A method of estimating and correcting for the effect of polarization leakage on the response of a heterodyne optical interferometer has been devised. In a typical application in which a heterodyne interferometer is used as a displacement or length gauge, the effect of the polarization leakage is a nonlinearity that typically gives rise to an error of the order of 1 nm in the displacement or length. By use of the present method, it should eventually be possible, in principle, to reduce the error to the order of 10 pm or less. The technique is primarily computational and does not require any additional interferometer hardware. Moreover, the computations can be performed on almost any modern computer in real time.

The figure schematically depicts a typical heterodyne interferometer. The interferometer utilizes two laser beams with frequencies ν1 and ν2 that differ by a known small amount (typically, | ν1 – ν2| is of the order of 1 to 100 kHz). Beams 1 and 2 are p- and s-polarized, respectively, in the reference frame of two polarizing beam splitters. Polarizers, polarization rotators, and a nonpolarizing beam splitter are used, along with the polarizing beam splitters, to separate and combine the beams at several junctions along the light path. The beams are made to interfere with each other at two photodetectors called the “reference” (R) and the “unknown” (U) photodetector, respectively. Ideally, all of the spolarized light that arrives at the lower beam splitter makes one and only one round trip along the lower horizontal interferometer arm between two corner-cube retroreflectors, and then impinges on the U photodiode.

The difference-frequency outputs of the two photodetectors are the desired heterodyne signals. Small variations in the distance between the two retroreflectors are what one usually seeks to measure. In the ideal case, one could compute these displacements precisely from variations in the difference between the phases of the heterodyne signals from the two photodetectors. In practice, the phase measurement is degraded by noise and by systematic effects caused by imperfections in the interferometer optics. The major systematic source of error in the phase measurement is polarization leakage; that is, impingement, on the U photodetector, of (1) a small portion of the s-polarized light that passes directly through the lower polarizing beam splitter without taking the round trip between the retroreflectors and (2) another small portion of the s-polarized light that takes two or more round trips.

This polarization-leakage phase error ( ε) is what gives rise to the nonlinearity in the interferometer response. The present method of estimating ε is based partly on the fact the polarization leakage is small and that εis much less than 1 radian (as it is in cases of practical interest). A mathematical analysis shows that ε is a periodic function of ideal single-round-trip phase ( ψ), and that given the aforementioned assumptions, it can be approximated by the following equation: ε( ψ) = Acos( ψ) + Bsin( ψ) + Ccos(2 ψ) + Dsin(2 ψ) + . . . , where A, B, C, and D are parameters to be determined in a calibration procedure summarized in the next paragraph. In order to suppress the nonlinearity to the 10-pm level, the dominant parameter in this equation must be determined to one part in 100, while the other parameters may be determined to lesser accuracy, depending on their relative amplitudes.

One of the retroreflectors is mounted on a piezoelectric transducer that can be driven by a suitable voltage to displace the retroreflector along the interferometer arm. The calibration procedure involves the use of a triangular waveform at a typical frequency of a few hertz to induce an oscillatory displacement that spans a useful number (typically ≈10) of phase cycles so that the periodic ε component of the uncorrected phase measurement becomes recognizable. The uncorrected phase measurements are sampled, filtered, and otherwise processed to extract A, B, C, and D. Then the instantaneous corrected phase measurement (which should now be a closer approximation of the desired phase ψ) is given by

ψcorrected = ψuncorrected – ε( ψuncorrected), where ε( ψuncorrected) is calculated by substitution of the appropriate quantities in the above equation for ε( ψ).

This work was done by Alex Abramovici and Randall Bartman of Caltech for NASA’s Jet Propulsion Laboratory. Under the Information Sciences category. NPO-20906

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This article first appeared in the April, 2002 issue of *NASA Tech Briefs* Magazine.

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