## Abstract

Free-space optical time and frequency transfer techniques can synchronize fixed ground stations at the femtosecond level, over distances of tens of kilometers. However, optical time transfer will be required to span intercontinental distances in order to truly unlock the performance of optical frequency standards and support an eventual redefinition of the SI second. Fiber dispersion and Sagnac uncertainty severely limit the performance of long-range optical time transfer over fiber networks, so satellite-based free-space time transfer is a promising solution. In pursuit of ground-to-space optical time transfer, previous work has considered a number of systematic shifts and concluded that all of them are manageable. One systematic effect that has not yet been substantially studied in the context of time transfer is the effect of excess optical path length due to atmospheric refraction. For space-borne objects, orbital motion causes atmospheric refraction to be imperfectly canceled even by two-way time and frequency transfer techniques, and so will require a temperature-, pressure-, and humidity-dependent correction. This systematic term may be as large as a few picoseconds at low elevations and remains significant at elevations up to ~35^{°}. It also introduces biases into previously-studied distance- and velocity-dependent corrections.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Accurate synchronization between remote locations is critical to modern science and industry, including for civil necessities such as the power grid, the banking system, and telecommunications networks, and also for scientific facilities such as particle accelerators and radio telescopes. Current and future improvements to optical lattice [1,2] and trapped ion [3] clocks will enable networks of clocks themselves to act as global-scale observatories with applications in metrology, astronomy, geodesy, and fundamental physics [4–6]. However, there are substantial challenges in constructing networks capable of transferring not only the sub-$10^{-18}$ fractional frequency uncertainty of these clocks but also a time scale capable of utilizing that precision. There are two main ways time is currently transferred across intercontinental distance scales: using global navigational satellite systems (GNSS; examples include GPS, GLONASS, Beidou, and Galileo) in common-view, all-in-view, or carrier-phase modes, and using two-way satellite time and frequency transfer (TWSTFT) over communication satellites. However, these techniques are both limited to total uncertainties $\gtrsim 0.5$ ns by receiver calibration, diurnal variations (in the case of TWSTFT), and GNSS code phase discontinuities [7]; this comes out to a time interval uncertainty of $\lesssim 2\times 10^{-16}/\tau$ with $\tau$ in months and limits real-time time-stamping against UTC to an uncertainty of a few nanoseconds.

Optical atomic clocks can now reach instabilities of $<10^{-18}$ in less than 1 day [1,2]. It is clear, however, that substantial improvements to long-range time transfer accuracy will be required to fully utilize them, including for a future redefinition of the second in the Systéme Internationale [6]. Terrestrial optical fiber networks support frequency transfer at $\delta \!f/f<10^{-18}$ [5,8] but are limited to picosecond-level time transfer accuracy due to uncertainties in the exact fiber routing, which then generate unknown Sagnac shifts. (If absolute synchronization is not required, such as for time interval transfers, sub-picosecond time transfer stability is achievable over optical fiber networks at 100-1000 km scales [9–11].) Thus, given the limitations of Sagnac uncertainty for submarine cables, as well as the more technical challenges of exponential path loss and fiber dispersion, free space optical time transfer using satellites appears to be the best path to achieving sub-picosecond synchronization over intercontinental distances.

A variety of techniques have been demonstrated for free-space optical time transfer, specifically including T2L2 (time transfer by laser link) [12,13], methods based on optically-transmitted pseudo-random ranging codes [11,14,15], and optical frequency comb two-way time and frequency transfer (OFC-TWTFT) [16–19]. The uncertainty of T2L2 using the JASON-2 satellite has been evaluated at 140 ps (coverage factor = 2) [20]. While OFC-TWTFT has so far been limited to terrestrial demonstrations, it has achieved instabilities below 1 fs over a 12 km link distance [16]. In order to achieve inaccuracies comparable to the instability of OFC-TWTFT, a number of corrections are required against the basic measured round-trip delays, including special- and general-relativistic effects [19,21–23], general relativistic tidal effects [24], and OFC-TWTFT instrument-specific calibrations such as delay-Doppler coupling [19]. It is important to note that while T2L2 is an incoherent technique, the highest-precision techniques require coherent optical receivers and single-optical-mode transmission to ensure maximal link reciprocity and effective path compensation through two-way synchronization.

The performance of optical time transfer also generically depends on the medium through which the laser pulses travel. For space-to-space applications, the medium is vacuum, while for terrestrial point-to-point experiments the medium is a turbulent but reciprocal air mass. The effects of atmospheric turbulence on coherent optical time transfer have been studied numerically and were found to not pose substantial performance limitations [25–27]; these results are supported experimentally by multiple >10 km performance demonstrations through the turbulent surface boundary layer [16,27,28]. (Coherent, single-mode, two-way optical time transfer is intrinsically much less susceptible to turbulence-induced timing noise than incoherent free-space optical communications, as the timing noise is limited only by the single-mode channel nonreciprocity over the optical time of flight rather than either long-time or multimodal atmospheric behavior.) However, while those studies focused on the dynamic timing noise due to turbulence, there is another atmospheric effect which they did not evaluate: if the elevation angle between the terminals is changing and the time of flight between them is substantial, the atmospheric refraction delay is asymmetric and therefore is not canceled even by two-way time transfer methods. Additionally, the static refractive and dispersive effects of the atmosphere on ground-to-space synchronization have been previously studied [29], but the motion-induced asymmetry in atmospheric refraction has not yet been considered. Even quantum (rather than classical) optical time transfer methodologies such as Refs. [4,30] require correction for link distance and therefore also require correction for atmospheric refraction.

## 2. Atmospheric refraction

The general phenomenon of atmospheric refraction is probably familiar to every astronomer who has used a telescope at low elevation angles [Fig. 1(a)]. In particular, it is well known within the satellite laser ranging (SLR) community, who correct for it using analytical [31,32] or ray-tracing [33] models. In addition, atmospheric refraction in SLR can be directly measured by taking advantage of the chromatic dispersion of air with a two-color laser ranging system [34]. In the context of time transfer, Samain *et al.* included atmospheric delays in their uncertainty budget for T2L2 but did not consider them in detail, because the total uncertainty budget of 140 ps is much larger than the atmospheric refraction correction for the JASON-2 satellite [20].

However, for time transfer techniques capable of achieving sub-picosecond or femtosecond precision, atmospheric refraction becomes a very significant correction: this Article reports a set of numerical simulation results which confirm the significance and quantify the expected correction magnitudes for several orbits and sample ground sites. The basic data flow of the simulation [Fig. 1(b)] is that realistic elevations and ground tracks were calculated, using two-line elements (TLEs) from `space-track.org`, for three artificial satellites and the Moon (Table 1), as viewed from three ground sites (Table 2) over a period of 364 days. (The chosen ground sites represent three national time and frequency laboratories with diversity in latitude and therefore satellite pass geometry.) For each time step with a satellite above the fixed 10$^{\circ }$ elevation mask from any site, historical weather data from the NOAA Integrated Surface Database [35] was interpolated to the relevant site and then used to calculate the atmospheric refraction delay $D(t)$ using the analytical formulas of Marini and Murray [36] and Gardner [31,37]. The one-way time of flight delay $L/c$ for geometric range $L(t)$ determined the delay between the upward- and downward-going laser pulses, and hence the refraction non-reciprocity was calculated as $\delta D(t)=D(t+\frac {L(t)}{c})-D(t)$. An operational time transfer system might choose to use ray tracing [33] rather than analytic approximations to achieve better accuracy in its atmospheric delay correction.

While the output of this model is a time series $\delta D(t)$ for each combination of satellite and site, it is more informative to view the results statistically. Figure 2 shows scatter plots of $\left |\delta D\right |$ versus instantaneous elevation for all four satellites as viewed from the Boulder, Colorado, USA site. Several features are immediately visible. First, the magnitude of $\delta D$ is generically large relative to the few-femtosecond precision of OFC-TWTFT and substantive even for picosecond-level time transfer methods. Second, the exact 2 cycles per sidereal day orbit of the GPS NAVSTAR constellation makes $E$, $dE/dt$, and $L/c$ extremely repeatable and so the delay points’ dispersion away from a single closed curve is due to the weather (temperature, surface pressure, and humidity) variations at the site. In contrast, the ISS and MMS-1 have turning points ($dE/dt=0\implies \delta D=0$) at every elevation, and so show filled curves under an upper bound. The Moon’s $dE/dt$ arises primarily from Earth rotation rather than satellite motion, and so while the curve is not as tight as the GPS NAVSTAR satellite, it does not show low-elevation points with $dE/dt=0$. (Note that the Moon’s orbital inclination varies with an 18.6 year cycle, much longer than the 364 day simulation window.) Third, while $\delta D$ must be zero for satellites in the geostationary belt — since $dE/dt$ is zero — the ever-increasing time of flight delays mean that $\left |\delta D\right |$ is actually larger for cis-lunar distances than at LEO, and may be a relevant correction at those distances for T2L2 as well as OFC-TWTFT.

## 3. Discussion

To better quantify the importance of $\delta D$, Fig. 3 shows the complementary cumulative distribution function defined as $\mathrm {CCDF}_{X}\left (x\right )=\Pr \left (X>x\right )$ — also called the exceedance — of the nonreciprocal atmospheric delay magnitude $\left |\delta D\right |$ for each site and satellite. The comparatively low value of $\frac {L}{c}\frac {dE}{dt}$ for the GPS NAVSTAR satellite gives it substantially lower values of $\left |\delta D\right |$ than the other three satellites. Nonetheless, 50% of all contacts with NAVSTAR 78 have $\left |\delta D\right |>200$ fs from Braunschweig and $\left |\delta D\right |>70$ fs for Boulder and New Delhi due to (a) their lower latitudes than Braunschweig and (b) Boulder’s substantial altitude of 1675 m. The ISS and MMS-1 satellites have 50% exceedances of 400 fs and 300 fs respectively, and the Moon has 900 fs from Boulder (due to its elevation) and 1 ps from the other two sites. For time transfer to the Moon, a 25$^{\circ }$ elevation mask is sufficient to keep $\left |\delta D\right |\lesssim 2.5$ ps, but this reduces the available contact windows by approximately 30%; even MMS-1 has $\left |\delta D\right |\gtrsim 2.5$ ps about 10% of the time.

In addition to the direct refraction nonreciprocity $\delta D$, the total atmospheric delay $D$ (Fig. 4) also biases other range- and range-rate-dependent corrections, if they are calculated using the time transfer system’s uncorrected measurement of optical path length and Doppler shift: the observed optical path length $L_{\mathrm {opt}}=L+c\cdotp D$ is longer than the geometric range by $c\cdotp D$, and the apparent velocity is shifted by $v_{\mathrm {opt}}=v+c\cdotp \frac {d}{dt}D\approx v+c\cdotp \frac {\delta D}{L/c}$ with $v\equiv \frac {dL}{dt}$, i. e. negative $v$ denotes the satellite moving towards the ground site. It should be noted that while $L_{\mathrm {opt}}$ is strictly greater than $L$, the analagous relationship $\left |v_{\mathrm {opt}}\right |\ge \left |v\right |$ is only true for circular orbits over a spherical Earth: if the satellite is also changing surface-relative altitude (e. g. due to nonzero eccentricity), the point of closest approach and the point of maximum elevation angle are not necessarily coincident. The exceedances of $\left |D\right |\equiv \left |\delta L/c\right |$ and $\left |\delta v\right |=\left |c^{2}\cdotp \frac {\delta D}{L}\right |$ are plotted in Figs. 4 and 5 respectively. The velocity bias $\delta v$ is much larger in LEO than higher orbits (even though $\delta D$ is comparable or even larger in those orbits) because $\delta v\propto \delta D/L$ is effectively a measurement of $dE/dt$.

Specializing the discussion of systematics to OFC-TWTFT, Sinclair *et al.* [19] highlighted four velocity- and path-length-dependent corrections, presented here in reverse order from that paper:

- 1. Velocity-dependent transceiver calibration, scaling as $v/c^{2}$,
- 2. Delay-Doppler coupling, scaling as $v/c$ times various hardware-fixed chromatic dispersion terms,
- 3. Bias due to asynchronous sampling, $\varepsilon _{\mathrm {async}}=\frac {v}{c}\frac {1}{2\thinspace \Delta f_\textrm{rep}}$ where $\Delta f_\textrm{rep}$ is the hardware-fixed difference in frequency comb repetition rates, and
- 4. The fundamental relativistic breakdown in reciprocity (i. e. a generalization of the Sagnac effect), $\varepsilon _{\mathrm {rel}}=v\cdotp L/c^{2}$

Of these, the effect of velocity bias on the first and second corrections are negligible as Sinclair *et al.* report their *total* magnitudes to be only 3 fs and 70 fs respectively for typical OFC-TWTFT system parameters and 30 m/s velocities; even the largest $\left |\delta v\right |\lesssim 15$ cm/s correction (for the ISS) only generates a sub-femtosecond bias in those two corrections. However, the remaining two corrections require more consideration.

The third correction has a scale factor of $(2\thinspace \Delta f_\textrm{rep} )^{-1}$, where $\Delta f_\textrm{rep}$ is the difference in repetition rate between the transfer comb and the remote comb; Sinclair *et al.* used $\Delta f_\textrm{rep} =2.2$ kHz. This gives corrections as large as 76 fs at 5% exceedance for the ISS and 30 fs at 50% exceedance. For objects above LEO the 50% exceedance is at sub-femtosecond levels.

The last (and for orbital distances also largest) correction is the relativistic breakdown in reciprocity. Inserting $L_{\mathrm {opt}}$ and $v_{\mathrm {opt}}$ into the definition of $\varepsilon _{\mathrm {rel}}$ and expanding to first order in $\delta L$ and $\delta v$ gives

The CCDFs of $\left |\delta \varepsilon _{\mathrm {rel}}\right |$ are shown in Fig. 6, and can reach multiple picoseconds at the 10% exceedance level.

## 4. Conclusions

As optical time transfer techniques simultaneously reach single-picosecond to single-femtosecond precision levels and also seek to expand from terrestrial to space-borne applications, it is clear that operational systems will have to account for atmospheric refraction. Given that the corrections can reach as high as 10 ps, it is also natural to ask how accurate they are — at what level does the uncertainty on the correction substantially limit the acheivable uncertainty of transatmospheric optical time transfer?

The total atmospheric delay $D$ is calculable to $\lesssim 30$ ps using ray-tracing techniques [33], or $1\times 10^{-3}$ fractional uncertainty. This translates to uncertainties as large as a few femtoseconds in $\delta D$. Furthermore, at LEO speeds of order 7 km/s, the relativistic correction bias term $\frac {v}{c}D$ also has an uncertainty on the order of 0.7 fs. (However, that can be avoided by using an independent method of precise orbital determination to calculate the relativistic correction.) Thus, the importance of the atmospheric uncertainty depends on the performance envelope of each time transfer method: T2L2, with other 10 ps-level uncertainty sources, likely will not be limited by atmospheric refraction uncertainty. OFC-TWTFT, by contrast, will need to address atmospheric refraction even to sustain its current performance (time deviations below 1 fs for periods of 0.1–50000 s [17]) over a ground-to-space link: atmospheric uncertainty could dominate for most or all of any individual ground-satellite contact period. One plausible option would be to take advantage of the wide optical spectrum available for OFC-TWTFT, and transmit two widely-spaced sections of optical spectrum. The measured delay between two spectral regions, combined with the known dispersion of air, would enable a direct measurement of the atmospheric refraction delay just as in two-color satellite laser ranging, although precision knowledge of atmospheric humidity profiles may be required [34]. For time transfer methods based on optically-transmitted ranging codes [11,14,15], atmospheric refraction corrections are also probably necessary for transatmospheric links. At the 1 ps level of precision, a static or seasonally-averaged correction tabulated as a function of elevation suffices for all but the most extreme weather conditions (e. g. attempting a satellite contact from the eye of a hurricane), but at the 100 fs precision level it may be necessary to calculate the atmospheric refraction correction based on current weather observations.

## Funding

Air Force Research Laboratory (FA9453-16-D-0004); Air Force Office of Scientific Research (21RVCOR078).

## Acknowledgements

Distribution Statement A. Approved for public release, distribution is unlimited. Approval no. AFRL-2020-0579.

## Disclosures

The author declares no conflicts of interest.

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