Femtosecond optical two-way time-frequency transfer in the presence of motion

Platform motion poses significant challenges to high-precision optical time and frequency transfer. We give a detailed description of these challenges and their solutions in comb-based optical two-way time and frequency transfer (O-TWTFT). Specifically, we discuss the breakdown in reciprocity due to relativity and due to asynchronous sampling, the impact of optical and electrical dispersion, and velocity-dependent transceiver calibration. We present a detailed derivation of the equations governing comb-based O-TWTFT in the presence of motion. We describe the implementation of real-time signal processing algorithms based on these equations and demonstrate active synchronization of two sites over turbulent air paths to below-1-fs time deviation despite effective velocities of $\pm 25\mathrm{m}/\mathrm{s}$, which is the maximum achievable with our physical setup. With the implementation of the time transfer equation derived here, we find no velocity-dependent bias between the synchronized clocks to within a two-sigma statistical uncertainty of 330 as.

Figure 1

Experimental setup to evaluate comb-based O-TWTFT with motion. (a) A multipassed moving retroreflector is located adjacent to clock $B$ to simulate clock $B$ motion with effective closing velocity, $V$. (b) A quadcopter-mounted-retroreflector is flown approximately equidistant from sites $A$ and $B$.

Figure 2

(a) Fundamental relativistic breakdown in reciprocity. Pulses launched simultaneously from static clock $A$ and moving clock $B$ in site $A$'s reference frame experience a nonreciprocal time of flight with $T_{\mathrm{A}\to \mathrm{B}}\ne T_{\mathrm{B}\to \mathrm{A}}$. (b) Breakdown in reciprocity due to asynchronous sampling. Pulses launched at different times (asynchronously) from static clock $A$ and moving clock $B$ (again relative to site $A$'s reference frame), e.g., at $t=0$ and at $t=\mathrm{\Delta }{t}_{\mathrm{async}}$, experience different time of flights.

Figure 3

(a) Definition of the “clock” or timescale at each site. A self-referenced frequency comb is phase locked to a cavity-stabilized laser at $n_{\mathrm{cw}}\sim 195\mathrm{THz}$. The comb produces a phase coherent pulse train with repetition period $1/f_r\sim 5\mathrm{ns}$ and femtosecond pulse-to-pulse timing jitter. A digital signal processor enables the labeling of each pulse, corresponding to the “ticks” of the clock. (b) Basic O-TWTFT configuration. Three heterodyne signals between the combs are measured at $D1, D2$, and $D3$. At $D4$ and $D5$, the phase-modulated DFB laser signal is detected for the communications and coarse timing signals. See text for additional details. PM, phase modulator; DFB, distributed feedback; light gray circles, 50:50 couplers; dark blue circles, wavelength division multiplexers (WDMs); $D$, photodetector.

Figure 4

Detailed schematic of master and remote sites. WDM, wavelength division multiplexer; ADC, analog-to-digital converter; FSO, free-space optical terminal; white ovals, fiber couplers; $D$, photodetector; black dots, injection sites for transceiver calibration via OTDR; gray shading, one example set of delays (see Sec. 3b3).

Figure 5

Diagram illustrating the relationship between the interferogram produced by (a) the interference of two combs and (b) the analogous heterodyne mixing of continuous-wave oscillators. (c) The peaks of the interferograms between the combs yield their associated phase difference with orders-of-magnitude higher precision than if the detected repetition rates were instead measured as in (b). This higher precision is a consequence of the shot-noise limited signal-to-noise ratio and the $\sim 1-\mathrm{THz}$ measurement bandwidth set by the optical pulse bandwidth. As indicated in the bottom graph, we can view the phase offset, alternately, against “oracle time,” which is mathematically convenient but experimentally inaccessible, or the local timebase, evaluated in terms of the local optical phase or the local ADC sample number. Note the $k_{\mathrm{AX}}$ are not integers.

Figure 6

Example digitized interferogram between comb $A$ and comb $X$ (blue line). To find its center, it is first filtered by a matched filter, then a Hilbert transform is applied to generate an envelope function (red dashed line) followed by a subsample interpolation to find its precise peak position. The other interferograms require a more involved procedure (cross-ambiguity function search) to find their peaks’ position due to Doppler shifts.

Figure 7

Measured time shift in the interferogram position as a function of Doppler shift, both without dispersion compensating fiber (DCF) and with DCF. The improvement with the addition of DCF is evident in the factor of 50 reduction in $y$ axis between the top and bottom plots. It is evident in the latter computation that a simple linear compensation for the delay-Doppler coupling is insufficient to maintain femtosecond-level timing.

Figure 8

Delay-Doppler coupling. The specific waveform dispersion causes coupling between the closing velocity (Doppler shift) and the measured arrival time of the interferogram as recorded before and after the insertion of the DCF. (a) High differential dispersion, $\mathrm{\Delta }{\beta }_2^{\mathrm{Combs}}$, leads to a chirped interferogram (top figure) and a large delay-Doppler coupling as illustrated by the cross-ambiguity function (bottom figure). The amplitude of the cross-ambiguity function is shown on an arbitrary linear scale with warmer colors indicating higher intensity (greater correlation). (b) Low differential dispersion achieved through the insertion of dispersion compensating fiber reduces the interferogram chirp (top figure) and the delay-Doppler coupling (bottom figure). A vertical line would reflect zero delay-Doppler coupling.

Figure 9

(a) Signal processing on site $B$. The heterodyne detection between the incoming comb $X$ and local comb $B$ yields the series of interferograms (blue trace). After digitization and filtering, the signal processor extracts the timestamps using a cross-ambiguity function search, as described in Appendix pp1. The coherent communication channel detects the heterodyne signal between the local cw laser and transmitted cw laser. As discussed in Ref. [27], it both generates the timestamps for the communication-based O-TWTFT, labeled $\left\{{\tilde{T}}_{\mathrm{AA}},{\tilde{T}}_{\mathrm{AB}},{\tilde{T}}_{\mathrm{BB}},{\tilde{T}}_{\mathrm{BA}}\right\}$, and transmits the timestamps recorded on site $A$ via data packets to site $B$. The signal processor computes the clock offset from the equations given in Sec. 3a Kalman-filter-based loop filter then applies the necessary feedback to comb $B$ via a direct digital synthesizer (DDS). (b) Image of the signal processor that uses an eight-core digital signal processor (DSP) and field programmable gate array (FPGA). (c) Image of the optical transceiver at site $B$. CAF, cross-ambiguity function; IGM, interferogram.

Figure 10

(a) Schematic of the Doppler simulator. The corner cube is on a 2-m-long rail. This six-pass design allows for bidirectional propagation, effective displacements of 24 m, and effective speeds of 24 m/s. (b) Detailed schematic of coupling to quadcopter-mounted-retroreflector.

Figure 11

(a) Out-of-loop clock offset sampled continuously (light blue) and only during active synchronization (gray dots) at the full $\sim 2-\mathrm{kHz}$ sampling rate over a 4-km open air path and at a 24-m/s effective speed by use of the Doppler simulator. This trace shows a short 1-s segment out of the full dataset to better illustrate the behavior. (b) Timing power spectral density (PSD) for the full 30-min duration dataset. (c) Fractional frequency instability (modified Allan deviation) both for continuous sampling (light blue squares) and only during active synchronization (gray circles). The strong white phase noise character at averaging times below 100 s is evident with the red dashed line of $5.2\times {10}^{-16}{t}^{-3/2}$ added as a guide to the eye. (d) Time deviation for the same data.

Figure 12

(a) Clock synchronization with the Doppler simulator operated at its maximum $\pm 24-\mathrm{m}/\mathrm{s}$ speed. The plot shows the instantaneous closing velocity (dark blue, top trace) as the retroreflector cycles back and forth on the Doppler simulator rail, the in-loop computed O-TWTFT clock offset (purple dots, center trace), and the out-of-loop verification clock offset during periods of active synchronization (gray dots, lower trace) and across signal fades (light blue line, lower trace). (b) Out-of-loop measured clock offset during periods of active synchronization for the following conditions: $V=0\mathrm{m}/\mathrm{s}$ and $L=0\mathrm{km}, V=\pm 24\mathrm{m}/\mathrm{s}$ and $L=0\mathrm{km}, V=\pm 24\mathrm{m}/\mathrm{s}$ and $L=2\mathrm{km}$, and $V=\pm 24\mathrm{m}/\mathrm{s}$ and $L=4\mathrm{km}$. All data are at the full 2-kHz sampling rate.

Figure 13

Clock synchronization during flight of quadcopter. Top panel: Three passes of the quadcopter showing the instantaneous closing velocity (blue) and out-of-loop clock offset during active synchronization (gray dots). Bottom panels: Expanded views containing continuously sampled clock offset (light blue) as well as only during active synchronization (gray dots) showing clock walk-off during signal fades and synchronization reacquisition.

Figure 14

Time series of closing velocity and clock offset for operation of the Doppler simulator in series with a (a) 0-, (b) 2-, and (c) 4-km open air path. The clock offset data are resampled to 10 Hz. (d), (e) Clock offset vs closing velocity (gray circles) for the 0-, 2-, and 4-km paths of (a)–(c). The red dashed line is a quadratic fit to the data which is used to estimate any residual bias.