## Plasmonic field enhancement and SERS in the effective mode volume picture

Optics Express, Vol. 14, Issue 5, pp. 1957-1964 (2006)

http://dx.doi.org/10.1364/OE.14.001957

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### Abstract

The controlled creation of nanometric electromag-
netic field confinement via surface plasmon polariton excitations in metal/insulator/metal heterostructures is described via the concept of an effective electromagnetic mode volume *V*_{eff}. Extensively used for the description of dielectric microcavities, its extension to plasmonics provides a convenient figure of merit and allows comparisons with dielectric counterparts. Using a one-dimensional analytical model and three-dimensional finite-difference time-domain simulations, it is shown that plasmonic cavities with nanometric dielectric gaps indeed allow for *physical* as well as *effective* mode volumes well below the diffraction limit in the gap material, despite significant energy penetration into the metal. In this picture, matter-plasmon interactions can be quantified in terms of quality factor *Q* and *V*_{
eff
}, enabling a resonant cavity description of surface enhanced Raman scattering.

© 2006 Optical Society of America

## 1. Introduction

*plasmonics*- the study of electromagnetic field confinement and enhancement via surface plasmon polaritons (SPPs) -have led to a number of important advances towards the goal of a nanophotonic infrastructure for confining and guiding electromagnetic radiation [1

1. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. **98**, 011,101 (2005). [CrossRef]

2. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mat. **2**, 229–232 (2003). [CrossRef]

3. W. L. Barnes, A. Dereux, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2002). [CrossRef]

4. 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 (1997). [CrossRef]

5. S. M. Nie and S. R. Emery, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science **275**, 1102 (1997). [CrossRef] [PubMed]

*designed*metal nanoscale cavities. The single molecule sensitivity relies on light localization in hot-spots on a roughened silver (Ag) surface where random, nanometer-sized junctions between surface protrusions are believed to form cavity-like structures for field enhancement [6

6. H. Xu, J. Aizpurua, M. Kaell, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E **62**, 4318–4324 (2000). [CrossRef]

7. A. Sundaramurthy, K. B. Crozier, G. S. Kino, D. P. Fromm, P. J. Schuck, and W. E. Moerner, “Field enhancement and gap-dependent resonance in a system of two opposing tip-to-tip Au nanotriangles,” Phys. Rev. B. **72**, 165409 (2005) [CrossRef]

9. D. J. Norris, M. Kuwata-Gonokami, and W. E. Moerner, “Excitation of a single molecule on the surface of a spherical microcavity,” Appl. Phys. Lett. **71**, 297–299 (1997). [CrossRef]

*Q*, being proportional to the cavity photon lifetime, and its effective mode volume

*V*

_{eff}, quantifying the electric field strength per photon.

*Q*and 1/

*V*

_{eff}can be thought of as the spectral and spatial energy density of the resonant mode, respectively. Prominent geometries include dielectric spheres [10

10. R. K. Chang and A. J. Campillo, eds., “*Optical Processes in Microcavities*” (World Scientific, Singapore, 1996). [CrossRef]

11. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. **23**, 247–249 (1998). [CrossRef]

12. 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]

13. A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room Temperature Operation of Submicrometre Radius Disk Laser,” Electron. Lett. **29**, 1666–1667 (1993). [CrossRef]

*Q*> 10

^{8}enabling

*Q*/

*V*¯

_{eff}~ 10

^{5}[12

12. 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]

*V*¯

_{eff}is the effective mode volume normalized to (λ

_{0}/

*n*)

^{3}, the cubic wavelength in the material. Photonic crystal micro-cavities on the other hand allow

*V*

_{eff}to approach the theoretical diffraction limit, corresponding to a cubic half wavelength in the material [15

15. O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science **284**, 1819–1824 (1999). [CrossRef] [PubMed]

*Q*and

*V*

_{eff}. The beauty of this approach lies in the ability to determine simple scaling laws for various processes such as spontaneous emission [17], strong matter-photon coupling [8] and non-linear optical thresholds [18

18. A. B. Matsko, A. A. Savchenkov, R. J. Letargat, V. S. Ilchenko, and L. Maleki, “On cavity modification of stimulated Raman scattering,” J. Opt. B: Quantum Semiclass. Opt. **5**, 272–278 (2003). [CrossRef]

*V*

_{eff}for metallic systems, where the generation of localized light volumes smaller than the diffraction limit in the dielectric space surrounding a metallic nanostructure does not in itself imply that

*V*

_{eff}is smaller than the diffraction limit. The applicability of the effective mode volume concept for metallic nanostystems requires careful account of the dispersive character of the plasmon-polariton excitations and the electromagnetic energy

*stored inside the metal*, both of which become significant for deep sub-wavelength confinement. After a discussion of a nanoplasmonic Fabry-Perot type resonator using both a onedimensional analytical model and threedimensional finite difference time domain (FDTD) simulations, the advantages of characterizing plasmonic energy confinement in terms of

*Q*and

*V*

_{eff}are demonstrated via a simple model for surface enhanced Raman scattering (SERS), which has typically been analyzed using scattering-type calculations [19

19. M. Kerker, D.-S. Wang, and H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles: errata,” Appl. Opt. **19**, 4159–4147 (1980). [CrossRef] [PubMed]

6. H. Xu, J. Aizpurua, M. Kaell, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E **62**, 4318–4324 (2000). [CrossRef]

## 2. Onedimensional model of gap plasmons

*z*-direction perpendicular to the plane of the dielectric core, the mode is confined via a coupled surface plasmon-polariton sustained by the metallic boundaries. Laterally, the physical extent of the cavity together with the increased wave vector of the surface plasmon mode propagating in the

*x*-direction will lead to confinement, while in propagation direction the mode can be confined using reflective walls or indeed a simple air boundary, leading to Fabry-Perot type oscillations. Before embarking on a numerical analysis of such a cavity, it is instructive to analytically consider the energy confinement properties of a canonical planar metal-air-metal heterostructure composed out of an air (ε

_{1}= 1) core of width 2

*a*surrounded by two metallic half-spaces (Fig. 1, right inset). As is well known, such a heterostructure can support two surface modes propagating in the

*x*-direction parallel to the interfaces that are set up by coupling of the surface plasmon-polariton modes of the individual air/metal boundaries [20

20. B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B **44**, 13,556–13,572 (1991). [CrossRef]

*vector*parity, which does not have a cut-off gap size and shows a symmetric scalar field distribution of the dominant electric field component,

*E*

_{z}, with respect to the symmetry plane, as depicted in the right inset of Fig. 1 (this is the lowest order

*capacitor*-type mode). The dielectric response of the metallic half-spaces is modeled using a Drude fit to the dielectric function ε(ω) for gold (Au) at visible and near-infrared frequencies [21

21. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*k*

_{x}and the dielectric response of the metal cladding and air gap via

22. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A **3**, 233–245 (1970). [CrossRef]

23. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A **299**, 309–312 (2002). [CrossRef]

_{1}(ω)-

*i*ε

_{2}(ω) is the complex dielectric function of a Drude model with damping contant γ.

24. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. **22**, 475 (1997). [CrossRef] [PubMed]

_{0}= 850nm Both parts are seen to increase with decreasing gap size, suggesting that the capacitor mode is becoming more electron-plasma in character, and that the electromagnetic energy is residing increasingly in the metal half-spaces. A plot of the fractional amount of energy inside the metal regions is shown in Fig. 2(b) for excitation at wavelengths λ

_{0}= 600 nm, 850 nm, 1.5 μm, 10 μm, and 100 μm (= 3THz), reaching e.g. 40% for a gap of 20 nm at λ

_{0}= 850 nm. For this and all following figures, the gap size is normalized to the respective free space wavelength, and the results for each wavelength are plotted over the range of convergenceof the analytical model. It can be seen then, that along with the increased localization of the field to the metal/air interface, either via small gap sizes or excitation closer to the surface plasmon frequency, comes a shift of the energy into the metal regions.

## 3. Electromagnetic energy density and the effective mode volume in plasmonics

*volume*

*V*

_{eff}of cQED [25

25. L. C. Andreani, G. Panzarini, and J.-M. Gérard, “Strong-coupling regime for quantum boxes in pillar microcavi-ties: Theory,” Phys. Rev. B **60**, 13,276 (1999). [CrossRef]

*length*

*L*

_{eff}for this one dimensional “resonator”,

*u*

_{E}(

*z*

_{0}) represents the electric field energy density at position

*z*

_{0}, corresponding to the position of interest within the cavity. For the structure of Fig. 1,

*z*

_{0}resides in the air gap where an object may be placed to interact with the field. Fig. 2(c) shows the variation of

*L*¯

_{eff}(normalized to the free space wavelength λ

_{0}) with normalized gap size for the capacitor mode.

_{0}/2, demonstrating that plasmonic metal structures do indeed sustain

*effective*as well as

*physical*mode lengths below the diffraction limit of light. The trend in

*L*

_{eff}with gap size tends to scale with the physical extent of the air gap. For large normalized gap sizes and low frequencies, this is due to the delocalized nature of the surface plasmon, leading to smaller mode lengths for excitation closer to the surface plasmon resonance frequency for the same normalized gap size. As the gap size is reduced to a point where the bandstructure of the capacitor mode turns over (see Fig. 1) and energy begins to enter the metallic half spaces, the continued reduction in mode length is due to an increase in field localization to the metal-air surface. In this regime, excitations with lower frequencies show smaller mode lengths for the same normalized gap size than excitations closer to the plasmon resonance, due to the fact that more energy resides inside the metal for the latter. Fig. 2(d) further elucidates this effect by showing the contributions of the electric field energy in air (continuous line) and in the metal (broken line) to the total effective mode length for two excitation wavelengths.

26. M. A. Ordal, L. L. Long, R. J. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. **22**, 1099–1119 (1983). [CrossRef] [PubMed]

27. I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, “Dipolar emitters at nanoscale proximity of metal surfaces: Giant enhancement of relaxation in microscopic theory,” Phys. Rev. B **69**, 121403(R) (2004). [CrossRef]

*L*

_{eff}which cannot be captured using the dielectric function approach.

*x*ʌ-direction, thus confining the propagation of the odd mode to a cavity length

*L*

_{x}. Assuming in a first approxi-mation perfectly reflecting cavity walls, the fundamental cavity mode will be excited when

*L*

_{x}= π/β(λ

_{0},

*a*). If one conservatively assumes a diffraction-limited lateral resonator width

*L*

_{y}= λ

_{0}/2, the cavity mode volume can thus be approximated as

*V*

_{eff}~

*L*

_{eff}(πλ

_{0}/2β).

*Q*, defined as the ratio of stored energy to loss per cycle. Assuming no radiative losses from the resonator edges and perfectly reflecting cavity walls, Q is limited by dissipative losses inside the metal alone:

*Q*=

*ω*

_{0}/(2

*v*

_{gr}Imag(β)), where

*v*

_{gr}is the group velocity of the guided mode between the metal plates. As a cavity figure of merit, Fig. 3 shows the analytically calculated

*Q*/

*V*¯

_{eff}. As can be seen,

*Q*/

*V*¯

_{eff}greatly increases for decreasing gap size, due to the fact that

*V*¯

_{eff}decreases much more strongly than

*Q*. For a resonator with 2

*a*= 50nm designed for excitation at λ

_{0}= 850nm, the analytic model predicts a

*Q*~ 51 and a normalized mode volume

*V*¯

_{eff}~ 0.015, leading to

*Q*/

*V*¯

_{eff}~ 3400. Excitation at mid-and far-infrared frequencies can yield

*Q*/

*V*¯

_{eff}≫ 10

^{4}for similar gap sizes.

_{0}= 850nm to 980nm due to the electric field penetration into the end mirrors. The calculated absorptive

*Q*

_{abs}~ 40 is of similar magnitude as the analytic result, and a radiative

*Q*

_{rad}~ 100 is found due to lateral leakage of radiation. The FDTD-calculated mode volume

*V*

_{eff}~ 0.009 is also in good agreement with the analytic model estimate when scaled by the cube of the increase in resonance wavelength. While these discussions have focused on a planar metal/insulator/metal heterostructure, the general conclusions regarding

*Q*/

*V*¯

_{eff}are also expected to hold for more complicated geometries involving two metallic surfaces separated by a nanoscale gap, such as e.g. nanoshells.

## 4. Application to surface-enhanced Raman scattering

**E**

_{i}/(ω

_{0})|

^{2}/2η (η the impedance of free space) and frequency ω

_{0}, leading to the emission of Stokes photons at frequency a) through a Raman active molecule. Due to the small Stokes emission shift, one usually assumes equal enhancement of the exciting field and the outgoing Stokes field, and a commonly used expression[19

19. M. Kerker, D.-S. Wang, and H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles: errata,” Appl. Opt. **19**, 4159–4147 (1980). [CrossRef] [PubMed]

*R*= |

**E**

_{loc}|

^{4}/|

**E**

_{i}|

^{4}, where |

**E**

_{loc}| is the local field amplitude at the Raman active site. For this treatment,

*Q*(ω

_{0})=

*Q*(ω)=

*Q*and

*V*

_{eff}(ω

_{0})=

*V*

_{eff}(ω)=

*V*

_{eff}, and it is thus assumed that both the incoming and the emitted photon are resonant with the cavity.

*s*

_{+}|

^{2}= |

**E**

_{i}|

^{2}

*A*

_{i}/2η being the power carried by the incident beam of cross section

*A*

_{i}, the evolution of the

*on-resonance*mode amplitude

*u*inside the cavity can be calculated from [28],

*u*

^{2}represents the total time-averaged energy in the cavity, γ = γ

_{rad}² γ

_{abs}is the energy decay rate due to radiation (γ

_{rad}) and absorption (γ

_{abs}), and κ is the coupling coefficient to external input which depends on the size and shape of the excitation beam. kκ can be expressed as κ = √γ

_{i}, where γ

_{i}is the contribution of the excitation channel to the total radiative decay rate [28]. For a symmetric two-sided cavity, in a first approximation one can estimate γ

_{i}= (γ

_{rad}/2)(

*A*

_{C}/

*A*

_{i}), with

*A*

_{c}corresponding to an effective radiation cross-section of the resonant cavity mode (its radiation field imaged back into the near-field of the cavity). Note that

*A*

_{i}has been assumed to be larger than

*A*

_{c}in the above relation, and that

*A*

_{c}can be no smaller than the diffraction limited area

*A*

_{d}, yielding

*A*

_{d}≤

*A*

_{c}≤

*A*

_{i}. In steady state, the mode amplitude can then be expressed as

*A*

_{c}=

*A*

_{i}). For a dielectric cavity (γ

_{rad}≫ γ

_{abs}), one thus gets

*u*∝1/√γ

_{rad}∝ √

*Q*, while for a metallic cavity (γ

_{abs}≫ γ

_{rad})

*u*∝ 1/γ

_{abs}∝

*Q*, explaining the different scaling laws for field enhancement in dielectric [29

29. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using spherical dielectric microcavity,” Nature **415**, 621–623 (2002). [CrossRef] [PubMed]

30. T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. **80**, 4249–4252 (1998). [CrossRef]

*u*= √ε

_{0}|

**E**

_{loc}|√

*V*

_{eff}, which gives for the enhancement of the incoming radiation in a metallic cavity

31. V. A. Shubin, W. Kim, V. P. Safonov, A. K. Sarychev, R. L. Armstrong, and V. M. Shalaev, “Surface-plasmon-enhanced radiation effects in confined photonic systems,” J. Lightwave Technol. **17**, 2183–2190 (1999). [CrossRef]

*A*

_{c}=

*A*

_{d}yields for our example Au resonator, with 2

*a*= 50 nm and λ

_{0}= 980 nm, an estimated SERS cross section enhancement of

*R*~ 1600.

*R*for a crevice between two Ag nanoparticles separated by a nanoscale gap, a configuration which is believed to sustain SERS hot-spots with

*R*~ 10

^{11}upon resonance, can also be obtained using the effective mode volume picture. The crevice can be approximately modeled using the capacitor-like cavity described above, but with a reduced lateral width

*L*

_{y}~

*L*

_{x}=π/β. For a 1 nm gap, with λ

_{0}= 400nm,

*A*

_{c}=

*A*

_{d}, and (

*Q*, γ

_{rad}) estimated from FDTD, eq. (5) yields

*R*~ 2.7 × 10

^{10}, in good agreement with full-field three-dimensional simulations of the enhancement for this coupled particle geometry [6

6. H. Xu, J. Aizpurua, M. Kaell, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E **62**, 4318–4324 (2000). [CrossRef]

*total*decay rate γ/γ

_{0}= (3/4π

^{2})(

*Q*/

*V*¯

_{eff})[32]. For collection of light emission outside the cavity, the overall cavity enhancement is weighted with an extraction efficiency,

*Q*/

*Q*

_{rad}[33

33. W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. **17**, 2170–2182 (1999). [CrossRef]

34. J. Vučković, M. Lončar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. **36**, 1131–1144 (2000). [CrossRef]

*peak*emission frequency of the Stokes line is (3/4π

^{2})(

*Q*

^{2}/

*V*¯

_{eff})(

*Q*/

*Q*

_{rad}). Incorporating the relation for incoming field enhancement from eq. (5), the overall enhancement is estimated to be 1.5 × 10

^{12}for the crevice example, similar to observed values[5

5. S. M. Nie and S. R. Emery, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science **275**, 1102 (1997). [CrossRef] [PubMed]

4. 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 (1997). [CrossRef]

## 5. Conclusion

*Q*/

*V*

_{eff}values can be achieved for nanometric gap sizes in metal/dielectric/metal het-erostructures. Furthermore, the field enhancement due to interactions between closely spaced metallic interfaces of more complicated geometries such as nanoparticles can be estimated in these terms. This unified description of enhancement effects in both dielectric and metallic resonators will help in establishing new design principles for nanophotonic devices.

## Acknowledgments

## References and links

1. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

2. | S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mat. |

3. | W. L. Barnes, A. Dereux, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

4. | 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. |

5. | S. M. Nie and S. R. Emery, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science |

6. | H. Xu, J. Aizpurua, M. Kaell, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E |

7. | A. Sundaramurthy, K. B. Crozier, G. S. Kino, D. P. Fromm, P. J. Schuck, and W. E. Moerner, “Field enhancement and gap-dependent resonance in a system of two opposing tip-to-tip Au nanotriangles,” Phys. Rev. B. |

8. | H. Kimble, ” |

9. | D. J. Norris, M. Kuwata-Gonokami, and W. E. Moerner, “Excitation of a single molecule on the surface of a spherical microcavity,” Appl. Phys. Lett. |

10. | R. K. Chang and A. J. Campillo, eds., “ |

11. | D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. |

12. | D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature |

13. | A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room Temperature Operation of Submicrometre Radius Disk Laser,” Electron. Lett. |

14. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, “ |

15. | O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science |

16. | W. Vogel and D.-G. Welsch, “ |

17. | E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

18. | A. B. Matsko, A. A. Savchenkov, R. J. Letargat, V. S. Ilchenko, and L. Maleki, “On cavity modification of stimulated Raman scattering,” J. Opt. B: Quantum Semiclass. Opt. |

19. | M. Kerker, D.-S. Wang, and H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles: errata,” Appl. Opt. |

20. | B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B |

21. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

22. | R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A |

23. | R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A |

24. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. |

25. | L. C. Andreani, G. Panzarini, and J.-M. Gérard, “Strong-coupling regime for quantum boxes in pillar microcavi-ties: Theory,” Phys. Rev. B |

26. | M. A. Ordal, L. L. Long, R. J. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. |

27. | I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, “Dipolar emitters at nanoscale proximity of metal surfaces: Giant enhancement of relaxation in microscopic theory,” Phys. Rev. B |

28. | H. A. Haus, “ |

29. | S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using spherical dielectric microcavity,” Nature |

30. | T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. |

31. | V. A. Shubin, W. Kim, V. P. Safonov, A. K. Sarychev, R. L. Armstrong, and V. M. Shalaev, “Surface-plasmon-enhanced radiation effects in confined photonic systems,” J. Lightwave Technol. |

32. | E. Hinds, “ |

33. | W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. |

34. | J. Vučković, M. Lončar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. |

**OCIS Codes**

(170.5660) Medical optics and biotechnology : Raman spectroscopy

(230.5750) Optical devices : Resonators

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 3, 2006

Revised Manuscript: February 28, 2006

Manuscript Accepted: February 28, 2006

Published: March 6, 2006

**Virtual Issues**

Vol. 1, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Stefan A. Maier, "Plasmonic field enhancement and SERS in the effective mode volume picture," Opt. Express **14**, 1957-1964 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-5-1957

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### References

- S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy inmetal/dielectric structures," J. Appl. Phys. 98, 011101 (2005). [CrossRef]
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