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The use of Allan variance to characterize the stability of optical signals affected by stochastic polarization fluctuations and the identification of the underlying power law noise processes is explored. Allan variance can ease the comparison in terms of polarization stability of optical systems affected by polarization fluctuations and define a near-optimum integration interval to reveal trends. Experiments with different optical sources show that white noise and random walk terms can be observed. This information can easily be used to systematically define, in real time, the denoising strategy in polarization-based sensing and for the optimization of polarization sensitive optical systems instead of the conventional approach relying on heuristics or information criteria.

In this paper, we perform a time-domain analysis of the higher-order Allan variance for atomic clock models of arbitrary order. Adopting a standard atomic clock model where the time series of the clock reading deviation is expressed as a Wiener or integrated Wiener process, we define the higher-order Allan variance as the mean squared higher-order difference of the clock reading deviation. The main results of this paper are threefold. First, we prove that the higher-order difference operation of the clock reading deviation, which can be interpreted as a linear aggregation with binomial coefficients, is not only sufficient but also necessary for a resulting aggregated time series to be an independent and identically distributed Gaussian process. Second, we derive a complete analytical expression of the higher-order Allan variance, which consists of both time-dependent and time-independent terms. Third, we prove that the higher-order Allan variance is time-independent if and only if the order of difference operation is greater than or equal to the order of the atomic clock model.

Allan variance has been widely utilized in evaluating the stability of the time series generated by atomic clocks and lasers, in time regimes ranging from short to extremely long. This multi-scale examination capability of the Allan variance may also be beneficial in evaluating the chaotic oscillating dynamics of semiconductor lasers, not just for conventional phase stability analysis purposes. Here we demonstrate Allan variance analysis of the complex time series generated by a semiconductor laser with delayed feedback, including low-frequency fluctuations (LFFs), which exhibit both fast and slow dynamics. Whereas the detection of LFFs is not easy with the conventional power spectrum analysis method in the low-frequency regime, we show that the Allan variance approach clearly captured the appearance of multiple time-scale dynamics, like LFFs. This study demonstrates that Allan variance can help in understanding and characterizing versatile laser dynamics, including LFFs, which involve dynamics spanning a wide range of time scales.

The Allan variance of signal and reference frequencies is measured by a least-squares fit of the output of two analog-to-digital converters (ADC's) to ideal sine waves. The difference in the fit phase of the two channels generates the timing data needed for the Allan variance. The fits are performed at the signal frequency ($\approx 10 $ MHz) without the use of heterodyning. Experimental data from a modified digital oscilloscope yields a residual Allan deviation of $3 \times 10^{-13}/\tau $, where $\tau$ is the observation time in sec. This corresponds to a standard deviation in time of $<$ 300 fs or 20 $\mu$Rad in phase. The experimental results are supported by statistical theory and Monte Carlo simulations which suggest that optimized devices may have one or two orders of magnitude better performance., Comment: 8 pages, 6 figures

The Allan Variance (AV) is a widely used quantity in areas focusing on error measurement as well as in the general analysis of variance for autocorrelated processes in domains such as engineering and, more specifically, metrology. The form of this quantity is widely used to detect noise patterns and indications of stability within signals. However, the properties of this quantity are not known for commonly occurring processes whose covariance structure is non-stationary and, in these cases, an erroneous interpretation of the AV could lead to misleading conclusions. This paper generalizes the theoretical form of the AV to some non-stationary processes while at the same time being valid also for weakly stationary processes. Some simulation examples show how this new form can help to understand the processes for which the AV is able to distinguish these from the stationary cases and hence allow for a better interpretation of this quantity in applied cases.

The instability of an atomic clock is characterized by the Allan variance, a measure widely used to describe the noise of frequency standards. We provide an explicit method to find the ultimate bound on the Allan variance of an atomic clock in the most general scenario where N atoms are prepared in an arbitrarily entangled state and arbitrary measurement and feedback are allowed, including those exploiting coherences between succeeding interrogation steps. While the method is rigorous and general, it becomes numerically challenging for large N and long averaging times., Comment: 17 pages, 2 figures

The Allan variance (AVAR) was introduced 50 years ago as a statistical tool for assessing of the frequency standards deviations. For the past decades, AVAR has increasingly being used in geodesy and astrometry to assess the noise characteristics in geodetic and astrometric time series. A specific feature of astrometric and geodetic measurements, as compared with the clock measurements, is that they are generally associated with uncertainties; thus, an appropriate weighting should be applied during data analysis. Besides, some physically connected scalar time series naturally form series of multi-dimensional vectors. For example, three station coordinates time series $X$, $Y$, and $Z$ can be combined to analyze 3D station position variations. The classical AVAR is not intended for processing unevenly weighted and/or multi-dimensional data. Therefore, AVAR modifications, namely weighted AVAR (WAVAR), multi-dimensional AVAR (MAVAR), and weighted multi-dimensional AVAR (WMAVAR), were introduced to overcome these deficiencies. In this paper, a brief review is given of the experience of using AVAR and its modifications in processing astro-geodetic time series.

Optical fiber links are known as the most performing tools to transfer ultrastable frequency reference signals. However, these signals are affected by phase noise up to bandwidths of several kilohertz and a careful data processing strategy is required to properly estimate the uncertainty. This aspect is often overlooked and a number of approaches have been proposed to implicitly deal with it. Here, we face this issue in terms of aliasing and show how typical tools of signal analysis can be adapted to the evaluation of optical fiber links performance. In this way, it is possible to use the Allan variance as estimator of stability and there is no need to introduce other estimators. The general rules we derive can be extended to all optical links. As an example, we apply this method to the experimental data we obtained on a 1284 km coherent optical link for frequency dissemination, which we realized in Italy.

The Allan variance was introduced fifty years ago for analyzing the stability of frequency standards. Beside its metrological interest, it is also an estimator of the large trends of the power spectral density (PSD) of frequency deviation. For instance, the Allan variance is able to discriminate different types of noise characterized by different power laws in the PSD. But, it was also used in other fields: accelerometry, geophysics, geodesy, astrophysics and even finances! However, it seems that up to now, it has been exclusively applied for time series analysis. We propose here to use the Allan variance onto spatial data. Interferometric synthetic aperture radar (InSAR) is used in geophysics to image ground displacements in space (over the SAR image spatial coverage) and in time thank to the regular SAR image acquisitions by dedicated satellites. The main limitation of the technique is the atmospheric disturbances that affect the radar signal while traveling from the sensor to the ground and back. In this paper, we propose to use the Allan variance for analyzing spatial data from InSAR measurements. The Allan variance was computed in XY mode as well as in radial mode for detecting different types of behavior for different space-scales, in the same way as the different types of noise versus the integration time in the classical time and frequency application. We found that radial AVAR is the more appropriate way to have an estimator insensitive to the spatial axis and we applied it on SAR data acquired over eastern Turkey for the period 2003-2011. Space AVAR allowed to well characterize noise features, classically found in InSAR such as phase decorrelation producing white noise or atmospheric delays, behaving like a random walk signal. We finally applied the space AVAR to an InSAR time series to detect when the geophysical signal, here the ground motion, emerges from the noise.

In this work, time series analysis techniques are used to analyze sequential, equispaced mass measurements of a Si density artifact, collected from an electromechanical transducer. Specifically, techniques such as Power Spectral Density, Bretthorst periodogram, Allan variance and Modified Allan variance can provide much insight regarding the stochastic correlations that are induced on the outcome of an experiment by the measurement system and establish criteria for the limited use of the classical variance in metrology. These techniques are used in conjunction with power law models of stochastic noise in order to characterize time or frequency regimes by pointing out the different types of frequency modulated (FM) or phase modulated (PM) noise. In the case of phase noise, only Modified Allan variance can tell between white PM and flicker PM noise. Oscillations in the system can be detected accurately with the Bretthorst periodogram. Through the detection of colored noise, which is expected to appear in almost all electronic devices, a lower threshold of measurement uncertainty is obtained and the white noise model of statistical independence can no longer provide accurate results for the examined data set., Comment: 8 pages, 7 figures

Allan variance (AVAR) was first introduced more than 40 years ago as a estimator of the stability of frequency standards, and now it is actively used for investigations of time series in astronomy, geodesy and geodynamics. This method allows one to effectively explore the noise characteristics for various data, such as variations of station and source coordinates, etc. Moreover, this technique can be used to investigate the spectral and fractal structure of the noise in measured data. To process unevenly weighted and multidimensional data, which are usual for many astronomy and geodesy applications, AVAR modifications are proposed by the author. In this paper, a brief overview is given of using of classical and modified AVAR method in astronomy and geodynamics., Comment: 10 pp., corresponds to the journal printed version with several misprints corrected

The Allan variance is a standard technique to characterise the stability of spectroscopic instruments used in astronomical observations. The period for switching between source and reference measurement is often derived from the Allan minimum time. We propose a new approach for the computation of the Allan variance of spectrometer data combining the advantages of the two existing methods into a unified scheme. Using the Allan variance spectrum we derive the optimum strategy for symmetric observing schemes minimising the total uncertainty of the data resulting from radiometric and drift noise. The unified Allan variance computation scheme is designed to trace total-power and spectroscopic fluctuations within the same framework. The method includes an explicit error estimate both for the individual Allan variance spectra and for the derived stability time. A new definition of the instrument stability time allows to characterise the instrument even in the case of a fluctuation spectrum shallower than 1/f, as measured for the total power fluctuations in high-electron-mobility transistors. We find a non-linear impact of the binning of spectrometer channels on the resulting noise and the Allan time deviating from the description in existing theoretical treatments., Comment: accepted for publication in Astronomy & Astrophysics

One limitation on the performance of optical traps is the noise inherently present in every setup. Therefore, it is the desire of most experimentalists to minimize and possibly eliminate noise from their optical trapping experiments. A step in this direction is to quantify the actual noise in the system and to evaluate how much each particular component contributes to the overall noise. For this purpose we present Allan variance analysis as a straightforward method. In particular, it allows for judging the impact of drift which gives rise to low-frequency noise, which is extremely difficult to pinpoint by other methods. We show how to determine the optimal sampling time for calibration, the optimal number of data points for a desired experiment, and we provide measurements of how much accuracy is gained by acquiring additional data points. Allan variances of both micrometer-sized spheres and asymmetric nanometer-sized rods are considered., Comment: 14 pages, 6 figures, presented at SPIE Optics+Photonics 2009 in San Diego, CA, USA