Optimizing the Nonlinear Optical Response of Plasmonic

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Optimizing the Nonlinear Optical Response of Plasmonic Metasurfaces Yael Blechman, Euclides Almeida, Basudeb Sain, and Yehiam Prior Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b03861 • Publication Date (Web): 12 Dec 2018 Downloaded from http://pubs.acs.org on December 12, 2018

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Optimizing the Nonlinear Optical Response of Plasmonic Metasurfaces Yael Blechman, Euclides Almeida, Basudeb Sain, Yehiam Prior Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100, Israel [email protected], [email protected]

Abstract Controlling the nonlinear optical response of nanoscale metamaterials opens new exciting applications such as frequency conversion or flat metal optical elements. To utilize the already well-developed fabrication methods, a systematic design methodology for obtaining high nonlinearities is required. In this paper we consider an optimization-based approach, combining a multiparameter genetic algorithm with 3D Finite-Difference Time Domain simulations. We investigate two choices of the optimization function: one which looks for plasmonic resonance enhancements at the frequencies of the process using linear FDTD, and another one, based on nonlinear FDTD, which directly computes the predicted nonlinear response. We optimize a FourWave-Mixing process with specific predefined input frequencies, in an array of rectangular nanocavities milled in a thin free-standing gold film. Both approaches yield a significant enhancement of the nonlinear signal. While the direct calculation gives rise to the maximum possible signal, the linear optimization provides the expected triply resonant configuration with almost the same enhancement, while being much easier to implement in practice. Keywords: Metamaterials, Optimization , Four-wave mixing, Plasmonics, Local field enhancement, Spatial overlap integral

Main Text From the early days of nonlinear optics, interactions such as Raman or wave mixing have been studied in gas-phase molecules or solid-state crystalline systems, in which the energy levels are fixed, and in order to achieve resonance enhancements one tunes the laser frequencies to the molecular resonances1. Advances in the design and fabrication of nano structures open up the possibility to do the reverse: namely for fixed laser frequencies, one tunes the ‘system’ for resonance enhancement. The resonant local field enhancements, induced by surface plasmons, greatly reduce the interaction length typically required for creating a strong nonlinear effect2. This significant scale down in the overall size of nonlinear devices enables their integration into a nanophotonic circuit, thus, leading the way for new nanoscale nonlinear optical components3–5. Nanoscale nonlinear devices can also be useful for applications such as sensing6, superresolution imaging7,8, light manipulation9–11, and in terahertz generation12.

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In arrays of metallic particles, once the separation between the particles exceeds 20-30 nm, the rapid decay of the evanescent field outside the metal causes the spectral response of the array to be essentially identical to that of a single particle. Metasurfaces consisting of arrays of nanometric cavities in thin metal films, however, are very different: the localized surface plasmons (LSP) generated at and near a cavity couple to propagating surface plasmons (SPPs) in the metal film, which, in turn, couple back to local fields in the neighboring cavity, giving rise to a strongly coupled set of individual sources. These nanohole arrays provide high versatility in designing their linear response, consisting of both LSPs, SPPs and cavity modes. Several works have investigated nonlinear enhancement in metallic nanohole arrays13–21, however, not many studied third-order processes, and even fewer considered the question of optimizing these processes. It is well known that the nonlinear interaction depends both on the frequency response of the designed structure 22, as well as the coupling efficiency of the different modes in the near field 23,24 . For a given set of metasurface parameters, we are able to calculate the nonlinear response of the structure directly using nonlinear 3D Finite-Difference Time Domain (FDTD) solution to Maxwell’s equations25. Such a calculation takes into account all the plasmonic interactions and parameter dependencies, and will yield the universal optimum. However, performing such a calculation repeatedly during an optimization process is both complicated to set up and quite demanding in terms or computational resources. Thus, one of our goals is to develop a simpler approach for optimization, using only linear FDTD calculations. The dependence of the linear optical properties of nanohole arrays on hole size, shape and array parameters has been extensively studied26-37, see Supporting Information. The Extra Ordinary Transmission (EOT) through a nanocavity array, as shown by Ebbesen26,27, when normalized to the cavities area, can even be larger than unity for certain structures and at well-defined frequencies. The high EOT at a given frequency results from plasmonic resonances (SPP or LSP), indicating high local field enhancement at this frequency. This relationship between far field transmission and local field enhancements suggests that in order to optimize a particular nonlinear optical process, it is a good idea to look for structures which exhibit high EOT at the relevant frequencies. Based on this approach, we have recently demonstrated an order of magnitude enhancement of the Four-Wave Mixing (FWM) response of an array of rectangular cavities by tuning a single parameter, the aspect ratio of the holes21. These results suggest a broader optimization approach, taking into account all the available degrees of freedom to obtain structures with resonant enhancements at all the relevant frequencies. In what follows, we define a multi-parameter objective function, which is based on the far-field linear transmission spectrum of the structure. In fact, this function is the product of the linear transmission coefficients at the frequencies corresponding to the FWM process. Using this function in a genetic algorithm optimization combined with linear FDTD computation, we find a configuration that gives rise to a triple resonance situation, where strong transmissions are found at all three relevant frequencies of the NL process. The obtained configuration is found to have a large FWM signal enhancement. In parallel, we introduce a different optimization function, which is based on our ability to directly calculate the nonlinear FWM signal for a given geometry. Here we show that the optimal set of parameters is

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different from the previous one and it gives rise to a configuration with slightly better nonlinear response, but which does not follow the intuitive triply resonant linear transmission spectral response. We discuss both approaches and compare the predictions to experiments. While in this paper we consider Four Wave Mixing from an array of nanocavities, the same methodology is applicable to other NL processes and other geometries.

Figure 1: Illustration of the FWM process in a nanohole array

Consider the FWM process 𝜔"#$ = 2𝜔' − 𝜔) , where the main input frequency 𝜔' is chosen to match the Ti:Sapphire laser frequency at 800 nm and the goal is to maximize the FWM signal in visible range, which was chosen arbitrarily to be at 622nm . This choice of frequencies dictates the second input beam to be at 1127nm. The process is illustrated in figure 1. Our metasurface (MS) consists of rectangular arrays of nano-sized rectangular holes milled in a thin freestanding gold film (see figure 2, and also the Supporting Information, Methods section). There are two reasons why we opted for free standing gold films. The first and more important one is the need to eliminate the nonresonant FWM signal from the much thicker substrate (i.e. glass), which, due to the quadratic dependence on the propagation length would swamp the signal from the metasurface. The second reason is the better mode overlap between the incoming/outgoing light and the surface modes when the dielectric layer is identical at the two sides of the film28. The hole dimensions are in the intermediate range of few hundred nanometers (50-700nm), which puts our metasurface in between the nanometer-size regime (much smaller than the wavelength) and photonic crystal (larger than the wavelength) regime. For a given film thickness, this system has 4 degrees of freedom (figure 2): hole dimensions (𝑥+ , 𝑦+ ) and array periodicities (𝑃/ , 𝑃0 ). These 4 parameters determine the resonant frequencies supported by the system - LSPs, SPPs, and cavity modes (because of non-negligible thickness) - and thus the entire optical response. Naturally, a much wider parameter space is possible where the individual cavity shape is also changed, but this is outside the scope of the current work. Beyond these four parameters, it is possible, of course, to change the film thickness, and this parameter will affect the tuning of the cavity mode resonance. In the Supporting Information we show such an investigation (including results of measurements) when changing also the thickness parameter.

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Figure 2 The definition of the parameters used in the optimization of the metasurface

For both optimization strategies, we are using Genetic Algorithm (GA) method29 combined with FDTD simulations for the candidate configurations. GA is used extensively in optics, for example pulse shaping for coherent control30–34, and more recently for plasmonic design35–39 in the linear regime. The GA is inspired by the biological evolution. The process is initialized by randomly selecting an initial population, the “first generation” of “individuals”, where each individual represents a point in the parameter space (the values of the parameters are its “genes”). The algorithm proceeds by evaluating each individual according to the specified objective function. Those individuals receiving the larger values (more “fit to survive”) are then used to construct the next generation of individuals, by using various “biological” operators: “mutation”, when the genes of a single individual are randomly modified within a predefined range; and “cross-breeding”, when the genes of two individuals (“parents”) are mixed to create a new individual (“offspring”). This creates a continuous process of improvement from one generation to the next, hopefully converging to an optimized individual (convergence is reached when there is no more improvement after moving to the next generation), which is presumably the best one within the defined parameter range. The genetic algorithm is an efficient optimization strategy in cases where the derivative of the objective function is not known, or does not exist, or when the function is not even known to be continuous as it is being numerically calculated at discrete points. As is known in optimization theory, any optimal configuration found by such an optimization process presents a local optimum over a specific range or parameters, and cannot be guaranteed to be the absolute global maximum. At each step of the GA, we calculated the optical response for every “individual” configuration by numerically solving the Maxwell’s equations by means of 3D FDTD method using the Lumerical commercial FDTD package40. We then evaluated each individual according to one of the two objective functions, as described below. For our FWM process, two photons at frequency 𝜔' interact with one photon at frequency 𝜔) to generate a photon at 𝜔"#$ . As is well-known from the theory of nonlinear optics, the generated

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power is proportional to the product of the generating fields |𝐸"#$ | ∝ |𝐸' |) |𝐸) | inside the nonlinear material. Strong EOT in the far-field at some frequency strongly hints at large local field enhancement inside the metasurface at this frequency. Thus, we substitute this product with the product of the transmission coefficients in the far field at those frequencies. Furthermore, in order for the FWM signal to be detectable in the far field and not decay inside the metasurface in the process of its generation, the corresponding transmission coefficient for this mode should also be high. So finally we arrive at the following optimization function, which should provide a reasonable prediction of the FWM enhancement: 𝐹(𝑥) = 𝑇 ) (𝑥, 𝜔' )𝑇(𝑥, 𝜔) )𝑇(𝑥, 𝜔"#$ ) , where 𝑇(𝑥, 𝜔) is the transmission at frequency 𝜔 for configuration 𝑥. As noted, the input frequencies are fixed, and the 4 component vector 𝑥 = (𝑥+ , 𝑦+ , 𝑃/ , 𝑃0 ) defines a set of parameters for a given geometry. The sample thickness was kept fixed and equal to 320nm. The GA optimization process finds the configuration giving the highest value of the target function 𝐹(𝑥). Following the above procedure, the best configuration, the one with the highest value of the target function, was found to be for X1=(400,100,860,620), where dimensions are listed in units of nm. The resulting normalized linear transmission spectrum is depicted in figure 3a.

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Figure 3 Optimization results for the two objective functions. First row – linear optimization (a) The calculated spectrum of the linearly optimized winning configuration: X1=(400,100,860,620). (b) Calculated FWM signal over a range of parameters around the winning configuration X1 (c) The values of the target function, 𝐹(𝑥) = 𝑇2 (𝑥, 𝜔1 )𝑇(𝑥, 𝜔2 )𝑇(𝑥, 𝜔𝐹𝑊𝑀 ), calculated over the same range of parameters, indicating strong similarity to the nonlinear prediction. Second row – direct optimization. (d) The linear transmission spectrum of the nonlinearly optimized winning configuration X2=(260,130,790,630) with two resonances and without any resonance at the third frequency. (e) The calculated FWM signal over a range of parameters around the configuration X2.

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One can immediately realize that the optimization process found a configuration that gives rise to a linear spectrum with pronounced resonances at the required frequencies. Note that it is not apriori obvious that a configuration that is triply resonant at the relevant frequencies even exists. In the Supporting Information we provide detailed analysis of the parameter dependence and the field distribution at each one of the resonant frequencies, and based on this analysis we can identify the peaks in the spectra as follows. The high-frequency mode at 622nm is a radiative SPP-Bloch mode, resulting from the coupling (via the cavities) between the SPPs from the opposite sides of the film. The sharp dip at 646nm is the Wood’s anomaly. The peak around 800nm is a cavity mode, also called Fabry-Perot type resonance. These three features create what is known as Fano-type resonant behavior. The lowest frequency resonance is an LSP mode. To substantiate the optimality of the obtained configuration, we calculated (and later fabricated and measured) sets of configurations in the neighborhood of the optimal one, changing only the hole dimensions (𝑥+ , 𝑦+ ). Figure 3c depicts the value of the target function in the linear optimization scheme for a range of parameters around the winning configuration X1 which gives the maximum. Figure 3b depicts the calculated FWM response for the configurations with the same range of parameters. One can clearly see that the calculated FWM response closely follows (even though it is not identical) the distribution of the values of the linear target function, strongly suggesting that the linear optimization scheme is a very rational approach to finding the maximal FWM response. As noted, an alternative approach to the optimization is to use the directly calculated FWM response as the target function that is to be optimized in the GA scheme. The nonlinear response can be calculated within the commercially available Lumerical FDTD package. Naturally, the nonlinear numerical calculation is more involved, and the optimization is more time and resource demanding. The best directly optimized configuration was found to be X2=(𝑥+ =260,𝑦+ =130,𝑃/ =790,𝑃0 =630) (in units of nm). Its calculated linear transmission spectrum is shown in figure 3(d). As with configuration X1 we investigated the nature of the resonances in the Supporting Information. Again, in order to demonstrate the optimality of the configuration X2, we calculated (and later fabricated and measured) sets of configurations in the neighborhood of the optimal one, changing the hole dimensions (𝑥+ , 𝑦+ ).The results are depicted in figure 3(e), where it can be seen that as expected, the GA indeed found the optimal configuration within this range of parameters. While the parameters of the two winning configurations do not seem to be very different, the linear transmission spectrum is very different: the linearly found configuration X1 gives rise to the expected triply resonant spectrum, whereas the directly nonlinearly optimized configuration X2 yields a spectrum that is only doubly resonant and does not exhibit a peak at the frequency 𝜔) . Thus, the intuitive expectation will be for a higher FWM signal for configuration X1. However, the theoretically calculated FWM intensity for X2 is 3 times higher than the one calculated for the

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configuration X1 (see table 1 for details). When we checked the value of the linear target function 𝐹(𝑥) for the configuration X2, it was found to be four orders of magnitude smaller than the value obtained for X1 (third row of table 1). To conclude this discussion of our optimization schemes: we have performed two different GA optimization procedures – linear and nonlinear. For both schemes we found and compared the winning configurations – the linear optimization yielded a triply resonant configuration which gave significant enhancement of the FWM signal whereas the direct nonlinear optimization resulted in a configuration that gives rise to a nonintuitive linear transmission spectrum with peaks only at two out of three relevant frequencies. This configuration, however, gives rise to a stronger FWM signal, a discrepancy that we explain below.

Figure 4: The SEM image of a fabricated array.

We fabricated the two configurations X1 and X2, as well as several nearby configurations with similar parameters, and measured their linear and nonlinear FWM response. As noted above, we used freestanding gold films to eliminate the non-resonant background generated in the much thicker substrate. Each fabricated configuration contains 40x40 holes array, a size which was experimentally found to provide a good approximation for an infinite array used in simulations. Figure 4 shows a SEM image of the center part of a particular array. The measurements (setup explained in detail in Supporting Information) were performed using white light source for the linear transmission measurements, and with Ti-Sapphire and OPA for the nonlinear measurements. The measured linear transmission spectra are depicted in figure 5(a) – for the linearly optimized configuration X1, and in figure 5(b) for the directly nonlinear optimized configuration X2. We find good agreement with the theoretically calculated spectra shown in figure 3(a) and 3(d). The generated FWM signal was measured to be similar for both optimized configurations X1 and X2, again in good agreement with the theoretical predictions, see figure 3(b,e).

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Figure 5 The measured linear (a,b) and nonlinear (c,d) signals. The linear response for (a) X1 and (b) X2 is in good agreement with theory, see figure 3(a,d). The nonlinear response was measured over a range of parameters near (c) X1 and (d) X2 .The measured intensity values are shown as vertical bars, superimposed on the same calculated values of the nonlinear signal as in figure 3(b,e), demonstrating good agreement.

The optimization that is based on the NL calculation improved the resulting FWM enhancement, and this is not surprising, as the calculation takes into account the actual NL FWM process, including both the frequency response and the mode structure. What was not anticipated is that the corresponding linear transmission spectrum does not match our expectations for triply resonant behavior. The linear optimization which is based on the far field transmission indeed found significant enhancement, but missed a less resonant configuration which yields still higher FWM signal. Additional pair of example configurations, X3 and X4, with the same behavior is investigated in the Supporting Information. These results can be explained by the difference in the linear response of the two structures, by analyzing the electromagnetic field distribution inside the metasurfaces. The nonlinear optical interaction between the three incoming waves via the material nonlinear susceptibility 𝜒 (=) generates the outgoing FWM. This interaction occurs exclusively in the active region at the metal-cavity interfaces and right next to them. As had already been observed many years ago, when several terms (pathways) contribute to a nonlinear process, their constructive (destructive) interference is important41. For molecules, it is

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the phase resulting from the quantum mechanically interfering wave functions for each transition. For nanostructures it is the Spatial Overlap Integral (SOI) of the field mode distribution inside the sample that determines the interaction. This factor is not present in Angstrom size objects such as molecules but could be relevant in structures of dimensions which are comparable to (even if smaller than) the optical wavelength in the material. The nonlinear signal generated inside a metasurface can be expressed42–44 using the Lorenz reciprocity principle, by the Spatial Overlap Integral (SOI) ZZZ (3) 2 ⇤ † NL ⇡ E1 E2 EF W M dV (1) Gold

> is the time-reversed field at the where 𝐸' , 𝐸) are the input beam local electric fields, and 𝐸"#$ FWM frequency. These values vary as function of the spatial coordinates throughout the metasurface. The integration is performed over the volume where the nonlinear susceptibility is nonzero, i.e. over the points in space inside the gold (and in particular excluding the nano holes). Using the expression for the SOI, our calculations show that the mode overlap for the NLoptimized configuration X2 is similar to that of the linearly optimized configuration X1. Thus, as can be seen in table 1, this overlap integral which is a result of the linear calculation, is a better linear predictor than the target function 𝐹(𝑥) which was used for the optimization, in the specific case of X1 and X2.

Inter alia, when looking at the field distribution in the three constituent frequencies of the FWM process for configurations 𝑋' and 𝑋) (figure 6), the near field at the anti-Stokes frequency (1127 nm) in 𝑋) is completely invisible in the far field spectrum. This critical importance of the contribution of dark modes and spatial mode overlap to nonlinear optical interactions had been noted before in the context of Fano resonances and their contribution to NL interactions45,46. Table 1 The values of the calculated nonlinear enhancement (using 3D FDTD), compared with SOI and the linear prediction function for the two configurations X1 and X2. The strength of the nonlinear signal is equivalent for the two configurations, the linear prediction function gives 4 orders of magnitude advantage to the linear far-field optimization, while the SOI metric gives the same order of magnitude value, which is much closer to the calculated nonlinear signal.

Calculated NL signal Spatial Overlap Integral (a.u) 𝐹(𝑥), linear prediction (a.u)

Linearly optimized (Configuration X1) 2.2 × 10C='

Directly optimized (Configuration X2) 7.2 × 10C='

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Figure 6 Cross-section at the hole center of the mode distribution for the two configurations at three different frequencies – 2 incoming and the FWM. The images are obtained via linear FDTD simulation. (a)-(d): linearly optimized configuration 𝑋' ; (e)-(h): direct FWM optimized configuration 𝑋) . The crosssection (XZ plane) shown is perpendicular to the direction of the polarization (Y axis), which is also the shorter side of the hole. The metal boundaries are shown by the black thick lines. The color code (different for each panel) indicates the local field enhancement factor relative to the input field.

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In order to investigate the overlap integral further, we have calculated its value in the neighborhood of the two configurations identified as optimal by the two methodologies, and compared it with the corresponding value of the NL enhancement. The results are shown in figure 7.

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Figure 7. Comparison of the SOI to the calculated FWM signal, in both cases the top plane is the SOI and the bottom plane is the calculated FWM signal intensity (a) for the neighborhood of X1, the optimal configuration as found by the linear optimization scheme. (b) for the neighborhood of X2, the optimal configuration as found by the direct calculation. All values are in normalized units (the normalization is done for each subfigure separately), because we are interested only in comparing the predictive power of the SOI.

Note the good match in figure 7b, which indicates that the SOI is a reasonable approximation for the full nonlinear calculation in this case. However, in other cases the SOI approximation is much less accurate. In our opinion, there is no real advantage to perform the optimization with the SOI objective function in place of the direct nonlinear calculation, both because they require similar computation resources (both time and memory), and also because the SOI results obtained will certainly be less accurate. To conclude, in this work we have developed an optimization approach to designing nanostructures with high nonlinear response, tailored to a specific nonlinear process with nonlinear signal in the visible regime. We proposed a specific target function, the product of the far-field transmission coefficients at the frequencies of the nonlinear process, and combined this novel function with a well-known evolutionary/genetic optimization strategy, obtaining a highly optimized structure which was triply resonant. Furthermore, the target function provided an accurate prediction of the nonlinear response in the neighborhood of the optimized configuration. A different approach was to actually calculate the FWM signal, and use this calculated result as our target function in the optimization procedure. This direct nonlinear optimization yielded a slightly higher FWM signal, but showed only a doubly resonant transmission spectrum. We explained these results in terms of local mode coherence (spatial mode overlap) within the metasurface, and derived reasonable agreement between predictions and observations. The optimization approach described in this paper is not restricted to a specific nonlinear process, and is also applicable to design of other types of nonlinear metasurfaces. Indeed, recently there is a

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surge of interest in nonlinear dielectric metamaterials. See in particular47,48 where efficient nonlinear nanoparticle structures have been suggested with signal in the visible and near IR range.

Associated Content Supporting Information Available: Description of optimization methodology, sample fabrication, optical measurements. Identification of resonant modes in the optimized configurations. Additional comparison between the optimization strategies in a larger parameter space.

Author Information The authors declare no competing financial interest.

Acknowledgement We thank Ori Avayu and Tal Ellenbogen for their help with the linear transmission measurements. We thank Peter Nordlander for useful discussions and for bringing relevant earlier work to our attention. EA acknowledges the support of the Koshland Foundation. This work was supported,

in part, by the Minerva Foundation and by the ICORE program of the Israel Science Foundation.

References (1) (2) (3) (4) (5) (6) (7) (8) (9)

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