Exact computation of lock-in amplifier outputs for arbitrary frequency modulations using Gauss-Chebyshev quadrature

The technique of lock-in amplification is widely used across many fields to separate a signal from a noisy background. The mathematical descriptions of frequency-modulated signals are, however, often approximated in a regime of small modulation deviation, potentially leading to inaccurate calculations of optimal modulation parameters. Here, we present an approximation-free numerical approach based on Gauss-Chebyshev quadrature that is able to calculate the lock-in amplifier output for any modulation type, modulation function, input signal, and modulation deviation. Our numerical technique is exact, straightforward, fast to compute, and general for any lock-in-amplified experiment. We apply our findings as an example to nitrogen-vacancy center-based magnetometry and use them to determine the optimal modulation parameters for this use case. Our analysis is accompanied by an open-source Python package containing a reference implementation of our numerical approach.