Formula sheet: Wave Optics

Wave Optics16 min readLesson 1 of 6

1.1 Huygens principle and the wavefront A wavefront is the surface on which every point has the same phase. A point source sends out spherical wavefronts and a distant source sends out plane wavefronts. Huygens built a construction on these surfaces: each point of a wavefront acts as a source of secondary wavelets, and the new wavefront is the envelope of those wavelets after a short time. This geometric construction explains reflection and refraction without new physics. The angle of incidence equals the angle of reflection because the two triangles in the construction are congruent. Refraction follows from the change in wave speed between the media, and the ratio of the sines equals the ratio of the speeds.

\frac{sin i}{sin r} = \frac{v _{1}}{v _{2}} = \frac{n _{2}}{n _{1}}

Snell's law derived from the wave picture

1.2 Interference and the double slit Two sources are coherent when they keep a constant phase difference. Light from a single source split into two paths is coherent, which is why Young's experiment uses one slit to illuminate two slits. At a point on the screen the two waves superpose, and the resultant intensity depends on the path difference. Constructive interference occurs where the path difference is a whole number of wavelengths, and destructive interference where it is an odd number of half wavelengths. The fringe width on the screen depends on the wavelength, the slit separation and the screen distance.

Δ = d sin θ, bright: Δ = mλ, dark: Δ = (2 m - 1) \frac{λ}{2}

Path difference conditions for bright and dark fringes

β = \frac{λ D}{d}

Fringe width in Young's double slit experiment

I = I_{1} + I_{2} + 2 I_{1} I_{2} cos ϕ

Resultant intensity for a phase difference

1.3 Diffraction and polarisation Diffraction is the bending of light around obstacles and through narrow openings. A single slit gives a central bright maximum with weaker secondary maxima on both sides. The condition for the minima and maxima involves the slit width and the wavelength. Polarisation shows that light is a transverse wave. A polariser transmits only the component of the electric field along its axis, and two crossed polarisers block the light. Malus's law gives the transmitted intensity when the analyser is rotated through an angle.

a sin θ = nλ (minima), a sin θ = (m + \frac{1}{2}) λ (secondary maxima)

Single slit diffraction conditions

I = I_{0} cos^{2} θ

Malus's law for polarised light

Exam

In every interference numerical, write the fringe width formula first and identify what changes. Between air and a liquid the wavelength changes, so the fringe width changes in the same ratio.

in hinglish

Formula sheet ka core hai Huygens principle aur interference. Wavefront par same phase ke points hote hain, aur secondary wavelets se naya wavefront banta hai. Path difference d sin theta hota hai; bright fringe ke liye path difference m lambda aur dark ke liye (2m-1) lambda/2. Fringe width beta = lambda D / d hoti hai, jisme D screen distance aur d slit separation hai. Diffraction single slit se hota hai: minima a sin theta = n lambda par. Polarisation batata hai light transverse wave hai, aur Malus law I = I0 cos^2 theta deta hai. Numerical me pehle formula likho, phir dekho kaunsi quantity badal rahi hai; liquid me wavelength badalne se fringe width bhi badalti hai. Wavefront aur ray me farq yaad rakho: ray wavefront ke perpendicular hoti hai, isliye direction ka answer ray se dete hain.