1. Introduction

Radio frequency (RF) antennas as well as nano-antennas are devices that improve the light-matter interaction of a sub-wavelength oscillator, such as a high-frequency circuit in the RF case or a quantum dot at frequencies of infrared and visible light, by increasing the absorption cross-section and impedance-matching the oscillator to free-space [1–3]. In addition, when the oscillator’s modes become quantized and the size of the antenna becomes sufficiently small to give rise to localized surface plasmon resonances (LSPR), the local density of electromagnetic states in the vicinity of the antenna changes [4] and can alter the radiation efficiency and relaxation rate of the oscillator [5]. For resonant cavities, this is known as the Purcell effect [6], where Purcell originally suggested the use of small metallic particles mixed with a nuclear-magnetic medium to achieve a significant rate enhancement [7]. Theoretical analysis has shown, however, that the Purcell factor derived for resonant optical cavities underestimates the achievable Purcell factors with nano-antennas [8]. This has spurred efforts in recent years to investigate antennas for ever shorter wavelengths and for applications such as energy harvesting [9], photo-detection [10] or higher harmonics generation [11].

To determine the LSPR frequency, where the highest rate enhancement is expected, one can measure the extinction cross-section. Typically, the extinction cross-section is measured by using a spectrometer attached to a bright- or dark-field microscope [12, 13], thus making it difficult to quantify the extinction cross-section. A further issue with this type of setup is the difficulty to characterize single nano-structures as the extinction signal can be very small compared to the incident intensity, intensity drift and noise of focused thermal light sources. In order to rectify this shortcoming and measure the extinction cross-section of few and single nanoparticles, various techniques that rely on thermal modulation of the host material’s refractive index [14], spatial modulation [15] combined with a common-path interferometer [16] and difference interference contrast microscopy [17] (amongst others) have been used. Another way to characterize linearly polarizing nano-antennas, is to switch polarization angles from a parallel to an orthogonal configuration in relation to the sample’s polarization and to measure the resulting difference in transmission. To this end, one could use a liquid crystal variable wave plate to modulate the polarization angle [18]. While modulating the polarization, however, this would also result in a simultaneous modulation of the beam intensity which could potentially mask the antenna’s signal.

We therefore propose to modulate the polarization angle of the incident wavelength-tunable monochromatic beam with a mechanically rotating λ/2 plate (RHWP) while optically chopping the beam for an intensity reference. Rotating the HWP at ω₀ modulates the polarization at ω₁ = 4ω₀, which has the benefit of separating mechanical disturbances at the rotation frequency ω₀ and unwanted intensity modulation caused by the wave plate itself at 2ω₀ from the intensity modulation caused by the antenna in the rotating polarization of the incident beam at ω₁. This makes it possible to measure very small values of the extinction cross-section down to single nano-antennas. In this article we will model a rotating polarization single nano-antenna spectroscopy setup as represented in Fig. 1(a) and demonstrate the measurement of few and single dipole nano-antennas as illustrated in Figs. 1(c) and 1(d).

 

Fig. 1 (a) shows the modulation of our Ti:Sapphire laser at frequency ω₂ by an optical chopper and rotation of the polarization at frequency ω₁. The λ/2 plate (HWP) is rotating at ω₀ and modulates the polarization at ω₁ = 4ω₀. The rotating HWP also causes an intensity modulation at the frequency ω₀ and at 2ω₀, respectively stemming from fabrication defects of the HWP and the birefringence of the HWP. The reference frequency input of the lock-in amplifier (LIA) is switched between ω₁ and ω₂ by a PC which then subsequently queries the phase and amplitude. A query at ω₂ returns the intensity of the beam while a query at ω₁ returns the amplitude and phase of the modulation caused by a polarizing nano-particle in the beam path. (b) shows a cartoon of the electronic spectrum measured by the photodiode and demonstrates the separation of interfering signals from the signals of interest. (c) shows the design geometry of our nano-antennas and (d) a scanning electron micrograph of a single nano-antenna we fabricated by e-beam lithography.

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2. Theory

We consider in the following the antenna geometry as shown in Figs. 1(c) and 1(d) with the setup as shown in Fig. 1(a). In principle, this method relies on generating an intensity modulation caused by the difference in beam extinction when the polarization is parallel or perpendicular to the antenna’s long axis. Assuming that the LSPR of the antenna’s short axis does not have a significant overlap with the LSPR of the long axis, the modulation caused by the antenna’s polarizability is P_ext = σ_extI₀, where I₀ is the incident beam intensity and σ_ext is the antenna’s extinction cross-section. According to our simulations, following the method described by Gluodenis and Foss [19], the LSPRs of the antenna’s long and short axes should have no significant overlap for aspect ratios greater than 2 to 2.5. Therefore, by measuring I₀ and P_ext one can determine the extinction cross-section. In essence, this technique is therefore only suited for characterizing linearly polarizing nano-structures.

In order to understand this technique in detail, the Jones formalism [20] is a suitable tool to model the measurement setup. A linearly polarized incident beam, described by the Jones vector ${\mathbf{J}}_{\text{in}}=\left(\begin{array}{c}1\\ 0\end{array}\right)$, that first passes through the RHWP, described by the matrix T_RHWP, and then passes through a polarizing nano-particle T_NP with an intensity attenuation of σ results in a Jones vector

Since J_in and J_out represent the polarization state of the electric field, the output intensity is given as $I_{\text{out}}={\mathbf{J}}_{\text{out}}{\mathbf{J}}_{\text{out}}^{*}$ and thus results in:

This is the reason why the use of a RHWP is advantageous when compared to a liquid crystal variable wave plate as a polarization angle modulator. The liquid crystal variable wave plate works by modulating the refractive index of its slow axis, thereby causing the necessary birefringence and consequent rotation in the transmitted polarization vector [18]. However, the modulation of the refractive index also causes a change in reflection and transmission coefficients, which leads to a modulation of the beam’s intensity, typically in the range of 1%, and follows the polarization change resulting in a constant background signal at the modulation frequency that could potentially mask weak extinction signals.

When measuring the intensity modulation caused by a rotating polarization, one has to carefully consider all the optical elements placed in the beam path after and including the modulator itself. Any beamsplitter, slightly tilted optical element, birefringent or polarizing material will cause an intensity modulation due to the difference in transmission and reflection coefficients of s and p polarized light and can potentially mask the signal of the antennas.

To reduce the effect of intensity fluctuations, we measured an intensity reference I_ref by optically chopping the beam at another frequency ω₂ and detecting the PD signal with a lock-in amplifier (LIA). This yields I_ref = |2A₀/π|, where the factor 2/π stems from measuring the first harmonic of the rectangularly modulated DC component of the beam, A₀. Due to the chopping of the beam, the signal at 4ω₀ is also reduced by 1/2, giving I_sig = |A_4ω0/2|. This results in the normalized signal amplitude

Since the normalized signal amplitude σ corresponds only to the losses caused by the antenna, σ is equivalent to the amount of light of a Gaussian beam passing through a circular aperture. Assuming a Gaussian beam with a full width at half maximum (FWHM) of d_FWHM and a circular extinction cross-section, σ_ext of a single antenna can be determined as follows

3. Experiment

Our setup essentially is a micro photoluminescence setup consisting of a Spectra Physics S3900 Ti:Sapphire laser that can be tuned from 1.25 eV to 1.46 eV. The output of the laser is spatially filtered and attenuated with a variable attenuator to ≈ 200 μW to reduce its dynamic range and prevent any saturation of the Thorlabs FDS1010 silicon photo-diode (Si PD). The polarization is rotated with a HWP mounted in a hollow shaft driven by a frequency stabilized motor at a frequency of 31 Hz resulting in a 124 Hz rotation of the polarization. To simultaneously measure an intensity reference, we optically chop the output of our laser at 320 Hz and switch the reference frequency input of our Stanford Research SR830 LIA with a Maxim DG419DJ analog switch. The measurement scheme is shown in Fig. 1(a). To ensure the least amount of signal offset caused by any tilted optics or slightly birefringent optical elements, we place the RHWP directly before a Carl Zeiss LD Achroplan 20×/0.40 objective and assure normal incidence of the beam centered on the objective. To characterize the residual intensity modulation of the optical setup, we place the Si PD in the beam path beneath the objective under normal incidence and measure the intensity at 124 Hz with an integration time of 1 s. The measurements show a 0.2% referenced intensity modulation which can be caused by the SiO₂ surface passivation of the Si PD or the quartz glass optics in the objective [22]. It should be noted, that the background signal is only limited by the tilt alignment accuracy and the birefringence of the optical elements after the RHWP and the sample. Therefore, the true limit of the setup is given by the combined noise of the light source, modulator, PD and LIA and can be controlled by the LIA integration time. Therefore, this technique should, in principle, allow the measurement of arbitrarily small extinction cross-sections of linearly polarizing nano-structures.

To demonstrate the functionality of the setup, we fabricate several arrays of Au dipole nano-antennas on a quartz glass substrate using e-beam lithography. The nominal arm lengths of the antennas are increased from 50 nm to 200 nm in 10-nm steps from array to array while keeping the nominal height and nominal width constant at 40 nm. The spacing between the antennas is kept constant at 500 nm while the antennas’ nominal feed gaps are kept at 20 nm. Between the Au and the substrate, 0.5 nm of Ti are deposited as an adhesion layer. The focus is set to d_FWHM ≈ 1.2 μm. Figure 2 shows the measured extinction spectra of antennas with nominal arm lengths of 140 nm to 200 nm that are resonant within the wavelength range of our Ti:Sapphire laser. The extinction cross-section is calculated from the normalized extinction signal amplitude using Eq. (13) and represents the extinction cross-section of a single antenna in the 4×4 array under the assumption that all antennas in the array are identical. In a further step we also fabricate single dipole nano-antennas on a quartz glass substrate with a constant nominal width and height of 40 nm and a 20-nm gap. Again we vary the nominal arm length from 50 nm to 200 nm in 10-nm steps. The results of the single antenna measurements are shown in Fig. 3. The extinction cross-section for the single antennas is calculated with Eq. (12). We also determine the geometry of the nano-antennas after processing using a scanning electron microscope. Thereby we are able to determine the actual aspect ratio and the corresponding LSPR energy for further evaluation (Fig. 4). The background signal shown in both measurements is caused by the birefringence of the quartz glass substrate used in the experiment that causes a 0.2% offset which corresponds to a ≈ 0.01 μm² extinction cross-section for single antennas and ≈ 0.001 μm² extinction cross-section for a 4×4 antenna array. The background signal is the lower detectable limit in this configuration and would be shot noise or incident-angle-offset limited if we removed all SiO₂ from the beam path. The extinction cross-section we observe is similar to other measurements [14, 16] and considering the antennas’ LSPR energies with respect to their corresponding aspect ratios, we also conclude that the 500 nm pitch does not significantly modify the LSPR energy.

 

Fig. 2 Measured extinction cross-section spectra of a 4×4 dipole nano-antenna arrays. The array pitch was kept constant at 500 nm for every array. The antenna’s nominal height and width were kept at 40 nm and the nominal gap was kept at 20 nm while only the antenna’s nominal arm length L was increased from 50 nm to 200 nm in 10-nm increments. Antennas that are resonant in our Ti:Sapphire laser’s emission wavelength window have nominal arm lengths of 140 nm to 190 nm.

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Fig. 3 Measured extinction cross-section spectra of single dipole nano-antennas. The antenna’s nominal height and width were kept at 40 nm and the nominal gap was kept at 20 nm while only the antenna’s nominal arm length L was increased from 50 nm to 200 nm in 10-nm increments. Antennas that are resonant in our Ti:Sapphire lasers emission wavelength window have nominal arm lengths of 140 nm to 170 nm.

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Fig. 4 LSPR energy vs. antenna aspect ratio for 4×4 antenna arrays and single antennas. The aspect ratios were determined from scanning electron micro-graphs (SEMs) with 6 nm resolution for the arrays and 2 nm resolution for the single antennas. The error-bars represent the standard deviation of the LSPR energy and the error range of the aspect ratio.

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4. Conclusion

In conclusion, we demonstrated a novel way to quantitatively measure very small extinction cross-sections of linearly polarizing antenna arrays and single nano-antennas. By using a rotating linear polarization we are able to inherently suppress interference caused by fabrication defects of the λ/2 plate as well as mechanical oscillations and birefringence interference signals from the modulator itself. Due to the low complexity of this modulation technique, we expect to simplify the process of characterizing linearly polarizing nano-antennas, thus providing an efficient and highly sensitive nano-antenna characterization tool for material sciences as well as nano-photonics. This technique can also be applied to a wide range of sample temperatures, thus enabling the investigation of single nano-antennas from cryogenic temperatures up to the antenna’s melting point, which could be relevant for the performance evaluation of localized surface plasmon based sensors at various operating temperatures or to optimize the antennas’ spectral response to match quantum dots which only function at cryogenic temperatures.