Coherence

8.1 Temporal Coherence

Coherence time $\tau_c$ : the time over which the wave maintains a well-defined phase.

Coherence length: $L_c = c\tau_c$ .

For a source with spectral width $\Delta\nu$ :

$\tau_c \approx \frac{1}{\Delta\nu}, \quad L_c \approx \frac{c}{\Delta\nu} = \frac{\lambda^2}{\Delta\lambda}$

A sodium lamp ( $\Delta\lambda \approx 0.6$ nm at $\lambda = 589$ nm) has $L_c \approx 0.6$ mm. A laser ( $\Delta\lambda \approx 10^{-6}$ nm) has $L_c \approx 300$ m.

Worked Example: Coherence length and fringe visibility

Problem. A mercury lamp emits light at $\lambda = 546.1$ nm with a spectral width $\Delta\lambda = 0.025$ nm. (a) Find the coherence length. (b) In a Michelson interferometer, at what Path difference does the fringe visibility drop to $1/e$ ? (c) How many fringes are visible before they Wash out?

Solution.

(a) $L_c = \lambda^2/\Delta\lambda = (546.1 \times 10^{-9})^2/(0.025 \times 10^{-9}) = 1.19 \times 10^{-2}$ m $= 11.9$ mm.

(b) For a Gaussian spectrum, visibility drops to $1/e$ when $\Delta x = L_c = 11.9$ mm.

(c) The number of fringes: $N_{\mathrm{fringes} = L_c/\lambda = (11.9 \times 10^{-3})/(546.1 \times 10^{-9}) = 2.18 \times 10^4}$ . Over 20000 fringes are visible — a large number, but far fewer than for a laser.

8.2 Spatial Coherence

The van Cittert-Zernike theorem states that the spatial coherence of light from an extended Incoherent source is given by the Fourier transform of the source intensity distribution.

For a circular source of angular diameter $\theta_s$ The transverse coherence length is:

$l_c \approx \frac{1.22\lambda}{\theta_s}$

8.3 Michelson Stellar Interferometer

Two separated mirrors direct light from a distant star into a single telescope. Fringes are observed When the mirror separation $d$ satisfies:

$d \lt \frac{1.22\lambda}{\theta_s}$

The first disappearance of fringes gives the angular diameter of the star: $\theta_s = 1.22\lambda/d$ .

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