Abstract

Enhancement in backscattering known as glory scattering results from geometric and material properties of spherically symmetric scatterers. The wave-front shape near the spherical scatterer is locally a circular torus. Radiation from a toroidal wave front is axially focused on the backward-directed axis. It is shown that the axial point caustic unfolds to an astroid caustic as the scatterer’s shape changes from spherical to slightly spheroidal. The wavefront pertinent for slightly spheroidal scatterers was modeled as a toroidal wave front with a superimposed harmonic angular perturbation. Experimental observations are displayed for cross-polarized backscattering by freely vertical rising, slightly oblate spheroidal air bubbles in water illuminated by a horizontally propagating laser beam. These patterns were recorded with a camera for two different incident-beam polarization directions relative to the axis of rotational symmetry of the bubble. Angular scattering patterns were also computed using a perturbation analysis based on use of the harmonically perturbed toroidal wave front and physical optics. Bubble oblateness was estimated from features of the angular scattering pattern and from hydrodynamic relations.

© 1991 Optical Society of America

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References

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  1. D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 130–141 (1984).
    [Crossref]
  2. W. P. Arnott, P. L. Marston, “Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
    [Crossref]
  3. D. S. Langley, (St. John’s University, Collegeville, Minn. 56321 (personal communication).
  4. W. P. Arnott, “Generalized glory scattering from spherical and spheroidal bubbles in water: unfolding axial caustics with angular perturbations of toroidal wavefronts,” doctoral dissertation (Washington State University, Pullman, Wash., 1988).
  5. W. P. Arnott, P. L. Marston, “Unfolding axial caustics of gloryscattering with harmonic angular perturbations of toroidal wavefronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
    [Crossref]
  6. W. P. Arnott, P. L. Marston, “Backscattering of laser light from freely rising spherical and spheroidal air bubbles in water,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 296–307 (1988).
  7. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  8. D. S. Langley, “Light scattering from bubbles in liquids,” doctoral dissertation (Washington State University, Pullman, Wash., 1984).
  9. H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc. Am. 37, 16–22 (1947).
    [Crossref]
  10. J. D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972).
  11. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  12. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [Crossref] [PubMed]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. S. K. Chaudhuri, F. B. Sleator, W. M. Boerner, “Analysis of internally reflected and diffracted fields in electromagnetic backscattering by dielectric spheroids,” Radio Sci. 19, 987–999 (1984). We have noted several mistakes in the formulas of Appendix B.
    [Crossref]
  15. M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [Crossref]
  16. P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226–232 (1987); “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [Crossref] [PubMed]
  17. J. F. Nye, “Rainbow scattering from spheroidal drops: an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [Crossref]
  18. R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).
  19. A. M. Gaudin, Flotation (McGraw-Hill, New York, 1957), pp. 340–354.
  20. D. W. Moore, “The rise of a gas bubble in a viscous liquid,” J. Fluid Mech. 6, 113–130 (1959).
    [Crossref]
  21. J. F. Harper, “The motion of bubbles and drops through liquids,” Adv. Appl. Mech. 12, 59–129 (1972).
    [Crossref]
  22. G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, New York, 1944).
  23. P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983); J. Acoust. Soc. Am. 78, 1128(E) (1985).
    [Crossref]
  24. F. S. Acton, Numerical Methods that Work (Harper & Row, New York, 1970), p. 52.

1989 (1)

W. P. Arnott, P. L. Marston, “Unfolding axial caustics of gloryscattering with harmonic angular perturbations of toroidal wavefronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
[Crossref]

1988 (1)

1987 (1)

P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226–232 (1987); “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[Crossref] [PubMed]

1984 (2)

J. F. Nye, “Rainbow scattering from spheroidal drops: an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[Crossref]

S. K. Chaudhuri, F. B. Sleator, W. M. Boerner, “Analysis of internally reflected and diffracted fields in electromagnetic backscattering by dielectric spheroids,” Radio Sci. 19, 987–999 (1984). We have noted several mistakes in the formulas of Appendix B.
[Crossref]

1983 (1)

P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983); J. Acoust. Soc. Am. 78, 1128(E) (1985).
[Crossref]

1981 (1)

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 130–141 (1984).
[Crossref]

1979 (1)

1976 (1)

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

1975 (1)

1972 (1)

J. F. Harper, “The motion of bubbles and drops through liquids,” Adv. Appl. Mech. 12, 59–129 (1972).
[Crossref]

1959 (1)

D. W. Moore, “The rise of a gas bubble in a viscous liquid,” J. Fluid Mech. 6, 113–130 (1959).
[Crossref]

1947 (1)

Acton, F. S.

F. S. Acton, Numerical Methods that Work (Harper & Row, New York, 1970), p. 52.

Arnott, W. P.

W. P. Arnott, P. L. Marston, “Unfolding axial caustics of gloryscattering with harmonic angular perturbations of toroidal wavefronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
[Crossref]

W. P. Arnott, P. L. Marston, “Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
[Crossref]

W. P. Arnott, “Generalized glory scattering from spherical and spheroidal bubbles in water: unfolding axial caustics with angular perturbations of toroidal wavefronts,” doctoral dissertation (Washington State University, Pullman, Wash., 1988).

W. P. Arnott, P. L. Marston, “Backscattering of laser light from freely rising spherical and spheroidal air bubbles in water,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 296–307 (1988).

Asano, S.

Berry, M. V.

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

Boerner, W. M.

S. K. Chaudhuri, F. B. Sleator, W. M. Boerner, “Analysis of internally reflected and diffracted fields in electromagnetic backscattering by dielectric spheroids,” Radio Sci. 19, 987–999 (1984). We have noted several mistakes in the formulas of Appendix B.
[Crossref]

Chaudhuri, S. K.

S. K. Chaudhuri, F. B. Sleator, W. M. Boerner, “Analysis of internally reflected and diffracted fields in electromagnetic backscattering by dielectric spheroids,” Radio Sci. 19, 987–999 (1984). We have noted several mistakes in the formulas of Appendix B.
[Crossref]

Clift, R.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).

Gaudin, A. M.

A. M. Gaudin, Flotation (McGraw-Hill, New York, 1957), pp. 340–354.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grace, J. R.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).

Harper, J. F.

J. F. Harper, “The motion of bubbles and drops through liquids,” Adv. Appl. Mech. 12, 59–129 (1972).
[Crossref]

Langley, D. S.

P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983); J. Acoust. Soc. Am. 78, 1128(E) (1985).
[Crossref]

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 130–141 (1984).
[Crossref]

D. S. Langley, (St. John’s University, Collegeville, Minn. 56321 (personal communication).

D. S. Langley, “Light scattering from bubbles in liquids,” doctoral dissertation (Washington State University, Pullman, Wash., 1984).

Lawrence, J. D.

J. D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972).

Marston, P. L.

W. P. Arnott, P. L. Marston, “Unfolding axial caustics of gloryscattering with harmonic angular perturbations of toroidal wavefronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
[Crossref]

W. P. Arnott, P. L. Marston, “Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
[Crossref]

P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226–232 (1987); “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[Crossref] [PubMed]

P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983); J. Acoust. Soc. Am. 78, 1128(E) (1985).
[Crossref]

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 130–141 (1984).
[Crossref]

W. P. Arnott, P. L. Marston, “Backscattering of laser light from freely rising spherical and spheroidal air bubbles in water,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 296–307 (1988).

Moore, D. W.

D. W. Moore, “The rise of a gas bubble in a viscous liquid,” J. Fluid Mech. 6, 113–130 (1959).
[Crossref]

Nye, J. F.

J. F. Nye, “Rainbow scattering from spheroidal drops: an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[Crossref]

Sleator, F. B.

S. K. Chaudhuri, F. B. Sleator, W. M. Boerner, “Analysis of internally reflected and diffracted fields in electromagnetic backscattering by dielectric spheroids,” Radio Sci. 19, 987–999 (1984). We have noted several mistakes in the formulas of Appendix B.
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc. Am. 37, 16–22 (1947).
[Crossref]

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

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, New York, 1944).

Weber, M. E.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).

Yamamoto, G.

Adv. Appl. Mech. (1)

J. F. Harper, “The motion of bubbles and drops through liquids,” Adv. Appl. Mech. 12, 59–129 (1972).
[Crossref]

Adv. Phys. (1)

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

Appl. Opt. (2)

J. Acoust. Soc. Am. (3)

P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226–232 (1987); “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[Crossref] [PubMed]

W. P. Arnott, P. L. Marston, “Unfolding axial caustics of gloryscattering with harmonic angular perturbations of toroidal wavefronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
[Crossref]

P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983); J. Acoust. Soc. Am. 78, 1128(E) (1985).
[Crossref]

J. Fluid Mech. (1)

D. W. Moore, “The rise of a gas bubble in a viscous liquid,” J. Fluid Mech. 6, 113–130 (1959).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nature (London) (1)

J. F. Nye, “Rainbow scattering from spheroidal drops: an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[Crossref]

Phys. Rev. Lett. (1)

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 130–141 (1984).
[Crossref]

Radio Sci. (1)

S. K. Chaudhuri, F. B. Sleator, W. M. Boerner, “Analysis of internally reflected and diffracted fields in electromagnetic backscattering by dielectric spheroids,” Radio Sci. 19, 987–999 (1984). We have noted several mistakes in the formulas of Appendix B.
[Crossref]

Other (11)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).

A. M. Gaudin, Flotation (McGraw-Hill, New York, 1957), pp. 340–354.

D. S. Langley, (St. John’s University, Collegeville, Minn. 56321 (personal communication).

W. P. Arnott, “Generalized glory scattering from spherical and spheroidal bubbles in water: unfolding axial caustics with angular perturbations of toroidal wavefronts,” doctoral dissertation (Washington State University, Pullman, Wash., 1988).

J. D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972).

W. P. Arnott, P. L. Marston, “Backscattering of laser light from freely rising spherical and spheroidal air bubbles in water,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 296–307 (1988).

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

D. S. Langley, “Light scattering from bubbles in liquids,” doctoral dissertation (Washington State University, Pullman, Wash., 1984).

F. S. Acton, Numerical Methods that Work (Harper & Row, New York, 1970), p. 52.

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, New York, 1944).

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Figures (11)

Fig. 1
Fig. 1

(a) Experimental and (b) computed scattering patterns of cross-polarized backscattering by a single spheroidal bubble that rose freely through a tank of water and passed through a horizontal laser beam. The incident beam was polarized in the bubble rise direction and a polarizer in front of the camera was oriented to pass light polarized at 90° to the bubble rise direction. The stationary bubble diameter measured when the bubble was trapped by a glass slide after passage through the beam was 654 μm. The computational parameters used are given in Table 1. In (b), the astroid caustic [Eq. (6)] associated with 3- and 4-chord glory ray wave fronts is overlaid on the computed pattern. The caustic farthest from the center results from the 3-chord glory ray. The angular scale in this and the other scattering patterns is calibrated for an observer in water.

Fig. 2
Fig. 2

(a) Wave front near the exit plane of Fig. 3 for a spherical bubble. The ring on top is the zero Gaussian curvature contour. An infinite number of rays, such as the ray labeled ∞, emanate from this contour and are focused in the backward direction. Two rays such as those labeled 1 and 2 contribute to the scattering at other angles. (b) Wave front near the exit plane of Fig. 3 for a slightly spheroidal bubble. Rays in the direction of the normal to the zero Gaussian curvature contour, shown as the solid black contour, give the astroid caustic shown in Fig. 3. In contrast to (a), only (now unfocused) rays labeled 1 through 4 contribute to the scattering in the exact backward direction.

Fig. 3
Fig. 3

Geometric arrangement of the bubble, coordinate system, and caustic. After interaction with the bubble, a plane wave incident along z on the oblate spheroidal bubble gives a backscattered wave front such as in Fig. 2(b). Patches of the wave front become focused to form a caustic in the form of an astroid curve in an observation plane along z. Four rays and two rays contribute to the scattering amplitude inside and outside the astroid caustic.

Fig. 4
Fig. 4

Solid curve is the oblateness (Γ) approximated from hydrodynamics as discussed in Appendix A. The experimental points are calculated from use of Eq. (12) and measurements of the distance from the center to cusp points on experimental scattering patterns. Error bars indicate the uncertainty in the experimental measurement of the bubble diameter (±10 μm) and the uncertainty in measuring the cusp location on an experimental angular scattering pattern.

Fig. 5
Fig. 5

(a) Coordinate systems used. (b) and (c) Paths of 3-, 4-, and 5-chord glory rays in the plane containing the Xs′ and Ys′ axes, respectively, and the center of the bubble. The Xs′ axis points in the bubble rise direction and the Xa′ axis is rotated 45° from the bubble rise direction. The Xs′, Ys′ coordinate system is defined as the symmetric coordinate system since the Xs′ axis is parallel to the axis of rotational symmetry of the bubble. Because of a lack of symmetry, the Xa′, Ya′ coordinate system is defined as the asymmetric coordinate system. In each experimental arrangement the incident laser beam was polarized in the X′ direction and the backscattered light that was photographed was polarized in the Y′ direction.

Fig. 6
Fig. 6

Experimental and computed scattering patterns for slightly oblate bubbles for the symmetric polarization case. The first quadrant of each composite pattern is a computed pattern and the remaining quadrants are experimental data. The oblateness and diameters of these bubbles were factors of ~10 and ~2 smaller than that in Fig. 1. The experimental and computational parameters measured and used are given in Table 1. The experimental pattern in (b) has more background scattering than the others because the exposure time was greater.

Fig. 7
Fig. 7

Experimental scattering pattern of a slightly oblate bubble from the symmetric polarization case. The bubble diameter and oblateness were intermediate between those in Figs. 6 and 1. The diameter measured for this bubble was 488 μm.

Fig. 8
Fig. 8

Experimental scattering patterns of a slightly oblate bubble for the asymmetric polarization case. The bubble diameter was measured to be 586 μm. In contrast to Fig. 7, the center of this pattern is bright. If the scatterers for this figure and Fig. 7 were spherical rather than oblate spheroidal, the form of the scattering patterns would have been the same.

Fig. 9
Fig. 9

(a) Experimental and (b) computed angular scattering patterns for the asymmetric polarization case. Although the bubble diameter measured was 660 μm, the bubble diameter for the computed pattern was taken to be 654.0 μm for comparison with Fig. 1(b). The other computational parameters used are given in Table 1. In (b), the astroid caustic [Eq. (6)] associated with 3- and 4-chord glory ray wave fronts is overlaid on the computed pattern. The caustic farthest from the center results from the 3-chord glory ray wave front.

Fig. 10
Fig. 10

Drag coefficient CD (solid curve) and Reynolds number (dashed curve) from Eqs. (A2) and (A3) in the bubble diameter domain relevant to this study.

Fig. 11
Fig. 11

Definition of the variables used for ray tracing. The path of a 3-chord glory ray is shown in an elliptical mirror symmetry plane of an oblate spheroidal bubble.

Tables (2)

Tables Icon

Table 1 Experimental and Computational Parameters of Figs. 1, 6, and 9

Tables Icon

Table 2 Computed Values of Ap for p = 3–16 from Use of Eqs.(C2) and (C3)a

Equations (45)

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k W ˜ ( s ) = - [ η ¯ + k ( s - b ) 2 2 α ] ,
k W ( s , ψ ) = k W ˜ ( s ) - f ( ψ ) ,
f ( ψ ) = Δ η ¯ 1 + cos 2 ψ 2 ,
K = W s W s s s + W s s W ψ ψ - W s ψ 2 s 2 + 2 W ψ W s ψ s 3 - W ψ 2 s 4 ,
s ( K = 0 ) s K b ( 1 + β cos 2 ψ ) + O ( β 2 ) ,             β 2 α Δ η ¯ k b 2 1 ,
( X s Z , Y s Z ) = ( cos ψ W s - sin ψ s W ψ , sin ψ W s + cos ψ s W ψ ) | s = s K .
U 2 / 3 + V 2 / 3 = ( 2 Δ η ¯ k b ) 2 / 3
V 2 ( γ , φ ) = π ( c 1 - c 2 ) exp ( i Δ η ¯ 2 ) B sym ( γ , φ ) ,
B sym ( γ , φ ) = j = 0 [ J j ( Δ η ¯ 2 ) + J j + 2 ( Δ η ¯ 2 ) ] exp ( i 3 j π 2 ) × sin [ 2 φ ( j + 1 ) ] J 2 ( j + 1 ) ( u ) .
B asym ( γ , ϑ ) = j = 0 [ J 2 j ( Δ η ¯ 2 ) - J 2 ( j + 1 ) ( Δ η ¯ 2 ) ] sin [ 2 ϑ ( 2 j + 1 ) ] J 2 ( 2 j + 1 ) ( u ) + i J 1 ( Δ η ¯ 2 ) J 0 ( u ) + i j = 1 { J 2 j + 1 ( Δ η ¯ 2 ) - J 2 j - 1 ( Δ η ¯ 2 ) } cos ( 4 ϑ j ) J 4 j ( u ) .
Δ η ¯ = k b U max 2 = 0.477 k a 2 μ X max L .
Δ η ¯ A 3 k a Γ 0 - 0.613 k a Γ 0 ,
Γ 0 = | b / a U max 2 A 3 | 0.389 U max 0.292 X max L .
Γ 0 = a 2 c 2 - 1 ,
C D = C D ( a , R e ) = 32 g a 3 3 R e 2 v 2 ,
w e = { 4 M R e 4 3 C D } 1 / 3 ,
C D ( R e ) = 24 R e ( 1 + 0.14 R e 0.687 ) .
V 2 ( γ , φ ) = π ( c 1 - c 2 ) exp ( i Δ η ¯ 2 ) B sym ( γ , φ ) ,
V 2 ( γ , ϑ ) = π ( c 1 - c 2 ) exp ( i Δ η ¯ 2 ) B asym ( γ , ϑ )
B sym ( γ , φ ) = - 1 2 π 0 2 π exp [ - i u cos ( ψ - φ ) ] × exp ( i Δ η ¯ 2 cos 2 ψ ) sin 2 ψ d ψ ,
B asym ( γ , ϑ ) = - 1 2 π 0 2 π exp [ - i u cos ( Ψ - ϑ ) ] × exp ( i Δ η ¯ 2 cos 2 Ψ ) sin 2 Ψ d Ψ
B sym ( γ , φ ) = - 1 π Δ η ¯ 2 d d ( 2 φ ) 0 2 π cos ( u cos ψ ) × exp ( i Δ η ¯ 2 cos ( 2 ψ + 2 φ ) ) d ψ ,
B asym ( γ , ϑ ) = - 1 π d d ( Δ η ¯ / 2 ) 0 2 π cos ( u cos Ψ ) × exp [ - i Δ η ¯ 2 sin ( 2 Ψ + 2 ϑ ) ] d Ψ ,
exp [ i Δ η ¯ 2 cos ( 2 ψ + 2 φ ) ] = j = 0 j ( i ) j J j ( Δ η ¯ 2 ) cos [ j ( 2 ψ + 2 φ ) ] , × exp [ - i Δ η ¯ 2 sin ( 2 ψ + 2 φ ) ] = j = 0 j J 2 j ( Δ η ¯ 2 ) cos [ 2 j ( 2 ψ + 2 φ ) ] - 2 i j = 0 J 2 j + 1 ( Δ η ¯ 2 ) sin [ ( 2 j + 1 ) ( 2 ψ + 2 φ ) ] ,
0 π cos ( 2 j ψ ) cos ( u cos ψ ) d ψ = ( - 1 ) j π J 2 j ( u ) .
η ¯ 3 ellipse = k [ 2 P 1 - P 0 + μ ( 2 P 2 - P 1 + P 3 - P 2 ) ] ,
Δ η ¯ p = η ¯ p ellipse - η ¯ p circle ,
Δ η ¯ p A p k a Γ 0 ,
sin θ i = μ sin θ t .
G ( x , z ) = x 2 c 2 + z 2 a 2 = 1 ,
n = x , z G ( x , z ) x , z G ( x , z ) = 1 Q ( x c 2 , z a 2 ) ,
Q ( x , z ) = [ ( x c 2 ) 2 + ( z a 2 ) 2 ] 1 / 2 .
n = 1 Q [ ± 1 c ( 1 - z 2 a 2 ) 1 / 2 , z a 2 ] ,
Q ( x , z ) = [ 1 c 2 + z 2 a 2 ( 1 a 2 - 1 c 2 ) ] 1 / 2 .
z i = a 2 c cos θ i [ 1 + cos 2 θ i ( a 2 c 2 - 1 ) ] 1 / 2 ,
x i = c sin θ i [ 1 + cos 2 θ i ( a 2 c 2 - 1 ) ] 1 / 2 .
P 0 = ( x i , a ) ,             P 1 = ( x i , z i ) ,
n i · r t = cos ( π - θ t ) ,
r i · r t = cos ( θ i - θ t ) .
l t = sin ( θ t - θ i ) ,
n t = - cos ( θ i - θ t ) .
P 2 - P 1 P 2 - P 1 = r i .
x r 1 = x i + l t n t ( z r 1 - z i ) .
z r 1 2 a 2 = 1 - x r 1 2 c 2 .
z r 1 = z i ( a 2 l t 2 c 2 n t 2 - 1 ) - 2 a 2 l t x i n t c 2 1 + a 2 l t 2 c 2 n t 2 .

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