Perfect absorption of the energy carried by a classical wave is an interdisciplinary research theme spanning areas as diverse as acoustics and mechanical waves1,2, to radio-frequency3, microwaves4,5 and optical wave settings6,7,8,9,10. At the core of this activity is the promise that perfect absorption can be beneficial to a variety of applications ranging from stealth technologies3,11,12, energy harvesting and photovoltaics13, sensing14,15, and photodetection16,17. 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 attention18,19,20. 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 cavity10,21,22. Although the scheme has been initially proposed in the framework of classical optics21,22, as the time-reversed process of a laser, it turns out that its implementation does not require time-reversal symmetry23,24,25. 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 optics8,18,20,22,26,27,28,29,30, spanning all areas of classical wave physics ranging from microwave25,31,32 and RF33, to acoustics34,35. 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 researchers36,37,38 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 non-linear CPA (NLCPA) protocols are two-fold: From the theoretical side, one needs to develop computational schemes since the well-established (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 broad-band 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 see39,40,41,42). 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 response43,44. 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 \(\overrightarrow{B}\) 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 non-linear 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 PVNA, 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 S21 and to port 3 S31 of the VNA give access to the reflected power of the system R = S212 + S312. The total absorbance is then given by A = 1 − R = 1 − S212 − S312 where the scattering matrix elements S212 and S312 have been normalized taking into account the absorbance due to the IQ-modulator, the cables, and the circulators (see “Methods” section).

Fig. 1: The experimental setup.
figure 1

a Photograph of the experimental system including the connections. The scale is given on the lower left. b Zoom showing the experimental non-linear system consisting of a cylindrical dielectric resonator (permittivity ϵr ≈ 36, radius = 4 mm, height = 5 mm) where a short-circuited diode is situated above separated by a 1 mm high Teflon spacer (permittivity ϵr = 2.1 with the two transmitting/receiving antennas, where the distance between the central parts of the antennas is 15 mm. The resonator is sandwiched between two aluminum plates with a distance 16 mm. The scale given corresponds to the diameter of 8 mm of the resonator. c Sketch detailing the setup. IQ-modulator stands for In-phase and Quadrature vector modulators, VNA for Vector Network Analyzer, PC for Personal Computer, and κ1 and κ2 for the coupling strengths of the antennas. The coaxial cables have a length of 50 cm.

The power of the incident signals PIQ1 and PIQ2, injected from each of the two antennas are controlled by the power injected from the VNA PVNA (maximal accessible PVNA = 10 dBm), and by a differential absorption \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}\) and \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}}\) associated to each of the two IQ-modulators adjoint to the two antennas. The same IQ-modulators can also tune the relative phases ϕIQ1 and ϕIQ2 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)

$$\omega {\psi }_{n}={H}_{n,n-1}{\psi }_{n-1}+{H}_{n,n+1}{\psi }_{n+1}+{H}_{n,n}{\psi }_{n},$$

where ω is the frequency of the incident monochromatic wave, ψn is the scattering magnetic field amplitude represented in the TE-modes of the individual resonator (Wannier basis) localized at the n-th resonator45, \({H}_{n,n+1}={H}_{n+1,n}^{* }\) describes the coupling between the n-th and n + 1-th resonators and Hn,n = ϵn 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 ϵn = ϵ ≈ 6.7 GHz and model the left and right antennas (leads). They are coupled with one another via a (real) coupling constant Hn,n+1 = ν giving rise to a dispersion relation \(\omega =\epsilon +2\nu \cos (k)\) (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 \({H}_{-1,0}={\kappa }_{1}=| {\kappa }_{1}| {{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}{\phi }_{1}}\) and \({H}_{0,1}={\kappa }_{2}=| {\kappa }_{2}| {{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}{\phi }_{2}}\), respectively. The non-linear resonator at site n = 0, has a resonant frequency ϵ0 = ϵ + Ω(ψ02). The complex function Ω(ψ02) = Ωr(ψ02) + iΩi(ψ02) depends on the local field intensity ψ02. Measurements of one-sided transmission measurements indicated that the best fit with the predictions of the model of equation (1) are achieved when

$${{\Omega }}(| {\psi }_{0}{| }^{2})\approx {\beta }_{0}+{\beta }_{1}| {\psi }_{0}{| }^{2},$$

where β0 = (67.74 + 3i) MHz, β1 = (−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

$${\psi }_{n}=\left\{\begin{array}{ll}{I}_{1}{{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}kn}+{R}_{1}{{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}kn}&n < 0\\ {I}_{2}{{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}kn}+{R}_{2}{{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}kn}&n > 0\end{array}\right.,$$

where I1,2 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:

$$\left[\omega -{{\Omega }}\left({\left|{\psi }_{0}\right|}^{2}\right)\right]{\psi }_{0}={\kappa }_{1}^{* }\left({I}_{1}{{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}k}+{R}_{1}{{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}k}\right)+{\kappa }_{2}\left({I}_{2}{{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}k}+{R}_{2}{{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}k}\right).$$

An additional relation between R1,2 and I1,2 is derived by substituting equation (3) back in equation (1) for n = ±1. We have that

$${\psi }_{0}=\frac{\nu }{{\kappa }_{1}}\left({I}_{1}+{R}_{1}\right)=\frac{\nu }{{\kappa }_{2}^{* }}\left({I}_{2}+{R}_{2}\right),$$

which allows us to express equation (4) in terms of the monochromatic frequency ω and the amplitudes I1,2 and R1,2 of the counter-propagating waves in each of the two leads.

A CPA protocol inhibits all outgoing waves, i.e., R1 = 0 = R2. Imposing these constrains in equation (5) allows us to express ψ0 as \({\psi }_{0}=\frac{\nu }{{\kappa }_{1}}{I}_{1}=\frac{\nu }{{\kappa }_{2}^{* }}{I}_{2}\). This relation indicates that the field ψ0 (and therefore the non-linear losses Ω(ψ02)) is controlled by the couplings and the incident amplitudes I1,2 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 \({I}_{2}=\frac{{\kappa }_{2}^{* }}{{\kappa }_{1}}{I}_{1}\) is satisfied.

For symmetric couplings with κ1 = κ2 = κ0 we have I2 = I1eiϕ where the relative phase is ϕ = (ϕ1 + ϕ2). When we substitute these expressions in equation (4), together with the CPA constraint R1 = R2 = 0, we arrive at a transcendental equation with respect to ω, whose (complex) roots are functions of the incident field intensity I12 and can be associated with an NLCPA. In fact, only their real value subset (if any!) of these ω-roots 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 I1 (or I2) is treated as a free parameter that can enforce an NLCPA.


To confirm the efficiency of the NLCPA protocol, we first analyze the total absorbance A(ω; I1, I2):

$$A(\omega ;{I}_{1},{I}_{2})\equiv 1-{{\Theta }}(\omega ;{I}_{1},{I}_{2});\;{{\Theta }}(\omega ;{I}_{1},{I}_{2})=\frac{| {R}_{1}{| }^{2}+| {R}_{2}{| }^{2}}{| {I}_{1}{| }^{2}+| {I}_{2}{| }^{2}},$$

where \(\omega \in {{{{{\mathcal{R}}}}}}\) are the real frequencies of an incident monochromatic wave with left/right amplitudes I1 and I2, respectively. For one-side incident waveforms (e.g. I2 = 0) the absorbance gets the expected form A = 1 − r2 − t2 where r2, t2 are the reflectance and transmittance respectively. In the multichannel case, the evaluation of A(ω; I1, I2) requires the knowledge of R1, R2 which, for a given set ω; I1, I2, can be calculated via equations (4, 5). Note that the absorbance A in equation (6) acquires the maximum value A = 1 whenever R1 = R2 = 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., κ1 = κ2 = κ0. 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 ϕIQ1 = 0 while varying the amplitude and phase \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}},{\phi }_{{{{{{\rm{IQ2}}}}}}}\) (via the second IQ-modulator) of the injected signal through the second antenna together with the total absolute power PVNA controlled by the VNA. Finally, we have scanned the residual parameter space and measured for each of the varying parameters the transmissions S21 and S31 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 PVNA. Of course, a necessary condition is that the extracted NLCPA frequency ωNLCPA 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 κ0. In the former case, we find a set of parameters PVNA = 0 dBm, \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}=5.0\) dB, \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}}=5.7\) dB, and ϕIQ2 = 92 for which A ≥ 99.99% at ωNC = 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 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 PVNA. In these measurements, the relative phases of the incident waves were kept fixed, given by ϕIQ2 = 92\(\deg\) and ϕIQ2 = 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 non-linear CPA which is its dependence on the total incident power, (Fig. 2, supplementary Figs. 2b and 3b).

Fig. 2: Absorbance A as a function of frequency ω and the injected power PVNA.
figure 2

In a the experimentally measured absorption for two weakly coupled antennas with coupling strength, κ0 = 0.23 GHz is shown. b shows the corresponding result of our modeling equation (1) with non-linear complex frequency shift Ω given by equation (2). The experimental maximal absorbance is A = 99.99% occuring at PNLCPA = 0 dBm, ωNLCPA ≈ 6.782 GHz. The theoretical NLPCA’s gives A = 1, at PNLCPA = − 0.486 dBm, ωNLCPA = 6.769 GHz. The results for two moderately coupled antennas (κ0 = 0.4 GHz) for the experiment (c) and the model (d) are shown. The experimental maximum absorbance is A ≈ 95% occurring at PNLCPA = 9.8 dBm, ωNLCPA = 6.747 GHz. The theoretical non-linear perfect coherent absorber (NLPCA) with A = 1, occurs at PNLCPA = 10.23dBm, ωNLCPA = 6.77GHz.

Next, we turn to the theoretical analysis of A(ω; I1, I2). In our modeling, κ1 = κ0 and κ2 = −iκ0 with κ0 = 0.23 GHz for the weak and κ0 = 0.4 GHz for the moderate couplings (Fig. 2b, d). We considered that I1 = I and I2 = eiϕI1 with relative phase ϕ = π/2. Using equations (4, 5) we extracted numerically the corresponding R1, R2 and evaluated the absorption A(ω; I) using equation (6). We have achieved perfect absorption A = 1 at (INLCPA, ωNLCPA) corresponding to (− 0.48 dBm, 6.769 GHz) and (10.23 dBm, 6.770 GHz) for weak and moderate coupling values κ0 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 κ0 approaches perfect coupling κ0 = ν. This domain appears to be in the vicinity of the NLCPA and tends to occur for larger INLCPA values as κ0 increase. For example, for κ0 = 0.4 GHz, one has that \(A\left(\omega \in [{\omega }_{{{{{{\rm{NLCPA}}}}}}}\pm 21\ {{{{{\rm{MHz}}}}}}]\, > \, 80 \% \right.\) occurring at incident powers around PVNA ≈ 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 R1 = R2 = 0 and combining equations (4, 5) we arrive to the transcendental equation for ωNLCPA

$$f(\omega )={{\Omega }}\left({\left|\frac{\nu }{{\kappa }_{1}}{I}_{1}\right|}^{2}\right)+\frac{{\kappa }^{2}}{\nu }{{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}k}-\omega =0;\;{\kappa }^{2}\equiv {\left|{\kappa }_{1}\right|}^{2}+{\left|{\kappa }_{2}\right|}^{2}\ ,$$

where \(\omega (k)=2\nu \cos (k)\). We re-iterate that physically acceptable NLCPA’s correspond to the case where the ωNLCPA roots of equation (7) are real, corresponding to propagating waves (supplementary note 3).

We consider the specific example of our system where the non-linear resonator takes the form of equation (2). The parametric evolution of the (complex) roots ωNLCPA of equations (7) vs. the intensity of the incident wave I = I1 = I2 are shown in Fig. 3 (top) for a weak (κ0/ν = 0.23) and moderate (κ0/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 INLCPA 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(ψ02). Another important conclusion of our analysis is that the critical coupling regime (κ0 = ν), enforces an ω − I parameter domain with high-absorbances. 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).

Fig. 3: Complex zeros dependence on the input power for different coupling strength.
figure 3

Trajectories (red and blue solid lines) of the complex zeros ω of the scattering matrix as a function of input power. The dashed and dotted lines are projections of these trajectories on the corresponding plane. The non-linearity Ω = β0 + β1ψ02 with β0 = (67.74 + 3.00i) × 10−3 GHz β1 = (− 0.141 + 0.410i) × 10−3 GHz mW−1. a For two weakly coupled antennas of relative coupling strength κ/ν = 0.23, where ν is the coupling strength inside the semi-infinite leads and κ the coupling strength to the non-linear resonator. b For two intermediately coupled antennas of κ/ν = 0.40. 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 PVNA injected by the vector network analyzer (VNA) using the same color scale is in Fig. 2.

The situation is more challenging when we consider perfect coupling, i.e., κ0/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 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 ωNLCPA “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 I2 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 β0 = 3.00i × 10−3 GHz and β1 = 0.410i × 10−3 GHz mW−1. In this case, the square-root degeneracy of the NLCPAs \({\omega }_{{{{{{\rm{NLCPA}}}}}}}-{\omega }_{{{{{{\rm{EP}}}}}}} \sim \sqrt{| {I}_{{{{{{\rm{EP}}}}}}}{| }^{2}-| I{| }^{2}}\) occurs on the real-ω plane with ωEP ≈ 6.7 GHz while IEP2 ≈ 27 dBm as in lower left of Fig. 3. Such behavior leads to abrupt frequency variations in the vicinity of the IEP, which span a frequency range as large as 90% of the available spectrum. Of course, once we move away from the critical IEP-value (say at I2 = 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 ≠ ωNLCPA.

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 ψ0, determines the local losses Ω0(ψ02). The corresponding scattering matrix is

$$S(\omega )=-\hat{1}+\frac{2{{{{{\rm{i}}}}}}\sin (k)}{\nu }{W}^{{{\dagger}} }\frac{1}{{H}_{{{{{{\rm{eff}}}}}}}-\omega }W=-\hat{1}+\alpha {W}^{{{\dagger}} }W,$$

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 − ωEP). In the last equality we assumed EP conditions, i.e., κ1 = κ2 = ν. Note, that the condition equation (7) for the existence of CPA’s is equivalent to the eigensolutions of the wave operator (described by Heff) with incoming boundary conditions (k → −k), i.e., \(\det [{H}_{{{{{{\rm{eff}}}}}}}(-k)-\omega (k)]=0\)23,25. Therefore, we associate the EP-CPAs with the formation of EPs in the spectrum of Heff(−k). From equation (8) we have calculated the absorption matrix

$$A =1-{S}^{{{\dagger}} }S=\frac{2\alpha }{1+\sqrt{1-\delta {\omega }^{2}}}{W}^{{{\dagger}} }W\\ \approx \frac{1}{2{\nu }^{2}}\left(1-\frac{1}{16}\delta {\omega }^{4}+{{{{{\mathcal{O}}}}}}(\delta {\omega }^{6})\right){W}^{{{\dagger}} }W.$$

indicating a quartic broadening of the absorption spectrum in the neighborhood of ωEP46 (\(\frac{1}{2{\nu }^{2}}{W}^{{{\dagger}} }W\) has eigenvalues 1 and zero). The latter prediction is nicely reproduced by the results shown in Fig. 3.


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.


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 non-linear system. In Fig. 4 the transmission intensities \(| {S}_{i1}^{(cal)}{| }^{2}\) are shown for both open and short termination circuits. The incident power PVNA has been set to 10 dBm while all IQ-Modulators are arranged to provide power reduction which is \({{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ1}}}}}}}={{{{{{\mathcal{P}}}}}}}_{{{{{{\rm{IQ2}}}}}}}=0\) dB together with an additional phase shift ϕIQ1 = ϕIQ2 = 0\(\deg\). We have found that the minimal transmission in the interrogating frequency range ω [6.5, 7] GHz is Tcal = 0.0442 which incorporates losses from the T-junction, microwave cables, insertion loss of the IQ-modulator, and the losses associated with the double-passing 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 \({S}_{i1}^{(cal)}\) 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.

Fig. 4: Transmission calibration.
figure 4

Transmission intensities \(| {S}_{21}^{(cal)}{| }^{2}\) and \(| {S}_{31}^{(cal)}{| }^{2}\) where the non-linear system has been replaced by open and short terminations at the two antennas, respectively.