Wave optics treats light as a wave and explains phenomena that cannot be accounted for by ray optics — particularly interference, diffraction and polarisation. The wave model was strongly supported by Young's double-slit experiment (1801) and later by Maxwell's theory.
Huygens' Principle
Every point on a wavefront acts as a secondary source of new spherical waves (secondary wavelets). The new wavefront is the forward tangent envelope of all secondary wavelets.
- Wavefront types:
- Spherical (from a point source)
- Cylindrical (from a line source)
- Plane (at very large distances from a source)
Huygens' principle correctly explains reflection and refraction, including the laws of reflection and Snell's law, using the geometry of wavefronts.
Superposition and Interference
When two coherent waves overlap, interference occurs.
Constructive interference: path difference = n x lambda (intensity maximum)
Destructive interference: path difference = (2n-1) x lambda/2 (intensity minimum)
- Young's Double Slit Experiment (YDSE):
- Fringe width: beta = lambda D / d, where D = distance to screen, d = slit separation.
- Position of n-th bright fringe: yn = n lambda D / d
- Position of n-th dark fringe: yn = (2n-1) lambda D / (2d)
- 1.Conditions for sustained interference:
- 2.Sources must be coherent (constant phase difference)
- 3.Sources must have same frequency
- 4.Amplitudes must be comparable
Intensity in interference:
If I1 = I2 = I0: I = 4 I0 cos2(phi/2), where phi is phase difference.
Diffraction
Diffraction is the bending of waves around corners or through slits, explained by Huygens' principle.
Single slit diffraction: Width of central maximum = 2 lambda D / a, where a = slit width.
Condition for minima: a sin(theta) = n lambda.
- Limit of Resolution:
- Rayleigh's criterion: two points are just resolved when the central maximum of one falls on the first minimum of the other.
- Resolving power of telescope: R = a / (1.22 lambda), where a = aperture diameter.
- Resolving power of microscope: R = 2 n sin(theta) / lambda (Abbe's criterion).
Polarisation
Ordinary light vibrates in all planes perpendicular to propagation — it is unpolarised. When restricted to one plane, light becomes plane polarised.
Polaroid: filters out all but one plane of vibration.
Malus's Law: I = I0 cos2(theta), where theta is angle between polariser and analyser.
Brewster's Angle (thetaB): tan(thetaB) = n. At this angle, reflected light is completely plane-polarised.
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In YDSE, D = 1.5 m, d = 0.3 mm, lambda = 600 nm. Find fringe width.
beta = lambda D / d = (600 x 10-9 x 1.5) / (0.3 x 10-3) = 900 x 10-9 / 3 x 10-4 = 3 x 10-3 m = 3 mm.
Find the minimum path difference for destructive interference in a YDSE with lambda = 500 nm.
Minimum path difference for dark fringe = lambda/2 = 250 nm.
Light of 600 nm falls on a single slit of width 1.2 mm. Find angular position of first minimum.
sin(theta) = lambda/a = 600 x 10-9 / 1.2 x 10-3 = 5 x 10-4. theta = sin-1(5 x 10-4) approx 0.029 degrees or 0.0005 rad.
Two polaroids are kept with transmission axes at 60 degrees. Intensity of unpolarised incident light is I0. Find transmitted intensity.
After first polaroid: I0/2. After second polaroid (Malus's law): I = (I0/2) cos2(60) = (I0/2)(1/4) = I0/8.
Calculate Brewster's angle for glass with n = 1.732.
tan(thetaB) = n = 1.732. thetaB = tan-1(1.732) = 60 degrees.
In YDSE, fifth bright fringe is at 2.5 mm from the centre. If D = 1 m and d = 1 mm, find wavelength.
y5 = 5 lambda D / d. lambda = y5 d / (5D) = (2.5 x 10-3 x 10-3) / (5 x 1) = 2.5 x 10-6 / 5 = 5 x 10-7 m = 500 nm.
In YDSE, d is doubled and D is halved. How does the fringe width change?
beta = lambda D/d. New beta' = lambda (D/2) / (2d) = lambda D / (4d) = beta/4. Fringe width reduces to 1/4.
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Common mistakes
- Confusing fringe width formula: beta = lambda D/d (not d/lambda or D/d alone).
- In Malus's law, forgetting to square the cosine: I = I0 cos2(theta), not I0 cos(theta).
- Thinking interference requires two identical sources — they must be coherent, not necessarily identical in all properties.
Summary
Wave optics explains light's wave behaviour. YDSE demonstrates interference with fringe width beta = lambda D/d. Diffraction occurs at edges and single slits. Polarisation confirms light is a transverse wave. Malus's law and Brewster's angle govern polarisation phenomena.