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.

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.

Diffraction by metallic planar gratings.

In this work, a kind of grating that, to our knowledge, has not yet been analyzed for diffractive purposes is proposed. The mentioned grating consists of metallic intercalated slits of two different metals on a glass substrate. The main characteristic and peculiarity of the proposed grating is that it is totally planar, without any slopes or grooves. We analyze the intensity distribution at the near- and far-field produced by the grating. The method used is rigorous-coupled wave analysis. We show how the metallic layer thickness is a crucial parameter to achieve the highest efficiency of the diffraction orders and, therefore, the highest contrast of the diffracted fringes. To conclude, we investigate how parameters such as the period, duty cycle, wavelength, or the used metals affect the diffracted field. Some nonexpected behaviors have been found. As we demonstrate by comparing with other kinds of gratings, the proposed grating would be useful in applications in which fringes are needed in both the front and back sides of the grating.

Phase-shifting Zernike phase contrast microscopy for quantitative phase measurement.

Zernike phase contrast microscopy is extended and combined with a phase-shifting mechanism to perform quantitative phase measurements of microscopic objects. Dozens of discrete point light sources on a ring are constructed for illumination. For each point light source, three different levels of point-like phase steps are designed, which are alternatively located along a ring on a silica plate to perform phase retardation on the undiffracted (dc) component of the object waves. These three levels of the phase steps are respectively selected by rotating the silica plate. Thus, quantitative evaluation of phase specimens can be performed via phase-shifting mechanism. The proposed method has low "halo" and "shade-off" effects, low coherent noise level, and high lateral resolution due to the improved illumination scheme.

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.

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.

Effect of fabrication errors on the diffraction pattern produced by sawtooth gratings.

In this work we investigate, analytically and numerically, the effect on the diffracted field produced by typical fabrication errors in sawtooth gratings. The analysis is carried out for the near and far field, showing the effects on the intensity and on the diffraction orders efficiency. When the grating profile is not perfect but presents a curved profile or overdevelopment error, some different diffraction orders appear, changing the intensity and the efficiency of each order. In addition, when roughness is present, a decreasing of efficiency is produced, but without generating different diffraction orders than the first one. We show the analytical dependence of these modifications in terms of the profile of the grating, corroborating the results with numerical methods.

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.

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.

Percutaneous tibial nerve stimulation in treatment of overactive bladder: when should retreatment be started?

OBJECTIVE: To study the response to posterior tibial nerve stimulation in patients with overactive bladder refractory to medical treatment. METHODS: A cohort of 53 patients were treated by posterior tibial nerve stimulation and followed up for a maximum of 24 months. All patients completed the International Consultation on Incontinence Modular Questionnaire-Short Form quality of life questionnaire and kept a urination diary to record the daytime urination frequency and night-time urination frequency. Urodynamic studies were also conducted. RESULTS: At 6 months of follow-up, a cure/improvement rate of 92.4% (49 of 53 cases) had been achieved. Ten patients were given additional treatment and were excluded from subsequent follow-up analysis. At 12 months of follow-up, a cure/improvement rate of 91.69% had been achieved (39 of 43). At 24 months of follow-up, of the 16 patients initially included during the first year, a cure/improvement rate of 62.5% had been achieved (10 of 16). The first sensation of bladder filling had increased by the end of treatment, with differences observed before and after posterior tibial nerve stimulation (P ≤ .001). The average post-treatment bladder capacity had increased by 72.7 mL compared with the initial value (P ≤ .001). At 24 months of follow-up, the group of 16 patients evaluated recorded a significant worsening of night-time urination frequency (P ≤ .05) and quality of life (P ≤ .01). CONCLUSION: Posterior tibial nerve stimulation is a good option for the treatment of overactive bladder. In our series, the optimal point to start retreatment would be at 24 months after therapy completion.

Self-imaging of gratings with rough strips.

We analyze the self-imaging process produced by a transmission grating whose strips present two different roughness levels. This kind of grating periodically modulates the transmitted light owing only to the different microtopographic properties of the strips. In spite of the fact that the grating is not purely periodic, it produces a kind of self-image at Talbot distances. These self-images gradually appear as light propagates, but they are not present just after the grating, as occurs in amplitude or phase gratings. There exists a distance from the grating, which depends on the stochastic properties of roughness, from which the contrast of the self-images becomes stable. Important cases are analyzed in detail, such as low- and high-roughness limits. We assume for the calculations that the grating can be used in a mobile system. Simulations using the Rayleigh-Sommerfeld regime have been performed, which confirm the validity of the theoretical approach proposed in this work.

Far field of gratings with rough strips.

In this work, we analyze the far-field pattern produced by a grating made of strips with two different random roughness levels. The efficiency and shape of the diffraction orders is obtained, which are shown to depend on the statistical properties of roughness. We assume for the calculations that the grating can be used in a mobile mechanical system. A preliminary experimental approach which partially corroborates the theoretical results is also performed.

Talbot effect with rough reflection gratings.

The Talbot effect is analyzed when steel tape gratings are used. These gratings are made on a steel substrate, and, because of the manufacture process, both levels of the grating are rough with different roughness parameters. A theoretical analysis based on Fresnel regime, which considers the statistical properties of roughness, is developed. Analytical formulas that show a decreasing exponential dependence on the intensity in terms of the distance between the grating and the observation plane are obtained, and an experimental verification is also performed.

Variogram-based method for contrast measurement.

We present a technique for determining the contrast of an intensity distribution in the presence of additive noise and other effects, such as undesired local amplitude or offset variations. The method is based on the variogram function. It just requires the measurement of the variogram at only four points and, as a consequence, it is very fast. The proposed technique is compared with other standard techniques, showing a reduction in the error of the contrast measurement.