Phase measurement accuracy limitation in phase shifting interferometry.

Abstract

In phase shift interferometry (PSI), several factors affect measurement accuracy, such as piezoelectric transducer (PZT) calibration (i.e. PZT slope error) and PZT nonlinearity, vibration, spurious reflection, source bandwidth, detector nonlinearity, and detector noise. The effects of these error sources on several algorithms to solve the phase of the wavefront are studied. When the simple arctangent formula is used, if the PZT slope is properly adjusted, the error due to the PZT quadratic nonlinearity can be tremendously reduced. An exact solution is derived to remove the error when the PZT quadratic nonlinearity is large. Although Carre's formula is insensitive to PZT slope, this formula is more sensitive to the detector nonlinearity than the simple arctangent formula. For most error sources, the error of the phase solved has a double-frequency characteristic. Thus, averaging two measured phases of two runs, which have a ninety degree phase shift related to each other, can effectively reduce the error. For a small vibration, the phase error has a very simple relation to the vibration amplitude, and a very complex relation to the vibration frequency. Although the error caused by vibration has this double-frequency characteristic, the averaging technique does not apply. The error caused by spurious reflection does not have such a characteristic. A new algorithm is proposed to eliminate the phase error caused by certain types of spurious reflection. When detector noise is concerned, the phase error is inversely proportional to the modulation of the intensity times the square root of the number of steps/buckets. For the shot noise, the phase error is inversely proportional to the fringe contrast times the square root of the total number of photons. In practice, the shot noise is very much smaller than the detector noise. In a practical environment, PZT calibration, vibration, and spurious reflection have much more prominent effects on the PSI than the source bandwidth, detector nonlinearity, and detector noise. When spurious reflection and vibration are under control, and the signal-to-noise ratio is about 20, the PSI has an accuracy of 2 degrees, i.e. 3.3nm at 633nm. Because vibration and detector noise are random error sources, the errors caused by them can be reduced by averaging many measurements. However, the error caused by the other discussed sources cannot be reduced by averaging many measurements.