Theoretical Zero-Thickness Broadband Holograms Based on Acoustic Sieve Metasurfaces

Abstract

Acoustic holography is an essential tool for controlling sound waves, generating highly complex and customizable sound fields, and enabling the visualization of sound fields. Based on acoustic sieve metasurfaces (ASMs), this paper proposes a theoretical design approach for zero-thickness broadband holograms. The ASM is a zero-thickness rigid screen with a large number of small holes that allow sound waves to pass through and produce the desired real image in the target plane. The hole arrangement rules are determined using a genetic algorithm and the Rayleigh–Sommerfeld theory. Because the wave from a hole has no extra phase or amplitude modulation, the intractable modulation dispersion can be physically avoided, allowing the proposed ASM-based hologram to potentially function in any frequency band as long as the condition of paraxial approximation is satisfied. Using a numerical simulation based on the combination of the finite element method (FEM) and the boundary element method (BEM), this research achieves broadband holographic imaging with a good effect. The proposed theoretical zero-thickness broadband hologram may provide new possibilities for acoustic holography applications.

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], particle manipulation [11,21,26,33,34], 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 [3,4,5,27,28,29,30,31,32,33,34] (such as loudspeakers in the air or piezoelectric transducers in water) 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. Using the numerical simulation based on the combination of the finite element method (FEM) and the boundary element method (BEM), broadband holographic imaging be achieved, and the excellent imaging effects confirm the feasibility of our design scheme.

2. Theories and Methods

2.1. Rayleigh–Sommerfeld Theory

The complex acoustic pressure at a point in the field due to a time–harmonic radiating source, with ${\mathrm{e}}^{\mathrm{i}{\omega }_{0}t}$ time dependence, is given by the Rayleigh–Sommerfeld integral [36]

$$p\left(\mathit{r}\right)=\frac{\mathrm{i}{\rho }_0{c}_0{k}_0}{2\mathsf{\pi }}{{\displaystyle \int }}_{S^{\prime }}^{}u\left({\mathit{r}}^{\prime }\right)\frac{{\mathrm{e}}^{-\mathrm{i}{k}_0\left|\mathit{r}-{\mathit{r}}^{\prime }\right|}}{\left|\mathit{r}-{\mathit{r}}^{\prime }\right|}\mathrm{d}{S}^{\prime }$$

where $\mathrm{i}=\sqrt{-1}$, ${\rho }_0$ and $c_0$ are, respectively, the density and the speed of sound in the background medium, $k_0={\omega }_0/c_0$ is the wave number, $S^{\prime }$ is the surface of the source, $u$ is the particle velocity normal to the surface of the source, and $\mathit{r}$ and ${\mathit{r}}^{\prime }$ are the observation and source point, respectively.

2.2. Modified Genetic Algorithm

A modified genetic algorithm [39] that only contains the mutation and evolution operations for a chromosome encoding the value (0 or 1) of every grid node is used to implement the design. The hologram information is encoded into one chromosome containing $N\times N$ genes, each of which is labeled by the value of a grid node. This algorithm has only mutation operations and no crossover operations. The mutation operation changes only one gene in the parent chromosome at each iteration. Offspring chromosomes are obtained after altering the gene. When calculating the offspring sound field, the sound field of the entire offspring chromosome is not calculated once, but only the influence of the changed gene on the sound field is calculated by using the Rayleigh–Sommerfeld theory, and then the influence is superimposed on the parent sound field. This superposition calculation method that reduces the amount of calculation significantly expedites an evolution operation by four orders of magnitude [39], which makes it more feasible in carrying out the design of a large-scale hologram.

2.3. Numerical Simulation

A combination of the FEM and the BEM is used for the numerical simulations and the software used is COMSOL Multiphysics. During the simulation of holographic imaging, a zero-thickness rigid boundary with a special arrangement of small holes is used to simulate the proposed hologram. The space below the rigid boundary is simulated by the FEM, and the space above the rigid boundary is simulated by the BEM. The plane where the rigid boundary is located is the coupling plane between the two methods. The plane 1/6 wavelength below the rigid boundary is the emission plane. The emitted sound wave is a plane wave with a sound pressure amplitude of 1 Pa and its propagation direction (i.e., z-direction) is perpendicular to the rigid boundary (the rigid boundary is located on the plane of z = 0 and its center is at the point of (0, 0, 0)). All boundaries in the calculation part of the FEM are set as plane-wave radiation boundaries (they can also be set as perfectly matched layers, and the calculation results of the two settings are basically the same) to achieve an anechoic environment, and the whole space in the calculation part of the BEM is set as free open space. For domains, the mesh is set to free tetrahedral, and for boundaries, the mesh is set to free triangular. The maximum element size of meshes is set to be no larger than 1/6 wavelength.

3. Results

3.1. Hologram Realized by the Array of Identical Point Sources

First, the holographic principle based on an array of identical point sources (PSs) is discussed. As shown in Figure 1, the PSs (represented by red dots) with source strength $Q_0$ are located at the grid nodes (represented by $G_{m,n}$) of an N × N array with a distance D between two neighboring nodes, where m, n = 1, 2, 3, …, N. Every grid node is assigned a value of 1 or 0, with 1 indicating the presence of a PS and 0 indicating the absence of a PS. According to the Rayleigh–Sommerfeld theory, the sound pressure of an array of PSs at any point $\left(x,y,z\right)$ in space can be described as a superposition of sound fields from all of these PSs

$$p\left(x,y,z\right)=\frac{\mathrm{i}{\rho }_0{c}_0{k}_0{Q}_0}{2\mathsf{\pi }}{\displaystyle \sum }_{m=1}^N{\displaystyle \sum }_{n=1}^N{A}_{m,n}\frac{{\mathrm{e}}^{-\mathrm{i}{k}_0{R}_{m,n}}}{R_{m,n}}$$

(1)

where $A_{m,n}=0 \mathrm{or} 1$ is the value of $G_{m,n}$, $R_{m,n}=\sqrt{{(x-x_{m,n})}^2+{(y-y_{m,n})}^2+z^2}$ is the distance from $G_{m,n}$ to the point $\left(x,y,z\right)$, and $\left(x_{m,n},y_{m,n},0\right)$ is the coordinate of $G_{m,n}$.

Combined with Equation (1), a desired real image (e.g., the four-pointed star pattern in this paper, as shown in Figure 2a; the black parts in the figure represent the presence of sound energy with equal magnitudes and the sound energy at these locations can be considered to be 1; the white parts represent the absence of sound energy, and the sound energy at these locations can be considered to be 0) in the target plane can be designed by using a modified genetic algorithm that only contains the mutation and evolution operations for a chromosome encoding the value (0 or 1) of every grid node. The specific operation process of the modified genetic algorithm is as follows:

(1)

Randomly generate a chromosome $A_1$ (each subsequent generation of the chromosome is named $A_2$, $A_3$, …, $A_{\mathrm{n}}$, …). $A_1$ contains all $A_{m,n}$, each $A_{m,n}$ as a gene, and the value of each gene is only 0 or 1. Moreover, let $A=A_1$.

(2)

Calculate the sound field $p_1\left(x,y,z_0\right)$ (each subsequent generation of the sound field is named $p_2$, $p_3$, …, $p_{\mathrm{n}}$, …) generated by chromosome $A$ in the desired image plane (the distance from the hologram is $z_0$) based on Equation (1), and let $p\left(x,y,z_0\right)=p_1\left(x,y,z_0\right)$.

(3)

Using $p\left(x,y,z_0\right)$ and the desired sound field $p_{\mathrm{desired}}\left(x,y,z_0\right)$ in the image plane, the root-mean-square error (RMSE) is calculated as the value of the fitness function $F_1$ (each subsequent generation of the fitness function is represented by $F_2$, $F_3$, …, $F_{\mathrm{n}}$, …), and let $F=F_1$.

$$\mathrm{RMSE}=\sqrt{{\displaystyle \sum }_{i=1}^I\left[{\displaystyle \sum }_{j=1}^J{\left[\frac{E\left(x_i,y_j,z_0\right)}{\mathrm{MAX}\left(E\right)}-\frac{E_{\mathrm{desired}}\left(x_i,y_j,z_0\right)}{\mathrm{MAX}\left(E_{\mathrm{desired}}\right)}\right]}^2\right]/\left(I·J-1\right)}$$

where $E\left(x_i,y_j,z_0\right)\propto p^2\left(x_i,y_j,z_0\right)$ and $E_{\mathrm{desired}}\left(x_i,y_j,z_0\right)\propto p_{\mathrm{desired}}^2\left(x_i,y_j,z_0\right),$ respectively, represent the sound energy generated by the hologram and the sound energy of the desired image at $\left(x_i,y_j,z_0\right)$ in the image plane; $\mathrm{MAX}\left(E\right)$ and $\mathrm{MAX}\left(E_{\mathrm{desired}}\right),$ respectively, represent the maximum values of $E$ and $E_{\mathrm{desired}}$; I and J, respectively, represent the numbers of samples along the x-direction and the y-direction. The RMSE is a sensitive method to show the profile differences between the sound energy generated by the hologram and the sound energy of the desired image and can be used to characterize the imaging quality in the image plane. The smaller the value of the RMSE, the closer the sound field of the image plane to the desired sound field. When the RMSE = 0, it means that the image generated by the hologram is the same as the desired image.

(4)

Perform a mutation operation ($1\to 0$ or $0\to 1$) for a random gene $A_{\mathrm{a},\mathrm{b}}$ whose coordinate is $\left(x_{m,n},y_{m,n},0\right)$ in the parent chromosome A to generate the offspring chromosome $A_{\mathrm{n}}$.

(5)

Calculate the effect of this mutation for the sound field in the desired image plane using the Rayleigh–Sommerfeld theory, represented by $p_{\mathrm{mutated}}\left(x,y,z_0\right)$. When the mutation operation is $1\to 0$, according to Equation (1), $p_{\mathrm{mutated}}\left(x,y,z_0\right)$ satisfies

$$p_{\mathrm{mutated}}\left(x,y,z_0\right)=-\frac{\mathrm{i}{\rho }_0{c}_0{k}_0{Q}_0}{2\mathsf{\pi }}\frac{{\mathrm{e}}^{-\mathrm{i}{k}_0\sqrt{{(x_{\mathrm{a},\mathrm{b}}-x)}^2+{(y_{\mathrm{a},\mathrm{b}}-y)}^2+z_0{}^2}}}{\sqrt{{(x_{\mathrm{a},\mathrm{b}}-x)}^2+{(y_{\mathrm{a},\mathrm{b}}-y)}^2+z_0{}^2}}$$

When the mutation operation is $0\to 1$, $p_{\mathrm{mutated}}\left(x,y,z_0\right)$ satisfies

$$p_{\mathrm{mutated}}\left(x,y,z_0\right)=\frac{\mathrm{i}{\rho }_0{c}_0{k}_0{Q}_0}{2\mathsf{\pi }}\frac{{\mathrm{e}}^{-\mathrm{i}{k}_0\sqrt{{(x_{\mathrm{a},\mathrm{b}}-x)}^2+{(y_{\mathrm{a},\mathrm{b}}-y)}^2+z_0{}^2}}}{\sqrt{{(x_{\mathrm{a},\mathrm{b}}-x)}^2+{(y_{\mathrm{a},\mathrm{b}}-y)}^2+z_0{}^2}}$$

(6)

Superimpose this effect on the parent sound field $p\left(x,y,z_0\right)$ to get the offspring sound field $p_{\mathrm{n}}\left(x,y,z_0\right)$ in the desired image plane, as follow

$$p_{\mathrm{n}}\left(x,y,z_0\right)=p\left(x,y,z_0\right)+p_{\mathrm{mutated}}\left(x,y,z_0\right)$$

(7)

Calculate the value (RMSE) of the offspring fitness function $F_{\mathrm{n}}$ by $p_{\mathrm{n}}\left(x,y,z_0\right)$ and $p_{\mathrm{desired}}\left(x,y,z_0\right)$.

(8)

Compare the magnitudes of $F_{\mathrm{n}}$ and $F$, and take the smaller one as the new parent fitness function. The chromosome with the smaller fitness function is taken as the new parent chromosome and the sound field with the smaller fitness function is taken as the new parent sound field. That is, if $F_{\mathrm{n}}<F$, then let $F=F_{\mathrm{n}}$, $A=A_{\mathrm{n}}$, $p\left(x,y,z_0\right)=p_{\mathrm{n}}\left(x,y,z_0\right)$; if $F<F_{\mathrm{n}}$, then let $F=F$, $A=A$, $p\left(x,y,z_0\right)=p\left(x,y,z_0\right)$. If $F$ is always less than $F_{\mathrm{n}}$ for a certain number (such as 100,000 times) of iterations, then record chromosome $A$ and end the iteration.

(9)

If the iteration does not end, go back to step 4.

Based on the desired image shown in Figure 2a, the overall map (as shown in Figure 2b) of the encoded PSs can be designed by the modified genetic algorithm at $z_0=120{\lambda }_0$, where ${\lambda }_0=c_0/f_0=10 \mathrm{mm}$ is the wavelength (the background medium is chosen as air with a sound velocity of $c_0=340 \mathrm{m}/\mathrm{s}$ and a density of ${\rho }_0=1.2 \mathrm{kg}/{\mathrm{m}}^3$, and the radiation frequency is set at $f_0=34 \mathrm{kHz}$). The distance between grid nodes is $D={\lambda }_0$, and there are 50 grid nodes in the horizontal and vertical directions, so the size of the hologram is $50{\lambda }_0\times 50{\lambda }_0$, which is the same as the size of the desired image. The BEM-based numerical simulation is used to confirm the performance of the PS array. Figure 2c shows the simulation results, which show a clear image that closely matches the desired image. The RMSE between this holographic image and the desired image shown in Figure 2a is 0.1737.

3.2. Theoretical Zero-Thickness Holograms Based on Acoustic Sieve Metasurfaces

To approximate the PS array, an acoustic sieve metasurface (ASM) that is a zero-thickness rigid screen with many small holes of subwavelength size is introduced. A sound field similar to that of a PS can be created by sound waves passing through a small hole. Despite the fact that a small hole in a rigid screen cannot be considered an ideal PS because it must radiate an ideal spherical wave, it is found that the sound field diffracted from a small hole is nearly identical to the sound field diffracted from an ideal PS (see the next paragraph for specific evidence), which is important because the reconstruction of a holographic image is largely dependent on the fields of diffraction units (such as holes or PSs) in the image plane. Furthermore, the transmitted wave from the small hole propagates forward, whereas the reflected wave has no bearing on image reconstruction. As a result, the concept of ASMs is introduced for the realization of holograms.

First, as shown in Figure 3a, the lateral profile of the sound field from a PS in the middle of a circular rigid boundary with a diameter of $30{\lambda }_0 \mathrm{is} \mathrm{investigated}$. On the red dashed line in Figure 3a, the lateral (x-y plane) energy profiles in the z-cut plane of $z=4{\lambda }_0$ are calculated. This distance ($4{\lambda }_0$) is beyond the evanescent region while remaining small enough that the simulation model does not consume excessive computational resources or memory space. Then, in the same model, the PS is replaced with holes of various diameters ($d=0.01{\lambda }_0, 0.02{\lambda }_0, 0.03{\lambda }_0, \dots , {\lambda }_0$) to allow plane sound waves to pass through. The lateral energy profiles of the same location (the red dashed line) generated by plane sound waves passing through holes of various diameters are calculated using the FEM-based numerical simulation. The lateral normalized energy profiles $E_{\mathrm{PS}/\mathrm{hole}}\left(x_k,0,4{\lambda }_0\right)/E_{\mathrm{PS}/\mathrm{hole}}\left(0,0,4{\lambda }_0\right)$ of the ideal PS and holes of various diameters are shown in Figure 3b. As can be seen, the smaller the hole’s diameter, the closer the energy profile approaches to the ideal PS. By calculating their RMSEs (shown by blue circles in Figure 3c), the degree of approximation between PS and holes of various diameters can be quantitatively verified

$$\mathrm{RMSE}=\sqrt{{\displaystyle \sum }_{k=1}^K{\left[\frac{E_{\mathrm{hole}}\left(x_k,0,4{\lambda }_0\right)}{E_{\mathrm{hole}}\left(0,0,4{\lambda }_0\right)}-\frac{E_{\mathrm{PS}}\left(x_k,0,4{\lambda }_0\right)}{E_{\mathrm{PS}}\left(0,0,4{\lambda }_0\right)}\right]}^2/\left(K-1\right)}$$

where $E_{\mathrm{hole}}\left(x_k,0,4{\lambda }_0\right)\propto p_{\mathrm{hole}}^2\left(x_k,0,4{\lambda }_0\right)$ and $E_{\mathrm{PS}}\left(x_k,0,4{\lambda }_0\right)\propto p_{\mathrm{PS}}^2\left(x_k,0,4{\lambda }_0\right)$ are the energy profiles of the hole and ideal PS, respectively, and $K=1500$ is the sampling number in the x-direction. The calculated RMSE values (blue circles) and relative transmitted energies (red dots) for holes of various diameters are shown in Figure 2c. Each blue circle represents an RMSE between $E_{\mathrm{hole}}\left(x_k,0,4{\lambda }_0\right)/\mathrm{MAX}\left(E_{\mathrm{hole}}\right)$ (color curves in Figure 2b) from a d-diameter hole and $E_{\mathrm{PS}}\left(x_k,0,4{\lambda }_0\right)/\mathrm{MAX}\left(E_{\mathrm{PS}}\right)$ (black curve in Figure 2b) from the ideal PS. Each red dot represents the logarithm of the simulated $E_{\mathrm{hole}}\left(0,0,4{\lambda }_0\right)/{E_{\mathrm{hole}}\left(0,0,4{\lambda }_0\right)|}_{d={\lambda }_0}$ from a d-diameter hole. The RMSE decreases as the hole size decreases, as shown in the figure, implying that the smaller the hole size, the better the effect of the hole simulating the PS. Furthermore, the figure shows that as the hole size decreases, the transmitted energy also decreases, and when the hole diameter is less than 0.2${\lambda }_0$, the energy decay rate is significantly accelerated. As a result, $d=0.2{\lambda }_0$ (i.e., 2 mm) is chosen as the diameter value of the ASM holes to balance the effect and efficiency.

Then, instead of the PSs in Figure 2b, the holes with a diameter of $d=0.2{\lambda }_0$ are employed to construct the ASM-based hologram shown in Figure 4a. The light gray area is the zero-thickness rigid screen and the small circles are the holes. Figure 4b shows the image of the ASM-based hologram calculated using numerical simulation based on a combination of the FEM and the BEM in the image plane. In the numerical simulation, all of the acoustic and geometric parameters of the holographic imaging system based on the ASM are the same as those of the holographic imaging system based on the PS array. The RMSE between this holographic image and the desired image shown in Figure 2a is 0.1755. As can be seen, this image closely resembles the desired image.

3.3. Theoretical Zero-Thickness Broadband Holograms

It is worth noting that the wave from a hole has no additional phase or amplitude modulations, which can physically remove the intractable modulation dispersion. As a result, even though it is designed for a specific frequency ($f_0=34 \mathrm{kHz}$), the proposed ASM should be feasible for a broadband hologram. Although the modulation dispersion has been removed, the intrinsic propagation dispersion [40] is provided by the propagation kernel ${\mathrm{e}}^{-\mathrm{i}{k}_0{R}_{m,n}}/R_{m,n}$ in Equation (1) describing the diffraction of a PS located at $G_{m,n}$. In principle, propagation dispersion cannot be eliminated, but it primarily affects a hologram’s imaging position and does not result in significant distortions or aberrations in the reconstructed holographic image [40,41,42]. The imaging distance ($z$) from a hologram varies with the radiation frequency ($f$) and follows the mathematical relationship [42]

$$\frac{2\pi f}{c_0}\left(\sqrt{{(x_{m,n}-x)}^2+{(y_{m,n}-y)}^2+z^2}-z\right)=\frac{2\pi f_0}{c_0}\left(\sqrt{{(x_{m,n}-x)}^2+{(y_{m,n}-y)}^2+z_0{}^2}-z_0\right)$$

Under the paraxial approximation, the mathematical relationship can be simply expressed as [42]

$$\frac{z}{f}=\frac{z_0}{f_0}$$

(2)

indicating that as the frequency increases, the imaging distance increases linearly.

The sound energy distributions in the image planes (according to Equation (2), the imaging distances ($z$) can be calculated as $60{\lambda }_0$, $80{\lambda }_0$, $100{\lambda }_0$, $144{\lambda }_0$, $180{\lambda }_0$ and $240{\lambda }_0$, respectively) of the ASM-based hologram are calculated using numerical simulation based on the combination of the FEM and the BEM when the incident frequencies ($f$) are, respectively, $f_0/2$, $f_0/1.5$, $f_0/1.2$, $1.2f_0$, $1.5f_0,$ and $2f_0$, as shown in Figure 5a–f. The figures show that the imaging effects in the six image planes are quite good. The RMSEs between these holographic images and the desired image shown in Figure 2a are 0.1942, 0.1813, 0.1771, 0.1743, 0.1849, and 0.1839, respectively. The graph (Figure 6) of the RMSE against frequency further shows that in the studied frequency range, all the RMSE values between the holographic images and the desired image shown in Figure 2a are modest and nearly constant.

4. Discussion

4.1. Selection of Geometric Parameters

Regarding the setting of the distance between the grid nodes, according to the Nyquist sampling theorem, the sampling condition in the calculation of the Rayleigh–Sommerfeld formula should satisfy [43]

$$\Delta x\le \frac{\lambda \sqrt{z^2+\mathsf{\Delta }{L}^2/2}}{\mathsf{\Delta }L}$$

where, $\Delta x$ is the sampling interval, which is the distance between the grid nodes (D), and $\mathsf{\Delta }L$ is the width of the diffraction field, which can be approximated as the diagonal length ($50\sqrt{2}{\lambda }_0$) of the proposed hologram. Substitute the relevant data into the above equation to obtain $D=\Delta x\le 23.9 \mathrm{mm}=2.39{\lambda }_0$.

Regarding the setting of the image size, if the desired image size is set too small, the image detail size may be smaller than the minimum resolution distance (according to the Rayleigh criterion, the minimum resolution distance should be $1.22\lambda z/a$, where $a$ can be approximately regarded as the diagonal length of the proposed hologram), resulting in a loss of image details. If the desired image size is set too large, the paraxial approximation will not be satisfied or the information content [21] of the hologram will be less than that of the image. Therefore, in general, for acoustic holographic imaging, the desired image size is set to be equal to or similar to the hologram size in most research [3,15,16,17,18,19,21,22,23,24,25,26,27,28].

4.2. Robustness

To verify the robustness of the proposed hologram, three cases that may cause errors are calculated using numerical simulation based on the combination of the FEM and the BEM. These three cases are, respectively, small displacements of holes from their ideal places, a dispersion of hole size, and an incidence wave with a wrong angle. In the case of small displacements of holes from their ideal places, every hole position is added a random error within $\pm 0.1{\lambda }_0$ in the x-direction and y-direction, respectively. Figure 7a–c show the calculated images for the three frequencies of $f_0/2$, $f_0,$ and $2f_0$ in this case. It can be seen from the figures that the images are almost identical to Figure 4c and Figure 5a,f but without the error. The RMSEs between the images and the desired image (Figure 2a) are 0.2012, 0.1788, and 0.1841, respectively. The RMSEs increase only slightly (rates of increase are less than 3.7%) compared to the non-error case. In the case of the dispersion of hole size, a random error within $\pm 0.1d$ is added to the diameter of every hole. Figure 7d–f show the calculated images for the three frequencies of $f_0/2$, $f_0,$ and $2f_0$ in this case. It can be seen from the figures that the images are almost identical to Figure 4c and Figure 5a,f but without the error. The RMSEs between the images and the desired image (Figure 2a) are 0.2009, 0.1780, and 0.1868, respectively. The RMSEs increase only slightly (rates of increase are less than 3.5%) compared to the non-error case. In the case of the incident wave with a wrong angle, the plane wave that was originally incident parallel to the z-axis is inclined to the y-direction by 5°. Figure 7g–i show the calculated images for the three frequencies of $f_0/2$, $f_0,$ and $2f_0$ in this case. Since the incident direction of the plane sound wave is inclined, the position of the image is shifted by $z\tan 5°$. The RMSEs between the images and the desired image (Figure 2a) are 0.2062, 0.1873, and 0.1868, respectively. The increases in RMSEs are also not significant (rates of increase are less than 6.8%) compared to the non-error case. To sum up, the proposed hologram offers a high level of robustness.

4.3. Actual Situation

To verify the effectiveness of the proposed hologram design method in practical applications, based on the combination of the FEM and the BEM, holographic imaging for the three frequencies of $f_0/2$, $f_0,$ and $2f_0$ is simulated in a case where the hologram is fabricated with a $0.1{\lambda }_0$-thick steel plate with holes, as shown in Figure 8a–c. It can be seen from the figures that the images are almost identical to Figure 4c and Figure 5a,f of the ideal rigid screen. The RMSEs between the images and the desired image (Figure 2a) are 0.2008, 0.1780, and 0.1853, respectively. The RMSEs increase only slightly (rates of increase are less than 3.4%) compared to the case of the ideal rigid screen.

It is worth noting that, the proposed hologram design method may not be well-suited for the background medium of water in practical applications, because most materials are not rigid enough relative to water.

5. Conclusions

In conclusion, a theoretical zero-thickness broadband hologram is proposed and comprehensively investigated. A design approach for an ASM capable of achieving the zero-thickness broadband hologram is offered in theory. The ASM is a hole-filled zero-thickness rigid screen. A genetic algorithm and the Rayleigh–Sommerfeld theory can be used to obtain the arrangement rules of holes in the ASM. It should have no thickness restrictions as long as the ASM-based hologram is sufficiently rigid in comparison to the background medium, and it can theoretically approach zero thickness. The intractable modulation dispersion could also be physically avoided because the wave from a hole has no extra phase or amplitude modulations, allowing the proposed ASM-based hologram to possibly function in any frequency band as long as the condition of paraxial approximation is satisfied. However, because propagation dispersion cannot be eliminated, one disadvantage is that the imaging distance from a hologram varies with the radiation frequency. Broadband holographic imaging with an excellent effect is achieved by using numerical simulation based on a combination of the FEM and the BEM. The proposed theoretical zero-thickness broadband hologram could open up new possibilities for expanding the application scenarios of acoustic holography.