Optics — Interference, Diffraction, and Geometric Optics

Interference occurs when two or more electromagnetic waves from coherent sources (same frequency, constant phase difference) overlap, producing an intensity pattern that is not simply the sum of individual intensities. At some points the waves add constructively (bright fringe), at others they cancel destructively (dark fringe).

The optical path difference $\Delta r$ between two rays determines the phase difference upon arrival. Condition for:

• Maximum (constructive): $\Delta r = m\lambda$ with $m=0,\pm1,\pm2,\ldots$ — crests coincide
• Minimum (destructive): $\Delta r = (m+\tfrac{1}{2})\lambda$ — crest meets trough

Young's experiment (two slits separated by $d$, screen at distance $L \gg d$): For small angles ($\sin\theta \approx \tan\theta = y/L$), the path difference is $\Delta r = d\sin\theta \approx d,y/L$. Maxima are located at: $$y_m = \frac{m\lambda L}{d} \quad (\text{maxima})$$ The fringes are equally spaced with fringe spacing $\Delta y = \lambda L / d$.

Thin films: when light strikes a film of thickness $t$ and refractive index $n$, part of the beam reflects off the first surface and part off the second. The path difference is $2nt$. Moreover, reflection at an interface from a lower to a higher $n$ introduces a phase shift of $\pi$ ($\lambda/2$). Consequently:
Minimum reflection (anti-reflective): $2nt = m\lambda$
Maximum reflection: $2nt = (m+\tfrac12)\lambda$

Real-world applications: anti-reflective coatings on eyeglasses and camera lenses, soap bubbles and oil slicks (interference colors), interference filters, CD/DVD readers.

Diffraction is the bending of waves around obstacles or through apertures. The simplest case is Fraunhofer diffraction (far-field, plane waves) by a single slit of width $a$.

Each point in the slit acts as a source of secondary wavelets (Huygens-Fresnel principle). Interference among all these sources produces the intensity pattern: $$I(\theta) = I_0\left(\frac{\sin\alpha}{\alpha}\right)^2,\quad \alpha = \frac{\pi a\sin\theta}{\lambda}$$ The central envelope (principal maximum) is twice as wide as the secondary maxima. Minima occur when the numerator vanishes ($\sin\alpha = 0$ with $\alpha \neq 0$): $$a\sin\theta = m\lambda \quad (m = \pm1, \pm2, \ldots)$$

Rayleigh criterion: two point sources are resolved (distinguishable) when the central maximum of one falls on the first minimum of the other: $$\theta_{min} = 1.22\frac{\lambda}{D}$$ where $D$ is the aperture diameter (e.g., the eye's pupil or a telescope objective).

Real-world applications: resolution limits of telescopes and microscopes, numerical aperture of optical fibers, X-ray diffraction by crystals, photography (aperture effects).

A diffraction grating is a periodic array of $N$ parallel slits with spacing $d$ (center-to-center distance between adjacent slits). It combines interference from $N$ sources with diffraction from each individual slit.

Principal maxima occur when the path difference between adjacent slits is an integer multiple of $\lambda$: $$d\sin\theta = m\lambda \quad (m = 0, \pm1, \pm2, \ldots)$$ As $N$ increases, the maxima become sharper and brighter, with $N-2$ negligible secondary maxima appearing between two principal maxima.

Resolving power: the ability to distinguish two close wavelengths: $$\mathcal{R} = \frac{\lambda}{\Delta\lambda} = mN$$ A grating with large $N$ can separate very close spectral lines.

Bragg's law: for X-ray diffraction by a crystal (three-dimensional grating with plane spacing $d$): $$2d\sin\theta = m\lambda$$ where $\theta$ is the angle of incidence relative to the crystal planes.

Real-world applications: optical spectroscopy (chemical analysis), fiber Bragg gratings (strain and temperature sensors), X-ray diffraction for crystal structure determination (DNA, proteins), CDs and DVDs acting as diffraction gratings.

Geometric optics describes light propagation in terms of rectilinear rays, neglecting diffraction effects. It is valid when obstacle dimensions are much larger than the wavelength.

Spherical mirrors: for a spherical mirror of radius of curvature $R$, the focal length is $f = R/2$ (concave mirror: $f>0$; convex: $f<0$). The mirror equation relates object distance $p$ and image distance $q$: $$\frac{1}{p} + \frac{1}{q} = \frac{1}{f} = \frac{2}{R}$$ The transverse magnification is $m = -q/p$ (negative: inverted image; $|m|>1$: magnified).

Thin lenses: the lensmaker's equation relates the focal length $f$ to the radii of curvature $R_1, R_2$ and the refractive index $n$ of the material: $$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ For a lens in air, $R_1>0$ if the first surface is convex, $R_2<0$ if the second is concave. The thin-lens equation is identical to the mirror equation: $1/p + 1/q = 1/f$. Converging lenses ($f>0$) focus parallel rays; diverging lenses ($f<0$) spread them out.

Real-world applications: prescription eyeglasses and contact lenses, microscopes and telescopes, cameras and video projectors, optical fibers (total internal reflection waveguides), parabolic mirrors for antennas and headlights.