We report on the polarization-dependent optical response of elongated nanoholes in optically thin gold films. We measured elastic scattering spectra of spatially isolated ellipsoidal nanoholes with varying aspect ratio and compared the results to electrodynamic simulations. Both experiments and theory show that the plasmon mode that is polarized parallel to the short axis of the ellipsoidal hole red-shifts with increasing aspect ratio. This behavior is completely opposite to the case of elongated metal particles. We present a simple analytical model that qualitatively explains the observations in terms of the different orientations of the induced dipole moments in holes and particles.
©2008 Optical Society of America
Nanoholes in metal films have been the subject of intense research since the discovery of extraordinary light transmission, an effect that arises due to a combination of surface plasmon polariton (SPP) generation and diffraction in periodic 2D hole arrays in optically thick metal films . However, as recently reported, the shape of the nanoholes also plays an important role in the extraordinary transmission phenomenon, and substantial polarization-dependent effects have been observed for 2D arrays of elongated holes [2–5]. In particular, a large increase in transmission has been shown to occur for incident polarization parallel to the short axis of elongated nanoholes. In some cases, this effect was accompanied by a red-shift of the transmission peaks . Similar effects were observed even for single nanoholes in thick metal films [6–8].
Whereas the optical properties of hole arrays and isolated holes in optically thick metal films can be well understood from applying arguments based on Babinet’s principle on sub-wavelength apertures in perfect conductor screens , the optical properties of nanoholes in optically thin metal films are less clear. A more accurate conformity with the Babinet’s principle is expected for thicker metal films and holes with resonances at much longer wavelengths, where the behavior of the metal is closer to that of a perfect conductor. Nanoholes in optically thin films interact via so-called anti-symmetric or “slow” SPPs, characterized by a symmetric charge distribution across the film, very short wavelength and a short propagation distance. If the edge-to-edge distance between holes ordered in pairs or short linear chains is tuned to half the SPP wavelength, a strong constructive SPP-mediated hole interaction occurs . This results in a more than ten-fold increase of the scattering efficiency per hole and a pronounced antenna-like emission and excitation pattern that may be utilized in novel nanophotonics architectures and applications . However, the interaction between nanoholes is also determined by the intrinsic optical properties of each individual hole. It has previously been shown that circular isolated holes in optically thin gold films exhibit a distinct resonance that red-shifts as the size of the hole increases [11–13]. This excitation can be attributed to a localized surface plasmon resonance (LSPR) of electric dipole character . An analogous size dependence and interpretation is well established also for the dipolar LSP in circular metal nanodisks [14, 15]. However, while studying the optical properties of elongated nanoholes in thin Au films, we found that a change in the hole aspect ratio produced spectral changes that are opposite to those found for metal particles. We discuss this effect, which is reproduced in electrodynamics simulations based on the Green’s dyadic method, based on a quasi-static model and argue that it originates in the opposite orientation of the induced electric dipole moment in holes vs. particles.
2. Sample fabrication & Optical characterization
Single nanoholes of various aspect ratios were produced in 20-nm gold films using electron-beam lithography (EBL). Negative resist (Shipley SAL601-SR7), diluted 1:1 in Microposit Thinner P, was spun onto a glass substrate at 6000 rpm and baked for 2 min at 90°C. A 10-nm gold film (the mask) was resistively evaporated to inhibit the charging of the glass surface during the exposure and the sample was then exposed to the electron beam (JEOL JBX9300FS). After the exposure, the mask was removed using Au etchant solution (4g KI, 1g I2, 200ml H2O) and the resist was developed in Shipley MF322 for 2 min. Finally, a 20-nm gold film was deposited using an electron-beam evaporator (AVAC HVC600) and a lift-off of the remaining resist was performed by tape-stripping. The short axes a of the elongated holes were kept constant at ca. 80 nm, while their long axes b were set to 80, 120 or 160 nm (see Fig. 1(a)).
Samples were characterized by scanning electron microscopy (SEM, FEI Quanta 200 FEG ESEM). The dimensions of the nanoholes in the SEM images were measured using ImageJ v1.36 software. The accuracy of these measurements is on the order of 10 nm, defined by the resolution of the electron beam and imaging parameters.
Elastic scattering spectra were measured using an inverted dark-field scattering microscope (Nikon TE300) fibre-coupled to a single-grating spectrometer (Shamrock SR- 303i, Andor) equipped with a CCD detector (iDus, Andor). The polarization state and direction of the incident light relative to the axes of the elongated holes were controlled in the same way as described in Ref. .
3. Experimental results
Scattering spectra of circular as well as elongated nanoholes for the two orthogonal polarization directions are shown in Figs. 1(b) and 1(c). When the incident light is polarized parallel to the short axis a, prominent spectral changes are observed for different b. The scattering intensity increases dramatically with b and this effect is accompanied by a substantial peak red-shift – more than 250 nm for b=160 nm compared to b=80 nm (Fig. 1(b)). In contrast, the intensity decreases slightly and the peak weakly blue-shifts for spectra recorded with incident polarization parallel to b. This variation is opposite to that well known for elongated metal nanoparticles, such as nanorods, which would have shown a strong redshift and intensity increase for the LSPR polarized parallel to the long axis instead of the short axis. In addition, unlike metal nanoparticles, the scattering intensity does not scale with the squared volume of the nanostructure, as can be deduced from Fig. 1, where the scattering intensity of the b=160 nm hole normalized by the squared volume is approximately 3 times larger than that of the circular a=b=80 nm hole.
4. Green’s tensor calculations
We compared the experimental results to theoretical scattering cross-sections of elongated nanoholes in a 20-nm gold film obtained using the Green’s tensor (GT) technique [16–18]. In this method, we consider a field E 0(r) incident from air on a gold film on top of a glass substrate. The hole in the gold film is modeled as a cylindrical void. The total electric field (incident field plus scattered field) satisfies the wave equation:
where k0 is the vacuum wave number, Δε(r)=ε(r)-εB(r) is the contrast between the object permittivity, ε(r), and that of the layered background, εB(r). Equation (1) has a Lippman- Schwinger type solution
where G(r,r’) is the Green’s function corresponding to the layered background, which satisfies:
and V runs over the volume of the hole. To solve Eq. (2), the volume V divided into small mesh elements, so that we obtain a system of equations from which we can solve for the electric field inside each of the elements. In order to increase the speed of the calculations, we utilized the top-down extended meshing algorithm (TEMA) .
Once the electric field is known in each mesh element, the electric field can be calculated anywhere in space. In particular, to calculate scattering cross-sections, we need the electric field in the far-field zone. To this end, we utilize the stationary phase method  and write the Green’s function on the asymptotic form
where R is the distance between the field point and the origin and k the wavevector in the medium. If one inserts this into Eq. (2), the scattered field is obtained as
If we label the air/gold/glass layers as layer 1/2/3, respectively, the Poynting vector in layer i reads:
where ni is the refractive index of layer i, c is the speed of light in vacuum and ε 0 the dielectric permittivity of vacuum. Thus, we get that the differential cross-section, or the power scattered into a solid angle dΩ at a distance R divided by the incoming power, is
Figure 2 shows the calculated differential forward scattering cross section vs. wavelength for elongated holes of the same shape and size as in the experiments (in 20 nm gold films on glass in air), assuming the gold dielectric constants of Ref. . The calculations are made for normal incidence, which simplifies comparison with published theoretical and experimental data for plasmonic nanoparticles (for which the forward scattering corresponds to extinction, according to the optical theorem). Although the experimental hole data is based on a dark-field illumination configuration, the trend in the calculated spectra with increasing b is clearly very similar to the experimental results shown in Fig. 1. Specifically, the calculations reproduce the enormous increase of the scattering cross section and the strong red-shift for polarization parallel to the short axis (Fig. 2(a)) while only weak variations are observed for the perpendicular polarization direction (Fig. 2(b)). This confirms the experimental observation of an essentially opposite shape-dependent LSPR behavior compared to elongated metal nanoparticles. However, when the b/a aspect ratio is kept constant and only the absolute size of the hole is changed, the trend in the LSPR position for both axes is the same as that of circular holes, i.e. the resonances red-shift with increasing dimension . This is illustrated in Fig. 2(c), which shows calculated forward scattering spectra for an elongated hole with the same aspect ratio (b/a=1.5) as the b=120 nm hole in Fig. 2(a) and 2(b), but with 25% shorter axes and, therefore, clearly blue-shifted resonance positions.
5. Quasi-static model
The polarization-dependent optical response of the elongated nanoholes can be discussed with the help of a quasi-static model for an ellipsoidal dielectric void embedded in a metallic medium. In the Rayleigh regime, the polarizability of an ellipsoidal structure (nanoparticle or nanovoid) in the i direction is proportional to :
where a, b and c are the three axes of the ellipsoid, and si=a,b,c.
We first analyze the case of an ellipsoidal metal particle embedded in a dielectric medium, in which εint=εm and εext=εd, being εm and εd the dielectric constants of the metal and the dielectric, respectively. The LSPR is induced when the real part of the denominator in Eq. (8) is zero, i.e.:
For a spherical particle Li=1/3, and the resonance condition is given by Re[εm]=-2εd. If the particle is elongated, however, the longer axis is characterized by Li<1/3 and the resonance condition is satisfied at more negative values of εm(Re[εm]<-2εd). This can only take place if the LSPR is red-shifted with respect to that of a spherical particle (see Fig. 3(a)). In contrast, the shorter axis of the ellipsoid corresponds to Li>1/3, and the corresponding resonance is therefore blue-shifted.
The situation is different in case of an ellipsoidal dielectric void embedded in a metal medium. The resonance condition is then defined by the following expression:
For a spherical void, the resonance is given by Re[εm]=-εd/2. Unlike elongated nanoparticles, the resonance in the direction of the short axis is induced when Re[εm]<-εd/2 and, consequently, is red-shifted with respect to the spherical case. Conversely, the long axis b corresponds to a slightly blue-shifted resonance, i.e., Re[εm]>-εd/2.
However, a nanohole in a thin metal film represents a special case of a void for two reasons: (1) the dipolar excitation of the hole can decay non-radiatively to SPPs, giving an additional LSPR broadening and peak-shift; (2) the hole cannot be polarized in the z-direction (See Fig. 3). The latter implies that a nanohole can be considered as an ellipsoidal void with a depolarization factor in the direction perpendicular to the film (Lz) equal to zero. The required condition Lx+Ly+Lz=1 yields a value of the depolarization factor larger than 1/2 in the direction of the short axis. Therefore, according to Eq. (11), the resonance condition is fulfilled for Re[εm] considerably more negative than -εd/2, which accounts for the strong red-shift of the scattering peaks of the elongated holes when the polarization is parallel to the short axis.
As discussed above, a rather simple analytical model can capture the opposite behavior of the LSPR peak position of elongated nanoholes and nanoparticles. To discuss the shape dependence of the scattering intensity in nanoholes, and to compare it with elongated nanoparticles, we visualize how the induced surface charges are distributed in the two cases of Fig. 4. An external electric field E0 separates positive and negative charges in the metal in such a way that the induced dipoles in holes and particles differ in orientation, as is seen comparing Figs. 4(a) and 4(c) (or 4(b) and 4(d)). In drawing these figures we have assumed that the frequency of the external field is smaller than any resonance frequency of the particles and holes. As can be deduced from in Eq. (8), the metal nanostructures (nanoparticles or nanovoids) suffer a change of sign in their electric dipole moment when the incident wavelength crosses the resonance wavelength, which is symbolized by the change of sign in the denominator of Eq. (8) (while the sign of the numerator keeps constant for all the wavelengths).
The relative strength of the scattered light in the different situations depicted in Fig. 4 can be understood by considering how the electric fields from the induced charges (Eind) act together with the external field. Inside the nanostructures such induced fields are proportional to the dipole moment of the nanostructure, i.e, Eind ∞ -p=- αE 0. Since nanoholes and nanoparticles show the opposite dipole moment orientation (opposite sign in the numerator of Eq. (8)), the induced fields will exhibit opposite signs as well. Thus, in the case of the nanohole in Fig. 4(a), Eind reinforces the external field: the negative charges at the lower side of the hole draws in more positive charges to the upper side, and vice versa. The net dipole moment, and hence the scattering intensity, is relatively stronger than it would be for a circular hole because the hole is elongated and the total charge drawn to the edges is larger. If instead the hole is illuminated by light polarized along the major axis, as in Fig. 4(b), the larger separation distance between induced charges leads to a smaller enhancement effect (weaker Eind) and, consequently, smaller scattering signal. Figures 4(c) and 4(d) illustrate and explain the opposite trend in the scattering intensity for an elongated particle. Here the positive and negative charges on opposite sides of the particle attract each other and thereby counteract, rather than reinforce, the external field. This effect is stronger when the field points along the minor axis of the particle (Fig. 4(c)) because the distance separating the charges is smaller and therefore Eind is stronger. As a result, in this case one finds the strongest scattering signal when the electric field is aligned with the major axis of the particle.
One may note that a quasi-static model can also be used to explain the spectral behavior of a pair of coupled nanoholes at very short separation distances [10, 12], where a substantial red-shift is observed if the polarization is perpendicular to the axis of the pair. This is because the external electric field and the induced field originating from the other hole add up and reinforce each other for low frequencies in this case. If the polarization is parallel to the pair axis, the resonance instead blue-shifts, and the scattering intensity decreases. The behavior is thus analogous to the case of elongated holes and we may, qualitatively, view the elongated hole as two merged interacting circular holes.
We have investigated the optical properties of elongated nanoholes in optically thin gold films. We found that the polarization-dependent optical response, characterized by the localized surface plasmon resonance peak positions and intensities, is reversed with respect to that of ellipsoidal metal nanoparticles of the same dimensions. The experimental results have been corroborated by rigorous electrodynamics simulations based on the Green’s tensor method and explained using a simple quasi-static model based on an ellipsoidal void embedded in a metallic medium. We believe that these results and understandings provide a firmer basis for the optimization and control of plasmonic nanostructures based on holes, which are highly interesting for biosensing [25, 26] and a range of other nanophotonic applications.
Financial support from the Swedish Research Council and the Swedish Foundation for Strategic Research are gratefully acknowledged.
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