Precision interferometry despite vibration

Interferometry can be extremely useful for testing optical components and optical systems as well as the metrology of many other components, such as the flatness and roughness of hard-disk-drive platters, the shape of magnetic recording heads and machined parts, and the deformations of diffuse surfaces. To make good use of interferometric data, the data must be analyzed by a computer to determine if the surface shape is correct, how well the surface being evaluated will perform if it is not corrected and, if necessary, to correct it. Until recently, a major limitation of the use of interferometry for precision metrology was its sensitivity to the environment.

In recent years, many techniques for performing high-quality interferometric measurements in less than ideal environments have been developed, and new techniques are constantly being introduced. However, some of these techniques are useful only in certain situations, or they may be too complicated or expensive to have wide commercial use. They must also work well with computer input methods so that computer analysis of the interferometric data can be performed.

Phase-shifting interferometry¹ is an excellent way to input interferometric data into computers. With interferometric data, there are three unknowns: the amplitude of the reference beam, the amplitude of the test beam, and the phase difference between the two interfering beams. Of these, the quantity of most interest is the phase difference between the two interfering beams (because the phase difference gives the shape of the surface being measured).

The phase difference between the two interfering beams can be determined by measuring the intensity of the interference fringes while the phase difference between the two interfering beams is changed in a known manner. Typically, the phase is changed by 90 degrees between consecutive intensity measurements. Since there are three unknowns, at least three intensity measurements must be made. To reduce the effects of vibration, all of the phase-shifting frames should be taken at once.

We have used an approach for obtaining all the phase-shifted frames of intensity data at once that works well over a large spectral bandwidth. It involves the use of a quarter waveplate (QWP) followed by linear polarizers at different angles to get the phase shifts.² In this method, the test and reference beams have circular polarization of the opposite sense. If these circularly polarized beams are transmitted through a linear polarizer, the result is a phase shift between the two interfering beams proportional to twice the rotation angle of the polarizer. Thus, if a phase mask is made of an array of four linear polarizer elements having their transmission axes at 0, 45, 90, and 135 degrees (where a polarizer element is placed over each detector element), the mask will produce an array of four 0, 90, 180, and 270 degree phase-shifted interferograms. We use a Twyman-Green interferometer that uses a polarization beam splitter (PBS) to obtain orthogonal polarizations for the test and reference beams and the micropolarizer phase-shifting array (see Figure 1). Figure 2 shows the results for measuring a 300mm diameter, 2m radius of curvature mirror, where the mirror and interferometer are on separate tables.³

Figure 1. Twyman-Green interferometer with the micropolarizer phase-shifting array. PBS: polarization beam splitter. QWP: quarter waveplate.

Figure 2. Interferogram and measured 3D map for measuring a 300mm diameter, 2m radius of curvature mirror, where the mirror and interferometer are on separate tables.

The micropolarizer phase-shifting array interferometer works very well in the presence of vibration and with a wide range of source wavelengths. The interferometer makes short exposures, so the vibration and the air turbulence are frozen. The effects of air turbulence can be reduced by taking many sets of data, arranging for the time between taking the data sets to be long compared to the time it takes for the turbulence to change, and then averaging the data.

Not only can a surface be measured in the presence of vibration, but if a surface is vibrating, the change in the surface shape can be measured and video can be taken of a surface as its shape changes. The results for a surface excited by a piezoelectric transducer vibrating at a frequency of 3069Hz are shown in Figure 3 and an online video.⁴

Figure 3. A frame from a movie showing the vibration of a hard disk platter excited by a piezoelectric transducer vibrating at a frequency of 3069Hz.⁴

In summary, the pixilated polarizer array is an excellent way to simultaneously obtain all the phase-shifted fringe patterns required for performing precise measurements in the presence of vibration and air turbulence, and this technique has greatly expanded the environments where interferometric measurements can be made. Once a person uses a simultaneous phase-shifting interferometer, it is hard to go back to using the vibration-sensitive temporal phase-shifting interferometers. It is exciting to combine modern electronics, computers, and software with old interferometric techniques to enhance measurement capabilities, and we plan to continue developments in this area to both improve measurement accuracy and make the measurements easier to perform.

The author acknowledges useful conversations with Michael B. North-Morris, James E. Millerd, Neal J. Brock, Brad Kimbrough and John Hayes at 4D Technology.