Phase-space tomography with a programmable Radon–Wigner display

Abstract

We show the adaptation of a multifunctional optical system consisting of two spatial light modulators for the optimal measurement of the Radon-Wigner transform of one-dimensional signals. The proposed Radon-Wigner display allows reconstructing the Wigner distribution and the phase or the mutual intensity of fully or partially coherent fields, respectively. It is also suitable for the analysis of two-dimensional rotationally symmetric or separable in Cartesian coordinates optical fields. 0 qÞ is the WD of the signal, ðx; qÞ are the space and frequency coordinates, Γðx 1 ; x 2 Þ ¼ hf ðx 1 Þf à ðx 2 Þi is the mutual intensity of the signal f ð·Þ, and s is a normalization parameter with dimension of length that depends on the system implementation. The transformation angle α varies in a π interval. The kernel of the fractional FT needed for the acquisition of the corresponding projections, Pðα; xÞ ¼ hj R duf ðuÞK α ðx; uÞj 2 i, is given bySeveral schemes for the RWT visualization-RadonWigner displays (RWDs)-have been proposed. They are based on the use of either combinations of Fresnel zone plates [3] or varifocal glass lenses [4,5]. In spite of important achievements, these setups have not found wide application in signal processing due to their complex design and the involvement of nonconventional optical elements. Moreover, in most of the schemes, each projection has a different magnification depending on the angle α.The use of spatial light modulators (SLMs) allows constructing an RWD that avoids the previous inconveniences and provides flexibility to the system. Here, we demonstrate how a multifunctional optical scheme consisting of two SLMs can be easily converted into an RWD. Its performance is verified experimentally. We also show that the RWD, primarily created for the description of 1D signals, can also be applied for the analysis of 2D optical fields.

Pseudo-differential representation of the metaplectic transform and its application to fast algorithms

“…= Q(y)|ψ 0 , where |Q(y) is the eigenvector ofQ corresponding to the eigenvalue y. We assume the usual normalization, q(x)|q(y) = Q(x)|Q(y) = δ(x − y), so dx |q(x) q(x)| = dx |Q(x) Q(y)| =1 (6) and |Q(x) =M † t |q(x) . Here1 is a unit operator.…”

“…This mapping is called the metaplectic transform (MT) [1,2]. It subsumes the Fourier transform as a special case and represents one of the pillars of modern phase space analysis used in many applications [3][4][5][6][7][8].…”

Optical Beam Characterisation via Phase-Space Tomography

“…Dichas medidas se pueden juntar posteriormente para incrementar la precisión de la WD reconstruida. Experimentalmente se probó esta ventaja para un haz rectángulo con modulación cuadrática de fase [14]:…”

Measurement of the Wigner distribution function of non-separable laser beams employing a toroidal mirror

“…which is defined as the two-dimensional Fouriertransform of the mutual coherence function. That method is well established for visible and UV wavelengths [15][16][17][18][19] and has been applied to synchrotron [20] and FEL sources [21,22] too.…”