Wave Optics: Interference and Diffraction

Shine light through two close slits and the screen shows stripes — places where light plus light equals darkness. Only waves can cancel. This experiment settled a century-long debate about light’s nature, and its logic (path difference → phase difference → interference) recurs from soap films to gravitational-wave detectors.

When light is treated as a wave, interference and diffraction emerge.

Young's double-slit — slits separated by $d$:

• Bright fringes: $d\sin\theta = m\lambda$
• Dark fringes: $d\sin\theta = (m+\tfrac{1}{2})\lambda$

Single-slit diffraction — slit width $a$:

$$a\sin\theta = m\lambda \;(\text{dark fringes})$$

Diffraction grating — $N$ slits give sharper maxima; resolution $R = mN$.

Thin-film interference

Phase shifts come from path difference $\Delta = 2nt$ and reflection at a higher-$n$ boundary (extra $\lambda/2$).

• One reflection shift: constructive at $2nt = (m+\tfrac{1}{2})\lambda$
• No or two shifts: constructive at $2nt = m\lambda$

Physics link: Telescope resolution is limited by diffraction: $\theta_{\min} = 1.22\lambda/D$.

Common pitfall: Interference does not create or destroy energy — dark fringes are matched by doubly-bright ones; energy is redistributed, never lost. If your fringe accounting seems to delete energy, the analysis (not physics) has a bug.