Wigner distribution and fractional Fourier transform for two-dimensional symmetric optical beams

A useful relationship between the fractional Fourier transform power spectra of a two-dimensional symmetric optical beam, on the one hand, and its Wigner distribution, on the other, is established. This relationship allows a significant simplification of the standard procedure for the reconstruction of the Wigner distribution from the field intensity distributions in the fractional Fourier domains. The Wigner distribution of a symmetric optical beam is analyzed, both in the coherent and in the partially coherent case.     

Modal decomposition of partially coherent beams using the ambiguity function

Phase-space representations of optical beams such as the ambiguity function or the Wigner distribution function have recently gained considerable importance for the characterization of coherent and partially coherent beams. There is growing interest in beam properties such as the beam propagation factor and the coherence and phase information that can be extracted from these phase-space representations. A method is proposed to decompose a partially coherent beam into Hermite-Gaussian modes by using the ambiguity function. The modal weights and the possible phase relations of the Hermite-Gaussian modes are retrieved. The method can also be applied for the decomposition of the Wigner distribution function. Some examples are discussed in the scope of beam characterization.     

Double Wigner distribution function of a first-order optical system with a hard-edge aperture

The effect of an apertured optical system on Wigner distribution can be expressed as a superposition integral of the input Wigner distribution function and the double Wigner distribution function of the apertured optical system. By introducing a hard aperture function into a finite sum of complex Gaussian functions, the double Wigner distribution functions of a first-order optical system with a hard aperture outside and inside it are derived. As an example of application, the analytical expressions of the Wigner distribution for a Gaussian beam passing through a spatial filtering optical system with an internal hard aperture are obtained. The analytical results are also compared with the numerical integral results, and they show that the analytical results are proper and ascendant.     

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 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.     

Propagation properties of a partially coherent radially polarized beam in atmospheric turbulence

Based on the extended Huygens–Fresnel integral, second-order moments of the Wigner distribution function of a partially coherent radially polarized beam propagating through atmospheric turbulence are derived. Besides, propagation properties such as the mean-squared beam width, angular width, effective radius of curvature, beam propagation factor and Rayleigh range can also be obtained and calculated numerically. It is shown that the propagation properties are dependent on the spatial correlation length, refraction index structure constant and propagation distance.     

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.     

Propagation properties of rectangular Laguerre–Gaussian-correlated Schell-model beams in atmospheric turbulence

Based on the extended Huygens–Fresnel principle and the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the cross-spectral density (CSD) and the propagation factor of a rectangular Laguerre–Gaussian-correlated Schell-model (LGCSM) beam propagating in atmospheric turbulence are derived. The statistical properties, such as the average intensity, the spectral degree of coherence (SDOC) and the propagation factor, of a rectangular LGCSM beam in free space and atmospheric turbulence are comparatively analysed. It is illustrated that a rectangular LGCSM beam exhibits self-splitting and combing properties on propagation in atmospheric turbulence, and the self-splitting properties of such beam are closely related to its beam orders m and n, which is quite different from other self-splitting beams. In addition, the rectangular LGCSM beam has an advantage for reducing the turbulence-induced degradation compared with the conventional partially coherent beams.