Abstract

A fiber-optic sensor that measures the particle volume fraction in dense suspensions is calibrated against a quantitative capacitance probe. For homogeneous, dense, random suspensions of smooth, monodisperse, transparent dielectric spheres, the calibration is simulated by using a ray-tracing Monte Carlo algorithm that predicts systematic uncertainties of the sensor’s output, the extent of its measurement volume, and the effects of changing its optical properties. The simulation shows that the output and accuracy of the sensor increase with a decreasing sphere diameter and with an increasing N.A. of the fiber. The output also increases when the ratio of the indices of refraction of the sphere and the suspending medium is increased. For small particles the measurement volume scales as the average interparticle distance.

© 1992 Optical Society of America

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References

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  1. C. W. Snoek, “A selection of new developments in multiphase flow measurement techniques,” Exp. Thermal Fluid Sci. 3, 60–73 (1990).
    [CrossRef]
  2. A. Cutolo, I. Rendima, U. Arena, A. Marzocchella, L. Massimilla, “Optoelectronic technique for the characterization of high concentration gas-solid suspension,” Appl. Opt. 29, 1317–1322 (1990).
    [CrossRef] [PubMed]
  3. M. Horio, K. Morishita, O. Tachibana, N. Murata, “Solid distribution and movement in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 147–154.
  4. E. U. Hartge, Y. Li, J. Werther, “Analysis of the local structure of the two-phase flow in a fast fluidized bed,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 153–160.
  5. E. U. Hartge, D. Rensner, J. Werther, “Solids concentration and velocity patterns in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 165–180.
  6. E. U. Hartge, D. Rensner, J. Werther, “Faseroptisches Meßsystem zur Erfassung lokaler Feststoff-Konzentrationen und-Geschwindigkeiten in höher konzentrierten Gas/Feststoff-Strömungen,” Synop. 1781/1989, Chem. Ing. Tech. 61, 744–745 (1989); complete manuscript on microfiche MS 1781/89 from Chem. Ing. Tech.
    [CrossRef]
  7. Y. Tung, J. Li, M. Kwauk, “Radial voidage profiles in a fast fluidized bed,” in Fluidization ‘88 Science and Technology, M. Kwauk, D. Kunii, eds. (Science Press, Beijing, 1988), pp. 139–145.
  8. Y. Matsuno, H. Yamaguchi, T. Oka, H. Kage, K. Higashitani, “The use of optical fiber probes for the measurement of dilute particle concentrations: calibration and application to gas fluidized bed carryover,” Powder Technol. 36, 215–221 (1988).
    [CrossRef]
  9. C. Brereton, L. Stromberg, “Some aspects of the fluid dynamic behaviour of fast fluidized beds,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 133–144.
  10. C. Acree Riley, M. Louge, “Quantitative capacitance measurements of voidage in gas-solid flows,” Particulate Sci. Technol. 7, 51–59 (1989).
    [CrossRef]
  11. M. Louge, M. Opie, “Measurements of the effective dielectric permittivity of suspensions,” Powder Technol. 62, 85–94 (1990).
    [CrossRef]
  12. H. Chang, M. Louge, in Fluidized Bed Combustion, E. J. Anthony, ed. (American Society of Mechanical Engineers, New York, 1991), Vol. 3, pp. 1215–1218.
  13. D. Deirmendjian, Electromagnetic Scttering on Spherical Polydispersions (American Elsevier, New York, 1969), p. 10.
  14. M. Kerker, The Scattering of Light (Academic, New York, 1969), p. 106–107.
  15. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981), p. 66.
  16. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. 1, 822–830 (1984).
    [CrossRef]
  17. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. 1, 831–835 (1984).
    [CrossRef]
  18. L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. 1, 836–839 (1984).
    [CrossRef]
  19. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), p. 354.
  20. B. Patrose, H. S. Caram, “Optical fiber probe transit anemometer for particle measurements in fluidized beds,” AIChE J. 28, 604–609 (1982).
    [CrossRef]

1990 (3)

M. Louge, M. Opie, “Measurements of the effective dielectric permittivity of suspensions,” Powder Technol. 62, 85–94 (1990).
[CrossRef]

C. W. Snoek, “A selection of new developments in multiphase flow measurement techniques,” Exp. Thermal Fluid Sci. 3, 60–73 (1990).
[CrossRef]

A. Cutolo, I. Rendima, U. Arena, A. Marzocchella, L. Massimilla, “Optoelectronic technique for the characterization of high concentration gas-solid suspension,” Appl. Opt. 29, 1317–1322 (1990).
[CrossRef] [PubMed]

1989 (2)

E. U. Hartge, D. Rensner, J. Werther, “Faseroptisches Meßsystem zur Erfassung lokaler Feststoff-Konzentrationen und-Geschwindigkeiten in höher konzentrierten Gas/Feststoff-Strömungen,” Synop. 1781/1989, Chem. Ing. Tech. 61, 744–745 (1989); complete manuscript on microfiche MS 1781/89 from Chem. Ing. Tech.
[CrossRef]

C. Acree Riley, M. Louge, “Quantitative capacitance measurements of voidage in gas-solid flows,” Particulate Sci. Technol. 7, 51–59 (1989).
[CrossRef]

1988 (1)

Y. Matsuno, H. Yamaguchi, T. Oka, H. Kage, K. Higashitani, “The use of optical fiber probes for the measurement of dilute particle concentrations: calibration and application to gas fluidized bed carryover,” Powder Technol. 36, 215–221 (1988).
[CrossRef]

1984 (3)

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. 1, 822–830 (1984).
[CrossRef]

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. 1, 831–835 (1984).
[CrossRef]

L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. 1, 836–839 (1984).
[CrossRef]

1982 (1)

B. Patrose, H. S. Caram, “Optical fiber probe transit anemometer for particle measurements in fluidized beds,” AIChE J. 28, 604–609 (1982).
[CrossRef]

Acree Riley, C.

C. Acree Riley, M. Louge, “Quantitative capacitance measurements of voidage in gas-solid flows,” Particulate Sci. Technol. 7, 51–59 (1989).
[CrossRef]

Arena, U.

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981), p. 66.

Brereton, C.

C. Brereton, L. Stromberg, “Some aspects of the fluid dynamic behaviour of fast fluidized beds,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 133–144.

Caram, H. S.

B. Patrose, H. S. Caram, “Optical fiber probe transit anemometer for particle measurements in fluidized beds,” AIChE J. 28, 604–609 (1982).
[CrossRef]

Chang, H.

H. Chang, M. Louge, in Fluidized Bed Combustion, E. J. Anthony, ed. (American Society of Mechanical Engineers, New York, 1991), Vol. 3, pp. 1215–1218.

Cutolo, A.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scttering on Spherical Polydispersions (American Elsevier, New York, 1969), p. 10.

Hartge, E. U.

E. U. Hartge, D. Rensner, J. Werther, “Faseroptisches Meßsystem zur Erfassung lokaler Feststoff-Konzentrationen und-Geschwindigkeiten in höher konzentrierten Gas/Feststoff-Strömungen,” Synop. 1781/1989, Chem. Ing. Tech. 61, 744–745 (1989); complete manuscript on microfiche MS 1781/89 from Chem. Ing. Tech.
[CrossRef]

E. U. Hartge, D. Rensner, J. Werther, “Solids concentration and velocity patterns in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 165–180.

E. U. Hartge, Y. Li, J. Werther, “Analysis of the local structure of the two-phase flow in a fast fluidized bed,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 153–160.

Higashitani, K.

Y. Matsuno, H. Yamaguchi, T. Oka, H. Kage, K. Higashitani, “The use of optical fiber probes for the measurement of dilute particle concentrations: calibration and application to gas fluidized bed carryover,” Powder Technol. 36, 215–221 (1988).
[CrossRef]

Horio, M.

M. Horio, K. Morishita, O. Tachibana, N. Murata, “Solid distribution and movement in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 147–154.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), p. 354.

Ishimaru, A.

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. 1, 831–835 (1984).
[CrossRef]

L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. 1, 836–839 (1984).
[CrossRef]

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981), p. 66.

Kage, H.

Y. Matsuno, H. Yamaguchi, T. Oka, H. Kage, K. Higashitani, “The use of optical fiber probes for the measurement of dilute particle concentrations: calibration and application to gas fluidized bed carryover,” Powder Technol. 36, 215–221 (1988).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969), p. 106–107.

Kuga, Y.

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. 1, 831–835 (1984).
[CrossRef]

Kwauk, M.

Y. Tung, J. Li, M. Kwauk, “Radial voidage profiles in a fast fluidized bed,” in Fluidization ‘88 Science and Technology, M. Kwauk, D. Kunii, eds. (Science Press, Beijing, 1988), pp. 139–145.

Li, J.

Y. Tung, J. Li, M. Kwauk, “Radial voidage profiles in a fast fluidized bed,” in Fluidization ‘88 Science and Technology, M. Kwauk, D. Kunii, eds. (Science Press, Beijing, 1988), pp. 139–145.

Li, Y.

E. U. Hartge, Y. Li, J. Werther, “Analysis of the local structure of the two-phase flow in a fast fluidized bed,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 153–160.

Louge, M.

M. Louge, M. Opie, “Measurements of the effective dielectric permittivity of suspensions,” Powder Technol. 62, 85–94 (1990).
[CrossRef]

C. Acree Riley, M. Louge, “Quantitative capacitance measurements of voidage in gas-solid flows,” Particulate Sci. Technol. 7, 51–59 (1989).
[CrossRef]

H. Chang, M. Louge, in Fluidized Bed Combustion, E. J. Anthony, ed. (American Society of Mechanical Engineers, New York, 1991), Vol. 3, pp. 1215–1218.

Marzocchella, A.

Massimilla, L.

Matsuno, Y.

Y. Matsuno, H. Yamaguchi, T. Oka, H. Kage, K. Higashitani, “The use of optical fiber probes for the measurement of dilute particle concentrations: calibration and application to gas fluidized bed carryover,” Powder Technol. 36, 215–221 (1988).
[CrossRef]

Morishita, K.

M. Horio, K. Morishita, O. Tachibana, N. Murata, “Solid distribution and movement in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 147–154.

Murata, N.

M. Horio, K. Morishita, O. Tachibana, N. Murata, “Solid distribution and movement in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 147–154.

Oka, T.

Y. Matsuno, H. Yamaguchi, T. Oka, H. Kage, K. Higashitani, “The use of optical fiber probes for the measurement of dilute particle concentrations: calibration and application to gas fluidized bed carryover,” Powder Technol. 36, 215–221 (1988).
[CrossRef]

Opie, M.

M. Louge, M. Opie, “Measurements of the effective dielectric permittivity of suspensions,” Powder Technol. 62, 85–94 (1990).
[CrossRef]

Patrose, B.

B. Patrose, H. S. Caram, “Optical fiber probe transit anemometer for particle measurements in fluidized beds,” AIChE J. 28, 604–609 (1982).
[CrossRef]

Probert-Jones, J. R.

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. 1, 822–830 (1984).
[CrossRef]

Rendima, I.

Rensner, D.

E. U. Hartge, D. Rensner, J. Werther, “Faseroptisches Meßsystem zur Erfassung lokaler Feststoff-Konzentrationen und-Geschwindigkeiten in höher konzentrierten Gas/Feststoff-Strömungen,” Synop. 1781/1989, Chem. Ing. Tech. 61, 744–745 (1989); complete manuscript on microfiche MS 1781/89 from Chem. Ing. Tech.
[CrossRef]

E. U. Hartge, D. Rensner, J. Werther, “Solids concentration and velocity patterns in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 165–180.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), p. 354.

Snoek, C. W.

C. W. Snoek, “A selection of new developments in multiphase flow measurement techniques,” Exp. Thermal Fluid Sci. 3, 60–73 (1990).
[CrossRef]

Stromberg, L.

C. Brereton, L. Stromberg, “Some aspects of the fluid dynamic behaviour of fast fluidized beds,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 133–144.

Tachibana, O.

M. Horio, K. Morishita, O. Tachibana, N. Murata, “Solid distribution and movement in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 147–154.

Tsang, L.

L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. 1, 836–839 (1984).
[CrossRef]

Tung, Y.

Y. Tung, J. Li, M. Kwauk, “Radial voidage profiles in a fast fluidized bed,” in Fluidization ‘88 Science and Technology, M. Kwauk, D. Kunii, eds. (Science Press, Beijing, 1988), pp. 139–145.

Werther, J.

E. U. Hartge, D. Rensner, J. Werther, “Faseroptisches Meßsystem zur Erfassung lokaler Feststoff-Konzentrationen und-Geschwindigkeiten in höher konzentrierten Gas/Feststoff-Strömungen,” Synop. 1781/1989, Chem. Ing. Tech. 61, 744–745 (1989); complete manuscript on microfiche MS 1781/89 from Chem. Ing. Tech.
[CrossRef]

E. U. Hartge, D. Rensner, J. Werther, “Solids concentration and velocity patterns in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 165–180.

E. U. Hartge, Y. Li, J. Werther, “Analysis of the local structure of the two-phase flow in a fast fluidized bed,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 153–160.

Yamaguchi, H.

Y. Matsuno, H. Yamaguchi, T. Oka, H. Kage, K. Higashitani, “The use of optical fiber probes for the measurement of dilute particle concentrations: calibration and application to gas fluidized bed carryover,” Powder Technol. 36, 215–221 (1988).
[CrossRef]

AIChE J. (1)

B. Patrose, H. S. Caram, “Optical fiber probe transit anemometer for particle measurements in fluidized beds,” AIChE J. 28, 604–609 (1982).
[CrossRef]

Appl. Opt. (1)

Exp. Thermal Fluid Sci. (1)

C. W. Snoek, “A selection of new developments in multiphase flow measurement techniques,” Exp. Thermal Fluid Sci. 3, 60–73 (1990).
[CrossRef]

J. Opt. Soc. Am. (3)

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. 1, 822–830 (1984).
[CrossRef]

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. 1, 831–835 (1984).
[CrossRef]

L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. 1, 836–839 (1984).
[CrossRef]

Particulate Sci. Technol. (1)

C. Acree Riley, M. Louge, “Quantitative capacitance measurements of voidage in gas-solid flows,” Particulate Sci. Technol. 7, 51–59 (1989).
[CrossRef]

Powder Technol. (2)

M. Louge, M. Opie, “Measurements of the effective dielectric permittivity of suspensions,” Powder Technol. 62, 85–94 (1990).
[CrossRef]

Y. Matsuno, H. Yamaguchi, T. Oka, H. Kage, K. Higashitani, “The use of optical fiber probes for the measurement of dilute particle concentrations: calibration and application to gas fluidized bed carryover,” Powder Technol. 36, 215–221 (1988).
[CrossRef]

Synop. 1781/1989, Chem. Ing. Tech. (1)

E. U. Hartge, D. Rensner, J. Werther, “Faseroptisches Meßsystem zur Erfassung lokaler Feststoff-Konzentrationen und-Geschwindigkeiten in höher konzentrierten Gas/Feststoff-Strömungen,” Synop. 1781/1989, Chem. Ing. Tech. 61, 744–745 (1989); complete manuscript on microfiche MS 1781/89 from Chem. Ing. Tech.
[CrossRef]

Other (10)

Y. Tung, J. Li, M. Kwauk, “Radial voidage profiles in a fast fluidized bed,” in Fluidization ‘88 Science and Technology, M. Kwauk, D. Kunii, eds. (Science Press, Beijing, 1988), pp. 139–145.

M. Horio, K. Morishita, O. Tachibana, N. Murata, “Solid distribution and movement in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 147–154.

E. U. Hartge, Y. Li, J. Werther, “Analysis of the local structure of the two-phase flow in a fast fluidized bed,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 153–160.

E. U. Hartge, D. Rensner, J. Werther, “Solids concentration and velocity patterns in circulating fluidized beds,” in Circulating Fluidized Bed Technology II, P. Basu, J. F. Large, eds. (Pergamon, New York, 1988), pp. 165–180.

H. Chang, M. Louge, in Fluidized Bed Combustion, E. J. Anthony, ed. (American Society of Mechanical Engineers, New York, 1991), Vol. 3, pp. 1215–1218.

D. Deirmendjian, Electromagnetic Scttering on Spherical Polydispersions (American Elsevier, New York, 1969), p. 10.

M. Kerker, The Scattering of Light (Academic, New York, 1969), p. 106–107.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981), p. 66.

C. Brereton, L. Stromberg, “Some aspects of the fluid dynamic behaviour of fast fluidized beds,” in Circulating Fluidized Bed Technology, P. Basu, ed. (Pergamon, New York, 1985), pp. 133–144.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), p. 354.

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Figures (7)

Fig. 1
Fig. 1

Simulated placement of the particle matrix in front of the optical fiber. Dimensions are not to scale. The darker square domain represents one of the concentric regions used in the estimation of the sensor’s measurement volume.

Fig. 2
Fig. 2

Schematic of the calibration setup. Dimensions are not to scale. The semitoroidal measurement volume shown is that of the capacitance probe. It penetrates approximately 2 mm into the suspension.

Fig. 3
Fig. 3

Experimental results and predictions of the simulation for an optical fiber with a core diameter of 200 μm and a N.A. of 0.37 in air. The experimental data are converted to F r by assuming that the simulation predicts F r correctly for 70-μm particles at (1 − ∊) = 50%. Crosses and open diamonds represent particles with mean diameters of 70 μm and 210 μm, respectively. These data points are obtained by pouring particles randomly along the probe assembly. Open squares are data for closely packed 210-μm particles with different random particle placements. The filled circles and triangles represent simulation results for 70- and 210-μm particles, respectively. The solid curves are least-squares fits to these results of the form F r = k (1 − ∊) m .

Fig. 4
Fig. 4

Results of the simulation for d p /d f = 1.05 (filled triangles), 0.35 (filled circles), and 0.10 (open squares) for suspensions of glass spheres in air and a fiber that has a N.A. of 0.37. Error bars represent the sample standard deviation of F r for 10 randomly simulated particle placements. The solids curves are least-squares fits of the form F r = k(1 − ∊) m

Fig. 5
Fig. 5

Exponent m and pre-exponential factor k versus d p /d f for a suspension of glass spheres in air with a fiber that has a N.A. of 0.37. The solid curves are visual fits to the data.

Fig. 6
Fig. 6

Simulated values of F r versus (1 − ∊) for d p /d f = 0.35. Filled circles and squares are for suspensions of glass spheres in air with fibers that have a N.A. of 0.37 and 0.28, respectively. Open circles are for water suspensions with a fiber that has a N.A. of 0.37.

Fig. 7
Fig. 7

Estimates of the measurement volume for an optical fiber that has a N.A. of 0.37 with d p /d f = 0.35. The numbers shown on the contours are particle volume fractions. The abscissa is the radial distance from the fiber’s axis and the ordinate is the height above the fiber’s face. The contours are the location where the number of particle hits per unit volume is one hundredth of its maximum value. Note that contours for 0.2 ≤ (1 − ∊) ≤ 0.5 are nearly identical. The shaded area below the x axis represents the lateral extent of the fiber core.

Equations (10)

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

I = K exp ( - 4 θ 2 / θ 0 2 ) ,
d 4 p = I d ω d A E tot = [ 2 r d r ( d f / 2 ) 2 ] × [ sin θ exp ( - 4 θ 2 / θ 0 2 ) d θ 0 θ 0 sin θ exp ( - 4 θ 2 / θ 0 2 ) d θ ] [ d ψ 2 π ] [ d ϕ 2 π ] ,
r = ( d f / 2 ) ξ 1 ,             ψ = 2 π ξ 2 ,             ϕ = 2 π ξ 3 ,
ξ 4 = 0 θ sin θ exp ( - 4 θ 2 / θ 0 2 ) d θ 0 θ 0 sin θ exp ( - 4 θ 2 / θ 0 2 ) d θ .
R TE = [ n 1 cos α - ( n 2 2 - n 1 2 sin 2 α ) 1 / 2 n 1 cos α + ( n 2 2 - n 1 2 sin 2 α ) 1 / 2 ] 2 ,
R TM = [ n 1 ( n 2 2 - n 1 2 sin 2 α ) 1 / 2 - n 2 2 cos α n 1 ( n 2 2 - n 1 2 sin 2 α ) 1 / 2 + n 2 2 cos α ] 2 ,
C r = F r 1 - F l .
Δ F r F r = F r - F r F r F l - F l 1 - F l F l ,
p [ F r ( N ) - F r ( N ) F r ( N ) δ ] = erf η 2 ,
η δ { N F r ( N ) [ 1 - F r ( N ) ] } 1 / 2 .

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