Abstract

Observations of an illuminated water droplet at a close distance are described mathematically by the Fourier transform of the Mie-scattering amplitude convolved with the aperture function of the observer’s eye. Most of the sharp enhancements found in the Fourier transform correspond to geometrical rays associated with the various terms in the Debye-series expansion of the Mie amplitude. However, there are some enhancements that cannot be ascribed to any individual Debye-series term. Instead, they arise from a constructive interference cooperation of the phase of a scattering resonance in a single partial wave with the region of the stationary phase corresponding to geometrical orbiting in the m-internal-reflection portion of the Fourier transform of the scattering amplitude. This phase cooperation amplifies the contribution that the scattering resonance makes to the Fourier-transform amplitude.

© 1988 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, 116–127 (1977).
    [CrossRef]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Sec. 13.23.
  3. C. W. Querfeld, “Mie atmospheric optics,” J. Opt. Soc. Am. 55, 105–106 (1965).
    [CrossRef]
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 4.3.3.
  5. 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]
  6. J. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  7. J. Lock, “The theory of the observations made of high order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  8. H. M. Nussenzveig, “High frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  9. Ref. 2, Sec. 12.35.
  10. Ref. 2, Sec. 12.32.
  11. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 9.6.
  12. J. R. Reitz, F. J. Milford, R. W. Christie, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.5.
  13. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  14. Ref. 2, Sec. 12.31.
  15. A. Messiah, Quantum Mechanics (Wiley, New York, 1968), Sec. 10.9.
  16. Ref. 2, Sec. 12.33.
  17. J. R. Probert-Jones, “Surface waves in backscattering and the localization principle,” J. Opt. Soc. Am. 73, 503 (1983).
    [CrossRef]
  18. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954) Sec. 125.
  19. K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (NY) 7, 259–286 (1959).
    [CrossRef]
  20. V. Khare, H. M. Nussenzveig, “The theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1977), pp. 723–764.
    [CrossRef]
  21. P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
    [CrossRef]
  22. P. Chýlek, J. D. Pendleton, R. G. Pinnick, “Internal and near-surface scattered fields of a spherical particle at resonant conditions,” Appl. Opt. 24, 3940–3942 (1985).
    [CrossRef]
  23. P. Chýlek, “Partial wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285–287 (1976).
    [CrossRef]
  24. P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
    [CrossRef] [PubMed]
  25. P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
    [CrossRef]
  26. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A. 1, 822–830 (1984).
    [CrossRef]
  27. E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), Sec. 6.8.
  28. J. A. Lock, “The temporary capture of light by a dielectric film,” Am. J. Phys. 53, 968–971 (1985).
    [CrossRef]
  29. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  30. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
    [CrossRef]
  31. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 3.6.1.
  32. J. A. Lock, J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” submitted to Appl. Opt.

1987 (1)

1985 (2)

1984 (2)

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
[CrossRef]

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A. 1, 822–830 (1984).
[CrossRef]

1983 (1)

1980 (1)

1979 (1)

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

1978 (2)

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

1977 (2)

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

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

1976 (2)

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

P. Chýlek, “Partial wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285–287 (1976).
[CrossRef]

1969 (2)

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]

1965 (1)

1959 (1)

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (NY) 7, 259–286 (1959).
[CrossRef]

Barber, P. W.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 4.3.3.

Christie, R. W.

J. R. Reitz, F. J. Milford, R. W. Christie, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.5.

Chýlek, P.

Conwell, P. R.

Ford, K. W.

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (NY) 7, 259–286 (1959).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 9.6.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 4.3.3.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 3.6.1.

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 Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1977), pp. 723–764.
[CrossRef]

Kiehl, J. T.

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Ko, M. K. W.

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

Lock, J.

Lock, J. A.

J. A. Lock, “The temporary capture of light by a dielectric film,” Am. J. Phys. 53, 968–971 (1985).
[CrossRef]

J. A. Lock, J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” submitted to Appl. Opt.

Merzbacher, E.

E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), Sec. 6.8.

Messiah, A.

A. Messiah, Quantum Mechanics (Wiley, New York, 1968), Sec. 10.9.

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christie, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.5.

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954) Sec. 125.

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]

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[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 Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1977), pp. 723–764.
[CrossRef]

Pendleton, J. D.

Pinnick, R. G.

Probert-Jones, J. R.

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A. 1, 822–830 (1984).
[CrossRef]

J. R. Probert-Jones, “Surface waves in backscattering and the localization principle,” J. Opt. Soc. Am. 73, 503 (1983).
[CrossRef]

Querfeld, C. W.

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christie, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.5.

Rushforth, C. K.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Sec. 13.23.

Walker, J.

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

Wheeler, J. A.

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (NY) 7, 259–286 (1959).
[CrossRef]

Wiscombe, W. J.

Woodruff, J. R.

J. A. Lock, J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” submitted to Appl. Opt.

Am. J. Phys. (2)

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

J. A. Lock, “The temporary capture of light by a dielectric film,” Am. J. Phys. 53, 968–971 (1985).
[CrossRef]

Ann. Phys. (NY) (1)

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (NY) 7, 259–286 (1959).
[CrossRef]

Appl. Opt. (4)

J. Math. Phys. (2)

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]

J. Opt. Soc. Am. (4)

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

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

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A. 1, 822–830 (1984).
[CrossRef]

Phys. Rev. A (1)

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Phys. Rev. Lett. (1)

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

Sci. Am. (1)

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

Other (14)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Sec. 13.23.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 4.3.3.

Ref. 2, Sec. 12.35.

Ref. 2, Sec. 12.32.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 9.6.

J. R. Reitz, F. J. Milford, R. W. Christie, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.5.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954) Sec. 125.

V. Khare, H. M. Nussenzveig, “The theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1977), pp. 723–764.
[CrossRef]

Ref. 2, Sec. 12.31.

A. Messiah, Quantum Mechanics (Wiley, New York, 1968), Sec. 10.9.

Ref. 2, Sec. 12.33.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 3.6.1.

J. A. Lock, J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” submitted to Appl. Opt.

E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), Sec. 6.8.

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Figures (7)

Fig. 1
Fig. 1

(a) Geometrical rays produced by a light ray incident upon the top of the droplet. The exiting rays appear on the right side of the droplet. The observation angle is θ0. (b) Geometrical rays produced by a light ray incident upon the bottom of the droplet. The exiting rays appear on the left side of the droplet.

Fig. 2
Fig. 2

Phase of the m = 11 portion of relation (3.11) as a function of the partial wave l. The arrows correspond to the locations of the geometrical rays R a ( 11 ) and R b ( 11 ) according to Eq. (4.1). The stationary-phase point R c ( 11 ) corresponds to an orbiting ray. The partial wave labeled E marks the geometric edge of the droplet.

Fig. 3
Fig. 3

The magnitude squared of the Fourier transform of the Mie electric field as a function of spatial frequency for θ0 = 90 deg, partial waves 7457 and 7470 on resonance, and initially unpolarized light.

Fig. 4
Fig. 4

Magnitude squared of the Fourier transform of the Mie electric field as a function of spatial frequency for θ0 = 90 deg, partial waves 7457 and 7470 off resonance, and initially unpolarized light.

Fig. 5
Fig. 5

Area under the non-Debye enhancement in the S1 polarization as a function of observation angle. The contribution provided by the 11th-order Debye term is denoted by R(11).

Fig. 6
Fig. 6

Area under the non-Debye enhancement in the S2 polarization as a function of observation angle.

Fig. 7
Fig. 7

Area under the non-Debye enhancement at θ0 = 90 deg as a function of droplet radius for the S2 polarization. The contribution provided by the 11th-order Debye term is denoted by R(11).

Tables (2)

Tables Icon

Table 1 Scattering Resonances for the al Polarizationa

Tables Icon

Table 2 Magnitudes and Phases of Various Contributions to the Fourier Transform of the Scattering Amplitudea

Equations (96)

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k = 2 π λ
E incident ( r ) = E 0 exp [ i ( k z ω t ) ] u ˆ x
lim r E scattered ( r ) = i E 0 k r exp [ i ( k r ω t ) ] × [ S 1 ( θ ) sin ϕ u ˆ ϕ S 2 ( θ ) cos ϕ u ˆ θ ] ,
S 1 ( θ ) = l = 1 2 l + 1 l ( l + 1 ) [ a l π l ( θ ) + b l τ l ( θ ) ] ,
S 2 ( θ ) = l = 1 2 l + 1 l ( l + 1 ) [ a l τ l ( θ ) + b l π l ( θ ) ] ,
π l ( θ ) = 1 sin θ P l 1 ( cos θ )
τ l ( θ ) = d d θ P l 1 ( cos θ ) .
J l ( w ) = w j l ( w ) ,
N l ( w ) = w n l ( w ) ,
x k a , y n k a
t l 1 = J l ( x ) J l ( y ) n J l ( x ) J l ( y ) , t l 2 = N l ( x ) N l ( y ) n N l ( x ) N l ( y ) , t l 3 = N l ( x ) J l ( y ) n N l ( x ) J l ( y ) , t l 4 = J l ( x ) N l ( y ) n J l ( x ) N l ( y ) ,
a l = t l 1 t l 1 + i t l 3 .
t l 1 = n J l ( x ) J l ( y ) J l ( x ) J l ( y ) , t l 2 = n N l ( x ) N l ( y ) N l ( x ) N l ( y ) , t l 3 = n N l ( x ) J l ( y ) N l ( x ) J l ( y ) , t l 4 = n J l ( x ) N l ( y ) J l ( x ) N l ( y ) ,
b l = t l 1 t l 1 + i t l 3 .
R l 11 = ( t l 1 t l 2 ) i ( t l 3 + t l 4 ) ( t l 1 + t l 2 ) + i ( t l 3 t l 4 ) ,
R l 22 = ( t l 1 t l 2 ) + i ( t l 3 + t l 4 ) ( t l 1 + t l 2 ) + i ( t l 3 t l 4 ) ,
T l 21 = 2 i / x 2 ( t l 1 + t l 2 ) + i ( t l 3 t l 4 ) ,
T l 12 = 2 i / n x 2 ( t l 1 + t l 2 ) + i ( t l 3 t l 4 ) .
t l 1 t l 1 + i t l 3 = 1 2 ( 1 R l 22 T l 21 T l 12 1 R l 11 ) ,
t l 1 t l 1 + i t l 3 = 1 2 [ 1 R l 22 m = 0 T l 21 ( R l 11 ) m T l 12 ] .
c l = i t l 1 + i t l 3
d l = i t l 1 + i t l 3
i t l 1 + i t l 3 = T l 21 1 R l 11 = m = 0 T l 21 ( R l 11 ) m ,
R l 11 = r l 11 exp ( i ϕ l 11 ) ,
T l 21 = n T l 12 = t l 21 exp ( i ϕ l 21 ) ,
R l 22 = r l 22 exp ( i ϕ l 22 ) .
r l 22 = r l 11 ,
t l 21 = { n [ 1 ( r l 11 ) 2 ] } 1 / 2 ,
ϕ l 22 = 2 ϕ l 21 ϕ l 11 π .
ϕ l 21 = arctan ( t l 1 + t l 2 t l 3 t l 4 )
ϕ l 11 = arctan [ 2 ( t l 1 t l 4 + t l 2 t l 3 ) ( t l 1 ) 2 + ( t l 3 ) 2 ( t l 2 ) 2 ( t l 4 ) 2 ] .
t l 1 t l 1 + i t l 3 = 1 2 { 1 + r l 11 exp [ i ( 2 ϕ l 21 ϕ l 11 ) ] m = 0 [ 1 ( r l 11 ) 2 ] ( r l 11 ) m × exp [ i ( 2 ϕ l 21 + m ϕ l 11 ) ] } .
l max x + 4.3 x 1 / 3
x 4.3 x 1 / 3 l x + 4.3 x 1 / 3
F i ( p , θ 0 ) = θ w / 2 θ w / 2 d ξ S i ( θ 0 + ξ ) e i p ξ ,
F 1 ( p , θ 0 ) = l = 1 l max 2 l + 1 l ( l + 1 ) [ a l U l ( p , θ 0 ) + b l V l ( p , θ 0 ) ] ,
F 2 ( p , θ 0 ) = l = 1 l max 2 l + 1 l ( l + 1 ) [ a l V l ( p , θ 0 ) + b l U l ( p , θ 0 ) ] ,
U l ( p , θ 0 ) = θ w / 2 θ w / 2 d ξ π l ( θ 0 + ξ ) e i p ξ
V l ( p , θ 0 ) = θ w / 2 θ w / 2 d ξ τ l ( θ 0 + ξ ) e i p ξ .
V l ( p , θ 0 ) θ w 2 ( 2 l 3 π ) 1 / 2 exp ( i π / 4 ) { exp ( i l θ 0 ) sinc [ ( p l ) θ w 2 ] + exp ( i l θ 0 ) sinc [ ( p + l ) θ w 2 ] } ,
U l ( p , θ 0 ) 1 l V l ( p , θ 0 ) ,
sinc ( w ) = sin w w .
F 1 ( p , θ 0 ) θ w 2 ( 2 π ) 1 / 2 exp ( i π / 4 ) l = 1 l max 2 l + 1 l ( l + 1 ) l 3 / 2 b l × { exp ( i l θ 0 ) sinc [ ( p 1 ) θ w 2 ] × exp ( i l θ 0 ) sinc [ ( p + 1 ) θ w 2 ] } ,
F 2 ( p , θ 0 ) θ w 2 ( 2 π ) 1 / 2 exp ( i π / 4 ) l = 1 l max 2 l + 1 l ( l + 1 ) l 3 / 2 a l × { exp ( i l θ 0 ) sinc [ ( p 1 ) θ w 2 ] × exp ( i l θ 0 ) sinc [ ( p + 1 ) θ w 2 ] } .
F i ( p , θ 0 ) K l = 1 l max 1 2 { 1 + r l 11 exp [ i ( 2 ϕ l 21 ϕ l 11 ) ] m = 0 [ 1 ( r l 11 ) 2 ] ( r l 11 ) m exp [ i ( 2 ϕ l 21 + m ϕ l 11 ) ] } × { exp ( i l θ 0 ) sinc [ ( p 1 ) θ w 2 ] + exp ( i l θ 0 ) × sinc [ ( p + 1 ) θ w 2 ] } .
θ w = 4.6 0
θ l 0 deviation = m π + 2 arcsin ( p x ) ( 2 m + 2 ) arcsin ( p n x )
θ 0 = { θ l 0 deviation [ mod 2 π ] if 0 θ l 0 deviation [ mod 2 π ] π θ l 0 deviation [ mod 2 π ] if π < θ l 0 deviation [ mod 2 π ] < 0 ,
p = l 0
p = l 0
F i ( m ) ( p , θ 0 ) K 2 l = 1 l max [ 1 ( r l 11 ) 2 ] ( r l 11 ) m exp [ i ( 2 ϕ l 0 21 + m ϕ l 0 11 ) ] × exp [ i z ( 2 Δ ϕ l 0 21 + m Δ ϕ l 0 11 ) ] { exp ( i l 0 θ 0 ) exp ( i z θ 0 ) × sinc [ ( p l 0 z ) θ w 2 ] + exp ( i l 0 θ 0 ) exp ( i z θ 0 ) × sinc [ ( p + l 0 + z ) θ w 2 ] }
ϕ l 21 = ϕ l 0 21 + z Δ ϕ l 0 21 +
ϕ l 11 = ϕ l 0 11 = z Δ ϕ l 0 11 + ,
z = l l 0 .
θ 0 = 2 Δ ϕ l 0 21 m Δ ϕ l 0 11
Δ ϕ l 0 21 = arcsin ( l 0 x ) + arcsin ( l 0 n x )
Δ ϕ l 0 11 = 2 arcsin ( l 0 n x ) π .
p = l 0
p = x cos ( θ 0 / 2 ) ,
ϕ l 21 ( n 2 x 2 l 2 ) 1 / 2 ( x 2 l 2 ) 1 / 2 l × [ arcsin ( l x ) arcsin ( l n x ) ] ,
ϕ l 11 2 ( n 2 x 2 l 2 ) 1 / 2 l [ π 2 arcsin ( l n x ) ] π
b a ( l / x ) .
( ϕ l 21 ) geometric = ( n 2 x 2 l 2 ) 1 / 2 ( x 2 l 2 ) 1 / 2 .
( ϕ l 11 ) geometric = 2 ( n 2 x 2 l 2 ) 1 / 2 .
λ = 0.6328 μ m ,
a = 750.007 μ m
n = 1.331.
lim l x [ 1 ( r l 11 ) 2 ] ( r l 11 ) 11 = 0.
ϕ l 11 = 0 ,
m = 0 T l 21 ( R l 11 ) m T l 12 = [ 1 ( r l 11 ) 2 ] exp [ i ( 2 ϕ l 21 + γ l ) ] [ ( 1 r l 11 ) 2 4 r l 11 sin 2 ( ϕ l 11 / 2 ) ] 1 / 2 ,
tan γ l = r l 11 sin ϕ l 11 1 r l 11 cos ϕ l 11 .
sin ( δ l 11 4 ) = ( 1 r l 11 ) 2 ( 3 r l 11 ) 1 / 2 .
( 1 + r l 11 ) exp ( 2 i ϕ l 21 ) .
r l 11 exp ( 2 i ϕ l 21 )
t l 3 = 0
ϕ l 11 = ϕ l 21 ± arccos ( cos ϕ l 21 r l 11 )
ϕ l 21 = π 2
t l 1 = ( n ) J l ( y ) N l ( x )
t l 2 = ( 1 ) N l ( x ) J l ( y ) ,
J l ( y ) N l ( y ) , J l ( y ) N l ( y ) ,
x l < y ,
t l 4 J l ( y ) ( 1 ) N l ( x ) [ ( 1 ) 2 J l ( x ) N l ( x ) + ( n ) 2 J l ( x ) N l ( x ) ] .
J l ( y ) ~ O ( 1 ) , J l ( x ) 1 2 ( 2 Δ x ) 1 / 4 exp [ x 3 ( 2 Δ x ) 3 / 2 ] , N l ( x ) ( 2 Δ x ) 1 / 4 exp [ x 3 ( 2 Δ x ) 3 / 2 ] , J l ( x ) ( 2 Δ x ) 1 / 4 J l ( x ) , N l ( x ) ( 2 Δ x ) 1 / 4 N l ( x ) ,
Δ = l x .
tan ϕ l 11 ~ ( 2 Δ x ) 1 / 2 exp [ 4 x 3 ( 2 Δ x ) 3 / 2 ]
tan ϕ l 21 ~ exp [ 2 x 3 ( 2 Δ x ) 3 / 2 ] .
= 2 x 2 l = 1 l max ( 2 l + 1 ) ( | a l | 2 + | b l | 2 ) ,
S 1 = S 2 = 1 2 l = 1 l max ( 2 l + 1 ) ( a l + b l ) ,
A i ( θ 0 ) = x σ x + σ d p | F i ( p , θ 0 ) | 2 ,
F i ( m ) ( p , θ 0 ) = K 2 l = 1 l max [ 1 ( r l 11 ) 2 ] ( r l 11 ) m exp [ i ( 2 ϕ l 21 + m ϕ l 11 π ) ] × { exp ( i l θ 0 ) sinc [ ( p l ) θ w 2 ] + exp ( i l θ 0 ) sinc [ ( p + l ) θ w 2 ] } ,
F i resonance ( p , θ 0 ) = K { exp ( i l r θ 0 ) sinc [ ( p l r ) θ w 2 ] + exp ( i l r θ 0 ) sinc [ ( p + l r ) θ w 2 ] } .
l = l r + z ,
2 π I = z Δ ϕ l 11 ,
θ 0 = ( L I ) 2 π | z | 2 Δ ϕ ave 21 ( γ l r γ l ) | z | ,
θ 0 = ( L + I ) 2 π | z | + 2 Δ ϕ ave 21 + ( γ l r γ l ) | z | ,
Δ ϕ ave 21 = ϕ l r 21 ϕ l 21 | z | .

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