Optical interferometry is the means by which real-world displacements are measured relative to a standard reference wavelength. Most interferometers use a helium-neon laser (633 nm) and generate a signal with a period of half the wavelength (316.5 nm). To resolve smaller distances, the system must perform fringe interpolation. For measurements in which every nanometer counts, it is important to establish the accuracy with which the interferometer can split fringes. Systematic errors can arise from imperfections in both the optics and the electronics.

In fact, it is simple to measure the accuracy with which an interferometer can interpolate optical fringes with no displacement reference at all. The key lies in the fact that errors made in interpolation repeat themselves with each successive fringe, which allows the powerful techniques of Fourier analysis to come into play. The only requirements for using the Fourier technique are the ability to generate a displacement that is nearly linear in time over a distance of several micrometers and the software analysis of the resulting displacement signal.

*The Fourier transform of the residuals of our measurement shows an rms fringe interpolation error of about 100 pm.*

**making the measurement** First, you must find a method to generate a linear displacement. The most effective way that we have found to generate the necessary linear displacement is to heat a thin-walled metal tube, to one end of which is glued a reflector (mirror or corner cube, depending on the interferometer). The other end of the tube is fixed. We use a 100-mm length of 25-mm-diameter aluminum tubing, wound with a single layer of copper magnet wire, and heat the tube by passing current through the wire. We typically dissipate 50 W to 200 W.

Once you have a linear displacement tool, the next step is aligning your interferometer to target the thermally scanned mirror, measuring the displacement as the mirror moves. From this measurement, you can estimate the fringe frequency. For example, to test a new interferometer that we are developing, we scanned about 11 mm in 4 s. Since we were using a helium-neon laser, 11 mm corresponds to about 36 fringes, so the fringe frequency was approximately 9 Hz. We sampled the displacement data at 1 kHz. In general, the sampling rate should be substantially higher than the fringe frequency.

Now fit the data to a line and look at the residuals from the fit. There will be some smooth, slow variation that will show that the thermal scan velocity is not exactly constant. More important, however, you will almost certainly see some periodic structure with the same periodicity as the fringe frequency (or perhaps twice the fringe frequency). Here is the key point: The scan itself, even if it is not perfectly linear, will not have any nonlinearity related to the optical wavelength. Any residuals at the fringe frequency must represent errors in fringe interpolation that are repeated with each successive fringe.

**working with the data** It often helps to fit the data to a low-order polynomial in order to better remove nonlinearities associated with the thermal scanner. We fit our data from the example above to a parabola. To really see what is going on, use the Fourier technique to transform the residuals, and you may find evidence of interpolation errors at multiples of the fundamental frequency as well (see figure). Our interferometer shows an rms fringe interpolation error of about 100 pm, corresponding to about one part in 3000. This is better than most off-the-shelf interferometers provide.

If it is necessary to improve the signal-to-noise ratio, try a faster scan, try fitting the raw data to a higher order polynomial, and use a windowing function (such as a Hanning window, available in most analysis packages) when doing the Fourier transform. It is also useful to minimize dead path and air currents in the interferometer.

We have found that most commercial interferometers are limited in useful resolution at the nanometer level. In the case of poor performance, possible culprits include stray reflections and optical feedback. The method we have outlined should allow you to measure the interpolation limit of your interferometer and possibly improve it. **oe**

**John Lawall**

*John Lawall is a physicist at the National Institute of Standards and Technology, Gaithersburg, MD.*