## 1 Introduction  
A beam of light is refracted at the interface of any two homogeneous materials, and the refraction angle follows the [PERSON]'s law. In natural materials, both the refracted and incident light lie on either side of the normal. Metamaterials can be engineered to exhibit negative refraction, i.e., the refracted light is located on the same side of the normal as the incident light [1]. Negative refraction has attracted a lot of attention because it has exciting applications in various fields [2; 3; 4; 5; 6], such as antenna technology, electromagnetic absorbers, phase compensation, subwavelength lithography, and planar focusing lenses. The practical implementation of Negative refraction can be achieved using a slab possessing both negative permittivity and negative permeability [7; 8; 9]. However, negative refraction based on bulk double negative (DNG) metamaterials suffer from inherent challenges such as losses, limited granularity and non-ideal anisotropy, which hinder their applicability. Therefore, the idea of negative refraction has inspired a large number of studies aimed at introducing alternative approaches, such as using phase-conjugating interfaces [10; 11], constructing phononic crystals [12; 13; 14; 15], employing a pair of parity-time (PT) symmetric metasurfaces [5; 16; 17], etc. In particular, the coexistence of CPA and laser modes in PT symmetric systems can be achieved by adjusting the material parameters and structural configuration. In this active scheme, the power flow and phase velocity are directed from the laser side to the CPA side, and negative refraction can be obtained without the need for a bulk negative refractive index material [5; 16]. With the study of PT-symmetric systems, it has been found that the imaginary part of the material parameters representing the loss and gain of materials plays an important role in the modulation of the wave field [5; 16; 18; 19; 20; 21; 22].  
In recent years, purely imaginary metamaterials (PIMs) were originally proposed in the field of optics [23; 24; 25; 26; 27], and since both light and sound are classical waves, similar materials were introduced into the field of acoustics [28; 29; 30]. Unlike the spatially distributed loss and gain in PT-symmetric systems, PIM has balanced loss and gain uniformly distributed in space, but in different material parameters. It is worth noting that CPA mode and laser mode can also be implemented in acoustic PIM structures [28; 29; 30]. Yet, they usually occur in different cases. That is, for a given PIM slab, the CPA mode and the laser mode have different wave vectors and cannot be excited at the same angle of incidence. To solve this problem, the acoustic CPA mode is usually obtained from a PIM slab, while the acoustic laser mode is implemented on another PIM slab with conjugate material parameters [28]. However, the negative refraction achieved using this structure has a requirement for the direction of incidence, i.e., the beam can only be incident from the CPA side.  
In this work, analytical solutions have been derived for the coexistence of acoustic CPA mode and acoustic laser mode in a single PIM slab. Such PIM slabs can freely switch states between the acoustic coherent perfect absorption and the acoustic laser depending on the direction of incidence. Further, by using a pair of identical PIM slabs, bidirectional negative refraction and planar focusing can be achieved. In this process, the illuminated PIM slab operates as a coherent perfect absorber, completely absorbing the incident wave, while the other identical PIM slab operates in the mode of a coherent laser, reproducing the incident acoustic waves on both sides. Moreover, the parameters in the appropriate range can be found by adjusting the mode order, which facilitates the experimental verification. To date, 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.