The Wigner Distribution Function of Two-dimensional Signals Coherent-optical Generation and Display

Redundancy of phase-space distribution functions in complex field recovery problems

Shows that the amplitude and phase of an optical field can be recovered from only a section of phase-space distribution functions, the information contained in other sections being redundant. Experimental implications of this result are discussed, and a physical interpretation of it is offered. In particular the sampling problem in tomography is solved for any field distribution.

One-step measurement of optical fields in multimode circular fibers

We propose to determine the optical field in multimode circular fibers by using a one-step method that measures the Wigner distribution function of a section of the field in the fiber. This method allows an estimation not only of the power carried by each mode but also of the relative phases of different modes in the fiber. An additional measurement with the same setup can even determine the propagation constants of different modes. An example is provided, and the connection of this method of field recovery to the coupling coefficient between fibers and light sources is also discussed.

Reconstruction of the Wigner transform of a rotationally symmetric two-dimensional beam from the Wigner transform of the beam's one-dimensional sample

A coherent rotationally symmetric two-dimensional beam is essentially one-dimensional in content: It is fully determined by the one-dimensional sample along a diagonal of the circularly symmetric field distribution in a transverse plane. The linear transform that reconstructs the four-dimensional Wigner distribution of the full two-dimensional beam from the two-dimensional Wigner distribution of the one-dimensional sample is presented.

Optical realization of the ambiguity function of real two-dimensional light sources

We propose a setup that can generate the sectional ambiguity function of a two-dimensional real light source. The setup is easy to implement; the theoretical analysis and experimental results are given.

Can the Wigner transform of a two-dimensional rotationally symmetric beam be fully recovered from the Wigner transform of its one-dimensional approximation?

It is shown that the full four-dimensional Wigner transform of a coherent, rotationally symmetric light beam can be completely recovered by measurement, in one step, of the Wigner transform of an equivalent one-dimensional light beam. The method of generating this equivalent light beam from a two-dimensional circular light beam is presented.

Phase-space measurements of micro-optical objects

The phase-space measurement of micro-optical objects with submillimeter dimensions is reported for the first time to our knowledge. The experimental data were compared with simulated results from interferometric measurements and were found to be in good agreement.

Implementation of the spatial and the temporal cross-ambiguity function for waveguide fields and optical pulses

On the basis of space-time duality, we propose experimental setups to implement the cross-ambiguity function optically in space and time in one and two dimensions. In space the cross-ambiguity is shown to be related to the coupling efficiency between butt-joined optical waveguides. In time it is related to the spectrogram or the frequency-resolved optical gating techniques for the characterization of optical pulses.

Wigner distribution function applied to third-order aberrations

The Wigner-distribution-function description of light beams is extended to aberrated optical systems. The simulations performed show that the different types of aberrations can be identified separately by the use of experimental devices that display projections of the Wigner distribution function of a two-dimensional beam.

White-light implementation of the Wigner-distribution function with an achromatic processor

A temporally incoherent optical processor that combines diffractive and refractive components is proposed for performing two different operations simultaneously: an achromatic image along an axis and an achromatic one-dimensional Fourier transformation along the orthogonal axis. These properties are properly employed to achieve the achromatic white-light display of the Wigner-distribution function associated with a one-dimensional real signal, with high redundancy and variable scale.

Real-time generation of the Wigner distribution of complex functions using phase conjugation in photorefractive materials

An optical processor for real-time generation of the Wigner distribution of complex amplitude functions is introduced. The phase conjugation of the input signal is accomplished by a highly efficient self-pumped phase conjugator based on a 45 degrees -cut barium titanate photorefractive crystal. Experimental results on the real-time generation of Wigner distribution slices for complex amplitude two-dimensional optical functions are presented and discussed.

Optical digital implementation of the Wigner distribution function: use in space variant filtering of real images

An optical digital processor based on the Wigner distribution function (WDF) has been implemented, permitting space variant filtering of 2-D real images. The processor sequentially generates all space samples of the distribution; thus, each sample can be modified by a different filter. The filtered Wigner distribution function is optically inverted, and a digital image processor selects the information associated with each sample of the image. Compared to a digital implementation, the required computer time has been reduced to the point where future practical use of Wigner filtering can be expected.

Space-variant filtering through the Wigner distribution function

Space-invariant and space-variant filtering of discrete images with unidimensional variation is performed in this paper through the Wigner distribution function (WDF). Low-pass, bandpass, and high-pass filtering is used in the Fourier domain and in the Wigner distribution, to compare their different behavior in both cases. A space-variant filter is generated to modify the WDF of an object to obtain, after inversion, a spatially variant filtered image. Finally, spatially variant defocused images generated through the Wigner distribution are restored by Wiener based filters applied in the Wigner domain. The method applied to recover the filtered images can easily be generalized to bidimensional images.

Image analysis through the Wigner distribution function

The Wigner distribution (WD) for discrete images is computed and applied for texture analysis. First, the WD for different test images with spatial frequency and gray level information content is computed, and the most important features of the WD are outlined. Likewise, the original image with 1-D variation is recovered from the discrete WD; local filtering operations have been performed on the distribution, and a filtered version of the original image is then obtained. Since the WD entails simultaneous representation of the intensity distribution and of the spatial interdependence, it seems specially adequate for processing of textured information. In this way, texture recognition is performed through the extraction of features from the WD, and the results are compared with other methods. Finally, to avoid the great computational effort that requires this kind of representation, a hybrid optical-digital processing system is proposed for the texture recognition process.

Generation of the Wigner distribution function of two-dimensional signals by a parallel optical processor

An optical system employing a reflective multipath scheme is implemented to generate a sectioned array of slices of the Wigner distribution function of a two-dimensional (2-D) signal. The optical system exhibits a parallel architecture, permitting the simultaneous generation and display of the N slices. Only a single transparency is needed, and no moving parts are required in the system. Experimental results on a few 2-D signals are presented.