Coherence Theory

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 $L_c$.

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 $I(\xi, \eta)$.

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.

The mutual coherence function quantifies the correlation between the optical field at two space-time points:

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

where the angle brackets denote a time average. The normalized form is the complex degree of coherence:

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

The magnitude $|\gamma_{12}(\tau)|$ satisfies $0 \leq |\gamma_{12}(\tau)| \leq 1$.

• $|\gamma| = 1$: fully coherent.
• $0 < |\gamma| < 1$: partially coherent.
• $|\gamma| = 0$: completely incoherent.

In a Michelson interferometer, the intensity at the output is:

$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\, |\gamma_{12}(\tau)| \cos(\Delta\phi)$

The fringe visibility (or contrast) is defined as:

$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} = \frac{2\sqrt{I_1 I_2}}{I_1 + I_2}\, |\gamma_{12}(\tau)|$

For equal intensities $I_1 = I_2$, the visibility equals $|\gamma_{12}(\tau)|$.

The Wiener-Khinchin theorem relates the power spectral density to the autocorrelation function:

$S(\nu) = \int_{-\infty}^{\infty} \Gamma_{11}(\tau)\, e^{2\pi i\nu\tau}\, d\tau$

$\Gamma_{11}(\tau) = \int_{-\infty}^{\infty} S(\nu)\, e^{-2\pi i\nu\tau}\, d\nu$

Thus the coherence time and spectral width satisfy the uncertainty relation:

$\tau_c \cdot \Delta\nu \sim 1$

This is a fundamental property linking temporal coherence to the source spectrum.

In Young’s experiment with partially coherent illumination, the fringe visibility is:

$V = |\gamma_{12}(0)| \cdot \left|\frac{2J_1(k a \theta)}{k a \theta}\right|$

where $a$ is the slit separation, $\theta$ is the angular source size, and $J_1$ is the Bessel function of the first kind. The first zero occurs when $k a \theta = 3.83$, giving the condition for the loss of spatial coherence fringes:

$a \approx \frac{1.22 \lambda}{\theta}$

Problem 1. A white-light source has bandwidth $\Delta\lambda \approx 300$ nm centered at $\lambda = 550$ nm. Calculate the coherence length.

Problem 2. An extended incoherent source of angular diameter $0.1$ mrad illuminates a double slit at $\lambda = 500$ nm. What is the maximum slit separation that yields visible fringes?

Problem 3. In a Michelson interferometer with equal beam intensities, the fringe visibility drops to $0.5$ at a path difference of $100\ \mu$m. Estimate the coherence length and bandwidth of the source.

Problem 4. Derive the relationship between the coherence area and the solid angle subtended by an extended source.

Solution. For a circular source of angular radius $\Delta\theta$, the coherence area is $A_c \approx \lambda^2 / (\pi (\Delta\theta)^2)$. If the source subtends a solid angle $\Omega = \pi (\Delta\theta)^2$, then $A_c \approx \lambda^2 / \Omega$. This expresses the fundamental trade-off: a source of larger angular extent produces light with smaller coherence area. $\blacksquare$

The Michelson stellar interferometer uses spatial coherence to measure the angular diameter of stars. By varying the baseline $d$ between two apertures until fringes disappear, the angular diameter $\theta$ is obtained from:

$\theta \approx 1.22 \frac{\lambda}{d_{\mathrm{max}}}$

where $d_{\mathrm{max}}$ is the maximum baseline at which fringes are visible. This technique enables angular resolution far beyond the diffraction limit of a single telescope.

Example. Betelgeuse ($\alpha$ Orionis) has angular diameter $\theta \approx 0.047$ arcseconds. At $\lambda = 500$ nm, this requires $d_{\mathrm{max}} \approx 1.22 \times 500 \times 10^{-9} / (0.047 \times \pi/648000) \approx 2.7$ m.

In quantum optics, coherence is described by the first-order correlation function $g^{(1)}(\tau)$ and second-order correlation function $g^{(2)}(\tau)$. For thermal light, $g^{(2)}(0) = 2$ (bunching). For coherent laser light, $g^{(2)}(\tau) = 1$. For non-classical light (photon antibunching), $g^{(2)}(0) < 1$.

Problem 5. Two slits separated by 0.5 mm are illuminated by a star of angular diameter 0.01 arcseconds at $\lambda = 550$ nm. Compute the fringe visibility and determine whether the fringes are observable.