Resolving capacity of the circular Zernike polynomials

Topography stitching in the spatial frequency domain for the representation of mid-spatial frequency errors

Sub-aperture fabrication techniques such as diamond turning, ion beam figuring, and bonnet polishing are indispensable tools in today's optical fabrication chain. Each of these tools addresses different figure and roughness imperfections corresponding to a broad spatial frequency range. Their individual effects, however, cannot be regarded as completely independent from each other due to the concurrent formation of form and finish errors, particularly in the mid-spatial frequency (MSF) region. Deterministic Zernike polynomials and statistical power spectral density (PSD) functions are often used to represent form and finish errors, respectively. Typically, both types of surface errors are treated separately when their impact on optical performance is considered: (i) wave aberrations caused by figure errors and (ii) stray light resulting from surface roughness. To fill the gap between deterministic and statistical descriptions, a generalized surface description is of great importance for bringing versatility to the entire optical fabrication chain by enabling easy and quick exchange of surface topography data between three disciplines: optical design, manufacturing, and characterization. In this work, we present a surface description by stitching the amplitude and unwrapped phase spectra of several surface topography measurements at different magnifications. An alternative representation of surface errors at different regimes is proposed, allowing us to bridge the gap between figure and finish as well as to describe the well-known MSF errors.

Development of Singular Points in a Beam Passed Phase Screen Simulating Atmospheric Turbulence and Precision of Such a Screen Approximation by Zernike Polynomials

This article addresses two issues. Firstly, it was shown that if the initial phase of a Gaussian beam is specified by the sum of Zernike polynomials or by a screen simulating atmospheric turbulence, in the process of propagation, singular points appear in the wavefront of such a beam. With the use of numerical simulation, the dependence of the vortices number on the distortion characteristics and on the distance traveled by the beam was determined. The second problem analyzed in the article is the problem of a phase screen approximation by a series formed by Zernike polynomials. The carried out numerical experiments made it possible to determine the dependence of approximation accuracy on the screen parameters and on the number of polynomials entering the basis of approximation.

Aberration characteristic correlation configuration optimization of a conformal dome based on the von Karman surface

In this paper, the von Karman surface is used in the configuration design of the infrared conformal dome to improve its aerodynamic performance. The principle of differential geometry is used to study the geometric characteristics of the von Karman dome. Additionally, by using ray tracing, the geometric aberrations and wave aberrations of the von Karman dome are analyzed. Further, considering the geometric characteristics and aberration characteristics, an optimization method for the configuration of the von Karman dome is proposed. To prove the effectiveness of the optimization method, the aberrations introduced by the conformal dome after the configuration optimization and the original von Karman dome are compared. The comparison showed that the geometric aberration of the optimized conformal dome is reduced by 43.68%. The optimization method can significantly correct the aberration introduced by the von Karman dome and improve the guidance capability of infrared detection technology.

Representation of complex optical surfaces with adaptive radial basis functions

For state-of-the-art optical elements in laser fusion applications, optical surfaces, especially those with residual fabrication error, are generally complex in symmetry, shape of aperture, and spatial frequency distribution. It is critical to represent the optical surface with high accuracy, efficiency, and flexibility during the stages of design, fabrication, and testing. For this purpose, in this paper, we propose that adaptive radial basis functions (ARBF) can be applied to represent the complex optical surface. As more degrees of freedom are harnessed by an adaptive algorithm, the proposed approach presents better accuracy and needs fewer basis functions than that of the classical radial basis functions. Both surfaces, with global and local variation, can be well represented by the proposed method. Furthermore, the fitting ability of ARBF is verified with the measured data from an element polished by magnetorheological finishing technology. Optimization frameworks of the shape parameters for practical use are also discussed. For good measure, spatial relevance between surface height and nodes location shows that ARBF is fairly flexible in representing complex optical surfaces.

Description and reimplementation of real freeform surfaces

Freeform surfaces are becoming an increasingly exciting opportunity in optical design, in particular when correcting systems with off-axis geometries. Nevertheless, especially when coming to commercial use, the challenges for manufacturing are difficult to handle. The optical quality of a system is perturbed by typical deformations, such as localized figure errors and regular mid-spatial-frequency ripples, that come from the diamond-turning process. In this proposal, we investigated a workflow for analyzing the impact of real optical surfaces on the optical performance for even complex systems. Based on a simple and robust description, the surface is implemented back into the design. While the more localized deviations are analytically described by radial basis functions, the residual ripple structures are covered by a new approach based on the power spectral density. The reimport of optical surfaces back into the design software allows simple estimations for the requirements on manufacturing and the analysis of the realistic impact on system performance.

Ion-beam polishing of fused silica substrates for imaging soft x-ray and extreme ultraviolet optics

We have studied the surface treatment of polished fused silica by neutralized Ar ions with energy of 500-1500 eV and incidence angles of 0-90°. We found the following regularities: for samples that passed the standard procedure of deep polishing (initial effective roughness σ(eff)∼0.5  nm), the effective roughness decreases to the ultrasmooth level (i.e., σ(eff)∼0.25  nm in the range of spatial frequencies q∈[4.9×10(-2)-63]  μm(-1)). The effect begins to be noticeable at the material removal of 150 nm and reaches saturation at depths of removal greater than 1 μm. For supersmooth samples (σ(eff)<0.3  nm), the effective roughness keeps the initial level at material removal down to tens of micrometers. The optimal ion energy range is 800-1300 eV (maximum smoothing effect); at higher energy some surface roughness degradation is observed. All the smoothing effects are observed at the incidence angle range θ(in)=0-35°. Increasing the ion energy above 1300 eV increases the etching rate by up to 4 μm per hour (J(ion)=0.8  mA/cm2), which allows for deep aspherization of sized substrates. The technique allows for manufacturing the optical elements for extreme ultraviolet and soft x-ray wavelength ranges with a numerical aperture of up to 0.6.

Problems in the application of a null lens for precise measurements of aspheric mirrors

Problems in the application of a null lens for surface shape measurements of aspherical mirrors are discussed using the example of manufacturing an aspherical concave mirror for the beyond extreme ultraviolet nanolithographer. A method for allowing measurement of the surface shape of a sample under study and the aberration of a null lens simultaneously, and for evaluating measurement accuracy, is described. Using this method, we made a mirror with an aspheric surface of the 6th order (i.e., the maximum deviation from the best-fit sphere is 6.6 μm) with the parameters of the deviations from the designed surface PV=5.3  nm and RMS=0.8  nm. An approximation of the surface shape was carried out using Zernike polynomials {Z(n)(m)(r,φ),m+n≤36}. The physical limitations of this technique are analyzed. It is shown that for aspheric measurements to an Angstrom accuracy, one needs to have a null lens with errors of less than 1 nm. For accurate measurements, it is necessary to establish compliance with the coordinates on the sample and on the interferogram.