Abstract

A dielectric sphere with an eccentric spherical dielectric inclusion and an incident amplitude-modulated plane electromagnetic wave constitute an exterior radiation problem, which is solved in this paper. A solution is obtained by combined use of the Fourier transform and the indirect-mode-matching method. The analysis yields a set of linear equations for the wave amplitudes of the frequency-domain expansion of the electric-field intensity within and outside the externally spherical inhomogeneous body; that set is solved by truncation and matrix inversion. The shape of the backscattered pulse in the time domain is determined by application of the inverse fast Fourier transform. Numerical results are shown for a pulse backscattered by an acrylic sphere that contains an eccentric spherical cavity. The effects of cavity position and size on pulse spreading and delay are discussed.

© 2012 Optical Society of America

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References

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  1. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic, 2000).
  2. N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A 11, 1859–1866 (1994).
    [CrossRef]
  3. M. P. Ioannidou, N. C. Skaropoulos, and D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
    [CrossRef]
  4. M. P. Ioannidou and D. P. Chrissoulidis, “Electromagnetic-wave scattering by a sphere with multiple spherical inclusions,” J. Opt. Soc. Am. A 19, 505–512 (2002).
    [CrossRef]
  5. A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. 24, 1695–1703 (2007).
    [CrossRef]
  6. A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. 24, 3437–3443 (2007).
    [CrossRef]
  7. B. Miu and W. Yawei, “Scattering analysis for eccentric-sphere model of single-nuclear cell,” in Proceedings of Symposium on Photonics and Optoelectronics (2011), pp. 1–4.
  8. J. G. Fikioris and N. K. Uzunoglou, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
    [CrossRef]
  9. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
    [CrossRef]
  10. J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
    [CrossRef]
  11. G. Han, Y. Han, J. Liu, and Y. Zhang, “Scattering of an eccentric sphere arbitrarily located in a shaped beam,” J. Opt. Soc. Am. B 25, 2064–2072 (2008).
    [CrossRef]
  12. F. Borghese, P. Denti, and R. Saija, “Optical properties of spheres containing several spherical inclusions,” Appl. Opt. 33, 484–493 (1994).
    [CrossRef]
  13. N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
    [CrossRef]
  14. A. P. Moneda, M. P. Ioannidou, and D. P. Chrissoulidis, “Radio-wave exposure of the human head: analytical study based on a versatile eccentric spheres model including a brain core and a pair of eyeballs,” IEEE Trans. Biomed. Eng. 50, 667–676 (2003).
    [CrossRef]
  15. E. E. M. Khaled and A. B. Alhasan, “Temporal behavior of short optical pulses scattered by small particles,” Phys. Scripta 54, 525–529 (1996).
    [CrossRef]
  16. L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
    [CrossRef]
  17. Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
    [CrossRef]
  18. G. Gouesbet and G. Gréhan, “Generic formulation of a generalized Lorenz-Mie theory for a particle illuminated by laser pulses,” Part. Part. Syst. Charact. 17, 213–224 (2000).
    [CrossRef]
  19. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  20. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1953), pp. 1864–1891.
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  22. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).
  23. S. Stein, “Addition theorems for spherical vector wave functions,” Quart. Appl. Math. 19, 15–24 (1961).
  24. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).
  25. J. D. Kanellopoulos and J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).
  26. Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
    [CrossRef]
  27. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, 1978).

2009 (1)

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

2008 (1)

2007 (2)

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. 24, 1695–1703 (2007).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. 24, 3437–3443 (2007).
[CrossRef]

2004 (1)

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

2003 (1)

A. P. Moneda, M. P. Ioannidou, and D. P. Chrissoulidis, “Radio-wave exposure of the human head: analytical study based on a versatile eccentric spheres model including a brain core and a pair of eyeballs,” IEEE Trans. Biomed. Eng. 50, 667–676 (2003).
[CrossRef]

2002 (1)

2000 (1)

G. Gouesbet and G. Gréhan, “Generic formulation of a generalized Lorenz-Mie theory for a particle illuminated by laser pulses,” Part. Part. Syst. Charact. 17, 213–224 (2000).
[CrossRef]

1998 (1)

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

1996 (2)

E. E. M. Khaled and A. B. Alhasan, “Temporal behavior of short optical pulses scattered by small particles,” Phys. Scripta 54, 525–529 (1996).
[CrossRef]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

1995 (2)

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

M. P. Ioannidou, N. C. Skaropoulos, and D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
[CrossRef]

1994 (2)

1992 (1)

1979 (2)

J. G. Fikioris and N. K. Uzunoglou, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
[CrossRef]

J. D. Kanellopoulos and J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical vector wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Alhasan, A. B.

E. E. M. Khaled and A. B. Alhasan, “Temporal behavior of short optical pulses scattered by small particles,” Phys. Scripta 54, 525–529 (1996).
[CrossRef]

Borghese, F.

Chrissoulidis, D. P.

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. 24, 3437–3443 (2007).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. 24, 1695–1703 (2007).
[CrossRef]

A. P. Moneda, M. P. Ioannidou, and D. P. Chrissoulidis, “Radio-wave exposure of the human head: analytical study based on a versatile eccentric spheres model including a brain core and a pair of eyeballs,” IEEE Trans. Biomed. Eng. 50, 667–676 (2003).
[CrossRef]

M. P. Ioannidou and D. P. Chrissoulidis, “Electromagnetic-wave scattering by a sphere with multiple spherical inclusions,” J. Opt. Soc. Am. A 19, 505–512 (2002).
[CrossRef]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

M. P. Ioannidou, N. C. Skaropoulos, and D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
[CrossRef]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A 11, 1859–1866 (1994).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

Denti, P.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1953), pp. 1864–1891.

Fikioris, J. G.

J. G. Fikioris and N. K. Uzunoglou, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
[CrossRef]

J. D. Kanellopoulos and J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

Gouesbet, G.

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

G. Gouesbet and G. Gréhan, “Generic formulation of a generalized Lorenz-Mie theory for a particle illuminated by laser pulses,” Part. Part. Syst. Charact. 17, 213–224 (2000).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Gréhan, G.

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

G. Gouesbet and G. Gréhan, “Generic formulation of a generalized Lorenz-Mie theory for a particle illuminated by laser pulses,” Part. Part. Syst. Charact. 17, 213–224 (2000).
[CrossRef]

Han, G.

Han, Y.

Han, Y. P.

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

Ioannidou, M. P.

A. P. Moneda, M. P. Ioannidou, and D. P. Chrissoulidis, “Radio-wave exposure of the human head: analytical study based on a versatile eccentric spheres model including a brain core and a pair of eyeballs,” IEEE Trans. Biomed. Eng. 50, 667–676 (2003).
[CrossRef]

M. P. Ioannidou and D. P. Chrissoulidis, “Electromagnetic-wave scattering by a sphere with multiple spherical inclusions,” J. Opt. Soc. Am. A 19, 505–512 (2002).
[CrossRef]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

M. P. Ioannidou, N. C. Skaropoulos, and D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
[CrossRef]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A 11, 1859–1866 (1994).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, 1978).

Kakogiannos, N. B.

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

Kanellopoulos, J. D.

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

J. D. Kanellopoulos and J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

Khaled, E. E. M.

E. E. M. Khaled and A. B. Alhasan, “Temporal behavior of short optical pulses scattered by small particles,” Phys. Scripta 54, 525–529 (1996).
[CrossRef]

Liu, J.

Méès, L.

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

Miu, B.

B. Miu and W. Yawei, “Scattering analysis for eccentric-sphere model of single-nuclear cell,” in Proceedings of Symposium on Photonics and Optoelectronics (2011), pp. 1–4.

Moneda, A. P.

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. 24, 1695–1703 (2007).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. 24, 3437–3443 (2007).
[CrossRef]

A. P. Moneda, M. P. Ioannidou, and D. P. Chrissoulidis, “Radio-wave exposure of the human head: analytical study based on a versatile eccentric spheres model including a brain core and a pair of eyeballs,” IEEE Trans. Biomed. Eng. 50, 667–676 (2003).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1953), pp. 1864–1891.

Ren, K. F.

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

Roumeliotis, J. A.

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Saija, R.

Sindoni, O. I.

Skaropoulos, N. C.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Stein, S.

S. Stein, “Addition theorems for spherical vector wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Uzunoglou, N. K.

Wu, Z. S.

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

Xu, Y. L.

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

Yawei, W.

B. Miu and W. Yawei, “Scattering analysis for eccentric-sphere model of single-nuclear cell,” in Proceedings of Symposium on Photonics and Optoelectronics (2011), pp. 1–4.

Zhang, Y.

Appl. Opt. (1)

IEEE Trans. Biomed. Eng. (1)

A. P. Moneda, M. P. Ioannidou, and D. P. Chrissoulidis, “Radio-wave exposure of the human head: analytical study based on a versatile eccentric spheres model including a brain core and a pair of eyeballs,” IEEE Trans. Biomed. Eng. 50, 667–676 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

J. Comput. Phys. (1)

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

J. Opt. Soc. Am. (3)

J. G. Fikioris and N. K. Uzunoglou, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. 24, 1695–1703 (2007).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. 24, 3437–3443 (2007).
[CrossRef]

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

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

Opt. Commun. (2)

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

Part. Part. Syst. Charact. (1)

G. Gouesbet and G. Gréhan, “Generic formulation of a generalized Lorenz-Mie theory for a particle illuminated by laser pulses,” Part. Part. Syst. Charact. 17, 213–224 (2000).
[CrossRef]

Phys. Scripta (1)

E. E. M. Khaled and A. B. Alhasan, “Temporal behavior of short optical pulses scattered by small particles,” Phys. Scripta 54, 525–529 (1996).
[CrossRef]

Quart. Appl. Math. (3)

S. Stein, “Addition theorems for spherical vector wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

J. D. Kanellopoulos and J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

Other (7)

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, 1978).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic, 2000).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1953), pp. 1864–1891.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

B. Miu and W. Yawei, “Scattering analysis for eccentric-sphere model of single-nuclear cell,” in Proceedings of Symposium on Photonics and Optoelectronics (2011), pp. 1–4.

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Figures (4)

Fig. 1.
Fig. 1.

Geometry of radiation problem.

Fig. 2.
Fig. 2.

Monochromatic excitation of acrylic sphere with spherical cavity (α1/α2=1/3, d1=α2α1, Θ1=0°): σmo/πα22 versus k0,cα2.

Fig. 4.
Fig. 4.

Pulsed excitation of acrylic sphere (α2=10cm, k0,cα2=14.1) with spherical cavity (d1=α2α1, Θ1=180°): effect of cavity size on backscattered pulse σmo(τb)/πα22, shown on the same scale as the (reference) incident pulse |f0(τ)|2.

Fig. 3.
Fig. 3.

Pulsed excitation of acrylic sphere (α2=10cm, k0,cα2=14.1) with spherical cavity (α1/α2=1/3, d1=α2α1, Φ1=0°): effect of look direction on backscattered pulse σmo(τb)/πα22, shown on the same scale as the (reference) incident pulse |f0(τ)|2.

Tables (2)

Tables Icon

Table 1. Generic IMM Integral I (Single Origin)a

Tables Icon

Table 2. Generic IMM Integral I (Dual Origin)a

Equations (42)

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

Mmn,e(ι)(kr)=θ^mzn(ι)(kr)Pnm(cosθ)sinθsin(mϕ)ϕ^zn(ι)(kr)dPnm(cosθ)dθcos(mϕ),
Nmn,e(ι)(kr)=r^n(n+1)zn(ι)(kr)krPnm(cosθ)cos(mϕ)+θ^[krzn(ι)(kr)]krdPnm(cosθ)dθcos(mϕ)ϕ^m[krzn(ι)(kr)]krPnm(cosθ)sinθsin(mϕ),
Mmn,o(ι)(kr)=θ^mzn(ι)(kr)Pnm(cosθ)sinθcos(mϕ)ϕ^zn(ι)(kr)dPnm(cosθ)dθsin(mϕ),
Nmn,o(ι)(kr)=r^n(n+1)zn(ι)(kr)krPnm(cosθ)sin(mϕ)+θ^[krzn(ι)(kr)]krdPnm(cosθ)dθsin(mϕ)+ϕ^m[krzn(ι)(kr)]krPnm(cosθ)sinθcos(mϕ).
Einc(r,ω)=f0(ωωc)n=1cn[M1n,o(1)(k0r)+jN1n,e(1)(k0r)],
Esca(r,ω)=f0(ωωc)n=1m=0n[Amn,eMmn,e(2)(k0r)+jBmn,eNmn,e(2)(k0r)+Amn,oMmn,o(2)(k0r)+jBmn,oNmn,o(2)(k0r)].
E1(r1,ω)=f0(ωωc)n=1m=0n[Gmn,eMmn,e(1)(k1r1)+jHmn,eNmn,e(1)(k1r1)+Gmn,oMmn,o(1)(k1r1)+jHmn,oNmn,o(1)(k1r1)],
E2(r,ω)=f0(ωωc)n=1m=0n[Cmn,eMmn,e(1)(k2r)+jDmn,eNmn,e(1)(k2r)+Cmn,oMmn,o(1)(k2r)+jDmn,oNmn,o(1)(k2r)+Emn,eMmn,e(2)(k2r1)+jFmn,eNmn,e(2)(k2r1)+Emn,oMmn,o(2)(k2r1)+jFmn,oNmn,o(2)(k2r1)].
S2(E0××QQ××E0)·r^ds2=S1(E1××QQ××E1)·r^1ds1,
S1(E1××QQ××E1)·r^1ds1=S1(E2××QQ××E2)·r^1ds1.
Un(ι,ι)(k,k,α)=2n(n+1)2n+1[kzn(ι)(kα)ηn(ι)(kα)kηn(ι)(kα)zn(ι)(kα)],
Vn(ι,ι)(k,k,α)=2n(n+1)2n+1[kzn(ι)(kα)ηn(ι)(kα)kηn(ι)(kα)zn(ι)(kα)],
[Mmn,e(ι)(kr)Nmn,e(ι)(kr)Mmn,o(ι)(kr)Nmn,o(ι)(kr)]=mn[A^mn,ι˜mn(kd)B^mn,ι˜mn(kd)jA^mn,ι˜mn(kd)jB^mn,ι˜mn(kd)B^mn,ι˜mn(kd)A^mn,ι˜mn(kd)jB^mn,ι˜mn(kd)jA^mn,ι˜mn(kd)Ǎmn,ι˜mn(kd)B̌mn,ι˜mn(kd)jǍmn,ι˜mn(kd)jB̌mn,ι˜mn(kd)B̌mn,ι˜mn(kd)Ǎmn,ι˜mn(kd)jB̌mn,ι˜mn(kd)jǍmn,ι˜mn(kd)][Mmn,e(1˜)(kr)Nmn,e(1˜)(kr)Mmn,o(1˜)(kr)Nmn,o(1˜)(kr)],
[A^mn,ιmn(kd)B^mn,ιmn(kd)Ǎmn,ιmn(kd)B̌mn,ιmn(kd)]=12[1(1)m(n+m)!(nm)!jj(1)m(n+m)!(nm)!][Amn,ιmn(kd)Bmn,ιmn(kd)Amn,ιmn(kd)Bmn,ιmn(kd)],
Ekl,eUl(2,1)(k2,k2,α1)=Gkl,eUl(1,1)(k1,k2,α1),
Ekl,oUl(2,1)(k2,k2,α1)=Gkl,oUl(1,1)(k1,k2,α1),
Fkl,eVl(2,1)(k2,k2,α1)=Hkl,eVl(1,1)(k1,k2,α1),
Fkl,oVl(2,1)(k2,k2,α1)=Hkl,oVl(1,1)(k1,k2,α1),
Esca(r,ω)f0(ωωc)1rexp{jωr/c0}F(r^,ω),
F=Fθθ^+Fϕϕ^=c0ωn=1jn+1m=0n[Fθ,mnθ^+Fϕ,mnϕ^]
Fθ,mn(r^,ω)=mPnm(cosθ)sinθ[Amn,o(ω)cos(mϕ)Amn,e(ω)sin(mϕ)]+dPnm(cosθ)dθ[Bmn,e(ω)cos(mϕ)+Bmn,o(ω)sin(mϕ)]
Fϕ,mn(r^,ω)=mPnm(cosθ)sinθ[Bmn,o(ω)cos(mϕ)Bmn,e(ω)sin(mϕ)]dPnm(cosθ)dθ[Amn,e(ω)cos(mϕ)+Amn,o(ω)sin(mϕ)].
σmoπα22=1(πα2)2f0(ω1ωc)f0(ω2ωc)F(i^,ω1)·F(i^,ω2)ej(ω1ω2)τbdω1dω2,
σmo(τb)πα22=c02(πα2)2n=1jn+1n=1(j)n+1f0(ω1ωc)f0(ω2ωc)×Fθ,1n(i^,ω1)Fθ,1n(i^,ω2)+Fϕ,1n(i^,ω1)Fϕ,1n(i^,ω2)ω1ω2ej(ω1ω2)τbdω1dω2=c02α22n=1(j)nn(n+1)n=1jnn(n+1)×14π2ωcωc+ωcωc+f0(ω1ωc)f0(ω2ωc)Fb,nn(ω1,ω2)ω1ω2ej(ω1ω2)τbdω1dω2,
Fb,nn(ω1,ω2)=[A1n,e(ω1)+B1n,o(ω1)][A1n,e(ω2)+B1n,o(ω2)]+[A1n,o(ω1)B1n,e(ω1)][A1n,o(ω2)B1n,e(ω2)]
σmo(τb)πα22=1(k0,cα2)2n=1(j)nn(n+1)n=1jnn(n+1)×14π2f0(ω1)f0(ω2)Fb,nn(ωc+ω1,ωc+ω2)(1+ω1ωc)(1+ω2ωc)ej(ω1ω2)τbdω1dω2,
-jn=1cn[n(n+1)A^1n,1kl(k2d1)+A^1n,1kl(k2d1)]Un(1,ι)(k0,k2,α2)jn=1cn[n(n+1)B^1n,1kl(k2d1)B^1n,1kl(k2d1)]Vn(1,ι)(k0,k2,α2)+n=1m=0nAmn,e[(n+m)!(nm)!A^mn,1kl(k2d1)+(1)mA^mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nAmn,o[(n+m)!(nm)!A^mn,1kl(k2d1)(1)mA^mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nBmn,e[(n+m)!(nm)!B^mn,1kl(k2d1)+(1)mB^mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)n=1m=0nBmn,o[(n+m)!(nm)!B^mn,1kl(k2d1)(1)mB^mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)=α12α22n=1m=0nGmn,eδnl[(n+m)!(nm)!δmk+(1)mδm,k]Un(1,ι)(k1,k2,α1),
-jn=1cn[n(n+1)Ǎ1n,1kl(k2d1)+Ǎ1n,1kl(k2d1)]Un(1,ι)(k0,k2,α2)jn=1cn[n(n+1)B̌1n,1kl(k2d1)B̌1n,1kl(k2d1)]Vn(1,ι)(k0,k2,α2)+n=1m=0nAmn,e[(n+m)!(nm)!Ǎmn,1kl(k2d1)+(1)mǍmn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nAmn,o[(n+m)!(nm)!Ǎmn,1kl(k2d1)(1)mǍmn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nBmn,e[(n+m)!(nm)!B̌mn,1kl(k2d1)+(1)mB̌mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)n=1m=0nBmn,o[(n+m)!(nm)!B̌mn,1kl(k2d1)(1)mB̌mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)=α12α22n=1m=0nGmn,oδnl[(n+m)!(nm)!δmk(1)mδm,k]Un(1,ι)(k1,k2,α1),
jn=1cn[n(n+1)B^1n,1kl(k2d1)+B^1n,1kl(k2d1)]Un(1,ι)(k0,k2,α2)jn=1cn[n(n+1)A^1n,1kl(k2d1)A^1n,1kl(k2d1)]Vn(1,ι)(k0,k2,α2)+n=1m=0nAmn,e[(n+m)!(nm)!B^mn,1kl(k2d1)+(1)mB^mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nAmn,o[(n+m)!(nm)!B^mn,1kl(k2d1)(1)mB^mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nBmn,e[(n+m)!(nm)!A^mn,1kl(k2d1)+(1)mA^mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)n=1m=0nBmn,o[(n+m)!(nm)!A^mn,1kl(k2d1)(1)mA^mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)=jα12α22n=1m=0nHmn,eδnl[(n+m)!(nm)!δmk+(1)mδm,k]Vn(1,ι)(k1,k2,α1),
jn=1cn[n(n+1)B̌1n,1kl(k2d1)+B̌1n,1kl(k2d1)]Un(1,ι)(k0,k2,α2)jn=1cn[n(n+1)Ǎ1n,1kl(k2d1)Ǎ1n,1kl(k2d1)]Vn(1,ι)(k0,k2,α2)+n=1m=0nAmn,e[(n+m)!(nm)!B̌mn,1kl(k2d1)+(1)mB̌mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nAmn,o[(n+m)!(nm)!B̌mn,1kl(k2d1)(1)mB̌mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nBmn,e[(n+m)!(nm)!Ǎmn,1kl(k2d1)+(1)mǍmn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)n=1m=0nBmn,o[(n+m)!(nm)!Ǎmn,1kl(k2d1)(1)mǍmn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)=jα12α22n=1m=0nHmn,oδnl[(n+m)!(nm)!δmk(1)mδm,k]Vn(1,ι)(k1,k2,α1),
-n=1m=0nCmn,e[(l+k)!(lk)!A^kl,1mn(k2d1)+(1)kA^kl,1mn(k2d1)]Ul(ι,1)(k2,k2,α1)n=1m=0nCmn,o[(l+k)!(lk)!Ǎkl,1mn(k2d1)+(1)kǍkl,1mn(k2d1)]Ul(ι,1)(k2,k2,α1)jn=1m=0nDmn,e[(l+k)!(lk)!B^kl,1mn(k2d1)+(1)kB^kl,1mn(k2d1)]Ul(ι,1)(k2,k2,α1)jn=1m=0nDmn,o[(l+k)!(lk)!B̌kl,1mn(k2d1)+(1)kB̌kl,1mn(k2d1)]Ul(ι,1)(k2,k2,α1)+n=1m=0nEmn,eδnl[(n+m)!(nm)!δmk+(1)mδm,k]Un(2,ι)(k2,k2,α1)=n=1m=0nGmn,eδnl[(n+m)!(nm)!δmk+(1)mδm,k]Un(1,ι)(k1,k2,α1),
-jn=1m=0nCmn,e[(l+k)!(lk)!A^kl,1mn(k2d1)(1)kA^kl,1mn(k2d1)]Ul(ι,1)(k2,k2,α1)jn=1m=0nCmn,o[(l+k)!(lk)!Ǎkl,1mn(k2d1)(1)kǍkl,1mn(k2d1)]Ul(ι,1)(k2,k2,α1)+n=1m=0nDmn,e[(l+k)!(lk)!B^kl,1mn(k2d1)(1)kB^kl,1mn(k2d1)]Ul(ι,1)(k2,k2,α1)+n=1m=0nDmn,o[(l+k)!(lk)!B̌kl,1mn(k2d1)(1)kB̌kl,1mn(k2d1)]Ul(ι,1)(k2,k2,α1)+n=1m=0nEmn,oδnl[(n+m)!(nm)!δmk(1)mδm,k]Un(2,ι)(k2,k2,α1)=n=1m=0nGmn,oδnl[(n+m)!(nm)!δmk(1)mδm,k]Un(1,ι)(k1,k2,α1),
-n=1m=0nCmn,e[(l+k)!(lk)!B^kl,1mn(k2d1)+(1)kB^kl,1mn(k2d1)]Vl(ι,1)(k2,k2,α1)n=1m=0nCmn,o[(l+k)!(lk)!B̌kl,1mn(k2d1)+(1)kB̌kl,1mn(k2d1)]Vl(ι,1)(k2,k2,α1)jn=1m=0nDmn,e[(l+k)!(lk)!A^kl,1mn(k2d1)+(1)kA^kl,1mn(k2d1)]Vl(ι,1)(k2,k2,α1)jn=1m=0nDmn,o[(l+k)!(lk)!Ǎkl,1mn(k2d1)+(1)kǍkl,1mn(k2d1)]Vl(ι,1)(k2,k2,α1)+jn=1m=0nFmn,eδnl[(n+m)!(nm)!δmk+(1)mδm,k]Vn(2,ι)(k2,k2,α1)=jn=1m=0nHmn,eδnl[(n+m)!(nm)!δmk+(1)mδm,k]Vn(1,ι)(k1,k2,α1),
-jn=1m=0nCmn,e[(l+k)!(lk)!B^kl,1mn(k2d1)(1)kB^kl,1mn(k2d1)]Vl(ι,1)(k2,k2,α1)jn=1m=0nCmn,o[(l+k)!(lk)!B̌kl,1mn(k2d1)(1)kB̌kl,1mn(k2d1)]Vl(ι,1)(k2,k2,α1)+n=1m=0nDmn,e[(l+k)!(lk)!A^kl,1mn(k2d1)(1)kA^kl,1mn(k2d1)]Vl(ι,1)(k2,k2,α1)+n=1m=0nDmn,o[(l+k)!(lk)!Ǎkl,1mn(k2d1)(1)kǍkl,1mn(k2d1)]Vl(ι,1)(k2,k2,α1)+jn=1m=0nFmn,oδnl[(n+m)!(nm)!δmk(1)mδm,k]Vn(2,ι)(k2,k2,α1)=jn=1m=0nHmn,oδnl[(n+m)!(nm)!δmk(1)mδm,k]Vn(1,ι)(k1,k2,α1).
-jn=1cn[n(n+1)A^1n,1kl(k2d1)+A^1n,1kl(k2d1)]Un(1,ι)(k0,k2,α2)jn=1cn[n(n+1)B^1n,1kl(k2d1)B^1n,1kl(k2d1)]Vn(1,ι)(k0,k2,α2)+n=1m=0nAmn,e[(n+m)!(nm)!A^mn,1kl(k2d1)+(1)mA^mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nAmn,o[(n+m)!(nm)!A^mn,1kl(k2d1)(1)mA^mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nBmn,e[(n+m)!(nm)!B^mn,1kl(k2d1)+(1)mB^mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)n=1m=0nBmn,o[(n+m)!(nm)!B^mn,1kl(k2d1)(1)mB^mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)=(1+δk0)α12α22Ekl,e(l+k)!(lk)!Ul(2,1)(k2,k2,α1)Ul(1,1)(k1,k2,α1)Ul(1,ι)(k1,k2,α1),
-jn=1cn[n(n+1)Ǎ1n,1kl(k2d1)+Ǎ1n,1kl(k2d1)]Un(1,ι)(k0,k2,α2)jn=1cn[n(n+1)B̌1n,1kl(k2d1)B̌1n,1kl(k2d1)]Vn(1,ι)(k0,k2,α2)+n=1m=0nAmn,e[(n+m)!(nm)!Ǎmn,1kl(k2d1)+(1)mǍmn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nAmn,o[(n+m)!(nm)!Ǎmn,1kl(k2d1)(1)mǍmn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nBmn,e[(n+m)!(nm)!B̌mn,1kl(k2d1)+(1)mB̌mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)n=1m=0nBmn,o[(n+m)!(nm)!B̌mn,1kl(k2d1)(1)mB̌mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)=(1δk0)α12α22Ekl,o(l+k)!(lk)!Ul(2,1)(k2,k2,α1)Ul(1,1)(k1,k2,α1)Ul(1,ι)(k1,k2,α1),
-jn=1cn[n(n+1)B^1n,1kl(k2d1)+B^1n,1kl(k2d1)]Un(1,ι)(k0,k2,α2)jn=1cn[n(n+1)A^1n,1kl(k2d1)A^1n,1kl(k2d1)]Vn(1,ι)(k0,k2,α2)+n=1m=0nAmn,e[(n+m)!(nm)!B^mn,1kl(k2d1)+(1)mB^mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nAmn,o[(n+m)!(nm)!B^mn,1kl(k2d1)(1)mB^mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nBmn,e[(n+m)!(nm)!A^mn,1kl(k2d1)+(1)mA^mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)n=1m=0nBmn,o[(n+m)!(nm)!A^mn,1kl(k2d1)(1)mA^mn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)=j(1+δk0)α12α22Fkl,e(l+k)!(lk)!Vl(2,1)(k2,k2,α1)Vl(1,1)(k1,k2,α1)Vl(1,ι)(k1,k2,α1),
-jn=1cn[n(n+1)B̌1n,1kl(k2d1)+B̌1n,1kl(k2d1)]Un(1,ι)(k0,k2,α2)jn=1cn[n(n+1)Ǎ1n,1kl(k2d1)Ǎ1n,1kl(k2d1)]Vn(1,ι)(k0,k2,α2)+n=1m=0nAmn,e[(n+m)!(nm)!B̌mn,1kl(k2d1)+(1)mB̌mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nAmn,o[(n+m)!(nm)!B̌mn,1kl(k2d1)(1)mB̌mn,1kl(k2d1)]Un(2,ι)(k0,k2,α2)+jn=1m=0nBmn,e[(n+m)!(nm)!Ǎmn,1kl(k2d1)+(1)mǍmn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)n=1m=0nBmn,o[(n+m)!(nm)!Ǎmn,1kl(k2d1)(1)mǍmn,1kl(k2d1)]Vn(2,ι)(k0,k2,α2)=j(1δk0)α12α22Fkl,o(l+k)!(lk)!Vl(2,1)(k2,k2,α1)Vl(1,1)(k1,k2,α1)Vl(1,ι)(k1,k2,α1),
-n=1m=0nCmn,e[(l+k)!(lk)!A^kl,1mn(k2d1)+(1)kA^kl,1mn(k2d1)]n=1m=0nCmn,o[(l+k)!(lk)!Ǎkl,1mn(k2d1)+(1)kǍkl,1mn(k2d1)]jn=1m=0nDmn,e[(l+k)!(lk)!B^kl,1mn(k2d1)+(1)kB^kl,1mn(k2d1)]jn=1m=0nDmn,o[(l+k)!(lk)!B̌kl,1mn(k2d1)+(1)kB̌kl,1mn(k2d1)]=(1+δk0)Ekl,e(l+k)!(lk)!Ul(1,2)(k1,k2,α1)Ul(1,1)(k1,k2,α1),
-jn=1m=0nCmn,e[(l+k)!(lk)!A^kl,1mn(k2d1)(1)kA^kl,1mn(k2d1)]jn=1m=0nCmn,o[(l+k)!(lk)!Ǎkl,1mn(k2d1)(1)kǍkl,1mn(k2d1)]+n=1m=0nDmn,e[(l+k)!(lk)!B^kl,1mn(k2d1)(1)kB^kl,1mn(k2d1)]+n=1m=0nDmn,o[(l+k)!(lk)!B̌kl,1mn(k2d1)(1)kB̌kl,1mn(k2d1)]=(1δk0)Ekl,o(l+k)!(lk)!Ul(1,2)(k1,k2,α1)Ul(1,1)(k1,k2,α1),
-n=1m=0nCmn,e[(l+k)!(lk)!B^kl,1mn(k2d1)+(1)kB^kl,1mn(k2d1)]n=1m=0nCmn,o[(l+k)!(lk)!B̌kl,1mn(k2d1)+(1)kB̌kl,1mn(k2d1)]jn=1m=0nDmn,e[(l+k)!(lk)!A^kl,1mn(k2d1)+(1)kA^kl,1mn(k2d1)]jn=1m=0nDmn,o[(l+k)!(lk)!Ǎkl,1mn(k2d1)+(1)kǍkl,1mn(k2d1)]=j(1+δk0)Fkl,e(l+k)!(lk)!Vl(1,2)(k1,k2,α1)Vl(1,1)(k1,k2,α1),
-jn=1m=0nCmn,e[(l+k)!(lk)!B^kl,1mn(k2d1)(1)kB^kl,1mn(k2d1)]jn=1m=0nCmn,o[(l+k)!(lk)!B̌kl,1mn(k2d1)(1)kB̌kl,1mn(k2d1)]+n=1m=0nDmn,e[(l+k)!(lk)!A^kl,1mn(k2d1)(1)kA^kl,1mn(k2d1)]+n=1m=0nDmn,o[(l+k)!(lk)!Ǎkl,1mn(k2d1)(1)kǍkl,1mn(k2d1)]=j(1δk0)Fkl,o(l+k)!(lk)!Vl(1,2)(k1,k2,α1)Vl(1,1)(k1,k2,α1).

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