Far field of binary phase gratings with errors in the height of the strips

Diffraction gratings are not always ideal but, due to the fabrication process, several errors can be produced. In this work we show that when the strips of a binary phase diffraction grating present certain randomness in their height, the intensity of the diffraction orders varies with respect to that obtained with a perfect grating. To show this, we perform an analysis of the mutual coherence function and then, the intensity distribution at the far field is obtained. In addition to the far field diffraction orders, a "halo" that surrounds the diffraction order is found, which is due to the randomness of the strips height

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

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

Far field diffraction of gratings with two roughness levels

We investigate the far field diffraction process produced by diffraction gratings with two roughness levels, such as steel tape gratings, which are used in environments where diffraction gratings made of glass are not possible. We obtain the analytical formulation describing the formation of diffraction orders in terms of the roughness parameters. Since roughness is stochastic, we use the mutual coherence function to propagate the field to the Fraunhofer regime. We conclude that diffraction orders exists when at least one correlation length of both roughness is larger than the period of the grating and the standard deviation on heights is around or larger than the illumination wavelength. Finally, we corroborate the analytical results with numerical simulations based on the fast Fourier transform, showing a high agreement

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 wor

Single-focus binary Fresnel zone plate

In this work, we propose and analyze a novel kind of binary Fresnel zone plate with single focus. It consists of a Fresnel zone plate whose zones have rough edges. We give analytical results for the intensity along the optical axis and demonstrate that lateral roughness of the zones produces the disappearance of secondary foci as a blurring of the edges. Besides, we corroborate its behavior by numerical simulations and experiments. This kind of Fresnel zone plate can be useful in a wide range of photonic applications, even for focusing with soft and hard X-rays or extreme ultraviolet radiation

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

Near field diffraction of cylindrical convex gratings

We analyze the field produced by a cylindrical convex diffraction grating at the Fresnel regime for several kinds of light sources, including a monochromatic quasipunctual source, finite size, and polychromatic sources. These results can help one understand the functioning of rotary optical encoder technology. A decrease in the self-image contrast is produced for finite nonpunctual sources. In addition, the polychromaticity of the source affects the smoothness of the self-images, making them quasicontinuous from a certain distance from the grating forward. Finally, we experimentally validate the obtained analytical predictions

Correlation technique for the compensation of diffraction widening of optical reference signals

Two-grating measurement systems are routinely employed for high-resolution measurements of angular and linear displacement. Usually, these systems incorporate zero reference codes (ZRCs) to obtain a zero reference signal (ZRS), which is used as a stage-homing signal. This signal provides absolute information of the position to the otherwise relative information provided by the two-grating incremental subsystems. A zero reference signal is commonly obtained illuminating the superposition of two identical pseudorandom codes and registering the transmitted light by means of a photodiode. To increase the resolution of the system, a reduction of the grating period and the ZRC widths is required. Due to this reduction, the diffractive effects produce a widening of the ZRS and, in turn, a loss of the measuring accuracy. In this work, we propose a method to narrow the distorted signal obtained with a Lau-based encoder, reinstating the accuracy of the ZRS. The method consists of the inclusion of a correlation mask on the detector. A theoretical model to design the mask has been developed, and experimental results have been obtained that validate the proposed technique