Three-dimensional omnidirectional acoustic illusion

: Acoustic waves are ubiquitous in human everyday experience, therefore, precise control over the deformation of acoustic waves is always extremely desirable, which can be used, for example, to transform or hide objects from incident waves. Acoustic illusion devices are generally implemented by transformation acoustics, which can deceive ears or sonar systems. extreme material parameters prescribed by coordinate transformation theory make the implementations particularly difficult, even with the help of acoustic metamaterials. Here, a novel method based on Fabry-Perot resonances offers a feasible solution omnidirectional passive acoustic illusion. We theoretically demonstrated perfect 3D acoustic illusion via Mie theory, reduced version is further designed numerically and implemented experimentally. In the future, our work opens new possibilities for the implementation of modern acoustic

implemented by transformation acoustics, which can deceive ears or sonar systems. Challenges remain, the complexed and extreme material parameters prescribed by coordinate transformation theory make the implementations particularly difficult, even with the help of acoustic metamaterials. Here, a novel method based on Fabry-Perot resonances offers a feasible solution for achieve three-dimensional (3D) omnidirectional passive acoustic illusion. We theoretically demonstrated perfect 3D acoustic illusion via Mie theory, reduced version is further designed numerically and implemented experimentally. In the future, our work opens new possibilities for the implementation of modern acoustic illusion devices, such as camouflage for anti-sonar detection.

Introduction:
Stories of desperate travelers being lured by a mirage of the miraculous pool among the dunes in a never-ending desert is familiar. The thirsty traveler runs forward madly, trying to throw himself into the bule pool.
Soon the shady water would vanish into thin air according to the regular version of the story, leaving him in the yellow dust under the burning sun again. Mirage here is the deceptive appearance of distant objects caused by the bending of light (refraction) in the air layers of varying density, i.e., optics illusion. They are tricks for the eyes, which make us think that there is something with nothing in reality. It's just like their optical cousins that made people go enamored, just with sound. The key of how to artificially realize the illusion effect lies in an object giving the detector some deceptive signals to hide itself or resemble another object, which was deemed impossible before the proposal of the coordinate-transformationbased technique.
The coordinate transformation method has been introduced in the field of electromagnetics (EM) based on the form-invariant of Maxwell's equations, linking spatial geometry with material parameters to calculate the electromagnetic parameters of devices with predefined functionalities [1,2], which is also especially applicable to acoustics. For instance, two-dimensional acoustic cloak is proposed by equivalence demonstration between the acoustic wave equation and the two-dimensional Maxwell's equations [3]. Simultaneously, three-dimensional acoustic cloak is established by the isomorphism of the acoustic equation and conductivity equation [4], which is verified by Mie theory [5]. The question is that the material parameters derived from coordinate transformation are often inhomogeneous, anisotropic, or of extreme values that do not exist in nature [6,7], which makes the applications of invisibility cloaks and more generic illusion devices very challenging even with the help of metamaterial approaches. Despite numerous schemes of acoustic cloaks have been proposed with physically feasible parameters [8][9][10][11][12], only a few have been experimentally verified so far [13][14][15][16][17][18][19]. Different from the cloak in free space, the cloak for hiding objects near reflecting boundaries [13][14][15][16][17] based on the quasi-conformal map was proposed, which doesn't require extreme material parameters for realization. However, there is a flip side to the so-called ground cloak, the huge size of cloak shell compared to the hidden object makes its application limited. In addition to the coordinate transformation, there are also other mechanisms for passive acoustic illusion and invisibility cloak, such as scattering cancellation [18] and metasurfaces [19]. Although some results are fascinating, the difficulty of the design and the complexity of the structures limit the practicality as well as the implementation of the current methods. Besides, it's noted that accompanied by various approaches for scattering manipulation proposed are the deficiencies of their respective functionalities, such as unidirectional or ground (near reflecting boundaries). Could there be a much simpler way of designing practical 3D cloak for acoustic illusion?
The Fabry-Pérot resonance [20][21] is a good candidate as it can maintain the phase constant while keeping perfect transmission.
In this paper, we develop a theoretical framework and experimental realization of acoustic omnidirectional illusion devices using holey structures along the radial directions in free space. Our approach requires neither the bulk metamaterials with negative indices nor ground, yet allows us to demonstrate a series of omnidirectional acoustic illusion effects.
For the cloak shell(region II), the pressure field can be written as [20] 0 0 max ln ln While that the total pressure field in the background is contributed by the incident wave and the scattered wave,  [20]. As a special case, the invisibility cloak hides an object by simply eliminating its acoustic scattering field, which can also be achieved when the relative index of the object B is equal to 1. In other words, there is nothing that can be detected in the background medium, which is theoretically illustrated in Supplemental Material Section 2 [20,24]. In fact, it is more challenging and intriguing to realize a general illusion cloak that detect the scattering pattern of a particular object A appears like some other object B of  Fig. 1(c), see also Supplemental Material Section 9 for detail description [24]. A square array of speakers is placed head-on to the sample to generate normal incident gaussian beam, then acoustic pressure field distribution inside the measured region, shown as a black dashed rectangle in Fig. 3(a), can be measured by a moving microphone mounted on a 3axis linear stages. The measured area here is 60 cm by 20 cm with a step of 8 mm, and located directly behind the sample. The whole experimental system is surrounded by sound-absorbing foams to reduce the noise and reflection. Hz( 1 m = ). In Fig. 3(a), we plot the pressure distribution for the original sphere in physical place covered by the cloak, and the inset displays the zoom-in view of the measured pressure distribution by experiment. The simulated results of pressure distribution for the equivalent bare sphere as shown in Fig.3(c), and that of the illusion device in the inset map as the reference. The observe plane here is located at XY plane with z = 0 cm (central plane of the device). In order to quantify the illusion effect, the corresponding near-field pattern of the Fig. 3(a) and 3(c) have been probed along half circle of radius 2.5λ with the object as the center and depicted in Fig.3(b) and 3(d), respectively. By multidimensional comparison, it can be observed in the numerical and experiment results that at the resonance frequency, one object covered by the cloak with reduced parameters mimics acoustically another object quite well. From this general method, we also take another example. The parameter setting here keeps consistent with the case in Fig.3, except that the inner sphere coated by the FP shell is replaced with an acrylic resin sphere. As mentioned above, due to the extreme impedance mismatch, the resin sphere can be regarded as rigid scattering object. Figure 4(a) and 4(c) shows the simulated field distribution of the illusion cloak and a single equivalent sphere, rectangular dotted regions depict location in which the experimental pressure distribution was mapped in the inset map.
Experimental pressure fields show similar scattering waves as in the numerical simulations. The corresponding near-field pattern of the Fig. 4(a) and 4(c) have also been depicted in Fig. 4(b) and 4(d), respectively. It's worth mentioning that the material of a single equivalent sphere in virtual space could still be approximated as rigid scatterer respect to the background medium. For simplicity, a solid resin sphere of radius c is used to replace the equivalent bare sphere in the experiment. We can see that the achieved illusion effects are quite good: the original sphere in Fig. 4(a) with the help of FP cloak has the same pressure distributions as the equivalent sphere in Fig. 4(c). So far, the different test results of simulation and experiment firmly validate our designs that the FP cloak is a simple and effective approach for transforming one object to another one. In general, the key of this method is: for an acoustic detector at a series specified frequencies and arbitrary incident angles, the detector will get the desired acoustic signals while we don't need to significantly change the original object. It's worth emphasizing that the direction and type of the incident waves are usually unpredictable in practical scenes, such an illusion effect is always valid (see Supplemental Material Section 4 to 8 [24]). Thus, the omnidirectional functionality of our proposed illusion device for any incoming waves would have unparalleled significance in practical applications. In addition, for different kinds of scatterer shapes, the related matrix of scattering coefficient and incoming coefficient could be determined by matching the boundary condition based on the so-called T matrix method and the above acoustic illusion effect is still valid [22][23].
In summary, we have developed a method to design 3D acoustic omnidirectional illusion cloak based on FP resonances. As a practical implementation of our strategy, we propose a reduced-version cloak design with simplified anisotropic parameters, which can be realized by densely perforating FP cavities with linearly shrinking cross section along the radial direction on a solid spherical shell. Good agreement is observed between the theoretical prediction and experiment measurement. Moreover, we further verified its generality for different source and objects of arbitrary shapes. Compared with transformation acoustics, this omnidirectional cloak greatly simplifies the material parameters, which would significantly facilitate practical applications of the designed structures. In the further, this unique but simpler method may find many applications in the design of modern acoustic illusion devices, such as camouflage for anti-sonar detection.