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# Aberrations of off-axis aspheric subapertures

The curvature of aspheres varies with the off-axis distance, and so does the aspheric departure. At the same time, how big the subaperture could be depends first on whether its departure is small enough to generate resolvable fringes. Consider an off-axis subaperture of a convex conic surface

where R is the radius of curvature at the vertex (negative for convex), and e is the eccentricity. The coordinates of the geometric center of the subaperture are (x0, 0, z0) in the parent frame. The optical axis of the testing system is approximately the surface normal at the geometric center when testing the subaperture. A local frame is, therefore, built as shown in Fig. 6.

By simple coordinate transformation, the local coordinates of the measuring points are related as follows:

where p is the off-axis angle between the optical axis of the testing system and that of the parent surface. Therefore, coordinate z can be solved as a function of (x, y). With spherical component zs subtracted from z, we then get the analytical description of the wavefront aberration. By using the Maclaurin series expansion up to third-order terms, the wavefront aberration is described as follows:

The other terms disappear because the off-axis is purely in the x-direction without loss of generality. The coefficients are explicitly related to Seidel aberrations56 and Zernike polynomial terms Z4 (astigmatism at 0 deg and focus), Z6 (coma and x-tilt), and Z9 (trefoil) in Cartesian coordinates:

The corresponding coefficients are obtained as follows:

Now we have

The trefoil is far less than the coma when e2sin2p is far less than 1. Moreover, P4, P*6, and P9 are quadratically, linearly, and cubically proportional to p, respectively, if p is small. Therefore, in the testing of off-axis subapertures, we may correct most of the astigmatism and coma while leaving the trefoil uncorrected.41

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