Schematic of TCG gap-detection method. (a) Checkerboard mark with a chirp in the (transverse) direction and a constant period in the (incident) direction. A second, adjacent chirped checkerboard produces fringes that move in opposite directions with changes in gap. (b) Schematic of diffracted beams in the transverse plane, showing an example of two incident beams that diffract, reflect, and rediffract, producing constructive interference. The distance between the two incident beams is called the interaction distance. (c) Schematic in the incident plane illustrating the path by which light is returned to the microscope.
Coarse gap detection demonstrated with TCG images at gaps of (a) 7, (b) 2, (c) 1.5, and (d) . Fine gap detection demonstrated with subregions of TCG images at successive gap increments of : (e) starting gap of , (f) , (g) , (h) , (i) , and (j) . Using a frequency-domain algorithm, the phase disparity across the midline can be measured with a sensitivity corresponding to gap increments.
Two-dimensional FDTD simulation of the Talbot effect with a grating of period , . The Talbot distance refers to the minimum distance between replications of the fundamental spatial frequency. Note that the first Talbot distance occurs at . Even multiples of the Talbot distance reproduce the original spatial frequency fundamental, while odd multiples reproduce the fundamental with a -phase shift. Doubling of the fundamental frequency occurs at integer multiples of . Surfaces of constant spatial frequency (Talbot surfaces) are parallel to the grating.
Two-dimensional FDTD simulation of the chirped-Talbot effect. A variation in grating period results in an inclination of the Talbot surfaces. With a chirped grating a number of Talbot surfaces can intersect a substrate plane that is parallel to the grating plane.
Talbot surfaces from (a) a fixed-period grating, , (b) a linear-chirped grating, , and (c) a quartic-chirped grating, , assuming . Note that each curve corresponds to a gap increment of . Compared to the linear-chirped grating, the quartic-chirped grating provides a narrower bandwidth of spatial frequencies at a given gap, which simplifies analysis. Plots are based on a model that assumes a locally constant grating period.
Two-dimensional FDTD simulation of intensity from a quartic-chirped grating, using the same constants and wavelength as in Fig. 5. Superimposed lines from Fig. 5(c) indicate the local-interaction approximation to Talbot surfaces. Deviations from the local approximation are evident for increasing gap and increasing chirp rate.
Chirped-Talbot effect FDTD simulation (a), and exposed resist at several gaps. Fringes in resist for gaps of (b) 2.0, (c) 4.9, and (d) .
Diagram of the chirped-Talbot effect FDTD for both sides of a chirped-checkerboard mark, and the associated phase disparity between the fringes exposed in resist.
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