Diffraction of electromagnetic plane waves by the gratings made by periodically corrugating the exposed planar boundaries of homogeneous, isotropic, linear dielectric--magnetic half--spaces is examined. The phase velocity vector in the diffracting material can be either co-parallel or anti-parallel to the time-averaged Poynting vector, thereby allowing for the material to be classified as of either the positive or the negative negative phase-velocity (PPV or NPV) type. Three methods used for analyzing dielectric gratings - the Rayleigh-hypothesis method, a perturbative approach, and the C formalism - are extended here to encompass NPV gratings by a careful consideration of field representation inside the refracting half--space. Corrugations of both symmetric as well as asymmetric shapes are studied, as also the diversity of grating response to the linear polarization states of the incident plane wave. The replacement of PPV grating by its NPV analog affects only nonspecular diffraction efficiencies when the corrugations are shallow, and the effect on specular diffraction efficiencies intensifies as the corrugations deepen. Whether the type of the refracting material is NPV or PPV is shown to affect surface wave propagation as well as resonant excitation of surface waves.
A description of an algorithm for a rather general modal grating calculation is presented. Arbitrary profiles, depth, and permittivity are allowed. Gratings built up from sub-gratings are allowed, as are coatings on the sidewalls of lines, and arbitrary complex structure. Conical angles and good conductors are supported.
The diffraction of electromagnetic plane waves by a rectangular grating formed by discrete steps in the interface of a homogeneous, isotropic, linear, negative phase--velocity (negative index) material with free space is studied using the semi--analytic C method. When a nonspecular diffracted order is of the propagating type, coupling to that order is significantly larger for a negative index material than for conventional material. The computed coupling strengths reported here are in agreement with recent experiments, and illustrate the role of evanescent fields localized at the grating inter…
We discuss the applicability of holographically recorded gratings in photopolymers and holographic polymer-dispersed liquid crystals as neutron optical elements. An experimental investigation of their properties for light and neutrons with different grating spacings and grating thicknesses is performed. The angular dependencies of the diffraction efficiencies for those gratings are interpreted in terms of a rigourous coupled wave analysis. Starting from the obtained results we work out the lines for the production of an optimised neutron optical diffraction grating, i.e., high diffraction effi…
We derive from first principles the expression for the angular/wavelength distribution of the intensity diffracted by a blazed reflective grating, according to a scalar theory of diffraction. We considered the most common case of a groove profile with rectangular apex. Our derivation correctly identifies the geometric parameters of a blazed reflective grating that determine its diffraction efficiency, and fixes an incorrect but commonly adopted expression in the literature. We compare the predictions of this scalar theory with those resulting from a rigorous vector treatment of diffraction fro…
The optical characterization of dielectric periodic line-gratings for which the period of the grating is significantly larger than the probing wavelength is investigated experimentally, numerically, and analytically. The experimental diffraction efficiencies are analyzed in transmission with the help of a closed-form approximate solution within a single scattering picture. In particular, the angular width of significant scattering and the haze are analyzed for samples of increasing grating profile amplitude.
A diffractive arrangement that allows imaging of an object without any intermediate or complementary element is presented. This optical system with only two diffraction gratings forms color images with white light.
Measurement and analysis of diffraction parameters of thick transmission holographic phase gratings recorded on PHC-488 photopolymer are presented. Precision determination of the spatial grating period is executed by Bragg angle measurement for two precisely known wavelengths in the first diffraction order and also by Bragg angle measurement on one wavelength in the first and second diffraction orders. Determination of actual grating thickness and depths of spatial modulation of a holographic recording, which are responsible for their main optical characteristics, is carried out with the appli…
A two-dimensional (2D) mathematical model of quadratically distorted (QD) grating is established with the principles of Fraunhofer diffraction and Fourier optics. Discrete sampling and bisection algorithm are applied for finding numerical solution of the diffraction pattern of QD grating. This 2D mathematical model allows the precise design of QD grating and improves the optical performance of simultaneous multiplane imaging system.
In the framework of the approximation of slowly varying amplitudes a multiwave dynamical theory of neutron diffraction on a moving phase grating was developed. The influence of the velocity of the grating, its period and height of the slits on the discrete energy spectrum and intensity of various diffraction orders was analyzed.
The increase of cell efficiency resulting from using a diffraction grating as a back reflector is investigated. An enhancement coefficient is introduced as a figure of merit that accounts for the ability of the rear grating to increase the generation of electric carriers. According to results in this paper, a diffraction-grating-like relief on the rear face of the photovoltaic cell may reduce the intensity of the specular reflection while favouring the intensity of high-angle rays. In this way, the path-length of reflected rays increases, and so does the efficiency of the cell.
Periodically patterned columnar thin films (PP-CTFs) were made by evaporating CaF2 and directing the vapor flux obliquely towards lithographically fabricated micrometer/sub-micrometer gratings. The growth of the PP-CTFs was controlled by the deposition rate to form prismatic air cavities within them and they function like blazed diffraction gratings with asymmetric diffraction patterns and diffraction efficiencies upto 52% in transmission at visible wavelengths. Scalar diffraction theory qualitatively explained the measured diffraction efficiencies.
The recent development of phase-grating moiré neutron interferometry promises a wide range of impactful experiments from dark-field imaging of material microstructure to precise measurements of fundamental constants. However, the contrast of 3 % obtained using this moiré interferometer was well below the theoretical prediction of 30 % using ideal gratings. It is suspected that non-ideal aspects of the phase-gratings was a leading contributor to this deficiency and that phase-gratings needed to be quantitatively assessed and optimized. Here we characterize neutron diffraction from phase-grating…
We experimentally characterize the positions of the diffraction maxima of a phase grating on a screen, for laser light at oblique incidence (so-called off-plane diffraction or conical diffraction). We discuss the general case of off-plane diffraction geometries and derive basic equations for the positions of the diffraction maxima, in particular for their angular dependence. In contrast to previously reported work [Jetty et al., Am. J. Phys. 80, 972 (2012)], our reasoning is solely based on energy- and momentum conservation. We find good agreement of our theoretical prediction with the experim…
Resonance diffraction of THz HCN laser radiation on a semiconductor (InSb) grating is studied both experimentally and theoretically. The specular reflectivity suppression due to the resonance excitation of the THz surface plasmon-polariton is observed on a pure semiconductor grating and on semiconductor gratings covered with a thin striped layer of the residual photoresist. Presence of a thin dielectric layer on the grating surface leads to the shift and widening of the plasmon-polariton resonance. A simple analytical theory of the resonance diffraction on a shallow grating covered with a diel…
We propose an Adaptive Perfectly Matched Layer (APML) to be used in diffraction grating modeling. With a properly tailored co-ordinate stretching depending both on the incident field and on grating parameters, the APML may efficiently absorb diffracted orders near grazing angles (the so-called Wood's anomalies). The new design is implemented in a finite element method (FEM) scheme and applied on a numerical example of a dielectric slit grating. Its performances are compared with classical PML with constant stretching coefficient.
A multi-reflection model of grating diffraction based on the simplified modal method is proposed. Simulation results for a guided mode resonance Brewster grating using our method and rigorous coupled wave analysis are presented to verify our model. The solution of our method is in good agreement with that of rigorous coupled wave analysis. Benefiting from its clear physical view, this model helps us to better understand the diffraction process inside subwavelength gratings. On the basis of the multi-reflection model, we developed a matrix Fabry-Perot (FP) resonance condition and a single-mode …
A high-accuracy solution of the diffraction problem has become necessary for the treatment of certain special questions of statistical physics. This article reports the creation of a computer program that serves as an instrumental method of calculating the parameters of diffraction phenomena when complex optical systems are being theoretically investigated. The program solves the diffraction problem by a rigorous method based on Maxwell"s equations under specified boundary conditions. An arbitrary for instance, diffuse configuration of the initial light field is allowed. Reflective gratings wi…
We report on the optical characterization of an ultra-high diffraction efficiency grating in 1st order Littrow configuration. The apparatus used was an optical cavity built from the grating under investigation and an additional high reflection mirror. Measurement of the cavity finesse provided precise information about the grating's diffraction efficiency and its optical loss. We measured a finesse of 1580 from which we deduced a diffraction efficiency of (99.635$\pm$0.016)% and an overall optical loss due to scattering and absorption of just 0.185 %. Such high quality gratings, including the …
The diffraction problem of a plane wave impinging on a grating formed by nested cavities is solved by means of the modal method, for $s$ and $p$ polarization modes. The cavities are formed by perfectly conducting sheets that describe rectangular profiles. The electromagnetic response of the grating is analyzed, paying particular attention to the generation of resonances within the structure. The dependence of the resonances on the geometrical parameters of the grating is studied, and results of far and near field are shown. The results are checked and compared with those available in the liter…
This document aims at presenting both theoretical and practical aspects of the grating2D.pro and grating3D.pro [1] ONELAB [2] models. The 2D model grating2D.pro applies to so-called mono-dimensional gratings, i.e. structures having one direction of invariance. Various geometries and materials can be handled or easily added. The two classical scalar polarization cases, denoted here E// and H//, are addressed as well as the conical vector case. The output consists in a full energy balance of the problem computed from the field maps. The 3D vector model grating3D.pro applies to crossed-gratings. …
In this article we present a theoretical study for Fraunhofer diffraction of a Laguerre-Gaussian laser beam with zeroth radial mode number and azimuthal mode number l by diffractive grating with embedded fork-shaped dislocations of integer order p. Analytical expressions describing the diffracted wave field amplitude and intensity distributions in the Fourier plane are deduced and analyzed. They are also followed by the vortex radii expressions.
The generation of induced radiation in volume resonator formed by metal threads is considered. It is shown that using of such volume diffraction grating allows increasing of lasing efficiency in terahertz range. The requirements on beam and grating parameters are obtained.
The equations providing to describe generation process in FEL with varied parameters of diffraction grating (photonic crystal) are obtained. It is shown that applying diffraction gratings (photonic crystal) with the variable period one can significantly increase radiation output. It is mentioned that diffraction gratings (photonic crystal) can be used for creation of the dynamical wiggler with variable period in the system. This makes possible to develop double-cascaded FEL with variable parameters changing, which efficiency can be significantly higher that of conventional system.
Diffraction gratings with large angular dispersion rates are central to obtaining high spectral resolution in grating spectrometers operating over a broad spectral range from infrared to soft-x-ray domains. The greatest challenge is of course to achieve large dispersion rates in the short-wavelength x-ray domain. Here we show that crystals in non-coplanar asymmetric x-ray Bragg diffraction can function as high-reflectance soft-x-ray diffraction gratings with dispersion rates that are at least two orders of magnitude larger than those that are possible with state-of-the-art man-made gratings. T…
We consider an optical diffraction grating in which the spatial distribution of open slits forms a fractal set. The Fraunhofer diffraction patterns through the fractal grating are obtained analytically for the simplest triad Cantor type and its generalized version. The resulting interference patterns exhibit characteristics of the original fractals and their scaling properties.
Diffraction gratings with high upward diffraction efficiency and large effective length are required for chip-scale light detection and ranging. In this paper, we propose a diffraction grating based on a multilayer silicon nitride waveguide, which theoretically achieves an upward diffraction efficiency of 92$\%$, a near-field effective length of 376 $μm$ and a far-field divergence angle of 0.105$^{\circ}$ at a wavelength of 850 nm. The diffraction grating has a high tolerance to process variations based on Monte Carlo Analysis. When the conditions are $\pm$5$\%$ layer thickness variation, $\pm…
Spectra of ultracold neutrons that appeared in experiments on neutron diffraction by a moving grating were measured using the time-of-flight Fourier spectrometer. Diffraction lines of five orders were observed simultaneously. The obtained data are in good agreement with the theoretical predictions based on the multiwave dynamical theory of neutron diffraction by a moving grating.
Optical gratings are a key component in many spectroscopy, communications, and imaging systems. While initially static elements, advances in optical materials have enabled dynamically tunable gratings to be designed. One common tuning strategy is relying on mechanical deformation of the grating pitch to modify the diffraction pattern. To date, most mechanically adaptive optical gratings consist of a hybrid system where rigid moieties are patterned on an elastomeric substrate. In the present work, we demonstrate an all-polymer tunable grating that is fabricated using replica molding from the po…
We have studied the optical properties of gratings micro-fabricated into semiconductor wafers, which can be used for simplifying cold-atom experiments. The study entailed characterisation of diffraction efficiency as a function of coating, periodicity, duty cycle and geometry using over 100 distinct gratings. The critical parameters of experimental use, such as diffraction angle and wavelength are also discussed, with an outlook to achieving optimal ultracold experimental conditions.
We report on the observation of emerging beam resonances, well known as Rayleigh-Wood anomalies and threshold resonances in photon and electron diffraction, respectively, in an atom-optical diffraction experiment. Diffraction of He atom beams reflected from a blazed ruled grating at grazing incidence has been investigated. The total reflectivity of the grating as well as the intensities of the diffracted beams reveal anomalies at the Rayleigh angles of incidence, i.e., when another diffracted beam merges parallel to the grating surface. The observed anomalies are discussed in terms of the clas…
The general features of the light scattering resulting in the so-called resonant Wood's anomalies in the reflection and transmission spectra are described using the effective parameters of a quasi-guided mode. The expression determining the spectral angular dependence of Wood's anomaly in the case of plasmonic grating structures is given and compared to the analogous expression for dielectric grating structures. Comparison of the resonant Wood's anomalies (RWA) with the surface plasmon-polariton resonances (SPPRs) is discussed as well.
A single-order transmission diffraction grating based on dispersion engineered all-dielectric metasurfaces is proposed and its wavelength discriminating properties have been theoretically described and confirmed using numerical simulations. The metasurface is designed using a 2D array of all-dielectric resonators, which emulates a Huygens source configuration to achieve a perfect match to free-space in broad bandwidth. Using a holey dielectric nanodisk structure as the unit cell, the resonant wavelength is tapered across the metasurface to engineer the wavelength dependent spatial phase gradie…
Symmetry and reciprocity constraints on polarization state of the field diffracted by gratings of quasi-planar particles are considered. It is shown that the optical activity effects observed recently in arrays of quasi-planar plasmonic particles on a dielectric substrate are due to the reflection of the field at the air-dielectric slab interface and are proportional to this reflection coefficient.
We introduce a method for using Fizeau interferometry to measure the intrinsic resolving power of a diffraction grating. This method is more accurate than traditional techniques based on a long-trace profiler (LTP), since it is sensitive to long-distance phase errors not revealed by a d-spacing map. We demonstrate 50,400 resolving power for a mechanically ruled XUV grating from Inprentus, Inc.
Analytical analysis of the decrease in the focal intensity of a femtosecond laser pulse caused by an imperfect surface of compressor diffraction gratings
This work theoretically investigates wide-spectrum and high-resolution diffraction optical elements (DOE) that are made of stacks of low-resolution binary phase gratings, whereby the two-dimensional grids in different grating layers are arranged with specified displacements. We remodel the common Kinoform algorithm for this multi-scale architecture. Numerical computations show that, by increasing the number of stacking layers, the resolution of diffraction fields can be improved and that the stability of optical elements within broad spectrums is significantly enhanced. Practical concern on la…
Implementing the modal method in the electromagnetic grating diffraction problem delivered by the curvilinear coordinate transformation yields a general analytical solution to the 1D grating diffraction problem in a form of a T-matrix. Simultaneously it is shown that the validity of the Rayleigh expansion is defined by the validity of the modal expansion in a transformed medium delivered by the coordinate transformation.
Under white light illumination, gratings produce an angular distribution of wavelengths dependent on the diffraction order and geometric parameters. However, previous studies of gratings are limited to at least one geometric parameter (height, periodicity, orientation, angle of incidence) kept constant. Here, we vary all geometric parameters in the gratings using a versatile nanofabrication technique, two-photon polymerization lithography, to encode hidden color information through 2 design approaches. The first approach hides color information by decoupling the effects of grating height and p…
In this paper, an analytical theory for the diffraction of a Bessel beam of arbitrary order $J_l(κr)$ on a 2D amplitude grating is presented. The diffraction pattern in the main and fractional Talbot planes under certain conditions is a lattice of annular microbeams, the diameters of which depend on the grating period, the illuminating beam diameter, the number of the Talbot plane, and the topological charge $l$. For the rings near the optical axis, the latter reproduces $l$ of the illuminating beam. Experiments carried out on the Novosibirsk free electron laser at a wavelength $λ= 141$$μ$m us…
We consider the spectral response of moving diffraction gratings, in which the incident light extends over a broad angular range and where the diffracted light is observed from a specific angle. We show that the dispersion relation between the frequency perceived by an observer who is looking at a moving grating and the incident frequency can exhibit some unique features, such as a flat band (i.e., a local minimum). An observer can see the light diffracted into a non-specular diffraction order from a multitude of incident light rays and the angle of incidence of each ray is frequency-dependent…
The time-dependent Bragg diffraction by multilayer gratings working by reflection or by transmission is investigated. The study is performed by generalizing the time-dependent coupled-wave theory previously developed for one-dimensional photonic crystal [Andr{é} J-M and Jonnard P, J. Opt. 17, 085609 (2015)] and also by extending the Takagi-Taupin approach of the dynamical theory of diffraction. The indicial response is calculated. It presents a time-delay with a transient time that is a function of the extinction length for reflection geometry and of the extinction length combined with the thi…
A theoretical approach for uniformly extracting light propagating through a light guide plate is developed here. Typically, liquid crystal display modalities utilize a backlit lighting system illuminated from the edge with light emitting diodes. The backlight acts as a light guide plate and is coupled with a diffuser sheet along the surface to uniformly extract the light. Our approach employs sub-micron diffraction gratings and eliminates the need for diffuser sheets, while ensuring the uniform extraction of light. The physical dimensions of the grating are varied along the surface of the ligh…
Volume Free Electron Laser (VFEL) was proposed in [1-4]. It can operate in the wide spectral range from microwaves to X-rays. To simulate the processes which take place in VFEL the superradiance from a short electron pulse moving in a volume diffraction grating is studied in wavelength range ~4 mm. It is supposed that Bragg condition for emitted photons is fulfilled and dynamical diffraction takes place. The spectral-angular distributions for transmitted and diffracted waves are derived. It is shown that the optimal geometry of diffraction exists and is determined by the energy and transverse …
We introduce a grating assisted tunneling scheme for tunable synthetic magnetic fields in photonic lattices, which can be implemented at optical frequencies in optically induced one- and two-dimensional dielectric photonic lattices. We demonstrate a conical diffraction pattern in particular realization of these lattices which possess Dirac points in $k$-space, as a signature of the synthetic magnetic fields. The two-dimensional photonic lattice with grating assisted tunneling constitutes the realization of the Harper-Hofstadter Hamiltonian.
Diffraction gratings are famous for their ability to exhibit, near a Wood anomaly, an arbitrarily large angular dispersion, e.g., with respect to the incidence angle or wavelength. For a diffraction grating under incidence by a plane wave at a fixed frequency, by rotating the incidence angle at a given angular velocity, the field propagated by a nonzero diffraction order will rotate at increasingly fast angular velocity as the incidence angle approaches the angle where Wood anomaly occurs. Such a fast rotating diffracted field has the potential to generate a substantial Doppler shift. Indeed, …
We analyze the diffraction characteristics of dielectric gratings that feature a high index grating layer, and devise, through rigorous numerical calculations, large bandwidth, highly efficient, high dispersion dielectric gratings in reflection, transmission, and immersed transmission geometry. A dielectric TIR grating is suggested, whose -1dB spectral bandwidth is doubled as compared to its all-glass equivalent. The short wavelength diffraction efficiency is additionally improved by allowing for slanted lamella. The grating surpasses a blazed gold grating over the full octave. An immersed tra…
We demonstrate the use of nanodiamond in constructing holographic nanoparticle-polymer composite transmission gratings with large saturated refractive index modulation amplitudes at both optical and slow-neutron wavelengths, resulting in efficient control of light and slow-neutron beams. Nanodiamond possesses a high refractive index at optical wavelengths and large coherent and small incoherent scattering cross sections with low absorption at slow-neutron wavelengths. We describe the synthesis of nanodiamond, the preparation of photopolymerizable nanodiamond-polymer composite films, the constr…
Expressions describing the vortex beams, which are generated in a process of Fresnel diffraction of a Gaussian beam, incident out of waist on a 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. T…
