Information advantage in sensing revealed by Fano-resonant Fourier scatterometry

Domain: physics
Published: 2025-12-10
Editorial outcome: minor_revision

Abstract
Fano resonances in nanophotonic structures are attractive for sensing due to their ultanarrow resonant linewidths and high local fields. Conventional read out schemes rely on measuring a frequency shift in Fano scattering spectra as function of perturbation. We experimentally demonstrate that angle-resolved analysis of the scattering of a Fano resonant structure is quantitatively more informative than measuring spectral shifts. We theoretically discuss how a perturbation affects fundamental nanophotonic properties of a Fano resonant metasystem, and how these are transduced to an observable far field response. We perform a rigorous experimental study in which we characterize deeply subwavelength perturbations in a Fano resonant dielectric metasurface using a conventional spectral approach, and a Fourier scatterometry based approach, and show that perturbations can lead to marked directional scattering in Fourier space. We finally quantitatively compare these two sensing methods in terms of their inherent Fisher information content, and show that an information advantage is obtained when the signal is resolved in Fourier space.

Introduction
Resonance based sensors are at the heart of many modern day scientific and commercial instruments to accurately detect minute forces or mass changes, characterize material properties and acquire high quality medical images. In these technologies, the strong responsivity of the system at resonance is exploited to retrieve precise estimations of the relevant physical property to be detected. In the field of nanophotonics, resonance based sensors are routinely exploited to detect perturbations in the form of biomolecules, gases, displacements and accelerations, where the resonant response originates from localized surface plasmon modes in noble metal nanoparticles or an interplay between optical and mechanical vibrations, to name a few.

In all these nanophotonics sensing protocols, a resonant spectral feature is usually identified in experimental observables like absorption, scattering and extinction cross sections in single nanoparticles, or reflection and transmission spectra in extended thin films or arrays of nanoparticles. Refractive index changes and geometry changes form a perturbation that imprints a detectable shift onto the resonance frequency. To achieve optimal sensing performance, which is usually captured in the sensing figure of merit (FOM), the shift of the resonance frequency per unit perturbation need to be large compared to the linewidth of the resonance, and systems which exhibit ultra high quality factor (Q-factor) resonant features are therefore highly sought after in the sensing community. The Fano resonance, which is a distinct asymmetric spectral line shape arising from interference effects between a broad superradiant and sharp subradiant state, has shown great potential in sensing applications due to their inherent high Q-factors. The recent advent of dielectric metasurfaces, which are two-dimensional (2D) sheets decorated by designer low-loss dielectric particles, has provided a powerful toolbox for engineering these Fano resonances through the physics of bound states in the continuum (BIC). A BIC is a phenomenon in which a state remains perfectly localized and therefore does not radiate, despite existing within the continuous energy spectrum of radiating states. True BICs therefore have a vanishing resonance linewidth and infinite quality factor, and can only exist within infinite and ideal lossless systems. In practical scenarios a BIC mode can be translated into a so-called quasi-BIC with finite and tunable quality factor by introducing a small asymmetry in the structural parameters of the metasystem. Sensors exploiting the excitation of these symmetry protected quasi-BIC modes have shown record high sensing FOM’s.

In parallel to these resonance based efforts, nanophotonic platforms have been designed that efficiently transduce information about a given perturbation into the far-field by monitoring the rich contents of angle resolved scattering patterns. Indeed, deeply subwavelength or “superresolution” estimation precision of a perturbation has been achieved by analyzing the Fourier space scattering of complex nanophotonic objects, and even single nanoparticles. These studies differ in experimental observable from the resonance based works in the sense that in the former, the far-field signal is resolved into its angular components (Fourier space), whereas in the latter the signal is usually acquired in an angle integrated modality on either a spectrometer or a single pixel bucket detector, thereby losing potentially useful information about the perturbation in Fourier space. In fact, Fano resonances have been shown to exhibit directionality properties, which could provide an information gain in a sensing based experiment. However, to the best of our knowledge, there are no reported sensing experiments in which the signal of a Fano resonant scattering sensor is analyzed in Fourier space, and in which the information advantage is quantified which such an angle resolved measurement modality may have relative to conventional spectroscopy.

In this work, we study how perturbations affect the far field scattering properties of a Fano resonant metasystem, and show that an information gain can be achieved when the resonant scattering is analyzed in Fourier space, as conceptually sketched in Fig. 1. We start with a theoretical discussion of all relevant nanophotonic response function changes a perturbation can induce onto a Fano resonant system, and how these are transduced into the far-field. We then perform a rigorous experimental study, in which we characterize the influence of a perturbation in dielectric Fano resonant metasystems in both an angle integrated spectral modality and a Fourier space measurement, and show that deeply subwavelength information about the perturbation is resonantly transduced into marked directional scattering in Fourier space. We finally compare both measurement modalities in terms of the Fisher information they contain about the perturbation, and show quantitatively that a Fourier space measurement has a higher Fisher information content than an angle integrated measurement.

Results
Theoretical motives
To study how a Fano resonant metasystem transduces information into far-field Fourier components, our first objective is to identify a Fano resonant metasurface that allows light scattering into higher order diffractive channels. This is not a trivial prerequisite, since the majority of Fano resonant metasurfaces studied in the literature are constructed by semi-infinite periodic lattices with subwavelength periodicity, thus only allowing coupling to the zero order transmission and reflection channels. Possible disturbances from the perfect subwavelength periodicity, that are necessary to generate diffraction channels, such as edge effects and fabrication imperfections generically lead to a reduction of the resonance quality factor. Besides semi-infinite non-diffractive lattices, other well know types of metasystems supporting Fano resonances are for instance plasmonic and dielectric oligomers of nanoparticles, and even single dielectric nanoparticles supporting so-called supercavity modes. In contrast to the sub-diffractive infinitely extended metal lattices, these finite sized systems are allowed to scatter light into a continuous angular spectrum, fully determined by the structure factor of the relevant mode profile at play. Drawbacks of many finite-sized Fano resonant structures are that often the quality factor is not easily tunable, or that complex structured illumination conditions are required to selectively excite the resonances. We identify symmetry protected quasi-BIC Fano resonances in finite sized annular metarings as particularly promising for our study. Such metarings were demonstrated in a recent study by Kühner et al. and have tuneable resonances that are accessible by simple plane wave illumination. While the interparticle spacing of the high refractive index meta-atoms in such rings is still subdiffractive to ensure strong collective mode coupling, the shape in which the meta-atoms are patterned allows light scattering into higher order Fourier channels. In the language of X-ray scattering or treatments of antenna directivity, the radiation pattern can be understood as the product of a ‘structure factor’, or ‘array factor’, that derives from the placement of units on a circle, and a ‘form factor’ or ‘element factor’ that accounts for the dipolar radiation pattern of each individual meta-atom. The structure factor of a circle, chosen as a few wavelengths across, provides concentric rings of scattered intensity in Fourier space. The flexible toolkit of symmetry protected quasi-BIC resonances allows relatively straightforward tuning of the Fano resonance by controlling structural symmetry parameters.

In the following, we will focus on sensing perturbations in the form of deeply subwavelength structural displacements between the individual building blocks in metarings as described above. This type of perturbation is frequently encountered in modern-day semiconductor chip manufacturing processes, where nanometer-scale misalignments between individual structures on a device layer can arise when the same wafer goes through several successive exposure/etch/exposure cycles where each exposure step only defines part of the structure. Accurately monitoring of these misalignments, which is termed “interlaced metrology”, is crucial to ensure high-quality chip performance. This is typically achieved by analyzing the diffracted signal of devoted scattering sensors that are printed during the same exposure as the actual devices in the chip. Currently, there is a large push within the semiconductor industry for small footprint scattering sensors containing an effective footprint of no more than 2×2 μm. Since the metarings contain a diameter of only 3 μm, which could be reduced even further if needed, we envision that these structures could be exploited as responsive, yet small footprint nanophotonic scattering sensors for diffraction based interlaced metrology applications. Figure 2a, b illustrate the scenario at hand on basis of electron microscopy images of resonant silicon metarings that we interrogate in the experimental part of this paper (see methods section for nanofabrication details). Figure 2a displays an unperturbed metaring: radially oriented silicon blocks are equidistantly placed on a ring, and blocks alternate in length. The asymmetry in the length of neighboring meta-atoms ΔL tunes the Fano resonance by a symmetry protected quasi-BIC mechanism. In Fig. 2b, the perturbed structure is shown. The perturbation corresponds to a collective upward shift Δy of the entire subring of short meta-atoms.

To explore the influence of perturbations in a Fano resonant sensing scenario, we tackle the problem from an eigenmode perspective. While one could derive quasinormal modes from a numerical eigensolver, we semi-analytically model the annular metasystem using coupled dipole theory, in which the individual meta-atoms are approximated as polarizable point dipoles, while all electrodynamic multiple scattering interactions are taken into account. Using the specific modeling parameters as discussed in the methods section, we construct the dipole-dipole interaction matrix of the system, from which we extract all relevant modes of the metarings by performing an eigenvalue decomposition. We obtain (essentially frequency-independent) eigenvectors p_m, which correspond to the natural coupled dipole oscillations in the metaring, along with corresponding eigenvalues λ_m(ω), describing the eigen-polarizabilities of the respective modes. The response of the metaring to incident light can be decomposed in the eigenmodes, and depends both on the magnitude of the eigenpolarizabilities and the overlap of the incident field with the eigenvector. We assume an externally applied electric field E_exc that correspond to a y-polarized plane wave normally incident onto the ring. This will lead to a driven dipole configuration p_driven = Σ α_m(ω)p_m, where α_m(ω) corresponds to the expansion coefficients of mode p_m. These coefficients depend on the overlap of the eigenmode with the external excitation field 〈p_m|E_exc〉 and the eigenvalue λ_m(ω) as α_m(ω) = λ_m(ω)〈p_m|E_exc〉.

The two most dominant eigenvector profiles in the expansion are depicted in Fig. 2c, d. From the plot in Fig. 2c, we see a collective dipolar eigenmode p_bright which is superradiant, whereas in the profile shown in Fig. 2d a subradiant eigenmode p_dark can be identified. These are exactly the bright and dark modes that are responsible for the Fano interference in our metasystem, and lay the foundation for the relevant physics in the sensing experiment.

We are now in a position to discuss the possible consequences of a perturbation to the modes of a Fano system. First of all, the eigenmode profiles themselves will be perturbed. This is highlighted for both the bright and dark modes in Figs. 2c and d, where the black and red arrows correspond to the unperturbed and perturbed eigenmodes respectively. We note that particularly the dark mode profile shows considerable sensitivity to perturbations. The perturbed eigenmode profiles will in turn lead to an altered overlap of the modes with the external driving field, along with perturbed eigenvalues. These two effects combined consequently lead to perturbed excitation coefficients α_m(ω), as shown in Fig. 2e, f for both the bright and dark mode, respectively. Here, the absolute value of the complex excitation coefficients is plotted as function of wavelength. Note here that α_bright(ω) has a relatively weak wavelength dependence, which is expected due to the broadband nature of a mode with large scattering losses. On the other hand α_dark(ω) shows a strong dispersion and a sharp peak around 780 nm where the Fano interference takes place, and again the largest sensitivity to perturbations. Note that the eigenmode perturbation and corresponding change in overlap with an external driving field is exactly the underlying mechanism that also allows coupling to a symmetry protected quasi BIC resonance, with the perturbation being a structural symmetry breaker (ΔL here). The prediction of Fig. 2c–f is that this same mechanism brings sensitivity to Δy.

We have so far discussed the influence of perturbations on eigenmodes only, but not yet on experimental observables with which an experimentalist would actually probe the perturbations. The conventional strategy in sensing is to analyze the spectral response of the Fano system, which would be a single scattering channel reflection or transmission spectrum in the case of nondiffractive metasurfaces, or an angle integrated scattering spectrum in finite sized diffractive structures. In Fig. 2g we show the angle integrated scattering cross section σ_scat of the unperturbed and perturbed driven dipole configuration p_driven for a y-polarized plane wave driving field. A Fano lineshape is evident, resulting from the modal interference of the eigenmodes p_bright and p_dark. A small shift and decrease in scattering amplitude is also visible for the perturbed cross section, which results from the perturbed excitation coefficient in Fig. 2f. This altered spectral lineshape can be used to for instance translate structural perturbations into measurable intensity differentials when the system is illuminated at resonance, thus providing information about the perturbation at hand. However, as the entire angular spectrum of the scattered signal is integrated, potentially rich angular dependent information is lost in the angle integration.

We now investigate how the perturbation influences directional scattering in Fourier space by resolving the scattered signal over all viewing angles k^. The acquired Fourier space radiation diagrams S_tot(k^, ω) can again be decomposed into modal contributions as S_tot(k^, ω) = α_bright(ω)S_bright(k^) + α_dark(ω)S_dark(k^), where S_bright(k^) and S_dark(k^) are the modal radiation patterns emitted by p_bright and p_dark, respectively. Intensity only measurements on a detector can be accounted for by taking the modulus squared of the complex radiation patterns, resulting in |S_tot(k^, ω)|^2 = |α_bright(ω)S_bright(k^) + α_dark(ω)S_dark(k^)|^2 = |α_bright|^2|S_bright|^2 + |α_dark|^2|S_dark|^2 + 2Re(α_bright α_dark* S_bright S_dark*).

From this expression it is evident that the total acquired radiation pattern is composed of individual contributions from the bright and dark modes, along with a modal interferometric term. Suppose there is complete transparency in extinction at a certain frequency and that there is no absorption. In that case, the optical theorem states that σ_scat = 0. Since σ_scat(ω) ∝ ∫ |S_tot(k^, ω)|^2 dk^, that would mean that the interferometric term in equation (4) fully cancels the first two terms for every k^. By conservation of photon flux it can then be concluded that the modal radiation patterns S_bright(k^) and S_dark(k^) should be very similar to each other, and that the expansion coefficients α_bright(ω) and α_dark(ω) tune the relative transparency. Perturbations imprinted onto the individual eigenmodes would in turn lead to corresponding changes in the modal radiation patterns and, through equation (4), on the total radiation pattern |S_tot(k^, ω)|^2 as well, which could lead to directional scattering in Fourier space. In Fig. 2h, i, we show Fourier space radiation patterns of the unperturbed and perturbed dipole configurations, while driving the system at resonance at a wavelength of 775 nm. In these figures, we observe concentric rings of bright intensity regions, which correspond to the spatial frequencies or structure factor of the annulus. These rings in k-space are further supplemented by dark intensity regions in the k_y/k_0 viewing direction, which are a result of a polarization selective form factor of the collective dipole along the y-direction. The perturbed radiation patterns shows broadly similar features, but marked differences can be spotted with respect to the unperturbed pattern upon closer inspection. To highlight the angular dependent differential signal in Fourier space, we subtract the unperturbed and perturbed Fourier images to arrive at a differential radiation pattern in Fig. 2j. A key advantage of the Fourier space diagram with respect to the angle integrated scattering spectrum in Fig. 2g is that in the former regions with both positive and negative intensity differentials may be resolved, which would otherwise cancel out when the signal would be integrated over all angles. This observation hints towards a potential gain in information about the perturbation when the scattered signal of a Fano resonant structure is dispersed in Fourier space, which we will set out to quantify experimentally in the next sections.

Spectral resonance characterization
To spectrally characterize the resonant properties of individual dielectric metarings experimentally, we perform dark-field scattering spectroscopy, with a schematic of the setup depicted in the Supplementary Information (SI). In brief, a single metaring (Si meta-atoms on a glass substrate, dimensions and fabrication methods listed in the methods section) is illuminated by a broadband supercontinuum laser with a low effective NA (NA = 0.18), after which the scattered signal is captured in reflection through the same high NA objective (NA = 0.95). Dark-field acquisition is ensured by filtering the low NA back reflection of the illumination out of the total signal by placing a beam stop in the conjugate Fourier plane of the imaging system. Examples of raw spectra are shown in Fig. 3a, where we show a series of dark field scattering spectra of individual rings with asymmetry parameters of ΔL = 0 nm in red and ΔL = 100 nm in blue, along with their respective averaged spectra in black. Different spectra with the same color indicate spectra scattered by different realizations of metarings with nominally identical fabrication parameters. A good overlap between these individual spectra is seen, indicating that metarings with nominally identical fabrication parameters have reproducible scattering properties. Spectra of the rings with ΔL = 100 nm contain a clear asymmetric feature around 720 nm with respect to the rings with ΔL = 0 nm, hinting towards a Fano lineshape in the scattered signal. To isolate the Fano resonant features within the scattering spectra, we normalize all raw spectra by the average spectrum of the metarings with asymmetry parameter ΔL = 0 nm, as these metarings are not expected to exhibit Fano resonant features due to symmetry protection of the dark mode. In Fig. 3b we show an exemplary normalized spectrum of an individual metaring with asymmetry ΔL = 100 nm. Indeed, an asymmetric Fano like lineshape is seen in the normalized scattering spectrum, and fitting the data in close vicinity around the resonance with a Fano function results in an excellent match. To validate that the Fano feature results from the quasi-BIC in our system, we analyze the normalized scattering spectra of the metarings as a function of the asymmetry parameter ΔL in Fig. 3c. We clearly observe a broadening of the resonance linewidth for increasing ΔL, along with a blueshift of the resonance wavelength, both characteristic of a quasi BIC due to symmetry breaking. These observations are substantiated by fit results in Fig. 3e, where we show resonance wavelengths and linewidths extracted by fitting Fano lineshapes to normalized scattering spectra of individual metarings. The datapoints and errorbars in this panel correspond to respectively the mean and standard deviation of the fit parameters retrieved from the spectra of multiple realizations of individual metarings.

We next analyze the influence of the perturbation on the scattering behavior, and show the normalized scattering spectra of the metarings with fixed asymmetry ΔL = 100 nm for increasing perturbation strength Δy in Fig. 3d. Performing the same fitting analysis on this set of normalized spectra results in extracted resonance wavelengths and linewidths versus Δy in Fig. 3f. These results show a broadening of the resonance for increasing Δy, which can be explained by the fact that the perturbation disturbs the otherwise perfect annular periodicity of the metaring, thereby degrading the resonance quality. This trend is in line with the results of studies on fabrication tolerances on resonant metasurfaces. Apart from the spectral broadening, a minor shift in resonance wavelength is seen as a function of Δy. These results show that the scattering spectra show deeply subwavelength sensitivity to structural perturbation within the metasurface, albeit that the spectral changes are small.

Fourier scatterometry
In this section, we investigate the sensitivity to perturbations of the scattered signal in Fourier space. For this, we utilize Fourier microscopy, (see SI for a sketch of the setup), where we disperse the scattered signal into angular contributions by imaging the back-focal-plane of a microscope objective onto a C-MOS camera. As we did for the spectroscopic measurements described in the previous section, we illuminate individual metarings with a low effective NA (NA = 0.18) and polarization parallel to the y-direction (parallel to the displacement), but now we band pass filter the excitation wavelength with a bandwidth of 10 nm. We vary the center wavelengths of these filters to spectrally overlap with the Fano resonances as measured in Fig. 2, and capture the corresponding Fourier images scattered by the individual metarings. As such, we acquire directional scattering patterns of individual rings both on and off-resonance, and investigate the sensitivity to perturbations in both spectral regimes.

In Fig. 4a, b, we show exemplary raw Fourier images of perturbed (Δy = 50 nm) and unperturbed (Δy = 0 nm) metarings, respectively, with fixed asymmetry parameter (ΔL = 80 nm), acquired at an on-resonance excitation wavelength of 730 nm. In all acquired images, we clip out in software a circle in the center of Fourier space, as it contains the direct reflection of the illumination by the substrate. The concentric rings originating from the structure factor of the annulus are evident, multiplied by the form factor of the collective dipole emission as we also saw in the calculations of Fig. 2h. We furthermore observe a clear asymmetry in scattered signal in the k_y/k_0 viewing direction for the perturbed ring as compared to the scattered signal of the unperturbed ring, indicating that structural perturbations are transduced into observable Fourier space asymmetries. Note that the symmetry axis in both the real and Fourier space is preserved, since displacements in the y-direction in real space lead to specific asymmetries along the k_y/k_0 viewing direction in Fourier space.

To better visualize this asymmetry in Fourier space, we subtract the angle resolved scattering patterns of an individual unperturbed ring from a perturbed ring to arrive at a differential Fourier space scattering image. To study the influence of the perturbation strength and the excitation wavelength on the asymmetry in Fourier space, we acquire such differential Fourier images for multiple perturbation strengths and for both on and off-resonance excitation. The results are shown in Fig. 4c–n. From left to right, the perturbation strength is gradually increased from Δy = 0 nm to Δy = 50 nm in steps of 10 nm, while in the top and bottom rows the excitation wavelength is set to respectively on (730 nm) and off-resonance (780 nm) excitation. We note that the differential images shown for the two wavelength are scattered by exactly the same pair of rings.

Differential images of unperturbed rings with nominally identical fabrication parameters scatter a relatively low differential contrast-to-noise ratio in Fourier space as can be seen in Fig. 4c, i. This observation underscores the highly reproducible scattering characteristics of metarings with nominally identical fabrication parameters, as was also evident from the scattering spectra from Fig. 2c. When the perturbation strength is gradually increased from Δy = 0 nm to Δy = 50 nm, the differential contrast in Fourier space at resonance in Fig. 4c–h scales accordingly, while the differential contrast off-resonance in Fig. 4i–n remains relatively unaltered. By comparing the calculated differential Fourier space image of Fig. 2j with the images of Fig. 4, we see that a simple dipole model can reproduce the salient features of the experiment. From these measurements, we conclude that deeply subwavelength structural misalignments are resonantly transduced by the Fano resonance into directional scattering asymmetries in Fourier space. It is important to note that the directional scattering effects are sensitive to the polarization state of the incident excitation, since the specific perturbation Δy breaks the degeneracy of the bright and dark modes along the x and y direction. An interesting question for future work is what polarization states would be optimal for retrieving information about perturbations, which could be tackled with the recently introduced framework of maximum information states.

Information advantage
In the previous two sections, we have presented two different experimental strategies for the detection of a perturbation in a Fano resonant metasystem. The intriguing question which of these two measurements most efficiently detects the perturbation then naturally arises. In this section, we will quantitatively answer this question by determining the information content that each measurement contains about the relevant perturbation. For this, we will use a concept from information theory termed the Fisher information, which defines a fundamental bound to the precision with which perturbations Θ can be estimated from measured data X, which are unavoidably corrupted by noise. Assuming that one has access to a theoretical description of the noise in terms of a probability density function p(X; Θ), which describes the probability of measuring a dataset X given a certain perturbation strength Θ, which in our specific sensing experiment corresponds to structural misalignments Δy, the Fisher information is defined as F = E[ (∂ ln p(X; Θ) / ∂ Θ)^2 ]. Assuming further that one can construct a so-called unbiased estimator, which is an operation that translates a dataset X into an estimation of the perturbation Θ^ that on the average should equate the true perturbation Θ, the fundamental precision bound of the estimation is defined by the Cramer-Rao bound σ^2_Θ ≥ 1 / F. From this expression, we see that the higher the Fisher information content, the lower the minimum standard deviation and thus the higher the precision with which the perturbation can be estimated from measured data. Maximization of the Fisher information has already been shown to positively impact the measurement precision in experimental contexts such as 3D localization and imaging, sensing perturbations in disordered environments and tracking single molecules.

To arrive at a closed form expression for the Fisher information in the context of our sensing experiment, we will now discuss the several noise factors that can corrupt our measured data and thus reduce the available Fisher information. The first noise source originates from the unavoidable fabrication errors that invoke additional shape perturbations besides the controlled perturbation that we want to detect. Although our structures show decent reproducibility in scattering properties, two individual metasystems with nominally identical fabrication parameters will never scatter in a perfectly identical way due to fabrication imperfections, as could already be observed from Figs. 4c and i, where small residues in scattering contrasts exist above the inherent noise floor of the measurement. The measurement noise floor in turn consists of photon shot noise and detector read noise, which we describe in the (SI). For simplicity, we have assumed that the only noise sources originate from photon shot noise and detector related noise sources in the derivation of the Fisher information, and describe how we correct for the fabrication noise later on in this section. For analytical tractability, we describe the noise in a single pixel on the detector as Gaussian p(X; Θ^) = 1 / sqrt(2πσ^2_tot(Θ^)) * exp(-(X - I_0(Θ^))^2 / 2σ^2_tot(Θ^)), which represents the probability of measuring X photons given a mean number of measured photons I_0(Θ^) and a standard deviation given by the sum of shot-noise and detector readout noise σ_read contributions σ^2_tot(Θ^) = I_0(Θ^) + σ^2_read. The choice of a Gaussian distribution instead of Poissonian is a convenient way to account for the mixed contributions from shot noise and detector readout noise, and for the count rates at hand the Gaussian distribution approaches the Poissonian one. Using this Gaussian noise model we can therefore make reasonable estimations of the available Fisher information in the data by using a closed form analytical expression, which is obtained by substituting equations (7) and (8) into equation (5): F = (1 / 2(I_0 + σ^2_read)) * (∂I_0 / ∂Θ^)^2. As the Fisher information is additive, we can approximate the total Fisher information as the sum of the Fisher information of the unperturbed ring I(0) and the difference I(Δy) - I(0). Using Eq. (9) with the sensitivity of an unperturbed ring I(0), we can estimate the Fisher information content of a sensing experiment from an unperturbed and perturbed measurement, for which we already acquired all the relevant data in the previous sections. Besides analytics, we note that model-free approaches exist to determine the available Fisher information from data where the noise model is inherently unknown.

We now quantitatively compare the Fisher information content between an angle integrated scattering measurement and Fourier space measurement. To keep all experimental parameters such as integration time, incident laser power and detector efficiency constant in the comparison, we estimate the Fisher information in the two different measurement strategies from our Fourier space data alone, constructing spectral data by software integration over angle. Figure 5 summarizes the workflow to arrive at quantitative values for the Fisher information content. We start with raw Fourier images scattered by an unperturbed (Δy = 0 nm) and perturbed ring (Δy = 50 nm) with fixed asymmetry (ΔL = 80 nm) and excited by a wavelength of 730 nm in Fig. 5a, b, respectively. Following the red connector lines, from these two images we calculate an angle resolved Fisher information image by applying Eq. (9) onto the individual non-zero count pixels of the images, with the result depicted in Fig. 5c. From this image we can see that Fisher information is directionally scattered in Fourier space, with the majority of the information scattering into the negative k_y/k_0 viewing direction. This can be explained by again inspecting Fig. 4h, which is the differential intensity image that corresponds to the Fisher information radiation pattern as shown in 5c, from which we can see that indeed the majority of the differential counts are scattered into the negative k_y/k_0 direction. A recent work has shown that the shape of these information patterns can be understood from a near-field description using so-called information sources, which radiate information instead of energy, and poses the interesting design question how antenna theory could be exploited to engineer these information patterns at will. To connect a single information value to this radiation pattern, we make use of the additivity of the Fisher information and sum all the values in the Fisher information radiation pattern. The retrieved value for the Fisher information will be generated by the intentional perturbation Δy, but also by unintentional nanoscale perturbations due to imperfections in nanofabrication ξ. To see how ξ can generate information in Fourier space, we again turn to Fig. 4c, i, where it can be seen that 2 meta rings with the same fabrication parameters still generate a differential count contrast due to the fabrication noise ξ, which will be added to the total Fisher information. To isolate the information that is generated purely from the intentional perturbation Δy that we want to probe, we apply the same workflow of Fig. 5a–c on metaring pairs with similar fabrication parameters (ring 1: Δy = 0 nm, ring 2: Δy = 0 nm). The corresponding Fisher information F_ξ defines the information generated by fabrication noise ξ. The Fisher information generated purely by Δy is then isolated by subtracting the information generated purely from fabrication noise from the information generated by the combination of ξ and Δy as F_Δy = F_{Δy+ξ} - F_ξ, which finally represent the red diamond data point in Fig. 5.

To estimate the information content of a simple spectral measurement (blue network) we first sum the individual Fourier space images in Figs. 5a, b. This operation simulates the measured count value a single pixel on a spectrometer would detect for the relevant excitation wavelength. We finally apply equation (9) onto this pair of values to arrive at the final information estimate represented by the blue circle. In Fig. 5d, we showcase the full comparison in information content between an angle integrated and angle resolved measurement. For this, we determined Fisher information values for a fixed perturbation strength of Δy = 50 nm, while varying the excitation wavelength and the asymmetry ΔL. We have used Fourier scatterometry data of four independent metaring pairs to determine average and standard deviations of the information content of both measurement modalities, indicated by data points and error bars respectively. In the same figure, the center and width of the gray vertical bars indicate the resonance wavelengths and linewidths of unperturbed metarings respectively. It can be seen that the information content of both the angle resolved and angle integrated measurements peak at the Fano resonance, where the peak follows the same blueshifted trend versus increasing asymmetry ΔL as we have observed in the spectroscopic measurements of Fig. 2, corroborating the fact that sensing at resonance more efficiently detects a perturbation than sensing off-resonance from an information perspective. We further see that the information content of a Fourier space measurement is always higher than that of an angle integrated measurement, both on and off-resonance, where the information gain can be as high as sevenfold. From the scalebar we can finally observe that the maximum information content increases with increasing asymmetry ΔL, ultimately saturating beyond ΔL = 80 nm. To clarify this behavior, we recall that the Fisher information performance metric is not only influenced by conventional factors such as resonance wavelength shifts and Q-factor, but also by the resonant amplitude that defines the modulation depth of the Fano lineshape. Indeed, a recent study has demonstrated that optimal sensing performance does not necessarily correspond to the scenario of highest Q-factor, but rather to balancing the Q-factor and resonance amplitude. Since the resonance amplitude increases versus increasing ΔL, as evident from Figure 3c, a similar argument can be applied to the metaring scattering sensor. Furthermore, the novelty of Fourier space analysis of resonant scattering requires a new performance metric that quantifies the degree of directional scattering induced by a perturbation, which would be an interesting topic for future investigations. All these factors combined influence the ultimate Fisher information plotted in Fig. 5d and are responsible for the saturation in Fisher information beyond ΔL = 80 nm.

We finally note that the information advantage of a Fourier space measurement with respect to an angle integrated measurement is generally valid for other Fano-resonant structures and perturbation types beyond those investigated in this paper. The crucial necessity is that the photonic structure should be able to scatter light into higher order diffraction angles. The Fano-resonance in that approach can be understood as a coupling effect between bright and dark modes: the optical theorem fundamentally implies that any near-field effect that gives rise to a Fano line also expresses as a far-field modal interferogram. This interferometric advantage can generate specific sensitivities as a function of viewing angle due to perturbations. To support this claim, we have performed coupled dipole calculations on different Fano-resonant structures and perturbation types in the SI, all of which demonstrate the information advantage of Fourier space readout. Since Fisher information (in shot-noise limited experiments) scales with the number of detected photons, a similar information content can be retrieved from a Fourier space measurement compared to conventional spectroscopy, yet for a lower photon budget. This means that the integration time, or incident power could be reduced in a Fourier space readout scheme, while not losing Fisher information with respect to conventional spectroscopy.

Methods
Coupled dipole calculations
The metarings discussed in the main text are theoretically modeled by coupled point dipoles. For an isolated dipole, we use the dyadic Green’s function of a dipole p positioned at r_0 in a homogeneous medium to describe the scattered fields at a probing position r as:
G_0(r, r_0) = (I + (1/k^2) ∇∇) * (exp(ik|r - r_0|) / 4π|r - r_0|), such that E(r, r_0) = ω^2 μ_0 G_0(r, r_0) * p.

Here, I is the unit dyad, ω is the driving frequency, k = nω/c is the wavenumber in the medium, μ the permeability of the medium and μ_0 the free space permeability. For an ensemble of N coupled dipoles {p_1…p_N} driven by an incident electric field E_exc, the solution of the n’th dipole configuration can be described by a self consistent set of equations as:
p_n = α_n * (E_exc(r_n) + Σ_{m≠n} G_0(r_n, r_m) * p_m).

This system of equations can be recast into a matrix form as
[p_1 ... p_N]^T = M^{-1} * [E_exc(r_1) ... E_exc(r_N)]^T

and
M_{i,j} = δ_{i,j} α_i^{-1} - (1 - δ_{i,j}) G_0(r_i, r_j).

Here, the (3N×3N) matrix M is the coupling matrix of the system, and describes the interactions between all the dipoles in the system. The scattered fields of the ensemble of dipoles, and the optical quantities that result from this can thus be calculated by a matrix inversion of the coupling matrix M. We analytically model the static polarizabilities of the dipolar particles in the metarings as prolate spheroids, with a long axis L_0 of 300 nm and short axis W of 100 nm. These particles are positioned in an annulus of radius 1.5 μm, and every other particle in the annulus is reduced in length by L_0 - ΔL, where ΔL = 100 nm. The dielectric permittivity of the particles ϵ is set to 12, approximating that of silicon, while the surrounding medium is set to vaccuum ϵ_m = 1. Radiation damping is included to the static polarizabilities α_0 as α^{-1} = α_0^{-1} - i * k^3 / 6πϵ_0. Finally, we compute the eigenvectors and corresponding eigenvalues of the coupling matrix M to retrieve the eigenmodes of the coupled system.

Sample design and nanofabrication
The dielectric metarings in the main text were fabricated using the following procedure. A glass coverslip (Menzel-Gläser) with a thickness of 170 μm was sonicated for 10 minutes in H_2O and cleaned in a solution of base piranha (NH_4OH/H_2O_2/H_2O = 1/1/5) at 75°C for 15 min. After thoroughly rinsing excess piranha with H_2O, the sample was blown dry with N_2. A 120 nm layer of amorphous silicon (a-Si) was evaporated onto the cleaned coverslips using electron beam evaporation (Polyteknik Flextura M508 E) at a deposition rate of 0.1 nm/s. After an oxygen plasma descum process of 2 min, a 60 nm layer of hydrogen silsesquioxane (HSQ) was spin coated (CEE Apogee 200 Spin Coater) for 60 s with a speed of 8000 rpm and acceleration of 1000 rpm/s and baked for 2 min at 180°C. A conductive aluminum layer of 10 nm was thermally evaporated (Polyteknik Flextura M508 E) at a rate of 0.05 nm/s. For alignment purposes of the electron beam (Raith), 50 nm gold nanoparticles were drop casted in the corners of the substrate, after which the sample was exposed at 50 kV acceleration voltage and an area dose of 1500 μC/cm^2. After exposure, the substrate was chemically developed in a solution of tetramethylammonium hydroxide (TMAH) for 60 s at 60°C and rinsed with H_2O for 15 s. Finally, the substrate was etched (Oxford Plasmalab 80+) using a mixture of 10 sccm sulfur hexafluoride (SF_6), 15 sscm trifluoromethane (CHF_3) and 3 sccm O_2. The length, width and height of the (ΔL = 0) rods in the ring corresponds to 335 nm, 115 nm and 120 nm, respectively.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper is based on several known concepts embedded into a new envelope shaped around “information advance in sensing”. In reality, the authors employ the so-called radial quasi-BIC modes (introduced and demonstrated earlier in Ref. 33 but termed here “Fano resonances”) for sensing (also demonstrated earlier, including the similar systems) and discuss how a perturbation affects fundamental properties of resonant systems. More specifically, they show that perturbations can lead to marked directional scattering in the Fourier space.\nThe real novelty of this paper is a comparison of two sensing methods in terms of their inherent Fisher information content, but this message and related results are not strong enough to justify publication in a highly ranked Nature Communications.", "reviewer_2": "This manuscript presents significant research elucidating, both theoretically and experimentally, the superior information acquisition capability of Fourier space analysis over conventional spectral analysis in sensing with Fano resonant metasurfaces. The introduction of Fisher information to quantitatively demonstrate this \"information advantage\" shows potential for proposing a new analytical paradigm in the sensing field. The study design is systematic, and the experimental results robustly support the claims. However, further clarification and discussion on a few points would enhance the paper's completeness and impact.\nThis work is judged to have sufficient potential for publication in Nature Communications after addressing the following concerns:\n1. The method described for correcting fabrication noise needs a clearer explanation of the statistical basis and specific calculation steps for subtracting the \"corresponding result.\" This elaboration on fabrication noise correction in Fisher Information Calculation is critical for the reliability of the results and requires detailed description.\n2. To bolster the current \"speculate\" statement, either add more convincing discussion by analyzing parameters like Q-factor, scattering intensity, and resonance shift sensitivity with varying ΔL, or clearly state this as a subject for future investigation. Strengthening the justification for maximum information gain at ΔL=80 nm would improve the manuscript.\n3. The direct interpretation and discussion of how theoretical eigenmode changes in Figure 2 manifest as experimental Fourier space asymmetries in Figure 4 should be strengthened. Specifically, an explicit explanation linking the perturbation of the pdark mode to the observed directional scattering is needed to enhance the connection between theory and experiment.\n4. The apparent author's note \"Radiation damping?\" in the Methods Section should be removed or revised into a complete statement.\n5. A brief consideration of whether the proposed methodology is a general principle applicable to other types of Fano resonant systems or perturbations beyond the specific system studied would be welcome when discussing the general applicability of Fourier space analysis.\n6. Consider adding brief explanations for terms like \"interlaced metrology\" that might be unfamiliar to readers outside that specific field to add a terminology explanation.\n7. The logical connection between the cancellation by the interference term and the similarity of modal radiation patterns should be clarified to reinforce the logic in Equation (4) discussion.\n8. Enhance the readability of the optical component list by adjusting text size or explicit mention in the caption to improve the readability of Figure S1 in Supplementary Information.\n9. A sentence or two commenting on the validity of the Gaussian noise assumption for Fisher information calculation would be appropriate when briefly discussing the Gaussian noise assumption.", "reviewer_3": "This manuscript presents a study demonstrating that angle-resolved measurements of light scattered from Fano-resonant dielectric metasurfaces provide significantly more information than conventional spectroscopic (angle-integrated) detection strategies. The authors quantify this improvement using Fisher information theory and validate the concept theoretically and experimentally.\nI find this work to be well executed in both its theoretical and experimental parts, and exploiting the angular (Fourier-space) information content of the far field appears to be a novel concept in resonant nanophotonic sensing, going beyond typical sensing schemes involving only the spectral resonance shift. The finding of a sevenfold increase in information demonstrates the advantage of this approach (under idealized conditions).\nIn the following section, I list a few points for the authors to consider:\n+ What are the prospects of generalising this approach to other common Fano-resonant structures?\n+ Both the perturbation and the excitation polarisation are aligned along the y-axis. How sensitive are the results with respect to this alignment? Would using different polarizations affect the directional sensitivity?\n+ Can the authors provide a prospect on the practical applications of their work? Could it be useful in semiconductor processing or biosensing with fewer photons, for example?\n+ Although this probably goes beyond the scope of this work, it could also be interesting to see if an artificial neural network can reach the detection limit resulting from the Fisher information analysis in this work (along the lines of Nature Photonics 19, 593–600 (2025)).\nIn summary, I think this is an interesting piece of work that should be published in Nature Communications once the above suggestions have been taken into account."}, "author_response": "We thank reviewer 1 for their time to read the manuscript and provide feedback. We would like to stress that we do not claim that Fano resonances per se, or radial quasi BIC modes, are novel for sensing. Instead the claim is that Fourier analysis is fundamentally quantitatively superior to spectral sensing. Fourier space analysis of the scattering of a Fano resonant structure has, to the best of our knowledge, hitherto never been considered in the sensing community, where perturbations have so far been determined by analyzing distortions in spectral lineshapes only. The superior information content of Fourier space analysis compared to conventional analysis of spectral lineshapes constitutes a general result, and is thereby valid for other Fano resonant structures and perturbation types beyond the specific meta ring geometry discussed in the main text. This is supported by revisions and additional calculations.\nModifications\nTo improve the message regarding the generality of our approach, we have added the following paragraph on page 24, starting from line 443:\n“We finally note that the information advantage of a Fourier space measurement with respect to an angle integrated measurement is generally valid for other Fano-resonant structures and perturbation types beyond those investigated in this paper. The crucial necessity is that the photonic structure should be able to scatter light into higher-order diffraction channels. The generality of the approach can be understood by viewing the Fano-resonance not only as a near-field coupling effect between bright and dark modes: the optical theorem fundamentally implies that any near-field effect that gives rise to a Fano line also expresses as a far-field modal interferogram. This interferometric advantage can generate specific sensitivities as a function of viewing angle due to perturbations. To support this claim, we have performed coupled dipole calculations on different Fano-resonant structures and perturbation types in the SI, all of which demonstrate the information advantage of Fourier space readout.”\nWe have also extended our study towards other types of Fano-resonant structures and perturbation types that are frequently encountered in the sensing community. To this end, we have added a new section “Different Fano structures and perturbation types” to the supplementary information in which we theoretically investigate the information advantage of Fourier space analysis over angle integrated spectral analysis for different scenarios of Fano resonant structures and perturbation types.\nWe appreciate the positive remarks of reviewer 2 on our work and would like to thank them for their detailed review and comments. In the following, we will address all comments of the reviewer in detail, together with the associated modifications made to the manuscript.\n[... (Author responses to Reviewer 2 and 3 omitted for brevity in this example, but would be included in full)]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors made a very serious revision and addressed clearly my major concerns. I did like that they extended the study by adding other types of Fano-resonant structures and perturbations that are frequently employed by the sensing community. Thes new section “Different Fano structures and perturbation types” added to the supplementary part is very useful, and it cover different scenarios of Fano resonant structures and perturbation types.\nI am happy to suggest publication in the current form", "reviewer_2": "As a reviewer specializing in devices utilizing Fano resonances, I would like to clarify that my expertise lies primarily in the development and characterization of such photonic structures, rather than in the field of sensing methodologies or information theory. While I appreciate the authors’ efforts to expand the sensing paradigm through the analysis of Fisher information in the Fourier space, I am not in a position to critically evaluate the novelty or impact of this approach from a sensing expert’s perspective.", "reviewer_3": "This manuscript NCOMMS-25-26949A has been seen by altogether three reviewers, who provided comments and suggestions.\nFrom my point of view, the authors have carried out a thorough revision of their manuscript and have also provided a very detailed rebuttal letter in which all concerns by the referees have been addressed. Apart from a number of technical clarifications, the authors have also addressed the concern of Referee 1 regarding novelty: I strongly agree here with the authors, that the information stored in a Fourier space analysis clearly goes beyond an analysis of the spectral lineshape only.\nIn light of the above, I strongly recommend this manuscript for publication in Nature Communications.\nMinor points to clarify at the proof stage:\n+ In line 434 there is a typo “…does not necessarily corresponds…”\n+ Typically the definition of the S-matrix does not only involve the scattered part, but also the incident part (in contrast to the transition matrix T). In this sense, the scattering cross section depends only on the scattered field, not on the incident one (see expressions below Eq. (4)). I assume this is just an issue of convention though, where the authors denote S_tot to be the matrix that describes “total radiated (scattered) field from all modes” rather than “total field = incident + scattered field”, correct? But maybe I also just misunderstood something here."}, "author_response": "We would like to thank all reviewers for their time and efforts to review our work. We feel that the message of our paper has improved substantially thanks to the critical comments of the reviewers, and are glad that all 3 reviewers were unanimously impressed with the quality of the manuscript and corresponding revisions."}]

Decision letter