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# Phase Hologram by Acoustic Metasurfaces

Xiao Guo, Haohan Zeng, Xinghao Hu, Youyu Mo, Xinzong Wang, Xiao Liang, Yusen Wu, Tingting Li, Haiyan Fan, Yifan Zhu, et al.

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## To cite this version:

Xiao Guo, Haohan Zeng, Xinghao Hu, Youyu Mo, Xinzong Wang, et al., Phase Hologram by Acoustic Metasurfaces. Small Structures, In press, 10.1002/sstr.202500354, http://dx.doi.org/10.1002/sstr.202500354

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https://hal.science/hal-05298905v1

Submitted on 6 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Phase Hologram by Acoustic Metasurfaces

Xiao Guo¹, Haohan Zeng¹, Xinghao Hu¹, Youyu Mo¹, Xinzong Wang¹, Xiao Liang¹, Yusen Wu¹, Tingting Li¹, Haiyan Fan¹, Yifan Zhu¹*, Hui Zhang¹*, Badreddine Assouar²*

Xiao Guo, Haohan Zeng, Xinghao Hu, Youyu Mo, Xinzong Wang, Xiao Liang, Yusen Wu,
Tingting Li, Haiyan Fan, Yifan Zhu, Hui Zhang

Jiangsu Key Laboratory for Design and Manufacturing of Precision Medicine Equipment,
School of Mechanical Engineering, Southeast University, Nanjing 211189, China

E-mail: yifanzhu@seu.edu.cn; seuzhanghui@seu.edu.cn

Badreddine Assouar

University of Lorraine, CNRS, Institut Jean Lamour, Nancy, F-54000, France.

E-mail: badreddine.assouar@univ-lorraine.fr

Keywords: phase hologram, acoustic hologram, acoustic metamaterials, amplitude
optimizing, acoustic holographic imaging, acoustic metasurfaces

Abstract

Conventional acoustic metamaterial-based holograms primarily focus on generating 3D images
through acoustic amplitude or intensity fields, which represent energy distributions perceivable
by an observer. In these earlier approaches, the phase is typically considered unimportant at the
image plane. However, from the perspective of information transmission, phase serves as a
valuable carrier, holding information equivalent to that of the amplitude channel. In this
research, we introduce the concept of an Acoustic Phase Hologram by metasurfaces. This
advanced approach decouples the phase from the amplitude distribution through an iterative
optimization design process. To validate our concept, we both numerically and experimentally
design and demonstrate a hologram designed in the phase channel. Compared to conventional
amplitude holograms, our approach offers significant advantages, including lower transmission
loss and strong anti-interference ability. This conceptual breakthrough holds strong potential
for applications across various domains, particularly in acoustic and optical communications,
and wave imaging.
# 1. Introduction

Holography is the reconstruction of the spatial distribution of phase and amplitude of wavefronts, achieving precise control and imaging of the sound field by reconstructing the diffraction limit sound pressure field. In recent years, this technology has demonstrated significant potential across a variety of applications, including particle manipulation,[1-5] medical imaging enhancement,[6,7] and ultrasound therapy,[8,9] which are of great importance with acoustic applications.[1-25] Using acoustic metamaterials to generate hologram is an efficient way for manipulating sound wave with high spatial resolution,[10,11] via pure phase modulation[12] or amplitude-phase modulations,[13-15] which have been considering as an important technology for sound field reconstruction, 3D acoustic holograms,[16,17] information encryptions,[18-22] etc.

Conventional acoustic meta-holograms focus on generating a 3D image with the acoustic or optical amplitude or intensity fields indicating the energy distributions which can be perceived by observer. In earlier cases, phase is a negligent physical quantity at the image plane. However, regarding the information communication aspect, phase is another valuable information carrier whose distribution contains information equivalent to amplitude channel. In previous studies on phase holograms, two-phase holograms designed using iterative Fourier transform algorithms were used to avoid independent modulation of the phase and amplitude of the incident sound wave, forming phase conjugation of the back-propagated target pattern on a predefined surface and achieving modulation of the phase and amplitude of the sound field.[23] However, there is an obvious coupling effect between phase and amplitude, which needs to be decoupled to study the independent modulation effect of phase on acoustic waves.

Meanwhile, phase holography has shown significant advantages in many aspects. Phase holography can be used in Holographic volumetric additive manufacturing. Phase encoding replaces traditional amplitude encoding, and millimeter scale complex cell scaffolds are printed in scattering hydrogels. The fidelity is improved and the cell survival rate is greater than 90%, breaking through the bottleneck of light efficiency and resolution in traditional volume manufacturing.[26] Simultaneously combining phase holography with generative AI, multi-channel virtual fluorescence images are generated from organoid phase images to quantify crypt formation and stem cell activity without the need for physical staining, promoting high-throughput drug screening.[27] In the recently reported metasurface-based dynamic acoustic holography, significant progress has been made in the application of programmable dynamic metasurfaces for precise manipulation of sound waves.[28]
Unlike previous methods that used monolithic acoustic holograms to reconstruct sound
fields,[1] we pre-designed a pure phase target image and achieved high-definition imaging of
the phase hologram by decoupling the amplitude and phase distributions in the hologram plane
and modulating the optimal amplitude compensation of the phase hologram. In this work, we
demonstrate the concept of “Acoustic Phase Hologram” (APH) enabled by metasurfaces. As
shown in **Figure 1**(a) and Figure 1(b), conventional amplitude hologram based on pure phase
modulation, require time-reversal technology with optimization algorithm such as Gerchberg–
Saxton(GS) method. Then, the proposed amplitude-phase modulation method can use time-
reversal technology with easy process. However, the proposed concept of APH here is totally
different since the predesigned and target images are all in the phase plane. Directly time-
reversal method cannot guarantee high fidelity image at phase plane owing to the fact that
amplitude and phase channel exists coupling effects, as shown in Figure 1(c). At the same time,
there is a periodic effect of phase, and constructing the target function of the algorithm in pure
phase results in errors and waste in the dimensionality of the function's solution space.
Therefore, the target function is constructed in terms of “-cos(phase)” (See Supplementary Note
1) and the initial amplitude is iteratively optimized to produce the final phase image with high
fidelity with the predesigned one, showing difference with conventional amplitude hologram.

By employing the Horned Lizard Optimization Algorithm (HLOA),[29] we iteratively seek
to optimize amplitude-compensation (AC) for phase hologram and utilize the time-reversal
method to calculate the required metasurface amplitude and phase distribution for the hologram
reconstruction. We encode the phase and compensate the amplitude information into designed
amplitude-phase acoustic metasurface[13] to realize our concept. Its mechanism relies on the
compensation of amplitude for phase holograms corresponding to complex environment, which
is different from the previous designs. In addition, we demonstrate that the hologram designed
in the phase channel has advantages compared to conventional amplitude hologram with robust
transmission and anti-interference ability. This could lead to applications in acoustic/optical
communications and imaging.
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Comparison between conventional amplitude hologram (AH) with the proposed phase hologram (PH). **a.** AH by phase modulation (PM). **b.** AH by amplitude and phase modulations (APM). **c.** PH by APM.</figcaption></figure>
## 2. Results

### 2.1. Amplitude Coupling/Decoupling and Phase Hologram.

The coupling between amplitude and phase holographic images is a common phenomenon in acoustic hologram.[20] As shown in Figure 2(a), the coupling between the phase field and amplitude field at the image plane is caused by the pre-designed amplitude image “A” and pre-designed phase image “P” using the time reversal method through acoustic metasurfaces at the hologram plane. When the pre-designed amplitude is set to be uniform, the phase image is imperfect, still influenced by the coupling effect as shown in Figure 2(b). The differential nature of the sound pressure can reveal the physical nature of the coupling between amplitude and phase. The fluctuation equation for the complex sound pressure can be separated into a real part $\frac{d^2A(x)}{dx^2}$ that controls the amplitude evolution and an imaginary part $\frac{d^2\phi(x)}{dx^2}$ that controls the dynamic phase amplitude-phase coupling (See Supplementary Note 2).

$$ \left\{ \begin{aligned} \frac{d^2 A(x)}{dx^2} &= A(x) \left( \left( \frac{d\phi(x)}{dx} \right)^2 - k^2 \right) \\ \frac{d^2 \phi(x)}{dx^2} &= - \frac{2}{A(x)} \frac{dA(x)}{dx} \frac{d\phi(x)}{dx} \end{aligned} \right. \qquad (1) $$

The amplitude is constant ($A(x) = A_0$) under plane wave excitation, and the real part of the equation holds constant. The linear variation of the phase under plane wave excitation can be expressed as $\phi(x) = kx + \phi_0$. Therefore, under plane wave excitation, the imaginary part dominates the real-time dynamic coupling between phase and amplitude. Through mathematical transformation, the imaginary part of the equation can be rewritten as $\frac{d}{dx}(A^2 \frac{d\phi}{dx}) = 0$. This indicates that acoustic energy remains constant along the propagation direction, so the conservation term of acoustic energy can be expressed as

$$ J(x) = A^2 \frac{d\phi}{dx} \qquad (2) $$

The linear dynamic coupling relationship between the amplitude $A(x)$ and the phase $\phi(x)$ in the imaginary part is described, so the coupling between the amplitude and the phase can be regulated by modulating the energy conservation relationship between the amplitude and the phase. In order to decouple the phase image with amplitude, the pre-designed amplitude needs to be iteratively optimized into a specific distribution compensation as shown in Figure 2(c). In this case, the final fine phase image is obtained at the image plane with a decoupling effect.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Amplitude coupling/decoupling and phase hologram. **a.** Time-reversal design for a pre-designed amplitude image “A” and pre-designed phase image “P” lead to coupling of phase and amplitude fields at image planes. **b.** When setting pre-designed amplitude uniformed, the phase image is imperfect, still influenced by the coupling effect. **c.** The fine phase holography image is obtained with decoupling effects by the iterative optimization of initial pre-designed amplitude compensation.</figcaption></figure>

## 2.2. Optimization of Amplitude Compensation and Phase Decoupling

The interaction between phase and amplitude in a hologram exhibits multidimensional coupling properties. The amplitude of the target pixel region in the hologram triggers significant local phase response (direct coupling), while adjacent pixels simultaneously generate secondary modulation through acoustic diffraction (indirect coupling). This coupling mechanism is manifested as a dual constraint mechanism in acoustic hologram. We construct a joint optimization model combining direct/indirect coupling to improve the energy focusing performance and image accuracy of the reconstructed acoustic field. This method (Figure 1 (c)) constructs an APM model through the time-reversal method, and compensates for the amplitude error in the time-reversal method using the HLOA algorithm combined with the principle of acoustic energy conservation (See Supplementary Note 3 for details). The resulting phase hologram has phase anti-interference characteristics and algorithm modulation ability, and its
performance is significantly better than traditional PM (Figure 1 (a)) and APM (Figure 1 (b)) methods.

Based on a pre-designed 87×87 pixels phase image of the letter “P”, we have constructed a multi-scale optimization framework that integrates the global diffraction efficiency and local fidelity to determine the optimal coupling AC for the phase distribution. At the same time, the HLOA algorithm was used to balance the performance between the global amplitude compensation of the hologram (Figure 3 (b1-b8)) and the local scale of the letter P region (Figure 3 (c1-c8)). The convergence curve shown in Figure 3(e) shows that the amplitude compensated gradually stabilizes during 1000 iterations, where the loss function of amplitude compensation is defined as

$$Loss = \frac{1}{N} \sum_{x,y} [1 - \cos(\phi_d(x,y) - \phi_a(x,y))] \quad (3)$$

Here $\phi_d(x, y)$ is the pre-designed phase distribution, $\phi_a(x, y)$ is the phase distribution after amplitude compensation, and $N$ is the total number of pixels in the image plane. Ultimately, we obtain phase holograms with optimal coupled amplitude compensation, as shown in Figure 3(b8).

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Optimization of amplitude compensation and phase decoupling. **a1.** Phase hologram. **a2-a8.** Phase hologram after coupling compensation amplitude. **b1.** Initial uniform amplitude. **b2-b8.** Iterative solution of coupling compensation amplitude. **c1-c8.** Compensate for local pixels at the position of letter P in amplitude holography. **d1.** Local pixel values at the position of letter P in the phase hologram of a1. **d2-d8.** Local pixel values at the position of letter P in the phase hologram after amplitude compensation coupling. **e.** Iterative convergence curve.</figcaption></figure>
## 2.3. Experimental Observation of Acoustic Phase Hologram with Metasurface

The measured phase fields of this research are accomplished in an anechoic chamber, where the desired acoustic metasurface shown in **Figure 4**(a) is prepared using a 3D printer with a printing accuracy of 0.1 mm (See Supplementary Note 4). The distance between Microphone #1 and the metasurface is 20 cm, and the scanning plane is set to the same size as the metasurface. In order to avoid the effect of the reflected wave, the incident wave is a pulse signal with a plane wave form. Microphone #1 is scanning to measure the acoustic field at hologram plane and referenced Microphone #2 is fixed. By using the two microphones method, [15] the phase fields are measured. Figure 4(b-d) and Figure 4(e-g) show simulated results of the phase hologram (PH) of the letters P and H, the optimized phase hologram (OPH) after amplitude compensation, and the experimental (Exp.) results, respectively. Figure 4(h-j) demonstrate the modulation of amplitude and phase of sound waves by metasurface structure. (See Supplementary Note 4 for the unit sizes. See Supplementary Note 5 for more complex holographic images.) The results show that the simulated phase image after the optimization (OPH) are finer than PH. The experimental measurements agree well with simulations, and validate the concept of the proposed phase hologram. By calculating the incident and reflected sound energy of the metasurface, we evaluate the total efficiency [30] of the phase holographic imaging, is about 80% for simulation.

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Experimental Observation of Acoustic Phase Hologram. **a.** Scanning the experimental platform. **b-d.** Phase hologram (PH), inset is predesigned phase image (11.5 × 11.5 cm²), Optimized</figcaption></figure>
Phase Hologram (OPH), and experimental results (Exp.) for the letter P, respectively. **e-g.** Corresponding results for letter H. **h, i.** Metasurface of encoding phase hologram coupling amplitude compensation and unit cell structure in metasurface. **e, f** Reflection amplitude and phase response of cells to parameters $h_1$ and $w$ under decoupling conditions.

## 2.4. Transmission Capability of Acoustic Phase Hologram

We have validated the transmission decay characteristics of the three hologram imaging methods, OPH, PH, and AH, over a range of imaging distances from 15 cm to 25 cm, using image of the letter “H”. Using the correlation with pre-designed images as an evaluation index, we have examined the image quality of each method with imaging distance. The correlation for evaluating the similarity between two images is calculated by [13]

$$ \text{Correlation} = \frac{\sum_j \sum_i (P_{i,j} - \bar{P})(H_{i,j} - \bar{H})}{\sqrt{((\sum_j \sum_i (P_{i,j} - \bar{P})^2)(\sum_j \sum_i (H_{i,j} - \bar{H})^2))}} \quad (4) $$

In the context where P represents the target image and H denotes the holographic image, $\bar{P}$ and $\bar{H}$ indicate the mean values of the respective images. The results show that the image correlation of OPH method in the pre-designed imaging range from 15 cm to 25 cm are kept above 90%, while the holograms of OPH at the imaging distances of 15 cm, 20 cm, and 25 cm are illustrated in Figure 5(f1-f3). The hologram obtained by the AH method shows a decreasing trend in image correlation as the imaging distance increases and the PH outperforms AH, showing better anti-decay performance. Figure 5(g1-g3) demonstrates the holograms of AH at 15 cm, 20 cm and 25 cm imaging distances. Further, at the imaging distance of 20 cm, we have experimentally validated the OPH method using a 3D-printed sample with a size of 12.5 × 12.5 × 2 cm³, as shown in Figure 4(h). In addition, Figure 5(c) and Figure 5(d) exhibit the OPH and AC obtained from the simulation, respectively. The results show that OPH has higher transmission capability than AH, which may be due to the fact that the diffraction effect accumulated with transmission distance influences more the acoustic wave amplitude than the phase.
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Transmission Capability of Acoustic Phase Hologram. **a.** Pre-designed phase image. **b.** Pre-designed uniform amplitude image. **c.** Optimized phase hologram. **d.** Compensated amplitude. **e.** Correlation between optimized phase hologram, phase hologram, and amplitude hologram with the pre-designed image at different imaging distances. **f (1-3).** Optimized phase hologram at 15 cm, 20 cm, and 25 cm imaging distances. **g (1-3).** Amplitude hologram at 15 cm, 20 cm, and 25 cm imaging distances.</figcaption></figure>
## 2.5. Anti-interference Capability of Acoustic Phase Hologram

We test the anti-interference capability of different hologram methods (PH, OPH, and AH) in complex noise environment. Preset noises with random amplitude and phase distributions are introduced into the pre-designed reference image, respectively, to simulate the actual noise interference environment and to evaluate the robustness of each method. The acoustic pressure $P$ after adding noise on the predesigned image can be calculated by

$$P = \sum_{l=1}^{N} [A_{0l}e^{i\phi_{0l}} + A_{nl}e^{i\phi_{nl}}] \quad (5)$$

Here $N$ represents the total number of pixels in the image. $A_{0l}$ and $\phi_{0l}$ denote the amplitude and phase of $l^{th}$ image pixel at the image plane. $A_{nl}$ and $\phi_{nl}$ denote the amplitude and phase of $l^{th}$ pre-set noise image pixel at the image plane.

The imaging results of AH (Figure 6(a)), PH (Figure 6(b)) and AH (Figure 6(c)) before and after adding noise were obtained numerically, respectively. The left side of Figure 6 shows the hologram without added noise, while the right side shows the hologram with added noise of the same intensity as the pre designed image. (See Supplementary Note 6 for holograms with different levels of noise intensity). To quantify the difference between the reconstructed image of each method and the reference image, we combined the correlation analysis and introduced the evaluation metric of Normalized Mean Square Error (NMSE) to compare the performance of each method under different noise intensity (Figure 6(e) and 6(f)). The NMSE is expressed as

$$\text{NMSE} = \frac{\|P-H\|_2^2}{\|P\|_2^2} \quad (6)$$

Here $P$ represents the theoretical hologram and $H$ denotes the fabricated hologram, with $\|\cdot\|_2$ denoting the Euclidean norm. Furthermore, the relative positioning between the image plane and the holographic plane also influences the quality of the image.

The results in Figure 6(d) and 6(e) show that the OPH method exhibits the best anti-noise performance under all noise levels, and its reconstructed image correlation always exceeds 90%. The correlation of the PH method is relatively stable, while the correlation of the AH method shows an obvious decreasing tendency with the increase of the noise intensity. This anti-noise performance of OPH, better than AH, shows the significant advantage of phase hologram in acoustic communications.
<figure><img src="image_9.png" /><figcaption>**Figure 6.** Anti-interference Ability of Acoustic Phase Hologram. **a.** Holograms before and after adding noise to AH. **b.** Holograms before and after adding noise to PH. **c.** Holograms before and after adding noise to the OPH. **d.** The distribution of correlation curves between three types of holograms and pre-designed images. **e.** The deviation curve distribution between three types of holograms and pre-designed images.</figcaption></figure>
### 3. Conclusion

In this research, we have reported on the coupling effects in holographic imaging between amplitude and phase channels. By introducing coupling amplitude compensation and iterative optimization, the phase holographic imaging is decoupled from the amplitude imaging. We have analytically, numerically and experimentally conceived and demonstrated the remarkable concept of the acoustic phase hologram by acoustic metasurface with simultaneous amplitude and phase modulations. By comparing the performances of OPH with PH and AH, we have shown that the phase hologram has significant advantages, compared to conventional amplitude holograms, with low loss when increasing the transmission distance, and low interference when increasing the noise intensity. These demonstrations show that phase meta-hologram could be a promising approach to tackle on the applications in the fields of wave imaging, information communication, etc. The current design requires continuous amplitude and phase modulations with many values. If we change the discretized phases and amplitudes into limited number, for example, for the amplitude of (0, 1) and phase (0, π) combination requires 2×2=4 different unit cells. Direct change continuous amplitude and phase modulations into discretized one will make the phase holographic imaging worse. Further optimization is needed for designing phase hologram by acoustic coding metasurfaces [28], which could be accomplished in the further works. Meanwhile, phase holographic imaging technology is expected to be applied to acoustic fields in different media simultaneously. However, it is necessary to redesign acoustic metamaterials for different media based on the differences in sound velocity, acoustic impedance, and attenuation of different media in order to achieve the expected results.

### 4. Experimental Section/Methods

**Numerical Simulation:** 3D numerical simulations are carried out by the finite element solver in commercial software COMSOL Multiphysics TM 6.2, using a high performance computing cluster. A full 3D geometry is used in the "Pressure Acoustics, Frequency Domain" module.

**Sample Fabrication:** All the samples are made of photosensitive resin, and are manufactured via 3D printing technique (SLA, 0.1 mm in precision). The scanning equipment is carried out using a scanning platform. The acquisition card model is VK-701H, and the probe model is BSWA MPA416. The illustrations in Figure 4 (b, e) are predesigned images with dimensions of 11.5 × 11.5 cm². The size of the metasurface is 11.5 × 11.5 × 2 cm³. The 3D printed sample size is 12.5 × 12.5 × 2 cm³, including a 1 cm thick size. The imaging distance between the target image plane and the 3D printed sample is 20 cm.
Experimental Setup:

(1) We first analyzed the amplitude phase coupling mechanism and derived the law of conservation of sound energy $\frac{d}{dx}(A^2 \frac{d\phi}{dx}) = 0$, indicating that the sound energy $J(x) = A^2 \frac{d\phi}{dx}$ remains constant along the propagation direction. The linear dynamic coupling relationship between the amplitude $A(x)$ and the phase $\phi(x)$ in the imaginary part is described, so the coupling between the amplitude and the phase can be regulated by modulating the energy conservation relationship between the amplitude and the phase (See Supplementary Note 2).

(2) This method constructs an APM model through the time-reversal method, and compensates for the amplitude error in the time-reversal method using the HLOA algorithm combined with the principle of acoustic energy conservation (See Supplementary Note 3 for details).

(3) Calculate the amplitude and phase that need to be modulated at each position of the hologram plane using the HLOA algorithm, and design a phase hologram metasurface. Under the incidence of frontal sound, the acoustic waves achieve a decoupling point within the internal structural space of the metasurface, enabling the unit cells to independently modulate the phase and amplitude of the metasurface reflected acoustic waves. The decoupling of amplitude and phase of sound waves on the metasurface allows for the modulation of all combinations of amplitude and phase across the entire range of [0, 1] and [0, $2\pi$] on the holographic plane (See Supplementary Note 4).

Taking the creation of a hologram of the letter "P" as an example, the process of constructing a phase hologram of the letter "P" can be decomposed into:

a. Design a phase image of the letter "P" on the pre-designed image plane, with the amplitude image is set to a full 1 amplitude.

b. Use time-reversal to calculate the amplitude and phase distribution on the hologram plane.

c. Using the amplitude and phase distribution designed by hologram plane to modulate the incident plane wave, achieving holographic imaging of the letter "P" on the imaging plane.

d. Using the pre-designed $-\cos(\text{phase})$ as the objective function, the HLOA algorithm combined with the principle of sound energy conservation is used to iteratively calculate the amplitude error of the pre-designed image plane. By compensating for the amplitude distribution of the pre-designed image, the minimum difference between the hologram of the image plane and the pre-designed image can be achieved.
**Acknowledgements**

This work was supported by the National Natural Science Foundation of China (No. 52272433), the Natural Science Foundation of Jiangsu Province (Grant No. BK20220798), Southeast University Interdisciplinary Research Program for Young Scholars (2024FGC1003).

**Data Availability Statement**

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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**Supporting Information**

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