Román Castañeda

Spatial coherence wavelets: Mathematical properties and physical features

Abstract This paper discusses a set of mathematical and physical properties regarding spatial coherence wavelets, a concept recently developed by Castaiieda and Garcia-Sucerquia. Some of these properties are inherited from the Wigner distribution function. The uncertainty principle that they obey, the superposition of uncorrelated spatial coherence wavelets and the effective diffraction aperture, are discussed.

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.

Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment

Theoretical considerations, simulations and experiments have been carried out to distinguish between Fresnel and the Fraunhofer diffraction regarding the formation of interference patterns by a conventional Youngs double-pinhole arrangement with variable separation, which is illuminated by a coherent plane wave. We show that the optical path difference introduced by this setup fits the Fresnels phase difference between the pinholes. Consequently, it is possible to determine the number of Fresnel zones subtended by a circle centered on one of the pinholes and with radius equals to the pinhole separation. Then, we propose a criterion for distinguishing between Fresnel and Fraunhofer diffraction based on this number of Fresnel zones, which is applicable for diffracting apertures with any shape.

Experimental evidence of the spatial coherence moiré and the filtering of classes of radiator pairs

Evidence of the physical existence of the spatial coherence moiré is obtained by confronting numerical results with experimental results of spatially partial interference. Although it was performed for two particular cases, the results reveal a general behavior of the optical fields in any state of spatial coherence. Moreover, the study of the spatial coherence moiré deals with a new type of filtering, named filtering of classes of radiator pairs, which allows changing the power spectrum at the observation plane by modulating the complex degree of spatial coherence, without altering the power distribution at the aperture plane or introducing conventional spatial filters. This new procedure can optimize some technological applications of actual interest, as the beam shaping for instance.

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.

Radiant, virtual, and dual sources of optical fields in any state of spatial coherence.

A novel description of interference and diffraction with fields in arbitrary states of spatial coherence is introduced in the framework of the phase-space representation. The field is modeled as produced by radiant and virtual point sources. The first ones emit the radiant power of the field, independently of its spatial coherence state, and the second ones emit the modulating energy in strong dependence on such state. This energy can take on positive and negative values that produce the interference and diffraction patterns after adding them to the radiant energy. Radiant and virtual point sources at a given plane can be arranged over two distinct layers, which can be brought together to provide a unified structure of point sources for the field at such plane. So, the coincidence of specific radiant and virtual sources at the same point induces a further type: the dual point source. Descriptions of diffraction arrangements, Youngs experiment with diffraction effects, and some implications of this model are discussed.

Spatial coherence modulation

The principles of amplitude and phase modulation of the spatial coherence of the optical field are discussed. They are based on the modification of the phase-space diagram of the field, provided by the marginal power spectrum, which allows synthesis of the modulating functions of the spatial coherence and the corresponding complex transmissions to be transferred onto a spatial light modulator for application purposes. Numerical and experimental results are presented. This novel technique can be applied in designing specific shapes of power distributions.

Determination of the spatial coherence of Schell-model beams with diffraction gratings

A technique for determining the complex degree of spatial coherence of Schell-model beams is proposed which uses uniformly illuminated binary amplitude gratings. The theoretical model is illustrated by simulation results.

The structured spatial coherence support

A novel concept for the description of the spatial coherence state of the optical field is introduced in the framework of the phase-space representation of the field, termed the structured spatial coherence support. It exhibits clear differences with the conventional coherence patch and spatial coherence region, and advantages (such as its individual accessibility and its representation by means of virtual point sources, for instance) that provide a robust basis for developing optimal algorithms for calculations and simulations. It also implies physical features that improve insight on the spatial coherence state of light.

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.