Coherence Length Formula Calculator

Q: What do the variables Lc, λ, and Δλ represent?

Lc = coherence length.λ = central wavelength.Δλ = spectral bandwidth (linewidth).

Q: What are the common pitfalls when applying the coherence length formula?

Confusing coherence length with coherence time.Forgetting that Δλ must be in the same units as λ.Assuming a laser has infinite coherence length (real lasers have finite linewidth).

Interpretation

Lc = λ²/Δλ. The distance over which a wave maintains a constant phase relationship. Used in interferometry and optical communications.

Lc = λ² / Δλ

Coherence Length

Variables

Symbol	Quantity	Unit
Lc	Coherence length	m
λ	Central wavelength	m
Δλ	Spectral bandwidth	m

What it means

Coherence length is a measure of the temporal coherence of a light source. It is inversely proportional to the spectral width (Δλ). A longer coherence length allows more interference fringes and is needed for high‑precision interferometry. Lasers have long coherence lengths; LEDs have short. Understanding this is essential for designing interferometers, holography systems, and in fibre optic communications where phase coherence matters.

Worked example

Coherence Length – Two Detailed Examples

Real‑World

Scenario: A HeNe laser has wavelength λ = 632.8 nm and spectral linewidth Δλ = 0.002 nm. The coherence length Lc = λ² / Δλ = (632.8²) / 0.002 = 400,000 nm? Actually 632.8 nm squared = 400,000 nm²? Let's compute: λ² = (632.8e-9)² = 4.00e-13 m², Δλ = 0.002e-9 m, so Lc = 4.00e-13 / 0.002e-9 = 2.00e-4 m = 0.2 m = 200 mm. The laser light remains coherent over a distance of about 20 cm, which is sufficient for many interferometric applications. The physicist uses this to design a Michelson interferometer with path differences less than the coherence length to observe interference fringes.

Parameter	Value
λ (nm)	632.8
Δλ (nm)	0.002

1Lc = (632.8e-9)² / (0.002e-9) = 4.00e-13 / 2e-12 = 0.2 m

Result 0.2 m ✓ Coherence length

Scenario: An LED emits at 850 nm with a spectral width of 10 nm. The coherence length Lc = (850²) / 10 = 72250 nm? Compute: λ² = (850e-9)² = 7.225e-13 m², Δλ = 10e-9 m, Lc = 7.225e-13 / 1e-8 = 7.225e-5 m ≈ 72 µm. This short coherence length means that the LED light is incoherent over even a short path difference, making it unsuitable for interferometry but excellent for illumination and communications.

Parameter	Value
λ	850
Δλ	10

1Lc = (850e-9)² / (10e-9) = 7.225e-13 / 1e-8 = 7.225e-5 m ≈ 72 µm

Result 72 µm ✓ Short coherence

Insight: Coherence length is inversely proportional to spectral linewidth. A narrow linewidth gives long coherence length, essential for high‑precision interferometry. Conversely, broad linewidth sources have short coherence and are used in coherence‑domain imaging.

Common mistakes

Coherence length: L_c = λ² / Δλ – where Δλ is the spectral width of the source.
Units: λ and Δλ in the same units (e.g., nm) – L_c in the same unit.
Interpretation: The distance over which the light remains coherent – for a laser, L_c can be long; for an LED, short.
Assumes: Gaussian or rectangular spectrum – this is an approximation.
Coherence time: τ_c = L_c / c – the corresponding time.

Applications

Coherence length, L_c = λ²/Δλ, is the distance over which a light wave maintains a predictable phase relationship, limited by the spectral width Δλ. This is important in interferometry, holography, and optical communications. Engineers use it to select light sources for coherence‑dependent applications. For example, laser sources have long coherence lengths, suitable for interferometry, while LEDs have short coherence lengths, reducing speckle. By understanding coherence length, they can design systems for optical coherence tomography (OCT), fibre sensors, and metrology. This concept is fundamental to understanding the temporal coherence of light.

Design of interferometers and holographic systems
Optical coherence tomography (OCT) for medical imaging
Selection of light sources for coherence‑sensitive applications
Speckle reduction in imaging and display systems
Understanding of laser and light source characteristics

Frequently Asked Questions

Q01What is the Coherence Length formula used for?

A01

It calculates the maximum path length difference over which light waves maintain a consistent phase relationship: Lc = λ² / Δλ.

Q02What do the variables Lc, λ, and Δλ represent?

A02

Lc = coherence length.
λ = central wavelength.
Δλ = spectral bandwidth (linewidth).

Q03What is the significance of coherence length?

A03

It determines the visibility of interference fringes; fringes are visible only when the path difference is less than Lc.

Q04How is the coherence length derived?

A04

From the Fourier transform of the spectrum; a narrow spectral width corresponds to a long coherence length.

Q05What is the difference between temporal coherence and spatial coherence?

A05

Temporal coherence is related to the spectral purity (Lc); spatial coherence relates to the uniformity of phase across a wavefront.

Q06Give a worked example using the coherence length formula.

A06

A laser with λ = 633 nm and Δλ = 0.01 nm has Lc = (633×10⁻⁹)² / (0.01×10⁻⁹) = 4.0×10⁻¹⁴ / 1.0×10⁻¹¹ = 0.04 m = 4 cm.

Q07What are the common pitfalls when applying the coherence length formula?

A07

Confusing coherence length with coherence time.
Forgetting that Δλ must be in the same units as λ.
Assuming a laser has infinite coherence length (real lasers have finite linewidth).

Q08How does coherence length affect interferometry?

A08

For an interferometer to produce fringes, the path difference must be less than the coherence length of the source.

Q09What is the coherence length of sunlight?

A09

Sunlight has a broad spectrum, so Lc is very short, on the order of microns, which is why white‑light interferometry requires careful path matching.

Q10How is coherence length used in optical coherence tomography (OCT)?

A10

OCT uses low‑coherence light; the coherence length determines the axial resolution of the imaging.