Analytic Spectra of Fabry–Pérot Interferometers for Hilbert Transform and its Application in Distance Monitoring

This research introduces how to find the analytic spectrum of a Fabry–Pérot interferometer (FPI) with arbitrary reflectivities of mirrors by giving the mathematical expression of the spectrum. It does not require the weak reflection assumption of the two mirrors which treats the spectrum as sinusoidal function instead of an Airy function. In this research, the relationships between the instantaneous frequency, the instantaneous phase, and the cavity length were given. The proposed method also illustrates that the conclusions given by the two-beam interference approximation are also applicable for high reflection conditions if the spectrum covers multiple complete free spectrum ranges. Compared to other methods of tracking interference maxima and minima, this method provides absolute cavity length measurements and does not encounter problems with losing track of the maxima and minima when working with unstable or noisy spectrum. Compared to the Fourier transform method, this approach considers the chirped nature of the spectrum instead of assuming it as a periodic signal. Also, this method has a higher resolution than the Fourier transform method if the spectrum covers a narrow wavelength range. We used experimental spectra from our previous research to illustrate the performance of this method. A precision at nanometer level and an accuracy at sub-micrometer level was achieved. Therefore, this method demonstrates strong potential for real-time and high-precision monitoring processes.