Impact of turbulence on high-precision ground-satellite frequency transfer with two-way coherent optical links

Bidirectional ground-satellite laser links suffer from turbulence-induced scintillation and phase distortion. We study the impact of turbulence on coherent detection and the related phase noise that restricts time and frequency transfer precision. We evaluate the capacity to obtain a two-way cancellation of atmospheric effects despite the asymmetry between up- and downlink that limits the link reciprocity. For ground-satellite links, the asymmetry is induced by point-ahead angle and possibly the use, for the ground terminal, of different transceiver diameters, in reception and emission. The quantitative analysis is obtained thanks to refined end-to-end simulations under realistic turbulence and wind conditions as well as satellite kinematics. These temporally resolved simulations allow characterizing the coherent detection in terms of time series of heterodyne efficiency and phase noise for different system parameters. We show that tip-tilt correction on ground is mandatory at reception for the downlink and as a pre-compensation of the uplink. Besides, thanks to the large tilt angular correlation, the correction is shown to be efficient on uplink despite the point-ahead angle. Very good two-way compensation of turbulent effects is obtained even with the asymmetries. The two-way differential phase noise is reduced to $1{\mathrm{rad}}^2$, with the best fractional frequency stability below $2\times {10}^{-17}$ after 1-s averaging time.

Figure 1

Principle of clock comparison using coherent optical links. The phase noises ${\phi }_1$ and ${\phi }_2$ are measured at intermediate frequency ${\omega }_{\mathrm{IF}}$ on both terminals $T_i$, where $i=1,2$ for ground and space, respectively. Received or transmitted electromagnetic fields are denoted ${\mathcal{E}}_{iS}$ and ${\mathcal{E}}_{iE}$. They pass through the aperture diameters denoted $D_i$ located in the planes ${\mathrm{\Pi }}_i$. Local oscillator fields are denoted ${\mathcal{E}}_{iL}$.

Figure 2

Turbulence and propagation modeling on ground-space optical link. The wavy black lines represent the moving phase screens with an apparent wind speed $V\left(z\right)$ where $z$ is the propagation axis. The red lines show the downlink plane wave. The blue Gaussian ones show the uplink wave. Line of sight is artificially set horizontal on the sketch. Photograph of the satellite from Ref. [37].

Figure 3

Turbulent downlink time series of heterodyne efficiency along an 8-s duration sampled every 1 ms. The red (lower) straight line shows the detection threshold. The blue (upper) straight line shows the mean heterodyne efficiency $\langle m\rangle$. $D_{1S}=0.40$ m. This case is labeled $\left(a\right)$ in Table 1.

Figure 4

Tip-tilt controlled downlink time series of heterodyne efficiency along an 8-s duration sampled every 1 ms. The red (lower) straight line shows the detection threshold. The blue (upper) straight line shows the mean heterodyne efficiency $\langle m\rangle$. $D_{1S}=0.40$ m. This case is labeled $\left(b\right)$ in Table 1.

Figure 5

Same description as in Fig. 4 for $D_{1S}=0.15$ m. This case is labeled $\left(c\right)$ in Table 1.

Figure 6

Downlink time series of phase noise along an 8-s duration sampled every 1 ms. Receiver aperture $D_{1S}=0.40$ m. Comparison between a turbulent case (dashed black curve, variance $61.9{\mathrm{rad}}^2$) and a tip-tilt control one (solid red curve, variance $65.2{\mathrm{rad}}^2$). These cases are, respectively, labeled $\left(a\right)$ and $\left(b\right)$ in Table 1.

Figure 7

Tip-tilt controlled downlink time series of phase noise along an 8-s duration sampled every 1 ms. Comparison between a small aperture case (dashed black curve, variance $66.7{\mathrm{rad}}^2$) and a large aperture one (solid red curve, variance $65.2{\mathrm{rad}}^2$). These cases are labeled $\left(b\right)$ and $\left(c\right)$ in Table 1, respectively.

Figure 8

Tip-tilt controlled uplink time series of heterodyne efficiency along an 8-s duration sampled every 1 ms. The red (lower) straight line shows the detection threshold. The blue (upper) straight line shows the mean heterodyne efficiency $\langle m\rangle$. $D_{1E}=0.40\mathrm{m}$. This case is labeled $\left(d\right)$ in Table 1.

Figure 9

Same description as in Fig. 8 for $D_{1E}=0.15\mathrm{m}$. This case is labeled $\left(e\right)$ in Table 1.

Figure 10

Tip-tilt controlled uplink time series of phase noise along an 8-s duration sampled every 1 ms. Comparison between a small aperture case (dashed black curve, variance $66.7{\mathrm{rad}}^2$) and a large aperture one (solid red curve, variance $65.2{\mathrm{rad}}^2$). These cases are labeled $\left(d\right)$ and $\left(e\right)$ in Table 1, respectively.

Figure 11

Piston time series of phase noise along an 8-s duration sampled every 1 ms. Comparison between two aperture diameters $D=0.15$ m (dashed black line) and $D=0.40$ m (solid red line). Variances are, respectively, $66.5{\mathrm{rad}}^2$ and $63.9{\mathrm{rad}}^2$. The variance of the difference [green (light gray) line] is $0.51{\mathrm{rad}}^2$.

Figure 12

Two-way compensation on phase noise in the case of symmetric apertures ($D_{1E}=D_{1S}=0.40$ m) with point ahead and with tip-tilt correction. Comparison between downlink (solid red line) and uplink (dashed black line) phase noise that correspond to labels $\left(b\right)$ and $\left(d\right)$ in Table 1, respectively. The two-way phase noise difference [green (light gray line)] has zero mean with a variance of $0.69{\mathrm{rad}}^2$.

Figure 13

Two-way compensation on phase noise in the case of asymmetrical apertures ($D_{1E}=0.15$ m, $D_{1S}=0.40$ m) without point ahead and with tip-tilt correction. Comparison between downlink (solid red line) and uplink (dashed black line) phase noise. Downlink corresponds to label $\left(b\right)$ in Table 1. Uplink mean heterodyne efficiency $\langle m\rangle =0.79$ and its relative fluctuation ${\sigma }_m^2/{\langle m\rangle }^2=0.03$. The two-way phase noise difference [green (light gray line)] has zero mean with a variance of $0.27{\mathrm{rad}}^2$.

Figure 14

Two-way compensation on phase noise in the case of asymmetrical apertures ($D_{1E}=0.15$ m, $D_{1S}=0.40$ m) with point ahead and with tip-tilt correction. Comparison between downlink (solid red line) and uplink (dashed black line) phase noise that corresponds to labels $\left(b\right)$ and $\left(e\right)$ in Table 1, respectively. The two-way phase noise difference [green (light gray line)] has zero mean with a variance of $1.0{\mathrm{rad}}^2$.

Figure 15

Modified Allan variance as a function of the averaging time, for one-way downlink (squares) and two-way compensated link (diamonds, triangles, asterisks) vis-a-vis the various link asymmetries. All results are obtained with tip-tilt control and $D_{1S}=0.40$ m. Two sigma error bars are displayed.

Figure 16

Two-way phase noise difference (standard deviation in rad.) as a function of the outer scale value $L_0$ in meters.