Abstract
Digital in-line holography is revisited to propose a mathematical model that describes the recording-reconstruction process as a linear shift-invariant system with a pseudo-point spread function even when the images are out of the optimal plane in the sense of signal processing. A particular case is treated to show that the optimal plane is the best focus plane in the sense of optics. Next, an exact solution of the holographic reconstruction by correlation is given. By means of the previous results, we study the behavior of the result of the correlation function between the diffraction pattern function produced by an opaque disk and a chirplet function and between the diffraction pattern produced by a phase disk and the same chirplet function.
Highlights
In nature, the class of chirp functions is one of the most important classes
The physical domain considered in this publication is physical optics and more precisely diffraction and interference phenomena where the linear frequency modulation (FM) chirp is preponderant [2, 3]
To describe such physical processes, the mathematical tool used is an integral operator with a scaling chirp kernel
Summary
The class of chirp functions is one of the most important classes. A wide ranged domain of applications such as seismic signals, sonar, speech signals and images, ECG and radar signals are concerned with such functions [1]. The physical domain considered in this publication is physical optics and more precisely diffraction and interference phenomena where the linear frequency modulation (FM) chirp is preponderant [2, 3] To describe such physical processes, the mathematical tool used is an integral operator with a scaling chirp kernel. In the far field approximation, spherical particles and opaque disks give the same diffraction patterns [9]. Let us recall that the intensity distribution may be represented as a bidimensional convolution between Π(r), and the chirplet from Eq (1), such as for the opaque disk, one has: Io(r).
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