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Abstract
We study a systematic formation of continuous exceptional points (EPs) in a fully-asymmetric optical microdisk. A parametric generation of chiral EP modes is investigated by analyzing asymmetricity-dependent coupling elements in an effective Hamiltonian. It is shown that given the external perturbation, the frequency splitting around EPs is scaled by the fundamental “strength” of EPs [J. Wiersig, Phys. Rev. Res. 4, 023121 (2022) [CrossRef] ] multiplied by the extra responding strength of the newly added perturbation. Our finding demonstrates that the sensitivity of EP-based sensors can be maximized by carefully examining the continuous formation of EPs.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A non-Hermitian degeneracy, the so-called exceptional point (EP) in open quantum (wave) systems [2], has attracted much attention for its non-trivial degenerate feature, which is a simultaneous coalescence of eigenvalues and corresponding eigenfunctions of the eigenvalues. Optical microcavities are the representative open quantum (wave) systems whose decaying eigenmodes are characterized by complex-valued eigenfrequencies. Up to now, various EPs have been found in this system and the EPs are distinguished by their characteristic mode couplings [3,4]. So far, such many EP-based applications have been reported as optical sensors [5], nanoparticle detection [6–9], gyroscopes [10–13], and coherent perfect absorption [14,15].
Amongst many different classes of EPs, the one generated due to the coupling of even $[\psi (x_1,x_2)=\psi (x_1,-x_2)]$ and odd parity $[\psi (x_1,x_2)=-\psi (x_1,-x_2)]$ standing wave modes has been of particular interest. Generally, the EP has been analyzed in terms of counter-clockwise (CCW) and clockwise (CW) traveling wave basis modes. When an even and an odd standing-wave mode are coupled, forming EPs, the coupled modes turn into propagating modes in a single direction. These are called chiral EP modes and were demonstrated in a series of asymmetric microcavities, such as a circular microcavity with two scatterers [8]. The studies following this observation discovered the continuous formation of chiral EPs in a chaotic microcavity with a single scatterer [16], a ring resonator with a wavegude [17], a microcavity with two triangle-shaped defects [18], and a quadrupole microcavity with a waveguide and with four surface roughness [19], which mimics the circular microcavity with two scatterers.
The underlying mechanism of the chiral EPs is based on the two-step symmetry-breaking process of cavity boundaries. As is well known, the doubly degenerate modes in a rotationally-symmetric circular cavity split into a pair of an even and an odd mode when the rotation symmetry is reduced to mirror symmetry (e.g., by introducing a single scatterer). The cavity boundary becomes fully asymmetric when a further asymmetric perturbation is applied to this mirror-symmetric cavity boundary (e.g., by introducing a second additional scatterer). Now, the modes in this fully asymmetric cavity can not be defined as either even or odd; instead, the modes are partially chiral. By fine-tuning the parameters (e.g., scatterer’s size and position), we can lead the partial chiral modes into fully chiral EP modes. The chiral modes were observed in such microcavities as a spiral [20–23], an asymmetric Limaçon [10], a Gutkin’s [24], and an asymmetric Reuleaux triangle microcavity [25]. Recently, the chiral EPs due to asymmetricity were studied in a stand-alone Reuleaux triangle microcavity [25,26]. So far, however, an explicit demonstration of a continuous formation of chiral EPs in stand-alone asymmetric microcavities is still lacking.
In this paper, we study the formation of continuous chiral EPs generated by coupling an even and an odd mode in asymmetric microdisks. Considering an oval-shaped dielectric microdisk, deformed sequentially as “rotational-symmetric circle” $\to$ “mirror-symmetric ellipse” $\to$ “fully-asymmetric oval”, the asymmetricity-dependent chirality and the non-orthogonality of modes are investigated. We then show the continuous 2nd-order chiral EPs (i.e., two modes coalesce) along the unique curve in a three-dimensional parameter space. Finally, we find that the optimized parameters for achieving a maximal spectral response against the external perturbation can be deduced in the continuous EP curves. The obtained optimized maximal response is crucial in practical applications aiming to build ultra-sensitive sensors.
The paper is organized as follows. In Sec. 2, we describe a fully asymmetric oval cavity and a generation of chiral EPs. In Sec. 3, the sensitivity of EPs is analyzed in terms of an effective model Hamiltonian. We show that the initial non-vanishing off-diagonal elements in a Jordan form multiplied by the additional external perturbation strength determine the sensitivity of EP sensors. Finally, we summarize our discussions in Sec. 4.
2. Exceptional points in asymmetric ellipse microdisks
To obtain a fully asymmetric microdisk, we apply a boundary shape perturbation to the elliptic-shaped cavity. The boundary of the asymmetric ellipse microdisk (AEM) is given as follows:
Fig. 1. The shape of an AEM for three deformation parameters and resonance modes. (a) is the ellipse at $\varepsilon _1 = 0.65$ and $\varepsilon _{2,3}=0.0$. (b) is the mirror-symmetric oval shape at $\varepsilon _{1,2} = 0.65$ and $\varepsilon _3 = 0.0$. (c) is the asymmetric shape at $\varepsilon _{1,2} = 0.65$ and $\varepsilon _3 = 0.25$. (d) and (e) are an odd mode and its paired even mode at $\varepsilon _1 = 0.65$, $\varepsilon _2=0.05712$, and $\varepsilon _3=0.0$, whose eigenvalues are $kR \sim 4.2550 - i 2.148 \times 10^{-3}$ and $\sim 4.2542 - i 1.892 \times 10^{-3}$, respectively. (f) is the EP mode at $\varepsilon _3 = 0.3623\cdots$, whose eigenvalue is $\sim 4.2530 - i 2.035 \times 10^{-3}$.
Given the asymmetric boundary shape, we obtain optical modes by employing the boundary element method (BEM) [27]. Under the assumption of time-harmonic fields ($e^{-i\omega t}$, $\omega =ck$, c: speed of light, $i=\sqrt {-1}$), the optical modes computed are the solutions of the Helmholtz equation, $-\nabla ^2\psi =n^2k^2\psi$, obeying the boundary condition of the transverse electric (TE)-polarized modes along the cavity boundary. At the same time, as we apply the Sommerfeld radiation condition [28], the modes are decaying states (quasi-normal modes) described by complex-valued wavenumbers $k\in \mathbb {C}$. Here, $n$ is the effective refractive index set to 1 for the outside of the cavity and 3.3 for the inside of the cavity. For the sake of convenience, a dimensionless wavenumber $kR$ is used with an average radius $R$ of AEM. Through the studies, we consider the cavity size $n kR \sim 10$.
2.1 Symmetric formation of EPs in AEM
We begin by analyzing doubly degenerate regular quasi-normal modes in a circular microdisk, which are well-defined by the modal number $(l,m)=(1,10)$. Here, $l$ and $m$ are the radial and the azimuthal number (the number of maxima in $|\psi |$ along $r$-direction and the half-number of maxima in $|\psi |$ along $\theta$-direction, respectively) in a circular microdisk. This mode pair can be reconfigured by controlling the deformation parameters $(\varepsilon _1,\varepsilon _2,\varepsilon _3)$. For instance, when $(\varepsilon _1,\varepsilon _2,\varepsilon _3)=(0.65,0.05712, 0)$, the mirror-symmetry of the cavity boundary about the $x$-axis is preserved. Then, we can characterize the mode parity; an odd $[\psi (x,y)=-\psi (x,-y)]$ and an even $[\psi (x,y)=\psi (x,-y)]$, as shown in Figs. 1(d) and (e), respectively. However, when $\varepsilon _3 \neq 0$, the parity becomes indefinable since the mirror-symmetry breaks, and the orthogonality of the two modes does not hold anymore. By further fine-tuning the parameters, we can obtain EPs of the two modes. As is shown in Fig. 1(f), we can obtain EP of the two modes at $\varepsilon _3 \sim 0.3623$, which exhibits a single-directional traveling wave.
The general property of parameter-dependent EP formation for the even and the odd mode of $(l,m)=(1,10)$ is examined. In Figs. 2(a) [Re($kR$)] and (b) [Im($kR$)], we obtain the modes in the parameter range: $0 \leq \varepsilon _3 \leq 1.0$ for $(\varepsilon _1,\varepsilon _2) = (0.65,0.05712)$. Here, for the sake of convenience, we define the modes in AEM as
Fig. 2. Real and imaginary eigenvalues, chirality, and non-orthogonality in the region $0.0\leq \varepsilon _3\leq 1.0$ for $\varepsilon _1 = 0.65$ and $\varepsilon _2 = 0.05712\cdots$. (a) and (b) are the real and the imaginary eigenvalues of the even mode (red dashed line) and its paired odd mode (black solid), respectively. (c) is the chirality of the two modes, whose parity is indefinable. (d) is the non-orthogonality obtained by overlaps. Insets (i) and (ii) are $\Psi _\mu$ and $\Psi _\nu$ at $\varepsilon _3=0.33$, respectively. Insets (iii) and (iv) are an even mode and its paired odd mode with respect to the $y$-axis at $\varepsilon _3=0.5$, respectively.
2.2 Chirality and non-orthogonality of modes around EPs
Now, the chirality variations of the modes depending on $\varepsilon _3$ are investigated. It is well known that when the even and the odd parity mode form EPs, they exhibit strong chirality, which can be conveniently computed using the boundary wave Husimi functions $\mathcal {H}(q,p)$ [29]. Here, $(q,p)$ is called the Birkhoff coordinate [30] spanning the phase space of ray-dynamics inside the cavity; $q$ is the boundary arc length where the ray bounces off, and $p=\sin \chi$ is the tangential component of the ray’s momentum incident with an angle $\chi$. Since $\mathcal {H}(q,p>0)$ and $\mathcal {H}(q,p<0)$ imply a quasi-probability of the CCW and the CW traveling wave, respectively, the chirality can be defined as follows:
When parameters are given as $\{\varepsilon _3: 0,0.5,1.0,1.5\}$, $\varepsilon _2\ne 0$, and $\varepsilon _1\ne 0$, the cavity boundary is uniaxially mirror-symmetric. Hence, $\alpha _\mathcal {H}$ is strictly “0” because the two modes constitute two orthogonal standing wave modes of even and odd parity; $\mathcal {H}(q,p)$ are distributed equally for $p > 0$ and $p < 0$. But, except for those specific parameters, $\alpha _\mathcal {H} \neq 0$ in general. More precisely, the paired modes are always chiral and non-orthogonal, as shown in Figs. 2(c) and (d). Particularly, the chiralities are maximized as “$\alpha _\mathcal {H}=1$” at the exact EPs ($\varepsilon _3 \approx 0.3623 \equiv \varepsilon _{3,EP}$). We emphasize that the peak of the CCW chirality appears in $0 < \varepsilon _3 < 0.5$, while that of the CW one is in $0.5 < \varepsilon _3 < 1$; i.e., EP$_{_{CCW}}$ and EP$_{_{CW}}$ are formed symmetrically with respect to $\varepsilon _3=0.5$. Straightforwardly, we can obtain the non-orthogonality between the two modes $\Psi _\mu$ and $\Psi _\nu$ by using the mode overlap $S$ depending on $\varepsilon _3$ as follows:
The exceptional points discussed so far are unambiguously clarified by examining the Riemann surface. In Figs. 3(a) and (b), we obtain the real and the imaginary parts of $kR$s in parameter space ($\varepsilon _2,\varepsilon _3$). As we can see in the figure, the self-intersecting Riemann surface reveals the clear branching point marked by “EP”. The dotted lines on the Riemann surfaces are branch cuts showing eigenvalue splitting at EP. The splitting point at $\varepsilon _{2,EP}$ and $\varepsilon _{3,EP}$ on the real and the imaginary branch cut is the very EP. In the region $\varepsilon _3 < \varepsilon _{3,EP}$, when the parameter $\varepsilon _2$ increases, the Riemann surfaces exhibit an avoided resonance crossing, that is, the real parts repulse and the imaginary ones cross due to strong coupling. In the region $\varepsilon _3 > \varepsilon _{3,EP}$, the real parts cross and the imaginary ones repulse due to weak coupling. At the border of the two coupling regimes, EPs are located.
Fig. 3. The real and the imaginary Riemann surface with branch cuts around the EP. (a) and (b) are the self-intersecting real and the imaginary Riemann surface, respectively. The dashed lines are the branch cuts showing the EP.
3. Continuous formation of EPs in asymmetric ellipse microdisks
In this section, we demonstrate the continuous formation of 2nd-order EPs (i.e., two modes coalesce) in a three-dimensional parameter space $(\varepsilon _1,\varepsilon _2,\varepsilon _3)$. Figures 4(a) and (b) exemplify the real and the imaginary part of the successive branch-cuts of modes $(l,m)=(1,10)$, which embed EPs (green-dots) on the branch cuts. For better visualization, we show Re($\kappa$) and Im($\kappa$) of the relative wavenumbers defined as $\kappa = [k_{\mu (\nu )}-\bar {k}]R$, where $\bar {k}=(k_{\mu }+k_{\nu })/2$, that are traced in the $(\varepsilon _2,\varepsilon _3)$ space with equidistantly spaced values of $\varepsilon _1$ in $[0.61,0.68]$. The dashed-curves in Figs. 4(a) and (b) that connect the green-dots substantiate $\kappa (\varepsilon _{1,EP},\varepsilon _{2,EP},\varepsilon _{3,EP})$ forming continuous EPs. Note that the one with $\varepsilon _1 = 0.65$ corresponds to the EP discussed mainly in the preceding sections. As is clearly observed here, we can always obtain 2nd-order EPs in our AEM by controlling any two of the three parameters when one parameter is fixed.
Fig. 4. Continuous formation of 2nd-order EPs (dashed-curve connecting green-dots) obtained by controlling three parameters $(\varepsilon _1,\varepsilon _2,\varepsilon _3)$. (a) and (b) are the branch cuts, respectively, of the real and the imaginary part of the relative wavenumbers defined as $\kappa = [k_{\mu (\nu )}-\bar {k}]R$, where $\bar {k}=(k_{\mu }+k_{\nu })/2$. The branch cuts for selected eight different values of $\varepsilon _1$ from $0.61$ to $0.68$ with the equidistant step size of $0.01$ are exemplified.
This continuous formation of EPs in our AEM is not restricted in the $(l,m)=(1,10)$ modes only. We have numerically confirmed this fact by examining other various cases; $l=1$, $2$, $3$. For example, when we fix $\varepsilon _1 = 0.67$ the even and the odd mode of $(l,m)=(1,25)$ coalesce to generate an EP at $(\varepsilon _2,\varepsilon _3) = (\sim 0.01179, \sim 0.5462)$, with $kR\approx 9.1106 - i 1.233 \times 10^{-4}$. For $(l,m)=(2,17)$, when we fix $\varepsilon _1 = 0.65$ the even and the odd mode coalesce at $(\varepsilon _2,\varepsilon _3) = (\sim 0.03193, \sim 0.2695)$ with $kR\approx 7.83414 -i 2.57467\times 10^{-3}$. And for $(l,m)=(3,28)$, when we fix $\varepsilon _1 = 0.65$, an EP is obtained at $(\varepsilon _2,\varepsilon _3) = (\sim 0.2366, \sim 0.4421)$ with $kR\approx 12.7636 - i 5.771\times 10^{-4}$.
Thus far, the continuous formation of the chiral EPs has been numerically investigated. Now, we analyze it analytically by studying the two-level effective non-Hermitian Hamiltonian described in terms of the CCW and the CW traveling wave basis mode [31]. The CCW and the CW basis mode in this paper are defined by the symmetric and the anti-symmetric superposition of the even and the odd eigenmode obtained in the mirror-symmetric oval cavity; $\psi _{_\text {CCW,CW}}\equiv (\psi _e\pm i\psi _o)/\sqrt {2}$. Note that we employ the bi-orthogonal normalization, $\int _\Gamma \psi ^2 dA =1$ consistently; the integration is performed over the whole interior region of the microdisk. The CCW and the CW traveling wave defined in this way are confirmed through the Husimi function distribution, i.e., $\mathcal {H}_{_{CCW}}(p<0)\approx 0$ and $\mathcal {H}_{_{CW}}(p>0)\approx 0$.
Assuming that our traveling wave basis modes are doubly degenerated such that $E_{_{CCW}} = E_{_{CW}} = E_0 = (k_\mu +k_\nu )R/2$ before coupling terms are introduced, the effective Hamiltonian for AEM reads as follows:
Now, the wavefunctions of the eigenmodes in AEM can be expanded in terms of our traveling wave basis modes as follows:
where $c_{_{CCW(CW)}}^{\mu (\nu )}=\int _\Gamma dxdy\psi _{_{CCW(CW)}}\Psi _{\mu (\nu )}$. Given the defined vectors $|\Psi _{\mu (\nu )} \rangle \equiv \left [c_{_{CCW}}^{\mu (\nu )}, c_{_{CW}}^{\mu (\nu )}\right ]^T$, we can compute off-diagonal elements of $\mathbf {H}$ by solving a system of linear equationsFigure 5 shows the obtained values of $\{V,\ W\}\in \mathbb {C}$ as a function of $\varepsilon _3$. In the figure, we can see the condition of $|W|=0$ and $|V|\ne 0$ at EP, which constitutes a lower-triangular non-scaled Jordan normal form. The physical interpretation for this lower-triangular Hamiltonian is that while the back-scattering from CCW to CW vanishes, only that from CW to CCW remains.
Fig. 5. The real and the imaginary values of the off-diagonal elements in the CCW-CW basis. (a) is the values of $V$ and $W$ depending on $\varepsilon _3$ for $\varepsilon _{1,2} = 0.65, \sim 0.05712$, respectively. The solid and the dashed line are the real and the imaginary eigenvalue of $W$, respectively, and the dot-dashed and the dotted line are the real and the imaginary eigenvalue of $V$, respectively. An EP takes place at $\varepsilon _3 \sim 0.3623$. (b) is the absolute values of $V$ and $W$ showing the EP.
4. Maximized sensitivity of frequency splittings around EPs
In this section, the sensitivity of the frequency splitting $\left [(\omega _\mu -\omega _\nu )R/c=\Delta \omega R/c=\Delta kR\in \mathbb {C}\right ]$ in the vicinity of EPs against external perturbation is discussed. The general spectral splitting responding to the external perturbation $\epsilon$ of the $N$-th order root is proportional to $\epsilon ^{1/N}$. Hence, the most crucial factor determining the splitting sensitivity is, of course, the number of coalescent modes $N$: the higher $N$ is, the more sensitive the splitting is. Unfortunately, although it is evident that the most efficient way to improve the sensitivity is to increase the number of coalescent modes, it is still challenging to obtain higher-order EPs in practice. Recent studies, however, revealed that besides this primary order-dependent spectral response against external perturbations, there is another intrinsic enhancing mechanism of the eigenvalue splitting in the vicinity of EPs [1,32]. Namely, the non-vanishing super (or sub)-diagonal element in the non-scaled Jordan form Hamiltonian determines the size of eigenvalue splittings. The element is thus called “strength” of EPs. In the case of our AEM, the non-vanishing sub-diagonal element in the lower-triangle Hamiltonian at EPs plays a vital role in amplifying the sensitivity of the spectral splitting.
4.1 Frequency splitting strength in the vicinity of EPs
For an explicit demonstration, we consider a generic Hamiltonian given in the non-scaled Jordan form at 2nd-order ($N=2$) EPs and an external perturbation as follows:
Note that while the variables $\{V_{EP}, c_1,\Delta n\}$ in Eq. (8) and $\{ A_0, B_1, \varepsilon \}$ in Eq. (4) in Ref. [1] have a one-to-one correspondence, here, we focus more on the impact of $c_1$.
4.2 Maximized frequency splittings
Now, we numerically examine $|\Delta kR|$ around EPs for the eight different values of $\varepsilon _1$ computed in Fig. 4. Our numerical experiments reveal that $\varepsilon _1 = 0.65$ exhibits the highest sensitivity among all the cases, as shown in Fig. 6(a). However, the maximum strength of EPs (i.e., $|V_{EP}|$) turns out to be produced at $\varepsilon _1\approx 0.635$. This result implies that both the strength of EPs and the opposite off-diagonal external perturbation term are crucial in determining the frequency splitting size. That is, the amplifying constant $\xi \equiv 2\sqrt {c_1V_{EP}}$ should be considered. In Fig. 6(b), we obtain eight cases of frequency splittings $|\Delta kR|$ around EPs formed at $\{\varepsilon _1:0.61,0.62,0.63,0.64,0.65,0.66,0.67,0.68\}$ by sampling five cases of external perturbations $\{\Delta n:5\times 10^{-6},1\times 10^{-5},5\times 10^{-5},1\times 10^{-4},1\times 10^{-3}\}$. To verify the consistent order of the splitting size, all the eight cases are normalized by the maximum splitting at $\varepsilon _1=0.65$ as follows:
Fig. 6. (a) is absolute eigenvalue difference depending on external refractive index for three cases of $\varepsilon _1$, $\varepsilon _1=0.61$ (black lines), $0.65$ (green lines), and $0.68$ (orange lines). (b) is perturbation (through overlapping) vs eigenvalues. The gray dashed line is the fitting line for $\Delta n=1\times 10^{-3}$, which shows a slope of 1. (c) is the off-diagonal element $\alpha$ of perturbation Hamiltonian matrix. Inset shows the other part $\beta$. The real parts of perturbation are illustrated by solid lines and the imaginary ones by dashed lines.
Here, it is emphasized that the slope $c_1$ [$c_2$] corresponding to the response ratio of $\alpha$ [$\beta$] to $\Delta n$ is dependent on $\varepsilon _1$, as we can see in Fig. 6(c) [and inset], while the strength of EPs (i.e., $V_{EP}$) is constant about $\Delta n$, since it is fixed before $\Delta n$ has applied afterward. Nevertheless, the values of $V_{EP}$ are still dependent on the basic parameter sets $(\varepsilon _1,\varepsilon _2,\varepsilon _3)$ of EPs, yet, they are determined prior to the external perturbation being applied. Once we get EPs at certain preliminary parameters $(\varepsilon _1,\varepsilon _2,\varepsilon _3)$, the strength of EPs, $(V_{EP})$ is fixed. In the EP situation, when an additional external perturbation is applied, the frequency splitting size in the vicinity of EPs is determined by $V_{EP}$ multiplied by the response ratio to the external perturbation, i.e., $c_1$ in our case and the splitting size can be predicted precisely. Figure 7 shows that the amplification constant $|\xi |$ has a maximum value at $\varepsilon _1=0.65$. On the other hand, as we can see in the figure, $|V_{EP}|$ (black-closed squares) does not have the maximum at $\varepsilon _1=0.65$ (gray-dashed line). Furthermore, the slope $c_1$ monotonically increases as a function of $\varepsilon _1$ and does not have any supremum point in the examined parameter range. However, their multiplication gives the maximum at the expected value of $\varepsilon _1=0.65$. In short, the strength of EPs, $|V_{EP}|$, plays a crucial role in determining the spectral sensitivity of EP-based sensors: the size of $|\Delta kR|$. Yet, we need to multiply it by the additional response strength against the external perturbations to obtain the optimized EP-based sensors. From the results, we can conclude that both physical properties of $|V_{EP}|$ and $c_1$ play a crucial role in determining the spectral sensitivity of EP-based sensors.
Fig. 7. Sensitivity comparison depending on $\varepsilon _1$. (a) is the amplification constant $|\xi |$ determining the sensitivity. The maximum value is at $\varepsilon _1=0.65$. Inset shows the frequency splittings $|\Delta kR|$ corresponding to $|\xi |$ in (a) for $\Delta n=1\times 10^{-3}$. (b) is perturbation strength $|V_{EP}|$ of EP and the response ratios $c_i$ to the external perturbation shown in Fig. 6(c). $|V_{EP}|$ is shown as black closed squares (left side $y$-axis). The external perturbations $|c_1|$ and $|c_2|$ of $\alpha$ and $\beta$ serve as closed red circles and open red circles (right side $y$-axis). The gray dashed line indicates the position guide of maximum value $|\xi |$ at $\varepsilon _1=0.65$. $|V_{EP}|$ is maximized in between $\varepsilon _1=0.63$ and $0.64$.
5. Summary
In summary, we have investigated the formation of chiral exceptional point (EP) modes in fully-asymmetric microdisks. We have obtained continuous EPs by controlling the multiple boundary-shape parameters of our microdisk. We have demonstrated that the spectral splitting sensitivity is crucially determined by the multiplication of the strength of EPs, the non-vanishing off-diagonal element in the non-scaled Jordan-form Hamiltonian and the counterpart off-diagonal element that arises due to the extra external perturbations. The method of implementing our multi-parameterized microcavities can have the advantage of flexibility in realizing EP-based sensors. Moreover, our findings have proven that systematic sensitivity maximization is possible for given 2nd-order EPs, which is a critical result, particularly in practical applications of extremely sensitive sensors.
Funding
Institute for Basic Science (IBS-R024-D1); National Research Foundation of Korea (2021R1A2C1095623).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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