Wave optics studies light as a wave, explaining phenomena that ray optics cannot: interference, diffraction, and polarization. The wave theory of light was proposed by Christian Huygens in the seventeenth century and firmly established by Thomas Young's interference experiment in 1801, which showed that light from two slits produces a pattern of bright and dark fringes.
The foundation of wave optics is Huygens' principle, which states that every point on a wavefront acts as a source of secondary wavelets, and the envelope of these wavelets gives the new wavefront. This principle explains reflection and refraction from the wave viewpoint and leads to the derivation of Snell's law.
Interference occurs when coherent waves from two sources superpose, producing alternate bright and dark fringes. Diffraction is the bending of light around obstacles, and polarization reveals the transverse nature of light. Together these phenomena confirm that light is a transverse electromagnetic wave.
Huygens' principle provides a geometrical construction for wave propagation. It states: 1. Every point on a wavefront is a source of secondary wavelets. 2. The secondary wavelets spread out in all directions with the speed of the wave in the medium. 3. The new wavefront is the forward envelope of all the secondary wavelets.
Using this principle, we can construct the reflected and refracted wavefronts when a plane wave meets a boundary. For reflection, applying the construction with the equality of speeds in the same medium gives the law of reflection: the angle of incidence equals the angle of reflection.
For refraction, the speed changes at the boundary. Applying the construction with different speeds in the two media gives Snell's law:
sin(i) / sin(r) = v1 / v2 = n2 / n1
Huygens' principle thus explains both laws of reflection and refraction from a single unified wave picture, and it also explains the propagation of plane, spherical, and cylindrical wavefronts.
Two waves are coherent if they have a constant phase difference. Thomas Young's double slit experiment produced two coherent sources from a single wavefront, allowing stable interference. When light from the two slits superposes, bright fringes appear where the path difference is an integral multiple of the wavelength, and dark fringes where it is an odd multiple of half a wavelength.
For constructive interference (bright fringes), the path difference is:
Delta = n lambda, n = 0, 1, 2, ...
For destructive interference (dark fringes):
Delta = (n + 1/2) lambda
In the double slit experiment with slit separation d and screen distance D, the fringe width, the distance between successive bright or dark fringes, is:
beta = lambda D / d
The intensity distribution on the screen is obtained by superposing the two waves. The fringe width depends on the wavelength, so white light gives coloured fringes, with the central fringe white.
Diffraction is the spreading of light around obstacles. In single slit diffraction, the diffraction pattern consists of a central bright maximum flanked by secondary maxima and minima. The condition for the minima on either side of the central maximum is:
a sin(theta) = n lambda, n = 1, 2, 3, ...
where a is the slit width. The width of the central maximum is twice that of the secondary maxima:
Width of central maximum = 2 lambda D / a
Diffraction becomes significant when the size of the obstacle or slit is comparable to the wavelength of light. The diffraction patterns of a single slit and the interference pattern of double slits differ in the intensities: in diffraction the secondary maxima fall off rapidly, while in interference all fringes have nearly equal intensity.
The resolving power of an optical instrument is its ability to distinguish two closely spaced objects. According to Rayleigh's criterion, two images are just resolved when the central maximum of one diffraction pattern falls on the first minimum of the other.
For a telescope, the angular resolution is:
theta = 1.22 lambda / D
where D is the aperture diameter. Larger apertures give better resolution. For a microscope, the resolving power is limited by the wavelength; smaller wavelengths give better resolution, which is why electron microscopes (using electron waves of very small wavelength) resolve far smaller details than optical microscopes.
Light is a transverse wave, so its electric field can vibrate in many directions perpendicular to the direction of propagation. In unpolarized light, the vibrations are in all directions. Polarization is the process of restricting the vibrations to a single plane; such light is called plane polarized.
When unpolarized light falls on a polaroid or a material with aligned molecules, only the component of the field vibrating in the transmission direction passes through. According to Malus' law, the intensity of the transmitted light through an analyser at angle theta to the polarizer is:
I = I0 cos^2(theta)
When theta = 90 degrees, the intensity is zero and the two polaroids are crossed, blocking all light.
Brewster's law states that when light is incident on a surface at the polarizing angle, the reflected and refracted rays are perpendicular. The polarizing angle ip satisfies:
tan(ip) = n2 / n1
At Brewster's angle, the reflected light is completely plane polarized.
Polarization proves that light is a transverse wave, because longitudinal waves cannot be polarized. Light is polarized by reflection, scattering, and by polaroid sheets.
Polarization has many applications. Polaroid sunglasses reduce glare by blocking the horizontally polarized light reflected from surfaces. Polarization is used in liquid crystal displays (LCDs), 3D cinema, and in analysing stresses in transparent materials through photoelasticity. It is also used to study the properties of materials by examining the rotation of the plane of polarization.
| Phenomenon | Key Condition | Formula |
|---|---|---|
| Constructive interference | Path diff = n lambda | Bright fringes |
| Destructive interference | Path diff = (n+1/2) lambda | Dark fringes |
| Fringe width (double slit) | beta = lambda D / d | Independent of n |
| Single slit minima | a sin theta = n lambda | n = 1, 2, 3, ... |
| Central maximum width | 2 lambda D / a | Twice secondary |
| Malus' law | I = I0 cos^2 theta | Analyser at angle theta |
| Brewster's law | tan ip = n2/n1 | Reflected light polarized |
| Rayleigh criterion | theta = 1.22 lambda/D | Telescope resolution |
| Wavefront type | Shape | Source |
|---|---|---|
| Plane wavefront | Plane | Distant point source |
| Spherical wavefront | Sphere | Point source |
| Cylindrical wavefront | Cylinder | Line source |
Wave optics explained light as a transverse wave through the phenomena of interference, diffraction, and polarization. Huygens' principle gave a wave-theoretic derivation of reflection and Snell's law. Young's double slit experiment demonstrated interference with fringe width beta = lambda D/d, and single slit diffraction produced a central maximum of width 2 lambda D/a with weaker secondary maxima. Rayleigh's criterion theta = 1.22 lambda/D defined the resolving power of instruments. Polarization, described by Malus' law and Brewster's law, confirmed that light is a transverse electromagnetic wave and enabled applications from polaroid sunglasses to LCDs. Wave optics complements ray optics to give a complete description of light.