<p> 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.</p><p> 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. </p><p> 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. </p><p> 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. </p><p> 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. </p><p> 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). </p><p> 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. </p><p> In summary, the SF is a useful tool to specify and characterize the spatial content of optical surfaces and wavefronts.</p>

<p> Aero-optical distortions are detrimental to airborne optical systems. To study distortion mechanisms, compressible large-eddy simulations are performed for a Mach 0.5 turbulent boundary layer and a separated shear layer over a cylindrical turret with and without passive control in the upstream boundary layer. Optical analysis is carried out using ray tracing based on the computed density field and Gladstone-Dale relation.</p><p> In the flat-plate boundary layer, the effects of aperture size, Reynolds number, small-scale turbulence, different flow regions and beam elevation angle are examined, and the underlying flow physics is analyzed. Three momemtum-thickness Reynolds numbers, <i>Re</i>_&thetas; = 875, 1770 and 3550, are considered. It is found that the level of optical distortions decreases with increasing Reynolds number within the Reynolds number range considered. The contributions from the viscous sublayer and buffer layer are small, while the wake region plays a dominant role followed by the logarithmic layer. By low-pass filtering the fluctuating density field, it is shown that small-scale turbulence is optically inactive. Consistent with previous experimental findings, the distortion magnitude is dependent on the propagation direction due to anisotropy of the boundary-layer vortical structures. Density correlations and length scales are analyzed to understand the elevation-angle dependence and its relation to turbulence structures. The applicability of Sutton's linking equation to boundary-layer flows is examined, and excellent agreement between linking equation predictions and directly integrated distortions is obtained when the density length scale is appropriately defined.</p><p> The second case studied involves a separated shear layer over a cylindrical turret with a flat window, with inflow from a flat-plate boundary layer with and without passive control devices. The flow and optical results show reasonable agreement with experimental data for the baseline case without control. Aperture size effect, frequency spectra of OPD and two-point spatial correlations of OPD are investigated. The similarities and differences of distortion characteristics compared to those induced by turbulent boundary layers are discussed. The distortions by a separated shear layer are much larger in magnitude and spatially less homogeneous than those induced by an attached boundary layer. It is found that pressure fluctuations are significant and play a dominant role in inducing density fluctuations and associated optical distortions in a separated shear layer, in contrast to the dominant role of temperature fluctuations in a turbulent boundary layer. When passive control is applied using a row of thin and tall pins in the upstream boundary layer, the numerical results confirm key experimental findings. The flow above the optical window is characterized by two distinct shear layers, whose combined effect leads to a significant reduction of density fluctuation magnitude in the main shear layer and associated optical distortions compared to the uncontrolled flow with a single strong shear layer.</p>