Fourier series

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Related to Fourier modes: Fourier series, Fourier theorem
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the sum of a series of trigonometric expressions

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To further investigate the convergence of the volume integral equation, we have increased the number of Fourier modes per direction to [+ or -]50, i.e., a total number of 10201 Fourier modes in the transverse plane, and the number of samples in the z-direction to 65.
The computations were performed using 15 Fourier modes in x and in y and eight sample points in the z-direction per layer of holes or pillars, i.e., a total of 48 sample points in the z-direction.
We have ascertained that field behavior in the distance of the slot (strip) edges is determined mainly by sums of Fourier components with low spatial frequencies, whereas the field asymptotics directly on the edges is caused by higher-order Fourier modes. An exponent of such power-type asymptotics has been computed with the help of the ratio of higher-order Fourier amplitudes.
In the next decade, SPM developments based on active control of the tip trajectory in ~MHz range, active feedbacks to lower thermomechanical noise, and advanced Fourier modes will emerge.
We find that just 16 Fourier modes in the computation of the complex contour integral are sufficient to determine the functions [[psi].sub.i] to the order of machine precision.
Owing to the structure of the separate matrices in the above equations, an efficient O([N.sub.z]M log M) matrix-vector product is obtained, where [N.sub.z] is the total number of unknowns, i.e., samples, in the aperiodic direction (z) and M is the total number of unknowns, i.e., Fourier modes, in the periodic direction (x).
8], by comparing the convergence of the error in the case of TM polarization with respect to the truncation order M, which is the upper index of the Fourier modes and corresponds to a total of 2M +1 Fourier modes in the periodic direction.