Temporal coherence

Temporal coherence describes how well the optical field at one point can be predicted from its value a time $\tau$ earlier. A wave with a narrow spectrum keeps a regular phase over many cycles; a broad spectrum is equivalent to a phase that wanders quickly. The time over which the correlation persists is the coherence time $\tau_c$, approximately the inverse of the spectral width $\Delta\nu$. A laser with a 1 MHz linewidth has $\tau_c \approx 1$ µs; a multimode helium-neon laser with about 1.5 GHz of oscillating bandwidth has about 0.7 ns; filtered white light spanning 300 nm around 550 nm has a coherence time of a few femtoseconds. Multiplied by the speed of light, $\tau_c$ gives the coherence length, which that entry tabulates for common sources.

Temporal coherence is separate from spatial coherence, which compares the field at two different points at the same instant.

Degree of temporal coherence

The quantity measured is the normalized first-order correlation function

$$g^{(1)}(\tau) = \frac{\langle E^*(t)\,E(t+\tau)\rangle}{\langle |E(t)|^2\rangle}.$$

By the Wiener-Khinchin theorem, $g^{(1)}(\tau)$ is the Fourier transform of the normalized power spectrum. Its magnitude equals the visibility of the fringes produced when a beam is split, delayed by $\tau$, and recombined with equal intensities. The spectral shape therefore fixes the shape of the visibility curve, and a single number $\tau_c$ is only a summary of it.

Definitions and numerical factors

A widely used definition (Goodman; Saleh and Teich) is

$$\tau_c = \int_{-\infty}^{\infty} |g^{(1)}(\tau)|^2\, d\tau .$$

For a Lorentzian spectrum of full width at half maximum $\Delta\nu$, typical of a single-frequency laser limited by phase noise, this gives $\tau_c = 1/(\pi\Delta\nu) = 0.318/\Delta\nu$. For a Gaussian spectrum, typical of Doppler-broadened gas lines and many broadband sources, it gives $\tau_c = \sqrt{2\ln 2/\pi}\,/\Delta\nu = 0.664/\Delta\nu$. Many authors drop these factors and write $\tau_c = 1/\Delta\nu$. For a 1 MHz Lorentzian laser the three conventions give 0.32 µs, 0.66 µs and 1 µs, and the corresponding coherence lengths in air are 95 m, 199 m and 300 m. Reported coherence lengths differ by factors of 2–3 for this reason, and a careful comparison states the definition used.

Measurement with a Michelson interferometer

The direct measurement uses a Michelson interferometer: one mirror is scanned, and the fringe visibility is recorded as a function of the path difference $\Delta L = c\tau$. For a single spectral line the visibility decays monotonically. For a spectrum with structure, the visibility reveals it. The sodium D lines at 589.0 and 589.6 nm, separated by 0.597 nm, make the visibility beat with a period in path difference of

$$\frac{\lambda^2}{\Delta\lambda} = \frac{(589.3\ \text{nm})^2}{0.597\ \text{nm}} = 0.58\ \text{mm}.$$

Michelson used such visibility curves in the 1890s to resolve fine structure in atomic lines, and the same principle, recording the whole interferogram and Fourier transforming it, is the basis of Fourier-transform spectroscopy. For ultrashort pulses the equivalent measurement is the field autocorrelation, whose width is set by the spectral bandwidth.

For narrow-linewidth lasers, whose coherence time exceeds any practical delay line, the linewidth is measured instead, for example with a delayed self-heterodyne setup, and $\tau_c$ is computed from it.

Where it matters in practice

Interferometric sensors, holography and coherent lidar require $\tau_c$ to exceed the largest round-trip delay in the system; for a frequency-modulated lidar with a 1 MHz Lorentzian laser, fringe contrast falls to 1/e at a range of about 48 m and is nearly gone by 150 m, half the 300 m coherence length. Short coherence also has uses. Optical coherence tomography relies on it to gate reflections by depth: for a Gaussian spectrum the axial resolution is

$$\Delta z = \frac{2\ln 2}{\pi}\,\frac{\lambda_0^2}{\Delta\lambda},$$

which for an 840 nm source with 50 nm bandwidth is 6.2 µm in air. Low temporal coherence also suppresses parasitic fringes from stray reflections, which is why superluminescent diodes and LEDs are preferred for some metrology and fiber-sensing tasks.

Pitfalls

Fringe contrast can fall for reasons unrelated to temporal coherence: unequal beam intensities, a polarization mismatch between the arms, or poor spatial overlap all reduce visibility. A test is to set the path difference near zero; if the visibility is still low there, the spectrum can be excluded as the cause. Mode hops and multimode operation also change $g^{(1)}(\tau)$ abruptly: a laser with two longitudinal modes has a visibility that revives periodically with path difference, with period equal to the cavity round-trip length, so a single short-delay measurement can greatly overstate or understate its coherence.

Common questions

How are coherence time and linewidth related?

They are inversely proportional: $\tau_c \approx 1/\Delta\nu$, with a numerical factor between 0.3 and 1 set by the spectral shape and the definition. A spectral width given in wavelength converts as $\Delta\nu = c\,\Delta\lambda/\lambda^2$; 1 nm at 1550 nm is 125 GHz.

Does a short pulse have low temporal coherence?

A transform-limited pulse has a spectrum as broad as its duration requires, so its coherence time is comparable to its duration. A train of phase-locked pulses from a mode-locked laser is a different case: its spectrum is a comb of narrow lines, and pulses far apart in the train can still interfere with each other.

Why does white light only give a few fringes?

Its coherence length is about 1 µm for a 300 nm band centred at 550 nm, so fringes appear only within a micrometre or so of zero path difference. White-light interferometers use this to locate the zero-path position precisely.

References: J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015); L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), Ch. 4; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 12; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 7 and 10; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 12.