Reconfigurable metalenses are compact optical components composed by arrays of meta-atoms that offer unique opportunities for advanced optical systems, from microscopy to augmented reality platforms. Although poorly explored in the context of reconfigurable metalenses, thermo-optical effects in resonant silicon nanoresonators have recently emerged as a viable strategy to realize tunable meta-atoms. In this work, we report the proof-of-concept design of an ultrathin (300 nm thick) and thermo-optically reconfigurable silicon metalens operating at a fixed, visible wavelength (632 nm). Importantly, we demonstrate continuous, linear modulation of the focal-length up to 21% (from 165 μm at 20 °C to 135 μm at 260 °C). Operating under right-circularly polarized light, our metalens exhibits an average conversion efficiency of 26%, close to mechanically modulated devices, and has a diffraction-limited performance. Overall, we envision that, combined with machine-learning algorithms for further optimization of the meta-atoms, thermally reconfigurable metalenses with improved performance will be possible. Also, the generality of this approach could offer inspiration for the realization of active metasurfaces with other emerging materials within field of thermo-nanophotonics.
Optical metasurfaces are two-dimensional (2D) arrangements of meta-atoms with sub-wavelength spacing that are engineered to precisely control and manipulate the properties of a light beam, such as its phase, amplitude, and polarization. Their evolution in the last two decades has enabled the miniaturization of numerous optical components, including holographic optical elements [1–3], beam deflectors [2, 4], and flat lenses [5–8]. More recently, reconfigurable metasurfaces have enabled active control of the optical properties revolutionizing the design, functionality and application domains of optical components and devices [9–12]. Phase-change materials [9, 12], , , stretchable substrates [15–17], strain-field engineering , thermo-optical effects [19–22], and free-carrier modulation [10, 11, 23, 24] are some of the engineering strategies used to achieve reversible changes of the metasurface function.
Reconfigurable (varifocal) metalenses (R-MLs) [14, 15, 20, 22, 24], in particular, have attracted increasing attention thanks to their enabling potential for tunable optics in microscopy systems , depth sensor devices [26, 27] as well as virtual and augmented reality platforms . Moreover, these R-MLs can uniquely combine focal length modulation with complex functionalities, such as multi-wavelength  operation, spectroscopy and polarization routing , that are critical for compact and smart imaging devices/sensors.
The realization of continuously reconfigurable MLs, however, presents significant challenges. Indeed, to generate a converging wavefront with a desired focal length, the nano-scatters constituting a ML must introduce a prescribed (parabolic) phase delay at each position along the ML radius (Figure 1a–c). To dynamically change the focal length, a distinct phase shift must be achieved at each lattice site. Thus, R-MLs require the non-trivial realization of both a parabolic spatial phase-profile and a spatially varying phase shift (Figure 1d and e).
To date, stretchable substrates  and strain-field engineering , which entail a mechanical modification of the ML structure, have demonstrated excellent reconfiguration capabilities with good efficiency (up to 30 and 60% respectively) and large focal length tuning (up to 20 and 100% respectively). Yet, there is a growing interest for non-mechanical modulation approaches such as phase-change materials, thermo-optical effects  as well as tuning of free-carriers and exciton-resonances . Indeed, these designs can offer unique opportunities in terms of fast modulation speed, including ultra-fast all-optical control , and reduced device thickness, i.e., ultra-thin lenses . Furthermore, achieving the desired modulation using a single uniform external control of the local optical properties (e.g., temperature or electrical potential), would greatly simplify device design, fabrication and integration, accelerating the deployment in real-word components.
Thermo-optic effects represent an attractive approach for the realization of dielectric R-MLs . Indeed, they entail a continuous and smooth change in the material optical properties (the refractive-index as shown in Supplementary Figure 1) and are robust against thermal-cycling. Yet, the low-magnitude of typical thermo-optical coefficients is generally assumed to limit the applicability of this strategy. Interestingly, silicon nano-resonators [5, 28] have been recently shown to exhibit pronounced shifts in their optical resonances (Figure 1b) both under external heating [19, 21] and for all-optical  modulation. Thus, by leveraging the amplification of thermo-optical effects by optical-resonance modes, silicon-based thermally reconfigurable dielectric metalenses (TR-MLs) could become a competitive and CMOS-compatible solution . However, a viable design for such an ultra-thin, tunable ML with fixed operation wavelength in the visible regime has not been demonstrated yet.
Here we report a proof-of-concept design of an ultrathin (300 nm thick) and thermo-optically reconfigurable silicon ML operating at a fixed wavelength in the visible regime (632 nm). We demonstrate that, using thermo-optic effects, it is indeed possible to achieve continuous modulation of the focal-length beyond the depth-of-focus of the lens. Specifically, operating under right-circularly polarized light, our TR-ML exhibits a change of 21% in the focal length, with a linear shift from 165 μm at 20 °C to 135 μm at 260 °C. The average conversion efficiency of the lens is 26%, close to the performance of mechanically modulated devices, while its Strehl ratio is 0.99, confirming a diffraction-limited performance. Importantly, in our design, we rely on a spatially uniform temperature increase of the structure, overcoming the need for a spatially varying modulation input and potentially enabling an all-optical photo-thermal modulation approach. Concurrently, in this work, we report an automatized methodology to design a reconfigurable metalens, compute its layout and verify the expected performance. Overall, although further optimization of the meta-atom design is needed to boost the performance of these components , our results demonstrate that TR-MLs can be a viable solution for active tuning of optical systems.
2.1 Thermally reconfigurable metalens phase profile and choice of design parameters
where is the temperature-dependent focal length and λ the wavelength of the incident light. Fixing the desired focal length at the initial temperature T 0, , and requiring f to be linearly dependent on the temperature T, it is possible to express as:
where Δf is the desired focal-length variation achievable at the maximum temperature variation ΔT. Using this assumption for , and omitting the dependence on the fixed parameters T 0, ΔT, f 0, Δf, the phase can be rewritten as follows:
where and the required phase shift at each temperature is:
We observe that in Eqn. (3) the required temperature-dependent phase profile ϕ(r,T) has been decomposed into a temperature-independent initial phase profile, ϕ 0(r), and a spatially varying phase shift, Δϕ, which depends on the temperature. Tuning of the phase shift at each lattice site can be achieved either by a structured external control  T(r) that generates different modulation inputs along the radius, i.e., , or by using meta-atoms that respond differently to the same uniform external control  T, i.e., . In our system, the phase shift must be generated by thermo-optical effects in dielectric nanofins. Applying different temperatures to nano-structures that are closely spaced is very challenging. We thus pursue the second strategy, and we consider a spatially uniform temperature input T (Figure 1a).
The analytical phase profile described in Eqn. (3) is uniquely defined once λ, ΔT, f 0, Δf, and the ML radius R are defined. We note that for each set of these constraints, a specific phase shift range Δmax ϕ must be attained with the meta-atoms (Figure 1c), where:
Seeking a favorable compromise between tunability and transmission efficiency of the ML in the visible regime, we set λ = 632 nm as the operating wavelength (see Supplementary Information S1, Figure 1b and Supplementary Figure 1 for further details). We also considered a maximum temperature variation ΔT = 240 °C, compatible with simple electric heaters (T max = 260 °C). Next, we quantified Δf/f 0 as a function of the lens numerical aperture (NA), its radius R as well as Δmax ϕ and we studied the maximum focal length variation versus the depth of focus (Figure 1f and Supplementary Figure 3). From this detailed analysis we observed that, in percentage, Δf/f 0 decreases for larger R and for smaller Δmax ϕ. Aiming for a ML with a numerical aperture (NA) of at least 0.08, comparable with commercial ultra-compact objectives, and with a focal length variation at least equal to the depth of focus, we defined the target phase profile of the TR-ML using the following parameters: , Δf = −45 μm and R = 15.75 μm. We thus obtain NA(f 0) = 0.08, NA(f 0 + Δf) = 0.10, Δmax ϕ = 102 deg, the average depth of focus z 0 ∼ 40 μm and the linear increase of the focal length with temperature is Δf/ΔT ∼ 1.9 μm/°C. The ideal ML phase profile and required phase change obtained for the chosen set of parameters are shown in Figure 1e and f.
2.2 Metalens design: silicon nanofin library and digitalization of the target phase profile
The response of individual meta-atoms must be engineered such that their phase parameter space (ϕ and Δϕ) satisfies the requirements imposed by the target phase and phase shift . Specifically, based on our design parameters discussed above, we require a phase range of , a maximum phase shift range of Δmax ϕ ∼ 100 deg (Figure 1d and e) and a linear variation of the phase shift Δϕ with temperature (Supplementary Figure 2).
Non-resonant silicon nano-pillars, which only support waveguide-modes [5, 39, 40], are insufficient to realize an ultra-thin, sub-μm thick ML (h = 300 nm) because, for ΔT = 240 °C, they can achieve a maximum phase shift (see Supplementary Note S1, S3 and Supplementary Figure 1). Ultra-thin, resonant meta-atoms are instead expected to provide a wider phase space thanks to the amplification of thermo-optical effects by the different optical modes [19, 22, 35] (Figure 1b). Yet, simple geometries supporting a single resonance cannot produce a phase modulation range exceeding π .
To introduce additional degrees of freedom for engineering the phase profile, we therefore adopted anisotropic meta-atoms (nanofin) composed of two silicon nano-pillars waveguides with different nano-pillar length and width that act as coupled waveguides (Figure 1c). Indeed, these anisotropic nanofins offer the opportunity to leverage the Pancharatnam-Berry phase [42, 43] and obtain an additional π phase accumulation  (see Methods for further details).
As shown in Figures 1c and 2a, the engineered nanofins are characterized by a set of geometrical parameters , where 1 and 2 indicate the first and the second nano-pillar, respectively, and by their rotation angle θ. The gap between the two nano-pillars (g = 60 nm) and their height (h = 300 nm) are fixed for all the nanofins (see Methods for further details). These are distributed with varying rotation angles in a sub-wavelength lattice on top of a sapphire substrate with a fixed period (p = 350 nm).
We used numerical simulations (COMSOL Multiphysics®, see Methods and Supplementary Note S3 for further details) to study the electromagnetic response of the nanofins. To obtain our nanofin library, we first performed a parameter sweep computing the transmission efficiency versus the maximum phase shift for every geometry # at a fixed angle θ = 0 and for ΔT = 240 °C. Interestingly, as shown in Supplementary Figure 14, there appears to be a trade-off between the maximum phase shift and the transmission efficiency. From this first study, we selected those geometries (Figure 2a) with an average transmission efficiency above 15% (Supplementary Table 1) that could also cover the required Δmax ϕ ∼100 deg. The geometric parameters of these selected nanofins are shown in Supplementary Figure 6a and listed in Supplementary Table 1. Next, for every selected geometry #, we performed a sweep over the rotation angle θ and input temperature T, obtaining the complete dataset of phase , phase shift Δϕ(θ, #) and transmission efficiency (Figure 2b, Supplementary Figure 4, Supplementary Figure 5). Finally, the nanofins must be distributed spatially based on the discretization of the analytical phase constraints. We thus constructed a routine that searched our dataset to identify, for every spatial position, the geometry (#) and rotation angle (θ) that satisfied the required phase-delay at both T = 20 °C and T = 260 °C as well as the linearity condition (Supplementary Note S2 and Supplementary Figure 2). With this approach we obtained the phase profile digitalization and the resulting layout for an optimal design of our TR-ML (Figure 2c and d, Supplementary Figure 5 and Supplementary Figure 6). In particular, we observe that, leveraging the PB phase, only five nanofin geometries are sufficient to satisfy well all the constraints.
Overall, we observed that the thermo-optical effect combined with the Pancharatnam-Berry (geometrical) phase effect in our engineered silicon nanofins provides a large range of phase and phase shift values enabling the design of a thermally reconfigurable metalens (TR-ML).
2.3 Thermally reconfigurable metalens
We simulated the propagation of a beam focused by our TR-ML obtaining the three-dimensional intensity profile of the focused beam (see Methods). From a qualitative inspection of the intensity profiles along the propagation direction, we verified that the focal length decreases as the temperature increases (Figure 3a and b). The focal length shifts of the metalenses were determined by measuring their point spread functions at increasing temperatures (from T = 20 °C up to T = 260 °C, with ΔT = 30 °C) along the propagation direction (z-axis) with 1 μm resolution. Furthermore, we observed that both the point-spread-function (PSF) of the beam focus and the depth of focus become narrower at higher temperatures. Thus, the lens NA increases with temperature, as expected (Figure 3c and d, f). From a quantitative analysis, we obtained that the ML focal length changes from f(20 °C) ∼ 165 μm to f(260 °C) ∼ 135 μm.
Due to intrinsic diffraction of low Fresnel number lens , our ML, exhibits an unavoidable deviation of its focal length from the ideal focal length defined with geometric optics. In particular, the peak irradiance position Z p derived from diffraction theory lies at an average distance δ = Z p − f ∼ 27 μm from the target focal length f over all temperatures. Nonetheless, the reported design satisfies all the tunability requirements. Indeed, the total focal length modulation, Δf ∼ 30 μm, is comparable to the final depth of focus, z 0 ∼ 35 μm. Furthermore, we observe that both the point-spread-function (PSF) of the beam focus and the depth of focus become narrower at higher temperatures as the NA increases from NA(20 °C) = 0.10 to (Figure 3e). In addition, the modulation follows the desired linear trend. In fact, from a linear fit of the focal length versus temperature we obtain a slope Δf/ΔT ∼ 1.3 μm/°C, a root mean square error RMSE = 3.7 μm and an R-squared value R 2 = 0.99.
Importantly, from the deviation of the discretized phase profile from the analytical one (Supplementary Figure 5d and e), we obtain a ML Strehl ratio of S ∼ 0.99 > 0.8, indicative of a diffraction-limited behavior of our design (see Methods for further details). Indeed, as shown in Figure 3f, the value of the full-width at half maximum of the Airy disk at the focal plane ( ) lies at less than 0.1 μm from the diffraction-limit one (σ ∼ 0.42λ/2NA in the Gaussian approximation).
We also characterized our TR-ML by measuring the focusing efficiency of the focal spot under RCP incident light. We defined the focusing efficiency as the focal spot power divided by transmitted power through an aperture with the same radius as the designed metalens. Our TR-ML exhibits an average focusing efficiency equal to 26%. Although, this value is limited by the optical losses and reduced transmission (intrinsically limited up to 50% by the polarization conversion and by the conversion-efficiency of the nanofin meta-atoms Supplementary Table 1), our result is comparable to that of mechanically actuated reconfigurable metalenses [30, 46, 47].
Similar to other adaptive optic approaches [46, 48], the TR-ML proposed in this work is affected by chromatic aberrations. Our metalens is designed to operate at one wavelength of λ = 0.632 μm with an effective focal length f ∼ 150 μm and an effective NA = 0.1. At this operating wavelength, focal length and NA, and assuming that the fractional change in the focal length is equal to the fractional change in the wavelength (Δf/f = Δλ/λ), the theoretical operation bandwidth [46, 49] should be . However, a linear dependence of the phase on the wavelength might be not a good approximation because of the presence of resonance modes. Therefore, to ensure achromatic tunable metalens, different approaches recently demonstrated [8, 50] should be specifically studied and combined for phase dispersion control over continuous bandwidth. In this regard, we note that for ML designs based on purely resonant meta-atoms, effective thermal modulation and broad achromatic bandwidth rely on contrasting design requirements (high- and low- quality factor modes, respectively).
Our design principle enables also higher NA thermally tunable metalens. As the proof of concept, we show in Supplementary Figure 16 the beam propagation resulting from the design of a thermally tunable metalens with NA = 0.4, radius R = 4.55 μm and a total focal length shift of 10% (from 10 μm up to 11 μm). This design is based on the same nano-resonators set identified for our initial design and has been obtained following the same methodological approach. In general, the maximum focal length shift is limited by the tradeoff between the metalens NA and radius as well as the tradeoff between the metalens NA and the maximum thermal phase shift provided by the nano-resonators (Supplementary Figure 1 and Supplementary Figure 2). Such trade-off also impacts the design of MLs with larger diameters, as they require a large maximum phase shift. However, selecting longer operation wavelengths, the ML design present a less stringent trade-off between maximum focal length shift and NA (see Supplementary Figure 16).
In this regard, although the thermally tunable metalens presented here has a small sub-millimeter aperture, our methods can be easily adapted to generate metalens designs with higher diameters while keeping the exact same set of nanofin resonators and parameter space. However, to reach high quality design while increasing the ML radius it is recommended either to decrease the total focal length variation requirement or to extend the parameter space by introducing other geometries to cover the range of the total phase thermal shift that will be required while increasing the ML radius. In particular, one possibility to increase the maximum phase shift is to combine both positive and negative phase shifts of the meta-atoms. Moreover, it would be interesting to explore the use of a material with a higher thermo-optic coefficient, such as germanium [51, 52], and to operate at infrared (telecom) wavelengths.
Concerning the TR-ML realization and operation, we envision the possibility of using a ring micro-heater to provide a uniform temperature bias to the lens. Indeed, this approach in combination with thin substrates can result in excellent temperature uniformity across the lens (Supplementary Figure S17). Implementation of a micro-heater underneath the lens would likely provide faster response . Yet, perturbations to the input polarization and optical losses could hinder the ML operation. More interestingly, we envision the possibility of leveraging optical heating for fast all-optical modulation of the metalens . In this case, collective heating effects are expected to result in a non-uniform temperature profile across the radius of the ML. While these should be carefully assessed in the design phase, the proposed methodology is already capable of accounting for an arbitrary temperature profile. From the long-term stability point of view, adverse effects of thermal stresses (i.e., delamination) are not expected for the low temperatures considered for this analysis. However, much higher operation temperatures as well as thermal fatigue over many thermal cycles should be carefully assessed experimentally towards fast and reliable modulation.
Overall, we have demonstrated a thermally reconfigurable metalens based on thermo-optic effects in silicon nanofin resonators. Our design enables a continuous modulation of the focal length with close-to diffraction limited optical performance and competitive focusing efficiency. Our approach combines a geometrical phase approach with a temperature dependent phase shift to tune the lens properties using a spatially uniform temperature input. Increasing the complexity of the meta-atom geometry and introducing more resonance modes  would offer a broader parameter space to search for TR-ML designs with advanced optical properties (high NA, high efficiency). Thus, we envision that coupling our approach with machine-learning algorithms [54, 55] capable of identifying non-intuitive meta-atom shapes, which cover a large phase shift range with high conversion efficiencies, will significantly improve the performance of thermally reconfigurable silicon metalenses. More broadly, the proposed design approach, based on geometric phase combined with thermo-optical effects, is expected to offer inspiration for the future realization of a broad class of active metasurfaces within the emerging field of thermo-nanophotonics .
5.1 Strehl ratio and focusing efficiency
Strehl ratio is defined as the ratio between the square of the electric field amplitude at the center of the designed intensity profile and the square of the electric field amplitude at the center of the ideal point spread function (PSF) [56–58]. It is given by the following equation:
where is the root mean square of the deviation of the designed phase from the theoretical phase. A lens is typically considered to correspond to diffraction limited performance if the Strehl ratio exceeds 0.8 [56–58].
The root mean square deviation of the designed ML phase profile from the analytical phase profile results to be σ[ϕ(T 0)] = 2.5 deg at 20 °C and at 260 °C (Supplementary Figure 5d and e). These values result in a Strehl ratio of S ∼ 0.99.
The efficiency is computed as the ratio between the total left circular polarized light intensity at the focal point upon conversion from right (RCP) to left (LCP) polarized light divided by the total light intensity (RCP plus LCP) which would be transmitted and focused assuming no losses due to absorption or reflection.
5.2 Simulation of the silicon nano-resonators
The geometry of all nano-resonators was simulated with a fixed gap between the two nano-pillars (g = 60 nm) and height (h = 300 nm). The gap has been minimized to maximize the coupling and its value has been chosen based on fabrication considerations such as the resolution of e-beam lithography. The nano-pillar high was fixed at about h ∼ λ/2 to ensure good mode coupling.
We simulated the interaction of circularly polarized light (electrical field E ) with our nano-scatterers to extract both the phase ϕ and the transmission T (coincident with the polarization conversion efficiency from RCP to LCP) of the electric field. We thus used the radio frequency (RF) module of COMSOL Multiphysics v5.5 to solve the Maxwell’s equation:
where , n and k are the real and the imaginary part of the refractive index n respectively, k 0 = 2π/λ is the wave number of free space and where only the constitutive relations for linear materials are considered.
Each silicon nanofins unit was simulated setting a periodic boundary condition along the transverse direction with respect to the propagation of light and a perfectly matched layer and input/output ports boundary conditions along the longitudinal direction.
We used the scattering parameter S 21, measured from the eigenmode expansion of the electromagnetic field at the output port 2 (see Supplementary Figure 7), to extract both the transmission T and phase ϕ of the electric field converted from right to left circularly polarized light:
and where the input RCP light incident on the input port 1 (S 1, S-parameter of incident wave) under sapphire substrate, and the output LCP component (S 2) at the exit port 2 are:
Note that, while performing COMSOL simulation, we inverted both the sign of output phase and the input/output polarization definition compared to the ones used in the formalism described above for the phase profile design of our metalens. We are inverting the phase and polarization because in COMSOL the phase convention is based on:
Thus, while in our formalism an increase of the phase value indicates retardation (delay accumulation) in phase profile, in COMSOL, a phase increase introduces an electrical field anticipation.
5.3 Thermally tunable metalens design method
The design of our metalens (ML) is based on three main steps written in MATLAB.
In the first step, the software computes the maximum focal length variation and numerical aperture (NA) for a metalens characterized by the inputs set by the user. At this stage, the input data simulated in COMSOL are loaded and shown and the analytical phase profile based on the user input parameters is created and displayed. The metalens design optimization and the distribution of the different NF# geometries over the ML surface, are performed during the second step where the metalens layout is created and the Strehl ratio is computed. In the last step the Beam Propagation Method (BPM) is used to retrieve the actual ML focus profile and to extract the ML focus position, its FWHM and its depth of focus and its efficiency at each temperature. Lastly, the ML performance is compared with the behavior of an ideal lens. Further details can be found in the Supplementary note S1 and in the following paragraph.
5.4 Beam propagation method (BPM)
In this section, we present the derivation of the theory supporting the beam propagation method here implemented to study the behavior of a beam focused by our designed metalens [59, 60]. All numerical calculations of the intensity profiles were performed in MATLAB environment.
When considering a plane wave incident on a homogeneous, isotropic, and linear medium, the electromagnetic wave equations reduce to the Helmholtz equation for E :
In the paraxial limit, the two-dimensional Fourier transform of the electric field E transverse to the propagation axis Z and at a certain fixed position z (i.e., the E field in the z = 0 plane) can be written as:
where k x , k y are the spatial frequencies coordinates of the Cartesian transverse coordinates x and y and ϕ the phase value in the position.
Then, the inverse Fourier transform becomes:
and replacing the E field expression in the Helmholtz equation with the Fourier representation just derived above, we find that the evolution of the field in the propagation direction Z, in the Fourier space corresponds to a multiplication for a factor :
In other words, the Fourier spectrum of E , in an arbitrary image plane at the z location, corresponds to the spectrum in the object plane (z = 0) multiply by the factor where the sign ‘+’ describes a wave propagating the forward positive direction z > 0 and ‘− ’ sign refers to a wave propagating in the negative half-space z < 0. Replacing the expression (13) into (11) we can express the electrical field as:
is the Fourier transform of the electrical field at the ML plane (z = 0) and ϕ and E 0 and are the phase and the electrical field amplitude of each nano-resonator at the position (x, y) on the z = 0 plane respectively. The intensity profile of the beam focused by the designed metalens can be finally retrieved from the squared of the electrical field propagation:
5.5 Geometrical phase
The phase delay induced by dielectric nano-structures on the incident light can be controlled by tuning either the refractive index, the relative rotation or the resonances of the nano-structures. Both on- and off-resonance based approaches are strictly related to the operating wavelength, the refractive index of the utilized materials and the geometry and location of the nano-antennas. On the contrary, the relative rotation of the nano-pillars introduces a geometrical phase delay which depends only on the geometrical asymmetry of the nano-structures and on the polarization of the incident light.
According to the Pancharatnam–Berry geometric effect [42, 43], circularly polarized light incident on a periodic layer consisting of subwavelength and anisotropic structures, with different orientations θ respect to the reference, is transmitted as the sum of two components: one with the same phase delay and the same handedness of the incident light and the other with a phase delay ϕ proportional to the rotation angle θ of the structure and with opposite handedness. In further detail, if the electric field incident on the nano-structure is circularly polarized, that is, , then the output electric field is given by [42, 43, 61], , :
where the entire operation is represented by the matrix:
where t l and t s are the complex transmittance coefficients corresponding to an incident light linearly polarized along the long and the short axis of the nanofin respectively, θ is the nano-structure rotation angle with respect to its long axis in the rotation, is the rotation matrix and the eigenvectors associated to the eigenvalues e ± i2θ . The resulting phase delay of the light component with opposite handedness, thus, is ϕ = ±2θ.
The design of our ML relies on the electric field component converted from right-handed circularly polarized light to left-handed circularly polarized light . The latter allows an average phase delay that can be exploited to cover the phase range needed for a thermally tunable focal length (Figure 2c, Supplementary Figure 6).
Funding source: Schweizerischer Nationalfonds zur Fӧrderung der Wissenschaftlichen Forschung
Award Identifier / Grant number: Eccellenza Grant #194181
Award Identifier / Grant number: Spark Grant #190809
Funding source: H2020 Marie Skłodowska-Curie Actions
Award Identifier / Grant number: 898315
A.A. acknowledges the support of the Marie Skłodowska-Curie Actions #898315. F.K. acknowledges the support of the Swiss National Science Foundation (Eccellenza Grant #194181). T.V. Tsoulos acknowledges the support of the Swiss National Science Foundation (Spark Grant # 190809).
Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: SNSF Eccellenza Grant #194181; SNSF Spark Grant #190809; H2020 Marie Sklodowska-Curie Actions #898315.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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