Abstract

A method for evaluating the intensity, degree of polarization, direction of polarization, and ellipticity of the scattered radiation emerging from a plane-parallel atmosphere containing large spherical particles is described. In this method, all the elements of the normalized phase matrix are represented by fourier series whose maximum required number of terms depend upon the zenith angles of the directions of the incident and scattered radiation. Some results are presented for an atmospheric model containing water spheres with size parameter 10.0 to show that this method can be used to evaluate reliably all the characteristics of the emergent radiation in a reasonable amount of computer time.

© 1970 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).
  2. K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).
  3. J. V. Dave, R. M. Warten, “Program for Computing the Stokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere,” (Rep. 320-3248, IBM Scientific Center, Palo Alto, California, 1968).
  4. B. M. Herman, J. Geophys. Res. 70, 1215 (1965).
    [CrossRef]
  5. B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
    [CrossRef]
  6. B. M. Herman, Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences (IBM Data Processing Division, White Plains, New York, 1967), pp. 211–237.
  7. G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
    [CrossRef] [PubMed]
  8. J. V. Dave, J. Gazdag, Appl. Opt. 9, 1457 (1970).
    [CrossRef] [PubMed]
  9. J. V. Dave, Appl. Opt. 9, 1888 (1970).
    [PubMed]
  10. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  11. Z. Sekera, “Investigation of Polarization of Skylight,” Final Report, Contract #AF19(122)-239, Department of Meteorology, University of California at Los Angeles, Los Angeles (1955).
  12. J. W. Hovenier, J. Atmos. Sci. 26, 488 (1969).
    [CrossRef]
  13. K. D. Abhyankar, A. L. Fymat, J. Math. Phys. 10, 1935 (1969).
    [CrossRef]
  14. J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Radiative Transfer 10, 557 (1970).
    [CrossRef]
  15. W. M. Irvine, J. Quant. Spectroso. Radiative Transfer 8, 471 (1968).
    [CrossRef]
  16. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier Publishing Company, Inc., New York, 1969).
  17. R. S. Fraser, “Scattering Properties of Atmospheric Aerosols,” Scientific Report No. 2, Contract #AF19 (604)-2429, Department of Meteorology, University of California at Los Angeles, Los Angeles (1959).

1970 (3)

J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Radiative Transfer 10, 557 (1970).
[CrossRef]

J. V. Dave, J. Gazdag, Appl. Opt. 9, 1457 (1970).
[CrossRef] [PubMed]

J. V. Dave, Appl. Opt. 9, 1888 (1970).
[PubMed]

1969 (2)

J. W. Hovenier, J. Atmos. Sci. 26, 488 (1969).
[CrossRef]

K. D. Abhyankar, A. L. Fymat, J. Math. Phys. 10, 1935 (1969).
[CrossRef]

1968 (2)

W. M. Irvine, J. Quant. Spectroso. Radiative Transfer 8, 471 (1968).
[CrossRef]

G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
[CrossRef] [PubMed]

1965 (2)

B. M. Herman, J. Geophys. Res. 70, 1215 (1965).
[CrossRef]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
[CrossRef]

Abhyankar, K. D.

K. D. Abhyankar, A. L. Fymat, J. Math. Phys. 10, 1935 (1969).
[CrossRef]

Armstrong, B. H.

J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Radiative Transfer 10, 557 (1970).
[CrossRef]

Browning, S. R.

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).

Coulson, K. L.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).

Dave, J. V.

J. V. Dave, Appl. Opt. 9, 1888 (1970).
[PubMed]

J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Radiative Transfer 10, 557 (1970).
[CrossRef]

J. V. Dave, J. Gazdag, Appl. Opt. 9, 1457 (1970).
[CrossRef] [PubMed]

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).

J. V. Dave, R. M. Warten, “Program for Computing the Stokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere,” (Rep. 320-3248, IBM Scientific Center, Palo Alto, California, 1968).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier Publishing Company, Inc., New York, 1969).

Fraser, R. S.

R. S. Fraser, “Scattering Properties of Atmospheric Aerosols,” Scientific Report No. 2, Contract #AF19 (604)-2429, Department of Meteorology, University of California at Los Angeles, Los Angeles (1959).

Fymat, A. L.

K. D. Abhyankar, A. L. Fymat, J. Math. Phys. 10, 1935 (1969).
[CrossRef]

Gazdag, J.

Herman, B. M.

B. M. Herman, J. Geophys. Res. 70, 1215 (1965).
[CrossRef]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
[CrossRef]

B. M. Herman, Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences (IBM Data Processing Division, White Plains, New York, 1967), pp. 211–237.

Hovenier, J. W.

J. W. Hovenier, J. Atmos. Sci. 26, 488 (1969).
[CrossRef]

Irvine, W. M.

W. M. Irvine, J. Quant. Spectroso. Radiative Transfer 8, 471 (1968).
[CrossRef]

Kattawar, G. W.

Plass, G. N.

Sekera, Z.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).

Z. Sekera, “Investigation of Polarization of Skylight,” Final Report, Contract #AF19(122)-239, Department of Meteorology, University of California at Los Angeles, Los Angeles (1955).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Warten, R. M.

J. V. Dave, R. M. Warten, “Program for Computing the Stokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere,” (Rep. 320-3248, IBM Scientific Center, Palo Alto, California, 1968).

Appl. Opt. (3)

J. Atmos. Sci. (2)

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
[CrossRef]

J. W. Hovenier, J. Atmos. Sci. 26, 488 (1969).
[CrossRef]

J. Geophys. Res. (1)

B. M. Herman, J. Geophys. Res. 70, 1215 (1965).
[CrossRef]

J. Math. Phys. (1)

K. D. Abhyankar, A. L. Fymat, J. Math. Phys. 10, 1935 (1969).
[CrossRef]

J. Quant. Spectrosc. Radiative Transfer (1)

J. V. Dave, B. H. Armstrong, J. Quant. Spectrosc. Radiative Transfer 10, 557 (1970).
[CrossRef]

J. Quant. Spectroso. Radiative Transfer (1)

W. M. Irvine, J. Quant. Spectroso. Radiative Transfer 8, 471 (1968).
[CrossRef]

Other (8)

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier Publishing Company, Inc., New York, 1969).

R. S. Fraser, “Scattering Properties of Atmospheric Aerosols,” Scientific Report No. 2, Contract #AF19 (604)-2429, Department of Meteorology, University of California at Los Angeles, Los Angeles (1959).

B. M. Herman, Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences (IBM Data Processing Division, White Plains, New York, 1967), pp. 211–237.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Z. Sekera, “Investigation of Polarization of Skylight,” Final Report, Contract #AF19(122)-239, Department of Meteorology, University of California at Los Angeles, Los Angeles (1955).

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).

J. V. Dave, R. M. Warten, “Program for Computing the Stokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere,” (Rep. 320-3248, IBM Scientific Center, Palo Alto, California, 1968).

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Figures (15)

Fig. 1
Fig. 1

Variations of the number of iterations, and CPU time for computations for a given frequency as a function of frequency n. Iterative procedure used: Gauss-Seidel.

Fig. 2
Fig. 2

Variations of the elements of the normalized scattering matrix as a function of the scattering angle; x = 10.0, m = 1.342.

Fig. 3
Fig. 3

Variations of the degree of polarization of the radiation scattered by a sphere with size parameter 10.0, m = 1.342. Incident radiation, unpolarized.

Fig. 4
Fig. 4

Variations of the intensity of the radiation diffusely reflected and transmitted by a plane-parallel, homogeneous atmosphere containing spherical monodispersion; x = 10.0, m = 1.342, θ0 = 0°.

Fig. 5
Fig. 5

Same as Fig. 4 but for the degree of polarization.

Fig. 6
Fig. 6

Variations as a function of nadir angle θ, of the intensity of the radiation diffusely reflected by a plane-parallel homogeneous atmosphere containing spherical nonodispersion; x = 10.0, m = 1.342, θ0 = 60°.

Fig. 7
Fig. 7

Same as Fig. 6 but for the degree of polarization.

Fig. 8
Fig. 8

Variations as a function of zenith angle θ, of the intensity of the radiation diffusely transmitted by a plane-parallel homogeneous atmosphere containing spherical monodispersion; x = 10.0, m = 1.342, θ0 = 60°.

Fig. 9
Fig. 9

Same as Fig. 8 but for the degree of polarization.

Fig. 10
Fig. 10

Variations as a function of the azimuth angle from the sun’s meridian plane φ0φ, of the intensity of the radiation diffusely reflected by a plane-parallel homogeneous atmosphere containing spherical monodispersion, x = 10.0, m = 1.342, θ0 = θ = 60° (antisolar almucantar).

Fig. 11
Fig. 11

Same as Fig. 10 but for the degree of polarization and the direction of polarization.

Fig. 12
Fig. 12

Same as Fig. 10 but for the ellipticity of polarization.

Fig. 13
Fig. 13

Variations as a function of the azimuth angle from the sun’s meridian plane φ0φ, of the intensity of the radiation diffusely transmitted by a plane-parallel homogeneous atmosphere containing spherical monodispersion; x = 10.0, m = 1.342, θ0 = θ = 60° (solar almucantar).

Fig. 14
Fig. 14

Same as Fig. 13 but for the degree of polarization and the direction of polarization.

Fig. 15
Fig. 15

Same as Fig. 13 but for the ellipticity of polarization.

Equations (55)

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I = ( I e , I r , I u , I v ) .
I = I e + I r ,
P = [ ( I e - I r ) 2 + I u 2 + I v 2 ] 1 2 / ( I e + I r ) .
tan 2 χ = I u / ( I e - I τ ) ,
tan β = b / a = - I v / { [ ( I e - I r ) 2 + I u 2 + I v 2 ] 1 2 + [ ( I e - I r ) 2 + I u 2 ] 1 2 } .
F ( cos Θ ) = | M 2 0 0 0 0 M 1 0 0 0 0 S 21 - D 21 0 0 D 21 S 21 | .
M [ μ , μ , ( φ - φ ) ] = L ( π - i 2 ) F ( cos Θ ) L ( - i 1 ) ,
L ( i ) = | cos 2 i sin 2 i 1 2 sin 2 i 0 sin 2 i cos 2 i - 1 2 sin 2 i 0 - sin 2 i sin 2 i cos 2 i 0 0 0 0 1 | .
cos Θ = μ μ + ( 1 + μ 2 ) 1 2 ( 1 - μ 2 ) 1 2 cos ( φ - φ ) ,
sin i 1 sin i 2 - cos i 1 cos i 2 cos Θ = ( 1 - μ 2 ) 1 2 ( 1 - μ 2 ) 1 2 + μ μ cos ( φ - φ ) ,
sin i 1 sin i 2 cos Θ - cos i 1 cos i 2 = cos ( φ - φ ) ,
sin i 1 cos i 2 cos Θ + cos i 1 sin i 2 = μ sin ( φ - φ ) ,
sin i 2 cos i 1 cos Θ + cos i 2 sin i 1 = μ sin ( φ - φ ) .
M 2 = R ( 1 ) + cos 2 Θ R ( 2 ) + 2 cos Θ R ( 3 ) , M 1 = R ( 2 ) + cos 2 Θ R ( 1 ) + 2 cos Θ R ( 3 ) , S 21 = cos Θ ( R ( 1 ) + R ( 2 ) ) + ( 1 + cos 2 Θ ) R ( 3 ) , and D 21 = ( 1 - cos 2 Θ ) R ( 4 ) . }
M i j [ μ , μ , ( φ - φ ) ] = n = 1 N ( μ , μ ) M i j ( n ) ( μ , μ ) cos ( n - 1 ) ( φ - φ ) ,
M i j [ μ , μ , ( φ - φ ) ] = n = 1 N ( μ , μ ) M i j ( n ) ( μ , μ ) sin ( n - 1 ) ( φ - φ ) .
M 11 ( n ) ( μ , μ ) = 1 2 F n ( 1 ) ( μ , μ ) + [ γ 2 ( μ , μ ) + 1 2 μ 2 μ 2 ] F n ( 2 ) ( μ , μ ) + μ μ F n ( 3 ) ( μ , μ ) + 2 γ ( μ , μ ) [ μ μ F n ( 2 ) ( μ , μ , 1 c ) + F n ( 3 ) ( μ , μ , 1 c ) ] + 1 2 F n ( 1 ) ( μ , μ , 2 c ) + 1 2 μ 2 μ 2 F n ( 2 ) ( μ , μ , 2 c ) + μ μ F n ( 3 ) ( μ , μ , 2 c ) ,
M 12 ( n ) ( μ , μ ) = 1 2 μ 2 F n ( 1 ) ( μ , μ ) + 1 2 μ 2 F n ( 2 ) ( μ , μ ) + μ μ F n ( 3 ) ( μ , μ ) - 1 2 μ 2 F n ( 1 ) ( μ , μ , 2 c ) - 1 2 μ 2 F n ( 2 ) ( μ , μ , 2 c ) - μ μ F n ( 3 ) ( μ , μ , 2 c ) ,
M 13 ( n ) ( μ , μ ) = γ ( μ , μ ) [ μ F n ( 2 ) ( μ , μ , 1 s ) + μ F n ( 3 ) ( μ , μ , 1 s ) ] + 1 2 μ [ F n ( 1 ) ( μ , μ , 2 s ) + μ 2 F n ( 2 ) ( μ , μ , 2 s ) ] + 1 2 μ ( 1 + μ 2 ) F n ( 3 ) ( μ , μ , 2 s ) ,
M 14 ( n ) ( μ , μ ) = μ γ ( μ , μ ) F n ( 4 ) ( μ , μ , 1 s ) - 1 2 μ ( 1 - μ 2 ) F n ( 4 ) ( μ , μ , 2 s ) ,
M 21 ( n ) ( μ , μ ) = M 12 ( n ) ( μ , μ ) ,
M 22 ( n ) ( μ , μ ) = [ γ 2 ( μ , μ ) + 1 2 μ 2 μ 2 ] F n ( 1 ) ( μ , μ ) + 1 2 F n ( 2 ) ( μ , μ ) + μ μ F n ( 3 ) ( μ , μ ) + 2 γ ( μ , μ ) [ μ μ F n ( 1 ) ( μ , μ , 1 c ) + F n ( 3 ) ( μ , μ , 1 c ) ] + 1 2 μ 2 μ 2 F n ( 1 ) ( μ , μ , 2 c ) + 1 2 F n ( 2 ) ( μ , μ , 2 c ) + μ μ F n ( 3 ) ( μ , μ , 2 c ) ,
M 23 ( n ) ( μ , μ ) = - γ ( μ , μ ) [ μ F n ( 1 ) ( μ , μ , 1 s ) + μ F n ( 3 ) ( μ , μ , 1 s ) ] - 1 2 μ [ μ 2 F n ( 1 ) ( μ , μ , 2 s ) + F n ( 2 ) ( μ , μ , 2 s ) ] - 1 2 μ ( 1 + μ 2 ) F n ( 3 ) ( μ , μ , 2 s ) ,
M 24 ( n ) ( μ , μ ) = - M 14 ( n ) ( μ , μ ) ,
M 31 ( n ) ( μ , μ ) = - 2 M 13 ( n ) ( μ , μ ) ,
M 32 ( n ) ( μ , μ ) = - 2 M 23 ( n ) ( μ , μ ) ,
M 33 ( n ) ( μ , μ ) = 3 2 γ 2 ( μ , μ ) F n ( 3 ) ( μ , μ ) + γ ( μ , μ ) [ F n ( 1 ) ( μ , μ , 1 c ) + F n ( 2 ) ( μ , μ , 1 c ) ] + 2 μ μ γ ( μ , μ ) F n ( 3 ) ( μ , μ , 1 c ) + μ μ [ F n ( 1 ) ( μ , μ , 2 c ) + F n ( 2 ) ( μ , μ , 2 c ) ] + 1 2 ( 1 + μ 2 ) ( 1 + μ 2 ) F n ( 3 ) ( μ , μ , 2 c ) ,
M 34 ( n ) ( μ , μ ) = [ γ 2 ( μ , μ ) - 1 2 ( 1 + μ 2 ) ( 1 - μ 2 ) ] F n ( 4 ) ( μ , μ ) + 2 μ μ γ ( μ , μ ) F n ( 4 ) ( μ , μ , 1 c ) - 1 2 ( 1 - μ 2 ) ( 1 + μ 2 ) F n ( 4 ) ( μ , μ , 2 c ) ,
M 41 ( n ) ( μ , μ ) = 2 M 14 ( n ) ( μ , μ ) ,
M 42 ( n ) ( μ , μ ) = - M 41 ( n ) ( μ , μ ) ,
M 43 ( n ) ( μ , μ ) = - M 34 ( n ) ( μ , μ ) ,
M 44 ( n ) ( μ , μ ) = μ μ [ F n ( 1 ) ( μ , μ ) + F n ( 2 ) ( μ , μ ) ] + [ γ 2 ( μ , μ ) + 1 2 ( 1 + μ 2 ) ( 1 + μ 2 ) ( 1 + μ 2 ) ] F n ( 3 ) ( μ , μ ) + γ ( μ , μ ) [ F n ( 1 ) ( μ , μ , 1 c ) + F n ( 2 ) ( μ , μ , 1 c ) ] + 2 μ μ γ ( μ , μ ) F n ( 3 ) ( μ , μ , 1 c ) + 1 2 γ 2 ( μ , μ ) F n ( 3 ) ( μ , μ , 2 c ) .
and             M i j ( n ) ( - μ , - μ ) = M j i ( n ) ( μ , μ ) M i j ( n ) ( μ , - μ ) = M j i ( n ) ( - μ , μ ) } ,
and             M i j ( n ) ( - μ , - μ ) = - M j i ( n ) ( μ , μ ) M i j ( n ) ( μ , - μ ) = - M j i ( n ) ( - μ , μ ) } .
μ d I ( τ ; μ , φ ) d τ = I ( τ , μ , φ ) - J ( τ ; μ , φ ) ,
J ( τ ; μ , φ ) = 1 4 e - τ / μ 0 M [ μ , - μ 0 , ( φ 0 - φ ) ] · F + 1 4 π - 1 + 1 0 2 π M [ μ , μ , ( φ - φ ) ] · I ( τ ; μ , φ ) d μ d φ .
F = ( 1 2 , 1 2 , 0 , 0 ) F ,
and             I ( 0 ; - μ , φ ) 0 I ( τ 1 ; + μ , φ ) 0 } ,
0 2 π cos n ( φ 0 - φ ) cos m ( φ - φ ) d φ = { 0 if n m π cos n ( φ 0 - φ ) if n = m 0 2 π if n = m = 0 ,
0 2 π cos n ( φ 0 - φ ) sin m ( φ - φ ) d φ = { 0 if n m π sin n ( φ 0 - φ ) if n = m ,
0 2 π sin n ( φ 0 - φ ) sin m ( φ - φ ) d φ = { 0 if n = m - π cos n ( φ 0 - φ ) if n = m 0.
0 1 [ I e ( n ) ( τ 1 ; - μ , - μ 0 ) + I r ( n ) ( τ 1 ; - μ , - μ 0 ) ] μ d μ
M 1 = K S 1 S 2 * , M 2 = K S 2 S 2 * , S 21 = K ( S 2 S 1 * + S 1 S 2 * ) / 2 , and D 21 = i K ( S 2 S 1 * - S 1 S 2 * ) / 2. } .
and             S 1 = T 2 + cos Θ T 1 , S 2 = T 1 + cos Θ T 2 } .
R ( 1 ) = K T 1 T 1 * R ( 2 ) = K T 2 T 2 * , R ( 3 ) = K ( T 2 T 1 * + T 1 T 2 * ) / 2 and R ( 4 ) = i K ( T 1 T 2 * - T 2 T 1 * ) / 2 } .
R ( j ) ( cos Θ ) = k = 1 N Λ k ( j ) P k - 1 ( cos Θ ) ,
C k ( x , m ) = ( 1 - 2 k ) ( 1 + 2 k ) b k ( x , m ) / ( k + 1 ) + ( 2 k - 1 ) i = 1 { [ p - 1 + ( p + 1 ) - 1 ] ( 2 k + 2 i - 1 ) i a p ( x , m ) - [ ( p + 1 ) - 1 + ( p + 2 ) - 1 ] ( k + i ) ( 2 i + 1 ) b p + 1 ( x , m ) } ,
D k ( x , m ) = ( 1 - 2 k ) ( 1 + 2 k ) a k ( x , m ) / ( k + 1 ) + ( 2 k - 1 ) i = 1 { [ p - 1 + ( p + 1 ) - 1 ] ( 2 k + 2 i - 1 ) i b p ( x , m ) - [ ( p + 1 ) - 1 + ( p + 2 ) - 1 ] ( k + i ) ( 2 i + 1 ) a p + 1 ( x , m ) } ,
R ( j ) ( cos Θ ) = n = 1 N ( μ , μ ) F n ( j ) ( μ , μ ) cos ( n - 1 ) ( φ - φ ) ,
F n ( j ) ( μ , μ ) = ( 2 - δ 1 n ) k = n N Λ k ( j ) Y k - 1 n - 1 ( μ ) Y k - 1 n - 1 ( μ ) .
R ( j ) ( cos Θ ) cos q ( φ - φ ) = - n = 1 N ( μ , μ ) + q F n ( j ) ( μ , μ , q c ) × cos ( n - 1 ) ( φ - φ ) ,
F 1 ( j ) ( μ , μ , 1 c ) = 1 2 F 2 ( j ) ( μ , μ ) , F 2 ( j ) ( μ , μ , 1 c ) = F 1 ( j ) ( μ , μ ) + 1 2 F 3 ( j ) ( μ , μ ) , F 1 ( j ) ( μ , μ , 2 c ) = 1 2 F 3 ( j ) ( μ , μ ) , F 2 ( j ) ( μ , μ , 2 c ) = 1 2 [ F 2 ( j ) ( μ , μ ) + F 4 ( j ) ( μ , μ ) ] , and F 3 ( j ) ( μ , μ , 2 c ) = F 1 ( j ) ( μ , μ ) + 1 2 F 5 ( j ) ( μ , μ ) . } .
R ( j ) ( cos Θ ) sin q ( φ - φ ) = n = 1 N ( μ , μ ) + q F n ( j ) ( μ , μ , q s ) sin ( n - 1 ) × ( φ - φ ) ,
F 1 ( j ) ( μ , μ , 1 s ) = 0 , F 2 ( j ) ( μ , μ , 1 s ) = F 1 ( j ) ( μ , μ ) - 1 2 F 3 ( j ) ( μ , μ ) , F 1 ( j ) ( μ , μ , 2 s ) = 0 , F 2 ( j ) ( μ , μ , 2 s ) = 1 2 [ F 2 ( j ) ( μ , μ ) - F 4 ( j ) ( μ , μ ) ] , and F 3 ( j ) ( μ , μ , 2 s ) = F 1 ( j ) ( μ , μ ) - 1 2 F 5 ( j ) ( μ , μ ) . }
F n ( j ) [ μ , μ , q ( c / s ) ] = 1 2 [ F n - q ( j ) ( μ , μ ) ± F n + q ( j ) ( μ , μ ) ] ,

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