# A Miniaturized and Fast System for Thin Film Thickness Measurement
## 3 Film Thickness Calculation Algorithm  
Figure 2 is the flow chart of the core algorithm in this paper. Firstly, aiming at the original interference signal collected by a spectrometer, EMD is used as a high-pass filter to extract high-frequency interference components, adaptively eliminating the influence of the non-uniform intensity distribution of the LED source. Second, to obtain the phase term introduced by film thickness, the interference component is transformed into the wavenumber \(\sigma\) domain through reciprocal of wavelength \(\lambda\). Third, through the spectral analysis method, we extract the phase term from the full spectrum to effectively avoid the interference of local noise. However, conventional spectral analysis methods, such as Fourier transform, cannot handle the problem of non-equal interval sampling of signals caused by the \(\lambda-\sigma\) transformation. In this paper, the phase term is solved by the LSP spectrum analysis and illustrated by phase power spectral density distribution. Finally, the peak value of the power spectral density function is extracted by the maximum value method to calculate the film thickness. The detailed flow of the proposed algorithm is as follows. EMD, as the preprocessing algorithm of Hilbert Huang Transform (HHT) [17], can adaptively decompose the complex nonlinear and non-stationary signal into a limited number of Intrinsic Mode Function (IMF). In this paper, EMD is used as a high-pass filter [18] to automatically filter out the high-frequency principal component of the reflectance spectrum signal. The mirror extending method [19] assists in processing the endpoint data of the original signal to eliminate the boundary oscillation effect caused by the third-order spline interpolation in EMD. Figure 1: Schematic diagram of the thin-film thickness measurement principle. The reflectance spectrum signal \(I_{SI}\) is a discrete function of the uniform sampled pixels for being collected by Charge Coupled Device (CCD) array in a micro spectrometer. After EMD filtering, \(I_{SI}\) is disassembled into a set of IMF components \(Imf_{k}\), \(k\) = 1, 2,..., \(m\) of different frequency ranges, and a monotonic trend function _Res_, as shown in Figure 3b. \(Imf_{1}\) is the first component to be filtered out with the highest signal frequency. In the case with no high-frequency noise, \(Imf_{1}\) is the film interference principal component. Then, according to the corresponding relationship between the pixel index and the wavelength of CCD array in the spectrometer, the intensity \(I_{i}\) of each \(\sigma_{i}\), \(i\) = 1, 2,..., \(N\), is obtained by converting \(Imf_{1}\) from the \(\lambda\) to the \(\sigma\) domain. \(N\) is the number of pixels of the spectrometer. Due to the wavelength drift of the spectrometer and the reciprocal transformation from the \(\lambda\) to the \(\sigma\) domain, the non-uniform sampled characteristic of \(\sigma_{i}\) made \(I_{i}\) unable to be directly analyzed by the conventional spectral analysis method. Therefore, it is necessary to preprocess \(I_{i}\) before performing spectral analysis to obtain film thickness information We adopt the cubic spline interpolation combined  
Figure 3: Empirical Mode Decomposition (EMD) high-pass filtering process. (**a**) EMD sifting process; (**b**) Intrinsic Mode Function (IMF) components of different frequency ranges. Figure 2: Flowchart of the proposed phase power spectrum (PPS) algorithm. fast Fourier transform (CIFFT) and LSP [20,21] as two representative methods to process \(I_{i}\), and finally choose the LSP after comparing the solution results. In particular, inspired by the time-frequency domain (\(t\)-\(f\)) transformation [22], we rewrite the LSP function into the wavenumber-phase factor domain (\(\sigma\)-\(z\)) transformation form, as Equations (2) and (3).