Abstract

A physical optics approximation based on Fresnel's laws is developed to calculate the intensity of light scattered by a three-axis ellipsoid of any orientation and any refractive index. Some results concerning totally reflecting spheres and dielectric spheroids are presented. An approach suitable for large scatterers is particularly good for small scattering angles. The angular intensities, i 1 and i 2, are then plotted versus θ for large axially oriented ellipsoids of various thicknesses. Theoretical small-angle light-scattering patterns are also presented and discussed. The data from one of them correspond to red cells in a shear flow.

© 1996 Optical Society of America

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  1. See, Scattering by Three-Dimensional Objects, J. Opt. Soc. Am. A 11(4) (1994).
  2. G. Mie, “Beiträge zur Optik trüber Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
    [CrossRef]
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    [PubMed]
  4. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–722 (1979).
    [CrossRef] [PubMed]
  5. F. Möglich, “Beugungserscheinungen an Körper von ellipsoidischer Gestalt,” Ann. Phys. 83, 601–735 (1927).
  6. A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer) wavelength,” J. Appl. Phys. 24, 1134–1142 (1953).
    [CrossRef]
  7. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [CrossRef]
  8. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  9. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  10. B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1993).
    [CrossRef]
  11. V. A. Erma, “An exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. I. Case of cylindrical symmetry,” Phys. Rev. 173, 1243–1257 (1968).
    [CrossRef]
  12. V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. II. General case,” Phys. Rev. 176, 1544–1553 (1968).
    [CrossRef]
  13. V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
    [CrossRef]
  14. P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [CrossRef] [PubMed]
  15. P. W. Barber, D-S. Wang, “Rayleigh–Gans–Debye applicability to scattering by nonspherical particles,” Appl. Opt. 17, 797–803 (1978).
    [CrossRef] [PubMed]
  16. P. Latimer, P. Barber, “Scattering by ellipsoids of revolution. A comparison between theoretical methods,” J. Colloid Interface Sci. 63, 310–316 (1978).
    [CrossRef]
  17. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, p. 172.
  18. G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, R. M. Heethar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. 32, 2266–2272 (1993).
    [CrossRef] [PubMed]
  19. G. J. Streekstra, A. G. Hoekstra, R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt. 33, 7288–7296 (1994).
    [CrossRef] [PubMed]
  20. P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
    [CrossRef]
  21. J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
    [CrossRef]
  22. J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
    [CrossRef]
  23. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, p. 432.
  24. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8, p. 434.
  25. P. Beckman, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968), Chap. 3, p. 76.
  26. E. E. Bell, “Optical constants and their measurement,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1965), Vol. XXV 2a, pp. 1–58.
  27. S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
    [CrossRef]

1994 (2)

1993 (2)

1983 (1)

J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
[CrossRef]

1982 (2)

S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
[CrossRef]

J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

1979 (1)

1978 (2)

P. W. Barber, D-S. Wang, “Rayleigh–Gans–Debye applicability to scattering by nonspherical particles,” Appl. Opt. 17, 797–803 (1978).
[CrossRef] [PubMed]

P. Latimer, P. Barber, “Scattering by ellipsoids of revolution. A comparison between theoretical methods,” J. Colloid Interface Sci. 63, 310–316 (1978).
[CrossRef]

1975 (2)

1973 (1)

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

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

1969 (1)

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
[CrossRef]

1968 (2)

V. A. Erma, “An exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. I. Case of cylindrical symmetry,” Phys. Rev. 173, 1243–1257 (1968).
[CrossRef]

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. II. General case,” Phys. Rev. 176, 1544–1553 (1968).
[CrossRef]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

1953 (1)

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer) wavelength,” J. Appl. Phys. 24, 1134–1142 (1953).
[CrossRef]

1927 (1)

F. Möglich, “Beugungserscheinungen an Körper von ellipsoidischer Gestalt,” Ann. Phys. 83, 601–735 (1927).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Asano, S.

Barber, P.

P. Latimer, P. Barber, “Scattering by ellipsoids of revolution. A comparison between theoretical methods,” J. Colloid Interface Sci. 63, 310–316 (1978).
[CrossRef]

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

Barber, P. W.

P. W. Barber, D-S. Wang, “Rayleigh–Gans–Debye applicability to scattering by nonspherical particles,” Appl. Opt. 17, 797–803 (1978).
[CrossRef] [PubMed]

P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
[CrossRef]

Beckman, P.

P. Beckman, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968), Chap. 3, p. 76.

Bell, E. E.

E. E. Bell, “Optical constants and their measurement,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1965), Vol. XXV 2a, pp. 1–58.

Draine, B. T.

Erma, V. A.

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
[CrossRef]

V. A. Erma, “An exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. I. Case of cylindrical symmetry,” Phys. Rev. 173, 1243–1257 (1968).
[CrossRef]

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. II. General case,” Phys. Rev. 176, 1544–1553 (1968).
[CrossRef]

Flatau, P. J.

Heethaar, R. M.

Heethar, R. M.

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
[CrossRef]

Hoekstra, A. G.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, p. 432.

Keller, S. R.

S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
[CrossRef]

Latimer, P.

P. Latimer, P. Barber, “Scattering by ellipsoids of revolution. A comparison between theoretical methods,” J. Colloid Interface Sci. 63, 310–316 (1978).
[CrossRef]

Mazeron, P.

J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
[CrossRef]

J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Möglich, F.

F. Möglich, “Beugungserscheinungen an Körper von ellipsoidischer Gestalt,” Ann. Phys. 83, 601–735 (1927).

Nijhof, E. J.

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]

Ravey, J. C.

J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
[CrossRef]

J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

Skalak, R.

S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
[CrossRef]

Stevenson, A. F.

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer) wavelength,” J. Appl. Phys. 24, 1134–1142 (1953).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8, p. 434.

Streekstra, G. J.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, p. 172.

Wang, D-S.

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Yamamoto, G.

Yeh, C.

Ann. Phys. (2)

G. Mie, “Beiträge zur Optik trüber Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

F. Möglich, “Beugungserscheinungen an Körper von ellipsoidischer Gestalt,” Ann. Phys. 83, 601–735 (1927).

Appl. Opt. (6)

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. Appl. Phys. (1)

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer) wavelength,” J. Appl. Phys. 24, 1134–1142 (1953).
[CrossRef]

J. Colloid Interface Sci. (1)

P. Latimer, P. Barber, “Scattering by ellipsoids of revolution. A comparison between theoretical methods,” J. Colloid Interface Sci. 63, 310–316 (1978).
[CrossRef]

J. Fluid Mech. (1)

S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
[CrossRef]

J. Opt. (Paris) (2)

J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

J. C. Ravey, P. Mazeron, “Light scattering in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
[CrossRef]

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

Phys. Rev. (3)

V. A. Erma, “An exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. I. Case of cylindrical symmetry,” Phys. Rev. 173, 1243–1257 (1968).
[CrossRef]

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. II. General case,” Phys. Rev. 176, 1544–1553 (1968).
[CrossRef]

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Other (6)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, p. 172.

P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, p. 432.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8, p. 434.

P. Beckman, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968), Chap. 3, p. 76.

E. E. Bell, “Optical constants and their measurement,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1965), Vol. XXV 2a, pp. 1–58.

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Figures (9)

Fig. 1
Fig. 1

Scattering geometry. An incident plane wave with wave vector k i gives rise to a scattered wave with wave vector k, also plane at large distances. S is the surface of the scatterer, n is the unit normal to the surface at dS pointing outward from the enclosed volume.

Fig. 2
Fig. 2

Ellipsoid orientation specified by Euler's angles (nutation ν, precession ψ, rotation φ). Only one half of the scatterer is displayed. Oxyz is the reference frame. Ox′y′z′ are the symmetry axes of the ellipsoid. The line of nodes u, perpendicular to vertical plane υz′z is also the intersection of the horizontal plane Oxy with the symmetry plane Ox′y′ of the scatterer. Precession angle ψ can be related to azimuth angle α (negative here) by ψ = α + π/2.

Fig. 3
Fig. 3

Dependence of the decimal logarithm of the backscattered intensity for perfectly conducting spheres on parameter size α according to Mie's theory and POA. The larger the size of the spheres, the better the agreement.

Fig. 4
Fig. 4

Angular variations of log i 1 for a perfectly conducting sphere with α = 10. The agreement between Mie's theory and POA is good for small and large scattering angles (0° < θ < 20° and θ > 80°).

Fig. 5
Fig. 5

Angular variations of log i 2 for a perfectly conducting sphere with α = 40. The agreement between Mie's theory and POA is excellent for θ to 30°. In the intermediate zone both curves give comparable intensity values but with oscillations out of phase.

Fig. 6
Fig. 6

Oblate spheroid with semiaxes 10, 10, and 10/3. The unpolarized incident light propagates parallel with the short axis of this dielectric spheroid (n = 1.33). The exact results (Asano) and approximate ones (POA) agree in this θ range. For a less favorable orientation of the spheroid, discrepancies arise from the curvature effects.

Fig. 7
Fig. 7

Angular variations of log νν(θ, ϕ) versus θ for dielectric (n = 1.065) ellipsoids with a = 90, c = 30, and b from 18 to 30(2) according to POA. The orientation is still favorable because the axes of the ellipsoids are oriented parallel with the axes of the reference frame, and the incident light propagate parallel with the b axis. The seven upper curves are for ϕ = 90°, and the seven lower curves are for ϕ = 0°. An increase in b results in an increase in the intensity and in a slight shift of the angular positions of the maxima toward smaller θ values.

Fig. 8
Fig. 8

(a) Sketch of a theoretical SALS pattern according to POA for one of the previous ellipsoids (a = 90, b = 18, c = 30, n = 1.065) corresponding to red cells deformed in a shear flow. The orientation and the shape of the central spot are directly related to those of the scatterer. The axis ratio of the central spot depends very little on n and b contrary to the location and shape of possible intensity reinforcements (polar here). (b) Experimental SALS pattern obtained with red cells under shear stress illuminated by the red light of a He–Ne laser.

Fig. 9
Fig. 9

Theoretical SALS patterns according to POA for n = 1.11. The incident light propagates parallel with the b axis. (a) Ellipsoid (a = 90, b = 18, c = 30); (b) Spheroid (a = 90, b = 30, c = 30). The reinforcements are now equatorial. Although the central spot has a smaller size than that in (a), its axis ratio is still ∼3.

Equations (24)

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exp ( ikOP ) 4 π iOP k × S f ( E , B ) exp ( i k r ) d s ,
f ( E , B ) = u ̂ × ( n × B ) n × E
E p = u 4 π × S f ( E , B ) exp ( i k r ) d s ,
E = E 0 exp i ( k r ω t ) ,
( Er ) = ( ) ( Ei ) , ( Et ) = ( ) ( Ei ) ,
x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 .
ν = Oz , Oz , ψ = Ox , Ou , φ = Ou , Ox ,
( M ) = ( ) ( M ) ,
( f 2 f 3 sin φ sin ν f 1 f 4 cos φ sin ν sin ψ sin ν cos ψ sin ν cos ν ) ,
f 1 = cos ν sin ψ cos φ + cos ψ sin φ , f 2 = cos ν sin ψ sin φ cos ψ cos φ , f 3 = cos ν cos ψ sin φ + sin ψ cos φ , f 4 = cos ν cos ψ cos φ sin ψ sin φ .
A x 2 + B y 2 + C z 2 + 2 Dxy + 2 Eyz + 2 Fzx 1 = 0 ,
A = ( a 2 b 2 sin 2 ψ sin 2 ν + b 2 c 2 f 2 2 + c 2 a 2 f 1 2 ) / a 2 b 2 c 2 , B = ( a 2 b 2 cos 2 ψ sin 2 ν + b 2 c 2 f 3 2 + c 2 a 2 f 4 2 ) / a 2 b 2 c 2 , C = [ a 2 b 2 cos 2 ν + c 2 sin 2 ν ( b 2 sin 2 φ + a 2 cos 2 φ ) ] / a 2 b 2 c 2 , D = ( a 2 b 2 sin 2 ν sin ψ cos ψ b 2 c 2 f 3 f 2 c 2 a 2 f 4 f 1 ) / a 2 b 2 c 2 , E = sin ν ( a 2 b 2 cos ν cos ψ + b 2 c 2 f 3 sin φ + c 2 a 2 f 4 cos φ ) / a 2 b 2 c 2 , F = sin ν ( a 2 b 2 cos ν sin ψ b 2 c 2 f 2 sin φ + c 2 a 2 f 1 cos φ ) / a 2 b 2 c 2 .
x 1 = n 1 , z 1 = x 1 × y 1 / | x 1 × y 1 | , y 1 = z 1 × x 1 .
A x 2 + B z 2 + 2 C xz = 1 ,
A = A D 2 / B , B = C E 2 / B , C = F DE / B .
S = π abc B , p 2 = A + B [ ( A B ) 2 + 4 C 2 ] 1 / 2 A + B + [ ( A B ) 2 + 4 C 2 ] 1 / 2 , tan 2 = 2 C A B .
± abc ( BC E 2 ) 1 / 2 , ( DE BF ) x ± [ B ( BC E 2 x 2 / a 2 b 2 c 2 ) ] 1 / 2 BC E 2 .
E υ υ = v ̂ E p ( z ) , E h υ = ĥ E p ( z ) , E υ h = v ̂ E p ( x ) , E hh = ĥ E p ( x ) .
υ υ = | E υ υ | 2 , h υ = | E h υ | 2 , υ h = | E υ h | 2 , hh = | E hh | 2 .
( ) = ( r m r p 0 0 0 r m + K 1 y 2 α s 0 s r m + K 1 z 2 α ) , ( ) = ( t m t p 0 0 0 t m + K 1 y 2 α s 0 s t m + K 1 z 2 α ) ,
r m = D K 21 / D K 32 , r p = P K 21 / P K 32 , t m = D K 31 / D K 32 , t p = P K 31 / P K 32 , α = 2 r m / P K 32 , s = K 1 y K 1 z α ,
D K ji = K jx K ix , P K ji = K jx K ix K 1 2 + K 1 x 2 .
( ) = ( ) , ( ) ( ) but commutation , ( ) ( ) + ( ) 2 = ( 1 ) .
k | sin θ cos ϕ cos θ sin θ sin ϕ h | cos θ / sin θ cos ϕ / 0 v | sin 2 θ sin ϕ cos ϕ / sin θ cos θ sin ϕ / ,

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