# Bidirectional Lambertian scatterers using acoustic holograms for the simultaneous control of reflected and transmitted wavefronts

Eric Ballestero, Josep Rodriguez-Sendra, Vicent Romero-García, Jean-philippe Groby, Noé Jiménez

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Eric Ballestero, Josep Rodriguez-Sendra, Vicent Romero-García, Jean-philippe Groby, Noé Jiménez. Bidirectional Lambertian scatterers using acoustic holograms for the simultaneous control of reflected and transmitted wavefronts. Advanced Functional Materials, 2025, 35 (30), 10.13039/501100011033. hal-05406082

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Submitted on 9 Dec 2025

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# Bidirectional Lambertian scatterers using acoustic holograms for the simultaneous control of reflected and transmitted wavefronts

Eric Ballestero<fnref n="1" /> | Josep Rodriguez-Sendra<fnref n="2" /> |

Vicent Romero-García<fnref n="3" /> | Jean-Philippe Groby<fnref n="1" /> |

Noé Jiménez<fnref n="2" />

### Correspondence

Eric Ballestero, Av. Olivier Messiaen, 72085 Le Mans, France.

Email: eric.ballestero[a]univ-lemans.fr

### Funding information

This work was supported by ANR projects IA-GS (ANR-17-EURE-0014) and METARoom (ANR-18-CE08-0021), Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital de la GVA project CIAICO/2022/052, and the Ministerio Español de Ciencia e Innovación grants RYC2021-034920-I, PID2022-142719OB-C21, PID2023-146237NB-I00, and CNS2023-145707 funded by MCIN/AEI/10.13039/501100011033 and

MICIU/AEI/10.13039/501100011033.
Equally contributing authors.

Acoustic holography enables the creation of tailored spa-tiotemporal wavefield distributions. In this study, we present the design of acoustic holograms that simultaneously con-trol both the reflected and the transmitted fields. This is accomplished through the use of a bidirectional metasur-face, which encodes the phase of a desired wavefront in a discrete spatial distribution of monolithic sub-elements. This approach is employed in particular to design an opti-mised hologram that behaves as an ideal Lambertian scat-ter for both reflection and transmission. The design fol-lows a quadratic-residue sequence, with theoretical results validated both numerically and experimentally. It has been observed that this kind of structure produces a uniform scattering response, with a high diffusion coefficient of ap-proximately 0.7 and a correlation scattering coefficient of 0.98 in both transmission and reflection. The hologram demonstrates the potential to significantly reduce imaging artefacts in ultrafast ultrasound techniques in strongly re-verberating environments, as evidenced by examples.

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<footnote n="1">¹Laboratoire d'Acoustique de l'Université du Mans (LAUM), Intitut d'Acoustique - Graduate School (IA-GS), UMR CNRS 6613, Le Mans, France.</footnote>

<footnote n="2">²Instituto de Instrumentación para Imagen Molecular, Consejo Superior de Investigaciones Científicas (CSIC), Universitat Politècnica de València, València, Spain.</footnote>

<footnote n="3">³Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Universitat Politècnica de València, Spain.</footnote>
Over the past decade, the use of acoustic holograms [1, 2] has been a key research topic for wavefront shaping in acoustics and ultrasonics. Acoustic holograms are able to generate arbitrary acoustic fields, typically producing complex images, by encoding specific phase and magnitude patterns. Usually, acoustic holograms are encoded in a passive structure, e.g., in a 3D-printed structure, and when an acoustic wave is reflected or transmitted through it, the target field is generated. At the beginning, phase-only holograms where proposed to synthesize fields in transmission [1, 2, 3, 4, 5] and, later, in reflection [6, 7]. Several strategies for amplitude and phase modulation have been proposed for acoustics in air [8, 9] and ultrasonics in water [10, 3, 11]. Multiple images can be produced by the same hologram using different frequencies [12].

Holograms have been used for many practical applications [13], like particle trapping, manipulation and patterning [14, 15, 16, 17], 3D-printing [18, 19], exotic and vortex beam generation [20, 21, 22], haptic devices [23] or volumetric displays [24]. Mainly triggered by the wide accessibility to modern 3D-printed technologies, holograms have also found practical application in biomedical ultrasound as a low-cost strategy for wavefront engineering. Traditionally, acoustic beamforming relied on phased-array systems, where the phases and amplitudes are controlled using multichannel transducers and electronics. While these devices are used in common medical practice, the poor channel scalability of their radio-frequency electronics, currently up to 1024 channels, limits their wavefront shaping performance [25]. In contrast, the pixel density in acoustic holograms can be three or four orders of magnitude higher [26], enabling complex image formation into biological tissues. Holograms have been tested for aberration compensation in transcranial ultrasound therapy, where the acoustic image can match therapeutic targets in the central nervous system [26], controlling the spatial distribution of thermal patterns for hyperthermia treatments [27, 28, 29], controlling cavitation patterns [30], produce localized drug delivery in the brain in arbitrary areas [31], neuromodulation [32] or transcranial vortex generation [33]. For imaging, holograms have been proposed for single-element imaging using compressive sensing [34], for local cavitation monitoring using holograms and passive cavitation beamforming [35], or high-resolution four-dimensional imaging [36].

Holograms can also be used to tailor scattering patterns. For example, an acoustic vortex can be encoded in a holographic metasurface to produce a scattering with no specular reflection [37]. Indeed, encoding a reflected wavefield in a passive structure has been applied for decades in concert hall acoustics using acoustic diffusers [38, 39]. Using phase grating acoustic diffusers, the phase of the reflected sound varies according to the depth of a series of tailored wells acting as quarter-wavelength resonators. Other approaches include the use of Helmholtz resonators and slow sound metamaterials to reduce the thickness of these structures [40]. Concert hall acoustic diffusers have been proven to reduce the specular reflection of objects [41], or to help distribute wave-fields evenly in space and time [42]. However, these structures only work in reflection due to the reflecting nature of manufacturing materials for audible sound. Recently, structures have been proposed to control the scattering of reflected and transmitted sound waves in the audible regime [43, 44, 45]. Such structures could also be used directly in reflection situations for reducing unwanted specular reflections, e.g., reflections from surgical instruments during ultrasound-guided medical procedures [46].

In this work, we present a bidirectional asymmetric ultrasonic hologram that can simultaneously control both the reflected and transmitted wave-fields. To do that, we design a metasurface made of a discrete spatial distribution of elastic slabs, as shown in Fig. 1. This approach allows encoding a given wavefield in transmission and reflection into a single monolithic 3D-printed structure. In particular, we designed a bidirectional hologram to produce a diffuse scattering pattern, evenly distributing energy along the whole angular spectrum, acting as a Lambertian metasurface in analogy with perfect optical diffusers in optics [47]. This acoustic structure can be useful to reduce unwanted specular reflections for biomedical ultrasound applications, as shown in the example in the last section of this paper
<figure><img src="image_2.png" /></figure>

**FIGURE 1** (a) Scheme of the bidirectional ultrasonic hologram and the uniform scattering pattern, an incident plane wave in arbitrary uniform scattering profiles for each direction; (b) Close-up of the lens; each monolithic block has a lateral size of $\Delta x \times \Delta y = 700 \ \mu m \times 700 \ \mu m$, and the total width of the structure is $2a$. (c) Representation of the 1D Transfer Matrix Method diagram of a monolithic block of total length $L$ composed of a solid slab (polymer) and the surrounding fluid layers (water).

using ultrafast ultrasound imaging.

## Results

### Bidirectional Monolith Optimization

We consider an ultrasonic hologram composed of a discrete spatial distribution of building blocks made of monolithic elastic blocks (polymer) of different height embedded in a fluid (water). An illustration of the hologram is shown in Fig. 1 and the mathematical derivations for asymmetric wave propagation in and out of each single monolithic block, along with material properties, can be found in the Materials & Methods section. However, one important operation before designing this hologram is to find a set of monolithic polymer block lengths, $h$, as well as the length of the water blocks $d_1$ (starting from L down to L/2) and $d_2$ (from 0 up to L/2), given a total system height $L = d_1 + h + d_2$, in order to cover all the combinations for the phase of the reflection, $\varphi_R$, and transmission, $\varphi_T$, coefficients. While the functions $R(d_1, h, d_2)$ and $T(d_1, h, d_2)$ are continuous, we propose to obtain a finite set of combinations [9] by discretizing the phase-space in $N_{\varphi}^2$ possible values.

In this way, we look for the geometrical parameters $d_1(\varphi_R, \varphi_T)$, $h(\varphi_R, \varphi_T)$, and $d_2(\varphi_R, \varphi_T)$ that generate all the combinations of the phases $\varphi_R(i)$, and $\varphi_T(j)$, where $i = 0, ..., N_{\varphi}$, $j = 0, ..., N_{\varphi}$, $\varphi_R(i) = -\pi + 2i\pi/N_{\varphi}$ and $\varphi_T(j) = -\pi + 2j\pi/N_{\varphi}$. We choose $N_{\varphi} = 21$ as the total amount of discrete radian steps, providing in this case resolution steps representing approximately 5% of the total angular unit circle. This choice is here justified by the fact that (i) in this work we are not really interested in finding the minimal length of material that would be required to provide any phase combination but rather one that could fit depending on a set of custom application-driven manufacturing constraints, and (ii), the phase discretization resulting from a very small manufacturing resolution could significantly increase the phase-space matrix, and thus proportionally lengthen the following optimization process for finding phase-space combinations, as discussed below.
<figure><img src="image_3.png" /><figcaption>FIGURE 2 (a) Multi-objective Pareto front for a single pixel of the hologram. The selected minima of the Pareto front is represented in black. (b) Cost function error $ε_t = ε_1 + ε_2$ at the end of the multi-objective minimization, as a function of the phase of the reflection and transmission coefficients. (c) Magnitude and (d) phase of the 1D reflection and transmission coefficients of an optimized polymer block length which satisfies $φ_R = φ_T = -1.886$ rad, obtained analytically (TMM) and numerically (FEM). (e-h) Phase-space representation of the optimized lengths $h(φ_R, φ_T)$, $d_1(φ_R, φ_T)$, $d_2(φ_R, φ_T)$, and $L = h + d_1 + d_2$, respectively.</figcaption></figure>

To find the geometrical parameters ($h$, $d_1$, $d_2$), a multi-objective minimization method based on genetic algorithms is used, with lower and upper boundaries on $d_1$ and $d_2$ such that $d_n \in [0, 0.95L]$ in order to maintain a common material section between blocks for manufacturing concerns. The multi-objective optimization algorithm would then yield a set of plausible solutions that can achieve the required couple of phase combinations $φ_R(i)$ and $φ_T(j)$. Indeed, more than one solution satisfying the optimization goals generally exists in this problem, i.e., the set of solutions composing the Pareto front. The multi-objective minimization algorithm would thus minimize the following cost functions for the i-th and j-th combination

$$ε_1(h, d_1, d_2) = |φ_R(i) - \arg[R_{TMM}(l, d_1, d_2)]| / 2π, \qquad (1)$$

$$ε_2(h, d_1, d_2) = |φ_T(j) - \arg[T_{TMM}(l, d_1, d_2)]| / 2π, \qquad (2)$$

where $arg(·) = tan(Im(·)/Re(·))$ symbolizes the phase angle, $φ_R(i)$ and $φ_T(j)$ are the objective phases, and $R_{TMM}$ and $T_{TMM}$ are the reflection and transmission coefficients obtained using the TMM model given by Eqs. (10-11). The 2π denominator is used for normalizing the cost function in an interval [0, 1]. Then, for each i-th and j-th element of the target phase-space, the multi-objective optimization problem can be formulated as

$$\min_{(h,d_1,d_2) \in \mathbb{R}} [ε_1(h, d_1, d_2), ε_2(h, d_1, d_2)]. \qquad (3)$$
Although the Pareto set represents a set of plausible geometrical parameters than can fit the minimization problem, only one solution per combination is needed to design the structure. Here, we will take the solution that presents a minimum distance from the ideal point ($ε_1, ε_2$) = 0, as shown in Fig. 2 (a). This selection can only be performed since the solutions and their relative cost functions are of the same type. Figure 2 (b) depicts the sum of the cost functions, $ε_t = ε_1 + ε_2$, for the retrieved combinations of $φ_R$ and $φ_T$. As the cost functions have been normalized on a [0, 1] interval, it can be seen that the sum of the cost functions is significantly low, with sparse maxima about two orders of magnitude below the maximum possible error (i.e., 1) and a vast amount of minima about three to six orders of magnitude lower. This demonstrates an excellent ability of the system to reproduce any kind of phase combinations.

A Finite Element Method (FEM) numerical validation using COMSOL Multiphysics® was performed for a selected phase-space matrix element, corresponding to $φ_R = φ_T = -1.886$ rad. Figures 2 (c,d), display the magnitude and phase of reflection and transmission coefficients, respectively. An excellent agreement can be observed between the TMM theory and the FEM simulation. Figure 2 (c) shows that the system presents maxima and minima of transmission coefficient (and conversely for the reflection coefficient). This implies that some hologram cells could potentially have a quasi-null reflection coefficient, letting all the ultrasonic waves pass. Due to the high variability of solutions that will be chosen, it is assumed that a relatively even distribution will be ultimately attained between full transmission and partial transmission blocks, i.e., blocks with marginal reflection magnitudes. Concerning the phases of both reflection and transmission coefficients, it can be seen in Fig. 2 (d) that both phases converge toward the desired value (-1.886 rad) at the design frequency (1.1 MHz).

Figures 2 (e-h) show the retrieved geometrical parameters $h, d_1, d_2$ and $L = h + d_1 + d_2$ obtained for each phase combination, respectively. A remarkable feature of Fig. 2 (e) can be seen, where the length of polymer blocks increases almost linearly in function of the gradient of the transmission coefficient phase, and drops to minimal values (similar to the ones at $φ_T = -π$) around a transmission phase of $π/2$, which is linked to a phase shit of the transmission coefficient around such lengths $h$. Figures 2(f,g) representing the lengths $d_1$ and $d_2$ show an inversely correlated pattern with $h$, i.e., large $h$ values lead to small $d_1$ and $d_2$ values. This is expected as the system was designed in a way that $L = h + d_1 + d_2$. This is shown in Fig. 2 (h) by a uniform total length $L$ distribution over the phase-space.

## Design of a Quadratic Residue Hologram

In order to design a hologram that can behave as an ideal bidirectional diffuser, we look for a spatial phase distribution for both reflection and transmission coefficients such that the spatial Fourier transform of said distribution present a flat magnitude, i.e., energy is evenly distributed along the whole angular spectrum. There are many number theoretic sequences that can help design such phase distribution, such as the maximum length or quadratic residue sequences [38]. In this work, we make use of a quadratic residue (QR) to generate evenly spaced diffraction lobes with even energy (see Materials & Methods for more details). A 2-dimensional QR-sequence composed of $N \times N$ elements can be expressed as [48]

$$s(n, m) = (n^2 + m^2)\mod(N), \quad (4)$$

where $n = 0, \dots, N$ and $m = 0, \dots, N$ are integers and indices of the sequence for the $n$-th and $m$-th element in the $x = n\Delta x$ and $y = m\Delta y$ directions, respectively, and mod refers to the modulo operator. When these sequences are proportional to the phase of a complex field in the form

$$\xi(x, y) = \exp(is(x, y)\pi/N), \quad (5)$$
130 the complex function $\xi(x, y)$ presents a flat spatial Fourier transform [39]. Therefore, these sequences can be used to design bidirectional acoustic diffusers when

$$ \varphi_T(x, y) = \varphi_R(x, y) = s(x, y)\pi/N, \qquad (6) $$

132 and N is a prime number.

133 The last step for designing the bidirectional hologram remains to go over each element $(n, m)$ of the hologram, located at coordinates $(x, y)$, and find the geometrical parameters $(h, d_1, d_2)$ associated to the phase combination element $\varphi_R(x, y)$ and $\varphi_T(x, y)$ given by Eq. (6). These geometrical parameters, $d_1(\varphi_R, \varphi_T)$, $h(\varphi_R, \varphi_T)$, and $d_2(\varphi_R, \varphi_T)$, are provided by the above-mentioned optimization procedure.

137 | Numerical validation

138 A numerical validation was performed using a pseudo-spectral time-domain (PSTD) simulations method with a k-space corrector [49]. This method provides a low numerical dispersion over coarse computational grids, and it is convenient for use in ultrasonics where the domains can cover hundreds of wavelengths. A bidirectional ultrasonic hologram using $N = 11$ and $M = 3$ repetitions was designed, considering water with properties $c_0 = 1500$ m/s and $\rho_0 = 1000$ kg/m³ for the surrounding media and a photo-polymer (Clear, FormLabs, USA) with properties $c_p = 2599$ m/s and $\rho_p = 1186$ kg/m³, measured experimentally [50]. This material choice gives a ratio of $Z_p/Z_w = 1.67$, validating the model assumptions. The lateral size of each pixel was set to $\Delta x = \Delta y = 700$ µm, and the frequency was set to 1.1 MHz.

146 Figure 3 (a) shows the 2D phase distribution in reflection, $\varphi_R(x, y)$. As shown previously in Fig. 2 (b), the minimal deviations of the multi-objective cost function mean that the designed phase profile for each side would almost not

<figure><img src="image_4.png" /><figcaption>FIGURE 3 Theoretical 2D surface phase (a,d) and magnitude (b,e) distributions of the reflected and transmitted fields, respectively, of a 11 × 11 2D QR sequence repeated $M = 3$ times. (c,f) Polar scattered fields of the QR sequence in reflection and transmission. (g-l) Filtering evanescent components out of the QR sequence. (m-r) Pseudo-spectral time-domain (PSTD) simulations.</figcaption></figure>
deviate from the target. The corresponding magnitude, as the result of the TMM model, is shown Fig. 3 (b). Ideally, the sequence presents a flat angular spectrum, and even considering variations in magnitude, the scattered field in the far field presents a flat spectrum distribution with high *N* peak values due to the periodicity of the repetitions, as shown in Fig. 3 (c). A similar behavior is observed in transmission mode, note $\phi_T(x, y) = \phi_R(x, y)$, but in this case the magnitude of the transmission coefficient is closer to the unity, see Figs. 3 (d-f), as a result of the chosen photo-polymer, whose properties are similar to water.

An important remark is that the theoretical phase distribution is discrete, and phase values given by the ideal sequence, Eq. (6), are constant along each unit block. Of course, due to the continuity of pressure and particle velocities all over the theoretical surface, the pattern of the phase distribution in the near field may not show such discrete profile, as those presented in Fig. 3 (a,d). In fact, this implies field discontinuities, that can only be achieved using evanescent components. If we filter the evanescent components using the angular spectrum method [50, 51], we obtain the field shown in Figs. 3 (g,h) for reflection and Figs. 3 (j,k) for transmission. At first, in the near field the filtered fields seem to differ from the theoretical QR sequence. However, evanescent components do not contribute to the far field and, therefore, the corresponding scattered field is quite close to the ideal, as shown in Figs. 3 (i,l).

Indeed, the simulated reflected and transmitted fields, shown in Figs. 3 (m-n,p-q), cannot describe the abrupt and discontinuous phase profile of the QR sequence. These simulated profiles are measured at the exit of the hologram, and many of the evanescent components may have lost their amplitude. However, the propagating modes of the simulated field actually capture the angular spectrum of the sequence. The near-field magnitude also presents the squared pattern, which emerged as a result of the interference of the multiple scattered waves traveling in $N^2$ directions covering the whole angular spectrum. As its angular spectrum is captured, the scattered far field also resembles the theoretical one as shown in Figs. 3 (o,r) for the reflected and transmitted modes. In addition, it has to be noted that the evanescent coupling at the interfaces and within the hologram caused by the coupling between neighboring reflected or transmitted waves, which are not accounted for in this model, may introduce some changes with respect to target values. This could explain the variations observed between theoretical and numerical phase distributions.

172 | Experimental Far-field scattering

For the experimental validation, a larger 3D-printed hologram, with $N = 53$ repeated $M = 3$ times (see Fig. 1 (a)) was designed for a frequency of 1.1 MHz. The hologram was built using stereolithography technique (SLA). A high *N* value was taken in order to produce a large structure for ultrasonic measurements, where no more than *M* = 3 repetitions were desired to reduce diffraction grating lobes due to periodicity. A larger *N* also helps design a hologram with a much more uniform spatial Fourier transform due to the nature of pseudo-random sequences. Acoustic pressure was measured in front of the hologram in the *x* and *y* directions in order to map the near-field pressure plane distribution, at *z* = 0.1 m. Note the Rayleigh diffraction distance of a disc is $ka^2/2 = 1.44$ m. A 2-dimensional spatial Fourier transform, using Eq. (12), was performed over the measured surface yielding the polar distribution of the energy in the far field (see Materials & Methods section for complete details).

Figures 4 (c-f) show the normalized scattered energy distribution from both sides of the structure. The video in the supplementary material 1 shows the complete polar distribution. First, Fig. 4 (c) shows the theoretical scattering of an ideal rectangular panel, where the scattering displays a narrow angular spectrum, i.e., most of the energy is reflected/transmitted in a beam at normal direction showing a narrow beam-width, as a result of the large value of the quantity $kd \gg 1$ and the absence of surface roughness. However, when the hologram is used, as shown in Figs. 4 (d-f) using the theory, simulation and experiment, respectively, the energy is scattered in all directions covering a broad angular spectrum. This result is expected as the design field distributions present a spatial Fourier transform
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

**FIGURE 4** Experimental water tank measurement setup of the 3D-printed diffusing hologram in (a) reflection and (b) transmission modes. (c) Scattering of a reference flat panel of same dimensions than the hologram. Scattered polar pressure distribution obtained (d) theoretically, (e) by simulation, and (f) experimentally, respectively. (g) Far-field polar pressure distribution diagram. See Supplementary Material Video 1 for a 3D representation of the whole polar response.

with uniform magnitude. There are some discrepancies, notably around ±25°, which may be caused by three factors. First, the aforementioned theoretical limitations impacting the surface phase distribution and ultimately the scattered pressure field. Second, the limitations of the experimental measurements on finite transverse plane located at some distance from the source, that results in limitations to capture the scattering at grazing angles. A detailed plot is given by Fig. 4 (g) for θ = 0 rad. Lastly, neither the simulation model nor the theory takes into account for transverse waves and elasticity of the solid material beyond pressure waves. However, one important remark is that since wavefront incidence angle is normal to the structure surface, the expected mode conversion from pressure waves to shear waves is low, and the contribution of transverse modes to the total scattered energy might remain low. In this sense, the most important contribution to the scattering comes from pressure waves and, as we observe, experimental results agree with the model. The scattered pressure fields behave as intended, despite the aforementioned discrepancies and measurement noise, where both sides of the hologram scatter waves uniformly; in similar fashion than the theoretical and numerical estimations.

To quantify the uniformity of the scattered energy, we estimated the diffusion coefficients $d^R$ and $d^T$, and the normalized diffusion coefficients, $d_n^R$ and $d_n^T$, and the correlation scattering coefficient $\delta^R$ and $\delta^T$, for the reflection
<table><caption>**TABLE 1** Scattering coefficients of the bidirectional hologram obtained by the theory, simulation and experiments. The diffusion coefficient of the flat panel, used for normalization, is $d_r^R = d_r^T = 0.022$ (note that $d_n^R = d_n^T = 0$ and $\delta^T = \delta^R = 0$ for the rectangular reference panel).</caption><thead><tr><th rowspan="2"></th><th colspan="3">Reflection</th><th colspan="3">Transmission</th></tr><tr><th>d<sup>R</sup></th><th>d<sub>n</sub><sup>R</sup></th><th>δ<sup>R</sup></th><th>d<sup>T</sup></th><th>d<sub>n</sub><sup>T</sup></th><th>δ<sup>T</sup></th></tr></thead><tbody><tr><td>Theory</td><td>0.672</td><td>0.665</td><td>0.939</td><td>0.677</td><td>0.669</td><td>0.982</td></tr><tr><td>Simulation</td><td>0.744</td><td>0.738</td><td>0.998</td><td>0.767</td><td>0.761</td><td>0.999</td></tr><tr><td>Experiment</td><td>0.658</td><td>0.651</td><td>0.999</td><td>0.726</td><td>0.721</td><td>0.975</td></tr></tbody></table>

and transmission problems. These parameters are defined in the Materials & Methods section. The results are listed in Table 1. First, the flat panel, used as a reference, results in a low diffusion coefficient, $d^R = 0.022$. This parameter is the unity when the energy of the scattered waves is uniform along all directions, and null when the scattering function is a Dirac delta function. The large $kd$ product results in a specular reflection. However, when the hologram is used, the diffusion coefficient is increased to about $d^R \approx d^T \approx 0.7$. The experimental and simulated values agree with the theory. Knowing that a diffusion parameter of unity is impossible to reach using periodic repetitions of the panel due to grating lobes [52], a value around 0.7 denotes a strong scattering behavior, akin to one obtained through a Lambertian scattering surface. As the scattering of the reference panel is highly directive, the normalized values are close to the former, $d_n^R \approx d_n^T \approx 0.7$, indicating that the hologram produces a scattering pattern much uniform than an ideal flat reflector of same dimensions.

However, the diffusion coefficient does not indicate in which particular direction the scattering is more uniform. Therefore, we calculated the correlation scattering coefficient, which indicates the dissimilarity of the scattering function of the structure and the reference scattering function of the flat reflector. We observe values close to $\delta^R \approx \delta^T \approx 0.98$, for experimental, simulated and theoretical scattering functions. If this value is close to zero, then the scattering function will be identical to the specular reflection of the reference. However, the retrieved values, close to the unity, indicates a strong reduction of the specular reflection has been produced. As can be seen in the polar responses, they are less uniform at grazing angles. In the following, we will use the structure to reduce the specular reflection in a ultrasound imaging application.

## Application for ultrasound imaging

Finally, the hologram was used to improve image quality in ultrafast ultrasound imaging systems. A 256-channel ultrasound acquisition platform (Vantage 256, Verasonics, Kirkland, USA) was used. A 128-channel, 2-7 MHz probe (IP-104, Sonic Concepts, USA), as shown in Fig. 5, was connected to the Vantage. An ultrafast plane wave imaging procedure was set using a frame rate of 10,000 frames per second, and radio-frequency signals were acquired using a 28-MHz sampling frequency. By using ultrafast ultrasound sequences, the dynamic processes can be studied in a fine temporal scale, enabling techniques such as functional ultrasound, ultrasound localization microscopy, or ultrasound elastography. However, these high frame rates can result in interferences when imaging in highly reverberating environments, such as a simple water tank. Here, experiments were set, one for multiple bubble imaging and localization, and a second for soft-tissue imaging.

First, bubbles were generated in a water tank using a 30-ml Luer slip syringe. Using ultrafast plane-wave imaging, echoes were acquired at 10,000 Hz and frames were composed using delay-and-sum. The resulting images, as
<figure><img src="image_10.png" /><figcaption>**FIGURE 5** (a) Example of an ultrasound image and (b) setup obtained in a water tank in which the reflections at the bottom of the tank produces interferences on the image. (c-d) Image by using the bidirectional hologram at the bottom of the tank to improve image quality (See supplementary material 2 for the video). (e-f) Individual bubble localization and tracking, (g-h) when reflections are present localization is hardly achieved. (i-j) Individual bubble localization and tracking using the metasurface, (k-l) using the bidirectional hologram, bubbles can be accurately localized and tracked. (m-n) Example of image of a soft tissue (tip of a finger) and surrounding bubbles without using the diffuser. (o-p) Same setup using the bidirectional hologram (See supplementary material 3 for the video).</figcaption></figure>

shown in Figs. 5 (a-b), show intense artifacts. These interferences are mainly produced by specular reflections on the reflecting walls, and cannot be efficiently filtered by singular-value-decomposition filters. When these data is used for bubble localization and tracking, the retrieved bubble localization image is noisy and these traces are not clear or even absent, as shown in Figs. 5 (e-f). To mitigate this, the diffusing metasurface is placed at the bottom of the tank. Then, energy is evenly reflected in all directions, greatly reducing the energy specular component. In this way, image quality is restored, and individual bubbles can be clearly imaged without interferences, as shown in Figs. 5 (c-d). This artifact-free images can then be used for bubble localization and tracking, as shown in Figs. 5 (i-l). Using the metasurface, bubble traces are retrieved with accuracy, enabling bubble position estimations in strong reverberating environments. Because motion is important in ultrafast imaging, a video is provided in the Supplementary material 2 where it is shown that bubble positions can be tracked in this reverberating environment only when using the hologram. It is worth noting here that the the scattered energy is given by $(1 - \alpha)\delta^R$, and the energy of the specular reflection is given by $(1 - \alpha)(1 - \delta^R)$, where $\alpha = 1 - |R|^2 - |T|^2$ is the absorption coefficient [53]. Therefore, the energy reflected in the specular reflection is lower than the energy of the total scattered field, i.e., the ratio of energies
is given by $(1 - \delta^R)/\delta^R$. For example, the ratio of specular to total scattered energy is –30 dB for the experiment ($\delta^R$ = 0.999), while it becomes –27.0 dB in the simulations and –12.1 dB in the theoretical calculations. In this way, an efficient diffuser have the potential to reduce artifacts from reverberation by reducing the energy of the specular reflections, as demonstrated by the imaging experiment.

The same procedure can be used to image in-vivo tissues in the absence (Figs. 5 (m-n)), and using the hologram (in Figs. 5 (o-p)) to produce diffuse reflections. Here, the tip of a finger is imaged. Note that the finger is a complex structure composed of soft (skin, muscles, tendons) and hard (bones) tissues. Therefore, ultrasound imaging using a standard delay-and-sum beamformer is only accurate to image the external soft-tissue layers of the finger. In this case, aberration free image can also be obtained in this environment just after introducing the hologram. In this case, some remaining bubbles can be also identified. Note that in ultrasound imaging the gray level usually indicates the energy of the echoes rather than their amplitude, and it is usually represented in a logarithmic (or highly compressed) color scale. Therefore, any small-amplitude echo, as those produced by the specular reflections, will be displayed. The absence of image artifacts indicates that the structure is indeed evenly scattering the energy. A video is provided in the Supplementary material 3 for the case of soft tissue imaging.

## Discussion

This work presents a bidirectional asymmetric ultrasonic hologram that produces a uniform scattering profile over the whole angular spectrum, where this flat structure is designed simultaneously for reflected and transmitted waves. The design framework of such hologram follows a multi-objective optimization procedure to find the geometrical parameters of the structure that produce the target reflection and transmission phase values. This method allows the pixel-wise tuning of the phase of the transmitted and reflected field at the two interfaces of the structure. In this way, the optimized building blocks can be used to replicate any desired reflection and transmission phase profiles. In the work herein, the case of an ideal holographic diffuser based on a quadratic residue sequence has been demonstrated. Theoretical, numerical and experimental results show that the scattering function is uniform. The results show a normalized diffusion coefficient around $d_n \approx 0.7$ and a correlation scattering coefficient of $\delta \approx 0.98$, indicating that the scattering function is uniform, and almost no energy is scattered in the specular direction.

An example application of this ultrasonic diffuser is provided to reduce the reverberation artifacts during ultrafast imaging in a water bath. Ultrafast ultrasound imaging has enabled groundbreaking ultrasound imaging techniques such as ultrasound localization microscopy for super-resolution imaging [54], functional ultrasound to image brain activity in real-time [55], or advanced elastography methods [56]. These techniques rely on the highly sensitive tracking of micro-bubbles, red blood cells, or tissue movements, respectively. However, in reverberating environments, such as a simple water tank, strong reflections can introduce artifacts, as shown in the application example. By using the acoustic hologram, energy is scattered uniformly by the diffusely reflecting surface, the acoustic intensity of the specular reflection is strongly reduced and, then, the image artifacts during ultrafast ultrasound imaging are minimized. In this way, by using this structure, the bubbles can be tracked with accuracy and tissues can be imaged without artifacts. This kind of tunable bidirectional hologram could be further optimized to create tailored scattering wave fields for one or both directions, with potential applications ranging from biomedical ultrasound therapy [26], to low-cost ultrasonic imaging systems by using compressive sensing [34].
## Materials and Methods

### Model and Theory

We consider an ultrasonic hologram composed of a discrete spatial distribution of building blocks made of a monolithic elastic blocks (polymer) with different height embedded in a fluid (water). The scheme is shown in Fig. 1.

On the one hand, we consider first a 1D problem for each constituting building block of the hologram. Unit cells are made of a monolithic polymer block of length $h$, surrounded by two water blocks of lengths $d_1$ and $d_2$, respectively as shown in Fig. 1 (c). The sound pressures, $p$, and normal particle velocities, $v_z$, at $z = 0$ and at $z = L$ can be modeled by means of the Transfer Matrix Method (TMM) [57]. Due to the lateral dimensions of the building blocks ($\Delta x = \Delta y = 700\mu m$), and normal incidence, we assume plane wave propagation for each element. This implies that we consider each building block is independent of each other in the structure and considered as an equivalent fluid. Such approximation can be made due to the small difference in impedance between the fluid and solid media, with a polymer-to-water impedance ratio $Z_p/Z_w \approx 1$. In addition, shear waves in the solid can also be neglected due to normal incidence. Therefore, the transfer matrix $T$ of the building block can be derived, and the reflection and transmission coefficients calculated. The system can thus be written as

$$ \begin{bmatrix} \rho \\ v_z \end{bmatrix}_{z=0} = \mathbf{T} \begin{bmatrix} \rho \\ v_z \end{bmatrix}_{z=L} = \begin{bmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{bmatrix} \begin{bmatrix} \rho \\ v_z \end{bmatrix}_{z=L}, \quad (7) $$

where $T$ is given by the product of the transfer matrices of the different media constituting each building block in the hologram, i.e.,

$$ \mathbf{T} = \mathbf{T}_{d_1} \mathbf{T}_h \mathbf{T}_{d_2}, \quad (8) $$

where $T_d$ and $T_h$ are the transfer matrices of the fluid and solid blocks, respectively, as shown in Fig. 1 (c). The total length $L = h + d_1 + d_2$ is constant for all the designed building blocks. Each transfer matrix block, $T_i$, can be written as a homogeneous and isotropic fluid, expressed as

$$ \mathbf{T}_n = \begin{bmatrix} \cos(k_n z_n) & i Z_n \sin(k_n z_n) \\ i 1/Z_n \sin(k_n z_n) & \cos(k_n z_n) \end{bmatrix}, \quad (9) $$

where $i = \sqrt{-1}$, $n = d_1, d_2, h$, $k_n = \omega/c_n$ and $Z_n = \rho_n c_n$ are the wavenumber and characteristic impedance of the medium, respectively, with $\omega$ the angular frequency, $c_n = \sqrt{K_n/\rho_n}$ the propagating sound speed, $K_n$ the bulk modulus of the medium, and $\rho_n$ the density, in the n-th media (water when $z_n = d_1$ or $z_n = d_2$, and polymer when $z_n = h$).

Considering the continuity of pressure and velocity at $z = 0$ and $z = L$ in a reciprocal transmission problem (meaning that the system is linear and time-invariant and the transmission does not depend on the direction of the incident wave) and accounting for the asymmetry of the hologram (meaning that there is no forced mirror symmetry with respect to $z = L/2$), the transmission and upward reflection coefficients, $T$ and $R$ (see Fig. 1(c)), can be expressed in function of the transfer matrix elements outlined in Eq. (7), as [57]

$$ T_{\text{TMM}} = \frac{2}{T_{11} + T_{12}/Z_0 + T_{21} Z_0 + T_{22}}, \quad (10) $$

$$ R_{\text{TMM}} = \frac{T_{11} + T_{12}/Z_0 - T_{21} Z_0 - T_{22}}{T_{11} + T_{12}/Z_0 + T_{21} Z_0 + T_{22}}. \quad (11) $$
It must be noted that the respective phases of the transmission and reflection coefficients are separated by a travel distance L, which implies that a reference phase origin for both phases must be chosen when writing down the fields from each side of the 1D building block. In this case a phase origin at z = 0 has been chosen for calibrating both coefficients.

Following Fraunhofer's diffraction theory, the far-field scattered sound pressure distribution can be obtained as a spatial Fourier transform of the pressure at the surface, e.g., the spatial Fourier transform of the reflection and transmission coefficients, respectively. In this case, the far-field pressure distribution, $p_s(\theta, \phi)$, where $\theta$ and $\phi$ are the azimuth and elevation angles, respectively, of a radiating rectangular surface producing a spatially dependent reflection and transmission coefficients, $\xi(x, y)$, of size $2a$ and $2b$ in the x and y directions, respectively, can be calculated using the Fraunhofer integral [58]

$$p_s(\theta, \phi) = \int_{-a}^{a} \int_{-b}^{b} \xi(x, y) e^{ik(x \sin \phi \sin \theta + y \sin \phi \cos \theta)} dx dy. \quad (12)$$

The subindex $s = R, T$ represents the reflection and transmission problems, respectively, and $\xi(x, y) = R_{TMM}(x, y)$ for the reflection and $\xi(x, y) = T_{TMM}(x, y)$ for the transmission problem.

## Experimental methods

For the experimental validation, a larger 3D-printed hologram, with $N = 53$ repeated $M = 3$ times (see Fig. 1 (a) in Sec. ), was designed for a frequency of 1.1 MHz. The hologram was built using stereolithography techniques (SLA) which used a transparent polymer previously characterized (See SI). A high $N$ value was taken in order to produce a large structure for ultrasonic measurements, where no more than $M = 3$ repetitions were desired to reduce diffraction grating lobes due to periodicity. A larger $N$ also helps design a hologram with a much more uniform spatial Fourier transform due to the nature of pseudo-random sequences. A custom 1.1-MHz piezoelectric transducer (PZT26, Ferroperm, Denmark) with aperture of $2a = 50$ mm, was used to generate ultrasound. The transducer was excited with a 20-cycle sinusoidal pulse burst at 1.1 MHz using a signal generator (model PXI-5412, 14-bit, 100 MS/s, National Instruments, USA) and amplified with a linear radio-frequency amplifier (model 1040L, 400 W, 55 dB, ENI, Rochester, NY, USA). Acoustic fields were measured with a polyvinylidene difluoride (PVDF) hydrophone (model HNR500, Onda, Castellon), calibrated from 500 kHz to 4 MHz, mounted on a 3D axis system (5-µm resolution, PI Micos GmbH, Germany). The hologram was immersed in a degassed and distilled water tank at controlled room temperature (22° C). Degassed water flow was used to eliminate the remaining bubbles trapped at the hologram surface. The hydrophone was placed on either sides of the hologram depending on the reflection or transmission measurement situation, as illustrated in Figs. 4 (a, b). Acoustic pressure was measured in front of the hologram in the x and y directions in order to map the far-field pressure plane distribution. In this way, the measurements were acquired in a transverse (x, y) square plane covering 140 mm, using a spatial step of 0.2 mm, located in the near-field at $z = 0.1$ m, note the Rayleigh diffraction distance of a disc is $ka^2/2 = 1.44$ m. On the one hand, the $z$ distance was chosen large enough to provide a temporal lag to eliminate the direct field by temporal windowing for reflection mode, and the reflections between the hydrophone-sample. On the other, this distance was chosen small enough to capture a broad angular spectrum. A 2-dimensional spatial Fourier transform, using Eq. (12), was performed over the measured surface yielding the polar distribution of the energy in the far field.
The calculation of the diffusion coefficient follows the standardized procedures described in Ref. [59]. The acquired waveforms were deconvolved and impulse response were obtained. Temporal windowing was applied to eliminate the direct field and the spectrum at each location was calculated using fast-Fourier transforms. Then, the scattering in the far-field $p_s(\theta, \phi)$ was calculated using Eq. (12). The diffusion coefficient, $d^s$, is given by

$$d^s = \frac{\left( \sum_{m=1}^{M} |p_s| \right)^2 - \sum_{m=1}^{M} |p_s|^2}{(M - 1) \sum_{m=1}^{M} |p_s|^2}, \quad (13)$$

were $s = (R, T)$ for the reflection and transmission problems, $M$ is the total number of measurements and $m$ is the index of each measurement. In order to compensate the non-uniformity of the grid in spherical coordinates, the following modification [59] is used

$$d^s = \frac{\left( \sum_{m=1}^{M} A_m |p_s| \right)^2 - \sum_{m=1}^{M} A_m |p_s|^2}{\left( \sum_{m=1}^{M} A_m - 1 \right) \sum_{m=1}^{M} A_m |p_s|^2}, \quad (14)$$

where

$$A_m = \begin{cases} \sin\left(\frac{\Delta\phi}{2}\right), & \text{for } \theta_m = 0, \\ 2\sin(\theta_m)\sin\left(\frac{\Delta\theta}{2}\right), & \text{for } 0 < \theta_m < \pi/2, \\ \frac{4\pi}{\Delta\phi}\sin\left(\frac{\Delta\theta}{4}\right), & \text{for } \theta_m = \pi/2. \end{cases} \quad (15)$$

The scattered field was estimated for both the hologram and a rectangular reference panel of same dimensions. For the reflection problem, a rigid rectangular reflector is used. For the transmission, a rectangular aperture is used as the reference. Then, and the corresponding diffusion coefficients were calculated. Finally, the normalized diffusion coefficient, $d_n$, was obtained as

$$d_n^s = \frac{d_s - d_r}{1 - d_r(\omega)}, \quad (16)$$

where $d_s$ and $d_r$ are the diffusion coefficient of the hologram and the rectangular reflector, respectively. This coefficient measures the uniformity of the scattering.
360 | Correlation-scattering coefficient

Finally, the correlation-scattering coefficient δ was calculated using the measured scattering as [53].

$$ \delta^s = 1 - \frac{\left| \sum_{m=1}^{M} \rho_s p_r^* \right|^2}{\sum_{m=1}^{M} |\rho_s|^2 \sum_{m=1}^{M} |\rho_r|^2}, \qquad (17) $$

where $ρ_s$ and $ρ_r$ are scattering of the hologram and the reference reflector for the m-th grid point, respectively, and (*) is the complex conjugate. This coefficient measures the correlation between the scattering of the structure and that of a flat panel of same dimensions.

365

366 | Quadratic Phase Modulation and Even Spacing of Diffraction Lobes

The phase distribution that corresponds to the quadratic residue sequence is given by

$$ \phi_n = \frac{2\pi r_n}{\rho}, \qquad (18) $$

where $r_n$ represents the n-th quadratic residue in the sequence and $\phi_n$ is the phase shift applied at the n-th position. The set of phase shifts corresponds to the quadratic residues modulo $\rho$ mapped to phase values. This set is periodic with period $\rho$, meaning that after $\rho$ elements, the sequence repeats. The diffraction pattern produced by this phase modulation can be understood through the diffraction angle $\theta$, which is related to the phase shift by the grating equation for diffraction:

$$ d \sin(\theta) = m\lambda, \qquad (19) $$

where $d$ is the spacing between the phase-modulating elements (e.g., a metasurface), $\lambda$ is the wavelength, and $m$ is the diffraction order. When the phase shifts follow the quadratic residue sequence, the resulting diffraction lobes are evenly spaced because the phase shifts themselves are distributed in a regular, cyclic manner due to the periodicity of the quadratic residue sequence. To quantify the angular spacing of the lobes, we consider the angular separation $\Delta\theta$ between two successive diffraction orders. Because the phase shifts are distributed according to the quadratic residues, the resulting angular separation is constant, leading to evenly spaced diffraction lobes. Specifically, if the phase sequence is applied to a surface with periodicity $d$, the resulting diffraction angles are separated by a constant value $\Delta\theta$, given by

$$ \Delta\theta = \frac{\lambda}{d}. \qquad (20) $$

Thus, the cyclic nature of quadratic residues not only governs the spacing between the diffraction lobes but also ensures a balanced energy distribution, making them ideal for applications where uniform scattering is desired.

382
References

383

384

1. Melde K, Mark A, Qiu T, Fischer P. Holograms for acoustics. Nature 2016 09;537:518-522.

385

386

2. Brown M, Nikitichev D, Treeby B, Cox B. Generating arbitrary ultrasound fields with tailored optoacoustic surface profiles. Applied Physics Letters 2017;110(9).

387

388

3. Brown MD, Cox BT, Treeby BE. Stackable acoustic holograms. Applied Physics Letters 2020;116(26):261901. https://doi.org/10.1063/5.0009829.

389

390

4. Fushimi T, Yamamoto K, Ochiai Y. Acoustic hologram optimisation using automatic differentiation. Scientific Reports 2021 06;11.

391

392

5. Zou HY, Ge Y, Zhao KQ, Lu YJ, Si Qr, Yuan SQ, et al. Acoustic Metagrating Holograms. Advanced Materials 2024;p. 2401738.

393

394

6. Sallam A, Meesala VC, Hajj MR, Shahab S. Holographic mirrors for spatial ultrasound modulation in contactless acoustic energy transfer systems. Applied Physics Letters 2021;119(14).

395

396

7. Ren X, Zhou Q, Huang J, Xu Z, Liu X. Holographic generation of arbitrary ultrasonic fields by simultaneous modulation of amplitude and phase. Ultrasonics 2023;134:107074.

397

398

8. Li Y, Liang B, Gu Z, Zou XY, Cheng JC. Reflected wavefront manipulation based on ultrathin planar acoustic metasurfaces. Scientific reports 2013 08;3:2546.

399

400

9. Memoli G, Caleap M, Asakawa M, Sahoo D, Drinkwater B, Subramanian S. Metamaterial bricks and quantization of meta-surfaces. Nature Communications 2017 01;8:14608.

401

10. Brown MD. Phase and amplitude modulation with acoustic holograms. Applied Physics Letters 2019;115(5).

402

11. Brown MD, Cox BT, Treeby BE. Binary Volume Acoustic Holograms. Physical Review Applied 2023;19(4):044032.

403

404

12. Brown MD, Cox BT, Treeby BE. Design of multi-frequency acoustic kineforms. Applied Physics Letters 2017;111(24):244101. https://doi.org/10.1063/1.5004040.

405

13. Zhang J, Tian Y, Cheng Y, Liu X. Acoustic holography using composite metasurfaces. Applied Physics Letters 2020;116(3).

406

407

14. Marzo A, Drinkwater BW. Holographic acoustic tweezers. Proceedings of the National Academy of Sciences 2019;116(1):84-89.

408

409

15. Ma Z, Holle AW, Melde K, Qiu T, Poeppel K, Kadiri VM, et al. Acoustic holographic cell patterning in a biocompatible hydrogel. Advanced Materials 2020;32(4):1904181.

410

411

16. Gu Y, Chen C, Rufo J, Shen C, Wang Z, Huang PH, et al. Acoustofluidic holography for micro-to nanoscale particle manipulation. ACS nano 2020;14(11):14635-14645.

412

413

17. Ghanem MA, Maxwell AD, Dalecki D, Sapozhnikov OA, Bailey MR. Phase holograms for the three-dimensional patterning of unconstrained microparticles. Scientific Reports 2023;13(1):9160.

414

415

18. Melde K, Choi E, Wu Z, Palagi S, Qiu T, Fischer P. Acoustic fabrication via the assembly and fusion of particles. Advanced Materials 2018;30(3):1704507.

416

417

19. Melde K, Kremer H, Shi M, Seneca S, Frey C, Platzman I, et al. Compact holographic sound fields enable rapid one-step assembly of matter in 3D. Science Advances 2023;9(6):eadf6182.

418

419

20. Jiménez-Gambín S, Jiménez N, Benlloch JM, Camarena F. Generating Bessel beams with broad depth-of-field by using phase-only acoustic holograms. Scientific reports 2019;9(1):20104.
420

421

21. Zhou Q, Zhang J, Ren X, Xu Z, Liu X. Multi-bottle beam generation using acoustic holographic lens. Applied Physics Letters 2020;116(13).

422

423

22. Gao S, Li Y, Ma C, Cheng Y, Liu X. Emitting long-distance spiral airborne sound using low-profile planar acoustic antenna. Nature Communications 2021;12(1):2006.

424

425

23. Reardon G, Dandu B, Shao Y, Visell Y. Shear shock waves mediate haptic holography via focused ultrasound. Science Advances 2023;9(9):eadf2037.

426

427

24. Hirayama R, Christopoulos G, Martinez Plasencia D, Subramanian S. High-speed acoustic holography with arbitrary scattering objects. Science advances 2022;8(24):eabn7614.

428

429

25. Cox L, Melde K, Croxford A, Fischer P, Drinkwater BW. Acoustic hologram enhanced phased arrays for ultrasonic particle manipulation. Physical Review Applied 2019;12(6):064055.

430

431

26. Jiménez-Gambín S, Jiménez N, Benlloch JM, Camarena F. Holograms to Focus Arbitrary Ultrasonic Fields through the Skull. Phys Rev Applied 2019 Jul;12:014016. https://link.aps.org/doi/10.1103/PhysRevApplied.12.014016.

432

433

27. Andrés D, Vappou J, Jiménez N, Camarena F. Thermal holographic patterns for ultrasound hyperthermia. Applied Physics Letters 2022;120(8):084102. https://doi.org/10.1063/5.0081565.

434

435

28. Andrés D, Rivens I, Mouratidis P, Jiménez N, Camarena F, Ter Haar G. Holographic focused ultrasound hyperthermia system for uniform simultaneous thermal exposure of multiple tumor spheroids. Cancers 2023;15(9):2540.

436

437

29. Cengiz C, Shahab S. Holographic thermal mapping in volumes using acoustic lenses. Journal of Physics D: Applied Physics 2024;57(36):365501.

438

439

30. Kim J, Kasoji S, Durham PG, Dayton PA. Acoustic holograms for directing arbitrary cavitation patterns. Applied Physics Letters 2021;118(5).

440

441

442

31. Jiménez-Gambín S, Jiménez N, Pouliopoulos AN, Benlloch JM, Konofagou EE, Camarena F. Acoustic holograms for bilateral blood-brain barrier opening in a mouse model. IEEE Transactions on Biomedical Engineering 2021;69(4):1359-1368.

443

444

32. Hu Z, Yang Y, Yang L, Gong Y, Chukwu C, Ye D, et al. Airy-beam holographic sonogenetics for advancing neuromodulation precision and flexibility. Proceedings of the National Academy of Sciences 2024;121(26):e2402200121.

445

446

33. Jiménez-Gambín S, Jiménez N, Camarena F. Transcranial Focusing of Ultrasonic Vortices by Acoustic Holograms. Phys Rev Applied 2020 Nov;14:054070. https://link.aps.org/doi/10.1103/PhysRevApplied.14.054070.

447

448

34. Kruizinga P, Meulen P, Fedjajevs A, Mastik F, Springeling G, Jong N, et al. Compressive 3D ultrasound imaging using a single sensor. Science Advances 2017 12:3:e1701423.

449

450

35. Lamothe N, Andrés D, Carrión A, Camarena F, Pineda-Pardo JA, Jiménez N. Monitoring holograms for therapeutic ultrasound using passive cavitation beamforming. Applied Acoustics 2024;224:110144.

451

452

36. Brown MD, Generowicz BS, Dijkhuizen S, Koekkoek SK, Strydis C, Bosch JG, et al. Four-dimensional computational ultrasound imaging of brain hemodynamics. Science advances 2024;10(3):eadk7957.

453

454

37. Jiménez N, Groby JP, Romero-García V. Spiral sound-diffusing metasurfaces based on holographic vortices. Scientific reports 2021;11(1):10217.

455

456

38. Schroeder MR. Diffuse Sound Reflection by Maximum Length Sequences. The Journal of the Acoustical Society of America 1975;57(1):149-150.

457

39. Cox T, D'Antonio P. Schroeder Diffusers: A Review. Building Acoustics 2003;10(1):1-32.
458

459

40. Jiménez N, Cox TJ, Romero-García V, Groby JP. Metadiffusers: Deep-subwavelength sound diffusers. Scientific reports 2017;7(1):5389.

460

461

462

41. Claes L, Zeipert H, Koppa P, Tröster T, Henning B. Additively manufactured acoustic diffuser structures for ultrasonic measurement applications. Proceedings of Meetings on Acoustics 2017;32(1):030004. https://asa.scitation.org/doi/abs/10.1121/2.0000688.

463

464

465

42. Vasan A, Allein F, Duque M, Magaram U, Boechler N, Chalasani SH, et al. Microscale Concert Hall Acoustics to Produce Uniform Ultrasound Stimulation for Targeted Sonogenetics in hSTRA1-Transfected Cells. Advanced NanoBiomed Research 2022;2(5):2100135. https://onlinelibrary.wiley.com/doi/abs/10.1002/anbr.202100135.

466

467

43. Tang Y, Zhang Y, Xie B, Cheng H, Tian J, Chen S. Transmission-reflection-integrated multifunctional continuously tunable metasurfaces for decoupled modulation of acoustic waves. Physical Review Applied 2022;17(4):044027.

468

469

44. Ju F, Zou X, Xue S, Qian SY. Simultaneous manipulation of acoustic waves in the reflected and transmitted regions with the full space metasurface. Applied Physics Express 2023;16(7):074003.

470

471

45. Cheng H, Guo J. Transmission-reflection-integrated multifunctional metasurfaces for multiplexed acoustic wave manipulation. Applied Physics Letters 2024;124(2).

472

473

46. Huang J, Dupont P, Undurti A, Friedman J, Cleveland R. Producing diffuse ultrasound reflections from medical instruments using a quadratic residue diffuser. Ultrasound in medicine & biology 2006 06;32:721-7.

474

475

476

47. Arslan D, Rahimzadegan A, Fasold S, Falkner M, Zhou W, Kroychuk M, et al. Toward Perfect Optical Diffusers: Dielectric Huygens' Metasurfaces with Critical Positional Disorder. Advanced Materials 2022;34(5):2105868. https://onlinelibrary.wiley.com/doi/abs/10.1002/adma.202105868.

477

48. Cox TJ, D'Antonio P. In: Acoustic Absorbers and Diffusers: Theory, Design and Application Crc Press; 2009. p. 295-306.

478

479

49. Treeby BE, Cox BT. Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian. The Journal of the Acoustical Society of America 2010;127(5):2741-2748.

480

481

50. Andrés D, Carrión A, Camarena F, Jiménez N. Methods to design and evaluate transcranial ultrasonic lenses using acoustic holography. Physical Review Applied 2023;20(4):044071.

482

483

51. Sapozhnikov OA, Tsysar SA, Khokhlova VA, Kreider W. Acoustic holography as a metrological tool for characterizing medical ultrasound sources and fields. The Journal of the Acoustical Society of America 2015;138(3):1515-1532.

484

485

52. Ballestero E, Jiménez N, Groby JP, Aygun H, Dance S, Romero-García V. Metadiffusers for quasi-perfect and broadband sound diffusion. Applied Physics Letters 2021 07;119(4). https://doi.org/10.1063/5.0053413, 044101.

486

53. Cox TJ, D'Antonio P. Acoustic absorbers and diffusers: theory, design and application. 3rd ed. CRC Press; 2016.

487

488

54. Errico C, Pierre J, Pezet S, Desailly Y, Lenkei Z, Couture O, et al. Ultrafast ultrasound localization microscopy for deep super-resolution vascular imaging. Nature 2015 11;527.

489

490

55. Macé E, Montaldo G, Cohen I, Baulac M, Fink M, Tanter M. Functional ultrasound imaging of the brain. Nature methods 2011;8(8):662-664.

491

492

56. Tanter M, Fink M. Ultrafast imaging in biomedical ultrasound. IEEE transactions on ultrasonics, ferroelectrics, and frequency control 2014;61(1):102-119.

493

494

495

496

57. Jiménez N, Groby JP, Romero-García V. The Transfer Matrix Method in Acoustics: Modelling One-Dimensional Acoustic Systems, Phononic Crystals and Acoustic Metamaterials. In: Jiménez N, Umnova O, Groby JP, editors. Acoustic Waves in Periodic Structures, Metamaterials, and Porous Media: From Fundamentals to Industrial Applications Cham, Switzerland: Springer; 2021. p. 103-164.
58. Rossing TD, Dunn F, Hartmann WM, Campbell DM, Fletcher NH. Springer Handbook of Acoustics. 1st ed. Springer Publishing Company, Incorporated; 2007.

59. ISO 17497-2:2012. Acoustics – Sound-scattering properties of surfaces – Part 2: Measurement of the directional diffusion coefficient in a free field. Geneva, Switzerland: International Organization for Standardization; 2012.

**Acknowledgments:**

E.B would like to gratefully acknowledge the financial support from Institut d'Acoustique – Graduate School (IAGS ANR-17-EURE-0014) for granting mobility during this project.

**Funding:**

The collaboration between the authors of this article was made possible thanks to Institut d'Acoustique – Graduate School (IAGS) project (ANR-17-EURE-0014) from the Laboratoire d'Acoustique du l'Université du Mans (LAUM) for the support and mobility. The authors gratefully acknowledge the ANR-RGC METARoom (ANR-18-CE08-0021) project and N.J and V.R-G acknowledges financial support from Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital de la Generalitat Valenciana through project CIAICO/2022/052 of the Programa para la promoción de la investigación científica, el desarrollo tecnológico y la innovación en la Comunitat Valenciana. This research has been supported by the Ministerio Español de Ciencia e Innovación grant RYC2021-034920-I and Agencia Estatal de Investigación grants PID2022-142719OB-C21 and CNS2023-145707 funded by MCIN/AEI/10.13039/501100011033 and grant PID2023-146237NB-I00 funded by MICIU/AEI/ 10.13039/501100011033.

**Author contributions:**

E.B, J-P.G, V.R-G, and N.J conceived the project, under the direction of N.J. E.B, V.R.-G and N.J wrote the code for the theoretical and numerical data. E.B performed the multi-objective optimization and provided the dimensions of the ultrasonic lens. J.R-S 3D-printed the sample and performed the experimental measurements along with N.J, as well as provided details on the experimental apparatus. N.J analyzed the experimental data. E.B, and N.J wrote the initial manuscript with input from all authors. E.B, V.R.-G and N.J prepared the revised version and the response to the referees with input from all authors. All authors have reviewed and commented on the final manuscript.

**Competing interests:**

There are no competing interests to declare.

**Data and materials availability:**

Data presented in this work is available upon reasonable request addressed to E.B and N.J.