Modelling and simulation of the second-order Doppler error of a laser dual-frequency interferometer

Abstract

Only the first-order Doppler frequency shift is considered in current laser dual-frequency interferometers; however; the second-order Doppler frequency shift should be considered when the measurement corner cube (MCC) moves at high velocity or variable velocity because it can cause considerable error. The influence of the second-order Doppler frequency shift on interferometer error is studied in this paper, and a model of the second-order Doppler error is put forward. Moreover, the model has been simulated with both high velocity and variable velocity motion. The simulated results show that the second-order Doppler error is proportional to the velocity of the MCC when it moves with uniform motion and the measured displacement is certain. When the MCC moves with variable motion, the second-order Doppler error concerns not only velocity but also acceleration. When muzzle velocity is zero the second-order Doppler error caused by an acceleration of 0.6g can be up to 2.5 nm in 0.4 s, which is not negligible in nanometric measurement. Moreover, when the muzzle velocity is nonzero, the accelerated motion may result in a greater error and decelerated motion may result in a smaller error.

Interference signal demodulation for nanopositioning and nanomeasuring machines

“…The ellipse equation coefficients A, B, C, D, E are determined by direct or recursive estimation using the least-squares method of the ADC values D A and D B [32]. The ellipse regression and correction can be achieved with different approaches [33][34][35][36][37][38][39][40]. The regression was used in the NMM-1 to correct the systematic deviations of the offsets, amplitudes and phases.…”

Retardation effects in laser-Doppler measurements

“…Although the Doppler effect can be considered to be well understood some recent publications [3,4,5] discuss the influence of the Doppler effect in displacement interferometers and it is claimed that relativistic corrections have to be taken into account to obtain correct results for high precision measurements. However, the first double-beam interferometer was realized by Albert Abraham Michelson to measure the ether drift [6] because the current light theory of that time claimed an ether as wave medium.…”

“…The results derived in this paper agree well with results from [4] that have been derived with the Lorentz transformation. Reference [3] derives the Doppler shift at reflection by combining optical-path-length considerations together with the Lorentz transformation. Consequently, a different equation for the Doppler shift at a moving reflector is derived in [3].…”