# Non-null subaperture layout design

It is increasingly convincing that subaperture stitching interferometry is flexible for the testing of a variety of surfaces, but what is the limit of its capability? In non-null testing, whether the subaperture can be tested or not is still determined by the aspheric departure that varies with the off-axis distance. Theoretically, the subaperture can be small enough to reduce the departure, but this is impractical because the number of subapertures increases sharply. Therefore, it is important to quickly estimate the applicability or complexity of non-null subaperture stitching interferometry for a given aspheric surface.^{43}

Non-null subaperture layout design includes the determination of subaperture location and interferometer parameters, and the calculation of subaperture departure from the best-fit sphere. The location of subapertures is basically determined by the requirements of lateral resolution, overlapping ratio, and full-aperturecovering capability.^{42} Here, we can still apply Eq. (1) as a simple approximation for the rough arrangement of subapertures, but the layout design is also related to the subaperture departure, which must not exceed the dynamic range of the interferometer. Therefore, the main problem involves calculating the aspheric departure of a subaperture whose location is given arbitrarily. If the departure exceeds the vertical range of an interferometer, we should change the subaperture location or change the interferometer parameters, such as the TS, the zoom, and so on. Fortunately, we do not need to meet any of the previous requirements accurately in designing the subaperture layout. For example, the overlapping ratio ranging from 20% to 50% is allowed and makes little difference to the stitching algorithm owing to its simultaneous stitching nature.

In 2006, we proposed a subtle method for layout design by minimizing the mean-square aspheric departure in the form of a surface integral.^{42} It is sophisticated since minimization of a surface integral is involved. However, a fast version of the method can be obtained with proper approximation.^{43}

Suppose the geometrical center (x0, 0, Z0) of subaperture i is on the generatrix of a rotationally symmetrical surface. The radius of curvature at the subaperture center is r. The angle between the normal vector and the global optical axis z is denoted by p. If *X**0** =* 0 or p = 0, we get the central subaperture. A local coordinate frame {*i*} is attached to the subaperture with the *z* axis parallel to the normal. The slight difference between the normal and the real optical axis of the interferometer is considered by the removal of piston, tilt and power of the subaperture data. The origin is set at the vertex of the TS (or its image about the focus of the test beam). A global model frame {M} is attached to the surface with the origin at the vertex (see Fig. 7).

The configuration of the model frame { *M*} with respect to the local frame { *i*} is denoted by *gi,* whose inverse is

where *r*ts is the radius of the TS.

**Figure 7 **Subaperture test configuration.

**Figure 8 **Two latitudinally adjacent subapertures.

For subaperture *k,* which is obtained by rotating subaperture i around the global optical axis z by an angle y, the configuration of the frame {M} with respect to the local frame {k} can be written as follows (see Fig. 8):

where *gi* is given in Eq. (10), and

where l_{x} = -[x_{0} -(r -r_{ts})sin p]cos p, and *l _{z}* = [x

_{0}-(r -r

_{ts})sin p]sin p. The rotation matrix R

*=*exp(6y) is described as an exponential equation, and ш = [ sin p 0 cos p ]

^{T}is the twist coordinate of rotation.

^{57}The matrix R represents rotation around the axis oriented with vector ш by the angle y. (Readers unfamiliar with the exponential equation of rigid body transformation should refer to a general robotics textbook.) Rotation around an arbitrarily spatial axis can be written as the compound of rotations around the coordinate axes.

Now the interferometric testing of subapertures can be simulated to calculate the aspheric departure. The measurement data of subaperture interferometry are triplets *W = (u,* v, ф). According to the test geometry, with a spherical interferometer as shown in Fig. 9, the object coordinates are related as follows (the triplets are inversed before they are used due to the inverted imaging property):

**Figure 9 **Test geometry of a spherical interferometer for concave surfaces.

For convex surfaces, the coordinate z, in Eq. (12) should be modified as follows:

The object coordinates are further transformed into the model frame {M}:

Applying the surface equation f(x, *y, z)* = 0 yields a closed-form equation of ф. Therefore, we obtain the complete triplets (u, v, ф) by solving the equation. Note the implied assumption that the subaperture is tested with the interferometer axis superposed on the normal at the geometrical center. This is usually not true due to the asymmetry of an off-axis subaperture. That is, the center and the radius of the best-fit sphere are usually not the curvature center and radius of curvature at the subaperture center, respectively. The actual size of the tested subaperture will probably affect the radius of the best-fit sphere, and vice versa. This is the major reason we proposed the minimization model in the form of a surface integral.^{42} In fact, we find the consequent difference of subaperture departure is small due to minor defocus and misalignment between the interferometer axis and the surface normal. It is negligible because precise calculation of the departure is not necessary for layout design. We further obtain the simulated measurement data set with piston, x-tilt, y-tilt, and power removed on the LS principle by minimizing the residual phases as follows:

where *j* indicates the measurement point j. Parameters a, *b, c,* and d are coefficients of the piston, x-tilt, y-tilt, and power, respectively. From the residuals, the peak-to-valley aspheric departure is calculated to estimate the number of fringes. The intensity function is then calculated as follows:

where *b/a* represents the contrast, and X is the wavelength. To visualize the subin- terferogram, we plot the two-level contour of the intensity function.

For example, we consider a convex hyperbolic surface with a clear aperture of 360 mm, where the radius of curvature at the vertex is 772.48 mm, and a square of eccentricity e^{2} = 2.1172. The full-aperture aspheric departure is about 150.7 pm. The surface equation is *(z -* a)^{2}/a^{2} - (x^{2} + y^{2})/b^{2} *=* 1, where *a = *691.4429, and b = 730.8391 mm. In addition to the central one, there are 141 off-axis subapertures distributed in six cycles (6, 12, 18, 24, 36, and 45 subapertures with p = 2.2 deg, 4.5 deg, 6.8 deg, 9.1 deg, 11 deg, and 12.7 deg, respectively). The TS radii of the central one and off-axis cycles are 1500, 1500, 1500, 1500, 1800, 2200, and 2200 mm, respectively. A converger off/15 is adopted with the appropriate zoom. The interferograms of off-axis subapertures in each cycle are plotted in Fig. 10, with 5.8, 21.3, 47.4, 58.6, 58.0, and 50.7

**Figure 10 **Subinterferograms of a convex hyperbolic surface.

**Figure 11 **Subaperture layout definition for a convex hyperbolic surface.

fringes, respectively. The subaperture layout is shown in Fig. 11. The large number of subapertures may increase the risk of subaperture stitching interferometry.