Coherence Theory

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

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.

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:

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

• Confusing temporal and spatial coherence. Temporal coherence depends on the source bandwidth; spatial coherence depends on the source size. A laser has high temporal coherence (narrow linewidth) but can have low spatial coherence if operated in multi-mode.
• Assuming a point source gives infinite coherence. A true point source gives perfect spatial coherence, but any real source has finite size. The van Cittert—Zernike theorem quantifies the trade-off.
• Forgetting that fringe visibility depends on both polarisation and coherence. Two beams with orthogonal polarisations produce no interference even if spatially and temporally coherent.
• Thinking the coherence length is the maximum path difference for fringes. While related, the visibility decreases gradually; the coherence length is typically defined as the path difference where visibility drops to $1/e$ or $1/2$.

• Holography: Requires high temporal and spatial coherence to record interference patterns between object and reference beams. Lasers are essential because of their long coherence length.
• Optical coherence tomography (OCT): Uses low-coherence interferometry to image subsurface tissue structure. The short coherence length of broadband light provides micron-scale axial resolution.
• LIDAR: Coherent LIDAR uses temporal coherence for Doppler velocity measurement of remote targets. The coherence length determines the maximum range.
• Radio astronomy: Very Long Baseline Interferometry (VLBI) uses spatial coherence across telescope arrays separated by thousands of kilometres to achieve angular resolution of micro-arcseconds.

Problem 1. A Michelson interferometer uses a sodium lamp ($\lambda = 589$ nm, $\Delta\lambda = 0.6$ nm). What is the maximum path difference for visible fringes?

Solution. Coherence length $l_c = \lambda^2 / \Delta\lambda = (589)^2 / 0.6 \approx 578,000$ nm $\approx 0.58$ mm. Fringes are visible for path differences up to roughly $l_c$, so the maximum path difference is about 0.58 mm. Beyond this, the temporal coherence is insufficient and fringe visibility drops to zero. $\blacksquare$

Problem 2. Two slits are separated by $d = 0.5$ mm and illuminated by a thermal source of width $w = 0.2$ mm at distance $D = 50$ cm ($\lambda = 550$ nm). Find the fringe visibility.

Solution. Using van Cittert—Zernike: $|\gamma| = |\sin(\pi w d/(\lambda D)) / (\pi w d/(\lambda D))|$. $\pi w d/(\lambda D) = \pi \times 2\times10^{-4} \times 5\times10^{-4} / (5.5\times10^{-7} \times 0.5)$ $= \pi \times 10^{-7} / (2.75\times10^{-7}) = \pi \times 0.364 = 1.143$ rad. $|\gamma| = |\sin(1.143)/1.143| = 0.81/1.143 = 0.709$. Fringe visibility $V = 0.71$ (71%). $\blacksquare$