Nonparaxial interference and diffraction under 3D spatial coherence.

The nonparaxial interference and diffraction by a planar array of emitters have been recently described in terms of the light energy confinement in Lorentzian wells, which are spatially structured by the geometric potential, activated in turn by the two-point correlation prepared at the array plane. Nevertheless, the use of nonplanar arrays of light emitters is of increasing interest in optical technology. Therefore, we extend the confinement model to include spatially structured Lorentzian wells by geometric potentials associated with nonplanar distributions of points. Such geometric potentials are activated by two-point correlations with 3D supports prepared at the nonplanar array. The theoretical analysis is supported and illustrated by numerical simulations.

Matrix algorithm for 3D nonparaxial optical field modeling under arbitrary spatial coherence.

Nonparaxial modeling of optical field propagation at distances comparable to the wavelength and under arbitrary spatial coherence is crucial for micro- and nano-optics. Fourier and Fresnel transform-based algorithms are unable to simulate it accurately because of their paraxial approach. A nonparaxial matrix algorithm, supported by the theoretical model that characterizes the optical field and the setup configuration in terms of sets of real and virtual point emitters, is capable of simulating the 3D optical field distribution in the volume delimited by the input and the output planes placed at a very short distance from each other by using experimental data as entries. The algorithm outcomes are accurate predictions of the power spectrum of interference and diffraction experiments. Simulations of specific experimental situations, including speckle phenomena, illustrate the algorithm's capabilities.

Three-dimensional nonparaxial characterization of physical point sources.

Optical point sources are considered theoretical idealizations. However, it is pertinent to characterize feasible physical point sources for the current micro- and nano-optics. It is shown that certain three-dimensional (3D) distributions of real and virtual point emitters provide a nonparaxial energy cone with the same geometry of the cone provided by the single ideal real point emitter placed at the midpoint of the 3D distribution. These cones fit along arbitrary distances from the 3D distribution up to very short ones (a fraction of the wavelength). A generalized criterion is proposed to characterize the physical point source as a feasible 3D object. Some implications are discussed.

Interaction description of light propagation.

The interaction between real and virtual point emitters is presented as the propagation principle of light under arbitrary spatial coherence. It is supported on a geometrical interpretation of the two-point correlation and the introduction of the geometrical potential. This principle and the spectrum of classes of point emitters constitute a complete theoretical model that offers a unified phenomenological framework for interference and diffraction not only of light but also of single particles, which is not possible in the conventional formalism based on the wave superposition principle.

Discreteness of the real point emitters as a physical condition for diffraction.

The discreteness of the zeroth-order class of the spectrum of classes of point emitters is shown as a necessary and sufficient physical condition for describing diffraction from the aperture plane to the far field. Although it does not contradict the continuity requirement of the cross-spectral density, it points out that this mathematical requirement is physically redundant. This feature gives new insight on the understanding of diffraction as well as novel tools for design and development of efficient non-paraxial propagation algorithms. It also leads to new criteria for characterizing physical point sources of light.

Spectrum of classes of point emitters of electromagnetic wave fields.

The spectrum of classes of point emitters has been introduced as a numerical tool suitable for the design, analysis, and synthesis of non-paraxial optical fields in arbitrary states of spatial coherence. In this paper, the polarization state of planar electromagnetic wave fields is included in the spectrum of classes, thus increasing its modeling capabilities. In this context, optical processing is realized as a filtering on the spectrum of classes of point emitters, performed by the complex degree of spatial coherence and the two-point correlation of polarization, which could be implemented dynamically by using programmable optical devices.

Spectrum of classes of point emitters: new tool for nonparaxial optical field modeling.

Numerical modeling of optical fields provides valuable support to both theoretical research and technological development in many optics fields. Fourier methods have been the most widely used tools of numerical modeling. However, important limitations have restricted their application in contemporary research that involve high numerical apertures, short propagation distances, and spatially partially coherent states of light, for instance. The spectrum of classes of point emitters is introduced as a numerical tool that overcomes such limitations for the design, analysis, and synthesis of nonparaxial optical fields in arbitrary states of spatial coherence. In this context, optical processing is realized as the filtering on the spectrum of classes of point emitters performed by the complex degree of spatial coherence that could be implemented dynamically by using programmable devices.

Three-dimensional micro-diffraction modeling.

Squared elementary cells with correlated radiant point sources are presented as basic structures for characterizing the propagation of the field emitted by two-dimensional planar sources of any shape and in arbitrary state of spatial coherence. The field is transported on a finite expansion of nonparaxial modes, whose propagation in the micro-diffraction domain is discussed under both the diffraction and the interference conditions.

Discreteness of the set of radiant point sources: a physical feature of the second-order wave-fronts.

The non-paraxial phase-space representation of diffraction of optical fields in any state of spatial coherence has been successfully modeled by assuming a discrete set of radiant point sources at the aperture plane instead of a continuous wave-front. More than a mere calculation strategy, this discreteness seems to be a physical feature of the field, independent from the sampling procedure of the modeling.

Phase-space non-paraxial propagation modes of optical fields in any state of spatial coherence.

The non-paraxial marginal power spectrum is decomposed in propagation modes, so that the zeroth-order mode is only emitted by the radiant point sources at the aperture plane, while the modes of higher orders than zero are only emitted by the virtual point sources. It allows representing the non-paraxial propagation of optical fields in arbitrary states of spatial coherence and along arbitrary distances from the aperture plane without approximations, by simply using the power distribution and the spatial coherence state at the aperture plane as entries. This modal expansion is potentially useful in micro-diffraction and spatial coherence modulation.

Analogies between classical scalar wave fields in any state of spatial coherence and some quantum states of light.

The border between the descriptions of the classical optical fields in any state of spatial coherence and the quantum coherence state of light is revisited in the framework of the phase-space representation. Although it is established that such descriptions are not completely equivalent, the exact calculation of the marginal power spectrum leads to new analogies that suggest that some features exclusively attributed to quantum states of light can be also shared by classical optical fields due to their spatial coherence state.

Non-approximated numerical modeling of propagation of light in any state of spatial coherence.

Due to analytical and numerical difficulties, the propagation of optical fields in any state of spatial coherence is traditionally computed under severe approximations. The paraxial approach in the Fresnel-Fraunhofer domain is one of the most widely used. These approximations provide a rough knowledge of the actual light behavior as it propagates, which is not enough for supporting applications, such as light propagation under a high numerical aperture (NA). In this paper, a non-approximated model for the propagation of optical fields in any state of spatial coherence is presented. The method is applicable in very practical cases, as high-NA propagations, because of its simplicity of implementation. This approach allows for studying unaware behaviors of light as it propagates. The light behavior close to the diffracting transmittances can also be analyzed with the aid of the proposed tool.

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, Young's experiment with diffraction effects, and some implications of this model are discussed.

Definition and invariance properties of the complex degree of spatial coherence.

Within the framework of the phase-space representation of random electromagnetic fields provided by electromagnetic spatial coherence wavelets, and by using the Fresnel-Arago laws for interference and polarization as an analysis tool, the meaning of the spatial coherence-polarization tensor and its invariance under transformations is studied. The results give new insight into the definition and properties of the complex degree of spatial coherence by showing that its invariance is not required for properly describing the behavior of random electromagnetic fields within the scope of physically measurable quantities.

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.

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.