Power Spectral Density (PSD) computation
Characterization of optical surfaces frequently involves the power spectral density (PSD). It can be calculated from surface profiles obtained via optical or a mechanical interference or AFM devices. The PSD is the square of the Fourier transform of the surface profile per spatial frequency unit or equivalently the Fourier transform of the autocovariance function that has been calculated from the surface profile. To improve the representatively of the PSD of a surface that has random roughness, we must take an ensemble average of PSD estimates calculated from profiles made at many different places on a surface. To obtain an ensemble average PSD from surface profile data, there are two options.
A large number of profiles can be taken at different places on the surface, the onedimensional PSD estimate calculated from each profile, and the resulting graphs averaged together to obtain a smooth, noise-free curve. The one-dimensional measured PSD can be modelled with an analytic function, typically a power-law, P(f) = Kn/Fn. If z(x) is the surface roughness height as a function of distance x, in the incidence plane of X-rays, a finite length PSD can be written as:
P(f) = 1/L | ∫0L dx z(x)e(-ikx)|2
where k is the wave number. Alternatively, the PSD can be computed from 2D data as a 2D-PSD and, under hypothesis of isotropy, integrated over the frequency. Measurements of z(x) yield digitized data, we assume that the surface roughness data set consists of N values for z(x) that are measured at equally spaced intervals Δx over a total length L = NΔx. If these discrete surface height data are adjusted to have a zero mean value, i.e., there are equal heights above and below a mean surface level, and are denoted by z(n), n = 0 → N − 1, then the PSD becomes:
P(fk) = L/N2 | Σ zj e-ikj |2 , with Fk = 0, 1/L, 2/L,…,Fc
This value is bandwidth limited, as the magnitude of the highest surface spatial frequency resolved in the measurement process is fc = (2Δx)−1, the Nyquist frequency.
In Figure 1 an example of PSD computed with different instruments that work in different spatial wavelength range is shown. Remarkable that the values obtained with the AFM, with the optical interferometer and from scattering measurements are extremely consistent, using three entirely different physical approaches.
The PSD of an X-ray mirror surface, as measured with various techniques, with different spatial wavelength range.
