Thermal Fluctuations of the Optical Properties of Nanomechanical Photonic Metamaterials

The combination of optical and mechanical resonances offers strong hybrid nonlinearities, bistability, and the ability to efficiently control the optical response of nanomechanical photonic metamaterials with electric and magnetic field. While optical resonances can be characterized in routine transmission and reflection experiments, mapping the high‐frequency mechanical resonances of complex metamaterial structures is challenging. Here, it is reported that high‐frequency time‐domain fluctuations in the optical transmission and reflection spectra of nanomechanical photonic metamaterials are directly linked to thermal motion of their components and can give information on the fundamental frequencies and damping of the mechanical modes. This is demonstrated by analyzing time‐resolved fluctuations in the transmission and reflection of dielectric and plasmonic nanomembrane metamaterials at room temperature and low ambient gas pressure. These measurements reveal complex mechanical responses, understanding of which is essential for optimization of such functional photonic materials. At room temperature the magnitude of the observed metamaterial transmission and reflection fluctuations is of order 0.1% but may exceed 1% at optical resonances.


Brownian Motion and Optical Properties of Nanostructured Materials: Theoretical Background
Classical Brownian motion experiments reveal how a microscopic particle is driven in chaotic motion externally by collisions with molecules of ambient gas or liquid. The majority of nanomechanical photonic metamaterials are constructed from cantilevers or doubly clamped beam-like components of microscopic length and nanoscopic cross section. The thermal motion of such structures under vacuum is driven internally by momentum transfer resulting from the interference, annihilation and creation of incoherent flexural phonons in the main mechanical modes. (The emission and absorption of thermal photons are not important due to the low momentum of such photons.) The combination of optical and mechanical resonances offers strong hybrid nonlinearities, bistability, and the ability to efficiently control the optical response of nanomechanical photonic metamaterials with electric and magnetic field. While optical resonances can be characterized in routine transmission and reflection experiments, mapping the high-frequency mechanical resonances of complex metamaterial structures is challenging. Here, it is reported that high-frequency time-domain fluctuations in the optical transmission and reflection spectra of nanomechanical photonic metamaterials are directly linked to thermal motion of their components and can give information on the fundamental frequencies and damping of the mechanical modes. This is demonstrated by analyzing time-resolved fluctuations in the transmission and reflection of dielectric and plasmonic nanomembrane metamaterials at room temperature and low ambient gas pressure. These measurements reveal complex mechanical responses, understanding of which is essential for optimization of such functional photonic materials. At room temperature the magnitude of the observed metamaterial transmission and reflection fluctuations is of order 0.1% but may exceed 1% at optical resonances.

Introduction
The thermal vibration frequencies of components of nanomechanical devices increase as objects decrease in size and the amplitude of such oscillations becomes increasingly important. In nanomechanical devices, they are of picometric scale in the Nano-components such as beams and cantilevers can be modeled as damped mechanical oscillators. [9] Considering structures located in the xy-plane, engaged in thermal motion in the z-direction, the Langevin equation for such motion can be written as [10] πγ ω π γ η ( ) ( ) where F th (t) is the time-dependent thermal force experienced by the oscillator related to the dissipation factor γ through the fluctuation-dissipation theorem, η(t) is a normalized white noise term, m eff is the effective mass of the object, k B is the Boltzmann constant, T is the temperature, is the natural angular frequency of oscillation, f 0 is the natural frequency, and k is the spring constant. The resonance quality factor Q = f 0 /γ in the limit of small damping, γ ≪ f 0 , as we assume here.
The origins of mechanical dissipation have been intensively studied over the past decades. [10,11] The most relevant loss mechanisms include clamping losses due to propagation of elastic waves into the substrate through the supports of the oscillator, fundamental anharmonic effects such as thermoelastic damping and phonon-phonon interactions, materialsinduced losses caused by the relaxation of intrinsic or extrinsic defect states in the bulk or surface of the resonator, and, where applicable, viscous damping caused by interactions with surrounding gas atoms or by compression of thin fluidic layers.
Thermomechanical fluctuations of a component's position z(t) are transduced to fluctuations in the intensity of light scattered on the component, ( ) where I 0 and I = μ(z, F)I 0 are the intensities of the incident and scattered light respectively, and μ(z, F) is, generally, a nonlinear function of the component's displacement z and optical frequency F. As, in a stochastic process, the power spectral density is equal to the Fourier transform of its autocorrelation function, [12] the scattered light amplitude spectral density S(f) resulting from small thermomechanical fluctuations in position δz(t) is In a nanomechanical photonic metamaterial, a non-diffracting array of identical oscillating components, the same formula will describe the spectra of fluctuations of the intensity of light reflected I r by and transmitted I t through the metamaterial ( Figure 1). Fluctuations S r,t of the metamaterial's specular reflectance/transmittance correspond to the ratio between the amplitudes of fluctuating and non-fluctuating components of the reflected/transmitted light, that is, Equation (2) divided by the incident intensity I 0 and optical reflectance/transmittance of the metamaterial without displacement, At the mechanical resonance, f = f 0 , this modulation depth spectral density reaches a peak value of Reflectance/transmittance fluctuations over a range of mechanical frequencies can be calculated by integration over the power spectral density of the fluctuations Integration from 0 to ∞, or at least over the whole resonance, gives the root-mean-square (RMS) fluctuations where the final term corresponds to the RMS beam displacement From here it is apparent that fluctuations of transmission and reflection will be largest at high temperatures, in metamaterials constructed from very light (low m eff ) building blocks, and at optical frequencies where transmission and reflection depend strongly on displacements of said building blocks. The largest fluctuations over a narrow spectral range δf will be seen at high quality mechanical resonances [Equation (4)]. As such, we should expect thermal fluctuations of optical properties to be strongest in highly optically dispersive nanomechanical metamaterials.
Assuming metamaterial beams, such as those shown in Figure 1, with an effective mass m eff = 1 pg and mechanical quality factor of Q = 1000 moving at a damped frequency  ∂ ≈ 1% nm −1 , one may expect to observe an RMS thermomechanical displacement amplitude of δz RMS = 160 pm at the center of the beams at room temperature, resulting in a 0.16% RMS fluctuation of optical properties. At the mechanical resonance, the corresponding spectral densities of displacement and optical property modulation reach peak values of 3 pm Hz −1/2 and 3 × 10 -5 Hz −1/2 , respectively. Thus, in nanomechanical photonic metamaterials, thermal fluctuations are transduced to fluctuations of optical properties that determine their functional noise floor and dynamic range. At the same time, observation of the spectra of thermal oscillations gives direct access to the resonant frequencies of natural mechanical modes at which the photonic metamaterial will be most responsive to external stimuli. This information on fluctuation and responsivity of the mechanical sub-system can help in the design of more efficient metadevices. [13] Moreover, nanomechanical photonic metamaterials typically comprise large arrays of nominally identical elements (e.g., metamolecules, beams, etc.), but in practice, with physical characteristics affected by fabrication tolerances, individual nanomechanical oscillators are endowed with slightly different natural frequencies. This results in the degradation of optical functionality. In the case where the natural frequencies of individual oscillators are closely spaced compared to the widths of their characteristically Lorentzian lines, the result will be an inhomogeneously broadened metamaterial mechanical resonance. Where the spacing is larger, the resonances of individual oscillators may be resolved. Measurements of the spectra of thermal oscillations can provide insight into the nature of such metamaterial resonance broadening and splitting. Previously, transduction of natural oscillations to modulation of light has been observed in cantilevers, [14] antennas, [15] microresonators, [16] and optomechanical systems, [17] and exploited in atomic force microscopy. [18] Here, we focus on the role of thermal oscillation in forming the optical properties of nanomechanical photonic metamaterials.

Brownian Motion and the Optical Properties of Metamaterials: Experimental Observation
Below we examine thermal fluctuations in the optical properties of two common types of photonic metamaterial: plasmonic and all-dielectric nanomechanical metamaterials fabricated on membranes of nanoscale thickness. [2,3,[5][6][7][8][19][20][21] Both the plasmonic and dielectric metamaterials consist of 2D arrays of nanoscale optical resonators supported by 1D arrays of mechanical resonators, which are flexible beams of microscale length cut from a silicon nitride membrane by focused ion beam (FIB) milling. (Due to their planar, monolayer geometry, such metamaterial structures are also sometimes known as metasurfaces.) The all-dielectric metamaterial shown in Figure 2a,  with a 115 nm layer of amorphous silicon by plasma-enhanced chemical vapor deposition. This bilayer was then structured by FIB milling to define an array of asymmetric nanorod pairs in the amorphous silicon layer, on 20.3 µm long silicon nitride beams. The structure supports a closed mode optical resonance [22] at a wavelength of 1530 nm, underpinned by the excitation of antiparallel displacement currents in the pair of amorphous silicon nanorods [23] (Figure 2a and Figure 3).
The plasmonic metamaterial shown in Figure 2b,d was manufactured on a 50 nm thick silicon nitride membrane coated with a 50 nm layer of gold by thermal evaporation. This bilayer was then structured by FIB milling to define gold nanorods, in groups of three per unit cell in a П-shaped arrangement split across two adjacent 14.4 µm long silicon nitride beams. The П-shaped nanorod arrangement supports a Fano-type optical resonance [7,24] at a wavelength of 1350 nm, resulting from the Adv. Optical Mater. 2022, 10, 2101591 interference between a "bright" dipole mode excited in the larger gold nanorod and an antisymmetric and "dark" magnetic dipole mode induced in the pair of smaller nanorods (Figures 2b and 3).
The dielectric and plasmonic metamaterial structures presented in Figure 2 and studied in this work are types that can be reconfigured: a) thermally, by changing the external temperature due to differential thermal expansion of neighboring beams; [21] b) optically, whereby the displacement of neighboring beams is driven by optical heating and the interaction between oscillating electric dipoles induced by the incident light; [7,8] c) acoustically, by ultrasonic vibrations; [4] d) electrostatically, by the Coulomb force between charged neighboring beams [2] or selected beams and another electrode; [20] and e) magnetically, by the Lorentz force acting in a static magnetic field on a current passing through the beams. [3] (In the latter two cases an electrically conductive layer must be added to the structure.) In both types of metamaterial, whatever the actuation mechanism, differential movements between neighboring beams change the unit cell geometry and thereby the optical properties of the array. In such structures, the frequencies of the fundamental mechanical oscillatory modes f 0 of the beams lie in the megahertz range. At the same time, the optical properties  reflected from the samples was monitored with a photodetector and a radio frequency spectrum analyzer.
As expected from Equation (3), in the spectra of transmitted and reflected light we observe a range of sharp peaks linked to the thermal movements of the silicon nitride beams at their natural frequencies in the megahertz range (Figure 4a,b). Here, the peaks are observed against a flat background of the photodetector noise floor. The magnitude of the underlying thermal RMS beam displacements is ≈50 pm (≈160 pm) for the dielectric (plasmonic) metamaterial beams at room temperature, according to Equation (7). The average quality factor of the observed mechanical resonances for the dielectric metamaterial is 1.6 × 10 3 , slightly higher than the quality factor of 1.0 × 10 3 for the plasmonic metamaterial.
In both cases, we see an isolated group of several peaks with Lorentzian profiles, which overlap in the case of closely spaced resonances (insets of Figure 4b). The collective mechanical response of the metamaterial is the sum of the individual responses of its mechanical elements. Due to a wide distribution of resonance frequencies over a small number of mechanical resonators, we resolve the individual resonances as a group of Lorentzian lines (i.e. each associated with a different beam within the structure), rather than spectral broadening of a single peak. (The beams are necessarily coupled through the supporting membrane, but we do not observe any evidence of coupling effects such as synchronization.) Here, the main cause of variations in the resonance frequencies of individual, nominally identical beams is most likely disparities in beam tension across the sample resulting from the non-uniformity of intrinsic stress in the membrane, rather than variations of their physical dimensions. These variations in tensile stress across the metamaterials, leading to the spread of individual peak frequencies, can be evaluated via Euler-Bernoulli beam theory, [12,25] which gives the stress-dependent fundamental frequency of a doubly clamped beam of rectangular cross section as where t is the beam thickness, E the Young's modulus, ρ the density of the material, and σ the tensile stress along the beam length. Figure 4c,d, illustrating the relationship between stress and fundamental frequency, shows that for both the plasmonic and dielectric metamaterials, the observed variations in fundamental mechanical resonance frequencies of beams are explainable by tensile stress variations from beam to beam of ≈30 MPa. This is not a surprizing number as stresses of between several hundred megapascals and a gigapascal are common in unstructured silicon nitride membranes, [26] which can result in post FIB-fabrication stress variations across metamaterial arrays.
Inhomogeneous illumination of several beams within the optical spot profile can also contribute to inhomogeneous shifts of beam frequencies through thermal expansion of the beam length, which reduces stress: σ = σ 0 − αEΔT, where σ 0 is the initial tensile stress, α the thermal expansion coefficient of the material, and ΔT the temperature change. Considering conductive cooling, in our experiments, the laser-induced temperature increases reach ≈10 K in the plasmonic metamaterial and ≈1 K in the dielectric metamaterial. For a silicon nitride beam with an initial stress of 30 MPa, such a temperature increase translates to around 20% stress reduction in the plasmonic metamaterial (2% for the dielectric metamaterial). In the present cases, this translates to a 6% (2%) decrease in resonance frequency (Figure 3c,d). We observe a significantly larger spread of resonance frequencies, with the lowest resonance frequency being 24% (7%) smaller than the largest, suggesting that the observed spread of resonance frequencies is mainly due to intrinsic stress variations of the beams.
Fluctuations in the optical properties of the metamaterials are presented in terms of S r,t , the relative spectral density of reflectance/transmittance modulation (Figure 4a,b). To evaluate the root-mean-square of relative fluctuation of optical properties / RMS r,t r ,t I I δ resulting from thermomechanical fluctuations we need to integrate, as shown by Equation (5), over the entire frequency range. If only fundamental mechanical modes of the metamaterial beams are taken into account, this integral can be approximated as 0 S N f ∆ , where N is the number of peaks/ modes and S 0 and Δf are the average amplitude and width of the peaks, respectively.
From the data presented in Figure 4a Finally, we shall note that thermal fluctuations can give rise to nonlinear effects. The nonlinearity of coupling between metamaterial optical properties μ r,t (z, F) and beam displacements z will result in fluctuations of the optical properties at harmonics of the fundamental mechanical oscillation frequency f 0 . Moreover, the mechanical motion itself may be thermally excited into the nonlinear regime. In the case of beams anchored at both ends, the oscillation becomes nonlinear when its amplitude becomes comparable to the beam thickness t divided by the square root of the mechanical quality factor Q: [27] a beam with length L, width w, and thickness t enters the nonlinear regime of thermal motion when As an example, at room temperature, a silicon nitride beam of rectangular 100 nm × 100 nm cross section and quality factor Q = 1000 will enter the nonlinear regime of Brownian motion when L > 100 µm. Such effects though lie beyond the scope of the present study, as the sub-200-pm RMS displacements of our metamaterial beams are too small. The emergence of nonlinearity in our structures would require displacements of several nanometers.

Conclusion
In conclusion, we have observed that the optical properties of dielectric and plasmonic nanomechanical photonic metamaterials at near-infrared (telecoms) wavelengths exhibit thermal fluctuations of order 0.1%., rising potentially to >1% at optical resonances. The spectra of fluctuations enable metamaterial characterization at the level of individual mechanical elements. They provide exact information on the frequencies at which a nanomechanical photonic metamaterial can be efficiently actuated by external stimuli, providing insight to mechanisms of broadening and splitting of mechanical resonances in metamaterials, and aiding in the optimization of their performance. Beyond metamaterials, our approach may be applied to the characterization of mechanical resonances of similar micro-/nanostructures, e.g., in nanoelectromechanical and biological systems.