Deterministic nanoantenna array design for stable plasmon-enhanced harmonic generation

3.1 Characterization of plasmon-enhanced THG

In Figure 3, we show the comparison of intensity and polarization dependences of the plasmon-enhanced THG for the Au nanoantenna array with a pitch distance of 400 nm and unpatterned bulk Si. Only the nanoantenna array of 400 nm-pitch-distance was used to characterize the THG since it gave the highest third-harmonic yield compared to other pitch distances. Si is a face-centered cubic (FCC) material that strongly generates odd-order harmonics at the air-Si interface [19]. The THG yield follows an I³ power law with increasing pump laser intensity, I, for both the 400 nm-pitch nanoantenna array and unpatterned bulk Si, when the laser polarization was aligned along the major axis of the nanoantennas (see Figure 3a). The laser focal spot was optimized to maximize the third-harmonic yield. Notably, the THG yield emitted from the nanoantenna array was ∼15% stronger than the emission from the unpatterned bulk Si due to field enhancement, even though the nanoantennas covered only ∼3% of the total area. Figure 3b depicts the first Brillouin zone of the FCC lattice of the Si substrate with four-fold symmetry (Γ–M and Γ–X directions). The third-harmonic yield in the bulk Si without nanoantennas was modulated with four-fold symmetry (a periodicity of 90°), which is the same as the periodicity of the crystal orientation in Si (black curve), as shown in Figure 3c. The third-harmonic yield was maximized when the laser polarization was oriented along the Γ–M direction of the Brillouin zone, but the yield was suppressed continuously when the laser polarization was rotated from the Γ–M to the Γ–X direction. Interestingly, a similar tendency but with a distinct deviation in the harmonic yield was seen in two Γ–M directions in the nanoantenna array (red curve); the harmonic yield at a polarization angle of 0° (parallel to the major axis of the nanoantenna) was stronger than that at a polarization angle of 90° (perpendicular to the major axis of the nanoantenna) for the same Γ–M direction [9, 20]. This is because the strongest field enhancement evidently occurs at a polarization angle of 0°, which is consistent with the simulation results (see Figure 2b and c).

Figure 3:

Laser intensity and polarization dependences of plasmon-enhanced THG for a 400 nm-pitch Au nanoantenna array on a Si film and unpatterned bulk Si. (a) Third-harmonic yield as a function of laser intensity on a log–log scale for the nanoantenna array on Si and for bulk Si without any nanoantennas with the laser polarization parallel to the Γ–M direction. The harmonic yield at every pump laser intensity was measured for five times with a fixed laser focal spot position. The dashed blue line represents an I³ fit where I is the laser intensity. (b) First Brillouin zone of the FCC lattice of Si. (c) Modulation of the third-harmonic yield as the laser polarization is rotated, with maximum emission when the polarization is parallel to the Γ–M directions in the bulk Si. The white arrows in the three FDTD simulation images represent different laser polarizations with respect to a nanoantenna. The blue arrow shows the difference in plasmon-enhanced harmonic yield between polarization angles of 0° and 90°. (d) Plasmonic THG yield enhancement, I_sp/I_bulk (red curve), and FDTD-simulated maximum plasmonic field enhancement (blue curve) as functions of the polarization angle.

This experimental data shows less polarization dependence in the field enhancement factor, while the polarization effect is rather distinct in the simulation. Physically, THG is induced inside the Si layer, not at the Au nanoantenna surface, which indicates that the plasmonic field enhancement at the Si surface may not effectively amplify the THG yield. We further simulated the plasmonic field distribution at a depth of 5 nm inside the Si surface (see Supporting Information Section 4). Interestingly, the enhanced field when the laser polarization is parallel to the major axis of the nanoantenna is distributed around the boundary of and underneath the Au nanoantenna, which could infer that the some of generated THG signal would be blocked by the Au nanoantenna. On the other hand, the enhancement field for the laser polarization perpendicular to the major axis of a nanoantenna is distributed outside of the Au nanoantenna, and the area of the enhancement field is slightly larger than that of the parallel polarization state. These effects, including imperfect numerical modeling of the fabricated nanoantenna, would result in lower THG yield ratio and less polarization dependence for the two polarization states.

3.2 Estimate of plasmonic THG yield enhancement

The plasmon contribution to the harmonic yield was further analyzed by calculating the harmonic yield per unit area in the bulk Si and nanoantenna array regions. The formula for extracting the third-harmonic power per unit area contributed by the bulk Si is

where P_bulk is the measured third-harmonic power of the bulk Si without a nanoantenna array and r is the radius of the focused beam. Here, when each nanoantenna is assumed to emit third-harmonic radiation of equal intensity, the plasmon-enhanced third-harmonic power per unit area can be extracted as follows:

where P_sp is the measured third-harmonic power of the nanoantenna array, and A_inactive and A_active denote the surface areas contributing to the bulk and plasmon-enhanced THG from the nanoantenna array, respectively (see Supporting Information Section 5 for the detailed calculations). The total number of nanoantennas located in the focused beam, N_total, was assumed to be

where d denotes the pitch distance of the fabricated nanoantenna array. Using Eqs. (1)–(3) above, the plasmon-enhanced THG yield enhancement, I_sp/I_bulk, which is the ratio between the plasmon-enhanced and bulk THG power per unit area, can be plotted as a function of laser polarization, as shown in Figure 3d. Note that we assume each nanoantenna within the focused beam area generates an equivalent third-harmonic emission here, which is common for evaluating the plasmonic field enhancement in nonlinear experiments [9, 15]. The THG yield enhancement has a maximum of ∼100 at a laser polarization angle of 0° and decreases to ∼25 when the polarization angle approaches 45°, then increases again up to ∼40 with some fluctuations in between for polarization angles of 45°–90° (red curve). On the other hand, the simulated maximum plasmonic enhancement decreases continuously as the polarization angle changes from 0° to 90° (blue curve). One would expect a continuously decreasing THG yield enhancement for polarization angles of 45°–90° due to decreasing plasmonic enhancement from 45° to 90°; however, we observe an increase in the THG yield enhancement above 45° (red curve in Figure 3d) due to the effective area change of plasmonic field enhancement with respect to the polarization (black curve in Figure 3c). In other words, the combination of both the reduced maximum plasmonic field enhancement and increased effective plasmonic THG area contribute to the plasmon-enhanced THG yield.

3.3 Instability of plasmon-enhanced THG

We observed fluctuations in the extracted plasmonic THG yield enhancement from all nanoantenna arrays (pitch distances of 400, 600, 800, and 1000 nm) upon laser illumination even with fixed laser focus conditions. Therefore, we systematically investigated the cause of the instability found in the THG yield by using two different focused laser beam radii on the sample (Figure 4a and b). Focused beam radii of 3.15 and 2.35 μm were obtained using objective lenses with different NA = 0.25 (Figure 4a) and NA = 0.4 (Figure 4b), respectively. The laser polarization was aligned parallel to the major axis of the nanoantenna arrays to achieve maximum plasmon-enhanced THG signals. We found that when the sample was constantly shifted in the horizontal direction with respect to the focused beam with an incremental step size of 1/10 of each pitch distance for 10 times, the THG yield enhancement at each position was different, providing a clear evidence that plasmon-enhanced harmonic generation can vary according to the focused beam position on a nanoantenna array. In these experiments, unpredictable mechanical vibrations can cause third-harmonic power fluctuations at each focused beam position. These vibrations can simultaneously induce a fluctuation of the laser beam position along the optical axis, which is not related to the pitch distance of the nanoantenna array. In our experiments, the experimental setup was built on a vibration isolation table and inside an enclosure to minimize external vibrational noises; we verified that the fluctuation of third-harmonic yield by mechanical vibration was negligible for a fixed laser beam position. In addition, the third-harmonic signal was averaged over 1 s for each laser beam position to minimize high-frequency noises; the resultant third-harmonic yield was obtained by accumulating 10⁸ successive pulses. Furthermore, the power instability of our femtosecond laser system did not significantly affect the plasmon-enhanced THG instability over time because the measured power instability of the laser was less than 0.17% over 1 h (Figure S4). If the mechanical vibration or laser stability affects the instability of the third-harmonic generation, this effect should also similarly influence all the experimental cases, such as all the pitch distances. Nevertheless, different increasing third-harmonic power instability was identified for pitch distances of 400, 600, 800, and 1000 nm.

Figure 4:

Plasmon-enhanced THG instability and evaluation of the ROA. Plasmonic THG yield enhancement measured at 10 different positions for each pitch distance with (a) radius = 3.15 μm and (b) radius = 2.35 μm. The dashed blue line represents the mean value of the THG yield enhancement for a nanoantenna array with a pitch distance of 400 nm. (c) Schematic illustration of the analytical die count estimation method. (d) Numerically calculated total number of nanoantennas excited by the laser beam (red curves) and the ROA values calculated based on the analytical die count estimation method (blue curves) for varying pitch distances with radius = 3.15 μm and radius = 2.35 μm. (e) CV of the THG yield enhancement as a function of the ROA with radius = 3.75, radius = 3.15 and radius = 2.35 μm. Note that radius = 3.75 μm (NA = 0.25*) resulted in a larger focused beam radius since a smaller laser beam diameter at input aperture was used with the same objective lens of radii = 3.15 μm (NA = 0.25). The dotted grey arrows are a guide to the eye, showing the ROA decreases with increasing focused beam radius for a pitch distance of 1000 nm, for instance.

The experimentally observed third-harmonic yield enhancement typically reflects the fabrication quality of nanoantennas regarding to a specified design. However, our results show that it is unclear whether the observed large variation of the extracted THG yield enhancement at different beam positions on the sample is caused by imperfections of the fabrication process or optical misalignment. For radius of 3.15 μm, as the pitch distance increases to 1000 nm, the THG yield enhancement exhibits a strong variation between 1 and 313, which is hard to quantify the plasmonic field enhancement of a nanoantenna. Such a large variation could be detrimental to the reliability and performance of plasmonic nanoantenna arrays. Experimentally, the total A_active and A_inactive areas can vary with the lateral beam position shift, which should be applied in calculating I_sp and I_bulk. Nevertheless, our analysis considered them as the fixed values for the same pitch distance and focused beam size. Because the precise measurement of the laser beam position on the nanoantenna array is complicated during the THG experiment, applying the different active areas for each THG data is practically not easy. In addition, the change of total active area is relatively less than the size of A_inactive (bulk surface area in nanoantenna array), which does not severely affect the third-harmonic power calculation using Eqs. (1)–(3), while this small active area change causes the total plasmon-enhanced THG yield instability. To investigate this, we calculated the nanoantenna array’s maximum and minimum active areas, (A_active × N_total) at a focused beam radius of 2.35 μm for each pitch distance (see Supporting Information Section 7). The A_active for each nanoantenna was acquired from the FDTD simulation, and the total active areas, A_active × N_total, were calculated using a graphical software, considering the area of a nanoantenna array where the focused laser beam overlaps. Figure S5 shows the calculated plasmonic THG yield enhancement for the maximum and minimum active areas at each pitch distance. The graph shows that the change in the total active area does not significantly affect the original analysis, as shown in Figure 4b; applying the exact active area for each measurement data does not significantly reduce the third-harmonic power instability in Figure 4a and b.

The method of counting the number of nanoantennas when extracting the THG yield enhancement using Eqs. (1)–(3) is only valid under particular conditions, such as when the pitch distance is sufficiently shorter than the focused beam diameter. When the focused beam size is reduced using a lens of higher NA, the THG yield enhancement instability starts to appear at a shorter pitch distance of 400 nm (Figure 4b), indicating that the pitch distance is closely correlated with the focused laser beam size in determining the onset of the instability. Interestingly, the mean values of the THG yield enhancement are very similar for all pitch distances, and tend to approach the mean value for the shortest pitch distance of 400 nm. Ideally, the measured THG yield enhancement should be the same for all pitch distances since the nanoantennas are not coupled with each other due to their far instance from each other. Moreover, the plasmonic THG yield enhancement is mainly determined by the geometry of a nanoantenna and the complex permittivities of the materials and environment, therefore the change of beam size or pitch distance should not play a role in the THG yield enhancement. When the pitch distance is short enough to minimize the instability effect, the THG yield enhancement should be the same for both NA values. However, since the third-order nonlinear process is highly sensitive to the peak intensity changes of the input laser, e.g., due to pulse broadening caused by dispersion in a thicker lens (higher NA), the THG yield enhancement was reduced for radius of 2.35 μm.

Surface lattice resonances, which constitute the diffractively coupled plasmon resonance modes in a nanoantenna array [21, 22], were not considered in the experimental analysis, although these could affect the efficiency of plasmon-enhanced THG. Surface lattice resonances require the incident light illumination to be spatially coherent over a large area to enable constructive interference of scattered light produced by all the nanoantennas that make up the array. Therefore, a focusing lens with a small NA (e.g., 0.1) is preferred for irradiation to achieve a uniform phase on the array. However, in our experiments, we used focusing lenses with relatively high NAs (0.25 and 0.4), hence did not satisfy this condition to excite the surface lattice resonance modes efficiently [23]. In fact, plasmon-enhanced THG is commonly known to originate from the Si substrate but strongly enlarged by the strong plasmonic fields generated at the interface between the nanoantennas and the substrate. On the other hand, we also considered the possibility of THG yield contributed by the third-order nonlinear responses of Au nanoantennas induced by interband transitions from the 5d band to the 6sp band in Au. It has been reported that the THG yield can be significantly increased by utilizing the third-order nonlinearity of Au which is strongly influenced by its interband transitions in Au nanoantennas fabricated on a glass substrate [24]. However, in our case, we believe that the THG contribution from third-order nonlinearity of Au nanoantennas is negligible since the third-order nonlinear susceptibility of Au is only approximately 1/10 that of Si, hence the THG power of Au is approximately 1/100 or even smaller compared to Si, since it scales with the nonlinear susceptibility squared [25, 26]. In addition, the fabricated Au nanoantennas are of polycrystalline nature, which should result in a lower THG efficiency than for single-crystalline materials.

Therefore, the instability of the THG yield enhancement could only have been induced by the spatial alignment between the focused laser beam and the nanoantenna array as the THG yield is predominantly originated from the field enhancement of the nanoantennas. Figure 4c illustrates the distribution of square unit cells containing nanoantennas in an array within a circular laser beam spot (dashed red line). Each nanoantenna is located at the center of its corresponding unit cell and the unit cell size is defined by the pitch distance. Nanoantennas which are located within ambiguous unit cells around the boundary of the beam spot (colored in gray) are more sensitive to beam position shift. Using this assumption, we further analyzed the effect of the ratio between the number of these ambiguous nanoantennas and the total number of nanoantennas within the focused beam diameter, which we found to be related to the instability of the THG yield enhancement.

To compute this ratio, we utilized the analytical die count estimation method, which is a well-known technique in the semiconductor industry for calculating the number of dies for a specific wafer size [27]. The analytical die count estimation is effective in counting the number of dies (square or rectangular shape) that can be made from a circular-shaped single wafer; the boundary pieces with non-square shapes cannot be used in a semiconductor process. Similarly, in our analysis, each nanoantenna can be assumed as a square unit cell (die) within a circular laser beam (wafer). We estimated that the increasing proportion of ambiguous unit cells of the total nanoantennas around the boundary of the beam spot (non-square shapes) increases the instability of THG. Here, the analytical die count estimation method is suitable for estimating the ratio between the number of ambiguous nanoantennas located at the beam boundary and the total number of nanoantennas within the beam diameter. The number of nanoantennas located at the boundary of the focused laser beam is given by

Using Eqs. (3) and (4), we define the ratio of ambiguity (ROA) as:

Equation (5) clearly shows that the pitch distance, d, of the nanoantenna array is closely correlated with the focused beam radius, r, in determining the ROA.

Figure 4d displays the total number of nanoantennas and the calculated ROA with radius = 3.15 and radius = 2.35 μm for varying pitch distances. The total number of nanoantennas decreases following the inverse square law according to Eq. (3) while the ROA increases linearly as the pitch distance increases for both radius values or focusing conditions. To further investigate the correlation between the ROA and the instability of the plasmonic THG yield enhancement, we calculated the coefficient of variation (CV) of the measured THG yield enhancement. The CV is a statistical measure of the spread of all measurements around the mean and represents the ratio between the standard deviation and the mean [28]. The THG enhancement instability under different focusing conditions cannot be directly compared because the experimental conditions, such as the input laser intensity and total number of excited nanoantennas, were different. On the other hand, the CV enables a quantitative comparison between different measurements regardless of the experimental conditions; therefore correlating the ROA with the CV is useful for investigating the instability of the THG yield enhancement. Figure 4e shows the correlation between the CV of I_sp/I_bulk and the ROA with radius = 3.75, radius = 3.15 and radius = 2.35 μm. A larger focused beam radius of 3.75 μm was obtained using the same objective lens of radius = 3.15 μm (corresponding to NA = 0.25) because a smaller laser beam diameter was used at the entrance of the lens. Particularly, both the CV of I_sp/I_bulk and ROA acquired from the three different focusing conditions coincide closely with each other. In addition, the CV of I_sp/I_bulk increases roughly linearly with increasing ROA while both the CV of I_sp/I_bulk and ROA tend to decrease with increasing focused beam radius for every pitch distance. The ROA is expressed by the ratio between pitch distance and beam radius, denoting the proportion of ambiguous unit cells (non-square shapes) of the total nanoantennas. The increasing ROA in Eq. (5) means the increase of ambiguous nanoantennas in the total number of nanoantennas. Thus, Figure 4e displays that the instability of THG is accelerated as the proportion of ambiguous nanoantenna increases. When the normalization by the active area is not considered for analyzing the THG power instability, the tendency of THG instability by the ROA increment is very similar. Figure S6 shows the power stability of plasmonic THG from the total third-harmonic power of nanoantenna array as a function of the ROA (see Supporting Information Section 8). Note that the conventional method for evaluating optical power stability was applied for displaying THG instability instead of the normalization by the area, still showing good agreement with our claim.

Thus, according to the above results, the stability of the plasmon-enhanced THG can be theoretically estimated in advance based on the ROA. For example, an ROA of less than 0.1 determines the maximum pitch distance of a nanoantenna array to be 100 nm if the beam radius is 1 μm, in order to yield a negligible CV for plasmon-enhanced THG experiments. Our results can support the deterministic design of a nanoantenna array to accomplish the requirements by considering scientific and non-scientific design factors. Conventional deterministic design means designing a system to satisfy the given requirements or risk-free systems [29, 30]. For example, when a steel sample can endure a certain force level for a steel tensile test, one can claim that this steel is deterministically designed to endure the applied force. In deterministic models, the model’s output is fully determined by the parameter and initial values to satisfy the requirement. Here, the requirement of a nanophotonic system based on the nanoantenna array could be a high photon yield, reliability, fabrication capability, or cost efficiency. The complete fulfillment of the aforementioned requirements is ideal, but it is practically not achievable. Our results provide a useful parameter for judging system reliability by revealing the maximum allowable pitch distance to satisfy the desired power stability. Then, the minimum pitch distance can be determined by considering other design requirements such as coupling effect, cost efficiency, or fabrication capability. For example, the unexpected coupling effect can occur when the pitch distance between nanoantennas is too short. Previous research reported that the plasmonic coupling effect decays over a length roughly 0.2 times the particle’s length [31, 32]. Further, dense nanoantenna fabrication requires state-of-the-art fabrication capability at a high cost. These design factors, including our technique, enable the deterministic design of pitch distance or optical configuration parameters (e.g., beam size) of the nanoantenna array to satisfy the tolerable stability of nanophotonic devices. Here, the ROA provides a useful and intuitive guideline for designing the nanoantenna arrays in terms of pitch distance and number of nanoantennas within a given focused beam size for maximizing plasmon-enhanced harmonic generation.