Coherence Theory

11.1 Temporal Coherence

A source has finite temporal coherence if the emitted light has a finite bandwidth $\Delta\nu$. The coherence time is

$\tau_c \sim \frac{1}{\Delta\nu}$

And the coherence length is

$L_c = c\,\tau_c \sim \frac{c}{\Delta\nu} = \frac{\lambda^2}{\Delta\lambda}$

For a Michelson interferometer, fringes are visible only when the path difference is less than $L_c$.

11.2 Spatial Coherence

The spatial coherence of a source is characterised by the coherence area $A_c$. For a circular source of angular radius $\Delta\theta$:

$A_c \approx \frac{\lambda^2}{\pi(\Delta\theta)^2}$

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

Theorem 11.1 (van Cittert-Zernike). The mutual coherence function of a quasi-monochromatic incoherent source with intensity distribution $I(\xi, \eta)$ is

$\Gamma(\Delta x, \Delta y) = \iint I(\xi, \eta)\, e^{-2\pi i(\xi\,\Delta x + \eta\,\Delta y)/(\lambda z)}\, d\xi\, d\eta$

This is proportional to the Fourier transform of $I(\xi, \eta)$.

11.3 Worked Example: Coherence Length of a Sodium Lamp

Problem. A sodium lamp emits the D line at $\lambda = 589$ nm with a linewidth $\Delta\lambda \approx 0.6$ nm. Find the coherence length and the maximum path difference for which fringes are visible in a Michelson interferometer.