## 2 Numerical Simulator  
### Amplitude Distribution of the Beam in the Plane of the Object  
The basic idea of Digital In-line Holography (DIH) is to record the intensity pattern diffracted by an object on an electronic array sensor. The object is then reconstructed numerically using image processing. Figure 1 is the typical set-up of a DIH experiment using an incident Gaussian beam. This specific case is well adapted to our experiments that use in most cases a single-mode laser beam. Part 1 allows an appropriate illumination of the object. Part 2 is the \"imaging\" system (leading to a \"transport\" of the diffraction pattern from the object to the sensor). In order to interpret precisely the experimental results, it is necessary to develop a general numerical model that can describe any set-up. We present here the simulator that has been developed, using the scalar theory of diffraction. At the beginning of the propagation, the waist of the incident Gaussian beam is located at the origin \((z=0)\). \(z\) is the longitudinal axis of propagation (horizontal axis on Figure 1). In the next equations, the transverse axes will be noted \(\mu\) and \(\
u\) in the plane of the laser source, \(\xi\) and \(\eta\) in the plane of the particle, \(x\) and \(y\) in the plane of the CCD sensor. The electric field amplitude in the incident plane can be written as follows:  
\[G(\mu,\
u,z=0)=G(\mu,\
u)=\exp\left(-\frac{\mu^{2}+\
u^{2}}{\omega_{0}^{2}}\right) \tag{1}\]  
where \(\omega_{0}\) 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:  
\[ \begin{split} G_{1}(\xi,\eta)&=\frac{\exp(i\frac{2 \pi}{\lambda}\ell_{1})}{i\lambda\sqrt{B_{1}^{x}B_{1}^{y}}}\int_{\mathbb{R}^{2} }G(\mu,\
u)\exp\left(i\frac{\pi}{\lambda B_{1}^{x}}[A_{1}^{x}\mu^{2}-2\xi\mu+ D_{1}^{x}\xi^{2}]\right)\\ &\exp\left(i\frac{\pi}{\lambda B_{1}^{y}}[A_{1}^{y}\
u^{2}-2\eta \
u+D_{1}^{y}\eta^{2}]\right)d\mu\ d\
u\end{split}  \tag{2}\]  
where \(\ell_{1}\) 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. \(\xi\) and \(\eta\) are the transverse coordinates in the object plane. Substituting Equation (1) into Equation (2), the complex amplitude \(G_{1}(\xi,\eta)\) in the object plane becomes:  
\[G_{1}(\xi,\eta)=\frac{\exp(i\frac{2\pi}{\lambda}\ell_{1})}{i\lambda\sqrt{B_{1 }^{x}B_{1}^{y}}}G_{0}K_{1}^{x}K_{1}^{y}\exp\left[-\left(\frac{\xi^{2}}{\omega_ {1_{x}}^{2}}+\frac{\eta^{2}}{\omega_{1_{y}}^{2}}\right)\right]\exp\left[-\frac {i\pi}{\lambda}\left(\frac{\xi^{2}}{R_{1x}}+\frac{\eta^{2}}{R_{1y}}\right)\right] \tag{3}\]  
with  
\[K_{1}^{x,y} = \left(\frac{\pi\omega_{0}^{2}}{1-i\frac{\pi\omega_{0}^{2}}{ \lambda}\left(\frac{A_{1}^{x,y}}{B_{1}^{x,y}}\right)}\right)^{1/2}\text{, }\omega_{1_{x,y}} = \left(\frac{\lambda B_{1}^{x,y}}{\pi\omega}\right)\left[1+\left(\frac{\pi \omega_{0}^{2}}{\lambda}\frac{A_{1}^{x,y}}{B_{1}^{x,y}}\right)^{2}\right]^{1 /2}\text{, }R_{1_{x,y}} = -\frac{\frac{B_{1}^{x,y}}{A_{1}^{x}\left(\frac{\pi\omega_{0}^{2}}{ \lambda}\frac{A_{1}^{x,y}}{B_{1}^{x,y}}\right)^{2}}}{\frac{D_{1}^{x,y}-\frac{ A_{1}^{x}\left(\frac{\pi\omega_{0}^{2}}{\lambda}\frac{A_{1}^{x,y}}{B_{1}^{x,y}} \right)^{2}}}{1+\left(\frac{\pi\omega_{0}^{2}}{\lambda}\frac{A_{1}^{x,y}}{B_{1 }^{x,y}}\right)^{2}}}\]  
Figure 1: Typical Digital In-Line Holography set-up. where \(\omega_{1_{x,y}}\) and \(R_{1_{x,y}}\) are respectively the beam waist radii and the beam curvatures in the object plane. They are deduced from the classical formula of Gaussian beams.