Rigorous coupled-wave analysis for practical planar dielectric gratings: 3. Increase of higher-order lights owing to degenerated complex diffraction

The diffraction efficiencies by practical dielectric holograms are evaluated with the rigorous coupled-wave analysis. As discussed in Parts 1 and 2, only the first reflection light can be diffracted by general sealed volume reflection holograms. Here we discuss the possibility that complex diffraction, which occurs in multiple-grating-storage holograms, makes diffraction efficiencies of higher-order lights increase drastically, i.e., the degenerated complex diffraction. Although evident in special cases only, they appear frequently in actual holographic use.     

Rigorous coupled-wave analysis for practical planar dielectric gratings: 2. Diffraction by a surface-eroded hologram layer

The diffraction efficiencies of practical dielectric holograms are evaluated with rigorous coupled-wave analysis. The cases of the hologram surfaces eroded in several shapes are treated and compared with those in which the surfaces are not eroded and with those in which there are only surface gratings. Eroding the surface will increase the higher-order reflection diffraction efficiencies and the transmissions, thus reducing the first reflection and the zeroth transmission. However, sealing the hologram with a cover plate, as is done in manufacturing many holograms, extinguishes the erosion effect.     

Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings

This paper presents an alternative approach called the hybrid-matrix or H-matrix algorithm for the stable and efficient implementation of the rigorous coupled-wave analysis of multilayered diffraction gratings. The H matrix is superior to both the T and R matrices because of its numerical stability across the whole range of large and small layer thicknesses. It is also more concise and simple than the S matrix in obviating the need to invoke auxiliary reference layers or two consecutive layers. For high efficiency, the stack H-matrix recursive algorithm is derived in terms of eigen-submatrices directly without any auxiliary layer matrix. Compared with the enhanced transmittance matrix (partial-solution) approach, the enhanced H-submatrix approach achieves 30% higher computation efficiency and requires less memory.     

Rigorous analysis of diffraction gratings of arbitrary profiles using virtual photonic crystals

A new approach is developed to calculate diffraction efficiency for a dielectric grating with an arbitrary refractive index profile. By treating a one-dimensional grating as a segment of a virtual two-dimensional (2D) photonic crystal, we exploit a rigorous theory of photonic crystal refraction and calculate the diffraction efficiencies. We expand, analytically in many cases, the dielectric function of the grating into 2D Fourier series. We find the eigenmodes for the virtual photonic crystal, and then use these eigenmodes to match the boundary conditions by solving a set of linear equations. In two such simple steps, the diffraction efficiencies can be computed rigorously without slicing the grating into thin layers.     

Off-Bragg analysis of the diffraction efficiency of transmission photorefractive holograms

Formulas for calculating the diffraction efficiency of gratings recorded in a photorefractive medium are given. The analysis uses coupled-wave theory for photorefractive hologram gratings and takes into account the photorefractive phase shift and fringe-bending effect. General solutions for diffracted (signal) and undiffracted (reference) waves are derived in a closed-form expression. By use of the derived formulas the diffraction efficiency for angle mismatch from the Bragg condition can easily be evaluated as compared with numerical-analysis methods. The diffraction efficiency is also quantified in terms of medium parameters and recording and reconstruction conditions.     

Off-Bragg analysis of the diffraction efficiency of reflection photorefractive holograms

Formulas are given for the calculation of diffraction efficiency of reflection-type gratings recorded in a photorefractive medium. The analysis incorporates the coupled-wave theory that was developed for photorefractive hologram gratings. This analysis takes into account grating slant with respect to the medium surface, light absorption during reconstruction, any incident angle of the reference beam, and any photorefractive phase shift. General solutions for signal and reference wave functions are given in a closed-form expression by use of a hypergeometric function. The optimum media parameters and recording conditions for high diffraction efficiency are obtained by the derived formulas. The diffraction properties for off-Bragg conditions are also discussed.     

Effects of process errors on the diffraction characteristics of binary dielectric gratings

The effects of fabrication errors on the predicted performance of surface-relief phase gratings are analyzed with a rigorous vector diffraction technique. For binary elements, errors in the dimensions of the profile [depth, linewidth (fill factor), and grating period], as well as errors in the shape of the profile, are investigated. It is shown that the dimension errors do not have a significant effect on grating performance when the grating is designed for either maximum or minimum diffraction efficiency. A trapezoid is used to model the shape error of the profile. For the first time, design rules that significantly reduce the effects of any shape error are presented.     

Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization

A rigorous modal theory for the diffraction of Gaussian beams from N equally spaced slits (finite grating) in a planar perfectly conducting thin screen is presented. The case of normal incidence and TE polarization state is considered; i.e., the electric field is parallel to the slits. The characteristics of the far-field diffraction patterns, the transmission coefficient, and the normally diffracted energy as a function of several optogeometrical parameters are analyzed within the so-called vectorial region, where the polarization effects are important. The diffraction pattern of an aperiodic grating is also considered. In addition, one diffraction property known to be valid in the scalar region is generalized to the vectorial region: the existence of constant-intensity angles in the far field when the incident beam wave is scanned along the N slits. The classical grating equation is tested for incident Gaussian beams under several conditions.     

Rigorous theory of the diffraction of Gaussian beams by finite gratings: TM polarization

Diffraction of TM-polarized Gaussian beams by N equally spaced slits (finite grating) in a planar perfectly conducting thick screen is treated. We extend to the TM polarization case the results of a previous paper where the TE polarization was considered. The far-field diffraction patterns, the transmission coefficient tau, and the normally diffracted energy E as a function of several optogeometrical parameters are analyzed within the so-called vectorial region. The existence of constant-intensity angles in the far field when the incident beam wave is scanned along the N slits is shown. In addition, the property E=Ntau/lambda, valid in the scalar region, is extended to the TM polarization case in the vectorial region, lambda being the wavelength. The coupling between slits is analyzed, giving an oscillating amplitude-decreasing function as the separation between slits increases, where the period for these oscillations is the wavelength lambda. Finally, the extraordinary optical transmission phenomena that appear when the wavelength is larger than the slit width (subwavelength regime) are analyzed.     

Fabry-perot multilayers for enhancing the diffraction efficiency of ion-implanted gratings

Enhancement of the free-space diffraction efficiency of gratings made by titanium-ion implantation is demonstrated both theoretically and experimentally. Indeed, by insertion of a grating into a multilayer dielectric Fabry-Perot cavity, the diffraction efficiency can be increased to as much as 24 times that of a single grating. The sensitivity of the diffraction efficiency to the optogeometrical parameters of the grating or of the Fabry-Perot cavity is discussed. Moreover, a process for performance of a phase grating inside a Fabry-Perot cavity is described, and experimental results concerning efficiency measurements are compared with computed values for various grating periods.     

Comparison of beam propagation method and rigorous coupled-wave analysis for single and multiplexed volume gratings

Multiplexed holographic structures have been suggested to provide large capacity and parallel access as three-dimensional storage media. One of the most widely used techniques in the literature for analyzing such structures has been the coupled-wave analysis and its variations. Another approach that is becoming increasingly popular because of the ease with which it can be implemented is the beam propagation method (BPM). The BPM is quantitatively compared with the rigorous coupled-wave analysis for the cases of single and multiplexed gratings. Normal and off-normal incidence as well as TE and TM polarizations are considered for single (slanted and unslanted) and multiplexed gratings. It is shown that the BPM, even in its most rudimentary form, is a powerful and accurate calculational method that is especially suited for analyzing the many multiplexed grating diffraction problem.     

Rigorous and approximate analysis of metallic gratings in soft X-ray and EUV regions

Abstract

In deep ultraviolet and soft X-ray regions the complex refractive indices of metals differ substantially from their values in the visible region of the spectrum. The implications of this fact are analysed for metallic gratings illuminated by electromagnetic fields with wavelengths ranging from soft X‐rays to near infrared, concentrating on short wavelengths. In particular, we study metal-stripe gratings (linear polarizers for visible light) by rigorous diffraction theory to determine the short-wavelength region in which a combination of the geometrical thin-element approximation and the theory of single-layer films can be applied. Then we study inductive grid filters for protection of X-ray detectors from infrared radiation in space applications.     

High-efficiency dielectric reflection gratings: design, fabrication, and analysis

We report on reflection gratings produced entirely of dielectric materials. This gives the opportunity to enhance the laser damage threshold over that occurring in conventional metal gratings used for chirped-pulse-amplification, high-power lasers. The design of the system combines a dielectric mirror and a well-defined corrugated top layer to obtain optimum results. The rules that have to be considered for the design optimization are described. We optimized the parameters of a dielectric grating with a binary structure and theoretically obtained 100% reflectivity for the -1 order in the Littrow mounting for a 45 degrees angle of incidence. Subsequently we fabricated gratings by structuring a low-refractive-index top layer of a multilayer stack with electron-beam lithography. The multilayer system was fabricated by conventional sputtering techniques onto a flat fused-silica substrate. The parameters of the device were measured and controlled by light scatterometer equipment. We measured 97% diffraction efficiency in the -1 order and damage thresholds of 4.4 and 0.18 J/cm(2) with 5-ns and 1-ps laser pulses, respectively, at a wavelength of 532 nm in working conditions.     

A model of coupled resonators for calculating the electromagnetic wave diffraction on planar one-dimensional Bragg gratings

The problem of excitation of a one-dimensional planar Bragg grating is studied. A model of coupled resonators is proposed for investigating the diffraction of electromagnetic waves on a waveguide surface with arbitrary corrugation profile. The model is based on the strict relationships derived from a solution to a two-dimensional boundary problem for the Helmholtz equation. A particular form of equations is presented for the case of a rectangular corrugation of plates in the grating.     

Control of morphology and diffraction efficiency of holographic gratings using siloxane-containing reactive diluent

Performance of holographic gratings, prepared by using epoxy-functionalized siloxane compound as a reactive diluent, was compared with that using N-vinylpyrrolidinone as a diluent in the presence of various contents of liquid crystal under different exposure beam intensity. Siloxane component strongly influenced the kinetics of polymerization, diffusion, and phase separation of the liquid crystalline compound, and high diffraction efficiency was obtained when 3-acryloxypropyltrimethoxysilane or 2-[(3,4-epoxycyclohexyl)ethyl]trimethoxysilane was used as a reactive diluent even at a very low concentration (10–25 wt%) of commercial liquid crystal mixture, E7 (Merk), in contrast to the case with N-vinylpyrrolidinone or benzyl glycidyl ether as the diluent without siloxane component.The phase-separated morphologies of gratings, such as spacing and surface topology, observed by SEM and AFM, were well controlled, and very regular and smooth morphologies were observed for the holographic gratings prepared with 3-acryloxypropyltrimethoxysilane and various contents of the liquid crystal.     

Simple measuring technique for the diffraction efficiency of slanted volume gratings at various wavelengths

To measure diffraction efficiencies of gratings as a function of wavelength, it is necessary to have quasi-monochromatic light sources of various wavelengths. We propose a method to measure the wavelength dependence of the grating diffraction efficiency by using a quasi-monochromatic light source. This method of estimating the real diffraction characteristics of the gratings for various wavelengths is very useful and simple. First the diffraction efficiency of the grating as a function of various incident-beam angles of monochromatic light is measured, then, using these data, we can obtain the diffraction efficiencies for various wavelengths of the same incident angle of light by virtue of a mathematical-conversion method. The mathematical-conversion results for two laminated differently slanted angle gratings of the same volume grating period are in good agreement with the experimental ones.     

Coupled-wave analysis of vector holograms. 2. Reflective gratings formed in photoanisotropic medium with uniaxial birefringence

The diffraction properties of reflective anisotropic gratings, which can be recorded in photoanisotropic media with uniaxial birefringence by three-dimensional vector holography, were characterized through the use of coupled-wave analysis (CWA). By investigating the perturbation of the dielectric tensor, we demonstrated that the gratings with sinusoidal distribution of the azimuthal angle of the optic axis diffract polarized light in which the ordinary and extraordinary components are converted for incident light. The polarization conversion was consistent with that calculated by a numerical method. In addition, it was shown that CWA enables highly accurate calculation of the diffraction efficiency with wavelength dispersion when the amplitude of the azimuthal angle is small.