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1 Supplementary Methods and Data 1. Apparatus Design The time-of-flight measurement apparatus built in this study is shown in Supplementary Figure 1. An erbium-doped femtosecond fibre oscillator (C-Fiber, MenloSystems GmbH) is used to produce measurement pulses whose output spectrum is centered at a 1550 nm wavelength with a bandwidth of 60 nm. The pulse repetition rate is initially set at 100 MHz and tuned in the range of ±200 khz. The output power is amplified to 240 mw using an erbium-doped fibre amplifier (EDFA) that is comprised of two diode pump lasers (450 mw, 980 nm) and a 1.5 m long erbium-doped fibre. The pulse width is 220 fs after amplification. The output pulse train is divided by a polarizing beam splitter (PBS) into two identical trains; one goes to the reference mirror (M REF ) and the other to the target mirror (M MEA ). A corner cube of retroreflector type is used as the target mirror. By combining a PZT actuator with a stepping motor, the target mirror is position-modulated along the optical axis with nanometre resolution over a 30 mm travel. The reference mirror is also of retroreflector type and fixed stationary. The returning pulses are recombined with the reference pulses, and directed to the balanced cross-correlation (BCC) unit that is comprised of a 4 mm long type-ii PPKTP (periodically poled KTiOPO 4 ) crystal, a balanced pair of photodetectors and two NATURE PHOTONICS 1

2 dichroic mirrors. The used PPKPT crystal has a poling period of 46.2 μm with a 100 nm phase-matching bandwidth centered at 1550 nm wavelength. The used photodetectors have a 0.3 MHz bandwidth. One dichroic mirror reflects light of 1550 nm wavelength and transmits light of 775 nm wavelength, and the other dichroic mirror does the opposite. The balanced cross-correlation (BCC) signal is detected at a sampling rate of 0.25 ms and fed back to a servo controller that regulates the piezoelectric actuator installed within the femtosecond laser oscillator to control the pulse repetition rate by varying the cavity length. The pulse repetition rate is finally precisely measured using a radio-frequency frequency counter in reference with to the Rb atomic clock. The returning pulses are not visible and on the edge of eye safety, thus a 632-nm HeNe laser of 3 mw power was installed in parallel to be used as the aiming beam for optical alignment between the femtosecond laser source and the target mirror. 2 NATURE PHOTONICS

3 SUPPLEMENTARY INFORMATION Supplementary Figure 1. Apparatus for time-of-flight measurement of femtosecond light pulses. a, Optical hardware configured based on the optical layout of Fig. 1. Red lines are the optical paths of the reference and returning pulses, and blue lines indicate NATURE PHOTONICS 3

4 two second-harmonic beams generated from the PPKTP crystal in the process of balanced optical cross-correlation. Abbreviations are; CL: collimation lens, PBS: polarization beam splitter, HWP: half-wave plate, QWP: quarter-wave plate, T: telescope beam expander, M: mirror, DM: dichroic mirror, L: lens, PC: polarization controller, SMF: single-mode fibre, EDF: Er-doped fibre, WDM: wavelength-division multiplexer, and I: isolator. b, The target mirror is located on the roof of the KAIST Satellite Research Center, which is ~0.7 km apart from the 7 th floor of the main building of the KAIST Department of Mechanical Engineering that accommodates the apparatus of Fig. 1. c, The target mirror is made of a corner cube of retroreflector type and it is mounted on a piezoelectric actuator to position-modulate the target mirror to verify the measurement precision. d, The back-reflected light seen in this photo is a He-Ne laser used as the aiming beam for optical alignment, which is visible and eye-safe. 2. Simulation for pulse broadening and intensity attenuation The simulation presented in Fig. 2 was intended to assess how significantly the zero-crossing point of the BCC signal tends to drift from its true position due to the pulse broadening as well as the intensity attenuation of the returning pulses. For the simulation, the process of intensity cross-correlation was analytically modeled in consideration of the quantum efficiency of the used PPKTP crystal for second-harmonic generation. Assuming that the returning pulses maintain the Gaussian shape, the intensity profile of the generated second-harmonic sub-pulses was reconstructed with random electric noise of 27.2 μv/hz 1/2. Then the BCC signal was numerically synthesized with a birefringence time delay of 150 fs. The zero-crossing point of the BCC signal was then computed while the pulse broadening was changed from 200 fs to 4 NATURE PHOTONICS

5 SUPPLEMENTARY INFORMATION 1 ps and independently the intensity attenuation of the returning pulses from 0 db to 1.5 db. In this simulation, only the second order dispersion was considered while the third order dispersion was neglected as the latter appears relatively minute for pulses traveling in air. 3. Experimental verification of the measurement precision Supplementary Figure 2 presents a representative experimental result that demonstrates the precision of the proposed locking control of the pulse repetition rate f r for time-of-flight measurement. In this case, a sinusoidal modulation of 120 nm amplitude was applied to the target mirror located at a 1.5 m distance. The mo The result shows that the BCC signal experiences a small amount of jitter of 0.63 fs in terms of the root-mean-square (RMS) value at an integration time of 0.5 ms. The jitter RMS level scales down to fs when the integration time is increased to 10 ms, which is converted to a precision of 10 nm (0.066 fs ( m/s)/2) in the measured distance. NATURE PHOTONICS 5

6 Supplementary Figure 2. Experimental data of the locking control of the pulse repetition rate f r with a 120 nm position modulation of the target mirror. a, Time-traced jitter level of the BCC signal. b, Locking control signal produced to regulate the pulse repetition rate of the femtosecond laser oscillator. c, Measured repetition rate using a frequency counter referenced to the Rb atomic clock. 4. Comparison with a conventional microwave time-of-flight distance meter The long-distance measuring capability of our time-of-flight method was verified in comparison to a conventional microwave time-of-flight instrument (Total Station TOPCON, Inc) widely used for terrestrial geodetic measurements. The target mirror was set up at a nominal distance of ~0.7 km in air, and it was moved back and forth in 20 mm steps. As shown in Supplementary Figure 3, the two measurement results well agree with each other the ± 2 mm precision of the TOPCON distance meter. 6 NATURE PHOTONICS

7 SUPPLEMENTARY INFORMATION This comparative test confirms that our time-of-flight method is capable of handing large distances with much better precision than the conventional microwave instrument. Supplementary Figure 3. Comparative test by measuring a long distance of ~0.7 km in air with the target mirror being moved back and forth in 20 mm steps. a, Target distances measured by a commercial microwave time-of-flight device (red) and our apparatus (blue). b, Difference between the two measurement results with a nominal offset of 1.52 m removed. Supplementary Figure 4 shows another measurement result in which the target mirror was position-modulated with small amplitudes of 5 μm, 2 μm and 700 nm, respectively. The microwave device was not capable of detecting the fine modulatedmovements of the target mirror at all due to its limited resolution. On the other hand, our method clearly shows the modulated movements while the measured nominal distance is subject to a slow-varying fluctuation of ~5 micrometre amplitude being NATURE PHOTONICS 7

8 caused by the air turbulence and building vibration. Supplementary Figure 4. Detection of fine movements of the target mirror at a ~0.7 km distance in air. The target mirror was position-modulated at 10 Hz with amplitude of a,5 μm b, 2 μm and c, 700 nm, respectively. 5. Performance verification by comparing with a commercial laser interferometer A direct comparison was made with a heterodyne He-Ne laser interferometer (HP5510A) that yields sub-wavelength resolution up to several metres. The target mirror was positioned at a 1.5 m distance and a sequence of incremental movements was given in 100 nm steps. As presented in Supplementary Figure 5, the two measurement results are in good agreement with a high level of linearity of The difference is less than 20 nm with a standard deviation of 7 nm. This test verifies that our method offers the same level of sub-wavelength precision as the cw laser 8 NATURE PHOTONICS

9 SUPPLEMENTARY INFORMATION interferometer is capable of only at short range. Supplementary Figure 5. Verification by comparing with an incremental laser interferometer (HP5510A). a, Plot of measured distances with 100 nm motion increments. b, Difference between two measurements. 6. Definition of the target distance Our measurement is based on a two-arm interferometer of Michelson type in which the reference mirror provides the reference point of zero-datum. Thus, the measured distance is defined as the path difference between the two arms: one armdistance to the target mirror from the polarization beam splitter (PBS) minus the other arm-distance to the reference mirror from the same PBS. In this respect, the reference NATURE PHOTONICS 9

10 mirror has been secured stationary during measurement so as not to affect the measurement precision. The birefringent wedge plates in Fig. 1b are not for providing the reference plane of zero-datum but its function has to do with the BCC signal. The principle of balanced optical cross-correlation requires pre-generating a certain amount of relative delay between the two input pulses so that the BCC signal becomes zero precisely when the two pulses are in complete overlap with each other. The required amount of delay is in fact half the time delay induced by the birefringence effect of the used PPKTP crystal. The inserted wedge plates enables precise adjustment of the time delay between the two input pulses. 7. Initial determination of the target distance The integer value of m is determined using the stroboscopic technique with the relation of m = f r2 /(f r2 - f r1 ) in which f r1 and f r2 are two consecutive pulse repetition rates satisfying the locking state. In the particular example shown in Fig. 3 with a target distance of ~695 m, f r1 and f r2 were respectively measured MHz and MHz, which subsequently enabled m to be worked out to be 463. The time needed to acquire m depends on the tuning speed of the pulse repetition rate, which is currently at the level of ~15 seconds. The most limiting factor is the control bandwidth of the PZT 10 NATURE PHOTONICS

11 SUPPLEMENTARY INFORMATION actuator that lies in the range of ~1 khz. For dual servo control, the control signal of the PZT actuator is continuously monitored. If the signal goes beyond the predetermined bound (lower threshold value: 10 V, upper one: 140 V), we give a command to move the stepping motor up or down and subsequently readjust the control range of the PZT actuator to maintain the locked state. 8. Dead zones due to limited tuning range Tuning of the pulse repetition rate over a finitely confined range results in dead zones for short distances in which the returning pulses cannot overlap with the reference pulses. The dead zone appears repeatedly at every length cycle of pulse separation and begin to disappear when the location D of the target mirror exceeds a certain threshold distance. The minimum tuning range required to avoid dead zones is given by f r =c/4d. In our current apparatus, the tuning range is limited to ±200 khz, so the threshold distance producing no dead zone is worked out to be m. Increasing the tuning range would enable us to shorten the threshold distance, for instance, down to ~3 m if the eight-fold technique is adopted to boost f r to 25.6 MHz at 800 MHz repetition S1. 9. Maximum closing velocity NATURE PHOTONICS 11

12 The closing velocity refers to the threshold velocity that the target mirror can sustain along the measurement direction while maintaining the phase-locking of the pulse repetition rate. The closing velocity is therefore limited by the control bandwidth of the PZT actuator installed to vary the cavity length of the femtosecond laser oscillator. The bandwidth of our current measurement system is 1 khz at a 10-μm stroke, which leads the closing velocity to be 2.3 m/s when measuring a 0.7-km distance. The closing velocity can be described in terms of the distance change rate v, i.e., v = dd/dt where D = mc/f r N. The tuning speed of the cavity length is given by dl/dt where L=c/f r. The closing velocity can therefore be expressed as v=d/l dl/dt. In our case of measurement, the closing velocity is worked out to be ~2.3 m/s with D = ~0.7 km and dl/dt = ~10 mm/s. The closing velocity could be further enhanced by fast cavity length control using an acousto-optics modulator or electro-optics modulator S2. Supplementary References: S1 Washburn, B. R. et al., Fiber-laser-based frequency comb with a tunable repetition rate. Opt. Express 12, (2004). 12 NATURE PHOTONICS

13 SUPPLEMENTARY INFORMATION S2 Baumann, E. et al., High-performance, vibration-immune, fiber-laser frequency comb. Opt. Lett. 34, (2009) NATURE PHOTONICS 13

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