Computer simulation of the speckle field propagation

Abstract

There is presented a simulation of the speckle propagation by means of a computer in this paper. The model is designed for in-plane translation of an object generating speckle field. There is used the Fresnel-Kirchhof diffraction theory for description of the speckle field propagation. The presented numerical model involves detection of speckle pattern at any observation angles. The model is verified through a method of correlation of speckle fields which results are compared to ones provided by theoretical relations.

Methods for Determination of Mean Speckle Size in Simulated Speckle Pattern

“…Hence, the main advantage of the optimized method rests not only on less computer time consuming algorithm, but also on the possibility of determining the mean speckle size from a smaller amount of the detected intensity values. This paper uses the simulation model ( Fig.1.) according to [6][7][8] and the way of reducing intensity values (decimation) presented in [6]. The aim of the paper [6] is to present an extension of the one-dimensional speckle correlation method, which is primarily intended for determination of one-dimensional object's translation, for detection of general in-plane object's translation.…”

“…For simulation of the speckle effect the numerical simulation model [7], [8] is used. The main part of the program code is comprised of the Fresnel-Kirchhoff diffraction integral calculation through the rectangle method.…”

“…The object's surface is represented by a matrix of values defining random variable surface roughness. The presented numerical model enables to select initial parameters of simulation independently, including nonzero angle of observation [8].…”

A Simulation Analysis of an Extension of One-Dimensional Speckle Correlation Method for Detection of General In-Plane Translation

“…The illuminated area of the object's surface is sampled by m i × n i points. The complex amplitude of the Gaussian beam in the object's surface with random variable surface roughness Δ( x , y ) can be determined from the following relation [16, 21]: $\begin{array}{c}U(x,y,\mathrm{normal\Delta }(x,y))\\ \hspace{1em}=\frac{{\omega }_o}{\omega \left(L_s+\mathrm{\Delta }\left(x,y\right)\right)}\exp \left[-\frac{x^2+y^2}{{\omega }^2\left(L_s+\mathrm{\Delta }\left(x,y\right)\right)}\right]\\ \hspace{1em}\hspace{1em}\times \exp [-ik\left(L_s+\mathrm{\Delta }\left(x,y\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}-ik\frac{x^2+y^2}{2R\left(L_s+\mathrm{\Delta }\left(x,y\right)\right)}+i\xi \left(L_s+\mathrm{\Delta }\left(x,y\right)\right)],\end{array}$ where R is a radius of curvature of the wavefront comprising the beam, ξ is a longitudinal phase delay of the beam, and ω o and ω are radii of the Gaussian beam at its waist and at a distance L s from the waist (Figure 1). …”

“…The resulting complex amplitude U ( x ′, y ′) in the detection plane ( x ′, y ′) at the distance L o from the object plane ( x , y ) is determined by the following relation [16, 21]: $\begin{array}{c}U\left(x^{\prime },y^{\prime }\right)=\frac{\exp \left(-ik{L}_{oN}\right)}{i\lambda L_{oN}}\\ \times \underset{-\infty }{\overset{\infty }{{\displaystyle false\int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle false\int }}}U\left(x,y\right)\exp \left(-ik\mathrm{\Delta }\left(x,y\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \left(-ik\frac{{\left(x_N-x^{\prime }\right)}^2+{\left(y-y^{\prime }\right)}^2}{2L_{oN}}\right)dxdy,\end{array}$ where $\begin{array}{c}{x}_N=x\cos {\theta }_o,\\ L_{oN}=L_o-x_N\tan {\theta }_o\end{array}$ represent transform relations of the x -coordinate in the object plane ( x , y ) and the distance L o for the case of the object plane is rotated around the y -axis by the angle θ o against the x -axis (Figure 1) [21]. The matrix detector of size V x ′ × V y ′ placed in the detection plane ( x ′, y ′) is sampled by m × n points.…”