Fundamental Limitations of Absolute Ranging via Deep Frequency Modulation Interferometry

Abstract

Deep frequency modulation interferometry (DFMI) resolves phase ambiguity in absolute distance measurements by jointly estimating two length-encoding parameters: the coarse and unambiguous effective modulation depth ($m$), and the fine but ambiguous interferometric phase ($φ$). We establish a comprehensive framework quantifying the fundamental precision limits and practical accuracy constraints of this technique. A Fisher-information analysis defines the intrinsic estimator precision for $m$ and $φ$, while the contribution of carrier frequency drift introduces an additional, time-dependent source of random error. Numerical simulations reveal a structured error landscape with previously unrecognized ``valleys of robustness,'' where systematic biases from common hardware imperfections are suppressed by orders of magnitude. An analytical model based on signal orthogonality explains their origin and predicts their locations. The results yield a consolidated error budget accounting for both random and systematic errors, providing a quantitative design paradigm for absolute length metrology via DFMI.

Figures (12)

• Figure 1: A typical DFMI setup based on a Michelson interferometer. A laser with sinusoidal frequency modulation is split into a reference arm and a measurement arm. The recombined light is detected by a photodiode, and the resulting voltage signal is digitized and processed. The setup is adapted from Rohr2023.
• Figure 2: Example DFMI signal for an interferometer with $\Delta l = 5\,\rm cm$, injected with a laser modulated by $6.87\,\rm GHz$ at $1\,\rm kHz$. The resulting configuration yields a modulation depth $m = 7.2\,\rm rad$.
• Figure 3: Statistical uncertainty of the modulation depth estimate ($\sigma_m$) as a function of the true modulation depth $m$ and the number of harmonics ($N_{h}$) used in the fit. The curves show the Cramér-Rao Lower Bound, calculated from the inverse of the full $4\times 4$ FIM, assuming a fixed raw SNR of $80\,\rm dB$. For low $N_{h}$, the precision exhibits sharp degradation in "dead zones". As more harmonics are included, these zones disappear and the precision becomes a smoother function of $m$, converging to an asymptotic limit.
• Figure 4: The trade-off between the raw voltage signal-to-noise ratio SNR$_v$ and the acquisition time $T_{\rm acq}$ required to meet a target precision for the modulation depth estimate, $\hat{m}$. The plot illustrates the design trade-space for a system sampled at $f_s = 200\,\rm kHz$, where the total number of samples is $N_v = T_{\rm acq} \cdot f_s$.
• Figure 5: Bias in the estimated modulation depth ($\hat{m}$) and interferometric phase ($\hat{\phi}$) due to second-harmonic distortion in the frequency modulation. The bias is shown as a function of the true modulation depth $m$ and the distortion amplitude $\epsilon$. The left and right panels show the mean and worst-case bias, respectively, from a Monte Carlo simulation over the unknown phases $\Phi$ and $\psi_2$. The plots reveal distinct vertical "valleys of robustness" where the bias is strongly suppressed. The $\hat{m}$ valleys align with the extrema of $J_2(2m)$, i.e., occurring at $m \approx 3.35, 4.98, 6.59, 8.17 \dots$, and the $\hat{\phi}$ valleys align with the zeros of $J_2(2m)$ starting at $m \approx 5.81$, i.e., occurring at $m \approx 5.81, 7.40, 8.98 \dots$. The dashed white lines indicate the theoretical predictions.