Summary

Huygens' principle -- every point on a wavefront is a source of secondary wavelets, and the new wavefront is their forward common tangent -- lets the laws of reflection ($i=r$) and refraction ($\sin i/\sin r=v_1/v_2$, Snell's law) be derived geometrically rather than assumed. Two sources are coherent when they share a constant phase relationship, which Young's double slit achieves by division of a single wavefront across two closely spaced slits $S_1,S_2$ (separation $d$). At a screen distance $D\gg d$, the path difference at a point $y$ from the centre is $\Delta\approx yd/D$; bright fringes occur at $\Delta=n\lambda$ and dark fringes at $\Delta=(n+\tfrac12)\lambda$, giving equally spaced fringes of width $\beta=\lambda D/d$.

Fraunhofer diffraction at a single slit of width $a$ instead gives minima at $a\sin\theta=n\lambda$ ($n\ne0$), producing a central maximum of total width $2\lambda D/a$ -- twice as wide and far brighter than the fainter secondary maxima flanking it, unlike the equally-bright fringes of double-slit interference. Because diffraction blurs every point image, an optical instrument's resolving power is limited: a microscope resolves down to $d_{\min}=1.22\lambda/(2n\sin\beta)$, and an astronomical telescope down to an angle $d\theta=1.22\lambda/D_{ap}$, both improving with shorter wavelength and, for the telescope, a larger aperture. …