Abstract

We present a semi-analytical formalism capable of handling the coupling of electromagnetic sources, such as point dipoles or free-propagating fields, with various kinds of dissipative resonances with radiation leakage, Ohmic losses or both. Due to its analyticity, the approach is very intuitive and physically-sound. It is also very economic in computational resources, since once the resonances of a plasmonic or photonic resonator are known, their excitation coefficients are obtained analytically, independently of the polarization, frequency or location of the excitation source. To evidence that the present formalism is very general and versatile, we implement it with the commercial software COMSOL, rather than with our in-house numerical tools.

© 2013 Optical Society of America

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References

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  1. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  2. R. K. Chang and A. J. Campillo, Optical Processes in Microcavities, Chap. 1 (World Scientific, 1996).
  3. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A49(4), 3057–3067 (1994).
    [CrossRef] [PubMed]
  4. P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev.2(6), 514–526 (2008).
    [CrossRef]
  5. P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen.39(1), 247–267 (2006).
    [CrossRef]
  6. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681 (1946).
  7. P. T. Kristensen, C. Van Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett.37(10), 1649–1651 (2012).
    [CrossRef] [PubMed]
  8. A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett.35(24), 4208–4210 (2010).
    [CrossRef] [PubMed]
  9. S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett.98(4), 47008 (2012).
    [CrossRef]
  10. C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett.110(23), 237401 (2013).
    [CrossRef]
  11. F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8(6), 223–225 (1998).
    [CrossRef]
  12. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994).
    [CrossRef]
  13. A COMSOL model file associated to the article can be found at http://www.lp2n.institutoptique.fr/Membres-Services/Responsables-d-equipe/LALANNE-Philippe .
  14. P. Nordlander and J. C. Tully, “Energy shifts and broadening of atomic levels near metal surfaces,” Phys. Rev. B Condens. Matter42(9), 5564–5578 (1990).
    [CrossRef] [PubMed]
  15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Pparticles (Wiley 1983).
  16. E. D. Palik, Handbook of Optical Constants of Solids, Part II (Academic Press, 1985).
  17. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1989).

2013 (1)

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett.110(23), 237401 (2013).
[CrossRef]

2012 (2)

S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett.98(4), 47008 (2012).
[CrossRef]

P. T. Kristensen, C. Van Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett.37(10), 1649–1651 (2012).
[CrossRef] [PubMed]

2010 (1)

2008 (1)

P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev.2(6), 514–526 (2008).
[CrossRef]

2006 (1)

P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen.39(1), 247–267 (2006).
[CrossRef]

1998 (1)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8(6), 223–225 (1998).
[CrossRef]

1994 (2)

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994).
[CrossRef]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A49(4), 3057–3067 (1994).
[CrossRef] [PubMed]

1990 (1)

P. Nordlander and J. C. Tully, “Energy shifts and broadening of atomic levels near metal surfaces,” Phys. Rev. B Condens. Matter42(9), 5564–5578 (1990).
[CrossRef] [PubMed]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681 (1946).

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994).
[CrossRef]

Bouhelier, A.

S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett.98(4), 47008 (2012).
[CrossRef]

Chew, W. C.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8(6), 223–225 (1998).
[CrossRef]

Colas des Francs, G.

S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett.98(4), 47008 (2012).
[CrossRef]

Derom, S.

S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett.98(4), 47008 (2012).
[CrossRef]

Hughes, S.

Hugonin, J. P.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett.110(23), 237401 (2013).
[CrossRef]

P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev.2(6), 514–526 (2008).
[CrossRef]

Koenderink, A. F.

Kristensen, P. T.

Lalanne, P.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett.110(23), 237401 (2013).
[CrossRef]

P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev.2(6), 514–526 (2008).
[CrossRef]

Leung, P. T.

P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen.39(1), 247–267 (2006).
[CrossRef]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A49(4), 3057–3067 (1994).
[CrossRef] [PubMed]

Liu, S. Y.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A49(4), 3057–3067 (1994).
[CrossRef] [PubMed]

Maksymov, I. S.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett.110(23), 237401 (2013).
[CrossRef]

Nordlander, P.

P. Nordlander and J. C. Tully, “Energy shifts and broadening of atomic levels near metal surfaces,” Phys. Rev. B Condens. Matter42(9), 5564–5578 (1990).
[CrossRef] [PubMed]

Pang, K. M.

P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen.39(1), 247–267 (2006).
[CrossRef]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681 (1946).

Sauvan, C.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett.110(23), 237401 (2013).
[CrossRef]

P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev.2(6), 514–526 (2008).
[CrossRef]

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8(6), 223–225 (1998).
[CrossRef]

Tully, J. C.

P. Nordlander and J. C. Tully, “Energy shifts and broadening of atomic levels near metal surfaces,” Phys. Rev. B Condens. Matter42(9), 5564–5578 (1990).
[CrossRef] [PubMed]

Van Vlack, C.

Vincent, R.

S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett.98(4), 47008 (2012).
[CrossRef]

Young, K.

P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen.39(1), 247–267 (2006).
[CrossRef]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A49(4), 3057–3067 (1994).
[CrossRef] [PubMed]

Europhys. Lett. (1)

S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett.98(4), 47008 (2012).
[CrossRef]

IEEE Microw. Guided Wave Lett. (1)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8(6), 223–225 (1998).
[CrossRef]

J. Comput. Phys. (1)

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994).
[CrossRef]

J. Phys. Math. Gen. (1)

P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen.39(1), 247–267 (2006).
[CrossRef]

Laser & Photon. Rev. (1)

P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev.2(6), 514–526 (2008).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681 (1946).

Phys. Rev. A (1)

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A49(4), 3057–3067 (1994).
[CrossRef] [PubMed]

Phys. Rev. B Condens. Matter (1)

P. Nordlander and J. C. Tully, “Energy shifts and broadening of atomic levels near metal surfaces,” Phys. Rev. B Condens. Matter42(9), 5564–5578 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett.110(23), 237401 (2013).
[CrossRef]

Other (6)

A COMSOL model file associated to the article can be found at http://www.lp2n.institutoptique.fr/Membres-Services/Responsables-d-equipe/LALANNE-Philippe .

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Pparticles (Wiley 1983).

E. D. Palik, Handbook of Optical Constants of Solids, Part II (Academic Press, 1985).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1989).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities, Chap. 1 (World Scientific, 1996).

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Figures (3)

Fig. 1
Fig. 1

(a) Excitation of a 30-nm-wide and 100-nm-long gold nanorod by either a plane wave or a dipole located on axis at a 10-nm distance above the rod. The dipole and the plane wave are polarized along the rod axis. The gold nanorod is embedded in a host medium of refractive index n = 1.5. (b) Dipole illumination: Total spontaneous decay-rate of the dipole normalized by the decay rate in a bulk material with n = 1.5. (c) Plane-wave illumination: Extinction cross-section. In (b), blue triangles and black circles are fully-vectorial computational results obtained with COMSOL for a coarse mesh with 41543 elements and a fine mesh with 875575 elements, respectively. The results obtained with the coarse mesh are in good agreement with the ones obtained with the fine mesh. In (c), black circles are fully-vectorial computational results obtained with COMSOL and a fine mesh. In (b) and (c), the red solid curves represent the predictions of the method proposed in the present work with Eq. (9) and (14), assuming that the excitation is due to a single resonance, namely the fundamental electric-dipole resonance of the nanorod. The gold permittivity is given by a Drude model (see text).

Fig. 2
Fig. 2

|Ez| of the normalized QNMs obtained at the frequencies ω ˜ 1 ( 1 10 5 ), ω ˜ 2 ( 1 10 5 )... ω ˜ 4 ( 1 10 5 ) . Note that, although it cannot compute the field distribution for ω= ω ˜ 4 (see Table 1), COMSOL can perform the calculation for the slightly detuned frequency ω= ω ˜ 4 ( 1 10 5 ) used for the normalization. In (a), the white dash line shows the nanorod boundaries.

Fig. 3
Fig. 3

Absorption and scattering cross-sections of the gold nanorod. (a) Sketch showing the plane wave illumination with a polarization parallel to the rod axis. (b) Absorption cross-section. (c) Scattering cross-section. Black circles are fully electromagnetic computational results obtained with COMSOL for the fine mesh. The solid-red curves are obtained with Eqs. (13)-(15). Note that the scattering cross-section obtained witrh the approximate method takes unphysical negative values for large wavelengths.

Tables (1)

Tables Icon

Table 1 Illustration of the iterative procedure for the nanorod electrical-dipole resonance

Equations (22)

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× E ˜ m =i ω ˜ m μ(r, ω ˜ m ) H ˜ m ,
× H ˜ m =i ω ˜ m ε(r, ω ˜ m ) E ˜ m ,
×E=iωμ(r,ω)H,
×H=iωε(r,ω)E+J.
α m ( ω )= i J(r) E ˜ m (r )d 3 r ( ω ω ˜ m ) + f m ( ω ),
( E ˜ m ( ωε ) / ω E ˜ m H ˜ m ( ωµ ) / ω H ˜ m ) d 3 r=1.
× E b =iωμ( r,ω ) H b and × H b =iω ε b ( r,ω ) E b +J( r ),
×E=iωμ(r,ω)H and ×H=iωε(r,ω)E+J( r ),
Ψ S ( r,ω ) iJ Ψ ˜ ( r 0 ) ω ω ˜ Ψ ˜ ( r ),
Ψ ˜ ( r )= ω ˜ ω iJ Ψ S ( r 0 ,ω ) Ψ S ( r,ω ).
P( ω )= 6π ε 0 c 3 ω 2 n | J | 2 Re[ α( ω ) J Ψ ˜ ( r 0 ) ],
× E s =iωμ( r,ω ) H s and × H s =iωε( r,ω ) E s +iωΔε( r,ω ) E b .
Ψ S ( r,ω ) m β m ( ω ) Ψ ˜ m ( r ).
β m ( ω )= ω Δε( r,ω ) E b ( r,ω ) E ˜ m ( r ) d 3 r ( ω ω ˜ m ) + g m ( ω ),
σ A = ω 2 S 0 V Im[ ε( r,ω ) ]| E S ( r,ω )+ E b ( r,ω ) | 2 d 3 r,
σ E = ω 2 S 0 V Im( Δε( r,ω )( E b ( r,ω )+ E s ( r,ω ) ) E b * ( r,ω ) ) d 3 r ,
σ S = σ E σ A .
Ψ( r,ω ) Ψ b ( r,ω )+ m ω Δε( r,ω ) E b ( r,ω ) E ˜ m ( r ) d 3 r ( ω ω ˜ m ) Ψ ˜ m ( r ),
P( ω )= 6π ε 0 c 3 ω 2 n | J | 2 Re[ J Ψ( r 0 ,ω) ].
×E=iωμ(r)H and ×H=iωε(r,ω)E+J( r )
×E=i ω L μ eff (r)H and ×H=i ω L ε eff (r)E+Jδ( r r 0 )
ω 4 = ω 1 ( ω 2 ω 3 ) / Z 1 + ω 2 ( ω 3 ω 1 ) / Z 2 + ω 3 ( ω 1 ω 2 ) / Z 3 ( ω 2 ω 3 ) / Z 1 + ( ω 3 ω 1 ) / Z 2 + ( ω 1 ω 2 ) / Z 3 .

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