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.

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:

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

The mutual coherence function quantifies the correlation between the wave field at two spacetime points:

$\Gamma_{12}(\tau) = \langle E^*(r_1, t) E(r_2, t + \tau) \rangle$

The complex degree of coherence is the normalised quantity:

$\gamma_{12}(\tau) = \frac{\Gamma_{12}(\tau)}{\sqrt{\Gamma_{11}(0) \Gamma_{22}(0)}}$

For quasi-monochromatic light, the visibility of interference fringes equals $|\gamma_{12}(\tau)|$. Fringes are visible when $0 < |\gamma| \leq 1$, with $|\gamma| = 1$ for perfectly coherent light and $|\gamma| = 0$ for incoherent light.

The Wiener-Khinchin theorem relates the power spectral density $S(\omega)$ of a stationary random process to the autocorrelation function via Fourier transform:

$\Gamma_{11}(\tau) = \int_{-\infty}^{\infty} S(\omega) e^{-i\omega\tau} d\omega$

The coherence time is inversely related to the spectral width: $\tau_c = \int_{-\infty}^{\infty} |\gamma_{11}(\tau)|^2 d\tau$. For a Lorentzian line shape, $\tau_c = 1/(\pi\Delta\nu)$.

Partially coherent light is described by the cross-spectral density function $W(r_1, r_2, \omega)$, which is the Fourier transform of the mutual coherence function:

$W(r_1, r_2, \omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \Gamma(r_1, r_2, \tau) e^{i\omega\tau} d\tau$

The Wolf equations govern the propagation of the cross-spectral density, generalising the Helmholtz equation to partially coherent fields.

Second-order coherence measures intensity correlations:

$g^{(2)}(\tau) = \frac{\langle I(t) I(t+\tau) \rangle}{\langle I(t) \rangle^2}$

• Thermal light (chaotic): $g^{(2)}(0) = 2$, exhibiting photon bunching.
• Coherent light (laser): $g^{(2)}(\tau) = 1$ for all $\tau$.
• Antibunched light (single-photon source): $g^{(2)}(0) < 1$.

The Hanbury Brown-Twiss (HBT) interferometer measures $g^{(2)}(\tau)$ and was originally used to measure the angular diameter of stars. The HBT effect demonstrated that intensity correlations contain information about source size even when first-order coherence is absent.

Problem. A source emits two equal-intensity spectral lines at $\lambda_1$ and $\lambda_2$ with $\Delta\lambda = |\lambda_2 - \lambda_1| \ll \lambda$. Find the fringe visibility in a Michelson interferometer as a function of path difference.

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: $\theta_s = 1.22\lambda/d$.

Problem. The sun has an angular diameter of approximately $0.53^\circ$ as seen from Earth. What is the transverse coherence length of sunlight at $\lambda = 550$ nm?