Abstract

The scattering phase matrices for finite hexagonal cylinders oriented randomly in space were computed by superposing the scattered intensities of Fraunhofer diffracted rays and geometrical optics rays. However, the effect of interference is considered when the optical path lengths for two rays, split by some obstacle and scattered in the same direction, are equal to each other. Single models (c/a = 2.5 and 0.4) for a hexagonal column and a plate, resembling the corresponding crystals in atmospheric clouds, are used in the computations. Our results showed different values for the phase matrix elements, P33 and P44, from those obtained previously by Cai and Liou. The backscattering linear depolarization ratios and the asymmetry factor for hexagonal plates oriented horizontally were then computed. The backscattering linear depolarization ratios exceeded 1.0 at certain orientations. Within the limitation of the use of single-crystal models for a hexagonal column and a plate, the results appear to agree well with most field and laboratory observations.

© 1985 Optical Society of America

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References

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  1. H. Jacobowitz, “A Method for Computing the Transfer of Solar Radiation through Clouds of Hexagonal Ice Crystals,” J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
    [CrossRef]
  2. K. N. Liou, H. Lahore, “Laser Sensing of Cloud Composition: A Backscattered Depolarization Technique,” J. Appl. Meteorol. 13, 257 (1974).
    [CrossRef]
  3. P. Wendling, R. Wendling, H. K. Weickmann, “Scattering of Solar Radiation by Hexagonal Ice Crystals,” Appl. Opt. 18, 2663 (1979).
    [CrossRef] [PubMed]
  4. Q. Cai, K.-N. Liou, “Polarized Light Scattering by Hexagonal Ice Crystals: Theory,” Appl. Opt. 21, 3569 (1982).
    [CrossRef] [PubMed]
  5. K.-N. Liou, Q. Cai, P. W. Barber, S. C. Hill, “Scattering Phase Matrix Comparison for Randomly Hexagonal Cylinders and Spheroids,” Appl. Opt. 22, 1684 (1983).
    [CrossRef] [PubMed]
  6. S. Asano, M. Sato, “Light Scattering by Randomly Oriented Spheroidal Particles,” Appl. Opt. 19, 962 (1980).
    [CrossRef] [PubMed]
  7. V. P. Dugin, S. O. Mirumyants, “The Light Scattering Matrices of Artificial Crystalline Clouds,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 12, 988 (1976).
  8. Y. Takano, M. Tanaka, “Phase Matrix and Cross Sections for Single Scattering by Circular Cylinders: a Comparison of Ray Optics and Wave Theory,” Appl. Opt. 19, 2781 (1980).
    [CrossRef] [PubMed]
  9. V. E. Derr, N. L. Abshire, R. E. Cupp, G. T. McNice, “Depolarization of Lidar Returns from Virga and Source Cloud,” J. Appl. Meteorol. 15, 1200 (1976).
    [CrossRef]
  10. K. Sassen, “Polarization Diversity Lidar Returns from Virga and Precipitation: Anomalies and the Bright Band Analogy,” J. Appl. Meteorol. 15, 292 (1976).
    [CrossRef]
  11. A. I. Carswell, in Clouds, Their Formation, Optical Properties, and Effects, P. V. Hobbs, A. Deepak, Eds. (Academic, New York, 1981), pp. 363–406.
    [CrossRef]
  12. K. Sassen, “Depolarization of Laser Light Backscattered by Artificial Clouds,” J. Appl. Meteorol. 13, 923 (1974).
    [CrossRef]
  13. K. Sassen, “Ice Crystal Habit Discrimination with the Optical Backscatter Depolarization Technique,” J. Appl. Meteorol. 16, 425 (1977).
    [CrossRef]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 44–55.
  15. Y. Takano, S. Asano, “Fraunhofer Diffraction by Ice Crystals Suspended in the Atmosphere,” J. Meteorol. Soc. Jpn. 61, 289 (1983).
  16. R. Greenler, Rainbows, Halos, and Glories (Cambridge U.P., New York, 1980), p. 80.
  17. K. N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: A Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
    [CrossRef]
  18. F. Pattloch, E. Tränkle, “Monte Carlo Simulation and Analysis of Halo Phenomena,” J. Opt. Soc. Am. A 1, 520 (1984).
    [CrossRef]
  19. J. B. Pollack, J. N. Cuzzi, “Scattering by Nonspherical Particles of Size Comparable to a Wavelength: A New Semi-Empirical Theory and Its Application to Tropospheric Aerosols,” J. Atmos. Sci. 37, 868 (1980).
    [CrossRef]
  20. C. M. R. Platt, “The Effect of Cirrus of Varying Optical Depth on the Extraterrestrial Net Radiative Flux,” Q. J. R. Meteorol. Soc. 107, 671 (1981).
    [CrossRef]
  21. K. Sassen, K. N. Liou, “Scattering of Polarized Laser Light by Water Droplet, Mixed-Phase and Ice Crystal Clouds. Part II: Angular Depolarizing and Multiple-Scattering Behavior,” J. Atmos. Sci. 36, 852 (1979).
    [CrossRef]
  22. R. S. McDowell, “Frequency Analysis of the Circumzenithal Arc: Evidence for the Oscillation of Ice-Crystal Plates in the Upper Atmosphere,” J. Opt. Soc. Am. 69, 1119 (1979).
    [CrossRef]
  23. C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some Micro-physical Properties of an Ice Cloud from Lidar Observation of Horizontally Oriented Crystals,” J. Appl. Meteorol. 17, 1220 (1978).
    [CrossRef]
  24. K. Sassen, “Deep Orographic Cloud Structure and Composition Derived from Comprehensive Remote Sensing Measurements,” J. Climate Appl. Meteorol. 23, 568 (1984).
    [CrossRef]
  25. S. Asano, “Transfer of Solar Radiation in Optically Anisotropic Ice Clouds,” J. Meteorol. Soc. Jpn. 61, 402 (1983).

1984 (2)

F. Pattloch, E. Tränkle, “Monte Carlo Simulation and Analysis of Halo Phenomena,” J. Opt. Soc. Am. A 1, 520 (1984).
[CrossRef]

K. Sassen, “Deep Orographic Cloud Structure and Composition Derived from Comprehensive Remote Sensing Measurements,” J. Climate Appl. Meteorol. 23, 568 (1984).
[CrossRef]

1983 (3)

S. Asano, “Transfer of Solar Radiation in Optically Anisotropic Ice Clouds,” J. Meteorol. Soc. Jpn. 61, 402 (1983).

K.-N. Liou, Q. Cai, P. W. Barber, S. C. Hill, “Scattering Phase Matrix Comparison for Randomly Hexagonal Cylinders and Spheroids,” Appl. Opt. 22, 1684 (1983).
[CrossRef] [PubMed]

Y. Takano, S. Asano, “Fraunhofer Diffraction by Ice Crystals Suspended in the Atmosphere,” J. Meteorol. Soc. Jpn. 61, 289 (1983).

1982 (1)

1981 (1)

C. M. R. Platt, “The Effect of Cirrus of Varying Optical Depth on the Extraterrestrial Net Radiative Flux,” Q. J. R. Meteorol. Soc. 107, 671 (1981).
[CrossRef]

1980 (3)

1979 (3)

1978 (1)

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some Micro-physical Properties of an Ice Cloud from Lidar Observation of Horizontally Oriented Crystals,” J. Appl. Meteorol. 17, 1220 (1978).
[CrossRef]

1977 (1)

K. Sassen, “Ice Crystal Habit Discrimination with the Optical Backscatter Depolarization Technique,” J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

1976 (3)

V. P. Dugin, S. O. Mirumyants, “The Light Scattering Matrices of Artificial Crystalline Clouds,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 12, 988 (1976).

V. E. Derr, N. L. Abshire, R. E. Cupp, G. T. McNice, “Depolarization of Lidar Returns from Virga and Source Cloud,” J. Appl. Meteorol. 15, 1200 (1976).
[CrossRef]

K. Sassen, “Polarization Diversity Lidar Returns from Virga and Precipitation: Anomalies and the Bright Band Analogy,” J. Appl. Meteorol. 15, 292 (1976).
[CrossRef]

1974 (2)

K. Sassen, “Depolarization of Laser Light Backscattered by Artificial Clouds,” J. Appl. Meteorol. 13, 923 (1974).
[CrossRef]

K. N. Liou, H. Lahore, “Laser Sensing of Cloud Composition: A Backscattered Depolarization Technique,” J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

1971 (2)

H. Jacobowitz, “A Method for Computing the Transfer of Solar Radiation through Clouds of Hexagonal Ice Crystals,” J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

K. N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: A Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Abshire, N. L.

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some Micro-physical Properties of an Ice Cloud from Lidar Observation of Horizontally Oriented Crystals,” J. Appl. Meteorol. 17, 1220 (1978).
[CrossRef]

V. E. Derr, N. L. Abshire, R. E. Cupp, G. T. McNice, “Depolarization of Lidar Returns from Virga and Source Cloud,” J. Appl. Meteorol. 15, 1200 (1976).
[CrossRef]

Asano, S.

Y. Takano, S. Asano, “Fraunhofer Diffraction by Ice Crystals Suspended in the Atmosphere,” J. Meteorol. Soc. Jpn. 61, 289 (1983).

S. Asano, “Transfer of Solar Radiation in Optically Anisotropic Ice Clouds,” J. Meteorol. Soc. Jpn. 61, 402 (1983).

S. Asano, M. Sato, “Light Scattering by Randomly Oriented Spheroidal Particles,” Appl. Opt. 19, 962 (1980).
[CrossRef] [PubMed]

Barber, P. W.

Cai, Q.

Carswell, A. I.

A. I. Carswell, in Clouds, Their Formation, Optical Properties, and Effects, P. V. Hobbs, A. Deepak, Eds. (Academic, New York, 1981), pp. 363–406.
[CrossRef]

Cupp, R. E.

V. E. Derr, N. L. Abshire, R. E. Cupp, G. T. McNice, “Depolarization of Lidar Returns from Virga and Source Cloud,” J. Appl. Meteorol. 15, 1200 (1976).
[CrossRef]

Cuzzi, J. N.

J. B. Pollack, J. N. Cuzzi, “Scattering by Nonspherical Particles of Size Comparable to a Wavelength: A New Semi-Empirical Theory and Its Application to Tropospheric Aerosols,” J. Atmos. Sci. 37, 868 (1980).
[CrossRef]

Derr, V. E.

V. E. Derr, N. L. Abshire, R. E. Cupp, G. T. McNice, “Depolarization of Lidar Returns from Virga and Source Cloud,” J. Appl. Meteorol. 15, 1200 (1976).
[CrossRef]

Dugin, V. P.

V. P. Dugin, S. O. Mirumyants, “The Light Scattering Matrices of Artificial Crystalline Clouds,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 12, 988 (1976).

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U.P., New York, 1980), p. 80.

Hansen, J. E.

K. N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: A Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Hill, S. C.

Jacobowitz, H.

H. Jacobowitz, “A Method for Computing the Transfer of Solar Radiation through Clouds of Hexagonal Ice Crystals,” J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

Lahore, H.

K. N. Liou, H. Lahore, “Laser Sensing of Cloud Composition: A Backscattered Depolarization Technique,” J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

Liou, K. N.

K. Sassen, K. N. Liou, “Scattering of Polarized Laser Light by Water Droplet, Mixed-Phase and Ice Crystal Clouds. Part II: Angular Depolarizing and Multiple-Scattering Behavior,” J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

K. N. Liou, H. Lahore, “Laser Sensing of Cloud Composition: A Backscattered Depolarization Technique,” J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

K. N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: A Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Liou, K.-N.

McDowell, R. S.

McNice, G. T.

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some Micro-physical Properties of an Ice Cloud from Lidar Observation of Horizontally Oriented Crystals,” J. Appl. Meteorol. 17, 1220 (1978).
[CrossRef]

V. E. Derr, N. L. Abshire, R. E. Cupp, G. T. McNice, “Depolarization of Lidar Returns from Virga and Source Cloud,” J. Appl. Meteorol. 15, 1200 (1976).
[CrossRef]

Mirumyants, S. O.

V. P. Dugin, S. O. Mirumyants, “The Light Scattering Matrices of Artificial Crystalline Clouds,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 12, 988 (1976).

Pattloch, F.

Platt, C. M. R.

C. M. R. Platt, “The Effect of Cirrus of Varying Optical Depth on the Extraterrestrial Net Radiative Flux,” Q. J. R. Meteorol. Soc. 107, 671 (1981).
[CrossRef]

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some Micro-physical Properties of an Ice Cloud from Lidar Observation of Horizontally Oriented Crystals,” J. Appl. Meteorol. 17, 1220 (1978).
[CrossRef]

Pollack, J. B.

J. B. Pollack, J. N. Cuzzi, “Scattering by Nonspherical Particles of Size Comparable to a Wavelength: A New Semi-Empirical Theory and Its Application to Tropospheric Aerosols,” J. Atmos. Sci. 37, 868 (1980).
[CrossRef]

Sassen, K.

K. Sassen, “Deep Orographic Cloud Structure and Composition Derived from Comprehensive Remote Sensing Measurements,” J. Climate Appl. Meteorol. 23, 568 (1984).
[CrossRef]

K. Sassen, K. N. Liou, “Scattering of Polarized Laser Light by Water Droplet, Mixed-Phase and Ice Crystal Clouds. Part II: Angular Depolarizing and Multiple-Scattering Behavior,” J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

K. Sassen, “Ice Crystal Habit Discrimination with the Optical Backscatter Depolarization Technique,” J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

K. Sassen, “Polarization Diversity Lidar Returns from Virga and Precipitation: Anomalies and the Bright Band Analogy,” J. Appl. Meteorol. 15, 292 (1976).
[CrossRef]

K. Sassen, “Depolarization of Laser Light Backscattered by Artificial Clouds,” J. Appl. Meteorol. 13, 923 (1974).
[CrossRef]

Sato, M.

Takano, Y.

Y. Takano, S. Asano, “Fraunhofer Diffraction by Ice Crystals Suspended in the Atmosphere,” J. Meteorol. Soc. Jpn. 61, 289 (1983).

Y. Takano, M. Tanaka, “Phase Matrix and Cross Sections for Single Scattering by Circular Cylinders: a Comparison of Ray Optics and Wave Theory,” Appl. Opt. 19, 2781 (1980).
[CrossRef] [PubMed]

Tanaka, M.

Tränkle, E.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 44–55.

Weickmann, H. K.

Wendling, P.

Wendling, R.

Appl. Opt. (5)

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

V. P. Dugin, S. O. Mirumyants, “The Light Scattering Matrices of Artificial Crystalline Clouds,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 12, 988 (1976).

J. Appl. Meteorol. (6)

V. E. Derr, N. L. Abshire, R. E. Cupp, G. T. McNice, “Depolarization of Lidar Returns from Virga and Source Cloud,” J. Appl. Meteorol. 15, 1200 (1976).
[CrossRef]

K. Sassen, “Polarization Diversity Lidar Returns from Virga and Precipitation: Anomalies and the Bright Band Analogy,” J. Appl. Meteorol. 15, 292 (1976).
[CrossRef]

K. N. Liou, H. Lahore, “Laser Sensing of Cloud Composition: A Backscattered Depolarization Technique,” J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

K. Sassen, “Depolarization of Laser Light Backscattered by Artificial Clouds,” J. Appl. Meteorol. 13, 923 (1974).
[CrossRef]

K. Sassen, “Ice Crystal Habit Discrimination with the Optical Backscatter Depolarization Technique,” J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some Micro-physical Properties of an Ice Cloud from Lidar Observation of Horizontally Oriented Crystals,” J. Appl. Meteorol. 17, 1220 (1978).
[CrossRef]

J. Atmos. Sci. (3)

J. B. Pollack, J. N. Cuzzi, “Scattering by Nonspherical Particles of Size Comparable to a Wavelength: A New Semi-Empirical Theory and Its Application to Tropospheric Aerosols,” J. Atmos. Sci. 37, 868 (1980).
[CrossRef]

K. Sassen, K. N. Liou, “Scattering of Polarized Laser Light by Water Droplet, Mixed-Phase and Ice Crystal Clouds. Part II: Angular Depolarizing and Multiple-Scattering Behavior,” J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

K. N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: A Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

J. Climate Appl. Meteorol. (1)

K. Sassen, “Deep Orographic Cloud Structure and Composition Derived from Comprehensive Remote Sensing Measurements,” J. Climate Appl. Meteorol. 23, 568 (1984).
[CrossRef]

J. Meteorol. Soc. Jpn. (2)

S. Asano, “Transfer of Solar Radiation in Optically Anisotropic Ice Clouds,” J. Meteorol. Soc. Jpn. 61, 402 (1983).

Y. Takano, S. Asano, “Fraunhofer Diffraction by Ice Crystals Suspended in the Atmosphere,” J. Meteorol. Soc. Jpn. 61, 289 (1983).

J. Opt. Soc. Am. (1)

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

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

H. Jacobowitz, “A Method for Computing the Transfer of Solar Radiation through Clouds of Hexagonal Ice Crystals,” J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

C. M. R. Platt, “The Effect of Cirrus of Varying Optical Depth on the Extraterrestrial Net Radiative Flux,” Q. J. R. Meteorol. Soc. 107, 671 (1981).
[CrossRef]

Other (3)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 44–55.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U.P., New York, 1980), p. 80.

A. I. Carswell, in Clouds, Their Formation, Optical Properties, and Effects, P. V. Hobbs, A. Deepak, Eds. (Academic, New York, 1981), pp. 363–406.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

(a) Scattering geometry for an incident light ray and a scattered light ray relative to a hexagonal ice crystal. (b) Scattering geometry for rotation of the coordinate system. All symbols are explained in the text.

Fig. 2
Fig. 2

(a) Schematic figure for the rays scattered at θ = 0,2α,π − 2α, and π in the plane containing the Z axis and the incident direction. (b)–(e) End views of a hexagonal cylinder and its light rays drawn in Fig. 2(a). Dot and circle represent reflection and refraction, respectively. Dot with parentheses denotes possible reflection. The left and right figures of (e), respectively, express a plane crystal and a columnar crystal.

Fig. 3
Fig. 3

Phase function averaged over the angles β and ϕ for randomly oriented hexagonal plates with vertical c axes (2-D plates) at the solar elevation angle of 40°.

Fig. 4
Fig. 4

Scattering phase matrix elements for hexagonal columns oriented randomly in space. The elements at the forward and backward scattering angles are indicated by short horizontal bars.

Fig. 5
Fig. 5

Same as Fig. 4 except for hexagonal plates oriented randomly in space.

Fig. 6
Fig. 6

(a) End view of a hexagonal column and a light ray causing the positive value of P43 at 116° ≲ θ ≲ 144° in Fig. 4. T denotes total reflection. (b) End view of a hexagonal column and a light ray causing the large values of δH,V at 2° ≲ θ ≲ 20° in Fig. 7. The digits 0 and 7 indicate the plane of incidence and plane of emergence, respectively.

Fig. 7
Fig. 7

Linear depolarization ratios δH and δV for 3-D columns and 3-D plates. The adopted parameters are the same as those in Figs. 4 and 5.

Fig. 8
Fig. 8

Geometrical paths of symmetrical spatial skew rays at an orientation of α = 60° and β = 30°: (a) in the case of column (L/2a = 2.5) and n = 4; (b) in the case of plate (L/2a = 0.4) and n = 6.

Fig. 9
Fig. 9

Backscattering linear depolarization ratios δH and δV for randomly oriented hexagonal plates with vertical c axes as a function of the incident light direction α. The lidar tilt angle from the zenith is given by π/2 − α.

Fig. 10
Fig. 10

Parallel-polarized backscattered intensities, M1 and M2, corresponding to Fig. 9 for randomly oriented hexagonal plates with vertical c axes as a function of the incident light direction α. The intensities at α = 0 and 90° are indicated by open circles. The hatched region is the estimated backscattered intensities for the unoriented crystals.23,24

Fig. 11
Fig. 11

Asymmetry factor for randomly oriented hexagonal plates with vertical c axes as a function of the solar elevation angle α. The dashed line ( cos θ ¯ ) R is obtained through the relation mentioned in Table I.

Tables (2)

Tables Icon

Table I Asymmetry Factors Computed from Geometrical Optics and Backscattering Linear Depolarization Ratios for 3-D Columns, 3-D Plates, and Spheresa

Tables Icon

Table II Main Contribution from Symmetrical Spatial Skew Rays to Backscatteringa

Equations (34)

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A = [ sin α 0 cos α 0 1 0 cos α 0 sin α ] [ cos β sin β 0 sin β cos β 0 0 0 1 ] .
[ cos χ n cos ψ n cos ω n ] = [ Φ 1 D 1 t Φ 2 D 2 r Φ n 1 D n 1 r ] * A [ cos α n cos β n cos γ n ] , n 2 .
B i j n = cos β i j n , i , j = 1 3 .
B n = [ Ξ n r , t ] * A .
sin ϕ s n = sin ( α n ρ n ) sin τ n i , t ,
cos ϕ s n = 1 tan β 33 n tan τ n i , t ,
ρ n = sgn ( cos β 32 n ) cos 1 ( cos β 31 n / 1 cos 2 β 33 n ) ,
τ i , t = { τ 1 , i for n = 1 , τ n , t for n 2 .
A ( n ) = { w 1 P s 1 R 1 P 1 , for n = 1 , w n P s n T n P n ( k = n 1 2 R k P k ) T 1 P 1 , for n 2 ,
[ w n ] 2 = { cos τ 1 i sin θ 1 , for n = 1 , cos τ 1 i sin θ n cos τ n t cos τ 1 t cos τ n i cos τ 1 i exp ( 2 k m i l = 1 n 1 d l + 1 , l ) , for n 2 .
k = n 1 2 R k P k
A ( n ) = P t n A ( n ) P - e n .
sin ϕ e n = sin β 33 n sin ( ρ n β ) sin θ n ,
cos ϕ e n = cos β 33 n + sin α cos θ n sin θ n cos α .
ϕ e n = ϕ t n = π / 2 for θ n = 0 and π .
S ( n ) = [ A 2 ( n ) A 3 ( n ) A 4 ( n ) A 1 ( n ) ] + [ A 2 ( n ) A 3 ( n ) A 4 ( n ) A 1 ( n ) ] = 2 [ A 2 ( n ) 0 0 A 1 ( n ) ] .
S ( n ) = [ A 2 ( n ) A 3 ( n ) A 4 ( n ) A 1 ( n ) ] + [ A 2 ( n ) A 4 ( n ) A 3 ( n ) A 1 ( n ) ] = [ 2 A 2 ( n ) A 3 ( n ) A 4 ( n ) [ A 3 ( n ) A 4 ( n ) ] 2 A 1 ( n ) ] .
G ( n ) = [ 1 2 [ M 2 ( n ) + M 3 ( n ) + M 4 ( n ) + M 1 ( n ) ] 1 2 [ M 2 ( n ) M 1 ( n ) ] 0 0 1 2 [ M 2 ( n ) M 1 ( n ) ] 1 2 [ M 2 ( n ) M 3 ( n ) M 4 ( n ) + M 1 ( n ) ] 0 0 0 0 S 12 ( n ) + S 34 ( n ) D 21 ( n ) 0 0 D 21 ( n ) S 12 ( n ) S 34 ( n ) ] ,
G 11 ( n ) = 1 2 [ M 2 ( n ) + M 3 ( n ) + M 4 ( n ) + M 1 ( n ) ] , G 22 ( n ) = G 33 ( n ) = 1 4 [ M 2 ( n ) M 3 ( n ) M 4 ( n ) + M 1 ( n ) ] + 1 2 [ S 12 ( n ) + S 34 ( n ) ] , G 44 ( n ) = S 12 ( n ) S 34 ( n ) , }
G 11 ( n ) = 1 2 [ M 2 ( n ) + 2 M 3 ( n ) + M 1 ( n ) ] , G 22 ( n ) = G 33 ( n ) = 1 4 [ M 1 ( n ) + M 2 ( n ) ] 1 2 S 12 ( n ) , G 44 ( n ) = M 3 ( n ) + S 12 ( n ) . }
G k l ( m , k a , L / 2 a ; α , β ; θ , ϕ ) = δ k l G D ( k a , L / 2 a ; α , β ; θ , ϕ ) + q [ n = 1 N δ ( θ θ n , ϕ ϕ n ) G k l ( n ) ( m , k a , L / 2 a ; α , β ; θ n , ϕ n ) ] , k , l = 1 4 .
G k l β ( m , k a , L / 2 a ; α ; θ ) = 6 π 0 π / 6 G k l ( m , k a , L / 2 a ; α , β ; θ , ϕ ) d β .
P k l ( m ; k a , L / 2 a ; θ ) = 0 π / 2 G k l β ( m ; k a , L / 2 a ; α ; θ ) cos α d α .
4 π P 11 d Ω / 4 π = 1 .
δ H = P 11 P 22 P 11 + 2 P 12 + P 22 ,
δ V = P 11 P 22 P 11 2 P 12 + P 22 .
δ H = n M 3 ( n ) / n M 1 ( n ) , δ V = n M 4 ( n ) / n M 2 ( n ) .
i = 1 n ϕ i + ϕ s n
i = 1 n ϕ i + ϕ s n
M 1 , 2 = n M 1 , 2 ( n )
A ( n ) / w n = P s n T n P n ( k = n 1 2 R k P k ) T 1 P 1 [ U 2 U 3 U 4 U 1 ] .
P * 1 T 1 ( k = 2 n 1 P * k R k ) P * n T n P * s n = [ U 2 U 4 U 3 U 1 ] .
P * n = P n
P 1 T 1 ( k = 2 n 1 P k R k ) P n T n P s n = [ U 2 U 4 U 3 U 1 ] .

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