Mj Martin Bastiaans

Fractional-Fourier-domain weighted Wigner distribution

A fractional-Fourier-domain realization of the weighted Wigner distribution (or S-method), producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms, is presented. The computational cost of this fractional-domain realization is the same as the computational cost of the realizations in the time or the frequency domain, since the short-time Fourier transform of the fractional Fourier transform of a signal corresponds to the short-time Fourier transform of the signal itself, with the window being the fractional Fourier transform of the initial one. The appropriate fractional domain is found from the analysis of the second-order fractional Fourier transform moments. Numerical simulations show a qualitative advantage in the time-frequency representation, when the calculation is done in the optimal fractional domain.

Propagation of the generating function of Hermite-Gaussian-type modes in first-order optical systems

Based on the common Hermite-Gaussian modes, a general class of orthonormal Hermite-Gaussian-type modes is introduced. Such modes can most easily be defined by means of their generating function. A propagation law for the generating function is formulated, when these modes propagate through first-order optical systems.

Radon-Wigner transform for optical field analysis

The Radon-Wigner transform, associated with the intensity distribution in the fractional Fourier transform system, is used for the analysis of complex structures of coherent as well as partially coherent optical fields. The application of the Radon-Wigner transform to the analysis of fractal fields is presented.

Wigner distribution moments measured as fractional Fourier transform intensity moments

It is shown how all global Wigner distribution moments of arbitrary order can be measured as intensity moments in the output plane of an appropriate number of fractional Fourier transform systems (generally anamorphic ones). The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived.

Noise analysis in Toeplitz and Hankel kernels for estimating time-varying spectra

An analysis of noise influence to time-varying spectra estimators is presented. Hankel and Toeplitz kernel forms of distributions are considered. It is shown that these two forms of kernels have the same variance, if the windows have the same form. Since the Hankel kernel results in a highly concentrated distribution for a noisy, frequency-modulated signal, this form can be more reliable for the estimation of signal parameters in noise. The results are generalized to multiplicative noise; in this case, the same conclusions hold for frequency-modulated signals.

Propagation law for Hermite- and Laguerre-Gaussian beams in first-order optical systems

Starting from Hermite-Gaussian beams, we generate a general class of rotationally symmetric beams. These beams are Laguerre-Gaussian beams, parameterized by two parameters h and g, representing the curvature and the width of the beam, respectively. The Wigner distribution of each member of this class is readily derived from the Wigner distribution of the Hermite-Gaussian beam from which it is generated. If these Laguerre-Gaussian beams propagate through an isotropic abcd-system, they remain in their class, while the propagation of the complex beam parameter h ± ig satisfies the well-known abcd-law.

Mapping of Hermite-Gaussian modes in ABCD systems

The orthonormal set of Hermite-Gaussian modes maps into another orthonormal set after propagation through a first-order optical system. A compact expression for these modes is derived. The introduction of a novel class of orthonormal modes is useful for non-interferometric phase recovery techniques and simplifies the design of mode converters and information processing systems.

New focus on Fourier optics techniques

We present a short overview on the application of fractional cyclic and linear canonical transformations to optical signal processing and dedicate some of the discussions to the particular features found in the fractional Fourier transform domain.

Transformation of the vortex part of the orbital angular momentum in first-order optical systems

The transformation of the vortex part of the orbital angular momentum of linearly polarized, partially coherent beams during their propagation through separable first-order optical systems is analyzed. We obtain that the evolution of the vortex part depends only upon the parameters ax, ay, bx, and by of the ray transformation matrix. Isotropic systems with the same ratio b/a produce the same change of the vortex part of the orbital angular momentum.

Powers of transfer matrices and cyclic cascades

The parameters of the transfer matrix describing a first- order optical system that is a cascade of k identical subsystems defined by the transfer matrix M, are determined from considering the subsystems eigenfunctions. A condition for the cascade to be cyclic is derived. Particular examples of cyclic first-order optical systems are presented.