Axial ratio of the spheroid: (a) ∊ = 0.5, (b) ∊ = 2. The semisize of the revolutional axis of the monosized spheroid is 1. Log normal-size distribution n(x) with a m = 1 and σ = 0.2 for the semisize of the revolutional axis of the spheroid is plotted as insets.

Fig. 5
Fig. 5

Mean and mean-square-root geometrical paths for a randomly oriented cylinder and spheroid with a common aspect ratio ∊ and a common surface areas of a sphere of radius a s .

Fig. 6
Fig. 6

Ratio of the ray path dispersion over mean geometrical paths for cylinders and spheroids.

Fig. 7
Fig. 7

Extinction and absorption efficiencies of cylinders (CYL) and spheroids (SPH) with aspect ratios (a) ∊ = 0.5 and (b) ∊ = 2. The equivalence size parameter is the size parameter of the sphere whose surface area is the same as that of the cylinder and the spheroid. Both the cylinder and the spheroid are oriented at a fixed orientation χ = π/4. Relative refractive index of both cylinders and spheroids, m = 1.05 - i0.0005.

Fig. 8
Fig. 8

Extinction and absorption efficiencies of cylinders (CYL) and spheroids (SPH) with aspect ratios (a) ∊ = 0.5 and (b) ∊ = 2. Both the cylinder and the spheroid are randomly oriented.

Fig. 9
Fig. 9

Extinction and absorption efficiencies of cylinders (CYL) and spheroids (SPH) with aspect ratios (a) ∊ = 0.5 and (b) ∊ = 2. The equivalence size parameter is the size parameter of the sphere of an equivalent surface area of the respective particle of size a m . The dispersion of the log-normal size distribution of the cylinder and the spheroid is σ = 0.2. Both the cylinder and the spheroid are polydisperse and oriented at a fixed orientation χ = π/4.

Fig. 10
Fig. 10

Extinction and absorption efficiencies of cylinders (CYL) and spheroids (SPH) with aspect ratios (a) ∊ = 0.5 and (b) ∊ = 2. Both the cylinder and the spheroid are polydisperse and randomly oriented.

Fig. 11
Fig. 11

Integration area for region II when β < 1. The area is given by ABC if β ≤ 1/2 and by ADEF if 1/2 < β < 1.

Fig. 12
Fig. 12

Spheroid whose revolutional axis makes an angle χ with the incident beam.

Tables (1)

Tables Icon

Table 1 Average Geometrical Cross Sections, Mean and Mean-Squared Geometrical Paths of Spheres, Randomly Oriented Spheroids, and Finite Circular Cylinders

Equations (60)

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|m-1|  1, x  1,
Qext=2P P1-exp-iklmr-1exp-klmidP,Qabs=1PP1-exp-2klmidP,Qsca=Qext-Qabs,
Qext=2  1-exp-iklmr-1×exp-klmipldl,Qabs=1-exp-2klmipldl,
ppoll=1/xp0l/xnxx2dx nxx2dx,
prnl= plΩdΩ ΩdΩ,
-x sin χ+z cos χ=L/2 for side I, -x sin χ+z cos χ=-L/2, for side II, x cos χ+z sin χ2a2+y2a2=1, for side III
ζ2=e+ξ sin χ cos χcos χ|ξ+β|1-η21-η2-ξ cos2 χcos χotherwise,ζ1=-e+ξ sin χ cos χcos χ|ξ-β|1-η2-1-η2-ξ cos2 χsin χotherwise.
l/a=ζ2-ζ1=ξ+β+1-η2sin χ|ξ+β|1-η2-ξ+β+1-η2sin χ|ξ-β|1-η221-η2sin χotherwise
l/a=ζ2-ζ1=ξ+β+1-η2sin χ-1-η2-βξ-1-η2+β2e/cos χ-1-η2+βξ1-η2-β-ξ+β+1-η2sin χ1-η2-βξ1-η2+β21-η2sin χotherwise
=2aL sin χ+πa2 cos χ=4a2 cos χβ+π/4.
tl sin χa=ζ2-ζ1sin χ=ξ+β+1-η2in I-ξ+β+1-η2in II2βin III21-η2in IV
qt=4-t22+t/2β-t/24-t2H2β-t+arccos β-β1-β22 δt-2ββ<14-t22+t/2β-t/24-t2H2-tβ1,
0+ qtdt=β+π/4,
pl=sin χql/sin χ/aαβ+π/4.
tan α=mimr-1,
ρ*=ρ1-i tan α,
ρ=2kamr-1sin χ,
Qext=2-4β+π/4 Gρ*; arcsin β+arccos β-β1-β24exp-iρ*ββ<12-4β+π/4 Gρ*; π/2β1,
Gu; ϕ=0ϕexp-iu sin θ32cos2 θ-12+β2sin θdθ=321-cos ϕ exp-iu sin ϕiu-12×0ϕexp-iu sin θdθ+12β-3iu×0ϕexp-iu sin θsin θdθ.
Gu; π2=β2-π4J0u-iH0u-π4β+i 3u×H1u+iJ1u,
Qabs=1-2β+π/4G-i2ρ tan α; arcsin β+arccos β-β1-β24exp-2ρβ tan αβ<11-2β+π/4 G-i2ρ tan α; π/2β1.
Qext=2kmil+k2mr-12-mi2l2, Qabs=2kmil-2k2mi2l2,
Qextχ=0=2-2 coskLmr-1exp-kLmi, Qabsχ=0=1-exp-2kLmi.
Qextχ=π2=π H1ρ*+iJ1ρ*, Qabsχ=π2=π2I12ρ tan α-L12ρ tan α,
prnl=0π/2dχ 1asin2 χ cos χqlasin χ0π/2dχ sin χ cos χe tan χ+π/4
prnl=1π/81+2eaHe-xDarctan e-1, x+Hx-eHe2+1-x×Darctan e-1, x-Darccosex, x+H1-xD π2, x-Darctan e-1, x+Hx-1He2+1-xDarcsin x-1, x-Darctan e-1, x+Hx-eHe2+1-xe24x3arccosx2-e2-x2-e21+e2-x21/2,
Dχ, x=dχ sin2 χ cos χΔ+x sin χe tan χ-x sin χ2Δ =6x2 sin2 χ+116x2sin χΔ-116x3arcsinx sin χ+e sin χ cos χΔ6x+e2+x26x3 Fχ, x-e1+x23x3 Eχ, x,
Fχ, x=0χda1-x2 sin2 a1/2, Eχ, x=0χ1-x2 sin2 a1/2da.
p0=sin χaβ+π/4.
prn0=83πa+L
ppol0=p0exp-3σ2/2,
ppol,rn0=prn0exp-3σ2/2
nr=12π1/2σ r-1 exp-ln2r/am2σ2,
pGaux=12πvexp-x-μ22ν2, μ=l, ν=l2-l2,
QextGau=2-2 coskmr-1μ-kν2miexp-kμmi-k2ν2mr-12-mi22, QabsGau=1-exp-2kmiμ-kν2mi
l=asin χπβ2β+π/4,
l2=a2sin2 χ1β+π/483 β-π4+121+4β2arccos β-16 β2β2+131-β2β<1a2sin2 χ8/3β-π/4β+π/4β1,
lrn=0π/2dχ sin χlcos χβ+π/40π/2dχ sin χcos χβ+π/4=4e1+2e a,
l2rn=0π/2dχ sin χl2cos χβ+π/40π/2dχ sin χ cos χβ+π/4=8a2e22e+143e+e6e+e2+1-13e2+112e+1+lne+e2+12e-14e2lne+e2+1=8a2e22e+14/3e-1-1/4ln 2e+5/4e-2-196 e-4+1768 e-6+Oe-7eln1/e+1-ln 2+2/3e-1/6e2+1/30e3-1/420e5+Oe6e0
lpol=l0 exp5σ2/2,
l2pol=l20 exp6σ2,
cos θ=1-t2-ρ22tρ, 0ρ1, 1-ρt1+ρ.
 ρdρdθ=q2tdt=dt×dρ ρ1+t2-ρ2t-ρ4+2t2+1ρ2-t2-121/2.
q2t=|t-1|1dρ ρ1+t2-ρ2t-ρ4+2t2+1ρ2-t2-121/2=4-t22, 0t2.
1-ρt2β, ρ|2β-1|, β<1, 1-ρt1+ρ, ρ<2β-1, 1/2<β<1.
|t-1|ρ1, 0t2β.
q3t=arccos β-β1-β22 δt-2β, β<1,
t=21-sin2 θ; 0ξβ+cos θ; π/2θπ-arccos β, β<1; π/2θπ, β1
sin θ=1-t2/4.
dξdη=q4tdt=-β+cos θdsin θ=t/2β-t/24-t2dt
q4t=t/2β-t/24-t2, 0t2 minβ, 1.
x cos χ+z sin χ2a2+y2a2+-x sin χ+z cos χ2b2=1.
l=2aba2 cos2 χ+b2 sin2 χ1/21-a2 cos2 χ+b2 sin2 χ-1 x2-a-2y21/2.
=πaa2 cos2 χ+b2 sin2 χ1/2=πb22 cos2 χ+sin2 χ.
l=2aba2 cos2 χ+b2 sin2 χ1/21-ρ21/2.
d=aa2 cos2 χ+b2 sin2 χ1/2 2πρdρ=π2 cos2 χ+sin2 χ3/22 ldl.
psphl=d=122b22 cos2 χ+sin2 χl×H2b2 cos2 χ+sin2 χ1/2-l, l0.
psphl=12b2 lH2b-l, =1
ppolsphl=-2 sin2 χ+cos2 χl4×erfc1/2σln-2 sin2 χ+cos2 χ1/2l/2amam2 exp2σ2
lsph=43b2 cos2 χ+sin2 χ1/2 l2sph=22b22 cos2 χ+sin2 χ

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