Mie scattering analysis of spherical Bragg “onion” resonators

Wei Liang; Yong Xu; Yanyi Huang; Amnon Yariv; J. G. Fleming; Shawn-Yu Lin

doi:10.1364/OPEX.12.000657

Optics Express
Vol. 12,
Issue 4,
pp. 657-669
(2004)
•https://doi.org/10.1364/OPEX.12.000657

Mie scattering analysis of spherical Bragg “onion” resonators

Wei Liang, Yong Xu, Yanyi Huang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin

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Wei Liang, Yong Xu, Yanyi Huang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, "Mie scattering analysis of spherical Bragg “onion” resonators," Opt. Express 12, 657-669 (2004)

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About this Article
History
- Original Manuscript: January 6, 2004
- Revised Manuscript: February 12, 2004
- Manuscript Accepted: February 15, 2004
- Published: February 23, 2004

Abstract

Combining the Mie scattering theory and a transfer matrix method, we investigate in detail the scattering of light by spherical Bragg “onion” resonators. We classify the resonator modes into two classes, the core modes that are confined by Bragg reflection, and the cladding modes that are confined by total internal reflection. We demonstrate that these two types of modes lead to significantly different scattering behaviors.

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References

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Year
Author
Publication

A. Ashkin and J. M. Dziedzic, “Observation of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351–1355 (1977).
[Crossref]
Petr Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A. 18, 2229–2233, (1978).
[Crossref]
V.B. Braginsky, M. L. Gorodetsky, and V.S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A. 137, 393–396 (1989).
[Crossref]
Craig F. Bohren and Donald R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, Inc, 1998).
[Crossref]
Kerry J. Vahala, “Optical microcavities,” Nature 424, 839–846, (2003).
[Crossref] [PubMed]
M. Cai, G. Hunziker, and K. Vahala, “Fiber-optic add/drop device based on a silica microsphere-whispering gallery mode system,” IEEE Photon. Technol. Lett. 11, 686–687, (1999).
[Crossref]
M. Cai, P.O. Hedekvist, A. Bhardwaj, and K. Vahala, “5-Gbit/s BER performance on an all fiber-optic add/drop device based on a taper-resonator-taper structure,” IEEE Photon. Technol. Lett. 12, 1177–1179, (2000).
[Crossref]
D.W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A. 57, R2293 (1998).
[Crossref]
J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A. 67, 033806 (2003).
[Crossref]
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]
Kevin G. Sullivan and Dennis G. Hall, “Radiation in spherically symmetric structure. I. the coupled-amplitude equations for vector spherical waves,” Phys. Rev. A. 50, 2701–2707, (1994).
[Crossref] [PubMed]
David D. Smith and Kirk A. Fuller, “Photonic bandgaps in Mie scattering by concentrically stratified spheres,” J. Opt. Soc. Am. B. 19, 2449–2455, (2002).
[Crossref]
Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “High quality factor Bragg onion resonators with omnidirectional reflector cladding,” Opt. Lett. 28, 2144–2146, (2003).
[Crossref] [PubMed]
Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “Modal analysis of spherically symmetric Bragg resonators,” to appear in Opt. Lett.
G. Mie, “Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[Crossref]
John David Jackson, Classical Electrodynamics, (John Wiley & Sons, Inc, Third Edition, 1999), Chap. 10.
G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1442, (1988).
[Crossref]
J. A Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515, (1994).
[Crossref]
M. L. Gorodetsky and V. S. IIchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154, (1999).
[Crossref]
U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878, (1961).
[Crossref]

2003 (3)

Kerry J. Vahala, “Optical microcavities,” Nature 424, 839–846, (2003).
[Crossref] [PubMed]

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A. 67, 033806 (2003).
[Crossref]

Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “High quality factor Bragg onion resonators with omnidirectional reflector cladding,” Opt. Lett. 28, 2144–2146, (2003).
[Crossref] [PubMed]

2002 (1)

David D. Smith and Kirk A. Fuller, “Photonic bandgaps in Mie scattering by concentrically stratified spheres,” J. Opt. Soc. Am. B. 19, 2449–2455, (2002).
[Crossref]

2000 (1)

M. Cai, P.O. Hedekvist, A. Bhardwaj, and K. Vahala, “5-Gbit/s BER performance on an all fiber-optic add/drop device based on a taper-resonator-taper structure,” IEEE Photon. Technol. Lett. 12, 1177–1179, (2000).
[Crossref]

1999 (2)

M. Cai, G. Hunziker, and K. Vahala, “Fiber-optic add/drop device based on a silica microsphere-whispering gallery mode system,” IEEE Photon. Technol. Lett. 11, 686–687, (1999).
[Crossref]

M. L. Gorodetsky and V. S. IIchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154, (1999).
[Crossref]

1998 (1)

D.W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A. 57, R2293 (1998).
[Crossref]

1997 (1)

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]

1994 (2)

Kevin G. Sullivan and Dennis G. Hall, “Radiation in spherically symmetric structure. I. the coupled-amplitude equations for vector spherical waves,” Phys. Rev. A. 50, 2701–2707, (1994).
[Crossref] [PubMed]

J. A Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515, (1994).
[Crossref]

1989 (1)

V.B. Braginsky, M. L. Gorodetsky, and V.S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A. 137, 393–396 (1989).
[Crossref]

1988 (1)

G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1442, (1988).
[Crossref]

1978 (1)

Petr 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)

A. Ashkin and J. M. Dziedzic, “Observation of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351–1355 (1977).
[Crossref]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878, (1961).
[Crossref]

1908 (1)

G. Mie, “Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[Crossref]

Ashkin, A.

A. Ashkin and J. M. Dziedzic, “Observation of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351–1355 (1977).
[Crossref]

Bhardwaj, A.

Bohren, Craig F.

Craig F. Bohren and Donald R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, Inc, 1998).
[Crossref]

Braginsky, V.B.

V.B. Braginsky, M. L. Gorodetsky, and V.S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A. 137, 393–396 (1989).
[Crossref]

Buck, J. R.

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A. 67, 033806 (2003).
[Crossref]

Cai, M.

M. Cai, G. Hunziker, and K. Vahala, “Fiber-optic add/drop device based on a silica microsphere-whispering gallery mode system,” IEEE Photon. Technol. Lett. 11, 686–687, (1999).
[Crossref]

Chylek, Petr

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

David Jackson, John

John David Jackson, Classical Electrodynamics, (John Wiley & Sons, Inc, Third Edition, 1999), Chap. 10.

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, “Observation of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351–1355 (1977).
[Crossref]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878, (1961).
[Crossref]

Fleming, J. G.

Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “Modal analysis of spherically symmetric Bragg resonators,” to appear in Opt. Lett.

Fuller, Kirk A.

David D. Smith and Kirk A. Fuller, “Photonic bandgaps in Mie scattering by concentrically stratified spheres,” J. Opt. Soc. Am. B. 19, 2449–2455, (2002).
[Crossref]

Furusawa, A.

D.W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A. 57, R2293 (1998).
[Crossref]

Georgiades, N. P.

D.W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A. 57, R2293 (1998).
[Crossref]

Gorodetsky, M. L.

M. L. Gorodetsky and V. S. IIchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154, (1999).
[Crossref]

V.B. Braginsky, M. L. Gorodetsky, and V.S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A. 137, 393–396 (1989).
[Crossref]

Gouesbet, G.

G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1442, (1988).
[Crossref]

Grehan, G.

G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1442, (1988).
[Crossref]

Guo, L. X.

Hall, Dennis G.

Hedekvist, P.O.

Huffman, Donald R.

Craig F. Bohren and Donald R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, Inc, 1998).
[Crossref]

Hunziker, G.

M. Cai, G. Hunziker, and K. Vahala, “Fiber-optic add/drop device based on a silica microsphere-whispering gallery mode system,” IEEE Photon. Technol. Lett. 11, 686–687, (1999).
[Crossref]

IIchenko, V. S.

M. L. Gorodetsky and V. S. IIchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154, (1999).
[Crossref]

Ilchenko, V. S.

D.W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A. 57, R2293 (1998).
[Crossref]

Ilchenko, V.S.

V.B. Braginsky, M. L. Gorodetsky, and V.S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A. 137, 393–396 (1989).
[Crossref]

Kiehl, J. T.

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

Kimble, H. J.

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A. 67, 033806 (2003).
[Crossref]

D.W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A. 57, R2293 (1998).
[Crossref]

Ko, M. K. W.

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

Liang, Wei

Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “Modal analysis of spherically symmetric Bragg resonators,” to appear in Opt. Lett.

Lin, Shawn-Yu

Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “Modal analysis of spherically symmetric Bragg resonators,” to appear in Opt. Lett.

Lock, J. A

Maheu, B.

G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1442, (1988).
[Crossref]

Mie, G.

G. Mie, “Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[Crossref]

Ren, K. F.

Smith, David D.

David D. Smith and Kirk A. Fuller, “Photonic bandgaps in Mie scattering by concentrically stratified spheres,” J. Opt. Soc. Am. B. 19, 2449–2455, (2002).
[Crossref]

Sullivan, Kevin G.

Vahala, K.

M. Cai, G. Hunziker, and K. Vahala, “Fiber-optic add/drop device based on a silica microsphere-whispering gallery mode system,” IEEE Photon. Technol. Lett. 11, 686–687, (1999).
[Crossref]

Vahala, Kerry J.

Kerry J. Vahala, “Optical microcavities,” Nature 424, 839–846, (2003).
[Crossref] [PubMed]

Vernooy, D.W.

D.W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A. 57, R2293 (1998).
[Crossref]

Wu, Z. S.

Xu, Yong

Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “Modal analysis of spherically symmetric Bragg resonators,” to appear in Opt. Lett.

Yariv, Amnon

Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “Modal analysis of spherically symmetric Bragg resonators,” to appear in Opt. Lett.

Ann. Phys. (1)

G. Mie, “Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[Crossref]

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (2)

M. Cai, G. Hunziker, and K. Vahala, “Fiber-optic add/drop device based on a silica microsphere-whispering gallery mode system,” IEEE Photon. Technol. Lett. 11, 686–687, (1999).
[Crossref]

J. Opt. Soc. Am. A (2)

G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1442, (1988).
[Crossref]

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

M. L. Gorodetsky and V. S. IIchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154, (1999).
[Crossref]

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

David D. Smith and Kirk A. Fuller, “Photonic bandgaps in Mie scattering by concentrically stratified spheres,” J. Opt. Soc. Am. B. 19, 2449–2455, (2002).
[Crossref]

Nature (1)

Kerry J. Vahala, “Optical microcavities,” Nature 424, 839–846, (2003).
[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Lett. A. (1)

V.B. Braginsky, M. L. Gorodetsky, and V.S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A. 137, 393–396 (1989).
[Crossref]

Phys. Rev. (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878, (1961).
[Crossref]

Phys. Rev. A. (4)

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

D.W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A. 57, R2293 (1998).
[Crossref]

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A. 67, 033806 (2003).
[Crossref]

Phys. Rev. Lett. (1)

A. Ashkin and J. M. Dziedzic, “Observation of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351–1355 (1977).
[Crossref]

Other (3)

Craig F. Bohren and Donald R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, Inc, 1998).
[Crossref]

Yong Xu, Wei Liang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin, “Modal analysis of spherically symmetric Bragg resonators,” to appear in Opt. Lett.

John David Jackson, Classical Electrodynamics, (John Wiley & Sons, Inc, Third Edition, 1999), Chap. 10.

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Fig. 1. A SEM image of the onion resonator.

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Fig. 2. Light scattering by a spherical Bragg “onion” resonator.

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Fig. 3. Radial dependence of the electrical field of TE mode and the magnetic field of TM mode.

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Fig. 4. Spectrum of the onion resonator modes (calculated with N_clad =4).

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Fig. 5. Spectrum of total scattering efficiency. The number of Bragg pairs is N_clad =4.

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Fig. 6. Comparison of scattering resonances numerically obtained using the Mie scattering method developed in section 2, and those given by a Lorentzian approximation. The number of Bragg pair is N_clad =4.

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Fig. 7. Scattering resonances of the TE₁₀ and TE₂₄ modes (with N_clad =4 and 5). The dashed lines represent the resonant scattering coefficient -Re(αl) (whose values are shown on the left y-axis), and the solid lines represent the total scattering efficiency

Q_{tot}^{scatt}

(whose values are shown on the right y-axis). The dashed vertical lines give the position of modal wavelength.

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Fig. 8. Comparison of the scattering resonance width Γ _res and the modal quality factor as a function of the Bragg pair number.

Γ_{res}^{L}

and

Γ_{res}^{tot}

represent, respectively, the resonance width of the scattering coefficient — Re(α) and the total scattering efficiency

Q_{tot}^{scatt}

(see Fig. 7(a)).

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Fig. 9. Effect of absorption on scattering resonances. The number of Bragg pairs is N_clad =4.

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Fig. 10. Comparison between plane and Gaussian incident wave. In both (a) and (b), the upper line is under the assumption the incident wave is a plane wave; the lower one is under the assumption the incident wave is a Gaussian beam with the waist width of 5µm with the sphere located at the center of the beam. In (a), the values of the upper and lower lines are shown on the left and right y-axis respectively. The number of Bragg pair is N_clad =4.

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Equations (28)

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

{\vec{E}}_{inc} = {\hat{e}}_{x} e^{ikz} = \sum_{l = 1}^{\infty} i^{l} \frac{\sqrt{4 π (2 l + 1)}}{2} [j_{l} (kr) ({\vec{X}}_{l, + 1} + {\vec{X}}_{l, - 1}) + \frac{1}{k} \vec{\nabla} \times j_{l} (kr) ({\vec{X}}_{l, + 1} - {\vec{X}}_{l, - 1})]

{\vec{E}}_{sc}^{x} = \sum_{l = 1}^{\infty} i^{l} \frac{\sqrt{4 π (2 l + 1)}}{2} [\frac{α_{l}}{2} h_{l}^{1} (kr) ({\vec{X}}_{l, + 1} + {\vec{X}}_{l, - 1}) + \frac{β_{l}}{2} \frac{\vec{\nabla}}{k} \times h_{l}^{1} (kr) ({\vec{X}}_{l, + 1} - {\vec{X}}_{l, - 1})]

{\vec{E}}_{sc}^{x} \approx \frac{{ie}^{ikr}}{kr} (\cos ϕ \cdot S_{2} (θ) {\hat{e}}_{θ} - \sin ϕ \cdot S_{1} (θ) {\hat{e}}_{ϕ})

S_{2} (θ) = \sum_{l = 1}^{\infty} \frac{(2 l + 1)}{l (l + 1)} (\frac{α_{l}}{2} \frac{P_{l}^{1}}{\sin θ} + \frac{β_{l}}{2} \frac{{dP}_{l}^{1}}{d θ})

S_{1} (θ) = \sum_{l = 1}^{\infty} \frac{(2 l + 1)}{l (l + 1)} (\frac{α_{l}}{2} \frac{{dp}_{l}^{1}}{d θ} + \frac{β_{l}}{2} \frac{P_{l}^{1}}{\sin θ})

\frac{d σ_{sc}}{d Ω} = \frac{{dP}_{sc}}{I_{inc} d Ω} = \frac{\frac{1}{2} \cdot c ε_{0} {∣ {\vec{E}}_{sc}^{x} ∣}^{2} \cdot r^{2}}{\frac{1}{2} \cdot c ε_{0} {∣ {\vec{E}}_{inc} ∣}^{2}}

= \frac{1}{k^{2}} ({∣ S_{2} ∣}^{2} \cos^{2} ϕ + {∣ S_{1} ∣}^{2} \sin^{2} ϕ)

σ_{total} = - \frac{π}{k^{2}} \sum_{l = 1}^{\infty} (2 l + 1) Re [α_{l} + β_{l}]

[\begin{matrix} \vec{H} \\ \vec{E} \end{matrix}] = [\begin{matrix} j_{l} (k_{n} r) {\vec{X}}_{l, m} & h_{l}^{1} (k_{n} r) {\vec{X}}_{l, m} \\ Z_{n} \frac{i}{k_{n}} \vec{\nabla} \times j_{l} (k_{n} r) {\vec{X}}_{l, m} & Z_{n} \frac{i}{k_{n}} \vec{\nabla} \times h_{l}^{1} (k_{n} r) {\vec{X}}_{l, m} \end{matrix}] \cdot [\begin{matrix} A_{n} \\ B_{n} \end{matrix}] (TM)

[\begin{matrix} \vec{E} \\ \vec{H} \end{matrix}] = [\begin{matrix} j_{l} (k_{n} r) {\vec{X}}_{l, m} & h_{l}^{1} (k_{n} r) {\vec{X}}_{l, m} \\ \frac{- i}{{Z_{n} k}_{n}} \vec{\nabla} \times j_{l} (k_{n} r) {\vec{X}}_{l, m} & \frac{- i}{{Z_{n} k}_{n}} \vec{\nabla} \times h_{l}^{1} (k_{n} r) {\vec{X}}_{l, m} \end{matrix}] \cdot [\begin{matrix} C_{n} \\ D_{n} \end{matrix}] (TE)

[\begin{matrix} j_{l} (k_{n} r_{n}) & h_{l}^{1} (k_{n} r_{n}) \\ \frac{Z_{n}}{k_{n}} \frac{\partial}{\partial r} [{rj}_{l} (k_{n} r)] ∣_{r_{n}} & \frac{Z_{n}}{k_{n}} \frac{\partial}{\partial r} [{rh}_{l}^{1} (k_{n} r)] ∣_{r_{n}} \end{matrix}] (\begin{matrix} A_{n} \\ B_{n} \end{matrix}) =

[\begin{matrix} j_{l} (k_{n + 1} r_{n}) & h_{l}^{1} (k_{n + 1} r_{n}) \\ \frac{Z_{n + 1}}{k_{n + 1}} \frac{\partial}{\partial r} [{rj}_{l} (k_{n + 1} r)] ∣_{r_{n}} & \frac{Z_{n + 1}}{k_{n + 1}} \frac{\partial}{\partial r} [{rh}_{l}^{1} (k_{n + 1} r)] ∣_{r_{n}} \end{matrix}] (\begin{matrix} A_{n + 1} \\ B_{n + 1} \end{matrix}) (TM)

[\begin{matrix} j_{l} (k_{n} r_{n}) & h_{l}^{1} (k_{n} r_{n}) \\ \frac{1}{Z_{n} k_{n}} \frac{\partial}{\partial r} [{rj}_{l} (k_{n} r)] ∣_{r_{n}} & \frac{\partial}{Z_{n} k_{n} \partial r} [r h_{l}^{1} (k_{n} r)] ∣_{r_{n}} \end{matrix}] (\begin{matrix} C_{n} \\ D_{n} \end{matrix}) =

[\begin{matrix} j_{l} (k_{n + 1} r_{n}) & h_{l}^{1} (k_{n + 1} r_{n}) \\ \frac{\partial}{Z_{n + 1} k_{n + 1} \partialr} [{rj}_{l} (k_{n + 1} r)] ∣_{r_{n}} & \frac{\partial}{Z_{n + 1} k_{n + 1} \partialr} [r h_{l}^{1} (k_{n + 1} r)] ∣_{r_{n}} \end{matrix}] (\begin{matrix} C_{n + 1} \\ D_{n + 1} \end{matrix}) (TE)

[\begin{matrix} A_{co} \\ B_{co} \end{matrix}] = M_{l}^{TM} \cdot [\begin{matrix} A_{out} \\ B_{out} \end{matrix}]

[\begin{matrix} C_{co} \\ D_{co} \end{matrix}] = M_{l}^{TE} \cdot [\begin{matrix} C_{out} \\ D_{out} \end{matrix}]

{\vec{E}}_{tot} = \sum_{l = 1}^{\infty} i^{l} \frac{\sqrt{4 π (2 l + 1)}}{2} [(j_{l} + \frac{α_{l}}{2} h_{l}^{1}) ({\vec{X}}_{l, + 1} + {\vec{X}}_{l, - 1}) + \frac{\vec{\nabla}}{k} \times (j_{l} + \frac{β_{l}}{2} h_{l}^{1}) ({\vec{X}}_{l, + 1} - {\vec{X}}_{l, - 1})]

{\vec{E}}_{tot}^{l, m} \propto \overset{⇀}{\nabla} \times ⌊ (A_{out} j_{l} (kr) + B_{out} h_{l}^{1} (kr)) {\vec{X}}_{l, m} ⌋ (TM)

{\vec{E}}_{tot}^{l, m} \propto (C_{out} j_{l} (kr) + D_{out} h_{l}^{1} (kr)) {\vec{X}}_{l, m} (TE)

A_{out} / B_{out} = 2 / β_{l}, C_{out} / D_{out} = 2 / α_{l}

B_{co} = {(M_{l}^{TM})}_{2, 1} A_{out} + {(M_{l}^{TM})}_{2, 2} B_{out} = 0

D_{co} = {(M_{l}^{TE})}_{2, 1} C_{out} + {(M_{l}^{TE})}_{2, 2} D_{out} = 0

β_{l} / 2 = - {(M_{l}^{TM})}_{2,1} / {(M_{l}^{TM})}_{2,2}

α_{l} / 2 = - {(M_{l}^{TE})}_{2,1} / {(M_{l}^{TE})}_{2,2}

g_{l} = \exp {- {[\frac{(l + 1 / 2) λ}{2 π W_{0}}]}^{2}}

Q_{tot}^{scatt} = \frac{σ_{total}}{π R^{2}} = - \frac{1}{k^{2} R^{2}} \sum_{l = 1}^{\infty} (2 l + 1) Re [α_{l} + β_{l}]

- Re (α or β) = \frac{2}{1 + {(Q \cdot 2 Δ λ / λ_{0})}^{2}}

Q_{tot}^{scatt} = Q_{bg}^{scatt} + \frac{2 l + 1}{k^{2} R^{2}} \frac{2}{1 + {(Q \cdot 2 Δ λ / λ_{0})}^{2}}

Abstract

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