Probing nonlinear systems with the Wigner distribution

A new way to study nonlinear systems under forced oscillations is presented. The approach is based on the analysis of the time-frequency spectrum of the output signal of the nonlinear system when a chirp (a frequency increasing in time) is applied as input signal. The time-frequency spectrum is a way to characterize the instantaneous frequency content of a signal. In this paper it is shown that this approach gives a physical explanation of the mechanism that generates forced nonlinear oscillations. Results with simulated and real data prove the validity of the method.     

Wigner distribution function in nonlinear optics

Transformation laws for the Wigner distribution function, the radiant intensity, the radiant emittance, and the first- and second-order moments of the Wigner distribution function through an inhomogeneous, Kerr-type medium have been derived as well as for the beam quality factor and the kurtosis parameter. It is shown that the inhomogeneous Kerr-type medium can be approximated from the Wigner-distribution-function transformation-law point of view with a symplectic ABCD matrix with elements depending on the field distribution.     

Wigner distribution for random systems

We use the Wigner distribution to study systems subjected to random forces. We define the instantaneous spectrum as the ensemble average of the Wigner distribution, and we write the differential equation whose solution gives us the time-varying spectrum of the state variable. We consider the cases of both constant and time-varying coefficients. We apply the method to study the instantaneous spectrum of a harmonic oscillator driven by Gaussian noise, with both constant and time-varying coefficients. In the latter case our method clearly reveals the nonstationarity of the power spectrum and we confirm our result by numerical simulations.     

Wigner distribution moments in fractional Fourier transform systems

It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.     

Wigner distribution function applied to twisted Gaussian light propagating in first-order optical systems

A measure for the twist of Gaussian light is expressed in terms of the second-order moments of the Wigner distribution function. The propagation law for these second-order moments between the input plane and the output plane of a first-order optical system is used to express the twist in one plane in terms of moments in the other plane. Although in general the twist in one plane is determined not only by the twist in the other plane but also by other combinations of the moments, several special cases exist for which a direct relationship between the twists can be formulated. Three such cases, for which zero twist is preserved, are considered: (i) propagation between conjugate planes, (ii) adaptation of the signal to the system, and (iii) the case of symplectic Gaussian light.     

Wigner distribution function for Gaussian-Schell beams in complex matrix optical systems

For Gaussian-Schell beam propagation through complex matrix optical systems, it is shown that, in some particular cases, an A B C D transformation law for the Wigner distribution function holds. For these situations, invariant quantities for the Gaussian-Schell beam propagation can be defined analogous to the real matrix case.     

Polychromatic axial behavior of aberrated optical systems: Wigner distribution function approach

We propose a new method for the computation of the tristimuli values that correspond to the impulse response along the optical axis provided by an imaging optical system working under polychromatic illumination. We show that all the monochromatic irradiance distributions needed for this calculation can be obtained from the Wigner distribution function associated with a certain version of the pupil function of the system. The use of this single phase-space representation allows us to obtain the above merit function for aberrated systems with longitudinal chromatic aberration and primary spherical aberration. Some numerical examples are given to verify the accuracy of our proposal.     

Sampling in the light of Wigner distribution

We propose a new method for analysis of the sampling and reconstruction conditions of real and complex signals by use of the Wigner domain. It is shown that the Wigner domain may provide a better understanding of the sampling process than the traditional Fourier domain. For example, it explains how certain non-bandlimited complex functions can be sampled and perfectly reconstructed. On the basis of observations in the Wigner domain, we derive a generalization to the Nyquist sampling criterion. By using this criterion, we demonstrate simple preprocessing operations that can adapt a signal that does not fulfill the Nyquist sampling criterion. The preprocessing operations demonstrated can be easily implemented by optical means.     

Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function

An analysis of the Strehl ratio and the optical transfer function as imaging quality parameters of optical elements with enhanced focal length is carried out by employing the Wigner distribution function. To this end, we use four different pupil functions: a full circular aperture, a hyper-Gaussian aperture, a quartic phase plate, and a logarithmic phase mask. A comparison is performed between the quality parameters and test images formed by these pupil functions at different defocus distances.     

Wigner distribution function of an Airy beam

We study the Wigner distribution function (WDF) of an Airy beam. The analytical expression of the WDF of an Airy beam is obtained. Numerical and graphical results of the WDF of an Airy beam provide an intuitive picture to explain the intriguing features of an Airy beam, such as weak diffraction, curved propagation, and self-healing. Our results confirm that these novel properties of an Airy beam are attributed to the continuum of sideways contributions to the field.     

The Wigner distribution function of self-Fourier functions

Abstract

Particular forms of the Wigner distribution function and its moments of arbitrary order are derived for a generalized definition of the self-Fourier functions. The general form of the optical system matrix for which an input self-Fourier function transforms to an output self-Fourier function is found. The bright soliton solution of the electromagnetic wave is treated as an example of a generalized self-Fourier function.     

Generalized Wigner function for the analysis of superresolution systems

The generalized Wigner function is able to represent light distributions that contain spatial and temporal information. The use of such a generalized Wigner distribution function for analysis and understanding of temporally restricted superresolving systems is demonstrated. These systems gain spatial resolution by conversion of the temporal degrees of freedom to spatial degrees of freedom.     

FEL beam characterization from measurements of the Wigner distribution function

The Free-Electron-Laser FLASH at DESY has been characterized by a quantitative determination of the Wigner distribution function. The setup, comprising an ellipsodial mirror and a moveable extreme UV sensitive CCD detector, enables the mapping of two-dimensional phase spaces corresponding to the horizontal and vertical coordinate axes, respectively. For separable beams this yields the entire Wigner distribution, offering comprehensive information about spatial coherence properties, wavefront, beam profiles, as well as beam propagation parameters.     

Opto-digital tomographic reconstruction of the Wigner distribution function of complex fields

An optical-digital method has been developed to obtain the Wigner distribution function of one-dimensional complex fields. In this technique an optical setup is employed to experimentally achieve the Radon-Wigner spectrum of the original signal through intensity measurements. Digital tomographic reconstruction is applied to the experimental spectrum to reconstruct the two-dimensional Wigner distribution function of the input. The validity of our proposal is demonstrated with experimental data, and the results are compared with computer simulations.     

Phase-space interferences as the source of negative values of the Wigner distribution function

It is shown that the negative values of the Wigner distribution function in classical optics are a consequence of the phase-space interference among the Gaussian beams into which an arbitrary light distribution (or a superposition of light distributions) can be decomposed. These elementary Gaussian beams partition the phase space in wave optics in adjacent, interacting, finite-area cells, in contrast to geometrical optics, where the phase space is continuous and a light beam can be decomposed into a number of perfectly localized, non-interacting rays.     

Three-dimensional matching by use of phase-only holographic information and the Wigner distribution

Matching of three-dimensional (3-D) objects is achieved by Wigner analysis of the correlation pattern between the phase-only holographic information of a reference object and that of a target object. First, holographic information on the reference object and on the target object is extracted by use of optical scanning holography as a form of electrical signal. This electrical information is then stored in a computer for digital processing. In the digital computer, the correlation between the phase-only information of the hologram of the reference object and that of the target object is calculated and analyzed by use of a Wigner distribution. The Wigner distribution yields a space-frequency map of the correlation pattern that indicates whether the 3-D image of the target object matches that of the reference object. When the 3-D image of the target object matches that of the reference object, the Wigner distribution gives a well-defined line that directly indicates the 3-D location of the matched target object. Optical experiments with digital processing are described to demonstrate the proposed matching technique.