Interference is the phenomenon that occurs when two or more coherent waves overlap in space, resulting in a new wave pattern that is the superposition of the individual waves.
For two waves of equal amplitude $A$, the resultant intensity $I$ at a point is given by
$$I = 4I_0\cos^2\!\left(\frac{\Delta\phi}{2}\right)$$where $I_0$ is the intensity of each individual wave and $\Delta\phi$ is the phase difference between them.
The type of interference depends on the path difference $\Delta r$ between the waves:
Coherence describes the ability of two waves to produce a stable interference pattern. It quantifies how well the phase relationship between the waves is maintained over time and space.
The coherence time $\tau_c$ is the time over which the phase remains predictable and is given by
$$\tau_c \approx \frac{1}{\Deltau}$$where $\Deltau$ is the spectral width of the source. The corresponding coherence length $L_c$ is
$$L_c = c\,\tau_c = \frac{c}{\Deltau}$$For a perfectly monochromatic source ($\Deltau \to 0$), $\tau_c$ and $L_c$ become infinite, allowing interference over arbitrarily large path differences.
| Aspect | Temporal Coherence | Spatial Coherence |
|---|---|---|
| Definition | Correlation of phase at different times for the same point | Correlation of phase at different points across a wavefront |
| Key Parameter | Coherence time $\tau_c$ (or length $L_c$) | Coherence area (or source size) |
| Effect on Interference | Limits maximum path difference that still yields visible fringes | Determines angular spread over which fringes remain sharp |
| Typical Sources | Lasers (large $\tau_c$), narrow‑band LEDs | Point sources, small apertures |
In experiments such as the double‑slit or Michelson interferometer, achieving sufficient coherence is essential. If the path difference exceeds $L_c$, the interference fringes wash out because the waves are no longer phase‑related.