# Fileset

[supplementary_APL23-APL23-AR-03798.docx](https://mdr.nims.go.jp/filesets/21bbdf30-e986-408f-8a9c-81669ba9984e/download)

## Creator

[Keisuke Watanabe](https://orcid.org/0000-0002-4285-2135), [Masanobu Iwanaga](https://orcid.org/0000-0002-8930-6940)

## Rights

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Keisuke Watanabe et al., Appl. Phys. Lett. 124, 111705 (2024) and may be found at https://doi.org/10.1063/5.0158793.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Optimum asymmetry for nanofabricated refractometric sensors at quasi-bound states in the continuum](https://mdr.nims.go.jp/datasets/e9f4db7b-b341-49ca-ad9d-5d9dac1c608f)

## Fulltext

Supplementary Material forOptimum asymmetry for nanofabricated refractometric sensors at quasi-bound states in the continuumKeisuke Watanabe1* and Masanobu Iwanaga21.* International Center for Young Scientists (ICYS), National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan, E-mail: watanabe.keisuke@nims.go.jp. https://orcid.org/0000-0002-4285-21352. Research Center for Electronic and Optical Materials, National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan. https://orcid.org/0000-0002-8930-6940S1. Derivation of equations and the critical coupling conditionEquation (2) in the main text describes the reflection amplitude from a resonance mode, which is dependent on the wavelength detuning λ−λ0 and is expressed as:        (S1)When there is no detuning of λ−λ0, meaning that the excitation wavelength of a tunable laser source is equal to the resonance peak wavelength, the amplitude reaches its maximum. In the main text, we consider this situation (i.e., λ = λ0); because we track peak wavelengths over time for the experiment. In this case, Equation (S1) can be expressed as:       (S2) Here, Q−1 = QR−1 + QNR−1. In the following, starting from the derivations of Equations (3) and (4), we derive the critical coupling condition.Derivation of Equation (3)When λ = λ0, the amplitude fluctuation δA caused by δλ is given by:24       (S3) This can be understood by considering the amplitude of the resonance mode when the peak wavelength is shifted by δλ, as depicted in Figure S1. First, A(λ0 + δλ, λ0) is expressed as:       (S4) Here, we define F = δλ/Δλ, where Δλ is the full width at half maximum (FWHM) of the resonance mode (thus Q = λ0/Δλ). Then we get:       (S5) Considering the approximation FQ−1 ~ 0, we obtain:       (S6)                                (S7)                                (S8)By substituting Equation (S8) into Equation (S3), we obtain:       (S9) which is the same as Equation (3) in the main text:       (S10) This equation indicates the amplitude fluctuation related to the frequency fluctuation.FIG. S1. Schematic showing the amplitude fluctuation δA in the resonance mode owing to the wavelength fluctuation δλ.Derivation of Equation (4)Solving Equation (S9) for F, we obtain:       (S11) Here, the approximation δA|δλ/A << 1 is used (i.e., 1− δA|δλ/A ~ 1). Recalling δλ = FΔλ = Fλ0/Q,       (S12) Finally, LOD = δλ/S can be expressed as:       (S13) which is the same as Equation (4) in the main text:       (S14) This equation clearly indicates that a large amplitude A yields a low LOD. Therefore, it is important to use a resonance mode with larger amplitudes to achieve a better sensing performance. Derivation of the critical coupling conditionIn the following, we consider the ideal condition of r = 0. In practice, a nonzero r slightly affects the LOD, but we have confirmed that this effect is sufficiently small. Now, considering Equation (5) in the main text, LOD is minimized when  is minimum. We rewrite this as f(x) with QR and QNR as x and y, respectively, we have:       (S15)Differentiating f(x) with respect to x, we obtain:       (S16)Therefore, one can find that the minimum f(x) is obtained when x = y (i.e., QR = QNR) by setting f’(x) = 0, which gives the critical coupling condition. This condition provides the balance between the Q factors and resonance amplitudes under the trade-off between them when asymmetries of the unit cell are changed, thus yielding the lowest LOD.S2. Resonance amplitude tracking at a fixed wavelengthFig. S2 compares the real-time refractive index measurement results between the resonance peak wavelength tracking and amplitude tracking at a fixed wavelength. Obviously, the amplitude change was nosier and did not provide clearer results than the wavelength change. This is because the amplitude contained a relatively large amount of noise. As stated in the main text, amplitude fluctuation includes several extrinsic noise sources, such as relative intensity noise, shot noise, and Johnson noise.25 Here, we divide the noise sources into two factors: δA triggered by wavelength fluctuations δA|δλ and all other sources of noise δA|O. This can be expressed as:24       (S17) The point here is that we are only interested in δA|δλ/A because the expression for the theoretical LOD in Eqs. (4) and (5) in the main text only accepts δA|δλ/A. Therefore, it is important to exclude δA|O/A to accurately evaluate the theoretical LOD, which can be achieved by obtaining δA|δλ/A by substituting the experimental δλ into Eq. (3). Note that the effect of real-time Fano fitting, which intuitively reduces the wavelength and amplitude fluctuations, has already been included in experimentally determined δλ. In practice, it is difficult to theoretically discuss the extent to which the fitting effect reduces δλ, which we consider to be beyond the scope of this study. Because the experimentally obtained δλ includes the result of fitting, δA|δλ obtained using Eq. (3) also takes into account the effect of fitting.FIG. S2. (a) Real-time measurement of resonance peak wavelength and (b) amplitude at a fixed wavelength (in this case, peak wavelength in water) for metasurface with α = 5%. The gray-shaded regions indicate the injection of different concentrations of index solutions, while the other white regions indicate the injection of water. (c) Representative spectra for different refractive index changes Δn. The Fano fitting curves are superimposed on the spectra. ΔA represents the change in amplitude compared with the resonance peak amplitude in water.S3. α-dependence of δA|δλBy substituting the δλ obtained in the experiment into Eq. (3), we obtain δA|δλ for different α. This result is shown in Fig. S3. This α-dependence reflects the fact that the radiation loss varies with α, and δλ/Δλ in Eq. (3) varies accordingly. To obtain a smooth curve for the LOD, we used the fitted function for the α-dependence of δA|δλ. Specifically, we plugged in δA|δλ = 1.56 × 10−7α−0.94 for δA in Eq. (4) to calculate of the theoretical LOD. FIG. S3. Experimentally determined δA|δλ as a function of α. The black line indicates the fitted function. 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