Non-linear coherent perfect absorption in the proximity of exceptional points

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 non-linearities and coherent perfect absorption is crucial. Here we show experimentally and theoretically the formation of non-linear 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 self-induced perfect absorption once its intensity exceeds a critical value. Note, that a (near) perfect absorption persists for a broad-band 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. Coherent perfect absorption allows control of light, with wide ranging applications. Here, this concept is experimentally implemented in the presence of non-linearities, where the formation of a self-induced exceptional point coherent perfect absorption mechanism is identified.

P erfect 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 low-cost/power all-optical 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 so-called 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 time-reversed process of a laser, it turns out that its implementation does not require time-reversal 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 above-mentioned cases, the CPA protocol demonstrated a narrow, resonant-based, (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 non-linear 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 two-fold: 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 non-linear 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 bi-stabilities and other non-linearity-driven 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 non-linear 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 self-induced CPA with a variety of photonic applications in areas like signal processing, non-linear 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 square-root frequency degeneracy in the neighborhood of a critical magnitude of the incident wave amplitudes-remnant of an exceptional point (EP) degeneracy occurring in linear non-hermitian systems. EP degeneracies have been documented in the non-Hermitian literature as a main source of wave phenomena ranging from loss-induced transparency and unidirectional invisibility, to parity-time-symmetric lasers, and hypersensitive sensors (for some recent reviews see [39][40][41][42] ). Here, we provide a paradigm of EP degeneracies associated with steady-state non-linear solutions of the wave operator, with incoming boundary conditions, that are responsible for a broad-band (near-) perfect absorption.

Results and discussion
Experimental setup. The experimental setup (Fig. 1a) consists of a dielectric resonator coupled to a short-circuited diode (Fig. 1b) sandwiched between two aluminum plates. This hybrid system demonstrates a non-linear response 43,44 . The resonator is excited via two kink antennas curved around it. The antennas excite the first TE-resonance mode of the resonator (around 6.7 GHz), where the magnetic field B ! has only a z-component and the electric field lies parallel to the aluminum plates. The magnetic field couples to the short-circuited 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 T-junction going through the In-phase and Quadrature (IQ) vector modulators into the circulators. The IQ-modulators 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 IQ-modulator, 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 P IQ1 and P IQ2 associated to each of the two IQ-modulators adjoint to the two antennas. The same IQ-modulators 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 time-independent 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 n-th resonator 45 , H n;nþ1 ¼ H Ã nþ1;n describes the coupling between the n-th and n + 1-th 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 semi-infinite 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 ω ¼ ϵ þ 2ν cosðkÞ (k ∈ [0, 2π] is the wave-vector of a propagating wave). The coupling strength between the non-linear resonator (n = 0) with the left (n < 0) and to the right (n > 0) leads are H À1;0 ¼ κ 1 ¼ jκ 1 je Àiϕ 1 and H 0;1 ¼ κ 2 ¼ jκ 2 je Àiϕ 2 , respectively. The non-linear 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 one-sided transmission measurements indicated that the best fit with the predictions of the model of equation (1) are achieved when 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 counter-propagating 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 ψ 0 ¼ ν This relation indicates that the field ψ 0 (and therefore the non-linear 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 re-arrangement 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 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 ): Aðω; I 1 ; I 2 Þ 1 À Θðω; I 1 ; I 2 Þ; Θðω; I 1 ; where ω 2 R are the real frequencies of an incident monochromatic wave with left/right amplitudes I 1 and I 2 , respectively. For one-side 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 P IQ1 ¼ 1 dB and ϕ IQ1 = 0 while varying the amplitude and phase P IQ2 ; ϕ IQ2 (via the second IQ-modulator) 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 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P IQ1 =P IQ2 q ¼ κ 1 =κ 2 ¼ 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, P IQ1 ¼ 5:0 dB, P IQ2 ¼ 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 κ 1 =κ 2 % 1. The same extreme absorption is found for moderate coupling constants, for which A ≈ 95%. The corresponding NLCPA parameters are ω NC ¼ 6:747 GHz; P VNA ¼ 9:8 dBm; P IQ1 ¼ 5:0 dB; ϕ IQ1 ¼ 0 ; P IQ2 ¼ 5:0 dB; ϕ IQ2 ¼ 83:5 . In both cases, a shift of the maximal absorbance as a function of the incident power is observed showing that the non-linearity 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 and ϕ IQ2 = 83. 5 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 non-linear 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 ω 2 ½ω NLCPA ± 21 MHz > 80% À 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 broad-band 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 ωðkÞ ¼ 2ν cosðkÞ. We re-iterate 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 non-linear 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 self-induced EP degeneracy of CPAs. Its formation demonstrates all the characteristics of an EP singularity, known from the physics of linear non-Hermitian operators with the most prominent being a square-root singularity. Since the EP occurs c For two perfectly coupled antennas κ/ν = 1. d As bottom left but the real part of the non-linearity is set zero, i.e., no real frequency shift. The absorbance A is reported as a density plot versus the frequency ω and the power P VNA injected by the vector network analyzer (VNA) using the same color scale is in Fig. 2. 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 broad-band absorption is counter-intuitive and challenges the understanding of CPA as a resonant phenomenon. It turns out that the broadband high-absorptivity is quite robust, forgiving small variations of the intensity |I| 2 with respect to the critical value.
To understand better the appearance of a broad-band high absorbance, we simplify further our critical coupling set-up by designing the non-linear diode characteristics in order to enforce the two complex-zero trajectories to fall onto the real plane. This is shown in Fig. 3 (lower right) where the non-linear 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 square-root degeneracy of the NLCPAs ω NLCPA À ω EP $ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jI EP j 2 À jIj 2 p occurs on the real-ω 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 broad-band absorption, by analyzing an effective "linear" model which describes the steady-state transport characteristics of our non-linear 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 eff ¼ Ω 0 þ e ik ν WW y ; W ¼ ðκ 1 ; κ 2 Þ and α ¼ 1 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., det½H eff ðÀkÞ À ωðkÞ ¼ 0 23,25 . Therefore, we associate the EP-CPAs 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 ( 1 2ν 2 W y 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 self-induced coherent perfect absorber by utilizing a simple microwave setup consisting of a dielectric resonator inductively coupled to a non-linear diode. In the weak coupling regime, we have observed sharp (resonant-like) 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 broad-band (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 self-induced 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 ðcalÞ 21 and S ðcalÞ 31 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 non-linear system. In Fig. 4 the transmission intensities jS ðcalÞ i1 j 2 are shown for both open and short termination circuits. The incident power P VNA has been set to 10 dBm while all IQ-Modulators are arranged to provide power reduction which is P IQ1 ¼ P IQ2 ¼ 0 dB together with an additional phase shift ϕ IQ1 = ϕ IQ2 = 0 . 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 T-junction, microwave cables, insertion loss of the IQ-modulator, 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 ZV-Z235). The measurements have been repeated for different IQ-modulator values and the measured S ðcalÞ i1 spectra showed the corresponding power reduction and phase shifts. The transmission used to evaluate the absorbance has then been normalized by S i1 ¼ S ðexpÞ i1 = ffiffiffiffiffiffiffi ffi T cal p . 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.