An analytic, efficient and optimal readout algorithm for compact interferometers based on deep frequency modulation

Compact laser interferometers with large dynamic range are one of the core emerging tools to improve low frequency performance in gravitational wave detectors by providing local displacement sensing with sub 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>pm</mml:mtext> <mml:mspace></mml:mspace> <mml:msup><mml:mtext>Hz</mml:mtext> <mml:mrow><mml:mo>-</mml:mo> <mml:mn>0.5</mml:mn></mml:mrow> </mml:msup> </mml:mrow> </mml:math> precision. Strong sinusoidal frequency modulations are used in such laser interferometers to create heterodyne-like photodetector signals from which the phase and other parameters, such as the absolute distance, can be extracted. The nested sinusoidal function in such signals is a challenge for the real-time parameter estimation in low-noise applications. In this article, we present an algorithm to calculate exact signal parameters in a non-iterative way from such interferometric signals. The algorithm makes use of a recurrence relation between Bessel functions to enable a direct extraction of modulation parameters from the signal. Additionally, the algorithm is capable of dealing with high phase dynamics where the Doppler-shift of the signal becomes relevant and can limit the range and precision of the parameter estimation, if not accounted for. Simulations show that the algorithm is computationally efficient, can be well parallelised and the phase estimation is close to optimal precision given by the Cramer-Rao lower bound of the signal parameters.