JEE Physics Wave Optics Notes Set 02

Huygens' Principle and Coherence

1. Every point of a wavefront acts as a source of secondary wavelets. The new wavefront is the envelope of these wavelets after a time t. This explains reflection, refraction and diffraction, but not the origin of spectra.
2. Two sources are coherent if their phase difference stays constant with time. Only coherent sources give a steady interference pattern. Two separate lamps are incoherent, so a single source is divided into two parts (two slits) to get coherent beams.
3. Phase difference and path difference are related by \( \phi=\dfrac{2\pi}{\lambda}\Delta \). Constructive interference needs \( \Delta=n\lambda \) and destructive interference needs \( \Delta=(n+\tfrac12)\lambda \).

Superposition and Intensity

For two waves with intensities \( I_1 \) and \( I_2 \) and phase difference \( \phi \), the resultant intensity is \( I=I_1+I_2+2\sqrt{I_1I_2}\cos\phi \).

Interference only redistributes energy. The average intensity over the pattern equals \( I_1+I_2 \).

Young's Double Slit Experiment

1. Whole set-up in a medium of index \( \mu \): \( \lambda\to\lambda/\mu \), so \( \beta\to\beta/\mu \).
2. Thin transparent sheet (index \( \mu \), thickness t) over one slit: the pattern shifts by \( \dfrac{(\mu-1)tD}{d} \) towards that slit, with no change in fringe width.
3. White light gives a white central fringe with coloured fringes on either side, because each colour has its own fringe width.
4. Two wavelengths: bright fringes coincide where \( n_1\lambda_1=n_2\lambda_2 \).
5. If one slit is partly covered or has an absorbing plate, the fringe positions stay the same but the dark fringes are no longer fully dark.

Single Slit Diffraction

1. Minima occur at \( b\sin\theta=n\lambda \) (n = 1, 2, 3, ...). Secondary maxima lie roughly at \( b\sin\theta=(n+\tfrac12)\lambda \).
2. Width of the central maximum on a screen at distance D is \( \dfrac{2\lambda D}{b} \); its angular width is \( 2\lambda/b \). It is twice as wide as each secondary maximum.
3. The intensity of secondary maxima falls rapidly: roughly \( 1:\dfrac{4}{9\pi^2}:\dfrac{4}{25\pi^2} \) for the central and first two secondary maxima.
4. A narrower slit gives a wider pattern. Diffraction fringes differ from interference fringes in that their widths are unequal and their intensities fall off.

Resolving Power

1. Rayleigh's criterion for a circular aperture of diameter D: the minimum angular separation is \( \Delta\theta=\dfrac{1.22\lambda}{D} \).
2. Resolving power of a telescope is \( \dfrac{D}{1.22\lambda} \). A larger objective and shorter wavelength resolve better.

Polarisation

1. Only transverse waves can be polarised. Light from ordinary sources is unpolarised. Sound waves and ultrasonic waves cannot be polarised.
2. Malus' law: plane polarised light of intensity \( I_0 \) through an analyser at angle \( \theta \) gives \( I=I_0\cos^2\theta \).
3. Unpolarised light of intensity I through an ideal polaroid gives I/2, polarised along the pass axis.
4. Brewster's law: at the polarising angle \( i_p \), \( \tan i_p=\mu \). Then the reflected and refracted rays are perpendicular, and the reflected light is completely polarised with the vibrations perpendicular to the plane of incidence. Its intensity is less than half of the incident intensity.
5. Three polaroids with the outer two crossed and the middle one at \( \theta \) to the first: \( I=\dfrac{I_0}{8}\sin^2 2\theta \) for unpolarised incident light of intensity \( I_0 \).

Thin Films

Light reflected from the two surfaces of a thin film interferes. A film that looks bright in reflected light looks dark in transmitted light, because the transmitted and reflected intensities add up to the incident intensity.