1 Introduction

Plasmonic nanoantennas efficiently convert propagating light to highly localized and strongly enhanced optical fields [1, 2]. Due to the strong field enhancement, they can efficiently enhance nonlinear light generation such as two- [3] and four-photon photoluminescence [4], second-harmonic [5], third-harmonic [6–8], and even fifth-harmonic generation [9]. In particular, the influence of the antenna volume on the third-harmonic intensity [6, 8] as well as the third-harmonic polarization [7] has been investigated. However, the angle dependence of the emitted third-harmonic signal has so far not been analyzed.

Here, we investigate the radiation pattern of the nanoantenna-enhanced third-harmonic signal. Furthermore, we analyze the third-harmonic emission of two nanoantennas, excited under the same laser focus.

2 Experimental setup and sample fabrication

2.1 Optical setup

The third-harmonic generation (THG) experiments are performed with an experimental setup based on a femtosecond Er:fiber laser (repetition rate: 40 MHz, center wavelength: 1.55 µm) depicted in Fig. 1a [6, 7]. Ultrabroadband light pulses in the near infrared are generated in a highly nonlinear fiber by four-wave-mixing. Subsequently, the laser pulses are shortened and spectrally filtered with a prism compressor. Pulses of the dispersive branch have a spectral bandwidth of 200 nm at a central wavelength of 1320 nm (Fig. 2a) and a pulse duration of 11.3 fs (Fig. 2c). The laser beam is expanded in a mirror telescope to fill the back aperture of a Cassegrain reflector (numerical aperture: NA = 0.65) to focus the laser light onto single gold nanoantennas in a confocal transmission microscope (Fig. 1b). Reflective optics are used to minimize chromatic aberration and dispersion. In addition, a half-wave plate can be inserted to match the linear polarization of the laser with the polarization response of the antenna. The third-harmonic emission of single gold nanoantennas is collected in transmission through the substrate with an oil immersion objective lens (NA = 1.45). The high numerical aperture allows for collection of angles up to 83° from the surface normal. Light at the fundamental wavelength is removed by a short-pass filter (BG39). The nonlinear emission is either focussed on a single-photon counting avalanche photodiode or a spectrometer equipped with a charge-coupled device (CCD) for spectral analysis or imaging.

Fig. 1
figure 1

Schematic drawing of the experimental setup consisting of a an Er:fiber laser system with an internal Si prism compressor (PC1), a highly nonlinear fiber (HNF), an external prism compressor (PC2), spectral filtering (SF), a telescope and b an inverse confocal transmission microscope. A Cassegrain objective is used to focus the laser light onto the nanoantennas. An oil immersion objective lens collects the emitted nonlinear radiation. A filter blocks the infrared light used for excitation. (inset) Sketch of the sample geometry. c Principle of back focal plane imaging to map the angular distribution of the nonlinear emission. Rays emitted from the sample plane under the angle θ 2 are focused in the back focal plane of the objective lens. A lens (Lens 2) is placed at the distance of its focal length (f 2 = 500 mm) with respect to the back focal plane of the objective lens. Using a second lens (Lens 1, f 1 = 200 mm), an image of the BFP forms on the CCD. Therefore, every image point x i on the monitor represents a specific radiation angle θ 2,i

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Fig. 2
figure 2

a Measured laser spectrum used for excitation (red line) and calculated scattering cross section of a typical dimer nanoantenna (arm length = 225 nm, width = 50 nm, height = 30 nm, gap = 20 nm) (black line). b Typical THG spectrum of a dimer nanoantenna with a plasmon resonance matching the laser spectrum. c Measured interferometric autocorrelation of the laser pulses used for excitation with a pulse length of 11.3 fs. d Cubic dependence of the measured spectrally integrated THG emission intensity on the laser fluence (black dots) and the corresponding cubic fit (red solid line). Above 60 kW/cm2, the gold antennas are destroyed. (left inset) Dimer antenna excited with intensities below the damage threshold. (right inset) Dimer antenna after excitation with intensities above the destruction limit

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To investigate the angular distribution of the emitted radiation, the back focal plane (BFP) of the objective lens is imaged by introducing an additional lens into the beam path [10–12]. Light emitted from the sample plane at an angle θ 2 to the optical axis is refracted at the reference sphere of the objective lens and is focused on a point in the back focal plane. The additional lens (Lens 2) can now be used to image the BFP onto the CCD camera (Lens 1) to obtain an angle-resolved radiation pattern of the investigated emitter (Fig. 1c). Care has to be taken when calibrating the angle axis of the radiation pattern, because when using high-numerical-aperture (NA) objective lenses paraxial optics does not apply anymore. For objective lenses satisfying Abbe’s sine condition, the principal plane is shaped into a reference sphere [13].

2.2 Nanofabrication of plasmonic antennas

The samples are fabricated using electron-beam lithography. Fused silica substrates of 170 µm thickness are used, because its amorphous structure generates only weak THG. Subsequently, the substrates are spin-coated with 50 nm Polymethylmethacrylate (PMMA) ebeam resist and a 10 nm thick Al layer is evaporated to avoid charging effects during the exposure with the electron beam. After Al etching and developing of the resist, 2 nm of Cr as adhesion layer and 30 nm of Au are thermally evaporated. Finally, the PMMA not exposed by the electron beam, along with the metallic layer on top, is removed in a lift-off process. Only the metal nanoantennas remain on the fused silica substrate. They are designed to match the laser spectrum with their plasmon resonance. The typical antenna size ranges from 200 to 300 nm in arm length and 20 to 80 nm in width, with antenna gaps of 20 to 50 nm in the case of dimer nanoantennas.

3 Results

3.1 Third-harmonic emission of single Au nanoantennas

Nanoantennas can efficiently enhance the third-harmonic generation [6–8]. The THG is only determined by the linear response of the antenna [8]. Consequently, it is essential to match the plasmon resonance with the laser spectrum used for excitation. The spectrum of our excitation laser closely matches the calculated plasmon resonance of the investigated gold antennas as shown in Fig. 2a. The simulations were carried out with a commercial finite-difference time-domain (FDTD) solver (Lumerical FDTD Solutions) using the measured antenna geometries and the dielectric function for Au by Johnson and Christy [14]. The excellent overlap of laser spectrum and plasmon resonance causes efficient third-harmonic generation and yields a typical THG spectrum centered at 430 nm (Fig. 2b). Next to the dominant THG maximum, a shoulder exists at 500 nm, which corresponds to weak multi-photon photoluminescence (MPPL) [15, 16]. The measured THG intensity of a dimer nanoantenna as a function of laser fluence is depicted in Fig. 2d. As expected, we find a cubic dependence [17] of the integrated THG signal. However, care has to be taken when choosing the laser intensity. For fluences exceeding 60 kW/cm2, the increase of the THG intensity becomes smaller. At high intensities, the heat introduced by the laser melts the Au structure and the antenna is destroyed [9]. This effect is clearly visible in the insets of Fig. 2d, where two scanning electron microscope images are shown after excitation with laser fluences below and above the damage threshold.

3.2 THG emission pattern of single Au nanoantennas

Nanoantennas consisting of a single rod possess Fabry–Pérot resonances [18, 19]. The fundamental mode exhibits a surface charge distribution with a single node in the center of the antenna [2]. Consequently, the angular distribution of the fundamental antenna mode is dipolar-like. This behavior has been demonstrated for light scattered by nanoantennas [20, 21], as well as for nanoantennas excited by single quantum dots [12]. However, the radiation pattern of the third-harmonic emission of Au nanoantennas has not been analyzed yet. We measure the angular distribution of the emitted third-harmonic radiation using back focal plane imaging. The emission is detected through the glass substrate, because only a small fraction of the radiation is emitted into the air [21]. A measured radiation pattern of a single-rod nanoantenna is presented in Fig. 3a. The maximum detectable angle is given by θ 2,max = arcsin(NA/n 2), where n 2 = 1.46 is the refractive index of the fused silica substrate and NA = 1.45 the numerical aperture of the used objective lens, which leads to an angle of θ 2,max = 83°. This angle is used to calibrate the emission angle axis θ 2, which is connected to the image pixels by \(\sin \theta_{2} \propto \sqrt {x^{2} + y^{2} }\) due to Abbe’s sine condition in non-paraxial optics. The THG emission pattern of the single-rod nanoantenna has two distinct areas, which are separated by the critical angle of total internal reflection of θ crit = 43° for the glass–air interface at which the antenna is located. The measured radiation pattern of the THG resembles that of a radiating dipole at the interface of two dielectrica. The radiated power from an electric dipole located at an interface of two dielectrica can be calculated analytically. For the case of two dielectrica with refractive indices of n 1 and n 2 with \(n = n_{2} /n_{1}\) and for a radiation angle 0° ≤ θ 2 ≤ θ 2,c , it is given by [22–24]

$$\begin{aligned} P_{p} \left( {\theta_{2} ,\varphi } \right) & = \frac{3}{2\pi }n^{3} \frac{{ \cos^{2} \theta_{1} \cdot \cos^{2} \theta_{2} }}{{(n\cos \theta_{1} + \cos \theta_{2} )^{2} }} \cdot \sin^{2} \varphi , \\ P_{s} \left( {\theta_{2} ,\varphi } \right) & = \frac{3}{2\pi }n^{3} \frac{{ \cos^{2} \theta_{2} }}{{(\cos \theta_{1} + n\cos \theta_{2} )^{2} }} \cdot \cos^{2} \varphi \\ \end{aligned}$$

and for θ c  ≤ θ 2 ≤ 90° by

$$\begin{aligned} P_{p} \left( {\theta_{2} ,\varphi } \right) & = \frac{3}{2\pi }\frac{{n^{3} }}{{n^{2} - 1}} \frac{{ \cos^{2} \theta_{2} \cdot (n^{2} \sin^{2} \theta_{2} - 1)}}{{\left( {n^{2} + 1} \right)\sin^{2} \theta_{2} - 1}} \cdot \sin^{2} \varphi , \\ P_{s} \left( {\theta_{2} ,\varphi } \right) & = \frac{3}{2\pi }\frac{{n^{3} }}{{n^{2} - 1}} \cdot \cos^{2} \theta_{2} \cdot \cos^{2} \varphi . \\ \end{aligned}$$
Fig. 3
figure 3

a Schematic drawing of the detection geometry. The antenna is excited from the top (air). The radiation pattern is detected from below (through the glass substrate). High angles are at the outer part of the radiation pattern (non-paraxial optics). b Measured THG emission pattern of a single-rod nanoantenna. The inset shows the antenna orientation and the laser polarization (red arrow). c Analytically calculated radiation pattern of an electric dipole at an air (n 1 = 1)—glass (n 2 = 1.46) interface. The inset depicts the orientation of the dipole. The red and white circles in b and c correspond to distinct angles of θ 2,crit = 43° and θ 2,max = 83°. d, e Linescans along the parallel (black line) and vertical (green line) direction of the experimentally measured (solid) radiation pattern shown in b as well as the calculated (dashed) ones shown in c

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Here, the indices s and p stand for s- and p-polarized waves. θ 1 is the emission angle in medium n 1, connected to θ 2 by Snell’s law n 1 sin θ 1 = n 2 sin θ 2. φ is the azimuthal angle. For n 1 = 1 and n 2 = 1.46, the equations lead to the radiation pattern depicted in Fig. 3c. The radiation of a dipole at the interface of two dielectrica is strongly different from a dipole radiating into free space [24, 25]. We can directly compare the experimental data to the analytic calculation. Figure 3d shows the angular distribution parallel (φ = 0°) to the long axis of the antenna. The distribution in direction perpendicular to the long axis (φ = 90°) is presented in Fig. 3e. We find an excellent agreement between the measured third-harmonic emission pattern of the single-rod nanoantenna and the analytically calculated radiation pattern of a single dipole at an air–glass interface.

3.3 THG radiation pattern of coherently emitting Au nanoantennas

We now investigate the radiation pattern of two coherently emitting antennas. In particular, we demonstrate that the resulting THG radiation patterns are fully compatible with a coherent emission of dipoles located at the center of the individual antennas. First, we simultaneously excite two single-rod antennas (length = 270 nm, width = 30 nm) separated by a distance of 2.55 µm with a slightly defocused laser beam. Coherent nonlinear emission is generated at the respective position of the two antennas. Since the radiated fields are in phase, distinct interferences are observed in the horizontal direction of the radiation pattern (Fig. 4a). Similar to the previous section, we can model the emission of each antenna by a single dipole placed at the center of the antenna. In that way, we calculate the radiation pattern of two dipoles radiating in phase using the FDTD solver. The dipoles are separated by a distance of 2.55 µm just as in the experiment and emit at the THG wavelength of 434 nm. The simulated THG radiation pattern (Fig. 4d) is in excellent agreement with the measured one. In general, the angular spacing caused by constructive and destructive interference is determined by the separation of the dipoles and the emission wavelength. For more closely spaced dipoles, it is expected that the first minimum of destructive interference is shifted to larger angles. We demonstrate this behavior by investigating an antenna consisting of two arms (arm length = 220 nm, width = 30 nm), which are separated by a gap of only 40 nm. The measured radiation pattern of the dimer antenna is shown in Fig. 4b. The radiation pattern is similar to that of a single dipole (Fig. 3b, c). However, destructive interference causes a minimum around 40° in the horizontal direction for the closely spaced dimer. Our results are in good agreement with simulated radiation patterns (Fig. 4e) of two dipoles positioned at the respective antenna centers (distance of the antenna arm centroids of 260 nm). Due to small differences in the geometry of the antenna arms caused by the fabrication, the THG signal of the respective antenna arms is slightly different. Therefore, the interference minimum is not exactly zero.

Fig. 4
figure 4

a–c Measured THG radiation patterns of single and multiple nanoantennas (a: arm length = 270 nm, width = 30 nm; b: arm length = 220 nm, width = 30 nm, antenna gap = 40 nm; c: arm length = 200 nm, width = 30 nm, antenna gap = 30 nm). d–e Calculated radiation pattern of point dipoles corresponding to those of the antennas shown in a–c. The chosen geometries are schematically drawn below the images. The red and white circles correspond to the distinct angles of θ 2,crit = 43° and θ 2,max = 83°

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To further demonstrate that the far-field THG radiation pattern can be understood as a coherent superposition of emitting dipoles, we analyze two dimer antennas (arm length = 200 nm, width = 30 nm, antenna gap = 30 nm) placed at a large distance of 2.55 µm as shown in Fig. 4c. The radiation pattern now is a superposition of two closely spaced and two far apart antennas. Again, the simulated radiation pattern (Fig. 4f) of dipoles placed at the center of each antenna arms (four dipoles in total) is in good agreement with the experimental result.

4 Conclusions

In conclusion, we have measured and calculated the third-harmonic radiation pattern of single and multiple gold nanoantennas. A dipole-like radiation pattern modified at the air–glass interface was observed for single-rod nanoantennas. The simultaneous excitation of two single-rod and dimer antennas resulted in interferences in the THG far-field radiation due to the coherent emission of the antennas. These interferences can be explained by modeling the antennas as dipoles located at the center of the antenna arms.