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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 16880–16891
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Ultrasmall metal-insulator-metal nanoresonators: impact of slow-wave effects on the quality factor

J. Yang, C. Sauvan, A. Jouanin, S. Collin, J.-L. Pelouard, and P. Lalanne  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 16880-16891 (2012)
http://dx.doi.org/10.1364/OE.20.016880


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Abstract

We study the quality factor variation of three-dimensional Metal-Insulator-Metal nanoresonators when their volume is shrunk from the diffraction limit (λ/2n)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

Optical nanoresonators with ultra-small volumes are a key ingredient for numerous nanophotonics applications. They are the building blocks (the so-called meta-atoms) of metamaterials [1

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

, 2

2. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).

], high performance sensors [3

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

5. 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]

], photovoltaic cells [6

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

], solid-state non-classical light sources [7

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. 105, 180502 (2010).

, 8

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]

] or nanolasers [9

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]

]. Confining light in three-dimensional (3D) volumes well below the diffraction limit can be achieved by taking advantage of the large wavevectors of surface plasmon polaritons (SPPs) that result from the coupling between light and free electrons in metals [10

10. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

]. Among numerous types of plasmonic resonators, Metal-Insulator-Metal (MIM) structures, i.e., alternating metal and dielectric layers with finite dimensions, show promising performance. In addition to ultra-small confinement volumes and high field enhancements [11

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

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

], MIM resonators (also known as cut-wire pairs in the metamaterials community [14

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]

]) can provide a magnetic response that is involved in the appearance of artificial magnetism at optical frequencies [14

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

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]

]. Consequently, arrays of MIM resonators have been studied for achieving either a negative effective index [14

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

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]

] or an efficient absorption for sensor or photodetector applications [5

5. 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]

, 19

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

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]

].

In view of potential applications of MIM resonators in a variety of areas ranging from metamaterials to photovoltaics, it is important to obtain a deep understanding of their optical properties, in particular when their size is scaled down to deep subwavelength dimensions. Indeed, ultra-small resonators are of major importance both for metamaterials, where meta-atoms should be much smaller than the wavelength, and for photodetector applications, where several resonators can be associated within a subwavelength cell to engineer multi-resonant structures [24

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]

]. The tunability of the resonance wavelength of MIM resonators with the resonator dimensions being well-known [11

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

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

, 25

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

27

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]

], we will mainly focus on the analysis of the quality factor (Q-factor) variation as the resonator size is scaled down.

We study the resonance with the lowest energy supported by a single 3D MIM resonator, see Fig. 1. This fundamental resonance is crucial in the context of ultra-small resonators because it has no cut-off and can be scaled down to deep subwavelength dimensions in the quasi-static regime. Moreoever, this resonance presents a ”magnetic” character and can be used to engineer negative-index metamaterials [14

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

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]

, 28

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 84, 139–148 (2006). [CrossRef]

, 29

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 83, 245119 (2011). [CrossRef]

]. We especially quantify the variation of the Q-factor when the resonator volume is shrunk from the diffraction limit V = (λ/2n)3 down to a deep subwavelength scale V = (λ/50)3. As shown by 3D fully-vectorial calculations, the main trend is a significant increase of the Q-factor by one order of magnitude when the size is reduced. To explain this increase, we use an approximate Fabry-Perot model, which allows us to derive semi-analytical expressions of the quality factor, of the absorption and of the radiation losses. The model thus provides a real understanding of the physics governing the mode lifetime at deep subwavelength scales. In particular, it evidences the crucial role played by the slowness of the SPP modes that are bouncing back and forth inside the resonator, a physical effect that is completely hidden in brute-force computations. Over the past ten years, Fabry-Perot models have been successfully applied to a variety of micro- and nanoresonators, including photonic-crystal microcavities and micropilars [30

30. C. Sauvan, P. Lalanne, and J.-P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B 71, 165118 (2005). [CrossRef]

, 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]

], and plasmonic nanowire resonators with a longitudinal length of a few wavelengths [32

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. 95, 257403 (2005). [CrossRef] [PubMed]

36

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. 11, 1020–1024 (2011). [CrossRef] [PubMed]

]. Hereafter we evidence that such a useful picture remains valid and helpful even for analyzing nanoresonators that are operating far below the diffraction limit in the quasi-static regime.

Fig. 1 Magnetic resonance of a single MIM nanoresonator. (a) The nanoresonator consists of a dielectric rectangular nanoparticle (width w, length L and thickness td) sandwiched between a metal substrate and a metal layer of thickness tm. The metal is silver and the dielectric material is a semiconductor (such as GaAs) with a high refractive index of 3.5. (b) Spectrum of the intensity enhancement |E|2/|Einc|2 at the point A in (c) for a resonator (w = 40 nm, td = 20 nm and L = 70 nm) illuminated by a plane wave impinging from air at normal incidence and polarized along the z-direction. (c) Distribution of the induced current J (see the text for its definition) and of the magnetic field |Hx|2 at resonance in the (y, z) plane (x = 0). Blue and red colors correspond to negative and positive values. The induced current forms a loop, which highlights the magnetic response of MIM resonators.

We note that a Fabry-Perot model has already been used for studying two-dimensional (2D) MIM resonators [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]

]. However, the authors 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]

] overlooked the importance of slow-wave effects and consequently derived a Q-factor expression that is not consistent with the asymptotic value derived in the quasi-static limit for metallic nanoparticles of arbitrary shape [37

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

]. We correct this discrepancy by properly taking into account slow-wave effects in the Fabry-Perot model. Indeed, as we show in Sections 3 and 4, a correct Fabry-Perot expression for the Q-factor is highly accurate and fully consistent with other asymptotic expressions derived in the quasi-static limit [37

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

]. In Section 3 we highlight three different regimes for the Q-factor variation with the resonator size. As the volume shrinks from the diffraction limit (λ/2n)3 to (λ/20)3, the Q-factor first increases because of a reduction of the radiation losses. Then, from (λ/20)3 to (λ/35)3, the radiation losses reduction is balanced by an increase of the absorption and one would intuitively expect a drop of the Q-factor as predicted 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]

]. However, the Q-factor keeps on increasing because of the slowdown of the SPP mode that bounces back and forth inside the resonator. This slow-wave effect, which is similar to the one classically encountered in photonic crystal microcavities [30

30. C. Sauvan, P. Lalanne, and J.-P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B 71, 165118 (2005). [CrossRef]

, 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]

], results in an increase of the cavity-mode lifetime. Finally, for ultrasmall resonators, the Fabry-Perot model correctly predicts the quality factor saturation toward the asymptotic value obtained in the quasi-static limit [37

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

].

2. Magnetic resonance of a single 3D MIM resonator

The 3D MIM resonator under study consists of a dielectric rectangular nanoparticle (width w, length L and thickness td) sandwiched between a metal substrate and a metal layer with the same width and length and with a thickness denoted by tm, see Fig. 1(a). Calculations have been performed for silver, whose permittivity has been taken from the data tabulated in [38

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

]. In order to maximize the confinement and the 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 tm of the top metallic layer is fixed in the following and we take tm = 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

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]

]. Stretching of the numerical space through coordinate transforms is additionally incorporated for improving the computational accuracy, see the method MM3 in the benchmark article [40

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]

] for more details.

We study the resonance with the lowest energy supported by the MIM resonator – the fundamental mode. Figure 1(b) presents the spectrum of the electric field enhancement inside the MIM resonator (dimensions are provided in the figure caption) when the latter is illuminated from air by a plane wave at normal incidence and polarized along the z-direction. Figure 1(c) shows the distribution of the magnetic field |Hx|2 and of the electric current density J(r) induced in the nanoresonator by the incident plane wave at resonance, λ = 947 nm. From the curl Maxwell equations, the induced current is proportional to the total electric field E(r),
J(r)=iωε0[ε(r)εref(r)]E(r),
(1)
with ω the frequency, ε0 the vacuum permittivity, ε(r) the relative permittivity distribution that defines the resonator and εref(r) the relative permittivity of a reference background. We have chosen a uniform background εref(r) = 1, which implies that the field radiated by the current density J(r) placed in air includes both the reflection of the incident plane wave by the metallic plane and the scattering by the resonator. As can be seen from Fig. 1(c), at the resonance wavelength, the electric current density J(r) induced by the incident plane wave is essentially localized around the resonator and forms a loop in the (y, z) plane, Re(Jx) ≪ Re(Jy) and Re(Jx) ≪ Re(Jz). Moreover, the resonant magnetic field is perpendicular to the loop. This highlights the magnetic character of the fundamental MIM resonance, as previously discussed in the literature [14

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

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]

, 28

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 84, 139–148 (2006). [CrossRef]

, 29

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 83, 245119 (2011). [CrossRef]

].

In the following, we thoroughly study the quality factor Q of this resonance. Note that, throughout the report, the resonance wavelength λ0 = 950 nm is maintained constant. In order to fulfill this condition as the two transverse resonator dimensions (the width w and the dielectric thickness td) are varied, one simply adjusts the length 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

Within the Fabry-Perot picture, the resonance is described as a standing wave pattern along the 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 β=k0neff+iα2 of the fundamental plasmonic mode supported by the MIM waveguide and its complex reflection coefficient r=Rexp(iϕr) at the air/MIM interface.

Fig. 2 Fundamental plasmonic mode of the MIM waveguide for λ = 950 nm. (a)–(c) Main field components |Ey|2, |Ez|2 and |Hx|2 of the mode for w = 40 nm and td = 20 nm. (d) Dependence on the dielectric thickness td of the effective index neff = Re(β)/k0, (e) of the attenuation α = 2Im(β) and (f) of the group index ng=neffλneffλ. In (d)–(f), four different widths are considered, w = 40 nm (solid blue), 100 nm (dashed red), 350 nm (dashed-dotted black) and ∞ (planar waveguide, thin solid line).

We first calculate and discuss these two important physical quantities in Section 3.1. Then we provide in Section 3.2 analytical expressions for the resonance length 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

The three main field components of the fundamental mode of the MIM waveguide, |Ey|2, |Ez|2 and |Hx|2, are shown in Figs. 2(a)–2(c) for w = 40 nm and td = 20 nm. For w → ∞ (planar MIM stack) and tm → ∞, this mode results from the coupling with a symmetric magnetic field Hx of two metal/insulator SPPs [41

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

]. The Hx-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 λ/(2n). Therefore, for the thicknesses of interest, td < 100 nm, the planar MIM stack is monomode. Reducing the top metal thickness to tm = 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 td decreases, the symmetric mode of the MIM stack interacts more strongly with the metal and presents a rapid increase of both its effective index neff = Re(β)/k0 and its attenuation α = 2Im(β), as shown by the thin-solid curves in Figs. 2(d)–2(e).

The fundamental mode of the MIM waveguide with a finite width follows a similar trend. As can be seen from Figs. 2(d)–2(e), the impact of the width w on the propagation constant is much weaker than the impact of the thickness td; 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 ng = c/vg, with vg 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 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).

Fig. 3 Reflectivity at the air/MIM interface for λ = 950 nm. (a) Map of the reflectivity as a function of the width w and the dielectric thickness td. (b) Reflectivity as a function of td for w = 40 nm (solid blue), 100 nm (dashed red), 350 nm (dashed-dotted black) and ∞ (planar waveguide, thin solid line). (c) Reflectivity as a function of w for td = 20 nm (solid blue), 35 nm (dashed red) and 50 nm (dashed-dotted black). The insets show the distribution of |Hx|2 in the (x, y) plane for the first two symmetric higher-order waveguide modes for td = 50 nm. The first two drops in the reflectivity curves identified by arrows correspond to the cut-off of these modes (w = 190 nm and w = 370 nm).

The large increase of the effective index is a key ingredient to build ultra-small resonators since the resonator length L corresponding to the fundamental resonance is of the order of λ0/(2neff). 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.

3.2. Fabry-Perot equations: phase-matching condition and quality factor

Fig. 4 Fabry-Perot model predictions for the resonance at λ0 = 950 nm. (a) Length of the nanoresonator predicted with Eq. (2) for w = 40 nm (solid blue), 100 nm (dashed red) and 350 nm (dashed-dotted black). (b) Quality factor predicted with Eq. (3), Q = k0ngLeff/[1 − Rexp(−αL)], for w = 40 nm (solid blue), 100 nm (dashed red) and 350 nm (dashed-dotted black). The markers represent the Q-factor values extracted from fully-vectorial calculations of intensity enhancement spectra [see Fig. 1(b)] for the same widths, w = 40 nm (blue circles), 100 nm (red squares) and 350 nm (black triangles). The horizontal arrow shows the quasi-static Q-factor Qs given by Eq. (4). (c) Quality factor predicted with the Fabry-Perot model for w = 40 nm (thick solid blue). The Q-factors predicted without radiation (R = 1 in Eq. (3), solid magenta), without absorption (α = 0 in Eq. (3), dashed-dotted black) and without slow light effect (ng = 5 in Eq. (3), dashed red) are also presented. The vertical dashed lines mark the different regimes of the Q-factor variation.

Within the Fabry-Perot picture, and under the assumption of a narrow resonance, the quality factor 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]

]
Q=k0ngLeff1Reff,
(3)
with k0 = 2π/λ0, ng=neffλneffλ the group index of the MIM waveguide mode, Reff = Rexp(−αL) the effective reflectivity that includes the absorption loss over one-half round-trip, and Leff = L + 2Lp the effective resonator length that includes the penetration length Lp=λ24πngϕrλ, which is due to the dispersive nature of the reflection. The penetration length represents less than 10% of the effective length Leff and it can be safely neglected for all values of w and td considered in the present work [44

44. The assumption LpL 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.

].

The predictions of Eq. (3) are shown in Fig. 4(b) for 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/2n)3 (w = 120 nm, td = 100 nm and L = 100 nm) down to a deep subwavelength scale V = (λ/50)3 (w = 40 nm, td = 5 nm and L = 30 nm), the Q-factor is significantly enhanced by one order of magnitude.

Strictly speaking, the Fabry-Perot model should be applied only when all higher-order modes are evanescent. For instance, for w = 100 nm, the MIM is monomode only for td > 15 nm, and for w = 350 nm, the MIM is multimode whatever the value of td. Hopefully, the presence of higher-order modes does not alter the model predictions, except close to their cut-off where a drop of 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

Because of its analytical treatment, the model allows us to provide a comprehensive analysis of the Q-factor limitations. As shown by Eq. (3), the cavity-mode lifetime is impacted both by absorption (α in Reff) 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 ng = 5. Taking 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

45. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

], we find that the Q-factor of large MIM resonators with V = (λ/2n)3 is mostly limited by radiation, QQα=0, whereas the Q-factor of small resonators in the quasi-static limit with V = (λ/50)3 is completely limited by absorption, QQR=1. Radiation and absorption losses are balanced for td = 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 ng = 5, a value corresponding to thick MIM waveguides with td = 100 nm (red dashed curve).

4. Fabry-Perot model in the quasi-static limit

In the quasi-static limit, i.e., for dimensions much smaller than the wavelength, the quality factor Qs of any metallo-dielectric resonant nanoparticle composed of a single lossy metal is completely determined from the relative permittivity of the metal εm = εm + m [37

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

],
Qs=ω0εmω2εm.
(4)
In this limit, the specific structure of the particle is not important and the 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]

] by assuming a nanoparticle built in a noble metal (|εm| ≫ εm) that supports a purely electrostatic resonance (∇ × 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

45. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

].

The quasi-static regime is usually associated to the absence of wave retardation effects because k0L ≪ 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, 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 ϕr ≈ 0. [CrossRef] [PubMed]

]. Thus, at first sight, the Fabry-Perot picture seems inappropriate to describe a resonance in the quasi-static limit. This intuitive conclusion is in contradiction with Fig. 4(b) that evidences the accuracy of the Fabry-Perot predictions even for ultrasmall resonators with 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]

], and hereafter, we additionally evidence that it applies also quantitatively to the quality factor of the resonance. In particular, we show that the closed-form expression of the Q-factor given by Eq. (3) reduces to Eq. (4) in the quasi-static limit.

We consider a Fabry-Perot resonance with a 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]

], we assume that the resonance damping in the quasi-static limit is purely due to absorption. This amounts to consider 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
Q=k0ngα.
(5)
To obtain this result, we have considered that LpL and LeffL [44

44. The assumption LpL 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.

]. 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]

].

The derivation of Eq. (7) leads to two important conclusions. First, it evidences that the Fabry-Perot resonator model can be appropriately used to describe a resonance in the quasi-static limit. In other words, the quasi-static regime does include wave retardation effects linked to the propagation inside the particle of plasmonic modes whose wavevector diverges as the length shrinks (actually the product k0neffL = 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
(k0ngα)s=ω0εmω2εm,
(8)
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

In summary, we have studied the quality factor of the fundamental resonance supported by MIM nanoresonators when their volume is shrunk from the diffraction limit V = (λ/2n)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.

A second important result of this work is to show that the Fabry-Perot model remains quantitatively valid down to very small dimensions far below the diffraction limit. This evidences that the localized plasmon resonances of MIM nanoparticles in the quasi-static limit can be quantitatively analyzed with the same wave retardation effects as the delocalized resonances of nanowires with a length of a few wavelengths. The nanoparticle is indeed an half-wavelength antenna but with a vanishing effective wavelength λ0/(2neff) [33

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

].

We hope that the Fabry-Perot analytical treatment of the magnetic resonance supported by MIM nanoparticles will be helpful for further ”bottom-up” approaches for the design of optical metamaterials starting from the optical properties of single meta-atoms.

Acknowledgments

The authors gratefully acknowledge Jean-Paul Hugonin for computational assistance. A. Jouanin acknowledges Saint Gobain Recherche for financial support.

References and links

1.

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

2.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).

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]

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. 10, 631–636 (2011). [CrossRef]

5.

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]

6.

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

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. 105, 180502 (2010).

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]

10.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

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]

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]

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]

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. 95, 137404 (2005). [CrossRef] [PubMed]

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. 101, 103902 (2008). [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]

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]

20.

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]

21.

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]

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. 96, 251104 (2010). [CrossRef]

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 84, 075102 (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]

25.

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

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]

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]

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 84, 139–148 (2006). [CrossRef]

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 83, 245119 (2011). [CrossRef]

30.

C. Sauvan, P. Lalanne, and J.-P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B 71, 165118 (2005). [CrossRef]

31.

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

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. 95, 257403 (2005). [CrossRef] [PubMed]

33.

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

34.

E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express 16, 16529–16537 (2008). [CrossRef] [PubMed]

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. 9, 2372–2377 (2009). [CrossRef] [PubMed]

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. 11, 1020–1024 (2011). [CrossRef] [PubMed]

37.

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

38.

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

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]

41.

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

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 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]

44.

The assumption LpL 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.

45.

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

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]

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]

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


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References

  1. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics1, 41–48 (2007). [CrossRef]
  2. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).
  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]
  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.10, 631–636 (2011). [CrossRef]
  5. 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]
  6. H. A. Atwater and A. Polman, “Plasmonic for improved photovoltaic devices,” Nature Mater.9, 205–213 (2010). [CrossRef]
  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.105, 180502 (2010).
  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. Photonics5, 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. Photonics4, 395–399 (2010). [CrossRef]
  10. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).
  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]
  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]
  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]
  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.95, 137404 (2005). [CrossRef] [PubMed]
  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.101, 103902 (2008). [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]
  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]
  20. 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]
  21. 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]
  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.96, 251104 (2010). [CrossRef]
  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. B84, 075102 (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]
  25. Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B75, 035411 (2007). [CrossRef]
  26. S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators,” Opt. Express15, 10869–10877 (2007). [CrossRef] [PubMed]
  27. 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]
  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. B84, 139–148 (2006). [CrossRef]
  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. B83, 245119 (2011). [CrossRef]
  30. 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]
  31. P. Lalanne, C. Sauvan, and J.-P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser and Photon. Rev.2, 514–526 (2008). [CrossRef]
  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.95, 257403 (2005). [CrossRef] [PubMed]
  33. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007). [CrossRef] [PubMed]
  34. 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]
  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.9, 2372–2377 (2009). [CrossRef] [PubMed]
  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.11, 1020–1024 (2011). [CrossRef] [PubMed]
  37. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006). [CrossRef] [PubMed]
  38. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370 (1972). [CrossRef]
  39. 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]
  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]
  41. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182, 539–554 (1969). [CrossRef]
  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 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]
  44. 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.
  45. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  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. B84, 195405 (2011). [CrossRef]
  47. G. Della Valle, T. Søndergaard, and S. I. Bozhevolnyi, “High-Q plasmonic resonators based on metal split nanocylinders,” Phys. Rev. B80, 235405 (2009). [CrossRef]

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