Setup: A monochromatic source illuminates two narrow parallel slits $S_1$ and $S_2$, separated by distance $d$, on a screen at distance $D$ (with $D \gg d$). An observation screen is placed at distance $D$.

Why coherent sources are needed:

• Coherent sources have the same frequency and a constant phase difference.
• In YDSE, $S_1$ and $S_2$ are derived from the same single source (wavefront division) → automatically coherent.
• Two independent sources (e.g., two bulbs) emit random phases → no sustained pattern.

Path difference for point P at height $y$ from center: $\Delta = S_2P - S_1P = d\sin\theta \approx \frac{dy}{D}$ (small angle approximation: $\sin\theta \approx \tan\theta = y/D$; valid when $y \ll D$)

Conditions:

• Bright fringe (constructive): $\Delta = n\lambda$, $n = 0, 1, 2, \ldots$
• Dark fringe (destructive): $\Delta = (2n-1)\lambda/2$, $n = 1, 2, 3, \ldots$