Systematic Frequency Error in Laser Synchronization Circuits for Fiber-Optic Time-Transfer Systems

This article addresses the problem of a systematic frequency error occurring in semiconductor-laser frequency synchronization circuits based on counting the beat note between the two lasers in a reference time interval using a high-frequency prescaler. Such synchronization circuits are suitable for operation in ultraprecise fiber-optic time-transfer links, used, e.g., in time/frequency metrology. The error occurs when the power of the light coming from the reference laser, to which the second laser is synchronized, is below about −50 to −40 dBm, depending on the details of particular circuit implementation. The error can reach tens of megahertz if left out of consideration and does not depend on the frequency difference between the synchronized lasers. Its sign can be positive or negative, depending on the spectrum of the noise at the prescaler input and the frequency of the measured signal. In this article, we present the background of the systematic frequency error, discuss important parameters allowing for predicting the error value, and describe the simulation and theoretical models being helpful for designing and understanding operation of discussed circuits. The theoretical models presented here show good agreement with the experimental data, which demonstrates the usefulness of the proposed methods. Implementing polarization scrambling to mitigate the effect of polarization misalignment of the lights of the lasers used was considered and the resulting penalty was determined.


Systematic Frequency Error in Laser Synchronization Circuits for Fiber-Optic
Time-Transfer Systems

I. INTRODUCTION
F IBER-OPTIC time and frequency (T&F) transfer starts to become a well-established technology that can be used not only to compare clocks in the optical domain but also to distribute timescales from nated universal time (UTC) laboratories to distant users. Apart from metrology laboratories and those involved directly in time, frequency, or metrology activities [1], [2], [3], the applications include a broad range of scientific cases, e.g., spectroscopy [4], astronomy [very long baseline interferometry (VLBI)] [5], [6], [7], or geodesy [8]. Due to the high stability and accuracy possible by fiber-optic transfer, there is also a growing number of sensitive commercial applications, such as in telecommunication [9], [10], navigation and positioning [11], and smart grids [12], or in the financial sector [13], where a certain amount of independency from global navigation satellite systems (GNSS), ubiquitously used for T&F transfer, is important [14].
A simplified schematic of a fiber-optic time-transfer link with stabilized propagation delay is shown in Fig. 1. Apart from a set of local and remote fiber-optic transmitters (TX) and receivers (RX), it includes a delay compensation system, which can be realized using a pair of matched variable electronic delay lines [15] or, in some cases, variable optical delays [16]. The optical TX needs to use different wavelengths (λ A and λ B ) to allow filtering out the undesired optical signals This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see ht. tps://creativecommons.org/licenses/by/4.0/ Highlights • A systematic frequency error occurs in the laser synchronization circuits because of a narrow-band noise present at the input of the prescaler used to divide down the beat note frequency.
• The models proposed in the text help to understand an analyze the problem, giving a tool to predict the value of the frequency error.
• Developed tools and theory will help design and implementation of picosecond-accurate long-distance fiber-optic time transfer systems using laser transmitters synchronized to weak incoming light. resulting from Rayleigh backscattering [17] and reflections.
In such a link, the timescale t 0 transferred to the remote end of the link can be expressed as where T RT is the round-trip propagation delay of the link and τ A↔B = τ A→B − τ B→A is the link asymmetry being the difference of the forward (τ A→B ) and backward (τ B A ) propagation delays of the link. The link asymmetry can be further decomposed into τ A↔B = λ A↔B D T + τ Sagnac + τ pol + τ C where the first term is related to the chromatic dispersion of the fiber ( λ A↔B is the wavelength difference of the local and remote lasers and D T is the chromatic dispersion accumulated along the fiber), τ Sagnac is the Sagnac correction [18], τ pol is related to the polarization dispersion, and τ C is the correction related to the asymmetry of propagation inside the local and remote terminals. For an accurate time transfer, it is essential to know both T RT and τ A↔B with low uncertainties. This is not a problem for the round-trip delay, which requires only a time interval measurement and can be realized with a few picoseconds uncertainty (see [19]). In the case of link asymmetry, however, the uncertainty depends also on the accuracy of the wavelengths of the lasers and the accuracy of determining the accumulated chromatic dispersion D T . The uncertainty contribution related to this term may be large and can reach tens of picoseconds for long links [9].
A possible solution to this problem is to use one of the lasers (e.g., from the local terminal of the transfer system) as a source of the reference light and stabilize the wavelength of the second laser to this wavelength with a desirable offset [20]. In hybrid systems, transferring jointly stable optical frequency and time [21], [22], [23], [24], this ultrastable optical frequency can also be used as a source of reference light. Using this approach, the uncertainty contribution can be reduced to a single picosecond level, even for 1000-km-long links, which requires uncertainty of the lasers' relative frequency lower than about 5 MHz (see [25] for detailed discussion).
To obtain good results, however, a level of care is required as it has already been shown that such a synchronization system is vulnerable to a systematic frequency error [20], [25], [26], which can be unacceptably large when the power of the reference light is not high enough. Thus, the main goal of this article is to thoroughly discuss the effects occurring in the laser frequency synchronization circuits under low optical levels, present the background of the observed systematic error, provide analytic and simulation models, as well as discuss available countermeasures to keep this error under control. First, we present a short description of the basic laser frequency synchronization circuit.

II. CONCEPT OF LASERS FREQUENCY SYNCHRONIZATION
The main idea behind the laser frequency synchronization circuit is shown in Fig. 2(a). Part of the reference light (ν R ) is combined with the light from the synchronized laser (ν S ) and produces a beat signal in the beat photodiode (PD).
The beat frequency f BEAT , equal to the difference f BEAT = |ν R − ν S |, is then processed in the laser stabilization subsystem, producing the signal controlling the frequency of the synchronized laser. Details of this block are sketched in Fig. 2(b). In practice, f BEAT must match the dense wavelengthdivision multiplex (DWDM) grid [27]. Due to the availability of suitable optical filters the frequency offset can be either 12.5, 25, or 50 GHz. In the first step, such high frequency is downconverted to a more convenient intermediate value (selected to be around 2.6 GHz in the circuits discussed hereinafter), then bandpass filtered, and next divided down by N M in a divider chain, where the initial division is carried out by a fast prescaler. This divided frequency needs to be low enough to enable its counting using a timer/counter implemented in a microcontroller, which is also used to run a proportional-integral-derivative (PID) controller.
The offset frequency between the reference and the synchronized lasers f BEAT = |ν R − ν S | can be found by analyzing the equilibrium condition in the feedback loop, which is formed by the synchronization system. It is equal to where K is the multiplication factor of the microwave synthesizer used for the downconversion of the original beat signal. The frequency of the synchronized laser may be either higher or lower than ν R , depending on the sign of the feedback loop gain.

III. ORIGIN OF FREQUENCY ERROR AT LOW OPTICAL LEVELS
The laser frequency synchronization circuit is a subsidiary of the entire fiber-optic transfer system, so it should be able to operate with a low optical input power, representing a small fraction of the reference light only. In such a case, the expected beat current is also low and the influence of the system noise becomes important. This noise can be assumed white and is composed of the shot noise of the light impinging the PD (mostly a relatively high power of the synchronized laser) and the thermal noise of the resistor (50 ) terminating the beat PD, with also a small contribution of the noise of the amplifiers and the mixer. It is assumed in all further considerations that the spectrum of the noise, which is distributed around f IF , is limited by the IF filter whose passband is fully covered by the bandwidth of the prescaler, making the operation of the circuit greatly independent on the prescaler bandwidth. The noise affects the operation of the fast prescaler (i.e., the N divider), influencing the number of cycles counted in a reference time interval, which is used to estimate f BEAT .
An example of such behavior is shown in Fig. 3, measured with the laser control fixed (i.e., with a feedback loop inactive). In Fig. 3(a), the case is shown when an optical power of around −40 dBm is applied as the reference light together with a power of about 5 dBm from the synchronized laser, producing substantial beat note, visible as a narrow peak in the spectrum at 2.6 GHz over the noise background. The spectrum of the noise was shaped by the filter with the noise bandwidth of about 1.5 GHz (BFCV-2610 + by Mini Circuits), which was followed by the prescaler rated for the operation up to 4 GHz (HMC988 by Hittite, configured as a by eight dividers). The output of the prescaler shows the square wave with the frequency of 325 MHz, which is an expected result. When the reference light is removed, however, leaving only noise as an input to the prescaler [see Fig. 3(b)], its output changes into a noisy waveform whose mean frequency is quite well defined (see Fig. 4 for the data for a few-day-long measurement, showing a mean frequency of about 322 MHz). One may thus expect that if the prescaler input power varies the mean, the output frequency of the prescaler also varies between two extremes-one corresponding to counting the pure signal (i.e., the desired frequency) and the other one corresponding to counting the pure noise, giving rise to a leveldependent systematic frequency error.

IV. ESTIMATION OF PRESCALER FREQUENCY ERROR
To get a deeper insight into the abovementioned error, we started from modeling the behavior of the prescaler treating it as a simple level-crossing detector driven by a timecontinuous random process with a zero mean. Defining the number of zero crossings in a time interval of duration T as n(T ), the mean frequency can then be expressed simply as f M = E{n(T )}/2T , where E{·} denotes an expectation (division by 2 is because both rising and falling edges are counted in this model). A known result from the level-crossing theory, which has been a subject of study for a long time, is that for a normal random process of power spectral density S( f ), the mean frequency is [29] f Applying (4) to the noise spectral density shown in Fig. 3(b) gives 2569 MHz, which is in a reasonably good agreement with the experimental value, equal to 2576 MHz (8 × 322 MHz). This result suggests that treating the prescaler as a level-crossing detector is a reasonable modeling approach, worth further investigation. Equation (4) can thus be used to determine the mean frequency corresponding to the bandlimited noise, hereinafter referred to as f Mn .
To tackle the problem when a sinusoidal component is also present, as such a process is no longer normal [29],  we decided to start with a simulation model developed in MATLAB. The model exploited is based on the same levelcrossing approach discussed above. We used 2 24

samples x[k]
(k is the sample number) taken from the continuous signal composed of a pure sinusoid with frequency f x and a bandlimited normal noise with a power spectral density flat in the frequency range between f L and f H , where f L < f x < f H . To determine the mean frequency, we counted a number of zero crossings in the entire vector x[k] by searching for the condition x[k]x[k + 1] < 0. The amplitude of the sinusoidal component was varied as well as its frequency and we chose the parameters of the noise density to mimic the conditions shown in Fig. 3.
The results of the simulations are shown in Fig. 5(a) where is plotted versus the signal-to-noise ratio (SNR) for a few different values of the prescaler input frequency. As it can be seen, both the value and the sign of the error depend on the difference between f x and f Mn . When f x > f Mn the error is negative and changes its sign to positive under the opposite condition. When f x is located far from f Mn and the SNR is low, the error can be huge-it can easily reach tens of megahertz. The error can be small, on the other hand, if ≈ f x f Mn is selected.
In Fig. 5(b) and (c), the results of simulations are compared with the experimental data obtained for a few high-speed prescalers (rated for the operation at a maximum frequency of 4, 6.5, and 12 GHz) for two different values of f x shown in Fig. 5(a), equal to 2.16 and 3.12 GHz. It is evident from the presented plots that the choice of particular prescaler is not critical as all examined circuits show similar behavior. The agreement between the measurement and the simulation is also satisfactory-the real-world data show a slightly greater error than predicted from the simulation model, which is not surprising noting the model's simplicity. This is, however, interesting that such a simple model can still give reasonable results in a relatively large range of SNRs. As mentioned previously, the acceptable level of error in laser frequency synchronization circuits for ultraaccurate time-transfer links is about 5 MHz [25] that calls for relative error at the level around 2 · 10 −3 [shown with a dashed line in Fig. 5

(b) and (c)].
To fulfill such a condition, the required SNR must be higher than some 8 dB in the presented examples and possibly even lower when f x closer to f Mn is selected (the question of the minimum required SNR will be addressed further in the text).
In addition to the simulation model (which requires substantial computation time), it would be desirable to have a formula allowing fast estimation of the systematic frequency error. The results available in the literature and known to the authors are, however, very general [28], [30] and cannot be directly applied. In this situation, an approximate solution may appear to be useful. A straightforward approach is to check whether (4) can be extended to the cases where both bandpass noise and a sinusoidal component are present in the spectrum entered in (4). Inserting the spectral density in the form (4), where S n ( f ) is the noise spectrum (continuous) and the Dirac delta represents the sinusoidal component with power P S and frequency f x , and performing the required calculations gives the approximated mean frequency f A M in the form where SNR =P S / S n ( f )d f is the SNR. The usefulness of the discussed approximation can be verified by comparing it with the results of the simulation. We assumed that S n ( f ) is uniformly distributed between the frequencies f L and f H . The comparison is shown in Fig. 6, where the simulation result f S M is marked with blue squares  and the approximation f A M is marked with red crosses. It is not surprising to see that the agreement is fairly good in a range of very low SNR where the noise dominates, making the prescaler input signal almost normal (thus, assumptions of (4) are almost satisfied). It deteriorates, however, in the more interesting range of the SNR where the sinusoidal component is dominating, even despite the fact that f A M → f x when SNR → ∞. This is because the signal driving the prescaler can now not be assumed to be normal, so the assumptions related to (4) are certainly violated. This deterioration is clearly visible in Fig. 6(b) where the logarithmic plot of the relative approximation accuracy, defined as f A M − f S M / f S M , is shown for f x = 2160MHz as an example. Fortunately, the approximation accuracy improves if f x is close to f Mn [see Fig. 6(c)].
Looking for a means to further improve the approximation accuracy for SNR > 0 dB, we discovered that substituting (exp(SNR) − 1) for SNR in (5), resulting in allows obtaining much better results. The corresponding curves are shown in Fig. 6 with black lines, in both linear and logarithmic scales, and display that the modified formula (6) gains at least an order of magnitude improvement for SNR > 0 dB. The change of the location of f x with respect to f Mn results in a change of the systematic frequency error, as it has already been mentioned [see Fig. 5(a)]. The curves presented in Fig. 6(a) appear to be universal so that, knowing the curve for some particular value f x1 , it can be converted into the curve for another value f x2 by scaling it by a factor equal . This feature is shown in Fig. 6(a) where the scale at the right side was drawn assuming f x = 2550 MHz (in this case, the abovementioned scaling factor equals about 0.05). In addition, (5) and (6) suggest that the discussed systematic frequency error can be expected to be independent on the specific shape of the noise spectrum S n ( f ), as the only noise-related parameter appears to be f Mn .

V. SYSTEMATIC ERROR IN A CLOSED FEEDBACK LOOP
To determine the frequency error remaining after closing the feedback loop, it is helpful to consider the incremental model of the circuit from Fig. 2. In addition to the already discussed error f N originating in the prescaler (defined here as f P = f M − f IFN , where f IFN is the nominal IF value, i.e., assuming no any prescaler frequency error), the model presented in Fig. 7(a) includes the inaccuracy of the electrical reference f REF , which appears in two different places as this reference is used in the microwave synthesizer supplying f LO , and also provides the time base for the beat note frequency counting by the timer/counter. All other error sources (as the laser noise ν L and the quantization noise ν Q , which occurs when a digitally tuned integrated tuneable laser assembly-ITLA [31]-is used as the synchronized laser) are omitted now as not contributing to the systematic frequency error. H L and H P are the tuning sensitivity of the synchronized laser and the gain of the PID controller, respectively.
The noise-dependent incremental gain of the prescaler is defined as α = N d f P_O /d f P_I , where f P_O and f P_I are the input and output prescaler frequencies, respectively. This parameter reflects the fact that the sensitivity of the prescaler to the change of its input frequency decreases when the SNR decreases. It can be calculated using (6) The relevant curves are shown in Fig. 7(b). It may be noted that α is relatively independent on f x in the considered frequency range and is very close to one for SNR higher than about 5 dB. Its value decreases toward zero when the SNR is lowered.
Based on the feedback theory the incremental change of the synchronized laser frequency, ν S can be cast in the form Assuming that the loop gain, equal to H P H L /M N , is high due to the integrating part of the PID controller, (7) reduces in the limit to The first term in (8) shows that in a closed feedback loop, the prescaler error, as shown in Fig. 5(a), scales inversely to the prescaler gain α and thus is effectively multiplied when the SNR is low. The inaccuracy of f REF scales in a similar way, although this effect is much less pronounced because usually M N f REF < f P . Assuming that a standard 10-MHz temperature compensated crystal oscillator (TCXO) is used with the inaccuracy of 2.5 ppm and that M N is in the order of 250 (as it is in the circuits, we investigated experimentally), M N f REF not greater than 2.5 kHz may be expected, whereas f N can be much larger, especially when the prescaler input frequency is far from f Mn (see Fig. 5). The last term in (8) shows that the inaccuracy of f REF transfers to the circuit output increased by the synthesizer's multiplication factor K . The error due to this can be expected not to be greater than about 10 kHz, as in the circuits with f LO ≈ 10 GHz considered here, K around 1000 is required. In practice, therefore, one can expect that the systematic error will be dominated by the first term of (8) related to the prescaler.
To verify the predictions shown above, we performed the measurement of the systematic frequency error in the complete synchronization circuit using an out-of-loop configuration shown in Fig. 8(a) (the PD, amplifiers, mixer, IF filter, and prescaler used in the out-of-loop path were exactly the same as used in the stabilization circuit). The optical power from the reference laser (Koheras Adjustic X15) was split into two unequal parts with 5% directed to the synchronization circuit via a variable optical attenuator (VOA) to control the SNR at the input of the prescaler. The remaining 95% of the light was combined with the light produced by the synchronized laser and applied to an out-of-loop beat PD. The resulting signal was then processed by the circuit analogous to this used inside the laser stabilization circuit (composed of a beat PD, microwave amplifiers, mixers, and frequency dividers) and finally applied to the frequency counter (Pik Time T4100). Due to the high power of the out-of-loop beat signal, one can be sure that no any systematic frequency error occurred in the out-of-loop divider chain. Such a setup can thus be used to detect any difference from the intended 12.5-GHz offset between the reference light and the synchronized laser.
The results of the measurement are shown in Fig. 8(b) for a few values of the nominal intermediate frequencies f IFN set by adjusting the microwave synthesizer inside the laser stabilization circuit. It is visible that when the SNR drops, the systematic offset starts to manifest-its expected value (marked with black dashed lines) was obtained using (6) taking the calculated value with a negative sign (this is because the feedback loop tends to cancel out the error introduced by the prescaler causing the sign inversion) and scaling it by α according to (8). The calculations agree well with the measurements in all investigated cases confirming the applicability of the theoretical approach developed so far. Fig. 8(b) (inset) shows an enlarged part of the curves where the error stays within ±5-MHz limit required for picosecond-accurate time-transfer systems. The required SNR values depend on the distance between f Mn and the nominal f IFN frequency. For f IFN equal to 2560 MHz (| f Mn − f IFN | = 16 MHz), an operation with an SNR down to about 2 dB is possible, whereas for larger distances, higher SNR is necessary.

VI. MINIMUM SNR AND REQUIRED SIGNAL LEVEL
The minimum signal level required for the operation of the discussed laser synchronization circuits can be, in principle, determined by defining the acceptable level of the frequency error ν Smax . For the considered application of ultra-accurate time transfer, a value of ν Smax = ±5 MHz is adequate (see [25]), but it can be different for other applications. Knowing this, and also the value of f Mn , which is specific for particular realization of the circuit, but can be quite easily measured with a frequency counter, one can choose a reasonable value of the nominal intermediate frequency f IFN reasonably close to f Mn (practical limit results from the long-term stability of f Mn -see Fig. 4 for an example). The opposite approach when f Mn is to be fit to the selected f IFN is also possible, although it seems less convenient in practice, especially when rational microwave synthesizers are easily available. Having these data, it is possible to determine the required SNR by solving (6) under condition f A M = f IFN , forcing the output of the dividers chain to keep the frequency at the reference value due to the integrating part of the PID controller in the feedback loop. The solution takes the form where f x was substituted with f IFN + ν Smax (to get meaningful results, the sign of ν Smax has to be positive for f IFN > f Mn and negative in the opposite case) and assuming typical working conditions ( ν Smax ≪ f IFN and f IFN ≈ f Mn ).
As visible from (9), the result depends mostly on the difference f Mn − f IFN , which is plotted in Fig. 9(a) for a few values of ν Smax . To get the optical power corresponding to this particular SNR, more details of particular circuit implementation are required, such as the frequency characteristics of the IF filter and noise levels of the amplifiers and mixers used. For the circuits evaluated in the laboratory, the optical powers are shown in Fig. 8(b) where the range of the plot corresponds to the values from about −60 to −40 dBm. This shows that the circuit investigated experimentally is capable to operate within the acceptable systematic error level of 5 MHz with the input powers below −50 dBm. The acceptable systematic error level, however, is not the only criterion that needs to be considered-a no less important factor is related to the deterioration of the stability of the output frequency ν S with decreasing SNR and its influence on the stability of the time transfer. One reason for decreasing the feedback loop efficiency is suppressing the noise of the synchronized laser ( ν L ) and its tuning quantization ( ν Q ) because of decreasing the loop gain and bandwidth. The other reason is related to the multiplication of the prescaler noise. Both these effects are due to decrease of the prescaler gain α. This is shown in Fig. 8(b) by the vertical bars showing the standard deviation of the measured frequency. It may be noted that the standard deviation of the frequency increases with decreasing SNR. Nevertheless, this increase is smaller when selected f IFN is closer to f Mn .
The effect of the mentioned noise on the stability of the time transfer in a 1000-km-long fiber link using standard G.652 single-mode fiber with the chromatic dispersion coefficient of 17 ps·nm −1 · km −1 is shown in form of the time deviation (TDEV) in Fig. 9(b). In general, the curves for the optical power larger than −51 dBm (corresponding to the SNR of about +6.5 dB) are all very similar and close to the noise floor of the stabilization circuit and resulting mostly from the tuning quantization of the ITLA used in the experiment, which can be controlled with a coarse resolution of 1 MHz only (conforming with ITLA multisource agreement). For lower optical powers, however, noticeable deterioration of the TDEV curves is observed, roughly about 2-3 times per each dB. The decrease of the synchronization circuit bandwidth is also visible observing the change of the position of the bump appearing at the averaging time around 1 s. Stability deterioration due to decreasing input power (so SNR) will manifest also as increased jitter. For the cases shown above, the jitter contribution will be about 250 fs, 520 fs, and 1 ps for the input power greater than −51, −52, and −55 dBm, respectively. This suggests that both systematic frequency error and stability should be considered together and fit to particular requirements. For example, the comparison of Fig. 9(a) and (b) reveals that for the lowest TDEV and jitter, the input power should not fall below −51 dBm, which corresponds to SNR ≈ +6.5 dB. For the systematic error not exceeding ±5 MHz, the minimum required SNR about 3 dB lower should be enough on the other hand, but this time elevated jitter is to be accepted. Nevertheless, in any case, a substantial improvement with respect to TDEV of a freerunning laser [shown with the dashed line in Fig. 9(b)] is well visible for almost all averaging times, which shows a clear advantage of the discussed laser stabilization circuits.

VII. PENALTY DUE TO POLARIZATION SCRAMBLING
All the results discussed so far assumed that the polarizations of the fields beating at the PD are aligned. It was assured in the examined circuits using a polarization controller shown in Fig. 2(a). However, such a solution cannot work reliably in practice, as standard fibers change their polarization state randomly due to external factors such as temperature and mechanical stress variations [16]. A simple solution may be to use a polarization scrambler [16], [32], which is a standalone fiber-optic component, not requiring any additional feedback loop or control algorithms (in contrast to an adaptive polarization controller). However, the polarization scrambler reduces the average power of the beat by 3 dB, resulting in a modulation of the IF signal level driving the prescaler [see Fig. 10(a)]. In this way, the instantaneous SNR is affected, resulting in a systematic frequency error, according to the discussion in Sections III, IV, and VI.
The effect of polarization scrambling is shown in Fig. 10(b) (solid lines) for two different f IFN values. It is noticeable that scrambling brings with it the disadvantage of increased SNR required to recover the same value of systematic frequency error without scrambling (dashed lines). In the examples shown, the penalty is either 4.4 or 6.6 dB depending on the difference between f IFN and f Mn . Once again, it can be seen that it is advantageous to choose f IFN close to f Mn in order to minimize the disadvantage.
The minimum scrambling penalty of 3 dB should be expected due to the decrease of IF signal level mentioned above. The theoretical prediction of the penalty can in principle be made by averaging the frequency over all instantaneous SNR values, ranging from the minimum for orthogonal polarizations to the maximum for parallel polarizations. However, it is difficult to obtain accurate and meaningful results because such a procedure is sensitive to the SNR distribution and cannot be held stable or approximated reliably.

VIII. CONCLUSION
The circuits developed here for synchronization of a semiconductor laser frequency to a weak optical reference from a remote laser allow a substantial reduction of the calibration uncertainty related to the chromatic dispersion in fiber-optic time-transfer systems. However, the systematic frequency error that occurs in such circuits must be kept under control. This error discussed in detail in the text arises in the first stage of the frequency divider chain (so-called prescaler) due to the bandlimited noise interfering with the sinusoidal component, whose frequency is estimated by counting the number of slopes in a reference time interval. Due to the experimental work carried out, the simulations, and theoretical models, we can state that the value of the prescaler error is directly related to the SNR at the prescaler input and the difference between the frequency of the sinusoidal component and the mean frequency corresponding only to the counting the bandpass noise (designated as f Mn in the text). Despite the noisy character of the prescaler output under such conditions (i.e., driven by the bandpass noise), the mean frequency observed at the prescaler output appears to be well defined and fairly stable. It is thus a reliable parameter, useful in characterizing circuits with prescalers. The value of this frequency can be easily determined, either based on the input noise spectrum or by direct measurement.
If the prescaler is part of a feedback loop used to bring the frequency of a stabilized laser closer to the frequency of the optical reference light, the systematic frequency error is multiplied by the reciprocal of the prescaler gain α, which is close to one for SNR higher than about 5 dB, but drops gradually to zero for lower values. At low SNR, the systematic frequency errors have thus a tendency to increase very rapidly. With the methods described in the text, it is possible to predict the error value or determine the required SNR to keep the error below an arbitrarily chosen value. According to the results presented, operation with an SNR down to a few dB is reliably possible in the experimentally examined circuits. The conditions are substantially relaxed when the circuit is operated near to the mean frequency determined by the prescaler input noise f Mn . A look at Fig. 4, showing the long-term behavior of f Mn under realistic conditions, shows that the difference between f Mn and the nominal IF f IFN can be chosen to be below 1 MHz.
Operation close to f Mn is also helpful when the polarization scrambling is implemented to avoid polarization alignment problems. In such a case, as shown experimentally, the scrambling penalty can be kept only slightly higher than the theoretical minimum of 3 dB.