A novel pitch evaluation of one-dimensional gratings based on a cross- correlation filter

If one-dimensional (1D), p –period and arbitrarily structured grating position-related topographical signals coexist with noise, it is difficult to evaluate the pitch practically using the centre-of-gravity (CG) method. The Fourier-transform-based (FT) method is the most precise to evaluate pitches; nevertheless it cannot give the uniformity of pitches. If a cross correlation filter ̶ a half period of sinusoidal waveform sequence (pT period), cross-correlates with the signals, the noise can be eliminated if pT ≈ p. After cross-correlation filtering, the distance between any two adjacent waveform peaks along the direction perpendicular to 1D grating lines is one pitch value. The pitch evaluation based on the cross-correlation filtering together with the detection of peaks position is described as the peak detection (PD) method in this paper. The pitch average and uniformity can be calculated by using the PD method. The computer simulation has indicated that the average of pitch deviations from the true pitch and the pitch variations are less than 0.2% and 0.2% for the sinusoidal and rectangular waveform signals with up to 50% uniform white noise, less than 0.1% and 1% for the sinusoidal and rectangular waveform signals and 0.6% and 2.5% for the triangular waveform signal if three waveform signals are mixed with Gaussian white, binomial and Bernoulli noise up to 50 % in standard deviation, one probability and trial probability respectively. As the examples, a highly oriented pyrolytic graphite (HOPG) with 0.246 nm distance between atoms and a 1D grating with 3000 nm nominal pitch are measured by a ultrahigh vacuum scanning tunneling microscope (UHV STM) and a metrological atomic force microscope (AFM) respectively. After the position-related topographical signals are cross-correlation filtered, the 0.240 nm and 3004.11 nm pitches calculated by using the PD method are very close to the 0.240 nm and 3003.34 nm results evaluated by the FT method.