Class 12 Physics · Chapter 10 NotesWave Optics

Early models of light included the corpuscular model, which treated light as a stream of particles and could explain reflection and refraction. However, it predicted that light would travel faster in a denser medium, which was later disproved. Christiaan Huygens proposed the wave theory in 1678, suggesting that light propagates as waves. Thomas Young's interference experiment in 1801 provided strong evidence for the wave nature of light. Maxwell's electromagnetic theory later showed that light is an electromagnetic wave that can travel through vacuum. Wave optics deals with phenomena such as interference, diffraction and polarisation, which cannot be explained by geometrical optics.

• Corpuscular modellight as particles, explained reflection and refraction but predicted wrong speed in denser media.
• Wave theorylight as waves, explained interference and diffraction.
• Maxwelllight is an electromagnetic wave, no medium needed.

A wavefront is a surface of constant phase. For a point source, the wavefront is spherical; at large distances, a small portion can be considered a plane wave. Huygens' principle states that every point on a wavefront acts as a source of secondary wavelets, which spread out with the speed of the wave. The new wavefront at a later time is the common tangent (envelope) to these secondary wavelets. This principle explains the propagation of waves and can be used to derive the laws of reflection and refraction. The absence of a backwave is explained by assuming that the amplitude of secondary wavelets is maximum in the forward direction and zero in the backward direction.

• Wavefrontlocus of points oscillating in phase.
• Spherical wavefront from a point source; plane wavefront at large distances.
• Huygens' principleeach point on a wavefront is a source of secondary wavelets.
• New wavefront is the forward envelope of secondary wavelets.

Using Huygens' principle, the laws of refraction and reflection can be derived. For refraction, if a plane wave is incident on a boundary between two media, the refracted wavefront is constructed by drawing secondary wavelets from points on the interface. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the speeds of light in the two media, which is also equal to the inverse ratio of their refractive indices. This gives Snell's law: n₁ sin i = n₂ sin r. When light goes from a denser to a rarer medium, there is a critical angle beyond which total internal reflection occurs. For reflection, the angle of incidence equals the angle of reflection. Huygens' principle also explains how prisms, lenses and mirrors affect wavefronts.

• Snell's lawn₁ sin i = n₂ sin r.
• Speed of light is less in a denser medium; wavelength decreases but frequency remains constant.
• Critical anglesin i_c = n₂ / n₁ for n₂ < n₁.
• Law of reflectionangle of incidence = angle of reflection.

When two waves superpose, the resultant displacement is the vector sum of the individual displacements. If the sources maintain a constant phase difference, they are coherent, and a stable interference pattern is observed. For two coherent sources of equal amplitude, the intensity at a point depends on the phase difference. Constructive interference occurs when the path difference is an integer multiple of the wavelength, giving maximum intensity. Destructive interference occurs when the path difference is an odd multiple of half the wavelength, giving zero intensity. If the phase difference changes rapidly with time, the sources are incoherent, and the average intensity is simply the sum of the individual intensities.

• Coherent sourcesconstant phase difference, stable interference pattern.
• Constructivepath difference = nλ, intensity = 4I₀.
• Destructivepath difference = (n + ½)λ, intensity = 0.
• Incoherentintensities add, average intensity = 2I₀.

Young's double-slit experiment demonstrates interference of light. Light from a single source illuminates two closely spaced pinholes, which act as coherent sources. The waves from these sources overlap on a screen, producing a pattern of bright and dark fringes. The position of the nth bright fringe from the central maximum is given by x = nλD/d, where D is the distance from the slits to the screen and d is the slit separation. Dark fringes occur at x = (n + ½)λD/d. The fringe width, which is the distance between consecutive bright or dark fringes, is β = λD/d. The experiment provides a direct measurement of the wavelength of light.

• Bright fringesx = nλD/d, n = 0, ±1, ±2, ...
• Dark fringesx = (n + ½)λD/d, n = 0, ±1, ±2, ...
• Fringe widthβ = λD/d.
• All fringes are equally spaced.

Diffraction is the bending of waves around obstacles or through narrow openings. When a parallel beam of light falls on a single narrow slit of width a, the pattern on a screen consists of a central bright maximum and alternating dark and bright fringes on either side. The minima (zero intensity) occur at angles θ = nλ/a, where n = ±1, ±2, ... The central maximum is at θ = 0. The intensity of secondary maxima decreases with increasing order. Diffraction limits the resolving power of optical instruments such as telescopes and microscopes. The phenomenon is consistent with the conservation of energy, as light energy is redistributed from dark to bright regions.

• Single-slit diffractionminima at θ = nλ/a, n = ±1, ±2, ...
• Central maximum is twice as wide as secondary maxima.
• Diffraction limits resolution of optical instruments.
• Energy is conserved; redistribution occurs.

Polarisation is a phenomenon that demonstrates the transverse nature of light waves. Natural light is unpolarised, meaning the electric field vector vibrates in all directions perpendicular to the direction of propagation. A polaroid sheet transmits only the component of the electric field parallel to its pass-axis, producing linearly polarised light. When unpolarised light passes through a polaroid, its intensity is reduced to half. If a second polaroid (analyser) is placed in the path, the transmitted intensity varies as I = I₀ cos²θ, where θ is the angle between the pass-axes of the two polaroids. This is Malus' law. Polaroids are used in sunglasses, cameras and 3D movie glasses.

• Unpolarised lightelectric field vibrates randomly in all transverse directions.
• Polaroid transmits only one component; intensity becomes half.
• Malus' lawI = I₀ cos²θ.
• Polarisation confirms light waves are transverse.

The resolving power of an optical instrument is its ability to distinguish between two closely spaced objects. Due to diffraction, even a perfect lens forms a blurred image of a point source, known as the Airy pattern. The limit of resolution is determined by the wavelength of light and the aperture of the instrument. For a circular aperture of diameter D, the minimum angular separation that can be resolved is approximately θ = 1.22 λ/D. This means that shorter wavelengths and larger apertures provide better resolution. The resolving power of telescopes and microscopes is fundamentally limited by diffraction, not by imperfections in the lenses.

• Resolving powerability to distinguish two close objects.
• Diffraction sets the limitθ = 1.22 λ/D for circular aperture.
• Larger aperture and shorter wavelength give better resolution.
• Explains why electron microscopes have higher resolution than optical microscopes.