Generalized formulation and symmetry properties of reciprocal nonabsorbing polarization devices: Application to liquid-crystal displays

We present a general formulation based on the Jones-matrix theory for reciprocal nonabsorbing polarization devices, including polarization interference filters and liquid-crystal displays. The development of this formulation is based on general symmetry conditions that relate the Jones matrix when the device is illuminated from the front side and from the back side. The application to liquid-crystal displays results in a constraint of the Jones-matrix elements, which represents a generalization of the existing models that explain their modulation properties.

Operational modes of a ferroelectric LCoS modulator for displaying binary polarization, amplitude, and phase diffraction gratings

We analyze the performance of a ferroelectric liquid crystal on silicon display (FLCoS) as a binary polarization diffraction grating. We analyze the correspondence between the two polarization states emerging from the displayed grating and the polarization and intensity of the diffracted orders generated at the Fourier diffraction plane. This polarization-diffraction analysis leads, in a simple manner, to configurations yielding binary amplitude or binary phase modulation by incorporating an analyzer on the reflected beam. Based on this analysis, we present two useful variations of the polarization configuration. The first is a simplification using a single polarizer, which provides equival…

Vortex beam generation and other advanced optics experiments reproduced with a twisted-nematic liquid-crystal display with limited phase modulation

In this work we propose the use of twisted-nematic liquid-crystal spatial light modulators (TN-LC-SLM) as a useful tool for training students in the manipulation of light beams with phase-only masks. In particular, we focus the work on the realization of phase-only gratings and phase-only spiral phases for the generation of vortex beams, beams carrying orbital angular momentum (OAM). Despite the extensive activity in this field, its experimental implementation for educational purposes is limited because it requires the use of very expensive high-resolution liquid-crystal on silicon (LCOS) SLMs. Here, we show that a low-cost experimental implementation can be done with older TNLC technology.…

Poincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratio

The Poincare sphere representation is used to analyze the polarization transformation achieved with a ferroelectric liquid crystal (FLC) optical modulator. This device acts as a switchable wave-plate, in which the orientation of the principal axes rotates under the action of an applied bipolar voltage. In the standard operational mode for intensity switching, the rotation angle of the principal axes is �θ = π/4 and the phase shift is φ = π (half-wave-plate). However, for wavelengths different from the design one, the FLC deviates from the half-wave-plate performance and the optical contrast is diminished. We use the Poincare sphere representation to perform a theoretical analysis of the int…

Experimental evaluation of Fourier transform holograms by a self-interferometric technique

We present a technique that combines an encoding method to display complex-valued holograms onto a phase-only spatial light modulator (SLM) with a phase-shifting interferometric (PSI) technique for experimentally evaluating the generated complex-valued optical fields. We demonstrate an efficient common-path polarization interferometer based on the SLM itself, not requiring any external additional element. The same setup can be used to simultaneously display the complex hologram and to apply the phase-shifting values required to retrieve the phase distribution of the optical field. A simple rotation of a polarizer allows to change from the intensity configuration to the interferometer config…

Achromatic linear retarder with tunable retardance

We present a universal design and proof-of-concept of a tunable linear retarder of uniform wavelength response in a broad spectral range. It consists of two half-wave retarders (HWR) between two quarter-wave retarders (QWRs), where the uniform retardance can be tuned continuously by simply rotating one of the HWRs. A proof-of-concept of this design is built by using commercially available Fresnel rhomb retarders that provide retardation with almost wavelength uniformity in the visible and near infrared from 450 to 1550 nm. The design is universal, since other achromatic QWRs and HWRs could also be employed. The system is experimentally demonstrated to control the state of polarization of a …

Using birefringence colors to evaluate a tunable liquid-crystal q-plate

Q-plates are geometrical phase elements that enable the realization of vector beams in simple and compact optical setups. In this work, we consider a tunable liquid-crystal commercial q-plate operative in the visible and near-IR range and study its spectral and color birefringence properties under broadband illumination. We first characterize the spectral retardance function of the device in a wide range from 400 to 1600 nm and determine how it changes upon applied voltage. Then we evaluate the color transmission characteristics when inserting the q-plate between crossed and parallel linear polarizers. These color properties agree with the trajectory in the CIExy chromaticity diagram as the…

Wavelength-compensated time-sequential multiplexed color joint transform correlator

We report a wavelength-compensated three-channel (RGB) joint transform correlator (JTC) for color pattern recognition using a ferroelectric liquid-crystal spatial light modulator (SLM) operating in binary pure phase modulation. We apply a previously reported time-multiplexing technique useful in creating wavelength-compensated diffraction patterns, based on the synchronization of properly scaled diffraction masks with the input wavelength selection obtained by applying a rotating RGB color-filter wheel to an Ar-Kr laser. The application of this technique to a JTC architecture permits real-time color object detection. In order to achieve a high light efficiency for the correlation process, w…

Optical System For Measuring The Spectral Retardance Function In An Extended Range

Optical retarders are key elements for the control of the state of polarization of light, and their wavelength dependance is of great importance in a number of applications. We apply a well-known technique for determinig the spectral retardance by measuring the transmission spectra between crossed or parallel polarizers. But we we develop an optical system to perform this measurement in a wide spectral range covering the visible (VIS) and near infrared (NIR) spectrum in the range from 400 to 1600 nm. As a result we can measure the spectral retardance of different retarders and easily identify the kind of reterder (multiple order, zero-order, achromatic). We show results with tunable liquid-…

Ray matrix analysis of anamorphic fractional Fourier systems

In this work we extend the application of the ray matrix approach to analyse anamorphic fractional Fourier systems, i.e., fractional Fourier optical systems where the fractional power is different for two orthogonal directions. The application of the ray matrix approach allows for easily obtaining the properties of the optical system, and it is therefore a powerful tool to design and simplify complicated systems. For simplicity we consider fractional Fourier systems with real orders and systems without apertures. We start by presenting the analysis of some previously reported anamorphic Fourier and fractional Fourier systems, and we end by proposing a simple optical system with tunable anam…

Fullin-situcharacterization of spatial light modulators in an optical correlator. Filter adaptation to operating curves

Abstract In this paper we present one method for the characterization of the spatial light modulators (SLMs) of a real-time Vander Lugt type of correlator. This correlator uses two SLMs: one to introduce the scene and a second to introduce the frequency‐matched filter. The SLM characterization methods are in situ, that is in the correlator set‐up. Illumination conditions are different for each SLM, and consequently different characterization techniques must be used in each case. For the characterization of the scene SLM a diffraction method is used, while for the characterization of the filter SLM an interferometric technique is more convenient. Finally, we take into account the operating c…

Generation of Bessel beam arrays through Dammann gratings.

In this work we apply the Dammann grating concept to generate an equal-intensity square array of Bessel quasi-free diffraction beams that diverge from a common center. We generate a binary phase mask that combines the axicon phase with the phase of a Dammann grating. The procedure can be extended to include vortex spiral phases that generate an array of optical pipes. Experimental results are provided by means of a twisted nematic liquid crystal display operating as a binary π phase spatial light modulator.

Vector modes generation using efficient on-axis system

Arbitrary polarized vector beam modes are efficiently generated. We use two flicker-free liquid-crystal on silicon (LCOS) spatial light modulators (SLM) in an on-axis common path architecture. Different superpositions of vector modes are obtained with high light efficiency.

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.

Time-multiplexed chromatic-controlled axial diffractive optical elements

Programmable diffractive optical elements DOEs with axial response have many interesting applications, including diffractive lenses, axicons, and optical tweezers. In all these cases, it is essential to properly select the modulation configuration of the spatial light modulator SLM where the DOE is displayed, in order to avoid the undiffracted zero order component that appears on axis and overlaps the desired axial response. However, in general, the chromatic dispersion in liquid crystal SLMs prevents the cancellation of the zero order for a broadband light source, thus limiting the possibilities for polychromatic program- mable axial DOEs. We operate a ferroelectric liquid crystal on silic…

Fractional Fourier transforms, symmetrical lens systems, and their cardinal planes

We study the relation between optical lens systems that perform a fractional Fourier transform (FRFT) with the geometrical cardinal planes. We demonstrate that lens systems symmetrical with respect to the central plane provide an exact FRFT link between the input and output planes. Moreover, we show that the fractional order of the transform has real values between 0 and 2 when light propagation is produced between principal planes and antiprincipal planes, respectively. Finally, we use this new point of view to design an optical lens system that provides FRFTs with variable fractional order in the range (0,2) without moving the input and output planes.

Solc filters in a reflective geometry

We present the realization of a bulk optics birefringent Solc filter in a reflective geometry. This geometry reduces by half the number of required retarders, ensures the same spectral retardance function in pairs of retarders, and helps to make more compact filters. The key element is a quarter-wave Fresnel rhomb located in between the set of retarders and a mirror. Two cases are considered: the first Solc filter uses multiple-order quartz retarders, and the second one uses two liquid-crystal retarders. The latter has the advantage of being tunable via an applied voltage. Experimental results show how to filter the spectral content of a supercontinuum laser.

Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders.

Unusually large phase modulation in a commercial liquid crystal spatial light modulator (LCSLM) is reported. Such a situation is obtained by illuminating with visible light a device designed to operate in the infrared range. The phase modulation range reaches 6π radians in the red region of the visible spectrum and 10π radians in the blue region. Excellent diffraction efficiency in high harmonic orders is demonstrated despite a concomitant and non-negligible Fabry–Perot interference effect. This type of SLM opens the possibility to implement diffractive elements with reduced chromatic dispersion or chromatic control.

Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM

Multiple internal reflection effects on the optical modulation of a commercial reflective parallel-aligned liquid-crystal on silicon (PAL-LCoS) spatial light modulator (SLM) are analyzed. The display is illuminated with different wavelengths and different angles of incidence. Non-negligible Fabry-Perot (FP) effect is observed due to the sandwiched LC layer structure. A simplified physical model that quantitatively accounts for the observed phenomena is proposed. It is shown how the expected pure phase modulation response is substantially modified in the following aspects: 1) a coupled amplitude modulation, 2) a non-linear behavior of the phase modulation, 3) some amount of unmodulated light…

Fractional Fourier Transforms and Geometrical Optics

Teaching stable two-mirror resonators through the fractional Fourier transform

We analyse two-mirror resonators in terms of their fractional Fourier transform (FRFT) properties. We use the basic ABCD ray transfer matrix method to show how the resonator can be regarded as the cascade of two propagation–lens–propagation FRFT systems. Then, we present a connection between the geometric properties of the resonator (the g parameters) and those of the equivalent FRFT systems (the FRFT order and scaling parameters). Expressions connecting Gaussian beam q-transformation with FRFT parameters are derived. In particular, we show that the beam waist of the resonator's mode is located at the plane leading to two FRFT subsystems with equal scaling parameter which, moreover, coincid…

Optical retarder system with programmable spectral retardance.

An optical system that works as a retarder waveplate with programmable spectral retardance is proposed. The system is based on a pixelated liquid crystal on silicon (LCoS) spatial light modulator (SLM). The input light beam is spectrally dispersed and different spectral components are projected onto different pixels of the LCoS-SLM. A different retardance is then addressed for each pixel, adapted to the incoming wavelength. Light reflected from the SLM is then recombined by the same setup. In this way a programmable polarization spectrum can be encoded. We illustrate the broadband characterization that is required for proper use of the system. Then several examples are shown, including spec…

Teaching Fourier optics through ray matrices

In this work we examine the use of ray-transfer matrices for teaching and for deriving some topics in a Fourier optics course, exploiting the mathematical simplicity of ray matrices compared to diffraction integrals. A simple analysis of the physical meaning of the elements of the ray matrix provides a fast derivation of the conditions to obtain the optical Fourier transform. We extend this derivation to fractional Fourier transform optical systems, and derive the order of the transform from the ray matrix. Some examples are provided to stress this point of view, both with classical and with graded index lenses. This formulation cannot replace the complete explanation of Fourier optics prov…

Phase joint transform sequential correlator for nonlinear binary correlations

We study the performance of nonlinear optical correlations using a joint transform correlator that operates in phase-only spatial light modulation at input joint transform plane. Amplitude input nonlinear optical time sequential correlations have shown better discrimination and noise robustness than conventional linear correlations. Those nonlinear correlations are based on decomposing the reference and the target into binary slices and to add the contribution of all linear correlations between them. Those correlations can be easily implemented using a conventional joint transform correlator. However, the system has poor efficiency and low cross-correlation peak intensity, mainly because of…

Microscope system with on axis programmable Fourier transform filtering

Abstract We propose an on-axis microscope optical system to implement programmable optical Fourier transform image processing operations, taking advantage of phase and polarization modulation of a liquid crystal on silicon (LCOS) display. We use a Hamamatsu spatial light modulator (SLM), free of flickering, which therefore can be tuned to fully eliminate the zero order component of the encoded diffractive filter. This allows the realization of filtering operation on axis (as opposed to other systems in the literature that require operating off axis), therefore making use of the full space bandwidth provided by the SLM. The system is first demonstrated by implementing different optical proce…

Efficient on-axis SLM engineering of optical vector modes

Abstract This work presents a method for the efficient experimental generation of arbitrary polarized vector beam modes. The optical system employs two liquid-crystal on silicon (LCOS) spatial light modulators (SLM) in a common path architecture, avoiding the use of beam-splitters. Each SLM displays a different phase-only mask, each one encoding a different pattern onto two orthogonal linear polarization components of the input beam. These phase-only masks are designed using a recently proposed random technique to encode complex amplitude values. This encoding technique reconstructs the complex function on-axis, thus avoiding incorporating carrier phases. By addressing such properly designe…

Programmable Supercontinuum Laser Spectrum Generator Based on a Liquid-Crystal on Silicon Spatial Light Modulator

Supercontinuum (SC) lasers combine a broadband light spectrum with the unique properties of single-mode lasers. In this work we present an optical system to spectrally filter a SC laser source using liquid-crystal on silicon (LCoS) spatial light modulators (SLM). The proposed optical system disperses the input laser and the spectrally separated components are projected onto the LCoS-SLM, where the state of polarization of each wavelength is separately modulated. Finally, recombining the modulated spectral components results in an output laser source where the spectrum can be controlled dynamically from a computer. The system incorporates two branches to independently control the visible (VI…

New key based on tilted lenses for optical encryption

A novel concept based on tilted spherical lenses for optical encryption using Lohmann’s type I systems is presented. The tilt angle of the spherical lenses is used as an encrypted key and the decryption performance is studied both qualitatively (visual image degradation) and quantitatively (mean squared error analysis) by numerical simulations. The paper presents a general mathematical framework in virtue of the dioptric power matrix formalism and oblique central refraction used in the optometry field. Computer simulations show that image information cannot be retrieved after a few degrees of tilt on both spherical lenses in the encryption system. In addition, a preliminary experiment is pr…

Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators

Recently we published theoretical expressions for the eigenvalues and eigenvectors for a twisted-nematic liquid crystal spatial light modulator (LCSLM) as a function of the twist angle and birefringence using the Jones matrix formalism. These polarization eigenvectors are of interest for newer video graphics adapter (VGA) resolution (640 3480 pixels) LCSLMs that are thinner and consequently have a lower birefringence. Although these devices can produce excellent amplitude modulation, they cannot achieve phase-only modulation using linearly polarized light. In these cases, the polarization eigenvectors are required to obtain phase-only modulation. We present experimental measurements for bot…

Programmable color tuning of a multiline laser by means of a twisted nematic liquid crystal display.

An optical system useful to tune in a controlled way the color of a triline argon krypton (Ar–Kr) laser by means of a twisted nematic liquid crystal display (TNLCD) is presented. The optical setup employs a 4f system and two blazed gratings to first separate and then recombine the spectrum of the light beam. The TNLCD is included in the intermediate focal plane operating in the amplitude modulation mode to control the relative transmission of each spectral line. The resulting color is accurately predicted by using a previously developed physical model of the spectral and voltage dependence of the TNLCD birefringence. By simply changing the gray level image addressed to the display, the Ar–K…

Accurate color predictability based on a spectral retardance model of a twisted-nematic liquid-crystal display

Abstract In this work we present the application of a simple physical model to accurately predict the broadband spectral transmittance and colorimetric properties of a twisted-nematic liquid crystal display (TNLCD). We spectroscopically calibrate the retardance parameters to evaluate the spectrum of the light transmitted by a TNLCD sandwiched between two linear polarizers. When the TNLCD is illuminated with a broadband light source, the full spectrum can be predicted as a function of the addressed gray level for any arbitrary orientation of the polarizers. Thus, the color of the transmitted light can be also be estimated with very good accuracy. As an example, a polarizers' configuration is…

Graphical representation of non-absorbing polarization devices

A graphical representation of general non-absorbing polarization devices operating under normal plane-wave incidence is presented. The representation is based on a four-dimensional spherical parametrization of the Jones matrix of this kind of polarization devices. The graphical representation takes the form of a solid cylinder. The projection of the point representing the device over the base of the cylinder gives the corresponding polarization eigenvectors represented in the complex plane, while the height of the point in the cylinder is the phase of its eigenvalue. Some simple examples like wave-plates and rotators are discussed. The representation may represent a useful tool to identify …

Second order diffractive optical elements in a spatial light modulator with large phase dynamic range

International audience; A study of the diffraction efficiency of a spatial light modulator with a large dynamic phase range is reported. We use a phase-only device that reaches 4π phase modulation depth for the wavelength of 454 nm. This allows operating phase-only diffractive optical elements in the second harmonic diffraction order, instead of in the usual first diffraction order. This type of implementation shows advantages in terms of resolution and diffraction efficiency. Experimental results are reported for blazed diffractive gratings and diffractive lenses.

Twist angle determination in liquid crystal displays by location of local adiabatic points

In this work we present a method for the determination of the twist angle of an arbitrary twisted nematic liquid crystal spatial light modulator. The method is based on the location of local adiabatic points, i.e., situations in which the liquid crystal SLM acts only as a rotation device. For these cases, the rotation induced on the polarization of the incident beam is equal to the twist angle. Consequently, the twist angle can be determined with high precision. We show that local adiabatic regime may be achieved in two ways, either by changing the incident beam wavelength, or by applying a voltage to the electrodes of the display. However, the simple model that describes the SLM in the off…