Abstract
Engineering lightmatter interactions using nonHermiticity, particularly through spectral degeneracies known as exceptional points (EPs), is an emerging field with potential applications in areas such as cavity quantum electrodynamics, spectral filtering, sensing, and thermal imaging. However, tuning and stabilizing a system to a discrete EP in parameter space is a challenging task. Here, we circumvent this challenge by operating a waveguidecoupled resonator on a surface of EPs, known as an exceptional surface (ES). We achieve this by terminating only one end of the waveguide with a tuneable symmetric reflector to induce a nonreciprocal coupling between the frequencydegenerate clockwise and counterclockwise resonator modes. By operating the system at critical coupling on the ES, we demonstrate chiral and degenerate perfect absorption with squaredLorentzian lineshape. We expect our approach to be useful for studying quantum processes at EPs and to serve as a bridge between nonHermitian physics and other fields that rely on radiation engineering.
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Introduction
Exceptional points (EPs) are generic degeneracies of nonHermitian systems, where two or more eigenvalues and the associated eigenvectors of a system coalesce, resulting in the reduction of the system’s dimensionality and in a severely skewed vector space^{1,2,3,4,5,6}. This is very different from Hermitian degeneracies known as diabolic points where eigenvectors stay orthogonal to each other although the eigenvalues are degenerate^{1,2,3,4,5,6,7,8}. This difference has created a variety of novel opportunities and attracted enormous attention from different scientific disciplines. Among the intriguing phenomena at or in the vicinity of EPs are chiral behavior^{9}, enhanced response to small perturbations^{10,11,12,13,14,15,16,17}, and enhanced transmission and lasing with increasing loss^{18,19,20}, just to name a few [see references^{2,3,4,5,6} for a complete list].
EPs emerge in systems through different routes, such as balanced loss and gain as in paritytime (PT) symmetric systems^{21,22,23,24}, judiciously engineered loss imbalance in lossonly systems^{18,19,25,26}, asymmetric coupling between modes of a system (e.g., clockwise CW and counterclockwise CCW modes)^{9,10}, postselection in quantum systems^{27,28}, and parametric modulation^{29}, all of which have been demonstrated in experiments. The latter three routes differ from the former two routes because they do not rely on introducing additional loss or gain into the system, and thus remain free from the associated noise contributions. As such they have the potential to be used in sensors^{11,30,31}, spontaneous emission control^{32,33}, and other fields where noise imposes stringent constraints.
The asymmetric coupling between CW and CCW modes of waveguidecoupled resonators has been achieved in experiments through the control of intermodal scattering by inserting two scatterers in the resonator mode field whose size and relative distance in the field can be tuned^{9,26}. This procedure gives rise to isolated EPs that are typically very difficult to stabilize against fabrication errors and fluctuations in the experimental environment. To overcome this problem, the concept of exceptional surfaces (ES) was recently introduced^{30,34,35,36} and experimentally implemented in magnon polaritons^{37} with potential applications in optical sensing^{30}, optical amplification^{38}, spontaneous emission control^{32}, and optical absorption^{39}: these hypersurfaces in parameter space consist of a continuous collection of EPs. An ES emerges in waveguidecoupled ring resonators through unidirectional (i.e., nonreciprocal) coupling between the modes such that the CW mode couples to the CCW but the CCW mode does not couple to the CW mode or vice versa (Fig. 1a, b). This feature provides a stability against unwanted perturbations (e.g., noise, fabrication imperfections, etc.) that typically drive the system away from discrete EPs and thereby deteriorate their most desirable features.
While, in principle, unidirectional coupling in microring resonators can be achieved via backreflection from a simple endmirror placed at one of the output ends of the waveguide^{28} (see Fig. 1a), we take a different route here that provides full experimental control over the strength and phase of the backreflected signal (i.e., we can tune the reflection magnitude and phase of the endmirror or the reflector), and hence allows us to steer the system on the ES. By critical coupling to the resonator on the ES we observe perfectly absorbing EPs^{40}, exhibiting the characteristic quartic absorption lineshape^{40} as the unique hallmark for perfect absorption that is both chiral (i.e., higher absorption for incidence from a specific direction) and degenerate (i.e., two purely incoming wave solutions coalesce at the ES). We emphasize that no additional loss (other than the typical resonator and component losses) is introduced to observe these signatures.
Results
Experimental setup and theoretical model
Our experimental system is composed of an onchip whispering gallery mode microsphere resonator coupled to a taperedfiber waveguide (Fig. 1d), which is used to couple light in and out of a resonant mode, and a fiber loop with a polarization controller acting as a tuneable endmirror (Fig. 1a). The resonator supports CW and CCW modes at the same frequency. We have identified a resonant mode with intrinsic quality factor of \(7.7\times {10}^{5}\) (measured in the deep undercoupling regime) in the \(1440\,{{{{{\rm{nm}}}}}}\) band and confirmed that in the absence of the endmirror, the transmission (\({T}_{{cw}}\) detected at \({D}_{2}\) and \({T}_{{ccw}}\) detected at \({D}_{1}\)) and reflection spectra (\({R}_{{cw}}\) detected at \({D}_{1}\) and \({R}_{{ccw}}\) detected at \({D}_{2}\)) are symmetric for light input in the CW direction (forward or left incidence) and CCW direction (backward or right incidence) (see Fig. 1c): both \({T}_{{cw}}\) and \({T}_{{ccw}}\) exhibit typical Lorentzian lineshapes; \({R}_{{cw}}={R}_{{ccw}}=0\), implying that there is no intermodal coupling between the CW and CCW modes (i.e., no mode splitting); and absorption defined as \({A}_{{cw}({ccw})}=1{T}_{{cw}\left({ccw}\right)}{R}_{{cw}\left({ccw}\right)}\) is symmetric (i.e., \({A}_{{cw}}={A}_{{ccw}}\)). We have also confirmed that the loading curve is the same for left and right incidence (inputs in the CW and CCW directions) (see Fig. 1c, inset). The introduction of the fiber loop acting as a tuneable endmirror breaks this symmetry by inducing unidirectional coupling between the CW and CCW modes: light transmitted in the forward direction (left incidence; CW mode) is reflected back and couples into the CCW mode, but light transmitted in the opposite direction (right incidence; CCW mode) travels directly to a detector without any backcoupling into the CW mode (i.e., no reflector at the output of the taper in the CCW direction).
Within the context of coupledmode theory, our experimental setup is described by \({\partial }_{t}A=i{H}_{{ES}}A\) where \(A={({a}_{{cw}},{a}_{{ccw}})}^{T}\), \({a}_{{cw}}\) and \({a}_{{ccw}}\) are the field amplitudes of the CW and CCW modes respectively, and \({H}_{{ES}}\) is the effective Hamiltonian given as
Here, \({\omega }_{0}i\Gamma\) are the complex frequencies of the degenerate CW and CCW modes, with \(\Gamma =\left({\gamma }_{0}+{\gamma }_{1}\right)/2\) corresponding to the cavity loss rate which consists of the waveguide coupling loss \({\gamma }_{1}\) and all other resonatorrelated losses (i.e., radiation, scattering and material absorption losses) \({\gamma }_{0}\), and \(\kappa\) denotes the CWtoCCW coupling strength. The zero value in the offdiagonal elements implies that the CCW mode does not couple to the CW mode. The unidirectional coupling strength in this system is defined as \(\kappa =r{\gamma }_{1}\) with \(r=\leftr\right{{\exp }}(i\phi )\) and \(\leftr\right\) and \(\phi\) corresponding to the magnitude and phase of backreflection from the endmirror (i.e., fiberloop mirror). Both the eigenvalues and the corresponding eigenvectors of this system are degenerate and given as \({\omega }_{{{{{\mathrm{1,2}}}}}}={\omega }_{0}i\Gamma\) and \({a}_{{{{{\mathrm{1,2}}}}}}={(0,1)}^{T}\), forming an EP with CW chirality at the complex frequency \({\omega }_{0}i\Gamma\). Clearly, the system is at an EP for any nonzero \(\kappa\) (i.e., \(\leftr\right \, \ne \, 0\) and \({\gamma }_{1} \ne \, 0\)) and for all values of \({\omega }_{0}\) and\(\,\Gamma\). Indeed, if \(\leftr\right\) and \(\phi\) are steered, the system will trace an ES surface formed by EPs at the complex ES frequency \({\omega }_{0}i\Gamma\). As such, the system will always stay on a surface even if there are variations both in the amplitude and the phase of \(r\). Variations in \({\gamma }_{0}\) and \({\gamma }_{1}\), on the other hand, will create a new ES at a new complex ES frequency differing only in the imaginary part. Similarly, any perturbation (e.g., by temperature) that affects \({\omega }_{0}\) will lead to a new ES frequency differing only in the real part. Thus, although experimental imperfections and fluctuations may shift the complex ES frequency, they will not be able to lift the nonHermitian degeneracy on the ES and the system will always remain on the surface. We note that any perturbation that will break unidirectional coupling between CW and CCW modes (e.g., a perturbation that changes the zero element in \({H}_{{ES}}\) to a nonzero value \(\delta\)) will lift the degeneracy, leading to split modes with \({\omega }_{{{{{\mathrm{1,2}}}}}}={\omega }_{0}i\Gamma \mp \sqrt{\kappa \delta }\), and push the system off the ES.
Exceptional surfaces and reflection spectra with squaredLorentzian lineshape
We investigated the formation and the stability of an ES in our system by monitoring the reflection and transmission spectra for left and right incidence by tuning the system parameters \(\leftr\right\) (reflection strength), \(\phi\) (reflection phase), and \({\gamma }_{1}\) (waveguideresonator coupling strength) (see Supplementary for theoretical model). We first set the system to critical coupling (\({\gamma }_{0}={\gamma }_{1}\)), confirmed with zero transmission at the resonance dip both for left and right incidence. Then we vary \(\leftr\right\) and \(\phi\), implementing a variable/tunable reflector, and monitor transmission \({T}_{{cw}({ccw})}\) and reflection \({R}_{{cw}({ccw})}\). We observe symmetric transmission spectra \({T}_{{cw}}\equiv {\left{t}_{{cw}}\right}^{2}\) and \({T}_{{ccw}}\equiv {\left{t}_{{ccw}}\right}^{2}\) with Lorentzian lineshapes for all values of \(\leftr\right\) and \(\phi\). These results agree well with coupledmode theory which predicts \({t}_{{cw}({ccw})}\propto \triangle /(\triangle i{\gamma }_{0})\) and hence \({T}_{{cw}({ccw})}\propto 1/\left(1+{{\gamma }_{0}}^{2}/{\Delta }^{2}\right)\), where \(\Delta\) is the lasercavity detuning. On the other hand, it is obvious that reflection spectra \({R}_{{cw}}\) and \({R}_{{ccw}}\) will demonstrate an asymmetric behavior because \({R}_{{ccw}}\) is constant at all frequencies (i.e., for right incidence reflection occurs from the mirror only and does not involve the resonator) while \({R}_{{cw}}\) exhibits a resonance. This asymmetric reflection with symmetric transmission for inputs in opposite directions already indicates the presence of an EP. According to coupledmode theory, the amplitude reflection coefficient for light incident from the left scales as \({r}_{{cw}}\propto {{t}_{{cw}}}^{2}\), leading to \({R}_{{cw}}\equiv {\left{r}_{{cw}}\right}^{2}\propto 1/{\left(1+{{\gamma }_{0}}^{2}/{\Delta }^{2}\right)}^{2}\). In other words, the reflection spectrum for left incidence features a squared Lorentzian response. We observe this asymmetry and quartic behavior (i.e., flattening) of the reflection lineshape in our experiments (see Fig. 2a and Supplementary Fig. S6). This type of response indicates the presence of a novel type of perfectly absorbing EP^{40}; lineshape modifications associated with such an EP have, however, not been observed up to date because, to the best of our knowledge, all previous experiments have been performed in the vicinity of an EP rather than exactly at an EP due to the difficulty of keeping a system at a discrete EP stably and continuously. Thus, the expected lineshape modification remained obscured and indiscernible in previous works. Our system, on the other hand, operates on an ES and is thus always exactly at an EP even in the presence of experimental imperfections and fluctuations.
To demonstrate that our system is, indeed, on the ES, we have collected \({R}_{{cw}}\) spectra at various values of \(\leftr\right\) and \(\phi\) when the system is at critical coupling (\({\gamma }_{0}={\gamma }_{1}\)) and extracted the frequency and linewidth of the resonance lineshape on the ES. We do this by fitting the experimental data with a function composed of the product of two Lorentzians, that is \(f={L}_{1}\cdot{L}_{2}\) with \({L}_{k}={{A}_{k}\triangle }_{k}/({\triangle }_{k}i{\Gamma }_{k})\), and by estimating \(\left\{{\triangle }_{k},{\Gamma }_{k}\right\}\) which should ideally satisfy \({\triangle }_{1}={\triangle }_{2}\) and \({\Gamma }_{1}={\Gamma }_{2}\) on the ES. Plotting the experimentally obtained \(\triangle \omega={\triangle }_{1}{\triangle }_{2}\) and \(\triangle \Gamma ={\Gamma }_{1}{\Gamma }_{2}\) as a function of \(\leftr\right\) and \(\phi\) has revealed the ES (see Fig. 2b, c). The \(\triangle \omega\) values are in the range \(\left[6.8\,{{\mathrm{MHz}}},3.4\,{{\mathrm{MHz}}}\right] \) and \(\triangle \Gamma\) are in the range \(\left[5.4\,{{\mathrm{MHz}}},8.8\,{{\mathrm{MHz}}}\right]\), which, when normalized with the frequency \({\omega }_{0}=207.3\,{{\mathrm{THz}}}\) and the linewidth \({\Gamma }_{0}=502\,{{\mathrm{MHz}}}\) of the resonance at the critical coupling without the endmirror, yield \(\left\triangle \omega/{\omega }_{0}\right \, \lesssim \, {10}^{8}\) and \(\triangle \Gamma /{\Gamma }_{0} \, \lesssim \, {10}^{2}\), implying that the system is, indeed, on an ES.
We have also performed experiments at undercoupling and overcoupling regimes by tuning \({\gamma }_{1}\) (i.e., varying the taperresonator gap), and observed that the system always stays on an ES and remains robust against changes and unwanted fluctuations in the waveguideresonator coupling strength (see Supplementary Figs. S7 and S8 for ES formed in the under and overcoupling regimes). Thus, steering the system in the 2D parameter space using \(\leftr\right\) and \(\phi\) always defines an ES regardless of the waveguideresonator coupling regime. The system will leave the ES only for perturbations that break unidirectionality and establish a symmetric or asymmetric coupling between the CW and CCW modes. We also note that the system is on an ES at all resonances across the spectrum (i.e., one can construct and probe multiple ES in parallel by simultaneously probing multiple resonances).
Chiral perfect absorption with quartic lineshape on an exceptional surface
Next, we study the absorption properties of the system (see Fig. 1a) operating on the ES for left incidence (input in CW direction) at various taperresonator coupling conditions when \(\leftr\right=1\) corresponding to a fully reflective endmirror (see Fig. 3), where \({T}_{{cw}}=0\). Under this condition, the absorption spectrum is calculated using \({A}_{{cw}}=1{R}_{{cw}}\) where the reflection spectrum is measured by detector \({D}_{1}\). The normalization is carried out by considering the losses \({L}_{{cw}}\), including the insertion and propagation losses when the left incident field travels from the input circulator to the reflector and then back along the same path to \({D}_{1}\) in the absence of the resonator (see Supplementary). The system stays on the ES as we vary the taperresonator coupling strength, but the absorption strongly depends on the coupling regime, achieving perfect absorption only at the critical coupling (see Fig. 3). This absorption behavior can be explained as follows: Our system operating at the critical coupling with \(\leftr\right=1\) (perfectly reflecting endmirror) represents a onechannel coherent perfect absorber (CPA) with \({T}_{{cw}}=0\) and \({R}_{{cw}}=1/{\left(1+{{\gamma }_{0}}^{2}/{\Delta }^{2}\right)}^{2}\to 0\) for \(\Delta \to 0\), and thus \({A}_{{cw}}=1\) at the ES frequency, thus we refer to this special onechannel CPA as a onechannel CPAES. Different from a conventional onechannel CPA (or critical coupling), the resonator is tuned here to an EP on the ES and hence the absorption lineshape is quartic as is the reflection lineshape (see Fig. 3). As the system moves away from the critical coupling point toward the undercoupling or the overcoupling regime, a gradual transition from a squared Lorentzian lineshape to a more Lorentzianlike lineshape is clearly seen (Fig. 3 and Supplementary Fig. S9). As discussed above, our analysis takes the losses into account in the normalization process. If the offresonance losses \({L}_{{cw}}\) are not accounted for, the absorption will be limited only by \({L}_{{cw}}\), that is \({A}_{{cw}}\to 1\) \({L}_{{cw}}\) as \(\Delta \to 0\), when the system is on the ES at critical coupling.
We now show that an ES in our system (see Figs. 1a, 4) leads to chiral and perfect absorption also for partially reflecting endmirrors (see Fig. 1a). The tuneable fiberloop reflector allows us to construct symmetric mirrors with varying reflection and transmission coefficients. As examples, here we present the results of experiments performed with a 10:90 endmirror (10% transmission and 90% reflection) and a 50:50 endmirror (50% transmission and 50% reflection) at different taperwaveguide coupling regimes. The spectra are normalized with the power input to the tapered waveguide for left incidence and with the power just before the reflector for right incidence. We note that varying the reflection phase \(\phi\) does not affect the observed features. Typical spectra obtained for the 10:90 endmirror at different taperwaveguide coupling regimes are shown in Fig. 4 (see Supplementary Fig. S10 for the 50:50 endmirror).
When the endmirror is not 100% reflecting, we have access to reflection and transmission spectra \(({T}_{{cw}},{R}_{{cw}})\) and \(({T}_{{ccw}},{R}_{{ccw}})\) for left (CW direction) and right incidence (CCW direction) from which the normalized absorption spectra \({A}_{{cw}}\) and \({A}_{{ccw}}\) can be calculated using the expression \({A}_{{cw}({ccw})}+{R}_{{cw}({ccw})}+{T}_{{cw}({ccw})}=1\). In the experiments, \({T}_{{cw}({ccw})}\) for left and right incidence exhibit typical resonance dips at the ES frequency with Lorentzian lineshapes. However, reflection spectra differ significantly: \({R}_{{cw}}\) has a squared Lorentzian lineshape with a flattened resonance dip around the ES frequency (see Fig. 4) whereas \({R}_{{ccw}}\) is constant (\(e.g.,{R}_{{ccw}}=0.9\) for the 10:90 endmirror and 0.5 for the 50:50 endmirror) at all frequencies because it does not involve the resonator (see Fig. 4). The chirality in this behavior (asymmetry in reflection) stems from the larger absorption for the left incidence compared to the right incidence and the degeneracy in the absorption is the result of the strong coupling between the CW and CCW modes of the resonator for left incidence (no coupling between them for right incidence). Indeed, the absorption spectrum \({A}_{{cw}}\) for CW input (left incidence) is a superposition of a Lorentzian term coming from the transmission \({T}_{{cw}}\) and a squaredLorentzian term from \({R}_{{cw}}\), whereas \({A}_{{ccw}}\) for CCW input is always a Lorentzian (see Supplementary). The weights of the Lorentzian and squaredLorentzian terms in the superposition are determined by the reflectivity of the endmirror (i.e., \({A}_{{cw}}\) is squared Lorentzian for a 100% reflecting mirror and it is Lorentzian for a 0% reflecting mirror) and hence the effective unidirectional coupling \(\kappa\) between the CW and CCW modes. Another parameter that affects \(\kappa\) and thereby the contribution of Lorentzian and squaredLorentzian terms to the final lineshape is the waveguideresonator coupling strength \({\gamma }_{1}\) through the expression \(\kappa =r{\gamma }_{1}\). Thus, when the taperresonator coupling or the reflectivity of the endmirror is varied, the system continues to be on an ES but the lineshape and the overall amount of the absorption are altered.
Interestingly, a gradual transition from a quartic (squaredLorentzian) form to a quadratic (Lorentzian) form in the \({A}_{{cw}}\) lineshape takes place as the taperresonator coupling moves from the critical coupling toward the undercoupling or overcoupling regime (see Fig. 4, upper panel) or the reflectivity of the endmirror is tuned. Perfect absorption with flattop squared Lorentzian lineshape is clearly seen when the system is at the critical coupling and the input is CW (see Fig. 4).
Tuning chiral absorption and its bandwidth on an exceptional surface
Finally, we steer the system on the ES and determine the absorption \({A}_{{cw}}\) and \({A}_{{ccw}}\) at the ES frequency for left (see Fig. 5a) and right (see Fig. 5b) incidence, respectively, by tuning the fiberloop mirror parameters and the taperresonator coupling. We present the results in Fig. 5 demonstrating that perfect absorption on an ES (\({A}_{{cw}}=1\) at resonance with squaredLorentzian lineshape) takes place for left incidence at the critical coupling for all reflectivity values \(\leftr\right\) of the endmirror (Fig. 5a). For right incidence, on the other hand, conventional perfect absorption (\({A}_{{ccw}}=1\) at resonance with Lorentzian lineshape) occurs at the critical coupling only for \(\leftr\right=0\) (i.e., completely transmitting endmirror), and \({A}_{{ccw}}\) decreases with increasing \(\leftr\right\). As the taperresonator gap increases from zero (i.e., overcoupling), \({A}_{{cw}}\) and \({A}_{{ccw}}\) first increase reaching their maximum value at the critical coupling, and then start decreasing as the gap increases (the system moves toward deep undercoupling regime). Chirality of absorption can be better seen in the ratio \(\xi ={A}_{{ccw}}/{A}_{{cw}}\) (Fig. 5c) which can be tuned in the range \(\left[{{{{\mathrm{0,1}}}}}\right]\): \(\xi =0\) denotes no absorption for right incidence (\({A}_{{ccw}}=0\)) corresponding to a fully reflecting endmirror; \(\xi =1\) denotes equal absorption for incidence in both directions (\({A}_{{ccw}}={A}_{{cw}}\)); and all other values of \(0 \, < \, \xi \, < \, 1\) imply chiral absorption, that is \({A}_{{cw}} > {A}_{{ccw}}\), with higher chirality at higher \(\leftr\right\) and at taperresonator gaps closer to the critical coupling condition.
It is worth noting that ES does not only enable chiral absorption at the ES frequency, but the involved degeneracy also provides a way to control the absorption bandwidth. We have observed that the absorption bandwidth defined as fullwidth at halfmaximum is different for left and right incidence (i.e., chirality in bandwidth). More interestingly, we have found that compared to the conventional onechannel CPA at critical coupling, the absorption bandwidth on the ES is \(1.26\), \(1.41\), and \(1.59\) times larger for 50:50, 10:90, and fully reflecting endmirrors, respectively. These values are close to the theoretically predicted values of \(1.27\), \(1.50\), and \(1.55\). The flattened absorption spectrum in the vicinity of critical coupling for left incidence may provide a remedy to the narrow absorption bandwidth of a conventional onechannel CPA (i.e., a waveguidecoupled resonator operating at critical coupling)—a problem that has hindered progress in technologies relying on CPA.
Discussion
In this work, we have demonstrated a nonHermitian optical device which exhibits an ES and chiral perfect absorption. Since the device operates always exactly at an EP when on the ES, it provides a stable and controllable platform to study EPrelated phenomena and processes, revealing chiral and degenerate perfect absorption on an ES with quartic reflection and absorption lineshapes as the defining hallmark. Our results will pave the way toward control of various optical processes and lightmatter interactions exactly at an EP (not limited to the spectra in the vicinity of an EP), with potential applications ranging from chiral lightmatter interaction, lasing, and emission to chiral nonlinear photonics and photovoltaics. Creating the ES through a simple unidirectional coupling route between two modes can be extended to other physical platforms where the coupling between modes and systems can be made unidirectional by electrical, optical, photonics or acoustic feedback or backreflection. Since no additional loss or gain is introduced into the system, the ES obtained through unidirectional coupling can also benefit studies of quantum dynamics in nonHermitian systems.
Data availability
The datasets that support the finding of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
S.K.O. and R.E. acknowledge support from Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Award No. FA95502110202. S.K.O. also acknowledges support from NSF (Grant No. ECCS 1807485) and AFOSR (Award no. FA95501810235). R.E. acknowledges support from ARO (Grant No.W911NF1710481), NSF (Grant No. ECCS 1807552), and the Alexander von Humboldt Foundation. S.R. acknowledges funding by the European Commission (MSCARISE 691209) and by the Austrian Science Fund (FWF) (P32300).
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S.K.O., R.E., and S.R. conceived the idea and supervised the research. S.K.O., S.S., and R.E. designed the experiments; S.S. performed the experiments with help from M.M. at the initial stages of the study; S.S. collected the experimental data and analyzed it together with Q.Z.; S.S. and Q.Z. performed numerical simulations with guidance from R.E., S.R., and S.K.O.; S.K.O. and R.E. wrote the paper with contributions from all authors. All authors read and agreed with the content and discussions in the paper.
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Soleymani, S., Zhong, Q., Mokim, M. et al. Chiral and degenerate perfect absorption on exceptional surfaces. Nat Commun 13, 599 (2022). https://doi.org/10.1038/s4146702227990w
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DOI: https://doi.org/10.1038/s4146702227990w
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