Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities

We present a comprehensive theoretical and numerical analysis of the physical mechanisms pertaining to the nonlinear pulsed excitation of optical modes in plasmonic cavities made of metallic nanowires. Our analysis is based on extensive numerical simulations carried out both in the frequency and time domains. The numerical algorithm used in our study is based on the multiple scattering method and allows us to include in our analysis the effects of both the surface and bulk nonlinear polarizations generated at the second harmonic (SH). In particular, we investigate the physical properties of plasmonic modes excited at the SH as the result of the interaction of femtosecond optical pulses with plasmonic nanocavities. We show that such cavities have two distinct types of modes, namely, plasmonic cavity modes and multipole plasmon modes generated via the hybridization of modes of single nanowires. Our analysis reveals that the properties of the latter modes depend only weakly on the cavity geometry, whereas the lifetime and quality factor of plasmonic cavity modes vary considerably with the system parameters. © 2010 Optical Society of America OCIS codes: (190.4350) Nonlinear optics at surfaces; (240.6680) Surface plasmons; (140.3945) Microcavities References and links 1. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004). 2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292, 77–79 (2001). 3. 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). 4. V. M. Shalaev, W. Cai, U. K. Chettiar, H-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. 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Introduction
In recent years, we have witnessed a paradigmatic shift in the research methodologies used in studying the physical properties of electromagnetic media.This development can be primarily traced to the advent of the research field of metamaterials and its applications to optics and materials science.Metamaterials are a new class of electromagnetic media, whose basic unit cell is artificially engineered so as the optical medium possesses pre-designed macroscopic physical properties tailored for specific applications.In particular, by engineering the material parameters and geometry of the unit cell at a scale that is comparable to or smaller than the operating wavelength, a suite of new materials with remarkable properties have recently emerged.Examples of such optical media include materials that are magnetically active at terahertz and optical frequencies [1], materials with negative index of refraction [2,3,4], and homogeneous or periodic media with low-index of refraction [5,6,7].
It is well known that metallic structures support localized and propagating electromagnetic excitations, also known as surface plasmon polariton (SPP) modes [8,9].As a consequence of the resonant excitation of these modes the electromagnetic field in the vicinity of metal surfaces is strongly enhanced and therefore the macroscopic properties of metal-based metamaterials are significantly affected [10,11,12,13,14,15].This effect has been successfully employed in several applications, i.e., surface-enhanced Ramman scattering [16], nanoscale optical antennae [17], and metallic nanotips for near-field optical microscopy [18,19].In addition, enhancement of the second harmonic (SH) field intensity as large as 10 8 has been observed when employing metallic nanoparticles [20].It is thus clear that achieving a complete characterization of nonlinear effects occurring at the nanoscale represents an important step towards designing future subwavelength active nanodevices, which can be operated at low optical power.
One important property of localized SPPs is that their physical properties, such as, resonance frequency, optical loss, field enhancement, are strongly dependent on the geometry and material parameters of the corresponding plasmonic nanostructures.Therefore, similar to the effective, macroscopic optical properties of metamaterials, the linear and nonlinear optical response of plasmonic nanostructures can be effectively tailored to specific applications by, e.g., molding the spatial distribution of the optical near-field so as to provide pre-defined functionalities.For example, previous work has shown that plasmonic nanostructures can be used to significantly enhance the optical absorption (efficiency) of solar cells, detectors, and other photovoltaic devices [21,22,23,24,25], as well as subwavelength, all-optical control of the optical power flow in active nanodevices [26,27].Moreover, because of the evanescent character of localized SPPs, they are ideal tools for achieving subwavelength confinement of the optical field, and as such, SPPs can be instrumental in designing ultra-compact devices, such as nanolasers or laser arrays [28,29,30,31] and optical microcavities [32,33,34,35].In this connection, a central issue is to devise plasmonic structures which support localized SPPs with low optical losses (modes with large Q factor).A promising approach to address this challenge, which we explore in this work, is to reduce the radiative losses by employing cavity-shaped nonlinear plasmonic structures rather than plasmonic nanoparticles.
In this paper we investigate the characteristics of linear and nonlinear localized SPPs excited upon the interaction of ultra-short optical pulses with plasmonic structures made of twodimensional (2D) distributions of metallic nanowires.In our study, we employ the multiple scattering matrix (MSM) algorithm for calculating the field dynamics and the spectral characteristics of the optical field, at both the fundamental frequency (FF) and the SH.Unlike our previous study [36], in which we have investigated the spectral optical response of plasmonic nanostructures, in this paper we analyze via time-domain techniques the optical properties of plasmonic cavities.Our study reveals the existence at the SH of two markedly different types of plasmon resonances, namely, geometry independent multipole plasmon modes, which correspond to the excitation of weakly interacting modes of single cylinders, and geometry dependent plasmonic cavity modes.In addition, we show that the main parameter characterizing the cavity modes, namely, the Q factor, strongly depends on the structure and shape of the cavity.Our results suggest that by carefully designing the system geometry, specifically, the separation distance between the scatterers, the radiative losses can be greatly reduced, thus being possible to design plasmonic cavities with extremely large Q factor.The paper is organized as follows.Section 2 presents the plasmonic structures we investigate and provides a brief overview of the numerical method employed in our analysis.Section 3 contains a detailed discussion of the results obtained for a series of plasmonic cavities with different geometries.The final section summarizes our findings and outlines the main conclusions of our work.

Description of the numerical method
We begin our brief outline of the numerical approach used in our study with a description of the geometry of the system we investigate.It consists of an ensemble of N parallel, infinitely long cylinders, C j , j = 1, 2, . . ., N, embedded in a background medium with electric permittivity ε b and magnetic permeability µ b (in our calculations we assume µ b = µ 0 ).The location and electric permittivity ε j (ω), magnetic permeability µ j , and surface second-order susceptibility, χ (2) s, j , their axis being perpendicular to the xy-plane (see Fig. 1).Furthermore, the position of the center of the k-th cylinder, specified in a coordinate system with the origin in O j , is defined by the polar coordinates (r j k , ϕ j k ), whereas the position of an arbitrary point P, defined with respect to the coordinate systems with the origin in O and O j , is specified by the polar coordinates (r P , ϕ P ) and (r j P , ϕ j P ), respectively.In what follows, we assume that the wave vector k 0 of the incoming pulse lies in the xy-plane, the angle between k 0 and the x-axis being φ 0 .Moreover, we consider only the case of TE polarized light since in the case of the TM polarization no SH is generated.Under these conditions, the only nonzero field components are H z , E x , and E y .
In order to determine the temporal evolution of the field generated upon the interaction between an incoming optical pulse and an assembly of metallic nanowires we use a three-step approach [37].First, the incoming pulse is Fourier transformed in a series of monochromatic plane waves, which have the same wave vector but different amplitude and frequency.Then, following the formalism introduced in Ref. [36], the linear and nonlinear scattering problems are solved for each monochromatic component of the input pulse.Finally, the optical field in the time domain is reconstructed by inverse Fourier transforming the results obtained at the previous step.Thus, following this algorithm, the magnetic field of the input pulse is given by where ω 0 is the pulse carrier frequency and Hinc z is the Fourier transform of the pulse envelope.In our calculations we have considered input pulses with Gaussian envelopes, where H 0 is the pulse amplitude and the pulse duration T 0 is related to the pulse full-width at half-maximum (FWHM) by T FW HM = 2 √ ln 2T 0 .With these definitions, the amplitude of the plane waves contained in the input pulse can be written as where the physical angular frequency ω = ω + ω 0 .For the sake of simplicity, we have assumed that the center of the pulse would reach the origin of the coordinate system at t = 0.The second step of our method consists in solving the scattering problem for a monochromatic plane wave, at both the FF and SH.To this end, we used a numerical method based on the MSM formalism, which has been successfully employed in the study of linear scattering from arrays of cylinders [38] and nonlinear scattering from a single cylinder [39,40], a 2D object of arbitrary shape [41], and arrays of cylinders embedded in a nonlinear medium [42].Thus, we use a modified version of the nonlinear MSM algorithm [36], which allows one to include in the theoretical formalism the effects induced by both the surface and bulk nonlinear polarizations.This feature is particularly important in the case of centrosymmetric media, such as metals, since in this case the local, bulk nonlinear polarization vanishes and therefore surface nonlinear effects become dominant [43,44].Thus, the numerical algorithm consists of two steps.First, the scattered field at the FF is calculated and it is then used to compute the total nonlinear polarization at the SH.This source polarization is then used to calculate the SH field.
Without being a constraint imposed by the MSM formalism, we assume in our study that all cylinders are made of the same material, namely, Ag.Moreover, we assume that the permittivity of the metal is described by the Drude model, ε(ω) = ε 0 1 − ω 2 p ω(ω+iν) , where ω p = 1.35 × 10 16 rad/s is the plasma frequency and ν = 2.73 × 10 13 s −1 is the damping frequency [45].In what follows we present a brief description of the MSM formalism, a comprehensive description of this numerical method being provided in Ref. [36].Thus, the magnetic field of an incoming plane wave can be expanded in a Fourier-Bessel series as, where J m denotes the Bessel functions of the first kind and a m = H ω e −im(π/2+φ 0 ) are the Fourier-Bessel coefficients, with H ω the field amplitude at the frequency ω.Similarly, the field scattered from the cylinder C j can be expressed as: m (k 0 r j P )e imϕ j P , where m is the Hankel function of the second kind and b m j are the expansion coefficients of the scattered field; they represent the main unknowns of the linear scattering problem.Once these coefficients are calculated, the total field at a point P can be determined by using: m (k 0 r j P )e imϕ j P .
By introducing the column vectors a j = {e −ik 0 r j cos(ϕ j −φ 0 ) a m } and b j = {b m j } we can relate the two sets of expansion coefficients, for the incoming and scattered fields, via the scattering matrix, S j , of the cylinder C j (for more details, see Ref. [38]): where I is the identity matrix and T jk is the coupling matrix between the cylinders C j and C k .
The matrices S j can be easily found by solving the scattering problem for an isolated cylinder, with the appropriate boundary conditions.Then, by introducing the single column vectors B = {b j } and G = {S j a j }, we can reduce the Eq. ( 7) to a single matrix equation: where S ω is the scattering matrix of the whole system; from the Eq. ( 7) it is found to be: Since G and T jk are known if the system characteristics, such as the material and geometrical parameters, are specified, the linear scattering problem has been reduced to the calculation of S ω and finding the solution of the Eq. ( 8).These calculations yield the coefficients b m j , which in turn can be used to determine the total field at the FF via the Eq. ( 6).
The second step in the nonlinear MSM algorithm consists of finding the field at the SH.In order to do this, one first has to determine the nonlinear polarization at the SH, which represents the source of the SH field.In the case of centrosymmetric media, such as noble metals considered in our study, the nonlinear polarization is the sum between a dipole-allowed surface contribution, P s (r; 2ω), which is strongly localized within a thin interfacial layer at the surface of the metal, and a nonlocal, bulk nonlinear polarization, P b (r; 2ω), generated inside the metal by electric quadrupoles and magnetic dipoles [46].The surface polarization is defined as where r s defines the surface of the metal, χ s is the surface second-order susceptibility tensor, and the Dirac function models the surface characteristic of this polarization.Due to the symmetries of a homogeneous and isotropic interface the surface nonlinear susceptibility tensor has only three independent components.In the case of Ag their values are

and χ(2)
s,⊥ = 0 [47].For an isotropic centrosymmetric medium the bulk nonlinear polarization is given by [48], where α, β , and γ are material dependent physical constants.Assuming that the electrons in the metal are described by the free-electron gas model, these bulk nonlinear coefficients are with e and m 0 being the electron charge and mass, respectively, and ε r (ω) = ε(ω)/ε 0 .
Using an approach similar to that used to determine the field at the FF, the total field at the SH, at a point P outside the cylinder C j , can be written as [36]: m (kr j P )e imϕ j P where k = √ µ 0 ε b Ω is the wave vector at the SH.In the Eq. ( 13) the index j signifies that H tot z, j is calculated in the coordinate system with the origin in O j , whereas the coefficients a Ω,m j and b Ω,m j are the Fourier-Bessel coefficients of the series expansion of the field generated by the nonlinear polarization and the scattered field, respectively.Note that the coefficients a Ω, j = {a Ω,m j } are fully determined by the nonlinear polarization and therefore the main unknowns of the nonlinear scattering problem are the coefficients b Ω, j = {b Ω,m j }.The two sets of expansion coefficients satisfy the following system of matrix equations: where g sel f Ω, j are the source coefficients of the nonlinear field generated by a single cylinder; their rather cumbersome expressions can be found in Ref. [36].By denoting the vectors A Ω = {a Ω, j }, B Ω = {b Ω, j }, and G sel f Ω = {g sel f Ω, j } we can reduce the system (14) to a single matrix equation: Finally, by solving this linear system one finds the coefficients b Ω, j and, subsequently, via the Eq. ( 13), the total field at the SH at any arbitrary position.
The final step of our algorithm consists in calculating the optical field in the time domain.To achieve this, one simply inverse Fourier transforms the field spectra obtained in the preceding step.Specifically, accurate description of the optical field dynamics has been achieved by dividing the total temporal domain, T = 70 √ 2T 0 , in N FT = 8192 equal time intervals.Typical execution time when employing 64 Intel® Xeon cores is about 1 hour, provided that the field variables are computed at a single spatial point.This procedure allows one to obtain a complete description of the dynamics of the optical near-and far-field, at both the FF and the SH.As a final remark on the numerical algorithm used here, we want to stress that it allows one to easily incorporate nonlinear surface effects, which would prove to be a challenging task for alternative methods based on the finite-difference time-domain discretization of the field variables.

Results and discussions
In this section we will illustrate how the numerical method just described can be applied to investigate the physical characteristics of linear and nonlinear localized SPP modes excited by sub-picosecond pulses upon their interaction with plasmonic nanocavities.In particular, we explore the relation between the geometry and material parameters of the plasmonic cavities and the main optical properties of the localized SPP modes.In the presentation of our main results we will focus on the optical properties of the localized modes excited at the SH as the main conclusions derived in this case also apply to the modes observed at the FF.In addition, since there is no incoming pulse at the SH, some physical quantities characterizing the localized SPP modes, such as their Q factor, can be much easier calculated if the optical field at the SH is analyzed.This approach can also be relevant for a series of potential technological applications, such as sensing or optical detection, as the optical signal generated at the SH is spectrally well separated from the incoming and scattered fields at the FF.
To begin with, we illustrate in Fig. 2 the generic characteristics of the linear and nonlinear interaction between an incoming optical pulse and a SPP cavity mode.One convenient approach for identifying these modes relies on the spectra of the absorption cross-section at the FF, σ a (ω), and the corresponding absorption spectra at the SH, Σ a (2ω).To be more specific, it is expected that at the resonance frequency of the localized SPP modes the spectra of the optical absorption presents resonances, as at these specific frequencies the interaction of the optical near-field with the metallic nanowires is enhanced.At the FF the absorption spectra are calculated by using the scattering coefficients b j , in conjunction with the optical theorem, whereas at the SH we simply compute the total Joule dissipation power by integrating over the metal regions the density of the dissipated power, P abs = 1  2 Re(J • E * ), with J = σ j E being the conduction current density and σ j the conductivity of the cylinder C j (see Ref. [36] for more details pertaining to these calculations).In the case of the Drude model, σ = ε 0 ω 2 p /(ν − iω).The top panels in Fig. 2, which corresponds to a plasmonic cavity made of 6 metallic nanowires, show that the spectra of the absorption cross sections at the FF and SH contain a series of spectral peaks.In addition, the spectral location of these peaks changes with the separation distance between adjacent nanowires, d, a variation of d of less than 100 nm leading to a spectral shift of the resonances at the FF of almost 300 nm.Additional information pertaining to the optical properties of plasmonic cavity modes is revealed by the temporal dynamics of the optical near-field.Thus, the bottom panels in Fig. 2 (and the accompanying movies) reveal that the field at the SH remains trapped in the cavity long after the initial pulsed excitation at the FF has passed through the cavity.Since the nonlinear response of the metal is assumed to be instantaneous, it can be inferred that a nonlinear cavity mode with a significant lifetime and, implicitly, large Q factor, is formed in the cavity.In addition, the plasmonic character of this mode is evident from the spatial distribution of the near-field, namely, the field has large values at the metal surfaces and decays steeply towards the center of the plasmonic cavity.In what  follows, we present a more detailed analysis of these plasmonic cavity modes, as well as modes of a different physical nature, namely, multipole plasmon modes.

Localized plasmon modes in coupled cylindrical nanowires
We begin by considering a series of different cavity geometries, namely, cylinder distributions containing 2, 3, and 4 cylinders.Figure 3 summarizes the main results pertaining to these three geometries.As in the previous case, the spectra of the absorption cross section at the SH show a series of sharp peaks, which suggests the existence of SPP modes.Importantly, the resonance frequency of these modes does not depend on the number of cylinders in the distribution of scatterers, which means that these modes are formed primarily due to the excitation of optical modes in each of the metallic nanowires.This conclusion is supported by the field distributions presented in the bottom panels of the Fig. 3. Thus, these plots clearly show that the modes at λ SH = 578 nm correspond to dipole (cut-off) modes of the nanowires.Similar field distribu- tions, shown in Fig. 4, demonstrate that the resonances at smaller wavelength (λ SH = 336 nm) correspond to quadrupole modes of the nanowires.Figure 4 also shows that, as expected, multipole resonances are not only excited in plasmonic structures containing a small number of scatterers but that in fact their existance is a generic phenomenon.However, when the number of scatterers increases the amplitude of the optical modes excited in each nanowire varies with its location in the 2D nanowire assembly, especially when the wavelength becomes comparable to the size of the plasmonic structure.It should be noted that similar resonant modes are exited at the FF, too, but they do not appear in the absorption spectra since they are "buried" in the background generated due to the absorption of the input pulse.These modes, however, can be identified as resonance peaks in the spectra of the scattering cross section [36].Moreover, note that the SH field in the region in-between adjacent cylinders is small, which means that for this separation distance the "hybridization" effects due to the interaction between the optical modes excited in adjacent cylinders are weak.
To characterize the influence of the system geometry on the resonance frequencies of the multipole plasmon modes we have determined the absorption spectra for different values of the angle of incidence φ 0 and separation distance d.The dispersion plots corresponding to the absorption spectra at the SH are presented in the Fig. 5.These absorption spectra clearly indicate that the resonance frequencies of the multipole plasmon modes are almost independent on the system parameters, supporting therefore the conclusion that they correspond to optical modes excited in each of the metallic nanowires.More specifically, the resonance frequencies of these modes remain unchanged even if the distance between the adjacent metallic nanowires is decreased to a value as small as 1 nm.On the other hand, a significant increase of the optical absorption occurs when the separation distance becomes smaller than a few nanometers, an effect explained by the field enhancement observed in the region separating adjacent nanowires.Figure 5 also shows that the spectral width of the resonances varies with the of incidence φ 0 .This dependence suggests that, as expected, the strength of the interaction between the input wave and the plasmonic structure and, consequently, the magnitude of the field generated at the SH, changes with the angle of incidence.In addition, it can be seen from Fig. 5 that as the separation distance between the nanowires increases the width of the spectral resonances decreases, and effect that is explained by the fact that the strength of the coupling between the modes excited in adjacent nanowires decreases with the separation distance.

Plasmonic cavity modes
While the analysis of localized multipole plasmon modes can provide a valuable insight into the contribution of each individual scatterer to the overall optical response of the plasmonic structure, it does not reveal the complete picture of the interaction between optical pulses and plasmonic cavities.To be more specific, our analysis reveals that plasmonic structures containing a larger number of metallic nanowires support additional plasmonic modes, which have a different physical origin as compared to that of the multipole plasmon modes.In order to illustrate this conclusion, we present in Fig. 6 the absorption spectra, at both the FF and the SH, of cavity-shaped plasmonic structures containing 4, 6, and 8 metallic nanowires.As in the previous cases these spectra present a series of spectral peaks, which correspond to resonances of the plasmonic system.By inspecting the field profiles corresponding to these spectral peaks we found that besides the multipole plasmon modes similar to those supported by plasmonic structures with a smaller number of metallic nanowires there are additional, markedly different type of modes, which we call plasmonic cavity modes.These localized plasmon modes are formed due to the coherent response of the whole cavity.For example, as expected, in the case of the four-cylinder cavity the absorption spectrum at the SH presents two resonance peaks at  λ SH = 578 nm and λ SH = 336 nm, spectral peaks that correspond to the dipole and quadrupole plasmon modes, respectively.The absorption spectra at the SH have, however, additional resonance peaks at λ SH = 321 nm, λ SH = 429 nm, and λ SH = 333 nm.Since the resonance wave- length changes significantly with the number of nanowires forming the plasmonic cavity, it can be concluded that these optical modes are determined by the coherent response of the whole structure.The field profiles presented in Fig. 6 further support this conclusion, by showing that at the corresponding resonance wavelengths the optical field is not confined only to the region surrounding each nanowire but spreads inside the plasmonic cavity.
We want to point out that the plasmonic cavity modes investigated here are similar to whispering-gallery modes recently observed in plasmonic structures with a different geometry [28], the main difference being that in our case the angular momentum of the plasmonic cavity modes is equal to zero.Indeed, the angular momentum of the incident beam is zero and therefore the angular momentum of the excited modes must be zero, too.One additional important feature of the plasmonic cavity modes presented in Fig. 6 is that they do not appear as resonances in the scattering cross section spectra.Therefore, they are dark-plasmon modes that do not couple with the radiation continuum [49], and as a result the corresponding radiative losses are suppressed.Nevertheless, in our case these modes are excited via the nonlinear polarization generated at the SH, which acts as localized dipole sources.These results suggest that the nonlinear polarization at the SH can be used to excite subradiant (low loss) propagating modes formed in chains of metallic nanoparticles, via the near-field resonant coupling of single-particle dark-plasmon modes.As we will prove later, the suppression of the radiative losses also leads to a considerable increase of the lifetime of the plasmonic cavity modes and, consequently, makes it possible to design plasmonic cavities with very large Q factor.
As it can be seen in Fig. 3, for a separation distance of d = 20 nm the four-cylinder cavity does not have a plasmonic cavity mode but such a mode exists for d = 60 nm.This observation provides further evidence that the characteristics of the plasmonic cavity modes are strongly influenced by the geometry of the cavity.In order to explore this dependence in more detail, we will focuss in what follows on the optical properties of the plasmonic cavity formed by placing metallic nanowires at the corners of a hexagon.In making this choice we were primarily guided by the fact that the cavity modes of this structure are very well defined and, as we will see in the next sub-section, they have a large Q factor.
Because the dispersion spectra of the absorption cross section represent a powerful tool for investigating the properties of localized SPP modes, we have calculated these spectra for the hexagonal plasmonic cavity.The results, plotted in Fig. 7, provide further insight into the specific properties of plasmonic cavity modes.As expected, because these modes do not depend on the optical coupling between the incoming wave and the plasmonic cavity, they are independent on the angle of incidence φ 0 .On the other hand, the separation distance between adjacent cylinders does have a notable effect on the spectral location of the resonance wavelength of the plasmonic cavity modes.Thus, the resonance wavelength of the cavity mode increases almost linearly with the separation distance, a wavelength shift of almost 150 nm being observed when the separation distance changes by about 100 nm.In addition, it can be seen that, as in the case of multipole plasmon modes, the spectral width of the resonance decreases as d increases; however, as we will demonstrate later, in the case of plasmonic cavity modes this behavior is determined by the interplay between the radiative and absorption losses.As we will show in the next sub-section, other parameters characterizing the plasmonic cavity modes, such as the Q factor, have a more intricate dependence on the geometry of the cavity.A general feature, however, of these modes is that their optical properties can be easily tailored by modifying the shape of the cavity.Moreover, similar to the case of plasmonic cavities containing a smaller number of metallic nanowires, high optical absorption is observed for a separation distance approaching 1 nm.Again, this effect is attributable to the strong electromagnetic field generated at the surface of the nanowires for small inter-cylinder separation distance.
The strong dependence of the resonance frequency of plasmonic cavity modes on the distance between cylinders or other geometrical parameters can have important applications to sensing or photovoltaic devices.Specifically, the plasmonic cavity can be viewed as playing the role of an optical antenna that collects and concentrates the signal carried by the input pulse into a reduced volume, making it possible to increase the signal-to-noise ratio and/or speed of a detector.These plasmonic cavities can also be employed in the design of lasers with subwavelength size, as has in fact been recently demonstrated [30,31].In particular, the Q factor of plasmonic cavities employed in laser applications plays a crucial role in determining the performance of such nanolasers.Consequently, in what follows we study in more detail the dependence of the Q factor of the plasmonic cavity modes on the parameters defining the plasmonic structure.

Time domain analysis
The main physical quantity that describes the temporal response of an optical mode is the Q factor (or, equivalently, its lifetime).In order to calculate this important parameter that characterizes a plasmonic cavity mode we proceeded as follows.First, we illuminated the cavity with an optical pulse of sub-picosecond duration and subsequently we recorded the optical field, at both the FF and SH, at an arbitrary location inside the cavity.If the carrier frequency of the input optical pulse is close to a resonance frequency of a plasmonic cavity mode the asymptotic temporal evolution of the optical field can be represented by an exponential dependence, where ω r is the resonance frequency of the plasmonic cavity mode.It should be noted that the relation ( 16) is independent of the location of the point in the cavity where the field measured, the value of the carrier frequency of the input optical pulse (as long as it is close to ω r ), and the duration of the incident optical pulse, a conclusion that is fully verified by our numerical simulations.Therefore, the Q factor of the optical mode can be easily determined by calculating the slope of the line representing the linear fit of the semi-logarithmic temporal dependence of the computed field inside the cavity.
We have calculated the Q factor of the plasmonic cavity mode of the the hexagonal plasmonic cavity and the main results are summarized in Fig. 8.As expected, when the carrier frequency of the input optical pulse is close to the resonance frequency of the cavity mode the temporal evolution of the normalized field inside the cavity follows the exponential decay described by the Eq. ( 16).From this asymptotic dependence we have derived the value of the Q factor of the cavity mode and the corresponding lifetime, τ = Q/ω r .
Our calculations show that the lifetime of the plasmonic cavity mode increases from τ = 380.23 fs at d = 20 nm to τ = 582.2fs at d = 100 nm.If the separation distance is further increased, the lifetime begins to decrease.Moreover, the Q factor of the optical mode follows the same dependence on the separation distance, its maximum value, Q max = 2294, being reached for d = 113 nm.Note that this extremely large value is more than twice as large as the Q factor of recently observed plasmonic whispering-gallery modes [28], although it should be mentioned that these modes were observed in three-dimensional plasmonic cavities.This dependence of the Q factor on the separation distance between the metallic nanowires is somewhat surprising because one would expect that the radiative losses increase with the separation distance and therefore the Q factor should monotonously decrease as d increases.In order to explain this apparent contradiction, we decompose the Q factor as where Q abs is determined by the absorption in the metallic nanowires and Q rad is due to radiative losses.The Q abs factor can be calculated from the absorption spectra by fitting with a Lorentzian the region the spectrum corresponding to the resonance peak.Thus, Q abs = ω r /∆ω, where ∆ω is the spectral width of the Lorentzian.By using this procedure we found that for small values of the separation distance Q abs ≈ Q, which means that the losses of the plasmonic cavity mode are primarily due to the absorption loss in the metal.This result also supports the conclusion that the plasmonic cavity mode is a dark-plasmon mode, whose radiative losses are suppressed.As d increases, the field confinement decreases, and therefore the absorption losses decrease.Consequently, the Q factor of the optical mode increases.If the separation distance is further increased, the optical field begins to leak more easily out of the cavity, the cavity effects become weaker, and consequently the radiative losses start to dominate.As a result, the Q factor of the mode begins to decrease.This scenario predicts that there is a separation distance for which the Q factor reaches a maximum value, a prediction which is fully verified by the results presented in Fig. 8b.This analysis also suggests that by minimizing the optical losses associated to plasmonic cavity modes it is possible to optimize considerably the efficiency of these plasmonic cavities, a property that can have important implications to the development of efficient subwavelength nanolasers.

Conclusions
In conclusion, we have presented and discussed the main optical properties of linear and nonlinear localized SPP modes excited upon the interaction between ultra-short optical pulses and nanocavities made of metallic nanowires.Our numerical analysis, based on the MSM formalism, has revealed that plasmonic cavities support two distinct types of localized SPP modes, namely, multipole plasmon modes that are the result of the hybridization of coupled plasmon modes supported by each metallic nanowire of the plasmonic cavity, and plasmonic cavity modes, which can be viewed as the coherent optical response of the entire assembly of metallic nanowires.We have also demonstrated that this dichotomy in the physical origin of these optical modes is responsible for their markedly different optical properties.For example, whereas the properties of the multipole plasmon modes depend almost exclusively on the size of the individual nanowires, the geometrical and material parameters of the plasmonic cavity strongly influence the characteristics of the plasmonic cavity modes.In particular, we have demonstrated that this feature can be used effectively to design plasmonic cavities with extremely large Q factors.As a consequence, we believe that the results presented in this work could lead to a better understanding of linear and nonlinear optical effects at the nanoscale and, equally important, will have significant technological applications to nano-sensors, ultra-small detectors, subwavelength lasers, and other active photonic nanodevices.

Fig. 2 .
Fig. 2. Top panels show logarithmic plots of the absorption cross sections, calculated for a plasmonic cavity containing 6 nanowires.The legend indicates the separation distance, in nanometers.Bottom panels show a snapshot of the temporal evolution of the intensity of the electric field.The left and right panels correspond to the FF (Media 1) and SH (Media 2), respectively.The plasmonic cavity consists of Ag cylinders with R = 200 nm and d = 60 nm.The wavelength at the FF is λ FF = 858 nm and the angle of incidence is φ 0 = 90 • .

Fig. 3 .
Fig. 3. Top panels show logarithmic plots of the absorption cross sections.The legends indicate the number of cylinders.Bottom panels present the amplitude of the electric field at the SH, for Ag cylinders with R = 200 nm and d = 20 nm, at λ SH = nm and φ 0 = 0 (for better visualization, we plot the fourth-order square root of the field amplitude).

Fig. 4 .(Fig. 5 .
Fig. 4. Distribution of the amplitude of the electric field (top panels) and the real part of the magnetic field (bottom panels) at the SH, calculated for three different plasmonic cavities made of Ag cylinders with R = 200 nm and separation distance d = 20 nm.The wavelength at the SH is λ SH = 336 nm and φ 0 = 0 (for better visualization, in the case of the electric field, we plot the fourth-order square root of the field amplitude).

Fig. 6 .
Fig. 6.Top panels show logarithmic plots of the absorption cross section at the FF and the SH.The legend indicates the number of cylinders forming the plasmonic cavity.Bottom panels present the distribution of the amplitude of the electric field at the SH, for Ag cylinders with R = 200 nm and separation distance d = 60 nm.From left to right, the wavelength at the SH is λ SH = 321 nm, λ SH = 429 nm, and λ SH = 333 nm.

Fig. 8 .
Fig. 8. a) Normalized electric field inside the cavity vs. time.The numbers in the legend refer to the separation distance in nanometers.b) The dependence of the Q factor on the separation distance d: circles represent simulation results whereas the dotted line is provided as a guide to the eye.The cavity consists of Ag cylinders with R = 200 nm and the incoming pulse has T 0 = 283 fs.