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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 16 — Jul. 30, 2012
  • pp: 18044–18065
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Periodic plasmonic nanoantennas in a piecewise homogeneous background

Saba Siadat Mousavi, Pierre Berini, and Derek McNamara  »View Author Affiliations


Optics Express, Vol. 20, Issue 16, pp. 18044-18065 (2012)
http://dx.doi.org/10.1364/OE.20.018044


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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 sab0 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

1. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011). [CrossRef]

,2

2. P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75(2), 024402 (2012). [CrossRef] [PubMed]

]. 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

3. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1(3), 438–483 (2009). [CrossRef]

].

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

4. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007). [CrossRef] [PubMed]

], where λ is the free-space incident wavelength. In [4

4. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007). [CrossRef] [PubMed]

] 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

5. W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220(1-3), 137–141 (2003). [CrossRef]

8

8. P. Biagioni, M. Savoini, J. Huang, L. Duó, M. Finazzi, and B. Hecht, “Near-field polarization shaping by a near-resonant plasmonic cross antenna,” Phys. Rev. B 80(15), 153409 (2009). [CrossRef]

]. 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

6. E. Cubukcu and F. Capasso, “Optical nanorods antennas as dispersive one-dimensional Fabry-Perot resonators for surface plasmons,” Appl. Phys. Lett. 95(20), 201101 (2009). [CrossRef]

]. In the case of dipoles and bowties the gap size and gap loading play an important role in determining the position of resonance [5

5. W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220(1-3), 137–141 (2003). [CrossRef]

12

12. A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008). [CrossRef] [PubMed]

]. 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

9. A. Alú and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008). [CrossRef]

, 12

12. A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008). [CrossRef] [PubMed]

]. 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

7. H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16(12), 9144–9154 (2008). [CrossRef] [PubMed]

]. High intensity fields in the dipole and bowtie gaps are strongly sensitive to the index of the material inside the gap [7

7. H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16(12), 9144–9154 (2008). [CrossRef] [PubMed]

, 9

9. A. Alú and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008). [CrossRef]

]. Also the effects of variations in the bow angle of a bowtie antenna on its spectral response have been investigated numerically and experimentally [7

7. H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16(12), 9144–9154 (2008). [CrossRef] [PubMed]

, 11

11. W. Ding, R. Bachelot, S. Kostcheev, P. Royer, and R. Espiau de Lamaestre, “Surface plasmon resonances in silver Bowtie nanoantennas with varied bow angles,” J. Appl. Phys. 108(12), 124314 (2010). [CrossRef]

], showing that the bow angle can be used as a tuning parameter. Aizpurua et al. [13

13. J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. Garcia de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005). [CrossRef]

] 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

14. W. Ding, R. Bachelot, R. Espiau de Lamaestre, D. Macias, A.-L. Baudrion, and P. Royer, “Understanding near/far-field engineering of optical dimer antennas through geometry modification,” Opt. Express 17(23), 21228–21239 (2009). [CrossRef] [PubMed]

17

17. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical Nanorod Antennas Modeled as Cavities for Dipolar Emitters: Evolution of Sub- and Super-Radiant Modes,” Nano Lett. 11(3), 1020–1024 (2011). [CrossRef] [PubMed]

]. Semi-analytical investigations have been done on nanodipole, nanobowtie, and hybrid dimer nanoantennas based on a microcavity model [14

14. W. Ding, R. Bachelot, R. Espiau de Lamaestre, D. Macias, A.-L. Baudrion, and P. Royer, “Understanding near/far-field engineering of optical dimer antennas through geometry modification,” Opt. Express 17(23), 21228–21239 (2009). [CrossRef] [PubMed]

], 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

18. P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant Optical Antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef] [PubMed]

], 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

3. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1(3), 438–483 (2009). [CrossRef]

], photovoltaics [3

3. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1(3), 438–483 (2009). [CrossRef]

], and biosensing [19

19. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009). [CrossRef]

]. With a growing range of applications, developing precise, yet practical design rules for nanoantennas seems essential.

2. Geometry and methods

Figure 1
Fig. 1 Geometry of a unit cell of the system under study: an Au rectangular dipole antenna on a silicon substrate covered by water. An x-polarized plane wave source illuminates the antenna in the z-direction from the substrate.
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

20. FDTD Solutions v. 7.5.6, Lumerical Solutions Inc., Vancouver, Canada.

], with a 0.5 × 0.5 × 0.5 nm3 mesh in the region around the antenna, was used for all simulations. Palik’s material data [21

21. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

] were used for gold and silicon, and Segelstein’s data [22

22. D. J. Segelstein, The complex refractive index of water, M.Sc. Thesis, University of Missouri – Kansas City, 1981.

] 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 nm3 cubic mesh to a 0.25 × 0.25 × 0.25 nm3 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

23. R. C. Boonton, Jr., Computational Methods for Electromagnetics and Microwaves (Wiley-Interscience, 1992)

]. Comparing this extrapolated wavelength to the wavelength obtained for a finite mesh of 0.5 × 0.5 × 0.5 nm3 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:
T(f)=SRe{Pm(f)}ds/SRe{Ps(f)}ds
(1)
where Pm,s is the Poynting vector at the monitor and source locations, f is the frequency and S is the surface of the reference plane where the transmittance is computed [20

20. FDTD Solutions v. 7.5.6, Lumerical Solutions Inc., Vancouver, Canada.

]. 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 (λres) 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 nm2. 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
Fig. 2 (a) Transmittance, (b) reflectance and (c) absorptance versus wavelength for monopoles of w = 20 nm, t = 40 nm and variable l (given in legend inset to (a)).
, 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)

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)

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
Fig. 5 (a) Transmittance, (b) reflectance and (c) absorptance versus wavelength for a dipole of g = 20 nm, w = 10 nm, t = 40 nm, p = q = 300 nm and variable l (given in legend inset to (a)). Part (d) shows |Ex|/|Einc| where Ex is taken at x = 0, y = 0, z = 3 nm on resonance and Einc is the incident field at the same location and wavelength in the absence of the antenna.
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 H2O) 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 λres 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
Fig. 6 |Ex| along the length of the antenna for different l values (given in legend in inset); |Ex| is also shown in the absence of the antenna.
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
Fig. 7 |E|=|Ex|2+|Ey|2+|Ez|2on the x-y plane 3 nm inside gold dipoles of dimensions (a) l = 190 nm, (b) l = 240 nm, (c) l = 280 nm, and w = 20 nm, t = 40 nm, p = q = 300 nm, g = 20 nm at λ = 1495 nm, which is the resonance wavelength of case (b).
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 λres = 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)
Fig. 8 (a) Transmittance, (b) reflectance and (c) absorptance versus wavelength for dipole antennas of l = 210 nm, w = 20 nm, t = 40 nm, p = q = 300 nm and variable g (given in legend inset to (a)). Part (d) plots |Ex|/|Einc| where Ex is taken at x = 0, y = 0, z = 3 nm on resonance and Einc is the incident field at the same location and wavelength but with the antenna removed.
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
Fig. 9 |E|=|Ex|2+|Ey|2+|Ez|2on the x-y plane 3 nm inside a gold dipole having l = 210 nm, w = 20 nm, t = 40 nm and p = q = 300 nm for (a) g = 4 nm, (b) g = 20 nm and (c) g = 44 nm.
.

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
Fig. 10 (a) Transmittance, (b) reflectance and (c) absorptance versus wavelength for dipole antennas of l = 210 nm, t = 40 nm, g = 20 nm, p = q = 300 nm and variable w (given in legend inset to (a)). Part (d) plots |Ex|/|Einc|, where Ex is taken at x = 0, y = 0, z = 3 nm on resonance and Einc is the incident field at the same location and wavelength in the absence of the antenna.
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 Δw/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
Fig. 11 |E|=|Ex|2+|Ey|2+|Ez|2on the x-y plane 3 nm inside a gold dipole having l = 210 nm, g = 20 nm, t = 40 nm and p = q = 300 nm for (a) w = 4 nm, (b) w = 20 nm and (c) w = 36 nm.
, 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
Fig. 12 λres=2Leffneffon the y-z plane at the middle of a gold dipole arm having l = 210 nm, g = 20nm, t = 40 nm and p = q = 300 nm for (a) w = 4 nm, (b) w = 20 nm and (c) w = 60 nm.
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
Fig. 13 (a) Transmittance, (b) reflectance, and (c) absorptance versus wavelength for dipole antennas of l = 210 nm, w = 20 nm, g = 20 nm, p = q = 300 nm and variable t (given in legend inset to (a)). Part (d) plots |Ex|/|Einc|, where Ex is taken at x = 0, y = 0, z = 3 nm on resonance and Einc is the incident field at the same location and wavelength in the absence of the antenna.
. The amount of shift decreases for decreasing Δt/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
Fig. 14 |E|=|Ex|2+|Ey|2+|Ez|2on the y-z plane at the middle of a gold dipole arm having l = 210 nm, g = 20 nm, w = 20 nm and p = q = 300 nm for (a) t = 30 nm, (b) t = 50 nm and (c) t = 70 nm.
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. 25, 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
Fig. 15 Transmittance, reflectance and absorptance of an array of dipoles, using (a) the full material properties of silicon, gold and water, and (b) forcing the imaginary part of the permittivity of gold to zero.
, 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 (Δλ) in Figs. 16(a)
Fig. 16 FWHM of the absorptance response as a function of (a) length l, (b) gap g, (c) width w, and (d) thickness t. The left axis shows the FWHM calculated as Δλ and the right axis shows the corresponding Δυ.
16(d) as a function of each design parameter. We convert the Δλ values to the frequency domain using
Δv=cλres2Δλ
(2)
where 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 (Δν) decreases.

3.3 Field decay from the monopole ends - effective length Leff

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

24. T. Itoh, “Analysis of microstrip resonators,” IEEE Trans. Microw. Theory Tech. 22(11), 946–952 (1974). [CrossRef]

]). Thus, a monopole appears to have an effective length (Leff) which is greater than its physical length. The longitudinal electric field component, Ex, is used to measure the 1/e field decay beyond the antenna ends. Figure 17
Fig. 17 Electric field Ex along the x-axis of a monopole of l = 210 nm, w = 20 nm, t = 40 nm and p = q = 300 nm, at y = 0 for several heights z (indicated in the legend). The average of Ex at the z locations of this figure (and for z = 15 and 25 nm) is also shown. The inset shows Ex at the physical end of the antenna, where the field is discontinuous.
shows Ex along the x-axis at y = 0 for several heights (z). The average of these decay lengths is denoted by δa (also termed the length correction factor) and used to determine the effective monopole length Leff as:

Leff=l+2δa
(3)

We could also take the electric displacement Dx or the polarization density Px 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 δa, which does not require FDTD modelling.

4. Surface plasmon mode of the antennas

4.1 Modal identification

Figures 18(a)
Fig. 18 (a)-(c) Electric field distribution of a surface plasmon mode plotted over the cross-section of a nanowire waveguide (w = 20 nm, t = 40 nm and λres = 2268 nm) computed using a mode solver. (d)-(f) Electric field distribution over a cross-section of the corresponding monopole antenna computed using the FDTD. (a) and (d) Re{Ey}, (b) and (e) Re{Ez}, (c) and (f) Im{Ex}.
-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 Ez we identify the mode as the sab0 mode [25

25. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

,26

26. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001). [CrossRef]

]. The longitudinal (Ex) 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.

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

4.3 Effective length of a monopole based on modal analysis

λres=2Leffneff
(5)

4.4 Transmission line model of dipoles

A transmission line model with a lumped element as shown in Fig. 21
Fig. 21 Transmission line model of the dipole where the gap is modeled as a parallel-plate capacitor.
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 Cg, which is connected to two open-circuited transmission lines, modelling the two arms of a dipole. ZIN(1) and ZIN(2) are the input impedances looking from the terminal port in Fig. 21, Z0 is the characteristic impedance of the transmission lines, β is the propagation constant and d + δm is the length of each transmission line (d = (l - g)/2). β is taken as the phase constant of the mode resonating in the antenna (i.e., based on neff, as computed in Sec. 4.2 near the expected resonance wavelength):
β=neffωε0μ0
Using a simple parallel-plate model, Cg is evaluated as
Cg=εH2OAd/g
(7)
where εH2O is the permittivity of water, Ad = 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 ZIN(1) and ZIN(2) add to zero [28

28. D. M. Pozar, Microwave Engineering (Wiley, 2005).

]. For our model we have:
ZIN(1)=jZ0cot(β(d+δm)) (8.a)
ZIN(2)=jωCgjZ0cot(β(d+δm)) (8.b)
which yields the following expression on resonance:
tan(neffωresε0μ0(d+δm))=2ωresCgZ0
(9)
where:
ωres=2πc/λres
Solving Eq. (9) for λres (numerically) provides a good estimate to this wavelength without requiring time-consuming 3D FDTD modeling.

5. Concluding remarks

References and links

1.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011). [CrossRef]

2.

P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75(2), 024402 (2012). [CrossRef] [PubMed]

3.

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1(3), 438–483 (2009). [CrossRef]

4.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007). [CrossRef] [PubMed]

5.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220(1-3), 137–141 (2003). [CrossRef]

6.

E. Cubukcu and F. Capasso, “Optical nanorods antennas as dispersive one-dimensional Fabry-Perot resonators for surface plasmons,” Appl. Phys. Lett. 95(20), 201101 (2009). [CrossRef]

7.

H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16(12), 9144–9154 (2008). [CrossRef] [PubMed]

8.

P. Biagioni, M. Savoini, J. Huang, L. Duó, M. Finazzi, and B. Hecht, “Near-field polarization shaping by a near-resonant plasmonic cross antenna,” Phys. Rev. B 80(15), 153409 (2009). [CrossRef]

9.

A. Alú and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008). [CrossRef]

10.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. L. Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photonics 2(4), 226–229 (2008). [CrossRef]

11.

W. Ding, R. Bachelot, S. Kostcheev, P. Royer, and R. Espiau de Lamaestre, “Surface plasmon resonances in silver Bowtie nanoantennas with varied bow angles,” J. Appl. Phys. 108(12), 124314 (2010). [CrossRef]

12.

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008). [CrossRef] [PubMed]

13.

J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. Garcia de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005). [CrossRef]

14.

W. Ding, R. Bachelot, R. Espiau de Lamaestre, D. Macias, A.-L. Baudrion, and P. Royer, “Understanding near/far-field engineering of optical dimer antennas through geometry modification,” Opt. Express 17(23), 21228–21239 (2009). [CrossRef] [PubMed]

15.

E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express 16(21), 16529–16537 (2008). [CrossRef] [PubMed]

16.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Perot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9(6), 2372–2377 (2009). [CrossRef] [PubMed]

17.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical Nanorod Antennas Modeled as Cavities for Dipolar Emitters: Evolution of Sub- and Super-Radiant Modes,” Nano Lett. 11(3), 1020–1024 (2011). [CrossRef] [PubMed]

18.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant Optical Antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef] [PubMed]

19.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009). [CrossRef]

20.

FDTD Solutions v. 7.5.6, Lumerical Solutions Inc., Vancouver, Canada.

21.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

22.

D. J. Segelstein, The complex refractive index of water, M.Sc. Thesis, University of Missouri – Kansas City, 1981.

23.

R. C. Boonton, Jr., Computational Methods for Electromagnetics and Microwaves (Wiley-Interscience, 1992)

24.

T. Itoh, “Analysis of microstrip resonators,” IEEE Trans. Microw. Theory Tech. 22(11), 946–952 (1974). [CrossRef]

25.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

26.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001). [CrossRef]

27.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. Stockman, “Plasmon Hybridization in Nanoparticle Dimers,” Nano Lett. 4(5), 899–903 (2004). [CrossRef]

28.

D. M. Pozar, Microwave Engineering (Wiley, 2005).

29.

H. G. Booker, Electromagnetism (Peter Peregrinus, 1982).

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(290.5850) Scattering : Scattering, particles
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 1, 2012
Revised Manuscript: July 10, 2012
Manuscript Accepted: July 11, 2012
Published: July 23, 2012

Citation
Saba Siadat Mousavi, Pierre Berini, and Derek McNamara, "Periodic plasmonic nanoantennas in a piecewise homogeneous background," Opt. Express 20, 18044-18065 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18044


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References

  1. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011). [CrossRef]
  2. P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75(2), 024402 (2012). [CrossRef] [PubMed]
  3. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1(3), 438–483 (2009). [CrossRef]
  4. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007). [CrossRef] [PubMed]
  5. W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220(1-3), 137–141 (2003). [CrossRef]
  6. E. Cubukcu and F. Capasso, “Optical nanorods antennas as dispersive one-dimensional Fabry-Perot resonators for surface plasmons,” Appl. Phys. Lett. 95(20), 201101 (2009). [CrossRef]
  7. H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16(12), 9144–9154 (2008). [CrossRef] [PubMed]
  8. P. Biagioni, M. Savoini, J. Huang, L. Duó, M. Finazzi, and B. Hecht, “Near-field polarization shaping by a near-resonant plasmonic cross antenna,” Phys. Rev. B 80(15), 153409 (2009). [CrossRef]
  9. A. Alú and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008). [CrossRef]
  10. L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. L. Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photonics 2(4), 226–229 (2008). [CrossRef]
  11. W. Ding, R. Bachelot, S. Kostcheev, P. Royer, and R. Espiau de Lamaestre, “Surface plasmon resonances in silver Bowtie nanoantennas with varied bow angles,” J. Appl. Phys. 108(12), 124314 (2010). [CrossRef]
  12. A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008). [CrossRef] [PubMed]
  13. J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. Garcia de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005). [CrossRef]
  14. W. Ding, R. Bachelot, R. Espiau de Lamaestre, D. Macias, A.-L. Baudrion, and P. Royer, “Understanding near/far-field engineering of optical dimer antennas through geometry modification,” Opt. Express 17(23), 21228–21239 (2009). [CrossRef] [PubMed]
  15. E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express 16(21), 16529–16537 (2008). [CrossRef] [PubMed]
  16. J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Perot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9(6), 2372–2377 (2009). [CrossRef] [PubMed]
  17. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical Nanorod Antennas Modeled as Cavities for Dipolar Emitters: Evolution of Sub- and Super-Radiant Modes,” Nano Lett. 11(3), 1020–1024 (2011). [CrossRef] [PubMed]
  18. P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant Optical Antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef] [PubMed]
  19. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009). [CrossRef]
  20. FDTD Solutions v. 7.5.6, Lumerical Solutions Inc., Vancouver, Canada.
  21. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
  22. D. J. Segelstein, The complex refractive index of water, M.Sc. Thesis, University of Missouri – Kansas City, 1981.
  23. R. C. Boonton, Jr., Computational Methods for Electromagnetics and Microwaves (Wiley-Interscience, 1992)
  24. T. Itoh, “Analysis of microstrip resonators,” IEEE Trans. Microw. Theory Tech. 22(11), 946–952 (1974). [CrossRef]
  25. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]
  26. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001). [CrossRef]
  27. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. Stockman, “Plasmon Hybridization in Nanoparticle Dimers,” Nano Lett. 4(5), 899–903 (2004). [CrossRef]
  28. D. M. Pozar, Microwave Engineering (Wiley, 2005).
  29. H. G. Booker, Electromagnetism (Peter Peregrinus, 1982).

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