Coherence length and bandwidth: the length over which the phase (and thus amplitude correlation) remains strong is related to the source bandwidth in what way?

Introduction / Context:
Coherence length (Lc) is a practical measure of how far an electromagnetic wave preserves a stable phase relationship, enabling high-contrast interference. It is fundamental in laser physics, spectroscopy, fiber optics, and synthetic aperture systems.

Given Data / Assumptions:

• Bandwidth Δν (or Δλ) characterizes spectral spread.
• Coherence time τc ≈ 1 / Δν for simple spectral shapes.
• Coherence length Lc = C * τc in a given medium (C is phase speed).

Concept / Approach:
If τc ≈ 1 / Δν, then Lc ≈ C / Δν: a narrower bandwidth (smaller Δν) means longer coherence time and longer coherence length. Conversely, broad-band sources have short coherence lengths, explaining why lasers (narrowband) support long-path interferometry.

Step-by-Step Solution:
Start from τc ≈ 1 / Δν.Compute Lc = C * τc, so Lc ≈ C / Δν.Therefore, Lc is inversely proportional to bandwidth.

Verification / Alternative check:
For a Gaussian spectrum: τc = 0.44 / Δν; the inverse dependence remains.

Why Other Options Are Wrong:

• Direct proportionality contradicts the physics of coherence.
• Square relation does not represent standard coherence models.
• 'None of these' is incorrect because the inverse proportionality is well established.

Common Pitfalls:

• Confusing radiometric bandwidth with detector electrical bandwidth.

Final Answer:
inversely proportional to the bandwidth