Generation of vector beams using synthetic phase holograms.

We discuss a class of synthetic phase holograms (SPHs) applied to the generation of vector fields. Each SPH encodes the transverse components of the vector field, modulated by different linear phase carriers. Such components, which are spatially separated by the carriers, are modulated by appropriate orthogonal polarizations. A final stage that makes the components collinear allows the generation of the vector field. We assess the efficiency and accuracy of the different SPHs, in the task of generating vector fields. The proposal is illustrated by the implementation of vector Bessel beams, which are experimentally generated in a setup based on a phase spatial light modulator.

Generation of vector Bessel beams with diffractive phase elements based on the Jacobi-Anger expansion.

We report the design and optimization of a diffractive phase element whose phase modulation is derived from the Jacobi-Anger relation, which allows the simultaneous generation of multiple scalar Bessel beams of different integer orders. In addition, by the appropriate treatment of a couple of such Bessel beams, we generate a vector Bessel beam of arbitrary order m. This beam is constructed by the collinear superposition of the scalar Bessel beam modes of orders m-1 and m+1, with circular orthogonal polarizations. We demonstrate experimentally both the simultaneous generation of the multiple scalar Bessel beams and the generation of vector Bessel beams of orders m=0 and m=1. These tasks are performed in an optical setup based on a pixelated liquid-crystal spatial light modulator.

Generalized revival and splitting of an arbitrary optical field in GRIN media.

Assuming a non-paraxial propagation operator, we study the propagation of an electromagnetic field with an arbitrary initial condition in a quadratic GRIN medium. We show analytically that at certain specific periodic distances, the propagated field is given by the fractional Fourier transform of a superposition of the initial field and of a reflected version of it. We also prove that for particular wavelengths, there is a revival and a splitting of the initial field. We apply this results, first to an initial field given by a Bessel function and show that it splits into two generalized Bessel functions, and second, to an Airy function. In both cases our results are compared with the numerical ones.

Revival and splitting of a Gaussian beam in gradient index media.

The short range revival of an arbitrary monochromatic optical field, which propagates in a quadratic GRIN rod, is a well-known effect that is established assuming the first-order approximation of the propagation operator. We discuss the revival and multiple splitting of an off-axis Gaussian beam propagating to relatively long distances in a quadratic GRIN medium. These effects are obtained assuming the second-order approximation of the propagation operator in this medium.

Optical realization of quantum Kerr medium dynamics.

We use the propagation of a conveniently shaped Gaussian beam in a GRIN media to mimic a quantum cavity filled with a Kerr medium. This is attained by introducing a second-order correction to the paraxial propagation of the beam. An additional result is that a Gaussian beam propagating in GRIN media may split into two Gaussian beams, corresponding to the generation of superposition of coherent states (Schrödinger cat states) in the cavity filled with the Kerr medium.

Holographic generation of a class of nondiffracting fields with optimum efficiency.

We discuss the accurate generation of complex optical fields using phase holograms that provide the optimum diffraction efficiency. In each considered case, the phase modulation of the employed hologram is identical to the phase of the desired optical field. We show that periodic and quasiperiodic nondiffracting optical fields, mathematically obtained through the superposition of multiple plane waves, can be generated with high fidelity using this approach.

Optical realization of quantum-mechanical invariants.

We study the problem of paraxial propagation in two grade-index media by using invariant techniques that allow a continuous solution of the problem. By using the well-known fact that this problem is analogous to the time-dependent harmonic oscillator in quantum mechanics, known methods there may be imported producing on the one hand a solution to the propagation problem and on the other hand a realization of a quantum-mechanical invariant.

Zero order synthetic hologram with a sinusoidal phase carrier for generation of multiple beams.

We discuss a phase synthetic hologram, for encoding arbitrary complex fields, whose design is based on a sinusoidal phase grating with a spatially modulated phase depth. An important feature of the hologram is that it encodes the complex field at the zero diffraction order of the carrier grating. The smoothness of this sinusoidal carrier grating facilitates the implementation of the hologram with a pixelated spatial light modulator. We take advantage of the hologram reconstruction at the zero-diffraction order for the simultaneous generation of a collection of complex beams.

Fractional Talbot field and of finite gratings: compact analytical formulation.

We present a compact analytical formulation for the fractional Talbot effect at the paraxial domain of a finite grating. Our results show that laterally shifted distorted images of the grating basic cell form the Fresnel field at a fractional Talbot plane of the grating. Our formulas give the positions of those images and show that they are given by the convolution of the nondistorted cells (modulated by a quadratic phase factor) with the Fourier transform of the finite-grating pupil.

Iterative optimization of phase-only diffractive optical elements based on a lenslet array.

We describe the design of Fourier-type phase-only array generators. The numerical optimization employs the Fienup algorithm, where the parageometric design of the phase retardation profile, with the form of a lenslet array, is used as the initial guess of the optimization process. This approach provides designs with high performance that can be obtained with comparatively low computing effort. This is particularly true for elements generating large spot arrays. For symmetric reconstruction fields, the optimized phase profile typically has the same symmetry as that for the reconstruction field and can be easily unwrapped.

Compact planar-integrated optical correlator for spatially incoherent signals.

A new, to our knowledge, approach for the planar integration of optical correlators is demonstrated. A VanderLugt-type architecture was used to allow the processing of the spatially incoherent signals of active optoelectronic smart-pixel-device arrays. In a folded optical system all passive components were implemented as a single multiple-phase-level element. The relations among the spatial resolution, the light efficiency, and the system design parameters are derived. High signal quality and low noise levels were achieved experimentally.

Reinterpretation and improvement of Talbot array illuminators.

We show that the transmittance of a finite Talbot array illuminator (TAI) can be expressed by the phase distribution of a pixelated lens, modulated by a discrete phase grating (G). Thus the TAI reconstruction field is given by the convolution of the grating's Fourier transform, with the point-spread function of the pixelated lens. On the basis of this approach we propose a method to improve the performance of a finite TAI by modifying the basic cell of the grating factor G.

Self-apodization of low-resolution pixelated lenses.

We show that a pixelated lens with appropriate parameters exhibits an apodized point-spread function that originates in the finite size of the pixel's pupil. We evaluate numerically the degree of apodization and the enlargement associated with the point-spread function in terms of the parameters that characterize the pixelated lens.

Efficient detour-phase encoding of one-dimensional multilevel phase diffractive elements.

We show that detour-phase encoding with a multilevel blaze structure as a carrier grating is especially suited to the implementation of diffractive elements with relatively high complexity in one axis. For our proposal the carrier grating is aligned perpendicularly to this axis. In this way the element can be encoded with a high space-bandwidth product and a high phase resolution by use of a moderate carrier frequency. Moreover, this frequency can be adjusted to isolate the reconstructed field from the noise resulting from high diffraction orders of the carrier grating or caused by etching errors during fabrication.

High-efficiency detour-phase holograms.

We suggest a new approach to the fabrication of detour-phase holograms. By combining the concept of the detour phase with highly efficient carrier gratings it is possible to overcome the major drawback of detour-phase holograms, i.e., their low efficiency. We call the new hologram type a kinoform detour-phase hologram (KDPH). KDPH's facilitate the implementation of diffractive elements with high phase complexity. Especially if off-axis reconstruction is desired, the number of degrees of freedom for the design of the phase component is increased significantly.

Efficiency limit of spatially quantized Fourier array illuminators.

We derive an upper limit for the efficiency of spatially quantized Fourier array illuminators. This complements a previously reported formula for the efficiency limit of diffractive elements constrained only in phase.

Talbot array illuminators with optimum compression ratio.

We describe Talbot array illuminators that generate, at fractions of the Talbot distance, arrays of bright spots with optimum compression ratio. We compare the performance of these Talbot array illuminators with that of an array of parabolic lenses.

Self-imaging of discrete gratings at fractions of the Talbot distance: an eigenvalue problem.

Using a matrix formulation of Fresnel diffraction, we describe discrete gratings that exhibit self-images at fractions of the Talbot distance. Such structures are obtained as solutions of an eigenvalue equation.

Fresnel diffraction of substructured gratings: matrix description.

We present a matrix formalism to describe the near-field diffraction pattern, at fractions of a Talbot distance, of a grating whose unit cell is composed of a discrete substructure. We show that this formalism is useful for designing Lohmann array illuminators.

Multilevel phase gratings for array illuminators.

We describe a variety of multilevel phase structures that can be used to generate Lohmann's array illuminators. We report several experimental verifications of the synthesis of such multilevel phase structures by using simple binary curves in a conventional optical processor.