Effect of atmospheric turbulence on timing instability for partially reciprocal two-way optical time transfer links

In this paper we analyze the limits of optical time transfer through atmospheric turbulence and relate those predictions to timing uncertainty analysis using the Allan timing variance (TVAR). The power spectrum of timing uncertainty due to atmospheric turbulence is expressed with the help of Taylor's frozen flow hypothesis, identifying a $f^{-8/3}$ and $f^{-2/3}$ power-law behavior for uncorrelated and partially correlated turbulence, respectively. The scaling of each power law is related to the geometry of the link and the turbulence profile. The power-law slopes are used to calculate two TVAR scaling coefficients relevant to turbulence timing noise, $c_{5/3}$ and $c_{-1/3}$, which can be applied to time-transfer analysis of timing data affected by turbulent fluctuations. Examples of a 2-km horizontal partially overlapping two-way link estimate the atmospheric contribution to timing fluctuations to be below $10\mathrm{fs}$, while a two-way link to a medium-Earth-orbit satellite experiences timing fluctuations on the order of $2\mathrm{fs}$. Comparison of turbulence theory to a recent two-way optical time transfer experiment shows good agreement with the expected power-law behavior and scaling factors.

Figure 1

Two-way optical time transfer setup through turbulent atmosphere along a folded path. The timing offsets $\mathrm{\Delta }{T}_A$ and $\mathrm{\Delta }{T}_B$ include slightly different turbulent path-length fluctuations due to the Tx and Rx separation $d$. The incomplete cancellation of path-length fluctuations will show up in the clock difference measurement $\mathrm{\Delta }{T}_{AB}$.

Figure 2

Two-way optical time transfer to a satellite in orbit. The long distance from ground to the satellite and high orbital velocity means the ground-to-satellite beam must be transmitted at point-ahead angle ${\theta }_{pa}$, which causes the uplink and downlink beams to sample different turbulence through the atmosphere.

Figure 3

Turbulence spatial power spectra ${\mathrm{\Phi }}_n\left(\kappa \right)$ using the Kolmogorov model [(3) orange solid], von Kármán model [(4) blue dotted], and Greenwood-Tarazano model [(5) red dash-dot]. Upper lines are $L_0=10\mathrm{m}$, and lower lines are $L_0=100\mathrm{m}$ for the VK and GT models.

Figure 4

Temporal power spectrum $S_x\left(f\right)$ of a two-way link computed for the Kolmogorov spectrum (orange solid), von Kármán spectrum (blue dotted), and Greenwood-Tarazano spectrum (red dash-dot). The transition from $f^{-8/3}$ uncorrelated turbulence [(12) purple dashed] to $f^{-2/3}$ partially reciprocal turbulence [(15) green dashed] can be seen just below $f=1\mathrm{Hz}$. The outer scale roll-off in the VK and GT models is around $f=0.1\mathrm{Hz}$ for $L_0=10\mathrm{m}$ and $f=0.01\mathrm{Hz}$ for $L_0=100\mathrm{m}$.

Figure 5

Numerical evaluation of timing noise spectrum for Earth to MEO one-way (triangles), MEO two-way (diamonds), Earth to LEO one-way (squares), and LEO two-way (circles) using the Greenwood-Tarazano atmospheric model. The higher slew rate from tracking a LEO satellite results in a shift of $S_x\left(f\right)$ to higher temporal frequencies. The one-way timing noise follows the $f^{-8/3}$ slope [(21) purple dashed/dotted] until the low-frequency roll-off from the turbulence outer scale begins. The two-way noise shows the change to $f^{-2/3}$ power-law slope [(22) green dashed] before experiencing the same outer scale roll-off.

Figure 6

Timing fluctuation power spectrum from 2-km partially reciprocal link with 0.5-m Tx Rx separation. Data from [13]. The one-way timing data (yellow) follows a $f^{-8/3}$ power-law slope (purple dashed-dot), while the two-way data (blue) shows the transition to larger scale correlated turbulence $f^{-2/3}$ (green dashed) near $f_c=0.6\mathrm{Hz}$. The broken power-law [(28) orange solid] fit to the two-way data gives the $C_n^2$ and wind speed $V$ values shown.

Figure 7

Atmospheric contribution to timing deviation (TDEV) in seconds, from 2-km horizontal experiment, including one-way time-of-flight uncertainty (yellow diamonds), two-way link uncertainty (blue squares), broken power-law fit using the Kolmogorov model (orange solid), Greenwood-Tarazano model (red dotted), and theoretical TDEV slopes from (26) (purple dashed dot) and (27) (green dashed). The total time transfer uncertainty stays below $10\mathrm{fs}$ out to $1000\mathrm{s}$ of averaging time where mechanical drift in the experimental terminals masks the contribution from turbulence.