A method for significant enhancement of the spectral resolution of a Fabry-Perot resonator in transmission and absorption measurements is proposed. In the method, a laser with ultrashort pulses is used as the optical source. A dispersive element is placed in front of the Fabry-Perot resonator and a phase modulator is incorporated into the resonator. The spectrum of the laser pulse transmitted through the system is approximately periodic with ultranarrow peaks. The sample transmission spectrum is measured by scanning the output pulse spectrum. It is demonstrated, in numerical simulations, that for realistic parameters of the phase modulator, the finesse of the Fabry-Perot resonator is increased from 72 to 1900 and a resolution of 1 MHz is achieved. A method for increasing the spectral range of measurements with scanning the periodic spectra is also proposed. The method is based on the use of a waveguide array of Mach-Zehnder interferometers or a single discretely tunable interferometer. The measurement of the sample transmission spectrum within 33 free spectral ranges of the resonator is numerically demonstrated. The spectral range of the measurement can be increased up to 10 THz resulting in the equivalent finesse of the system of 10~7 for a 100 fs laser pulse.
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