# Theoretical Zero-Thickness Broadband Holograms Based on Acoustic Sieve Metasurfaces
## 1 Introduction  
Acoustic holography [1; 2] is an essential tool for controlling sound waves, generating highly complex and customizable sound fields, and enabling the visualization of sound fields. Due to its powerful and flexible capability of arbitrary sound beam shaping and pre-designed sound field reconstruction, acoustic holography has recently received increased attention in many interdisciplinary fields related to acoustics, including medicine [3; 4; 5; 6], engineering [7; 8], and biology [9; 10; 11], in addition to the pure acoustic field. Many applications have been made possible by acoustic holography, including acoustic fabrication [7], beam shaping [12; 13; 14; 15; 16; 17; 18; 19; 20], acoustic holographic imaging [3; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28], volumetric display [29; 30], volumetric haptics [31; 32], volumetric haptics [33; 34], volumetric haptics [30], volumetric haptics [32; 31], particle manipulation [33; 21; 26; 34; 35], 3D ultrasound imaging [8], cell manipulation [9; 10], characterizing medical ultrasounds [4; 5], neurostimulation and neuromodulation [6], and energy transfer [25]. Instead of acoustically recording a hologram from a physical scene with an exacting recording environment and a complex process, modern computer-generated holography directly calculates the required phase and/or amplitude maps in the acoustic holographic plane before rendering them for reconstruction [35]. The algorithms used include the iterative angular spectrum approach [21], the iterative backpropagation algorithm [33], the weighted Gerchberg-Saxton algorithm [20], the pseudo-inverse method [36], the conjugate field method [37], the deep learning method [35], and so on. Physically, these calculated maps are realized by either discrete and independently driven elements of active phased-array emitters [3; 4; 5; 27; 33; 28; 34; 29; 30; 31; 32; 33; 34] (such as loudspeakers in the air or piezoelectric transducers inwater) or meta-pixels of passive acoustic metasurfaces [6; 7; 8; 9; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 38], or a combination of both [26]. In the reconstruction process, these acoustic holographic devices with the required phase and/or amplitude distributions diffract sound waves to form the desired real image in the pre-designed image plane. Metasurface-based acoustic holograms, which have become a research hotspot in the field of acoustic holography in recent years, can overcome the disadvantages of active phased-array holograms, such as a complex structure and circuit, high power consumption, and the need for careful calibration and tuning. That is, holography can be realized with low costs, a large aperture, on a large scale, with high precision, and with high frequency using metasurface-based acoustic holograms [18]. However, due to the structural complexity of metasurface units, their thickness is still not ultra-thin, especially in high-frequency bands, which makes expanding the application range of metasurface-based acoustic holograms difficult. To address the aforementioned issues, this paper proposes a design concept for an acoustic sieve metasurface (ASM) that can theoretically realize a zero-thickness broadband hologram. The ASM is a rigid screen with many small holes, through which the sound wave can pass to obtain the desired real image in the target plane. The arrangement rules of the holes on the ASM can be found using the Rayleigh-Sommerfeld theory [36] and optimization algorithms (a genetic algorithm [39; 40] in this paper). There is no thickness limit as long as the ASM-based hologram is sufficiently rigid relative to the background medium, so it can theoretically achieve zero thickness. Furthermore, because the wave from a hole has no additional phase or amplitude modulations, the intractable modulation dispersion can be physically eliminated, allowing the proposed ASM-based hologram to potentially work in any frequency band as long as the condition of paraxial approximation is satisfied.