Abstract

When the refractive index m of a sphere is such that rainbows occur in the forward or backward direction, the glory scattering becomes exceptionally strong. A number of these refractive-index values have been determined from the geometry of ray paths. A physical-optics model of the scattering leads to an a2x4/3 dependence in the scattered irradiance, where a is the radius of the sphere, and x = ka is the size parameter. Normal glory scattering gives an irradiance proportional to x. Mie theory computations illustrate the presence of rainbow glories at predicted m values and the x4/3 irradiance factor. As in normal glory scattering, the rainbow-enhanced glory light contains a strong cross-polarized component. Experiments using single glass spheres immersed in liquids show the predicted cross-polarized scattering with a sensitive dependence on m.

© 1991 Optical Society of America

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References

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  1. H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236(4), 116–127 (1977).
    [CrossRef]
  2. W. J. Humphreys, Physics of the Air, 2nd ed. (McGraw-Hill, New York, 1929), pp. 458–482.
  3. H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974).
    [CrossRef]
  4. H. M. Nussenzveig, W. J. Wiscombe, “Forward optical glory,” Opt. Lett. 5, 455–457 (1980).
    [CrossRef] [PubMed]
  5. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  6. G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  7. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.
  8. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 145.
  9. J. McK. Ellison, C. V.Peetzc Peetzc, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London, Sect. B 74, 105–123 (1959).
  10. G. E. Davis, “Scattering of light by an air bubble in water,” J. Opt. Soc. Am. 45, 572–581 (1955).
    [CrossRef]
  11. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  12. V. Khare, “Surface waves and rainbow effects in the optical glory,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 417–464.
  13. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
    [CrossRef]
  14. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  15. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  16. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
    [CrossRef]
  17. D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).
    [CrossRef]
  18. 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]
  19. 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). [Note: the numerical factor in Eq. (21b) should be 0.0235, not 0.042, but the calculations in Figs. 8 and 9 were performed correctly.]
    [CrossRef]
  20. 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).
    [CrossRef]
  21. D. E. Barrick, “Spheres,” in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, New York, 1970), Vol. 1, Chap. 3.
  22. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  23. P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).
    [CrossRef]
  24. The derivatives are best performed using an operator approach. From the angle relations, we form the operator d/dρ = ∂/∂ρ + tan θ cot ρ∂/∂θ + 2(tan θ cot ρ − P)∂/∂δ. Successive applications of the operator for high-order derivatives can be automated with any of several symbolic math software packages.
  25. D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).
    [CrossRef] [PubMed]
  26. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  27. P. L. Marston, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816–1818 (1983).
    [CrossRef]
  28. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 9.
  29. S. D. Mobbs, “Theory of the rainbow,” J. Opt. Soc. Am. 69, 1089–1092 (1979).
    [CrossRef]
  30. G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
    [CrossRef] [PubMed]

1988 (2)

1987 (1)

1984 (2)

1983 (2)

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).
[CrossRef]

P. L. Marston, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816–1818 (1983).
[CrossRef]

1982 (1)

1981 (1)

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).
[CrossRef]

1980 (2)

1979 (3)

1977 (2)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236(4), 116–127 (1977).
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

1976 (1)

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

1974 (1)

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974).
[CrossRef]

1959 (1)

J. McK. Ellison, C. V.Peetzc Peetzc, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London, Sect. B 74, 105–123 (1959).

1955 (1)

1908 (1)

G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Arnott, W. P.

Barrick, D. E.

D. E. Barrick, “Spheres,” in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, New York, 1970), Vol. 1, Chap. 3.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.

Bryant, H. C.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974).
[CrossRef]

Davis, G. E.

de Boer, J. H.

Ellison, J. McK.

J. McK. Ellison, C. V.Peetzc Peetzc, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London, Sect. B 74, 105–123 (1959).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 9.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air, 2nd ed. (McGraw-Hill, New York, 1929), pp. 458–482.

Jarmie, N.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 145.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

V. Khare, “Surface waves and rainbow effects in the optical glory,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 417–464.

Können, G. P.

Langley, D. S.

D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).
[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).
[CrossRef]

P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).
[CrossRef]

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).
[CrossRef]

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). [Note: the numerical factor in Eq. (21b) should be 0.0235, not 0.042, but the calculations in Figs. 8 and 9 were performed correctly.]
[CrossRef]

Lock, J. A.

Marston, P. L.

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]

D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).
[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).
[CrossRef]

P. L. Marston, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816–1818 (1983).
[CrossRef]

P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).
[CrossRef]

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).
[CrossRef]

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). [Note: the numerical factor in Eq. (21b) should be 0.0235, not 0.042, but the calculations in Figs. 8 and 9 were performed correctly.]
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Mobbs, S. D.

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Forward optical glory,” Opt. Lett. 5, 455–457 (1980).
[CrossRef] [PubMed]

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
[CrossRef]

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236(4), 116–127 (1977).
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Peetzc, C. V.Peetzc

J. McK. Ellison, C. V.Peetzc Peetzc, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London, Sect. B 74, 105–123 (1959).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 9.

Probert-Jones, J. R.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 9.

van de Hulst, H. C.

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

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 9.

Walker, J. D.

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

Wiscombe, W. J.

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)

G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (4)

J. Acoust. Soc. Am. (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).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).
[CrossRef]

Proc. Phys. Soc. London, Sect. B (1)

J. McK. Ellison, C. V.Peetzc Peetzc, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London, Sect. B 74, 105–123 (1959).

Sci. Am. (2)

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974).
[CrossRef]

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236(4), 116–127 (1977).
[CrossRef]

Other (9)

W. J. Humphreys, Physics of the Air, 2nd ed. (McGraw-Hill, New York, 1929), pp. 458–482.

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

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 145.

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). [Note: the numerical factor in Eq. (21b) should be 0.0235, not 0.042, but the calculations in Figs. 8 and 9 were performed correctly.]
[CrossRef]

V. Khare, “Surface waves and rainbow effects in the optical glory,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 417–464.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 9.

The derivatives are best performed using an operator approach. From the angle relations, we form the operator d/dρ = ∂/∂ρ + tan θ cot ρ∂/∂θ + 2(tan θ cot ρ − P)∂/∂δ. Successive applications of the operator for high-order derivatives can be automated with any of several symbolic math software packages.

D. E. Barrick, “Spheres,” in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, New York, 1970), Vol. 1, Chap. 3.

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Figures (11)

Fig. 1
Fig. 1

Example ray paths: (a) a single-ray (2,1) glory for m ≈ 1.848; (b) a double-ray (3,1) glory for m = 1.17; (c) the (3,1) rainbow–glory ray that occurs at m ≈ 1.18; (d) the (4,2) rainbow–glory ray that occurs at m ≈ 1.465.

Fig. 2
Fig. 2

Wave-front curvature parameter Q as a function of refractive index m for the (2,1), (3,1), and (4,2) glory waves. Q = 0 when the rainbow coincides with the glory, indicating a cubic wave front.

Fig. 3
Fig. 3

Rainbows in the backward glory. The upper portion shows the geometrically determined limits on m for single or double (P, L) rays. The lower portion shows the maximum of I2 (normalized as discussed in Subsection II.C) near the backward direction according to Mie theory. Sharp peaks and interference effects occur near mP,L* values, at the upper limits of the double-ray ranges. The size parameter x ∝ 1/m with x = 18,318 at m = 1.

Fig. 4
Fig. 4

Mie results show rainbow effects in the forward glory I2 near mP,L* values, at the upper limit of double-ray regions of m. Here x varies as in Fig. 3.

Fig. 5
Fig. 5

Mie theory and physical-optics approximation (POA) for irradiance I2 near the backward direction. The computations correspond to conditions of part of the experimental study: m = m3,1* ≈ 1.18 and x = 15,525. The two curves are nearly indistinguishable.

Fig. 6
Fig. 6

Mie results for the maximum in I2 near the optic axis. When m corresponds to m3,1* or m4,2*, the scattering enhancement ≈x4/3, as predicted by physical optics.

Fig. 7
Fig. 7

(a) Apparatus for measuring backward glory scattering as m is varied. For forward scattering the beam splitter is removed and a mirror above the sphere sends light to the camera. The polarizers are set to select the cross-polarized light I2. (b) Detail of the temperature cell containing the sphere surrounded by a liquid to approximate m.

Fig. 8
Fig. 8

Photographs of the backward glory I2 from a 2-mm-diameter glass sphere with varying relative index. From left to right, m ≈ 1.1770, 1.1776, 1.1790, 1.1806. The irradiance oscillates because of interference of two (3,1) glory waves for m ≤ 1.180, and these vanish at higher m.

Fig. 9
Fig. 9

Comparison of Mie and measured I2 near the center of the backward glory pattern, showing interference effects as m varies near m3,1*.

Fig. 10
Fig. 10

Photographs of the near-zone (to view) and far-zone (bottom row) forward glory I2 from a 2.5-mm-diameter glass sphere. From left to right, the refractive index m ≈ 1.4608, 1.4614, 1.4633, 1.4664. Two (4,2) glory waves interfere for m ≤ 1.4652, and these vanish at higher m.

Fig. 11
Fig. 11

Mie theory results for I2 near the center of the forward glory pattern, showing interference effects as m varies near m4,2*.

Tables (1)

Tables Icon

Table 1 Refractive Indices, Incident Angles, and Refractive Angles, for Several (P, L) Rainbow–Glory Rays

Equations (29)

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

sin θ = m sin ρ .
ϕ = 2 θ - 2 P ρ + ( P - 1 ) π .
m sin ρ = cos [ P ρ - π ( P - L ) / 2 ] .
( P - L - 1 ) π / 2 P ρ ( P - L ) π / 2 P .
1 Ray :             sec ( L π / 2 P ) > m > 0 ,             P - 1 > L ,
csc ( π / 2 P ) m P ,             P - 1 = L ,
2 Rays :             sec ( L π / 2 P ) m m P , L * ,             P - 1 > L > 0.
tan θ = P tan ρ .
P tan ρ = B ( tan P ρ ) B ,
E j ( s , ψ ) = E i F j ( ψ ) q - 1 / 2 exp ( i μ + i η ) ,
c 1 = r 1 P - 1 ( 1 - r 1 2 ) ,             c 2 = ( - 1 ) P + L - 1 r 2 P - 1 ( r - r 2 2 ) ,
F 1 ( ψ ) = c 1 sin 2 ψ + c 2 cos 2 ψ ,             F 2 ( ψ ) = ( c 2 - c 1 ) ( sin 2 ψ ) / 2.
q = 1 + A csc θ ,             A = 2 ( P tan ρ - tan θ ) .
η = k a { 1 - cos θ + 2 m P cos ρ + [ 1 - cos ( θ - δ ) ] sec δ } .
η ( s ) = η 0 + ( s - b ) d η d s + ( s - b ) 2 2 d 2 η d s 2 + ( s - b ) 3 6 d 3 η d s 3 + ,
η ( s ) = η 0 + k ( s - b ) 2 2 α ,             α = a ( 1 + sin θ A ) .
η ( s ) = η 0 - k ( s - b ) 3 Λ a 2 ,             Λ = ( P 2 - 1 ) 2 ( P 2 - m 2 ) 1 / 2 3 P 2 ( m 2 - 1 ) 3 / 2 .
E j ( R , γ , ξ ) k E i exp ( i μ + i k R ) 2 π i R q 1 / 2 D j ( γ , ξ ) ,
D j ( γ , ξ ) = 0 s W j exp ( i k η ) d s ,
W 1 ( γ , ξ , s ) = π ( c 1 + c 2 ) J 0 ( k s sin γ ) + π ( c 1 - c 2 ) J 2 ( k s sin γ ) cos 2 ξ ,
W 2 ( γ , ξ , s ) = π ( c 1 - c 2 ) J 2 ( k s sin γ ) sin 2 ξ
D j b λ 0 α 1 / 2 W j ( γ , ξ , b ) exp [ i π / 4 - i k α ( 1 - cos γ ) ] .
D j 2 π a b 3 k a Λ - 1 / 3 W j ( γ , ξ , b ) A i ( 0 ) ,
I 2 2 π x b 2 ( α - a ) a 3 I R [ ( c 1 - c 2 ) J 2 ( k b sin γ ) sin 2 ξ ] 2 ,
I 2 ( 0.2424 ) x 4 / 3 b a I R [ ( c 1 - c 2 ) J 2 ( k b sin γ ) sin 2 ξ ] 2 .
Δ γ λ / 2 b ,
I 2 max ( P = 2 , L = 1 , m = 1.8 ) 0.077 x I R ,
I 2 max ( P = 3 , L = 1 , m = 1.17996 ) 0.005 x 4 / 3 I R ,
I 2 max ( P = 4 , L = 2 , m = 1.46521 ) 0.0007 x 4 / 3 I R .

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