Relativistic corrections in displacement measuring interferometry

Displacement measuring interferometry is based on measuring the Doppler fre quency shift that a beam of radiation undergoes upon reflection off a mirro r 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 usin g the classical expression for the Doppler shift. We calculate relativistic corrections to the Doppler frequency shift, considering arbitrary stage mo tion in two dimensions and multiple passes through the moving interferomete r arm. Changes in optical path lengths due to the varying stage displacemen t are explicitly taken into account. For stage velocities on the order of o nly 1 m/s the resulting corrections to the classically derived stage displa cement can amount to nanometers. We discuss model velocity profiles similar to those currently employed in industrial step-and-scan systems for integr ated circuit manufacturing, and for recently proposed scanning-beam interfe rence lithography schemes. Expected future increases in stage speed and waf er sizes will necessitate the inclusion of relativistic corrections to the Doppler shift to maintain pattern placement accuracy at the nanometer level . (C) 2000 American Vacuum Society. [S0734-211X(00)01506-7].