Time transfer by laser link: a complete analysis of the uncertainty budget

T2L2 [1–3] is a two-way time transfer mission based on the timing of optical pulses emitted by a network of laser stations and detected by a dedicated space instrument. It allows for the synchronization of remote clocks with a time stability in the range of a few picoseconds over hundreds of seconds. After four unsuccessful proposals, in 2005 T2L2 was accepted as a passenger instrument on the Jason 2 altimetry satellite. It was launched in June 2008 and has been working without any major interruption since that time. It was put in place by a Delta launcher at an altitude of 1336 km and an inclination of 66°.

The elementary T2L2 link is a ground-space time transfer between a given laser station and the space instrument. For a time comparison between two laser stations, several ground to space elementary links are used for each ground station. To realize the ground-space transfer, the laser station emits asynchronous short light pulses towards the satellite. A reflector array on board the satellite returns some of the light pulses to the ground station. The epoch of the laser emission t_E and the epoch of the reception t_R are measured according to the ground station's timescale. The T2L2 payload records the arrival time t_B of laser pulses according to the timescale of the space clock. Schematically, the time offset Δ_AS between a ground clock A and the space clock S is computed for each laser pulse, where the difference between the start and the arrival epochs is shifted by the time of flight divided by two. Δ_AS is given by:

$$\begin{eqnarray}&&{{{\scr{\Delta}}}_{\text{AS}}}={{t}_{\text{E}}}+\frac{{{t}_{\text{R}}}-{{t}_{\text{E}}}}{2}-{{t}_{\text{B}}}+{{C}_{\text{AS}}}\end{eqnarray} \tag{ 1 }$$

where C_AS is a correction term that takes into account instrumental calibrations, atmospheric phenomena, speed aberrations and relativistic effects. The T2L2 data are very interesting for both time and frequency metrology and for fundamental physics. For instance, T2L2 is able to calibrate and validate the best available microwave time transfer techniques [4, 5] and will be able to compare the time transfer systems embedded on the new mission ACES (Atomic Clock Ensemble in Space) that will be launched on the ISS (International Space Station) in 2017 [6]. The possibilities given by an accurate time transfer between remote clocks is also of interest for the search for a potential anisotropy of the speed of light [7] or for the validation of one-way laser ranging for some future tests involving solar orbit [8, 9]. All these scientific objectives rely on a rigorous uncertainty budget for the whole experiment that is governed by the characteristics of both the space instrument and the laser stations network, as well as the calibration process used on the ground.

In section 2 we give a brief description of the whole instrumentation (ground and space) involved in a T2L2 time transfer. In sections 3 and 4, we describe the time transfer equations for both the ground to space and the ground to ground links. Then, from these equations, we give in section 5 the uncertainty budget of a typical T2L2 time transfer.

The ground-level part of the T2L2 time transfer is based on the international laser station network [10]. This gathers data from more than 40 laser ranging stations over the world and is driven by the International Laser Ranging Service (ILRS) which provides global satellite and lunar laser ranging data to support research in many scientific domains such as geodesy, geophysics, lunar science, and fundamental physics.

A typical laser station is composed of the following elements (figure 1):

• A clock to give the station a frequency reference (5 or 10 MHz). From this frequency reference a numerical divider generates a one pulse per second signal (PPS) for timescale purposes. For laser ranging, this clock is used to measure the time interval between the start and the return; for T2L2, it is the clock which has to be synchronized.
• A neodymium YAG doubled laser to generate pulses at a rate of between 10 Hz and 2 kHz. The wavelength is usually 532 nm and the full width at half maximum is between 50 ps to 200 ps.
• A telescope to transmit and receive laser pulses. Some stations have the same telescope for both transmission and reception, while others have two distinct apertures.
• A start detector to generate the start reference at the output of the laser.
• A return detector to detect pulses which have been reflected by the satellite.
• An event timer that has two independent channels to time tag the electrical pulses of both start t_E and return t_R detectors. The time setting of the event timer is carried out at the first initialization with the PPS signal delivered by the clock of the time and frequency standard laboratory.

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The T2L2 space equipment is an instrument able to time with picosecond resolution the arrival of all laser pulses arriving from the laser stations. It comprises a photo-detection device and an event timer. This equipment is associated with an ultra-stable quartz oscillator used as the T2L2 onboard clock and a retro-reflector array. These two components are used by T2L2 but are not integral to T2L2: the oscillator is the frequency reference of the DORIS System [11] and the LRA is used for satellite laser ranging (figure 2).

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The photo detection device includes two avalanche photodiode photo detectors. One of them works in a nonlinear mode for time tagging purposes [12], the other works in a linear mode to trig the non linear detector and to measure the received laser energy. The most important function of the nonlinear photo detector is to generate an electrical pulse from a very low level of light. The propagation delay inside the photo detector depends on the photon number. To avoid the temporal noise that would be introduced by an uncontrolled energy variation of the laser pulse (atmosphere, pointing error, laser variation and so on) this internal delay is compensated for. This is achieved with the linear photo detector giving the energy received. This energy together with the arrival time is recorded in real time onboard and downloaded to ground every two hours. The compensation for the internal delay of the detector is achieved by means of a post treatment process [13, 18]. The laser energy received at the satellite plane is sent to the photo-detectors through two distinct optics assemblies. This allows the geometry to fit the field of view of the instrument, the photon number to be adjusted and the spectral band width to be fixed. All this instrumentation is divided on the Jason 2 payload in two units A and B.

The offset Δ_AS between a ground clock A and the space clock S is given by:

$$\begin{eqnarray}&&{{{\scr{\Delta}}}_{\text{AS}}}=\frac{{{t}_{\text{E}}}+{{t}_{\text{R}}}}{2}-{{t}_{\text{B}}}+\frac{{{C}_{\text{Sag}}}}{2}+{{C}_{\text{Rel}}}+{{C}_{\text{Atm}}}+{{C}_{\text{ICal}}}+{{C}_{\text{ECal}}}\end{eqnarray} \tag{ 2 }$$

where t_E and t_R are respectively the emission and reception epochs of laser pulses at the laser station, t_B the reception epoch on board, C_Sag a correction due to the speed aberration, C_Rel a term illustrating the relativistic frequency shift [13], C_Atm an atmospheric correction and C_ICal (internal calibration), C_ECal (external calibration) are two calibration terms to obtain respectively the time of flight between the reference point of the station and the satellite and the epoch of laser events at the reference point of the laser station. In order to minimize the noise associated with the measurement of the time of flight (computed from t_R and t_E in equation (2)), some reception epochs t_R' are computed from an interpolation calculated on the actual measure t_R. This process also allows the use of laser events which have been detected on board the satellite but not on the ground. It is especially important for laser stations using a single photon mode detector (where the detection probability is in the range of 10%). In this case, the number of echoes detected by the laser station may be several times lower than the number of detected events on board.

C_Sag is associated with the computation of the time coordinates to take into account the propagation of laser pulses. C_Sag is given by:

$$\begin{eqnarray}&&{{C}_{\text{Sag}}}=\frac{2}{{{c}^{2}}}(x-X)\cdot \dot{X}.\end{eqnarray} \tag{ 3 }$$

It is computed from the coordinates x from the satellite and X from the station for each emission epoch.

C_Rel is associated with the transformation from the proper time of the onboard clock to coordinate time. It is a frequency shift of the space oscillator integrated as a function of time. C_Rel is given by:

$$\begin{eqnarray}&&{{C}_{\text{Rel}}}={\int}^{}\frac{1}{{{c}^{2}}}\left(\frac{{{{\dot{x}}}^{2}}}{2}-\frac{\text{G}{{\text{M}}_{E}}}{x}\right)\text{}\text{ d}t\end{eqnarray} \tag{ 4 }$$

where GM_E is the geocentric gravitational constant. This relativistic frequency shift includes a constant term, which can be neglected in the ground to ground time transfer, and also a periodic term. The amplitude of the periodic frequency shift integrated over an orbit is in the range of 100 ps.

C_Atm is a correction reflecting the optical path difference introduced by the atmosphere between the uplink and the downlink. The worst case occurs with a laser station located near the equator and an observation of the satellite at a very low elevation angle (typically 5°). In that case, the time delay for the one-way link is roughly 40 ns and the time correction C_Atm between the uplink and the downlink is 1.5 ps.

C_ICal is obtained from epochs measured by means of laser pulses sent to the reception chain of the station by means of a calibration corner cube located in the laser beam (figure 1) at a set distance d_cc from the spatial reference of the station (cross axes of the telescope mount). This is measured at the same time as the acquisition of echoes on the satellite from the mean value of an ensemble of N_ICal epoch differences:

$$\begin{eqnarray}&&{{C}_{\text{ICal}}}={{\langle t_{\text{E}}-{{t}_{{\tilde{\text{R}}}}}\rangle}_{{{N}_{\text{ICal}}}}}-{{\delta}_{\text{cc}}},\end{eqnarray} \tag{ 5 }$$

${{t}_{{\tilde{\text{R}}}}}$ being the reception epochs of laser pulses coming from the retro reflector and δ_cc a correction term taking into account the distance between the reference corner cube and the cross axis of the telescope.

C_ECal is measured by means of a dedicated calibration process. Laser stations are basically designed to measure the time difference between a start and a stop with an accuracy of the start pulse of typically 100 ns. In order to obtain the picosecond accuracy required for T2L2, laser stations are carefully calibrated with a dedicated calibration station. This calibration is based on a set of simultaneous measurements made of the habitual chronometry of the laser station and the dedicated calibration station. It permits measurement of the delay between the optical pulse at the cross axis of the telescope and the electrical reference coming from a given output of the PPS distribution unit of the time and frequency laboratory. The calibration station gathers all the metrology required to perform that measurement in a unique piece of equipment: this comprises a SigmaTime STX301 sub-picosecond event timer (figure 3), an optical module to grab laser pulses from the laser station and an optical fiber (figure 4). The event timer has two independent channels connected to optical and electrical inputs. The optical module comprises a collimation optic connected to a 50 m mono mode optical fiber (532 nm).

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Characteristics of the calibration station are given in table 1.

The term C_ECal is computed from the mean value of an ensemble of N_ECal measurements. The time equation that allows the accurate time-tagging of laser pulses may be written as:

$$\begin{eqnarray}&&{{C}_{\text{ECal}}}={{\langle t_{\text{E}}^{{}}-{{t}_{{\tilde{\text{E}}} e}}\rangle}_{{{N}_{\text{ECal}}}}}-{{\delta}_{\text{ocx}}}+{{\delta}_{\text{PPS}}}-\left({{\delta}_{\text{ocf}}}+{{\delta}_{f}}+{{\delta}_{\text{det}}}+{{\delta}_{\text{stx}}}\right)={{\langle t_{\text{E}}^{{}}-{{t}_{{\tilde{\text{E}}}}}\rangle}_{{{N}_{\text{ECal}}}}}-{{\delta}_{\text{ocx}}}+{{\delta}_{\text{prg}}},\end{eqnarray} \tag{ 6 }$$

where ${{t}_{{\tilde{\text{E}}}}}$ is the emission epoch measured by the calibration station, δ_ocx the free space delay propagation between the station reference and the optical input of the calibration station, and δ_prg the global internal propagation inside the calibration station. δ_PPS is the delay in the PPS cable, δ_f the delay in the optical fiber, δ_ocf the delay of the collimation optic, δ_det the propagation constant induced by the optical to electrical conversion inside the detector [14] and δ_stx the delay in the cable between the detector and the event timer. By using the same calibration station to calibrate the different laser stations, the delays ${{\delta}_{\text{prg}}}={{\delta}_{\text{PPS}}}-\left({{\delta}_{\text{ocf}}}+{{\delta}_{f}}+{{\delta}_{\text{det}}}+{{\delta}_{\text{stx}}}\right)$ in equation (6) do not need to be known accurately. It must be emphasized that in this case, the calibration becomes relative since we do not know the real emission epoch of laser pulses. It is perfectly well suited for an accurate comparison of T2L2 with any other time transfer techniques from the temporal reference of each laboratory but it would become a difficulty if it was necessary to compare a given technique directly from the laser pulses.

The term $\langle t_{\text{E}}^{{}}-{{t}_{{\tilde{\text{E}}}}}\rangle$ is determined from the time setting of the SigmaTime STX301 event timer and from a simultaneous acquisition of an ensemble of laser pulses from the laser station and the calibration station. The time setting process allows the acquisition of measurements directly referenced to the time reference of the laboratory. It is carried out using a preliminary measurement of the PPS reference signal of the laboratory. The PPS threshold level is always set at 1 V. The time stability over one day of a typical PPS signal is in the range of 20 ps. To minimize the uncertainty of that time setting process, the PPS measurement is carried out over a set of N_TS = 30 PPS events. At the same time, a scan of that PPS signal is also performed by the event timer to determine the actual shape of the PPS signal. For that measurement, the event timer is used as an oscilloscope. It generates a graph where the abscissa axis is time and the ordinate axis is voltage. One channel of the event timer is used as the trigger for the time reference and the other is used to measure the signal. Figure 5 is an example of such a scan measured by the STX301 event timer.

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The ground to space time transfer computed from equation (2) can be used directly on individual echoes. It allows monitoring of the time transfer with the optimum temporal resolution (the typical time interval between consecutive laser event is roughly 1 s). The same computation, integrated over a few tens of seconds or over an entire satellite pass in order to minimize the white noise of the time transfer, is also of interest.

Ground to ground time transfer can be realized in either common-view or non-common-view mode. In the first case, the satellite is seen during a common period while in the other, the satellite is observed alternatively. For common-view, the maximum distance to the ground is roughly 6000 km.

Ground to ground time transfer depends very significantly on the visibility of the satellite. In common-view mode, the space oscillator is only solicited over the time interval of laser pulses (typically between 0.1 s to a few seconds). Over a longer period, noise is common for both stations and the difference vanishes. In the non–common-view mode, the noise of the space oscillator has to be considered over the time interval during which the satellite is not visible, and the noise of the space oscillator can become the main source of noise. That noise can be significantly reduced with a correction model C_osc taking into account parameters such as temperature, radiation or time. Up to now this model has not been operational. In what follows, we will only discuss ground to ground time transfer in common-view.

A ground to ground time transfer Δ_AB between two ground clocks A and B in common-view is computed from the difference between the individual time transfers x_AS and x_BS individually acquired by the stations. One has:

$$\begin{eqnarray}&&{{{\scr{\Delta}}}_{\text{AB}}}={{{\scr{\Delta}}}_{\text{BS}}}-{{{\scr{\Delta}}}_{\text{AS}}}+{{C}_{\text{osc}}}.\end{eqnarray} \tag{ 7 }$$

Since the laser emissions for each station are not synchronous, data from the ground to space time transfer of each station are first interpolated on the full seconds in the timescale of the satellite. The final time transfer x_AB is computed from the individual time transfer interpolated on the common full seconds.

Considering that the terms of equation (2) are independent, the combined uncertainty of a time transfer $u_{c}^{{}}\left({{{\scr{\Delta}}}_{\text{GS}}}\right)$ between a ground clock and the space clock may be written as:

$$\begin{eqnarray}&&u_{c}^{2}\left({{{\scr{\Delta}}}_{\text{GS}}}\right)=\frac{{{u}^{2}}\left({{t}_{\text{E}}}\right)}{4}+\frac{{{u}^{2}}\left({{t}_{\text{R}}}\right)}{4}+{{u}^{2}}\left({{t}_{\text{B}}}\right)+{{u}^{2}}\left({{C}_{\text{ICal}}}\right)+{{u}^{2}}\left({{C}_{\text{ECal}}}\right)+{{u}^{2}}\left({{C}_{\text{Atm}}}\right)+\frac{{{u}^{2}}\left({{C}_{\text{Sag}}}\right)}{4}+{{u}^{2}}\left({{C}_{\text{Rel}}}\right).\end{eqnarray} \tag{ 8 }$$

The combined uncertainty of a ground to ground time transfer in common view is given by:

$$\begin{eqnarray}&&u_{c}^{2}\left({{{\scr{\Delta}}}_{\text{AB}}}\right)=u_{c}^{2}\left({{{\scr{\Delta}}}_{\text{AS}}}\right)+u_{c}^{2}\left({{{\scr{\Delta}}}_{\text{BS}}}\right)+{{u}^{2}}\left({{C}_{\text{osc}}}\right)\end{eqnarray} \tag{ 9 }$$

Each uncertainty term in equations (8) and (9) is evaluated from other parameters. Some of them are evaluated using a statistical analysis of a series of observations (Type A) and others are arrived at using other methods (Type B).

Each uncertainty term should be determined on the basis of the precise experimental setup for a given link between two laser stations. In order to give a more general uncertainty budget, we will consider in the following a typical setup reflecting the characteristics of a large number of laser stations well suited for time transfer. The main uncertainties of such a station are summarized in table 2.

The final combined uncertainty depends also on the quantity of data acquired. Of this data, the number of laser events emitted N_E, echoes acquired N_R and events recorded on board N_B are especially important for the final uncertainty budget. The technical specifications of the laser station, together with observation conditions, of course have a strong influence on the outcome. For the sake of simplicity, these numbers are chosen to correspond to data acquired after a complete pass of the satellite above a ground station when observation conditions are difficult, with minimum on board events N_B equal to 200. For the MeO laser station, this requirement represents roughly 90% of recorded passes. This value ensures also a satisfactory signal to noise ratio. The number of measurements for the internal and external calibration process and time setting is selected so that there is a negligible uncertainty associated with the mean value of each of those terms. Table 3 gives the values for all these measurements.

5.1. Instrumental uncertainties u(t_E), u(t_R), u(t_B), u(C_ICal), u(C_ECal)

5.2. Atmospheric uncertainty u(C_Atm)

5.3. Sagnac and relativity uncertainties u(C_Sag), u(C_Rel)

5.4. Ground to space and ground to ground extension of uncertainties u_E(Δ_GS), u_E(Δ_GG)

Observatoire de la Côte d'Azur (OCA) has two laser stations on the same site (MeO and FTLRS) that can be used together to realize a time transfer in a co-location configuration (figure 6). MeO is one of the biggest laser stations in the world with a 1.54 m telescope and a frequency doubled Nd:YAG laser (rate = 10 Hz, FWHM = 150 ps, Energy = 100 mJ) [29]. FTLRS is a transportable system using a very small telescope (13 cm) and also a doubled frequency Nd:YAG laser (rate = 10 Hz, FWHM = 35 ps, Energy = 30 mJ) [30]. The distance between the two SLR stations is 35 m.

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Since the beginning of the T2L2 mission on Jason 2, five co-location experiments have been made at OCA. Among them, one was conducted in spring 2010, with a unique H-Maser connected to both MeO and FTLRS [31]. Another was performed between March and April 2012 with two independent clocks connected to FTLRS (Caesium) and to MeO (H-Maser) and also two independent GPS receivers (Dicom GTR50) and an ultra stable event timer (STX301) to inter-compare T2L2 results with GPS and direct comparison [32].

In the first experiment made with a unique clock for both MeO and FTLRS, the comparison is made directly based on the differences between calibration and T2L2 results. One gets from nine common passes acquired in four days (filtered @ ± 3σ):

$$\begin{eqnarray}&&{{{\scr{\Delta}}}_{\text{Meo}.\text{FTLRS}}}=\text{}\text{ 37}\text{ps}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&u\left({{{\scr{\Delta}}}_{\text{Meo}.\text{FTLRS}}}\right)\text{}\text{}=\text{}\text{ 6}0\text{ps};\text{}\text{}k=1.\end{eqnarray} \tag{ 11 }$$

This result is in close agreement with the uncertainty budget given in table 10.

For the second experiment, conducted with an independent clock for each station, a comparison was made between the calibrated T2L2 link (MeO-FTLRS) and a direct comparison was based on the STX301 event timer. The computation took into account the important drift existing between the caesium and the H-Maser in order to avoid any error in the epochs of realization of each comparison. The global uncertainty budget of the comparison depends on the uncertainties of the ground to ground time transfer and also on the uncertainties introduced by the direct comparison. It includes uncertainties in cables, output to output skew introduced by pulse distribution units and errors generated by the interpolation process. The combined uncertainty of that direct comparison is estimated at u_c(Δ_Direct) = 50 ps [33], giving an extend uncertainty of u_E(Δ_Direct) = 100 ps with a coverage factor of 2. One gets for a given pass:

$$\begin{eqnarray}&&\text{T2L2}:{{{\scr{\Delta}}}_{\text{Meo}.\text{FTLRS}}}=\text{}\text{ 95}910\text{ps}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\text{STX3}01:{{{\scr{\Delta}}}_{\text{Meo}.\text{FTLRS}}}=\text{}\text{ 96}076\text{ps}\text{.}\end{eqnarray} \tag{ 13 }$$

The difference between T2L2 and direct comparison is 166 ps which is compatible with the quadratic sum of u_E(Δ_GG) given in table 10 and u_E(Δ_Direct).

In the typical laser station summarized in table 2, the most significant uncertainty terms come from the delay variation (thermal) of the cable between the time and frequency laboratory and the event timer of the laser station and from the laser pulse width variation. To improve the former, we designed an ultra stable time signal generator including an event timer able to monitor the time delay variation in the propagation of the signals (SigmaTime STS201) [32]. This monitoring is performed by means of a double propagation of the signal emitted by the distributor and repeated by the user (laser station). The overall standard deviation of the instrument (distribution and monitoring) is better than 1.5 ps RMS. With such a performance and with a laser having a pulse width uncertainty of 10 ps, the ground to ground expanded uncertainty becomes smaller than 100 ps (table 11).

Another way to improve the global uncertainty of T2L2 is to improve our knowledge of the duration of the laser pulses emitted by the laser station. Up to now, the pulse width has been determined using a dedicated process performed by the calibration station using high speed photo detection triggered alternatively from positive to negative slope. This process is perfectly well suited to our needs but requires good laser energy stability. Some pulse widths in the range of 20 ps were measured using this process. When the energy stability is not high enough, the measurement can be biased by a significant factor. In this case a single photon measurement may be preferable. The pulse width is determined from data acquired by a dual channel event timer triggered by a multi photo detector and stopped by a single photon detector. This process would permit the use of a single photon detector as a detection device, permitting the simultaneous measurement of pulse width and time delay. This single photon detector has been built and will be used in the next calibration process [34].

Today, the T2L2 experiment launched in 2008 on the satellite Jason 2 is fully operational. It gives a unique opportunity to perform time transfers in common-view between remote sites with an uncertainty better than 140 ps (coverage factor = 2). It allows accurate comparisons with other microwave time transfer systems using a totally independent technique. Some direct comparisons of collocated observations and other dedicated experiments have also enabled us to validate the uncertainty budget described in section 5.

The T2L2 space mission has been extended until 2017. This extension will allow for non-common-view time transfer realization, link budget analysis and other optical time transfer comparisons [35]. If the timetable is compatible with the launch of the ACES mission, some time transfer comparisons between the optical time transfer of ACES (European Laser Timing ELT) and T2L2 should be scheduled at the mission start-up together with some comparisons with the microwave link MWL also embedded on ACES.

The authors would like to thank the Centre National d'Etudes Spatiales (CNES) for its global support for the T2L2 mission.