Abstract

The development of a fast and reliable whispering gallery mode (WGM) simulator capable of generating spectra that are comparable with experiment is an important step forward for designing microresonators. We present a new model for generating WGM spectra for multilayer microspheres, which allows for an arbitrary number of concentric dielectric layers, and any number of embedded dipole sources or uniform distributions of dipole sources to be modeled. The mode excitation methods model embedded nanoparticles, or fluorescent dye coatings, from which normalized power spectra with accurate representation of the mode coupling efficiencies can be derived. In each case, the emitted power is expressed conveniently as a function of wavelength, with minimal computational load. The model makes use of the transfer-matrix approach, incorporating improvements to its stability, resulting in a reliable, general set of formulae for calculating whispering gallery mode spectra. In the specific cases of the dielectric microsphere and the single-layer coated microsphere, our model simplifies to confirmed formulae in the literature.

© 2017 Optical Society of America

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2017 (1)

J. M. M. Hall, A. François, S. Afshar, N. Riesen, M. R. Henderson, T. Reynolds, and T. M. Monro, “Determining the geometric parameters of microbubble resonators from their spectra,” J. Opt. Soc. Am. B 34(1), 2699–2706 (2017).
[Crossref]

2016 (2)

G. Tao, J. J. Kaufman, S. Shabahang, R. RezvaniNaraghi, S. V. Sukhov, J. D. Joannopoulos, Y. Fink, A. Dogariu, and A. F. Abouraddy, “Digital design of multimaterial photonic particles,” Proc. Nat. Acad. Sci. USA 113, 6839–6844 (2016).
[Crossref] [PubMed]

C. Qu and E. C. Kinzel, “Polycrystalline metasurface perfect absorbers fabricated using microsphere photolithography,” Opt. Lett. 41, 3399–3402 (2016).
[Crossref] [PubMed]

2015 (8)

W. Liang, A. A. Savchenkov, Z. Xie, J. F. McMillan, J. Burkhart, V. S. Ilchenko, C. W. Wong, A. B. Matsko, and L. Maleki, “Miniature multioctave light source based on a monolithic microcavity,” Optica 2, 40–47 (2015).
[Crossref]

J. M. M. Hall, S. Afshar, M. R. Henderson, A. François, T. Reynolds, N. Riesen, and T. M. Monro, “Method for predicting whispering gallery mode spectra of spherical microresonators,” Opt. Express 23, 9924–9937 (2015).
[Crossref] [PubMed]

N. Riesen, S. Afshar, A. François, and T. M. Monro, “Material candidates for optical frequency comb generation in microspheres,” Opt. Express 23, 14784–14795 (2015).
[Crossref] [PubMed]

T. Reynolds, M. R. Henderson, A. François, N. Riesen, J. M. M. Hall, S. Afshar, S. J. Nicholls, and T. M. Monro, “Optimization of whispering gallery resonator design for biosensing applications,” Opt. Express 23, 17067–17076 (2015).
[Crossref] [PubMed]

D. Farnesi, A. Barucci, G. C. Righini, G. N. Conti, and S. Soria, “Generation of hyper-parametric oscillations in silica microbubbles,” Opt. Lett. 40, 4508–4511 (2015).
[Crossref] [PubMed]

T. C. Preston and J. P. Reid, “Determining the size and refractive index of microspheres using the mode assignments from Mie resonances,” J. Opt. Soc. Am. A 32, 2210–2217 (2015).
[Crossref]

N. Riesen, T. Reynolds, A. François, M. R. Henderson, and T. M. Monro, “Q-factor limits for far-field detection of whispering gallery modes in active microspheres,” Opt. Express 23, 28896–28904 (2015).
[Crossref] [PubMed]

A. François, T. Reynolds, and T. M. Monro, “A fiber-tip label-free biological sensing platform: A practical approach toward in-vivo sensing,” Sensors 15, 1168–1181 (2015).
[Crossref] [PubMed]

2014 (3)

N. V. Kryzhanovskaya, M. V. Maximov, and A. E. Zhukov, “Whispering-gallery mode microcavity quantum-dot lasers,” IEEE J. Quant. Electron. 44, 189–200 (2014).
[Crossref]

D. Farnesi, A. Barucci, G. C. Righini, S. Berneschi, S. Soria, and G. Nunzi Conti, “Optical frequency conversion in silica-whispering-gallery-mode microspherical resonators,” Phys. Rev. Lett. 112, 093901 (2014).
[Crossref] [PubMed]

Y. Yang, J. Ward, and S. N. Chormaic, “Quasi-droplet microbubbles for high resolution sensing applications,” Opt. Express 22, 6881–6898 (2014).
[Crossref] [PubMed]

2013 (5)

2012 (1)

J. J. Kaufman, G. Tao, S. Shabahang, E.-H. Banaei, D. S. Deng, X. Liang, S. G. Johnson, Y. Fink, and A. F. Abouraddy, “Structured spheres generated by an in-fibre fluid instability,” Nature 487, 463–467 (2012).
[Crossref] [PubMed]

2011 (4)

E. Nuhiji, F. G. Amar, H. Wang, N. Byrne, T.-L. Nguyen, and T. Lin, “Whispering gallery mode emission generated in tunable quantum dot doped glycerol/water and ionic liquid/water microdroplets formed on a superhydrophobic coating,” J. Mater. Chem. 21, 10823–10828 (2011).
[Crossref]

J. Geng, R. W. Ziolkowski, R. Jin, and X. Liang, “Numerical study of the near-field and far-field properties of active open cylindrical coated nanoparticle antennas,” IEEE Photon. J. 3, 1093–1110 (2011).
[Crossref]

G. Kozyreff, J. Dominguez-Juarez, and J. Martorell, “Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes,” Laser Photon. Rev. 5, 737–749 (2011).
[Crossref]

A. François, K. J. Rowland, and T. M. Monro, “Highly efficient excitation and detection of whispering gallery modes in a dye-doped microsphere using a microstructured optical fiber,” Appl. Phys. Lett. 99, 141111 (2011).
[Crossref]

2010 (2)

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010).
[Crossref] [PubMed]

2008 (1)

D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008).
[Crossref]

2007 (2)

2006 (3)

2005 (2)

H. Quan and Z. Guo, “Simulation of whispering-gallery-mode resonance shifts for optical miniature biosensors,” J. Quant. Spectrosc. Radiat. Transfer 93, 231–243 (2005).
[Crossref]

A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Annals of Physics 315, 352–418 (2005).
[Crossref]

2004 (3)

2003 (4)

2002 (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

1999 (1)

1988 (1)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[Crossref]

1987 (1)

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[Crossref]

1985 (1)

R. G. Barrera, G. A. Estevez, and J. Giraldo, “Vector spherical harmonics and their application to magnetostatics,” Eur. J. Phys. 6, 287 (1985).
[Crossref]

1978 (1)

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 65 (1978).

1976 (2)

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

H. Chew, M. Kerker, and P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
[Crossref]

1965 (1)

Abouraddy, A. F.

G. Tao, J. J. Kaufman, S. Shabahang, R. RezvaniNaraghi, S. V. Sukhov, J. D. Joannopoulos, Y. Fink, A. Dogariu, and A. F. Abouraddy, “Digital design of multimaterial photonic particles,” Proc. Nat. Acad. Sci. USA 113, 6839–6844 (2016).
[Crossref] [PubMed]

J. J. Kaufman, G. Tao, S. Shabahang, E.-H. Banaei, D. S. Deng, X. Liang, S. G. Johnson, Y. Fink, and A. F. Abouraddy, “Structured spheres generated by an in-fibre fluid instability,” Nature 487, 463–467 (2012).
[Crossref] [PubMed]

Afshar, S.

Amar, F. G.

E. Nuhiji, F. G. Amar, H. Wang, N. Byrne, T.-L. Nguyen, and T. Lin, “Whispering gallery mode emission generated in tunable quantum dot doped glycerol/water and ionic liquid/water microdroplets formed on a superhydrophobic coating,” J. Mater. Chem. 21, 10823–10828 (2011).
[Crossref]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

Arnold, S.

Ashili, S. P.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[Crossref]

Astratov, V. N.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[Crossref]

Banaei, E.-H.

J. J. Kaufman, G. Tao, S. Shabahang, E.-H. Banaei, D. S. Deng, X. Liang, S. G. Johnson, Y. Fink, and A. F. Abouraddy, “Structured spheres generated by an in-fibre fluid instability,” Nature 487, 463–467 (2012).
[Crossref] [PubMed]

Barrera, R. G.

R. G. Barrera, G. A. Estevez, and J. Giraldo, “Vector spherical harmonics and their application to magnetostatics,” Eur. J. Phys. 6, 287 (1985).
[Crossref]

Barucci, A.

D. Farnesi, A. Barucci, G. C. Righini, G. N. Conti, and S. Soria, “Generation of hyper-parametric oscillations in silica microbubbles,” Opt. Lett. 40, 4508–4511 (2015).
[Crossref] [PubMed]

D. Farnesi, A. Barucci, G. C. Righini, S. Berneschi, S. Soria, and G. Nunzi Conti, “Optical frequency conversion in silica-whispering-gallery-mode microspherical resonators,” Phys. Rev. Lett. 112, 093901 (2014).
[Crossref] [PubMed]

Belov, V. N.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Berneschi, S.

D. Farnesi, A. Barucci, G. C. Righini, S. Berneschi, S. Soria, and G. Nunzi Conti, “Optical frequency conversion in silica-whispering-gallery-mode microspherical resonators,” Phys. Rev. Lett. 112, 093901 (2014).
[Crossref] [PubMed]

Bierwagen, J.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Blatt, J.

J. Blatt and V. Weisskopf, Theoretical Nuclear Physics, Dover Books on Physics Series (Dover Publications, 1991).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering by a Sphere (Wiley-VCH Verlag GmbH, 2007), pp. 82–129.

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Burkhart, J.

Byrne, N.

E. Nuhiji, F. G. Amar, H. Wang, N. Byrne, T.-L. Nguyen, and T. Lin, “Whispering gallery mode emission generated in tunable quantum dot doped glycerol/water and ionic liquid/water microdroplets formed on a superhydrophobic coating,” J. Mater. Chem. 21, 10823–10828 (2011).
[Crossref]

Chance, R.

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 65 (1978).

Chew, H.

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[Crossref]

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[Crossref]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

H. Chew, M. Kerker, and P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
[Crossref]

Chormaic, S. N.

Conti, G. N.

Deng, D. S.

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Figures (6)

Fig. 1
Fig. 1

The geometry of a spherical resonator with N layers. (a) A single layer contains a uniform distribution of dipoles, to represent an active layer. (b) One or more individual dipoles can be placed in a given layer, to represent one or more embedded nanoparticles.

Fig. 2
Fig. 2

Spectra obtained from a simulated silica microsphere, incorporating dispersion, coated with a high refractive index layer (n2 = 1.7) with a diameter of 25 µm, surrounded by water. A single electric dipole is oriented (a) in the radial direction and (b) in the tangential direction.

Fig. 3
Fig. 3

Spectra for a silica microsphere coated with a polymer layer (PMMA), both of which include dispersion. The polymer layer functions as an active layer. (a) Both TE and TM modes are excited. (b) A zoomed in plot showing the FWHM of several peaks and their Q-factors.

Fig. 4
Fig. 4

The sensitivity of silica microspheres coated with PMMA as a function of the surrounding refractive index, for two example layer thicknesses. (a) d = 10 nm, (b) d = 50 nm. Inset: the figure of merit (FOM), Q.S, in units of 105 nm/R.I.U., as a function of λ.

Fig. 5
Fig. 5

The execution time T of the formulae P/P0(λ) as a function of wavelength. The results are shown for a numbers of layers N = 1, 2 and 3. The results for a single-dipole excitation oriented (a) parallel, and (b) perpendicular to the surface of the sphere are similar in magnitude. (c) The results of a single uniform distribution of dipoles within the center of the sphere begin to plateau as the wavelength becomes small. The tolerance selected is τ = 1 × 1012.

Fig. 6
Fig. 6

Demonstration that the multilayer model reproduces the microsphere model results, for D = 6 µm, n1 = 1.59, and n2 = 1.33. (a) Spectra for single-dipole excitation in both tangential and radial orientations. (b) Spectra for a uniform distribution of dipoles are also shown.

Equations (110)

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Y l m = Y l m ( θ , ϕ ) r ^ , Ψ l m = ( 1 i l ( l + 1 ) ) r Y l m ( θ , ϕ ) , Φ l m ( θ , ϕ ) = ( 1 i l ( l + 1 ) ) r × Y l m ( θ , ϕ ) ,
E ( r , θ , ϕ ) = l = 0 m = l m = 1 [ E l m r ( r ) Y l m ( θ , ϕ ) + E l m ( 1 ) ( r ) ψ l m ( θ , ϕ ) + E l m ( 2 ) ( r ) Φ l m ( θ , ϕ ) ] ,
E = i c ω ε ( × H ) , H = i c ω μ ( × E ) ,
E j = l , m ( i c n j 2 ω ) A j × [ j l ( k j r ) Φ l m ( θ , ϕ ) ] + ( i c n j 2 ω ) B j × [ h l ( 1 ) ( k j r ) Φ l m ( θ , ϕ ) ] + C j j l ( k j r ) Φ l m ( θ , ϕ ) + D j h l ( 1 ) ( k j r ) Φ l m ( θ , ϕ ) ,
H j = l , m ( i c μ j ω ) c j × [ j l ( k j r ) Φ l m ( θ , ϕ ) ] ( i c μ j ω ) D j × [ h l ( 1 ) ( k j r ) Φ l m ( θ , ϕ ) ] + ( 1 μ j ) A j j l ( k j r ) Φ l m ( θ , ϕ ) + ( 1 μ j ) B j h l ( 1 ) ( k j r ) Φ l m ( θ , ϕ ) .
E j = l , m [ ( i c n j 2 ω ) l + ( l + 1 ) i [ A j 1 r j l ( k j r ) + B j 1 r h l ( 1 ) ( k j r ) ] Y l m ( θ , ϕ ) ( i c n j 2 ω ) { A j 1 r d d r [ r j l ( k j r ) ] + B j 1 r d d r [ r h l ( 1 ) ( k j r ) ] } ψ l m ( θ , ϕ ) + { C j j l ( k j r ) + D j h l ( 1 ) ( k j r ) } Φ l m ( θ , ϕ ) ] ,
H j = l , m [ ( i c μ j ω ) l + ( l + 1 ) i [ C j 1 r j l ( k j r ) + D j 1 r h l ( 1 ) ( k j r ) ] Y l m ( θ , ϕ ) + ( i c μ j ω ) { C j 1 r d d r [ r j l ( k j r ) ] + D j 1 r d d r [ r h l ( 1 ) ( k j r ) ] } Ψ l m ( θ , ϕ ) + ( 1 μ j ) { A j j l ( k j r ) + B j h l ( 1 ) ( k j r ) } Φ l m ( θ , ϕ ) ] .
E H j = M j ( r ) A j ,
M j ( r ) = 1 k j r ( ( i n j ) ψ l ( k j r ) ( i n j ) χ l ( k j r ) 0 0 ( 1 μ j ) ψ l ( k j r ) ( 1 μ j ) χ l ( k j r ) 0 0 0 0 ψ l ( k j r ) χ l ( k j r ) 0 0 ( i n j μ j ) ψ l ( k j r ) ( i n j μ j ) χ l ( k j r ) ) ,
E H j ( r ) = ( E j ( 1 ) ( r ) H j ( 2 ) ( r ) E j ( 2 ) ( r ) H j ( 1 ) ( r ) ) , A j = ( A j B j C j D j ) .
det ( M j T M ( r ) ) = i μ j n j k j 2 r 2 W k     j r [ ψ l ( k j r ) , χ l ( k j r ) ] = 1 μ j n j k j 2 r 2 ,
det ( M j T E ( r ) ) = 1 k j 2 r 2 ( i n j μ j ) W k     j r     j [ ψ l ( k j r ) , χ l ( k j r ) ] = n j μ j k j 2 r 2 ,
W x [ f ( a x ) , g ( a x ) ] f ( a x ) g ( a x ) f ( a x ) g ( a x ) , for the derivative with respect to x .
E H j d ( r ) = θ ( r j r ) M j ( r ) a j   L + θ ( r r j ) M j ( r ) a j H .
E H j d = ( E j d ( 1 ) H j d ( 2 ) E j d ( 2 ) H j d ( 1 ) ) , a j L = ( a j E L 0 a j M L 0 ) , a j H = ( 0 a j E H 0 a j M H ) ,
a j E L ( r j ) = 4 π k j 2 μ j ϵ j P . j × [ h l ( 1 ) ( k j r j ) Φ l m * ( θ j , ϕ j ) ] , a j M L ( r j ) = 4 π i k j 3 1 ϵ j h l ( 1 ) ( k j r j ) P . Φ l m * ( θ j , ϕ j ) ,
a j E H ( r j ) = 4 π k j 2 μ j ϵ j P . j × [ j l ( k j r j ) Φ l m * ( θ j , ϕ j ) ] , a j M H ( r j ) = 4 π i k j 3 1 ϵ j j l ( k j r j ) P . Φ l m * ( θ j , ϕ j ) .
E H j T ( r ) = M j ( r ) A j + θ ( r j r ) M j ( r ) a j L + θ ( r r j ) M j ( r ) a j H ,
E H j T ( r j , θ , ϕ ) = E H j + 1 T ( r j , θ , ϕ ) ,
M j ( r j ) [ A j + a j H ] = M j + 1 ( r j ) [ A j + 1 + a j   + 1 L ] ,
A j + 1 = M j + 1 1 ( r j ) M j ( r j ) A j + M j + 1 1 ( r j ) M j ( r j ) a j H a j + 1 L .
A N + 1 = T ( N + 1 , 1 ) A 1 + D ,
T ( N + 1 , j ) = M N + 1 1 ( r N ) M N ( r N ) M N 1 ( r N 1 ) M N 1 ( r N 1 ) M N 1 1 ( r N 2 ) M N 2 ( r N 2 ) M j + 2 1 ( r j + 1 ) M j + 1 ( r j + 1 ) M j + 1 1 ( r j ) M j ( r j ) , for 1 j < N + 1 ,
T ( N + 1 , j ) = I 4 × 4 , for j = N + 1 .
M j + 1 1 ( r j ) M j ( r j ) = i n j + 1 μ j k j + 1 k j G ( j + 1 , j ) = i n j + 1 μ j k j + 1 k j ( G 11 G 12 0 0 G 21 G 22 0 0 0 0 G 33 G 34 0 0 G 43 G 44 ) ( j + 1 , j ) .
( G L ψ l ( k j r j ) χ l ( k j   + 1 r j ) G R ψ l ( k j r j ) χ l ( k j + 1 r j ) G L χ l ( k j r j ) χ l ( k j   + 1 r j ) G R χ l ( k j r j ) χ l ( k j   + 1 r j ) G L ψ l ( k j   + 1 r j ) ψ l ( k j r j ) G R ψ l ( k j   + 1 r j ) ψ l ( k j r j ) G L ψ l ( k j   + 1 r j ) χ l ( k j r j ) G R ψ l ( k j   + 1 r j ) χ l ( k j r j ) )
D = j = 1 N + 1 T ( N + 1 , j ) ( 1 δ j , N + 1 ) a j H T ( N + 1 , j ) ( 1 δ j , 1 ) a j L = ( D 1 , D 2 , D 3 , D 4 ) .
A 1 = T 1 ( N + 1 , 1 ) A N + 1 + T 1 ( N + 1 , 1 ) D .
S = ( T 11 T 12 0 0 T 21 T 22 0 0 0 0 T 33 T 34 0 0 T 43 T 44 ) 1 = ( S 11 S 12 0 0 S 21 S 22 0 0 0 0 S 33 S 34 0 0 S 43 S 44 ) .
A N + 1 = D 1 + S 22 ( S 21 S 12 + S 11 S 22 ) A 1 ,
B N + 1 = D 2 S 21 ( S 21 S 12 + S 11 S 22 ) A 1 ,
C N + 1 = D 3 S 44 ( S 43 S 34 S 33 S 44 ) C 1 ,
D N + 1 = D 4 + S 43 ( S 43 S 34 S 33 S 44 ) C 1 .
B N + 1 = S 21 S 22 A N + 1 ,
D N + 1 = S 43 S 44 C N + 1 .
T 11 = 0 for TM resonances ,
T 33 = 0 for TE resonances .
E H N + 1 T ( r ) = M N + 1 ( r ) A N + 1 + θ ( r r N + 1   ) M N + 1 ( r ) a N + 1 H ( r N + 1   ) .
E s c = l , m ( i n N + 1 ) 1 k N + 1 r χ l   ( k N + 1 r ) Ψ l m ( θ , ϕ ) [ B N + 1 + a N + 1 E H ( r N + 1   ) ] + l , m Φ l m ( θ , ϕ ) 1 k N + 1 r χ l ( k N + 1 r ) [ D N + 1 + a N   + 1 M H ( r N + 1   ) ] ,
H s c = l , m ( 1 μ N + 1 ) Φ l m ( θ , ϕ ) 1 k N + 1 r χ l ( k N + 1 r ) [ B N + 1 + a N   + 1 E H ( r N + 1   ) ] + l , m ( i n N + 1 μ N + 1 ) 1 k N + 1 r χ l   ( k N + 1 r ) Ψ l m [ D N + 1 + a N + 1 M H ( r N + 1   ) ] .
P total = r 2 S s c . r ^ d Ω = c 8 π μ N + 1 r 2 ( E s c × μ N + 1 H s c * ) . r ^ d Ω = c 8 π μ N + 1 l , m ( i n N + 1 k N + 1 2 ) χ l   k N + 1 r χ l * ( k N + 1 r ) | B N + 1 + a N + 1   E H ( r N + 1   ) | 2 + i n N + 1 k N + 1 2 χ l ( k N + 1 r ) χ l * ( k N + 1 r ) | D N + 1 + a N + 1   M H ( r N + 1   ) | 2 ,
P total = c 8 π ϵ N + 1 μ N + 1 1 k N + 1 2 l , m [ ( 1 n N + 1 2 ) | B N + 1 + a N + 1 E H ( r N + 1   ) | 2 + | D N + 1 + a N + 1 M H ( r N + 1 ) | 2 ] .
A 1 = ( S 21 S 12 + S 11 S 22 ) S 22 D 1 , C 1 = ( S 43 S 34 S 33 S 44 ) S 44 D 3 , B N + 1 = D 2 + S 21 S 22 D 1 , D N + 1 = D 4 + S 43 S 44 D 3 .
D = T ( N + 1 , j ) ( 1 δ j , N + 1 ) a j H T ( N + 1 , j ) ( 1 δ j , 1 ) a j L = ( T 12 j ( 1 δ j , N + 1 ) a j E H T 11 j ( 1 δ j , 1 ) a j E L T 22 j ( 1 δ j , N + 1 ) a j E H T 21 j ( 1 δ j , 1 ) a j E L T 34 j ( 1 δ j , N + 1 ) a j M H T 33 j ( 1 δ j , 1 ) a j M L T 44 j ( 1 δ j , N + 1 ) a j M H T 43 j ( 1 δ j , 1 ) a j M L ) ,
B N + 1 + a N + 1 E H ( r N + 1   ) = ( T 22 j + S 21 S 22 T 12 j ) ( 1 δ j , N + 1 ) a j E H ( r j   ) ( T 21 j + S 21 S 22 T 11 j ) ( 1 δ j , 1 ) a j E L ( r j   ) + δ j , N + 1 a j E H ( r j   ) = α l a j E H ( r j   ) β l a j E L ( r j   ) ,
α l = ( T 22 j + S 21 S 22 T 12 j ) ( 1 δ j , N + 1 ) + δ j , N + 1 and β l = ( T 21 j + S 21 S 22 T 11 j ) ( 1 δ j , 1 )
D N + 1 + a N + 1 M H ( r N + 1   ) = ( T 44 j + S 43 S 44 T 34 j ) ( 1 δ j , N + 1 ) a j M H ( r j   ) ( T 43 j + S 43 S 44 T 33 j ) ( 1 δ j , 1 ) a j M L ( r j   ) + δ j , N + 1 a j M H ( r j   ) = γ l a j M H ( r j   ) ζ l a j M L ( r j   ) ,
γ l = ( T 44 j + S 43 S 44 T 34 j ) ( 1 δ j , N + 1 ) + δ j , N + 1 and ζ l = ( T 43 j + S 43 S 44 T 33 j ) ( 1 δ j , 1 )
B N + 1 + a N + 1 E H ( r N + 1   ) = 4 π k j 2 μ j ϵ j P . j   × { [ α l j l ( k j r j   ) β l h l ( 1 ) ( k j r j   ) ] Φ l m * ( θ j   , ϕ j   ) } ,
D N + 1 + a N + 1 M H ( r N + 1   ) = 4 π i k j 3 1 ϵ j [ γ l j l ( k j r j   ) ζ l h l ( 1 ) ( k j r j   ) ] P . Φ l m * ( θ j   , ϕ j   ) .
l , m | B N + 1 + a N + 1 E H ( r N + 1 ) | 2 = 16 π 2 k j 6 ( μ j ϵ j ) l { 2 l + 1 4 π l ( l + 1 ) | ] α l j l ( k j r j ) β l h l ( 1 ) ( k j r j ) [ | 2 k j 2 r   2 | P r | 2 + 2 l + 1 8 π | { α l d d r j ] r j j l ( k j r j ) [ β l d d r j ] r j h l   ( 1 ) ( k j r j ) [ } | 2 k j 2 r j 2 ( | P θ | 2 + | P ϕ | 2 ) } ,
l , m | D N + 1 + a N + 1 M H ( r N + 1 ) | 2 = 16 π 2 k j 6 ( 1 ϵ j 2 ) l 2 l + 1 8 π | [ γ l m j l ( k j r j ) ζ l m h l ( 1 ) ( k j r j ) ] | 2 ( | P θ | 2 + | P ϕ | 2 ) .
P t o t a l = P + P = c 2 ϵ N + 1 μ N + 1 k j 4 n j 2 n N + 1 2 1 ϵ j 2 l ( 2 l + 1 ) { ( n j 2 n N + 1 2 ) l ( l + 1 ) | [ α l j l ( k j r j ) β l h l ( 1 ) ( k j r j ) ] | 2 k j 2 r j 2 | P r | 2 + [ ( n j 2 n N + 1 2 ) | { α l d d r j [ r j j l ( k j r j ) ] β l d d r j [ r j h l ( 1 ) ( k j r j ) ] } | 2 k j 2 r j 2 + | [ γ l j l ( k j r j ) ζ l h l ( 1 ) ( k j r j ) ] | 2 ] [ | P θ | 2 + | P ϕ | 2 2 ] } .
P P 0 = 1 2 ϵ N + 1 μ N + 1 n j 2 n N + 1 2 3 n j ϵ j l ( n j 2 n N + 1 2 ) ( 2 l + 1 ) l ( l + 1 ) | [ α l j l ( k j r j ) β l h l ( 1 ) ( k j r j ) ] | 2 k j 2 r j 2
P P 0 = 1 4 ϵ N + 1 μ N + 1 n j 2 n N + 1 2 3 n j ϵ j × l ( 2 l + 1 ) { [ ( n j 2 n N + 1 2 ) | { α l d d r j [ r j j l ( k j r j ) ] β l d d r j [ r j h l ( 1 ) ( k j r j ) ] } | 2 k j 2 r j 2 + | [ γ l j l ( k j r j ) ζ l h l ( 1 ) ( k j r j ) ] | 2 ] }
P P 0 = 1 2 ϵ N + 1 μ N + 1 n j 2 n N + 1 2 3 n j ϵ j l ( n j 2 n N + 1 2 ) ( 2 l + 1 ) l ( l + 1 ) j | [ α l j l ( k j r j ) β l h l ( 1 ) ( k j r j ) ] | 2 k j 2 r j 2 d 3 r j / j d 3 r j
= 1 2 ϵ N + 1 μ N + 1 n j 2 n N + 1 2 3 n j k j 2 ϵ j V j shell 4 π l ( n j 2 n N + 1 2 ) l ( l + 1 ) I l ( 1 ) ,
I l ( 1 ) = ( 2 l + 1 ) j | [ α l j l ( k j r j ) β l h l ( 1 ) ( k j r j ) ] | 2 d r j ,
P P 0 = 1 4 ϵ N + 1 μ N + 1 n j 2 n N + 1 2 3 n j k j 2 ϵ j ( n j 2 n N + 1 2 ) 4 π l ( 2 l + 1 ) { [ j ( n j 2 n N + 1 2 ) | { α l d d r j [ r j j l ( k j r j ) ] β l d d r j [ r j h l ( 1 ) ( k j r j ) ] } | 2 + k j 2 r j 2 | [ γ l j l ( k j r j ) ζ l h l ( 1 ) ( k j r j ) ] | 2 d r j / j d 3 r j ] } ,
= 1 4 ϵ N + 1 μ N + 1 n j 2 n N + 1 2 3 n j K j 2 ϵ j V j shell 4 π l ( n j 2 n N + 1 2 ) I l ( 2 ) + I l ( 3 ) ,
I l ( 2 ) = ( 2 l + 1 ) j shell | { α l d d r j [ r j j l ( k j r j ) ] β l d d r j [ r j h l ( 1 ) ( k j r j ) ] } | 2 d r j ,
I l ( 3 ) = ( 2 l + 1 ) j shell k j 2 r j 2 | [ γ l j l ( k j r j ) ζ l h l ( 1 ) ( k j r j ) ] | 2 d r j .
P t o t a l P 0 = 1 3 P P 0 + 2 3 P P 0 = 1 2 ϵ N + 1 μ N + 1 n j 2 n N + 1 2 n j k j 2 ϵ j V j shell 4 π l [ ( n j 2 n N + 1 2 ) l ( l + 1 ) I l ( 1 ) + ( n j 2 n N + 1 2 ) I l ( 2 ) + I l ( 3 ) ] .
Ψ [ p l q l ] ( z ) = p l ( z ) q l ( z ) d z = constant + 1 2 { [ z l ( l + 1 ) z ] p l q l 1 2 ( p l q l + p l q l ) + z p l q l } ,
[ l ( l + 1 ) I l ( 1 ) + I l ( 2 ) ] = 1 k j { | α l | 2 ( l + 1 ) ( Ψ [ | ψ l 1 | 2 ] ( k j r j ) Ψ [ | ψ l 1 | 2 ( k j r j 1 ) ) + | α l | 2 l ( Ψ [ | ψ l + 1 | 2 ] ( k j r j ) Ψ [ | ψ l + 1 | 2 ] ( k j r j 1 ) ) + | β l | 2 ( l + 1 ) ( Ψ [ | χ l 1 | 2 ] ( k j r j ) Ψ [ | χ l 1 | 2 ] ( k j r j 1 ) ) + | β l | 2 l ( Ψ [ | χ l + 1 | 2 ] ( k j r j ) Ψ [ | χ l + 1 | 2 ] ( k j r j 1 ) ) ( l + 1 ) ( α l β l * ( Ψ [ ψ l 1 χ l 1 * ] ( k j r j ) Ψ [ ψ l 1 χ l 1 * ) ( k j r j 1 ) ] + α l * β l ( Ψ [ ψ l 1 * χ l 1 ] ( k j r j ) Ψ [ ψ l 1 * χ l 1 ] ( k j r j 1 ) ) ) l ( α l β l * ( Ψ [ ψ l + 1 χ l + 1 * ] ( k j r j ) Ψ [ ψ l + 1 χ l + 1 * ] ( k j r j 1 ) ) + α l * β l ( Ψ [ ψ l + 1 * χ l + 1 ] ( k j r j ) Ψ [ ψ l + 1 * χ l + 1 ] ( k j r j 1 ) ) ) } ,
I l ( 3 ) = ( 2 l + 1 ) j shell k j 2 r j 2 | [ γ l j l ( k j r j ) ζ l h l ( 1 ) ( k j r j ) ] | 2 d r j = 1 k j ( 2 l + 1 ) { | γ l | 2 ( Ψ [ | ψ l | 2 ] ( k j r j ) Ψ [ | ψ l | 2 ] ( k j r j 1 ) ) + | ζ l | 2 ( Ψ [ | χ l | 2 ] ( k j r j ) Ψ [ | χ l | 2 ] ( k j r j 1 ) ) ( γ l ζ l * ( Ψ [ ψ l χ l * ] ( k j r j ) Ψ [ ψ l χ l * ] ( k j r j 1 ) ) + γ l * ζ l ( Ψ [ ψ l * χ l ] ( k j r j ) Ψ [ ψ l * χ l ] ( k j r j 1 ) ) ) } .
| ( P / P 0 ) l = l max + 1 ( P / P 0 ) l = l max | ( P / P 0 ) l = l max + 1 < τ .
Y l m Ψ l m = Φ l m Ψ l m = Φ l m Y l m = 0 ,
Y l m Y l m * d Ω = Ψ l m Ψ l m * d Ω = Φ l m Φ l m * d Ω = δ l l δ m m , Y l m Ψ l m * d Ω = Ψ l m Φ l m * d Ω = Y l m Φ l m * d Ω = 0 ,
× ( f ( r ) Y l m ) = 1 r f ( r ) Φ l m ; × ( f ( r ) Ψ l m ) = ( d f d r + 1 r f ( r ) ) Φ l m , × ( f ( r ) Φ l m ) = l ( l + 1 ) i r f Y l m ( d f d r + 1 r f ) Ψ l m .
m = l m = l Φ l m Φ l m * = m = l m = l Ψ l m Ψ l m * = 2 l + 1 8 π ( e θ e θ + e ϕ e ϕ ) ; m = l m = l Y l m Y l m * = 2 l + 1 4 π ( e r e r ) ; m = l m = l Φ l m Y l m * = m = l m = l Φ l m Ψ l m * = m = l m = l Y l m Ψ l m * = 0 .
X l m ( θ , ϕ ) = ( 1 i ) ( 1 l ( l + 1 ) ) r × Y l m ( θ , ϕ ) [ 23 , 51 ]
X l l m ( θ , ϕ ) = ( 1 i ) ( 1 l ( l + 1 ) ) r × Y l m ( θ , ϕ ) Eq . ( 5.9.14 ) of [ 54 ]
Y L ( m ) = ( 1 i ) ( 1 l ( l + 1 ) ) r × Y l m ( θ , ϕ ) ; Y L ( e ) = ( 1 i ) ( 1 l ( l + 1 ) ) r Y l m ( θ , ϕ ) ; ( 1 i ) Y l m ( θ , ϕ ) r ^ = Y L ( o ) [ 2 ]
Y l l m = X l m = Y L ( m ) = Φ l m , Y L ( e ) = Ψ l m , ( 1 i ) Y L ( o ) = Y l m .
T ( 2 , 1 ) = M 2 1 ( r 1 ) M 1 ( r 1 ) ,
T 11 = i n 2 μ 1 G 11 = i n 2 μ 1 k 2 k 1 n 2 n 1 [ n 2 μ 1 ψ l   ( k 1 r 1 ) χ l ( k 2 r 1 ) n 1 μ 2 ψ l ( k 1 r 1 ) χ l   ( k 2 r 1 ) ] ,
T 33 = i n 2 μ 1 k 2 k 1 G 33 = i n 2 μ 1 k 2 k 1 [ n 1 μ 2 ψ l   ( k 1 r 1 ) χ l ( k 2 r 1 ) n 2 μ 1 ψ l ( k 1 r 1 ) χ l   ( k 2 r 1 ) ] .
n 2 ψ l   ( k 1 r 1 ) ψ l ( k 1 r 1 ) = n 1 χ l   ( k 2 r 1 ) χ l ( k 2 r 1 ) TM resonance condition ,
n 1 ψ l   ( k 1 r 1 ) ψ l ( k 1 r 1 ) = n 2 χ l   ( k 2 r 1 ) χ l ( k 2 r 1 ) TE resonance condition .
D = T 2 ( 2 , 2 ) a 2 L = I 4 × 4 a 2 L
B 2 + a 2 E H ( r 2 ) = α l α 2 E H β l α 2 E H L ; where α l = 1 and β l = ( T 21 2 + S 21 S 22 T 11 2 ) = S 21 S 22
D 2 + a 2 M H ( r 2 ) = γ l α 2 M H ζ l α 2 M L ; where γ l = 1 and ζ l = ( T 43 + S 43 S 44 T 33 ) = S 43 S 44
P P 0 = 3 2 l ( 2 l + 1 ) l ( l + 1 ) | [ j l ( k 2 r 2 ) + S 21 S 22 h l ( 1 ) ( k 2 r 2 ) ] | 2 k 2 2 r 2 2 ,
P | | P | | 0 = 3 4 l ( 2 l + 1 ) { [ | { d d r 2 [ r 2 j l ( k 2 r 2 ) ] + S 21 S 22 d d r 2 [ r 2 h l ( 1 ) ( k 2 r 2 ) ] } | 2 k 2 2 r 2 2 + | [ j l ( k 2 r 2 ) + S 43 S 44 h l ( 1 ) ( k 2 r 2 ) ] | 2 ] } .
S 21 S 22 = [ n 1 μ 2 ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) n 2 μ 1 ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) ] [ n 2 μ 1 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) n 1 μ 2 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ]
= [ ( ε 2 / n 2 ) ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) ( ε 1 / n 1 ) ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) ] [ ( ε 2 / n 2 ) ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ( ε 1 / n 1 ) ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ] ,
S 43 S 44 = n 2 μ 1 ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) n 1 μ 2 ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) n 1 μ 2 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) n 2 μ 1 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 )
= ( ε 1 / n 1 ) ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) ( ε 2 / n 2 ) ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) ( ε 1 / n 1 ) ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ( ε 2 / n 2 ) ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) .
B N + 1 + a N + 1 E H ( r 2 ) = ( T 22 + S 21 S 22 T 12 ) a j E H
= α l a 1   E H β l a 1 E L ; where α l = ( T 22 + S 21 S 22 T 12 ) and β l = 0
D N + 1 + a N + 1 M H ( r 2 ) = ( T 44 + S 43 S 44 T 34 ) a 1 M H
= γ l a 1 M H ζ l a 1 M L ; where γ l = ( T 44 + S 43 S 44 T 34 ) and ζ l = 0
S 22 = [ det ( M 2 ) 1 det ( M 2 ) ] 1 i n 2 μ 1 k 2 k 1 n 2 n 1 [ n 2 μ 1 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) n 1 μ 2 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ] ,
= i k 1 n 1 ϵ 2 ϵ 1 ϵ 2 μ 2 n 1 n 2 [ ϵ 1 r 1 j l ( k 1 r 1 ) χ ( k 2 r 1 ) ϵ 2 ψ l ( k 1 r 1 ) r h l ( 1 ) ( k 2 r 1 ) ] .
S 44 = [ det ( M 2 ) 1 det ( M 1 ) ] 1 i n 2 μ 1 k 2 k 1 [ n 1 μ 2 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) n 2 μ 1 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ] ,
= i ( n 1 μ 2 ) [ n 2 ϵ 2 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) n 1 ϵ 1 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ] .
P | | P | | 0 = 1 4 ϵ 2 μ 2 3 n 1 ϵ 1 ( k 1 r 1 ) 2 l ( 2 l + 1 ) { | d d r 1 [ r j j l ( k 1 r 1 ) ] | 2 k 1 2 r 1 2 | D l | 2 + μ 1 μ 2 ϵ 2 ϵ 1 | j l ( k 1 r 1 ) | 2 | D l | 2 } ,
P P 0 = 1 2 ϵ 2 μ 2 3 n 1 ϵ 1 ( k 1 r 1 ) 2 l ( 2 l + 1 ) l ( l + 1 ) | j l ( k 1 r 1 ) | 2 k 1 2 r 1 2 | D l | 2 ,
D l = [ ϵ 1 j l ( k 1 r 1 ) χ l ( k 2 r 1 ) ϵ 1 ψ l ( k 1 r 1 ) h l ( 1 ) ( k 2 r 1 ) ] , D l = D l ( ϵ μ ) .
1 ( 2 l + 1 ) [ l ( l + 1 ) I l ( 1 ) + I l ( 2 ) ] = α l 2 ( 2 l + 1 ) k 1 { ( l + 1 ) Ψ [ ψ l 1 2 ] ( k 1 r 1 ) + l 2 Ψ [ ψ l + 1 2 ] ( k 1 r 1 ) } .
I l ( 3 ) = 1 k 1 ( 2 l + 1 ) γ l 2 Ψ [ ψ l 2 ] ( k 1 r 1 )
P t o t a l P 0 = 1 3 P P 0 + 2 3 P | | P | | 0 = 3 2 ϵ 2 μ 2 n 1 2 n 2 2 n 1 k 1 2 ϵ 1 r 1 3 l [ l ( l + 1 ) I l ( 1 ) + I l ( 2 ) + I l ( 3 ) ] .
T = M 3 1 ( r 2 ) M 2 ( r 2 ) M 2 1 ( r 1 ) M 1 ( r 1 ) = i k 3 n 2 μ 2 i k 2 n 1 μ 1 G ( 3 , 2 ) G ( 2 , 1 ) .
T 11 = i k 3 n 2 μ 2 i k 2 n 1 μ 1 [ G 11 ( 3 , 2 ) ] G 11 ( 2 , 1 ) + G 12 ( 3 , 2 ) G 21 ( 2 , 1 ) ] = 0
T 33 = i k 3 n 2 μ 2 i k 2 n 1 μ 1 [ G 33 ( 3 , 2 ) ] G 33 ( 2 , 1 ) + G 34 ( 3 , 2 ) G 43 ( 2 , 1 ) ] = 0
μ 2 n 3 χ l ( k 3 r 2 ) n 2 μ 3 χ l ( k 3 r 2 ) = C D ψ l ( k 2 r 2 ) + χ l ( k 2 r 2 ) C D ψ l ( k 2 r 2 ) + χ l ( k 2 r 2 ) } ( TM )
n 2 μ 3 χ l ( k 3 r 2 ) n 3 μ 2 χ l ( k 3 r 2 ) = E F ψ l ( k 2 r 2 ) + χ l ( k 2 r 2 ) E F ψ l ( k 2 r 2 ) + χ l ( k 2 r 2 ) ( TE )
C D = [ n 2 μ 1 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) n 1 μ 2 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ] [ n 1 μ 2 ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) n 2 μ 1 ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) ]
E F = [ n 1 μ 2 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) n 2 μ 1 ψ l ( k 1 r 1 ) χ l ( k 2 r 1 ) ] [ n 2 μ 1 ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) n 1 μ 2 ψ l ( k 2 r 1 ) ψ l ( k 1 r 1 ) ] .

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