Method for Determining The Difference of Delays of Laser Radiation Pulse Propagation Trough Optical Fiber in The Time Scale Comparison & Synchronization Systems
Time scale (TS) comparison and synchronization of spatially distant references of units of time and frequency are necessary to meet the country’s needs for accurate time and reference frequency signals, in particular, for the operation of the global navigation satellite system GLONASS, communication systems and information transmission, during scientific research and solving a number of other problems [1, 2]. In metrological practice, the comparison of TS of remote standards is carried out during the comparison of standards of various metrological laboratories with the aim of their mutual control in order to maintain specified accuracy characteristics [3, 4]. Modern requirements for the accuracy of comparisons of TS of remote standards provide for an error in determining the divergence of TS of no more than 100–200 ps when the distance between the references is up to 100 km. In a number of works, for example, [5–7], it was shown that providing such a level of accuracy with distances between references up to hundreds of kilometers is possible only with the help of comparison systems and synchronization of Wi-Fi using fiber optic communication lines (FOCL) to transmit signals between points of reference location.
The difference in propagation delays in fiber-optic signals from two laser transmitters arises due to the non-identity of the spectral characteristics of lasers (mainly differences in the central wavelengths of radiation) and the effect of chromatic dispersion of an optical fiber (OF).
To estimate the difference between the propagation delays of optical radiation with different wavelengths in typical OFs with a length of 100 km, the values of the refractive index of OFs for specified wavelengths were calculated using the Sellmeyer formula and the coefficients of the Sellmeyer series [10, 11] and the corresponding delay values were calculated. The calculation results showed that the difference between the propagation delays of signals from two lasers with a difference of their central wavelengths of 10 and 20 pm is from 30 to 90 ps, and with a difference of wavelengths of 50 pm – 150–200 ps or more. These values are not only commensurate with the maximum permissible error of the systems of time scales comparison and synchronization, but in some cases exceed it. Therefore, considering the difference in the propagation delay of signals through an optical fiber is necessary to achieve the required accuracy of comparisons and synchronization of time scales.
METHOD FOR DETERMINING THE DIFFERENCE OF DELAYS FOR THE OPTICAL PULSES PROPAGATION
In the calculations, the values of the difference in wavelengths of lasers were chosen in accordance with the level of error of most existing optical spectrum analyzers, which can be used as control means for the TS comparison and synchronization systems. The results of the calculations show that determining the difference in signal delays in the FOCLs by an indirect method (according to the measurement of wavelengths with an optical spectrum analyzer and data on chromatic dispersion of optical substances) will not provide the required accuracy required to determine corrections to the readings of the comparison and synchronization systems.
On the other hand, the use of lasers with high-precision tuning and stabilization of radiation wavelengths equipment as a part of TS comparison and synchronization systems will lead to a significant increase in the cost of systems (2–3 times or more) and additional restrictions on the operating conditions of the equipment (thermal stabilization, vibration insulation, etc.). Similar restrictions imposed on the use of such systems led us to the idea of using a different scheme for measuring the difference in the propagation delays of signals through an optical fiber.
It follows from the above that the preferred way to determine the difference in the propagation delays through the FOLs of the system of TS comparison and synchronization are experimental studies of the propagation of laser optical pulses though the FOCLs on which the system is being deployed.
The basis of our proposed method is of the difference in the propagation delays of optical pulses in FOCL and time parameters of pulses is the following procedure of actions. First, using a digital storage oscilloscope with a photodetector, we register a pair of pulses from two lasers from the system at the FOCL input, then the same pair of pulses is recorded at the FOCL output.
The difference in time intervals between the centers of two pulses at the FOCL input and output, corresponding to the difference in the propagation delays of pulses along fiber optic links, is determined by the subsequent mathematical processing of the recorded data.
A waveform view of the pulses at the FOCL input and output is shown in Fig. 1, where u(t) is the signal at the output of the photoreceiver, Δtp is the difference of the propagation delays of the pulses through the FOCL. A scheme proposed for the experimental studies is presented in Figure 2.
The measurement procedure is proposed to be carried out in the following order. The equipment from the system of time scale comparison (pulse generator GI and laser modules L1 and L2), optical delay line ODL (optical fiber several meters long, necessary for introducing initial delay between pulses), fiber optic combiner O and digital storage oscilloscope DSO with photoreceiver PR are placed at the FOCL input. Then, applying electric pulses to the two laser modules from the generator, register with the oscilloscope an oscillogram of a pair of optical pulses from two lasers at the output of the combiner (oscillogram of the pulses before the FOCL input). Next, connect the output of the combiner to the FOCL input and register the oscillogram of a pair of optical pulses from two lasers at the FOCL output. The received oscillograms should be stored as files with digital data for subsequent mathematical processing.
The oscillograms recorded by the digital oscilloscope are a sequence of discrete in time and level of readings. This sequence is determined by the shape of the input pulses, but it is also influenced by the noise of the PR and the oscilloscope, the sampling frequency of the oscilloscope. Therefore, when processing waveforms to determine the difference in the propagation delays of pulses, it is necessary to approximate the experimental data – find a functional relationship describing the shape of each pulse. It is known that the optical pulse of a Gaussian form (typical for pulsed lasers of time-comparison systems that do not use solitons) expands due to the influence of dispersion, but its shape approximately remains Gaussian . This allows you to select the Gaussian function as an approximation for the pulse shape.
For determining the parameters of the approximating function, the least squares method is widely used, however, for functions of a complex shape (including Gaussian), its direct application is difficult and leads to unacceptable time costs in processing the experimental results. In such cases, it is advisable to use iterative methods that use a consistent refinement of the values of the coefficients of the approximating function by calculating corrections to the approximate values of these coefficients. In particular, the use of an iterative method to approximate the Gaussian function of the peaks of the reflection spectrum of fiber-optic Bragg sensors is known .
This iterative method can also be applied to approximate the shape of an optical pulse from digital samples of a registered waveform. In this case, as a functional dependence, the voltage will be considered as a function of time u(t), which characterizes the instantaneous optical power in a pulse. In this case, the approximating function will be of the following form:
where ka is the amplitude coefficient; Φ (t, tm, σ) is the Gaussian function; σ is the parameter characterizing the pulse width; tm is the time point at which the function reaches its maximum value.
The parameters of the approximating function ka, tm, σ are determined by calculating the corrections to their initial values using the least squares method. The initial values ka0, tm0, σ0 can be chosen as follows:
• as tm0, take the time corresponding to the maximum value u(t);
• calculate σ0 using the formula σ0 ≈ 0.42 · Δτ0.5, where Δτ0.5 is the duration impulse level 0.5 amplitude Umax;
• calculate the value of ka0using the formula ka0= Umax · σ0 · (2π) 1 / 2.
Corrections δka0, δtm0, δσ0 to clarify the initial values of the parameters of the function ka0, tm0, σ0 are determined by solving a system of equations:
a11 · δka0 + a12 · δtm0 + a13 · δσ0 = b1,
a21 · δka0 + a22 · δtm0 + a23 · δσ0 = b2, (2)
a31 · δka0 + a32 · δtm0 + a33 · δσ0 = b3.
To calculate the coefficients a11, a12, …, a33, b1, b2, b3 the experimental data u(ti), the values of the function Φi = Φ (ti, tm0, σ0), calculated at the points ti (i = 1…N, N is the number of samples) using approximate coefficients, as well as the values of partial derivatives of this function in points indicated:
The refined values of the parameters of the function ka1, tm1, σ1 are determined by the formulas:
ka1 = ka0 + δka0; tm1 = tm0 + δtm0; σ1 = σ0 + δσ0. (5)
Next, ka1, tm1, σ1 are taken as approximate values of the coefficients of the approximating function and the calculation cycle is repeated.
The following operations are performed to calculate the parameters of each recorded pulse (two pulses at the FOCL input and two pulses at the FOCL output). The difference in pulse propagation delays through the FOCL Δτp is determined from the calculated tm parameters of four approximating functions:
Δτр = (tmII out – tmI out) – (tmII in – tmI in), (6)
where tmI in and tmI out are the parameters of the Gaussian function for laser pulses module L1 at the input and output of fiber optic lines, respectively; tmII in and tmII out are the parameters of the Gaussian function for the pulses of the laser module L2 at the input and output of the FOCL, respectively.
Taking into account the calculated value of Δτp as an amendment to the readings of one of the time interval gauges that will be used to exclude the error component of the scale comparisons time caused by the influence of chromatic dispersion of optical fiber.
The practical testing of the proposed method was carried out using a 47.5 km optical fiber coil. Oscillograms of pulses from two laser modules at the input and output of the optical fiber coil were recorded during the experiment. For the recorded data, an approximation was carried out using the above iterative method and, using the obtained functional dependencies, the values of the time intervals between pulses at the fiber coil input and at the coil exit were calculated. The time interval between the pulses at the input of the fiber coil was 52.325 ns, and the output of the coil was 52.635 ns. The difference of these time intervals, corresponding to the difference of the propagation delays of pulses from the two laser modules, was 0.310 ns. To confirm the results obtained, direct measurements of the propagation delays of pulses in this optical fiber coil were carried out using an electron-counting frequency meter. The difference between the delays obtained was 0.33 ns. The difference in the delay differences determined by the two methods was 0.02 ns.
Due to the difference in the spectral characteristics of the lasers and the effect of the chromatic dispersion of the optical fiber, the propagation delay through the FOCL from the two laser transmitters is not equal to each other. This effect reduces the accuracy of comparisons and synchronization of time scales by the systems using FOCL. We consider the way to determine the difference in propagation delays of signals through the FOCL of the system of TS comparison and synchronization, which provides for experimental studies of the propagation of pulses from the lasers of the system through the FOCL, on which the system is deployed, as more preferable one.
To prove the correctness of the idea, the research was carried out. In the experimental setup, the difference in the time intervals corresponding to the difference in the propagation delays of the pulses from the two laser modules was 0.310 ns. Direct measurements of the propagation delays of pulses in this optical fiber coil were carried out using an electron-counting frequency counter. The difference between the delays obtained was 0.33 ns. The difference in the delay differences determined by the two methods was 0.02 ns. The good convergence of the results of the two experiments confirms the possibility of applying the proposed method to determine the corrections to the readings of the systems of TS comparison and synchronization using FOCL.