Quasi-Talbot effect of a grating in the deep Fresnel diffraction region

Based on the theory of scalar diffraction, the diffraction of gratings in the deep Fresnel diffraction region is developed, and the general formula of the diffraction intensity of the one-dimensional grating is presented by using the Hankel function. Through numerical calculations, some interesting diffraction phenomena are found. In the deep Fresnel diffraction region, the dominant effects, with increasing propagation distance from the grating, are, in order, the geometrical effect, the quasi-geometrical effect, and the interference and diffraction effects. Furthermore, the diffraction intensities vary periodically in the diffraction effect region with increasing propagation distance. Quasi-Talbot imaging of the grating exists in the interference and diffraction regions, and the intensity distributions most similar to the structure of the grating are not at the exact Talbot distances. These phenomena in the deep Fresnel diffraction region are distinct from those in the Fresnel diffraction region. The formation origin of quasi-Talbot imaging of the grating is also discussed, and the numerical calculations powerfully verify the theoretical results.     

泰伯衍射光场的三维分布特征

采用菲涅尔近似公式和傅里叶级数展开的方法，分析了罗奇光栅、正统强度光栅、正统振幅光栅的泰伯衍射光场的三维强度分布特征，给出了计算机模拟的三维强度分布图，并讨论了一些在三维传感领域有应用前景的性质。     

Uniform scattering patterns from grating-diffuser cascades for display applications

A cascade of a thick grating and a thin diffuser is shown to scatter radiation efficiently and uniformly over a wide angle. Cascading the grating with a diffuser causes the single-beam power spectrum of the diffuser to be replicated at each diffraction angle of the grating. The grating period is chosen so that the first diffraction order falls near the one-half point of the power-spectrum peak of the diffuser. The relative strengths of the diffraction orders are optimized to obtain uniformity of the resulting intensity distribution in the plane of the diffraction orders. The intensity distribution in the perpendicular plane is governed solely by the diffuser. Such a cylindrical system is considered on the basis of the requirements for projection TV's of a large horizontal span (100 degrees ) and a narrower vertical span (~15 degrees ). Broadband illumination is studied by consideration of three simultaneously illuminating wavelengths. Experimental results are given for a cascade of a grating formed in photoresist and an etched-glass diffuser.     

Talbot effect by a photorefractive volume phase grating

In our proposal a light intensity distribution generated by an incoherently illuminated planar amplitude grating is projected into a photorefractive crystal. This 3D distribution is mapped as an index refractive perturbation via the photorefractive effect thereby generating a volume phase grating. The self-imaging phenomenon in the Fresnel field of this volume phase grating coherently illuminated is theoretically and experimentally analyzed. A model to simulate this volume grating that considers the 3D light intensity distribution formed in the crystal combined with the photorefractive grating formation theory is proposed. A path-integral approach to calculate the self-image patterns which account for the inhomogeneous propagation through the photorefractive grating is employed. The experimental and theoretical results show that the self-images location coincides with that of the self-images generated by planar phase grating of the same period. Moreover, the self-images visibility depends on three parameters: the exit pupil diameter of the incoherent recording optical system, the external electric field applied on the crystal, and the crystal thickness. To study the visibility behavior, a phase parameter which includes the three mentioned parameters is proposed. The self-images visibility shows the typical sinusoidal dependence found in planar phase grating. A good agreement between theoretical and experimental results is observed.     

Color separation of high-density dielectric rectangular grating in the Fresnel diffraction region

A high-density dielectric rectangular grating is designed for color separation in a Fresnel diffraction field. The Fresnel field distribution is analyzed and the optimization conditions for color separation are given. The process of the modes propagating and energy exchanging with the diffraction orders are expressed by modal method. The color separation for different polarizations can be realized. The energy efficiency is 96.3% at the 633 nm wavelength and 86.9% at the 488 mm wavelength for both TE polarizations, while the energy efficiency is theoretically 96.3% at the 633 nm wavelength for TE polarization and 90.6% at the 488 nm wavelength for TM polarization. The field distributions are scanned by the near-field scanning optical microscopy, and the efficiency is 71.2% for the 633 nm wavelength and 67.3% for the 488 nm wavelength for both TE polarizations experimentally.     

Computer Generated Diffractive Multi-focal Lens

The method has been proposed for computing Fresnel-type multi-focal lenses on the basis of special-type phase nonlinearity. A multi-focal lens is represented as a mathematical superposition of a thin lens and nonlinearity distorted Fresnel lens. Selection of the nonlinearity type is reduced to the problem of the groove form determination for the phase diffraction grating with pre-set energy distribution between orders. In particular, bi-focal lens and seven-focal lens have been investigated.     

微透镜列阵衍射效应所致的串扰

本文分析了微透镜列阵衍射效应的影响因素,推导出了微透镜焦平面上光强分布的解析表达式,对菲涅尔数评价衍射效应的物理含义给予了合理的解释.并利用ZEMAX软件对微透镜列阵进行仿真,基于惠更斯子波直接积分的算法计算得到了微透镜列阵焦平面上的光场强度分布.通过比较不同条件下所得到的计算结果,验证了以菲涅尔数作为微透镜列阵衍射效应评价依据的的合理性,同时验证了以菲涅尔数判断焦斑间串扰的可行性.     

Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings

Expressions describing the vortex beams that are generated by the process of Fresnel diffraction of a Gaussian beam incident out of waist on fork-shaped gratings of arbitrary integer charge p, and vortex spots in the case of Fraunhofer diffraction by these gratings, are deduced. The common general transmission function of the gratings is defined and specialized for the cases of amplitude holograms, binary amplitude gratings, and their phase versions. Optical vortex beams, or carriers of phase singularity with charges mp and -mp, are the higher negative and positive diffraction-order beams. The radial part of their wave amplitudes is described by the product of the mpth-order Gauss-doughnut function and a Kummer function, or by the first-order Gauss-doughnut function and the difference of two modified Bessel functions whose orders do not match the singularity charge value. The wave amplitude and the intensity distributions are discussed for the near and far fields in the focal plane of a convergent lens, as well as the specialization of the results when the grating charge p=0; i.e., the grating turns from forked into rectilinear. The analytical expressions for the vortex radii are also discussed.     

Computer Reconstruction of a Random Rough Surface in the Presence of Higher Diffraction Orders

Abstract

The profile of a random rough surface (RRS), whose mean roughness R_a is greater than the light wavelength, is visualized by computer processing. The surface is presented as a sum of sinusoidal gratings. The light diffracted from this surface field is registered by a photodiode array. The second and third diffraction orders from each grating are taken into account in computer processing of the diffracted field and the mixing field–the field obtained at the mixing of the reference and the diffraction fields. The criterion for taking into account higher diffraction orders is the asymmetry of the diffraction pattern to the left and to the right relative to the central peak (the field of zero diffraction orders obtained from each grating) The number of the diffraction orders higher than the first is defined from the average intensity distribution between the central peak and the diffraction orders to the left and to the right at arbitrary light wavelength. The surface profile is reconstructed by a computer program and the mean roughness R_a is calculated. The obtained value of R_a is in satisfactory agreement with that measured by the contact pin method.     

Fabrication of switchable liquid crystal devices using surface relief gratings in photopolymer

Holographically recorded surface relief gratings in dry, self-developing acrylamide based photopolymer were used to fabricate two types of switchable liquid crystal (LC) device. One is an electrically switchable LC diffraction grating and the other is an electrically switchable polarization rotator. The electrically switchable diffraction grating was characterized by measuring the dependence of the intensity in the first diffracted order on the applied electric field. The polarization rotator was characterized by studying the influence of the applied electric field on the twist angle and the variation of intensity in the zero and the first orders of diffraction.     

Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate

We deduce and study an analytical expression for Fresnel diffraction of a plane wave by a spiral phase plate (SPP) that imparts an arbitrary-order phase singularity on the light field. Estimates for the optical vortex radius that depends on the singularity's integer order n (also termed topological charge, or order of the dislocation) have been derived. The near-zero vortex intensity is shown to be proportional to rho2n, where p is the radial coordinate. Also, an analytical expression for Fresnel diffraction of the Gaussian beam by a SPP with nth-order singularity is analyzed. The far-field intensity distribution is derived. The radius of maximal intensity is shown to depend on the singularity number. The behavior of the Gaussian beam intensity after a SPP with second-order singularity (n = 2) is studied in more detail. The parameters of the light beams generated numerically with the Fresnel transform and via analytical formulas are in good agreement. In addition, the light fields with first- and second-order singularities were generated by a 32-level SPP fabricated on the resist by use of the electron-beam lithography technique.     

Image structure from mirrors with print-through undulations: theoretical analysis

The image structure produced by a periodic hexagonal pattern of mirror surface undulations has been analyzed. Such undulations form a two-dimensional phase grating that can result from the polishing of honeycomb mirrors or, for example, meniscus mirrors with a hexagonal pattern of axial supports. For monochromatic light of wavelength lambda, undulations having uniform peak-to-valley amplitude H ? lambda and period L cause a decrease in the central intensity of the point spread function (PSF), and a fraction, ~13(H/lambda)(2), of the total power is diffracted into an infinite hexagonal array of satellite images. These have angular separations of 2lambda/ radical3L and intensity profiles in the form of perfect diffraction limited PSF's, but with intensities decreasing with increasing diffraction order. The six innermost (first-order) satellites each have central intensities approximately 2(H/lambda)(2) times that of the central image. If the amplitudes of the surface bumps are of random size with a normal frequency distribution, then the intensity of the diffracted orders decreases, and an additional weak structure appears over the image plane; the positions and heights of the peaks in this grasslike structure depend on the particular two-dimensional distribution of the random bumps. When the input is polychromatic, the diffracted orders other than zero give images that are elongated radially and decrease outward in intensity with a 1/lambda(4) dependence.     

Fresnel Images of a Binary Diffraction Grating with Open Ratio 0·5

Abstract

Analysis of the Fresnel image of the binary-amplitude diffraction grating with open ratio 0·5 is presented. Fresnel images situated at the distance z = Nd²/8λ (N = 1,2…) from the grating which was illuminated by a plane wave were calculated and investigated by experiment. Forms of intensity distributions in the second grating plane are presented, the second grating cooperating with the self-imaging grating.     

Linear displacement measurement with a grating and speckle pattern illumination

We present a study of the variations of a speckle pattern passing through a grating that can be displaced. This study is described theoretically by a simple model based on the scalar diffraction theory in the Fresnel zone. The intensity correlation of the modified speckle as a function of the grating displacement is obtained and compared with experimental results. The possibilities of metrological applications in optical encoders are suggested.     

Pitch-modulated phase grating and its application to displacement encoder

Abstract

A binary phase grating with modulated pitch is investigated for a simple displacement encoder. The grating consists of a binary phase grating to eliminate the zeroth-order diffraction, and the pitch of the grating is modulated to compensate the higher harmonics of the encoder displacement signal. Therefore, an undistorted sinusoidal signal as a function of displacement is obtained by simply superimposing a conventional binary grating on the pitch-modulated phase gratings for any air gap between the gratings. The characteristics of the proposed gratings and the encoder signal are investigated by the Fresnel diffraction theory. The proposed grating has been fabricated lithographically, and the signal was examined experimentally. Considering these results, the proposed technique can suppress interpolation error and will be useful for an encoder in precision machining.     

Four-level phase grating generated by combination of two binary gratings through coherent imaging

With the aim of steering as much incident light power in the +1 diffraction order as possible and suppressing the power in the -1 order, we generated an effective four-level grating by coherently imaging a 180 degrees binary-phase grating onto a 90 degrees phase grating. The periods of the gratings were well matched. The measured overall power fractions in the +1 and -1 orders were 54% and 2%, respectively.     

Amplitude modulation and beam-steering properties of active binary phase gratings with reconfigurable absorption areas

By combining a binary phase grating and materials with a controllable absorption, it is shown theoretically that it is possible to modulate the light in a zero-order diffraction beam and that a high contrast level for the beam modulation can be obtained. The intensity of higher diffraction orders also changes, but it is calculated that high contrasts cannot be achieved for these higher-order beams with the active gratings that we examine. This specific modulator design that we use can be applied both for transmitted and for reflected light. Using the same ideas, one may build a beam-deflection device by varying the period of the grating by selectively changing the absorption levels in the grating. The deflection efficiency of the device can be improved compared with other designs by use of a grating with a reduced intensity of the nondeflected zero-order beam.     

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.     

Diffraction properties of quasi-random pinhole arrays: suppression of higher orders and background fluctuations

A transmission grating can be formed from quasi-random pinhole arrays. We show here that this type of grating can reduce the higher diffraction orders effectively when an appropriate arrangement is used. A simple but accurate analytical model is provided to describe the physical mechanism that is responsible for suppression of the higher orders, and this model is verified through numerical simulations. In particular, the background intensity enhancement and the fluctuations in this intensity that are introduced by the quasi-random arrangement of pinholes are also assessed. The consistent one-to-one match produced between the diffraction directions and the wavelengths means that this type of grating has considerable potential for use in spectroscopic measurement and monochromator applications.     

Theoretical Analysis of the Lau Effect with a Sinusoidal Phase Grating

Abstract

By means of both partial coherence theory and diffraction theory, the Lau effect with a sinusoidal phase grating in a double-grating configuration is analysed. The optical intensity distribution at the observation plane is obtained. This distribution is related to the modulation of the phase grating and the separation of the two gratings. The results are somewhat different from the case of double amplitude gratings.