Abstract
Coherent perfect absorption is one of the possibilities to get high absorption but typically suffers from being a resonant phenomena, i.e., efficient absorption only in a local frequency range. Additionally, if applied in high power applications, the understanding of the interplay of nonlinearities and coherent perfect absorption is crucial. Here we show experimentally and theoretically the formation of nonlinear coherent perfect absorption in the proximity of exceptional point degeneracies of the zeros of the scattering function. Using a microwave platform, consisting of a lossy nonlinear resonator coupled to two interrogating antennas, we show that a coherent incident excitation can trigger a selfinduced perfect absorption once its intensity exceeds a critical value. Note, that a (near) perfect absorption persists for a broadband frequency range around the nonlinear coherent perfect absorption condition. Its origin is traced to a quartic behavior that the absorbance spectrum acquires in the proximity of the exceptional points of the nonlinear scattering operator.
Introduction
Perfect absorption of the energy carried by a classical wave is an interdisciplinary research theme spanning areas as diverse as acoustics and mechanical waves^{1,2}, to radiofrequency^{3}, microwaves^{4,5} and optical wave settings^{6,7,8,9,10}. At the core of this activity is the promise that perfect absorption can be beneficial to a variety of applications ranging from stealth technologies^{3,11,12}, energy harvesting and photovoltaics^{13}, sensing^{14,15}, and photodetection^{16,17}. Along these lines, the quest for lowcost/power alloptical switching and modulation schemes that simultaneously utilize coherent interaction of light beams and absorbing matter for extreme absorption is recently gaining a lot of attention^{18,19,20}. One such protocol is the socalled coherent perfect absorption (CPA).
Coherent perfect absorption (CPA) is a multichannel waveform shaping protocol that leads to the complete extinction of a monochromatic radiation when it enters a weakly lossy cavity^{10,21,22}. Although the scheme has been initially proposed in the framework of classical optics^{21,22}, as the timereversed process of a laser, it turns out that its implementation does not require timereversal symmetry^{23,24,25}. It rather solely relies on wave interference effects that entrap the incident radiation inside the lossy cavity, leading to its complete absorption. Such mechanism allows us to control light with light in a linear fashion, through just the relative phases and amplitudes of the multiple inputs. Subsequent studies nicely demonstrated the CPA implementation, beyond the original platform of optics^{8,18,20,22,26,27,28,29,30}, spanning all areas of classical wave physics ranging from microwave^{25,31,32} and RF^{33}, to acoustics^{34,35}. In all the abovementioned cases, the CPA protocol demonstrated a narrow, resonantbased, (perfect) absorption with very sharp characteristics around the frequency of perfect absorbance. Obviously, addressing this “deficiency” will open up a whole range of possibilities for the CPA scheme including solar photovoltaic or stealth applications.
At the same time, most of the CPA studies and their implementations have been performed that underlying wave systems were linear, i.e., under the assumption of scale invariance. The absence of implementation of CPA protocols in wave systems that lack scale invariance comes as a surprise, specifically since nonlinear mechanisms are abundant in nature and they offer additional degrees of freedom for light manipulation. Only recently, some researchers^{36,37,38} have put forward the question of the applicability of a CPA scheme in cases where scale invariance is violated. The reasons for this lack of effort to identify nonlinear CPA (NLCPA) protocols are twofold: From the theoretical side, one needs to develop computational schemes since the wellestablished (linear) scattering formalism is not anymore applicable. Moreover, in the presence of nonlinear interactions one needs to control, not only the relative phases and amplitudes of the incident waves but also their absolute magnitude. From the experimental side, one might question the viability of such protocols due to bistabilities and other nonlinearitydriven phenomena which might destroy the delicate interferences between various propagating waves, or result in modulation instabilities making the NLCPA concept unrealistic. It is therefore imperative to test the implementation of a nonlinear CPA protocol (if at all realizable) under experimental conditions.
In this paper, we introduce an electromagnetic platform, consisting of microwave resonators, where an NLCPA with broadband characteristics can be investigated theoretically and implemented experimentally. The absence of scale invariance and the dependence of the scattering process from the absolute magnitude of the incident wave amplitudes result in a selfinduced CPA with a variety of photonic applications in areas like signal processing, nonlinear interferometry, and sensing. In the case of perfect coupling of the resonators with the interrogating antennas, the system supports a different type of NLCPA modes which demonstrate a squareroot frequency degeneracy in the neighborhood of a critical magnitude of the incident wave amplitudes– remnant of an exceptional point (EP) degeneracy occurring in linear nonhermitian systems. EP degeneracies have been documented in the nonHermitian literature as a main source of wave phenomena ranging from lossinduced transparency and unidirectional invisibility, to paritytimesymmetric lasers, and hypersensitive sensors (for some recent reviews see^{39,40,41,42}). Here, we provide a paradigm of EP degeneracies associated with steadystate nonlinear solutions of the wave operator, with incoming boundary conditions, that are responsible for a broadband (near) perfect absorption.
Results and discussion
Experimental setup
The experimental setup (Fig. 1a) consists of a dielectric resonator coupled to a shortcircuited diode (Fig. 1b) sandwiched between two aluminum plates. This hybrid system demonstrates a nonlinear response^{43,44}. The resonator is excited via two kink antennas curved around it. The antennas excite the first TEresonance mode of the resonator (around 6.7 GHz), where the magnetic field \(\overrightarrow{B}\) has only a zcomponent and the electric field lies parallel to the aluminum plates. The magnetic field couples to the shortcircuited diode thus inducing a nonlinear behavior of the system. The incident and reflected waves of the system are separated by circulators connected to the source cable, the antenna, and the measuring port of the vector network analyzer (VNA). The excitation, with power P_{VNA}, is injected from port 1 of the VNA and it is splitted equally by a Tjunction going through the Inphase and Quadrature (IQ) vector modulators into the circulators. The IQmodulators allow to vary the power and phase difference between the excitation lines. Thus the measured complex transmission amplitudes from port 1 to port 2 S_{21} and to port 3 S_{31} of the VNA give access to the reflected power of the system R = ∣S_{21}∣^{2} + ∣S_{31}∣^{2}. The total absorbance is then given by A = 1 − R = 1 − ∣S_{21}∣^{2} − ∣S_{31}∣^{2} where the scattering matrix elements ∣S_{21}∣^{2} and ∣S_{31}∣^{2} have been normalized taking into account the absorbance due to the IQmodulator, the cables, and the circulators (see “Methods” section).
The power of the incident signals P_{IQ1} and P_{IQ2}, injected from each of the two antennas are controlled by the power injected from the VNA P_{VNA} (maximal accessible P_{VNA} = 10 dBm), and by a differential absorption \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}\) and \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}}\) associated to each of the two IQmodulators adjoint to the two antennas. The same IQmodulators can also tune the relative phases ϕ_{IQ1} and ϕ_{IQ2} of each of the two incident waves. Finally, the couplings between the resonator and the two antennas have been treated as free parameters and they have been adjusted by curving the horizontal part of the kink antennas (Fig. 1) and/or by appropriately positioning the dielectric resonator in their proximity (see supplementary note 1 and supplementary fig. 1).
Theoretical model
The transport characteristics of a system of coupled resonators are modeled using a timeindependent coupled mode theory (CMT)
where ω is the frequency of the incident monochromatic wave, ψ_{n} is the scattering magnetic field amplitude represented in the TEmodes of the individual resonator (Wannier basis) localized at the nth resonator^{45}, \({H}_{n,n+1}={H}_{n+1,n}^{* }\) describes the coupling between the nth and n + 1th resonators and H_{n,n} = ϵ_{n} is the resonant frequency of the n − th resonator. For the specific case of Fig. 1a the scattering system consists of only one resonator at n = 0. The two semiinfinite chains of coupled resonators with n ≠ 0 have the same resonant frequency ϵ_{n} = ϵ ≈ 6.7 GHz and model the left and right antennas (leads). They are coupled with one another via a (real) coupling constant H_{n,n+1} = ν giving rise to a dispersion relation \(\omega =\epsilon +2\nu \cos (k)\) (k ∈ [0, 2π] is the wavevector of a propagating wave). The coupling strength between the nonlinear resonator (n = 0) with the left (n < 0) and to the right (n > 0) leads are \({H}_{1,0}={\kappa }_{1}= {\kappa }_{1} {{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{\phi }_{1}}\) and \({H}_{0,1}={\kappa }_{2}= {\kappa }_{2} {{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{\phi }_{2}}\), respectively. The nonlinear resonator at site n = 0, has a resonant frequency ϵ_{0} = ϵ + Ω(∣ψ_{0}∣^{2}). The complex function Ω(∣ψ_{0}∣^{2}) = Ω_{r}(∣ψ_{0}∣^{2}) + iΩ_{i}(∣ψ_{0}∣^{2}) depends on the local field intensity ∣ψ_{0}∣^{2}. Measurements of onesided transmission measurements indicated that the best fit with the predictions of the model of equation (1) are achieved when
where β_{0} = (67.74 + 3i) MHz, β_{1} = (−0.141 + 0.41i) MHz mW^{−1} while ν = 0.1 GHz.
The most general solution of equation (1) in the left and right leads, can be written as
where I_{1,2} are given by the scattering boundary conditions and represent the amplitudes of a left/right incident waves. Substituting the above expressions in equation (1) for n = 0, we get:
An additional relation between R_{1,2} and I_{1,2} is derived by substituting equation (3) back in equation (1) for n = ±1. We have that
which allows us to express equation (4) in terms of the monochromatic frequency ω and the amplitudes I_{1,2} and R_{1,2} of the counterpropagating waves in each of the two leads.
A CPA protocol inhibits all outgoing waves, i.e., R_{1} = 0 = R_{2}. Imposing these constrains in equation (5) allows us to express ψ_{0} as \({\psi }_{0}=\frac{\nu }{{\kappa }_{1}}{I}_{1}=\frac{\nu }{{\kappa }_{2}^{* }}{I}_{2}\). This relation indicates that the field ψ_{0} (and therefore the nonlinear losses Ω(∣ψ_{0}∣^{2})) is controlled by the couplings and the incident amplitudes I_{1,2} of the incoming waves while it remains unaffected by ω. A rearrangement of the above relation into an expression for the relative amplitudes leads us to the conclusion that a potential CPA occurs only if the condition \({I}_{2}=\frac{{\kappa }_{2}^{* }}{{\kappa }_{1}}{I}_{1}\) is satisfied.
For symmetric couplings with ∣κ_{1}∣ = ∣κ_{2}∣ = κ_{0} we have I_{2} = I_{1}e^{iϕ} where the relative phase is ϕ = (ϕ_{1} + ϕ_{2}). When we substitute these expressions in equation (4), together with the CPA constraint R_{1} = R_{2} = 0, we arrive at a transcendental equation with respect to ω, whose (complex) roots are functions of the incident field intensity ∣I_{1}∣^{2} and can be associated with an NLCPA. In fact, only their real value subset (if any!) of these ωroots are physically admissible CPA solutions as they are the only ones that satisfy the incoming boundary conditions (i.e., propagating waves). We point out that as opposed to the linear CPA, here the I_{1} (or I_{2}) is treated as a free parameter that can enforce an NLCPA.
Absorbance
To confirm the efficiency of the NLCPA protocol, we first analyze the total absorbance A(ω; I_{1}, I_{2}):
where \(\omega \in {{{{{\mathcal{R}}}}}}\) are the real frequencies of an incident monochromatic wave with left/right amplitudes I_{1} and I_{2}, respectively. For oneside incident waveforms (e.g. I_{2} = 0) the absorbance gets the expected form A = 1 − ∣r∣^{2} − ∣t∣^{2} where ∣r∣^{2}, ∣t∣^{2} are the reflectance and transmittance respectively. In the multichannel case, the evaluation of A(ω; I_{1}, I_{2}) requires the knowledge of R_{1}, R_{2} which, for a given set ω; I_{1}, I_{2}, can be calculated via equations (4, 5). Note that the absorbance A in equation (6) acquires the maximum value A = 1 whenever R_{1} = R_{2} = 0, i.e., when we have an NLCPA condition.
In order to minimize the available parameter space, we strived to achieve symmetric coupling amplitude configurations, i.e., ∣κ_{1}∣ = ∣κ_{2}∣ = κ_{0}. The latter has been guaranteed via weak power measurements (linear scattering regime), when the reflected signals at each antenna individually was measured to be the same. We have further simplified our interrogation scheme by setting \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}=1\) dB and ϕ_{IQ1} = 0 while varying the amplitude and phase \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}},{\phi }_{{{{{{\rm{IQ2}}}}}}}\) (via the second IQmodulator) of the injected signal through the second antenna together with the total absolute power P_{VNA} controlled by the VNA. Finally, we have scanned the residual parameter space and measured for each of the varying parameters the transmissions S_{21} and S_{31} from which we extracted the total reflected power and absorbance A.
From the previous discussion, we expect that when \(\sqrt{{{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}/{{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}}}=\left{\kappa }_{1}/{\kappa }_{2}\right=1\) the system might support an NLCPA at some critical incident power P_{VNA}. Of course, a necessary condition is that the extracted NLCPA frequency ω_{NLCPA} is real. To this end, we proceed with the analysis of the absorbance measurements for two settings associated with a weak and moderate coupling constants κ_{0}. In the former case, we find a set of parameters P_{VNA} = 0 dBm, \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}=5.0\) dB, \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}}=5.7\) dB, and ϕ_{IQ2} = 92^{∘} for which A ≥ 99.99% at ω_{NC} = 6.782 GHz. The slight difference between left and right incident wave powers is attributed to the fact that \(\left{\kappa }_{1}/{\kappa }_{2}\right\approx 1\). The same extreme absorption is found for moderate coupling constants, for which A ≈ 95%. The corresponding NLCPA parameters are \({\omega }_{{{{{{\rm{NC}}}}}}}=6.747\ {{{{{\rm{GHz}}}}}},{P}_{{{{{{\rm{VNA}}}}}}}=9.8\ {{{{{\rm{dBm}}}}}},\,{{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}=5.0\ {{{{{\rm{dB}}}}}},\,{\phi }_{{{{{{\rm{IQ1}}}}}}}={0}{\deg },{{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}}=5.0\ {{{{{\rm{dB}}}}}},{\phi}_{{{{{{\rm{IQ2}}}}}}}=83.{5}{\deg }\). In both cases, a shift of the maximal absorbance as a function of the incident power is observed showing that the nonlinearity induces also a slight frequency shift. In Fig. 2a, c we report the measured absorbance as a function of frequency ω and incident power P_{VNA}. In these measurements, the relative phases of the incident waves were kept fixed, given by ϕ_{IQ2} = 92\(\deg\) and ϕ_{IQ2} = 83. 5\(\deg\) for weak and moderate couplings respectively. The supplementary note 2 provides additional experimental evidence that variations of the relative phase (supplementary Figs. 2d and 3d) and amplitudes (supplementary Figs. 2c and 3c) of the incident waves result in an abrupt deterioration of the absorbance—thus underlying the delicate interferometric process between the two left and right monochromatic waves, also typical for linear CPAs. We, however, underline an additional feature of our nonlinear CPA which is its dependence on the total incident power, (Fig. 2, supplementary Figs. 2b and 3b).
Next, we turn to the theoretical analysis of A(ω; I_{1}, I_{2}). In our modeling, κ_{1} = κ_{0} and κ_{2} = −iκ_{0} with κ_{0} = 0.23 GHz for the weak and κ_{0} = 0.4 GHz for the moderate couplings (Fig. 2b, d). We considered that I_{1} = I and I_{2} = e^{iϕ}I_{1} with relative phase ϕ = π/2. Using equations (4, 5) we extracted numerically the corresponding R_{1}, R_{2} and evaluated the absorption A(ω; I) using equation (6). We have achieved perfect absorption A = 1 at (I_{NLCPA}, ω_{NLCPA}) corresponding to (− 0.48 dBm, 6.769 GHz) and (10.23 dBm, 6.770 GHz) for weak and moderate coupling values κ_{0} respectively. These results are in quantitative agreement with the experiment. A surprising feature of our calculations, which reproduces the behavior of the measured absorbance A(ω), is the broadening of the frequency range over which large absorption values are achieved as κ_{0} approaches perfect coupling κ_{0} = ν. This domain appears to be in the vicinity of the NLCPA and tends to occur for larger I_{NLCPA} values as κ_{0} increase. For example, for κ_{0} = 0.4 GHz, one has that \(A\left(\omega \in [{\omega }_{{{{{{\rm{NLCPA}}}}}}}\pm 21\ {{{{{\rm{MHz}}}}}}]\, > \, 80 \% \right.\) occurring at incident powers around P_{VNA} ≈ 10 dBm which is the maximum power of our VNA.
Absorption broadening due to EP degeneracies of NLCPAs
To understand the origin of the broadband high absorptivity we analyze the parametric evolution of the NLCPA frequencies versus the incident power I. By imposing the CPA conditions R_{1} = R_{2} = 0 and combining equations (4, 5) we arrive to the transcendental equation for ω_{NLCPA}
where \(\omega (k)=2\nu \cos (k)\). We reiterate that physically acceptable NLCPA’s correspond to the case where the ω_{NLCPA} roots of equation (7) are real, corresponding to propagating waves (supplementary note 3).
We consider the specific example of our system where the nonlinear resonator takes the form of equation (2). The parametric evolution of the (complex) roots ω_{NLCPA} of equations (7) vs. the intensity of the incident wave I = I_{1} = I_{2} are shown in Fig. 3 (top) for a weak (κ_{0}/ν = 0.23) and moderate (κ_{0}/ν = 0.4) coupling constants. In the frequency range ω ∈ [6.5, 6.9] GHz (passband of the leads), the equation (7) has only one complex root ω which crosses the real plane at the incident intensity I_{NLCPA} resulting to maximum absorbance A = 1 (see density plots). This is a direct confirmation that the perfect absorption that we have found in our experiment is indeed associated with an NLCPA condition. From Fig. 3 (top) we see that higher (experimentally inaccessible) intensities of the incident waves suppress the absorbance as they lead to an enhanced impedance mismatch of the resonator. The redshift of the maximum absorbance is associated with the real part of the nonlinear component of Ω(∣ψ_{0}∣^{2}). Another important conclusion of our analysis is that the critical coupling regime (κ_{0} = ν), enforces an ω − I parameter domain with highabsorbances. This result has been already demonstrated in Fig. 2 but now it is more prominent since we are able to analyze incident powers above 10 dBm (power limit of our VNA).
The situation is more challenging when we consider perfect coupling, i.e., κ_{0}/ν = 1. In this case, equation (7) has two complex roots within the propagating band, (Fig. 3c). One of them (red trajectory) approaches the real plane from above, while the other one (blue trajectory) crosses the real plane from below as the intensity of the incident wave increases. At the crossing point (marked with a black cross) the two roots degenerate forming a selfinduced EP degeneracy of CPAs. Its formation demonstrates all the characteristics of an EP singularity, known from the physics of linear nonHermitian operators with the most prominent being a squareroot singularity. Since the EP occurs at the real ω − plane it constitutes a physically allowable CPA with absorbance A = 1. The latter, together with the fact that the two coalescing complex ω_{NLCPA} “trajectories” have small imaginary part (see dashed lines projected in the imaginary ω − plane), results to the appearance of a (near)perfect (i.e. A ≥ 95%) absorption for a broad frequency range covering approximately 90% of the allowed propagation band of the leads. We stress that the broadband absorption is counterintuitive and challenges the understanding of CPA as a resonant phenomenon. It turns out that the broadband highabsorptivity is quite robust, forgiving small variations of the intensity ∣I∣^{2} with respect to the critical value.
To understand better the appearance of a broadband high absorbance, we simplify further our critical coupling setup by designing the nonlinear diode characteristics in order to enforce the two complexzero trajectories to fall onto the real plane. This is shown in Fig. 3 (lower right) where the nonlinear parameters in equation (2) are taken to be β_{0} = 3.00i × 10^{−3} GHz and β_{1} = 0.410i × 10^{−3} GHz mW^{−1}. In this case, the squareroot degeneracy of the NLCPAs \({\omega }_{{{{{{\rm{NLCPA}}}}}}}{\omega }_{{{{{{\rm{EP}}}}}}} \sim \sqrt{ {I}_{{{{{{\rm{EP}}}}}}}{ }^{2} I{ }^{2}}\) occurs on the realω plane with ω_{EP} ≈ 6.7 GHz while ∣I_{EP}∣^{2} ≈ 27 dBm as in lower left of Fig. 3. Such behavior leads to abrupt frequency variations in the vicinity of the I_{EP}, which span a frequency range as large as 90% of the available spectrum. Of course, once we move away from the critical I_{EP}value (say at ∣I∣^{2} = 15 dBm), we recover the typical sharp CPA features with respect to detuning ω which are reflected in an abrupt drop in absorption for frequency detuning ω ≠ ω_{NLCPA}.
We can quantify the broadband absorption, by analyzing an effective “linear” model which describes the steadystate transport characteristics of our nonlinear setting Fig. 1 in the proximity of the EP. Specifically, the EP scattering field ψ_{0}, determines the local losses Ω_{0}(∣ψ_{0}∣^{2}). The corresponding scattering matrix is
where \({H}_{{{{{{\rm{eff}}}}}}}={{{\Omega }}}_{0}+\frac{{{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}k}}{\nu }W{W}^{{{\dagger}} },W=({\kappa }_{1},{\kappa }_{2})\) and \(\alpha =\frac{1}{{\nu }^{2}}\frac{\sqrt{1\delta {\omega }^{2}}}{1+\sqrt{1\delta {\omega }^{2}}}\) (where δω = ω − ω_{EP}). In the last equality we assumed EP conditions, i.e., κ_{1} = κ_{2} = ν. Note, that the condition equation (7) for the existence of CPA’s is equivalent to the eigensolutions of the wave operator (described by H_{eff}) with incoming boundary conditions (k → −k), i.e., \(\det [{H}_{{{{{{\rm{eff}}}}}}}(k)\omega (k)]=0\)^{23,25}. Therefore, we associate the EPCPAs with the formation of EPs in the spectrum of H_{eff}(−k). From equation (8) we have calculated the absorption matrix
indicating a quartic broadening of the absorption spectrum in the neighborhood of ω_{EP}^{46} (\(\frac{1}{2{\nu }^{2}}{W}^{{{\dagger}} }W\) has eigenvalues 1 and zero). The latter prediction is nicely reproduced by the results shown in Fig. 3.
Conclusions
We demonstrated the viability of a selfinduced coherent perfect absorber by utilizing a simple microwave setup consisting of a dielectric resonator inductively coupled to a nonlinear diode. In the weak coupling regime, we have observed sharp (resonantlike) absorption up to 99.99% at the frequency of the NLCPA. As the coupling with the interrogating antennas is approaching the critical coupling regime, the frequency range where extreme absorption (>95%) occurred, increases dramatically leading to a broadband (near)perfect absorption in the neighborhood of the NLCPA. A CMT model describes the experimental findings nicely and shows that this broadening is attributed to the formation of selfinduced exceptional point degeneracies associated with the NLCPA frequencies. It will be interesting to investigate more complex scenarios where the EP of the NLCPA frequencies is of higher order resulting (probably) in an even broader (near)perfect absorption. Our results pave the way to applications of CPA in microwave/RF and optical regimes.
Methods
The details of the experimental setup are given in the corresponding subsection. The experimental data have been acquired using a standard vector network analyzer using a full 12 term calibration. We additionally calibrated the measured transmission by measuring the scattering matrix elements \({S}_{21}^{(cal)}\) and \({S}_{31}^{(cal)}\) for the system shown in Fig. 1, where we attached either an open or a short terminator to port 2 of each circulator, thus replacing the connection to the nonlinear system. In Fig. 4 the transmission intensities \( {S}_{i1}^{(cal)}{ }^{2}\) are shown for both open and short termination circuits. The incident power P_{VNA} has been set to 10 dBm while all IQModulators are arranged to provide power reduction which is \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}={{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}}=0\) dB together with an additional phase shift ϕ_{IQ1} = ϕ_{IQ2} = 0\(\deg\). We have found that the minimal transmission in the interrogating frequency range ω ∈ [6.5, 7] GHz is T_{cal} = 0.0442 which incorporates losses from the Tjunction, microwave cables, insertion loss of the IQmodulator, and the losses associated with the doublepassing through the circulators. Another source of energy loss (which is already included in these measurements) is associated with the open and short terminators. We have used the terminators from a VNA calibration kit (Rhode & Schwarz ZVZ235). The measurements have been repeated for different IQmodulator values and the measured \({S}_{i1}^{(cal)}\) spectra showed the corresponding power reduction and phase shifts. The transmission used to evaluate the absorbance has then been normalized by \({S}_{i1}={S}_{i1}^{(exp)}/\sqrt{{T}_{{{{{{\rm{cal}}}}}}}}\). This choice of normalization underestimates the experimental absorbance, thus guaranteeing a conservative estimation of our measurements.
Data availability
The data presented in this paper are available from the corresponding author upon reasonable request.
Code availability
Codes used in this paper are available from the corresponding author upon reasonable request.
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Acknowledgements
Y.T. and T.K acknowledge useful discussions with Do Hyeok Jeon. Y.T., S.S., and T.K. acknowledge partial support from the Office of Naval Research (Grant No. N000141912480) and a grant from Simons Foundation for Collaboration in MPS No. 733698.
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T.K. and U.K. designed and supervised the study. S.S. and Y. T. contributed equally to the development of the theory and numerical simulations. S.S. modeled the experimental data. T.K. supervised the theoretical work and modeling of experimental data. U.K., F.M., and M.R. designed the experimental setup, performed the experiments, and the experimental data analysis. T.K. and U.K. wrote the manuscript with input from all authors.
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Suwunnarat, S., Tang, Y., Reisner, M. et al. Nonlinear coherent perfect absorption in the proximity of exceptional points. Commun Phys 5, 5 (2022). https://doi.org/10.1038/s42005021007822
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DOI: https://doi.org/10.1038/s42005021007822
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