The Linear Shift Invariant Approximation for Phase Shifting Interferometers

Abstract

Phase shifting interferometry is a common method to measure wavefronts affected by optical elements. Phase shifting interferometers are not linear shift-invariant (LSI), yet they are often approximated to be when using the resulting phase as an equivalent of the surface. This is because they can be approximated as LSI under the proper circumstances, which is when most measurements are taking place. This thesis uses the fact that a sine wave is the eigenfunction of a linear shift invariant system. The output for a sinusoidal input is calculated, and the conditions under which the output is not sinusoidal are determined. The sinusoidal quality, or “sininess error” of the output is based on the output sinusoid’s frequency, as that should not change based on eigenfunction properties. When the sininess error is large, this indicates a non sinusoidal output, which means the LSI approximation is not a good approximation to use. The results from the above calculation are checked using a numerical simulation based on imaging. Both methods supported the same conclusion. In addition, a simplified calculation was derived by making some assumptions and approximations. While the quality of this approximation degrades under certain circumstances, it helps approximate the sininess error in a quick manner without cumbersome equations. Finally, an experimental verification was performed by measuring a sinusoidal phase object with a phase shifting interferometer that was configured to violate the LSI approximation. The experimental results match the predictions, demonstrating that interferometers cannot always be approximated as LSI. The results of the experiment also prove that the calculations are correct. While this thesis proves that phase shifting interferometers are not LSI, they are still usually approximated as LSI. Most who use phase shifting interferometers are measuring well polished optical surfaces, which are in a region where sininess error is extremely low. Therefore, the LSI approximation works unless a person is measuring an irregular surface, such as a deep sinusoidal grating.