## Ultrasmall metal-insulator-metal nanoresonators: impact of slow-wave effects on the quality factor |

Optics Express, Vol. 20, Issue 15, pp. 16880-16891 (2012)

http://dx.doi.org/10.1364/OE.20.016880

Acrobat PDF (3364 KB)

### Abstract

We study the quality factor variation of three-dimensional Metal-Insulator-Metal nanoresonators when their volume is shrunk from the diffraction limit (*λ*/2*n*)^{3} down to a deep subwavelength scale (*λ*/50)^{3}. In addition to rigorous fully-vectorial calculations, we provide a semi-analytical expression of the quality factor *Q* obtained with a Fabry-Perot model. The latter quantitatively predicts the absorption and radiation losses of the nanoresonator and provides an in-depth understanding of the mode lifetime that cannot be obtained with brute-force computations. In particular, it highlights the impact of slow-wave effects on the *Q*-factor as the size of the resonator is decreased. The Fabry-Perot model also evidences that, unexpectedly, wave retardation effects are present in metallic nanoparticles, even for deep subwavelength dimensions in the quasi-static regime.

© 2012 OSA

## 1. Introduction

1. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics **1**, 41–48 (2007). [CrossRef]

3. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. **7**, 442–453 (2008). [CrossRef]

5. A. Cattoni, P. Ghenuche, A. M. Haghiri-Gosnet, D. Decanini, J. Chen, J.-L. Pelouard, and S. Collin, “*λ*^{3}/1000 plasmonic nanocavities for biosensing fabricated by soft uv nanoimprint lithography,” Nano Lett. **11**, 3557–3563 (2011). [CrossRef] [PubMed]

6. H. A. Atwater and A. Polman, “Plasmonic for improved photovoltaic devices,” Nature Mater. **9**, 205–213 (2010). [CrossRef]

8. J. T. Choy, B. J. M. Hausmann, T. M. Babinec, I. Bulu, M. Khan, P. Maletinsky, A. Yacoby, and M. Loncar, “Enhanced single-photon emission from a diamond-silver aperture,” Nat. Photonics **5**, 738–743 (2011). [CrossRef]

9. M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics **4**, 395–399 (2010). [CrossRef]

11. H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. **96**, 097401 (2006). [CrossRef] [PubMed]

13. M. Kuttge, F. J. G. de Abajo, and A. Polman, “Ultrasmall mode volume plasmonic nanodisk resonators,” Nano Lett. **10**, 1537–1541 (2010). [CrossRef]

14. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. **30**, 3198–3200 (2005). [CrossRef] [PubMed]

14. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. **30**, 3198–3200 (2005). [CrossRef] [PubMed]

18. J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett. **107**, 043903 (2011). [CrossRef] [PubMed]

14. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. **30**, 3198–3200 (2005). [CrossRef] [PubMed]

15. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. **30**, 3356–3358 (2005). [CrossRef]

5. A. Cattoni, P. Ghenuche, A. M. Haghiri-Gosnet, D. Decanini, J. Chen, J.-L. Pelouard, and S. Collin, “*λ*^{3}/1000 plasmonic nanocavities for biosensing fabricated by soft uv nanoimprint lithography,” Nano Lett. **11**, 3557–3563 (2011). [CrossRef] [PubMed]

19. L. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. **93**, 123308 (2008). [CrossRef]

24. C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. **99**, 241104 (2011). [CrossRef]

24. C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. **99**, 241104 (2011). [CrossRef]

11. H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. **96**, 097401 (2006). [CrossRef] [PubMed]

12. G. Leveque and O. J. F. Martin, “Tunable composite nanoparticle for plasmonics,” Opt. Lett. **31**, 2750–2752 (2006). [CrossRef] [PubMed]

25. Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B **75**, 035411 (2007). [CrossRef]

27. J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B **79**, 035401 (2007). [CrossRef]

*Q*-factor) variation as the resonator size is scaled down.

## 2. Magnetic resonance of a single 3D MIM resonator

*w*, length

*L*and thickness

*t*) sandwiched between a metal substrate and a metal layer with the same width and length and with a thickness denoted by

_{d}*t*, see Fig. 1(a). Calculations have been performed for silver, whose permittivity has been taken from the data tabulated in [38

_{m}38. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370 (1972). [CrossRef]

*Q*-factor, the dielectric material has been chosen to be a semiconductor (such as GaAs) with a high refractive index (

*n*= 3.5). Since it has only a weak impact on the resonator properties as long as it remains larger than the skin depth, the thickness

*t*of the top metallic layer is fixed in the following and we take

_{m}*t*= 25 nm. All rigorous fully-vectorial calculations are performed with a 3D frequency-domain fully-vectorial modal method known as the aperiodic Fourier Modal Method (a-FMM) [39

_{m}39. E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A **18**, 2865–2875 (2001). [CrossRef]

40. M. Besbes, J.-P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. V. Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Europ. Opt. Soc. Rap. Public. **2**, 07022 (2007). [CrossRef]

*Q*of this resonance. Note that, throughout the report, the resonance wavelength

*λ*

_{0}= 950 nm is maintained constant. In order to fulfill this condition as the two transverse resonator dimensions (the width

*w*and the dielectric thickness

*t*) are varied, one simply adjusts the length

_{d}*L*of the MIM particle. The quality factor can be directly derived from the full-width-at-half-maximum of a Lorentzian fit of the intensity enhancement spectrum shown in Fig. 1(b). However, in order to get more physical insight, we analyze the MIM resonator as a Fabry-Perot cavity.

## 3. Fabry-Perot model of the resonance

*z*-direction created by the bouncing between the nanoparticle facets at

*z*= −

*L*/2 and

*z*=

*L*/2 of a single mode, the fundamental plasmonic mode of the MIM waveguide, whose cross-section in the (

*x*,

*y*) plane is represented in Fig. 2. The single mode approximation amounts to assume that all higher-order modes of the MIM waveguide play a negligible role to build the resonance. The validity of the assumption is discussed in more details in the following; as will be shown, it depends mostly on the transversal size of the MIM waveguide. The key parameters of the Fabry-Perot model are the propagation constant

*L*and the quality factor

*Q*. These formulae are validated against fully-vectorial calculations and the gained physical understanding for the

*Q*-factor variation with the resonator volume is discussed in Section 3.3.

### 3.1. Fabry-Perot parameters: fundamental mode of the MIM waveguide and its reflectivity

*E*|

_{y}^{2}, |

*E*|

_{z}^{2}and |

*H*|

_{x}^{2}, are shown in Figs. 2(a)–2(c) for

*w*= 40 nm and

*t*= 20 nm. For

_{d}*w*→ ∞ (planar MIM stack) and

*t*→ ∞, this mode results from the coupling with a symmetric magnetic field

_{m}*H*of two metal/insulator SPPs [41

_{x}41. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**, 539–554 (1969). [CrossRef]

*H*-symmetric mode of a planar MIM stack has no cut-off as the insulator thickness decreases, unlike the antisymmetric mode that is cut-off at roughly

_{x}*λ*/(2

*n*). Therefore, for the thicknesses of interest,

*t*< 100 nm, the planar MIM stack is monomode. Reducing the top metal thickness to

_{d}*t*= 25 nm does not change the nature of the fundamental mode, except that the symmetry is slightly broken, see Figs. 2(a)–2(c). As the dielectric thickness

_{m}*t*decreases, the symmetric mode of the MIM stack interacts more strongly with the metal and presents a rapid increase of both its effective index

_{d}*n*

_{eff}= Re(

*β*)/

*k*

_{0}and its attenuation

*α*= 2Im(

*β*), as shown by the thin-solid curves in Figs. 2(d)–2(e).

*w*on the propagation constant is much weaker than the impact of the thickness

*t*; the propagation constant of the 2D waveguide is almost equal to that of the planar stack, especially for small thickness. Let us emphasize that the mode group index

_{d}*n*=

_{g}*c/v*, with

_{g}*v*the group velocity, also increases as the dielectric thickness is reduced, see Fig. 2(f). As will be shown hereafter, the slowdown of the plasmonic mode supported by thin MIM waveguides directly impacts the resonator lifetime. Finally, it is noteworthy that for finite widths higher-order modes with the same profile along

_{g}*y*as the fundamental plasmonic mode may exist. They are determined by a total-internal-reflection condition at the air/dielectric interfaces and possess an increasing number of nodes along

*x*, see the insets in Fig. 3(c).

*L*corresponding to the fundamental resonance is of the order of

*λ*

_{0}/(2

*n*

_{eff}). However, the price to pay for a reduction of the resonator volume is an enhanced dissipation in the metal. In order to accurately quantify the impact on the quality factor of the absorption increase, we need to thoroughly evaluate the variation of the radiation losses with the size. In the Fabry-Perot picture, the normalized power dissipated by radiation per round-trip is simply equal to 1 −

*R*, with

*R*the reflectivity of the fundamental mode at the air/MIM interfaces,

*z*= −

*L*/2 and

*z*=

*L*/2.

*w*and

*t*with the a-FMM [39

_{d}39. E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A **18**, 2865–2875 (2001). [CrossRef]

40. M. Besbes, J.-P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. V. Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Europ. Opt. Soc. Rap. Public. **2**, 07022 (2007). [CrossRef]

*w*=

*t*= 100 nm, the facets only weakly reflect the fundamental plasmonic mode,

_{d}*R*= 0.7, whereas for

*w*= 40 nm and

*t*= 10 nm

_{d}*R*is as large as 0.999. Figure 3(b) shows the dependence of the reflectivity with the thickness

*t*for

_{d}*w*= 40, 100, 350 nm and ∞. The four curves show the same main trend, namely a significant enhancement of

*R*as

*t*is reduced. The width dependence is highlighted in Fig. 3(c) for

_{d}*t*= 20, 35 and 50 nm. It is important to note the non-monotonous variation of the reflectivity. The narrow dips correspond to a strong backscattering into higher-order MIM waveguide modes that are cutoff at the dip wavelength. The insets in Fig. 3(c) show the profile of the first two

_{d}*x*-symmetric higher-order modes [42]. When a higher-order mode passes its cut-off and becomes propagative, the total density of states is modified by the appearance of a new mode (often with a small group velocity) and this causes a sharp dip in the modal reflectivity of the fundamental mode. The dip is especially pronounced for the appearance of the first higher-order mode. Finally, it is noteworthy that even extremely tiny waveguides with transverse dimensions as small as

*w*< 100 nm and

*t*< 10 nm are not monomode. This unexpected observation can be intuitively understood from the large value of the effective index

_{d}*w*infinite) with a small dielectric thickness

*t*; the cut-off width of higher-order modes is indeed linked to the value of

_{d}*t*is decreased.

_{d}### 3.2. Fabry-Perot equations: phase-matching condition and quality factor

*λ*

_{0}= 950 nm, the length

*L*of the resonator is easily obtained from the phase-matching condition with

*m*= 1, 2,...,

*n*

_{eff}the effective index of the waveguide mode given in Fig. 2(d) and

*ϕ*the phase of the modal reflection coefficient

_{r}*r*. We are interested hereafter in the fundamental resonance

*m*= 1. The reflection phase varies between

*ϕ*=

_{r}*π*/8 for the largest waveguide cross-section (

*w*= 350 nm and

*t*= 100 nm) and

_{d}*ϕ*= −

_{r}*π*/20 for the smallest cross-section (

*w*= 40 nm and

*t*= 5 nm) and therefore, with a good approximation, one may consider that Eq. (2) leads to

_{d}*L*=

*λ*

_{0}/(2

*n*

_{eff}), corresponding to the classical half-wavelength condition for the resonator length [43

43. Unusual Fabry-Perot resonances with *m* = 0 may exist in subwavelength metallic structures, see E. Feigenbaum and M. Orenstein, “Ultrasmall volume plasmons, yet with complete retardation effects”, Phys. Rev. Lett. **101**, 163902 (2008). For this situation to occur, the positive propagation phase has to be fully compensated by a negative reflection phase. Such a resonance does not exist in the MIM resonators under study since *ϕ _{r}* ≈ 0. [CrossRef] [PubMed]

*t*for

_{d}*w*= 40, 100 and 350 nm. Because of the strong increase of

*n*

_{eff},

*L*rapidly decreases as

*t*decreases. On the other hand, the resonator width

_{d}*w*is found to weakly impact the length.

*Q*can be derived analytically as [31

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

*k*

_{0}= 2

*π*/

*λ*

_{0},

*R*

_{eff}=

*R*exp(−

*αL*) the effective reflectivity that includes the absorption loss over one-half round-trip, and

*L*

_{eff}=

*L*+ 2

*L*the effective resonator length that includes the penetration length

_{p}*L*

_{eff}and it can be safely neglected for all values of

*w*and

*t*considered in the present work [44].

_{d}*w*= 40, 100 and 350 nm together with fully-vectorial calculations of the

*Q*-factor, extracted from Lorentzian fits of the intensity enhancement spectra and shown with various markers. The analytical formula of Eq. (3) accurately predicts the

*Q*-factor increase. When the resonator volume is shrunk from roughly the diffraction limit

*V*= (

*λ*/2

*n*)

^{3}(

*w*= 120 nm,

*t*= 100 nm and

_{d}*L*= 100 nm) down to a deep subwavelength scale

*V*= (

*λ*/50)

^{3}(

*w*= 40 nm,

*t*= 5 nm and

_{d}*L*= 30 nm), the

*Q*-factor is significantly enhanced by one order of magnitude.

*w*= 100 nm, the MIM is monomode only for

*t*> 15 nm, and for

_{d}*w*= 350 nm, the MIM is multimode whatever the value of

*t*. Hopefully, the presence of higher-order modes does not alter the model predictions, except close to their cut-off where a drop of

_{d}*Q*is inaccurately predicted in Fig. 4(b) because of the rapid change of the facet reflectivity. Note that the deviation between the model predictions and rigorous calculations vanishes as the waveguide width increases.

### 3.3. Analysis of the Q-factor increase

*Q*-factor limitations. As shown by Eq. (3), the cavity-mode lifetime is impacted both by absorption (

*α*in

*R*

_{eff}) and radiation (1 −

*R*). In order to separate the contributions of these two loss channels and to clarify the impact of the SPP slowdown, Fig. 4(c) shows the

*Q*-factor predicted by Eq. (3) for

*w*= 40 nm (thick solid blue curve) together with the

*Q*-factor values that would have been achieved for

*α*= 0,

*R*= 1 and

*n*= 5. Taking

_{g}*α*= 0 in Eq. (3) amounts to remove the absorption; the black dashed-dotted curve thus gives an approximation of the radiation-limited

*Q*. Then, considering the particle facets as perfect reflectors,

*R*= 1 in Eq. (3), amounts to neglect the radiation losses and gives an approximation of the absorption-limited

*Q*. Consistently with the fact that the absorption and the scattering of small particles respectively scale as the particle volume and as the particle volume squared [45], we find that the

*Q*-factor of large MIM resonators with

*V*= (

*λ*/2

*n*)

^{3}is mostly limited by radiation,

*Q*≈

*Q*

_{α}_{=0}, whereas the

*Q*-factor of small resonators in the quasi-static limit with

*V*= (

*λ*/50)

^{3}is completely limited by absorption,

*Q*≈

*Q*

_{R}_{=1}. Radiation and absorption losses are balanced for

*t*= 35 nm. Finally, the impact on the resonance lifetime of the slowdown of the MIM waveguide mode is unraveled by considering Eq. (3) with a constant group index

_{d}*n*= 5, a value corresponding to thick MIM waveguides with

_{g}*t*= 100 nm (red dashed curve).

_{d}*Q*-factor variation can be considered. They are marked by vertical dashed lines in Fig. 4(c). The first regime concerns large resonator thicknesses,

*t*> 35 nm. The absorption

_{d}*αL*and the group index

*n*are roughly constant and the increase of the quality factor is purely due to a reduction of the radiation losses (increase of the reflectivity

_{g}*R*). In the second regime, 10 <

*t*< 35 nm, the

_{d}*Q*-factor increases (solid thick blue curve) due to the slowdown of the SPP mode bouncing inside the resonator. Indeed, it is noticeable that a resonator with the actual absorption and radiation but with a constant group index (red dashed curve) would see its

*Q*-factor decreasing because the radiation-loss reduction is balanced by an increase of the absorption. Finally, for extremely thin dielectric layers,

*t*< 10 nm, we enter the quasi-static regime where the resonator performance is purely limited by absorption. The validity of the Fabry-Perot model in this last regime is discussed in the next Section.

_{d}## 4. Fabry-Perot model in the quasi-static limit

*Q*of any metallo-dielectric resonant nanoparticle composed of a single lossy metal is completely determined from the relative permittivity of the metal

^{s}*ε*=

_{m}*ε*′

*+*

_{m}*iε*″

*[37*

_{m}37. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. **97**, 206806 (2006). [CrossRef] [PubMed]

*Q*-factor depends neither on the geometric shape nor on the dielectric media, provided that the latter are dispersionless and lossless. This universal result has been derived in [37

37. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. **97**, 206806 (2006). [CrossRef] [PubMed]

*ε*′

*| ≫*

_{m}*ε*″

*) that supports a purely electrostatic resonance (∇ ×*

_{m}**E**= 0) whose damping is solely due to absorption (no radiation losses). This last assumption can be justified by the fact that the scattering of small particles scales as the particle volume squared whereas the absorption scales only as the particle volume [45].

*k*

_{0}

*L*≪ 1 for small size-to-wavelength ratios. On the other hand, the Fabry-Perot model describes a resonance as the result of a round-trip phase accumulation of 2

*mπ*,

*m*= 1, 2,... [43

43. Unusual Fabry-Perot resonances with *m* = 0 may exist in subwavelength metallic structures, see E. Feigenbaum and M. Orenstein, “Ultrasmall volume plasmons, yet with complete retardation effects”, Phys. Rev. Lett. **101**, 163902 (2008). For this situation to occur, the positive propagation phase has to be fully compensated by a negative reflection phase. Such a resonance does not exist in the MIM resonators under study since *ϕ _{r}* ≈ 0. [CrossRef] [PubMed]

*V*= (

*λ*/50)

^{3}. In order to clarify whether a nanoparticle can be described by a non-vanishing wave retardation in the quasi-static limit, we now analytically study the asymptotic behavior of the Fabry-Perot model as resonator sizes tend toward zero. We emphasize that retardation effects are present in very small objects and that MIM nanoparticles, even in the quasi-static limit, are equivalent to half-wavelength antennas. This equivalence has been recently discussed for the resonance wavelength [46

46. S. B. Hasan, R. Filter, A. Ahmed, R. Vogelgesang, R. Gordon, C. Rockstuhl, and F. Lederer, “Relating localized nanoparticle resonances to an associated antenna problem,” Phys. Rev. B **84**, 195405 (2011). [CrossRef]

*Q*-factor given by Eq. (3) reduces to Eq. (4) in the quasi-static limit.

*Q*-factor given by Eq. (3). As in [37

37. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. **97**, 206806 (2006). [CrossRef] [PubMed]

*R*= 1 in Eq. (3), an assumption legitimated for the geometry under study by the numerical results of Fig. 4(c). Moreover, the resonator length is small and the absorption term in the

*Q*-factor can be approximated by exp(−

*αL*) ∼ 1 −

*αL*. With these two approximations, Eq. (3) leads to To obtain this result, we have considered that

*L*≪

_{p}*L*and

*L*

_{eff}≈

*L*[44]. Equation (5) emphasizes that, in the absence of radiation losses, the

*Q*-factor is purely driven by the guided SPP mode bouncing back and forth inside the particle. The important parameter is the group-index-to-attenuation ratio, and not only the attenuation as concluded in [26

26. S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators,” Opt. Express **15**, 10869–10877 (2007). [CrossRef] [PubMed]

*t*/

_{d}*λ*→ 0 of the normalized propagation constant

26. S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators,” Opt. Express **15**, 10869–10877 (2007). [CrossRef] [PubMed]

*ε*the relative permittivity of the dielectric layer. We note from Fig. 2 that the normalized propagation constant of the fundamental mode of 2D MIM waveguides with finite widths w becomes independent of the width as

_{d}*t*/

_{d}*λ*→ 0 and asymptotically tends toward

*w*, we can have

*t*is reduced. Thus Eq. (6) can be safely used to derive the limit of the

_{d}*n*/

_{g}*α*ratio for any 2D MIM waveguide. The expressions of

*Q*-factor in the quasi-static limit is found to be given by which is exactly the

*Q*value of Eq. (4) derived in [37

^{s}**97**, 206806 (2006). [CrossRef] [PubMed]

*ε*′

*| ≫*

_{m}*ε*″

*. The prediction of Eq. (7) is represented by the horizontal arrow in Fig. 4(b). It is noteworthy that the 3D MIM nanoparticle under study enters the quasi-static regime only for very small dielectric thicknesses,*

_{m}*t*< 10 nm.

_{d}*inside*the particle of plasmonic modes whose wavevector diverges as the length shrinks (actually the product

*k*

_{0}

*n*

_{eff}

*L*=

*π*is fixed by the Fabry-Perot phase-matching condition). The second important conclusion concerns the properties of plasmonic waveguides in the quasi-static limit. Indeed, the

*Q*-factor expressions of Eqs. (4) and (5) should be identical as the transverse cross-section shrinks, whatever the shape of the waveguide cross-section. We can therefore deduce that, in the limit of small transverse dimensions compared to the wavelength, the propagation constant of any plasmonic waveguide should satisfy regardless of the geometric shape of the waveguide cross-section or of the dielectric media composing it, provided that they are dispersionless and lossless. Note that Eq. (8) holds in a regime where the scattering of the plasmonic mode at an interface can be neglected. The waveguide dimensions corresponding to this regime can be different from one waveguide to the other [47

47. G. Della Valle, T. Søndergaard, and S. I. Bozhevolnyi, “High-Q plasmonic resonators based on metal split nanocylinders,” Phys. Rev. B **80**, 235405 (2009). [CrossRef]

## 5. Conclusion

*V*= (

*λ*/2

*n*)

^{3}down to a deep subwavelength scale

*V*= (

*λ*/50)

^{3}. The ten-fold increase of the

*Q*-factor given by rigorous fully-vectorial calculations has been accurately predicted with a Fabry-Perot model, which provides analytical expressions of the quality factor, of the absorption and of the radiation losses. These expressions allow for a comprehensive analysis of the increase of the mode lifetime as the dimensions of the resonator are shrunk. The main mechanisms are a reduction of the radiation losses and a slowdown of the plasmonic mode bouncing back and forth inside the resonator. The slow-wave effect is responsible for the

*Q*-factor increase when the radiation-loss decrease is fully balanced by an increase of the absorption.

*λ*

_{0}/(2

*n*

_{eff}) [33

33. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. **98**, 266802 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

2. | C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics |

3. | J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. |

4. | N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nature Mater. |

5. | A. Cattoni, P. Ghenuche, A. M. Haghiri-Gosnet, D. Decanini, J. Chen, J.-L. Pelouard, and S. Collin, “ |

6. | H. A. Atwater and A. Polman, “Plasmonic for improved photovoltaic devices,” Nature Mater. |

7. | I. S. Maksymov, M. Besbes, J.-P. Hugonin, J. Yang, A. Beveratos, I. Sagnes, I. Robert-Philip, and P. Lalanne, “Metal-coated nanocylinder cavity for broadband nonclassical light emission,” Phys. Rev. Lett. |

8. | J. T. Choy, B. J. M. Hausmann, T. M. Babinec, I. Bulu, M. Khan, P. Maletinsky, A. Yacoby, and M. Loncar, “Enhanced single-photon emission from a diamond-silver aperture,” Nat. Photonics |

9. | M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics |

10. | H. Raether, |

11. | H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. |

12. | G. Leveque and O. J. F. Martin, “Tunable composite nanoparticle for plasmonics,” Opt. Lett. |

13. | M. Kuttge, F. J. G. de Abajo, and A. Polman, “Ultrasmall mode volume plasmonic nanodisk resonators,” Nano Lett. |

14. | G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. |

15. | V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. |

16. | S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. |

17. | A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. |

18. | J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett. |

19. | L. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. |

20. | J. Le Perchec, Y. Desieres, and R. Espiau de Lamaestre, “Plasmon-based photosensors comprising a very thin semiconducting region,” Appl. Phys. Lett. |

21. | N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. |

22. | J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. |

23. | C. Wu, B. Neuner, G. Shvets, J. John, A. Milder, B. Zollars, and S. Savoy, “Large-area wide-angle spectrally selective plasmonic absorber,” Phys. Rev. B |

24. | C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. |

25. | Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B |

26. | S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators,” Opt. Express |

27. | J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B |

28. | F. Garwe, C. Rockstuhl, C. Etrich, U. Hübner, U. Bauerschäfer, F. Setzpfandt, M. Augustin, T. Pertsch, A. Tünnermann, and F. Lederer, “Evaluation of gold nanowire pairs as a potential negative index material,” Appl. Phys. B |

29. | C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B |

30. | C. Sauvan, P. Lalanne, and J.-P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B |

31. | P. Lalanne, C. Sauvan, and J.-P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser and Photon. Rev. |

32. | H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. |

33. | L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. |

34. | E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express |

35. | J. Dorfmuller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-perot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. |

36. | T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: evolution of sub- and super-radiant modes,” Nano Lett. |

37. | F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. |

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

39. | E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A |

40. | M. Besbes, J.-P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. V. Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Europ. Opt. Soc. Rap. Public. |

41. | E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. |

42. | Because of symmetry reasons, anti-symmetric higher-order modes do not impact the reflectivity of the symmetric fundamental mode. |

43. | Unusual Fabry-Perot resonances with |

44. | The assumption L is valid, since for large waveguide cross-sections (w = 100 nm and t = 100 nm) _{d}L ≈ 5 nm ≪ _{p}L ≈ 100 nm and for small cross-sections (w = 40 nm and t < 25 nm) _{d}L < 1 nm ≪ _{p}L ≈ 30 nm. |

45. | C. F. Bohren and D. R. Huffman, |

46. | S. B. Hasan, R. Filter, A. Ahmed, R. Vogelgesang, R. Gordon, C. Rockstuhl, and F. Lederer, “Relating localized nanoparticle resonances to an associated antenna problem,” Phys. Rev. B |

47. | G. Della Valle, T. Søndergaard, and S. I. Bozhevolnyi, “High-Q plasmonic resonators based on metal split nanocylinders,” Phys. Rev. B |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(240.6680) Optics at surfaces : Surface plasmons

(160.3918) Materials : Metamaterials

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: March 23, 2012

Manuscript Accepted: June 22, 2012

Published: July 11, 2012

**Citation**

J. Yang, C. Sauvan, A. Jouanin, S. Collin, J.-L. Pelouard, and P. Lalanne, "Ultrasmall metal-insulator-metal nanoresonators: impact of slow-wave effects on the quality factor," Opt. Express **20**, 16880-16891 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16880

Sort: Year | Journal | Reset

### References

- V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics1, 41–48 (2007). [CrossRef]
- C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).
- J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater.7, 442–453 (2008). [CrossRef]
- N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nature Mater.10, 631–636 (2011). [CrossRef]
- A. Cattoni, P. Ghenuche, A. M. Haghiri-Gosnet, D. Decanini, J. Chen, J.-L. Pelouard, and S. Collin, “λ3/1000 plasmonic nanocavities for biosensing fabricated by soft uv nanoimprint lithography,” Nano Lett.11, 3557–3563 (2011). [CrossRef] [PubMed]
- H. A. Atwater and A. Polman, “Plasmonic for improved photovoltaic devices,” Nature Mater.9, 205–213 (2010). [CrossRef]
- I. S. Maksymov, M. Besbes, J.-P. Hugonin, J. Yang, A. Beveratos, I. Sagnes, I. Robert-Philip, and P. Lalanne, “Metal-coated nanocylinder cavity for broadband nonclassical light emission,” Phys. Rev. Lett.105, 180502 (2010).
- J. T. Choy, B. J. M. Hausmann, T. M. Babinec, I. Bulu, M. Khan, P. Maletinsky, A. Yacoby, and M. Loncar, “Enhanced single-photon emission from a diamond-silver aperture,” Nat. Photonics5, 738–743 (2011). [CrossRef]
- M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics4, 395–399 (2010). [CrossRef]
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).
- H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett.96, 097401 (2006). [CrossRef] [PubMed]
- G. Leveque and O. J. F. Martin, “Tunable composite nanoparticle for plasmonics,” Opt. Lett.31, 2750–2752 (2006). [CrossRef] [PubMed]
- M. Kuttge, F. J. G. de Abajo, and A. Polman, “Ultrasmall mode volume plasmonic nanodisk resonators,” Nano Lett.10, 1537–1541 (2010). [CrossRef]
- G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett.30, 3198–3200 (2005). [CrossRef] [PubMed]
- V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett.30, 3356–3358 (2005). [CrossRef]
- S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett.95, 137404 (2005). [CrossRef] [PubMed]
- A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett.101, 103902 (2008). [CrossRef] [PubMed]
- J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107, 043903 (2011). [CrossRef] [PubMed]
- L. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett.93, 123308 (2008). [CrossRef]
- J. Le Perchec, Y. Desieres, and R. Espiau de Lamaestre, “Plasmon-based photosensors comprising a very thin semiconducting region,” Appl. Phys. Lett.94, 181104 (2009). [CrossRef]
- N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett.10, 2342–2348 (2010). [CrossRef] [PubMed]
- J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett.96, 251104 (2010). [CrossRef]
- C. Wu, B. Neuner, G. Shvets, J. John, A. Milder, B. Zollars, and S. Savoy, “Large-area wide-angle spectrally selective plasmonic absorber,” Phys. Rev. B84, 075102 (2011). [CrossRef]
- C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett.99, 241104 (2011). [CrossRef]
- Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B75, 035411 (2007). [CrossRef]
- S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators,” Opt. Express15, 10869–10877 (2007). [CrossRef] [PubMed]
- J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B79, 035401 (2007). [CrossRef]
- F. Garwe, C. Rockstuhl, C. Etrich, U. Hübner, U. Bauerschäfer, F. Setzpfandt, M. Augustin, T. Pertsch, A. Tünnermann, and F. Lederer, “Evaluation of gold nanowire pairs as a potential negative index material,” Appl. Phys. B84, 139–148 (2006). [CrossRef]
- C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83, 245119 (2011). [CrossRef]
- C. Sauvan, P. Lalanne, and J.-P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B71, 165118 (2005). [CrossRef]
- P. Lalanne, C. Sauvan, and J.-P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser and Photon. Rev.2, 514–526 (2008). [CrossRef]
- H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett.95, 257403 (2005). [CrossRef] [PubMed]
- L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007). [CrossRef] [PubMed]
- E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express16, 16529–16537 (2008). [CrossRef] [PubMed]
- J. Dorfmuller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-perot resonances in one-dimensional plasmonic nanostructures,” Nano Lett.9, 2372–2377 (2009). [CrossRef] [PubMed]
- T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: evolution of sub- and super-radiant modes,” Nano Lett.11, 1020–1024 (2011). [CrossRef] [PubMed]
- F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006). [CrossRef] [PubMed]
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370 (1972). [CrossRef]
- E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A18, 2865–2875 (2001). [CrossRef]
- M. Besbes, J.-P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. V. Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Europ. Opt. Soc. Rap. Public.2, 07022 (2007). [CrossRef]
- E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182, 539–554 (1969). [CrossRef]
- Because of symmetry reasons, anti-symmetric higher-order modes do not impact the reflectivity of the symmetric fundamental mode.
- Unusual Fabry-Perot resonances with m = 0 may exist in subwavelength metallic structures, see E. Feigenbaum and M. Orenstein, “Ultrasmall volume plasmons, yet with complete retardation effects”, Phys. Rev. Lett.101, 163902 (2008). For this situation to occur, the positive propagation phase has to be fully compensated by a negative reflection phase. Such a resonance does not exist in the MIM resonators under study since ϕr ≈ 0. [CrossRef] [PubMed]
- The assumption Lp ≪ L is valid, since for large waveguide cross-sections (w = 100 nm and td = 100 nm) Lp ≈ 5 nm ≪ L ≈ 100 nm and for small cross-sections (w = 40 nm and td < 25 nm) Lp < 1 nm ≪ L ≈ 30 nm.
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
- S. B. Hasan, R. Filter, A. Ahmed, R. Vogelgesang, R. Gordon, C. Rockstuhl, and F. Lederer, “Relating localized nanoparticle resonances to an associated antenna problem,” Phys. Rev. B84, 195405 (2011). [CrossRef]
- G. Della Valle, T. Søndergaard, and S. I. Bozhevolnyi, “High-Q plasmonic resonators based on metal split nanocylinders,” Phys. Rev. B80, 235405 (2009). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.