Abstract

The parallel iteration procedure for computing scattering by a multilayer sphere is described. The procedure uses a successive doubling strategy applied to four sets of multiple-scattering amplitudes, which is reminiscent of the fast Fourier transform (FFT) algorithm. The procedure is then used to calculate scattering of a plane wave by a modified Luneburg lens. The evolution of the transmission rainbow for the Luneburg lens parameter f>1 into an orbiting ray for f=1 and into a series of morphology-dependent resonances for f<1 is studied, and various features of the scattered intensity as a function of scattering angle are commented on. It is found that some resonances are formed without the presence of an exterior centrifugal barrier to confine them.

© 2008 Optical Society of America

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References

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  1. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. I. Ray theory,” J. Opt. Soc. Am. A 25, 2971-2979 (2008).
    [CrossRef]
  2. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A 25, 2980-2990 (2008).
    [CrossRef]
  3. O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657-3660 (1981).
    [CrossRef] [PubMed]
  4. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960-1967 (1985).
    [CrossRef] [PubMed]
  5. D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551-1559 (1990).
    [CrossRef] [PubMed]
  6. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for a multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
    [CrossRef]
  7. L. Kai and P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501-511 (1994).
    [CrossRef] [PubMed]
  8. B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286-3296 (1996).
    [CrossRef] [PubMed]
  9. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Grehan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188-5198 (1997).
    [CrossRef] [PubMed]
  10. J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594-5603 (2005).
    [CrossRef] [PubMed]
  11. R. C. Gonzalez, Digital Image Processing, 3rd ed. (Pearson/Prentice Hall, 2008), pp. 299-302.
  12. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 124-125.
  13. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 365, Eq. (9.3.1); p. 437, Eq. (10.1.1).
  14. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505-1509 (1980).
    [CrossRef] [PubMed]
  15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 127, 128, and 478.
  16. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82-124 (1969).
    [CrossRef]
  17. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 176-178.
  18. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 209-210.
  19. P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285-287 (1976).
    [CrossRef]
  20. P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229-2233 (1978).
    [CrossRef]
  21. C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585-1592 (1992).
    [CrossRef]
  22. P. Chylek, J. D. Pendleton, and R. G. Pinnick, “Internal and near-surface scattered field of a spherical particle at resonant conditions,” Appl. Opt. 24, 3940-3942 (1985).
    [CrossRef] [PubMed]
  23. M. Schneider and E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379-2388 (1994).
    [CrossRef] [PubMed]
  24. J. P. A. J. van Beeck and M. L. Reithmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259-2266 (1996).
    [CrossRef] [PubMed]
  25. P. Massoli, “Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227-3235 (1998).
    [CrossRef]
  26. D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).
    [CrossRef]
  27. K. M. Lee, P. T. Leung, and K. M. Pang, “Iterative perturbation scheme for morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. A 15, 1383-1393 (1998).
    [CrossRef]
  28. P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
    [CrossRef]
  29. W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
    [CrossRef]
  30. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
    [CrossRef] [PubMed]
  31. J. A. Adam and P. Laven, “Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach,” Appl. Opt. 46, 922-929 (2007).
    [CrossRef] [PubMed]
  32. C. L. Brockman and N. G. Alexopoulos, “Geometrical optics of inhomogeneous particles; glory ray and the rainbow revisited,” Appl. Opt. 16, 166-174 (1977).
    [CrossRef] [PubMed]
  33. A. Y. Perelman, “Scattering by particles with radially variable refractive indices,” Appl. Opt. 35, 5452-5460 (1996).
    [CrossRef] [PubMed]

2008 (2)

2007 (1)

2005 (1)

1998 (2)

1997 (1)

1996 (3)

1994 (2)

1992 (1)

1991 (2)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for a multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).
[CrossRef]

1990 (2)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551-1559 (1990).
[CrossRef] [PubMed]

1989 (1)

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

1985 (2)

1984 (1)

P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

1981 (1)

1980 (1)

1978 (1)

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

1977 (1)

1976 (1)

1969 (1)

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

Ackerman, T. P.

Adam, J. A.

Alexopoulos, N. G.

Altenkirch, R. A.

Arnott, W. P.

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

Barber, P. W.

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).
[CrossRef]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Bhandari, R.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 127, 128, and 478.

Brockman, C. L.

Chowdhury, D. Q.

Chylek, P.

Gonzalez, R. C.

R. C. Gonzalez, Digital Image Processing, 3rd ed. (Pearson/Prentice Hall, 2008), pp. 299-302.

Gouesbet, G.

Grehan, G.

Guo, L. X.

Hill, S. C.

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).
[CrossRef]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Hirleman, E. D.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 127, 128, and 478.

Johnson, B. R.

Kai, L.

Kiehl, J. T.

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

Ko, M. K. W.

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

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Lam, C. C.

Laven, P.

Lee, K. M.

Leung, P. T.

Lock, J. A.

Mackowski, D. W.

Marston, P. L.

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

Massoli, P.

Menguc, M. P.

Nussenzveig, H. M.

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

Pang, K. M.

Pendleton, J. D.

Perelman, A. Y.

Pinnick, R. G.

Reithmuller, M. L.

Ren, K. F.

Schneider, M.

Toon, O. B.

Trinh, E.

P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

van Beeck, J. P. A. J.

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 176-178.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 209-210.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 124-125.

Wang, Y. P.

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for a multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

Wiscombe, W. J.

Wu, Z. S.

Young, K.

C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585-1592 (1992).
[CrossRef]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Appl. Opt. (15)

O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657-3660 (1981).
[CrossRef] [PubMed]

R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960-1967 (1985).
[CrossRef] [PubMed]

D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551-1559 (1990).
[CrossRef] [PubMed]

L. Kai and P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501-511 (1994).
[CrossRef] [PubMed]

B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286-3296 (1996).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Grehan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188-5198 (1997).
[CrossRef] [PubMed]

J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594-5603 (2005).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505-1509 (1980).
[CrossRef] [PubMed]

P. Chylek, J. D. Pendleton, and R. G. Pinnick, “Internal and near-surface scattered field of a spherical particle at resonant conditions,” Appl. Opt. 24, 3940-3942 (1985).
[CrossRef] [PubMed]

M. Schneider and E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379-2388 (1994).
[CrossRef] [PubMed]

J. P. A. J. van Beeck and M. L. Reithmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259-2266 (1996).
[CrossRef] [PubMed]

P. Massoli, “Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227-3235 (1998).
[CrossRef]

J. A. Adam and P. Laven, “Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach,” Appl. Opt. 46, 922-929 (2007).
[CrossRef] [PubMed]

C. L. Brockman and N. G. Alexopoulos, “Geometrical optics of inhomogeneous particles; glory ray and the rainbow revisited,” Appl. Opt. 16, 166-174 (1977).
[CrossRef] [PubMed]

A. Y. Perelman, “Scattering by particles with radially variable refractive indices,” Appl. Opt. 35, 5452-5460 (1996).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (1)

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

J. Math. Phys. (1)

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. (1)

J. Opt. Soc. Am. A (4)

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

Nature (1)

P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

Phys. Rev. A (2)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

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

Radio Sci. (1)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for a multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

Other (6)

R. C. Gonzalez, Digital Image Processing, 3rd ed. (Pearson/Prentice Hall, 2008), pp. 299-302.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 124-125.

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 365, Eq. (9.3.1); p. 437, Eq. (10.1.1).

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 176-178.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 209-210.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 127, 128, and 478.

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Figures (6)

Fig. 1
Fig. 1

TE (solid curve) and TM (dashed curve) intensity as a function of the scattering angle θ for a Luneburg lens with f = 1.0 , a = 8.117 μ m , and λ = 0.51 μ m computed for an M = 128 layer sphere using the parallel iteration procedure.

Fig. 2
Fig. 2

TE intensity as a function of the scattering angle θ for a modified Luneburg lens with f = 0.9 (dashed curve), f = 1.0 (solid curve), and f = 1.1 (dotted–dashed curve) for a = 8.117 μ m and λ = 0.51 μ m computed for an M = 128 layer sphere using the parallel iteration procedure.

Fig. 3
Fig. 3

(a) Scattered efficiency ε as a function of sphere radius for a modified Luneburg lens with f = 0.9 and λ = 0.51 μ m . The efficiency for the TM polarization has been vertically offset by Δ ε = 0.1 for clarity. (b) TE scattered efficiency ε for a smaller range of sphere radii illustrating the S = 0 resonances in the partial waves n = 48 through n = 53 .

Fig. 4
Fig. 4

Effective radial potential of the partial wave n = 50 as a function of r a for k a = 51.584 corresponding to the S = 0 resonance. The effective energy of this size parameter is denoted by the horizontal line U eff = 2661 .

Fig. 5
Fig. 5

(a) TE scattered efficiency ε as a function of sphere radius for a modified Luneburg lens with f = 0.75 and λ = 0.51 μ m . (b) TE scattered efficiency ε for a smaller range of sphere radii illustrating the S = 0 resonances in the partial waves n = 50 through n = 56 .

Fig. 6
Fig. 6

Effective radial potential of the partial wave n = 50 as a function of r a for (a) k a = 49.403 corresponding to the S = 0 resonance and (b) k a = 51.510 corresponding to the S = 1 resonance. The effective energy of this size parameters is denoted by the horizontal line U eff = 2441 in (a) and U eff = 2653 in (b).

Tables (1)

Tables Icon

Table 1 TE and TM Resonant Size Parameters for f = 0.9 , λ = 0.51 μ m , and M = 128 as a Function of the Partial Wave Number n a

Equations (58)

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

X j , j + 1 = N j + 1 k a j ,
Y j , j + 1 = N j k a j .
N ( r ) = [ 1 + f 2 ( r a ) 2 ] 1 2 f
E n ( w ) = ψ n ( w ) ψ n ( w ) ,
F n ( w ) = χ n ( w ) χ n ( w ) ,
G n ( w ) = ψ n ( w ) χ n ( w ) ,
E n ( w ) ( n + 1 ) w ,
F n ( w ) n w ,
G n ( w ) ( 1 2 ) ( e 2 ) 2 n + 1 [ w ( n + 1 2 ) ] 2 n + 1 .
G n ( w ) G n ( w + ε ) ( 1 ε w ) 2 n + 1 ,
α = N j for TE ,
= N j + 1 for TM ,
= N j + 1 for TE ,
= N j for TM .
N j , j + 1 = α ψ ( X j , j + 1 ) ψ ( Y j , j + 1 ) β ψ ( X j , j + 1 ) ψ ( Y j , j + 1 ) ,
D j , j + 1 = α χ ( X j , j + 1 ) ψ ( Y j , j + 1 ) β χ ( X j , j + 1 ) ψ ( Y j , j + 1 ) ,
P j , j + 1 = α ψ ( X j , j + 1 ) χ ( Y j , j + 1 ) β ψ ( X j , j + 1 ) χ ( Y j , j + 1 ) ,
Q j , j + 1 = α χ ( X j , j + 1 ) χ ( Y j , j + 1 ) β χ ( X j , j + 1 ) χ ( Y j , j + 1 ) .
N j , j + 1 = χ ( X j , j + 1 ) χ ( Y j , j + 1 ) G ( X j , j + 1 ) G ( Y j , j + 1 ) n j , j + 1 ,
D j , j + 1 = χ ( X j , j + 1 ) χ ( Y j , j + 1 ) G ( Y j , j + 1 ) d j , j + 1 ,
P j , j + 1 = χ ( X j , j + 1 ) χ ( Y j , j + 1 ) G ( X j , j + 1 ) p j , j + 1 ,
Q j , j + 1 = χ ( X j , j + 1 ) χ ( Y j , j + 1 ) q j , j + 1 ,
n j , j + 1 = α E ( Y j , j + 1 ) β E ( X j , j + 1 ) ,
d j , j + 1 = α E ( Y j , j + 1 ) β F ( X j , j + 1 ) ,
p j , j + 1 = α F ( Y j , j + 1 ) β E ( X j , j + 1 ) ,
q j , j + 1 = α F ( Y j , j + 1 ) β F ( X j , j + 1 ) .
a n , b n = N 12 ( N 12 + i D 12 ) .
a n , b n = G ( X 12 ) n 12 [ G ( X 12 ) n 12 + i d 12 ] .
N 123 = D 12 N 23 N 12 P 23 = G ( Y 12 ) G ( Y 23 ) G ( X 23 ) n 123 ,
D 123 = D 12 D 23 N 12 Q 23 = G ( Y 12 ) G ( Y 23 ) d 123 ,
P 123 = Q 12 N 23 P 12 P 23 = G ( Y 23 ) G ( X 23 ) p 123 ,
Q 123 = Q 12 D 23 P 12 Q 23 = G ( Y 23 ) q 123 ,
n 123 = d 12 n 23 n 12 p 23 G ( X 12 ) G ( Y 23 ) ,
d 123 = d 12 d 23 n 12 q 23 G ( X 12 ) G ( Y 23 ) ,
p 123 = q 12 n 23 p 12 p 23 G ( X 12 ) G ( Y 23 ) ,
q 123 = q 12 d 23 p 12 q 23 G ( X 12 ) G ( Y 23 ) .
a n , b n = N 123 ( N 123 + i D 123 ) .
a n , b n = G ( X 23 ) n 123 [ G ( X 23 ) n 123 + i d 123 ] .
a n , b n = N 12 ... M + 1 ( N 12 ... M + 1 + i D 12 ... M + 1 ) .
a n , b n = G ( X M , M + 1 ) n 12 ... M + 1 [ G ( X M , M + 1 ) n 12 ... M + 1 + i d 12 ... M + 1 ] .
n max = 1 + X 128 , 129 + 4.3 ( X 128 , 129 ) 1 3 = 80 .
E n 1 ( w ) = ( n w ) 1 [ E n ( w ) + ( n w ) ]
n start = n max + 15 ,
E n start + 1 ( w ) = 0 .
F 0 ( w ) = tan ( w ) ,
F n ( w ) = ( n w ) + [ ( n w ) F n 1 ( w ) ] .
G 0 ( X j 1 , j ) G 0 ( Y j , j + 1 ) = tan ( X j 1 , j ) tan ( Y j , j + 1 ) ,
G n ( X j 1 , j ) G n ( Y j , j + 1 ) = G n 1 ( X j 1 , j ) [ ( n X j 1 , j ) + F n ( X j 1 , j ) ] [ ( n Y j , j + 1 ) + E n ( Y j , j + 1 ) ] { G n 1 ( Y j , j + 1 ) [ ( n X j 1 , j ) + E n ( X j 1 , j ) ] [ ( n Y j , j + 1 ) + F n ( Y j , j + 1 ) ] } .
j = 2 T 1 + 1 + K ( 2 T ) .
I ( θ ) { F ( ) F [ Δ ( k a π ) 1 2 ] } 2 ,
F ( w ) = 0 w d v exp ( i π v 2 2 )
θ = ( π 2 ) + Δ .
φ trans = k a + k a [ ( f 2 + 1 ) f ] arcsin [ ( f 2 + 1 ) 1 2 ] π ,
φ diff = 2 k a + π 2 .
φ trans φ diff = 2 π P ,
k a = ( n + 2 S + 3 2 ) [ 2 f ( f 2 + 1 ) ] ,
X = n ( n + 1 ) ( k a ) 2 .
N ( r ) = { [ ( f 2 + 1 ) f 2 ] ( g f 2 ) ( r a ) 2 } 1 2

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