Temporal Coherence
Temporal coherence is the degree to which the electric field of a light wave at a given point in space remains correlated with itself at later times. In lasers and photonics it is the property that lets a beam interfere with a delayed copy of itself, and it is what distinguishes a narrow-linewidth laser from broadband or thermal light. It is distinct from spatial coherence (correlation across the beam cross-section).
A perfectly monochromatic wave with a fixed frequency and phase would be temporally coherent for all time. Real sources have a finite optical bandwidth and random phase diffusion, so the correlation decays after a characteristic coherence time τcoh.
Quantitative measures:
The first-order temporal coherence function is the normalized field autocorrelation:
γ(τ)=⟨E∗(t) E(t+τ)⟩
⟨∣E∣2⟩.
∣γ(τ)∣ starts at 1 and falls as the delay τ grows. τcoh is the delay over which this correlation drops significantly (definitions vary: 1/e, FWHM of ∣γ∣, etc.).
Coherence length is the corresponding path length in vacuum:
Lcoh=c τcoh.
It is the maximum optical-path difference in an interferometer that still produces high-contrast fringes.
Temporal coherence and linewidth are Fourier-transform partners (Wiener–Khinchin theorem). For a Lorentzian spectrum of FWHM Δν,
τcoh≈1π Δν, Lcoh≈cπ Δν.
A convenient wavelength form often used for broadband sources is:
Lcoh≈λ2/Δλ
(exact prefactor depends on the spectral shape). Narrower spectrum ⇒\Rightarrow⇒ longer coherence time and length.
Typical orders of magnitude:
White light / LED: Lcoh of micrometers.
Superluminescent diode / ASE: tens of micrometers.
Multimode diode or HeNe without stabilization: millimeters to meters.
Single-frequency HeNe: tens to hundreds of meters.
Stabilized single-frequency solid-state or fiber laser (kHz or sub-Hz linewidth): kilometers to 10/5 km.
Lasers achieve long temporal coherence because stimulated emission and the cavity filter a narrow set of frequencies; further linewidth narrowing comes from high cavity QQQ and frequency stabilization. Multimode or pulsed lasers can have short coherence despite high spatial coherence.
How it is measured:
Fringe visibility versus path delay in a Michelson or Mach–Zehnder interferometer; delayed self-heterodyne / self-homodyne beat-note spectroscopy for narrow-linewidth lasers; Fourier transform of the measured optical spectrum when the spectrum is well known.
Applications:
High temporal coherence is required when:
Interferometry (Michelson, Mach–Zehnder, Fabry–Pérot) must maintain fringe contrast over large path imbalances.
Holography and holographic interferometry (object and reference paths must stay within Lcoh).
Coherent communications, optical phase-locked loops, and heterodyne/homodyne detection.
Coherent beam combining and precision metrology (length, frequency, gravity).
Laser Doppler velocimetry and coherent lidar (phase-stable beat notes).
Low temporal coherence is required when:
Optical coherence tomography (OCT): axial resolution is set by Lcoh (or equivalently by source bandwidth); broadband SLD, swept lasers, or supercontinuum sources give micrometer-scale depth resolution.
White-light / low-coherence interferometry for surface profiling.
Speckle reduction in displays, projectors, and some imaging systems.
Avoiding parasitic interference and stimulated Brillouin scattering in fibers.
Temporal coherence tells you how long (in time or path length) a laser’s phase remains predictable. Single-frequency lasers maximize it for precision interference; broadband sources minimize it for ranging and tomography.