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.     

Broadband transmission masks, gratings and filters for extreme ultraviolet and soft X-ray lithography

Lithography and patterning on a nanometre scale with extreme ultraviolet (EUV) and soft X-ray radiation allow creation of high resolution, high density patterns independent of a substrate type. To realize the full potential of this method, especially for EUV proximity printing and interference lithography, a reliable technology for manufacturing of the transmission masks and gratings should be available. In this paper we present a development of broadband amplitude transmission masks and gratings for extreme ultraviolet and soft X-ray lithography based on free-standing niobium membranes. In comparison with a standard silicon nitride based technology the transmission masks demonstrate high contrast not only for in-band EUV (13.5 nm) radiation but also for wavelengths below Si L-absorption edge (12.4 nm).The masks and filters with free standing areas up to 1000 × 1000 μm² and 100 nm to 300 nm membrane thicknesses are shown. Electron beam structuring of an absorber layer with dense line and dot patterns with sub-50 nm structures is demonstrated. Diffractive and filtering properties of obtained structures are examined with EUV radiation from a gas discharge plasma source.     

Design of antireflection gratings with approximate and rigorous methods

High-spatial-frequency gratings can be used as an alternative to thin-film antireflection coatings to reduce the reflectivity at the boundary between two different media. In the case of one-dimensional gratings, the conditions on the grating structure can be approximately determined by the effective medium theory (EMT) in combination with the thin-film theory. For two-dimensional gratings, which can be used to reduce the polarization sensitivity, a corresponding EMT does not exist. We present an estimation of the effective permittivity of two-dimensional gratings. The range of validity of the antireflection grating design by the EMT is determined by the use of rigorous electromagnetic theory. Beyond the validity of EMT, rigorous theory is used to design antireflection gratings with a maximized feature size.     

Internal symmetries in conical diffraction from metallic gratings

Abstract

Previous investigations in conical diffraction have shown that for a metallic grating illuminated under conditions of resonant excitation of surface plasmons, the maximum efficiency of polarization conversion is obtained when the plane of incidence forms an angle φ = 45% with the main section of the grating. This fact has been understood as a consequence of the evident symmetry imposed by the incident wave and it has not been questioned up to now. Here we show that, apart from this highly symmetrical configuration, the polarization conversion efficiency has local maxima also in other cases which do not exhibit such an evident symmetry. We demonstrate that the occurrence of both the local maxima for the new configurations and the maximum corresponding to φ 45% are a consequence of internal symmetries imposed by the propagation direction of the resonantly excited surface plasmons.     

Surface gradient integrated profiler for X-ray and EUV optics

A new ultraprecise profiler has been developed to measure, for example, asymmetric and aspheric profiles. The principle of our measuring method is that the normal vector at each point on the surface is determined by making the incident light beam on the mirror surface and the reflected beam at that point of coincident. The gradient at each point is calculated from the normal vector, and the surface profile is then obtained by integrating the gradients. The measuring instrument was designed in accordance with the above principle. In the design, four ultraprecise goniometers were applied to adjust the light axis for normal vector measurement. The angle-positioning resolution and accuracy of each goniometer are, respectively, 0.018 and 0.2 μrad. Thus, in the measuring instrument, the most important factor is the accuracy of the normal vectors measured by the goniometers. Therefore, the rotating angle-positioning errors were measured and calibrated. An elliptical profile mirror for nanometer hard-X-ray focusing was measured, and compared with the measured profile using a stitching interferometer. The absolute measurement accuracy of approximately 5 nm (peak-to-valley) was achieved. Then the measurements of 1000-mm-long flat, spherical and parabolic mirrors were demonstrated. The surface profiles of the mirrors were obtained by integrating the interpolated gradient.     

Study of broadband polarization conversion with metallic surface-relief gratings by rigorous coupled-wave analysis

The conical rigorous coupled-wave analysis (RCWA) is employed to calculate the polarization conversion through the excitation of surface plasmons on metallic gratings. Various examples are examined with this numerical scheme. Our calculated results are consistent with those obtained from experiment and from other numerical methods. Three types of subwavelength surface-relief gratings are studied for the capability of broadband polarization conversion in the visible region. For wide-angle applications, various incident angles are studied and high polarization conversion efficiency is achieved.     

Quantitative X-ray Rietveld analysis of metallic aluminum content in nano-aluminum powders

Accurate analysis of the content of metal aluminum (Al) in nano-Al powders is a difficult task due to the high reactivity of nano-Al powders. Here, the X-ray Rietveld quantitative phase analysis (QPA) method was applied to analyze the content of metal aluminum in nano-Al powders produced by three methods. A good agreement between the observed and calculated diffraction pattern was obtained and the conventional Rietveld factors (R_p, R_wp and GOF) converged to satisfactory values, which suggested that Al content was evaluated accurately. At the same time, the mixture sample of 89.77 wt.% micro-Al + 10.23 wt.% α-Al₂O₃ was study and the fitting pattern and R factors were satisfactory, which further verified the reliability of the X-ray Rietveld QPA method.     

Rigorous coupled-wave analysis for practical planar dielectric gratings: 1. Thickness-changed holograms and some characteristics of diffraction efficiency

A new and useful method for obtaining diffraction efficiencies from holograms manufactured practically is presented. Applying the rigorous coupled-wave analysis, we express each difference between the practical and the ideal as a mathematical component that can be easily integrated. In Part 1 the effects due to thickness change in the hologram layer (observed frequently after the development process) are treated. Although uniform swelling or shrinking causes a simple reconstruction wavelength or incidence-angle shift, nonuniform thickness extends the capacity of the Bragg condition matching, creating a diffraction efficiency curve in the asymmetric profile. Other characteristics of diffraction are also maintained. A refractive-index change has an effect that is similar to the thickness change. Higher-order terms in permittivity modulation create negligible effects in general holograms when used at or near the simple first-order Bragg condition.     

Properties of TM resonances on metallic slit gratings

Electromagnetic resonances on metallic slit gratings induced by TM polarized incident light have been investigated and physically interpreted. We have developed an electromagnetic model imposing surface impedance boundary conditions on the metallic grating surface from which we derive simple formulas explaining all physical properties of these resonances. It is demonstrated that Fabry-Perot (or cavity) resonances are generated by the zeroth slit mode yielding extraordinary transmission. For very narrow slits, the resonant H-field is squeezed to the slit walls and causes enhanced power losses. The excitation of surface plasmon polaritons (SPPs), however, is generated by two mode coupling. SPPs are linked to sharp absorption peaks and dips in transmittance. It is shown that these phenomena are primarily caused by the interaction of the electromagnetic fields with the finite conducting slit walls. These findings have been confirmed by measured transmittance data of gold gratings with periods of 0.5 μm, 1 μm, and 2 μm.     

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.     

Polarization conversion in conical diffraction by metallic and dielectric subwavelength gratings

Subwavelength metallic and dielectric diffraction gratings which rotate the linear polarization of incident light by 90 degrees are examined. Using rigorous diffraction theory in total-internal-reflection configuration, it is shown that full conversion from incident transverse electric field to transverse magnetic zero-order field can be achieved with both dielectric and metallic elements, but dielectric gratings provide higher efficiency and are thus preferable. The fabrication aspects and constraints are discussed in detail and the behavior of the gratings over broad wavelength bands is presented.     

High-fidelity replication of Dammann gratings using soft lithography

We report the experimental results of using the soft lithography method for replication of Dammann gratings. By using an elastomeric stamp, uniform grating structures were transferred to the UV-curable polymer. To evaluate the quality of the replication, diffraction images and light intensity were measured. Compared with the master devices, the replicas of Dammann gratings show a slight deviation in both surface relief profile and optical performance. Experimental results demonstrated that high-fidelity replication of Dammann gratings is realized by using soft lithography with low cost and high throughput.     

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 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 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.     

Influence of morphologic texture on stress analysis by X-ray and neutron diffraction in single-phase metallic materials

In this work, a study on the influence of morphologic texture on the residual stress determination by diffraction in metallic materials with cubic and hexagonal symmetry is proposed. To this end, elastic self-consistent model has been developed to properly take into account the morphologic texture. Extreme crystallites morphologies (sphere, disc and fibre) were studied, and coupled with the crystallographic texture to reflect the combined effect of morphologic and crystallographic texture in elasticity. In the case of morphologic texture, a stronger influence than the crystallographic texture on the estimated residual stresses (several tens of MPa difference) was observed. We propose a methodology through a scale transition model to take into account the influence of these different morphologies in the stress analysis by diffraction methods. The main purpose of this work was to make the best choice for lattice planes (hkl) used for residual or internal stress analysis, in elasticity, depending on the morphologic (and crystallographic) texture of the polycrystal, especially when the usual X-ray Elasticity Constants (XECs) are used instead of the stress factors.     

Grating-coupled transmission-type surface plasmon resonance sensors based on dielectric and metallic gratings

We investigated grating-coupled transmission-type surface plasmon resonance (SPR) for sensing applications. In the transmission-type SPR structure, propagating surface plasmons are outcoupled to radiation modes by dielectric and metallic gratings on a metal film. The results calculated in air and water suggest that the proposed structures present extremely linear sensing characteristics. In terms of a figure of merit, a metallic grating-based structure performs 5.4 and 3.7 times better than that of a dielectric grating in air and water, respectively.