Coherence Theory

16.1 Temporal Coherence

Temporal coherence describes the correlation of a wave with itself at different times. The coherence time $\tau_c$ is the time over which the phase relationship is maintained.

For a quasi-monochromatic source with bandwidth $\Delta\omega$:

$\tau_c \sim \frac{2\pi}{\Delta\omega} = \frac{1}{\Delta\nu}$

The coherence length: $l_c = c\tau_c = \lambda^2/\Delta\lambda$.

16.2 Spatial Coherence

Spatial coherence describes the correlation of a wave at different points in space at the same time. 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:

$\gamma(\Delta x) = \frac{\iint I(\xi, \eta)\,e^{-ik(\xi\Delta x)/(R)}\,d\xi\,d\eta}{\iint I(\xi, \eta)\,d\xi\,d\eta}$

Where $I(\xi, \eta)$ is the source intensity distribution and $R$ is the distance to the source.

Michelson stellar interferometer: Uses two separated apertures to measure the spatial coherence of starlight, from which the angular diameter of the star can be determined. The first fringe visibility minimum occurs at:

$d = \frac{0.61\lambda}{\alpha}$

Where $\alpha$ is the angular diameter and $d$ is the aperture separation.

16.3 Degree of Coherence

The complex degree of coherence $\gamma_{12}(\tau)$ between fields at points 1 and 2 with time delay $\tau$:

$\gamma_{12}(\tau) = \frac{\langle E_1^*(t)E_2(t+\tau)\rangle}{\sqrt{\langle|E_1|^2\rangle\langle|E_2|^2\rangle}}$

This satisfies $0 \leq |\gamma_{12}| \leq 1$. The visibility of interference fringes is:

$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} = |\gamma_{12}|$