Coherent vs incoherent light

Light is classified as coherent or incoherent based on the temporal and spatial relationships between its phase at different points and times. This distinction fundamentally affects how the light interferes, focuses, propagates, and interacts with measurement systems.

Temporal coherence. Light is temporally coherent at time scale $\tau$ if the phase relationship between $E(t)$ and $E(t + \tau)$ remains predictable. The coherence length $L_c = c \tau_c$ where $\tau_c$ is the coherence time. Temporal coherence is determined by the source spectral width:

$$\tau_c \;\approx\; \frac{1}{\Delta\nu}.$$

Source	Coherence time	Coherence length
Sunlight (visible)	$\sim 5$ fs	$\sim 1.5$ μm
White-light LED	$\sim 20$ fs	$\sim 6$ μm
Halogen lamp + interference filter (10 nm BW)	$\sim 0.2$ ps	$\sim 60$ μm
HeNe laser (single-mode)	$\sim 1$ ms	$\sim 300$ km
Multimode laser diode	$\sim 1$ ps	$\sim 300$ μm
DFB laser (single-mode, 1 MHz linewidth)	$\sim 1$ μs	$\sim 300$ m
Frequency-stabilized HeNe (10 Hz linewidth)	$\sim 0.1$ s	$\sim 30000$ km

Spatial coherence. Light is spatially coherent across a transverse region if the phase relationship between $E(\mathbf{r}_1)$ and $E(\mathbf{r}_2)$ is well-defined for points $\mathbf{r}_1, \mathbf{r}_2$ within that region. Laser TEM₀₀ output is fully spatially coherent; extended thermal sources are not.

For a thermal source of diameter $d$ at distance $L$, the transverse coherence radius at the observation point is:

$$r_c \;\approx\; \frac{\lambda L}{d}$$

(van Cittert-Zernike theorem). A point-like source produces fully-coherent illumination at any distance; an extended source produces partially-coherent illumination with limited coherent area.

Implications of coherence.

Phenomenon	Coherent light	Incoherent light
Interference	Strong fringes; possible	Weak or no fringes (extended pathlength averages out)
Speckle	Yes, prominent	No
Focusable spot	Diffraction-limited, single-spot	Limited by source size (geometric image of source)
Beam quality	$M^2$ approaches 1	$M^2 \gg 1$
Holography	Possible	Not possible (no fringes)
Diffraction patterns	Sharp, high contrast	Diffuse
Polarization	Well-defined	Often randomized in thermal sources
Phase contrast imaging	Works	Limited
Coherence-based gating (OCT)	Possible	Not possible

Coherent imaging. A coherent source illuminates an imaging system with light that has consistent phase across the entrance pupil. The system's coherent point-spread function (the field, not intensity) governs image formation. Speckle and ringing artifacts result from any phase-disturbing features in the sample.

Incoherent imaging. An incoherent source (e.g., LED, lamp, sun) illuminates with light whose phases are random. The intensity PSF governs imaging. No speckle; smoother appearance; this is the "natural" microscopy imaging condition.

Partially coherent light. Most practical sources are between fully coherent and fully incoherent. The degree of coherence is characterized by the complex degree of coherence:

$$\gamma_{12}(\tau) \;=\; \frac{\langle E^*(\mathbf{r}_1, t) E(\mathbf{r}_2, t + \tau) \rangle}{\sqrt{\langle |E(\mathbf{r}_1, t)|^2 \rangle \langle |E(\mathbf{r}_2, t + \tau)|^2 \rangle}}.$$

Magnitude in $[0, 1]$ characterizes coherence: $|\gamma| = 1$ fully coherent, $|\gamma| = 0$ fully incoherent. The Wiener-Khinchin theorem relates $\gamma(\tau)$ to the source spectrum through Fourier transform.

Coherence and laser linewidth. A laser's linewidth $\Delta\nu$ characterizes its temporal coherence. Standard linewidth specs:

Laser	$\Delta\nu$	$L_c$
Free-running DFB	1 – 10 MHz	30 – 300 m
External-cavity DFB	$\sim 100$ kHz	$\sim 3$ km
Frequency-stabilized DFB	$\sim 100$ Hz	$\sim 3000$ km
Stabilized HeNe	$\sim 10$ Hz	$\sim 30000$ km
State-of-the-art atomic clock laser	$< 1$ Hz	$> 3 \times 10^5$ km
Multi-longitudinal-mode FP laser	spread $\sim 10$ GHz	$\sim 30$ mm

Why coherence matters for telecom.

• Modulator extinction: requires phase-stable signal; achievable only with single-mode lasers
• Coherent detection: requires LO with $\sim$ kHz or narrower linewidth; standard DFB sources work
• Self-homodyne effects in fiber: long fiber + temporal coherence produces interferometric noise
• Mode partition noise: multimode lasers exhibit large RIN due to mode-to-mode power transfer

For high-symbol-rate coherent transmission (PDM-16QAM at 64 Gbaud), source linewidth must be $< 100$ kHz to avoid significant phase noise penalty.

Coherence for OCT. Optical coherence tomography uses limited coherence as an imaging tool — the longitudinal resolution is approximately $L_c/2$:

OCT class	Source	$L_c$	Axial resolution
Time-domain OCT	SLD ($\sim 30$ nm BW)	$\sim 30$ μm	$\sim 15$ μm
Spectral-domain OCT	Same	$\sim 30$ μm	$\sim 7$ μm (with broader BW source)
Swept-source OCT	Tunable laser	$\sim 60$ nm sweep	$\sim 8$ μm (in tissue)
Ultra-high-res OCT	Ti:sapphire ($\sim 100$ nm BW)	$\sim 3$ μm	$\sim 1$ μm

Speckle and laser coherence. A coherent beam illuminating a rough surface produces a speckle pattern with high contrast (intensity variance equal to mean). For imaging applications where speckle is undesirable:

• Reduce spatial coherence: rotate diffuser, use multimode fiber
• Reduce temporal coherence: broaden source spectrum (SLD instead of laser)
• Polarization scrambling: speckle uncorrelated in two polarizations; combining reduces by $\sqrt{2}$
• Time-averaging: speckle pattern depends on configuration; varying position averages it out

Quantum coherence vs classical coherence. "Quantum coherence" refers specifically to the quantum-mechanical phase coherence between quantum states. A single-photon stream from an attenuated laser remains classically coherent (well-defined classical phase) and the photon-counting statistics agree with the classical theory. True quantum-coherent (squeezed, entangled) light states are exceptions, not the rule.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 12 (statistical optics, coherence theory); Born & Wolf, Principles of Optics (7th ed., 1999), Ch. 10 for the canonical treatment; Goodman, Statistical Optics (2nd ed., 2015) for the comprehensive engineering reference.