Coherence and Superposition

Interference patterns appear when two (or more) waves overlap and their amplitudes add. To get a stable pattern the sources must be coherent.

Concept

Principle of superposition. When two waves $y_1$ and $y_2$ pass through the same point, the resultant displacement is

$y = y_1 + y_2$

For sinusoidal waves of the same frequency,

$y_1 = a_1 \sin(\omega t),\quad y_2 = a_2 \sin(\omega t + \phi)$

the resultant has amplitude

$A = \sqrt{a_1^2 + a_2^2 + 2 a_1 a_2 \cos\phi}$

and the intensity

$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$

Constructive interference ( $\phi = 2n\pi$ ): $I_{\max} = I_1 + I_2 + 2\sqrt{I_1 I_2}$ . For equal amplitudes, $I_{\max} = 4 I_0$ .

Destructive interference ( $\phi = (2n+1)\pi$ ): $I_{\min} = I_1 + I_2 - 2\sqrt{I_1 I_2}$ . For equal amplitudes, $I_{\min} = 0$ .

Coherent Sources

For an observable interference pattern, two sources must:

Have the same frequency (and hence the same wavelength in vacuum).
Have a constant phase difference (no random jumps).
Have comparable amplitudes (for maximum contrast).

Two independent ordinary lamps do not produce sustained interference because their atomic emissions are uncorrelated; the phase relation changes faster than the eye can register, washing the pattern out.

Ways to obtain coherent sources from a single source:

Slits illuminated by a single primary source (Young's experiment).
Beam splitters (Michelson interferometer).
Reflection off a thin film (Fizeau, Newton's rings).

Path Difference and Phase Difference

If two waves from coherent sources reach a point along paths differing by $\Delta x$ , the phase difference is

$\phi = \frac{2\pi}{\lambda}\Delta x$

Constructive: $\Delta x = n\lambda$ (integer wavelengths).
Destructive: $\Delta x = (n + \tfrac12)\lambda$ (odd half-wavelengths).

Worked Example

Two coherent sources emit in phase with equal amplitudes. At a point, the path difference is $0.4\lambda$ . Find the intensity ratio $I/I_{\max}$ .

$\phi = 2\pi \cdot 0.4 = 0.8\pi$

$I = 4 I_0 \cos^2(\phi/2) = 4 I_0 \cos^2(0.4\pi)$

$\cos(0.4\pi) \approx 0.309 \Rightarrow I \approx 4 I_0 \cdot 0.0955 = 0.382 I_0$

So $I/I_{\max} = 0.382 / 4 \approx 0.0955$ or about 10%.

Common Confusions

"Coherent" doesn't mean "in phase" — it means the phase relationship is constant.
$I_{\max} = 4 I_0$ for equal sources, not $2 I_0$ . (Amplitudes add, then square.)
Phase difference of $\pi$ gives destructive interference, not $\pi/2$ .

Key Takeaways

Superposition: amplitudes add, intensities depend on phase.
For coherence, sources must share frequency and have a fixed phase relation.
$\phi = (2\pi/\lambda)\Delta x$ links path difference and phase.
Equal coherent sources: $I = 4 I_0 \cos^2(\phi/2)$ .