Optical phase space, Wigner representation, and invariant quality parameters

Wigner distribution and entropy of partially coherent light generated by perfect optical vortices

We developed analytical expressions for the Wigner distribution function of partially coherent fields generated by the scattering of beams with a particular phase structure, namely perfect optical vortex beams. In addition, we provide the modal decomposition of the field correlations and evaluate the evolution of Shannon entropy associated with the partially coherent field.

Estimation of dislocated phases in wavefronts through intensity measurements using a Gerchberg-Saxton type algorithm

Estimation of the phase of a singular paraxial light field from experimentally measured intensities using a Gerchberg-Saxton type algorithm is demonstrated. A combination of cylindrical lenses which does not conserve the orbital angular momentum of the light field is used in obtaining the measured intensities. Consistent extraction of the phases in regard of the orbital angular momentum is demonstrated both at the input and output transverse planes, using the measured intensities.

Radial-angular entanglement in Laguerre-Gaussian mode superpositions

Classical optic entanglement between the radial and angular degrees of freedom in Laguerre-Gaussian mode superpositions is explored within the framework of symmetric first-order optical systems. The Gouy phase picked by a Laguerre-Gaussian mode on free propagation is seen to be of consequence to the radial-angular entanglement in the mode superpositions. We illustrate examples of mode superpositions for which radial-angular entanglement is preserved on passage through symmetric first-order optical systems. An indicator of radial-angular entanglement in two-mode Laguerre-Gaussian superpositions is demonstrated to be a robust free space signaler in the presence of atmospheric turbulence, through examples.

Free space optical communication using beam parameters with translational and transverse rotational invariance

Two natural requirements on a measurable quantity possessed by a paraxially propagating light-field to be suitable for free space optical communication are invariance under free space propagation and invariance under transverse plane rotation. While the former invariance ensures that the measurable quantity is robust while signalling through free space, the latter invariance ensures that a detector measuring the quantity can be oriented at any angle in the transverse plane, and a measurement by the detector yields the same value for the quantity irrespective of the transverse angle, thus avoiding alignment issues. The variance matrix of a paraxially propagating light-field is analyzed from the perspective of the aforementioned invariances. That the "charge" of a paraxial light-field, which is contained in the variance matrix, and which has been previously well studied for its suitability toward free space optical communication, possesses these two invariance properties, emerges naturally in the analysis. Seven functionally independent quantities other than charge, which are derived from the variance matrix, and which share these invariances, are presented and studied for their suitability toward signalling through turbulent atmosphere using the low-order Hermite-Gaussian modes. It is found that the spot size of a Gaussian light-field can be effectively used as a switch, to communicate through short distances in a turbulent atmosphere.

Wave propagation analysis using the variance matrix

The propagation of a coherent laser wave-field through a pseudo-random phase plate is studied using the variance matrix estimated from Shack-Hartmann wavefront sensor data. The uncertainty principle is used as a tool in discriminating the data obtained from the Shack-Hartmann wavefront sensor. Quantities of physical interest such as the twist parameter, and the symplectic eigenvalues, are estimated from the wavefront sensor measurements. A distance measure between two variance matrices is introduced and used to estimate the spatial asymmetry of a wave-field in the experiment. The estimated quantities are then used to compare a distorted wave-field with its undistorted counterpart.

Classical light beams and geometric phases

We present a study of geometric phases in classical wave and polarization optics using the basic mathematical framework of quantum mechanics. Important physical situations taken from scalar wave optics, pure polarization optics, and the behavior of polarization in the eikonal or ray limit of Maxwell's equations in a transparent medium are considered. The case of a beam of light whose propagation direction and polarization state are both subject to change is dealt with, attention being paid to the validity of Maxwell's equations at all stages. Global topological aspects of the space of all propagation directions are discussed using elementary group theoretical ideas, and the effects on geometric phases are elucidated.

Transverse energy circulation and the edge diffraction of an optical vortex beam

Edge diffraction of a circular Laguerre-Gaussian beam represents an example of the optical vortex symmetry breakdown in which the hidden "vortex" energy circulation is partially transformed into the visible "asymmetry" form. The diffracted beam evolution is studied in terms of the irradiance moments and the moment-based parameters. In spite of the limited applicability of the moment-based formalism, we show that the "vortex" and "asymmetry" parts of the orbital angular momentum can still be reasonably defined for the hard-edge diffracted beams and retain their physical role of quantifying the corresponding forms of the transverse energy circulation.

Degree of paraxiality of a partially coherent field

We extend the concept of the degree of paraxiality, introduced recently for monochromatic fields, to the domain of stochastic fields. As an example we analytically evaluate the degree of paraxiality for a broad class of model stochastic fields, the Gaussian Schell-model fields, without and with truncation and twist phase. The dependence of the degree of paraxiality on the size and the state of coherence of the source as well as on the truncation parameter and the magnitude of twist phase is analyzed by a number of numerical examples.

Phase-space rotations and orbital Stokes parameters

We introduce the orbital Stokes parameters as a linear combination of a beam's second-order moments. Similar to the ones describing the field polarization and associated with beam energy and its spin angular momentum, the orbital Stokes parameters are related to the total beam width and its orbital angular momentum. We derive the transformation laws for these parameters during beam propagation through first-order optical systems associated with phase-space rotations. The values of the orbital Stokes parameters for Gaussian modes and arbitrary fields expressed as their linear superposition are obtained.

Interference in phase space

The phase-space representation of interference based on the marginal power spectrum gives new insight on interference, enlarging its potential applications by means of the principle of spatial coherence modulation. Carrier and (0,pi)-rays produced by three different types of supports are introduced for describing interference as the result of adding the radiant energy propagated by the carriers and the modulating energy (which can be positive or negative) propagated by the (0,pi)-rays. Numerical examples are presented.

Phase-space representation and polarization domains of random electromagnetic fields

The phase-space representation of stationary random electromagnetic fields is developed by using electromagnetic spatial coherence wavelets. The propagation of the field's power and states of spatial coherence and polarization results from correlations between the components of the field vectors at pairs of points in space. Polarization domains are theoretically predicted as the structure of the field polarization at the observation plane. In addition, the phase-space representation provides a generalization of the Poynting theorem. Theoretical predictions are examined by numerically simulating the Young experiment with electromagnetic waves. The experimental implementation of these results is a current subject of research.

Phase space representation of spatially partially coherent imaging

The phase space representation of imaging with optical fields in any state of spatial coherence is developed by using spatial coherence wavelets. It leads to new functions for describing the optical transfer and response of imaging systems when the field is represented by Wigner distribution functions. Specific imaging cases are analyzed in this context, and special attention is devoted to the imaging of two point sources.

Spatial coherence wavelets and phase-space representation of diffraction

The phase-space representation of the Fresnel-Fraunhofer diffraction of optical fields in any state of spatial coherence is based on the marginal power spectrum carried by the spatial coherence wavelets. Its structure is analyzed in terms of the classes of source pairs and the spot of the field, which is treated as the hologram of the map of classes. Negative values of the marginal power spectrum are interpreted as negative energies. The influence of the aperture edge on diffraction is stated in terms of the distortion of the supports of the complex degree of spatial coherence near it. Experimental results are presented.

Holographic features of spatial coherence wavelets

The behavior of the marginal power spectrum as a two-channel-multiplexed hologram is analyzed. Its "negative energies" make it quite different from the conventional holograms, i.e., it is not recordable in general and the objects to be reconstructed (the cross-spectral densities at both the aperture and the observation planes) are virtual. The holographic reconstruction results from the superposition of the spatial coherence wavelets that carry the marginal power spectrum. These features make the marginal power spectrum a powerful tool for analysis and synthesis of optical fields, for instance, in optical information processing (signal encryption) and beam shaping for microlithography.

Overall degree of coherence for vectorial electromagnetic fields and the Wigner function

We elaborate the overall degree of coherence for vectorial electromagnetic waves within paraxial approximation, expressing it in terms of a polarization Wigner function and the spatial-angular Stokes parameters.

Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, we derived an approximate analytical formula for a partially coherent twisted anisotropic Gaussian Schell-model (AGSM) beam propagating through an apertured paraxial general astigmatic (GA) optical system by use of a tensor method. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. Our formulas avoid time-consuming numerical integration and provide a convenient and effective way for studying the propagation and transformation of a partially coherent twisted AGSM beam through an apertured paraxial GA optical system.

Electromagnetic spatial coherence wavelets

The recently introduced concept of spatial coherence wavelets is generalized to describe the propagation of electromagnetic fields in the free space. For this aim, the spatial coherence wavelet tensor is introduced as an elementary amount, in terms of which the formerly known quantities for this domain can be expressed. It allows for the analysis of the relationship between the spatial coherence properties and the polarization state of the electromagnetic wave. This approach is completely consistent with the recently introduced unified theory of coherence and polarization for random electromagnetic beams, but it provides further insight about the causal relationship between the polarization states at different planes along the propagation path.

Wigner representation and geometric transformations of optical orbital angular momentum spatial modes

An exact Wigner representation of optical spatial modes carrying orbital angular momentum is found in closed form by exploiting the underlying SU(2) Lie-group algebra of their associated Poincaré sphere. Orthogonality relations and observables of these states are obtained within the phase space picture. Development of geometric phases on mode transformations is also elucidated.

Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian-Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FRT plane are studied in detail. The results show that the fractional order of the FRT has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FRT of twisted anisotropic GSM beams.

Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media

By adopting a new tensor method, we derived an analytical propagation formula for the cross-spectral density of partially coherent twisted anisotropic Gaussian Schell-model (GSM) beams through dispersive and absorbing media. Using the derived formula, we studied the evolution properties and spectrum properties of twisted anisotropic GSM beams in dispersive and absorbing media. The results show that the dispersive and absorbing media have strong influences on the propagation properties of twisted anisotropic GSM beams and their spectrum evolution. Our method provides a simple and convenient way to study the propagation of twisted anisotropic GSM beams in media with complex refractive index.

Fractional Fourier transform for partially coherent Gaussian-Schell model beams

The fractional Fourier transform (FRT) is applied to partially coherent twisted anisotropic Gaussian-Schell model (GSM) beams based directly on the cross-spectral density. An analytical and concise formula is derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through a FRT system in terms of the tensor method. The corresponding tensor ABCD law for performing a FRT is obtained. The connection between the FRT formula and the generalized Collins formula for partially coherent beams is discussed. The formulas derived provide a powerful tool for analyzing and calculating the FRTs of partially coherent beams.

Spatial coherence and information entropy in optical vortex fields

We show how a vortex structure manifests itself in the one-dimensional projection of a vortex field. We calculate the extent of spatial coherence and entropy of such projections. We quantify the spatial coherence and discuss the properties of the Wigner functions for the projected field.

Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams

A 4 x 4 complex curvature tensor M>(-1) is introduced to describe partially coherent anisotropic Gaussian-Schell model (GSM) beams. An analytical propagation formula for the cross-spectral density of partially coherent anisotropic GSM beams is derived. The propagation law of M(-1) that is also derived may be called partially coherent tensor ABCD law. The analytical formulas presented here are useful in treating the propagation and transformation of partially coherent anisotropic GSM beams, which include previous results for completely coherent Gaussian beams as special cases.