Characterizing d-dimensional quantum channels by means of quantum process tomography.

In this Letter, we propose a simple optical architecture based on phase-only programmable spatial light modulators, in order to characterize general processes on photonic spatial quantum systems in a d>2 Hilbert space. We demonstrate the full reconstruction of typical noises affecting quantum computing, such as amplitude shifts, phase shifts, and depolarizing channels in dimension d=5. We have also reconstructed simulated atmospheric turbulences affecting a free-space transmission of qudits in dimension d=4. In each case, quantum process tomography was performed in order to obtain the matrix χ that fully describes the corresponding quantum channel, E. Fidelities between the states are experimentally obtained after going through the channel, and the expected ones are above 97%.

Preparing arbitrary pure states of spatial qudits with a single phase-only spatial light modulator.

Spatial qudits are D-dimensional (D ≥ 2) quantum systems carrying information encoded in the discretized transverse momentum and position of single photons. We present a proof-of-principle demonstration of a method for preparing arbitrary pure states of such systems by using a single phase-only spatial light modulator (SLM). The method relies on the encoding of the complex transmission function corresponding to a given spatial qudit state onto a preset diffraction order of a phase-only grating function addressed at the SLM. Fidelities of preparation above 94% were obtained with this method, which is simpler, less costly, and more efficient than those that require two SLMs for the same purpose.

The minimum Euclidean distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial light modulators.

Digital addressing of the electrical signal in spatial light modulators, as it is the case in present liquid crystal on silicon (LCoS) displays, may lead to temporal phase fluctuations in the optical beam. In diffractive optics applications a reduction in the modulation diffraction efficiency may be expected. Experimental work is done characterizing the fluctuations amplitude and phase depth for three different digital addressing sequences. We propose a diffractive model to evaluate the modulation diffraction efficiency of phase diffractive optical elements (DOEs) in the presence of phase fluctuations. Best results are obtained for the most stable electrical sequence even though its phase depth is as small as 280 degrees . The results show good agreement with the numerical calculation given by the model.

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.

Digital holography with a point diffraction interferometer.

We propose a technique to implement digital holography by means of a point diffraction interferometer. The device uses a liquid crystal television display and works under partially coherent illumination which makes it useful for 3D microscopy. We present the theory on which the method is based and the obtained results.

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.

Gray-level computer-generated hologram filters for multiple-object correlation.

Synthesis of gray-level computer-generated holograms allows for an increase of the information storage capability that is usually achieved with conventional binary filters. This is mainly because more degrees of freedom are available. We propose to profit from this feature by synthesizing complex filters formed by many superimposed holograms, each with a different carrier frequency. We apply these gray-level filters to perform multichannel correlation and in this way enhance the capability of optical correlators to process the information in parallel and simultaneously. First, we analyze the behavior of some performance criteria on the impulse response and on the correlation as a function of the number of holograms that are multiplexed. Then we show the results of two experiments: In the first a composed phase-only filter is used in a multiple-object recognition process. In the second a composed synthetic discriminant function filter is used to implement an object classification by means of a binary code.

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.

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.

Multiphase homodyne interferometry: analysis of some error sources.

Some sources of error with multiphase homodyne interferometry are reviewed. A major advantage over the classic two-channel approach is that the inaccuracies that originate from laser-power fluctuations and drifts are shown to be automatically compensated for by proper adjustment of the light beams.

Three-channel Homodyne Interferometer.

A three-channel homodyne interferometer based on a structured delay plate is described. Detection and data acquisition are performed with standard optoelectronics in a PC environment. The device, which is used as a displacement sensor, features sub-nanometric sensitivity.

General compressor for ultrashort pulses with nonlinear chirp.

A new type of grating compressor capable of compensating, in principle, for any arbitrary group-delay dispersion is described. The system relies on a specially designed grating with a variable groove spacing along its surface as well as a variable angle of incidence. In this manner each wavelength actually sees a different grating, always at Littrow incidence, that gives rise to distinct group delays. The design equations are derived, and the particular case of a cubic compressor is discussed. It is shown that, within practical limits, both quadratic and cubic terms can be simultaneously compensated for in typical optical fibers.