Generalized data reduction approach for aspheric testing in a non-null interferometer

Data reduction in non-null tests is difficult due to the presence of retrace error. We propose a simple yet effective data reduction approach for aspheric testing in a non-null interferometer. The new approach gives figure error of the aspheric by just subtracting the theoretical wavefront and first-order errors from the real wavefront obtained in the non-null interferometer. Precise prediction of the theoretical wavefront can be achieved by accurate calibration of the partial compensation system. The approach can be considered a generalization of the traditional data processing method in null tests, and errors that may affect its accuracy are discussed. A set of experiments have been carried out to demonstrate its validity and feasibility.     

Fabrication error analysis and experimental demonstration for computer-generated holograms

Aspheric optical surfaces are often tested using computer-generated holograms (CGHs). For precise measurement, the wavefront errors caused by the CGH must be known and characterized. A parametric model relating the wavefront errors to the CGH fabrication errors is introduced. Methods are discussed for measuring the fabrication errors in the CGH substrate, duty cycle, etching depth, and effect of surface roughness. An example analysis of the wavefront errors from fabrication nonuniformities for a phase CGH is given. The calibration of these effects for a CGH null corrector is demonstrated to cause measurement error less than 1 nm.     

Adaptive control in an adaptive optics experiment

This paper presents results from an adaptive optics experiment in which an adaptive control loop augments a classical adaptive optics feedback loop. Closed-loop wavefront errors measured by a self-referencing interferometer are fed back to the control loops, which drive a membrane deformable mirror to correct the wavefront. The paper introduces new frequency-weighted deformable mirror modes used as the control channels and new wavefront sensor modes for analyzing the performance of the control loops. The corrected laser beam also is imaged by a diagnostic target camera. The experimental results show reduced closed-loop wavefront errors and correspondingly sharper diagnostic target images produced by the adaptive control loop as compared with the classical AO loop.     

Estimation of phase shifts linked only to the coating for a dielectric mirror

The wavefront is an important characteristic for a dielectric mirror. Its measurement is usually performed with interferometers. We introduce a new method to evaluate only the coating wavefront distortion due to nonuniform thickness errors by using a reflectometer. This method uses some reflectance or transmittance maps at a wavenumber σ(m) for which the reflectance or transmittance factor variation is high. These variations are translated into some central wavenumber σ(c) variations, which enables the determination of a phase map from experimental treatment.     

Primary Aberrations for a Refracting Surface of Revolution

Abstract

Primary aberration coefficients are evaluated for a refracting surface of revolution between two homogeneous media when it is illuminated by a monochromatic plane wavefront whose propagation direction makes a small angle with the symmetry axis of the surface.     

Shock front analysis for axial shear wave propagation in a hyperelastic incompressible solid

Summary A wavefront analysis is employed to study the propagation of axial shear waves in an incompressible hyperelastic solid, whose strain energy function is expressible as a truncated power series in terms of the basic invariants of the left Cauchy-Green tensor. Waves are generated by the application of an axial shear stress at the surface of a cylindrical cavity in an unbounded medium. Depending on the nature of the boundary condition, an acceleration front or a shock front propagates from the boundary of the cavity. For an acceleration front, the coefficients in the wavefront expansion satisfy a sequence of transport equations which can be solved analytically. For a shock front, a wavefront analysis gives approximate formulas for the wave speed, shock front and intensity of the various field variables at the front. As well, our shock front analysis is used to devise a method of estimating the breaking distance of a shock front. In order to test the validity of the results of our wavefront analysis, numerical solutions are obtained for waves initiated by a step function or by a finite duration pulse at the boundary. Our numerical solutions are found by using a recently proposed relaxation scheme for systems of conservation laws.     

高精度球面检测中调整误差的精确校正

针对高精度球面检测中球面的调整误差影响,分别研究了波前离焦和倾斜所引入的Zernike像差项,并以此为基础,提出了基于Zernike系数的高精度球面调整误差校正方法.利用Zygo干涉仪对大数值孔径的待测球面进行了检测,实验中对待测球面同时引入波前离焦和倾斜误差,再分别运用所提出的校正方法和传统的消倾斜离焦校正方法得到波面数据并进行对比,进而验证了所提出的校正方法的可行性.实验结果表明,利用该方法可有效地校正大数值孔径球面存在调整误差时所引入的球面像差,并且实验中的校正精度达到了0.002λ（λ为检测光波长）.提出的基于Zernike系数的球面调整误差校正方法可降低对于调节机构的精度以及检测人员的经验要求,在高精度球面检测中具有非常高的实际应用价值.     

Absolute interferometric testing based on reconstruction of rotational shear

A method, believed to be new, for the absolute interferometric testing of flat or spherical surfaces is presented. It is based on the classic three-flat test, combined with additional measurements of one test piece in different rotational positions. Full-surface absolute maps for each test piece are determined with a data-processing technique based on the rotationally sheared maps of the rotated surface. An optimized numerical reconstruction algorithm employing linear filtering and superposition of the different rotational shear spectra in the angular frequency domain is used to reconstruct the rotationally sheared data. The technique does not require any assumptions about the surfaces under test; has low error propagation, even in the case of high spatial resolution; and is computationally efficient.     

信号时域平均处理的新算法

时域平均是从混有噪声的复杂周期信号中提取感兴趣周期分量的常用方法,但在其实施时往往由于周期截断误差的存在而不能取得满意的效果。本文提出了信号时域平均处理的新算法,基本上解决了周期截断误差对平均结果的影响问题。分析验证说明：新算法比过去的旧算法效果好得多     

Differential Shack-Hartmann curvature sensor: local principal curvature measurements

The concept of a differential Shack-Hartmann (DSH) curvature sensor was recently proposed, which yields wavefront curvatures by measuring wavefront slope differentials. As an important feature of the DSH curvature sensor, the wavefront twist curvature terms can be efficiently obtained from slope differential measurements, thus providing a means to measure the Monge-equivalent patch. Specifically, the principal curvatures and principal directions, four key parameters in differential geometry, can be computed from the wavefront Laplacian and twist curvature terms. The principal curvatures and directions provide a "complete" definition of wavefront local shape. Given adequate sampling, these measurements can be useful in quantifying the mid-spatial-frequency wavefront errors, yielding a complete characterization of the surface being measured.     

Wave-optical evaluation of interference fringes and wavefront phase in a hard-x-ray beam totally reflected by mirror optics

The intensity flatness and wavefront shape in a coherent hard-x-ray beam totally reflected by flat mirrors that have surface bumps modeled by Gaussian functions were investigated by use of a wave-optical simulation code. Simulated results revealed the necessity for peak-to-valley height accuracy of better than 1 nm at a lateral resolution near 0.1 mm to remove high-contrast interference fringes and appreciable wavefront phase errors. Three mirrors that had different surface qualities were tested at the 1 km-long beam line at the SPring-8/Japan Synchrotron Radiation Research Institute. Interference fringes faded when the surface figure was corrected below the subnanometer level to a spatial resolution close to 0.1 mm, as indicated by the simulated results.     

Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors

In both temporal and spatial carrier phase shifting interferometry, the primary source of phase calculation error results from an error in the relative phase shift between sample points. In spatial carrier phase shifting interferometry, this phase shifting error is caused directly by the wavefront under test and is unavoidable. In order to minimize the phase shifting error, a pixelated spatial carrier phase shifting technique has been developed by 4D technologies. This new technique allows for the grouping of phase shifted pixels together around a single point in two dimensions, minimizing the phase shift change due to the spatial variation in the test wavefront. A formula for the phase calculation error in spatial carrier phase shifting interferometry is derived. The error associated with the use of linear N-point averaging algorithms is presented and compared with those of the pixelated spatial carrier technique.     

Microstructural features and mechanical properties of AM60 and AZ31 friction stir spot welds

The microstructural features and mechanical properties of AM60 and AZ31 friction stir spot welds are investigated in joints made using different tool designs (threaded and three-flat/threaded tools) and dwell time settings. Since the hook regions are curved inwards towards the keyhole periphery in AM60 friction stir spot welds made using threaded and three-flat/threaded tools and different dwell time settings, the distance from the tip of the hook region to the keyhole periphery mainly determines their failure load properties. In contrast, the hook regions are curved outwards from the axis of the rotating tool in AZ31 friction stir spot welds and their failure strength properties are determined by the bonded width, the distance from the tip of the hook region to the sheet intersection, the depth of tool shoulder penetration into the surface of the upper sheet and the distance from the tip of the hook region to the top of the welded joint.     

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.     

A Model for Non-Rayleigh Scattering Statistics

The independent-scatterer K -distribution model, which has been introduced to describe a variety of scattering situations, is extended to include the effects of correlation between scatterers and finite illumination size. A model in which the array of scatterers is represented by a Γ -lorentzian cross-section fluctuation is proposed. In the appropriate limits this reduces to the independent K -distribution model. Following scattering by the Γ -lorentzian surface, the autocorrelation function and moments of the detected intensity for radiation of arbitrary beamwidth and wavefront curvature are derived. The results are compared with the predictions of the independent K -distribution model and the implications of the differences, which reflect the fact that the independent model cannot represent spatial averaging over the correlations within the surface, are discussed.     

How are glories formed?

Mie theory can be used to generate full-color simulations of atmospheric glories, but it offers no explanation for the formation of glories. Simulations using the Debye series indicate that glories are caused by rays that have suffered one internal reflection within spherical droplets of water. In 1947, van de Hulst suggested that backscattering (i.e., scattering angle theta = 180 degrees) could be caused by surface waves, which would generate a toroidal wavefront due to spherical symmetry. Furthermore, he postulated that the glory is the interference pattern corresponding to this toroidal wavefront. Although van de Hulst's explanation for the glory has been widely accepted, the author offers a slightly different explanation. Noting that surface waves shed radiation continuously around the droplet (not just at theta = 180 degrees), scattering in a specific direction theta = 180 degrees - delta can be considered as the vector sum of two surface waves: one deflecting the incident light by 180 degrees - delta and the other by 180 degrees + delta. The author suggests that the glory is the result of two-ray interference between these two surface waves. Simple calculations indicate that this model produces more accurate results than van de Hulst's model.     

Wavefront error measurement of high-numerical-aperture optics with a Shack-Hartmann sensor and a point source

We developed a new, to the best of our knowledge, test method to measure the wavefront error of the high-NA optics that is used to read the information on the high-capacity optical data storage devices. The main components are a pinhole point source and a Shack-Hartmann sensor. A pinhole generates the high-NA reference spherical wave, and a Shack-Hartmann sensor constructs the wavefront error of the target optics. Due to simplicity of the setup, it is easy to use several different wavelengths without significant changes of the optical elements in the test setup. To reduce the systematic errors in the system, a simple calibration method was developed. In this manner, we could measure the wavefront error of the NA 0.9 objective with the repeatability of 0.003 lambda rms (lambda = 632.8 nm) and the accuracy of 0.01 lambda rms.     

Adaptive Control of a Micromachined Continuous Membrane Deformable Mirror for Aberration Compensation

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Aperture size effect on ultrasonic wavefront distortion correction

The influences of aperture size on wavefront distortion correction are investigated both theoretically and numerically. A multilayer, phase-screen model is assumed to be the underlying, distorting medium. Numerical simulations were performed using three wavefront distortion correction methods: time-shift compensation (TSC), backpropagation followed by time-shift compensation (BP+TSC), and the previously proposed, multilayer, phase-screen compensation (MPSC) method. The distorted wavefronts were generated by propagating a planar wavefront through a multilayer, phase-screen model constructed with a two-dimensional (2-D) scanned map of a real abdominal slice. Performances were evaluated by L2 errors between the corrected wavefronts and the undistorted planar wavefront. Point spread functions also were calculated to evaluate the relative image quality. Theoretical analysis shows L2 error will decrease as aperture size grows when exact phase compensation (EPC) is applied, although finite errors will always exist along the edges of the corrected wavefront. Three different aperture sizes, 14.24 mm (64 elements), 28.48 mm (128 elements), and 56.96 mm (256 elements) are considered in this study. Numerical results show that the quality of wavefront with EPC is essentially limited by the aperture size, and the correction methods considered are relatively robust against the aperture size. It also shows that, for low aberration, results with MPSC and EPC are comparable. However, for high aberration, MPSC significantly outperforms EPC in suppression of L2 error and sidelobes. This study suggests that, for most medical ultrasound imaging systems, the exact structure of the distorting medium may not be necessary to be known a priori for optimal distortion correction because of the limitation imposed by finite aperture size.