Linear and third-order nonlinear optical properties of self-assembled plasmonic gold metasurfaces

2.1 Lithography

Lithography is a top-down approach to metasurface fabrication in which patterned structures are etched from a thin layer of plasmonic material such as gold or other noble metals. An advantage to lithographic techniques is the ease with which a variety of structures can be created. However, only small areas can be patterned at significant cost and time. There are two primary lithographic techniques that are useful in creating metasurfaces: electron beam lithography (capable of sub-10 nm resolution) and focused ion beam lithography (resolution of tens of nanometers). Electron beam lithography uses a beam of electrons to write patterns into substrates point by point, whereas focused ion beam lithography uses focused ions to ablate substrates at a faster, although still slow, rate [7]. This makes lithographic methods suitable for proof-of-concept demonstrations [39].

A recent variation of the lithographic approach for the formation of plasmonic metasurfaces is shadow-sphere lithography [40], [41]. In this technique, a self-assembled monolayer of polystyrene spheres is deposited onto a substrate and then isotropically etched to reduce the diameter of the spheres, creating tunable interparticle gaps (50–120 nm). A metal vapor deposition source is then directed at an angle toward the surface using the polystyrene spheres as a mask. Multiple layers of material can be deposited from different angles to form increasingly diverse structures. The polystyrene spheres are then removed from the film, yielding an ordered array of similar structures. This method has a spatial resolution of 10 nm and is limited to structures compatible with 6-fold rotational symmetry due to the hexagonal packing of polystyrene spheres.

2.2 Block copolymers

In the block copolymer approach, a substrate is patterned with a block copolymer layer, for instance by spin coating micelles of poly(styrene-block-2-vinylpyridine) (PS-b-P2VP) [42]. These substrates are then incubated in a nanoparticle solution, resulting in clusters of nanoparticles with <5 nm interparticle spacing within a cluster and down to 10 nm gaps between clusters. These interparticle distances have been further decreased to as small as 2.8 nm by patterning the block co-polymers onto thermal shrinkage film [43], suggesting a method of overcoming a top-down limitation.

2.3 Evaporative methods

Direct evaporation of a nanoparticle suspension onto a substrate is perhaps the simplest method of metasurface fabrication. Nanoparticle assembly is achieved through a collective process of evaporation-induced capillary action, and interparticle and interfacial forces. Assembly may also be facilitated through substrate or nanoparticle functionalization. Surfaces may be electrostatically charged by coating with a polyelectrolyte [44], thiolated-polymers [45], alternating layers of polymer such as polyallylamine HCl (PAH), and polystyrene sulfonate (PSS) [46], for example. Alternatively, the nanoparticles can be functionalized to promote advantageous electrostatic or steric interactions among themselves or with the substrate. Alkanethiol functionalization of gold nanoparticles has produced dense nanoparticle films (2 nm separation) over millimeter-sized uncharged substrates [47]. Our own method, described at the end of this section, is similar to this technique at a larger scale. Gold nanoparticles have also been synthesized in the presence of alkane-thiols [48], yielding similar results to nanoparticles functionalized post-synthesis. Other non-thiol stabilizers have been used, such as 4-dimethylaminopyridine [44], which can be advantageously removed with water rinsing.

2.4 Dip coating

Dip coating is similar to evaporative drying, except the substrate is immersed in a nanoparticle solution and evaporation occurs during substrate withdrawal. In one example of this method using nanocubes, a gold-coated substrate was first coated in alternate layers of PAH and PSS [49], similar to [46]. Two possible routes could then be taken: either a droplet of nanocubes was deposited between the substrate and a coverslip with a 50 μm spacer or the sample was completely submerged and then withdrawn from a nanocube solution. The resulting metasurface in this case was sparse, randomly distributed, and with no structural control.

2.5 Layer-by-layer assembly

Layer-by-layer assembly is an alternative to the dip-coating method, which involves alternating cycles of modifying the surface chemistry and nanoparticle exposure in order to build up layers. Thiolate-stabilized nanoparticles form a denser metasurface than the nonfunctionalized nanoparticles used in the dip-coating method mentioned above and can result in strong interlayer plasmonic coupling over a centimeter-sized region [50]. In some cases, the procedure is carried out in an organic solvent and the nanoparticles are converted to Janus particles, i.e. particles in which one side exhibits a positive charge and the other side exhibits a negative charge, using a plasma-based surface modification technique between layers [50]. Alternatively, alkanedithiols can be used between each monolayer to functionalize the surface of the most recent monolayer [51]. These multilayer metasurfaces tend to show strong interlayer plasmonic coupling, an even thickness, but a random placement of particles in this cited example.

2.6 Langmuir-Blodgett

The Langmuir-Blodgett preparation of metasurfaces is related to the dip-coating technique. In this method, a dense nanoparticle monolayer with interparticles distances as low as 2 nm forms at the air/water interface following isothermal compression [52]. The substrate is inserted vertically into the liquid and withdrawn very slowly (~1 mm/min). This allows the monolayer to be carefully transferred onto the substrate. A dielectric spacer layer functionalized with alkanethiols can also be used to help control size, shape, and arrangement of the resulting monolayer [53].

2.7 Langmuir-Schaefer

Instead of inserting the substrate vertically into the liquid and slowly withdrawing it, the monolayer can be formed and then transferred horizontally onto a transmission electron microscopy (TEM) grid or a polydimethylsiloxane stamp pad by lightly contacting the surface of the water [54]. Depending on the concentration of nanoparticles in the film, the resulting surface can be small islands of monolayers, a monolayer with patches of bilayers, or a uniform bilayer. The density of packing can be improved by covering most of the trough with glass, leaving a defined space for film formation and substrate transfer. Solvent evaporation in the uncovered region fluxes the nanoparticles to the confined space, forcing a more densely ordered layer over an area of 50 μm² [55]. Alkane-thiols can also aid in increasing the packing density. If a bilayer is desired, multiple monolayers can be transferred after drying each layer.

2.8 Patterned surfaces

A hybrid approach that uses both bottom-up and top-down approaches is the use of patterned surfaces to guide the self-assembly process. A common method is to pattern a surface with electron beam lithography, followed by wetting the surface with nanoparticle solution. The nanoparticle droplet can then be dragged across the surface with a coverslip, causing the nanoparticles to fall into a predefined arrangement [56], [57]. Either the wells [58] or the nanoparticles [59] can be functionalized with thiols or complimentary DNA strands [60] to assist in aligning the nanoparticles in their wells. In these cases, metasurfaces comprised of well-separated nanoparticles are achieved. Alternatively, patterning larger wells can encourage the formation of efficiently packed small clusters of nanoparticles [61], [62]. Other templates include stretched elastomeric substrates, which, upon plasma oxidation, form wrinkles that the nanoparticles fall into [63].

2.9 Field assisted assembly

One final technique in ordering nanoparticles into metasurfaces is to use electric and/or magnetic fields to align the nanoparticles. These methods can only be used on nanoparticles that have a dipole, such as metal nanorods. Gold nanorods can be functionalized with nematic liquid crystals and drop cast onto a TEM grid [64]. In the presence of a weak magnetic field, a highly ordered three-dimensional superstructure is formed with mixed morphology and domains extending to several μm². Using a 1.3 T electromagnet results in an hcp array over an area of several hundred nm². Electric fields can also be used by trapping a liquid suspension between two electrodes with an applied DC field [65]. As the liquid evaporates, the ordering imposed by the electric field is maintained. This also has the advantage of being able to align nanorods perpendicular to the surface, which is difficult to achieve through other means.

The distinctive property of plasmonic nanoparticles is their ability to efficiently couple to light and confine it into subwavelength volumes. This results in a large local electric field enhancement on the surface of the nanoparticles, giving rise to unique optical properties. In order to maximize these properties in an assembly, the interparticle gap between adjacent plasmonic nanoparticles needs to become subnanometric. Such confinement of the local electric field leads to enhanced linear and NLO optical properties. As discussed previously, top-down approaches are currently constrained to smaller scale samples and gap distances greater than 5 nm, preventing the ability to maximize the optical properties through near-field enhancements. Bottom-up approaches have the potential to alleviate these issues, offering scalable, broadly tunable, and accessible materials. Film-coupled nanosphere geometry (and variants) is a prominent strategy in the literature for studying the fundamental optical properties of sub-nm gaps [66], [67], [68]. In this geometry, a mirror is coated with a monolayer of ligand spacers of defined lengths, typically less than 1 nm. Nanospheres are sparsely distributed on top of the ligands, creating nanometer-scale gaps between the nanospheres and their image particle created by the mirror, effectively creating an isolated dimer geometry. This technique works well for studying the optical properties at nanometer- and micrometer-length scales; however, it is not amenable to larger-length scales since the dimer mode is generally perpendicular to the substrate, limiting response to only obliquely incident light, or the nanosphere’s density is relatively dilute, constraining optical interactions.

The self-assembly technique used in this paper efficiently creates macroscale hexagonally packed films, enabling billions of tunable sub-nm gaps to be probed simultaneously, leading to larger changes in the optical response with respect to isolated dimer geometry. Gold metasurface films are fabricated by evaporation-induced self-assembly of gold nanoparticle mixtures onto partially submerged substrates. Pairs of alkane-thiols and alkane-dithiols assist in the assembly, the lengths of which determine the interparticle spacing, allowing for spacing as small as ~0.5 nm (atomic lattice of gold is 0.41 nm).

Glass slides serve as the substrate onto which the films are assembled. The glass slide is prepared by sonicating for 30 min in a 5% (w/v) potassium hydroxide and methanol solution and triple-rinsed with sonication in fresh 18.2 MΩ water. The prepared slides are stored in new 18.2 MΩ water until use.

Citrate-stabilized 15-nm-diameter gold nanosphere solutions with an optical density of approximately 1 are concentrated by a factor of 10 by centrifuging 50 ml at 13,000×g. 45 ml of supernatant is discarded and the nanospheres are redispersed in the remaining liquid.

Film assembly occurs in a 20-ml glass vial using a pair of alkane monothiols and dithiols of the same length. The resulting gap widths from using thiols consisting of alkane chains of 2, 3, 6, 8, 10, and 12 carbons (C2, C3, C6, C8, C10, and C12, respectively) are listed in Ref. [69]. For C4 and C6, 10 μl of each monothiol and dithiol is combined with 1 ml of tetrahydrofuran in the glass vial. For C2, C3, and C5, 20 μl of each monothiol and dithiol is used instead. 1 ml of the concentrated nanospheres is added to the THF/thiol vial and shaken for 10 s. A glass slide is added to the vial and briefly shaken in order to wet the slide. The substrate is leaned vertically against the side of the vial and the vial is then left uncapped and covered with an inverted 125-ml beaker. The film assembles for 3 h before the beaker is removed and the film is grown for another 3 h. The slides are then carefully removed from the remaining colorless solutions and the slides and vials are ambiently dried overnight. After drying, the side of the glass slide with the lower quality film is cleaned with THF to remove the film. Excess thiols are then removed from the higher-quality film side by gently pipetting THF over the film. The slides are again ambiently dried.

These metasurfaces are characterized optically using absorbance spectroscopy and structurally using TEM imaging. The absorbance spectrum of the film is taken by placing the film-coated slide into a UV-Vis spectrometer using a clean slide as the reference. TEM samples are prepared by adding 1 ml of water to the dried vial. Carefully tilting the vial to the side dislodges the gold film, causing it to float. A TEM grid is then submerged in the solution, and a piece of film is carefully lifted out of the vial using the TEM grid. The grid is ambiently dried for 1–2 h, then dipped in THF and allowed to dry before imaging.

3.1 Optical characterization

Linear absorption measurements generally employ commercially available spectrometers, which give highly accurate absorbance information on materials including thin films. Refractive index, on the other hand, can be measured by different methods, including ellipsometry [72], interferometry [73], spectral reflectance [74], and m-line and Brewster angle measurements [75], which are related to prism-coupling methods [76]. It should be noted that all methods require knowledge of the film thickness. In addition, the absorbance over the desired wavelength range determines which method can be employed. To illustrate the linear optical characterization of metasurfaces, we refer the reader to Refs. [27], [69] which also will be discussed below.

Figure 1 shows how the absorbance of a self-assembled gold metasurface comprised of nanoparticles redshifts with decreasing interparticle distance. This distance is determined by the length of the alkanethiol ligands used in the assembly process as described in Section 2. With gap sizes varying from 0.5 to 3.0 nm, a peak wavelength shift of 200 nm is observed, demonstrating the ease and sensitivity of absorption measurements in characterizing metasurfaces. The absorption measurements were performed using a commercially available spectrometer with an unpolarized white light source. For more detail, see Ref. [69].

Figure 1:

Metasurface absorbance peak as a function of gap.

(A) Interparticle gap as a function of the number of carbon atoms, C, in the alkane chains. The straight black line is a fit: gap (nm)=0.12 C+0.48 nm. (B) Normalized experimental absorbance spectra of the metasurfaces for each ligand length. (C) Experimental absorbance peak wavelength (black dots) of the metasurfaces as a function of interparticle gap. Finite element simulations of the metasurface with hexagonally close packing of the nanospheres with a local response (thick solid black line), a nonlocal response (thin dotted black, β=1×10⁶ m/s; thin dotted gray, β=3×10⁶ m/s lines), and local response from isolated dimers (thick solid gray line) as a function of interparticle gap. (From Ref. [69], with permission).

As mentioned before, several methods are available for refractive index measurement. One of the oldest and very reliable methods for thin films is ellipsometry, with review papers as early as 1963 [77]. The characteristic ellipsometric parameters are defined by an amplitude (Ψ) and a phase shift difference (Δ), which are related by the complex reflectance ratio ρ(λ, θ)=R^p/R^s = tan(Ψ)e^iΔ [27], [72]. R^p and R^s are the reflection coefficients for the parallel and perpendicular optical polarizations with respect to the film surface, respectively. This method was applied to the measurements described in Refs. [27], [69], as well as in Ref. [43] and an earlier work on gold nanoparticle thin films fabricated by layer-by-layer self-assembly [78]. As reported in [27], [69], ellipsometry measurements were performed in the vis-near infrared range (400 to 1500 nm), in 10-nm steps at an angle of 60° for the gold metasurfaces. A commercially available ellipsometer was used (J.A. Woollam Co., V-VASE) and the probed area on the metasurface was close to 7 mm². Secondary reflections from the glass-air interface were avoided. After collecting the ellipsometry measurements, a theoretical model is used to deconvolve the optical response of the material under study. In Refs. [27], [69], a three-layer model was used for the metasurface, consisting of the glass substrate, gold nanospheres, and an ambient air layer. As the metasurface is deeply sub-wavelength, it was modeled as being a homogenous layer. From independent measurements, the metasurface thickness was set at 16.6 nm in the model (15.4-nm-diameter gold nanospheres+0.59 nm ligand layer, from Ref. [27]). The dielectric function was modeled as several Gaussian oscillators in order to give a line shape consistent with the Kramers-Kronig formulation. Figure 2A and B shows the experimental (black solid line) and fitted (red dashed line) ellipsometry angles Ψ and Δ. Using the theoretical models indicated above, Figure 2C and D shows the real (C) and imaginary (D) parts of the linear index of refraction. The result is an average over five samples.

Figure 2:

Metasurface ellipsometry and linear refractive index spectra.

Spectroscopic ellipsometry data (solid black lines) of Ψ (A) and Δ (B) averaged over five metasurface samples composed of a monolayer 15.4±1.8 nm gold nanospheres capped with 1-propanethiol shells on a SiO₂ (Corning 1737) substrate. The dashed red lines are the three-layer model fitted to the experimental data. Real (C) and imaginary (D) parts of the linear index of refraction of the averaged metasurfaces obtained from the fits from Figure 2(A) and (B). (From Ref. [27], with permission).

As indicated in Ref. [27], a figure of merit (FOM), given by the ratio between the absolute maximum of the real part of the refractive index to its imaginary part (FOM=n/k), should be close to unity for plasmonic metals at visible frequencies. From the gold metasurfaces reported here, the average FOM is ~1.3.

3.2 NLO characterization

After its introduction in 1989 [79], [80], the single-beam Z-scan has been the most exploited method for measurements of third-order nonlinearities, giving the absolute value and sign for both NLR and NLA coefficient, from which Re(χ⁽³⁾), Im(χ⁽³⁾), and |χ⁽³⁾| can be inferred using the appropriate equations [35]. From the single-beam Z-scan, one can also obtain information on higher-order susceptibilities, χ⁽²ⁿ⁺¹⁾, n=2, 3…. A recent review on techniques for NLO characterization of materials, including nanomaterials, can be found in Ref. [36]. Although variations of the Z-scan can give time resolved information, as in the two-color Z-scan technique introduced in Ref. [81], one of the most used methods for time resolved measurements is the optical Kerr gate (OKG) technique, first introduced in the late 1960s [82] and reviewed in [35], [36]. Briefly, we will describe only those two methods, as they have been used in the work reviewed here. For the details and applications of other methods, Ref. [36] is recommended. Combining the Z-scan and OKG techniques under the same spectro-temporal conditions allows for the origin of the nonlinearity to be inferred from the theoretical models.

Figure 3 shows a typical experimental setup for simultaneous measurement of NLR and NLA. The measurement is performed by placing the sample in a focused beam and translating it along its z-axis through the focal region. The transmitted intensity versus the sample position allows the determination of the material’s NLO coefficients: the beam that goes to the detector with a small aperture in front of it will measure the NLR coefficient, whereas the beam that goes to the detector without an aperture in front of it, thereby collecting all of the incident light, measures the material’s NLA coefficient. The reference detector is used to normalize the pulse-to-pulse fluctuations that arise from the pump beam.

Figure 3:

Typical Z-scan setup, indicating the optical elements.

The insets show the Z-scan signatures for open and closed aperture. The vertical axes in the insets represent (1-normalized transmission).

Theoretically, the Z-scan method relies on the analysis of the phase distortion of the wavefront of the incident beam upon propagation through the NL medium, whereby the self-action of the beam leads to an index change, ∆n(r, I), and absorption change, ∆α(r, I), where r is the radial distance from the laser beam axis. A positive NLR (n₂>0) results in a self-focusing effect, and conversely, n₂<0 results in self-defocusing of the beam. The beam intensity detected through a small aperture (small compared to the beam cross-section) at the far-field varies with the sample position, and the experimental Z-scan signature obtained is shown in the inset of Figure 3. On the other hand, the detector in the far-field that collects the entirety of the incident beam will depict either NLA (which can be due to multiphoton absorption) or saturated absorption (SA). The typical signatures in this case are also shown in the inset of Figure 3. It is straightforward to qualitatively analyze the curves shown in the inset of Figure 3: for closed-aperture Z-scan, by scanning from the laser source, before the focal plane, (−z), to the detector after the focal plane, (+z), the dispersion-type curve has a valley followed by a peak for a self-focusing material, and conversely for a self-defocusing material. Far from the focus and at the focal plane, the transmittance measured through the aperture does not change. For the open aperture Z-scan, moving the sample in the same way as above, from −z to +z, a valley indicates NLA, and a peak indicates SA. If only negligible NLA occurs or if the signal is below the detection limit of the setup, the signature will be horizontal line will be the signature.

Quantitatively, it has been shown that the peak-to-valley transmittance variation, ΔT_pv, at a given wavelength, λ, and for a sample of length L, in the absence of NLA, is given by [36], [80]:

In Eq. (1), Δn₀ is the refractive index change at the focal point considering that third-order nonlinearity is dominant and is determined as Δn₀=n₂I₀, where I₀ is the irradiance at the focus. S is the aperture linear transmittance parameter, which is given by [80]:

The practical equations to infer the NLR and the NLA coefficients can be written as [36], [43], [71]:

T_{open (closed)} is the measured intensity dependent transmissivity when the aperture in front of the detector is fully open (or partially open), with a value defined by S [Eq. (2)], which typically varies from 0.1 to 0.8. α₂ and n₂ are the NLA and NLR coefficients, respectively.

ΔΦ0(3) is the nonlinear phase shift induced by the intensity dependence of the refractive index, where k is the wavenumber and L_eff is an effective length.

Figure 4 shows a typical closed- and open-aperture Z-scan measurement of CS₂ in a 1 mm cuvette at 800 nm, 100 fs, 1 kHz input beam. CS₂ is a well-studied NLO material, with a self-focusing NLR and negligible NLA at this wavelength [83]. Its NLR coefficient is n₂≈3.6×10⁻¹⁶cm²/W. It is common in experiments to use this value as standard and infer the intensity at the focal point.

Figure 4:

Closed (A) and open (B) aperture Z-scan for CS₂.

Closed (A) and open (B) aperture Z-scan for CS₂ (1-mm-thick cuvette). The laser beam intensity used was I₀=35 GW/cm².

As an example of a typical open-aperture Z-scan, Figure 5 shows the result for a thin (1.5 μm) lead-germanate amorphous film on a quartz substrate prepared by RF sputtering [84], obtained using 150 fs pulse duration from a Ti-sapphire laser at 800 nm. From the data, the value of the NLA due to TPA was found to be α₂=(3±1)×10³ cm/GW. Further discussion on the origin of such a high value of the NLA can be found in Ref. [84].

Figure 5:

Open aperture Z-scan for thin (1.5 μm) lead-germanate amorphous film on a quartz substrate at 800 nm (150 fs input pulse duration). (From Ref. [84], with permission).

The typical experimental setup for the OKG method is shown in Figure 6A. The pump and probe beams are linearly polarized, and the angle between them is 45°. The transmitted pump beam, with an intensity I₀, is prevented from reaching the detector by adjusting the analyzer. Due to the optical Kerr effect, the probe beam polarization is rotated, and a fraction of its intensity, I_P, is transmitted by the analyzer. The transmitted intensity through the analyzer is given by I_s=I₀sin²(ϕ/2) with ϕ=2πLn₂I_P/3λ where L is the sample length. In the limit of small induced rotation (I_P≈ 0), I_s is considered to be negligible, indicating that the signal detected is proportional to I0IP2.

Figure 6:

Experimental setup and typical signature of optical Kerr gate (OKG) signal.

(A) Typical experimental setup for OKG (adapted from Ref. [36]) (B) Normalized Kerr shutter signals for CS₂ (black solid squares) and the lead-germanium film (LGF, red solid circles). The inset shows the Kerr shutter signal for the film in an expanded scale (from Ref. [84], with permission).

Figure 6B shows the results for CS₂ and a lead-germanate amorphous film on quartz substrate, both obtained at 800 nm, 150 fs (from Ref. [84]).

The asymmetric temporal double decay for CS₂ has two decay times: a fast one, ~10’s fs (due to the bound electrons contribution), and a slower one, ~1–2 ps (due to molecular reorientation) [85]. On the other hand, the signal due to the LGF is symmetric, implying that the nonlinear response time is shorter than the pulse duration (for further discussion see Ref. [84]). Additionally, the OKG method can be used to measure short pulse widths, provided the nonlinear medium employed has a known and very short response time for the nonlinearity.

In sections 4 and 5, the methods described here will be applied to the study of nonlinearities of the gold metasurfaces described in Section 2.

4.1 On-resonance studies

As described in detail in Ref. [27], to measure the NLR and NLA coefficients of the gold metasurface, we exploited the conventional Z-scan technique with a Ti-sapphire regenerative amplifier (800 nm, 1 kHz repetition rate, 100 fs pulse duration) as the excitation source. The calibration was performed using CS₂, as described above. Figure 7 shows the Z-scan signatures for the NLA and NLR averaged over four samples. As a test to ensure that the system has no unexpected nonlinear effect, a blank SiO₂ sample substrate was characterized. At high intensities (5.87 GW/cm²), no nonlinear signal was detected, as seen in Figure 7A. At an intensity of 10 GW/cm², Figure 7B shows a typical signature for one of the metasurface samples, where SA and negative (self-defocusing) nonlinear refraction is directly observed.

Figure 7:

NLA (red squares – “open aperture Z-scan”) and NLR (green circles – “closed aperture Z-scan”) measurements.

(A) Glass substrate at I₀=5.87 GW/cm² and (B) gold metasurface Z- at I₀=10 GW/cm². In (B), the yellow up triangles are the result of the point-by-point ratio between NLR and NLA measured values. (C–D) Experimentally determined average values of NLA and NLR coefficients for all samples, showing saturation behavior as a function of incident power density. The black continuous lines are fits to the experimental data (adapted from Ref. [27], with permission).

In parts C and D of Figure 7, the intensity dependence for the NLA and NLR are shown, respectively, where saturation of the NLA and saturation of the NLR as a function of incident intensity is measured (dots in both figures). The lines in Figure 7C, D are a theoretical fit to the equations:

As reported in [27], the strong one-photon absorption arising from the plasmonic resonant excitation at ~753 nm explains the saturated behavior of the NLA coefficient (see Figure 2D). As a consequence of the Kramers-Kronig relation, saturation for the NLR index is also observed. Table 1 summarizes the literature values for NLR and NLA of some gold nanostructures measured with 800 nm, 100 fs pulses. The high n₂ value for the gold metasurface is notable, as well as the NLA coefficient α₂.

4.2 Off-resonance studies

The off-resonance studies were motivated by the enhancement in the already defined FOM (=n/k), at the telecom wavelength of 1500 nm (S-band in telecom systems). As pointed out in Ref. [71], the FOM is ~17 times higher at this wavelength than at around 750 nm. Therefore, we studied the nonlinear response of the gold metasurface at 1500 nm, as described in detail in Ref. [71]. The optical source for the Z-scan measurements consisted of an optical parametric amplifier operating at 1 kHz, 70 fs pulse duration, tuned to 1500 nm. Maximum intensities up to I=50 GW/cm² could be obtained. The closed- and open-aperture Z-scan signatures are shown in Figure 8, with the dots being the experimental data and the red solid lines are fits resulting from the application of Eqs. (3–5).

Figure 8:

Z-scan measurements at 1500 nm, 70 fs, 1 kHz for the gold MS.

(A) Closed aperture and (B) open aperture. Excitation intensity: 45 GW/cm². The red solid lines are fits resulting from the application of Eqs. (3.3–3.5). (Adapted from Ref. [71].)

From these data, the NLR index is calculated to be n₂=−1.05×10⁻¹⁰ cm²/W and the NLA coefficient is α₂=3×10⁻⁶ cm/W at 1500 nm. Although the linear absorption at 1500 nm is relatively small given the sample thickness, the linear absorption coefficient is inferred to be α₀=9.2×10³ cm⁻¹ (see Ref. [71] for details). del Coso and Solis [89] have shown that there is a relationship between the NLR and the third-order susceptibility in absorbing media. Using their equations on our results (see [71] for details), the corrected values for the real and imaginary part of the third-order susceptibility are given in Table 2, where, for comparison, the same calculations at 800 nm for a similar gold metasurface sample (from Ref. [27]) are also shown.

To further characterize the gold metasurface at 1500 nm, Figure 9A shows the OKG results in the fused silica substrate (blue curve) and the gold metasurface (red curve) measured using the same experimental conditions. The experiment was performed such that we could probe just the silica substrate or the metasurface-on-substrate by translating the sample. It was remarkable to observe that the magnitudes of the signals from both probed regions (substrate and gold metasurface on the substrate) were of the same order of magnitude. This implies a large NLR at 1500 nm for the gold metasurface, confirming the Z-scan measurements. However, it should be noted that the fused silica substrate is 1 mm thick, while the gold metasurface is only 15 nm thick, leading to an interaction length ratio (L(gold)/L(substrate)) of ~10⁻⁵. The 1/e characteristic decay time of nonlinearity was measured to be 1.42 ps.

Figure 9:

(A) OKG signal of the Si substrate (blue) and gold MS (red); (B) Intensity dependence of OKG in gold MS.

(A) OKG signal of the silica substrate (blue) and gold MS on the silica substrate (red), showing a characteristic decay time (1/e)=1.42 ps. The inset shows a diagram of the OKG setup. (B) OKG signal measured at τ=0.67 ps versus total input intensity, showing a slope of (2.3±0.1).

An analysis of the origin of the nonlinearity at 1500 nm was performed based on theoretical and experimental data related to the band structure of gold by Rangel and coworkers [90]. The 5d band in gold, which is associated with the valence electrons, is ~2.4 eV below the Fermi energy level. Therefore, at 1500 nm (0.83 eV), we do not expect one-photon interband absorption (5d→6sp) when the gold metasurface is excited at low laser intensity. Following the band labels as in [90], one-photon absorption involving electrons in the 6sp band [L₆⁻(6)] is expected, around the L direction. Conversely, for higher laser intensities (in the range exploited in the present experiments), a two-photon absorption transition [L_4,5⁺ (5)]→[L₆⁻(6)] can occur when exciting the gold metasurface at 1500 nm, as inferred from the open-aperture Z-scan results.

The self-defocusing NLR (negative n₂) measured in the closed-aperture Z-scan experiment indicates that the main contribution arises from free carriers (electrons in the 6sp band). The same mechanism also leads to electron-phonon scattering, therefore explaining the 1.42 ps decay time observed from the OKG measurements. Further discussion can be found in Ref. [71], where a COMSOL calculation has been carried out to qualitatively explain the electric field enhancement due to the self-assembly of the gold metasurface.