**1. Introduction**

O ff-axis aspheric optical elements are often used in modern optical systems such as o ff-axis TMA (Three Mirror Anastigmatic), telescope (e.g., GMT (Giant Magellan Telescope), TMT (Thirty Meter Teloscope), E-ELT (European Extremely Large Telescope)), and so on [1,2]. In order to achieve high quality, the o ff-axis aspheric optical element has become the focus of research in optical manufacturing. To evaluate the quality of o ff-axis aspheric surface, the related optical measurement technique is necessary to gain the shape accuracy, which is the key parameter of the optical element. Usually, interferometry is commonly used to test the shape accuracy of the flat and the spherical optical elements. However, until now, there has been no unified method for aspheric surfaces, especially o ff-axis aspheric surfaces, to test their profile with a nano-precision.

To measure the shape error of the o ff-axis aspheric surface at a high accuracy, researchers have o ffered several useful methods [3–13]. Wang Xiao-kun tested an o ff-axis ellipsoid mirror with sub-aperture stitching interferometry and applied least-squares fitting to process the test data, resulting in a 1.275λ of PV and 0.113λ of RMS [4]. Similarly, Yongfu Wen tested an o ff-axis hyperboloid mirror with o ff-axis annular sub-aperture stitching interferometry and obtained the results by a complex calculation [5]. Obviously, this sub-aperture stitching technology requires more measuring time, and the more complex data processing method is not a null test technique [4–6]. Jan Burke used a flat mirror as the aiding element to detect a 90◦ off-axis paraboloid mirror, which obtained the results of PV = 343 nm and RMS = 50 nm. Although this method belongs to the auto-collimation measurement method, the adjustment processing is di fficult [7]. Similarly, Ki-Beom Ahn used a spherical convex reference mirror as the aiding element to detect the ellipsoid mirror, which was the secondary mirror of the Giant Magellan Telescope (GMT) [8]. Due to the diameter of this ellipsoid (up to 1.06 m), the aperture size of the spherical convex reference mirror would reach 0.99 m, leading to serious di fficulties in the fabrication of this aiding element. Additionally, the adjustment processing is tough. As a null test technique, the computer generated hologram (CGH) method is a suitable compensator for o ff-axis aspheric surface measurement [9–11]. M. M. Talha measured a freeform surface with the CGH method with a result of PV = 0.0479λ [12]. Similarly, Li Fa-zhi used CGH as the aiding element to measure an off-axis high order aspheric surface [13].

When the aperture of the o ff-axis aspheric surface is up to the meter level and the asphericity rises to the millimeter level, the CGH method will not be suitable because the di fficulty and cost of CGH fabrication increase dramatically. In this instance, the application of the two aiding elements, fold mirror together with GGH, could be a good choice. J. H. Burge used this method to measure the segmen<sup>t</sup> of primary mirror of the GMT, which was an o ff-axis aspheric surface with an aperture size of 8.4 m [14]. In the same way, Chang Jin Oh measured an o ff-axis paraboloid with a 4.2 m aperture diameter and 9 mm asphericity [15].

In order to obtain accurate measurement results of the o ff-axis conic aspheric surface, three null test methods are introduced in Section 2. After that, an o ff-axis paraboloid (OAP) was chosen to be tested in Section 3. The aiding elements from the three methods were designed and fabricated respectively to measure the OAP. Finally, the correctness of three methods were mutually cross-checked by their experimental results.

### **2. The Shape Measurement Methods**

Interferometry is frequently reckoned as an e ffective tool to test the shape accuracy of the optical surface; however, it is limited to directly measuring flat or spherical surfaces. Due to the inherent aberration of the o ff-axis aspheric surfaces, an auxiliary optical element is required for the null test. Moreover, the aiding element changes with the variation of the measurement method.

Due to having perfect image points, a conic aspheric surface such as paraboloid, ellipsoid, and hyperboloid can be measured using the classical null test method of auto-collimation. Within this method, a flat mirror (for paraboloid) or a spherical mirror (for ellipsoid or hyperboloid) is required as the aiding optical element. As shown in Figure 1, two optical layouts for measuring an o ff-axis paraboloid with flat mirror and one layout for measuring an o ff-axis ellipsoid with a convex sphere mirror are illustrated.

**Figure 1.** Schematic diagram of auto-collimation: (**a**) o ff-axis paraboloid located at o ff-axis; (**b**) o ff-axis paraboloid located at on-axis; (**c**) o ff-axis ellipsoid located at o ff-axis.
