# Acoustic Equivalent Lasing and Coherent Perfect Absorption Based on a Conjugate Metamaterial Sphere  
[PERSON]  
1 Jiangsu Key Laboratory of Opto-Electronic Technology, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China; [EMAIL_ADDRESS] (X.-X.M.); [EMAIL_ADDRESS] (X.-F.Z.); [EMAIL_ADDRESS] (J.X)  
[PERSON]  
1 Jiangsu Key Laboratory of Opto-Electronic Technology, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China; [EMAIL_ADDRESS] (X.-X.M.); [EMAIL_ADDRESS] (X.-F.Z.); [EMAIL_ADDRESS] (J.X)  
[PERSON]  
1 Jiangsu Key Laboratory of Opto-Electronic Technology, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China; [EMAIL_ADDRESS] (X.-X.M.); [EMAIL_ADDRESS] (X.-F.Z.); [EMAIL_ADDRESS] (J.X)  
[PERSON]  
1 Jiangsu Key Laboratory of Opto-Electronic Technology, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China; [EMAIL_ADDRESS] (X.-X.M.); [EMAIL_ADDRESS] (X.-F.Z.); [EMAIL_ADDRESS] (J.X)  
[PERSON]  
1 Jiangsu Key Laboratory of Opto-Electronic Technology, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China; [EMAIL_ADDRESS] (J.X)  
Footnote 1: [[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
Footnote 2: Key Laboratory of Modern Acoustics of MOE, School of Physics, Nanjing University, Nanjing 210093, China  
Footnote 3: [[https://www.mdpi.com/journals/article/pdf/1](https://www.mdpi.com/journals/article/pdf/1)]([https://www.mdpi.com/journals/article/pdf/1](https://www.mdpi.com/journals/article/pdf/1))  
enables the out-coupling of topological acoustic equivalent lasing (AEL) modes with the desired handedness [28], and nonreciprocity energy transmission can be realized by lossy materials [29; 30; 31; 32].  
A conjugate metamaterial, whose effective relative permittivity and permeability are complex conjugates of each other, was proposed in the field of electromagnetism more than a decade ago [33]. Based on the electromagnetic conjugate metamaterials, many excellent phenomena were realized, such as perfect absorption and amplification [34; 35; 36]. Inspired by this, acoustic conjugate metamaterials (ACMs) have recently been proposed, where the phases of the effective mass density and bulk compressibility are complex conjugates [37]. The ACMs possess a real refractive index, indicating the existence of undecaying propagating waves in the ACMs, although the materials exhibit the elements of gain and loss simultaneously. Based on an ACM slab, AEL and coherent perfect absorption (CPA) modes were realized for plane wave incidence. However, research on spherical ACMs has seldom been reported. In this paper, we explore the extraordinary scattering properties of a three-dimensional (3D) ACM sphere under spherical wave incidence with different topological orders. Analytical derivations with rigorous scattering theory and consistent simulations demonstrate that the AEL mode or (and) CPA mode, where incident spherical waves with specific topological orders are extremely scattered or (and) totally absorbed, could be realized with a small (huge) monolayer ACM sphere. Meanwhile, a multilayered ACM sphere is proposed to realize the AEL and/or CPA modes for spherical wave incidence with multiple orders.  
This paper is organized as follows. In Section 2, we first propose an ACM sphere and build its scattering model. In Sections 3 and 4, we demonstrate the AEL and CPA modes, respectively, based on a small monolayer ACM sphere. In Section 5, we demonstrate the simultaneous realization of AEL and CPA modes based on a huge monolayer ACM sphere. In Section 6, a multilayered ACM sphere is proposed to realize the AEL and/or CPA modes with multiple orders. Section 7 contains a conclusion.