The DeepFMKit Python package: A toolbox for simulating and analyzing deep frequency modulation interferometers

TL;DR

DeepFMKit delivers an open-source, end-to-end Python framework for simulating and analyzing Deep Frequency Modulation Interferometry (DFMI). It unifies a high-fidelity physics engine—modeling time-of-flight delays, arbitrary modulation waveforms, and 1/f^α noise—with flexible parameter-estimation back-ends, including a parallelized nonlinear least squares fitter and two time-domain EKFs. The framework supports high-throughput studies via a declarative Experiment module and Factory-based parallelization, enabling systematic exploration of design choices and error sources. This tool accelerates prototyping and optimization of precision interferometers, with validated performance and a pathway for extending noise models and readout algorithms.

Abstract

Deep Frequency Modulation Interferometry (DFMI) is an emerging laser interferometry technique for high-precision metrology, offering picometer-level displacement measurements and the potential for absolute length determination with sub-wavelength accuracy. However, the design and optimization of DFMI systems involve a complex interplay between interferometer physics, laser technology, multiple noise sources, and the choice of data processing algorithm. To address this, we present DeepFMKit, a new open-source Python library for the end-to-end simulation and analysis of DFMI systems. The framework features a high-fidelity physics engine that rigorously models key physical effects such as time-of-flight delays in dynamic interferometers, arbitrary laser modulation waveforms, and colored noise from user-defined $1/f^α$ spectral densities. This engine is coupled with a suite of interchangeable parameter estimation algorithms, including a highly-optimized, parallelized frequency-domain Non-linear Least Squares (NLS) for high-throughput offline analysis, and multiple time-domain Extended Kalman Filter (EKF) implementations for real-time state tracking, featuring both random walk and integrated random walk (constant velocity) process models. Furthermore, DeepFMKit includes a high-throughput experimentation framework for automating large-scale parameter sweeps and Monte Carlo analyses, enabling systematic characterization of system performance. DeepFMKit's modular, object-oriented architecture facilitates the rapid configuration of virtual experiments, providing a powerful computational tool for researchers to prototype designs, investigate systematic errors, and accelerate the development of precision interferometry.

Figures (11)

• Figure 1: Example DFMI optical configuration. A laser undergoing strong sinusoidal frequency modulation is injected into a Michelson interferometer. A beam splitter (BS) divides the incoming light and recombines the beams returning from the two interferometer arms. The reference arm is integrated within the BS assembly via a high-reflectivity mirror, while the measurement arm reflects from a target mirror. The recombined beam is directed to a quadrant photoreceiver. The photocurrents are converted to voltages by transimpedance amplifiers and digitized. The resulting signals are processed by DeepFMKit to provide high-resolution displacement, angular, and absolute range measurements of the target mirror.
• Figure 2: Simulated voltage signals for the interferometer depicted in Figure \ref{['fig:setup']}. The configuration has an OPD of $\Delta l = 5\,\rm cm$ and is injected with a laser modulated by $6.87\,\rm GHz$ at $1\,\rm kHz$, resulting in an effective modulation depth of $m = 7.2\,\rm rad$. The quadrant photoreceiver is sampled at 200 kHz for one modulation cycle ($1/f_m = 1\,\rm ms$), producing the four time series shown.
• Figure 3: DeepFMKit's basic "simulate-and-analyze" workflow.
• Figure 4: Validation of the colored noise generation engine. The plot shows the amplitude spectral densities (ASDs) of generated time series for three different noise colors: white noise ($\alpha=0$), flicker noise ($\alpha=1$), and random walk noise ($\alpha=2$). The solid colored lines show the ASDs of the generated noise, which closely match the theoretical $1/f^{\alpha/2}$ slopes (dashed black lines), confirming the fidelity of the noise model.
• Figure 5: Comparison of the StandardNLS and IntegratedEKF fitters in tracking a highly dynamic phase signal. A strong random walk phase noise ("Groundtruth") was simulated by injecting a large amount of random walk laser frequency noise (100 MHz/$\sqrt{\rm Hz}$ at 1 Hz) into an interferometer model with a 20 cm OPD. The EKF output is shown for two different tunings of the process noise covariance: a responsive filter (red) that closely tracks the high-frequency dynamics at the cost of higher estimate variance, and a more heavily filtered configuration (blue) that provides a smoother, but slightly sluggish, estimate. The batch-processing NLS fitter (black dots), configured to provide an estimate of the average phase over 100 ms data buffers, shows similar lag as the sluggish EKF filter with low process noise covariance.