Entropy, purity and optical hysteresis in markovian optical modes.

We describe the synthesis of optical modes whose axial structure follows a random tandem array of Bessel beams of integer order. The array follows fluctuations of Markov-chain type and the amplitude values for each beam are linked to a sequence of random vectors. As a prototype, we describe the synthesis of optical fields for Markov-chain type Ehrenfest. This process models the thermodynamic equilibrium and then it can be related to the evolution and stability of optical systems, in this way, it offers a similitude with partially coherent processes where the coherence degree is now distributed between all the compounds of the resulting random vector. The matrix representation for the stochastic process allows incorporating entropy properties and the calculus of the purity for the optical field. This constitutes the basis to describe the interference between markovian modes. When the set of markovian modes type Ehrenfest reaches a stable configuration they become indistinguishability non-conservative optical field having associated hysteresis features. Computer simulations are presented.

Stability of focusing regions and its vortex-solitonic properties.

Focusing regions, also known as caustic regions, are the singular solutions to the amplitude function of optical fields. Focusing regions are generated by the envelope curve of a set of critical points, which can be of attractor or repulsor type. The nature of the critical point depends on the refractive index. An important property of the critical points is that they present charge-like features. When a focusing region is generated in media with a random refractive index, current-like effects appear, and the evolution of the focusing regions follows a diffusion behavior. The morphology of the focusing regions may generate vortices or "eternal solutions" of solitonic type in a nonlinear medium. Herein, the condition under which these effects occur is analyzed and experimentally corroborated.

Invariant correlated optical fields driven by multiplicative noise.

We describe the evolution of a linear transmittance when it is perturbed with multiplicative noise; the evolution is approximated through an ensemble of random transmittances that are used to generate diffraction fields. The randomness induces a competition mechanism between noise and transmittance, and it is identified through the self-correlation function. We show that the geometry of the self-correlation function is a single peak preserved in the diffraction field that can be matched with localization-like effects. To corroborate the theoretical predictions, we perform an experiment using a linear grating where the noise is approximated by a stochastic Markov chain. Experimental results are shown.

Generation of long-range curved-surface plasmonic modes and their propagation through thin metal films in a tandem array.

We describe the generation of plasmonic modes that propagate in a curved trajectory. This is performed by masking a metal surface with two screens containing a randomly distributed set of holes that follow a Gaussian statistic. The diameter of the holes is less than the wavelength of the illuminating plane wave. By implementing scaling and rotations on each screen, we control the correlation trajectory and generate long-range curved plasmonic modes. The study is generalized for the transmission of a plasmonic mode propagating in a tandem array of thin metal films using the evanescent character of the electric field.

Topological properties of the interaction between focusing regions kind cusped.

We study here the cusped-cusped interaction between two kinds of Pearcey optical fields by analyzing its topological structure. We do it in two steps; the first one is an irradiance interaction that allows us to identify organization regions. The second one is an amplitude interaction, where it is shown that the interference fringes are organized around the irradiance distribution. The topological behavior of the optical field is analyzed identifying regions with different phase functions, one of them, corresponds with a catastrophe function which has associated a focusing region, the other region can be approximated by a quadratic function. The main consequence heritage from the phase structure is interference fringes emerge from focusing regions having similar features like a topological charges. We show computational and experimental results which are in very well agreement with the theoretical model.

Diffraction by three-dimensional slit-shape curves: decomposition in terms of Airy and Pearcey functions.

We analyze the diffraction field generated by coherent illumination of a three-dimensional transmittance characterized by a slit-shape curve. Generic features are obtained using the Frenet-Serret equations, which allow a decomposition of the optical field. The analysis is performed by describing the influence of the curvature and torsion on osculating, normal, and rectifying planes. We show that the diffracted field has a decomposition in three optical fields propagating along three optical axes that are mutually perpendicular. The decomposition is in terms of the Pearcey and Airy functions, and the generalized Airy function. Experimental results are shown.

Markovian optical modes.

We describe the transition of a set of optical modes following a Markov chain process, where the mean value of the amplitude converge to a new type of partially coherent mode, with the property that the coherence features are easily tunable with the parameters of the chain. The amplitude of the resulting mode depends on the probability transition of the chain. As a prototype, we establish an analogy with gambler's chain ruin, using as a basis for the vector space the Bessel modes of integer order. Computer simulations are shown.

Simple correction factor for laser speckle imaging of flow dynamics.

One of the major constraints facing laser speckle imaging for blood-flow measurement is reliable measurement of the correlation time (τ(C)) of the back-scattered light and, hence, the blood's speed in blood vessels. In this Letter, we present a new model expression for integrated speckle contrast, which accounts not only for temporal integration but spatial integration, too, due to the finite size of the pixel of the CCD camera; as a result, we find that a correction factor should be introduced to the measured speckle contrast to properly determine τ(C); otherwise, the measured blood's speed is overestimated. Experimental results support our theoretical model.

Description of phase singularities and their application to focusing design.

We describe the focusing region associated with transmittances, analyzing its associated phase function. We show that generic features can be studied from the differential equation for focusing geometry, which is obtained through angular representation for diffraction fields. With the treatment, we recover the results for circular zone plates, and by introducing a linear transformation into the transmittance function we generate structures that keep the ability to generate focusing. According to the choice of the parameters involved, the diffraction field presents new focusing regions, whose three-dimensional geometry and spatial evolution can be described in a selective fashion with analysis of only the phase singularities associated with the diffraction field and avoidance of the integral representation. The treatment is also applied to a simple lens. We recover the theoretical predictions obtained by Berry and Upstill [M. V. Berry and C. Upstill, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259], and these predictions are corroborated experimentally. The results obtained are shown.

Interference between caustics of diffraction fields.

The caustics of diffraction fields are structures that present a nonlinear phase distribution. The superposition of two caustics presents interesting irradiance distributions, which we analyze within the framework of catastrophe theory. This treatment permits the inclusion of the optical path difference involved in a parametric family, and by employing the geometric theory of diffraction we can analyze the interference patterns in a selective fashion. The theoretical predictions are verified experimentally with a Michelson interferometer.

Design of projective zone plates and their focusing properties.

Transmittances are designed by generation of a family of curves whose local geometry is similar to that of zone plates. The family is obtained by means of projective mapping in the normal direction of an initial curve, so that all the curves of the family have a common evolute. This property allows the prediction of the geometry of the focusing region of the transmittance. Experimental results with an ellipse as the initial curve are shown. We show that the proposed zone plates can be considered a generalization of conventional zone plates.