High precision optical cavity length and width measurements using double modulation

We use doubly phase modulated light to measure both the length and the linewidth of an optical resonator with high precision. The first modulation is at RF frequencies and is set near a multiple of the free spectral range, whereas the second modulation is at audio frequencies to eliminate offset errors at DC. The light in transmission or in reflection of the optical resonator is demodulated while sweeping the RF frequency over the optical resonance. We derive expressions for the demodulated power in transmission, and show that the zero crossings of the demodulated signal in transmission serve as a precise measure of the cavity linewidth at half maximum intensity. We demonstrate the technique on two resonant cavities, with lengths 16 m and a 4 km, and achieve an absolute length accuracy as low as 70 ppb. The cavity width for the 16 m cavity was determined with an accuracy of approximately 6000 ppm. Through an analysis of the systematic errors we show that this result could be substantially improved with the reduction of technical sources of uncertainty.