Chapter 10 - Wave Optics

Think of dropping a stone in a still pond. Ripples spread out as circles. Light also spreads like that. A wavefront is a line joining all points that have same phase, like same point in ripple.

Three types: Spherical wavefront from a point source nearby like a bulb, plane wavefront far away like sun rays coming almost straight, cylindrical from a line source.

Huygens Principle says every point on a wavefront acts as a new source of tiny ripples called secondary wavelets. These spread in all directions at speed of light c. The new wavefront is the outer boundary touching all these tiny ripples.

Daily picture: Crowd doing a Mexican wave in stadium. Each person standing creates next wave. Envelope of all small actions becomes big wavefront.

Using Huygens we can prove laws we learnt in ray optics. For reflection, a wavefront hits a mirror. Time taken for one end to hit and form wavelet must equal time for other end. Geometry gives angle in equals angle out i = r. No magic, just equal time.

For refraction, wavefront goes from faster medium to slower. Imagine soldiers marching from road to mud. Those who enter mud first slow down, so line turns. If speed in first medium is v1 and second is v2, time matching gives n1 sin i = n2 sin r where n = c/v. If v2 less than v1, ray bends towards normal. Exactly Snell law.

Key point: Frequency never changes, only wavelength and speed change because n = c/v = lambda_vac / lambda_medium.

When two waves meet, they add up. This is superposition. If crest meets crest, bigger crest bright. Crest meets trough, they cancel dark. This is interference.

For clear pattern, sources must be coherent meaning same frequency and constant phase difference, like twins marching in step. If phase keeps changing randomly, pattern washes out.

Two conditions: path difference and phase difference link: phase diff = (2pi / lambda) * path diff. Constructive bright when path diff = n * lambda, phase diff = 2n pi. Destructive dark when path diff = (2n-1) lambda/2, phase diff = (2n-1) pi.

Coherent sources made by splitting same source: Young's double slit, Fresnel biprism, Lloyd mirror. Two independent bulbs cannot give stable interference because random phase jumps.

Young made two tiny slits S1 and S2 close together, lit by same source. They act like coherent twins. Light from both goes to screen and interferes making alternate bright and dark bands called fringes.

Geometry: slit separation d, distance to screen D, wavelength lambda. At point P path difference = d sin theta ≈ d * y / D. Bright central max at y=0 where path diff zero. Other bright at d sin theta = n lambda. Dark at d sin theta = (2n-1) lambda/2.

Fringe width beta = distance between two bright = lambda D / d. All bright fringes equally spaced. If you use white light, central white but others coloured because lambda different - VIBGYOR spreads.

Intensity pattern: all bright equally bright in ideal YDSE, but in real single slit envelope makes central brightest. Intensity I = 4 I0 cos^2(phi/2) where phi phase diff.

Diffraction is bending of waves around corners. Sound bends easily around door you hear around corner, light bends little so shadow looks sharp but actually faint light in shadow.

When light passes through narrow opening, it spreads. Single slit diffraction pattern has central bright wide and double width of others, plus dimmer secondary maxima.

Difference vs interference: Interference from two sources many equal bright fringes. Diffraction from single slit one opening spreading, central brightest wide, others dimmer. YDSE actually includes both - diffraction envelope limits interference.

Condition for bending more: gap size comparable to wavelength. Light lambda ~ 500 nm very small so needs very narrow slit ~ microns to see bending. Sound lambda ~ meters so bends around door easily.

Single slit width a. Minima dark at a sin theta = n lambda n=1,2... Maxima approx at a sin theta = (2n+1) lambda/2 but not exact. Width of central max = 2 lambda D / a. So narrower slit gives wider central spread, wider slit gives narrower but brighter.

Because of diffraction, lens cannot make point image perfectly sharp, it makes Airy disc. Two close points blur together. Rayleigh criterion says two points just resolved when central max of one falls on first minima of other. Angular separation theta = 1.22 lambda / D for circular aperture D.

Resolving power: Telescope = D / 1.22 lambda, bigger diameter sees finer stars. Microscope = 2 n sin beta / 1.22 lambda, oil immersion n larger better. Electron microscope uses tiny lambda so huge resolution.

Light is transverse wave - vibration perpendicular to direction. Ordinary light vibrates in all directions around travel. Polarized light vibrates only in one plane.

Polaroid sheet is like a fence with vertical bars - only rope shaking vertical passes, horizontal blocked. First Polaroid makes polarized light, second rotates intensity changes via Malus law I = I0 cos^2 theta where theta angle between polaroids.

Brewster angle: when reflected and refracted rays are 90 degrees, reflected is fully polarized. tan ip = n. Polarization by reflection useful for sunglasses.

Daily uses: Polaroid sunglasses block horizontal glare from water road, 3D glasses have different polarization for each eye, LCD screens use polarizers.

Only transverse waves can be polarized, so polarization proves light is transverse, not longitudinal like sound.

Wave optics is everywhere. Thin film colours soap bubble oil on water - light reflects from top and bottom surfaces interferes, 2 n t = m lambda bright, (m+1/2) lambda dark gives rainbow colours without dye.

Hologram 3D image uses interference of reference and object beam, reconstructs 3D light field.

Doppler effect for light - if source moving away wavelength stretches red shift, towards blue shift. Used to know galaxies moving away universe expanding.

Diffraction limits camera lens sharpness aperture diffraction, so large aperture lenses better but diffraction still limit. CD rainbow is diffraction grating - grooves spaced ~ micron split colours.

Difference between rainbow from ray optics and diffraction colours: rainbow from refraction+TIR in drops, diffraction colours from bending around grating.