Coherence | Physics - Wyatt's Notes

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?

flowchart TD
    A[8_Coherence] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Coherence is the predictability of a wave’s phase over time and space. A laser maintains phase relationships for microseconds, allowing interference over meters. A light bulb’s atoms emit independently, so phases randomize in femtoseconds, limiting coherence to micrometers. The coherence length is the maximum path difference where interference fringes remain visible, like the distance over which two runners stay in step. Spatial coherence depends on source size: a point source is perfectly coherent across space, while an extended source like the sun has limited spatial coherence. The van Cittert-Zernike theorem connects source angular size to coherence area.

Mistake 1: Assuming $g^{(2)}(0) = 2$ means thermal light has “twice the coherence” The second-order correlation function $g^{(2)}(0) = 2$ for thermal light reflects photon bunching — photons from a chaotic source tend to arrive in pairs. This is an intensity correlation effect, not a measure of coherence in the traditional sense. Students sometimes interpret $g^{(2)}(0) > 1$ as enhanced coherence, when it actually indicates the opposite: the light is more chaotic.

Mistake 2: Confusing the coherence time with the correlation time of the source The coherence time $\tau_c \approx 1/\Delta\nu$ characterises how long the field maintains a definite phase, while the source correlation time relates to the atomic emission process. For a laser, $\tau_c$ can be microseconds or longer despite atomic transitions occurring on nanosecond timescales. Students often assume these timescales must be comparable.

• Diffraction: The formation of diffraction fringes depends on the coherence of the source; partially coherent light produces reduced-visibility fringes.
• Fourier Optics: Uses the Wiener-Khinchin theorem to relate the power spectrum of the source to the mutual coherence function via Fourier transforms.
• Polarization: The coherence matrix extends to vector fields, connecting coherence theory to the polarisation state of partially polarised light.

Mistake 3: Assuming that spatial coherence requires a point source The van Cittert-Zernike theorem shows that even an extended incoherent source produces spatially coherent light over a finite area $A_c \approx \lambda^2/\Omega$. The coherence area increases as the source becomes more compact. Students sometimes think only lasers or point sources produce spatially coherent light, when in fact any source produces some degree of spatial coherence over sufficiently small transverse distances.

Coherence is the predictability of a wave’s phase over time and space. A laser maintains phase relationships for microseconds, allowing interference over meters. A light bulb’s atoms emit independently, so phases randomize in femtoseconds, limiting coherence to micrometers. The coherence length is the maximum path difference where interference fringes remain visible, like the distance over which two runners stay in step. Spatial coherence depends on source size: a point source is perfectly coherent across space, while an extended source like the sun has limited spatial coherence. The van Cittert-Zernike theorem connects source angular size to coherence area.

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