Chapter 10: Wave Optics | Summary

Using Huygens principle, Snell's law of refraction (n1 sin i = n2 sin r) is derived geometrically. When light passes from medium 1 to medium 2, the ratio of the sines of the angles equals the ratio of the wave speeds (v1/v2). In a denser medium, the wavelength and speed decrease, but the frequency remains unchanged. Total internal reflection occurs when light in a denser medium strikes the interface at an angle greater than the critical angle ic, defined by sin ic = n2/n1.

For reflection, Huygens construction proves the law of reflection (angle of incidence equals angle of reflection) using congruent triangles. The behaviour of wavefronts through prisms, convex and concave lenses, and concave mirrors can all be explained: a prism tilts the wavefront, a convex lens converts a plane wavefront into a converging spherical one, and a concave mirror similarly converges reflected wavefronts to a focal point. The total travel time from object to image is the same along any ray.

The superposition principle states that the resultant displacement at a point is the vector sum of displacements due to individual waves. Two sources are coherent if the phase difference between them does not change with time. For coherent sources vibrating in phase, constructive interference occurs when the path difference is an integer multiple of the wavelength (S1P ~ S2P = nλ), giving resultant intensity 4I0. Destructive interference occurs when the path difference is a half-integer multiple of wavelength, giving zero intensity.

For an arbitrary phase difference φ between two coherent sources, the resultant intensity is I = 4I0 cos²(φ/2). For incoherent sources, the phase difference changes rapidly with time and the average intensity is simply 2I0 — the intensities add without any stable interference pattern. This is why two separate sodium lamps illuminating the same screen produce no fringes.

Young achieved coherent sources by illuminating two closely spaced pinholes S1 and S2 from a single source S. Since both slits derive from the same source, any phase change in S appears identically in both, keeping them phase-locked. The spherical waves from S1 and S2 overlap on a screen, creating alternating bright and dark fringes. Bright fringes occur at positions x = nλD/d and dark fringes at x = (n + 1/2)λD/d, where D is the slit-to-screen distance and d is the slit separation.

Dark and bright fringes are equally spaced, with fringe width β = λD/d. The double-slit pattern is actually a superposition of single-slit diffraction from each slit and the double-slit interference pattern. The experiment provided the first reliable method to measure the wavelength of visible light; yellow light has a wavelength of about 0.6 μm.

Diffraction is the bending of waves around obstacles and through apertures, exhibited by all wave types. For a single slit of width a illuminated by monochromatic light, a broad central maximum appears at θ = 0, with secondary maxima at θ ≈ (n + 1/2)λ/a and minima at θ = nλ/a (n = ±1, ±2, …). Secondary maxima become progressively weaker with increasing n. The double-slit interference pattern is a modulation of the single-slit diffraction envelope from each individual slit.

Diffraction limits the resolving power of optical instruments: the minimum angular separation that a telescope or microscope can resolve is set by the wavelength of light used. Richard Feynman noted that interference and diffraction are physically the same phenomenon — the distinction is merely one of usage, with interference typically referring to few-source superposition and diffraction to many-source superposition. In both, light energy is redistributed without net gain or loss, consistent with energy conservation.

Light waves are transverse electromagnetic waves — the electric field oscillates perpendicular to the direction of propagation. In unpolarised light, the electric vector randomly takes all directions in the transverse plane. A polaroid consists of long-chain molecules aligned in one direction; it absorbs electric field components along the molecular axis and transmits only the component along its pass-axis. When unpolarised light passes through a polaroid, the transmitted intensity is half the incident intensity.

When polarised light of intensity I0 passes through a second polaroid whose pass-axis makes an angle θ with the first, the transmitted intensity is I = I0 cos²θ — Malus' law. When θ = 90° (crossed polaroids), the transmitted intensity is zero. Polaroids have practical applications in sunglasses, windowpanes, photographic cameras, and 3D movie cameras. Polarisation phenomena are unique to transverse waves and do not occur for longitudinal waves like sound.

Wave Optics establishes light as a transverse electromagnetic wave through four interconnected phenomena: Huygens principle provides the geometrical framework for propagation and derives the laws of reflection and refraction; the principle of superposition explains interference, culminating in Young's double-slit experiment that allows precise wavelength measurement; single-slit diffraction reveals the limits of geometrical optics and sets the resolution of optical instruments; and polarisation confirms the transverse nature of light waves, governed quantitatively by Malus' law.

For the CBSE Class 12 board examination, this chapter is highly important as it involves both conceptual understanding and numerical problem-solving. Key areas tested include derivation of Snell's law using Huygens principle, fringe width calculations in Young's experiment, conditions for constructive and destructive interference, single-slit diffraction minima positions, and application of Malus' law to polaroid systems. Students should pay special attention to the distinction between coherent and incoherent sources, the formula I = 4I0 cos²(φ/2), and the physical significance of each optical phenomenon.