Optimal Pattern Location in Modulation Enhanced Microscopy

Abstract

Super-resolution fluorescence microscopy techniques enable researchers to observe biological processes at the nanometer scale, surpassing the diffraction limit of light. Among these, methods that combine illumination patterns with single-molecule localization offer high spatial resolution while maintaining photon efficiency. A leading example is MINFLUX microscopy, which achieves nanometer-scale localization by positioning a doughnut-shaped excitation beam, with a central intensity minimum, close to the emitter. This configuration allows for highly precise triangulation of the emitter’s position using only a few detected photons. However, theoretical models often assume perfect modulation with a zero-intensity center, whereas in practical implementations, optical aberrations and alignment errors introduce residual intensity at the beam’s center, degrading the localization precision. To systematically investigate how deviations from ideal modulation affect localization performance, we introduce a modulation contrast parameter m ∈ (0, 1], where m = 1 represents perfect modulation and values below one reflect increasing residual intensity at the excitation minimum. We extend the Cramér–Rao Lower Bound (CRLB) framework to incorporate this parameter, allowing us to quantify how imperfect modulation reduces the Fisher information and consequently increases the theoretical lower bound on localization precision. We show that decreasing modulation contrast not only worsens achievable precision but also shifts the optimal illumination spacing L, challenging previously established scaling laws. We derive and validate a predictive formula for this optimal spacing, L_opt ≈ 1.30 σ_illum√ 1 − m, which is experimentally accessible and maintains high precision. This relationship remains valid as long as the emitter lies within 40% of the pattern diameter. We further extend the framework to account for uncertainty in emitter position by incorporating prior information, showing that the optimal spacing increases with prior uncertainty σ_prior. In addition, we evaluate iterative MINFLUX and show that under non-ideal conditions m < 1, the standard multi-step narrowing strategy becomes suboptimal. Instead, performing repeated measurements at the optimized spacing L_opt achieves significantly better precision over 50% improvement at m = 0.95. Our results could provide a foundation for MINFLUX single-particle tracking, where selecting L_opt based on system contrast and prior uncertainty can maximize localization precision in each frame while minimizing photon budget. Finally, experimental validation under non-ideal modulation conditions will be crucial to confirm the practical relevance of these predictions and to further refine theoretical models.