Freeform concentrator design for IR wireless-to-fiber link communications.

Airplanes use heavy wired harnesses to provide multimedia services to the seats. Optical wireless communications (OWC) are a natural choice to reduce the amount of weight, reduce the wiring complexity, and avoid possible spurious electromagnetic radiation that risks affecting the airplane's navigation systems. The light's dual use as lighting and optical communications functionalities allows for providing light and multimedia content through the reading lamp. Thus, an optical system using optical fibers to replace wires and a reading lamp can provide a cabin seat with lighting and onboard connectivity. However, changing shielded harnesses by optical fibers is-from an optical design point of view-a challenging task as the reading lamp must also meet the stringent requirements to link the optical wireless transmissions to the optical fiber. The difficulty up to now lies in injecting the light emitted from the passenger's device into the optical fiber using the reading lamp as the receiving antenna and light injector. Here, we describe a proof-of-concept device that experimentally allowed for establishing a link between a transmitter and a photodetector coupled to an optical fiber-end, i.e., the link consisted of an optical wireless communication and the launching of the light modulated signal into an optical fiber. Additionally, from the experimental experience, we will describe the optical design strategies permitting designing a compound freeform concentrator to allow optical free space-to-fiber links.

General formula to design a freeform singlet free of spherical aberration and astigmatism: comment.

In their paper [Appl. Opt.58, 1010 (2019)APOPAI0003-693510.1364/AO.58.001010] González-Acuña et al. claimed: "an analytical closed-form formula for the design of freeform lenses free of spherical aberration and astigmatism." However, as we show here, their formula can only be applied when the object and image are both real, and the image is inversed; additionally, the refractive index in the object and image media is the same. Here, we present the complete solution of this particular formula.

Influence of protein and energy level in finishing diets for feedlot hair lambs: growth performance, dietary energetics and carcass characteristics.

Forty-eight Pelibuey×Katahdin male intact lambs (23.87±2.84 kg) were used in an 84-d feeding trial, with six pens per treatment in a 2×2 factorial design arrangement. The aim of the study was to evaluate the interaction of two dietary energy levels (3.05 and 2.83 Mcal/kg ME) and two dietary protein levels (17.5% and 14.5%) on growth performance, dietary energetics and carcass traits. The dietary treatments used were: i) High protein-high energy (HP-HE); ii) High protein-low energy (HP-LE); iii) Low protein-high energy (LP-HE), and iv) Low protein-low energy (LP-LE). With a high-energy level, dry matter intake (DMI) values were 6.1% lower in the low-protein diets, while with low-energy, the DMI values did not differ between the dietary protein levels. Energy levels did not influence the final weight and average daily gain (ADG), but resulted in lower DMI values and higher gain efficiencies. No effects of protein level were detected on growth performance. The observed dietary net energy (NE) ratio and observed DMI were closer than expected in all treatments and were not affected by the different treatments. There was an interaction (p<0.03) between energy and protein level for kidney-pelvic and heart fat (KPH), KPH was higher in lambs fed high energy and high protein diet but not in high energy and low protein diet. The KPH was increased (20.2%, p = 0.01) in high-energy diets, while fat thickness was increased (21.7%, p = 0.02) in high-protein diets. Therefore, it is concluded that dietary energy levels play a more important role in feed efficiency than protein levels in finishing lambs with a high-energy diet (>2.80 Mcal/kg ME). Providing a level of protein above 14.5% does not improves growth-performance, dietary energetics or carcass dressing percentage.

Coherent digital demodulation of single-camera N-projections for 3D-object shape measurement: co-phased profilometry.

Fringe projection profilometry is a well-known technique to digitize 3-dimensional (3D) objects and it is widely used in robotic vision and industrial inspection. Probably the single most important problem in single-camera, single-projection profilometry are the shadows and specular reflections generated by the 3D object under analysis. Here a single-camera along with N-fringe-projections is (digital) coherent demodulated in a single-step, solving the shadows and specular reflections problem. Co-phased profilometry coherently phase-demodulates a whole set of N-fringe-pattern perspectives in a single demodulation and unwrapping process. The mathematical theory behind digital co-phasing N-fringe-patterns is mathematically similar to co-phasing a segmented N-mirror telescope.

Synchronous phase-demodulation and harmonic rejection of 9-step pixelated dynamic interferograms.

We propose a novel synchronous phase-demodulation of pixelated interferograms using squared 3x3 phase-shifted unit-cells. This 3x3 unit-cell is tiled over the CCD image sensor to create a two-dimensional (2D) pixelated carrier. Our synchronous phase-demodulation uses this 2D carrier to demodulate the pixelated interferogram as in the standard 2x2 unit-cell case. The main motivation behind the use of a 3x3 pixelated carrier (instead of the usual 2x2) is its higher harmonic robustness, allowing one to demodulate intensity-distorted fringe patterns. The harmonic rejection robustness of our spatial 3x3 configuration equals the robustness of the temporal least-squares 9-step phase-shifting algorithm (PSA). In other words, extending from the usual 2x2 phase-shifting unit-cell to 3x3 unit-cells, one extends the harmonic rejection of the demodulation algorithm. Finally we also prove that our proposed 9-step, 3x3 pixelated carrier uses the 2D available spectral space more efficiently than using these 9-steps in a linear spatial-carrier configuration.

Two-step demodulation based on the Gram-Schmidt orthonormalization method.

This Letter presents an efficient, fast, and straightforward two-step demodulating method based on a Gram-Schmidt (GS) orthonormalization approach. The phase-shift value has not to be known and can take any value inside the range (0,2π), excluding the singular case, where it corresponds to π. The proposed method is based on determining an orthonormalized interferogram basis from the two supplied interferograms using the GS method. We have applied the proposed method to simulated and experimental interferograms, obtaining satisfactory results. A complete MATLAB software package is provided at http://goo.gl/IZKF3.

Multiplicative phase-shifting interferometry using optical flow.

Fringe patterns with a multiplicative phase shift among them appear in experimental techniques as photoelasticity and RGB shadow moiré, among others. These patterns cannot be processed using standard phase-shifting demodulation techniques. In this work, we propose to use a multiframe regularized optical flow algorithm to obtain the interesting modulating phase. The proposed technique has been applied to simulated and experimental interferograms obtaining satisfactory results.

Shack-Hartmann centroid detection using the spiral phase transform.

We present a Shack-Hartmann (SH) centroid detection algorithm capable to measure in presence of strong noise, background illumination and spot modulating signals, which are typical limiting factors of traditional centroid detection algorithms. The proposed method is based on performing a normalization of the SH pattern using the spiral phase transform method and Fourier filtering. The spot centroids are then obtained using global thresholding and weighted average methods. We have tested the algorithm with simulations and experimental data obtaining satisfactory results. A complete MATLAB package that can reproduce all the results can be downloaded from [http://goo.gl/o2JhD].

Phase-shifting interferometry corrupted by white and non-white additive noise.

The standard tool to estimate the phase of a sequence of phase-shifted interferograms is the Phase Shifting Algorithm (PSA). The performance of PSAs to a sequence of interferograms corrupted by non-white additive noise has not been reported before. In this paper we use the Frequency Transfer Function (FTF) of a PSA to generalize previous white additive noise analysis to non-white additive noisy interferograms. That is, we find the ensemble average and the variance of the estimated phase in a general PSA when interferograms corrupted by non-white additive noise are available. Moreover, for the special case of additive white-noise, and using the Parseval's theorem, we show (for the first time in the PSA literature) a useful relationship of the PSA's noise robustness; in terms of its FTF spectrum, and in terms of its coefficients. In other words, we find the PSA's estimated phase variance, in the spectral space as well as in the PSA's coefficients space.

Harmonics rejection in pixelated interferograms using spatio-temporal demodulation.

Pixelated phase-mask interferograms have become an industry standard in spatial phase-shifting interferometry. These pixelated interferograms allow full wavefront encoding using a single interferogram. This allows the study of fast dynamic events in hostile mechanical environments. Recently an error-free demodulation method for ideal pixelated interferograms was proposed. However, non-ideal conditions in interferometry may arise due to non-linear response of the CCD camera, multiple light paths in the interferometer, etc. These conditions generate non-sinusoidal fringes containing harmonics which degrade the phase estimation. Here we show that two-dimensional Fourier demodulation of pixelated interferograms rejects most harmonics except the complex ones at {-3(rd), +5(th), -7(th), +9(th), -11(th),…}. We propose temporal phase-shifting to remove these remaining harmonics. In particular, a 2-step phase-shifting algorithm is used to eliminate the -3(rd) and +5(th) complex harmonics, while a 3-step one is used to remove the -3(rd), +5<(th), -7(th) and +9(th) complex harmonics.

Design of phase-shifting algorithms by fine-tuning spectral shaping.

To estimate the modulating wavefront of an interferogram in Phase Shifting Interferometry (PSI) one frequently uses a Phase Shifting Algorithm (PSA). All PSAs take as input N phase-shifted interferometric measures, and give an estimation of their modulating phase. The first and best known PSA designed explicitly to reduce a systematic error source (detuning) was the 5-steps, Schwider-Hariharan (SH-PSA) PSA. Since then, dozens of PSAs have been published, designed to reduce specific data error sources on the demodulated phase. In Electrical Engineering the Frequency Transfer Function (FTF) of their linear filters is their standard design tool. Recently the FTF is also being used to design PSAs. In this paper we propose a technique for designing PSAs by fine-tuning the few spectral zeroes of a PSA to approximate a template FTF spectrum. The PSA's spectral zeroes are moved (tuned) while gauging the plot changes on the resulting FTF's magnitude.

Two-step self-tuning phase-shifting interferometry.

A two-step self-tuning phase-shifting method is presented. The phase-step between the two interferograms is not known when the experiment is performed. Our demodulating method finds, in a robust way, this unknown phase-step. Once the phase-step is estimated we proceed to phase demodulate the interferograms. Moreover our method only requires the fringe patterns to have a constant unknown phase-shift between them. As a consequence, this technique can be used to demodulate open and closed-fringed patterns without phase-sign ambiguity. The method may be regarded as a self-tuning quadrature filter, which determines the phase-shift between the two fringe patterns and finally estimates the demodulated phase map. The proposed technique has been tested with simulated and real interferograms obtaining satisfactory results.

Two-step interferometry by a regularized optical flow algorithm.

A two-step phase-shifting method, that can demodulate open- and closed-fringed patterns without local sign ambiguity is presented. The proposed method only requires a constant phase-shift between the two interferograms. This phase-shift does not need to be known and can take any value inside the range (0, 2π), excluding the singular case where it corresponds to π. The proposed method is based on determining first the fringe direction map by a regularized optical flow algorithm. After that, we apply the spiral phase transform (SPT) to one of the fringe patterns and we determine its quadrature signal using the previously determined direction. The proposed technique has been applied to simulated and experimental interferograms obtaining satisfactory results. A complete MATLAB software package is provided in [http://goo.gl/Snnz7].

Error-free demodulation of pixelated carrier frequency interferograms.

Recently, pixelated spatial carrier interferograms have been used in optical metrology and are an industry standard nowadays. The main feature of these interferometers is that each pixel over the video camera may be phase-modulated by any (however fixed) desired angle within [0,2pi] radians. The phase at each pixel is shifted without cross-talking from their immediate neighborhoods. This has opened new possibilities for experimental spatial wavefront modulation not dreamed before, because we are no longer constrained to introduce a spatial-carrier using a tilted plane. Any useful mathematical model to phase-modulate the testing wavefront in a pixel-wise basis can be used. However we are nowadays faced with the problem that these pixelated interferograms have not been correctly demodulated to obtain an error-free (exact) wavefront estimation. The purpose of this paper is to offer the general theory that allows one to demodulate, in an exact way, pixelated spatial-carrier interferograms modulated by any thinkable two-dimensional phase carrier.

Fourier transform demodulation of pixelated phase-masked interferograms.

Recently a new type of spatial phase shifting interferometer was proposed that uses a phase-mask over the camera's pixels. This new interferometer allows one to phase modulate each pixel independently by setting the angle of a linear polarizer built in contact over the camera's CCD. In this way neighbor pixels may have any desired (however fixed) phase shift without cross taking. The standard manufacturing of these interferometers uses a 2x2 array with phase-shifts of 0, pi/2, pi, and 3 pi/2 radians. This 2x2 array is tiled all over the video camera's CCD. In this paper we propose a new way to phase demodulate these phase-masked interferograms using the squeezing phase-shifting technique. A notable advantage of this squeezing technique is that it allows one the use of Fourier interferometry wiping out the detuning error that most phase shifting algorithms suffers. Finally we suggest the use of an alternative phase-mask to phase modulate the camera's pixels using a linear spatial carrier along a given axis.

The general theory of phase shifting algorithms.

We have been reporting several new techniques of analysis and synthesis applied to Phase Shifting Interferometry (PSI). These works are based upon the Frequency Transfer Function (FTF) and how this new tool of analysis and synthesis in PSI may be applied to obtain very general results, among them; rotational invariant spectrum; complex PSI algorithms synthesis based on simpler first and second order quadrature filters; more accurate formulae for estimating the detuning error; output-power phase noise estimation. We have made our cases exposing these aspects of PSI separately. Now in the light of a better understanding provided by our past works we present and expand in a more coherent and holistic way the general theory of PSI algorithms. We are also providing herein new material not reported before. These new results are on; a well defined way to combine PSI algorithms and recursive linear PSI algorithms to obtain resonant quadrature filters.

Spectral analysis of phase shifting algorithms.

Systematic spectral analysis of Phase Shifting Interferometry (PSI) algorithms was first proposed in 1990 by Freischlad and Koliopoulos (F&K). This analysis was proposed with the intention that "in a glance" the main properties of the PSI algorithms would be highlighted. However a major drawback of the F&K spectral analysis is that it changes when the PSI algorithm is rotated or its reference signal is time-shifted. In other words, the F&K spectral plot is different when the PSI algorithm is rotated or its reference is time-shifted. However, it is well known that these simple operations do not alter the basic phase demodulation properties of PSI algorithms, except for an unimportant piston. Here we propose a new way to analyze the spectra of PSI algorithms which is invariant to rotation and/or reference time-shift among other advantages over the nowadays standard PSI spectral analysis by F&K.

Phasorial analysis of detuning error in temporal phase shifting algorithms.

Phase error analysis in Temporal Phase Shifting (TPS) algorithms due to frequency detuning has been to date only performed numerically. In this paper, we show an exact analytical expression to obtain this phase error due to detuning using the spectral TPS response. The new proposed method is based on the phasorial representation of the output of the TPS quadrature filter. Doing this, the detuning problem is reduced to a ratio of two symmetrical spectral responses of the quadrature filter at the detuned frequency. Finally, some popular cases of TPS algorithms are analyzed to show the usefulness of the proposed method.

Noise in phase shifting interferometry.

We present a theoretical analysis to estimate the amount of phase noise due to noisy interferograms in Phase Shifting Interferometry (PSI). We also analyze the fact that linear filtering transforms corrupting multiplicative noise in Electronic Speckle Pattern Interferometry (ESPI) into fringes corrupted by additive gaussian noise. This fact allow us to obtain a formula to estimate the standard deviation of the noisy demodulated phase as a function of the spectral response of the preprocessing spatial filtering combined with the PSI algorithm used. This phase noise power formula is the main result of this contribution.

Spatial carrier interferometry from M temporal phase shifted interferograms: Squeezing Interferometry.

It is well known that having 3 temporal phase shifting (PS) interferograms we do not have many possibilities of using an algorithm with a desired frequency spectrum, detuning, and harmonic robustness. This imposes severe restrictions on the possibilities to demodulate such set of temporal interferograms. It would be nice to apply for example a 7 step PS algorithm to these 3 images in order to have more possibilities to phase demodulate them; even further, it would be even better to apply a quadrature filter having a spatial spread given by a real number to these 3 interferograms. In this paper we propose to do just that; namely we show how to demodulate a set of M-steps phase shifting images with a quadrature filter having a real-number as spatial spread. The interesting thing in this paper is to use a higher than M spread quadrature filter to demodulate our interferograms; in traditional PS interferometry one is stuck to the use of M step phase shifting formula to obtain the searched phase. Using a less than M PS formula is not interesting at all given that we would not use all the available information. The main idea behind the "squeezing" phase shifting method is to re-arrange the information of the M phase shifted fringe patterns in such a way to obtain a single carrier frequency interferogram (a spatio-temporal fringe image) and use any two dimensional quadrature filter to demodulate it. In particular we propose the use of Gabor quadrature filters with a spread given by real-numbers along the spatial coordinates. The Gabor filter may be designed in such way that we may squeeze the frequency response of the filter along any desired spatio-temporal dimension, and obtain better signal to noise demodulation ratio, and better harmonic rejection on the estimated phase.