Optical surface characterization with the structure function
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Abstract
It is important to characterize surface and transmitted wavefront errors in terms of the spatial content. The errors are typically analyzed in three spatial domains: figure, ripple (or mid-spatial frequency) and roughness. These errors can affect optical system performance. For example, mid-spatial frequency errors can lead to self-focusing and power loss in a high-power laser system. Currently, power spectral density (PSD) is used for the spatial content characterization in high-end optics, although there are potential pitfalls. For example, the low spatial content is removed before calculation, only a small fraction of surface data are used, and the results are sensitive to details like the windowing. As an alternative, the structure function (SF) does not have such problems. It is the expectation of the squared height difference as a function of separation. The linear SF has been used in astronomy and captures data of all spatial frequencies. However, it does not capture anisotropy on the surface. The two-quadrant area SF introduced in this dissertation obviates this problem. It is computationally correct for any arbitrary aperture over all spatial content with anisotropic information. This dissertation discusses some computational issues of the SF, which includes the calculation of the linear / area SF, sliding sampling method for large numbers of points within the aperture, analysis of periodic errors, and connection between the linear SF and area SF. Moreover, the relationships between the SF and other surface characterization techniques (Zernike polynomials, autocorrelation function (ACF), PSD, and RMS gradient) have been investigated. It turns out that the linear SF of the sum of the Zernike terms only equals to the sum of the linear SF of each of the Zernike polynomials with different azimuthal frequencies. However, this theorem does not apply to the area SF. For stationary surfaces, the SF contains similar information as ACF, but it provides better visualization. The SF is computationally correct for any arbitrary aperture shape without extra processing, while the PSD always needs additional mathematical processing. After connecting the SF to the RMS gradient, the SF slope at the origin has been evaluated. Use of a SF to specify optical surfaces over the full range of spatial frequencies of interest implies the combination of data from instruments with substantially different lateral resolutions. This research shows the combination of data from a Fizeau and a coherence scanning interferometer (CSI) for various precision surfaces. The investigation includes the connection method of the coordinate systems between the Fizeau data and the CSI sub-aperture data, the convergence of the averaged SF of sub-aperture samples, the uncertainty analysis, and the effect of the instrument transfer function (ITF). In addition, the SF was used to explore two typical noise contributions (electronic noise and air turbulence) in phase shifting interferometry. Based on dynamic measurements, the SF was used to analyze the spatial components of a diamond turned surface after the compensation machining. In summary, the SF is a useful tool to specify and characterize the spatial content of optical surfaces and wavefronts.