OPD Demodulation Based on Spectrum Interrogation and Fourier Transform
A universal method for the OPD demodulation is to use the fast Fourier transform (FFT) [21-28], since the reflected spectrum of the FFP cavity is a periodic curve in frequency domain. The reflected spectrum could be expressed as
where k = 2n/A (A is wavelength), L is the OPD, D(k) is the light source background, C is the contrast factor, and ф0 is the initial phase.
After Fourier transformation, Equation 5.4 changes to
where * denotes a complex conjugate and the upper-case letters denote the Fourier spectrum. Here, n is the digital frequency in the FFT spectrum. If the cavity length is long enough, the digital frequency n0 is much larger than the spread of the spectrum caused by the variations of D(v). For a general FFT method to interrogate the cavity length, the peak position of carrier frequency n0 is detected, as shown in Figure 5.12, and then the optical path L can be calculated by using 
where 8k is the optical spectrum sampling frequency interval, N is the points of FFT, and c is the light velocity in vacuum.
Because n0 is an integer, one solution for achieving high OPD resolution is to interpolate massive data points using zero padding during FFT, which makes the density of the data points high enough so that the phase information can be retrieved from any given peak rocation. The drawback of this technique is the greatly increased amount of c alculation since the zero padding usually needs to expand the length of the FFT data to 10 times or more to achieve high data point density for reducing the peak position reading error.
For improving demodulation performance, the real peak nf could be calculated from the FFT data using the Buneman frequency
Figure 5.13 Schematic of the positive FFT peak of an envelope-removed interferometric fringe.
estimation, where n0 is not an integer any more. As shown in Figure 5.13, the real peak nf locates between the data points with index numbers of n and n + 1, and complex data values of rne^n and rn+1eфп+1.
nt could be calculated by
The optical path L can be calculated by using Equation 5.6, where nf substitutes for n0. Furthermore, from the first estimated nf, type II peak index can be calculated by 
where k0 and k1 are the wave numbers of the spectrum’s first and last data points, and INT is a rounding calculation. The optical path L can be calculated by using Equation 5.8, where n^ substitutes for n0. Based on Equation 5.8, the OPD demodulation performance is much better than direct peak tracking; 1-nm demodulation resolution and 10-kHz demodulation speed have been achieved .
A similar method based on the combination of the Fourier transform method and the minimum mean-square error estimation- based signal processing method was presented by Zhou and Yu , which is capable of providing subnanometer resolution and absolute measurement over a wide dynamic range.
Another approach to demodulate the OPD using the FFT is to select B(n — n0) by filtering, and to obtain the inverse Fourier transform of B(n — n0) as
where ф(А) = 2n/AL. The imaginary part gives the principal value of the phase change with modulo 2n, and the phase ф(А) is wrapped into the range [-П, +п]. So, there are discontinuities with 2n phase jumps in ф(А). This wrapped phase is corrected by using a phase unwrapping algorithm. Then, a measurement of a large phase change exceeding 2n can be realized. The optical path d, can be calculated when the phase ф(А) is obtained. However, when we scan the wavelength from A0 to X1, we obtain a phase change Дф(A), so the optical path L is actually obtained from the following equation :
Based on Equation 5.10, in order to get better performance, the light source envelope should be removed to eliminate the low- frequency cross talk. Resolution of 10 nm could be realized using this method .