## Abstract

Abstract

General properties of retardation-based resonances involving slow surface plasmon-polariton (SPP) modes supported by metal nanostructures are considered. Explicit relations for the dispersion of SPP modes propagating along thin metal strips embedded in dielectric and in narrow gaps between metal surfaces are obtained. Strip and gap subwavelength resonant structures are compared with respect to the achievable scattering and local-field enhancements lending thereby their distinction as nano-antennas and nano-resonators, respectively. It is shown that, in the limit of extremely thin strips and narrow gaps, both structures exhibit the same *Q* factor of the resonance which is primarily determined by the complex dielectric function of metal.

©2007 Optical Society of America

## 1. Introduction

Resonance interaction in nanoscale metal structures involving excitation of surface plasmons has been a subject of intensive and extensive research in recent years [1–3]. It has been shown that the metal nanostructures at resonance can dramatically enhance both scattered and local electric fields by concentrating electromagnetic energies into subwavelength volumes. Various configurations and ingenious designs of metal nanostructures have been proposed and explored to realize the field enhancement effects. Probably the most intensively investigated direction is associated with the exploitation of electrostatic resonances in electron oscillations in metal nanostructures of different topologies [3–5]. It has been generally accepted that once the structure sizes are sufficiently small, i.e. at nanoscale, their consideration can be carried out in the quasistatic (electrostatic) approximation [3–5]. We have recently put forward a concept of retardation-based resonances involving slow surface plasmon-polariton (SPP) modes supported by metal nanostructures [6–8]. It should be emphasized that the slow SPPs do not exhibit cutoff for small cross sections [9] making thereby possible truly nanoscale confinement of propagating fields, so that, similar to electrostatic resonances [3–5], the slow-plasmon resonances can be found for extremely small metal structures, i.e. at nanoscale [6–8].

In this work, general properties of retardation-based resonances involving slow SPPs supported by metal nanostructures are considered. Since the spatial extent of resonant structure scales with the wavelength of the corresponding slow SPP mode, approximate explicit relations for the dispersion of slow SPPs propagating along thin metal strips and in narrow gaps between metal surfaces are obtained and compared with the exact dispersion solutions. Strip and gap subwavelength resonant structures are then qualitatively compared with respect to the achievable scattering and local-field enhancements lending thereby their distinction as nano-antennas and nano-resonators, respectively. It is shown that, in the limit of extremely thin strips and narrow gaps, both structures exhibit the same *Q*-factor of the resonance which is mainly determined by the complex dielectric function of metal.

## 2. Characteristics of slow SPP modes

SPPs are electromagnetic waves that are bound to and propagate along metal-dielectric interfaces [10]. For an individual metal surface, the SPP propagation constant (wave number) *k _{sp}* is given by

*k*=(2

_{sp}*π/λ*) [

*ε*

_{m}

*ε*

_{d}/(

*ε*

_{m}+

*ε*

_{d})]

^{0.5}, where

*λ*is the wavelength in air,

*ε*

_{m}and

*ε*

_{d}are the dielectric functions of metal and dielectric, respectively [10]. In the long-wavelength part of the visible spectrum and infrared, SPPs are quite close to the light line (i.e., to free propagating light waves) and unwieldy to be directly used for constructing strongly localized and enhanced fields. However, placing two (or more) metal-dielectric interfaces close to each other introduces the coupling between the SPPs of individual interfaces, which in turn slows down the appropriate resulting SPP-mode making accessible nm-sized field localization at any wavelength [9]. Many different SPP modes can be found in multiple-interface systems, when the SPPs associated with individual metal-dielectric interfaces start interacting with each other. Considering the SPP modes associated with two metal-dielectric interfaces one finds that the SPP modes can be supported by either a thin metal film embedded in dielectric or a thin dielectric layer surrounded by metal, often called the insulator-metal-insulator (IMI) or metal-insulator-metal (MIM) structures, correspondingly [11].

Let us first consider the SPP modes in the symmetric IMI configuration of a thin metal film with the thickness *t* being embedded in the dielectric (extending indefinitely on both sides of the film). When two identical SPP modes start overlapping with each other for small thicknesses of the metal film, the propagation constants of the symmetric and anti-symmetric (individual) mode combinations become different [9]. Note that the symmetric (with respect to the film mid-plane) configuration of the transverse electric field component corresponds to the anti-symmetric one for the longitudinal electric field component and vice versa. Hereafter the symmetric transverse-field configuration is called the symmetric SPP mode. Since the SPP damping is determined by the longitudinal SPP component, the symmetric SPP mode exhibiting the odd symmetry of longitudinal field (which thereby crosses zero changing its sign) at the mid-plane of the metal film experiences considerably smaller attenuation than the anti-symmetric SPP mode. Conversely, the anti-symmetric SPP mode exhibits the even symmetry of longitudinal electric field, maximizing the electromagnetic energy density within the metal (see, for example, Fig. 1), thereby slowing down the mode propagation and also increasing its attenuation. The symmetric SPP mode is therefore conventionally called the long-range SPP (LR-SPP) and the anti-symmetric one — the short-range SPP (SR-SPP).

Using the appropriate boundary conditions for the normal and tangential electric field components and the aforementioned symmetry of the corresponding field component distributions, allows one to obtain the following dispersion relation for LR(SR)-SPPs:

where the hyperbolic tangent at the left-hand side of Eq. (1) should be put to the power of -1 in the case of SR-SPP dispersion. It can be seen [6] that, in the limit of very thin films, the SR-SPP propagation constant *k*
_{srsp} increases rapidly with its real and imaginary parts: *k*
_{srsp}≈-(2*ε*
_{d})/(*tε*
_{m})→∞, when the metal film thickness *t* decreases to zero. This means that both the SR-SPP wavelength and propagation length decrease, approaching zero for infinitely thin films. It can also be shown that, in this limit, their ratio tends to a constant determined by the ratio between the real and imaginary parts of the metal susceptibility (a ratio that can be substantial for noble metals [12]) as follows:*L _{srsp}/λ_{srsp}*→|Re(

*ε*)|/[4

_{m}*π*Im(

*ε*)].

_{m}Let us now obtain approximate explicit expressions for the LR- and SR-SPPs that can be rather useful when designing resonant nanostructures [6–8]. It turns out that a very good approximation can be found in the closed form by making use of the following simplification: $\sqrt{{k}_{l\left(s\right)\mathrm{rsp}}^{2}-{\epsilon}_{m}{k}_{0}^{2}}\approx {k}_{0}\sqrt{{\epsilon}_{d}-{\epsilon}_{m}}$ that can be used down to reasonably small film thicknesses. The corresponding *explicit* dispersion relations can then be written down:

Considering the properties of the LR- and SR-SPP supported by a thin gold film surrounded by air at the excitation wavelength of 775 nm and using the gold permittivity *ε _{m}*=-23.6+1.69

*i*[12], the effective mode index,

*N*

_{eff}=Re(

*k*

_{sp})/

*k*

_{0}, and propagation length,

*L*=[2Im(

*k*

_{sp})]

^{-1}, were calculated for both SPP modes using the exact [Eq. (1)] and explicit [Eq. (2)] dispersion relations (Fig. 1). It is seen that the explicit solutions given by Eq. (2) result in accurate values of the mode effective index and propagation lengths for both LR- and SR-SPPs in the whole range of film thicknesses, down to the thickness of 10 nm (Fig. 2).

We now consider the SPP modes in the *symmetric* MIM configuration of a thin dielectric layer with the thickness *t* sandwiched between two metal surfaces (with metal extending indefinitely on both sides of the dielectric). The only SPP mode surviving for all values *t* of gap between metals, the so called gap SPP (G-SPP), is the mode exhibiting odd symmetry of the longitudinal electric field component and, consequently, even symmetry of the transverse field (see, for example, Fig. 3).

The G-SPP dispersion can be written as follows [9,11]:

where *k*
_{gsp} denotes the G-SPP propagation constant. For sufficiently small gap widths (*t*→0), one can use the approximation tanhx ≈ *x* resulting in the following expression:

Here, *k*
^{0}
_{gsp} represents the G-SPP propagation constant in the limit of very narrow gaps (*t*→0), when the real part of the correspondent effective index becomes much larger than the dielectric refractive index. Note that, in this limit, the propagation constants of the SR-SPP and G-SPP modes become *equal* for *complementary* structures (i.e. obtained one from another by exchanging metal and dielectric).

For relatively large gaps, when the two SPP modes are only starting to interact, one can use the approximation tanhx ≈1-2exp(-2x) resulting in the following first-order corrected expression for the G-SPP propagation constant:

$$\mathrm{where}\phantom{\rule{.5em}{0ex}}{\alpha}_{0}=\sqrt{{k}_{\mathrm{sp}}^{2}-{\epsilon}_{d}{k}_{0}^{2}}=\frac{{k}_{0}{\epsilon}_{d}}{\sqrt{-{\epsilon}_{m}-{\epsilon}_{d}}}\phantom{\rule{.5em}{0ex}}.$$

Considering the properties of the G-SPP guided in an air gap between gold (infinitely thick) layers at the excitation wavelength of 775 nm, the effective mode index and propagation length were calculated using the exact [Eq. (3)] and approximate [Eqs. (4) and (5)] dispersion relations (Fig. 4). It is seen that the small-gap approximation given by Eq. (4) results in accurate values of the G-SPP mode effective index and propagation length for gap widths already below 300 nm (Fig. 4).

## 3. Retardation-based resonant nanostructures

Let us now consider the above described nanostructures of *finite* extension in the direction of (slow) SPP mode propagation. Analyzing the SR-SPP field distribution (Fig. 1) one immediately realizes that, contrary to conventional SPPs, the SR-SPPs can be efficiently reflected at the film termination, because the transverse field component is out of phase on both sides of a film whereas the longitudinal field should radiate out of a thin film end (i.e., out of a subwavelength aperture). Moreover, it is expected that the reflectivity should increase with the decrease in the film thickness, also because the SR-SPP effective index increases drastically (Fig. 2) leading to the increase in pure Fresnel reflection. A thin metal strip represents thereby a resonator, in which the counter-propagating SR-SPPs being efficiently reflected by the strip ends form a resonant standing-wave pattern under the condition of constructive interference. Taking into account that any mode reflection introduces the phase shift *φ*, the resonance condition (for a strip of width *w*) can be written as follows:

Here, *N*
_{eff}=Re(*k*
_{srsp})/*k*
_{0} is the effective index of the corresponding SR-SPP [Eq. (2)]. The reflection phase *φ*, which can in principle be calculated following the procedure used for subwavelength slits in metal [13], determines the shortest resonant strip for given wavelength, strip material and thickness. For example it has been found [6] that, for 10-nm-thick silver and gold strips in air, the shortest strip width is quite close to one third of the SR-SPP wavelength: *w*
_{0}≈*λ*/3*N*
_{eff}, allowing thereby for resonators featuring spatial dimensions *considerably* smaller than the light wavelength [6–8].

The G-SPP field distribution exhibiting the symmetry opposite to that of the SR-SPP [cf. Figs. (1) and (3)] suggest its efficient reflection at the gap termination as well, because the (strong) transverse field component in the gap should be matched to free propagating field components across a very narrow (subwavelength) gap. One also expects that the reflectivity should increase with the decrease in the gap width, since the G-SPP effective index increases drastically (Fig. 4) leading to the increase in pure Fresnel reflection. It has indeed been shown [13] that the reflectivity tends to unity in the limit of very narrow gaps. Finally, Eq. (6) can be considered as expressing the resonance condition for a G-SPP structure of length *w*, the only modification being that *N*
_{eff}=Re(*k*
_{gsp})/*k*
_{0} represents, in this case, the effective index of the corresponding G-SPP [Eq. (3)]. Examples of resonant enhancement by nanometer-sized strip and gap structures are shown in Fig. 5. Resonant enhancement of scattering cross sections is well pronounced for both structures reaching the values larger than geometrical extensions.

It should be noted that the resonances due to constructive interference of counter-propagating G-SPPs have been previously considered in several publications (see Ref. 14 and references therein). On the other hand, it has apparently escaped the attention that the tunability properties of a plasmonic subwavelength particle deposited on a metallic slab [15] have the *same* physical origin and explanation, because it is not needed to have a finite size of the structure as a whole — it is sufficient to have a finite size of the gap (due to the finite size of the upper metal cladding). The calculated resonances can be easily identified with the help of the above resonance condition [Eq. (6)] accounting for the main observed features, such as the resonant wavelength shift with the spacer thickness and its susceptibility [15].

Further insight into the physics of the resonance in strip and gap structures can be gained by considering the electric field distributions in these structures at the wavelength of the first-order (fundamental) resonance (Fig. 6), i.e., the first resonance occurring at this wavelength when the structure extension increases from small values. First of all, it is seen that this resonance should be ascribed to the resonance in the *longitudinal* electric field component, since it is this component that reaches its maximum at the structure center (in the propagation direction of slow SPP modes), i.e., at the center of the metal strip [Fig. 6(a)] or metal cladding layers [Fig. 6(b)]. The 0.5*π*-phase lag between the longitudinal and transverse SPP field components [10] results in the corresponding standing-wave intensity patterns being out of phase (Fig. 6). Furthermore, the structures of scattered fields (dominating at resonance in the displayed total field distributions) resemble that of an *electric dipole* (in the figure plane) for the strip [Fig. 6(a)] and that of a *magnetic dipole* (perpendicular to the figure plane) for the gap structure [Fig. 6(b)]. These features originate directly from the corresponding field distributions of the SR-SPP (Fig. 1) and G-SPP (Fig. 3). Note that the *magnetic* response of parallel metal films is of great importance in the field of optical *negative-index metamaterials* [16]. Again, it has apparently escaped the attention [17] that the resonance in these structures is related to the constructive interference of counter-propagating slow SPPs reflected at the structure terminations.

Let us now consider these structures from the perspective of using them as efficient nano-antennas and nano-resonators. First of all, it should be stressed that both strip and gap structures are rather efficient in resonantly enhancing scattered (Fig. 5) and local (Fig. 6) electromagnetic fields. On the other hand, the scattering maximum is better pronounced (albeit somewhat wider) for the strip than gap structure (Fig. 5). One reason is that an electric dipole associated with an oscillating current inside the strip [Fig. 6(a)] is much better coupled to free propagating field components than a magnetic dipole associated with an effective induction current loop formed in the gap structure [Fig. 6(b)]. Stronger scattering exhibited by the strip configuration implies also that the radiation loss of the corresponding resonator is larger, resulting thereby in lower levels of the local electric field [cf. Figs. 6(a) and 6(b)]. Consequently, we would suggest using (and referring to) the strip structures as nano-antennas and the gap structures as nano-resonators.

It is instructive to examine the behavior of these structures in the limit of extremely thin strips and narrow gaps. It transpires from the discussion in Section 2 that, in this limit, the slow-SPP propagation constants are given by the *same* expression: *k*
_{srsp}≈*k*
_{gsp}≈-(2*ε*
_{d})/(*tε*
_{m}). Note that, in this case, the metal cladding layers in the gap configuration are assumed being infinitely thick. Since it is expected that, in this limit, the reflectivity at the structure termination approaches unity, the resonance characteristics should become very close for both strip and gap structures. It is indeed seen (Fig. 5) that the 10-nm-thick strip and 10-nm-gap structures have close values of resonant lengths with the tendency of becoming closer for thicker cladding layers of gap structures. Assuming the resonator loss being dominated by the slow-SPP propagation loss occurring only in the metal, the resonance *Q* factor has the expression:

The *Q* factor of the resonance is thereby determined by the complex dielectric function of metal (explicitly) and by that of dielectric (implicitly) via the slow-SPP mode effective index, *N*
_{eff}=Re(*k*
_{sp})/*k*
_{0}. It is interesting to note that the *Q*
_{st} factor of electrostatic resonances shown to be independent on the nanostructure form depends on the metal susceptibility, and thereby on the wavelength, in a different manner, i.e., *Q*
_{st}=*ω*[*d*Re(*ε*
_{m})/*dω*]/[2Im(*ε*
_{m})] with *ω*being the light frequency [3]. On the other hand, in the limit of long wavelengths and Drude response for the metal susceptibility *ε*
_{m}(*ω*), one obtains for both *Q* factors the same scaling: *Q*~1/*λ*.

Considering the silver nanostructures whose scattering spectra are shown in Fig. 5, one can evaluate the corresponding *Q* factor [Eq. (7)] as *Q*≈5 using *N*
_{eff}≈1.5 (Figs. 2 and 4) and the silver susceptibility at *λ*≈650 nm [12]. It is seen that while this value of *Q* fits well to the spectral width of resonance for the strip resonator (~130 nm), the resonances for the gap resonators are considerably narrower (<50 nm). The latter discrepancies reflect the fact that the characteristics of gap resonators with finite thickness of metal cladding can be rather different from those for infinitely thick cladding layers. The influence of the metal cladding thickness can be seen from the spectra shown in Fig. 5: thicker cladding layers result in smaller mode effective indexes (and thereby longer resonators) and broader but more pronounced resonances. These tendencies (also found for other parameters and for gold structures) indicate that the optimization of the gap resonator structure is a complicated task, because of a nontrivial relationship between the propagation and reflection losses and the structural parameters involved [8].

The strip and gap resonant structures analyzed insofar are essentially two-dimensional, extending indefinitely along the third dimension perpendicular to the plane of light incidence (i.e., along the *z*-axis in Fig. 5). In practice, extension of these structures along the z-axis is finite complicating the situation. However, if the extension length *L* is much larger than the structure width *w* (*L*≫*w*), one can still use all the aforementioned results since any additional resonances associated with slow SPP propagation along the *z*-axis would be well separated in the wavelength domain being displaced toward long wavelengths. In the opposite limit, i.e. *L*≪*w*, one should consider strip SPP waveguiding along the *x*-axis, whose effect would result in a (slight) decrease of the effective mode index to be used in Eq. (6). The situation becomes more complicated in the intermediate case (i.e., for *L*~*w*), when different resonant configurations would correspond to different polarizations and the possibility of whispery-gallery resonances should be considered as well, requiring a dedicated analysis.

Three-dimensional analogues of the strip and gap nanostructures are correspondingly thin metal nanorods and annular apertures [6]. Similarly to the SR-SPP and G-SPP, cylindrical and coaxial SPPs feature the propagation constants increasing rapidly with the decrease in the structure cross sections (without cutoff) and, being efficiently reflected at structure terminations, result in resonances found for nanometer-sized (in the propagation direction) structures [18,19]. It is also expected that nanorods and annular nano-apertures exhibit similar differences with respect to the achievable scattering and local-field enhancements lending their distinction as nano-antennas and nano-resonators, respectively. We would like to emphasize that, in both cases, the exact location of resonances requires accurate *evaluation* of the *reflection phase* [Eq. (6)], a circumstance that is often overlooked.

Finally, it should be borne in mind that, similarly to the electrostatic resonance in metallic nanospheres [20], it is possible to significantly enhance the resonance if the surrounding dielectric medium exhibits optical gain. It can be shown that the slow-SPP propagation loss (in the limit of very thin strips and narrow gaps) will be compensated once the following condition is satisfied: Re(*ε*
_{d})Im(*ε*
_{m})=Re(*ε*
_{m})Im(*ε*
_{d}). The corresponding estimation of the intensity gain results in the value *α*~10^{3}cm^{-1}, which is within the limits, e.g. of concentrated laser dyes [20]. When the propagation loss will be compensated, the reflection loss would become dominant similarly to the situation in conventional lasers. However, it is clear that the *coherent radiation generation is feasible in the gap structure* filled with highly concentrated laser dyes, e.g. by following the route used to observe the plasmon loss compensation in silver aggregates [20]. Such a device generating a tightly confined (subwavelength) SPP mode would be ideal for future applications, e.g. for plasmonic interconnects and bio-sensors.

## 4. Conclusion

In summary, general properties of retardation-based resonances involving slow SPP modes supported by thin metal strips embedded in dielectric and by narrow gaps between metal surfaces have been considered. Since the spatial extent of resonant structures scales with the wavelength of the corresponding slow SPP mode, explicit relations for the dispersion of the corresponding slow SPPs, i.e. SR-SPPs and G-SPPs respectively, were obtained, comparing approximate and exact dispersion solutions. Strip and gap subwavelength resonant structures were qualitatively compared with respect to the scattered field distributions (at resonance) and achievable enhancements in scattered and local fields. It was then argued that, even though both types of structures are efficient in resonantly enhancing scattered and local fields, the strip structures exhibit properties corresponding better to nano-antennas whereas the gap structures should be rather referred to as nano-resonators. It has also been shown that, in the limit of extremely thin strips and narrow gaps, both structures exhibit the same *Q*-factor of the resonance determined primarily by the complex dielectric function of metal. Finally, the possibility of enhancing the resonance by using the dielectric medium having optical gain was considered as well concluding that the compensation of the mode propagation loss is within the limits of concentrated laser dyes. We believe that the findings reported can help in designing nanoplasmonic structures dedicated not only to reaching high local field enhancements but also to perfecting optical negative-index metamaterials.

## Acknowledgments

The authors gratefully acknowledge financial support (SIB) from the European Network of Excellence, PLASMO-NANO-DEVICES (FP6-2002-IST-1-507879), (TS) from the Danish Research Council for Technology and Production, and (SIB and TS) from the NABIIT project financed by the Danish Research Agency (contract No. 2106-05-033).

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