Calibrating Retrace Error for Interferometric Testing of Asphere

@Salsbury. I have been testing an aspheric component without a custom null, so I am getting retrace error in my results and I need to calibrate the error out.

What method is preferred for calibrating retrace error? I have read regarding a random steel ball calibration process, and I have done this in the past. However, I have recently read two papers stating that a ray trace method [1], or a point characteristic function (PCF) method [2] may be preferable. However, my concern with both of these methods is that they seem more suited for a custom or ‘home-built’ interferometer, as opposed to a commercial system which I am using. Any thoughts would be much appreciated!

[1] Dong Liu, Yongying Yang, Chao Tian, Yongjie Luo, and Lin Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17 , 7025-7035 (2009)
[2] He Yiwei, Xi Hou, Quan Haiyang, and Weihong Song, “Retrace error reconstruction based on point characteristic function,” Opt. Express 23 , 28216-28223 (2015)

Retrace error is commonly ignored and overlooked in the practice of interferometry via the argument of double-path or null-path conditions. However, in reality, these conditions are often difficult to achieve with commercial optics let alone aspheric and freeform optics. These complicated optics have large shape deviations from spheres which produce high fringe densities. For this reason, it is critical to understand the limitations of measurement accuracy of the interferometer that is being used and how this accuracy may degrade as a function of fringe density.
The random steel ball calibration process is a great way to characterize the transmission sphere and provides a repeatable way to measure manufacturing errors, but this process is usually done close to the null condition and doesn’t provide any insight into retrace errors that will be present when measuring more complicated shapes. As a general rule of thumb, the retrace errors will scale with NA of the transmission spheres.
Other methods for calibrating retrace error do exist and [1] and [2] are just two examples of the many outlined methods in the literature. However, as you pointed out, these methods require careful consideration and detailed knowledge of the optical design of the interferometer in order to provide an accurate and representative model of the retrace errors that are to be expected in the measurements—making “home-built” interferometers the only viable platform for these techniques.
Currently there are very few commercial instruments which are capable of reliably measuring aspheric optics and custom null optics are usually only feasible for large scale research projects or high throughput commercial manufacturing.

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Wow, this is great help thank you!

This really helps to explain the key benefit of the steel ball calibration. While it may not be great for retrace calibration, the need for calibrating the transmission sphere makes a lot of sense. In you experience how many ball rotations would you perform for calibration, i.e., is measuring 5 points on the ball sufficient, or do you need say 20?

I guess we are still stuck with using profilometery coupled with interferometry for the time being to make some of the aspheres. If only someone could figure out a good commercially viable way to do freeform metrology…(that isnt limited by calibration issues)

Good question regarding number of measurements on the steel ball. This has more to do with the sources of noise in the interferometer and the surface statistics of the steel ball than the transmission sphere itself. If the steel ball has loosely controlled surface roughness and surface figure more measurements will be required to reduce the speckle variation and surface figure errors from the steel ball. A good rule of thumb is to take an average of N measurements and then repeat this experiment M times. If the variation between M and M+1 results is less than the required measurement accuracy (I.e. lambda/20) than N is sufficient.