Relativistic corrections in displacement measuring interferometry

Displacement measuring interferometry is based on measuring the Doppler frequency shift that a beam of radiation undergoes upon reflection off a mirror connected to a moving stage. Usually the velocity of the reflecting stage is very small compared to the speed of light and is therefore deduced using the classical expression for the Doppler shift. We calculate relativistic corrections to the Doppler frequency shift, considering arbitrary stage motion in two dimensions and multiple passes through the moving interferometer arm. Changes in optical path lengths due to the varying stage displacement are explicitly taken into account. For stage velocities on the order of only 1 m/s the resulting corrections to the classically derived stage displacement can amount to nanometers. We discuss model velocity profiles similar to those currently employed in industrial step-and-scan systems for integrated circuit manufacturing, and for recently proposed scanning-beam interference lithography schemes. Expected future increases in stage speed and wafer sizes will necessitate the inclusion of relativistic corrections to the Doppler shift to maintain pattern placement accuracy at the nanometer level.