Phase noise matching in resonant metasurfaces for intrinsic sensing stability

Interferometry offers a precise means of interrogating resonances in dielectric and plasmonic metasurfaces, surpassing spectrometer-imposed resolution limits. However, interferometry implementations often face complexity or instability issues due to heightened sensitivity. Here, we address the necessity for noise compensation and tolerance by harnessing the inherent capabilities of photonic resonances. Our proposed solution, termed “resonant phase noise matching,” employs optical referencing to align the phases of equally sensitive, orthogonal components of the same mode. This effectively mitigates drift and noise, facilitating the detection of subtle phase changes induced by a target analyte through spatially selective surface functionalization. Validation of this strategy using Fano resonances in a 2D photonic crystal slab showcases noteworthy phase stability ( σ<10−4π ). With demonstrated label-free detection of low-molecular-weight proteins at clinically relevant concentrations, resonant phase noise matching presents itself as a potentially valuable strategy for advancing scalable, high-performance sensing technology beyond traditional laboratory settings.


Supplementary Text Figs. S1 to S16
Other Supplementary Materials for this manuscript include the following: Movies S1

SI 1. Phase-sensitivity estimation based on spectral resonance information
In the process of designing a resonant interferometry platform, it is possible to estimate the expected phase sensitivity of a resonance without having to simulate or measure the phase response to refractive index changes if the spectral sensitivity is known.The goal is to show the approximate relationship between the typically well-known spectral information and phase information to facilitate the design of phase-based sensing platforms.A simulated or experimentally obtained spectrum of the resonance peak allows to determine the FWHM.It is possible to estimate the phase sensitivity from the FWHM and spectral sensitivity of a resonance: Eq. S1 The phase dynamic range (approximately linear range) of a single resonance can be defined as the index change Δn that results in a wavelength shift of Δλ = 1/2 FWHM: Eq. S2 These equations relate to the approximately linear range of a Lorentz resonance (Fig. S1) and together also illustrate the trade-off between phase sensitivity and dynamic range.

SI 2. Phase response of Fano resonances -explanation
According to temporal coupled mode theory, the reflection channel is given by: The Fabry-Perot reflection coefficient  is given by: Where  is the phase due to propagation inside the Fabry-Perot slab, and  is the Fresnel coefficient between the slab and the external medium.
The Fabry-Perot transmission coefficient  is given by: Finally, the Lorentzian channel (), which represents the GMR, is given by: Notice that, for  ≪  0 , () ≈ - , which is a phasor with phase equal to  2, whereas for  ≫  0 , , which is a phasor with phase equal to - 2. Therefore, () sweeps a phase shift of  across the resonance.
The Fano resonance results from interference between the channel  and the channel ()( ± ).Since the only frequency dependent term is the Lorentzian term (), and we have seen that () sweeps a  phase shift, we should expect that () will also sweep a  phase shift, since all other terms are constant.That is not the case, however, because apart from the phase sweep, the Lorentzian channel also changes amplitude, which affects the phase of ().
We can identify two paradigmatic examples of phase sweep setting  = 0.2 and  =  2. The figures below show the phase sweep associated with the ± sign in the equation for ().The sign depends on the symmetry of the modes, and TE modes give positive sign, and TM modes give negative sign.For reference, the Lorentzian phase is also shown (the Lorentzian is obtained by setting  = 2, which results in  = 0).

Fig. S3. 'negative' phase response of Lorentzian
As can be seen from the Figures S2 a and b, there are two distinct phase behaviors.In the case when the "plus" sign is used, the Fano resonance makes a 2π phase sweep, whereas when the negative sign is used, the Fano resonance sweep is only modest (these behaviors are not inherent to the plus or minus signs and may be inverted depending on the  values used).We can understand these behaviors by inspecting the interference between the channels.First, focus attention on the green plots (positive) sign of Figure S4.The inset shows the corresponding phasors at different frequencies.The orange phasor is the reflection (Fabry-Perot) channel.Notice that, in this case, the Lorentzian channel (the green phasors) grows as it tends towards a p phase shift with respect to the reflection channel (the orange phasor).For example, at point 5 the phase of the Lorentzian (green) phasor is almost at a p angle with the FP (orange) phasor.Since the magnitude of the Lorentzian (green) phasor is large at this point, the phase of the Fano (blue) phasor is almost p shifted with respect to the FP.Since away from the resonance, the Lorentzian contribution is negligible, it follows that the Fano (blue) phase must eventually go back to the FP (orange) phase.Thus, it completes a 2p phase shift across the resonance.This behavior pushes the phase of the total channel (the blue phasors) towards a full 2p phase loop across the resonance.Now contrast this behavior with the negative sign condition (black curves in Fig. S5).Now there are two different features.First, the phase already starts at pi phase shift (see point 1), and then evolves TOWARDS the FP channel.As we move closer to resonance (around point 3), the contribution of the Lorentzian (black) channel increases, which pushes the Fano phase (blue) towards it.But since the Lorentzian channel now is moving towards the FP channel, the Fano phase (blue) quickly reduces, as the contribution of the Lorentzian channel diminishes.Notice that the phase at point 4 is already closer to the FP phase than the phase at point 3.That explains the "blip" in the phase of the negative condition (dashed black line).
The derivations above are meant to explain the phase response shown in Figure 3 of the main manuscript (Fig. S2) and going beyond this, highlight that different trends of the Fano phase response (i.e., Fig. S3) are also possible.

SI 6. SLED spectrum
The SLED used for all the experiments shown in this work displays the spectrum below with a maximum intensity around 845 nm.To estimate the performance of any biosensor, the bulk refractive index sensitivity is typically investigated by introducing solutions of known refractive index and measuring the response, here the phase.Although the performance a sensor designed for protein sensing depends on the surface sensitivity rather than the bulk sensitivity, the bulk LOD is a first indicator of the expected biosensing performance.We here diluted glucose in H 2 O and measured the corresponding refractive index with a handheld commercial refractometer (Reichert, Brix/RI-Chek) for calibration purposes.We took images of the corresponding interferogram over time for each concentration.

Fig. S11. Bulk sensitivity.
To characterize the experimental bulk phase sensitivity, we exposed the sensor to glucose solutions of varying concentrations, which we calibrated using a commercial refractometer, and the corresponding phase responses are plotted together for ease of visualization.This is the data leading to the plot in Figure 3 B in the main manuscript.

SI 8. Fano fit and Q-factor extraction
To extract the FWHM of the Fano peaks shown in Figure 2 in the main manuscript, we fit a Fano function: 2 + ( - 0 ) 2 +  Here, a is the amplitude, q the Fano parameter (4, 58),  the peak half-width,  0 the peak wavelength and e an offset value.The FWHM is then approximated as FWHM = 2  and Q =  0 / FWHM.

SI 9. IgG sensing based on antibody spotting
Although the protein sensing results shown in the main manuscript are to be seen as proof-of-principle sensing to test the novel concepts of degenerate metasurface interferometry, we are mentioning the possibility of antibody spotting here, because spotting will allow highly multiplexed biomarker detection for clinical studies in the future.

Fig. S12. Antibody spotting.
Phase in response to flowing IgG over a surface (starting ~ 6 min after PBS baseline) functionalized with anti-IgG antibodies.Antibodies are spotted on the sensor surface using the sciFLEXARRAYERs S3 ultra-low volume dispensing system.The concentration of the antibody solutions in PBS was 500 ug/mL and both the specific anti-IgG antibodies and non-specific isotype antibodies were spotted next to each-other for the self-referenced interferometric read-out.

SI 11. Interferogram contrast characterization
In the degenerate mode interferometry approach, the signal and reference beam have the same amplitude since they correspond to the same resonant mode, and the fringe visibility is therefore high with C ~ 80%.The resonant mode can also be interfered with the background rather than itself, which results in an unreferenced interferometry approach (mode -background, "sharp -flat").The resulting contrast is lower (~ 40%) since the background intensity (reflection from the air-glass interface of the sample substrate and Fabry-Perot background from the a-Si:H slab) and the resonantly reflected light intensity are different.Note that the background signal (reflectance of sample area without nanopatterning) in an a-Si:H slab with its high refractive index of ~ 3.5 is higher than, e.g., a Si 3 N 4 slab, which is useful since the background fringes can be used to monitor the existence of non-resonant drift in the same image as the resonant phase information.

SI 12. Table S1: Resonant interferometric sensing -literature review summary
The comparison presented in the table below is restricted to platforms with a similar degree of complexity and similar footprint as well as multiplexing capability and therefore limited to out-of-plane coupling platforms such as plasmonic nanohole arrays and dielectric photonic crystal slabs employing phase detection.We did not include waveguide-based platforms, e.g. ( 52), because there the superior overall LOD is based on the considerably higher phase sensitivity which is enabled by large-footprint waveguide arrangements without imaging capability.

Dielectric grating
HeNe Laser Detection limit stated to be 1° 1.9 x 10 -3 π X * We were unable to replicate this value of phase sensitivity when simulating the author's structure ** 3.43 × 10 −7 was obtained with system resolution, not 3σ noise, and is therefore not the relevant LOD based on 3σ.*** authors quoted statistical error phase of lock-in amplifier 0.01•,  2.68 × 10 −7 RIU.This is not the LOD based on 3σ.

SI 13. Noise reduction due to self-referencing in degenerate interferometry case
The schematic below illustrates how self-referencing based on mode degeneracy (phase noise matching), the referencing of equal resonances, results in lower noise compared to typical system of two resonances that are each referenced with a broad background and then referenced with each other, where the noise adds up due to signal subtraction.

SI 14. Material index effect on surface sensitivity
The simulations are based on the assumption of a protein refractive index of 1.45 and homogeneous, dense protein layer.The goal is not to quantify expected resonance shifts upon protein binding, but rather to illustrate the effect of the material index on the surface sensitivity to explain why a-Si:H was chosen for this work.6) for more information.The period is tuned in space leading to a tuned resonance condition, which is now fulfilled where the resonance is observed as bright lines.A lifting of the mode degeneracy results in 'splitting' of this resonance line as seen here.Therefore, the chirped approach can be used to ensure an alignment leading to an equal excitation of the degenerate modes at the Γ-point for degenerate mode interferometry.

SI 16. Noise analysis -"pink" camera noise indication
The self-referencing based on degenerate mode interferometry reduces the system-noise to a low level of σ = 9.2 • 10 -5 π.To characterize the nature of this remaining noise, we perform a Fourier transform of the noise signal and compare this signal to the Fourier transform of both white noise and pink noise, where pink noise is the white noise divided by the frequency f.The white noise shows a flat response, as expected.Plotted on a loglog scale, it is evident from Fig. S17 C, that the remaining resonant noise ("sharp -sharp") does not show a white noise behavior, but rather a frequency dependent behavior, similar to the pink noise.This result is an indication that the remaining noise might limited by the CMOS itself (44, 45).

SI 17. Movie S1 -caption
Movie S1 displays how the interferogram fringes respond to wavelength tuning of the incident light as the resonance goes through an almost complete cycle.On the left, the video is showing the spectrum of the excitation beam measured with a spectrometer.The tuning of the peak wavelength is achieved via rotation of the bandpass filter (SI 3) and the well-known Fabry-Perot resonance dependence on the angle of incidence.On the right, the video shows the corresponding interferogram for each of these incidence peak wavelengths recorded with the CMOS camera.Specifically, the main, centered part of the displayed image represents a nanohole array going through the resonance cycle, while the top and bottom parts of the image show the background (un-patterned silicon surface).Since the phase noise matching approach that we employ for specific protein sensing would result in zero detectable phase shifts when tuning the incident wavelength (both mode components depend equally on the wavelength), we here sheared the reflected beams such that the resonant nanohole array overlaps with the non-resonant background to extract the phase-wavelength dependence.It is evident that the background fringes drift in the opposite direction with respect to the resonant phase of the nanohole array.We therefore reference the resonant phase wavelength dependence with these background fringes to extract the phase behavior (i.e., Fig. 3 A).Note that we did not adjust the analyzer orientation throughout the wavelength tuning, meaning that the contrast of the interferogram reduces while going through the peak of the resonance in this case, since the reflectance increases while the background remains constant.When using the phase noise matching approach, the contrast is optimal since both orthogonal components have similar reflectance values even throughout a protein sensing experiment, since very small resonance changes are detected that do not result in large changes in relative resonance amplitudes of the two components.Further note that we have rescaled all images contained in the video separately such that the absolute intensities are her not representative of the resonance amplitude behavior.

Fig. S1 .
Fig. S1.Phase and magnitude of Lorentz oscillator model.For the illustration of the approximately linear range of the phase response within a range of FWHM/2.

Fig. S4 .
Fig. S4.Loretz versus Fano phase.Phase of the Lorentzian (GMR) channel in solid lines, and the total (Fano) phase in dashed lines.The green lines refer to the positive sign, and the black lines refer to the negative sign.Inset refers to the green curves.

Fig. S5 .
Fig. S5.Loretz versus Fano phase.Phase of the Lorentzian (GMR) channel in solid lines, and the total (Fano) phase in dashed lines.The green lines refer to the positive sign, and the black lines refer to the negative sign.Inset refers to the black curves.
Figure S6 illustrates how resonances supported by the specific structure used in this work (Fig. 2 in main manuscript) with resonance wavelengths at different points of the FP continuum show interesting phase behaviors based on the above described interference effects.

Fig. S6 .
Fig. S6.Simulated phase of various Fano resonances.As a result of different overlap with the Fabry-Perot continuum.The middle inset at around 840 nm corresponds to the structure used in this work, see Fig. 2 in main manuscript.

Fig. S8 .
Fig. S8.RCWA simulations of resonance wavelength shift in response to bulk refractive index alterations of the upper cover medium (compare Fig. 2 in main manuscript) for the TM and the TE mode.

Fig. S9 .
Fig. S9.Isotherm fit.Same data as shown in Fig. 4D in the main manuscript.

Fig. S10 .
Fig. S10.Spectrum of the SLED used throughout this work

Fig. S13 .
Fig. S13.Interferogram contrast.Image of 3 cases of interferograms on the same sample in the same FOV at 45° analyzer orientation.1. Mode-background interference results in low contrast.2. Selfinterference results in high contrast because signal and reference beam are the orthogonal components of the same resonant mode and therefore have the same amplitude.3. The background interferogram has an overall lower signal, but the contrast is also high since the orthogonal background components have approximately the same amplitude.
Fig. S14.Resonant phase noise matching.Schematic explanation of noise reduction due to selfreferencing of two equal resonance components versus a typical referencing case of a resonance with a flat background or broad resonance.

Fig. S15 .
Fig.S15.Surface sensitivity.Simulation of resonance shift in response to increasing the thickness of a 'protein layer', approximated as homogeneous layer with refractive index of 1.45.The TM modes supported by a a-Si:H and a Si3N4 nanohole array at the same wavelength are compared in terms of their surface sensitivities.

Fig. S16 .
Fig.S16.'Chirped' nanohole array for alignment.Image of nanohole array at resonance with constant period leading to same reflectance over the whole sensor area (top left).This configuration is used for the interferometry work.In the same FOV (bottom right) a chirped nanohole array is used for alignment purposes to ensure mode degeneracy for interferometry.Note that the intensity scale is adjusted to show the chirped nanohole array clearly, leading to an overexposure of the top left resonance.

Fig. S17 .
Fig. S17.Noise characterization.White noise with gaussian profile was generated (MATLAB randn function) to compare this to the remaining noise of the self-referenced phase information (Fig.3main manuscript).The Fourier transform of the white noise displays a flat response, while pink noise (white noise multiplied with 1/f) shows a linear frequency dependence when plotted on a loglog scale.The resonant "sharp-sharp" noise shows a similar trend.