Abstract

The finite spatial coherence width of sunlight at the Earth imposes restrictions on the production of scattering phenomena based on the interference of light waves. With the spatial coherence properties of sunlight taken into account, the visibility of the supernumerary rainbow sequence adjacent to the primary rainbow and the radii of the water droplets that produce the optimum glory intensity were calculated. A substantial reduction was found in the contrast of all the supernumeraries beyond the first few, and the peak observability of the glory occurred for water droplets with radii between 10 and 20 μm.

© 1989 Optical Society of America

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References

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  1. K. Sassen, “Angular scattering and rainbow formation in pendant drops,”J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  2. P. L. Marston, “Rainbow phenomena and the detection of non-sphericity in drops,” Appl. Opt. 19, 680–685 (1980).
    [CrossRef] [PubMed]
  3. D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).
    [CrossRef]
  4. 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]
  5. W. D. Wright, The Measurement of Colour (Hilger, London, 1969), Chap. 4, App. 2, Table 4.
  6. J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de regenboogkleuren,” Physica 11, 49–60 (1944).
    [CrossRef]
  7. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Secs. 12.1, 12.4.1.
  8. R. W. Wood, Physical Optics, 3rd. ed. (Macmillan, New York, 1934), pp. 186–187.
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 6-3.
  10. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  11. J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
    [CrossRef]
  12. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?”J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [CrossRef]
  13. A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–289 (1983).
    [CrossRef]
  14. R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), p. 181.
  15. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 13.23.
  16. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Art. 10.4.32.
  17. C. W. Querfeld, “Mie atmospheric optics,”J. Opt. Soc. Am. 55, 105–106 (1965).
    [CrossRef]
  18. G. P. Können, “Appearance of supernumeraries of the secondary rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
    [CrossRef]
  19. S. D. Gedzelman, “Rainbows in strong vertical atmospheric electric fields,” J. Opt. Soc. Am. A 5, 1717–1721 (1988).
    [CrossRef]
  20. D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), color plate 2.3.
  21. H. C. van de Hulst, “A theory of the anti-coronae,”J. Opt. Soc. Am. 37, 16–22 (1947).
    [CrossRef]
  22. See Ref. 15, Sec. 13 .33 .
  23. V. Khare, H. M. Nussenzveig, “The theory of the glory,” in U. Landman, ed., Statistical Mechanics and Statistical Methods in Theory and Application (Plenum, New York, 1977), Sec. 8.3, pp. 723–765.
    [CrossRef]
  24. See Ref. 15, Secs. 13.32, 17.42.
  25. J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
    [CrossRef] [PubMed]
  26. T. S. Fahlen, H. C. Bryant, “Optical back scattering from single water droplets,”J. Opt. Soc. Am. 58, 304–310 (1968).
    [CrossRef]
  27. H. C. Bryant, A. J. Cox, “Mie theory and the glory,”J. Opt. Soc. Am. 56, 1529–1532 (1966).
    [CrossRef]
  28. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,”J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  29. 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]
  30. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  31. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,”J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
    [CrossRef]
  32. H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,”IEEE Trans. Antennas Propag. AP-18, 89–99 (1970).
    [CrossRef]
  33. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
    [CrossRef]
  34. H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974), Fig. 2.
    [CrossRef]
  35. B. J. Mason, The Physics of Clouds (Clarendon, Oxford, 1971), Fig. 3.9.
  36. T. A. Cerni, “Determination of the size and concentration of cloud drops with an FSSP,”J. Clim. Appl. Meteor. 22, 1346–1355 (1983).
    [CrossRef]
  37. E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976), Fig. 3.19.
  38. R. G. Pinnick, D. M. Garvey, L. D. Duncan, “Calibration of Knollenberg FSSP light-scattering counters for measurement of cloud droplets,”J. Appl. Meteorol. 20, 1049–1057 (1981).
    [CrossRef]

1988

1987

1984

1983

T. A. Cerni, “Determination of the size and concentration of cloud drops with an FSSP,”J. Clim. Appl. Meteor. 22, 1346–1355 (1983).
[CrossRef]

A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?”J. Opt. Soc. Am. 73, 1626–1628 (1983).
[CrossRef]

A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–289 (1983).
[CrossRef]

1981

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

R. G. Pinnick, D. M. Garvey, L. D. Duncan, “Calibration of Knollenberg FSSP light-scattering counters for measurement of cloud droplets,”J. Appl. Meteorol. 20, 1049–1057 (1981).
[CrossRef]

1980

P. L. Marston, “Rainbow phenomena and the detection of non-sphericity in drops,” Appl. Opt. 19, 680–685 (1980).
[CrossRef] [PubMed]

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

1979

1977

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

1976

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

1974

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974), Fig. 2.
[CrossRef]

1970

H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,”IEEE Trans. Antennas Propag. AP-18, 89–99 (1970).
[CrossRef]

1969

J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef] [PubMed]

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]

1968

1966

1965

1947

1944

J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de regenboogkleuren,” Physica 11, 49–60 (1944).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Art. 10.4.32.

Arnott, W. P.

Brill, D.

D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), color plate 2.3.

Bryant, H. C.

Cerni, T. A.

T. A. Cerni, “Determination of the size and concentration of cloud drops with an FSSP,”J. Clim. Appl. Meteor. 22, 1346–1355 (1983).
[CrossRef]

Cox, A. J.

Dave, J. V.

Duncan, L. D.

R. G. Pinnick, D. M. Garvey, L. D. Duncan, “Calibration of Knollenberg FSSP light-scattering counters for measurement of cloud droplets,”J. Appl. Meteorol. 20, 1049–1057 (1981).
[CrossRef]

Fahlen, T. S.

Falk, D.

D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), color plate 2.3.

Fraser, A. B.

Garvey, D. M.

R. G. Pinnick, D. M. Garvey, L. D. Duncan, “Calibration of Knollenberg FSSP light-scattering counters for measurement of cloud droplets,”J. Appl. Meteorol. 20, 1049–1057 (1981).
[CrossRef]

Gedzelman, S. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 6-3.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Secs. 12.1, 12.4.1.

Inada, H.

H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,”IEEE Trans. Antennas Propag. AP-18, 89–99 (1970).
[CrossRef]

Jarmie, N.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974), Fig. 2.
[CrossRef]

Khare, V.

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

V. Khare, H. M. Nussenzveig, “The theory of the glory,” in U. Landman, ed., Statistical Mechanics and Statistical Methods in Theory and Application (Plenum, New York, 1977), Sec. 8.3, pp. 723–765.
[CrossRef]

Können, G. P.

Langley, D. S.

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

Marston, P. L.

Mason, B. J.

B. J. Mason, The Physics of Clouds (Clarendon, Oxford, 1971), Fig. 3.9.

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976), Fig. 3.19.

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,”J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

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

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. Khare, H. M. Nussenzveig, “The theory of the glory,” in U. Landman, ed., Statistical Mechanics and Statistical Methods in Theory and Application (Plenum, New York, 1977), Sec. 8.3, pp. 723–765.
[CrossRef]

Pinnick, R. G.

R. G. Pinnick, D. M. Garvey, L. D. Duncan, “Calibration of Knollenberg FSSP light-scattering counters for measurement of cloud droplets,”J. Appl. Meteorol. 20, 1049–1057 (1981).
[CrossRef]

Plonus, M. A.

H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,”IEEE Trans. Antennas Propag. AP-18, 89–99 (1970).
[CrossRef]

Prins, J. A.

J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de regenboogkleuren,” Physica 11, 49–60 (1944).
[CrossRef]

Probert-Jones, J. R.

Querfeld, C. W.

Reesnick, J. J. M.

J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de regenboogkleuren,” Physica 11, 49–60 (1944).
[CrossRef]

Sassen, K.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Art. 10.4.32.

Stork, D.

D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), color plate 2.3.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), p. 181.

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 (Dover, New York, 1981), Sec. 13.23.

Walker, J. D.

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

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

Wood, R. W.

R. W. Wood, Physical Optics, 3rd. ed. (Macmillan, New York, 1934), pp. 186–187.

Wright, W. D.

W. D. Wright, The Measurement of Colour (Hilger, London, 1969), Chap. 4, App. 2, Table 4.

Am. J. Phys.

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

Appl. Opt.

IEEE Trans. Antennas Propag.

H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,”IEEE Trans. Antennas Propag. AP-18, 89–99 (1970).
[CrossRef]

J. Appl. Meteorol.

R. G. Pinnick, D. M. Garvey, L. D. Duncan, “Calibration of Knollenberg FSSP light-scattering counters for measurement of cloud droplets,”J. Appl. Meteorol. 20, 1049–1057 (1981).
[CrossRef]

J. Clim. Appl. Meteor.

T. A. Cerni, “Determination of the size and concentration of cloud drops with an FSSP,”J. Clim. Appl. Meteor. 22, 1346–1355 (1983).
[CrossRef]

J. Math. Phys.

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.

J. Opt. Soc. Am. A

Phys. Rev. Lett.

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]

Physica

J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de regenboogkleuren,” Physica 11, 49–60 (1944).
[CrossRef]

Sci. Am.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231(7), 60–71 (1974), Fig. 2.
[CrossRef]

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

Weatherwise

A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–289 (1983).
[CrossRef]

Other

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), p. 181.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 13.23.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Art. 10.4.32.

D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), color plate 2.3.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Secs. 12.1, 12.4.1.

R. W. Wood, Physical Optics, 3rd. ed. (Macmillan, New York, 1934), pp. 186–187.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 6-3.

W. D. Wright, The Measurement of Colour (Hilger, London, 1969), Chap. 4, App. 2, Table 4.

See Ref. 15, Sec. 13 .33 .

V. Khare, H. M. Nussenzveig, “The theory of the glory,” in U. Landman, ed., Statistical Mechanics and Statistical Methods in Theory and Application (Plenum, New York, 1977), Sec. 8.3, pp. 723–765.
[CrossRef]

See Ref. 15, Secs. 13.32, 17.42.

B. J. Mason, The Physics of Clouds (Clarendon, Oxford, 1971), Fig. 3.9.

E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976), Fig. 3.19.

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Figures (6)

Fig. 1
Fig. 1

Intensity in the supernumerary rainbow region of coherent light (λ = 0.55 μm) scattered by a water droplet as a function of the scaled position Y0 defined in Eq. (2.20). The primary rainbow intensity is normalized to unity.

Fig. 2
Fig. 2

Intensity in the supernumerary rainbow region of sunlight (λ = 0.55 μm) scattered by a 250-μm-radius water droplet as a function of the scaled position Y0. The primary rainbow intensity is normalized to unity. The difference from Fig. 1 arises from the partial spatial coherence of sunlight.

Fig. 3
Fig. 3

Intensity in the glory region of coherent light (λ = 0.55 μm) light scattered by a water droplet as a function of the scaled position R0 defined in Eqs. (1.7) and (3.1). The backscattered intensity is normalized to unity.

Fig. 4
Fig. 4

Glory observability factor U of Eq. (3.10) for the first glory ring (solid curve), the second glory ring (dashed curve), and the third glory ring (dotted curve) as a function of the water droplet radius.

Fig. 5
Fig. 5

Intensity in the glory region of sunlight (λ = 0.55 μm) scattered by a 15- μm-radius water droplet as a function of the scaled position R0. The backscattered intensity is normalized to unity. The difference from Fig. 3 arises from the partial spatial coherence of sunlight.

Fig. 6
Fig. 6

Intensity in the glory region of sunlight (λ = 0.55 μm) scattered by a 30-μm-radius water droplet as a function of the scaled position R0. The backscattered intensity is normalized to unity. The difference from Fig. 3 arises from the partial spatial coherence of sunlight.

Tables (1)

Tables Icon

Table 1 Characteristics of Supernumerariesa

Equations (42)

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

d 2 λ π ( z i 2 s ) .
I 0 ( r 0 ) = I inc λ 2 z 0 2 G 2 ( a ) F ( K x m r 0 z 0 ) ,
x = 2 π a λ .
F = 4 J 1 2 ( x r 0 z 0 ) ( x r 0 z 0 ) 2 .
G = π a 2 .
I 0 ( r 0 ) = I inc λ 2 z 0 2 G 2 ( a ) S d 2 r i z i 2 F [ K x m ( r 0 z 0 + r i z i ) ] .
R 0 = K x m r 0 z 0
W = - K x m r i z i ,
I 0 ( R 0 ) = I inc λ 2 z 0 2 G 2 ( a ) ( K x m ) - 2 d 2 W A S ( W ) F ( R 0 - W ) ,
A S ( W ) = { 1 if W K x m s / z i 0 if W > K x m s / z i .
I 0 ( R 0 ) G 2 ( a ) π s 2 F ave ( R 0 ) ,
θ R = π + 2 arcsin ( 4 - n 2 3 ) 1 / 2 - 4 arcsin ( 4 - n 2 3 n 2 ) 1 / 2 ,
ξ R = arcsin ( 4 - n 2 3 ) 1 / 2 .
L R = 2 a [ 1 + 3 ( n 2 - 1 3 ) 1 / 2 ]
ξ = ξ R + ,
θ = θ R + α 2 + β 3 + γ 4 + O ( 5 ) ,
α = 3 4 ( 4 - n 2 n 2 - 1 ) 1 / 2 ,
β = n 2 + 8 16 ( n 2 - 1 ) ,
γ = 17 n 2 - 8 256 ( n 2 - 1 ) ( 4 - n 2 n 2 - 1 ) 1 / 2 .
± = ± ( ψ α ) 1 / 2 - β 2 α 2 ψ ± 1 2 α 5 / 2 ( 5 4 β 2 α - γ ) ψ 3 / 2 + O ( ψ 2 )
ψ = θ - θ R .
L ( ξ ± ) = L R + a ( 4 - n 2 3 ) ψ ± a 2 ( 4 - n 2 3 ) 1 / 2 ( ψ α ) 3 / 2 + a 36 ( 11 n 2 - 56 4 - n 2 ) ( n 2 - 1 3 ) 1 / 2 ψ 2 + O ( ψ 5 / 2 ) .
L ( ξ + ) - L ( ξ - ) = N λ
ψ = 3 4 ( 3 ) 1 / 3 ( 4 - n 2 ) 1 / 6 ( n 2 - 1 ) 1 / 2 ( N λ a ) 2 / 3 .
δ = a sin ξ + - a sin ξ - .
G = ( Δ r ) ( r Δ ϕ ) a 7 / 6 .
F ( x 0 , y 0 ) = Ai 2 ( - K x 2 / 3 y 0 z 0 ) ,
K = ( 2 3 ) 2 / 3 ( n 2 - 1 ) 1 / 2 ( 4 - n 2 ) 1 / 6 ,
Ai ( - u ) = ( 3 2 π 2 ) 1 / 3 0 cos π 2 [ v 3 - ( 12 π 2 ) 1 / 3 v u ] d v .
y 0 z 0 = ψ 3 4 ( 3 ) 1 / 3 ( 4 - n 2 ) 1 / 6 ( n 2 - 1 ) 1 / 2 [ ( N + 1 4 ) λ a ] 2 / 3
Y 0 = K x 2 / 3 y 0 / z 0 ,
V = I max - I min I max + I min ,
F = ( c 1 + c 2 ) 2 J 0 2 ( x r 0 z 0 ) + ( c 1 - c 2 ) 2 J 2 2 ( x r 0 z 0 )
c 1 = - 0.2 , c 2 = 1.0.
G = 2 π a ( Δ r ) a 4 / 3
c 1 c 2 exp ( - 0.4 x 1 / 3 ) .
I 0 x 8 / 3 exp ( - 0.8 x 1 / 3 ) ,
x = 1000
a 88 μ m .
c 1 = - 0.2 exp ( - 0.4 x 1 / 3 ) ,
c 2 = 1.0 exp ( - 0.4 x 1 / 3 ) .
U = x 8 / 3 exp ( - 0.8 x 1 / 3 ) V ,

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