On the measurement of frequency and of its sample variance with high-resolution counters

Abstract

A frequency counter measures the input frequency ν averaged over a suitable time τ , versus the reference clock. High resolution is achieved by interpolating the clock signal. Further increased resolution is obtained by averaging multiple frequency measurements highly overlapped. In the presence of additive white noise or white phase noise, the square uncertainty improves from σ 2 ν ∝ 1/τ 2 to σ 2 ν ∝ 1/τ 3 . Surprisingly, when a file of contiguous data is fed into the formula of the two-sample (Allan) variance σ 2 y (τ ) = E{ 1 2 (y k+1 − y k ) 2 } of the fractional frequency fluctuation y, the result is the modified Allan variance mod σ 2 y (τ ). But if a sufficient number of contiguous measures are averaged in order to get a longer τ and the data are fed into the same formula, the results is the (non-modified) Allan variance. Of course interpretation mistakes are around the corner if the counter internal process is not well understood.

In-line extraction of an ultrastable frequency signal over an optical fiber link

“…With the Λ-type data, noise is averaged out in a narrower bandwidth and we can probe both the various phase noise dependence and the noise floor. Detailed discussions on the properties of overlapping and modified Allan deviation can be found in [23,24]. Fig.…”

Hybrid fiber links for accurate optical frequency comparison

“…The frequency of the beat-note signal between the local laser beam and the beam transmitted at the output of the 86-km long fiber was measured simultaneously using two deadtime-free counters [28] (where one was Π-type and the other was Λ-type [29,30]) with a gate time of 1 s. No cycle slip was observed in the frequency data throughout the total measurement time of 66 691 s. The fractional frequency instabilities of the ANC transfer method in terms of the Allan deviation [29,30] are shown in Fig. 2.…”

End-to-end topology for fiber comb based optical frequency transfer at the 10−21level

“…For this purpose, the RF frequencies are counted using synchronous dead-time free frequency counters (K&K FXE) [35] referenced to a hydrogen maser. The counters with a fundamental gate time of τ 0 = 1 ms are operated in Λ frequency averaging mode with an averaging time of 1 s [36,37]. We post-process the recorded frequency time series on a 1 s grid to get the time-resolved frequency ratios between the source and target frequency measured at each of the two frequency combs, which we then plug into Eq.…”