Abstract

We provide numerical demonstrations of the applicability and accuracy of the quasi-static method and the finite-element method in the investigation of the modifications of the spontaneous emission rate and the energy level shift of an emitter placed near a silver-air interface or a silver nano-sphere. The analytical results are presented as a reference. Our calculations show that the finite element method is an accurate and general method. For frequency away from the radiative mode, the quasi-static method can be applied more effectively for calculating the energy level shift than the spontaneous emission rate. But for frequency around, there is a blue shift for both and this shift increases with the increasing of emitter-silver distance. Applying the theory to the nanosphere dimmer, we see similar phenomenon and find extremely large modifications of the spontaneous emission rate and energy level shift. These findings are instructive in the fields of quantum light-matter interactions.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (11)

A. B. Taylor and P. Zijlstra, “Single-Molecule Plasmon Sensing: Current Status and Future Prospects,” ACS Sensors 2, 1103–1122 (2017).
[Crossref] [PubMed]

H. Imada, K. Miwa, M. Imai-Imada, S. Kawahara, K. Kimura, and Y. Kim, “Single-Molecule Investigation of Energy Dynamics in a Coupled Plasmon-Exciton System,” Phys. Rev. Lett. 119, 013901 (2017).
[Crossref] [PubMed]

R. M. Liu, Z. K. Zhou, Y. C. Yu, T. W. Zhang, H. Wang, G. H. Liu, Y. M. Wei, H. J. Chen, and X. H. Wang, “Strong Light-Matter Interactions in Single Open Plasmonic Nanocavities at the Quantum Optics Limit,” Phys. Rev. Lett. 118, 237401 (2017).
[Crossref] [PubMed]

Y. Zhang, Q. S. Meng, L. Zhang, Y. Luo, Y. J. Yu, B. Yang, Y. Zhang, R. Esteban, J. Aizpurua, Y. Luo, J. L. Yang, Z. C. Dong, and J. G. Hou, “Sub-nanometre control of the coherent interaction between a single molecule and a plasmonic nanocavity,” Nature Communications 8, 5225 (2017).

J. J. Ren, Y. Gu, D. X. Zhao, F. Zhang, T. C. Zhang, and Q. H. Gong, “Evanescent-Vacuum-Enhanced Photon-Exciton Coupling and Fluorescence Collection,” Phys. Rev. Lett. 118, 073604 (2017).
[Crossref] [PubMed]

R. Jurga, S. D’Agostino, F. Della Sala, and C. Ciraci, “Plasmonic Nonlocal Response Effects on Dipole Decay Dynamics in the Weak and Strong-Coupling Regimes,” The Journal of Physical Chemistry C 121, 22361–22368 (2017).
[Crossref]

C. J. Yang and J. H. An, “Suppressed dissipation of a quantum emitter coupled to surface plasmon polaritons,” Phys. Rev. B 95, 161408 (2017).
[Crossref]

S. C. Zhu, Z. G. Yu, L. Liu, C. Yang, H. C. Cao, X. Xi, J. M. Li, and L. X. Zhao, “Enhancing the spontaneous emission rate by modulating carrier distribution in GaN-based surface plasmon light-emitting diodes,” Opt. Express 25, 9617–9627 (2017).
[Crossref] [PubMed]

E. Lassalle, A. Devilez, N. Bonod, T. Durt, and B. Stout, “Lamb shift multipolar analysis,” J. Opt. Soc. Am. B 34, 1348–1355 (2017).
[Crossref]

G. M. Das, A. B. Ringne, V. R. Dantham, R. K. Easwaran, and R. Laha, “Numerical investigations on photonic nanojet mediated surface enhanced Raman scattering and fluorescence techniques,” Opt. Express 25, 19822–19831 (2017).
[Crossref] [PubMed]

M. K. Dezfouli, C. Tserkezis, N. A. Mortensen, and S. Hughes, “Nonlocal quasinormal modes for arbitrarily shaped three-dimensional plasmonic resonators,” Optica 4, 1503–1509 (2017).
[Crossref]

2016 (5)

C. Tserkezis, N. Stefanou, M. Wubs, and N. A. Mortensen, “Molecular fluorescence enhancement in plasmonic environments: exploring the role of nonlocal effects,” Nanoscale 8, 17532–17541 (2016).
[Crossref] [PubMed]

Y. Li and C. Argyropoulos, “Controlling collective spontaneous emission with plasmonic waveguides,” Opt. Express 24, 26696–26708 (2016).
[Crossref] [PubMed]

A. Vagov, I. A. Larkin, M. D. Croitoru, and V. M. Axt, “Role of nonlocality and Landau damping in the dynamics of a quantum dot coupled to surface plasmons,” Phys. Rev. B 93, 195414 (2016).
[Crossref]

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref] [PubMed]

Y. Zhang, Y. Luo, Y. Zhang, Y. J. Yu, Y. M. Kuang, L. Zhang, Q. S. Meng, Y. Luo, J. L. Yang, Z. C. Dong, and J. G. Hou, “Visualizing coherent intermolecular dipole–dipole coupling in real space,” Nature 531, 623–627 (2016).
[Crossref] [PubMed]

2015 (1)

M. Li, S. K. Cushing, and N. Q. Wu, “Plasmon-enhanced optical sensors: a review,” Analyst 140, 386–406 (2015).
[Crossref]

2014 (2)

A. González-Tudela, P. A. Huidobro, L. Martín-Moreno, C. Tejedor, and F. J. García-Vidal, “Reversible dynamics of single quantum emitters near metal-dielectric interfaces,” Phys. Rev. B 89, 041402 (2014).
[Crossref]

A. Delga, J. Feist, J. Bravo-Abad, and F. J. Garcia-Vidal, “Quantum Emitters Near a Metal Nanoparticle: Strong Coupling and Quenching,” Phys. Rev. Lett. 112, 253601 (2014).
[Crossref] [PubMed]

2013 (2)

C. Belacel, B. Habert, F. Bigourdan, F. Marquier, J.-P. Hugonin, S. Michaelis de Vasconcellos, X. Lafosse, L. Coolen, C. Schwob, C. Javaux, B. Dubertret, J.-J. Greffet, P. Senellart, and A. Maitre, “Controlling Spontaneous Emission with Plasmonic Optical Patch Antennas,” Nano Letters 13, 1516–1521 (2013).
[Crossref] [PubMed]

D. Punj, J. de Torres, H. Rigneault, and J. Wenger, “Gold nanoparticles for enhanced single molecule fluorescence analysis at micromolar concentration,” Opt. Express 21, 27338–27343 (2013).
[Crossref] [PubMed]

2012 (6)

C. Van Vlack and S. Hughes, “Finite-difference time-domain technique as an efficient tool for calculating the regularized Green function: applications to the local-field problem in quantum optics for inhomogeneous lossy materials,” Opt. Lett. 37, 2880–2882 (2012).
[Crossref] [PubMed]

Y. J. Lu, J. Kim, H. Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C. Y. Wang, M. Y. Lu, B. H. Li, X. G. Qiu, W. H. Chang, L. J. Chen, G. Shvets, C. K. Shih, and S. J. Gwo, “Plasmonic Nanolaser Using Epitaxially Grown Silver Film,” Science 337, 450–453 (2012).
[Crossref] [PubMed]

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[Crossref] [PubMed]

C. Van Vlack, P. T. Kristensen, and S. Hughes, “Spontaneous emission spectra and quantum light-matter interactions from a strongly coupled quantum dot metal-nanoparticle system,” Phys. Rev. B 85, 075303 (2012).
[Crossref]

Y. G. Huang, G. Y. Chen, C. J. Jin, W. M. Liu, and X. H. Wang, “Dipole-dipole interaction in a photonic crystal nanocavity,” Phys. Rev. A 85, 053827 (2012).
[Crossref]

U. Hohenester and A. Trügler, “MNPBEM–A Matlab toolbox for the simulation of plasmonic nanoparticles,” Computer Physics Communications 183, 370–381 (2012).
[Crossref]

2011 (2)

N. J. Halas, S. Lal, W. S. Chang, S. Link, and P. Nordlander, “Plasmons in Strongly Coupled Metallic Nanostructures,” Chem. Rev 111, 3913–3961 (2011).
[Crossref] [PubMed]

R. M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Materials 10, 110–113 (2011).
[Crossref]

2010 (2)

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Materials 9, 707–715 (2010).
[Crossref]

Y. T. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[Crossref]

2009 (2)

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3, 654–657 (2009).
[Crossref]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (4)

L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, “Design of plasmonic nanoantennae for enhancing spontaneous emission,” Opt. Lett. 32, 1623–1625 (2007).
[Crossref] [PubMed]

O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Strong Enhancement of the Radiative Decay Rate of Emitters by Single Plasmonic Nanoantennas,” Nano Letters 7, 2871–2875 (2007).
[Crossref] [PubMed]

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402–406 (2007).
[Crossref] [PubMed]

K. A. Willets and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy and Sensing,” Annu. Rev. Phys. Chem. 58, 267–297 (2007).
[Crossref]

2006 (3)

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum Optics with Surface Plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and Quenching of Single-Molecule Fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[Crossref] [PubMed]

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2006).
[Crossref]

2005 (1)

K. Okamoto, I. Niki, A. Scherer, Y. Narukawa, T. Mukai, and Y. Kawakami, “Surface plasmon enhanced spontaneous emission rate of InGaN/ GaN quantum wells probed by time-resolved photoluminescence spectroscopy,” Appl. Phys. Lett. 87, 071102 (2005).
[Crossref]

2004 (1)

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Materials 3, 601–605 (2004).
[Crossref] [PubMed]

1999 (1)

H. X. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, “Spectroscopy of Single Hemoglobin Molecules by Surface Enhanced Raman Scattering,” Phys. Rev. Lett. 83, 4357 (1999).
[Crossref]

1998 (1)

A. Campion and P. Kambhampati, “Surface-enhanced Raman scattering,” Chemical Society Reviews 27, 241–250 (1998).
[Crossref]

1997 (2)

S. M. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single Molecule Detection Using Surface-Enhanced Raman Scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
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1996 (1)

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Y. Zhang, Y. Luo, Y. Zhang, Y. J. Yu, Y. M. Kuang, L. Zhang, Q. S. Meng, Y. Luo, J. L. Yang, Z. C. Dong, and J. G. Hou, “Visualizing coherent intermolecular dipole–dipole coupling in real space,” Nature 531, 623–627 (2016).
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A. B. Taylor and P. Zijlstra, “Single-Molecule Plasmon Sensing: Current Status and Future Prospects,” ACS Sensors 2, 1103–1122 (2017).
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Analyst (1)

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C. Belacel, B. Habert, F. Bigourdan, F. Marquier, J.-P. Hugonin, S. Michaelis de Vasconcellos, X. Lafosse, L. Coolen, C. Schwob, C. Javaux, B. Dubertret, J.-J. Greffet, P. Senellart, and A. Maitre, “Controlling Spontaneous Emission with Plasmonic Optical Patch Antennas,” Nano Letters 13, 1516–1521 (2013).
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C. Tserkezis, N. Stefanou, M. Wubs, and N. A. Mortensen, “Molecular fluorescence enhancement in plasmonic environments: exploring the role of nonlocal effects,” Nanoscale 8, 17532–17541 (2016).
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P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2006).
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B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Materials 9, 707–715 (2010).
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R. M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Materials 10, 110–113 (2011).
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K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Materials 3, 601–605 (2004).
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Nat. Photonics (1)

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3, 654–657 (2009).
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Nature (5)

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402–406 (2007).
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M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
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Y. Zhang, Y. Luo, Y. Zhang, Y. J. Yu, Y. M. Kuang, L. Zhang, Q. S. Meng, Y. Luo, J. L. Yang, Z. C. Dong, and J. G. Hou, “Visualizing coherent intermolecular dipole–dipole coupling in real space,” Nature 531, 623–627 (2016).
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Opt. Lett. (2)

Optica (1)

Phys. Rev. A (1)

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C. J. Yang and J. H. An, “Suppressed dissipation of a quantum emitter coupled to surface plasmon polaritons,” Phys. Rev. B 95, 161408 (2017).
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Figures (5)

Fig. 1
Fig. 1 Schematic diagrams. An emitter ‘A’ is placed near (a) a silver-air interface or (b) a silver nano-sphere. For simplicity, the dipole moment of the emitter ‘A’ is normal to the surface. ε1 and ε2 are the permittivities of air and silver, respectively. The distance between the emitter and the surface of metal is l. The radius for the sphere is a.
Fig. 2
Fig. 2 The modified spontaneous emission rate and the energy level shift. The red solid line, dashed blue line and the black circles are the results by the methods of analytical, quasi-static and FEM, respectively. (a) – (d) are for short emitter-interface distance l = 2nm. (a) Normalized spontaneous emission rate Γ/Γ0 with Γ0 and (b) energy level shift. The insets are for frequency away from ωsp. There are no visible differences for both the quasi-static method and the FEM method. (c) and (d) are for ΓR_error ≡ (Γi − Γanaly)/Γanaly and Δerror ≡ Δi − Δanaly, respectively, where the subscript i represents the quasi-static (Eq. (2)) or the FEM method (Eq. (4)) and subscript analy is the analytic (Eq. (5)). The inset in (d) shows a slight blue-shift for Γ/Γ0 by the quasi-static method to explain the relative large error around ωsp. (e) – (f) are for longer emitter-interface distance l = 5nm.
Fig. 3
Fig. 3 The modified spontaneous emission rate and the energy level shift for emitter around a silver nano-sphere. The red solid line, dashed blue line and the black circles are the results by the methods of analytical, quasi-static and FEM, respectively. (a) and (b) are for short emitter-interface distance l = 2nm. (a) Normalized spontaneous emission rate Γ/Γ0 with Γ0 and (b) energy level shift. The insets are for frequency around the dipole mode of the LSPP. (c) and (d) are for longer emitter-interface distance l = 5nm.
Fig. 4
Fig. 4 Schematic diagram. A perfect electric dipole emitter with parallel orientation is located at the center of silver nanosphere dimer of radius a = 20nm and gap L.
Fig. 5
Fig. 5 The modified spontaneous emission rate and the energy level shift for emitter with parallel orientation located at the center of nanosphere dimmer. The dashed blue line and the black circles are the results by the methods of quasi-static and FEM, respectively. (a) and (b) are for longer dimmer gap L = 4nm. (a) Normalized spontaneous emission rate Γ/Γ0 with Γ0 and (b) energy level shift. (c) and (d) are the same as (a) and (b) except for a shorter dimmer gap L = 2nm.

Equations (10)

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Γ ( r ; ω ) = 2 d 2 [ n G ( r ; r ; ω ) n ] ε 0 , Δ ( r ; ω ) = d 2 [ n G ( r ; r ; ω ) n ] ε 0 ,
G static z z ( r , r ) z G static ( r , r ) z = ε 2 ε 1 16 π ε 1 ( ε 2 + ε 1 ) l 3 ,
G static r r ( r , r ) = 1 4 π ε 1 a 3 n = 1 ( n + 1 ) 2 ( 1 + ξ ) 2 n + 4 ε 2 ε 1 ε 2 + n + 1 n ε 1 ,
G FEM n n ( r , r ) n G ( r , r ) n = ε 0 E s ( r ) / d ,
G planar ( s ) ( r , r , ω ) = i 4 π p = e , o n 0 ( 2 δ n , 0 ) λ h 1 [ R A M ¯ λ n p ( h 1 r ) M ¯ λ n p ( h 1 r ) ] + [ R B N ¯ λ n p ( h 1 r ) N ¯ λ n p ( h 1 r ) ] d λ ,
R A = h 1 h 2 h 1 + h 2 , R B = k 2 2 h 1 k 1 2 h 2 k 2 2 h 1 + k 1 2 h 2 .
M ¯ λ n e ( h 1 r ) = [ n J n ( λ r ) r sin n ϕ r ^ J n ( λ r ) r cos n ϕ ϕ ^ ] e i h 1 z , M ¯ λ n o ( h 1 r ) = [ n J n ( λ r ) r cos n ϕ r ^ J n ( λ r ) r sin n ϕ ϕ ^ ] e i h 1 z , N ¯ λ n e ( h 1 r ) = 1 λ 2 + h 1 2 [ i h 1 J n ( λ r ) r cos n ϕ r ^ i h 1 n r J n ( λ r ) sin n ϕ ϕ ^ + λ 2 J n ( λ r ) cos n ϕ z ^ ] e i h 1 z , N ¯ λ n o ( h 1 r ) = 1 λ 2 + h 1 2 [ i h 1 J n ( λ r ) r sin n ϕ r ^ + i h 1 n r J n ( λ r ) cos n ϕ ϕ ^ + λ 2 J n ( λ r ) sin n ϕ z ^ ] e i h 1 z .
G sphere ( s ) ( r , r , ω ) = i k 1 3 4 π p = e , o n = 1 m = 0 n ( 2 δ m , 0 ) 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! × [ R H M n m p ( k 1 r ) M n m p ( k 1 r ) + R V N n m p ( k 1 r ) N n m p ( k 1 r ) ] .
R H = k 2 τ 2 τ 1 k 1 τ 1 τ 2 k 2 τ 2 κ 1 k 1 κ 1 τ 2 , R V = k 2 τ 2 τ 1 k 1 τ 1 τ 2 k 2 τ 2 κ 1 k 1 κ 1 τ 2 ,
M nme ( k r ) = m sin θ h n ( 1 ) ( k r ) P n m ( cos θ ) sin m ϕ θ ^ h n ( 1 ) ( k r ) d P n m ( cos θ ) d θ cos m ϕ ϕ ^ , M nmo ( k r ) = m sin θ h n ( 1 ) ( k r ) P n m ( cos θ ) cos m ϕ θ ^ h n ( 1 ) ( k r ) d P n m ( cos θ ) d θ sin m ϕ ϕ ^ , N nme ( k r ) = n ( n + 1 ) k r h n ( 1 ) ( k r ) P n m ( cos θ ) cos m ϕ r ^ + 1 k r d ( r h n ( 1 ) ( k r ) ) d r [ d P n m ( cos θ ) d θ cos m ϕ θ ^ m sin θ P n m ( cos θ ) sin m ϕ ϕ ^ ] , N nmo ( k r ) = n ( n + 1 ) k r h n ( 1 ) ( k r ) P n m ( cos θ ) sin m ϕ r ^ + 1 k r d ( r h n ( 1 ) ( k r ) ) d r [ d P n m ( cos θ ) d θ sin m ϕ θ ^ m sin θ P n m ( cos θ ) cos m ϕ ϕ ^ ] ,

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