Mirror surface autocovariance functions and their associated visible scattering

Image degradation due to surface scatter in the presence of aberrations

Image analysis in the presence of surface scatter due to residual optical fabrication errors is often perceived to be complicated, nonintuitive, and achieved only by computationally intensive nonsequential ray tracing with commercial optical analysis codes such as ASAP, Zemax, Code V, TracePro, or FRED. However, we show that surface scatter can be treated very similarly to conventional wavefront aberrations. For multielement imaging systems degraded by both surface scatter and aberrations, the composite point spread function is obtained in explicit analytic form in terms of convolutions of the geometrical point spread function and scaled bidirectional scattering distribution functions of the individual surfaces of the imaging system. The approximations and assumptions in this formulation are discussed, and the result is compared to the irradiance distribution obtained using commercial software for the case of a two-mirror telescope operating at an extreme ultraviolet wavelength. The two results are virtually identical.

Advanced laser-backlit Grazing-Incidence X-Ray Imaging Systems for Inertial Confinement Fusion Research. II. Tolerance Analysis

Two example ultrahigh-spatial-resolution laser-backlit grazing-incidence x-ray microscope designs for inertial confinement fusion (ICF) research have been described [Appl. Opt. 40, 4570 (2001)]. Here details of fabrication, assembly, and optical surface errors that are characteristic of present state-of-the-art superpolished multilayer-coated spherical mirrors are given. They indicate that good image qualities can be expected; in particular, <0.5-mum spatial resolution at very high x-ray energies (up to 25 keV) appears to be feasible: Existing ICF imaging diagnostics approach ~2 mum spatial at low (<2 keV) energy. The improvement in resolution compared with that of other grazing-incidence devices is attributed to a fortuitous residual on-axis aberration dependence on short wavelengths; recent advances in mirror fabrication, including a new thin-film deposition technique to correct figure errors precisely in one dimension; and novel design. For even higher resolutions, a means of creating precise aspherical mirrors of spheric-quality microroughness may be possible by use of the same deposition technique.

Imaging performance of telescope mirrors for far-ultraviolet astronomy

We describe image testing, surface metrology, and modeling of telescope mirrors (0.5 m in diameter, f/4.3) for the Far Ultraviolet Spectroscopic Explorer (FUSE) satellite. Laboratory image testing of wavelengths in the visible, vacuum, and midultraviolet validated a theoretical analysis by use of the Optical Surface Analysis Code (OSAC). Our modeling is based on surface metrology, including measurements of figure, midfrequency error, and microroughness. This combination of metrology, out-of-band performance testing, and modeling verified that the mirrors would meet mission requirements. We use OSAC to predict the FUSE telescope's far-ultraviolet (90-120-nm) point-spread function and assess its effect on instrument efficiency. The mirrors have a 90% encircled energy diameter of 1.5 arc sec at lambda = 100 nm. Including the effects of spacecraft pointing error, the mirrors have a predicted average slit transmission at lambda = 100 nm of approximately 87% and 96% for the 1.25- and 4-arc sec-wide spectrograph slits, respectively, where the required transmissions are 50% and 95%.

Effects of surface scatter on the optical performance of x-ray synchrotron beam-line mirrors

The increased availability of advanced synchrotron radiation sources is resulting in a resurgence of activity in the field of x-ray and extreme ultraviolet imaging systems. However, scattering effects caused by residual optical fabrication errors frequently dominate geometrical design errors in the degradation of image quality at these very short wavelengths. Traditional optical design and analysis techniques (geometrical ray tracing) are therefore inadequate for predicting the performance of high-resolution synchrotron beam-line optics. A surface-scattering theory must be implemented to model the image degradation effects of residual surface irregularities over the entire range of relevant spatial frequencies. This includes small-angle scattering effects caused by mid-spatial-frequency surface errors that fall between the traditional figure and finish specifications. Performance predictions are presented parametrically to provide insight into the optical fabrication tolerances necessary to meet the requireme nts of a specific application.

Prediction of the bidirectional reflectance-distribution function from atomic-force and scanning-tunneling microscope measurements of interfacial roughness

Computer codes that are based on Elson's theory for light scattering by interfacial roughness in multilayer coatings were used to predict the bidirectional reflectance-distribution function (BRDF) of several opaque coatings from surface-roughness profiles measured by either a scanning-tunneling microscope or an atomic-force microscope. The predictions usually agreed with measured BRDF values to within a factor of 2. The coatings consisted of single layers of Ag or Ni and dielectric stacks with up to three layers.

Extreme UV and x-ray scattering measurements from a rough LiF crystal surface characterized by electron micrography

XUV and x-ray scattering by a LiF crystal is measured. The angular distribution of the scattered radiation (ADSR) reveals characteristic features, side peaks or asymmetry. The surface of the sample is statistically characterized by a microdensitometer analysis of electron micrographs resolving the short spatial wavelengths of the surface roughness. This analysis shows that the surface has a large microroughness with an autocovariance function which is Gaussian in its initial portion. The first-order perturbation vector theory of the roughness-induced scattering leads to an interpretation of the ADSR features in terms of the modulation of the surface power spectral density function associated with the microroughness by an optical factor. The possibility of obtaining short scale roughness characterization from XUV or x-ray measurements is discussed.

Performance prediction of the AXAF Technology Mirror Assembly using measured mirror surface errors

We have developed a math model relating the measured parameters of the Technology Mirror Assembly (TMA) to its final performance. This scalar scattering model is valid for large and small amplitude features. It allows the user to specify power spectral densities and/or autocovariance functions within any spatial bandwidth, including microroughness. We present new TMA data in the bandwidth of ~0.1-1000 mm(-1), predicting performance and comparing them with x-ray test data. We also account for assembly, alignment. and particulate contamination. Finally, we comment on improved performance expected after repolishing.

Fractal surface finish

Surface finish measurements are usually fitted to models of the finish correlation function which are parametrized in terms of root-mean-square roughnesses, delta, and correlation lengths, l. Highly finished optical surfaces, however, are frequently better described by fractal models, which involve inverse power-law spectra and are parametrized by spectral strengths, K(n), and spectral indices, n. Analyzing measurements of fractal surfaces in terms of sigma and l gives results which are not intrinsic surface parameters but which depend on the bandwidth parameters of the measurement process used. This paper derives expressions for these pseudoparameters and discusses the errors involved in using them for the characterization and specification of surface finish.

Soft x-ray scattering from rough surfaces: experimental and theoretical analysis

Specularly reflected and scattered radiation was measured at lambda = 50, 100, 200 A from artificially roughened flat mirrors and from thick vacuum evaporated Au samples. The results of the roughened samples are in reasonable agreement with Beckmann's scalar theory using an exponential autocorrelation function. The angle-dependent scattering distributions at different angles of incidence and different wavelengths are described with a unique mean roughness and autocorrelation length. Some small discrepancies remain because the scalar theory does not take into account the optical constants of the scattering surface. The use of vector theory in its simplest analytical form is not successful; however, it provides a means to qualitatively correct the scalar theory for the influence of the actual optical constants in agreement with our experiments. The thick Au films are roughened by surface crystallization and yield completely different scattering distributions. These results could not be fitted to scalar theory, either with an exponential or a Gaussian autocorrelation function.