Electronic limitations in phase meters for heterodyne interferometry N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague National Institute of Standards and Technology, U.S. Department of Commerce, Technology Administration, Gaithersburg, MD, USA Limitations imposed by the phase meters used in heterodyne interferometers are evaluated. These instruments measure the phase relationship between electrical signals generated by the heterodyning process, allowing the inter- ferometers to resolve fractions of an optical fringe. Measurements indicate that the phase meters used in currently available heterodyne interferometers probably limit achievable accuracy to a greater extent than barriers imposed by the optics. We show that a new class of time interval counters offers a means of greatly improving accuracy in these instruments. Keywords: Phase meters; heterodyne interferometry Introduction Commercial heterodyne interferometers claim dis- placement measurement accuracies of approxi- mately 10 nm, and because the interferometers are based on well-known optical wavelengths, such ac- curacies have been accepted with little questioning. However, more demanding requirements, which push displacement measurement uncertainties down to 1 nm and below, have led to the investiga- tion of remaining sources of error. Reasonable at- tention has been given to the fidelity of the hetero- dyning process, which converts the optical path difference between beams that have traversed the test and reference legs to a phase difference be- tween electrical signals from the test and reference photodetectors. 1 - 9 This article deals with the errors introduced in the next step: the measurement of the electrical phase angle. Heterodyne interferometers , A heterodyne interferometer, shown schematically in Figure 1, uses a dual-frequency laser to produce two electrical signals: a reference signal, VR, which is at the laser difference frequency (at = f 1 - f 2 ), and a measurement signal, VM, which is at the dif- ference frequency but Doppler-shifted by the mo- tion of the test reflector (of± M). This Doppler shift Address reprint requests to E. C. Teague, National Institute of Standards and Technology, U.S. Department of Commerce, Technology Administration, Metrology Building, Room A 117, Building 220, Gaithersburg, MD 20899, USA. © 1993 (typography and design) Butterworth-Heinemann U.S. Government work not protected by U.S. copyright PRECISION ENGINEERING can equivalently be thought of as a time-varying phase shift, cp(t) = 211' J Af (t) dt. If v(t) is the test mirror or reflector speed, then Af (t) = 2nv(t) A where n is the index of refraction of the medium along the light path and A is the wavelength of the light. For commercial heterodyne interferometers, the difference frequency is typically between 250 kHz and 20 MHz. At rest, the reference and measure- ment signals have the same frequency and their phase relationship is proportional to the position of the test reflector. The ability to measure this phase angle determines how accurately one knows the change in position of the test reflector. In a simple Michelson interferometer, the phase is related to displacement by cp(t) = 41Tnl(t)IA, where L(t) is the displacement of the test reflector relative to the reference. From the above relation- ship, 1° of phase represents a test reflector displace- ment of A/720, or about 0.9 nm for He-Ne laser wavelengths. Angular resolution Most phase meters measure phase angle by mea- suring time ratios. In interferometry, absolute accu- racy is not necessary, but stability, linearity, and resolution are. The angular resolution obtainable depends on the signal frequency and the timing resolution: 173