Time of flight (ToF)

Time of flight (ToF) is the time light takes to cover a path; in ranging it is the round-trip time $t$ from a source to a target and back to a detector beside the source, and the distance follows from

$$d = \frac{c\,t}{2},$$

where the factor 2 accounts for the return trip. In air, where $c/n$ differs from the vacuum value by about 0.03%, a round trip of 1 ns corresponds to 0.1499 m, 1 µs to 149.9 m, and a timing uncertainty of 10 ps to 1.5 mm. Time of flight is the principle behind pulsed lidar, depth cameras, laser rangefinders and fiber reflectometry.

Direct time of flight

A direct ToF system emits a short pulse, starts a clock, and stops it when the echo crosses a threshold at the detector. The detector is a fast photodiode, an avalanche photodiode or, for weak returns, a single-photon avalanche diode (SPAD) whose output edge is timed by a time-to-digital converter. With SPADs the measurement is repeated over many pulses and the arrival times are histogrammed, as in time-correlated single-photon counting; the peak of the histogram gives the range and the background of uncorrelated counts sets the noise.

Two quantities are easily confused. The range resolution, the smallest separation at which two targets along the same line give separate echoes, is set by the pulse duration $\tau$:

$$\Delta d = \frac{c\,\tau}{2},$$

so a 1 ns pulse resolves 15 cm and a 5 ns pulse 0.75 m. The precision with which a single isolated echo is located can be much finer, limited by the timing jitter of the detector and electronics and by the signal-to-noise ratio. A system with 100 ps of jitter has a single-shot spread of 1.5 cm, and averaging 100 independent shots reduces it by a factor of 10, to 1.5 mm, if the jitter is random.

The pulse repetition rate $f_\text{rep}$ limits the unambiguous range: an echo must arrive before the next pulse leaves, so the maximum is $c/(2f_\text{rep})$, which is 150 m at 1 MHz.

Indirect time of flight

Indirect, or phase-shift, ToF modulates the intensity of a continuous source at a frequency $f_\text{mod}$ and measures the phase delay $\varphi$ of the returned modulation. The distance is

$$d = \frac{c\,\varphi}{4\pi f_\text{mod}},$$

and since the phase repeats every $2\pi$, distances are unambiguous only up to

$$d_\text{max} = \frac{c}{2 f_\text{mod}}.$$

At 20 MHz that is 7.49 m, and a phase error of 1° corresponds to 2.1 cm. Raising the frequency improves the depth precision for a given phase noise and shortens the unambiguous range in proportion; at 100 MHz it is 1.50 m. Depth cameras therefore often combine two or more modulation frequencies and resolve the ambiguity from the pair. Multipath, light reaching a pixel by more than one route, biases the phase and is the main source of systematic error indoors.

Frequency-domain alternative

FMCW lidar chirps the optical frequency of a narrow-linewidth laser and measures range as the beat frequency between the echo and a local copy of the transmitted light. It is a time-of-flight measurement read out in the frequency domain, and it gives the radial velocity from the Doppler shift as well. The trade-offs are compared in Lidar: Time of Flight vs FMCW.

Time of flight in fiber

In a fiber the pulse travels at the group velocity, so $c$ is replaced by $c/n_g$, where $n_g$ is the group index. With $n_g$ = 1.4682 for standard single-mode fiber at 1550 nm, one nanosecond of round trip corresponds to 10.2 cm of fiber and a microsecond to 102 m; the speed of light in fiber entry gives the one-way latency. Optical time-domain reflectometry applies this to locate splices, connectors and breaks along a fiber, and an error in the entered group index scales every distance on the trace.

Pitfalls

• Using the vacuum speed of light for a path through glass or fiber, or the phase index where the group index applies, and forgetting the factor 2 of the round trip.
• Fixed delays in cables, drivers and detectors, typically nanoseconds, which must be calibrated against a target at a known distance.
• Range walk: a threshold detector fires earlier on a strong echo than on a weak one, shifting the apparent range with target reflectivity unless constant-fraction timing or amplitude correction is used.

In mass spectrometry "time of flight" means something unrelated to optics: ions of different mass-to-charge ratio, accelerated through the same voltage, are separated by their arrival times at a detector.

Common questions

What is the difference between direct and indirect ToF?

Direct ToF times short pulses; indirect ToF measures the phase shift of a continuously modulated signal. Direct ToF reaches long ranges, while indirect ToF suits short-range depth cameras with dense pixel arrays.

What limits the accuracy of a ToF measurement?

Timing jitter and signal-to-noise ratio set the precision, the pulse duration sets the resolution between neighboring targets, and calibration of fixed delays and of the propagation speed sets the absolute accuracy.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); D. Derickson (ed.), Fiber Optic Test and Measurement (Prentice Hall, 1998); W. Becker, Advanced Time-Correlated Single Photon Counting Techniques (Springer, 2005).