#### Date of Award

2020

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Optical Science and Engineering

#### Committee Chair

Patrick J. Reardon

#### Committee Member

Lingze Duan

#### Committee Member

Seyed Sadeghi

#### Committee Member

James B. Hadaway

#### Committee Member

Robert Lindquist

#### Subject(s)

Optical measurements., Interferometry., Metrology.

#### Abstract

This dissertation employs a graphical method of laser beam propagation analysis, the yybar diagram, to develop analytical solutions for a difficult optical metrology challenge: modeling of the behaviour of an interferometric system when testing small radius objects. The method presented utilizes the yybar diagram for Gaussian beams to model interferometric systems. The method enables rapid, intuitive layout and understanding of the complex properties of beam propagation, yielding algebraic calculations producing quantitative results. We first derive a relation for minimum test radius; the smallest radius part that can produce a perfect fluffed fringe, wherein the test and reference waves are identical in curvature and size. It also has only one test position that produces a fluffed fringe. For larger radii, there are two positions that yield fluffed fringes, commonly designated the “null” and “cat’s eye” positions. The separation between these two positions are typically used for measuring the radius of curvature (ROC) of the test part. Analysis reveals a rapidly increasing systematic radius measurement error as the test part radius approaches the minimum; the two positions converge to one implying that the radius is zero. Below the minimum radius, only a single position creates fluffed fringes, but with reduced contrast. An experiment was run that verifies the analysis. A Twyman-Green interferometer was built and well calibrated. A series of test balls were acquired and the glass ones were coated with aluminium, then measured using callipers to confirm their specified radius. These were measured on the interferometer and the data verify the preceding analyses. The same parts were measured for RoC on a commercial Fizeau interferometer; there is a similar increase in error with decreasing test radius. As expected, it does not directly match our derivation, however, a simple scaling relation is empirically determined that fits our analytical results to this data. Further analyses were performed to show the usefulness of the method including modeling a phase shifting Fizeau interferometer, revealing a systematic error for small radius parts. Finally, the analysis is extended to a cylindrical wave test, revealing an unavoidable residual astigmatism error.

#### Recommended Citation

Alsalamah, Reem, "Improved modeling and verification of metrology on small radius optics" (2020). *Dissertations*. 214.

https://louis.uah.edu/uah-dissertations/214