Beam Coherence Time in Optics & Communications

• Beam Coherence Time is the duration over which a beam’s phase or spatial alignment remains stable, crucial for high-performance optical and communication systems.
• It is defined via statistical autocorrelation functions, using methods like 1/e decay and integral definitions to link temporal coherence with spectral properties.
• Measurement approaches range from interferometric setups to deep learning predictions in near-field arrays, enabling optimized beamforming and system reliability.

Beam coherence time quantifies the temporal scale over which the phase or spatial alignment of a beam, array, or field remains sufficiently well-defined to maintain key system performance—such as interference, beamforming gain, or holographic fidelity. In contemporary research, “beam coherence time” (or more generally, “coherence time”) is rigorously defined in terms of the statistical autocorrelation functions of the electromagnetic field, array phases, or spatial field distributions. The parameter plays a foundational role in quantum optics, laser physics, nonlinear photonics, free-space and fiber communications, millimeter-wave networks, and radar systems. Its precise determination dictates the achievable visibility in optical interference, limits information rates via channel dynamics, and constrains the update rates of beamforming weights in distributed and near-field arrays.

1. Formal Definitions and Theoretical Foundations

The coherence time $\tau_c$ is fundamentally tied to the first-order field autocorrelation function. For a classical or quantum field $E(t)$, the normalized first-order temporal coherence function is

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

The coherence time is then the characteristic timescale over which $|g^{(1)}(\tau)|$ decays:

• Integral definition (Goodman/Mandel–Wolf convention):

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

• 1/e decay definition: $\tau_c$ is the value for which $|g^{(1)}(\tau_c)| = 1/e$.

For Gaussian or Lorentzian spectra, the coherence time is inversely proportional to the spectral width:

$$\tau_c \approx \frac{1}{\Delta \omega} \approx \frac{\lambda^2}{c\,\Delta\lambda}$$

In beamforming and array contexts, “beam coherence time” $T_B$ is operationally defined as the interval over which the array gain (SNR or directivity) associated with a fixed set of beamforming weights remains within a specified threshold (e.g., within 3 dB of its initial value) (Chafaa et al., 3 Nov 2025, Silbernagel et al., 17 Sep 2025, Zheng et al., 11 May 2024).

For distributed coherent systems, the mutual phase relationship among nodes dictates $T_B$, governed by the time for the aggregate relative phase drift $\mathrm{Var}[\Delta \phi(\tau)]$ to reach unity (radian squared) (Silbernagel et al., 17 Sep 2025).

2. Experimental and Measurement Methodologies

Measurement of beam coherence time is diverse, reflecting context and the physical observable of interest:

• Direct field or intensity autocorrelation: Split-pulse or interferometric setups (e.g., Michelson interferometer or split-and-delay units) directly access $g^{(1)}(\tau)$ via fringe visibility as a function of delay. For instance, free-electron laser (FEL) pulses at FLASH were characterized with a split-and-delay apparatus, yielding $\tau_c = 1.75$ fs at $\lambda = 8$ nm (Singer et al., 2012).
• Statistical phase drift tracking: In distributed beamforming (e.g., 60 GHz meshes), $T_B$ is tracked using “beamformer-halt” experiments. Array SNR gain degradation, $G(\tau)$, measured after freezing weights, is mapped to the phase variance via

$$G(\tau) \leq (N^2-N)\,e^{-\mathrm{Var}[\Delta\phi(\tau)]} + N$$

The time to a 3 dB drop directly specifies beam coherence time (Silbernagel et al., 17 Sep 2025).

• Weak projective measurement: Single-photon temporal coherence can be measured using controlled two-photon interference at a polarization beam splitter, reconstructing $g^{(1)}(\tau)$ via frequency-resolved coincidence counts (Hofmann et al., 2012).
• Spectroscopic and interferometric methods: In holography and light source characterization, coherence time and length are extracted from spectral linewidth ($\Delta\lambda$) by emission spectroscopy and from the maximum path difference supporting fringes in a Michelson interferometer (Escarguel et al., 15 Oct 2024).

3. Quantum, Laser, and Nonlinear Photonic Regimes

In quantum optics and laser physics, coherence time manifests fundamentally in the decay of quantum field correlations, with ramifications for photon statistics and quantum interference.

• For continuous-wave laser models, the beam coherence time $\tau_{\mathrm{coh}}$ is dictated by phase diffusion,

$g^{(1)}(\tau) = e^{-(\ell/2)|\tau|}$

so $\tau_{\mathrm{coh}} = 2/\ell$, where $\ell$ is the phase diffusion constant. In Heisenberg-limited lasers (sub-Poissonian photon statistics), $\ell$ is suppressed, yielding $\tau_{\mathrm{coh}} \propto \mu^4$ (mean photon number), four orders faster than the conventional Schawlow–Townes limit $\tau_{\text{ST}} \propto \mu$ (Ostrowski et al., 24 Feb 2025).

• In spontaneous four-wave mixing, the temporal coherence of each beam is extracted using second-order correlation $g^{(2)}(0)$ measurements. Chirp in the pump broadens spectral entanglement, reducing the single-beam $\tau_c$, while post-generation chirp affects only mode-matching, not intrinsic coherence (Ma et al., 2011).

4. Coherence Time in Distributed, mmWave, and Near-Field Arrays

In contemporary mmWave and THz networks, “beam coherence time” encompasses the temporal integrity of phase alignment among distributed nodes or the spatial match of a fixed beam in the presence of high mobility:

• Phase coherence in distributed arrays: The beam coherence time is the interval over which all radio frequency (RF) chains maintain phase alignment within a specified margin (e.g., $\leq 1\,\mathrm{rad}^2$ drift), considering oscillator phase noise, synchronization jitter, and environmental drift. Optical time synchronization (OTS) networks can extend $T_B$ to several hundred seconds, supporting stable multi-node beamforming (Silbernagel et al., 17 Sep 2025).
• Adaptive beamforming in mobile/near-field THz systems: In near-field (NF) conditions with spherical wave effects, beam coherence time $T_B$ is the operational timespan until beamforming gain falls below a threshold $\xi$ due to user mobility and steering mismatch. Deep learning models, trained on simulated trajectory and system parameters, can predict $T_B$ within 10% accuracy, enabling beam update scheduling at milliseconds-to-hundred-milliseconds intervals, reducing training overhead by orders of magnitude compared to updating at the much shorter channel coherence time $T_C$ (Chafaa et al., 3 Nov 2025).

5. Channel and Beam Coherence for Wireless Communications

In wireless systems, coherence time bounds the reliability of channel estimates, the duration of valid precoders, and the integrity of fixed beam alignment. In non-terrestrial networks (NTNs), such as low Earth orbit (LEO) satellite-to-ground links, high mobility of the base station (BS) dramatically compresses both channel and beam coherence times.

• The classical channel coherence time, $T_c$, is determined by the lag where the normalized channel autocorrelation falls below a threshold. For directive beams, a separate “beam coherence time” arises due to beam misalignment.
• Beamwidth tuning enhances $T_c$ in terrestrial or low-mobility contexts, but in NTNs with orbital velocities, misalignment dominates decoherence, and $T_c$ becomes nearly independent of beamwidth (Zheng et al., 11 May 2024).
• A dominant line-of-sight (LoS) channel component extends coherence time by shifting the autocorrelation’s decay from nanoseconds (pure NLoS) to microseconds or longer (high Rician $K$-factor).

6. Applications in Holography and Nonlinear Instability

The requirement for beam coherence time arises in practical and diagnostic scenarios:

• Holography: Successful recording of three-dimensional relief in holography necessitates that the optical path difference across the object does not exceed the coherence length $L_c = c\tau_c$ of the source:
  • He–Ne lasers ($L_c \sim 30$ cm, $\tau_c \sim 1$ ns) enable high-depth, high-fidelity holograms.
  • LEDs with 1 nm filters ($L_c \sim 0.4$ mm, $\tau_c \sim 1$ ps) or unfiltered LEDs ($L_c \sim 20$ μm, $\tau_c \sim 0.07$ ps) restrict hologram formation to very shallow reliefs or contact recording only (Escarguel et al., 15 Oct 2024).
• Nonlinear laser–plasma interactions: In stimulated Brillouin backscatter (BSBS), the collective instability threshold becomes almost independent of $T_c$ (collective regime) once the light’s path during $T_c$ exceeds the laser speckle length, shifting instability control from random-phase amplification to diffraction (Korotkevich et al., 2013).

7. Summary Table of Contexts and Key Findings

Domain	Definition/Measurement	Scaling/Regime/Impact
Quantum/Laser Physics	$1/e$ decay of $g^{(1)}(\tau)$, phase diffusion	$\tau_c\propto \mu^4$ (Heisenberg-limited), or $\propto\mu$ (Schawlow–Townes); sub-Poissonian stats extend $\tau_c$ (Ostrowski et al., 24 Feb 2025)
FEL/Optical Pulses	Field autocorrelation; split-delay interferometry	$\tau_c$ set by bandwidth ($\sim 1/\Delta\nu$), e.g. 1.75 fs at 8 nm (Singer et al., 2012)
Distributed Arrays	SNR drift in “beamformer-halt” test; phase noise analysis	$T_B$ governed by residual multi-node phase drift; OTS allows $T_B > 200$ s at 60 GHz (Silbernagel et al., 17 Sep 2025)
Terahertz/NF Beamforming	Operational SNR drop; deep FNN regression	$T_B$ ranges from 0.5 ms (vehicular) to 200 ms (pedestrian); FNN matches numerical $T_B$ within 10% (Chafaa et al., 3 Nov 2025)
NTN/High-Mobility	Channel and beam autocorrelation decay	$T_c$ compressed to $\sim$μs or less; beamwidth ineffective as $v_\mathrm{BS}\rightarrow$ km/s; strong LoS extends $T_c$ (Zheng et al., 11 May 2024)
Holography	Maximum path difference for interference	Efficacy of source determined by $L_c$: ns-scale lasers for deep relief, ps-scale for shallow (Escarguel et al., 15 Oct 2024)
Nonlinear Instability	Temporal decorrelation in pump field	Beyond a threshold, instability threshold governed by diffraction, independent of $T_c$ (Korotkevich et al., 2013)

• “Distributed Coherent Beamforming at 60 GHz Enabled by Optically-Established Coherence” (Silbernagel et al., 17 Sep 2025)
• “Deep Learning Prediction of Beam Coherence Time for Near-FieldTeraHertz Networks” (Chafaa et al., 3 Nov 2025)
• “A Statistical Evaluation of Coherence Time for Non-Terrestrial Communications” (Zheng et al., 11 May 2024)
• “Analytical results for laser models producing a beam with sub-Poissonian photon statistics and coherence scaling as the Heisenberg limit” (Ostrowski et al., 24 Feb 2025)
• “Spatial and temporal coherence properties of single free-electron laser pulses” (Singer et al., 2012)
• “Effect of chromatic dispersion induced chirp on the temporal coherence property of individual beam from spontaneous four wave mixing” (Ma et al., 2011)
• “An exploration of temporal coherence of light through holography” (Escarguel et al., 15 Oct 2024)
• “Beyond the random phase approximation: Stimulated Brillouin backscatter for finite laser coherence times” (Korotkevich et al., 2013)
• “Direct observation of temporal coherence by weak projective measurements of photon arrival time” (Hofmann et al., 2012)

Beam coherence time therefore serves as a unifying metric for temporal stability, directly governing system performance across quantum optics, advanced wireless networks, and nonlinear photonics.