Collimation technique and testing applied to finite size polychromatic sources.

Highly collimated beams are required in numerous applications and techniques. Different methods have been proposed for collimating monochromatic point light sources during the recent years. In this work, we analyze how a finite size and polychromatic light source can be collimated using only one diffraction grating and a CMOS camera placed after the source and the collimating lens. For this, we determine the period of the fringes diffracted by the grating and compare it with the period of the grating. Analytical equations are obtained to predict the amplitude of the fringes and their period. Since self-images disappear for finite size polychromatic sources at long distances from the grating, the period has to be measured close to the grating. In addition, we give an analytical equation to determine the error in the positioning of the source in terms of the source size and the setup parameters. Finally, we experimentally corroborate the obtained analytical formalism using a white LED of size s=0.6 mm collimated by a lens with focal length f=25 mm, and a Ronchi binary grating of period d=250 µm. In this case, we achieve an experimental error in the positioning of the source with respect to the focal plane of the lens of δzexp=92 µm.

Diffraction by gratings with random fill factor.

In this work, we analyze the diffraction produced by Ronchi gratings where the fill factor is not constant, but presents random fluctuations around its nominal value. This effect can be produced while developing the grating with etchers since the process can be slightly unpredictable. We obtain the theoretical formalism to describe the intensity produced by the grating at near and far field, showing that smoothing of the self-images is produced at the near field and, consequently, cancellation of higher diffraction orders is obtained at the far field. In addition, different nominal fill factors produce different diffraction behaviors in terms of the randomness. We corroborate the analytical formalism using a direct integration method based on the Rayleigh-Sommerfeld formula and conclude that the numerical results are in high agreement with the theoretical predictions.

Near-field diffraction of chirped gratings.

In this Letter, we analyze the near-field diffraction pattern produced by chirped gratings. An intuitive analytical interpretation of the generated diffraction orders is proposed. Several interesting properties of the near-field diffraction pattern can be determined, such as the period of the fringes and its visibility. Diffraction orders present different widths and also, some of them present focusing properties. The width, location, and depth of focus of the converging diffraction orders are also determined. The analytical expressions are compared to numerical simulation and experimental results, showing a high agreement.

Diffraction by random Ronchi gratings.

In this work, we obtain analytical expressions for the near-and far-field diffraction of random Ronchi diffraction gratings where the slits of the grating are randomly displaced around their periodical positions. We theoretically show that the effect of randomness in the position of the slits of the grating produces a decrease of the contrast and even disappearance of the self-images for high randomness level at the near field. On the other hand, it cancels high-order harmonics in far field, resulting in only a few central diffraction orders. Numerical simulations by means of the Rayleigh-Sommerfeld diffraction formula are performed in order to corroborate the analytical results. These results are of interest for industrial and technological applications where manufacture errors need to be considered.

Design of superachromatic quarter-wave retarders in a broad spectral range.

A superachromatic quarter-wave retarder using an arbitrary number of waveplates in a broadband spectral range has been proposed. Their design is based on the optimization of a merit function, the achromatism degree (AcD), which represents a global behavior metric for the retardation. By means of this technique, the thickness and azimuth of each waveplate is determined. The achromatism degree is a measure of the distance between the overall retardation and a target retardation weighted by the spectrum of the incident light. We report on a particular case where all waveplates are made of quartz. As application examples, the design of a quarter-wave retarder using two, three, and four waveplates in the spectral ranges of 500-700 nm and 400-1000 nm was studied. The numerical results show that for these ranges, the best designs obtained present a maximum difference of 0.013° and 0.010° with respect to the target retardation, respectively. In addition, an analysis of their achromatic stability is presented. These results can be applied in the aerospace industry, spectroscopic ellipsometry, and spectrogoniometry, among others.

Self-imaging technique for beam collimation.

A simple collimation technique based on measuring the period of one self-image produced by a diffraction grating is proposed. Transversal displacement of the grating is not required, and then automatic single-frame processing can be performed. The self-image is acquired with a CMOS camera, and the period is computed using the variogram function. Analytical and experimental results are obtained, which show the simplicity and accuracy of the proposed technique.

Circularly polarized light with high degree of circularity and low azimuthal error sensitivity.

The generation of circularly polarized light with a high circularity degree and low azimuthal error sensitivity was analyzed using a system composed by two waveplates. It is shown how the high circularity degree is achieved using a combination of a half- (λ/2) and a quarter- (λ/4) waveplate λ/2+λ/4 configuration. However, the lowest azimuthal sensitivity under small variations in the azimuths of the waveplates is obtained by employing a λ/4+λ/2 configuration. Analytical calculus particularized for quartz and MgF2 waveplates is presented.

Optimal achromatic wave retarders using two birefringent wave plates.

Two plates of different birefringence material can be combined to obtain an achromatic wave retarder. In this work, we achieve a correction for the overall retardation of the system that extends the relation to any azimuth. Current techniques for the design of achromatic wave retarders do not present a parameter that characterizes its achromatism on a range of wavelengths. Thus, an achromatic degree has been introduced, in order to determine the optimal achromatic design composed with retarder plates for a spectrum of incident light. In particular, we have optimized a quarter retarder using two wave plates for the visible spectrum. Our technique has been compared to previous results, showing significant improvement.

Optimal achromatic wave retarders using two birefringent wave plates: reply.

This reply attempts to cast some more light on the achromatic systems composed by wave plates, in particular to the calculus of the overall retardation and the use of the Jones matrix equivalence theorem. An equivalent expression for the overall retardation of the system in terms of the trace is also given.

Gaussian-Schell-model beams propagating through rough gratings.

In this work we analyze the near-field intensity distribution produced by a rough grating illuminated with a Gaussian-Schell-model beam. This kind of grating is formed by rough and smooth slits. Statistical techniques are used to describe the grating, and the Fresnel approach is used to perform the propagation of light. Two kinds of coherence affect the light propagation. One of them comes from the light beam, since it is not totally coherent. The other one comes from the rough topography of the grating surface. We have found that the Talbot effect is not present just after the grating, but it gradually increases. In addition, the contrast of the self-images decreases from a certain distance due to the coherence properties of the illumination beam. Then, the self-imaging process is only present between two specific distances from the grating. To corroborate the analytical results, we have performed numerical simulations for the mean intensity distribution based on the Sommerfeld-Rayleigh approach, showing their validity.

Double diffractive optical element system for near-field shaping.

Iterative algorithms based on Fourier transform are used for the design of diffractive optical elements (DOEs), which produce a given intensity distribution, usually at the far field. For the near field, these algorithms can also be used by changing the Fourier transform for the Fresnel transform. However, when the distance between the DOE and the observation plane is short, the results obtained with this modification are not always valid. In the present work, we develop a technique for obtaining the desired intensity distribution in the near field using two DOEs in tandem. We have designed an algorithm based on the standard Gerchberg-Saxton algorithm to determine the modulation of the two DOEs. The best results are obtained when the first DOE modulates the amplitude and the second DOE modulates the phase.

Near-field diffraction of gratings with surface defects.

Diffraction gratings produce self-images in the near field. Defects on the surface of the grating may occur due to the manufacturing process. These devices are often placed in dirty industrial environments. Dust particles or drops of liquid can be deposited over their surface. In this work, we analyze the effect of surface defects placed over the grating on the self-imaging process. We analytically show how the self-images gradually recover as we separate from the grating when one defect is present. Also a random distribution of surface defects over the grating is analyzed. In particular, we focus on how the contrast of the self-images decreases in terms of the density of the defects. Analytical expressions for the near field are derived, considering a stochastic description of the spatial distribution of defects. In addition, numerical simulations based on the Rayleigh-Sommerfeld formulation are performed to validate the analytical results.

Rough Fresnel zone plates over metallic surfaces.

We analyze the focusing properties of Fresnel zone plates fabricated over steel tapes using laser ablation. Our intention is to implement the use of micro-optical elements when the use of conventional chrome-glass elements is not indicated. Because of the manufacture process, the surface presents a certain anisotropic roughness, which reduces the focusing properties. First, we develop numerical simulations by means of the Rayleigh-Sommerfeld approach, showing how roughness in both levels of the Fresnel zone plate affects the focalization of the lens. We also manufacture Fresnel zone plates over steel tape, and perform experimental verification that corroborates the numerical results.

Collimation method using a double grating system.

We present a collimation technique based on a double grating system to locate with high accuracy an emitter in the focal plane of a lens. Talbot self-images are projected onto the second grating producing moiré interferences. By means of two photodetectors positioned just behind the second grating, it is possible to determine the optimal position of the light source for collimation by measuring the phase shift between the signals over the two photodetectors. We obtain mathematical expressions of the signal in terms of defocus. This allows us to perform an automated technique for collimation. In addition, a simple and accurate visual criterion for collimating a light source using a lens is proposed. Experimental results that corroborate the proposed technique are also presented.

Binary gratings with random heights.

We analyze the far-field intensity distribution of binary phase gratings whose strips present certain randomness in their height. A statistical analysis based on the mutual coherence function is done in the plane just after the grating. Then, the mutual coherence function is propagated to the far field and the intensity distribution is obtained. Generally, the intensity of the diffraction orders decreases in comparison to that of the ideal perfect grating. Several important limit cases, such as low- and high-randomness perturbed gratings, are analyzed. In the high-randomness limit, the phase grating is equivalent to an amplitude grating plus a "halo." Although these structures are not purely periodic, they behave approximately as a diffraction grating.

Continuous self-imaging regime with a double-grating mask.

We analyze the Talbot effect produced by a mask composed of two diffraction gratings. Combinations with phase and amplitude gratings have been studied in the near-field regime. For a two-phase-gratings configuration, the Talbot effect is canceled, even when using monochromatic light; that is, the intensity distribution is nearly independent of the distance from the mask to the observation plane. Therefore, the mechanical tolerances of devices that use the Talbot effect may be improved. In addition, the spatial frequency of the fringes is quadrupled, which improves the accuracy of devices that employ this mask. An experimental verification for the best case two phase gratings, has also been performed, validating the theoretical results.

Self-images location of amplitude/phase binary gratings.

We analyze the near-field behavior of binary amplitude/phase diffraction gratings, which modulate at the same time the amplitude and phase of the incident light beam. As it is expected, the distance between two consecutive self-images of the grating depends only on the period of the grating and the wavelength of the illumination. However, the location of the self-images depends on the specific properties of the grating. In this work, we analyze the location of the self-images in terms of the Fourier coefficients of the grating, obtaining analytical expressions. This analysis can be useful in applications in which the position of the self-images must be at certain fixed distances from the grating. Finally, an experimental and numerical verification of the proposed theory is performed.

Numerical model of the inhomogeneous scattering by the human lens.

We present in this work a numerical model for characterizing the scattering properties of the human lens. After analyzing the scattering properties of two main scattering particles actually described in the literature through FEM (finite element method) simulations, we have modified a Monte Carlo's bulk scattering algorithm for computing ray scattering in non-sequential ray tracing. We have implemented this ray scattering algorithm in a layered model of the human lens in order to calculate the scattering properties of the whole lens. We have tested our algorithm by simulating the classic experiment carried out by Van der Berg et al for measuring "in vitro" the angular distribution of forward scattered light by the human lens. The results show the ability of our model to simulate accurately the scattering properties of the human lens.

Diffraction of gratings with rough edges.

We analyze the far field and near field diffraction pattern produced by an amplitude grating whose strips present rough edges. Due to the stochastic nature of the edges a statistical approach is performed. The grating with rough edges is not purely periodic, although it still divides the incident beam in diffracted orders. The intensity of each diffraction order is modified by the statistical properties of the irregular edges and it strongly decreases when roughness increases except for the zero-th diffraction order. This decreasing firstly affects to the higher orders. Then, it is possible to obtain an amplitude binary grating with only diffraction orders -1, 0 and +1. On the other hand, numerical simulations based on Rayleigh-Sommerfeld approach have been used for the case of near field. They show that the edges of the self-images are smoother than the edges of the grating. Finally, we fabricate gratings with rough edges and an experimental verification of the results is performed.

Analytical determination of the uncertainty and the optimum sampling frequency for one-dimensional images with noise.

Kriging is an estimation technique that has been proved useful in image processing since it behaves, under regular sampling, as a convolution. The uncertainty obtained with kriging has also been shown to behave as a convolution for the case of regular sampling. The convolution kernel for the uncertainty exclusively depends on the spatial correlation properties of the image. In this work we obtain, first, analytical expressions for the uncertainty of 1D images with noise using this convolution procedure. Then, we use this uncertainty to propose a new criterion for determining whether a 1D image with noise is correctly sampled.