Two different measure- and-correct algorithms, the Chu-Ray algorithm and the Eom algorithm, were simulated to estimate their effectiveness for correcting periodic nonlinearity applied a simulated wafer positioning system. Both algorithms, which demonstrated no periodic nonlinearity in some example constant velocity and constant acceleration datasets, and their specific implementations are described in detail. The algorithms were tested for both slit and scan directions using estimated wafer stage trajectories for ramping up to a constant scanning velocity with no velocity in the slit direction. Three different heterodyne frequencies and three different sampling frequencies were simulated to estimate the optical system and data acquisition parameters. The initial phase offset was also considered in the simulations and had a significant effect on both the scan and slit offset errors. The offset error appears to increase for certain initial phase values which manifests from compensating the periodic errors combined with the high stage dynamics. High noise levels and displacement jumps were observed when high velocity changes during a correction period suggesting increasing the minimum velocity threshold and limiting the maximum acceleration can improve the effectiveness of these algorithms.@en

Heterodyne interferometry is a widely applied technique for length and displacement measurements in precision systems. Free-space optical beam delivery couples the source directly to the interferometer location. Conversely, fiber delivery is desired to decouple the source and interferometer. Drawbacks to using fibers are either time consuming alignment (PM fiber) or time varying polarization output (MM fiber). Jones matrices were used to analyze Joo-type interferometers to simulate the effects of using MM fiber inputs. The results showed loss of interference at some instances from the rotating polarization states. Measurements with a MM fiber coupled interferometer showed minimal polarization rotation and no interference loss.@en

Displacement interferometers are widely used in precision engineering and metrology applications. For multi-axis systems, free space delivery of the optical beams requires high tolerance pointing stability and couples the source to the interferometer location. Fiber delivery is desired to decouple the source and interferometer but configurations which have to be used for this, contribute typical errors such as periodic nonlinearity. In this paper, we describe a fiber-coupled Joo-type interferometer. Spatial separation of the input beams in this type of interferometer eliminate periodic nonlinearities. This is contrasted by typical heterodyne interferometer systems which have two orthogonal, coaxial beams with a difference frequency. We present the interferometer design, discuss the fiber deployment, and compare free-space and fiber-coupled versions. The results indicate that fiberinduced disturbances are rejected as theory predicts and no periodic nonlinearity was detected.@en

Displacement interferometry is widely used for accurately characterizing nanometer and subnanometer displacements in many applications. In many modern systems, fiber delivery is desired to limit optical alignment and remove heat sources from the system, but fiber delivery can exacerbate common interferometric measurement problems, such as periodic nonlinearity, and account for fiber-induced drift. In this Letter, we describe a novel, general Joo-type interferometer that inherently has an optical reference after any fiber delivery that eliminates fiber-induced drift. This interferometer demonstrated no detectable periodic nonlinarity in both free-space and fiber-delivered variants.@en