Measurement of a free spectral range of a Fabry-Perot cavity using frequency modulation and null method under off-resonance conditions

In this paper, we discuss a method to measure the free spectral range (FSR) of a Fabry-Perot cavity (FP-cavity) using frequency modulation with one electric optical modulator (EOM) and the null method. A laser beam modulated by the EOM, to which a sine wave signal is supplied from a radio frequency (RF) oscillator, is incident on the FP-cavity. The transmitted or reflected light from the FP-cavity is observed and converted to an RF signal by a high-speed photodetector, and the RF signal is synchronously demodulated with a lock-in amplifier by referring to a cosine wave signal from the oscillator. We theoretically and experimentally demonstrate that the lock-in amplifier signal for the transmitted or reflected light becomes null with a steep slope when the modulation frequency is equal to the FSR under the condition that the carrier frequency of the laser is slightly detuned from the resonance of the FP-cavity. To reduce the measurement uncertainty for the FSR, we also discuss a selection method for laser power, a modulation index and the detuning shift of the carrier frequency, respectively.