A simple derivation of the cyclic error due to optical mixing is proposed for the cancelable circuit design. R and M beatings are collected by two photodiodes and conveniently converted by transimpedance amplifiers, such that the output signals are turned into ac-coupled voltages. The detected phase can be calculated as a function of the real phase (a change in optical path difference) in the case of zero-crossing detection. What turns out is a cyclic non-linearity which depends on the actual phase and on the amount of optical power leakage from the R channel into the M channel and vice versa. We then applied this result to the prototype of displacement gauge we are developing, which implements the cancelable circuit design with wavefront division. The splitting between R and M is done with a double coated mirror with a central hole, tilted by 45° with respect to the surface normal. The interferometer features two removable diffraction masks, respectively located before the merging point (a circular obscuration) and before the recombination point (a ring obscuration). In order to predict the extent of optical mixing between R and M, the whole layout was simulated by means of the Zemax ® Physical Optics Propagation (POP) tool. After the model of our setup was built and qualitatively verified, we proceeded by calculating the amount of optical leakages in various configurations: with and without the diffraction masks as well as for different sizes of both the holey mirror and the diffraction masks. The corrisponding maximum displacement error was then calculated for every configuration thanks to the previously derived formula. The insertion and optimization of the diffraction masks greatly improved the expected optical isolation inside the system.
Data acquisition from our displacement gauge has just started. We plan to experimentally verify such results as soon as our prototype gauge will reach the desired sub-nanometer sensitivity.