Fundamental Limitations of Absolute Ranging via Deep Frequency Modulation Interferometry

Abstract: High-precision absolute length metrology using laser interferometry faces inherent challenges from interferometric ambiguity. Deep Frequency Modulation Interferometry (DFMI) addresses this by simultaneously extracting two distinct length-encoding parameters: the precise, ambiguous interferometric phase ($\Phi$) and the coarser, unambiguous effective modulation depth ($m$). This paper establishes the fundamental statistical and systematic limitations governing the absolute accuracy of DFMI. We detail two primary readout algorithms: a frequency-domain Non-Linear Least Squares (NLS) fit and a time-domain Extended Kalman Filter (EKF). We derive the Cram\'er-Rao Lower Bound (CRLB) for the estimation of $m$ and $\Phi$, identifying intrinsic dead zones'' where precision degrades, and asymptotic limits as a function of signal quality and integration time, achievable by both NLS and EKF. Critically, we develop analytical models for dominant systematic biases, revealing previously unchartedvalleys of robustness'' -- specific operating points where errors from modulation non-linearity and residual amplitude modulation are strongly suppressed. This analysis clarifies critical trade-offs between different design parameters, integration time, and total error, informing instrument design and underscoring the necessity of advanced compensation techniques for high-accuracy applications.