*2.1. Auto-Collimation Method*

The auto-collimation method is inexpensive and does not need a complex design or a long time to prepare. To make this method simple and useful, the aperture and the F-number of the aiding element should be considered. As shown in Equation (1), there are three errors occurring in the test results of this method.

$$Error\_{total} = Error\_{TS} + Error\_{\text{aiding}} + Error\_{\text{asphalré}} \tag{1}$$

In this equation, *Errortotal* is the total error of the measurement system, *ErrorTS* is the error of the TS (transmission sphere, a standard lens used on the interferometer), and *Erroraiding* is the error of the aided element (such as flat mirror or convex sphere mirror). *Erroraspheric* is the error of the o ff-axis aspheric under testing. Usually, the quality of TS is so high that *ErrorTS* is very small. If the *Erroraiding* is also much smaller than *Erroraspheric*, the test result *Errortotal* is approximately equal to *Erroraspheric*. Otherwise, when the *Erroraiding* is not small enough, we can subtract *Erroraiding* from *Errortotal* to obtain *Erroraspheric* after calibrating the *Erroraiding*. However, for o ff-axis aspheric surfaces, it is di fficult to adjust all the components to the correct location. If there exists alignment error, there will be unavoidable misalignment errors such as astigmatism and coma in the test results. As a result, the adjustment turns into the most di fficult process within the auto-collimation method.

The limitation of the auto-collimation method is attributed to the surface type and aperture of the off-axis aspheric under testing. As the aperture of the aiding element should be larger than the o ff-axis aspheric, this is a di fficult and expensive mission if the aperture is up to meter level.

### *2.2. Single Computer Generated Hologram (CGH) Method*

As a di ffractive optical element, CGH can produce any shape wavefronts [9]. Figure 2 shows the principle of this method. The interferometer, CGH, and o ff-axis aspheric under testing are all on the same axis, so the aberration that should be compensated will be reduced [16].

**Figure 2.** Schematic diagram of single computer generated hologram (CGH) method for o ff-axis aspheric surface.

In Figure 2, the output wavefront after TS is the convergen<sup>t</sup> sphere wavefront. However, the CGH changes the sphere wavefront into the o ff-axis aspheric wavefront to match the theoretical shape of the off-axis aspheric surface under testing, where this o ff-axis aspherical wavefront vertically illuminates on the surface. After that, the wavefront with the information of the o ff-axis aspheric surface is reflected again into CGH via the same path it comes from, and ultimately back to the interferometer to form interference fringes, from which we eventually gain the test results. The test result with the single CGH method is also involved in several errors, as shown in Equation (1), by replacing *Erroraiding* with *ErrorCGH*. Before the CGH basement reaches the precision requirement, it should be processed to achieve a high quality in surface shape and parallelism, and then processed by photoetching to ensure the quality of the CGH [17]. As a result, the *ErrorCGH* is too small to have an e ffect on the testing results.

The CGH includes three parts: the test CGH, reflection CGH, and crosshair CGH. The test CGH is the transmission di ffraction part (the red part in Figure 2), which creates the same wavefront of the o ff-axis aspheric surface under testing. When the o ff-axis aspheric surface is di fferent, the corresponding CGH varies according to the customized design. Reflection CGH is used to align the interferometer and CGH (the yellow part in Figure 2), and crosshair CGH is applied to align the CGH and off-axis aspheric under testing (the blue part in Figure 2).

Within the single CGH method, the CGH design is the most important part. The geometry parameters in Figure 2 determine the size of the CGH and the aberration to be compensated. As it is a key parameter in the single CGH method, the aberration should be compensated by the CGH diffraction wavefront that decides the measurement accuracy and can be designed with Zemax (Version June 9, 2009). After obtaining the CGH phase function, the fringe etching position can be calculated with a special MATLAB program (R2016b). The design process is shown in Figure 3.

**Figure 3.** CGH design process.

The disadvantage of the single CGH method lies in the minimum fringe spacing and the size of the whole CGH. When the minimum fringe spacing is less than the ability of photoetching, this customized CGH cannot be fabricated successfully.
