Application of the random ball test for calibrating slope-dependent errors in profilometry measurements

Abstract

Optical profilometers such as scanning white light interferometers and confocal microscopes provide high-resolution measurements and are widely utilized in many fields for measuring surface topography. Slope-dependent systematic errors can be present in the measurement and can be the same order of magnitude as features on the surface to be measured. We propose a self-calibration technique, the random ball test (RBT), for calibrating slope-dependent errors of such instruments. The calibration result can be used to compensate future measurements of similar spherical geometries such as profiles of refractive microlenses. A simulation study validates the approach and shows that the RBT is effective in practical limits. We demonstrate the calibration on a 50× confocal microscope and find a surface slope-dependent bias that increases monotonically with the magnitude of the surface slope and is as large as ∼800 nm at a surface slope of 12°. The uncertainty of the calibration is smaller than the observed measurement bias and is dominated by residual random noise. Effects such as drift and ball radius uncertainty were investigated to understand their contribution to the calibration uncertainty.

Absolute interferometric test of cylindrical wavefront with a fiber optic

“…The random ball test assumes a large number of interferograms of a good quality sphere with errors that are statistically distributed such that the average of the errors goes to zero. [3][4][5][6][7] The fiber optic reference test utilizes a specially processed optical fiber to provide a high quality reference wave from an incident line focus from the cylindrical wave under test. [13][14][15][16] By taking measurements at different rotations and translations of the fiber, an analogous procedure can be employed to determine the quality of the converging cylindrical wave with high accuracy-a "random fiber" test.…”

“…This is a onedimensional (1-D) analogy to a Cali ball. [3][4][5][6][7] The fiber optic is translated along its length various Δy j distances to get additional information about the fiber. The measured wavefront for each shift distance is…”

“…[3][4][5][6][7] In this step, the fiber is rotated around its axis Δϕ i , and each time it is rotated, an arbitrary new patch of the fiber surface is illuminated and produces a different wavefront measurement:…”

“…This is the complete cylindrical analogue of the Cali ball test [3][4][5][6][7] where the fiber is shifted and rotated randomly. The resulting wavefront measurement of each patch is…”

“…Examples of the techniques to calibrate a spherical reference are the even/odd decomposition, 1 symmetric/asymmetric decomposition, 2 and the Cali ball averaging method. [3][4][5][6][7] Techniques to calibrate the test part are rotationally and nonrotationally symmetric decomposition 8 and also even/odd decomposition. 9,10 From the review of the absolute testing method of spherical surfaces, it is apparent that these methods cannot be directly employed for absolute testing for cylindrical surfaces.…”