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# Object-image mapping with ray tracing

Suppose the measured data are a set of triplets W, = {wj,t=(uj,i, v,,,, ф,,,,)}, i = 1,2,... ,s, and j = 1,2,... ,Ni, where фj is the measured height on pixel (%, v,,,), s is the number of subapertures, and N is the number of measuring points in subaperture i. The triplets are first mapped into a Cartesian frame CM, which is attached to the nominal surface and thereafter called the model frame (also the global frame). The origin of the model frame is the surface vertex, and the z axis is the optical axis, as shown in Fig. 19. By ray tracing of the null or near-null optics, the lateral coordinates (x0,,,, jy0,,,) on the nominal surface are related to the interferometer CCD pixels (u,,,, v,,,). This object-image mapping can also be derived analytically for flat or spherical subapertures because the test geometry is exactly known.32

It is then easy to calculate the z coordinate zj with the lateral coordinates by applying the surface equation. We further get the coordinates (x,,,, jj-,„ z,,,) of the measuring point by adding the amount of ф,,, and subtracting the residual aberration along the normal direction:

where e,,, is the residual aberration after compensation by near-null optics. It is given by the nominal near-null subaperture layout design. This term is ideally zero for null subaperture stitching. n,,, is the unit vector of the surface normal, and r, is the defocus coefficient for subaperture i. As per the sign convention of surface normal, ф,,, is positive for peaks on the test surface, corresponding to a negative optical path difference. Subaperture piston, tip, and tilt can be similarly incorporated in Eq. (32).

We further include the systematic error in Eq. (32), which is identical for all subapertures, e.g., the reference surface error of the TF or TS. For null or nearnull tests, the systematic error induced by misalignment of null or near-null optics is identical for those subapertures of the same ring on the asphere. Such a systematic error can be optimally recognized and then separated from the surface error by virtue of the overlapping redundant information. Self-calibration is realized simultaneously during stitching optimization. Suppose the systematic error is described by Zernike polynomials

where cq is the polynomial coefficient, and Q is the number of polynomial terms. The coordinates (xp, у zp) of the measuring point are now calculated by offsetting the point (х/, y0p, z0p) on the nominal surface along the normal direction:

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