Interference & Coherence
Interference & Coherence
When coherent waves overlap, amplitudes add and intensity depends on the phase difference $\delta$. Constructive interference ($\delta=2m\pi$) produces bright fringes; destructive ($\delta=(2m+1)\pi$) produces dark fringes. Young\'s double-slit experiment (1801) demonstrated the wave nature of light.
Coherence quantifies the ability to interfere: temporal coherence length $l_c=\lambda^2/\Delta\lambda$ governs path-difference range; spatial coherence governs the lateral source extent over which fringes remain visible.
Fringe Spacing (Young\'s Experiment)
Slits separated by $d$ at distance $L$: $$\Delta y=\frac{\lambda L}{d}.$$ Phase difference $\delta=2\pi d\sin\theta/\lambda$. Bright at integer $m$, dark at half-integer.
Fringe Visibility
$$\mathcal{V}=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}=|\gamma_{12}(\tau)|,$$ where $\gamma_{12}$ is the complex degree of coherence.
Example 1
Double slit: $d=0.5\,\text{mm}$, $\lambda=550\,\text{nm}$, $L=1\,\text{m}$. Find fringe spacing.
Solution: $\Delta y=550\times10^{-9}\times1/(5\times10^{-4})=1.1\,\text{mm}$.
Example 2
Na lamp: $\lambda=589\,\text{nm}$, $\Delta\lambda=0.6\,\text{nm}$. Find coherence length.
Solution: $l_c=\lambda^2/\Delta\lambda=589^2/0.6\approx578\,\mu\text{m}$.
Practice
- Distinguish temporal and spatial coherence.
- If $d$ doubles, what happens to fringe spacing?
- Why do white-light fringes show color at the edges?
- Define the complex degree of coherence $\gamma_{12}(\tau)$.
Show Answer Key
1. Temporal coherence: correlation of a wave with itself at different times (related to spectral bandwidth, $\tau_c \sim 1/\Delta\nu$). Spatial coherence: correlation between different points on the wavefront at the same time (related to source angular size, $l_c \sim \lambda/\Delta\theta$).
2. Young's fringes: $\Delta y = \lambda L/d$. If $d$ doubles, fringe spacing halves (fringes get closer together).
3. White light has short coherence length ($\tau_c \sim 1/\Delta\nu$, only a few fringes visible). Away from zero path difference, different wavelengths constructively interfere at different positions, creating colored bands at the edges of the pattern. Central fringe is white (all wavelengths in phase at zero OPD).
4. $\gamma_{12}(\tau) = \langle E_1^*(t)E_2(t+\tau)\rangle / \sqrt{\langle|E_1|^2\rangle\langle|E_2|^2\rangle}$. $|\gamma_{12}| = 1$: fully coherent. $|\gamma_{12}| = 0$: incoherent. $0 < |\gamma_{12}| < 1$: partially coherent. Fringe visibility $\mathcal{V} = |\gamma_{12}|$ in a two-beam interference experiment (van Cittert-Zernike theorem relates spatial coherence to source intensity distribution).