## 4 Discussion and Conclusions  
It is worth noting that when the laser mode is implemented, the energy of the transmitted and reflected waves is greater than the energy of the incident waves, so this mode generates energy, or to put it another way, requires energy input. The input of energy can be achieved by means of acoustic gain media. The imaginary part of the material parameters represents the loss and gain of the material. Thus, due to the opposite sign of the imaginary part of the bulk compressibility and mass density, the material of the PIM slabs contains both gain and loss elements.  
At present, the experimental research of acoustic PIM system is still in the early stage. A major challenge lies in the design and implementation of acoustic gain media. Loudspeakers with appropriate gain circuitry [20; 31] or acoustoelectric effects using piezo  
Figure 5: (**a**) The relative material parameters \(\rho_{r}\) and \(\beta_{r}\) versus y-coordinate. (**b,d**) the acoustic field patterned with point source placed at (\(-12\)r, 0) and (\(12\)r, 0), respectively. (**c**) The acoustic field intensity amplitudes at the imaging planes. The red and black curves represent the intensity fields in the focus planes indicated by the red dashed lines in (**b,d**), respectively.  
electric semiconductors [32; 33] have been widely used to provide wave amplification in experiments, despite the complex circuitry and external energy involved. Due to the low heat capacitance and thermal inertia of carbon nanotubes (CNTs), a more flexible and tunable approach is proposed by employing thermoacoustics, where the fluctuating Joule heating is converted to sound [34; 35; 36]. So far, PIM systems have mostly been discussed theoretically [25; 26; 27; 28; 30]. Therefore, practical applications in this work may require further sophisticated design and engineering.  
In conclusion, we have derived the explicit expressions of material parameters required for the coexistence of laser and CPA modes in the acoustic PIM slab. Numerical simulations confirm that coexistence of odd laser and even CPA modes or coexistence of even laser and odd CPA modes can be achieved with different orders. Such a PIM slab can freely switch states between coherent complete absorber and acoustic laser depending on the direction of incidence. We further demonstrate that planar focusing can be achieved with gradient refractive index materials. These findings may find applications in acoustic amplification, absorption and imaging. In addition, the parameters in the appropriate range can be found by adjusting the mode order, which facilitates experimental verification. So far, the required acoustically pure imaginary metamaterial (PIM) has not been experimentally demonstrated, so the results presented in the manuscript are theoretical speculations on its existence.  
Conceptualization, L.-L.Z. and X.-J.L.; funding acquisition, L.-L.Z., Y.J., P.C., Q.D. and X.-J.L.; investigation, L.-L.Z., Y.J., P.C., Q.D. and X.-J.L.; project administration, L.-L.Z. and X.-J.L.; writing--original draft, L.-L.Z.; writing--review and editing, L.-L.Z. and X.-J.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Natural Science Foundation of Jiangsu Province (under grant number BK20190111), the Fundamental Research Funds for the Jinling Institute of Technology (under grant number jit-b-201644), and the Natural Science Research Project of Higher Education Institutions of Jiangsu Province (under grant number 20 KJB510013).  
Not applicable.  
The authors declare no conflict of interest.