# Numerical Models for Exact Description of in-situ Digital In-Line Holography Experiments with Irregularly-Shaped Arbitrarily-Located Particles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulator

#### 2.1. Amplitude Distribution of the Beam in the Plane of the Object

_{0}is the waist of the beam. Then the laser beam propagates through the first optical system from the incident plane to the object plane. This part of the optical set-up is described by an optical transfer matrix M

_{1}as can be seen in Figure 1. The complex field amplitude of the wave in the plane of the object is noted G

_{1}. It is obtained from a generalized Fresnel integral [19,25,26,27]. G

_{1}is given by:

_{1}is the optical path between the incident plane and the object plane. ${A}_{1}^{x,y}$, ${B}_{1}^{x,y}$, and ${D}_{1}^{x,y}$ are the matrix elements of the matrix ${M}_{1}^{x,y}$, where ${M}_{1}^{x}={M}_{1}^{y}={M}_{1}$ for an axisymmetric system. ${M}_{1}^{x}$ and ${M}_{1}^{y}$ will be different for a cylindrical system. ξ and η are the transverse coordinates in the object plane. Substituting Equation (1) into Equation (2), the complex amplitude G

_{1}(ξ, η) in the object plane becomes:

_{1x,y}and R

_{1x,y}are respectively the beam waist radii and the beam curvatures in the object plane. They are deduced from the classical formula of Gaussian beams. It is important to note that this formulation allows the description of cylindrical systems, by separation of the two transfer matrices along both transverse axes : ${M}_{1}^{x}$ and ${M}_{1}^{y}$. The details of this calculus can be found in the appendix of reference [19].

#### 2.2. Definition of the Objects

^{iϕ}and equal to 1 outside the object. The procedure is slightly longer. We realize a collection of circular opaque objects. The transmission coefficient of the object is then modified. We can further create an object where ϕ is not constant. For both opaque and phase objects, we can adjust the characteristic dimensions of the object. Some examples will be presented in the next sections.

#### 2.3. Amplitude and Intensity Distributions in the Plane of the CCD Sensor

_{2}. It is given by the following relation:

_{2}is the optical path between the object plane and the CCD plane. ${A}_{2}^{x,y}$, ${B}_{2}^{x,y}$, and ${D}_{2}^{x,y}$ are the coefficients of the matrices ${M}_{2}^{x,y}$ which describe the second part of the optical system (see Figure 1). We have ${M}_{2}^{x}={M}_{2}^{y}={M}_{2}$ for an axisymmetric system. T is the transmittance of the object. x and y are the transverse coordinates in the sensor plane. This second integral is evaluated numerically. The intensity distribution recorded by the CCD sensor is then obtained from:

#### 2.4. Hologram Analysis by Fractional Fourier Transform

_{p}| ≤ π/2. The kernel of the fractional operator is defined by:

_{p}) defined by:

_{p}) enables conservation of energy. Parameters s

_{p}(p = x or y) are defined from the experimental set-up according to [32]:

_{p}are the pixel sizes along both axes (p = x or y). In practice δ

_{x}= δ

_{y}. To reconstruct the image of the particle, we evaluate the 2D-FRFT of the diffraction pattern. The diffraction pattern contains a term with a linear chirp φ. This linear chirp is easily recovered through 2D-FRFT operation. The kernel of the 2D-FRFT operator is indeed composed of a linearly-chirped function, leading to applications in spatial optics when considering diffraction patterns (as here) or in temporal optics when considering chirped pulses [33]. The best reconstruction plane is reached when [19]:

_{a}± φ = 0

_{a}is the phase contained in 2D-FRFT and φ is the phase contained in the diffraction pattern. From this condition, the optimal fractional orders of reconstruction ${\alpha}_{x}^{opt}$ and ${\alpha}_{y}^{opt}$ can be obtained.

## 3. Simulation of Two Objects in Different Longitudinal Planes: Mixing of Opaque and Phase Objects

_{0}= 2.3 µm. The wavelength is λ = 642 nm. The simulated configuration respects e

_{1}= 58.8 mm, f

_{1}= 42.8 mm, e

_{2}= 342.84 mm, f

_{2}= 5.5 mm. The first object is located at distance Δ

_{1}= 9.5 mm and the second object is located at distance Δ

_{2}= 10 mm from the second lens as described in Figure 3. The distance between the second lens and the CCD sensor equals 18 mm.

_{1,2x,y}are the magnification factors between Plane 1 (or Plane 2) and the CCD plane. w

_{1,2x,y}are the beam waist dimensions (along x and y, respectively) of the unperturbed Gaussian beam in Plane 1 (or in Plane 2 respectively). w

_{totx,y}are the beam waist dimensions of the unperturbed Gaussian beam in the plane of the CCD sensor. They are given by:

^{2}. The magnification factor introduced by the system for the object in Plane 1 is equal to 6.8, while it is equal to 5.3 for the object in plane 2.

**Figure 4.**(

**a**) Simulated hologram considering both phase and opaque objects located in two different planes; (

**b**) Reconstructed phase object with optimal order ${\alpha}_{x,y}^{opt}$ = −0:600 π/2 (red arrow); (

**c**) Reconstructed opaque object with optimal order ${\alpha}_{x,y}^{opt}$ = −0:686 π/2 (red arrow).

**Figure 5.**Transverse intensity profile of the reconstructed phase (

**a**) and opaque (

**b**) object: $\left|{\mathcal{F}}_{{\alpha}_{x}^{opt},{\alpha}_{y}^{opt}}\left[I\left(x,y\right)\right]\left({x}_{\alpha},{y}_{\alpha}\right)\right|$.

_{A}, y

_{B}and y

_{c}are normalized intensities measured on the three points A, B and C of the transverse profile defined in Figure 6. The present simulator allows to evaluate this parameter for any complex phase object. The method is efficient when 0.2 π < φ < 1.6 π. In other cases, the discontinuity is too small to be observed experimentally. Parameter φ can be recovered from estimated parameter $\frac{{y}_{A}-{y}_{B}}{{y}_{c}}$. A comparison of experimental results with simulation allows to validate the determination of the phase shift introduced by the object.

**Figure 6.**Transverse intensity profiles of the reconstructed phase objects $\left(\left|{\mathcal{F}}_{{\alpha}_{x}^{opt},{\alpha}_{y}^{opt}}\left[I\left(x,y\right)\right]\left({x}_{\alpha},{y}_{\alpha}\right)\right|\right)$ when φ = 0.25 π (solid line), φ = 0.5 π (dotted line) and φ = π (dashed line).

## 4. Two Examples of Application

#### 4.1. Simulation of Phase Objects in a Droplet for Detection of Nanoparticles

_{1}and M

_{2}. Let us consider the experimental set-up of Figure 7. The beam emitted by a fibered-laser diode is collimated using a first lens and then focused in the vicinity of a suspended water droplet with a microscope objective. Calibrated microspheres (diameter 20 µm) are introduced in the droplet. Figure 8 (on the left) shows optimal reconstruction of some of the microspheres. The wavelength of the incident beam is 642 nm. The diameter of the water droplet is 1.5 mm and its refractive index is 1.33. The other parameters are e

_{1}= 42.8 mm, f

_{1}= 42.8 mm, e

_{2}= 135.6 mm, f

_{2}= 5.5 mm, e

_{3}= 10 mm, z

_{1}= 0.825 mm, z

_{2}= 1.125 mm, and z = 9.7 mm. Let us now present in Figure 8 (on the right) the optimal reconstruction issued from a simulated hologram. In the case of this reconstruction, we have considered 7 circular particles: 4 are opaque objects, 3 are transparent phase objects (phase shift introduced π). The fact that the particles are located in a droplet is not a problem. The different optical elements of the droplet are considered through the appropriate definition of matrices M

_{1}and M

_{2}. The different inclusions are clearly reconstructed. The opaque and non-opaque objects are easily identified. The possibility to simulate this experiment is particularly attractive for us. This set-up can indeed be used to detect 50 nm nanoparticles [22]: the droplet is seeded with nanoparticles. The droplet is then heated using a second laser (a frequency-doubled Nd:YAG laser emitting 4 ns, 10 mJ pulses in our case). Heating a nanoparticle creates a bubble which is reconstructed using Digital In-Line Holography [22]. For more details concerning the description of spherical micro-bubbles in this experiment, please refer to [22]. This indirect detection method is particularly efficient. With this new simulator that can describe arbitrarily-shaped phase objects as arbitrarily-shaped bubbles, we should be able to obtain information on morphology of nanoparticles through analysis of the shape of microbubbles in the future [35]. Our numerical model allows this.

**Figure 8.**(

**a**) Experimental reconstruction of inclusions in a droplet; (

**b**) Simulated reconstruction of opaque and transparent inclusions in a droplet.

#### 4.2. Library of Ice Crystal Objects for the Calibration of Interferometric Airborne Instruments

**Figure 9.**Some examples from our library of transparent phase objects describing ice crystals. A dendritic crystal with sector-like ends (

**top left**); A malformed crystal (

**top right**); An ordinary dendritic crystal (

**bottom left**) and a crystal with sectorlike branches (

**bottom right**).

**Figure 10.**Transparent phase particle whose form is the one of a dendritic crystal with sector-like ends (

**top**); Example of Digital In-line Hologram generated by this particle (

**bottom left**) and its optimal reconstruction using 2D-FRFT (

**bottom right**).

## 5. Conclusions

## Acknowledgments

^{3}. These results have further received funding from the European Union Seventh Framework Programme (FP7/2007-2013): project EUFAR (EUropean Facility for Airborne Research), and project HAIC (High Altitude Ice Crystal) under grant agreement n°ACP2-GA-2012-314314.

## Author Contribution

## Conflicts of Interest

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**MDPI and ACS Style**

Brunel, M.; Wichitwong, W.; Coetmellec, S.; Masselot, A.; Lebrun, D.; Gréhan, G.; Edouard, G.
Numerical Models for Exact Description of in-situ Digital In-Line Holography Experiments with Irregularly-Shaped Arbitrarily-Located Particles. *Appl. Sci.* **2015**, *5*, 62-76.
https://doi.org/10.3390/app5020062

**AMA Style**

Brunel M, Wichitwong W, Coetmellec S, Masselot A, Lebrun D, Gréhan G, Edouard G.
Numerical Models for Exact Description of in-situ Digital In-Line Holography Experiments with Irregularly-Shaped Arbitrarily-Located Particles. *Applied Sciences*. 2015; 5(2):62-76.
https://doi.org/10.3390/app5020062

**Chicago/Turabian Style**

Brunel, Marc, Wisuttida Wichitwong, Sébastien Coetmellec, Adrien Masselot, Denis Lebrun, Gérard Gréhan, and Guillaume Edouard.
2015. "Numerical Models for Exact Description of in-situ Digital In-Line Holography Experiments with Irregularly-Shaped Arbitrarily-Located Particles" *Applied Sciences* 5, no. 2: 62-76.
https://doi.org/10.3390/app5020062