Influence of the incident angle in the performance of liquid crystal on silicon displays.

In this paper we experimentally analyze the performance of a twisted nematic liquid crystal on silicon (LCoS) display as a function of the angle of incidence of the incoming beam. These are reflective displays that can be configured to produce amplitude or phase modulation by properly aligning external polarization elements. But we demonstrate that the incident angle plays an important role in the selection of the polarization configuration. We performed a Mueller matrix polarimetric analysis of the display that demonstrates that the recently reported depolarization effect observed in this type of displays is also dependant on the incident angle.

Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics.

In this paper we provide evidence of the temporal fluctuations of the phase modulation property of a liquid crystal on silicon (LCoS) display, and we analyze its effect when the device is used for displaying a diffractive optical element. We use a commercial twisted nematic LCoS display configured to produce a phase-only modulation, and we provide time resolved measurements of the diffraction efficiency that show rapid fluctuations of the phase modulation, in the millisecond order. We analyze how these fluctuations have to be considered in two typical methods for the characterization of the phase modulation: two beam interference and diffraction from a binary grating. We finally provide experimental results on the use of this device for displaying a computer generated hologram. A reduction of the modulation diffraction efficiency results from the phase modulation fluctuation.

Mueller-Stokes characterization and optimization of a liquid crystal on silicon display showing depolarization.

In this paper we characterize the polarimetric properties of a liquid crystal on silicon display (LCoS), including depolarization and diattenuation which are usually not considered when applying the LCoS in diffractive or adaptive optics. On one hand, we have found that the LCoS generates a certain degree (that can be larger than a 10%) of depolarized light, which depends on the addressed gray level and on the incident state of polarization (SOP), and can not be ignored in the above mentioned applications. The main origin of the depolarized light is related with temporal fluctuations of the SOP of the light reflected by the LCoS. The Mueller matrix of the LCoS is measured as a function of the gray level, which enables for a numerical optimization of the intensity modulation configurations. In particular we look for maximum intensity contrast modulation or for constant intensity modulation. By means of a heuristic approach we show that, using elliptically polarized light, amplitude-mostly or phase-mostly modulation can be obtained at a wavelength of 633 nm.

Combined Mueller and Jones matrix method for the evaluation of the complex modulation in a liquid-crystal-on-silicon display.

We apply the polar decomposition of the Mueller matrix describing a liquid-crystal-on-silicon display to identify the diattenuator, depolarizer, and retarder contributions as a function of the gray level. The retarder contribution is expressed in terms of the equivalent Jones matrix to apply previous techniques to evaluate the phase modulation. This allows searching for optimized polarization configurations for phase- or amplitude-only modulation responses. We present results for lambda=633 nm showing a phase-only modulation up to 2 pi rad and flat intensity modulation.

Achromatic diffractive lens written onto a liquid crystal display.

We propose a programmable diffractive lens written onto a liquid crystal display (LCD) that is able to provide equal focal lengths for several wavelengths simultaneously. To achieve this goal it is necessary that the LCD operate in the phase-only regime simultaneously for the different wavelengths. We design the appropriate lens for each wavelength, and then the lenses are spatially multiplexed onto the LCD. Various multiplexing schemes have been analyzed, and the random scheme shows the best performance. We further show the possibility of finely tuning the chromaticity of the focal spot by changing the relative weights of the multiplexing among the various wavelengths.

Depth of focus increase by multiplexing programmable diffractive lenses.

A combination of several diffractive lenses written onto a single programmable liquid crystal display (LCD) is proposed for increasing the Depth of Focus (DOF) of the imaging system as a whole. The lenses are spatially multiplexed in a random scheme onto the LCD. The axial irradiance distribution produced by each lens overlaps with the next one producing an extended focal depth. To compare the image quality of the multiplexed lenses, the Modulation Transfer Function (MTF) is calculated. Finally we obtain the experimental Point Spread Functions (PSF) for these multiplexed lenses and experimental results in which an extended object is illuminated under spatially incoherent monochromatic light. We compare the images obtained in the focal plane and in some defocused planes with the single lens and with three multiplexed lenses. The experimental results confirm that the multiplexed lenses produce a high increase in the depth of focus.

Programmable apodizer to compensate chromatic aberration effects using a liquid crystal spatial light modulator.

Programmable apodizers written on a liquid crystal spatial light modulator (LCSLM) offer the possibility of modifying the point spread function (PSF) of an optical system in monochromatic light with a high degree of flexibility. Extension to polychromatic light has to take into account the liquid crystal response dependence on the wavelength. Proper control of the chromatic properties of the LCSLM in combination with the design of the correct apodizer is necessary for this new range of applications. In this paper we report a successful application of a programmable amplitude apodizer illuminated with polychromatic light. We use an axial apodizing filter to compensate the longitudinal secondary axial color (LSAC) effects of a refractive optical system on the polychromatic PSF. The configuration of the LCSLM has been optimized to obtain a good amplitude transmission in polychromatic light. Agreement between experimental and simulated results shows the feasibility of our proposal.

Symmetry properties with pupil phase-filters.

Pupil filters can modify the three dimensional response of an optical system. In this paper, we study different pupil symmetries that produce a predictable image behavior. We show that different pupil-filters that satisfy certain symmetry conditions can produce axial responses which are either identical or mirror reflected. We also establish the differences in the symmetry properties between amplitude-only filters and phase-only filters. In particular, we are interested in phase filters that produce transverse superresolution with axial superresolution or high depth of focus.

Integration in the Fourier domain for restoration of a function from its slope: comparison of four methods.

In some measurement techniques the profile, f(x), of a function should be obtained from the data on measured slope f(?)(x) by integration. The slope is measured in a given set of points, and from these data we should obtain the profile with the highest possible accuracy. Most frequently, the integration is carried out by numerical integration methods [Press et al., Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987)] that assume different kinds of polynomial approximation of data between sampling points. We propose the integration of the function in the Fourier domain, by which the most-accurate interpolation is automatically carried out. Analysis of the integration methods in the Fourier domain permits us to easily study and compare the methods' behavior.

Amplitude Apodizers Encoded onto Fresnel Lenses Implemented on a Phase-Only Spatial Light Modulator.

We show that both a lens and a nonuniform amplitude transmission filter can be encoded simultaneously onto a twisted nematic liquid-crystal spatial light modulator (SLM) working in the phase-only mode. The inherent equivalent apodization that is due to the pixelated structure of the SLM is compensated for. In addition, different types of nonuniform transmission pupil such as transverse apodizing, transverse hyperresolving, and axial hyperresolving (multifocusing) filters are implemented. The excellent agreement between numerical and experimental results shows the capability of this method to encode amplitude apodizers on a phase-only SLM.

Fully complex synthetic discriminant functions written onto phase-only modulators.

Synthetic discriminant functions (SDF's) are an effective tool for pattern-recognition applications. However, their experimental implementation is difficult because of the difficulty in writing full complex modulation functions onto spatial light modulators (SLM's) with restricted coding domains. Iterative methods are required for the implementation of SDF filters in real SLM's. A great deal of experimental research has been done with phase-only filters because they can be successfully implemented with liquid-crystal SLM's. We have recently introduced a technique for encoding arbitrary amplitude information onto the phase-only filter, thus allowing us to encode an arbitrary complex function onto a phase-only SLM. We apply this technique to the generation of arbitrary complex SDF filters, thus avoiding the necessity of iterative algorithms. We examine the discrimination capabilities of fully complex SDF filters designed with different parameters and constraints. Experimental results obtained with liquid-crystal SLM's are included.

Inherent apodization of lenses encoded on liquid-crystal spatial light modulators.

Focusing diffractive optical elements encoded in liquid-crystal spatial light modulators yields an inherent equivalent apodization of the focused spot as a result of the pixelated nature of these devices and the finite extent of each pixel. We present a theoretical explanation for and experimental evidence of this effect. We demonstrate an experimental procedure for measuring the apodization and a method to compensate for this effect.

Programmable axial apodizing and hyperresolving amplitude filters with a liquid-crystal spatial light modulator.

Amplitude-transmitting filters for apodizing and hyperresolving applications can be easily implemented by use of a two-dimensional programmable liquid-crystal spatial light modulator operating in a transmission-only mode. Experimental results are in excellent agreement with theoretical predictions. This approach can permit the analysis of various filter designs and can allow the filters to be changed rapidly to modify the response of an optical system.

Encoding amplitude information onto phase-only filters.

We report a new, to our knowledge, technique for encoding amplitude information onto a phase-only filter with a single liquid-crystal spatial light modulator. In our approach we spatially modulate the phase that is encoded onto the filter and, consequently, spatially modify the diffraction efficiency of the filter. Light that is not diffracted into the first order is sent into the zero order, effectively allowing for amplitude modulation of either the first-order or the zero-order diffracted light. This technique has several applications in both optical pattern recognition and image processing, including amplitude modulation and inverse filters. Experimental results are included for the new technique.

Bessel function output from an optical correlator with a phase-only encoded inverse filter.

We report on a technique for producing a Bessel function correlation output for an arbitrary input pattern. The central dark spot at the center of the Bessel function correlator output is narrower than the width of the normal correlation spot and can be extremely useful for locating the center of the correlation signal. The Bessel function is produced by convolution of the extremely sharp correlation produced by an inverse filter with the Bessel function and is encoded with a single phase-only liquid-crystal spatial light modulator. To encode amplitude information on the filter, we spatially modulate the phase encoded on the filter. Amplitude modulation is obtained by modulation of the diffraction efficiency of the phase grating. Experimental results are presented.

One-step multichannel pattern recognition based on the pixelated structure of a spatial light modulator.

We present an architecture in which a multichannel correlator can perform simultaneous optical pattern recognition. Processing in parallel is made possible by use of the different diffraction orders produced by the pixelated structure of the liquid-crystal spatial light modulator employed to display the input scene. We codify additional quadratic phases in the filters to separate the correlation information corresponding to each channel. We demonstrate that the system can recognize different targets simultaneously. Good agreement between experimental and numerically simulated results is obtained.

Real-time binary-amplitude phase-only filters.

A real-time binary-amplitude phase-only filter (BAPOF) implemented in available phase-only modulators is presented. The BAPOF has an amplitude transmission equal to one only in a region of support, while the transmission is equal to zero in the complementary region. To implement zero transmission in a phase-only modulator we propose to add a linear phase to the region of support. In this way the correlation desired is obtained off axis. Computer simulations and experimental results obtained with this technique are given.

Polychromatic axial behavior of aberrated optical systems: Wigner distribution function approach.

We propose a new method for the computation of the tristimuli values that correspond to the impulse response along the optical axis provided by an imaging optical system working under polychromatic illumination. We show that all the monochromatic irradiance distributions needed for this calculation can be obtained from the Wigner distribution function associated with a certain version of the pupil function of the system. The use of this single phase-space representation allows us to obtain the above merit function for aberrated systems with longitudinal chromatic aberration and primary spherical aberration. Some numerical examples are given to verify the accuracy of our proposal.

Color pattern recognition with circular component whitening.

Polychromatic object recognition based on circular whitening preprocessing of red-green-blue components and multichannel matched filtering is described. Computer simulations and experimental results are provided to facilitate recognizing a color target among objects of similar shape but with different color contents. Experimental results are obtained with an optical correlator with two spatial light modulators, one to introduce the scene and the second one to introduce the filter.