Ultra-thin Acoustic Metasurface-Based Schroeder Diffuser

Schroeder diffuser is a classical design, proposed over 40 years ago, for artificially creating optimal and predictable sound diffuse reflection. It has been widely adopted in architectural acoustics and it has also shown substantial potential in noise control, ultrasound imaging, microparticle manipulation, among others. The conventional Schroeder diffuser, however, has a considerable thickness on the order of one wavelength, severely impeding its applications for low frequency sound. In this paper, a new class of ultra-thin and planar Schroeder diffusers are proposed based on the concept of acoustic metasurface. Both numerical and experimental results demonstrate satisfactory sound diffuse reflection produced from the metasurface-based Schroeder diffuser despite it being one order of magnitude thinner than the conventional one. The proposed design not only offer promising building blocks with great potential to profoundly impact architectural acoustics and related fields, but also constitutes a major step towards real-world applications of acoustic metasurfaces.


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In the 1970s, Schroeder published two seminal papers on sound scattering from maximum length sequences and quadratic residue sequences diffusers [1,2]. For the first time, a simple recipe was proposed to design sound phase grating diffusers with defined acoustic performance.
These two papers opened a brand new field of sound diffusers with applications in architectural acoustics [3][4][5], noise control [6][7][8], ultrasound imaging [9], microparticle separation [10] and have inspired other disciplines such as energy-harvesting photodiodes [11]. D'Antonio and Konnert [12] presented one of the most accessible review papers examining the theory behind Schroeder's diffusers (SDs). Most importantly, they commercialized SDs and promoted them to be widely adopted in architectural acoustics, where the diffusers can be used to spread the reflections into all directions, reducing the strength of the undesired specular reflection and echo, as well as preserving the sound energy in the space [3]. In contrast to diffusers, sound absorbers reduce the energy in the room, which can be problematic for unamplified performances in concert halls, opera houses, and auditoria. Sound diffusers are also used to promote desired reflections in order to enhance spaciousness in auditoria, to improve speech intelligibility, and to reduce the noise in urban streets [3,[13][14]. Instead of using a surface with random or geometric reflectors, Schroeder innovatively designed a family of diffusers based on number theory sequences, with the ultimate goal to produce predicable and optimal scattering (i.e., the sound is scattered evenly in all directions regardless of the angle of incidence). In spite of the great success that SDs have achieved, they are conventionally designed to have a grating structure with a thickness that is half of the wavelength at the operating frequency in order to achieve the desired phase delays. To put this into perspective, 4 the thickness of an SD reaches a remarkable value of 69 cm at 250 Hz, which is in the range of human voices, truck noises, etc. Figure 1(a) shows a simple one-dimensional (1-D) SD to illustrate the basic concept of SDs. The bulky size of conventional SDs poses a fundamental limitation on their applicability, i.e., SDs are typically limited to mid-and high frequencies because they are too large to be accommodated at low frequencies, which is a very important part of sound that human perceive. In addition, SDs usually do not complement the visual appearance of a space due to their large size and irregular surface. Although active methods may offer a solution to this limitation [15], they are much more expensive and complicated and therefore less practical compared to their passive counterpart.
In this paper, we revisit the SD and redesign it using the concept of acoustic metasurface [16][17][18][19][20][21][22][23][24][25]. Despite the considerable efforts dedicated to the research on acoustic metamaterials and acoustic metasurfaces , they are still at an embryonic stage from the real-world application perspective. Metasurfaces, in particular, are thin structures having subwavelength thickness consisting of unit cells that could give rise to numerous intriguing phenomena such as super sound absorption [16][17], wavefront shaping [18][19][20][21][22], dispersion-free phase engineering [23], and asymmetric acoustic transmission [24][25]. Here we show the potential of using acoustic metasurfaces to break down the fundamental physical barrier in designing ultra-thin SDs. As will be demonstrated in this paper, the metasurface-based SD (MSD) has a comparable performance to the conventional SD that has already been commercialized and widely used in practice. More importantly, the MSD is one order of magnitude thinner with a planar configuration and therefore is more suitable for low frequency applications in 5 architectural acoustics or other related fields. This paper will present the theoretical design, numerical simulation, and experimental demonstration of the ultra-thin MSDs with a thickness that is 1/20 of the center frequency wavelength 0  . The unit cell of the proposed MSD is a locally resonant element having a relatively simple geometry and its acoustical response can be engineered flexibly and precisely by adjusting a single geometrical parameter, which enables convenient analytical prediction of its acoustical phase response. The metasurface is designed in a way that the thickness is minimized while the performance is not significantly affected by the viscosity effect [40][41]. This is in contrast to the widely studied space-coiling structure-based metasurfaces which may be liable to suffer from large viscous losses at a comparable thickness [19][20][21]23]. Our initial design is further improved by the broadened frequency band introduced by a hybrid structure containing units operating at multiple optimal frequencies. The experimental and simulation results were in good agreement and both showed that the MSD yielded a performance on a par with the conventional SD, despite it being one order of magnitude thinner. This study, for the first time, attempts to bridge the gap between acoustic metasurfaces and their applications to real-world problems.
First we briefly review the conventional design of SDs and elucidate the fundamental limitation of this design. In order to generate diffuse reflection for different incident acoustic waves, the phase shift at the surface of a SD must yield a specific profile such as a special number sequence [42]. Conventionally, the desired phase delay in a SD is achieved by controlling the sound path in a grating structure, resulting in the fact that the maximum depth of individual unit of grating, also referred to as the "well", must reach a half of the wavelength 6 to ensure that the phase changes within a 2 range. Figure 1(a) shows schematically the 1-D model of a SD formed by a series of wells, which is for generating diffuse reflections in a twodimensional (2-D) plane and is called a single plane diffuser [42]. To generate diffuse reflections in three-dimensional (3-D) space, one needs to use a 2-D model shown in Fig. 1 In the 1-D case , the depths of the wells are dictated by a mathematical number sequence, such as a quadratic residue sequence (QRS) as shown in Fig. 1(a) for which the sequence number for the nth well, n S , is given by [42]: where Modulo indicates the least non-negative remainder, N is the number of wells per period. One example of quadratic residue diffusers with 7 N= shown in Fig. 1 The phase delay that a SD needs to yield is previously considered unattainable by a simple structure with a deep-subwavelength size. We will revisit this problem from the perspective of acoustic metasurfaces and demonstrate that it is possible to realize such a phase profile by using properly-designed metasurface units at a deep-subwavelength scale in the thickness direction.
The schematic diagram of the proposed MSD is illustrated in Fig. 1(c). The ultra-thin MSD is The corresponding 2-D QRS is shown in Fig. 2(b). This QRS is obtained with indexes n and m starting from 4 (in order to place the zero depth well at the center of the diffuser) in Eq. (3).
The photograph of a MSD sample with 2×2 periods (one period is defined as 7×7 unit cells corresponding to one full QRS) is shown in Fig. 2  in the y-z plane is in theory identical to that in the x-z plane. The acoustic energy is scattered into different directions after impinging upon the sample. Numerous side-lobes with similar magnitudes can be observed and diffuse reflection can be effectively realized by the sample. This is more pronounced in Fig. 3(c), which shows the simulated and the measured far-field scattering directivity of the sample (polar response). The reflected fields of the flat plate in Figs.  11 We have demonstrated that MSDs can be designed to achieve efficient acoustic diffuse reflection in the vicinity of the center frequency. This initial design suffers from the relatively narrow bandwidth due to the resonance nature of the unit cell. We will further enhance the MSD by broadening the operating frequency range which is crucial for certain practical applications. A broadband MSD (BMSD) has a hybrid structure comprising components designed for generating the desired phase delay at multiple frequencies. The multi-frequency QRS is shown in Fig. 5(a), in which n A , n B , n C , and n D represent four different target frequencies and the subscript n represents the number in QRS.
In this manner, the staggered units for four operating frequencies lead to the BMSD design that targets different frequencies and yields a 14 14  array. Figure 5