In-phase sub-Nyquist lenslet arrays encoded onto spatial light modulators.

When encoding diffractive lenses onto a spatial light modulator (SLM), there is a Nyquist limit to the smallest focal length that can be formed. When this limit is surpassed, a two-dimensional array of lenslets is formed. There have been very few discussions on the performance of these lenslets. In this work, we focus on the phase distribution of these lenses in the array. We show that, for certain values of the focal length, the lenslets are all in perfect phase. We show that this situation happens for a total number of N/4 different discrete equidistant sub-Nyquist focal lengths, where N×N is the number of pixels in the SLM. We find other distances in between where the array is composed of two sets of lenslets with a relative π phase among them. Finally, we illustrate these phase distributions in the application to generate an array of vortex producing lenses. We expect that these results might be useful for high-accuracy interferometric or multiple imaging where this phase must be exactly the same for each replica.

Encoding lenses with focal lengths lower than the Nyquist limit using high phase-modulation displays.

Liquid crystal displays allow the easy implementation of diffractive optical elements. However, the shortest focal lengths for lenses are limited by Nyquist conditions. In this work, we show that focal lengths much lower than this Nyquist limit can be encoded onto devices having a large phase-dynamic range. Experimental results are included with a display showing 10π phase modulation reducing the Nyquist limit by a factor of about 1/10.

Simulation of multimode optical fibers using the angular spectrum algorithm and a Fourier analysis.

Multimode optical fibers are gaining increased interest for higher data rate transmissions. In this work, we examine the propagation of beams in multimode optical fibers using the angular spectrum method for several values for the propagation constant V. We prepare a sequence of 4096 images over 200,000 steps, each representing a wavelength within the fiber. We perform a one-dimensional Fourier transform of these data and obtain the propagation constant for each transmitted mode. We then obtain the electric field profile for each transmitted mode. We find excellent agreement with weakly guided mode theory. This work represents another step in the use of this algorithm for analyzing more complicated optical fiber structures.

Gouy phase effects on propagation of pure and hybrid vector beams.

The robustness of the polarization spatial distribution of vector beams upon propagation is crucial for a number of applications, including optical communications and materials processing. This study has been commonly centered on Gouy phase effects on focused vector beams. In this work, we present a theoretical and experimental analysis of the Gouy phase's effects on the propagation of pure and hybrid vector beams. Experimental results at various axial planes, before and past the focus, are obtained by using a simplified liquid-crystal spatial light modulator-based optical system that allows the easy generation of these beams. Furthermore, a new alternative optical set-up that is devoid of moving elements is demonstrated, which simplifies this study. We experimentally verify the differences between pure and hybrid vector beams upon propagation. While the first ones remain stable, hybrid vector beams show Gouy phase effects that demonstrate an optical activity where the local polarization states rotate by an angle that depends on the propagation distance. Experimental results agree with the theory.

Simulation of optical fiber couplers using the angular spectrum algorithm.

We examine single-mode optical fiber transmission using the angular spectrum method. We find excellent agreement with the theoretical solutions for the cylindrical single-mode optical fiber. Next, we examine fiber coupler configurations with 1×2, 1×3, and 1×5 designs. The input within the central fiber core is broadcast equally into the surrounding fiber cores and then returns with propagation distances as expected. Then we examine supermode theory for the 1×3 case. The initial energy is sent into different combinations of fiber cores where their phase relationship governs the output. Surprisingly, we find evidence of an unexpected mode in some of the multicore designs. We expect these results might be useful for exploration of more complicated fiber core arrays. However, computational times are short using this algorithm.

Q-plates with a nonlinear azimuthal distribution of the principal axis: application to encoding binary data.

We encode q-plates where the angular orientation of the principal axis is varied spatially with a nonstandard distribution. In the usual q-plate design, the orientation of the optical axis depends linearly on the azimuthal angle. In this work, we examine cases where this azimuthal dependence is nonlinear. We consider two cases: first, where the principal axis distribution is like an inverse-tangent function of the azimuth; and second, where it displays linear and flat segments. This last case is proposed as a new method for encoding binary data into the azimuthal lobes of the vector beam. We encode these patterns using a spatial light modulator system that allows new and exotic q-plate designs without the difficulty of fabricating individual plates. Experimental results are presented.

Vector Beam Polarization State Spectrum Analyzer.

We present a proof of concept for a vector beam polarization state spectrum analyzer based on the combination of a polarization diffraction grating (PDG) and an encoded harmonic q-plate grating (QPG). As a result, a two-dimensional polarization diffraction grating is formed that generates six different q-plate channels with topological charges from -3 to +3 in the horizontal direction, and each is split in the vertical direction into the six polarization channels at the cardinal points of the corresponding higher-order Poincaré sphere. Consequently, 36 different channels are generated in parallel. This special polarization diffractive element is experimentally demonstrated using a single phase-only spatial light modulator in a reflective optical architecture. Finally, we show that this system can be used as a vector beam polarization state spectrum analyzer, where both the topological charge and the state of polarization of an input vector beam can be simultaneously determined in a single experiment. We expect that these results would be useful for applications in optical communications.

Nondiffracting vector beams where the charge and the polarization state vary with propagation distance.

We generate nondiffracting vector beams where the charge and the polarization state vary with the propagation distance. We use reflective geometry where a parallel-aligned spatial light modulator is used to spatially modulate two orthogonal linear polarizations. We encode spiral phases with equal charge but with opposite signs onto the two polarization directions to encode a vector beam and add two axicon phases. Both the charge and the phase shift between the two axicons can be varied along the focus line. We provide experimental results that demonstrate both features.

Generation of integer and fractional vector beams with q-plates encoded onto a spatial light modulator.

We generate programmable vector beams with arbitrary q-plates encoded using a spatial light modulator system. Consequently, we can analyze new and exotic q-plate designs without the difficulty of fabricating individual plates. We show experimental results for positive and negative integer and new fractional vector beam values.

Diffraction gratings generating orders with selective states of polarization.

We propose specially designed double anisotropic polarization diffraction gratings capable of producing a selective number of diffraction orders and with selective different states of polarization. Different polarization diffraction gratings are demonstrated, including linear polarization with horizontal, vertical and ± 45° orientations, and circular R and L polarization outputs. When illuminated with an arbitrary state of polarization, the system acts as a complete polarimeter where the intensities of the diffraction orders allow measurement of the Stokes parameters with a single shot. Experimental proof-of-concept is presented using a parallel-aligned liquid crystal display operating in a double pass architecture.

Nondiffracting Bessel beams with polarization state that varies with propagation distance.

We generate nondiffracting Bessel beams whose polarization state varies with propagation distance. We use a reflective geometry where a single parallel-aligned spatial light modulator device is used to spatially modulate two orthogonal linear polarizations with two axicon phase profiles. Then, by adding an extra phase retardation radial profile between these linear states, we are able to modulate the state of polarization along the line focus of the axicon. We provide experimental results that demonstrate the polarization axial control with zero-order and higher order Bessel beams.

Azimuthal multiple-beam interference effects with combinations of vortex beams.

We create a series of vortex beams consisting of positive and negative topological charges and develop a geometry where they interfere creating azimuthal multiple-beam interference effects. Usually the combination of two beams with opposite charges creates an azimuthal two-beam interference effect where the intensity varies sinusoidally with azimuthal angle. We combine several of these beams to create patterns where the interference becomes more sharply defined in the azimuthal direction and where destructive interference eliminates some of the intensity peaks. The process is complicated by the fact that the radii of the different vortex beams depend on the topological charge and the focal length of the Fourier lens. We generate a series of patterns where the different charges are focused with different focal lengths such that their radii agree. However to encode these, we encode patterns from the lower charges that have been Fresnel diffracted to the plane of the highest charge. These complicated patterns are encoded onto a liquid crystal display (LCD). Experimental results agree well with theory.

Encoding high-order cylindrically polarized light beams.

In this work we present a setup for the experimental production of cylindrically polarized beams, as well as other variations of polarized light beams. The optical system uses a single transmissive phase-only spatial light modulator, which is used to apply different spatial phase modulation to two output collinear R and L circularly polarized components. Different cylindrically polarized light beams can be obtained by applying different phase shifts to these two circularly polarized components. The system is very efficient since modulation is directly applied to the light beam (as opposed to other common methods operating in the first order of encoded diffraction gratings). Different variations to the cylindrically polarized light beams are also reported, obtained by adding linear or quadratic relative phase shifts between the two circular polarization components of the light beam. Experimental results are provided in all cases.

Arithmetic of focused vortex beams in three-dimensional optical lattice arrays.

In this work, we present a method to generate a 3D lattice of vortex beams. We apply phase look-up tables (LUTs) designed to generate gratings having an arbitrary content of diffraction orders. This phase LUT can be applied to a variety of diffraction optical elements, such as linear phase gratings, blazed diffractive lenses, and spiral phase patterns. We concentrate on combinations of all of these to create 3D structures of vortex beams. In particular, we generate all of these elements in the first output quadrant and eliminate the zero-order diffraction that often unavoidably accompanies these patterns. We discuss different ways of producing these 3D vortex gratings, and how the various output beams are related to the arithmetic of the 3D distribution of topological charges. Experimental results are provided by means of a liquid crystal spatial light modulator.

Analysis of the propagation dynamics and Gouy phase of Airy beams using the fast Fresnel transform algorithm.

There is great interest in Airy beams because they appear to propagate in a curved path. These beams are usually generated by inserting a cubic phase mask onto the input plane of a Fourier transform system. Here, we utilize a fast Fresnel diffraction algorithm to easily derive both the propagation dynamics and the Gouy phase shift for these beams. The trajectories of these beams can be modified by adding additional linear and quadratic phase terms onto the cubic phase mask. Finally, we have rewritten the equations regarding the propagating Airy beams completely in laboratory coordinates for use by experimentalists. Experimental results are included. We expect that these results will be of great importance in applications of Airy beams.

Accelerating light beams with arbitrarily transverse shapes.

Accelerating beams are wave packets that preserve their shape while propagating along curved trajectories. Their unique characteristics have opened the door to applications that range from optical micromanipulation and plasma-channel generation to laser micromachining. Here, we demonstrate, theoretically and experimentally, that accelerating beams can be generated with a variety of arbitrarily chosen transverse shapes. We present a general method to construct such beams in the paraxial and nonparaxial regime and demonstrate experimentally their propagation in the paraxial case. The key ingredient of our method is the use of the spectral representation of the accelerating beams, which offers a unique and compact description of these beams. The on-demand accelerating light patterns described here are likely to give rise to new applications and add versatility to the current ones.

Generalized diffractive optical elements with asymmetric harmonic response and phase control.

We report a method to generate phase-only diffractive beam splitters allowing asymmetry of the target diffracted orders, as well as providing a tailored phase difference between the diffracted orders. We apply a well-established design method that requires the determination of a set of numerical parameters, and avoids the use of image iterative algorithms. As a result, a phase lookup table is determined that can be used for any situation where a first-order (blazed) diffractive element is modified to produce higher orders with desired intensity and/or phase relation. As examples, we demonstrate the phase difference control on triplicators, as well as on other generalized diffractive elements like bifocal Fresnel lenses and phase masks for the generation of vortex beams. Results are experimentally demonstrated by encoding the calculated phase pattern onto parallel-aligned liquid crystal spatial light modulators.

Direct generation of abruptly focusing vortex beams using a 3/2 radial phase-only pattern.

Abruptly focusing Airy beams have previously been generated using a radial cubic phase pattern that represents the Fourier transform of the Airy beam. The Fourier transform of this pattern is formed using a system length of 2f, where f is the focal length of the Fourier transform lens. In this work, we directly generate these abruptly focusing Airy beams using a 3/2 radial phase pattern encoded onto a liquid crystal display. The resulting optical system is much shorter. In addition, we can easily produce vortex patterns at the focal point of these beams. Experimental results match theoretical predictions.

Generalized phase diffraction gratings with tailored intensity.

We report the generation of continuous phase masks designed to generate a set of target diffraction orders with defined relative intensity weights. We apply a previously reported analytic calculation that requires resolving a single equation with a set of parameters defining the target diffraction orders. Then the same phase map is extended to other phase patterns such as vortex generating/sensing gratings. Results are demonstrated experimentally with a parallel-aligned spatial light modulator.

Abruptly autofocusing vortex beams.

We generate abruptly autofocusing beams that produce vortices at the focus. We give explicit equations for the phase-only Fourier masks that generate these beams including explanations for controlling the focal distance and numerical aperture. We show experimental results for the focal distance, the vortex pattern and show that the diameter of the focused beam can be made smaller than the size of a comparable Airy beam from a lens. Finally we show how to move the focus spot in three dimensions by encoding additional optical elements onto the phase pattern.