Axial shift mapping: a self-referencing test for measuring the axial figure of near-cylindrical surfaces.

Lateral shearing self-referencing interferometry methods shift the surface under test between measurements to separate its topography from that of the reference surface. However, rigid body errors occur during shifting, creating an ambiguity in the quadratic term of the extracted surfaces. We present axial shift mapping, a lateral shearing self-referencing interferometry method for cylinders, in which the quadratic ambiguity is resolved by measuring the rigid body errors using known artifact mirrors residing in the interferometer's field of view. First, one-dimensional lines of a flat mirror are measured with 2.8 nm RMS difference compared to a three flat test. Then, axial shift mapping is extended to cylindrical surfaces using a computer generated hologram. We find that axial shift mapping results in full surface extraction of cylindrical optics, along the axial direction, with a repeatability of 4.4 nm RMS. We also find that the reference surface extracted through axial shift mapping is within 4.5 nm RMS of the transmitted wavefront error of the computer generated hologram substrate, which was expected to be the largest contribution of reference wavefront error.

Optical design of diffraction-limited x-ray telescopes.

Astronomical imaging with micro-arcsecond (µas) angular resolution could enable breakthrough scientific discoveries. Previously proposed µas x-ray imager designs have been interferometers with limited effective collecting area. Here we describe x-ray telescopes achieving diffraction-limited performance over a wide energy band with large effective area, employing a nested-shell architecture with grazing-incidence mirrors, while matching the optical path lengths between all shells. We present two compact nested-shell Wolter Type 2 grazing-incidence telescope designs for diffraction-limited x-ray imaging: a micro-arcsecond telescope design with 14 µas angular resolution and 2.9m2 of effective area at 5 keV photon energy (λ=0.25nm), and a smaller milli-arcsecond telescope design with 525 µas resolution and 645cm2 effective area at 1 keV (λ=1.24nm). We describe how to match the optical path lengths between all shells in a compact mirror assembly and investigate chromatic and off-axis aberrations. Chromatic aberration results from total external reflection off of mirror surfaces, and we greatly mitigate its effects by slightly adjusting the path lengths in each mirror shell. The mirror surface height error and alignment requirements for diffraction-limited performance are challenging but arguably achievable in the coming decades. Because the focal ratio for a diffraction-limited x-ray telescope is extremely large (f/D∼105), the only important off-axis aberration is curvature of field, so a 1 arc sec field of view is feasible with a flat detector. The detector must fly in formation with the mirror assembly, but relative positioning tolerances are on the order of 1 m over a distance of some tens to hundreds of kilometers. Although there are many challenges to achieving diffraction-limited x-ray imaging, we did not find any fundamental barriers.

Thermal oxide patterning method for compensating coating stress in silicon substrates.

We introduce a novel method for correcting distortion in thin silicon substrates caused by coating stress. Thin substrates, such as lightweight mirrors for x-ray or optical imaging, and semiconductor wafers or flat panel substrates, are easily distorted by stress in thin film coatings. We report a new method for correcting stress-induced distortion in flat silicon substrates which utilizes a micro-patterned silicon oxide layer on the back side of the substrate. Due to the excellent lithographic precision of the patterning process, we demonstrate stress compensation control to a precision of ~0.2%. The proposed process is simple and inexpensive due to the relatively large pattern features on the photomask. The correction process has been tested on flat silicon wafers that were distorted by 30 nm-thick compressively-stressed coatings of chromium, achieving RMS surface height and slope error reductions of a factor of 68 and 50, respectively.

Compensating film stress in thin silicon substrates using ion implantation.

Future space telescopes, especially X-ray telescopes, will require thin mirrors to achieve high optical throughput. Thin mirrors are more difficult to fabricate than thick mirrors, but recent advances have made accurate fabrication of thin mirrors possible. However, mirrors must have a reflective coating, which typically has non-repeatable and non-uniform intrinsic stress that deforms a thin mirror. Reducing coating stress by controlling deposition parameters typically reduces reflectivity. Non-uniform integrated stress compensation (NISC) methods, in which spatially controlled stress is applied to the mirror substrate backside to balance the frontside coating stress, decouple the film stress from the reflectivity. Ion implantation is one NISC method, where high-energy ions are implanted into a glass or silicon substrate to generate stress near the substrate surface. In this paper, we demonstrate the use of ion implantation for stress compensation of 30 nm thick chromium films applied to the front of five silicon wafers. The reflective films have mean integrated stress between -8 and -35 N/m, which cause deformations between 400 and 1600 nm RMS. We demonstrate that these wafers can be restored to the pre-coating shape to within 60 nm RMS, in most cases within 1/20th of the coating deformation.

Demonstration of resolving power λ/Δλ > 10,000 for a space-based x-ray transmission grating spectrometer.

We present measurements of the resolving power of a soft x-ray spectrometer consisting of 200 nm period lightweight, alignment-insensitive critical-angle transmission (CAT) gratings and a lightweight slumped-glass Wolter-I focusing mirror pair. We measure and model contributions from source, mirrors, detector pixel size, and grating period variation to the natural linewidth spectrum of the Al-K α 1 α 2 doublet. Measuring up to the 18th diffraction order, we consistently obtain small broadening due to gratings corresponding to a minimum effective grating resolving power Rg>10,000 with 90% confidence. Upper limits are often compatible with Rg=∞. Independent fitting of different diffraction orders, as well as ensemble fitting of multiple orders at multiple wavelengths, gives compatible results. Our data leads to uncertainties for the Al-Kα doublet linewidth and line separation parameters two to three times smaller than values found in the literature. Data from three different gratings are mutually compatible. This demonstrates that CAT gratings perform in excess of the requirements for the Arcus Explorer mission and are suitable for next-generation space-based x-ray spectrometer designs with resolving power five to 10 times higher than the transmission grating spectrometer onboard the Chandra X-ray Observatory.

Correcting flat mirrors with surface stress: analytical stress fields.

Thin mirrors, important for next-generation space telescopes, are difficult to accurately fabricate. One approach is to fabricate a mirror using traditional methods, then to bend the mirror using surface stress to correct residual height errors. We present two surface stress fields that correct any height error field in thin flat plates. For round plates, we represent these as linear combinations of Zernike polynomials. We show that equibiaxial stress, a common and easy-to-generate state of stress, cannot generally be used to make exact corrections. All three components of the surface stress are needed for exact corrections. We describe a process to design an equibiaxial stress field to make approximate corrections in round plates. Finally, we apply the three stress fields to simulate flattening of a measured glass wafer with 3.64 µm root-mean-squared (RMS) height error. Using our chosen equibiaxial stress field, the residual error is 0.34 µm RMS. In comparison, using all three stress components, the correction is exact and the required RMS stress is about 2.5× smaller than when using equibiaxial stress only. We compare the deformation with a finite element model and find agreement within 10 nm RMS in all three cases.

Diffraction efficiency of 200-nm-period critical-angle transmission gratings in the soft x-ray and extreme ultraviolet wavelength bands.

We report on measurements of the diffraction efficiency of 200-nm-period freestanding blazed transmission gratings for wavelengths in the 0.96 to 19.4 nm range. These critical-angle transmission (CAT) gratings achieve highly efficient blazing over a broad band via total external reflection off the sidewalls of smooth, tens of nanometer thin ultrahigh aspect-ratio silicon grating bars and thus combine the advantages of blazed x-ray reflection gratings with those of more conventional x-ray transmission gratings. Prototype gratings with maximum depths of 3.2 and 6 µm were investigated at two different blaze angles. In these initial CAT gratings the grating bars are monolithically connected to a cross support mesh that only leaves less than half of the grating area unobstructed. Because of our initial fabrication approach, the support mesh bars feature a strongly trapezoidal cross section that leads to varying CAT grating depths and partial absorption of diffracted orders. While theory predicts broadband absolute diffraction efficiencies as high as 60% for ideal CAT gratings without a support mesh, experimental results show efficiencies in the range of ∼50-100% of theoretical predictions when taking the effects of the support mesh into account. Future minimization of the support mesh therefore promises broadband CAT grating absolute diffraction efficiencies of 50% or higher.

High-efficiency 5000 lines/mm multilayer-coated blazed grating for extreme ultraviolet wavelengths.

Volume x-ray gratings consisting of a multilayer coating deposited on a blazed substrate can diffract with very high efficiency, even in high orders if diffraction conditions in-plane (grating) and out-of-plane (Bragg multilayer) are met simultaneously. This remarkable property, however, depends critically on the ability to create a structure with near atomic perfection. In this Letter we report on a method to produce these structures. We report measurements that show, for a 5000l/mm grating diffracting in the third order, a diffraction efficiency of 37.6% at a wavelength of 13.6nm. This work now shows a direct route to achieving high diffraction efficiency in high order at wavelengths throughout the soft x-ray energy range.

Blazed high-efficiency x-ray diffraction via transmission through arrays of nanometer-scale mirrors.

Diffraction gratings are ubiquitous wavelength dispersive elements for photons as well as for subatomic particles, atoms, and large molecules. They serve as enabling devices for spectroscopy, microscopy, and interferometry in numerous applications across the physical sciences. Transmission gratings are required in applications that demand high alignment and figure error tolerances, low weight and size, or a straight-through zero-order beam. However, photons or particles are often strongly absorbed upon transmission, e.g., in the increasingly important extreme ultraviolet (EUV) and soft x-ray band, leading to low diffraction efficiency. We demonstrate the performance of a critical-angle transmission (CAT) grating in the EUV and soft x-ray band that for the first time combines the advantages of transmission gratings with the superior broadband efficiency of blazed reflection gratings via reflection from nanofabricated periodic arrays of atomically smooth nanometer-thin silicon mirrors at angles below the critical angle for total external reflection. The efficiency of the CAT grating design is not limited to photons, but also opens the door to new, sensitive, and compact experiments and applications in atom and neutron optics, as well as for the efficient diffraction of electrons, ions, or molecules.

Near-normal-incidence extreme-ultraviolet efficiency of a flat crystalline anisotropically etched blazed grating.

We have measured the extreme-ultraviolet (EUV) efficiency at an angle of incidence of 10 degrees of a flat crystalline anisotropically etched blazed grating. The measured efficiencies are high for uncoated gratings and agree well with a calculated model derived from a reasonable estimate of the groove profile. The highest groove efficiencies derived from the measurements are 48.8% at 19.07 nm and 64.1% at 16.53 nm for the -2 and -3 orders, respectively, which are comparable to the best values obtained yet from a holographic ion-etched blazed grating. This presents opportunities to instrument designs for high-resolution EUV spectroscopy in astrophysics where high efficiency in high orders is desirable.

Monolayer/bilayer transition in Langmuir films of derivatized gold nanoparticles at the gas/water interface: an x-ray scattering study.

The microscopic structure of Langmuir films of derivatized gold nanoparticles has been studied as a function of area/particle on the water surface. The molecules (AuSHDA) consist of gold particles of mean core diameter D approximately 22 angstroms that have been stabilized by attachment of carboxylic acid terminated alkylthiols, HS-(CH2)15-COOH. Compression of the film results in a broad plateau of finite pressure in the surface pressure versus area/particle isotherm that is consistent with a first-order monolayer/bilayer transition. X-ray specular reflectivity (XR) and grazing incidence diffraction show that when first spread at large area/particle, AuSHDA particles aggregate two dimensionally to form hexagonally packed monolayer domains at a nearest-neighbor distance of a = 34 angstroms. The lateral positional correlations associated with the two-dimensional (2D) hexagonal order are of short range and extend over only a few interparticle distances; this appears to be a result of the polydispersity in particle size. Subsequent compression of the film increases the surface coverage by the monolayer but has little effect on the interparticle distance in the close-packed domains. The XR and off-specular diffuse scattering (XOSDS) results near the onset of the monolayer/bilayer coexistence plateau are consistent with complete surface coverage by a laterally homogeneous monolayer of AuSHDA particles. On the high-density side of the plateau, the electron-density profile extracted from XR clearly shows the formation of a bilayer in which the newly formed second layer on top is slightly less dense than the first layer. In contrast to the case of the homogeneous monolayer, the XOSDS intensities observed from the bilayer are higher than the prediction based on the capillary wave model and the assumption of homogeneity, indicating the presence of lateral density inhomogeneities in the bilayer. According to the results of Bragg rod measurements, the 2D hexagonal order in the two layers of the bilayer are only partially correlated.

Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations.

The analysis of many systems in optical communications and metrology utilizing Gaussian beams, such as free-space propagation from single-mode fibers, point diffraction interferometers, and interference lithography, would benefit from an accurate analytical model of Gaussian beam propagation. We present a full vector analysis of Gaussian beam propagation by using the well-known method of the angular spectrum of plane waves. A Gaussian beam is assumed to traverse a charge-free, homogeneous, isotropic, linear, and nonmagnetic dielectric medium. The angular spectrum representation, in its vector form, is applied to a problem with a Gaussian intensity boundary condition. After some mathematical manipulation, each nonzero propagating electric field component is expressed in terms of a power-series expansion. Previous analytical work derived a power series for the transverse field, where the first term (zero order) in the expansion corresponds to the usual scalar paraxial approximation. We confirm this result and derive a corresponding longitudinal power series. We show that the leading longitudinal term is comparable in magnitude with the first transverse term above the scalar paraxial term, thus indicating that a full vector theory is required when going beyond the scalar paraxial approximation. In spite of the advantages of a compact analytical formalism, enabling rapid and accurate modeling of Gaussian beam systems, this approach has a notable drawback. The higher-order terms diverge at locations that are sufficiently far from the initial boundary, yielding unphysical results. Hence any meaningful use of the expansion approach calls for a careful study of its range of applicability. By considering the transition of a Gaussian wave from the paraxial to the spherical regime, we are able to derive a simple expression for the range within which the series produce numerically satisfying answers.