Abstract

On the basis of our previous work on the extension of the geometrical-optics approximation to Gaussian beam scattering by a spherical particle, we present a further extension of the method to the scattering of a transparent or absorbing spheroidal particle with the same symmetric axis as the incident beam. As was done for the spherical particle, the phase shifts of the emerging rays due to focal lines, optical path, and total reflection are carefully considered. The angular position of the geometric rainbow of primary order is theoretically predicted. Compared with our results, the Möbius prediction of the rainbow angle has a discrepancy of less than 0.5° for a spheroidal droplet of aspect radio κ within 0.95 and 1.05 and less than 2° for κ within 0.89 and 1.11. The flux ratio index F, which qualitatively indicates the effect of a surface wave, is also studied and found to be dependent on the size, refractive index, and surface curvature of the particle.

© 2006 Optical Society of America

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References

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    [PubMed]
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    [CrossRef] [PubMed]
  3. N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
    [CrossRef]
  4. Y. P. Han and Z. S. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
    [CrossRef]
  5. J. P. Barton, "Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination," Appl. Opt. 34, 5542-5551 (1995).
    [CrossRef] [PubMed]
  6. L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K.-Y. Tan, "Computations of spheroidal harmonics with complex arguments: a review with an algorithm," Phys. Rev. 58, 6792-6806 (1998).
  7. H. M. Al-Rizzo and J. M. Tranquilla, "Electromagnetic wave scattering by highly elongated and geometrically composite objects of large size parameters: the generalized multipole technique," Appl. Opt. 34, 3502-3521 (1995).
    [CrossRef] [PubMed]
  8. J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
    [CrossRef]
  9. T. Wriedt, "A review of elastic light scattering theories," Part. Part. Syst. Charact. 15, 67-74 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. J. A. Lock, "Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects," Appl. Opt. 35, 515-531 (1996).
    [CrossRef] [PubMed]
  13. J. D. Walker, "Multiple rainbows from single drops of water and other liquids," Am. J. Phys. 44, 421-433 (1976).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. F. Xu, K. F. Ren, and X. Cai, "Extended geometrical-optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle," Appl. Opt. 45, 4990-4999 (2006).
    [CrossRef] [PubMed]
  18. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, 1976).
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    [CrossRef] [PubMed]
  20. J. P. Chevaillier, J. Fabre, G. Gréhan, and G. Gouesbet, "Comparison of diffraction theory and generalized Lorenz-Mie theory for a sphere located on axis of a laser beam," Appl. Opt. 29, 1293-1298 (1990).
    [CrossRef] [PubMed]
  21. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  22. D. Marcuse, "Light scattering from elliptical fibers," Appl. Opt. 13, 1903-1905 (1974).
    [CrossRef] [PubMed]
  23. A. Ghatak, Optics (McGraw-Hill, 1977).
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    [CrossRef] [PubMed]
  25. W. Möbius, "Zur Theorie des Regenbogens und ihrer experimentellen Prüfung," Ann. Phys. (Leipzig) 33, 1493-1558 (1910).
    [CrossRef]
  26. D. Yildiz, J. P. A. J. van Beeck, and M. L. Riethmuller, "Global rainbow thermometry applied to a flashing two-phase R134-A jet," in Proceedings of the Eleventh International Symposium on Application of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 2002).
    [PubMed]
  27. J. P. A. J. van Beeck and M. L. Riethmuller, "Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity," Appl. Opt. 35, 2259-2266 (1996).
    [CrossRef] [PubMed]

2006 (1)

2004 (1)

2001 (1)

1998 (3)

C. L. Adler, J. A. Lock, and B. R. Stone, "Rainbow scattering by a cylinder with a nearly elliptical cross section," Appl. Opt. 37, 1540-1550 (1998).
[CrossRef]

T. Wriedt, "A review of elastic light scattering theories," Part. Part. Syst. Charact. 15, 67-74 (1998).
[CrossRef]

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K.-Y. Tan, "Computations of spheroidal harmonics with complex arguments: a review with an algorithm," Phys. Rev. 58, 6792-6806 (1998).

1997 (1)

1996 (3)

1995 (2)

1993 (1)

N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
[CrossRef]

1991 (1)

1990 (1)

1988 (1)

J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

1986 (1)

1983 (1)

1979 (1)

1976 (1)

J. D. Walker, "Multiple rainbows from single drops of water and other liquids," Am. J. Phys. 44, 421-433 (1976).
[CrossRef]

1975 (1)

1974 (1)

1910 (1)

W. Möbius, "Zur Theorie des Regenbogens und ihrer experimentellen Prüfung," Ann. Phys. (Leipzig) 33, 1493-1558 (1910).
[CrossRef]

Adler, C. L.

Al-Rizzo, H. M.

Asano, S.

Barton, J. P.

Cai, X.

Chevaillier, J. P.

Fabre, J.

Farafonov, V. G.

N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
[CrossRef]

Fraster, A. B.

Ghatak, A.

A. Ghatak, Optics (McGraw-Hill, 1977).

Gouesbet, G.

Gréhan, G.

Hamelin, P.

Han, Y. P.

Hovenac, E. A.

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, 1976).

Kooi, P. S.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K.-Y. Tan, "Computations of spheroidal harmonics with complex arguments: a review with an algorithm," Phys. Rev. 58, 6792-6806 (1998).

Leong, M. S.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K.-Y. Tan, "Computations of spheroidal harmonics with complex arguments: a review with an algorithm," Phys. Rev. 58, 6792-6806 (1998).

Li, L. W.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K.-Y. Tan, "Computations of spheroidal harmonics with complex arguments: a review with an algorithm," Phys. Rev. 58, 6792-6806 (1998).

Lock, J. A.

Marcuse, D.

Möbius, W.

W. Möbius, "Zur Theorie des Regenbogens und ihrer experimentellen Prüfung," Ann. Phys. (Leipzig) 33, 1493-1558 (1910).
[CrossRef]

Peden, I. C.

J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

Ren, K. F.

Riethmuller, M. L.

J. P. A. J. van Beeck and M. L. Riethmuller, "Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity," Appl. Opt. 35, 2259-2266 (1996).
[CrossRef] [PubMed]

D. Yildiz, J. P. A. J. van Beeck, and M. L. Riethmuller, "Global rainbow thermometry applied to a flashing two-phase R134-A jet," in Proceedings of the Eleventh International Symposium on Application of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 2002).
[PubMed]

Schneider, J. B.

J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

Stone, B. R.

Tan, K.-Y.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K.-Y. Tan, "Computations of spheroidal harmonics with complex arguments: a review with an algorithm," Phys. Rev. 58, 6792-6806 (1998).

Tranquilla, J. M.

van Beeck, J. P. A. J.

J. P. A. J. van Beeck and M. L. Riethmuller, "Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity," Appl. Opt. 35, 2259-2266 (1996).
[CrossRef] [PubMed]

D. Yildiz, J. P. A. J. van Beeck, and M. L. Riethmuller, "Global rainbow thermometry applied to a flashing two-phase R134-A jet," in Proceedings of the Eleventh International Symposium on Application of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 2002).
[PubMed]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Voshchinnikov, N. V.

N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
[CrossRef]

Walker, J. D.

J. D. Walker, "Multiple rainbows from single drops of water and other liquids," Am. J. Phys. 44, 421-433 (1976).
[CrossRef]

Wriedt, T.

T. Wriedt, "A review of elastic light scattering theories," Part. Part. Syst. Charact. 15, 67-74 (1998).
[CrossRef]

Wu, Z. S.

Xu, F.

Yamamoto, G.

Yeo, T. S.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K.-Y. Tan, "Computations of spheroidal harmonics with complex arguments: a review with an algorithm," Phys. Rev. 58, 6792-6806 (1998).

Yildiz, D.

D. Yildiz, J. P. A. J. van Beeck, and M. L. Riethmuller, "Global rainbow thermometry applied to a flashing two-phase R134-A jet," in Proceedings of the Eleventh International Symposium on Application of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 2002).
[PubMed]

Am. J. Phys. (1)

J. D. Walker, "Multiple rainbows from single drops of water and other liquids," Am. J. Phys. 44, 421-433 (1976).
[CrossRef]

Ann. Phys. (1)

W. Möbius, "Zur Theorie des Regenbogens und ihrer experimentellen Prüfung," Ann. Phys. (Leipzig) 33, 1493-1558 (1910).
[CrossRef]

Appl. Opt. (15)

J. P. Chevaillier, J. Fabre, G. Gréhan, and G. Gouesbet, "Comparison of diffraction theory and generalized Lorenz-Mie theory for a sphere located on axis of a laser beam," Appl. Opt. 29, 1293-1298 (1990).
[CrossRef] [PubMed]

C. L. Adler, J. A. Lock, and B. R. Stone, "Rainbow scattering by a cylinder with a nearly elliptical cross section," Appl. Opt. 37, 1540-1550 (1998).
[CrossRef]

Y. P. Han and Z. S. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

J. A. Lock, "Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects," Appl. Opt. 35, 515-531 (1996).
[CrossRef] [PubMed]

F. Xu, X. Cai, and K. F. Ren, "Geometrical-optics approximation of forward scattering by coated particles," Appl. Opt. 43, 1870-1879 (2004).
[CrossRef] [PubMed]

F. Xu, K. F. Ren, and X. Cai, "Extended geometrical-optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle," Appl. Opt. 45, 4990-4999 (2006).
[CrossRef] [PubMed]

D. Marcuse, "Light scattering from elliptical fibers," Appl. Opt. 13, 1903-1905 (1974).
[CrossRef] [PubMed]

S. Asano and G. Yamamoto, "Light scattering by a spheroidal particle," Appl. Opt. 14, 29-49 (1975).
[PubMed]

S. Asano, "Light scattering properties of spheroidal particles," Appl. Opt. 18, 712-722 (1979).
[CrossRef] [PubMed]

J. P. Chevaillier, J. Fabre, and P. Hamelin, "Forward scattered light intensities by a sphere located anywhere in a Gaussian beam," Appl. Opt. 25, 1222-1225 (1986).
[CrossRef] [PubMed]

E. A. Hovenac, "Calculation of far-field scattering from nonspherical particles using a geometrical optics approach," Appl. Opt. 30, 4739-4746 (1991).
[CrossRef] [PubMed]

H. M. Al-Rizzo and J. M. Tranquilla, "Electromagnetic wave scattering by highly elongated and geometrically composite objects of large size parameters: the generalized multipole technique," Appl. Opt. 34, 3502-3521 (1995).
[CrossRef] [PubMed]

J. P. Barton, "Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination," Appl. Opt. 34, 5542-5551 (1995).
[CrossRef] [PubMed]

J. A. Lock, "Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection," Appl. Opt. 35, 500-514 (1996).
[CrossRef] [PubMed]

J. P. A. J. van Beeck and M. L. Riethmuller, "Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity," Appl. Opt. 35, 2259-2266 (1996).
[CrossRef] [PubMed]

Astrophys. Space Sci. (1)

N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Part. Part. Syst. Charact. (1)

T. Wriedt, "A review of elastic light scattering theories," Part. Part. Syst. Charact. 15, 67-74 (1998).
[CrossRef]

Phys. Rev. (1)

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K.-Y. Tan, "Computations of spheroidal harmonics with complex arguments: a review with an algorithm," Phys. Rev. 58, 6792-6806 (1998).

Other (4)

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, 1976).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

A. Ghatak, Optics (McGraw-Hill, 1977).

D. Yildiz, J. P. A. J. van Beeck, and M. L. Riethmuller, "Global rainbow thermometry applied to a flashing two-phase R134-A jet," in Proceedings of the Eleventh International Symposium on Application of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 2002).
[PubMed]

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Figures (15)

Fig. 1
Fig. 1

Schema of ray tracing in a homogeneous spheroid that is located on the z axis of the incident Gaussian beam. Rotation axis of the spheroid is on the z axis. (a) In a prolate spheroid, (b) in an oblate spheroid.

Fig. 2
Fig. 2

Scheme for the calculation of phase shifts.

Fig. 3
Fig. 3

Comparison of the scattering intensity calculated by GLMT for a sphere and by the EGOA for a spherical prolate (κ = 1.0005) and oblate (κ = 0.9995) spheroid. The projection radius of the spheroid R is equal to the radius of the sphere (R = a = 100 μm). The particle of refractive index m = 1.33 is located at the center of a Gaussian beam of waist radius w 0 = 50 μm and wavelength λ = 0.6328 μm. The curves of the EGOA have been, respectively, offset by the factors of 102, 104, and 106 for clarity.

Fig. 4
Fig. 4

Same parameters as Fig. 3, but with m = 1.33 + 0.001i.

Fig. 5
Fig. 5

Forward-scattering intensity of a spheroidal droplet of projection radius R = b = 10 μm and aspect ratio κ=1.5 illuminated by a plane wave. The scattering pattern is the same as that given by Hovenac in Fig. 8 of Ref. 10.

Fig. 6
Fig. 6

Scattering intensity calculated by the EGOA for an oblate droplet of projection radius R = 100 μm, different aspect ratios, and located at the center of a Gaussian beam of w 0 = 50 μm and λ = 0.6328 μm. The results of the cases κ = 0.8, 0.95, and 0.99 have been, respectively, offset by the factors of 102, 104, and 106 for clarity.

Fig. 7
Fig. 7

Scattering intensity calculated by the EGOA for a prolate droplet of projection radius R = 100 μm, different aspect ratios, and located at the center of a Gaussian beam of w 0 = 50 μm and λ = 0.6328 μm. The results of the cases κ = 1.05, 1.2, and 2.0 have been, respectively, offset by the factors of 102, 104, and 106 for clarity.

Fig. 8
Fig. 8

Impact factor as a function of aspect ratio calculated for a spheroid of projection radius R = 100 μm illuminated by a Gaussian beam of waist radius w 0 = 50 μm (dashed line) and a quasi-plane wave of w 0 = 10 cm (solid curve).

Fig. 9
Fig. 9

Primary-order rainbow position versus aspect ratio, predicted by the EGOA method for a spheroidal droplet of projection radius R = 100 μm and illuminated by the Gaussian beam of waist radii w 0 = 25, 75, 200, and 3000 μm, respectively.

Fig. 10
Fig. 10

Ray tracing in three prolate spheroids with the same projection radius of R = 100 μm, but with aspect ratio κ = 1.35, κ = 1.5, and κ = 1.65, respectively. The three incident rays are parallel to the z axis.

Fig. 11
Fig. 11

Deviation angle θ p of the emergent rays ( p = 2) from three prolate spheroids with the same projection radius of R = 100 μm, but with aspect ratios κ = 1.35, κ = 1.5, and κ = 1.65, respectively (2000 equidistant incident rays are used).

Fig. 12
Fig. 12

Intensity of the incident ray associated with the rainbow position θrg in Fig. 9.

Fig. 13
Fig. 13

Comparison of primary-order rainbow position (versus aspect ratio) predicted by the EGOA method and the Möbius formula when a spheroidal droplet of projection radius R = 100 μm is illuminated by a plane wave.

Fig. 14
Fig. 14

Geometric rainbow position versus refractive index for a prolate spheroid illuminated by a plane wave. The results indicated by the curves are calculated by the EGOA and those indicated by symbols are obtained by the Möbius formula.

Fig. 15
Fig. 15

Geometric rainbow position θrg versus particle position d for a prolate droplet of aspect ratio κ = 1.1 and projection radius R = 100 μm and illuminated by a Gaussian beam of waist radius w 0 = 100 μm.

Equations (26)

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S j = S d + p = 0 S j , p ,
I j = I 0 ( k r ) 2 i j ( θ ) = I 0 ( k r ) 2 | S j ( θ ) | 2 ,
ϕ p = π 2 + ϕ p , PH + ϕ p , FL + ϕ G .
ϕ 0 , PH = k ( R O P ¯ H S 0 ¯ + O P R 0 ¯ S 0 H 0 ¯ ) .
ϕ 1 , PH = k ( R O P ¯ H S 0 ¯ ) k m r S 0 S 1 ¯ + k ( O P R 1 ¯ S 1 H 1 ¯ ) .
ϕ p , PH = k ( R O P ¯ H S 0 ¯ ) k m r L p + k ( O P R p ¯ S p H p ¯ ) ,
O P R p ¯ S p H p ¯ = z p 2 + y p 2 H p R p ¯ 2 = z p 2 + y p 2 ( y p z p tan θ p ) 2 tan 2 θ p + 1 .
R O P ¯ H S 0 ¯ = z 0     2 + y 0     2 ( y 0 z 0 tan β ) 2 tan 2 β + 1 ,
ϕ G = ϕ H S 0 ϕ P Q = ϕ H S 0 [ k ( d + a ) ϕ i ] = k [ a z 0     2 + y 0     2 ( y 0 z 0 tan β ) 2 tan 2 β + 1 ] [ k ( d + a ) ϕ i ] .
ϕ 0 , PH = k z 0     2 + y 0     2 ( y 0 z 0 tan β ) 2 tan 2 β + 1 + k z 0     2 + y 0     2 ( y 0 z 0 tan θ 0 ) 2 tan 2 θ 0 + 1 ,
ϕ p , PH = z 0 2 + y 0 2 ( y 0 z 0 tan β ) 2 tan 2 β + 1 k m r L p + k [ z p 2 + y p 2 ( y p z p tan θ p ) 2 tan 2 θ p + 1 ] ,
L p = 2 p a cos θ r ,
k z 0 2 + y 0 2 ( y 0 z 0 tan β ) 2 tan 2 β + 1 = k a cos θ i ,
k z p 2 + y p 2 ( y p z p tan θ p ) 2 tan 2 θ p + 1 = k a cos θ i .
Δ ϕ p , T , 1 = 2 tan 1 [ ( sin 2 θ i , p 1 / m r 2 ) 1 / 2 cos θ i , p ] ,            
Δ ϕ p , T , 2 = 2 tan 1 [ m r     2 ( sin 2 θ i , p 1 / m r 2 ) 1 / 2 cos θ i , p ] .    
ϕ p , j = π 2 + ϕ G + ϕ p , PH + ϕ p , FL + Δ ϕ p , T , j .
r 1 , p = cos θ i , p m r cos θ r , p cos θ i , p + m r cos θ r , p ,
r 2 , p = m r cos θ i , p cos θ r , p m r cos θ i , p + cos θ r , p ,
ε j , p = { r j , p for p = 0 ( 1 r j , 0           2 ) 1 / 2 ( 1 r j , p 2 ) 1 / 2 Π n = 1 p 1 ( r j , n ) for p 1 .
D G = cos θ i sin τ sin θ p | d θ p d τ | ,
τ = tan 1 ( y 0 z 0 ) .
| d θ p d τ | = | θ p , l + 1 θ p , l 1 | [ r l - 1 2 + r l + 1 2 2 r l 1 r l + 1 cos ( τ l + 1 τ l 1 ) r l 2 ] l / 2 ,
r l = ( y 0 , l 2 + z 0 , l 2 ) 1 / 2 .
ξ p = k m i L p .
S j , p = k R | S G | ε j , p D G       1 / 2 exp ( ξ p ) exp ( i ϕ p ) ,

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