# One-dimensional optical compression and backpropagation of particle digital in-line holograms

Paul Bresson, Fabrice Lamadie, Lilian Chabrol, Fabrice R. A. Onofri

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Paul Bresson, Fabrice Lamadie, Lilian Chabrol, Fabrice R. A. Onofri. One-dimensional optical compression and backpropagation of particle digital in-line holograms. Applied optics, In press. cea-05318643

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Distributed under a Creative Commons Attribution 4.0 International License
To be published in *Applied Optics*:

**Title:** One-dimensional optical compression and backpropagation of particle digital in-line holograms

**Authors:** Paul Bresson, Fabrice Lamadie, Lilian Chabrol, Fabrice Onofri

**Accepted:** 15 October 25

**Posted:** 16 October 25

**DOI:** https://doi.org/10.1364/AO.578014

Published by Optica Publishing Group under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/). Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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# One-dimensional optical compression and backpropagation of particle digital in-line holograms

PAUL BRESSON,¹,* FABRICE LAMADIE¹, LILIAN CHABROL,² AND FABRICE R.A. ONOFRI,²,**

1: CEA, DES, ISEC, DMRC, Univ Montpellier, 30207 Bagnols-sur-Ceze, Marcoule, France

2: Aix-Marseille Université, CNRS, IUSTI, UMR 7343, 13453, Marseille, France

*paul.bresson@cea.fr; **fabrice.onofri@univ-amu.fr

**Abstract:** This work explores the possibility of using cylindrical lenses in digital in-line holography to reduce particle hologram imprint and cross-interference on the camera sensor. Experimental results on sprayed and levitated droplets, together with simulations using Rayleigh-Sommerfeld integrals and ABCD analysis, confirm the feasibility of this compression method for spherical particles. However, it turns out that, when using a back-propagation method to reconstruct the particle field, reducing the fill rate of the camera sensor does not in itself improve the quality of the particle field reconstruction. It is the asymmetric magnification of the optical system that enables this improvement, transforming the twin particle problem into a two half-twin particles problem. The latter have less impact on the reconstruction process, allowing the statistics to be improved. This advantage increases with the magnification factor of the optical system. The authors believe that this optical method could be even more interesting for parametric inversion methods used for full volumetric reconstruction of particle fields.

OCIS codes: (090.1995) Digital holography; (120.3940) Metrology, (280.2940) Flow diagnostics, (290.5850) Scattering, particles

## 1. Introduction

Digital in-line holography (DIH, e.g. [1-3]) allows the simultaneous characterization of the 3D positions, trajectories, velocities, shapes and sizes of static or moving particles in flows. Its ability to probe a wide range of particles - including droplets, bubbles, aggregates, and bacteria - over a wide range of sizes - from micrometers to millimeters - makes it valuable in fields ranging from soft matter physics [4, 5] to fluid mechanics [6-11] or biology [12-15]. For spherical particles, DIH also allows the estimation of the real part of their relative refractive index [16-19], which is particularly useful for characterizing multiphase flows [20, 21]. In a basic free-space configuration – characterized by the absence of gradients, complex interfaces, or optics – the accuracy of DIH for particle sizing and positioning is directly influenced by the size and resolution of the image sensor and its distance from the particulate medium. This configuration is primarily used in multiphase flow studies, where DIH setups simply collect the light scattered by the particles in the forward direction using high-resolution CCD or CMOS cameras. In this case, DIH is a lens-free imaging system with no optical aberrations. Conversely, in biological applications notably, holographic microscopes [22, 23] are often used to obtain highly magnified holograms [24]. High magnification improves object positioning and shape analysis. But it introduces optical aberrations [25] and it decreases the size of the observable field as well as the working distance. Reducing the field of view degrades instantaneous statistics, while reducing the working distance increases the risk of damage to the optical system, especially with reactive flows [21]. Advantages and disadvantages also exist in other aspects of DIH, such as the choice of a direct [3, 9, 10, 23, 26, 27] rather than an inverse method [4, 28-32] for hologram analysis. Indeed, direct methods, using backpropagation as in the present work for instance, are easier to implement and faster. However, the resolution they provide on the particle field is limited by the sensor characteristics
(pixel size and number). Conversely, inverse methods require much more sophisticated algorithms and computational resources but can achieve subpixel resolutions. Another example is the use of a Gabor configuration [33, 34] in combination with an intensity sensor. While this setup reduces the sensitivity of the DIH to mechanical vibrations, as well as its overall size and cost, the hologram phase loss induces the well-known twin-particle (also called twin-image) problem, e.g. [35].

<figure><img src="image_2.png" /><figcaption>Fig. 1 Schematic diagram of particle hologram formation in a (a) free space configuration and with a (b) one-dimensional optical compression optics built with two plano-convex cylindrical lenses (with focal length, $f_x$). In the present 4F- configuration, the particle-to-sensor distance is 4f_x.</figcaption></figure>

The maximum number of particles that can be analyzed simultaneously is another key issue in DIH, and certainly one of the most challenging and least studied. This issue is related to several assumptions of DIH, such as that the particulate medium must remain optically diluted, that the intensity scattered by the particles must be negligible with respect to that of the incident wave, or that multiple scattering effects are negligible [3, 31, 36, 37]. It is precisely the aim of this paper to present and investigate a straightforward optical method to reduce the fill rate of the camera sensor (SFR) and as it will be shown, although not initially expected, to partially mitigate the twin-particle problem. For the sake of clarity, the principle of the proposed method is illustrated in Fig. 1 for two particles ($p = 1, 2$) with diameters $D_p$, refractive indices $m_p$ and coordinates $(x_p, y_p, z_p)$. They are illuminated by a plane wave. Essentially, the two individual holograms are optically compressed in one direction using plano-convex cylindrical lenses, with aperture $\phi_a$, width $w$, separation distance $e$, refractive index $m_2$ and focal length $f_x$, placed between the particle to be analyzed and the camera. By compression, we mean that the initial holograms made up of circular diffraction fringes are compressed by the optics to form elliptical fringe patterns on the camera sensor. Their imprint and cross-interference on the sensor are clearly reduced, see Fig. 1. This one-dimensional compression is novel in the context of particle sizing with DIH, although it has been considered for holographic displays applications [38] and multiplex record of holograms [39]. It has also been suggested to improve the lateral resolution of shadowgraphs when sizing capillary jets or fibers, e.g. [40], or to compress droplet interferograms analyzed by the Interferometric Laser Imaging Droplet Sizing (ILIDS) technique [41, 42]. For the DIH, we can start by using simple considerations to assess the method's gain in terms of SFR. In free space, the hologram of a spherical particle with radius $a$ exhibits a rotationally symmetric pattern, with a radius $\rho$, filling the sensor area $S = \pi$
$\rho^2$. To further simplify, let consider a 4F optical system with magnification factor $\gamma = -1$, where $F = f_x/(2 - E/(m_1f_x))$ is the effective focal length of the doublet of plano-cylindrical lenses (with spacing $E$) centered between the sensor and the particle, $z = L = 4F = 2f_x$ (see Fig. 1). The hologram of the spherical particle is compressed to half-width $\gamma a$ along the $x$-direction and remains almost unchanged with half-width $\rho$ along the $y$-direction. The area filled by the compressed hologram on the camera sensor is assumed to be elliptical, so that $C = \pi(|\gamma|a)\rho$. For small Fresnel numbers, $Fr = a^2/(\lambda_1|z_p|) \approx a^2/(\lambda_1L) \ll 1$, where $\lambda_1$ is the wavelength of the illuminating plane wave in the surrounding medium, the half-width of the hologram in free space can be approximated by the radius of the corresponding Airy disk [34, 43], which is given by $\rho \approx (1.22\lambda_1/2a)L$. Finally, compared to a free-space system, the gain of the proposed method in terms of hologram imprint on the sensor is $R_1 = C/S \approx 1.64|\gamma|Fr$. For typical experimental parameters, e.g. $L = 4 \times 0.075$ m, $\lambda_1 = 0.6328$ µm and $a = 48$ µm, we obtain $\epsilon_1 \approx 0.02$. This indicates that the SFR could be reduced by a factor of fifty simply by adding two lenses to a standard DIH setup. This was deemed sufficient to carry out this study.

Following this introduction, the paper is organized as follows. Section 2 provides details of the materials and methods used. It begins with an overview of the experimental setup for the analysis of droplets, either trapped in an acoustic field or sprayed. The section then describes the model used to simulate and backpropagate holograms in free space and through compression optics. This model is based on Rayleigh-Sommerfeld integrals and ABCD analysis. The model is presented first in its general formulation, then for cylindrical optics, together with a discussion of the corresponding cutoff frequencies. The four-step algorithm used for direct particle identification and characterization is also briefly presented. Section 3 presents a comparison and discussion of the main experimental and numerical results. Section 4 concludes with a summary of the advantages and disadvantages of the method as well as prospects.

## 2. Material and methods

### *Experimental setup and conditions*

The optical setup, the droplet generation and control systems are depicted in Fig. 2. The emission of the DIH system consists of a fiber-coupled and unpolarized HeNe laser ($\lambda = 0.6328$ µm, 10 mW), and a collimation optics, resulting in a total beam diameter of 48 mm. The detection unit consists of a high-resolution CCD camera (Imager ProX4M from LaVision GmbH), which has a 14-bit sensor with an area $\Delta x \times \Delta y = N_x\delta x \times N_y\delta y$ (with $N_x \times N_y = 2048^2$ square pixels with a width $\delta x = \delta y = 7.4$ µm) [44]. The emission and the detection are configured in a Gabor setup, with the camera in air $m_1 = 1$, facing the incident beam. The compression optics consists of two back-to-back plano-convex cylindrical lenses, made of BK7 glass with a *visible antireflective coating*, from Newport Corporation. The spherical aberrations introduced by this optics are minimal for a 4F configuration (see below). The lenses are rectangular with dimensions $\phi_a^2 \approx 50.8 \times 50.8$ mm$^2$, width $w = 5.094$ mm, and are separated by a distance $e = 80$ µm. Two focal lengths were tested, with $f_x = 150$ mm and $f_x = 300$ mm. In addition, a shadowgraph system was positioned at a 90° angle to the DIH system in the $yz$-plane. Its emission unit consists of a fiber-coupled high-power blue LED with collimation optics, while its detection unit uses an identical camera to the DIH system with a magnification optics (7.3 to 11.6 µm/pixel) allowing simultaneous observation of multiple droplets. To minimize blurring effects, the exposure time for both synchronized cameras was limited to 30 µs. In summary, this setup allows simultaneous recording of shadowgraphy and DIH images (with or without compression optics), but not two DIH images simultaneously.
Two types of droplets were characterized in this study. First, DEHS (di-Ethyl-Hexyl-Sebacic-Acid-Ester) droplets were suspended between the transducer and the reflector of a standing-wave acoustic trap [44]. This setup achieved a maximum sound pressure level of 140 dB for a wavelength in air of approximately 3.4 mm and a piezoelectric frequency of 100 kHz. The position of the acoustic trap was fine-tuned using 3D micro-displacements, allowing precise centering of the droplets in the illumination beams, even when positioned away from the optimal focusing position (OFP), defined as $(x_p, y_p, z_p) = (0,0, -L)$. The extremely low vapor pressure of DEHS ensures that droplet evaporation can be neglected during experiments lasting several minutes. The second type of droplet studied consists of water droplets sprayed from a full cone nozzle positioned twenty centimeters above the zone being probed. In this case, the spray system replaces the levitation trap.

<figure><img src="image_3.png" /><figcaption>Fig. 2 Schematic diagram of experimental setup. The spray system and the levitation trap are used alternately.</figcaption></figure>

### 3. Computational models for hologram simulation and backpropagation through the optical system

#### General formulation for hologram simulation

Consider an unpolarized electromagnetic plane wave of wavelength in vacuum $\lambda$ and with electric field amplitude $E_0$ propagating along the z-axis in a continuous medium of refractive index $m_1$, see Fig. 1. In free space, the classical Rayleigh-Sommerfeld diffraction integrals (e.g. [34]) allow to estimate the amplitude $E_z$ of the electric field diffracted by a particle. Under paraxial conditions and for relatively low Fresnel numbers, the diffraction of this particle, viewed through an optical system, can be modeled with Collins' integrals [45] using ABCD transfer matrices (see [46, 47] for details and more advanced approaches dealing with structured beams).

Any centrosymmetric optical system can be represented by a single two-dimensional ABCD transfer matrix, e.g. [34]. Its coefficients – labeled A, B, C and D – account for propagation effects between the incoming and outgoing fields. For systems with astigmatism, such as those with cylindrical optics (or when observing a pipe flow [9, 48]), a more general four-dimensional matrix is required. Let $(u,v,z_p)$ and $(x,y,z)$ be the initial and final positions of a ray respectively, with initial tilt angles $\beta_u \approx du/dz$ and $\beta_v \approx dv/dz$, and final angles $\beta_x \approx dx/dz$ and $\beta_y \approx dy/dz$. Note that these rays are perpendicular to the geometric wavefronts [49, 50] that are propagated with Collins' integrals. The four-dimensional ABCD matrix for the optical system shown in Fig. 1 is defined as follows:

$$ \begin{pmatrix} x \\ m_1\beta_x \\ y \\ m_1\beta_y \end{pmatrix} \equiv ABCD \begin{pmatrix} u \\ m_1\beta_u \\ v \\ m_1\beta_v \end{pmatrix} \qquad (1) $$
where:

$$ABCD = \begin{pmatrix} A_x & B_x & 0 & 0 \\ C_x & D_x & 0 & 0 \\ 0 & 0 & A_y & B_y \\ 0 & 0 & C_y & D_y \end{pmatrix} = \begin{pmatrix} ABCD_x & 0 \\ 0 & ABCD_y \end{pmatrix}. \quad (2)$$

Expressions of $ABCD_x$ and $ABCD_y$ in terms of the optical parameters introduced in Fig. 1 are given in what follows. Note that the positions and derivatives are reversed from Collins' convention.

The electric field $E_z$ in the plane at $z$ (with $B_x B_y \neq 0$) is then derived from the electric field in the plane at $z = z_p$ as follows [45]:

$$E_z(x,y) = \frac{e^{ik\xi_p}}{i\lambda\sqrt{B_x B_y}} \iint_{R^2} E_0(u,v) e^{\frac{ik}{2B_x}[A_x u^2 - 2xu + D_x x^2]} e^{\frac{ik}{2B_y}[A_y v^2 - 2yv + D_y y^2]} \mathrm{d}u\mathrm{d}v \quad (3)$$

or with its convolution form (with $A_x A_y \neq 0$) [9]:

$$E_z(x,y) = \frac{e^{ik\xi_p}}{i\lambda\sqrt{B_x B_y}} e^{\frac{ikC_x}{2A_x}x^2} e^{\frac{ikC_y}{2A_y}y^2} [E_0 * h] \left(\frac{x}{A_x}, \frac{y}{A_y}\right), \quad (4)$$

where $i$ is the imaginary unit, $*$ is the two-dimensional convolution operator, $\xi_p = \int_{z_p}^z m(z) dz$ is the optical path along $z$ in the propagation medium where $m(z)$ is assumed to be constant, and $h$ is the impulse response with:

$$h(u,v) = e^{\frac{ik}{2}\left(\frac{A_x}{B_x}u^2 + \frac{A_y}{B_y}v^2\right)} \quad (5)$$

Eq. (3) generalizes the free space diffraction integral to accommodate an optical system characterized by an ABCD matrix under a small-angle approximation. With the appropriate values for the coefficients $ABCD_x$ and $ABCD_y$, this propagation equation is equivalent to that derived analytically by Xiao et al [38] for the wavefront passing through a single plano-convex lens. We preferred to use the ABCD matrix formalism because of its simplicity and power. In fact, the simple multiplication of successive matrices makes it easy to describe more complex optical systems, such as the two-lens system used in the experimental setup to minimize optical aberrations.

Based on this framework, the formation of a hologram can be described in three sequential steps:

1. Free space propagation of the incident field to the location of the first particle.

2. Successive free space propagations between particles, where the field at each stage is modified by the transmission $T$ of each particle encountered [51].

3. Final propagation governed by an ABCD transfer matrix that takes into account for the optical properties of the media before and after the optical system as well as the system itself (see Fig. 1).

For a single particle, omitting the phase terms for simplicity, the amplitude expression in Eq. (4) can be reformulated as $E(x,y) \propto [(1-T) * h](x/A_x, y/A_y)$, where $T$ denotes the transmission of the particle. In the subsequent analysis, the particle is conventionally modeled [9] as an opaque disk, where $1-T=0$ on the disk and $T=0$ elsewhere, although more sophisticated approaches have recently been proposed [52]. The intensity measured by the quadratic sensor of the camera is $I \propto E\bar{E}$, where $\bar{E}$ is the complex conjugate operator:
$$I(x,y) \approx 1 - [T * h](\frac{x}{A_x}, \frac{y}{A_y}) - [T * \bar{h}](\frac{x}{A_x}, \frac{y}{A_y}) + \left| [T * h](\frac{x}{A_x}, \frac{y}{A_y}) \right|^2. \quad (6)$$

Classically, it consists of four contributions, from left to right: the unitary intensity of the incident wave minimally perturbed by the particle (assuming an optically dilute medium); the intensity of the field diffracted by the real particle perturbing the incident wave; the intensity of the field diffracted by a "virtual" particle perturbing a virtual incident wave propagating in the opposite direction; and the intensity of the field diffracted by the real particle itself. In the DIH framework, the last contribution is considered negligible compared to the first three. The virtual particle, often referred to as a "twin" or "ghost" particle, arises due to the loss of phase information in the field recorded by the camera sensor (e.g. [31, 53, 54]). To reduce or eliminate this artifact, several approaches have been proposed in the literature. Among them we find, off-axis reconstructions (e.g., [55]); phase-shifting methods requiring multiple exposures (e.g., [56]), which are hardly applicable to particle flows; methods based on differences in the intensity distributions of weak scatterers (e.g., [57]), that are limited to small scatterers; wave front filtering and iterative phase retrieval techniques (e.g., [35, 58-60]), which require large computational resources; or more recently, deep learning solutions or networks (e.g., [61-63]).

## Hologram backpropagation

The experimentally recorded or simulated holograms, with intensity $I(x,y)$, are backpropagated using the inverse of the forward propagation model. For simplicity, we consider the screen in the plane $z = 0$. Let the reconstructed plane be at a distance $d$ from the optical system, so that it is in the plane $z = -d - (d_s + 2w + E)$, where $w$ is the width of a cylindrical lens, $d_s$ is the distance from the screen to the optical system (equal to $f_x$ in our case), and $e$ is the distance between the two lenses. The reconstructed electric field $E_d$ is then:

$$E_d(u,v) = \frac{1}{i\lambda} \left( \frac{1}{B_x(d)B_y(d)} \right)^{1/2} [I(x,y) * h_{\text{Back,d}}(u,v)] (A_x u, A_y v), \quad (7)$$

where the impulse response $h_{\text{Back,d}}(u,v)$ is

$$h_{\text{Back,d}}(u,v) = e^{-\frac{ik}{2} \left( \frac{1}{A_x B_x(d)} u^2 + \frac{1}{A_y B_y(d)} v^2 \right)}, \quad (8)$$

where for the x-direction:

$$\begin{align*}
C_x &= \frac{e}{f_x^2 m_1} - \frac{2}{f_x} \equiv -\frac{1}{f_e}, \\
A_x &= -1 + \frac{f_x}{f_e} - \frac{1}{f_e} \left( \frac{w}{m_2} + \frac{d_s}{m_1} \right), \\
D_x(d) &= -1 + \frac{f_x}{f_e} - \frac{1}{f_e} \left( \frac{w}{m_2} + \frac{d}{m_1} \right), \tag{9} \\
B_x(d) &= \frac{A_x D_x - 1}{C_x},
\end{align*}$$

and, the y-direction:

$$ABCD_y = \begin{pmatrix} A_y & B_y \\ C_y & D_y \end{pmatrix} = \begin{pmatrix} 1 & \frac{d+d_s+e}{m_1} + \frac{2w}{m_2} \\ 0 & 1 \end{pmatrix}. \quad (10)$$

When $d = d_s = f_x$ and $e = w = 0$, we obtain a standard 4F system with a unit magnification factor $\gamma = -1$ in the x-direction, maintaining the probed volume dimensions as in the case of free-space propagation.
It is important to note that in the setup shown in Fig. 1, the rays reaching the sensor diverge. To be able to backpropagate the hologram under plane wave illumination, a virtual lens (with a matrix denoted by $L_{//}$) must be introduced on the sensor surface to convert the recorded diverging field into a collimated one:

$$ABCD_x(d, f_x, w, m_2) \equiv L_{//} ABCD_x = \begin{pmatrix} A_x & B_x(d) \\ 0 & \frac{1}{A_x} \end{pmatrix}, \quad (11)$$

where:

$$L_{//} \equiv \begin{pmatrix} \frac{1}{C_x} & 0 \\ -\frac{1}{A_x} & 1 \end{pmatrix}. \quad (12)$$

This virtual lens transformation preserves the intensity of the hologram on the sensor: it simply redresses the light so that an incoming plane wave is still a plane wave on the screen. The derivation of the corresponding transfer matrix for the y − direction is straightforward

$$ABCD_y = P_{d_s}^{m_1} P_{w}^{m_2} P_{E}^{m_1} P_{w}^{m_2} P_{d}^{m_1} = \begin{pmatrix} 1 & \frac{d+d_s+e}{m_1} + \frac{2w}{m_2} \\ 0 & 1 \end{pmatrix}. \quad (13)$$

<figure><img src="image_4.png" /><figcaption>Fig. 3 Flowchart of the backpropagation algorithm.</figcaption></figure>

## Particle field characterization algorithm

During backpropagation, the 3D position and size of the particles are determined by a four-step algorithm [64]. Its flowchart is shown in Fig. 3 and it can be summarized as follows.

First, after an initialization procedure, the number of particles and their centers in the sensor plane are identified using a method inspired by the HYBRID approach [65]. This method uses the minimum intensity $I_{min}$ of each reconstruction and the maximum of the Tenengrad operator ($G$, see Fig. 11 c) [66], calculated as:

$$I_{\min}(u, v) = \min_z (|E_z(u, v)|^2), \quad G_{\max}(u, v) = \max_z (G_z(u, v)). \quad (14)$$

These quantities are employed to detect particle edges by applying a threshold $t$, optimized to maximize the overall normalized sharpness $S(t)$, with the mean Tenengrad applied to the
edge of the binarized map $I_{\min}(u,v) < t$ (i.e. the sum of pixels corresponding to the binarization of image $I_{\min}$ by a threshold $t$, see Ref. [64] for more details]:

$$S(t) = \frac{\sum_{u,v} I_{\min}(u,v) < t}{G_{\max}(u,v)} \frac{I_{\min}(u,v) < t}{\sum_{u,v} I_{\min}(u,v) < t} \quad (15)$$

Second, the lateral positions and number of the particles in the measurement volume are determined from the particle edge pixels, producing binarized images of the particles.

Third, to resolve overlapping particle images, a watershed algorithm [67] is applied to the binarized images to resolve overlapping particle images. In other words, the algorithm involves calculating a focus metric for each particle at regular intervals throughout the measurement volume.

Finally, the particle positions along the optical axis are determined using a focusing function based on the variance of the real part of the reconstructed field near the edges of each particle [64, 68]. To put it differently, the algorithm refines this metric around the previously identified maxima by gradually decreasing the step size to determine the optimal position of the focus for each particle.

### Evaluation of convolution product and sampling criteria

To speed up the computation, the convolution in Eqs. (6) and (7) are computed with a standard Fast Fourier Transform (FFT) algorithm [69]. To do this, consider a sensor of size $\Delta x = N_x \delta x$, where $N_x$ and $\delta x$ represent the number and size of pixels respectively. The reconstructed field is also assumed to have the same size and sampling interval [69]. To satisfy the Shannon-Nyquist condition for accurate sampling of the impulse response $h_{\text{Back},d}$, the following must hold:

$$\delta x' < \min\left(\frac{\lambda|A_x B_x(d)|}{\Delta x}, \delta x\right), \quad (16)$$

where $x'$ is the spatial variable in the evaluation of $h_{\text{Back},d}$. The condition $\delta x' < \frac{\lambda|A_x B_x(d)|}{\Delta x}$, necessary to obtain an adequate sampling of the quadratic phase in Eq. (7) is detailed in [9]. A similar constraint applies along the $y$-axis, with the variables $N_y$, $\delta y$ and $\Delta y$.

The Fourier transform of the impulse response defines the transfer function $H_{\text{Back},d}$:

$$H_{\text{Back},d}(\delta v_{x'}, \delta v_{y'}) = i\lambda (A_x B_x(d) A_y B_y(d))^{1/2} e^{-i\pi\lambda(A_x B_x(d)\delta v_{x'}^2 + A_y B_y(d)\delta v_{y'}^2)}, \quad (17)$$

where $v_{x'}$ and $v_{y'}$ are the spatial frequencies in the $x-$ and $y-$ directions used to evaluate the transfer function. Thus, the sampling condition for spatial frequencies is:

$$\delta v_{x'} < \min\left(\frac{\delta x}{\lambda|A_x B_x(d)|}, \frac{1}{\Delta x}\right). \quad (18)$$

The sampling conditions depend on the distance $d$ of the reconstruction plane. For short distances, it is preferable to use and thus sample $H_{\text{back},d}$, while for larger distances it is preferable to use and sample $h_{\text{Back},d}$, e.g. [69].

For thin lenses with $E = 0$, one can use a good approximation that can be bound by the extremal values of $d$:

$$B_x(d) = \frac{1}{m_1}\left(d_s + d - \frac{2d_s d}{f_x}\right), \quad B_y(d) = \frac{1}{m_1}(d_s + d). \quad (19)$$

Finally, the limited aperture of the optical system acts as a low-pass filter, reducing the amount of scattered information reaching the sensor. For our setup, the minimum spatial dimensions $L_x$ and $L_y$ are required to ensure that the lenses do not limit the information collected by the sensor, with:
$$L_{x,\min} = \max\left(\frac{\Delta x}{A_x}, \frac{\lambda B_x(0)}{\delta x}\right), \quad L_{y,\min} = \max\left(\frac{\Delta y}{A_y}, \frac{\lambda B_y(0)}{\delta y}\right). \qquad (20)$$

## 4. Results and discussions

Let's begin by examining some experimental findings. Fig. 4 (b, c) shows holograms generated by DHES droplets inside the acoustic trap. These holograms were recorded under two conditions: (b) free-space and (c) using cylindrical optics. Note that the corresponding shadowgraph images (a, d) were taken approximately one minute after the holograms, ensuring that the droplets have the same size but not necessarily the same position (due to small instabilities in the acoustic trap and the surrounding atmosphere). The droplet diameters range from 90 ± 5 to 460 ± 23 µm, with a mean deviation between shadowgraph and DIH estimates of (a) ± 5 µm and (b) ± 7 µm for the sizes, and (a) ± 420 µm and (b) ± 665 µm for the axial positions. Several observations can be made from these experimental recordings. First, cylindrical lenses effectively reduce the footprint and interference between individual holograms without changing the observed field. To quantify the gain of the method in terms of SFR, the holograms in Fig. 4 were binarized after applying an 8x8 pixel Gaussian smoothing filter to minimize background noise. The binarization criteria was based on a 7% deviation from the background level, resulting in a pixel classification as either H (signal) or B (background).

<figure><img src="image_5.png" /><figcaption>Fig. 4 DHES droplets in an acoustic trap. (a) Side view shadowgraph image and corresponding (b) free-space hologram. (c) Hologram with cylindrical compression optics and (d) shadowgraph</figcaption></figure>
image. The orange frame indicates the region of interest (ROI) used to estimate the SFR, R₂.
The contrast of the images has been optimized for clarity.

A secondary SFR measure $R_2 = (H/B)_{cyl}/(H/B)_{free}$ was computed for the ROI (highlighted in orange in Fig. 4) containing nearly two full holograms, yielding $R_2 \approx 0.05$. Thus, for the corresponding background noise level and camera settings, the gain of the proposed method in terms of SFR is close to 20 for droplets of 320 ± 16 and 460 ± 23 µm in size. Fig. 4 also shows that the cylindrical optics significantly reduces the interference region (so-called edge fringes) between the fields diffracted by the emitter of the acoustic trap and the droplets. This behavior should be particularly beneficial for studies focusing on the characterization of particles near interfaces.

Experiments were also performed by moving the entire acoustic trap to record holograms for larger axial positions $z_p = (0,0,\Delta z - L)$ of the same DHES droplet, see Fig. 5, Fig. 6 and Fig. 7. Defocusing effects are shown in Fig. 5 for $\Delta z = -40, -30,..., +40$ mm, i.e. the maximum displacement range of the optical bench on which was fixed the trap. At $\Delta z = 0$, i.e. optimal focusing length (OFL), the hologram height is minimized and closely matches the droplet size due to the unitary magnification of the optical system. Overall, for $\Delta z = 0$, the hologram envelope tends to be elliptical (or rectangular) in shape. As the droplet moves closer to or further away from the compression optics, the hologram height gradually increases, while its lateral spread (corresponding to free-space propagation) remains constant overall. For larger distances ($z_p < -L$) the hologram fringes exhibit a double-egg (or lemniscate) shape, while for smaller distances ($z_p > -L$) they exhibit a hyperbolic shape. These distinct fringe patterns offer a reliable visual indicator of the droplet's axial position.

Fig. 6 compares the corresponding experimental and simulated hologram intensity profiles along the x-axis and the y-axis. The overall agreement is good, despite the noise in the experimental profiles and the fact that refraction and edge effects [52] are not considered in the particle transmission model. We can also conclude that optical aberrations induced by both lenses do not seem to be significant under current conditions. As shown in Fig. 7, under these laboratory conditions, the shadowgraph and DIH estimates for $\Delta z$ and $D$ agree with less than 5% deviation. Fig. 8 compares the typical particle size distributions obtained for a water spray, with the corresponding lognormal fits. The mean particle size and corresponding root mean square (RMS) are 189 ± 148 µm for the DIH and 181 ± 182 µm for the Shadowgraph systems respectively. This is a good agreement for systems with different viewing angles and depths of field.
<figure><img src="image_6.png" /><figcaption>Fig. 5 Comparison of (a) reconstructed droplet images from (b) experimental holograms of DHES droplets in levitation; (d) reconstructed droplet images from (c) simulated holograms. The latter are generated with the synthetic (e) droplet images synthesized using the shadowgraph estimates for the experimental droplet axial positions (0, 0, $\Delta z-L$) with $\Delta z=(-40,-30,...,+40) \pm 0.1$ mm and diameter $D_p=490 \pm 25$ µm.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6 Comparison of the normalized intensity profiles along the *x*-axis and *y*-axis for the experimental and simulated holograms of Fig. 5. For drawing considerations, a constant offset was added between each case.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7 Experimental comparison of axial positions and sizes, measured with shadowgraph images and the DIH system with compression optics, for different axial positions (0, 0, $Δ-L$) of the same DHES droplet (parameters of Fig. 5 and Fig. 6).</figcaption></figure>

Numerical simulations make it easier to evaluate certain properties compared to experiments. In this context, Fig. 9 shows numerical holograms of water droplets for increasing magnifications $|γ| = 0.37$ to 1.90 obtained by adjusting the ratio $z_2/z_1$. Clearly, the field of view is increased at lower magnifications, but it is limited by the optical aperture of the system. Higher magnifications reduce the field of view and increase the hologram compression. That's why, in this first work on the optical compression of particle holograms, a magnification factor close to one was chosen as a good compromise for the characterization of particle fields as well as for the minimization of spherical aberrations.

<figure><img src="image_9.png" /><figcaption>Fig. 8 Comparison of normalized particle size distributions for a spray of droplets. The dashed and dotted-dashed lines represent the corresponding lognormal distribution fits. The inset image shows a typical hologram processed to derive these statistics.</figcaption></figure>
Fig. 10 compares simulated holograms (a, c, e) and reconstructed particle images (b, d, f) for a DIH system operating (a, b) in free-space, with (c, d) cylindrical optics or with (e, f) spherical optics. In the latter scenario, the cylindrical lenses were replaced by spherical plano-convex lenses with the same focal length and aperture. For these simulations, an effective focal length of 150 mm and a distance $z_1 = 120$ mm were used to minimize the twin-particle problem within the probed zone.

These complete holograms were generated by summing the complex amplitudes of individual holograms formed by $P = 64$ monodispersed water droplets of 50 µm in diameter, randomly distributed in a volume $(\Delta x_p, \Delta y_p, \Delta z_p) = (\pm N_x \delta x, \pm N_y \delta y, \pm (3/2) N_z \delta x)$ of nearly 3.5 cm³. While these calculations account for interference between individual holograms on the sensor, they do not account for multiple scattering effects that may occur [37]. Consistent with all calculations presented here, the reconstruction increment is iteratively reduced from $\delta z = 1$ mm, at a distance from the particles down to $\delta z = 1$ µm, to accurately capture the position of the peak of the focusing function.

All the hologram morphologies observed in Fig. 5. for the cylindrical compression optics, namely, elliptical, double-egg, and hyperbolic shapes, are also observed in Fig. 10 (c). In contrast, all free-space and spherically compressed holograms exhibit identical holograms and reconstructed image shapes, differing only in scale factors. This distinction also applies to the artifacts (halos) associated with the twin-particle problem, as shown in the reconstructions shown in Fig. 10 (b, d, f). Spherical compression yields holograms with the smallest footprint, effectively minimizing the SFR. However, the reconstructed particle images show the opposite trend. The latter are sharper in the free space case than in the spherical compression case, showing artifacts positioned further away from the particle edges. The cylindrical compression case shows intermediate behaviors.

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<figure><img src="image_10.png" /><figcaption>Fig. 9 Numerical comparisons of the field of view of the DIH system for different magnification factors, water droplets of $D_p=300$ µm in diameter. Each image is 2048 x 2048 pixels.</figcaption></figure>

In free space, the twin image problem is well documented in the literature (e.g. [35, 53]). The standard involves two particles: a real particle and a virtual twin particle whose positions along the z-axis are symmetric with respect to the camera sensor plane. They produce identical diffraction patterns on the sensor. On the other hand, the backpropagation method is based on the direct reconstruction of the three-dimensional field at distances $z, z - \delta z, z - 2\delta z, ...$ where the real particle is assumed to be located. Since the real and virtual particles are on opposite sides of the camera sensor, they are easy to distinguish. However, each reconstructed particle image is affected by the diffraction $1 - T - [[T * \bar{h}_z] * \bar{h}_z]$ caused by its own twin particle along with the influence of all other twin particles. This reduces the signal-to-noise ratio in the reconstructed particle field. When spherical lenses are used for the magnification ($\gamma > 1$) of holograms, it is as if the camera sensor had been positioned closer to the particle flows, as shown in Fig. 11 (a).

Then, following the same reasoning as for free space, a real particle and its virtual twin are symmetrical with respect to this virtual sensor (i.e., the real sensor that is imaged by the optical system, Fig. 11 (a)). The twin particle is therefore closer to its real particle, and its diffraction has a greater detrimental effect on the reconstruction process. This twin particle may even be in the optical zone of the flow being probed, where it is indistinguishable from a real particle. Both effects are exacerbated at higher magnifications. A peculiarity of cylindrical lenses compared to spherical ones is that they split the classical twin particle into two “half-twin”
particles. Along the y-axes, the half-twin particle is located symmetrically behind the real sensor, as in the free-space configuration. Its diffraction contribution is weaker due to the larger distance, and this half-twin is always distinguishable from the real particle inside the probe volume, Fig. 11 (c).

<figure><img src="image_11.png" /><figcaption>Fig. 10 (a, c, e) Simulated holograms and corresponding (b, d, f) reconstructed particle images for P=64 monodisperse water droplets with Dp=50 µm dispersed in a volume of 3.5 cm³ around the OFL (=300 mm), under (a, b) free-space, (c, d) cylindrical and (e, f) spherical compression conditions. The image orientations have been adjusted to facilitate comparison.</figcaption></figure>

Along the x-axes, the half-twin is located symmetrically behind the virtual sensor, just as it would be with a spherical lens of the same focal length. However, its diffraction intensity is reduced due to the presence of its twin, and it cannot be confused with a real particle because it is associated with its half-twin which is easily recognizable. The typical focusing functions shown in Fig. 11 (b) clearly illustrate this effect. When cylindrical lenses are used instead of spherical lenses, the peak of the focusing function that identifies a (virtual) particle near or inside the probe volume is eliminated. This reduces the risk of confusing real and virtual particles. It also increases the signal-to-noise ratio of the reconstructed field, improving particle detectability and statistical confidence.

Fig. 12 shows contour plots of the relative error in estimating particle diameter and particle distance from the camera sensor. For clarity, the scale is linear for diameters, while distances are plotted on a logarithmic scale. The holograms analyzed were generated numerically for a probe volume similar to that in Fig. 10, containing P = 1, 2, 4, ..., 256 water droplets with
diameters uniformly distributed among the size classes of 25, 50, 100, 200, 400, and 800 µm, simulating a water spray with a broadband size distribution (uniform between 25 and 800 µm, mean diameter 307 µm and standard deviation 282 µm) and increasing concentrations. Between 50 and 100 holograms were generated for each particle count $P$.

<figure><img src="image_12.png" /><figcaption>Fig. 11 Illustration of half-twin formation and separation along (a) the x-axis and (c) y-axis when using spherical and cylindrical optics. (b) Evolution of the corresponding focusing functions: with cylindrical optics, the peak along the x-axis that identifies a virtual particle near the probe volume disappears almost completely, making it easier to distinguish from the real particle. The second half-twin, along the y-axis remains behind the camera sensor, like in the free-space case. Its contribution to the reconstructed field noise is thus reduced.</figcaption></figure>

The same Gaussian white noise (mean = 0, σ = 0.0485) was added to the different holograms to better replicate experimental conditions. Fig. 13 shows the corresponding mean value and RMS (error bars) of the SFR of type $R_2$ for a particle number going up to $P=2048$. Cylindrical compression provides an intermediate SFR compared to free-space and spherical compression. As a concrete example, an SFR of 50% corresponds to approximately 70, 200, and 700 particles per hologram for the free-space, cylindrical, and spherical configurations respectively. Note that the SFR produced by the droplets projected images (with and without overlapping) is also shown in Fig. 13, as this criterion is often used in literature.

The estimates obtained with cylindrical compression optics are globally less variable for droplet diameters and, more importantly, for axial positions, see Fig. 12. The improvement over spherical compression is evident, with most errors attributable to twin particles located too close to the real particles. The improvement over the free space configuration is less pronounced, but still significant. With cylindrical compression optics, the average error in particle axial positions is statistically reduced by a factor of ~20 (i.e. $10^{-4.32}/10^{-5.58}$) for the axial positions and ~2.8 (i.e. $0.11/0.04$) for diameters. These improvements are due to the reduced propagation of artifacts associated with the twin-particle problem, as illustrated by the reconstructions shown in Fig. 10 b and Fig. 10 d. Since the same noise was added to the different holograms, its relative influence should be moderated on our comparisons of the different optical configurations (free space, cylindrical and spherical compression). Note that although holograms with more than 256 particles were generated (as in Fig. 13), they were not
included in the statistics presented in Fig. 12 because all results were clearly biased by the performances of the HYBRID algorithm.

<figure><img src="image_13.png" /><figcaption>Fig. 12 Numerical evaluation of the relative error of the DIH system in estimating particle diameter (linear scale) and axial distance (logarithmic scale) from the camera sensor. Configurations: (a) free space, (b) cylindrical compression, and (c) spherical compression, with the same field of view ($\gamma=-1$) and Gaussian white noise (mean=0, $\sigma=0.0485$). The processed holograms contained 1, 2, 4, ..., 256 water droplets of uniformly distributed sizes $D=25, 50, 100, 200, 400$ or 800 µm (i.e. mean diameter 307 µm, standard deviation 282 µm). The holograms shown in the lower right corners correspond to 256 particles in the probe volume.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 13 Evolution of the SFR from holograms and particle projected images for an increasing number of the particles within the probed zone (3.5 cm³ in volume). Same parameters as in Fig. 12, except for the number of particles, which goes to *P*=2048. The purple area highlights all cases used for the statistics shown in Fig. 12.</figcaption></figure>

## 5. Conclusion

The one-dimensional compression method effectively reduces the SFR by producing holograms that appear egg-shaped, elliptical, or hyperbolic for rotationally symmetric particles, depending on the distance of the particle from the optical plane. It significantly reduces overall error in both axial positioning and diameter measurement by limiting the influence and detection of twin particles. The corresponding statistics are much better than with a spherical optics at the same magnification. One-dimensional compression also mitigates cross-interference caused by diffraction from walls perpendicular to the compression axis, a feature valuable in the study of interfacial phenomena. However, for several reasons, the reduction in SFR and the resulting benefits are less than the authors themselves originally expected. On the one hand, the *e*₁ estimator overestimates the benefits of the method in terms of SFR, since it does not consider noise levels and blurring effects. The hologram imprint reduction rate also decreases with increasing |Δz|. This phenomenon is amplified if the focal length of the compression optics is reduced. On the other hand, we have shown that the free space and spherical compression configurations maintain good accuracy at high SFRs, due to the ability of the backpropagation method to reconstruct individual holograms along with their multi-wave interferences (among 2, 3, 4, ..., *P* − 1 holograms). It turns out that interferences appear to have surprisingly weak effects on the reconstruction quality, provided that the basic assumptions of the DIH are still met. This is also a reason why the authors believe that the advantages of the one-dimensional compression method should be greater for parametric and inverse reconstruction methods based on the analysis of individual hologram intensity profiles [4, 28-32] or phase retrieval methods [70, 71] for instance, a subject that is already under investigation [52, 72]. The analysis and correction of noise sources and optical aberrations induced by the magnification optics (as for example in ILIDS [73] for instance), as well as more quantitative arguments for the selection of the optimal focal length, will be the subject of future work. In any case, the prospects opened up by the results and guidelines presented here are already promising.

## Acknowledgments
The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) and Bpifrance, under grants ANR-23-CE51-0023 (project ATICS) and PIA-MIRAGE-DOS0196786/00.

## Disclosures

The authors declare no conflicts of interest

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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