Abstract

Scattered light patterns produced by spherical transparent particles of a wide range of diameters (1–100 μm) and for a useful range of forward scattering angles (0–20°) are calculated by using both Lorenz-Mie theory and geometrical optics theory. A detailed comparison of the results leads to a definitive assessment of the accuracy of geometrical optics theory in the forward direction. Emphasis is put on simultaneous sizing and velocimetry of particles using pedestal calibration methods.

© 1981 Optical Society of America

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References

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  1. G. Grehan, G. Gouesbet, “The Computer Program supermidi for Mie Theory Calculations Without Practical Size or Refractive-Index Limitations,” Internal Report TTI/GG/79/03/20.
  2. A. Ungut, “Particle Size and Velocity Measurements by Laser Anemometry,” Ph.D. Thesis, U. Sheffield (1978).
  3. A. Ungut, A. J. Yule, D. S. Taylor, N. A. Chigier, AIAA J. Energy 2, 6 (1978).
  4. A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, J. Energy 1, 4 (1977).
    [CrossRef]
  5. L. Lorenz, Vidensk. Selsk. Skr. 6, No. 6, 1 (1980).
  6. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  7. G. Grehan, G. Gouesbet, Appl. Opt. 18, 3489 (1979).
    [CrossRef] [PubMed]
  8. W. G. Tam, R. Corriveau, J. Opt. Soc. Am. 68, 763 (1978).
    [CrossRef]
  9. N. Morita, T. Tanaka, T. Tomohisa, N. Yoshiro, IEEE Trans. Antennas Propag. AP-16, 724 (1968).
    [CrossRef]
  10. W. Tsai, Pogorzelski, J. Opt. Soc. Am. 65, 1427 (1975).
    [CrossRef]
  11. G. Gouesbet, G. Grehan, “A Formalism to Compute the Scattered Intensities from An Isotropic, Homogeneous, Spherical, Nonmagnetic Particle Located on the Axis of an Axisymmetric Incident Light Profile, Using Bromwich Functions,” submitted to J. Opt. Soc. Am.
  12. M. Kerker, The Scattering of Light and Other Electro-Magnetic Radiation (Academic, New York, 1969).
  13. W. J. Lentz, Appl. Opt. 15, 668 (1976).
    [CrossRef] [PubMed]
  14. G. Grehan, Nouveaux progrès en théorie de Lorenz-Mie. Application à la mesure de diamètres de particules dans les écoulements. Thèse de 3eme cycle, U. Rouen (1980).
  15. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  16. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).
  17. J. R. Hodkinson, J. Greenleaves, J. Opt. Soc. Am., 53, 724 (1963).
    [CrossRef]
  18. A. Ungut, G. Grehan, G. Gouesbet, “A Definitive Assessment of Geometrical Optics Light Scattering Theory in Near Forward Directions with Application to Particle Sizing, Joint Sheffield/Rouen, Internal Report TTI,UGG/80/06/III.
  19. Circe, Bibliothèque, SSP, IBM du Circe (1977).
  20. L. Robin, Fonctions spériques de Legendre et fonctions spéroīdales, Vols, 1, 2, 3 (Gauthier-Villars, Paris, 1957, 1958, 1959).
  21. G. Petiau, La théorie des Fonctions de Bessel (CNRS, Paris, 1953).
  22. W. D. Ross, Appl. Opt. 11, 1919 (1972).
    [CrossRef] [PubMed]
  23. P. Debye, Ann. Phys. 4, No. 30, 57 (1909).
    [CrossRef]
  24. T. Hawksley, B.C.U.R.A. 25, 4, part 1, 105 (1951) : 26, 4, part 2, 117 (1951): 5, part 3, 181 (1952).
  25. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

1980 (1)

L. Lorenz, Vidensk. Selsk. Skr. 6, No. 6, 1 (1980).

1979 (1)

1978 (2)

W. G. Tam, R. Corriveau, J. Opt. Soc. Am. 68, 763 (1978).
[CrossRef]

A. Ungut, A. J. Yule, D. S. Taylor, N. A. Chigier, AIAA J. Energy 2, 6 (1978).

1977 (1)

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, J. Energy 1, 4 (1977).
[CrossRef]

1976 (1)

1975 (1)

1972 (1)

1968 (1)

N. Morita, T. Tanaka, T. Tomohisa, N. Yoshiro, IEEE Trans. Antennas Propag. AP-16, 724 (1968).
[CrossRef]

1963 (1)

J. R. Hodkinson, J. Greenleaves, J. Opt. Soc. Am., 53, 724 (1963).
[CrossRef]

1951 (1)

T. Hawksley, B.C.U.R.A. 25, 4, part 1, 105 (1951) : 26, 4, part 2, 117 (1951): 5, part 3, 181 (1952).

1909 (1)

P. Debye, Ann. Phys. 4, No. 30, 57 (1909).
[CrossRef]

1908 (1)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Atakan, S.

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, J. Energy 1, 4 (1977).
[CrossRef]

Chigier, N. A.

A. Ungut, A. J. Yule, D. S. Taylor, N. A. Chigier, AIAA J. Energy 2, 6 (1978).

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, J. Energy 1, 4 (1977).
[CrossRef]

Corriveau, R.

Debye, P.

P. Debye, Ann. Phys. 4, No. 30, 57 (1909).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Gouesbet, G.

G. Grehan, G. Gouesbet, Appl. Opt. 18, 3489 (1979).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, “The Computer Program supermidi for Mie Theory Calculations Without Practical Size or Refractive-Index Limitations,” Internal Report TTI/GG/79/03/20.

G. Gouesbet, G. Grehan, “A Formalism to Compute the Scattered Intensities from An Isotropic, Homogeneous, Spherical, Nonmagnetic Particle Located on the Axis of an Axisymmetric Incident Light Profile, Using Bromwich Functions,” submitted to J. Opt. Soc. Am.

A. Ungut, G. Grehan, G. Gouesbet, “A Definitive Assessment of Geometrical Optics Light Scattering Theory in Near Forward Directions with Application to Particle Sizing, Joint Sheffield/Rouen, Internal Report TTI,UGG/80/06/III.

Greenleaves, J.

J. R. Hodkinson, J. Greenleaves, J. Opt. Soc. Am., 53, 724 (1963).
[CrossRef]

Grehan, G.

G. Grehan, G. Gouesbet, Appl. Opt. 18, 3489 (1979).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, “The Computer Program supermidi for Mie Theory Calculations Without Practical Size or Refractive-Index Limitations,” Internal Report TTI/GG/79/03/20.

G. Grehan, Nouveaux progrès en théorie de Lorenz-Mie. Application à la mesure de diamètres de particules dans les écoulements. Thèse de 3eme cycle, U. Rouen (1980).

G. Gouesbet, G. Grehan, “A Formalism to Compute the Scattered Intensities from An Isotropic, Homogeneous, Spherical, Nonmagnetic Particle Located on the Axis of an Axisymmetric Incident Light Profile, Using Bromwich Functions,” submitted to J. Opt. Soc. Am.

A. Ungut, G. Grehan, G. Gouesbet, “A Definitive Assessment of Geometrical Optics Light Scattering Theory in Near Forward Directions with Application to Particle Sizing, Joint Sheffield/Rouen, Internal Report TTI,UGG/80/06/III.

Hawksley, T.

T. Hawksley, B.C.U.R.A. 25, 4, part 1, 105 (1951) : 26, 4, part 2, 117 (1951): 5, part 3, 181 (1952).

Hodkinson, J. R.

J. R. Hodkinson, J. Greenleaves, J. Opt. Soc. Am., 53, 724 (1963).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electro-Magnetic Radiation (Academic, New York, 1969).

Lentz, W. J.

Lorenz, L.

L. Lorenz, Vidensk. Selsk. Skr. 6, No. 6, 1 (1980).

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Morita, N.

N. Morita, T. Tanaka, T. Tomohisa, N. Yoshiro, IEEE Trans. Antennas Propag. AP-16, 724 (1968).
[CrossRef]

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

Petiau, G.

G. Petiau, La théorie des Fonctions de Bessel (CNRS, Paris, 1953).

Pogorzelski,

Robin, L.

L. Robin, Fonctions spériques de Legendre et fonctions spéroīdales, Vols, 1, 2, 3 (Gauthier-Villars, Paris, 1957, 1958, 1959).

Ross, W. D.

Tam, W. G.

Tanaka, T.

N. Morita, T. Tanaka, T. Tomohisa, N. Yoshiro, IEEE Trans. Antennas Propag. AP-16, 724 (1968).
[CrossRef]

Taylor, D. S.

A. Ungut, A. J. Yule, D. S. Taylor, N. A. Chigier, AIAA J. Energy 2, 6 (1978).

Tomohisa, T.

N. Morita, T. Tanaka, T. Tomohisa, N. Yoshiro, IEEE Trans. Antennas Propag. AP-16, 724 (1968).
[CrossRef]

Tsai, W.

Ungut, A.

A. Ungut, A. J. Yule, D. S. Taylor, N. A. Chigier, AIAA J. Energy 2, 6 (1978).

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, J. Energy 1, 4 (1977).
[CrossRef]

A. Ungut, “Particle Size and Velocity Measurements by Laser Anemometry,” Ph.D. Thesis, U. Sheffield (1978).

A. Ungut, G. Grehan, G. Gouesbet, “A Definitive Assessment of Geometrical Optics Light Scattering Theory in Near Forward Directions with Application to Particle Sizing, Joint Sheffield/Rouen, Internal Report TTI,UGG/80/06/III.

van de Hulst, H. C.

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

Yoshiro, N.

N. Morita, T. Tanaka, T. Tomohisa, N. Yoshiro, IEEE Trans. Antennas Propag. AP-16, 724 (1968).
[CrossRef]

Yule, A. J.

A. Ungut, A. J. Yule, D. S. Taylor, N. A. Chigier, AIAA J. Energy 2, 6 (1978).

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, J. Energy 1, 4 (1977).
[CrossRef]

AIAA J. Energy (1)

A. Ungut, A. J. Yule, D. S. Taylor, N. A. Chigier, AIAA J. Energy 2, 6 (1978).

Ann. Phys. (2)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

P. Debye, Ann. Phys. 4, No. 30, 57 (1909).
[CrossRef]

Appl. Opt. (3)

B.C.U.R.A. (1)

T. Hawksley, B.C.U.R.A. 25, 4, part 1, 105 (1951) : 26, 4, part 2, 117 (1951): 5, part 3, 181 (1952).

IEEE Trans. Antennas Propag. (1)

N. Morita, T. Tanaka, T. Tomohisa, N. Yoshiro, IEEE Trans. Antennas Propag. AP-16, 724 (1968).
[CrossRef]

J. Energy (1)

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, J. Energy 1, 4 (1977).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. R. Hodkinson, J. Greenleaves, J. Opt. Soc. Am., 53, 724 (1963).
[CrossRef]

Vidensk. Selsk. Skr. (1)

L. Lorenz, Vidensk. Selsk. Skr. 6, No. 6, 1 (1980).

Other (12)

G. Grehan, G. Gouesbet, “The Computer Program supermidi for Mie Theory Calculations Without Practical Size or Refractive-Index Limitations,” Internal Report TTI/GG/79/03/20.

A. Ungut, “Particle Size and Velocity Measurements by Laser Anemometry,” Ph.D. Thesis, U. Sheffield (1978).

G. Gouesbet, G. Grehan, “A Formalism to Compute the Scattered Intensities from An Isotropic, Homogeneous, Spherical, Nonmagnetic Particle Located on the Axis of an Axisymmetric Incident Light Profile, Using Bromwich Functions,” submitted to J. Opt. Soc. Am.

M. Kerker, The Scattering of Light and Other Electro-Magnetic Radiation (Academic, New York, 1969).

A. Ungut, G. Grehan, G. Gouesbet, “A Definitive Assessment of Geometrical Optics Light Scattering Theory in Near Forward Directions with Application to Particle Sizing, Joint Sheffield/Rouen, Internal Report TTI,UGG/80/06/III.

Circe, Bibliothèque, SSP, IBM du Circe (1977).

L. Robin, Fonctions spériques de Legendre et fonctions spéroīdales, Vols, 1, 2, 3 (Gauthier-Villars, Paris, 1957, 1958, 1959).

G. Petiau, La théorie des Fonctions de Bessel (CNRS, Paris, 1953).

G. Grehan, Nouveaux progrès en théorie de Lorenz-Mie. Application à la mesure de diamètres de particules dans les écoulements. Thèse de 3eme cycle, U. Rouen (1980).

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

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

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Figures (18)

Fig. 1
Fig. 1

Scattering geometry, coordinate systems.

Fig. 2
Fig. 2

Geometrical optics components of the scattered light.

Fig. 3
Fig. 3

Comparison of geometrical optics results with Lorenz-Mie calculations for perfectly spherical particles; 1.0-μm diam.

Fig. 4
Fig. 4

As Fig. 3; 2.0-μm diam.

Fig. 5
Fig. 5

As Fig. 3; 4.0-μm diam.

Fig. 6
Fig. 6

As Fig. 3; 6.0-μm diam.

Fig. 7
Fig. 7

As Fig. 3; 9.0-μ diam.

Fig. 8
Fig. 8

As Fig. 3; 10.0-μm diam.

Fig. 9
Fig. 9

As Fig. 3; 15.0-μm diam.

Fig. 10
Fig. 10

As Fig. 3; 25.0-μm diam.

Fig. 11
Fig. 11

As Fig. 3; 50.0-μm diam.

Fig. 12
Fig. 12

As Fig. 3; 100.0-μm diam.

Fig. 13
Fig. 13

Comparison of geometrical optics results with Lorenz-Mie calculations for a small spread of particle diameters. Δd ≃ λ (average solutions); 1.0-μm diam.

Fig. 14
Fig. 14

As Fig. 13; 5.0-μm.

Fig. 15
Fig. 15

As Fig. 13; 10.0-μm diam.

Fig. 16
Fig. 16

As Fig. 13; 25.0-μm diam.

Fig. 17
Fig. 17

As Fig. 13; 50.0-μm diam.

Fig. 18
Fig. 18

As Fig. 13; 100.0-μm diam.

Equations (77)

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m = n ( 1 - i k ) ,
I θ = λ 2 4 π 2 r 2 i 2 cos 2 ϕ ,
I ϕ = λ 2 4 π 2 r 2 i 1 sin 2 ϕ .
i j = S j 2 .
S 1 = n = 1 2 n + 1 n ( n + 1 ) [ a n π n ( cos θ ) + b n τ n ( cos θ ) ] ,
S 2 = n = 1 2 n + 1 n ( n + 1 ) [ a n τ n ( cos θ ) + b n π n ( cos θ ) ] .
π n ( cos θ ) = P n 1 ( cos θ ) / sin θ ,
τ n ( cos θ ) = d d θ P n 1 ( cos θ ) ,
a n = ψ n ( α ) ψ n ( μ ) - m ψ n ( μ ) ψ n ( α ) ξ n ( α ) ψ n ( μ ) - m ψ n ( μ ) ξ n ( α ) ,
b n = m ψ n ( α ) ψ n ( μ ) - ψ n ( μ ) ψ n ( α ) m ξ n ( α ) ψ n ( μ ) - ψ n ( μ ) ξ n ( α ) ,
α = π d / λ ,
μ = m α .
ψ n ( z ) = π z 2 J n + 1 / 2 ( z ) ,
ξ n ( z ) = ψ n ( z ) + i χ n ( z ) ,
χ n ( z ) = ( - 1 ) n π z 2 J - n - 1 / 2 ( z )
S D 1 ( θ ) = S D 2 ( θ ) = k 2 d 2 4 · J 1 [ k d sin ( θ ) / 2 ] k d sin ( θ ) / 2 .
S 1 , 2 ( θ ) = p , q k d 2 E 1 , 2 [ sin ( 2 ) 2 sin θ d θ d ] 1 / 2 · exp [ i δ + i π 2 ( p + 1 - ½ q - ½ s - 2 l ] ,
m = sin / sin β .
θ = π - 2 - ρ π + 2 p β ,
θ = 2 π l + q θ ,
d θ d = 2 p tan β tan - 2.
E 1 , 2 = γ 1 , 2 for p = 0 ,
E 1 , 2 = ( 1 - γ 1 , 2 2 ) ( - γ 1 , 2 ) p - 1 for p > 0 ,
γ 1 = ( cos - m cos β ) / ( cos + m cos β ) ,
γ 2 = ( m cos - cos β ) / ( m cos + cos β ) ,
I θ , ϕ = S 2 , 1 ( θ ) · S * 2 , 1 ( θ ) { cos 2 ϕ sin 2 ϕ } ( 1 / k 2 r 2 ) .
S 2 , 1 ( θ ) · S * 2 , 1 ( θ ) = A D 2 , 1 2 + A R 2 , 1 2 + A X 2 , 1 2 + 2 A D 2 , 1 A R 2 , 1 · sin δ R - 2 A D 2 , 1 A X 2 , 1 sin δ X - 2 A R 2 , 1 A X 2 , 1 cos ( δ R - δ X ) ,
A R 1 , 2 = E R 1 , 2 ( d / 2 ) · ( sin 2 2 sin θ | d θ d | ) 1 / 2 · k A X 1 , 2 = E X 1 , 2 ( d / 2 ) · ( sin 2 2 sin θ | d θ d | ) 1 / 2 · k
I av = 1 2 λ d - λ d + λ I ( d ) d ( d )
π n ( cos θ ) = 2 cos θ π n - 1 ( cos θ ) - π n - 2 ( cos θ ) + [ cos θ π n - 1 ( cos θ ) - π n - 2 ( cos θ ) ] 1 n - 1 .
π 1 ( cos θ ) = 1             π 2 ( cos θ ) = 3 cos θ .
τ n ( cos θ ) = n cos θ π n ( cos θ ) - ( n + 1 ) π n - 1 ( cos θ ) .
a n = ψ n ( α ) A n ( μ ) - m ψ n ( α ) ξ n ( α ) A n ( μ ) - m ξ n ( α ) ,
b n = m ψ n ( α ) A n ( μ ) - ψ n ( α ) m ξ n ( α ) A n ( μ ) - ξ n ( α ) ,
A n ( μ ) = ψ n ( μ ) ψ n ( μ ) .
ψ n ( μ ) ψ n ( μ ) = - n μ + J n - 1 / 2 ( μ ) J n + 1 / 2 ( μ ) .
J n - 1 / 2 ( μ ) J n + 1 / 2 ( μ ) = 2 ( n + 1 / 2 ) μ - 1 + 1 - 2 ( n + 3 / 2 ) μ - 1 + 1 2 ( n + 5 / 2 ) μ - 1 + .
a 1 + 1 a 2 + 1 a 3 + .
a 1 + 1 a 2 + 1 a 3 +
a m = ( - 1 ) m + 1 2 ( n + m - 1 / 2 ) μ - 1 .
a 1 , a 2 , , a p = a 1 + 1 a 2 + + 1 a p .
J n - 1 / 2 ( μ ) J n + 1 / 2 ( μ ) = a 1 a q - 1 , , a 1 a q , , a 1 a 2 a q - 1 , , a 2 a q , , a 2 ,
ψ n + 1 ( α ) = 2 n α ψ n ( α ) - ψ n - 1 ( α ) ,
ψ n ( α ) = - n α ψ n ( α ) + ψ n - 1 ( α ) ,
ψ 0 ( α ) = sin α ,
ψ 1 ( α ) = sin α α - cos α .
ξ 0 ( α ) = sin α + i cos α ,
ξ 1 ( α ) = ( sin α α - cos α ) + i ( cos α α + sin α ) .
ψ n + 1 ( α ) = ψ n ( α ) · [ ψ n + 1 ( α ) ψ n ( α ) ] .
ψ n + 1 ( a ) ψ n ( α ) = J n + 3 / 2 ( α ) J n + 1 / 2 ( α ) ,
N = integer part of ( α ) .
N = integer part of ( m α ) .
N = 1 , 2 [ integer part of ( α ) ] + 9.
Min [ Re ( σ n ) , Re ( b n ) , Im ( a n ) · Im ( b n ) ] < 10 - 30 ,
p = 0 ; q = + 1 ; l = 0 ; s = - 1
θ = θ = π - 2 .
| d θ d | = 2 ,
sin ( 2 ) sin θ = 1.
δ x = k . d . sin θ / 2 ,
E x 1 = { sin ( θ / 2 ) - [ m 2 - 1 + sin 2 ( θ / 2 ) ] 1 / 2 } / { sin ( θ / 2 ) + [ m 2 - 1 + sin 2 ( θ / 2 ) ] 1 / 2 } ,
E x 2 = { m 2 sin ( θ / 2 ) - [ m 2 - 1 + sin 2 ( θ / 2 ) ] 1 / 2 } / { m 2 sin ( θ / 2 ) + [ m 2 - 1 + sin 2 ( θ / 2 ) ] 1 / 2 } .
p = 1 ; q = - 1 ; l = 0 ; s = - 1 ,
θ = - θ = 2 β - 2 .
sin = m sin ( θ / 2 ) / [ m 2 + 1 - 2 m cos ( θ / 2 ) ] 1 / 2 , sin β = sin ( θ / 2 ) / [ m 2 + 1 - 2 m cos ( θ / 2 ) ] 1 / 2 ,
sin ( 2 ) 2 sin θ = ( m / 2 ) [ m cos ( θ / 2 ) - 1 ] / { [ m 2 + 1 - 2 m cos ( θ / 2 ) ] cos ( θ / 2 ) } ,
| d θ d | = 2 [ m 2 + 1 - 2 m cos ( θ / 2 ) ] / [ m ( m - cos ( θ / 2 ) ] ,
δ R = - k . d . [ m 2 + 1 - 2 m cos ( θ / 2 ) ] 1 / 2 .
E R 1 = 4 m [ m cos ( θ / 2 ) - 1 ] [ m - cos ( θ / 2 ) ] / ( m 2 - 1 ) 2 ,
E R 2 = E R 1 / cos 2 ( θ / 2 ) .
e θ = ( - i / k r ) S 2 ( θ ) cos ϕ exp ( - i k r + i ω t ) ,
e ϕ = ( - i / k r ) S 1 ( θ ) sin ϕ exp ( - i k r + i ω t ) ,
S 2 ( θ ) = S D 2 ( θ ) + S R 2 ( θ ) + S X 2 ( θ ) ,
S 1 ( θ ) = S D 1 ( θ ) + S R 1 ( θ ) + S X 1 ( θ ) ,
S D 1 , 2 = A D 1 , 2 ,
S R 1 , 2 = - i A R 1 , 2 exp ( i δ R ) ,
S X 1 , 2 = i A X 1 , 2 exp ( i δ x ) .
I θ , ϕ = e θ , ϕ · e θ , ϕ * ,

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