Wave Optics — Notes

• A wavefront is the locus of points oscillating in the same phase; rays are normal to wavefronts. Point source → spherical; line source → cylindrical; distant source → plane.
• Huygens’ principle: every point of a wavefront is a source of secondary wavelets spreading at the wave speed; the new wavefront after time $t$ is the forward envelope of wavelets of radius $vt$.

Reflection: plane wavefront AB meets the mirror at A. In time $t$, B moves to C ($BC = vt$) and the wavelet from A grows to $AE = vt$. The reflected wavefront CE is tangent to it. Right triangles ABC and CEA share hypotenuse AC with $BC = AE$, so they are congruent and $i = r$.

Refraction ($v_2 < v_1$): in time $t$, $BC = v_1 t$ in medium 1 and $AE = v_2 t$ in medium 2. Then $\sin i = BC/AC$, $\sin r = AE/AC$, so $\dfrac{\sin i}{\sin r} = \dfrac{v_1}{v_2} = \dfrac{n_2}{n_1} = n_{21}\quad\Rightarrow\quad n_1\sin i = n_2\sin r.$ On refraction the frequency stays the same; speed and wavelength both fall by the factor $n$: $\lambda' = \lambda/n$.

• Superposition: the resultant displacement is the sum of the individual displacements.
• Coherent sources: same frequency and a constant phase difference. Ordinary independent sources change phase randomly (each atom emits for about $10^{-9}$ s), so no steady pattern forms. Coherent sources are derived from one source (dividing a wavefront, as in Young’s slits) or are lasers.
• Phase difference $\varphi = \dfrac{2\pi}{\lambda}\Delta$ for path difference $\Delta$ (sources in phase).
• Constructive: $\Delta = n\lambda$, $\varphi = 2n\pi$. Destructive: $\Delta = (n + \tfrac12)\lambda$, $\varphi = (2n + 1)\pi$.
• Intensity: $I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\varphi$; for equal $I_0$, $I = 4I_0\cos^2(\varphi/2)$. $I_\text{max}/I_\text{min} = (a_1 + a_2)^2/(a_1 - a_2)^2$. Incoherent: $I = I_1 + I_2$. Energy is redistributed, not destroyed.
• Conditions for a clear, sustained pattern: coherent sources; nearly equal amplitudes; narrow sources close together; monochromatic light.

Slits S₁, S₂ (separation $d$) lit from one narrow slit act as coherent sources; screen at distance $D \gg d$. Path difference at height $y$: $\Delta = yd/D$.

• Bright: $y_n = \dfrac{n\lambda D}{d}$; dark: $y = \left(n + \tfrac12\right)\dfrac{\lambda D}{d}$ ($n = 0, 1, 2, \dots$).
• Fringe width $\beta = \dfrac{\lambda D}{d}$ (bright and dark fringes equally wide); angular fringe width $\lambda/d$.
• $\beta$ increases with $\lambda$ and $D$, decreases with $d$; in a liquid $\beta' = \beta/n$.
• White light: central fringe white, a few coloured fringes (blue nearer the centre), then uniform illumination.

• Diffraction: bending of light round edges into the geometrical shadow; noticeable when the slit width $a$ is comparable to $\lambda$.
• Wavelets from all points of the slit superpose. At $\theta = 0$ all are in phase: bright central maximum.
• Minima: $a\sin\theta = n\lambda$ ($n = \pm1, \pm2, \dots$): split the slit into $2n$ parts whose wavelets cancel in pairs. Weak secondary maxima lie roughly midway.
• Central maximum: angular width $2\lambda/a$, linear width $2\lambda D/a$; twice as wide as the secondary maxima and much brighter.
• Narrower slit or longer wavelength → wider pattern.
• Interference (two sources, equal fringes of equal brightness) vs diffraction (one slit, wide bright centre, fading side maxima).