Abstract

A sequence of rainbows is produced in light scattering by a particle of high symmetry in the short-wavelength limit, and a supernumerary interference pattern occurs to one side of each rainbow. Using both a ray-tracing procedure and the Debye-series decomposition of first-order perturbation wave theory, I examine the spacing of the supernumerary maxima and minima as a function of the cylinder rotation angle when an elliptical-cross-section cylinder is normally illuminated by a plane wave. I find that the supernumerary spacing depends sensitively on the cylinder-cross-section shape, and the spacing varies sinusoidally as a function of the cylinder rotation angle for small cylinder ellipticity. I also find that relatively large uncertainties in the supernumerary spacing affect the rainbow angle only minimally.

© 2000 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  23. M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
    [CrossRef]
  24. C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964).
    [CrossRef]
  25. 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]
  26. 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]
  27. V. A. Erma, “Exact solution for scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
    [CrossRef]
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    [CrossRef]
  30. R. Schiffer, “Light scattered by perfectly conducting statistically irregular particles,” J. Opt. Soc. Am. A 6, 385–402 (1989).
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  31. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 605.
  32. P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light-scattering,” in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 43–53, Figs. 4 and 5.
    [CrossRef]
  33. J. P. Barton, “Electromagnetic-field calculations for irregularly shaped, layered cylindrical particles with focused illumination,” Appl. Opt. 36, 1312–1319 (1997).
    [CrossRef] [PubMed]
  34. H.-B. Lin, J. D. Eversole, A. J. Campillo, J. P. Barton, “Excitation localization principle for spherical microcavities,” Opt. Lett. 23, 1921–1923 (1998).
    [CrossRef]
  35. J. P. Barton, “Effects of surface perturbations on the quality and the focused-beam excitation of microsphere resonance,” J. Opt. Soc. Am. A 16, 1974–1980 (1999).
    [CrossRef]
  36. Ref. 11, Table 10.13, p. 478.

1999 (2)

1998 (3)

H.-B. Lin, J. D. Eversole, A. J. Campillo, J. P. Barton, “Excitation localization principle for spherical microcavities,” Opt. Lett. 23, 1921–1923 (1998).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[CrossRef]

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

1997 (2)

1992 (1)

1991 (1)

1989 (1)

1987 (1)

1980 (1)

1979 (1)

1976 (2)

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

A. Nelson, L. Eyges, “Electromagnetic scattering from dielectric rods of arbitrary cross section,” J. Opt. Soc. Am. 66, 254–259 (1976).
[CrossRef]

1975 (1)

1974 (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

1969 (3)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

V. A. Erma, “Exact solution for scattering of electromagnetic waves from bodies 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)

C. Yeh, “Perturbation method in the diffraction of electromagnetic waves by arbitrarily shaped penetrable obstacles,” J. Math Phys. 6, 2008–2013 (1965).
[CrossRef]

1964 (1)

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964).
[CrossRef]

1910 (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

1908 (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering, Vol. MS 89 of the SPIE Milestone Series (SPIE, Bellingham Wash., 1994), pp. 198–204.

1881 (1)

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881); reprinted in Scientific Papers by Lord Rayleigh (Dover, New York, 1964), Vol. 1, pp. 518–536.

1838 (1)

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 (1838); reprinted in Geometrical Aspects of Scattering, P. L. Marston, ed., Vol. MS 89 of the SPIE Milestone Series (SPIE, Bellingham Wash., 1994), pp. 298–309.

Adler, C. L.

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 (1838); reprinted in Geometrical Aspects of Scattering, P. L. Marston, ed., Vol. MS 89 of the SPIE Milestone Series (SPIE, Bellingham Wash., 1994), pp. 298–309.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 605.

Barber, P. W.

P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light-scattering,” in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 43–53, Figs. 4 and 5.
[CrossRef]

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

Barton, J. P.

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in OpticsE. Wolf, ed. (Elsevier, Amsterdam, 1980), Vol. 18, pp. 257–346.
[CrossRef]

Campillo, A. J.

de Boer, J. H.

Debye, P.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering, Vol. MS 89 of the SPIE Milestone Series (SPIE, Bellingham Wash., 1994), pp. 198–204.

Erma, V. A.

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

Eversole, J. D.

Eyges, L.

Hill, S. C.

P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light-scattering,” in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 43–53, Figs. 4 and 5.
[CrossRef]

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

Hovenac, E. A.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Können, G. P.

Lin, H.-B.

Lock, J. A.

Marcuse, D.

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[CrossRef]

Möbius, W.

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105–254 (1907–1909).

Nelson, A.

Nussenzveig, H. M.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

Presby, H. M.

Rayleigh, Lord

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881); reprinted in Scientific Papers by Lord Rayleigh (Dover, New York, 1964), Vol. 1, pp. 518–536.

Schiffer, R.

Stone, B. R.

Takano, Y.

Tanaka, M.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[CrossRef]

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in OpticsE. Wolf, ed. (Elsevier, Amsterdam, 1980), Vol. 18, pp. 257–346.
[CrossRef]

van de Hulst, H. C.

R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
[CrossRef] [PubMed]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 13.23, pp. 243–246.

Walker, J. D.

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

Wang, R. T.

Yeh, C.

C. Yeh, “Perturbation method in the diffraction of electromagnetic waves by arbitrarily shaped penetrable obstacles,” J. Math Phys. 6, 2008–2013 (1965).
[CrossRef]

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964).
[CrossRef]

Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105–254 (1907–1909).

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 experimentallen Prufung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

Appl. Opt. (5)

J. Math Phys. (1)

C. Yeh, “Perturbation method in the diffraction of electromagnetic waves by arbitrarily shaped penetrable obstacles,” J. Math Phys. 6, 2008–2013 (1965).
[CrossRef]

J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

J. Quant. Spectrosc. Radiat. Transfer (1)

M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[CrossRef]

Opt. Lett. (2)

Philos. Mag. (1)

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881); reprinted in Scientific Papers by Lord Rayleigh (Dover, New York, 1964), Vol. 1, pp. 518–536.

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 scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
[CrossRef]

Phys. Rev. A (1)

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964).
[CrossRef]

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Phys. Z. (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering, Vol. MS 89 of the SPIE Milestone Series (SPIE, Bellingham Wash., 1994), pp. 198–204.

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 (1838); reprinted in Geometrical Aspects of Scattering, P. L. Marston, ed., Vol. MS 89 of the SPIE Milestone Series (SPIE, Bellingham Wash., 1994), pp. 298–309.

Other (7)

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Sect. 10.4, pp. 446–452.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 13.23, pp. 243–246.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in OpticsE. Wolf, ed. (Elsevier, Amsterdam, 1980), Vol. 18, pp. 257–346.
[CrossRef]

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

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 605.

P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light-scattering,” in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 43–53, Figs. 4 and 5.
[CrossRef]

Ref. 11, Table 10.13, p. 478.

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Figures (3)

Fig. 1
Fig. 1

Far-zone scattered intensity as a function of scattering angle in the vicinity of the p = 2 rainbow for a circular-cross-section cylinder with n = 1.333 and x = 1000.0 calculated by exact wave theory (solid curve) and the p = 2 Debye-series portion of exact wave theory (dashed curve).

Fig. 2
Fig. 2

Deviation of the p = 2 supernumerary spacing parameter from its average value, h 2(ξ) - h ave, as a function of the cylinder rotation angle ξ for an elliptical-cross-section cylinder with n = 1.333, x = 1000.0, and ∊ = 0.0001. The solid curve is the prediction of ray theory, the dashed curve is the m = 2 Fourier component of the ray theory result, and the open and solid circles are the prediction of first-order perturbation theory along with the modeling of the supernumerary intensity minima with Airy theory and CAM theory, respectively.

Fig. 3
Fig. 3

Deviation of the p = 2 rainbow angle from its average value, θ2 R (ξ) - θave R , as a function of the cylinder rotation angle ξ for an elliptical-cross-section cylinder with n = 1.333, x = 1000.0, and ∊ = 0.0001. The solid curve is the prediction of ray theory, and the open circles are the prediction of first-order perturbation theory along with the modeling of the supernumerary intensity minima with Airy theory.

Equations (63)

Equations on this page are rendered with MathJax. Learn more.

cosϕiD=n2-1/p2-11/2, sinϕtD=1/nsinϕiD, θpD=p-1π+2ϕiD-2pϕtD.
Y=-hpX3/3a2+OX4/a3,
hp=p2-12p2-n21/2/p2n2-13/2.
x2/a2+y2/b2=1,
=b/a-1.
θ2Rξ=θ2D-8 sinϕtDcos3ϕtD×cos2ξ+θ2D+O2, θ3Rξ=θ3D+32 sinϕtDcos3ϕtD×cos2ϕtDcos2ξ+θ3D+O2.
Iθ=2πI0F/rax1/3/hp2/3Ai2-x2/3Δ/hp1/3,
Δ=θ-θpD,
Eθ  Ai-x2/3Δ/hp1/3uΔ-ivΔx-1/3Ai-x2/3Δ/hp1/3uΔ,
uΔ=1+u1Δ+u2Δ2+, vΔ=v0+v1Δ+v2Δ2+.
L/a=LR/a+A+X/a2-h+X/a3/3+B+X/a4,
L/a=LR/a+A-X/a2+h-X/a3/3+B-X/a4.
hpξ=h++h-/2.
h2ξ=e0+m=1 em cosmξ+m=1 fm sinmξ,
h3ξ=g0+m=1 gm cosmξ+m=1 jm sinmξ.
h2ξe0+19sinϕtD3/4cosϕtD-10/3 cos2ξ+Φ,
Φ250°n-285°.
rθ=a1+δfθ
fθO1.
δ  1,
kaδ  2π.
α=n for TE,1 for TM, β=1 for TEn for TM
y=nka,
El=αHl1xJly-βHl1xJly,
Iθ=2I0/πkr|Fθ|2,
Fθ=l=- bl expilθ
Fθ=l=- al expilθ
al0bl0=αJlxJly-βJlxJly/El,
cl0dl0=2i/n2πxEl.
al=al0+2in2-1δ/π×l=- il-lal,l1+Oδ2, bl=bl0+2in2-1δ/π×l=- il-lbl,l1+Oδ2, cl=cl0-2in2-1δ/n2π×l=- il-lcl,l1+Oδ2, dl=dl0-2in2-1δ/n2π×l=- il-ldl,l1+Oδ2,
al,l1=l2/x2JlyJly+JlyJlyIl,l-il/x2JlyJlyIl,l/ElEl, bl,l1=JlyJlyIl,l/ElEl, cl,l1=l2/xyJlyHl1x+JlyHl1xIl,l-il/xyJly×Hl1xIl,l/ElEl, dl,l1=JlyHl1xIl,l/ElEl,
Il,l=1/2π02πdθfθexpil-lθ, Il,l=1/2π02πdθdf/dθexpil-lθ.
fθ=A0+q=1 Aq cosqθ-ξ+q=1 Bq sinqθ-ξ,
Il,l=A0δl,l+1/2q=1Aq-iBqexp-iqξδl,l-q+1/2q=1Aq+iBqexpiqξδl,l+q, Il,l=1/2q=1 iqAq-iBqexp-iqξδl,l-q-1/2q=1 iqAq+iBqexpiqξδl,l+q,
l,l21=Tl21δl,l-4in2-1δ/πil-lTl,l21+Oδ2, l,l22=Rl22δl,l-4in2-1δ/πil-lRl,l22+Oδ2,
Tl21=4i/nπxDl, Rl22=-αHl2xHl2y-βHl2xHl2y/Dl,
Dl=αHl1xHl2y-βHl1xHl2y.
Tl,l21=l2/xyHl2yHl1x+Hl2yHl1xIl,l-il/xy×Hl2yHl1xIl,l/nDlDl, Rl,l22=l2/x2Hl2yHl2y+Hl2yHl2yIl,l-il/x2Hl2y×Hl2yIl,l/DlDl
Tl,l21=Hl2yHl1xIl,l/nDlDl, Rl,l22=Hl2yHl2yIl,l/DlDl
l,l12=Tl12δl,l-4in2-1δ/πil-lTl,l12+Oδ2, l,l11=Rl11δl,l-4in2-1δ/πil-lRl,l11+Oδ2,
Tl12=4in/πxDl, Rl11=-αHl1xHl1y-βHl1xHl1y/Dl,
Tl,l12=l2/xyHl1xHl2y+Hl1xHl2yIl,l-il/xyHl1x×Hl2yIl,ln/DlDl, Rl,l11=l2/y2Hl1xHl1x+Hl1xHl1xIl,l-il/y2Hl1x×Hl1xIl,l/DlDl
Tl,l12=Hl1xHl2yIl,ln/DlDl, Rl,l11=Hl1xHl1xIl,l/DlDl
al0bl0=1/21-Rl22-Tl211-Rl11-1Tl12=1/21-Rl22-p=1 Tl21Rl11p-1Tl12,
al,l1bll1=Rl,l22+Tl,l211-Rl11-1Tl12+Tl211-Rl11-1Rl,l111-Rl11-1Tl12+Tl211-Rl11-1Tl,l12, =Rl,l22+p=1 Tl,l21Rl11p-1Tl12+s=0t=0 Tl21Rl11sRl,l11Rl11tTl12+p=1 Tl21Rl11p-1Tl,l12,
cl0dl0=1/nTl211-Rl11-1=1/np=1 Tl21Rl11p-1,
cl,l1dl,l1=2nTl,l211-Rl11-1+2nTl21×1-Rl11-1Rl,l111-Rl11-1=2n p=1 Tl,l21Rl11p-1+2n s=0t=0 Tl21Rl11sRl,l11Rl11t,
Fθ=F0θ+2in2-1δ/πA0Q0θ+2in2-1δ/πq=1 iqQ1q, θ×Aq cosqξ-Bq sinqξ-2in2-1δ/πq=1 iqQ2q, θ×Aq sinqξ+Bq cosqξ.
F0θ=b00+2 l=1 bl0 coslθ,
Q0θ=U0+2 l=1 Ul coslθ, Q1q, θ=l=0 Vl,q+coslθ+-1q cosl+qθ+l=1q-1-1lVl,q- coslθ, Q2q, θ=l=0 Vl,q+sinlθ--1q sinl+qθ-l=1q-1-1lVl,q- sinlθ
Ul=Jl2y/El2, Vl,q+=JlyJq+ly/ElEq+l, Vl,q-=JlyJq-ly/ElEq-l
Ul=Tl,l21Rl11Tl12+Tl21Rl,l11Tl12+Tl21Rl11Tl,l12, Vl,q+=Tq+l,l21Rl11Tl12+Tq+l21Rq+l,l11Tl12+Tq+l21Rq+l11Tq+l,l12, Vl,q-=Tq-l,l21Rl11Tl12+Tq-l21Rq-l,l11Tl12+Tq-l21Rq-l11Tq-l,l12.
rθ=a1+1+2+2cos2θ-1/2,
fθ=rθ-a.
A0=/2-32/16+33/32+, A2=-/2+53/64+, A4=32/16-33/32+, A6=-53/64+, B2=B4=B6=0.
δ=/2, A0=1, A2=-1
δ=/2,  A0=1.
2.338 107=π/180x2/3θamin-θ2R/h21/3, 4.087 949=π/180x2/3θbmin-θ2R/h21/3,
θ2R=2.336 182θamin-1.336 182θbmin, h2=0.992 28310-6x2θbmin-θamin3.
Iθ, ξ  Ai2-xave2/3θ-θpRξ/hpξ1/3,
δh20.029 923δθminh2xave2/3
δθ2Rδθmin
θ2Rξ=θaminξ-2.338 107180/πh2ξ1/3/xave2/3,

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