Abstract: An alignment simulation method simulates the signal waveform for an alignment mark using various alignment methods as well as the signal strength for an alignment mark, which is useful in optimizing the thickness of one or more layers as well as the geometry of the mark. The simulation of signal strength is also useful in optimizing the thickness of a layer for artifact wafers. The alignment simulation method includes accurately modeling the alignment mark, including one or more layers of various thicknesses and materials. The accurate modeling of the alignment mark includes such things as smoothing regions of the alignment mark and generating lateral shifts of the layers. The model of the alignment mark is a series of small pixels, each including the thickness of the layers and the layers indices of refraction. Using scalar diffraction, a complex reflectivity is generated for each pixel and a fast fourier transform is performed on the series of pixels.

Abstract: A command trajectory drives a stage to produce a smooth motion while minimizing any vibrations or oscillations of the structure. The command trajectory provides an acceleration and a deceleration that have derivatives, known as the jerk, equal to zero at the beginning and end of the acceleration and deceleration periods. Because the jerk is equal to zero at the beginning and end of acceleration and deceleration, the influence of the reactive forces on the positioning system's structure is reduced, thereby minimizing oscillation of the structure. Moreover, the jerk is continuous throughout the acceleration and deceleration periods, resulting in a smooth continuous motion of the stage. The jerk on the stage during acceleration and deceleration has an adjustable duration to reduce the structural disturbances and decrease settling time.