Lasing is a well-established field in optics with several applications. Yet, having lasing or huge amplification in other wave systems remains an elusive goal. Here, we utilize the concept of coherent perfect absorber-laser to realize an acoustic analog of laser with a proven amplification of more than $104$ in terms of the scattered acoustic signal at a frequency of a few kHz. The obtained acoustic laser (or the coherent perfect absorber-laser) is shown to possess extremely high sensitivity and figure of merit with regard to ultra-small variations of the pressure (density and compressibility) and suggests its evident potential to build future acoustic pressure devices such as precise sensors.

## I. INTRODUCTION

The field of acoustic metamaterials started with the seminal work on locally resonant sonic materials.^{1} The main purpose was to build new materials to control and tailor sound propagation and its interaction with matter in an unprecedented manner.^{2} Several innovative applications were put forward, ranging from negative refractive parameters,^{3–6} lensing,^{7,8} transformation acoustics and cloaking,^{9} and zero-index metamaterials.^{10} Later, the concept of acoustic metamaterials was extended to acoustic metasurfaces.^{11–14} Several disruptive applications were proposed such as Parity–Time ($PT$) symmetry and non-reciprocal propagation,^{15–17} topological acoustics,^{18–20} enhanced sensing,^{16} Fano resonators,^{21,22} sound perfect absorption,^{23,24} insulation of buildings,^{25,26} time-modulated metasurfaces, and moving media.^{27,28}

In the same vein, the re-discovery of non-Hermitian Hamiltonian operators, in particular, quantum systems exhibiting the $PT$-symmetry, has given rise to several new physical and technological applications.^{29} The extension to nanophotonics and other wave systems takes a particularly attractive form,^{30} insofar as the initial time-dependent Schrödinger equation is simply replaced by an equation of the spatial wave evolution, along the axis of propagation. In fact, operators satisfying the $PT$-symmetry were first thought of as a mean to replace Hermitian (self-adjoint) operators in the realm of quantum mechanics. The fact that the eigenvalues (spectrum) of these operators are real-valued is important from a physics perspective, as the eigenvalues correspond to measurable variables. The reality of the spectrum occurs in the exact (unbroken) phase of the non-Hermitian $PT$-symmetric operators.^{31,32}

For instance, one spectacular feature of the $PT$-symmetric system is the unidirectional wave propagation at an exceptional point (EP) that stems from the coalescence of the eigen-spectrum.^{30,33} Most of the intriguing physical phenomena such as unidirectional invisibility,^{34} electromagnetic teleportation,^{35} Bloch oscillations,^{36} negative refraction,^{37} and field localization^{38} are attributed to the singular characteristics of the EP. Quite interestingly, and as was done for some other wave phenomena, $PT$-symmetry and EPs were demonstrated for acoustic waves.^{15,39} The used structures were similar to those in optics, but the balance of gain and loss is enforced to mass density. Soon after these theoretical propositions, the concept was subsequently verified along with some intriguing applications, e.g., acoustic invisible sensors,^{16} signal manipulation,^{40} cloaking,^{15} and beam splitting.^{20}

Recently, a new kind of singularity associated with the CPA-laser (coherent perfect absorber-laser: CPAL) effect gained substantial attention in the field of $PT$-symmetric optics.^{41} CPA-lasers exhibit eigenvalues to be either zero or infinity, which allows for coherent perfect absorption and lasing, altogether. This peculiar behavior was shown to possess unprecedented features, especially in the realm of electronics where loss and gain were realized using positive and negative resistors, respectively.^{40} In particular, the CPAL was employed as a method to enhance the sensitivity of radio-frequency sensors by operating a $PT$-symmetric system at its CPAL point.^{40,42} The duality of CPA and lasing (at the same frequency) in such non-Hermitian systems has attracted much attention for sensing applications owing to their potential to amplify the system response to a small perturbation; yet, the means of measuring the system’s spectral response and the associated effects of noise remain challenging.^{42,43} By translating the CPAL concept into acoustic systems, we can observe similar singularity in $PT$-symmetric acoustics, i.e., CPA-Saser. Saser^{44} is analogous to laser in the acoustic domain, which may exhibit perfect absorption and amplification at the CPAL point for acoustic waves. Hence, by exploiting transitions between acoustic absorption and acoustic lasing at a CPAL point, we demonstrate that small perturbations can be translated into large changes in the output intensity (which may be detected easily), provided that the phase between two coherent input waves can be precisely controlled.

In this work, we propose a CPAL-based acoustic sensing platform, composed of three layers, where the passive layer is sandwiched between gain and loss layers. The arrangement of acoustic materials with complex parameters (gain/loss) determines the existence of the CPAL point in the proposed sensor to induce ultrahigh sensitivity for the incident acoustic waves. The rest of this paper is organized as follows: Sec. II provides the mathematical formulation and background of the acoustic CPAL effect through the $PT$-symmetry using a multi-layered geometry. Section III gives the main results of this work and details the origin of acoustic CPAL and its intriguing scattering properties, while Sec. IV gives details on the sensing mechanism and its robustness to detect a small background pressure variation as well as the possibility of non-destructive characterization. Finally, Sec. V gives some concluding remarks.

## II. MATERIALS AND METHODS

The equations governing the propagation of acoustic (pressure) waves in the limit of small perturbation in incompressible and inviscid fluids can be derived from the momentum and mass conservation equations of fluids at rest;^{45} i.e.,

where $\rho $ denotes the density and $\beta $ is the compressibility of the background medium (here, $\beta =1/\kappa $, with $\kappa $ being the bulk modulus of the medium). These equations are complemented by the continuity of the normal component of the velocity field $n\u22c5u$, with $n$ being the normal unit vector to the considered boundary as well as the pressure field $p$.

The system of Eq. (1) leads to the wave equation (assuming piece-wise homogeneous physical properties),

where $1/c2=\rho \beta $ is the speed of sound in the given medium. The structures we consider in this study are multi-layered (with one dimension infinitely extended) as illustrated in Fig. 1, as example. Hence, a convenient transfer matrix formalism is developed for this purpose to compute the transmitted and reflected signals (arrows of Fig. 1) and is compared to the results from the finite element method (FEM, from COMSOL Multiphysics^{46}). Moreover, Fig. 1 shows layers with alternating gain and loss (positive/negative imaginary parts in the mass density, assuming a time-harmonic dependence of $e\u2212i\omega t$) as this is needed to achieve the so-called $PT$-symmetry and the subsequent exceptional point (EP) when this symmetry is broken at specific points in the spectral regime. This effect was already proposed both theoretically and experimentally in the realm of pressure waves.^{15,16} However, in order to set up our problem, we wish to re-emphasize some key points and notations. First, the scattering matrix relating the outgoing (scattered) signals to the incoming (incident) ones can be expressed as

with $t$ being the transmission coefficient (identical in both incidences due to reciprocity) and $rR$ and $rL$ being the reflection coefficients from right and left incidences, respectively. All these coefficients are complex valued, and they are related to the symmetry of the $S$-matrix; i.e., $S\u22121(\omega )=S\u2217(\omega )$ due to the $PT$-symmetry (a generalized form of energy conservation) with $\u2217$ denoting the conjugate transpose of a matrix (or the complex-conjugate of a complex number),

## III. RESULTS

### A. $PT$-symmetry and the CPAL effect

#### 1. EP and $PT$-symmetry

As shown in Ref. 15, when $T>1$, $rR$ and $rL$ have a phase difference of $\pi $, while they are in phase for $T<1$. It was further shown that the $PT$-symmetric (or exact) phase corresponds to $T<1$, while the broken phase is obtained for $T>1$. The transition between these two phases occurs when $T=1$, at the so-called EP, i.e., where the coalescence of the two eigenvalues occurs. The correspondence between the eigenvalues and transmittance $T$ is further demonstrated in Fig. S2 of the supplementary material. In order to quantitatively characterize these $PT$-symmetry effects, we need to compute the eigenvalues of the $S$-matrix given in Eq. (3), which we should denote by $s\xb1$ (so as to have $s+\u2265s\u2212$), which are given by

Thus, when $T<1$, $s\xb1$ are complex-conjugate, i.e., are unit-modular ($|s\xb1|=1$). For $T>1$ (the broken phase), $s\xb1$ have different modulus, namely, $|s\xb1|=T(1\xb11\u22121/T)$. The amplitude of these eigenvalues is given in Fig. 2(a) as well as their phases in Fig. 2(b). The regions where $s\xb1$ are split stand for the broken phase. The low frequency region spanning frequencies $f0<3$ kHz describes regular EP and $PT$-symmetry breaking. This behavior is confirmed in Figs. 3(a) and 3(b) where the T/R spectra are plotted and where a zero left reflection can be observed at 2.859 kHz.

#### 2. CPAL effect

Yet, we know that using gain/loss media, above a certain threshold may result in another related effect, namely, coherent perfect absorber-laser (CPAL) reminiscent of a singular point in the spectrum. In fact, as can be seen in Fig. 2, at a frequency of $f=4.383$ kHz (approximately 7.8 cm or so in the range of the size of the device. It can be shown that the CPAL at a higher frequency/smaller wavelength is possible; i.e., the wavelength is approximately one fourth of the total width of the device; see the supplementary material), one of the eigenvalues $s+$ diverges (plotted in the logarithmic scale) as shown by the black line, while the second eigenvalue $s\u2212$ vanishes as shown by the dashed red line. $s+$ thus corresponds to lasing, while $s\u2212$ corresponds to coherent perfect absorption (CPA). It is to be emphasized that the CPAL event is observed in the broken phase, i.e., corresponding to the non-unimodular eigenvalues.

To understand this effect, we consider analyzing the terms of the transfer matrix and the incoming ($pLin$, $pRin$) and outgoing pressure waves ($pLout$, $pRout$) (both amplitudes and phases). We can thus easily derive the conditions of resonant scattering (amplification or lasing) and of CPA. For instance, for lasing to occur, we have to impose zero incoming signals, i.e., $pLin=pRin=0$, and finite outgoing ones, i.e., $pLout\u2248pRout\u22481$. By inspection of the coefficients of the transfer matrix, i.e., $(pLin,pLout)T=M(pRout,pRin)$, we obtain $m11pRout=0$ and $m21pRout=pLout$. Since we want finite scattering, this just means $m11=0$ and $m21=pLout/pRout$ or, in other words, a condition on the $M$-matrix through its components $m11$, which depend on the geometrical and physical parameters of the structure and on frequency. The second condition means that the outgoing waves are related to $m21$, which is not a constraint as the outgoing signals cannot be directly controlled (tuned) and rather just obey this condition at specific frequencies (both amplitude and phase). Let us consider CPA operation occurring at the same frequency of lasing. We have to enforce finite incident signals but zero outgoing ones, as all the energy is absorbed by the device when perfect absorption occurs. Specifically, this leads to $m22=0$ and $m12=pLin/pRin$. The first condition depends on the device itself, while the second is of crucial difference, as it means that the incoming pressure waves (from both sides) should be tailored in both amplitude and phase, i.e., being coherent, as $m12$ is a complex number; i.e., $\varphi (pLin)=\varphi (pRin)+\varphi (m12)$ and $|pLin|=|m12\Vert pRin|$. This shows that CPA is more stringent than lasing that only requires $m11=0$. Moreover, these two conditions should occur at the same frequency, which means that $m11$ $m12$ should vanish simultaneously for CPAL to take place. Hence, CPAL is a complex and rather a more constrained effect than EP and $PT$-symmetry breaking.

From another perspective, the $S$-matrix terms are inversely proportional to $m22$ (namely, $s11=\u2212m21/m22$, $s22=m12/m22$, $s12=1/m22$, and $s21=m11\u2212m12m21/m22$), suggesting that CPAL is equivalent to a pole (and a zero) in the $S$-matrix and reflecting onto the eigenvalues $s\xb1$. These eigenvalues are plotted in Figs. 2(a) and 2(b) for the amplitude and phase, respectively. As per the structure we consider (shown in Fig. 1), CPAL occurs with the same physical properties and geometry as before (i.e., the regular EP and $PT$-symmetry breaking occurring around 0.6 kHz) but for a higher frequency, i.e., around 4 kHz. As can be seen in Fig. 2(a), $s\xb1$ undergoes a resonance, reaching extremely high amplitude ($\u2248102$) by jumping from 1, i.e., the unit-modular or exact $PT$-symmetric phase. Thus, during this broken $PT$ phase, CPAL takes place. This fact is further demonstrated by the phase behavior at this point; i.e., $\varphi (s+)$ jumps from $\u2212\pi /2$ to $\pi /2$ at the CPAL point. The CPA is described in Fig. 2(a) by the red curve that reaches a value near-zero at exactly the same frequency 4.383 kHz and its phase undergoes a similar jump. As we are interested in the practical applications of this feature, we define the output coefficient of the acoustic device as the ratio between the outgoing energy divided by the ingoing one; i.e.,

If $\Psi =0$, we have CPA, and if $\Psi \u2192\u221e$ (or $\Psi \u226b1$), we have lasing. This coefficient is plotted in Fig. 3(a) along with transmission and reflection spectra. At 4.383 kHz, $\Psi $ diverges as expected for CPAL operation ($T\u226b1$ and $R\u226b1$, as demonstrated by both FEM simulations and TMM calculations). The phases of both $t$ and $rL$ undergo jumps at the EP. The near-field pressure field is plotted in Fig. 3(c) at the CPAL frequency, and its amplitude $|p|2$ reaches extremely high values of $2.2\xd7104$ Pa for an incident wave of only 1 Pa, demonstrating the amplification (lasing) abilities of this $PT$-based device. Here, it should be emphasized that $R\u226bT$ ($\u226b1$). However, if we interchange the position of gain and loss layers (or equivalently switch from left incidence to right incidence), we will have $T>R$. However, due to reciprocity, $T$ must remain the same, and, hence, the scenario shown in Fig. 3 is the best-suited for lasing as it provides the highest $\Psi $ [see the inset of Fig. 3(a)].

### B. Acoustic sensing mechanism

Figures 2 and 3 clearly show that the resonance associated with the CPAL point from intertwined gain and loss possesses a high $Q$-factor ($\u2248103$) that is important for sensing capabilities purposes. In order to use our device as an acoustic sensor, we first investigate its output amplitude behavior under small perturbations on density or compressibility of the surrounding or of the middle passive layer (see the supplementary material for a discussion on the role of the thickness of the passive layer) (if the sensor is operated for non-destructive characterization). Figures 4(a) and 4(b) show the frequency responses of $\Psi $ for the proposed sensor as functions of changes in $\u211c(\rho 0)$ and $\u211c(\beta 0\u22121)$, respectively. For the former case, $\u211c(\rho 0)$ is varied between 1.29 and 1.889 $kg/m3$, and in the latter, $\u211c(\beta 0\u22121)$ is varied between 0.1518 and 0.2222 MPa, as we first consider an airborne situation (see the remainder of this section for the discussion on the use of other media). Due to the extremely high $Q$-factor of the CPAL resonance and its extreme sensitivity, we can see that $\Psi $ is greatly shifted for small changes in the surrounding acoustical properties. In Fig. 4(a), we can observe a redshift, while in Fig. 4(b), a blueshift is observed with respect to increasing $\beta 0\u22121$ (i.e., $\kappa 0$) (yet, if we choose $\beta 0$, this corresponds actually to a redshift, too, as what matters really here is the acoustic refractive index in analogy to refractive index sensing,^{47} if we define $n=\rho r\beta r$, and the subscript $r$ denotes relative quantity with respect to the surrounding). These figures demonstrate the potential of CPAL acoustic sensors. It is instructive to check the effect of a change in the imaginary parts of $\rho 0$ and $\beta 0$, keeping their real parts constant, as in some scenarios, the presence of the analyte causes some absorption (change in imaginary parts). Figures 4(c) and 4(d) are the same as in Figs. 4(a) and 4(b) but for variation of the imaginary parts. The main difference here is that there is no shift of the CPAL frequency, but the amplitude of the resonance and its $Q$-factor is dramatically reduced. Therefore, sensing by scanning frequency shifts is not possible in this case.

Next, we will quantitatively evaluate the performance of the CPAL-based acoustic sensor in the active mode [$\u211c(\rho 0)$]. The reactive mode with analogy to electromagnetism would correspond to [$\u2111(\rho 0)$]. To this end, we use the classical definition of sensitivity $S$ and figure of merit (FoM); i.e.,

and

where $\Delta X$ denotes the (positive) variation of the given parameter $X$, FWHM denotes the full width at half maximum (of the corresponding resonance or peak), and MDU is the abbreviation for mass density unit (in analogy with RIU or refractive index unit, in optics). $S$ is thus the ratio of the unit change if the frequency (or more precisely the peak-shift in the CPAL resonance) for a unit change in $\u211c(\rho 0)$ of the surrounding medium. The FoM is another quantity that further and comprehensively evaluates the performance of the CPAL sensor. To have high FoM, not only $S$ should be maximized but also the FWHM should be minimized. Generally, these two extremums do not occur at the same frequency. Therefore, a sort of trade-off should be sought for.

Figure 5(a) plots the main sensing parameters $S$ and FoM vs the gain/loss parameters [$\u2111(\rho )/\rho 0$, normalized with $\rho 0$, the density of the surrounding medium, here, air] from 0.05 to 0.75. The sensitivity $S$ reaches extremely high values in the order of $104$ Hz/MDU, with MDU referring to the mass density unit, i.e., 1 $kg/m3$. Hence, the smaller $\u2111(\rho )$, the higher the resonance frequency $f0$ and thus $S$. Therefore, if the interest is on CPAL sensing, as in our case, we should maximize both $S$ and FoM. For this, we could define the function $\xi (\alpha )=(S(\alpha )/Smax\u22121)2+(FOM(\alpha )/FOMmax\u22121)2$, with $\alpha =\u2111(\rho )/\rho 0$, and we seek the minimum of this function. (Please note that here, we give the same weight, in the optimization process, for $S$ and $FOM$.) This occurs, in our scenario, for $\u2111(\rho )/\rho 0=0.1$. To get an idea of the sensor capability at this specific operation point, we could observe a 2.5 Hz frequency-shift if the mass density changes by only 100 $mg/m3$ or 0.008$%$ of the surrounding’s (air) density. If the focus is on lasing (amplification), then a higher gain/loss is needed to increase $\Psi $, but $S$ is one order of magnitude lower, as can be seen in Fig. 5. The regime of lasing is shown in Fig. 5(b) for varying acoustic frequencies and loss/gain parameter (in the logarithmic scale).

Another point we wish to emphasize is that all the results in this work were done assuming airborne structures. However, we verified that all our conclusions remain unchanged in case another medium other than air (so denser and stiffer) was chosen. In the case of water (density $\rho 0=997$ $kg/m3$ and bulk modulus $1/\beta 0=2.2$ GPa, i.e., a speed of sound of 1480 m/s, therefore 4.3 times faster than that in air), the only change we observe is a blueshift of the CPAL and EP (by a factor $\u22484.3$) to 18.95 kHz (the wavelength of the CPAL effect is itself constant and equal to 7.81 cm for both media). This effect is thus robust and equally sensitive underwater as in air, provided the loss and gain are enforced in a balanced manner.

It can also be emphasized that it is further possible to make an acoustic active Fabry–Pérot sensor with only active layers. Yet, different from the passive Fabry–Pérot CPA, it achieves only a lasing behavior. The exotic acoustic CPA-laser property proposed in this work is not found in both active and passive acoustic Fabry–Pérot interferometers. Moreover, the error bars are expected to be large for an active Fabry–Pérot sensor, as the external circuit used to enforce the gain (in density) could be highly unstable and nonlinear at a large output voltage. The gain saturation of the amplifiers also forbids its practical implementation. This demonstrates the superiority of the concept of acoustic CPAL, with promising sensing and amplification evident applications. For these applications to be realistically implemented, it is required to investigate the robustness of the device with respect to unavoidable noise in the physical parameters.

## IV. DISCUSSION

### 1. Noise effect

Distinguishing signal changes caused by small perturbations from inevitable background noises is of paramount importance for acoustic sensor applications. We should point out that phase and flicker noises, which were reported to influence the performance and resolvability of EP-based sensors, may be insignificant in our proposed acoustic sensing platform.^{48–50} This is because a CPAL-based sensor monitors at a fixed frequency the scattered power of a monochromatic signal instead of tracking resonance frequency shifts due to an eigenvalue bifurcation nearby the second- or higher-order EPs. Consequently, phase and flicker noises, modal interference, and limited spectral density issues, which are regarded as relevant issues in EP-based sensors, can be addressed using our proposed new sensing paradigm.

In this paper, we mainly focus on demonstrating the ultrahigh sensitivity and large modulation depth of CPAL-based acoustic (pressure) sensors. Thus, the measurement of the level of noise, in particular, the noise due to the exciting sources, as well as the spectral resolution that includes the tunable exciting spectrometer and laser requires an experimental realization. With no loss of generality, only the sensitivity (via the output coefficient $\Psi $) was calculated to estimate the sensor’s performance.

The effects of the background noise (dominantly contributed by the thermal noises) on the sensing capabilities are displayed in Fig. 6, where two levels of white noise are enforced. We found that these levels of noise may only slightly deteriorate the CPAL-locked $PT$-symmetric acoustic sensor performance, as evidenced by the persistence of the CPAL peak, that is used to calculate the sensitivity.

### 2. Non-destructive sensing

Next, an important use of our sensor would be to non-destructively characterize the density and/or compressibility of some region (as in pipes). In our situation, this would be to excite the structure of Fig. 1 and to detect changes in $\Psi $ due to slight changes of $\rho $ or $\beta $ of the middle (passive) layer. Figure 7(a) gives this validation, similar to Fig. 4(a), and it can be clearly seen that here also, $\psi $ undergoes a redshift of the CPAL point for a slight increase in the density of the analyte, demonstrating its high sensitivity for acoustic non-destructive characterization, with some promising applications such as in the oil and gas industry. For instance, the inset of Fig. 7(a) plots the computed sensitivity calculated using Eq. (7) as before. It shows a reduced sensitivity compared to the previous results of Fig. 5; yet, these levels of *S* are acceptable for our purpose of non-destructive characterization. It can also be noted that for small gain/loss parameter $\u2111(\rho )/\rho 0$, there is no sensitivity of the device. This is due to the fact that for small values of $\u2111(\rho )/\rho 0$, there is no CPAL effect (no lasing or CPA). For higher values of this parameter, the device can be used for detecting the content of a pipe, for example, in a non-invasive manner. We also perform simulations demonstrating the CPA operation as can be seen in Fig. 7(b). These results show the difference between the CPAL (laser) and CPA (absorber) operations. It should also be emphasized that for sensing, CPAL is more convenient as with CPA all the energy is absorbed; therefore, sensing is not straightforward in this scenario.

## V. CONCLUDING REMARKS

A CPAL-based acoustic sensor using the paradigms of $PT$-symmetry breaking has been numerically investigated in this contribution. Compared with classical acoustic sensors, the CPAL sensor with only three layers (gain, loss, and passive) exhibited a much higher sensitivity in the order of $104$ Hz per mass density unit. The robustness of the sensing mechanism with respect to noise is analyzed, and its use to non-destructively sense the density is demonstrated. To our best knowledge, this is the first time that CPAL acoustic sensors were reported. The obtained results show the potential of this device to improve acoustic sensors by balancing gain and loss. Moreover, the advantage of the CPAL-based laser on classical acoustic lasers or SASERs lies mainly in the fact that a Fabry–Pérot sensor with double active/passive acoustic layers will only achieve lasing/absorption behavior (see Fig. S4 in the supplementary material). The dual and exotic operation (CPA and CPAL) cannot thus be found in regular acoustic or optical lasers.

## SUPPLEMENTARY MATERIAL

See the supplementary material for complete details on the effect of the dimension of the device and the thickness of the passive layer. We also explain multiple exceptional points (EPs) by a Fabry–Pérot like interference. We further give the CPA operation of the device and discuss its advantages over a classical laser and/or absorber (e.g., SASER).

## ACKNOWLEDGMENTS

The research reported in this paper was supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Grant No. OSR-2016-CRG5-2950 and KAUST Baseline Research Fund No. BAS/1/1626-01-01.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Acoustics*(Multidisciplinary Digital Publishing Institute, 2019), Vol. 1, pp. 590–607.