Ultrathin Acoustic Holography

How to realize acoustic holography via ultradeep‐subwavelength structures is a challenging problem in the past decades, which is thought impossible due to the linear proportional relationship between the structural thickness and acoustic wavelength. In this article, the methodology of ultrathin holography by patterning holes in an acoustic insulation plate with an ultradeep‐subwavelength thickness is introduced. The transmitted sound field can be manipulated arbitrarily to form a desired shape by designing the ultrathin pattern based on the nonlocal wave interaction theory. The physical mechanism of the nonlocal behavior to achieve a sophisticated hologram is revealed due to the interaction among the sound wave components themselves. Furthermore, the experiments are designed to map out the pressure amplitude field of a “sun” pattern in air and water, respectively. The work demonstrates the advantage of nonlocal ultrathin holography in the applications of ultrathin acoustic devices and provides inspiration for the holographic wave manipulation.


Introduction
Wave manipulation using artificial materials is a hot topic in recent years. In the past decades, great progress has been achieved in manipulating acoustic waves through novel structures, such as phononic crystals, [1][2][3] acoustic metamaterials, [4][5][6][7][8][9][10][11][12][13][14] and acoustic metasurfaces. [15][16][17][18][19][20][21] Acoustic metasurfaces, as 2D materials with subwavelength thickness, can provide the desired www.advmatinterfaces.de perforated theoretical zero-thickness plane model, in which the incident acoustic waves are forced to pass the plane only through the holes on it. The incident, reflected, and transmitted acoustic waves interact with each other and change their wave fields collaboratively via a nonlocal effect between the different holes. Based on this model, we can control the amplitudes and delays of acoustic waves on both reflection and transmission surfaces of the UTH to form an arbitrary acoustic field pattern. It can be found that the UTH could have the same ability to control acoustic waves as the traditional material structures but with near zero thickness, which indicates that it can powerfully manipulate acoustic waves independent of the wavelength. In the following, we will derive the theoretical model for the UTH, and experimentally demonstrate its effectiveness in realizing focusing and other sophisticated holograms in air and water.

Analysis of the Ultrathin Holography Model
To introduce the UTH better, let us start with traditional metasurface as Figure 1a shows. For convenience, we code every unit cell composing the metasurface with number 1 to N, in which N is the total number of unit cells in this case. Considering the I th unit cell, we denote the normal vibration velocity of its incident and transmitted wave by i ( ) u I and , t ( ) u I respectively. Then, the relationship of them can be deduced as where u i and u t represent the normal vibration velocity vector of incident and transmitted waves, respectively, which can be expressed as , ,…, , …, , ,…, , …, And the local behavior and nonlocal coupling are characterized by matrix D and X, respectively. Here, , where A (I) and φ (I) denote the amplitude modulation and the phase shift of I th unit cell, respectively. In matrix X, the nondiagonal element X(I, J) is the coupling coefficient between the I th and J th units, and the distinct form of X varies with each structure. For traditional structures, D plays the key role in the sound wave manipulation. In some recently proposed nonlocal metasurfaces, the effects of D and X are both considered. [15,21,39] Furthermore, when we compress unit-cell thickness to near zero, the manipulation ability brought by unit structure itself will disappear, and the nonlocal effect characterized by X plays a leading role. This condition can be expressed as Figure 1b shows our nonlocal hologram with a theoretical zero thickness. We divide the whole plate into N square units numbered from 1 to N, and assume the side length of a unit is far less than acoustic wavelength. Some units are replaced by holes that sound waves can pass through freely. For each through-hole unit on the plane in Figure 1b, the air inside the hole could be considered as an ultrathin air plate that will be forced to vibrate by the incident, reflected, and transmitted acoustic waves. Since the scale of each hole is much smaller than the wavelength, the sound wave excited by the ultrathin air plate can be regarded as a hemispherical wave radiating into semi-unbounded space. Because of mutual radiation, these hemispherical waves will interact with other through-hole units and change their acoustic conditions, which couple all the units at different locations. Because there is no acoustic structure but www.advmatinterfaces.de only a perforated plane, the nonlocal effect X will become the main influence to regulate the acoustic field.
First, we deduce the calculation method for the transmitted and reflected sound fields of the theoretical zero-thickness plane, and explain the derivation of nonlocal effect. Each unit on the plane satisfies the acoustic boundary conditions, where both the sound pressure and normal vibration velocity should be continuous. For a rigid-wall unit (Σ (I) ∈ Σ w ), normal vibration velocity on two sides of the unit cell is zero. While for a through-hole unit (Σ (I) ∈ Σ t ), acoustic boundary condition can be expressed as u I are the vibration velocities of the incident, reflected, and transmitted waves that radiated from the I th unit; i ( ) p I is the incident acoustic pressure on the I th unit, r are the reflected and transmitted acoustic pressures that radiated from the J th unit to the I th unit, respectively. The sound pressure at I th unit will be affected by sound waves radiated from all other through-hole units, which lead to the nonlocal coupling between units at different locations. As the size is much smaller than the sound wavelength, a through-hole unit is equivalent to a point source, whose radiation characteristics can be expressed by the Greens function where i is the imaginary unit, ρ 0 and c 0 are the density and the acoustic speed of the medium respectively, S 0 is the area per unit on the plane, r (I) and r (J) are the location vectors of the I th and J th units, respectively, and g(r (I) ; r (J) ) is the Greens function between these two positions. It can be found that, for certain incident sound waves, r where r (I) − r (J) denotes the relative distance between units I, J and determines the strength of coupling. When the specific pattern of the hologram is designed, the coupling between units is determined. Here, we define is the characteristic impedance of incident waves on the I th unit. We use the matrix Z xr to completely describe the interaction between all units and rewrite Equation (8), and then to obtain By solving Equation (10), the vibration velocity can be expressed as This is the specific form of Equation (4) for the proposed hologram. When the incident sound field and hologram pattern are given, the transmission and reflection sound fields can be derived by the calculated vibration velocity distributions. More derivation details can be found in note S1 (Supporting Information). Because of the existence of Z xr , the vibration velocity of each unit is determined by the mutual coupling of all units, not only by the local characteristics. Therefore, the nonlocal effect is an important factor in holographic imaging by UTH with theoretical zero thickness, and its influence cannot be ignored when calculating holographic sound field.
Next, we will explain the design method of hologram for transmission field holography. Our task is to design a preset hologram pattern for a given incident wave, so that the transmitted sound field approaches the target sound field distribution. The UTHs pattern is described by a vector v = (1, 1, 0, …, 0), whose I th element represents the state of I th unit on the pattern: a through-hole (1) or a hard boundary (0). Then, the elements of Z xr in Equation (9) are associated with v, because Σ (I) ∈ Σ t when v(I) = 1 and Σ (I) ∈ Σ w when v(I) = 0. As a specific acoustic wave manipulation problem, we are mainly concerned about the acoustic field at the image plane, which can be obtained as where where r (I) and r (K) are the location vectors of the I th unit on the hologram and the K th point on the image plane, respectively. Assume that the target acoustic field is p t t . The task is to design a UTH with pattern v to minimize the deviation between the target acoustic field and calculated acoustic field, which can be described by It is difficult to obtain an analytical solution of v, so we employ genetic algorithm to obtain an approximate solution. The optimization details are given in Figure S2 (Supporting Information). When v is obtained, the hologram is accordingly determined.

Acoustic Focusing via Ultrathin Holography
In order to demonstrate the acoustic wave manipulation capability of the UTH, we propose an experiment to investigate the focus scenario first of all. The schematic diagram of the experiment is shown in Figure 2a. The UTH used in experiment is a 1-mm thick square steel plate with a side length of 0.6 m, and the pattern where drilled holes locate is a 0.3-m side length square area located at the center of the UTH. We divided each side of this area into 60 parts, and then 3600 units can be obtained. The side length of each unit is 0.5 cm. The 8.5 kHz incident wave was radiated by a point source, which was located at 15 cm away from the UTH on its center axis in Figure 2a (see more experiment details in note S3, Supporting Information). In the expected acoustic field as we design, the focus is located on the center axis and is 14.75 cm away from the plate. The considered acoustic field is a square area perpendicular to the rigid plane, whose side length is 0.4 m. Figure 2b,c gives the theory and simulation results of the acoustic field, respectively.
Besides, we scanned the acoustic field on x-z image plane, and the result of the experiment is shown in Figure 2d. The acoustic fields measured on the cross-section line and the focal axis are shown in Figure 2e,f, respectively. The experimental results agree well with the theory, which shows that the UTH works well in the acoustic focusing situation. Meanwhile, the theoretical result of acoustic focusing without considering the nonlocal effect is also given in Figure S4a (Supporting Information), which shows that the nonlocal effect is an important factor to manipulate acoustic waves for the UTH.

Ultrathin Holography in Air
Next, we designed an experiment to verify the acoustic imaging ability of the UTH in air. The schematic diagram of experimental setup is shown in the Figure 3a. In this case, a 1-mm thick square steel plate with a side length of 0.64 m was used as the UTH. The pattern area has 14 400 units with 120 pieces on each side, and the side length of each unit is 0.4 cm. The incident acoustic wave at 8 kHz is radiated by a point source, which is 12 cm far away from the plane and is located on the center axis perpendicular to the plane. The image plane is parallelized to the steel plate and the distance between them is 12 cm. The desired acoustic field on the transmission side is a pattern of the "sun" with a side length of 0.28 m as shown in Figure 3b. The theory, simulation and experimental results of the transmitted acoustic field on the hologram plane are shown in Figure 3c-e, respectively. It can be found that the simulation and measurement

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results fit well with the corresponding theoretical calculation result. The slight discrepancy between experiment and theory is caused by the experimental errors, such as the fabrication errors and inaccurate source position. Therefore, we conducted simulations with these errors to prove the robustness of the design method (see the simulation results in Figure S5, Supporting Information). In addition, the theoretical result without considering nonlocal coupling is shown in Figure S4b (Supporting Information), which confirms that the nonlocal effect cannot be ignored in holographic imaging by UTH. So far, from the above acoustic focusing and holography examples, we can achieve the conclusion that our proposed UTH with the nonlocal effect can be successfully used for different acoustic applications. Our method is also suitable for manipulating reflected waves, for which the acoustic holography examples of letters and sophisticated "panda" patterns for the reflection case are given in note S6 (Supporting Information).

Ultrathin Holography Underwater
In order to expand the UTHs application scenarios, we explored its imaging ability in water. For this case, an important problem is to find a perfect sound insulation surface because the characteristic impedance of water is much greater than that of air. For example, a tungsten plate can be regarded as rigid sound insulator in air due to the large impedance mismatch. However, in the circumstance of underwater, the tungsten plate is not a hard boundary because the characteristic impedances of tungsten and water are not that different. In this case, sound waves would transmit through the wall and interfere with holographic imaging. To solve this problem, we applied a superhydrophobic coating on the surface of tungsten plate. The superhydrophobic coating induces an inherent air/water interface referred to as the Cassie-Baxter states, rendering a giant acoustic impedance mismatch, which provides a perfect soft boundary and leads to all-angle and wide-spectrum total reflection in water environment. [40] Therefore, we used the perforated tungsten plates coated with the nanosilicon dioxide superhydrophobic films to fabricate the UTH for underwater applications. The side length of the UTH used in this ultrasonic holography experiment is 76 mm, and the thickness is only 0.05 mm. As in the case of focus experiment, we divided each side of the central area into 120 parts, and then 14 400 units with a side length of 0.38 mm can be readily obtained. We carried out the experiment in a water tank, and the schematic experimental setup is shown in Figure 4a. The photo of the experiment scene can be found in Figure S3 (Supporting Information). The transducer transmits ultrasonic pulse of 500 kHz, and the hydrophone measures the sound field behind the UTH. Figure 4b shows the UTH sample used in this experiment. The results of theoretical calculation, simulation, and experiment are shown in Figure 4c-e, respectively. By comparison, the theoretical calculation and simulation results are in good agreement with the pattern shown in Figure 3b. It should be mentioned that the distortion of imaging in Figure 4e is caused by fabrication errors of UTH Figure 3. Results of ultrathin holography in air. a) Illustration of the experiment. A pattern of the "sun" was designed as the desired acoustic field pattern on the holographic image plane. b) The "sun" pattern designed in the acoustic holography. c) Theoretically calculated pressure amplitude field on the image plane. d) Simulated pressure amplitude field on the image plane. e) Measured pressure amplitude field in the experiment.

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and pulse echoes from water tank walls. In general, the results show that our hologram design method can also be applied in underwater environment.

Conclusion
In summary, we propose a new perspective and methodology to manipulate acoustic wave propagation through an ultrathin holography technique. The theoretical design method is given, and the feasibility is verified by experiments. The genetic algorithm is used to find the optimal hologram pattern for a desired sound field. We have demonstrated that the sound field modulation capability stems from the nonlocal effect between subwavelength units at different locations, which can be regarded as the interaction among the acoustic waves themselves. This helps UTH break the thickness limit and reach the theoretical zero-thickness. In addition, the hologram experiments in air and water prove its feasibility and applicability in diverse media. This work might provide a new perspective for the structured wave field manipulation and inspire further researches on the ultrathin-integrated acoustic devices or new mechanisms in the-state-of-art wave-field engineering techniques. Moreover, it provides more possibilities for manipulating waves with ultralong wavelength, such as seismic waves.

Experimental Section
Fabrication and Measurement Methods: In the air-environment experiment, the UTH was made of a 304 stainless steel plane with 1 mm of the thickness, and the authors had used the laser beam cutting to make holograms or focusing pattern on it (0.1 mm in the cutting precision). The experiments of acoustic focusing as well as hologram were performed in an anechoic chamber (see Figure S3, Supporting Information). The UTH was set in the center of the anechoic chamber and the acoustic point source (a piezoelectric ceramic sheet) was located on the axis of the UTH and the microphone was fixed on the other side of the plane. The Brüel and Kjaer PULSE testing system (3560) was utilized as the measurement instrument, which recorded the acoustic pressure distribution in space. In measurement, a Brüel and Kjaer microphone (3160) was used to collect acoustic wave signals on the image place. In hologram measurement, the hologram image plane was divided into square girds (the length of each grid edge was much smaller than the acoustic wavelength), the acoustic pressure was measured on the grid points to achieve the acoustic field on the image plane. For the case underwater, a tungsten plate with the thickness of 0.05 mm was cut into a hologram pattern by laser-cutting technique, and the superhydrophobic film was coated by spraying the nanosilica + absolute ethanol solution. The experiment was carried out www.advmatinterfaces.de in a water tank. Olympus transducer and PA hydrophone were used to transmit ultrasonic pulse and measure ultrasound field, respectively.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.