We have performed comprehensive electromagnetic simulations and preliminary experiments to explore the effects of geometrical and material parameters on the extraordinary optical transmission (EOT) through periodic arrays of subwavelength holes in a bilayer stack consisting of a gold or silver film atop a vanadium dioxide film (Au/Ag + VO2), where the latter undergoes a semiconductor-to-metal phase transition. Using the finite-difference time-domain (FDTD) and finite-element methods (FEM), we vary iteratively the array periodicity, VO2 film thickness and hole diameters, as well as the refractive index inside the VO2-layer holes and the VO2 optical constants. For each variation, we compare the metallic-to-semiconducting ratios of the zero-order transmission (T00) peaks and find sharp maxima in these ratios within narrow parameter ranges. The maxima arise from Fabry-Perot and Fano-type resonances that minimize T00 in the semiconducting phase of the perforated bilayers. At a fixed array period, the primary factors controlling the VO2-enabled EOT modulation are the VO2 thickness, diameter of the VO2-layer holes, and absorption in the two VO2 phases. Besides uncovering the origins of the higher metallic-phase T00, this study provides a protocol for optimizing the performance of the bilayer hole arrays for potential uses as dynamically tunable nano-optical devices.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Transmission of light through narrow apertures is one of the paradigms of nanoscale optics [1–5]. Unforeseen by standard diffraction theory [6,7], the so-called extraordinary or enhanced optical transmission (EOT) [8–10] manifests itself as a sequence of sharp dips and asymmetric peaks in the far-field spectra of electromagnetic waves transmitted through arrays of subwavelength apertures in metallic films. The current understanding of EOT through nanohole arrays [9,11,12] attributes the primary transmission mechanism to resonant tunneling  of evanescent modes boosted by multiple scattering of hybrid waves consisting of surface-plasmon polaritons  and quasi-cylindrical  or Norton  surface waves. Hole arrays have found many uses [5,10,17]—for example, in sensing [18,19], nanochemistry , plasmonic nano-tweezers , optofluidics , thermoplasmonics , surface-enhanced Raman spectroscopy [24–26], plasmonic color generation and control [27,28], surface-plasmon lasing , and tunable optical devices . For subwavelength apertures, several methods have been devised for active modulation of the transmission, generally by altering the optical constants of the adjacent materials via external stimuli: photoinduced [31–35], magnetic [36,37], electrical , electronic [39–41], electrochemical , electrochromic , and thermochromic [44–47]. The latter is the focus of this paper, which systematically investigates the modulation of the EOT effect by means of a reversible, temperature-induced, semiconductor-to-metal phase transition (SMPT).
The paper describes the parameters that affect the EOT switching for bilayer hole-array nanostructures comprising a plasmonic metal (gold, Au, or silver, Ag) and thermochromic vanadium dioxide (VO2). Section 2 presents preliminary experimental results for gold + VO2 bilayer hole arrays of periodicities 650 nm and 845 nm. Section 3 introduces the numerical and analytical methods employed in this work and maps out the parameter space explored via electromagnetic simulations. Section 4 presents simulations results for zero-order optical transmission (T00) peaks, ratios, and spectra for variations in geometrical and materials parameters: Au + VO2 array period, VO2-layer thickness, Au + VO2 thru-hole diameter, VO2-hole diameter, VO2 absorption, VO2-hole refractive index, VO2 complex refractive index, Ag + VO2 array period with and without VO2 holes, and Ag + VO2 thru-hole diameter. Section 5 includes image plots of simulated power flow and electric-field intensity in some representative scenarios: unperforated vs. perforated VO2 layer, three VO2-hole diameters, two VO2-layer thicknesses, and three Au + VO2 array periods. Section 6 summarizes our findings. Supplement 1 includes some of the details, figures, and discussions.
Our method of modulating the optical transmission through plasmonic nanohole arrays utilizes vanadium dioxide, a canonical correlated-electron material that undergoes phase transitions driven by temperature or light [48–50]. Above a critical temperature Tcrit = 340 K (67 °C), bulk VO2 switches from a monoclinic semiconductor to a rutile metal, and vice versa upon cooling. These coupled electronic and structural phase transitions are typically manifested by hysteresis loops in the electrical, crystallographic and optical properties of the VO2 sample as a function of temperature; for example, see the transmittance measurement in Fig. 1(a). A planar, undecorated VO2 film is more opaque to infrared (IR) light in the metallic (MetVO2, T > Tcrit) than in the semiconducting (SemiVO2, T < Tcrit) state and the contrast increases at longer wavelengths, as shown by the calculated transmittance in Fig. 1(b). The relative permittivity [Fig. 1(c, d)] and index of refraction [Fig. 1(e, f)] have non-negligible imaginary components, exhibit significant dispersion (esp. for MetVO2) and differ substantially in the two states. It is these large changes in optical constants, induced close to room temperature during the SMPT, that make VO2 a prime candidate for (ultrafast ) tunable optoelectronic devices. Examples of VO2-enabled plasmonic modulators and switches can be found in Refs. [47,52–59].
In principle, the EOT effect can be tuned across the SMPT by dwelling at different sample temperatures in the transition regions where the optical constants of VO2 change continuously from their SemiVO2 to MetVO2 values on heating and vice versa on cooling [60,61]. The hysteresis enables bistable switching for thermal excursions of 15 °C or more in the two end states, while within the hysteresis loop the EOT magnitude depends on whether a given temperature is reached by ramping up or down. These latter scenarios are beyond the scope of the current work, for which we performed optical measurements first at room temperature (SemiVO2) and then at a stable 85 °C (MetVO2). Since the VO2 phase transition is fully reversible, the order of the final-state measurements does not affect the switching functionality.
We have performed three-dimensional electromagnetic simulations and preliminary experiments, aiming to optimize the structural parameters of bilayer nanohole arrays in a gold (Au) or silver (Ag) film atop a VO2 film on a semi-infinite glass substrate (nglass = 1.50; see inset in Fig. 3) or freestanding in air (nair = 1.00), for potential applications in nanoplasmonic optical switching. We obtain closely matching computational results with two different fully vectorial Maxwell solvers. As we vary the geometry of the perforated Au + VO2 or Ag + VO2 bilayers, we search the parameter space for combinations of array period, hole diameter and VO2 film thickness that maximize the ratio of the zero-order transmission at the longest-wavelength EOT peak in the MetVO2 state, peak-T00, to the zero-order transmission at the same wavelength in the SemiVO2 state. For the remainder of this paper, we refer to this Met-to-Semi peak-T00 ratio, which quantifies the EOT switching or modulation, as the M2S-ratio. After several iterative optimizations, the simulations of gold + VO2 hole arrays on glass (i.e., Air-Au + VO2-Glass) produced a maximum M2S-ratio = 188 at a peak wavelength λpeak = 778 nm with these optimal parameters: array period Parray = 720 nm, thru-hole diameter Dthru-hole = 302 nm, and combined Au and VO2 bilayer thickness tAu+VO2 = 200 + 245 nm.
Furthermore, we seek to elucidate the electromagnetic origins of the effect we term reverse optical switching, observed in the current experiments and simulations [e.g., Fig. 3(a)] and previously in Refs. [44,62], whereby the EOT through Au + VO2 and Ag + VO2 bilayer hole arrays is counterintuitively higher in the MetVO2 state than in the SemiVO2 state. Such increase in light transmission through a perforated VO2 film across the SMPT is ‘reverse’ in the sense that ‘ordinary’ IR transmission through a plain (unperforated) VO2 film is lower in the metallic state [e.g., Fig. 1(a, b)].
We find the highest M2S-ratios occur when transmission at λpeak through the perforated Au + SemiVO2 bilayer decreases sharply within narrow ranges of the explored parameters—array period, VO2 film thickness, thru-hole diameter, VO2-hole diameter and VO2 absorption —while the corresponding peak-T00 for Au + MetVO2 follows a monotonic trend for all but one parameter sweeps. When varying VO2 thickness or hole diameter, the dips in peak-T00 for Au + SemiVO2 likely originate from Fabry-Perot (FP) resonances [4,9,63–65] that build up standing waves vertically inside the SemiVO2 film and holes or horizontally along the film’s planar interfaces, thus diverting optical energy away from the zero-order transmission channels. These FP modes arise almost exclusively in the SemiVO2 phase because its optical constants, unlike those of MetVO2, are almost non-dispersive and mostly real-valued within a wide spectral range [Fig. 1(c–f)]. At vacuum wavelengths in the range λ = 600–1000 nm, SemiVO2 acts as a slightly lossy dielectric with a nearly constant refractive index nsemi = 2.9–3.0 [Fig. 1(e), dashed-line box] and a relatively small extinction coefficient κsemi = 0.35–0.50 [Fig. 1(f)]. As the hole diameter and VO2 thickness are swept through values that fulfill FP (anti-)resonance conditions, in-plane and vertical modes in the perforated SemiVO2 film “trap” some of the light emerging from the perforated Au (or Ag) film, thus diminishing the EOT through the bilayer hole array in the semiconducting state. Since the refractive index of MetVO2 varies substantially (nmet = 1.7–2.5) and the extinction coefficient is larger (κmet = 0.46–1.4) in this spectral range, FP-type light-trapping modes are suppressed and the holes in the VO2 film “funnel” the transmitted light more effectively in the metallic state. In addition to FP (anti-)resonances with symmetric Lorentzian profiles, our perforated bilayers exhibit the usual Fano-profile spectral shapes expected of EOT peaks [2,4,66–68]. Curiously, Fano-like profiles also emerge in the dependence of the M2S-ratios on array period or peak wavelength [Fig. 4 and Fig. 10(a)]. Fano lineshapes are discussed in Section 3.3.
2. Experimental measurements
The VO2 films were fabricated by a combination of pulsed laser deposition (PLD) and thermal post-annealing, as previously reported in Ref. . Upon switching, stoichiometrically correct VO2 films show typically a 95% relative change in transmission intensity at 1550 nm and a ∼5 °C hysteresis [Fig. 1(a)]. The desired VO2 thickness was confirmed by measuring the step height by profilometry. A 200-nm-thick Au layer was deposited on top of the VO2 layer via RF sputtering. Subsequently, 100×100 µm2 arrays of holes were milled through the bilayer using a single-column focused ion beam (FIB). Hole arrays with periodicities of 650/845 nm and Au-hole diameters of 230/220 nm, respectively, were milled with an ion-beam diameter of 60 nm.
The optical setup for measuring the zero-order transmission (T00) is shown schematically in Fig. S1(f, inset) and Fig. 2(e, inset). A 10× objective lens simultaneously focused incident light onto the sample and reflected light onto the charge-coupled-device (CCD) sensor of an imaging camera. Incident polarized white light was sent towards the sample through a 90:10 beamsplitter. Transmitted light was collected with an output lens and sent through an optical fiber to a grating spectrometer. T00 was measured in the semiconducting and metallic states of the VO2 layer by varying and holding the temperature of the sample with a heating stage. The transmission hysteresis of the plain VO2 film shown in Fig. 1(a) was measured with a similar setup, except for replacing the spectrometer with an optical power detector and a 1550-nm bandpass filter.
Scanning electron microscope (SEM) images at different magnifications (10k× and 35k×) of the Au + VO2 hole array with Parray ≈ 650 nm are shown in Fig. S1(a). The spiral scanning method used in FIB milling causes the inner part of the hole to have a higher exposure than the outside resulting in a conical hole shape that narrows with depth, with the top Au surface having larger-diameter opening apertures than the top surface of the underlying VO2 layer. Grain analysis on one of the higher-magnification SEM images gave an average opening-aperture diameter DAu(-holes) ≈ 230 nm in the Au layer [Fig. S1(b)] and DVO2(-holes) ≈ 125 nm in the VO2 layer [Fig. S1(c)]. The average periodicity in the xy-plane (i.e., sample surface) was determined by a two-dimensional fast-Fourier transform (2D-FFT) [Fig. S1(d)]; the reciprocal-space line profiles in the x- and y-directions are extracted in Fig. S1(e).
The experimental [Fig. S1(f)] and FDTD-simulated [Fig. S1(g, h)] spectra show good agreement for the EOT peak position, even as Fig. S1(a) reveals variations in hole shapes and diameters. The peak wavelength is primarily determined by the Bragg coupling condition for exciting surface plasmon polaritons on a periodically decorated metal-dielectric interface, which depends only on the lattice constant and indices of refraction of the materials . The linewidths of the experimental transmission curves are larger than the simulated ones because the incident beam in the experiments includes non-zero components of in-plane wavevectors around normal incidence. The measured spectra exhibit asymmetric Fano-like profiles as expected from the phenomenological theory of EOT [2,4,66–68] and the numerical simulations. On the other hand, Fig. S1 makes it clear that a mismatch between hole diameters in the Au vs. VO2 layer causes a significant difference in the modulation of the transmission between the metallic and semiconducting states of VO2. The simulated M2S-ratio is 8.74 at λpeak = 698 nm for DVO2 = DAu = 230 nm (not shown), almost an order of magnitude higher than 1.15 (λpeak = 697 nm) for DVO2 = 125 nm < DAu = 230 nm [Fig. S1(g)]. The experimentally measured T00 peaks [Fig. S1(f)] have a ratio of 1.75 (λpeak = 778 nm), which lies between these two simulated cases. In fact, by allowing for a margin of error in choosing the threshold pixel intensity in the image analysis of VO2 aperture sizes in Fig. S1(c), where the scaling is ∼7 nm/pixel, we simulate an intermediate case of DVO2 = 165 nm that yields M2S-ratio = 1.74 (λpeak = 697 nm) [Fig. S1(h)], almost equal to the experimental value. (See Supplement 1 for supporting content.)
A similar analysis of the larger-period Au + VO2 hole array is presented in Fig. 2. The electron micrographs in Fig. 2(a) were taken at the same magnifications as those in Fig. S1(a). Grain [Fig. 2(b)] and 2D-FFT [Fig. 2(c, d)] analyses of the SEM images yield DAu ≈ 220 nm and Parray ≈ 845 nm. Unlike Fig. S1(a), Fig. 2(a) does not reveal the underlying VO2 layer, so we assume DVO2 = DAu ≈ 220 nm. It appears that the larger dose and nominally specified hole diameter (310 nm) resulted in more uniform milling through the Au and VO2 layers. The experimental results [Fig. 2(e)] are in better agreement with both the FDTD [Fig. 2(f)] and FEM [Fig. 2(g)] simulations in terms of EOT switching: M2S-ratio = 2.21 (λpeak = 893 nm) for the experimental, 2.60 (λpeak = 866 nm) for the FDTD, and 2.58 (λpeak = 869 nm) for the FEM spectra. Here too the larger linewidth and spectral shift of the measured peaks can be attributed to off-angle components of in-plane wavevectors contained in the focal volume of the incident light. A secondary peak corresponding to a higher-order SPP mode is observed on the high-energy side of the main EOT peak.
Altogether, the experimental measurements on the two Air-Au + VO2-Glass hole arrays of different periods reproduced well the reverse switching of EOT across the VO2 phase transition. We also observed that the area fraction of the holes in the VO2 layer affects the modulation ratio, an effect which we investigate computationally in Fig. 7.
3. Numerical simulations and Fano model
We used two different numerical methods , FDTD and FEM, in order to cross-check the computed results as well as exploit the advantages inherent in each technique. The FDTD and FEM results in this work are in good mutual agreement: within 2–3 nm for the peak positions and 5–15% differences in peak-transmission and switching-ratio values. For example, the FDTD spectra in the third panel of Fig. 3 (Parray = 720 nm) have peak-T00 values at λpeak = 757 nm of 4.50×10−3 for MetVO2 and 4.72×10−4 for SemiVO2, resulting in M2S-ratio = 9.53; the corresponding FEM spectra (not shown) have peak-T00 values at λpeak = 759 nm of 4.81×10−3 for MetVO2 and 5.47×10−4 for SemiVO2, and hence M2S-ratio = 8.79. Crucially, the overall EOT spectra as well as peak-T00 and M2S-ratio trends as a function of the various hole-array parameters are robustly consistent between the two simulation methods, warranting our treatment of the FDTD and FEM results as essentially interchangeable [e.g., cf. Figure 4(a, b)]. (See Supplement 1 for supporting content.)
3.1 Finite-difference time-domain (FDTD) method
The FDTD method [71,72] solves Maxwell’s time-dependent curl equations directly on a discretized numerical grid. It approximates the derivatives of the electromagnetic field vectors as finite differences, sampled at discrete spatial and temporal points in a specific arrangement called a Yee cell . We used the commercial software FDTD Solutions (v8.18.1262). The simulation domain spans an xyz-volume of ½Parray × ½Parray × 2200 nm3, where the overall ¼-factor stems from using the anti-symmetric condition (zero tangential electric field) at the yz-boundaries and the symmetric condition (zero normal electric field) condition at the xz-boundaries. Both z-boundaries are terminated with 128 stretched-coordinate perfectly matched layers (PML) to absorb the reflected and transmitted waves. The domain is discretized globally with a non-uniform conformal orthogonal mesh, which is overridden locally, within a volume that fully encloses all materials interfaces, with a finer uniform mesh of 2.5-nm increments. The incident illumination is an x-polarized plane wave, launched from the air side at normal incidence to the Au layer as a broadband pulse (400–1500 nm). Material dispersion curves [Fig. 1(c–f) for VO2] are obtained within the software by fitting ‘multi-coefficient models’ to interpolated experimental data extracted from Verleur et al.  for VO2 and Johnson and Christy  for Au/Ag. (See Supplement 1 for supporting content.)
3.2 Finite-element method (FEM)
Unlike the FDTD method, the FEM leaves Maxwell’s equations intact but approximates the solution space by subdividing the computational domain into ‘finite elements’—small geometric patches with locally defined polynomial approximations (interpolation functions) of the solution—and stitching the elements together under conditions of continuity of the tangential electric and magnetic fields . The FEM results in this work were obtained with COMSOL Multiphysics (v126.96.36.199). The computational xyz-domain is ½Parray × ½Parray × ∼1500 nm3, with perfect electric conductor walls at the yz-boundaries and perfect magnetic conductor walls at the xz-boundaries. The air (vacuum) input medium, Au film, VO2 film, glass substrate and hole subdomains are meshed adaptively with tetrahedral elements, which within the films and holes are set to grow no larger than 50 nm. Each z-boundary is capped with five sweep-meshed PMLs for absorbing the transmitted and reflected waves with minimal boundary reflections. Image plots of the electric-field intensity and power flow in different planes intersecting the unit cell are presented in Section 5. The Au and VO2 optical constants used in the COMSOL simulations are extracted directly from the FDTD Solutions fits to ensure identical materials responses. (See Supplement 1 for supporting content.)
3.3 Fano-profile fits
Although a quantitative microscopic interpretation of EOT [9,11,12,15,75,76] began to emerge a full decade after the initial discovery, it had been recognized early on [66,67] that the dip-peak EOT spectra can be modeled very well [2,4,68,77–81] with a Fano-type formalism . In the general Fano theory, a system’s response function to an external perturbation acquires a characteristic Fano profile when a discrete resonant state interferes destructively (sharp dip) and constructively (asymmetric peak) with a continuum of states (or a broader spectral line) [83,84]. Specifically for EOT, discrete states can be any resonant surface electromagnetic modes (e.g., SPPs), while the continuum can include the (typically weak) direct transmission through the holes as well as any spectrally broad features (e.g., localized surface-plasmon resonances at the aperture rims ). A variation of the Fano-profile function for hole-array transmission [68,77] is given below, expressed in terms of the vacuum wavelength λ (in nm) rather than the customary frequency or energy:
The crucial Fano parameter q determines the asymmetry of the resonance. Being the cotangent of the phase shift between discrete and continuum modes, q is related to their coupling strength as well as to the relative excitation strengths—i.e., the ratio of resonant to non-resonant transmission amplitudes [66,79,84,86]. The term Tb is associated with the (background) portion of the direct transmission that is uncoupled from the discrete state, while Tc is associated with the zero-order continuum transmission that is coupled to and mixes with the discrete state. The resonance linewidth and position (both in nm) are given by Λ and λres, respectively. In general, the higher the |q| value, the more symmetric, Lorentzian-like the lineshape becomes, signaling that the external perturbation (i.e., incident illumination) couples less efficiently to the continuum of scattering states. Conversely, as q → 0, the external perturbation decouples from the discrete state and the Fano lineshape turns into an inverted-Lorentzian anti-resonance . The most asymmetric lineshapes arise when |q| = 1.
The phenomenological Fano model does not reveal the microscopic origins of the different transmission channels (e.g., SPPs and quasi-cylindrical waves), but it does provide reasonably good fits to the EOT peaks, as Fig. 3(a) demonstrates. The solid lines are the best Fano fits [Eq. (1)] to the longest-wavelength EOT peaks simulated by FDTD for four representative Air-Au + VO2-Glass hole arrays of different periods and thru-hole diameters (Dthru-hole ≡ DAu = DVO2). (The spectra in this figure are scaled by the indicated factors to equalize the SemiVO2 peak-T00 values for better visualization of the EOT modulation.) The extracted Fano asymmetry parameter q in Fig. 3(b, bottom panel) exhibits a different trend in each VO2 phase with increasing Parray ( = 3Dthru-hole): qmet (open squares) grows almost monotonically from ∼3 to ∼6, whereas qsemi (solid circles) starts from ∼6, dips through a minimum of ∼3 at Parray = 690 nm—close to 720 nm, where the highest M2S-ratio occurs [see Fig. 4(a)]—and then approaches ∼6 at larger periods.
As mentioned above, the Fano q-parameter encodes the phase shift between the discrete (resonant) and continuum (direct, non-resonant) transmission channels [86,87]. Since q is related to the ratio of discrete-to-continuum excitation strengths, large resonant amplitudes and/or small continuum amplitudes should yield large |q| values, and vice versa. The spectral position and intensity of the EOT peak depend on the lattice constant: λpeak redshifts and peak-T00 increases with increasing Parray. The peak redshifts to satisfy momentum conservation and grows because good metals like Au and Ag allow SPPs to propagate with less dissipation at IR frequencies due to greater absorption lengths . The resonant transmission channel (SPP modes) thus gets enhanced at larger array periods corresponding to longer resonant wavelengths (λres). The direct transmission channel, however, literally narrows at longer wavelengths as the skin depth of plasmonic metals decreases, which in turn shrinks the effective diameters of the holes . Therefore, the combination of these effects tends to increase the Fano q-parameter as a function of periodicity, as reported in Ref. .
As seen in Fig. 3(b, bottom panel), qmet in the MetVO2 state of the Air-Au + VO2-Glass hole arrays also (mostly) increases with Parray, in line with the above reasoning. However, qsemi in the SemiVO2 state varies non-monotonically with Parray, exhibiting a pronounced dip around Parray = 690 nm (λpeak = 730 nm). Since the Fano q-parameter represents the ratio of resonant (SPPs) to continuum (direct evanescent transmission) contributions to T00, the qsemi dip in Fig. 3(b, bottom panel) —and the corresponding qmet/qsemi peak in Fig. 3(b, top panel)—could be caused by one or more interactions that decrease qsemi at specific values of the structural parameters: weaker coupling of the external illumination to the resonant channel; stronger coupling of the external illumination to the continuum transmission channel; and/or a more effective coupling between this (SPPs) or another (e.g., FP modes) resonant channel and the reflection continuum . In other words, when geometry (e.g., Parray = 690 nm, Dthru-hole = 230 nm, tAu = tVO2 = 200 nm) and the VO2 optical properties (nsemi + iκsemi ≈ 3 + 0.4i for λ = 600–1000 nm) “conspire” to make the EOT lineshapes more Fano-like (i.e., more asymmetric) as qsemi → 3, interference between the discrete and continuum channels reduces the zero-order transmission. The results in Section 4 seem to point to FP-type anti-resonances as the discrete modes that produce the recurring dips (“valleys”) observed in the SemiVO2 phase as a function of array period [Fig. 4 and Fig. 10(a)], VO2 film thickness [Fig. S2(a) and Fig. 5(a)], thru-hole diameter [Fig. S2(b) and Fig. 6(a)], VO2-hole diameter [Fig. 7(a)], VO2 absorption [Fig. S3(a)] and SemiVO2 refractive index (Fig. 9).
3.4 Parameter space explored via FDTD and FEM simulations
We have performed the following parameter sweeps (listed in Supplement 1) of hole-array geometry and VO2 optical constants in order to optimize the Met-to-Semi EOT switching (i.e., maximize the M2S-ratio) and intuit why T00 is generally higher in the MetVO2 phase of the perforated bilayers. The “(Au)” or “(Ag)” designation after a Roman numeral means that the given optimization sweep applies to gold + VO2 hole arrays on a glass substrate or freestanding silver + VO2 hole arrays. The thickness of the Au and Ag layers in all simulations is kept constant at 200 nm.
The geometric iterations of Au + VO2 hole arrays produced an optimized perforated bilayer with M2S-ratio = 196 [Fig. 7(a)] at λpeak = 778 nm for Parray = 720 nm, DAu = 302 nm, DVO2 = 304 nm, tAu = 200 nm, tVO2 = 245 nm. A close second Au + VO2 hole array with M2S-ratio = 188 [Fig. 6(a)] differs only in DVO2 = DAu = 302 nm. Since it would be difficult in practice to tune DVO2 with 2-nm precision, we take the latter (equal diameters) as the “gold standard” in terms of EOT modulation, which is shown in Fig. 6(c), Fig. 12(c, d) and Fig. 13(c, d, h). Although we explored the parameter space of perforated Au + VO2 on glass in more detail than that of perforated freestanding Ag + VO2, we simulated enough of the latter structures to observe the same trends in EOT modulation —for instance, compare Fig. 4(a) and Fig. 10(a). Therefore, the results shown below for Au + VO2 hole arrays apply just as well to Ag + VO2 hole arrays. (See Supplement 1 for supporting content.)
4. Simulation results: zero-order transmission peaks, ratios, and spectra
4.1 Varying array period for Air-Au + VO2-Glass hole arrays
Optimization I(Au): max(M2S-ratio) ≈ 10 at λpeak = 758 nm and Parray = 720 nm.
The periodic separation of the holes in the xy-plane, Parray, is the key parameter that determines the spectral location of the EOT peak, λpeak, associated with the air-metal interface under normal-incidence illumination. Therefore, it is reasonable to start the optimization process of maximizing the EOT switching by varying Parray for typical values of film thickness (tAu = tVO2 = 200 nm) and hole diameter (DAu = DVO2 = Parray/3). The simulated peak-T00 in each VO2 phase and the corresponding M2S-ratios are shown in Fig. 4(a) for the FDTD and Fig. 4(b) for the FEM optimization runs.
To a first approximation, λpeak is governed by the SPP dispersion relation for an unperforated metal-dielectric interface, augmented by integer multiples of the reciprocal lattice vectors of the periodic hole array in order to bridge the momentum mismatch between the freely propagating incident light and the bound SPP modes [3,90]. When the Fano-type interferences  described in Section 3.3 are considered, λpeak is always redshifted by some tens of nanometers with respect to Parray, as Fig. 3(a) demonstrates visually and the inset in Fig. 4(b, top panel) quantifies via a linear fit. Intriguingly, not only do T00 spectra exhibit Fano lineshapes [e.g., see Fig. 3(a)], but peak-T00 ratios vs. Parray (or vs. λpeak) also follow closely the Fano-profile function [solid lines in Fig. 4 and Fig. 10(a), top panels], in a generalized form of Eq. (1). Since peak-T00 for MetVO2 increases monotonically with array period, the Fano-like shapes of these M2S-ratio curves stem from the SemiVO2 peak-T00 “valley” in the 550–700 nm spectral region [Fig. 4 and Fig. 10(a), bottom panels], which overlaps the largely non-dispersive region of nsemi(λ = 600–1000 nm) mentioned in Section 1 and highlighted in Fig. 1(e).
4.2 Varying VO2 thickness and thru-hole diameter for Air-Au + VO2-Glass hole arrays
Optimizations II(Au) & III(Au): max(M2S-ratio) = 12 at λpeak = 759 nm and tVO2 = 220 nm; max(M2S-ratio) = 36 at λpeak = 774 nm and Dthru-hole = 290 nm. (See Supplement 1 for supporting content.)
After determining the optimal array period, we fix Parray (720 nm), Dthru-hole (240 nm = 720 nm/3) and tAu (200 nm), and then sweep the thickness of the VO2 layer, tVO2. Peak-T00 values decrease in both phases as the VO2 layer gets thicker [Fig. S2(a)], as expected from Beer’s law for light traversing a lossy dielectric film. (The air-filled holes do not contribute to the dissipative absorption.) The FEM results reveal pronounced dips in SemiVO2 peak-T00 that bring about a primary peak and a smaller secondary peak in the M2S-ratio at tVO2 = 220 nm and tVO2 = 570 nm, respectively. It is not obvious if these tVO2 values fulfill specific FP conditions.
A puzzling feature is the very sharp dip in the MetVO2 state at tVO2 = 510 nm—unique because the simulated MetVO2 peak-T00 curves in all the other Au + MetVO2 hole arrays behave monotonically as a function of the geometrical parameters. To better visualize this dip, the top panel of Fig. S2(a) plots the inverse 1/M2S-ratio (diamond markers, right-axis scale): Note that this steep, inverted (q1/M2S < 0) Fano profile reaches a maximum at tVO2 = 508 nm, where the Fano profile of the secondary M2S-ratio peak (qM2S > 0) has its minimum.
The next step is to sweep Dthru-hole at fixed Parray (720 nm), tAu (200 nm) and the just-optimized tVO2 (220 nm). The FEM results are shown in Fig. S2(b). As expected [6,7,79,91], the transmission is very weak for deeply subwavelength holes—e.g., T00 < 10−7 at λ = 740 nm for Dthru-hole = 50 nm—but grows rapidly as the aperture is widened towards the (material-dependent) cutoff diameter for propagating guided modes, and then saturates thereafter. The transmission peaks also broaden [cf. Figure 6(b–e)] and redshift with increasing diameter, caused by, respectively, increased radiative damping and the nonlinear dependence of transmission on wavelength below cutoff [4,79,91]. The inset in Fig. S2(b) empirically quantifies the redshift of the peak position via a quadratic fit. The M2S-ratio as a function of Dthru-hole [Fig. S2(b), top panel] exhibits a primary peak at Dthru-hole = 290 nm and a smaller secondary peak at Dthru-hole = 580 nm = 2 × 290 nm. The Lorentzian lineshapes and integer scaling strongly suggest that these two peaks arise from consecutive standing-wave modes of a Fabry-Perot (anti-)resonance.
4.3 Varying VO2 thickness (2nd iteration) for Air-Au + VO2-Glass hole array
Optimization IV(Au): max(M2S-ratio) = 105 at λpeak = 774 nm and tVO2 = 245 nm.
The iterative process of maximizing the EOT switching continues by varying the thickness of the VO2 layer, tVO2, once again at fixed Parray (720 nm), tAu (200 nm) and Dthru-hole, but this time with the latter having the newly found optimal value of 290 nm. The FEM results for the peak-T00 and M2S-ratio are shown in Fig. 5(a), while Fig. 5(b–e) display the pairs of simulated T00 spectra, in each VO2 phase, for the four representative open-circle markers in Fig. 5(a, top panel). The MetVO2 peak-T00 curve in Fig. 5(a, bottom panel) also trends downwards with increasing tVO2, as it does in Fig. S2(a), but instead of a pronounced dip around tVO2 = 510 nm, this curve gently plateaus. The SemiVO2 peak-T00 trend resembles that in Fig. S2(a), but with notable differences: the two dips in Fig. 5(a) are much sharper and the second one is located at almost exactly double the tVO2 value of the first one: 494 nm vs. 245 nm. Owing to the greater depth of these SemiVO2 peak-T00 minima, the two prominent M2S-ratio peaks in Fig. 5(a, top panel) exceed 100 [cf. Fig. S2(a, top panel)].
The two tVO2 values, 245 nm and 494 nm ≈ 2 × 245 nm, for which the M2S-ratio maxima and corresponding SemiVO2 peak-T00 minima occur in Fig. 5(a), suggest the involvement of (anti-)resonant FP modes spaced in tVO2 by an integer number of half wavelengths λsemi/2 inside the SemiVO2 material, where λsemi = λvac, peak/nsemi(λvac, peak) = 773 nm/2.89 = 267 nm ∼ 245 nm. As to why these presumably consecutive modes are spaced by nearly two half wavelengths rather than one, we speculate that non-zero phase changes on reflection at the materials interfaces also contribute to the FP resonance conditions [64,65], in addition to the accumulated optical-path-length phase due to the waves traversing the thickness of the VO2 layer.
The SemiVO2 T00 spectrum in Fig. 5(d), which is largely responsible for the narrow M2S-ratio peak at tVO2 = 494 nm in Fig. 5(a, top panel), has a distinctly different shape from the other spectra in this set. Upon comparing Fig. 5(d) with Fig. 5(b, c, e), it appears that the SemiVO2 spectrum for tVO2 = 494 nm should have had a peak around λpeak = 773 nm, more or less aligned with the peak of the corresponding MetVO2 spectrum, but instead has a small blueshifted peak (λ = 737–780 nm) and a very broad redshifted “hill” (λ > 780 nm), with a deep valley in between (T00 < 10−7 at λ = 780 nm). Curiously, this SemiVO2 spectrum resembles the MetVO2 spectrum (not shown) associated with the feature at tVO2 = 510 nm in Fig. S2(a, bottom panel) (see Section 4.2). Such an abrupt change from a peak to a dip at tVO2 = 494 nm is further evidence of an FP-type anti-resonance of waves undergoing destructive interference in the z-direction in the VO2 film and within the holes that perforate it, after reflections at the Au-VO2 and VO2-glass xy-interfaces. Additional FEM simulations (not shown) confirm that the holes in the VO2 layer are crucial for the resonant mechanism, since varying tVO2 as in Fig. 5 but without perforating the VO2 layer generates an approximately exponential decay of the M2S-ratio that never exceeds 1.45 for any VO2 thickness in this range.
4.4 Varying thru-hole diameter (2nd iteration) for Air-Au + VO2-Glass hole arrays
Optimization V(Au): max(M2S-ratio) = 188 at λpeak = 778 nm and Dthru-hole = 302 nm.
An additional increase in the Met-to-Semi switching ratio is achieved by a second tuning of the diameter Dthru-hole (also labeled DAu+VO2) of the thru-holes (Fig. 6), after setting the VO2 thickness to the optimal tVO2 value (245 nm) from Fig. 5(a). The new optimal value Dthru-hole = 302 is somewhat larger than the one found in Fig. S2(b), 290 nm, and the dips in the SemiVO2 peak-T00 curve are sharper, indicating that the vertical (z-direction) FP resonances as a function of VO2 thickness are coupled in some way with the lateral (xy-plane) FP resonances as a function of hole diameter . In other words, different tVO2 values lead to a different position, width and depth of the primary M2S-ratio peak that emerges during a Dthru-hole sweep [cf. Fig. S2(b) and Fig. 6(a), top panels]. As already mentioned in Section 4.2 and demonstrated in Fig. 6(b–e), T00 spectral peaks broaden and redshift with increasing Dthru-hole. In the semiconducting state, the T00 spectra also change shape at values of Dthru-hole where the M2S-ratio has primary and secondary maxima: 302 nm [Fig. 6(c)] and 590 nm ≈ 2 × 302 nm [Fig. 6(e)]. The near-integer scaling points again to lateral FP-type anti-resonances in the SemiVO2 transmission.
4.5 Varying hole diameter only in VO2 layer for Air-Au + VO2-Glass hole arrays
Optimization VI(Au): max(M2S-ratio) = 196 at λpeak = 778 nm and DVO2 = 304 nm.
It was already demonstrated in Fig. S1(g, h) that DVO2, the diameter of holes in the VO2 layer alone, affects the M2S-ratio. When DVO2 << DAu, the VO2 layer optically resembles an unperforated film in that most of the light emerging from the holes in the Au layer traverses VO2 material rather than air-filled waveguides embedded in it, and transmission in the semiconducting state exceeds that in the metallic state. In Fig. 7(a), FEM simulations of T00 through hole arrays with a fixed Au-hole diameter (DAu = 302 nm) and variable VO2-hole diameter show M2S-ratio < 1 for DVO2 ≤ 150 nm. In the limit of vanishing holes in the VO2 layer, the Fresnel-calculated transmission ratio at λ = 773 nm for a plain VO2 film on glass (i.e., no Au layer) in Fig. 1(b) and the M2S-ratio for a half-perforated bilayer on glass (i.e., holes only in the Au layer) in Fig. 7(b) are quite similar: TPlainMetVO2/TPlainSemiVO2 = 0.39 vs. M2S-ratio(DVO2 = 0 nm) = 0.45. Conversely, as DVO2 is increased beyond DAu, less and less of the light emerging from the Au holes interacts with VO2 material and thus the EOT modulation diminishes: for DVO2 > 480 nm, M2S-ratio = 1.0–1.4.
Between the two regimes, the switching ratio rises to 196 at DVO2 = 304 nm as the SemiVO2 peak-T00 drops two orders of magnitude. This (anti-)resonant DVO2 hole diameter is only 2 nm larger than the fixed DAu = 302 nm hole diameter and M2S-ratio = 196 is only marginally higher than the “gold standard” M2S-ratio = 188 optimized in Section 4.4 by varying the Dthru-hole diameter of holes in both layers. However, there is a difference of several orders of magnitude in how peak-T00 scales with varying Dthru-hole vs. varying only DVO2. For the thru-hole diameter sweep, the maximum transmission spans ∼10−8–1 [Fig. 6(a), bottom panel], while for the VO2-hole diameter sweep the ∼10−4–10−1 range is much smaller [Fig. 7(a), bottom panel]. The widths, spectral positions and shapes of the T00 peaks also differ between the two sweeps. Peaks broaden and redshift significantly with increasing Dthru-hole, and even begin to split for SemiVO2 [e.g., see Fig. 6(e)], whereas they remain qualitatively unchanged with increasing DVO2, as seen in the T00 spectra in Fig. 7(b–e).
4.6 Varying (hypothetically) VO2 absorption for Air-Au + VO2-Glass hole array
Optimization VII(Au): max(M2S-ratio) = 220 at λpeak = 778 nm and εimag × 1.05. (See Supplement 1 for supporting content.)
The parameter sweep presented in Fig. S3 is “hypothetical” since the VO2 permittivity cannot be easily tuned experimentally; besides, the imaginary part εimag ≡ Im[εVO2(λ)] cannot be varied independently of the real part as the two parts of the dielectric response function are linked via causality-imposed Kramers-Kronig (K-K) relations . Nevertheless, it is informative to simulate EOT spectra through perforated bilayers consisting of a regular Au film and a hypothetical “VO2” film with artificially modified absorption, i.e., α εimag, where α is a real-valued positive scaling factor. By leaving the real part of the VO2 permittivity unchanged, we manually override the K-K relations in this set of FEM simulations. As Fig. S3(a) demonstrates, when the εimag functions of both SemiVO2 and MetVO2 are multiplied by a factor α substantially lower or higher than unity, the M2S-ratio drops steeply from its maximum value of 220 at α = 1.05 [Fig. S3(d)] down to about 20 at α = 0.10 [Fig. S3(b)] and α = 0.50 [Fig. S3(b)] or about 2 at α = 3.00 [Fig. S3(e)]. The M2S-ratio peak around α ≈ 1 once again appears to be caused by an FP-type anti-resonance in the SemiVO2 transmission, whereas the MetVO2 transmission decreases slowly and monotonically with increasing artificial absorption (i.e., α). For low absorption (α << 1), both MetVO2 and SemiVO2 become nearly lossless dielectrics, and there is not a pronounced dip in the SemiVO2 peak-T00, probably because the lower values of α εimag modify the reflected waves’ phase changes on reflection  at the Au-SemiVO2 and SemiVO2-glass interfaces in ways that render the FP anti-resonance condition unfulfilled. In the high-absorption limit (α >> 1), the electromagnetic fields are strongly attenuated as the waves traverse either MetVO2 or SemiVO2, the differential interfacial reflections diminish, and hence the M2S-ratio approaches unity (e.g., M2S-ratio = 1.62 for α = 4).
4.7 Varying (hypothetically) refractive index of material inside holes in VO2 layer
Optimization VIII(Au): max(M2S-ratio) = 188 at λpeak = 778 nm and nHoleVO2 = 1.00.
The parameter sweep in Fig. 8 is also hypothetical because filling only the holes perforating the VO2 layer with different dielectrics (e.g., index-matching fluids) would be experimentally unfeasible. While filling the Au + VO2 thru-holes may be feasible, it would redshift the subwavelength regime of the holes by a factor equal to the refractive index of the filling material and thus unnecessarily complicate the interpretation of the EOT switching. Filling only the VO2-layer holes keeps λpeak nearly constant, as the four representative pairs of FEM-simulated (hypothetical) T00 spectra in Fig. 8(b–e) demonstrate. Simulating different refractive indices nHoleVO2 of the material filling the VO2-layer holes illustrates the idea that the VO2 holes act as “light funnels” for the EOT emerging from the plasmonic hole array. As shown in Fig. 8(a, bottom panel), increasing nHoleVO2 enhances peak-T00 for both SemiVO2 and MetVO2, but it does so more effectively in the semiconducting state because the waves tend to penetrate deeper into the plane of the perforated SemiVO2 film, as opposed to the perforated MetVO2 film where the light is more concentrated inside the holes.
Filling in the VO2 holes with a dielectric of nHoleVO2 > 1 results in an exponential decrease—with two rate constants—of the M2S-ratio towards unity [Fig. 8(a, top panel)]. An exponential dependence can be understood from the fact that the peak wavelength inside the VO2 holes is reduced to λpeak/nHoleVO2, which roughly equates to enlarging the hole diameter by the same factor. This increase in the effective DVO2 in turn leads to the tails of the evanescent waves, which are exponentially weaker away from the hole centers, to penetrate the cylindrical sidewalls with diminished amplitudes. Why the empirical fit works better with two rate constants instead of one is not entirely clear at this point, though it seems reasonable that this dependence may stem from the different trends of SemiVO2 peak-T00 for 1 < nHoleVO2 < 1.6 vs. 1.6 < nHoleVO2 < 3 [Fig. 8(a, bottom panel)]. In terms of the maximum M2S-ratio(nHoleVO2 = 1.00) = 188, this optimization affirms the “gold standard” of Optimization V(Au).
4.8 Varying (hypothetically) complex refractive index of semiconducting VO2
Optimization IX(Au): max(M2S-ratio) = 3766 at λpeak = 787 nm and (nsemi + iκsemi) × 1.20.
Another hypothetical parameter sweep is shown in Fig. 9(a, b). We start with the results from Fig. 4(b) for Air-Au + VO2-Glass hole arrays with different periods (Parray) and corresponding thru-hole diameters (Dthru-hole = Parray/3), plotted as a function of peak wavelength (λpeak) [see inset in Fig. 4(b) for Parray-to-λpeak conversion]. We then perform a series of FEM simulations with the same geometries but with the refractive index, nSemiVO2(λ) [Fig. 9(c)], and extinction coefficient, κSemiVO2(λ) [Fig. 9(d)], of semiconducting VO2 scaled by an artificial multiplier, β—also manually overriding the K-K relations for this set of FEM simulations. The components of the complex refractive index of MetVO2, also displayed in Fig. 9(c, d), are not modified. When β < 1, the SemiVO2 transmission curve as a function of λpeak (or Parray) gradually loses the pronounced dip at λpeak = 758 nm (or Parray = 720 nm), which occurs for β = 1, and rises above the MetVO2 peak-T00 curve [Fig. 9(a, bottom panel)]. The corresponding M2S-ratios decrease from about 10 to less than 1 [Fig. 9(a, top panel)]. Conversely, when β > 1, the SemiVO2 dip first deepens and sharpens substantially, before rising and flattening out again for β > 1.20 [Fig. 9(b, bottom panel)]. The dips also shift to longer peak wavelengths and larger array periods. At β = 1.20, the maximum M2S-ratio = 3766 at λpeak = 787 nm and Parray = 750 nm [Fig. 9(b, top panel)].
The strong non-monotonic dependence of the SemiVO2 dip and M2S-ratios on the SemiVO2 refractive index provides further evidence that a Fabry-Perot (anti-)resonance is involved in the SMPT-induced EOT switching. In short, when the right geometrical and (hypothetical) material parameters combine in the perforated SemiVO2 layer, transmission emerging from the adjacent plasmonic-layer hole array is attenuated as optical energy is channeled into FP-type modes.
4.9 Varying array period for Air-Ag+/–VO2-Air hole arrays: with vs. without VO2 holes
Optimization I(Ag): max(M2S-ratio) = 12 at λpeak = 720 nm and Parray = 690 nm.
This section demonstrates that the SMPT-induced EOT switching mechanism is robust with respect to changing the plasmonic metal that generates the EOT effect, as well as dispensing with the glass substrate. The FDTD simulation results in Fig. 10(a) for freestanding silver + VO2 hole arrays are very similar to those for gold + VO2 arrays on glass in Fig. 4: (i) both M2S-ratio curves are fitted quite well to Fano profiles with q ≈ 3 and linewidths of 170 nm (Ag + VO2) and 200 nm (Au + VO2); (ii) the maximum M2S-ratios are almost equal (12 vs. 10) and occur in the same 600–800 nm spectral window; (iii) both SemiVO2 peak-T00 curves have “valleys” around 550–700 nm, while the MetVO2 curves rise monotonically with increasing array period; and (iv) peak wavelengths in both cases scale linearly with array period. A minor difference is that the maximum M2S-ratio occurs at Parray = 690 nm (λpeak = 720 nm) for Ag + VO2 but at Parray = 720 nm (λpeak = 758 nm) for Au + VO2, although, for the former at Parray = 720 nm (λpeak = 749 nm), M2S-ratio = 11 is hardly different from the maximum value.
Since the real part of the Ag permittivity is more negative than that of Au throughout the visible and infrared region (e.g., εAg, real = –29 and εAu, real = –22 at 778 nm), for the same array period the Ag EOT peak is less redshifted than the Au EOT peak with respect to λRayleigh = Parray, the so-called Rayleigh wavelength where the air-side (1,0) diffraction mode changes from radiative to evanescent (i.e., grazing to the surface). If we take the “gold standard” Air-Au + VO2-Glass hole array—with maximum M2S-ratio ≈ 190 at λpeak = 778 nm, Parray = 720 nm, Dthru-hole = 302 nm and tVO2 = 245 nm [Section 4.4 and Fig. 6(c)]—and only swap the gold layer for a silver layer, then the simulated Air-Ag + VO2-Glass EOT peaks (not shown) are closer to λRayleigh, the Ag + MetVO2 and Ag + SemiVO2 peak-T00 values are, respectively, higher and lower than their Au + VO2 counterparts, and the Ag + VO2 M2S-ratio = 511 at λpeak = 767 nm, although the overall T00 spectra for the two perforated bilayers are qualitatively the same. Perhaps coincidentally, at λ = 778 nm, the M2S-ratio for the Ag + VO2 T00 spectra is also 190. The main takeaway is that hole arrays in gold + VO2 and silver + VO2 bilayer films behave equivalently with regard to EOT modulation effected by the phase transition of the VO2 layer. Furthermore, comparing Fig. 10(a) to Fig. 4(a) and Fig. 11(a) to Fig. S2(b), along with other simulations not shown here, demonstrates that the glass substrate does not affect the EOT switching.
An interesting case is presented in Fig. 10(b). The bottom panel shows the peak-T00 curves in the SemiVO2 and MetVO2 states of a half-perforated freestanding Air-Ag–PlainVO2-Air hole array, which has holes only in the Ag layer while the VO2 layer is unperforated (“plain”). [Note: The peak-T00 points and corresponding M2S-ratios are plotted as a function of λpeak rather than Parray, but the conversion is largely linear, as seen in the inset in Fig. 10(a).] In comparison with the fully perforated Ag + VO2 hole arrays [Fig. 10(a), bottom panel], the SemiVO2 peak-T00 curve of the Ag–PlainVO2 hole arrays lacks a pronounced “valley” and the MetVO2 curve rises at first but then decreases for λpeak > 720 nm. The simulated M2S-ratios (square markers) are plotted in Fig. 10(b, top panel), together with an analytical curve (solid line) calculated with the Fresnel equations as a Met-to-Semi ratio of normal-incidence transmittance through a 200-nm-thick VO2 film without holes and without an Ag overlayer. (Note: The “Fresnel-ratio” curve is plotted as a function of λvac rather than Parray because there are no EOT peaks in the case of a stand-alone plain VO2 film.) The fact that the M2S-ratio (square markers) of the perforated Ag film sitting on a plain VO2 film resembles so closely the Fresnel-ratio of the plain VO2 film alone means that the holes in the VO2 layer play a critical role in the mechanism responsible for the reverse-switching EOT modulation. Without holes in the VO2 layer of the bilayer structure, the plasmonic layer (Ag or Au) becomes superfluous for optical switching as the unperforated VO2 film can modulate the transmitted light just as effectively on its own. Only when the VO2 layer is also perforated with hole arrays do FP-like resonant effects emerge and enhance the EOT modulation, as described in the preceding sections.
4.10 Varying thru-hole diameter for Air-Ag + VO2-Air hole arrays
Optimization III(Ag): max(M2S-ratio) = 32 at λpeak = 727 nm and Dthru-hole = 260 nm.
We wrap up the optimization sequence with another example of Fano-like behavior [Eq. (1)], this time relating to freestanding Ag + VO2 hole arrays in air as a function of Dthru-hole (≡ DAg+VO2 = DAg = DVO2). The results plotted in Fig. 11(a) are indeed similar to those in Fig. S2(b) for Au + VO2 hole arrays on glass. A Fano-profile curve [Fig. 11(a, top panel), solid line) fits the M2S-ratios very well, although the Fano parameter qM2S = 41 is rather large, signifying that the asymmetry is low and the M2S-ratio peak has a nearly-Lorentzian shape. On the other hand, Fig. 11(b, bottom panel) shows the Fano parameters in each VO2 state, extracted from the best-fit Fano profiles of the same EOT spectra represented in Fig. 11(a), while Fig. 11(b, top panel) plots the Met-to-Semi ratios of these q-parameters (triangle markers) as well as a “meta-Fano” fit (solid line)—i.e., a Fano-profile curve fitted to a ratio of Fano parameters. Although this fit is not as good as the one in Fig. 11(a), it yields a “meta” qq-ratios = 0.11 that corresponds to a more pronounced asymmetry.
The “valley” in the SemiVO2 q-parameter curve Fig. 11(b, bottom panel) occurs near Dthru-hole = 260 nm, which is where in Fig. 11(a) the M2S-ratio has its maximum (top panel) and the SemiVO2 peak-T00 has its “valley” (bottom panel). Recalling that the Fano parameter characterizes the ratio of resonant (discrete) to continuum (direct, non-resonant) contributions to the transmission, the q-parameter curve for SemiVO2 apparently encodes some changes, relative to the MetVO2 state, in the coupling strengths of and/or interferences between the resonant and non-resonant transmission channels. The nature of these changes is beyond the scope of the current work, although we are intrigued by the resemblance between the Fano-parameter curve of our Ag + SemiVO2 hole arrays in Fig. 11(b, bottom panel) and the Fano-parameter curve of Au hole arrays in Fig. 7(c) in Ref. .
5. Simulation results: 2D plots of power flow and electric-field intensity
The purpose of this section is to visualize the funneling vs. spreading of optical energy into, respectively, the holes and the xy-plane of the VO2 layer. It shows several representative 2D color images of the intensity, E2, i.e., the square of the magnitude (norm in COMSOL) of the electric field, which is directly proportional to the wave’s irradiance. The image plots are overlain with arrows of the time-averaged real part of the Poynting vector, i.e., the power flow, 〈S〉. The lengths of the 〈S〉 arrows are logarithmically scaled in units of W/m2. Except for Fig. 12, the E2 colors and 〈S〉 arrows are plotted in the two orthogonal half-planes of symmetry that cut through the center of the hole: xz-plane of polarization parallel to the incident wave’s electric-field vector Einc, and yz-plane perpendicular to the incident polarization.
In each part of Fig. 12, four xy-plane projections plus the two above-mentioned xz- and yz-plane projections display the magnitude of the power flow, |〈S〉|, normalized by the initial irradiance at the input port, I0 = (Power in)/(Port area), which injects 1 W of power into an area of (Parray/2)2. The logarithmic color scale represents |〈S〉|/I0 as a unitless quantity in dB and is the same in Fig. 12(a, b, c, d): dark blue corresponds to –39 dB, yellow to - 9 dB, and dark red to + 11 dB. Additionally, Fig. 12 delineates the 3D ¼-geometry used in the FEM simulations.
In Fig. 13, Fig. S4 and Fig. S5, E2 is normalized by E02 = (377 V/m)2, the squared magnitude of the input port’s incident electric field. The E2/E02 ratios are also plotted as unitless quantities in dB, on another consistent logarithmic color scale where dark blue corresponds to + 30 dB, yellow to + 90 dB, and dark red to + 130 dB.
5.1 Unperforated vs. perforated VO2 layer
The images in Fig. 12 compare the power-flow magnitude in different planes of the half-perforated Air-Gold–PlainVO2-Glass hole array (i.e., no holes in the VO2 layer) [Fig. 12(a, b); see also Fig. 7(b)] vs. the “gold standard” fully perforated Air-Gold + VO2-Glass hole array [Fig. 12(c, d); see also Fig. 6(c)], in each state of the VO2 material. Considering the half-perforated case, the power flow spreads farther into the PlainVO2 layer in the metallic state [Fig. 12(b)] than it does in the semiconducting state [Fig. 12(a)]. Consequently, more detector-bound light enters the glass substrate from the plain SemiVO2 layer and M2S-ratio = 0.45 < 1 [see Fig. 7(a, top panel, diamonds)]. In the fully perforated, fully optimized case, the situation is reversed: The power flow is relatively more concentrated (“funneled”) within the hole in the MetVO2 state [Fig. 12(d)] than in the SemiVO2 state [Fig. 12(c)]. Therefore, more light enters the glass substrate in the metallic state and M2S-ratio = 188 >> 1 [see Fig. 6(a, top panel, circles)].
5.2 Diameter of hole in VO2 layer
Varying only DVO2 (also labeled DHoleVO2) while holding the other geometrical parameters constant, was discussed in Section 4.5 and Fig. 7. The plots in Fig. 13 compare the electric-field intensity and power flow for Air-Au + VO2-Glass hole arrays of three different DVO2 values.
When DVO2 = 160 nm < DAu = 302 nm [Fig. 13(a, b, g)], there is almost no EOT modulation since M2S-ratio = 1.01 at λpeak = 774 nm [see Fig. 7(a, top panel, diamonds) and Fig. 7(c)]. The intensity images (Fig. 13) in the SemiVO2 state (left panels) look qualitatively similar to their MetVO2 counterparts (right panels), with two exceptions: (i) a region of relatively low (greenish) intensity localized at the SemiVO2 output aperture in Fig. 13(a, b); and (ii) a spatially sharp dip in the SemiVO2 intensity, followed by recovery, at the VO2-Glass interface about 1/3 of the way in from the left edge in Fig. 13(b). These features are clearly observed in the intensity line profiles extracted along the VO2-Glass interface: feature (i) spans the regions –150 nm < x < 0 nm in Fig. 13(g, lower panel) and –80 nm < y < 0 nm in Fig. 13(g, upper panel); and feature (ii) appears around y = –270 nm in Fig. 13(g, upper panel). Although the MetVO2 y-profile intensity in Fig. 13(g, upper panel) drops even lower than the SemiVO2 intensity around y = 265–280 nm, the dip is not as sharp as feature (ii) and the MetVO2 intensity does not recover. A cautionary lesson can be drawn from Fig. 13(g) in conjunction with Fig. 7(c), namely that the relative light intensities registered at the output apertures of the bilayer hole array do not always predict the relative amounts of T00 reaching the far field in the two VO2 phases. To state the obvious, the VO2 material between the holes plays a critical role in the EOT modulation.
The intensity images [Fig. 13(c, d)] and line profiles [Fig. 13(h)] for DVO2 = 302 nm = DAu qualitatively resemble those described above, but there is clearly a much higher overall intensity above this MetVO2 layer, as seen in the upper right regions of the right panels of Fig. 13(c, d), in comparison with the same regions above the SemiVO2 layer (left panels). It was shown in Section 4.4 that this so-dubbed “gold standard” geometric configuration has one of the highest (non-hypothetical) switching ratios, M2S-ratios = 188 [see Fig. 6(a, c)], so it is not surprising that the holes in the MetVO2 layer appear to better funnel the light through than the holes in the SemiVO2 layer—in other words, with less “leakage” into the bulk of the VO2 film [cf. left vs. right panels in Fig. 13(c, d)]. Interestingly, the sharp dip in the SemiVO2 intensity around y = –270 nm [Fig. 13(h, upper panel)] is a narrower and deeper version of the same feature described above in the case of DVO2 = 160 nm [Fig. 13(g, upper panel)].
At DVO2 = 540 nm > DAu = 302 nm, the holes in the VO2 layer are, to a first approximation, no longer subwavelength since the cutoff diameter below which waveguiding in a cylindrical hole in a perfect electric conductor becomes evanescent  would be Dc = λpeak/1.71 = 457 nm < 540 nm. Because most of the light emerging from the holes in the Au layer then propagates through the empty (air) space of the holes in the VO2 layer [Fig. 13(e, f)], the interaction with the VO2 material is relatively weak and so is the EOT modulation: M2S-ratio = 0.90 at λpeak = 782 nm [see Fig. 7(a, e)]. The intensity images [Fig. 13(e, f)] and line profiles [Fig. 13(i)] look quite similar in the two VO2 phases—again except for the dip in the SemiVO2 intensity around y = –270 nm, now even sharper [Fig. 13(i, upper panel)]. In this particular case, the dip spatially coincides with the rim of the SemiVO2 hole. The origin of this persistent intensity dip is unclear; it may belong to a lateral (xy-plane) resonance generated at the Au-VO2 interface for the Au-hole diameter (DAu = 302 nm) common to the three cases discussed above.
5.3 Thickness of VO2 layer
(See Supplement 1 for content.)
5.4 Array period with scaled thru-hole diameter
(See Supplement 1 for content.)
We have performed FDTD and FEM electromagnetic simulations and preliminary experiments in order to optimize the geometry of bilayer gold + VO2 and silver + VO2 nanohole arrays towards a large modulation of the zero-order optical transmission within narrow spectral bands. The highest feasible switching ratio obtained in the simulations is close to 200, which makes such perforated bilayer structures potentially useful for high-speed tunable switching [93,94] of optical signals in hybrid nanophotonic-nanoplasmonic devices. For example, the relative spectral purity, deep modulation and ultrafast phase transition of the bilayer hole array may make it a competitive component of active display technologies based on plasmonic structural colors . In the course of the optimization process, we uncovered Fabry-Perot and Fano (anti-)resonances for variations of individual geometrical and optical parameters. The resonances appear to be coupled: for example, changing the periodicity of the hole array also affects the value of the VO2 film thickness that maximizes the switching ratio. The FP-type anti-resonances arise from the peculiar index of refraction of semiconducting VO2 in the 600–1000 nm wavelength range, where the real part has a nearly non-dispersive value (∼3) and the extinction coefficient is small (∼0.4) and also nearly constant. The simulations have mapped the parameter space of the bilayer hole arrays and are currently guiding more extensive experiments to test the robustness of the EOT modulation against real-world fabrication and measurement conditions.
Office of Science (CNMS2018-161, CNMS2019-171).
E. U. Donev gratefully acknowledges the expert support of ORNL/CNMS staff scientists Christopher Rouleau, Jason Fowlkes, Dayrl Briggs and Dale Hensley. Fabrication of some of the VO2 thin films was conducted at the Center for Nanophase Materials Sciences (CNMS2018-161, CNMS2019-171), which is a DOE Office of Science User Facility.
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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