## Abstract

Periodic rectangular gold nanomonopoles and nanodipoles in a piecewise inhomogeneous background, consisting of a silicon substrate and a dielectric (aqueous) cover, have been investigated extensively via 3D finite-difference time-domain simulations. The transmittance, reflectance and absorptance response of the nanoantennas were studied as a function of their geometry (length, width, thickness, gap) and found to vary very strongly. The nanoantennas were found to resonate in a single surface plasmon mode supported by the corresponding rectangular cross-section nanowire waveguide, identified as the *sa _{b}^{0}* mode [Phys. Rev. B

**63**, 125417 (2001)]. We determine the propagation characteristics of this mode as a function of nanowire cross-section and wavelength, and we relate the modal results to the performance of the nanoantennas. An approximate expression resting on modal results is proposed for the resonant length of nanomonopoles, and a simple equivalent circuit, also resting on modal results, but involving transmission lines and a capacitor (modelling the gap) is proposed to determine the resonant wavelength of nanodipoles. The expression and the circuit yield results that are in good agreement with the full computations, and thus will prove useful in the design of nanoantennas.

© 2012 OSA

## 1. Introduction

Nanoantennas have been widely investigated, both experimentally and numerically, during the past decade. Nanomonopoles, nanodipoles and nanobowties have been of particular interest. Although there are some conceptual similarities between optical nanoantennas and classical microwave antennas, the physical properties of metals at optical frequencies dictate applying a different scaling scheme [1,2]. Moreover, feeding procedures are very different between classical and optical nanoantennas, since driving nanomonopole, nanodipole, and nanobowties nanoantennas using galvanic transmission lines is not an option, due to their small size. Instead, localized oscillators or incident beams are often used to illuminate nanoantennas [3].

One of the primary differences between classical monopoles and dipoles and their optical counterparts is their resonant length which is considerably shorter than λ/2 [4], where λ is the free-space incident wavelength. In [4] Novotny introduces useful analytical expressions for wavelength scaling of free-standing cylindrical nanomonopoles of different radii surrounded by a dielectric medium. The results, however, are not easily applicable to other nanoantenna geometries such as dipoles and bowties, or non-cylindrical nanoantennas. Adding a substrate, which is often required in practice, necessitates modifications to wavelength scaling rules.

As in classical antennas, the spectral position of the resonance in the optical regime depends strongly on the geometry of the antennas. Antenna length has been investigated as a crucial tuning parameter in nanomonopoles, nanodipoles, and nanobowties [5–8]. Capasso and Cubukcu proposed a resonant length scaling model for free-standing cylindrical nanomonopoles by using the decay length of the surface plasmon mode excited in the corresponding plasmonic waveguide as the scaling factor [6]. In the case of dipoles and bowties the gap size and gap loading play an important role in determining the position of resonance [5–12]. Alu and Engheta have looked at nanoantennas as lumped nanocircuit elements and investigated some of their properties such as optical input impedance, optical radiation resistance, and impedance matching [9, 12]. In these studies the gap is considered as a lumped capacitor, which is connected in parallel to the nanodipole. This model identifies the gap length and gap loading as additional tuning parameters of nanodipoles. Fischer and Martin show that in nanodipoles, decreasing the gap shifts the resonance towards the red region of the spectrum, whereas in the case of bowties the resonance hardly shifts as a result of changing the gap [7]. High intensity fields in the dipole and bowtie gaps are strongly sensitive to the index of the material inside the gap [7, 9]. Also the effects of variations in the bow angle of a bowtie antenna on its spectral response have been investigated numerically and experimentally [7, 11], showing that the bow angle can be used as a tuning parameter. Aizpurua *et al.* [13] have studied the properties of nanomonopole and nanodipoles and showed that not only the geometry of the nanorods, but also the coupling between the nanorods dramatically increase the field enhancement in the gap.

Fabry-Perot and cavity models were proposed to determine the properties of nanomonopoles and nanodipoles [14–17]. Semi-analytical investigations have been done on nanodipole, nanobowtie, and hybrid dimer nanoantennas based on a microcavity model [14], and analytical expressions were suggested for surface and charge densities corresponding to the SPP mode propagating in the nanoantenna, which in turn represent near- and far-field properties of these nanoantennas.

One of the first experimental demonstrations of optical antennas was reported by Muhlschlegel *et al.* [18], where gold nanodipoles were illuminated with picosecond laser pulses. The field enhancement in the antenna gap was considered as was the system response to relate the resonant length of the nanodipoles to the illuminating wavelength. Nanoantennas, in general, and the three above-mentioned types in particular, have found many applications in imaging and spectroscopy [3], photovoltaics [3], and biosensing [19]. With a growing range of applications, developing precise, yet practical design rules for nanoantennas seems essential.

Although many theoretical and experimental studies have been carried out on various aspects of nanoantennas and their applications, a systematic study of their spectral response to variations in design parameters is lacking in the literature. In this paper, we present a full parametric study of the spectral response of infinite arrays of rectangular gold nanomonopoles and nanodipoles on a silicon substrate covered by water. (The materials were selected in anticipation of an eventual biosensing application to be described elsewhere; however, the study remains otherwise generic.) We vary the nanoantenna length (*l*), width (*w*), thickness (*t*), and, in the nanodipole case, the gap length (*g*), as well as the vertical and horizontal distance (*p, q*) between any two adjacent nanoantennas in an infinite array. Physical insight into the resonant response of arrays of nanoantennas is then provided through modal analysis of the corresponding plasmonic nanowire waveguides. A simple rule is proposed to determine the effective length of a nanomonopole in a piecewise homogeneous background from the modal properties of the corresponding nanowire. An equivalent circuit using transmission lines and a capacitor is proposed for the nanodipoles, with the capacitor taking into account the effects of the gap. This simple rule and model should become helpful aids in the design of such nanoantennas. (In the remainder of this paper we refer to nanoantenna, nanomonopole and nanodipole simply as antenna, monopole and dipole.)

The antenna geometry and the method used in its study are discussed in Section 2. The parametric study of monopole and dipole arrays is presented in Section 3. Section 4 discusses the operation of the antennas from a modal viewpoint and gives expressions for the resonant length of monopoles and the equivalent circuit of dipoles. Section 5 gives our conclusions.

## 2. Geometry and methods

Figure 1
gives a sketch of the dipole geometry under study. The array cell is symmetric about the *x* and *y* axes. An infinite array is constructed by repeating the cell along *x* and *y* with pitch dimensions *p* and *q* (respectively). An *x*-polarized plane wave source having an electric field magnitude of 1 V/m, located in the silicon substrate, illuminates the array from below at normal incidence.

The finite difference time domain (FDTD) method [20], with a 0.5 × 0.5 × 0.5 nm^{3} mesh in the region around the antenna, was used for all simulations. Palik’s material data [21] were used for gold and silicon, and Segelstein’s data [22] for water. Transmittance and reflectance reference planes were located 2.5 µm above and below the silicon-water interface, respectively, parallel to the interface. (A convergence analysis was performed where the resonant wavelength of an array was tracked as a function of mesh dimensions in the neighborhood of the antenna. Mesh dimensions were halved successively, starting from a 2 × 2 × 2 nm^{3} cubic mesh to a 0.25 × 0.25 × 0.25 nm^{3} cubic mesh, over which the resonant wavelength was observed to trend monotonically. The wavelength of resonance for mesh dimensions of zero (infinitely dense) could thus be extrapolated using Richardson’s extrapolation formula [23]. Comparing this extrapolated wavelength to the wavelength obtained for a finite mesh of 0.5 × 0.5 × 0.5 nm^{3} reveals a ~2% error, which considering the broad spectral response of the structures of interest, was deemed acceptable.)

The transmittance *T* was calculated as a function of frequency (wavelength) using:

**is the Poynting vector at the monitor and source locations,**

*P*^{m,s}*f*is the frequency and

*S*is the surface of the reference plane where the transmittance is computed [20]. Equation (1) was also used to compute the reflectance

*R*of the system by changing

*S*to the appropriate reference plane. The absorptance is then determined as

*A =*1

*-T-R*.

Throughout this paper the resonant wavelength (*λ _{res}*) refers to the free-space wavelength at which the transmittance curve reaches its minimum value. Reflectance resonance and absorptance resonance refer to the wavelengths at which reflectance and absorptance reach their minima, respectively (in general these three resonant wavelengths are different).

## 3. Parametric study of an array of antennas

A rigorous analysis of the design parameters of the two types of antennas is carried out by varying one design parameter at a time and monitoring the response of the system. Results are presented for monopole and dipole antennas in Sec. 3.1 and Sec. 3.2, respectively. These results will be useful as a guideline for design and to relate the antenna performance to its geometry. The minimum values of *g*, *t* and *w* reflect approximate limitations of an eventual fabrication process.

#### 3.1 Periodic array of monopoles

An array of monopoles (*g* = 0) with fixed pitch is a relatively simple, yet, effective resonant structure. Here, we investigate such an array by determining the influence of changing each design parameter (independently) on the system response, while keeping the other parameters fixed, including the pitch (*p* × *q*)which is maintained to 300 × 300 nm^{2}. We consider variations of length *l*, width *w* and thickness *t* of the monopoles.

### 3.1.1 Length (*l*)

Length is one of the main design parameters of antennas. As shown in Fig. 2 , increasing the length of a monopole shifts its transmittance, reflectance, and absorptance resonances to longer wavelengths. The red shift is expected by analogy to classical antennas, where resonance occurs when the antenna length is roughly half a wavelength. Increasing the length thus increases the wavelength at which the antenna is resonant. Increasing the length also decreases slightly the absorptance in the monopoles and broadens the absorptance response.

Field enhancements (not shown here) very much depend on the location along the monopoles; the only regions with field enhancement are the ends of the monopole. In contrast, as discussed in Sec. 3.2, the gap region of dipoles generates highly enhanced fields, making dipoles very sensitive to changes in the gap region.

### 3.1.2 Width (*w*)

Monopole width is another design parameter. Considering the system response to changes in width *w*, shown in Fig. 3
, we note that increasing the latter blue-shifts the transmittance, reflectance and absorptance resonances. We also note that the amount of shift decreases as Δ*w*/*w* decreases. The reasons for this behaviour will become clear in Sec. 4, where we examine the modal characteristics of the corresponding nanowire waveguides.

The absorptance level, as shown in Fig. 3(c), does not follow a linear trend with increasing monopole width. A maximum value of absorptance is evident, unlike the linearly decreasing trend that we observed as a result of increasing the length of the monopoles.

### 3.1.3 Thickness (*t*)

The response of the system to variations in thickness *t* is shown in Fig. 4
. As with the width, increasing the thickness causes a blue-shift in the resonant wavelengths. The absorptance peaks at *t* = 30 nm, while the amount of shift of the resonant wavelength decreases as Δ*t*/*t* decreases. This behavior will also be explained in Sec. 4 where we discuss the modal characteristics of the corresponding nanowire waveguides.

#### 3.2 Periodic array of dipoles

A region of highly localised, enhanced fields is one of the main attractions of dimer antennas, such as dipoles and bowties. Here we chose to study a periodic array of dipoles not only because of its similarities to an array of monopoles, but also for its advantage over monopoles, namely, having a gap region with highly concentrated fields, and its sharper wavelength response compared to bowties and monopoles. In this section we study the response of an array of dipoles to variations in individual dipole length, gap, width and thickness.

### 3.2.1 Length (*l*)

Figure 5
shows the response of a periodic array of dipoles to changes in the length from *l =* 190 to 280 nm in steps of 10 nm, while *w*, *t*, *g*, *p* and *q* remain fixed. As is evident from Fig. 5(a), increasing the length of the dipole red-shifts the resonant wavelength, decreases the amount of power transmitted at resonance, and increases the level of reflectance of the system, but the absorptance remains almost unchanged.

Elongating the dipole while keeping the pitch constant means increasing the proportion of gold surface area covering the silicon-water interface (and thus intercepting a greater fraction of the incident wave) resulting in more reflection and less transmission. The electric field enhancement on resonance is calculated at the center of the gap, 3 nm above the silicon-water interface (a representative location in the gap in H_{2}O) with respect to the electric field at the same location in the absence of the antenna, and is shown in Fig. 5(d). The electric fields in the absence of the antenna are computed at the *λ _{res}* of each corresponding dipole. The difference between the electric fields at different wavelengths in the absence of the dipole is, however, negligible, as expected. The field enhancement, although slightly decreasing as

*l*increases, is relatively constant over the range of lengths, which implies the electric field distribution in the gap does not change much by increasing the length of the dipoles. However, as the antenna length increases, the coupling between any two adjacent antennas (along the

*x*-axis) increases. This means higher field localization at antenna ends, which leads to slightly less field localization in the gap as the antenna length increases, explaining the trend of Fig. 5(d). This is also confirmed in Fig. 6 which shows the magnitude of the

*x*-component of the electric field along the antenna length, taken at the silicon-gold interface at

*y*= 0 at the resonant wavelength for each case. This figure clearly shows coupling between two neighboring dipoles for this pitch

*p*. As the length increases the distance between the ends of any two adjacent dipoles becomes smaller. When the dipole is long enough, such that the distance between two neighboring antennas is the same as the gap length, the electric fields at the gap are equal to those at the ends.

Figure 7
shows the electric field distribution of dipoles over the *x*-*y* cross-section close to the silicon-gold interface, slightly inside the gold. Figure 7(b) shows the field distribution on resonance at *λ _{res}* = 1495 nm, while Figs. 7(a) and 7(c) show the fields at the same wavelength for dipoles that are shorter and longer, respectively. The magnitude of the electric field is clearly enhanced on resonance.

### 3.2.2 Gap (*g*)

Figures 8(a)
–8(c) show the transmittance, reflectance and absorptance of the system as a function of wavelength for different antenna gap lengths *g*. While the transmittance increases with increasing gap length, the reflectance decreases, as the proportion of gold covering the surface becomes smaller. Increasing the gap of an array of dipoles moves the response from the limit *g* = 0, corresponding to an array of monopoles of length *l*, to the limit *g* = *p* - *l*, corresponding to an array of monopoles of length (*l* - *g*)/2. At this limit, the length of the antennas are reduced to less than a half of their original value, which according to classical antenna theory implies a blue-shift in the resonance. This is indeed observed in the results of Fig. 8. There is also a capacitance associated with the gap that explains in part the wavelength shift, as discussed in Sec 4.4. The field enhancements, shown in Fig. 8(d), are calculated as described in Sec. 3.2.1. A steep decrease in the field enhancement is evident as the gap increases, which is corroborated by the field distributions of Fig. 9
.

We note from Fig. 9 that in dipoles with a small gap, fields are appreciably larger in the gap than at the ends. However, as the gap gets larger the fields are almost equally distributed at both ends of a single arm of the dipole. Localized fields in the gap region of dipoles make small-gap antennas highly sensitive to changes in the gap region.

### 3.2.3 Width (*w*)

Figure 10
shows the transmittance, reflectance and absorptance of the array of dipoles as a function of wavelength for different antenna widths *w*. From Fig. 10(a) one can clearly see that increasing the width of the antenna from 4 to 60 nm blue-shifts the position of the resonance and lowers the level of transmittance on resonance. A similar shift in the absorptance peak is observed in Fig. 10(c). Figure 10(b), however, does not show any particular trend in the reflectance, since for *w* > 36 nm we cannot clearly identify the resonance peak. The amount of shift decreases as Δ*w*/*w* decreases. This property is explained in terms of the characteristics of the mode excited in the dipole arms, as will be discussed in Sec.4. Figure 10(d) shows field enhancements calculated at the center of the dipole gap, 3 nm above the silicon-water interface. One can clearly see that *w* = 20 nm gives the maximum enhancement of the electric field, while *w* = 4 nm yields the minimum enhancement.

From Fig. 11
, which shows the total electric field over the *x*-*y* cross-section of the antenna where the field enhancements are calculated, one can clearly see that at *w* = 20 nm the fields at the center of the gap reach their largest value, resulting in the strongest field enhancement. However, for very small and very large widths, although fields are strongly localized at the extremities, the non-uniform distribution of fields over the gap yields smaller field values at the center of the gap.

In Fig. 12
the total electric field is shown over a *y*-*z* cross-sectional plane taken at the middle of a dipole antenna arm. As the antenna gets wider, the field becomes less intense across the *y*-*z* plane, and less coupling occurs between the fields localized at the left and right edges. This establishes two separate localized field regions (Fig. 12(c)), as opposed to one high intensity region that exists in narrow-width dipoles (Fig. 12(a)). Clearly, the antenna fields are strongly dependent on *w*.

### 3.2.4 Thickness (*t*)

A similar study was performed to understand the effects of changing the thickness *t* of the dipole. Increasing the thickness causes a blue-shift in the resonances, as shown in Fig. 13
. The amount of shift decreases for decreasing Δ*t*/*t*. This property is explained in terms of the characteristics of the mode excited in dipole arms, as will be discussed in Sec. 4. Figure 13(d) shows field enhancements calculated at the center of the dipole gap, 3 nm above the silicon-water interface. One notes that the field enhancement does not depend strongly on thickness.

In Fig. 14
the total electric field is shown over a *y*-*z* cross-sectional plane taken at the middle of a dipole antenna arm. As the antenna gets thicker, the field becomes more localised near the silicon-water interface but does not change appreciably in character (compared to changes in width - Fig. 12).

### 3.2.5 Alignment of transmittance, reflectance and absorptance extrema

From Figs. 2–5, 8, 10 and 13, we note that the wavelength corresponding to the minimum of the transmittance curve does not lineup with extrema in the reflectance or the absorptance curves for a given array of dipoles. To determine the reason, the imaginary parts of the permittivity of gold and water were made zero, one at the time. Results are shown in Fig. 15 , where one could clearly see that the positions of the minima in the transmittance and reflectance curves are aligned if the permittivity of gold is purely real, which implies that the misalignment is due to the absorption of gold. As the imaginary part of the permittivity of gold is forced to zero, no energy is absorbed by the antennas, which makes the absorptance close to zero (small losses remain in water); thus the illuminating beam is partially transmitted and reflected. This also shows that most of the energy in the system is absorbed by the gold and not by the water. In fact, water absorption is negligible over most of the wavelength range shown (for the reference planes adopted). Figure 15 shows the transmittance, reflectance and absorptance curves on the same scale, which clearly demonstrates their relative values.

### 3.2.6 Full width at half maximum

In this section we consider the full-width-at-half-maximum (FWHM) of the absorptance response of the arrays. We determine the FWHM by finding the difference between the two wavelengths corresponding to the half value of the peak of each response curve. These results are shown on the left vertical axes (Δ*λ*) in Figs. 16(a)
–16(d) as a function of each design parameter. We convert the Δ*λ* values to the frequency domain using

*c*is the speed of light in free-space and

*ν*=

*ω*/(2π)is the frequency, and we plot this on the right axis of each figure.

Figure 16(a) shows an increasing FWHM as the length of the dipoles increases. This is caused by an increase in the loss of the antennas due to their longer length, resulting in broadening (see also Sec. 4.2). The same argument holds for the change in FWHM as a function of gap length: by increasing the length of the gap in a dipole of fixed length, we effectively make the dipole arms shorter, thus decreasing the loss and the FWHM, as shown in Fig. 16(b). The FWHM is shown in Figs. 16(c) and 16(d) as a function of dipole width and thickness. Here, as the length of the dipoles is fixed, the only contributing factor is the change in the attenuation of the mode resonating in the antenna - it increases as *λ* decreases (see Sec 4.2). Thus, by increasing the width and thickness of dipoles, their FWHM (Δ*ν*) decreases.

#### 3.3 Field decay from the monopole ends - effective length L_{eff}

Generally, resonance depends on a balance of stored electric and magnetic energies. Energy is stored not only over the physical length of a monopole but also in the fields decaying at its ends (as in microstrip resonators at microwave frequencies *viz*. the end fringing fields [24]). Thus, a monopole appears to have an effective length (*L _{eff}*) which is greater than its physical length. The longitudinal electric field component,

*E*, is used to measure the 1/

_{x}*e*field decay beyond the antenna ends. Figure 17 shows

*E*along the

_{x}*x*-axis at

*y =*0 for several heights (

*z*). The average of these decay lengths is denoted by

*δ*(also termed the length correction factor) and used to determine the effective monopole length

_{a}*L*as:

_{eff}We could also take the electric displacement *D _{x}* or the polarization density

*P*to measure the length correction factor. Not surprisingly, the measures yield essentially the same value regardless of their definition. The results show that the antenna fields (which can be thought of as equivalent current densities) penetrate the background a non-negligible distance (~20 nm) beyond the ends of the monopole. In Sec. 4, we propose an alternative method to estimate

_{x}*δ*, which does not require FDTD modelling.

_{a}## 4. Surface plasmon mode of the antennas

The antennas investigated in the previous section are formed from rectangular cross-section Au nanowires in a piecewise homogeneous background. It is known that a thin, wide metal stripe in a symmetric [25] or asymmetric [26] background supports several surface plasmon modes. The nanowire comprising the antennas is very similar in structure to the asymmetric stripe [26]. It is thus surmised that it operates (and resonates) in a surface plasmon mode of the nanowire, compatible with the geometry and the excitation scheme (an *x*-polarized plane wave.) In this section we identify the mode of operation of the antenna, we relate its propagation characteristics on the nanowire to the performance of the antennas, and we propose design models resting on modal results to predict the performance of the antennas. One must note that only the “bonding” (dipole) resonant mode can be excited in the antennas of this study given the symmetry of the excitation [27].

#### 4.1 Modal identification

The surface plasmon mode that is excited on the antennas must first be identified. To this end, we found the modes and their fields on a nanowire waveguide of the same cross-sectional configuration as one of the monopoles analyzed in Sec. 3.1.1 (*l* = 210 nm, *w =* 20 nm, *t =* 40 nm). The wavelength for the modal analysis was set to *λ* = *λ _{res}* = 2268 nm, which is the resonant wavelength of the aforementioned monopole. In order to remain consistent with the FDTD computations, the finite-difference mode solver in Lumerical was used and the same mesh as in the antenna cross-sectional plane (

*y*-

*z*plane, 0.5 nm mesh) was adopted. The same material properties for gold, silicon and water were retained.

Figures 18(a)
-18(b) show the real part of the transverse electric fields (which are at least 10 × larger than their corresponding imaginary parts) and Fig. 18(c) shows the imaginary part of the longitudinal electric field (which is significantly larger than its real part) of the nanowire mode of interest computed using the mode solver. These field components are compared to the corresponding electric field components distributed over a *y*-*z* cross-section taken near the center of the monopole antenna in Figs. 18(d)-18(f) computed using the FDTD method. A very close resemblance between all corresponding field distributions is apparent from the results, suggesting that the mode of operation is correctly identified and that the antenna operates in only one surface plasmon mode (monomode operation). Based on the distribution of *E _{z}* we identify the mode as the

*sa*mode [25,26]. The longitudinal (

_{b}^{0}*E*) field component of the monopole (Fig. 18(f)) has a large background level because it consists of the sum of the surface plasmon mode field and the incident (plane wave) field.

_{x}In general, the surface plasmon modes that are excited on an antenna depend on the polarisation and orientation of the source. Modes that share the same symmetry as the source, and that overlap spatially and in polarisation with the latter, can be excited.

#### 4.2 Effective index and attenuation

Now that we have identified the mode resonating in the monopoles, or each arm of the dipoles, we can evaluate the effective refractive index *n _{eff}* (real part of the complex effective index, Re{

*N*}) and the attenuation

_{eff}*α*(

*α*= 20log

_{10}(Im{

*N*}2π/λ)) of this mode as a function of the nanowire cross-section. For this purpose, the incident wavelength was fixed to

_{eff}*λ*= 1400 nm and

_{0}*w*and

*t*were changed, one at the time, to determine their influence on

*n*and

_{eff}*α*. From these results we then explain some of the trends observed in the parametric study of Sec. 3.

Figures 19(a) and 19(b) give the computed results. Evidently, the effective index and the attenuation decrease with increasing nanowire width and thickness. Therefore, increasing the width or thickness of an antenna, while keeping its length fixed, results in a blue-shift of its resonant wavelength, because:

This behaviour is observed in our FDTD computations in Figs. 3 and 10, and in Figs. 4 and 13. It is worth noting here that the rate of change of*n*and

_{eff}*α*decrease as

*w*and

*t*increase. Applying this observation to Eq. (4), we expect a smaller shift in the position of resonance as

*w*and

*t*increase. This is also observed in the trends of Figs. 3, 4, 10 and 13.

Next, *w* and *t* were fixed to representative values (*w* = 20 nm, *t* = 40 nm) and the incident wavelength was varied to determine its influence on *n _{eff}* and

*α*. As is observed from Figs. 20(a) and 20(b),

*n*and

_{eff}*α*decrease with increasing wavelength.

Returning to Figs. 16(c) and 16(d), for a fixed dipole arm length *d* = (*l* - *g*)/2, the FWHM decreases as *w* and *t* decrease because *α* decreases with the latter (recall that *λ _{res}* red-shifts with decreasing

*w*and

*t*- see Figs. 10 and 13 or the previous paragraph). However, if

*d*varies, one needs to consider the product

*dα*as the total loss. Given the slow rate of change of

*α*with wavelength (Fig. 20(b)), the total loss changes mostly with

*d*. Increasing the length of the dipole while the gap is fixed thus increases the FWHM, as shown in Fig. 16(a) (

*d*increases). However, increasing the gap while the antenna length is fixed decreases the FWHM, as shown Fig. 16(b) (

*d*decreases). These observations also explain the results of Fig. 8 (broader response for smaller gaps).

#### 4.3 Effective length of a monopole based on modal analysis

We wish to use the results of the modal analysis to estimate *δ _{a}* and the physical length

*l*of the monopole required for resonance at a desired

*λ*. An estimate, inspired from RF antenna theory, of the required physical length would be

_{res}*l = λ*, where

_{res}/2n_{eff}*n*is obtained from modal analysis at the desired

_{eff}*λ*. However, this is incorrect because we know from the parametric study (section 3) that the monopole operation is such that it appears longer than its physical length due to fields extending beyond its ends (Fig. 17),

_{res}*i.e.*,

*L*>

_{eff}*l*. We thus propose the following alternative relation:

The nanowire of Sec. 4.1 (*w* = 20 nm, *t* = 40) was analyzed at different wavelengths, corresponding to the resonance wavelengths *λ _{res}* of monopoles of length

*l*(Fig. 2) in a pitch large enough to eliminate the coupling effects between neighboring monopoles (

*p = q =*700 nm). (The values of

*n*were obtained from the data of Fig. 20 by interpolation at the required wavelengths). The decay of main transverse field component of the mode, |

_{eff}*E*|, away from the nanowire along the

_{z}*z*-axis was evaluated. In the positive

*z*-direction (into the H

_{2}O) |

*E*| falls to 1/

_{z}*e*of its value a distance

*δ*from the nanowire surface. In the negative

_{w}*z*-direction (into the Si) |

*E*| falls to 1/

_{z}*e*of its value a distance

*δ*from the nanowire surface. The decays thus obtained from |

_{s}*E*| of the mode along with

_{z}*n*are summarized in Table 1 . (In general

_{eff}*δ*≠

_{s}*δ*, in which case we use a weighted decay length correction factor

_{w}*δ*(1

_{m}=*-τ*)

*δ*+

_{w}*τδ*, where

_{s}*τ*/(1

*-τ*) is defined as the ratio of |

*E*| at the Au-Si interface to |

_{z}*E*| at the Au-H

_{z}_{2}O interface.)

Using *n _{eff}* and

*λ*for every value of physical length

_{res}*l*from Table 1 into Eq. (5) yields

*L*. One can then estimate the physical length of the monopole

_{eff}*l*by analogy with Eq. (3) as:

_{est}*δ*from Table 1, we find the estimated physical length of the monopoles

_{m}*l*as summarized in Table 2 . Our estimated lengths are within 20% of the physical length, which is acceptable considering the broad response of the monopoles. Part of this error may be caused by numerical inaccuracies due to the finite mesh size used in the FDTD and modal analyses.

_{est}#### 4.4 Transmission line model of dipoles

A transmission line model with a lumped element as shown in Fig. 21
is proposed to account for the effect of the gap on the position of the resonant wavelength of dipoles observed in Fig. 8. In this model the gap is represented by capacitor *C _{g}*, which is connected to two open-circuited transmission lines, modelling the two arms of a dipole.

*Z*

_{IN}^{(1)}and

*Z*

_{IN}^{(2)}are the input impedances looking from the terminal port in Fig. 21,

*Z*is the characteristic impedance of the transmission lines,

_{0}*β*is the propagation constant and

*d*+

*δ*is the length of each transmission line (

_{m}*d*= (

*l*-

*g*)/2).

*β*is taken as the phase constant of the mode resonating in the antenna (

*i.e.*, based on

*n*, as computed in Sec. 4.2 near the expected resonance wavelength):

_{eff}*C*is evaluated aswhere ${\epsilon}_{H2O}$ is the permittivity of water,

_{g}*A*=

_{d}*wt*is the cross-sectional area of the dipole arms, and

*g*is the length of the gap.

By Analogy to a lumped-element LC circuit, resonance occurs when the input impedances *Z _{IN}*

^{(1)}and

*Z*

_{IN}^{(2)}add to zero [28]. For our model we have:

*λ*(numerically) provides a good estimate to this wavelength without requiring time-consuming 3D FDTD modeling.

_{res}However, we must first determine the characteristic impedance *Z _{0}* to be used. We assume

*Z*to be equal to the wave impedance

_{0}*Z*of the mode, which is given in general as [29]:

_{w}**and**

*E***and are the modal electric and magnetic fields of the nanowire corresponding to the dipole of interest. Figure 22 shows the real part of**

*H**Z*over the

_{w}*y-z*cross-section (

*w =*20 nm,

*t =*40 nm) of the nanowire, computed using Eq. (10). The nanowire waveguide has an inhomogeneous cross-section, so the mode does not have a unique

*Z*. We therefore determine

_{w}*Z*as a weighted average over

_{0}*Z*:

_{w}*f*(

*y*,

*z*) is taken as the magnitude of the transverse modal electric fields:The integrals in Eq. (11) were taken as double sums over the nodes where the discretized fields and wave impedance are known. Note that the small region inside the metal where values of

*Z*become large (Fig. 22, z ~25 nm,

_{w}*y*= 0) is due to the denominator of Eq. (10) becoming close to zero -

*Z*is non-physical here so this region was removed from the averaging calculations.

_{w}Going back to Eq. (9), we now solve for *λ _{res}* and plot the results in Fig. 23
as a function of

*g*,

*w*and

*t*, (using modal computations for each case of

*w*and

*t*), as well as the values obtained from the FDTD analysis. The pitch was set to 700☓700 nm

^{2}to eliminate coupling effects between antennas in the FDTD analysis, thus making the results directly comparable to the results of modal analysis. Very good agreement is noted between Eq. (9) and the FDTD computations.

## 5. Concluding remarks

We performed a full parametric study of periodic plasmonic monopoles and dipoles in a piecewise homogeneous environment consisting of a silicon substrate and an aqueous cover. The study considered three system responses: transmittance, reflectance and absorptance. The responses were evaluated numerically and the results interpreted. Increasing the length red-shifts the resonance of monopoles and dipoles, whereas increasing the width, thickness and gap causes a blue-shift in their responses. We show that such trends are expected by identifying the surface plasmon mode that is excited in the antennas (the *sa _{b}^{0}* mode [25,26]) and computing its effective index and attenuation as a function of geometry and wavelength. The field enhancement (|

*E*|/|

_{x}*E*|) and FWHM of dipoles were also computed, yielding values of up to ~100 in

_{inc}*g*= 4 nm gaps, and 30 to 40 THz, respectively.

We proposed an expression resting on modal results (for the surface plasmon mode excited in the antennas) to predict the resonant length of a monopole given its cross sectional dimensions and the required resonant wavelength. The expression, which takes into account field extension beyond the antenna ends, estimates the physical length of monopoles to within ~20% when compared with the FDTD results. Finally, we proposed a simple equivalent circuit, also resting on modal results, but involving transmission lines and a capacitor which models the gap, to determine the resonant wavelength of dipoles. This circuit successfully estimates the resonant wavelength of a dipole to within ~10% when compared to the FDTD results. The expression and the equivalent circuit should prove useful as design guidelines for optical monopole and dipole antennas.

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