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Abstract
Second harmonic generation is enhanced at the surface lattice resonance in plasmonic nanoparticle arrays. We carried out a parametric investigation on two-dimensional lattices composed of gold nanobars where the centrosymmetry is broken at oblique incidence. We study the influence of the periodicity, the incidence angle and the direction of the linear input polarization on the second harmonic generation. Excitation of the surface lattice resonance either at the fundamental or second harmonic wavelength, achieved by varying the incidence angle, enhance the conversion efficiency. As a special case, we demonstrate that both the wavelengths can be simultaneously in resonance for a specific period of the lattice. In this double resonant case, maximum second harmonic power is achieved.
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1. Introduction
Thanks to the fast development in manufacturing of subwavelength structures in a controlled and reliable way, metamaterials allow to manipulate the light-matter interaction beyond the possibilities of conventional materials. Plasmonic metasurfaces [1] are bidimensional structures consisting of periodically arranged metallic nanoparticles. Plasmons are the collective oscillation of conduction band electrons in metals, and they have been largely studied due to intriguing properties, such as dramatic subwavelength localization and extreme electromagnetic field enhancement [2]. The optical properties depend on the electromagnetic mode supported by the elemental nanoparticles (localized surface plasmon resonance, LSPR) and on the long-range coupling determined by the periodic arrangement (surface lattice resonance, SLR) [3]. SLRs can emerge when the scattered field of a single nanoparticle is in phase with the emitted fields from the surrounding nanoparticles, exciting a collective plasmonic wave propagating along the lattice. Accordingly, the conditions for SLR are closely related to the presence of a diffraction mode propagating along the lattice, also known as Rayleigh anomaly (RA) [4,5]. The spectral position of the SLR depends on the nanoparticle periodicity and the surrounding refractive index, but can be tuned with the angle of incidence. At the SLR the oscillation damping is partially compensated [3,6], thus tuning the interplay between LSPR and SLR enables to design ultra narrow resonances [7–9]. The associated strong field enhancement indeed boosts nonlinear effects. In 1985 Couatz et al. showed that, as the incidence angle is changed, SHG undergoes strong enhancement in metallic mono-dimensional gratings when the impinging beam is resonant with the nonlocal surface plasmon [10], what is today known as SLR. However, in the similar structure the SHG is also enhanced when the SLR condition is fullfield at the second harmonic frequency [11]. Similarly, for two-dimensional (2D) nanoparticle arrays the SHG is enhanced for the SLR at the fundamental frequency [12,13] and at the second harmonic [14]. The primary role of collective response yields surprising and counter-intuitive phenomena, such as larger SHG for less dense metallic gratings [15,16] or the strong dependence on the structure of the unit cell [17]. Even in the nonlinear regime, plasmonic metasurfaces provide a versatile way to shape light [18]. The conversion process can be further improved by utilizing multiple-resonant nanoparticles, where the LSPR enhancement occurs simultaneously at the pump and signal wavelengths [19–25]. Recently, the relevance of double-resonance in the presence of SLRs along orthogonal directions has been stressed out theoretically [26] and was experimentally demonstrated [27].
In this paper, we perform a parametric investigation to study the influence of the SLR on the SHG, exploring various double resonant conditions, where the SLR of the fundamental and second harmonic are simultaneously excited. Therefore, multiple 2D rectangular arrays composed of centrosymmetric gold nanobars on fused silica are fabricated, whereby just the lattice period in one direction is varied while the LSPR value is kept constant. We measured the SHG from these metasurfaces at different angles of incidence and linear polarization directions. The required non-centrosymmetry for SHG is provided by the left-right symmetry breaking associated with a tilted incident wavefront. Relatively high average output powers are achieved, thus allowing the experimental investigation of the nonlinear effects using general purpose and cheap equipment such as CMOS cameras and compact spectrometers. Finally, we model our experimental results using the nonlinear inverse scattering method [28], which indeed connects the SHG to the the linear response of the structure at the two involved wavelengths [29].
2. Sample fabrication and linear characterization
We study several samples, where gold nanobars are arranged into 2D lattices on a 1 mm thick fused silica substrate. The metasurface template is shown in Fig. 1(a). In all the fabricated samples, the shape of the bars working as nanoantennas is preserved. The elemental antenna is 400 nm long ($w_x$), 300 nm wide ($w_y$), and 50 nm high (t). The period in the $y$-direction $P_y$ is fixed to 500 nm. We vary solely the period $P_x$ along the $x$-direction, the latter spanning the interval from 520 to 1200 nm with steps of 20 nm. The structured area of each lattice is 3x3 mm2. The wide area nanopatterning is accomplished using electron-beam lithography and standard metal lift-off technique (see Appendix 6.1). The scanning electron microscope (SEM) image in Fig. 1(b) shows that our gold bars possess a large degree of uniformity, at the same time featuring very smooth edges on the nanometric scale.
Fig. 1. (a) Sample design: 2D lattice of gold nanobars on a 1 mm fused silica substrate. The geometrical parameters are $P_x\in \left [{520}\;\textrm{nm},\ {1200}\;\textrm{nm}\right ]$, $P_y$=500 nm, $w_x$=400 nm, $w_y$=300 nm, $t$=50 nm. (b) SEM image of the sample featuring $P_x$=800 nm.
To characterize the linear response of the fabricated metasurfaces, a supercontinuum laser is used as broadband light source in the setup (see Supplement 1, Fig. S1), featuring a beneficial larger spatial coherence [9,30]. The light is weakly focused on the metasurface to a beam diameter of approx. 300 µm with an achromatic ${400}\;\textrm{mm}$ focal length lens. In all the experiments presented here the sample is rotated around the $y$-axis at the metasurface, changing the optical angle of incidence $\theta _{AOI}$ on the structure. Thus, we are exploiting the angular dependence of the SLR only along the $x$-direction. Light propagates towards positive $z$-direction, and crosses the metasurface before entering the glass substrate.
The numerically computed LSPR for one isolated nanobar is at $\lambda$ =1276 nm and at $\lambda$ =978 nm for light polarized along the x-direction (TM or p-polarization) and along the y-direction (TE or s-polarization), respectively (see Supplement 1, Fig. S2). The corresponding bandwidth is very broad, approx. 700 nm. The experimental linear transmission spectrum as a function of the angle of incidence reveals the location of the plasmon resonances [6], exemplary shown for one lattice in Fig. 2(a). The broad transmission minimum at about $\lambda$ =1150 nm (for TE at $\lambda$ =900 nm, not shown here) emerges from the hybridization between LSPR and the $P_y$ lattice mode [31]. Its position does not change with $P_x$ (see Supplement 1, Fig. S3). The narrow SLRs are visible as transmission anomalies [32,33], which closely follow the wavelength and the angle-depending RA conditions, represented by the superimposed white lines. The incidence angle to achieve the RA condition either for a reflection grating (diffracted light propagating in air) or transmission grating (diffracted light propagating in the glass substrate) is given by [6]:
where $P_x$ is the period along $x$, $m$ the diffraction order, $\lambda$ the vacuum wavelength, and $n_{air}$ and $n_{glass}$ are the wavelength-dependent refractive indices of the air and fused silica, respectively. At the RA, the diffraction order $m$ is diffracted by ±90° and travels along the metasurface (see pictorial sketch in Fig. 2(b)). The asymmetric index environment for the metasurface introduces two families of RA curves, thus introducing an additional degree of freedom to achieve the double resonance. However, this comes at the price of a reduction of the SLR Q-factor [34].Fig. 2. (a) Experimental linear transmission spectrum with TM-polarized light for the $P_x$=800 nm structure. Superposed white lines show the angle of the RAs at the (solid) air and (dashed) fused silica interfaces. The branches partially visible at shorter wavelengths stem from the RA at $|m|=2$. (b) Sketch of the diffracted orders when passing through the structured interface with a period of $P_x$=770 nm and an incidence angle $\theta _{AOI}={19.5}^{\circ}$ for $\lambda ={1032}\;\textrm{nm}$ and $\lambda = {516}\;\textrm{nm}$.
Simulated spectra stemming from RCWA (Rigorous Coupled Wave Analysis) match very well with the experimental results (see Supplement 1, Fig. S4). The good agreement shows that small technological imperfections (surface roughness, particle shape deformation, edge and corner rounding, etc.) have a negligible effect on the long distance interaction [35] and that our metasurfaces contain enough nanoparticles to be approximated with an infinite array [36].
Beyond the RA condition stemming from the momentum conservation, nonlinear effects are strongly dependent on the optical field distribution. We computed the optical field at the metasurface by using a commercial FDTD tool (see Appendix 6.2). At normal incidence $\theta _{AOI}={0}^{\circ}$, TM and TE polarized light excite an electric dipole in the nanobar along the $x$ and the $y$-direction, respectively. A rotation of the metasurface around the $y$-axis leads to an asymmetric field distribution along $x$ (Fig. 3). In the TE case, the dipole changes into an electric quadrupole distribution. Note that the field enhancement is stronger for TE than for TM. We confirmed the presence of a nonlocal plasmon oscillation with the field extending well beyond the unit cell [31] by measuring the power transported along $x$-direction on the metasurface plane depending on $\theta _{AOI}$. Peaks and discontinuities in the power are observed whenever a SLR is excited.
Fig. 3. Simulated E-field amplitude in a unit cell plotted on the $xy$-plane at the nanobar mid-section when $P_x$=760 nm. The input beam at $\lambda$= 1032 nm satisfies the SLR condition being $\theta _{AOI}$= 17°. $E_0$ is the field amplitude in the absence of the metasurface.
We now focus on the two wavelengths corresponding to the fundamental (or pump, $\lambda _{FF}$=1032 nm) and its second harmonic ($\lambda _{SH}$=516 nm) in the frequency conversion experiment. Figure 4(a)-d show the simulated linear transmission at $\lambda _{FF}$ (top row) and at $\lambda _{SH}$ (second row) versus $\theta _{AOI}$ and $P_x$, for TM (left column) and TE (right column) input polarizations. Dips, peaks and jumps are mostly visible in proximity of the RA conditions, confirming that they originate from the SLR. For TE polarized light, the SLR in air is prevented by the reflection of the substrate [34]. Due to the smaller fill factor, the absolute transmission increases for longer periods. The average transmission at $\lambda _{FF}$ is lower than at $\lambda _{SH}$ due to the shorter spectral distance from the LSPR.
Fig. 4. (a-d) Simulated linear transmission in the 0th diffraction order for different incidence angles $\theta _{AOI}$ and periods $P_x$ for the (a-b) pump (1032 nm) and the (c-d) second harmonic wavelength (516 nm), for (left) TM and (right) TE-polarized pump light respectively. (e-f) SHG power measured on the transmitted 0th diffraction order normalized with respect to the absolute maximum in the plot. The superposed lines show conditions where the RA for $|m|=1$ is fulfilled (solid) in air and (dashed) in fused silica; the corresponding wavelengths are 1032 nm (red) and 516 nm (green). The branches partially visible at longer periods stem from the RA at $|m| = 2$.
3. Experiments in the nonlinear regime
3.1 Second harmonic generation
For the nonlinear investigations we used an ultrafast laser delivering 200 fs (FWHM) short pulses at a wavelength of 1032 nm with a repetition rate of 200 kHz and a variable average power. The linearly polarized beam is focused onto the structured area to a 250 µm beam diameter with a 400 mm focal length lens. The metasurface leads to a frequency conversion of the impinging light. We investigate the SHG light which propagates in the direction of the incidence light (0th transmission order). We experimentally verified that SHG from the pure glass surface [37] is not detectable in the range of powers we employ (more information see Appendix 6.3 and Supplement 1).
While the centrosymmetry of gold is broken at the nanoparticle surface, the SHG is still prohibited due to the spatial symmetry. However, oblique incidence angles break the spatial symmetry and allow SHG from the structure [38], see Fig. 3. The angular dependency of the SHG signal for different periods $P_x$ is shown in Fig. 4(e)-(f). By a direct comparison with the linear response (Fig. 4(a)-(d)), the strict connection between linear and nonlinear regime becomes immediately clear [14]. At the RAs, a fast change versus the incident angle is observed in the SHG signal, including even the 2nd diffraction order. It is obvious that the SLRs have a significant impact on the frequency conversion. Pronounced SLR related anomalies in the linear transmission also cause higher SHG efficiency. The strong linear dependence on the input polarization is transferred to the SHG. In fact, the TE case is mostly sensitive to SLR in glass (dashed red line), whereas for TM inputs SLRs are excited on both surfaces. While lattice resonances of $\lambda _{FF}$ are more relevant in determining the SHG, surprisingly even the barely visible $\lambda _{SH}$ SLR in the linear regime has an influence on the frequency conversion strength for both input polarizations.
Especially interesting are the multiple intersections between the $\lambda _{FF}$ and $\lambda _{SH}$ SLRs, corresponding to the double resonance case. To see the behavior approaching the double resonance in more detail, we plotted in Fig. 5 the average SHG power versus the incidence angle in the region of interest. For TM, the most prominent peak is at ($P_x$=760 nm, $\theta _{AOI}$=20.5°), see Fig. 4(e) and Fig. 5(a), corresponding to the double resonance. Close to this condition, the linear diffracted light in air travels for both wavelengths along the nanoparticle array, schematically shown in Fig. 2(b). For slightly detuned periods, the two lattice resonances are clearly distinguishable. The region ${520}\;\textrm{nm}\leq P_x\leq {620}\;\textrm{nm}$, $\theta _{AOI} < {32}^{\circ}$ (Fig. 4(e)) shows a SHG enhancement with a peak at the SLR for $\lambda _{FF}$ in glass. For increasing periods the SHG peak undergoes a sudden increase reaching a local maximum in $P_x={620}\;\textrm{nm}, \theta _{AOI}={11.5}^{\circ}$ (Fig. 5(b)), corresponding to the double resonance with $\lambda _{SH}$ in air and $\lambda _{FF}$ in glass. The other double resonance of these two SLR at $(P_x={1140}\;\textrm{nm}, \theta _{AOI}={33.5}^{\circ})$ leads also to a small signal peak (not shown). On the other hand, the double resonance with the $\lambda _{SH}$ SLR in glass as well as the simultaneous excitation of both $\lambda _{FF}$ SLRs does not show an enhancement. In the TE case for ${520}\;\textrm{nm}<P_x<{700}\;\textrm{nm}$ (Fig. 4(f)), the SHG rises for increasing periods, reaching a global maximum in $P_x={620}\;\textrm{nm}, \theta _{AOI}={11.5}^{\circ}$ (Fig. 5(c)). The effect of the double SLR in glass manifests as a fast decay of the peak for longer periods. In the TE case there is another enhancement region $P_x\geq {1080}\;\textrm{nm}$ (Fig. 4(f)), where two very close SLR peaks merge into a single peak for several periods after crossing with a third resonance (second order at $\lambda _{FF}$ in glass). Other double resonances show minor effects: near the fundamental SLR ($P_x={860}\;\textrm{nm}, \theta _{AOI}={13.5}^{\circ}$), a relative SHG maximum is observed. The TE polarized light provides higher conversion efficiencies than TM polarized light, related to the stronger field enhancement. The highest conversion efficiency is achieved for TE polarized light at the double resonance. On the other side, for TM inputs the strongest signal is observed at large angles and short periods along the SLR in air (Fig. 4(e) and 5(b)), where the SHG is slightly lower than in the double resonant case. We ascribe the high efficiency to the larger E-field asymmetry and the longer coupling length between the incident beam and the metasurface. We observed a saturation at large incidence angles $\theta _{AOI}>$45°. At normal incidence, the signal is usually below the detectable limit, except when the SLR is close to normal incidence, see Fig. 4(f) at $P_x\approx {700}\;\textrm{nm}$. This is probably caused by a weak quadrupolar mode [13] or small fabrication imperfections [39] enhanced by the resonance.
Fig. 5. Measured average second harmonic power versus the incidence angle $\theta _{AOI}$ for different periods (different colored curves in each panel) and input polarizations (different rows).
An analyzer at the output is used to determine the polarization of the SHG. Independently from the incident polarization, the SHG component along the $x$-direction (TM output) is always dominant. Indeed, the SHG polarization is determined by the spatial symmetry breaking, which occurs along the $x$-direction due to the tilted input wavefront [38], see Fig. 3.
3.2 SHG characterization
To verify that the origin of the detected signal is a second order nonlinear process, the emitted spectrum is analyzed via a compact back-thinned CCD spectrometer. In the bandpass filter limited range from 500 nm to 789 nm, only a narrow peak is observed around the second harmonic, as shown in Fig. 6(a). Thus, two-photon luminescence is negligible in our case [40]. The center wavelength of the SHG (516.6 nm) is slightly red-shifted with respect to the half excitation wavelength, probably due to the asymmetry between forward and backward scattered light [41]. Furthermore, the central wavelength of the SHG does not change noticeably with the incidence angle or the lattice period $P_x$ (not shown here). A typical angular dependence of the SHG spectrum is shown in Fig. 6(b). Sharp discontinuities are clearly visible at the indicated RAs, their positions changing with both the angle $\theta _{AOI}$ and the wavelength of the nonlinear signal. Noticeably, a spectrally-resolved measurement permits to reveal the sharpness of the transition. The sharp features get indeed dimmed when using a detector uncapable of distinguishing the different wavelengths.
Fig. 6. (a) Normalized spectra of the pump beam (red curve, top horizontal axis) and the corresponding SHG (green curve, bottom horizontal axis) at $P_x={760}\;\textrm{nm}$ and $\theta _{AOI}={20}^{\circ}$. (b) Normalized SHG spectrum for $P_x=800$nm at different angles of incidence $\theta _{AOI}$; the superposed solid lines show the RA for the fundamental (red color) and the second harmonic (green color).
The origin of the SHG signal is also verified by measuring the input-output power, see the double-log plot in Fig. 7(a) for the case of TM polarization and double resonance. The output power ramps up with the average input power, reaching a threshold around ${180}\;\textrm{mW}$ (corresponding to a peak intensity of ≈15 GW cm−2) where the sample gets permanently damaged. The maximum average output power is 190 pW, corresponding to a maximum SHG conversion efficiency of 1.06×10−9. Taking the metasurface thickness into account, the maximum second order nonlinearity is around 1.0 pm V−1 (for more details, see Supplement 1), which is of the same order of magnitude with respect to similar plasmonic metasurfaces [11,27,42]. Polynomial best-fitting finds a linear slope in the double log scale of 2.35. The exact value of the slope depends on the incidence angle and on the period. Slightly steeper slopes than 2 have been already reported for nonlinear processes in metallic nanostructures [27,43,44]. Such a deviation from a pure square dependence can originate from the spatial and temporal nonlocality associated with the excitation of electrons in an energetic band [45]. We finally investigated what happens when the sample is damaged. The recorded images in Fig. 7(b) reveal that above 180 mW average input power a black area is formed in the center of the SHG spot, where the nano-structure is permanently modified. Increasing the excitation power above this threshold leads to a wider modified area but also to a larger SHG spot. The output signal thus gradually decreases above the permanent damage threshold.
Fig. 7. (a) Measured average SHG output power (black dots) and its fit (red dashed line) versus the average pump input power; a double logarithmic scale is used. Fitted power function: $P_{out} = c_0 P_{in}^{a}$, where $a=2.35$ and $c_0=1.30\cdot 10^{-8}\text {W}^{-1.35}$. (b) Pictures of the emitted beam at different pump powers. The gray scale is normalized to unity. Here $P_x={760}\;\textrm{nm}$, $\theta _{AOI}={21}^{\circ}$, and the input is TM polarized.
4. Interpretation of the results
The simplest way to describe SHG from metallic metasurfaces is the nonlinear scattering technique first described in Ref. [28]. Using the Lorentz reciprocity theorem, the electric field at the second harmonic $\mathbf {E}_\mathrm {SH}$ in the far field is given by the following surface integral extended to all the gold interfaces [29]
A more physical interpretation can be given in correspondence to the lattice resonances. Owing to the momentum conservation, the pump beam efficiently excites a propagating delocalized surface plasmon at $\omega$. The second-order nonlinearity of the metasurface then generates a plasmon at 2$\omega$. The last step is the coupling of the plasmon at $2\omega$ with the free space: this is maximized when a RA in the linear regime is observed at 2$\omega$.
We finally compare the predictions of the nonlinear inverse scattering model with the measured data. Specifically, in Fig. 8 the theoretical (red lines) and experimental (black lines) normalized SHG in correspondence to the calculated RA in air (Eq. (1)) for $|m|= 1$ and $\lambda _{FF}={1032}\;\textrm{nm}$. For TM polarized pump (Fig. 8(a)), the experiment and simulation have a peak near the double resonant condition $P_x={760}\;\textrm{nm}$. For lower periods the SHG power decreases, and the experiment features a lower depression than predicted. The signal reaches another maximum at $P_x={600}\;\textrm{nm}$. For even shorter periods and higher incident angles the signal suddenly drops because the calculated RA deviates from the SHG peak. The experimental results in Fig. 5(b) show that the maximum signal stays constant for higher incidence angles. In the TE case plotted in panel Fig. 8(b), the experiment shows a broad peak due to the interaction between the double resonances at $P_x={760}\;\textrm{nm}$ and the crossing with the SLR of the pump wavelength in glass at $P_x={840}\;\textrm{nm}$. Interestingly, the latter does not cause any visible effect in the TM case. The simulations just predict a narrower peak at $P_x={840}\;\textrm{nm}$, not the double resonance. Limitation of the simulation is the prediction of the SHG enhancement at the second harmonic SLR. Finally, we stress out that the numerical simulations also confirm that i) the main contribution of SHG is TM polarized, no matter what the polarization state of the pump is; ii) The SHG is stronger for TE than for TM polarization input.
Fig. 8. Experimental (black lines) and numerical (red lines) SHG power versus the period $P_x$ along a SLR at the fundamental frequency. The chosen incidence angle $\theta _{AOI}$ for each $P_x$ obeys the SLR condition for $\lambda _{FF}={1032}\;\textrm{nm}$ in air for $|m|=1$. Both the quantities are normalized to unity. Left and right panel correspond to TM and TE input, respectively.
5. Conclusion
We experimentally demonstrated the role of double surface lattice resonance in maximizing the SHG in plasmonic metasufaces. Our parametric study on centrosymmetric 2D nanobar arrays confirms [14] that local maxima of the SHG occur in correspondence to the SLR either at $\omega$ or 2$\omega$, thus leading to a further enhancement of frequency conversion when both pump and second harmonic SLR get excited simultaneously. Such a result has been predicted based upon the discrete dipole approximations [26] and recently experimentally confirmed by tilting the sample along two orthogonal directions [27]. Here we created the double resonance by tilting the metasurface with a specific period in just one direction. The close connection between the linear and the nonlinear response of the metasurface has been theoretically confirmed using the nonlinear inverse scattering approach, providing a generalization of the Miller’s rule to the SLR case [29]. Finally, due to the high quality of our samples and the fact that we are exploiting the double resonance, we achieved a maximum SHG conversion efficiency of 1.06×10−9. Such a comparatively large efficiency, in combination with high average excitation power, allowed us the direct measurement of both the spatial profile and spectrum of the generated second harmonic.
Straightforward extensions of our work include nano-antennas of different shape, with specific attention to nano-gapped resonators. A deeper theoretical treatment will be pursued in order to fully understand the nonlinear coupling between the nonlocal surface plasmons propagating across the metasurface, for example concentrating on the role played by the multipolar response of the elemental antenna.
6. Appendix
6.1. Fabrication
To fabricate 2D gold nanobar arrays, we have used a standard metal lift-off process. First, we took the clean fused silica substrate and pattern the nanostructures using electron-beam lithography on a bi-layer of positive tone resists (300 nm ARP617.06 and 100 nm ARP6200.04) using a 10 nm gold layer as a conductive surface on top. The sample was then developed in AR600-546 for 30 s to remove the ARP6200.04 resist layer. The remaing ARP617.06 resist was then developed in MIBK:IPA (1:1) for 80 s. A 50 nm Au layer was then deposited using an electron beam evaporator at 1 nm s−1 deposition rate at a pressure lower than 2×10−5 mbar, using a 3 nm adhesion layer made of titanium. After the deposition process, a lift-off based on acetone and isopropanol is performed to realize the gold nanobar arrays on the fused silica substrate.
6.2. Numerical simulations
Linear optical transmission spectra of a 2D array of nanobars were simulated using a commercial Finite Difference Time Domain (FDTD) solver by Ansys Lumerical and Rigorous Coupled Wave Analysis (RCWA) [46]. The gold optical constants used for simulations were taken from Ref. [47]. Transmission computed from FDTD and RCWA results are in very good agreement with each other, regardless of the geometric parameters of the structure. Due to the larger computation time required by the FDTD simulations, RCWA is a better choice for investigating the parametric dependence of the optical response of the nanostructures. In RCWA-based simulations, 225 spatial harmonics were enough to obtain a good match with the experimental results for the 2D grating. After the simulation, the reflection from the backside of the substrate was taken into account in order to be comparable with the experiment. The map of the electric fields on the surface of the nanostructures is simulated using FDTD solver. FDTD solutions are also used in the computation of the overlap integral providing the SHG according to the nonlinear inverse scattering technique. In all FDTD simulations, Bloch boundary conditions were used in $x$ and $y$-directions, whereas PML (Perfectly Matched Layer) boundary conditions were applied along $z$. The nanobar object was meshed 5 nm on the transverse plane $xy$-directions and 2 nm on the $z$-direction using mesh override region.
6.3. Nonlinear detection
The optical intensity at the sample plane is imaged onto a standard CMOS camera to investigate the emitted SHG power. Bandpass filters limit the detectable spectral range from 500 nm to 540 nm. The signal detected on the camera is calibrated versus the SHG of a nonlinear crystal, once the losses in the system were taken into account. The linear behavior of the camera in the range of interest is verified. The validity of the camera measurements at low signal is also cross-checked with a photo-multiplier tube. We verified that the input-power dependency of the SHG from a bulk nonlinear crystal (BBO) show a quadratic behavior (slope 2.04). We use a camera because several SHG output beams, parallel to the original 0th transmission order, can arise at high incidence angles and large periods, due to multiple reflections inside the glass substrate. The camera allows an easy distinction of these few millimeters separated replicas. In addition, the beam is laterally displaced when the whole substrate is rotated around the $y$-axis. Finally, for better visibility we only showed the response for positive angles, but we scanned from −70° to 70° to ensure that the SHG response under the employed illumination conditions is equal for negative and positive angles within our experimental accuracy. Further details regarding the experiment can be found in the Supplement 1.
Funding
Deutsche Forschungsgemeinschaft (CRC 1375 NOA, 398816777).
Acknowledgements
We acknowledge the valuable support of Werner Rockstroh, Natali Sergeev, Detlef Schelle, Holger Schmidt and Daniel Voigt in the fabrication of the nano-structured samples.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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