Distortion mapping correction in aspheric null testing

Abstract

We describe methods to correct both symmetric and asymmetric distortion mapping errors induced by null testing elements such as holograms or null lenses. We show experimental results for direct measurement and correction of symmetric mapping distortion, as well as an example result for analytical mapping performed using an orthogonal set of vector polynomials for asymmetric correction. The empirical determination of symmetric distortion is made via calculation from predicted and measured changes to aberrations induced via known changes to the testing point.

Rectangular domain curl polynomial set for optical vector data processing and analysis

“…Although most imaging systems suffer from some amount of distortion, one case where it is very significant is interferometric null testing of steep aspheric surfaces. 24,25 In most cases of imaging distortion, it is relatively easy to quantify and find a mapping correction for the distortion. The most common approach is to place a set of fiducial markers on the optic being tested, which have known, measured locations.…”

“…Since we are dealing with discrete data, we can write Eq. (24) as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 5 ; 3 2 6 ; 3 5 3 R ¼Ca; (25) where R is a column vector of P data values, a is a column vector containing the N expansion coefficients, and C is a P × N matrix representing the values of the C polynomials at the locations of the data points. The coefficients can then be found using a pseudoinverse E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 6 ; 3 2 6 ; 2 6 6 a ¼ ðC TC Þ −1CT R: Based on the polynomial terms, which stood out from all the processed measurements and showed a steady change (increase or decrease) as the misclocking between the two slopes increased, we constructed a model that predicted the slope clocking mismatch between the x and y slopes when an "unknown" misclocking was introduced.…”

Imaging issues for interferometry with CGH null correctors

“…This is done by placing reference marks on the optic under test and locating their position in the image. 8 Mapping relations are derived by fitting polynomial functions to the data to provide a transformation of the distorted data back to real coordinates. We have developed an orthonormal vector basis 3 , defined by derivatives of Zernike polynomials, for this purpose.…”

Parametric registration of cross test error maps for optical surfaces

“…It is assumed that all error maps project the distribution of errors in height onto a certain plane, e.g., the plane perpendicular to the geometrical axis of the test surface. Possible image distortion in the interferometric null test [5], for example, is well corrected before generating the error map. Therefore the misalignment between two error maps we need to consider includes only the piston, tip-tilt, lateral shift in X and Y directions, clocking (rotation around the normal of the image plane) and scaling.…”