Interference frequency estimation is essential in spectral-domain interferometric sensing and imaging, and its performance determines system sensitivity. To date, an objective and practical criterion is still absent for the proper evaluation of fundamental sensitivity limit in a given system. Here, we report the derivation of Cramer-Rao bound (CRB) for unbiased estimators of spectral interference frequency, which imposes a theoretical limit on measurement sensitivity. Results based on a shot noise-limited model are presented. A more complete model, including dark and read noises, is also discussed and its approximate CRB is obtained. Poisson statistics were used in both cases. Asymptotic behaviors and simplified forms of CRB are studied for fringe visibility approaching 0 or 1. Further, we show that the current Fourier transform-based estimation algorithm achieves CRB only for low visibility, but is inferior by as much as √2 times for high visibility. This performance gap may potentially permit a sensitivity gain if better algorithms can be devised. Finally, we introduce a practical means to accurately estimate CRB from experimentally acquired spectral data. The results are verified in both simulation and experiments. Although spectral interference is discussed here, the same derivation can be applied to spatial or temporal interference as well, provided they can be similarly modelled.
Interference frequency estimation is essential in spectral-domain interferometric sensing and imaging, and its performance determines system sensitivity. To date, an objective and practical criterion is still absent for the proper evaluation of fundamental sensitivity limit in a given system. Here, we report the derivation of Cramer-Rao bound (CRB) for unbiased estimators of spectral interference frequency, which imposes a theoretical limit on measurement sensitivity. Results based on a shot noise-limited model are presented. A more complete model, including dark and read noises, is also discussed and its approximate CRB is obtained. Poisson statistics were used in both cases. Asymptotic behaviors and simplified forms of CRB are studied for fringe visibility approaching 0 or 1. Further, we show that the current Fourier transform-based estimation algorithm achieves CRB only for low visibility, but is inferior by as much as √2 times for high visibility. This performance gap may potentially permit a sensitivity gain if better algorithms can be devised. Finally, we introduce a practical means to accurately estimate CRB from experimentally acquired spectral data. The results are verified in both simulation and experiments. Although spectral interference is discussed here, the same derivation can be applied to spatial or temporal interference as well, provided they can be similarly modelled.
Parameter estimation, especially frequency estimation, from noisy observations of interference is essential in optical interferometric sensing and metrology. The Cramer-Rao bound (CRB) of such estimation determines measurement sensitivity limit. Unlike the well-studied complex sinusoids in communication theory, an optical interference signal is distinctly different in its model parameters and noise statistics. The connection between these parameters and their estimation bounds has not been well understood. Here we propose a complete, realistic multiparameter interference model corrupted by a combination of shot noise, dark noise, and readout noise. We derive the Fisher information matrix and the CRBs for all model parameters, including intensity, visibility, optical path length (frequency), and initial phase. We show that the CRBs of frequency and phase are coupled but not affected by the knowledge of intensity and visibility. Knowing the initial phase offers significant sensitivity advantage, which is verified by both theoretical derivations and numerical simulations. In addition to the complete model, a shot noise-limited case is studied, permitting the calculation of the CRBs directly from measured data.
Bounding the optimal precision in parameter estimation tasks is of central importance for technological applications. In the regime of a small number of measurements, or low signal-to-noise ratios, the meaning of common frequentist bounds, such as the Cramér-Rao bound (CRB), becomes questionable. Here, we discuss a different (Barankin-like) approach to derive the conditions for the existence of optimal estimators and calculate their variance, for any number of measurement repetitions. We elaborate on connections to the shot-noise limit and the Kitaev phase estimation algorithm and derive an extended Cramér-Rao bound that is compatible with a finite variance in situations where the Barankin bound (BaB) is undefined. Finally, we show an exemplary numerical confrontation between frequentist and Bayesian approaches to parameter estimation. Published by the American Physical Society 2024
Precision measurement has been an important research area in sensing and metrology. In classical physics, the Fisher information determines the maximum extractable information from statistically unknown signals, based on a joint probability density function of independently and identically distributed random variables. The Cramer-Rao lower bound (CRLB) indicates the minimum error of the Fisher information, generally known as the shot-noise limit. On the other hand, coherence has pushed the resolution limit further overcoming the diffraction limit using many-wave interference strictly confined to the first-order intensity correlation. However, practical implementation is limited by the lithographic constraints in, e.g., optical gratings. Recently, a coherence technique of superresolution has been introduced to overcome the diffraction limit in phase sensitivity using higher-order intensity correlations of a phase-controlled output field from an interferometer. Here, the superresolution is adopted for precision metrology in an optical spectrometer, whose enhanced frequency resolution is linearly proportional to the intensity-product order, overcoming CRLB. Unlike quantum sensing using entangled photons, this technique is purely classical and offers robust performance against environmental noises, benefiting from the interferometer's scanning mode for fringe counting.
We derive a quantum Cramér-Rao bound (QCRB) on the error of estimating a time-changing signal. The QCRB provides a fundamental limit to the performance of general quantum sensors, such as gravitational-wave detectors, force sensors, and atomic magnetometers. We apply the QCRB to the problem of force estimation via continuous monitoring of the position of a harmonic oscillator, in which case the QCRB takes the form of a spectral uncertainty principle. The bound on the force-estimation error can be achieved by implementing quantum noise cancellation in the experimental setup and applying smoothing to the observations.
The widely used quantum Cramér–Rao bound (QCRB) sets a lower bound for the mean square error of unbiased estimators in quantum parameter estimation, however, in general QCRB is only tight in the asymptotical limit. With a limited number of measurements biased estimators can have a far better performance for which QCRB cannot calibrate. Here we introduce a valid lower bound for all estimators, either biased or unbiased, which can serve as standard of merit for all quantum parameter estimations.
The photonic lantern is an all-photonic wavefront sensor native to single-mode-fibre-fed instruments, but its performance is almost always quoted through a specific reconstruction algorithm, obscuring how much wavefront information the device itself encodes. We develop, from first principles, the Fisher-information and Cramer-Rao theory of the photonic-lantern wavefront sensor, benchmark it against the quantum Cramer-Rao bound via an explicit multi-parameter quantum-Fisher-information calculation, and relate it to two established frameworks: the Fourier-filtering noise-propagation model of Chambouleyron et al (2023) and the classical/quantum sensitivity limit of Haffert et al (2023). Treating the lantern as a deterministic map from aberration coefficients to N output intensities, we derive the Poisson and read-noise Fisher information matrices (FIM), the per-mode CRLB, the per- photon Fisher-Rao geometry on the intensity simplex, and a flux- and estimator-independent sensitivity metric beta with quantum ceiling beta = 2. The lantern CRLB scales as N_ph^(-1/2) and is bounded, mode by mode, by the quantum limit of 1/2 rad rms per photon. That multi-parameter bound is jointly saturable: the phase generators are real and commuting, so the mean Uhlmann curvature vanishes and wavefront sensing carries no quantum incompatibility between simultaneously estimated modes, extending Haffert et al's single-mode ceiling to all low-order modes at once. We further show that the photon-noise sensitivity s_gamma of Chambouleyron et al is exactly the diagonal of our per- photon FIM, whereas the CRLB uses the diagonal of its inverse; the two coincide only for a diagonal FIM, so s_gamma is optimistic for a mode-mixing lantern. The framework is device-agnostic: it returns estimator-independent sensitivities comparable on a common beta <= 2 scale with pyramid, Zernike and PIAA-ZWFS sensors.