interference of light When two light wave trains cross a region of space, energy is redistributed within the region, as a result of interference between the waves. Under certain conditions a steady intensity distribution pattern is obtained with maxima and minima which are called fringes. The resultant intensity at any point depends upon the amplitudes and phase relationships of the component waves. The resultant displacement at any point is given by the vector sum of the individual displacements, that each one would produce independently. This is called the principle of superposition.

In order to obtain a permanent interference pattern, it is necessary that the sources producing the waves are coherent; which means that the phase relationships between the waves produced by the sources remain constant. In addition to get a good contrast between the maxima (where the amplitudes are in phase) and the minima (where the amplitudes are out of phase), the amplitudes of two waves must be nearly the same. These conditions can only be satisfied if the wave trains are obtained from the same source. There are only two possible ways to achieve this. The first is called the division of wave front and the second is called the division of amplitude.

An example of the division of wave front method of obtaining interference is the Fresnel's biprism shown in the fig. i1. A biprism kept in front of a narrow slit, illuminated with monochromatic light of sodium lamp produces two virtual images, with the distance between them 2d. On a screen kept at a distance, D the interference pattern can be observed. The condition of maxima at any point on the screen (at a distance y from centre O) is the path difference between the waves coming from two virtual sources should be nl (where n is an integer). Similarly for minima the path difference should be (2n+1) l /2. The path difference is given by,

S2P - S1P = 2yd / D = n l

for nth bright fringe from the centre O. The fringe width, b the distance between any two consecutive maxima is,

b = Dl / (2d)

From this relation the wavelength of light can be obtained.

The division of amplitude producing coherent wave trains is observed in thin films, shown in fig. i2. An incident ray of light (i) on meeting the interface, is partly reflected and partly transmitted. The transmitted ray will produce successive rays (1),(2),(3) etc. suffering partial reflection and transmission at bottom and top surfaces as shown in fig. i2. The intensity of rays decrease rapidly for the successive rays. If now the rays are collected by a lens, they will interfere constructively or destructively depending upon the phase differences. The path difference, D between the successive rays can be obtained geometrically (see fig. i2).

D = 2m d cos q r

where d is the thickness of the film, and m the refractive index of the film material. The product of the geometric path difference (2m d cos q r ) with m gives the equivalent path difference in vacuum.The ray (1) suffers a phase change at the interface. Therefore the condition of constructive interference is,

2m d cos q r = (2n + 1) l /2

and the condition of destructive interference is,

2m d cos q r = n l

When this condition is fulfilled the rays (2), (3), (4) . . . etc. are all out of phase with (1). It can be shown that the total amplitude of (2) (3) (4) . . is just equal to that of (1). Therefore the condition (i7) gives a complete darkness.

When the condition of maximum (i6 ) is satisfied, rays (2), (4), (6) etc. are in phase with (1), but rays (3), (5), (7) etc. will be out of phase with (2), (4), (6) etc.. But since (2) is stronger than (3), (4) is stronger than (5) and so on, the pairs cannot cancel each other and the result is a maximum intensity.

If the incident light is falling vertically on the film the angle q r~ 0° . The variation of thickness of the film gives rise to bright and dark fringes. This class of fringes has important practical application in the testing of optical flatness of surfaces.

The Newton's ring observed by a plano-convex lens and a flat glass plate (fig. i3) are also similar fringes. In this case since the locus of equal thickness are circles, circular fringes are observed. When the contact between the glass plate and convex surface is perfect, the central spot is dark. This is a direct evidence of phase change in reflection from rare to dense medium. In Newton's ring the relationship between the wavelength of light, the diameter of the nth dark ring, Dn and radius of curvature of the plano-convex lens is,

Dn2 = 4n l R

Michelson interferometer (fig. i4a) is another important example of division of amplitude method, for obtaining interference. Light from an extended source is divided in to two beams by using a partially silvered glass plate G1. The two beams are again brought together after reflecting on plane mirrors M1 and M2. If the optical distances from the rear face of G1 for the two rays 1 and 2 are the same the image M2' coincides with M1. If the distance between the mirrors is d, then the path difference between the rays D is,

D = 2d cos a

where a is the angle between the direction of observation and vertical axis (see fig. i4b). The condition of constructive interference is,

2d cos a = m l

The fringes are circular in shape, which are the locus of equal inclination. These are called Haidinger fringes.

The Michelson interferometer can be used for precise determination of length of an object. The meter was standardised in terms of wavelength of light using this instrument by A. Michelson himself. The famous experiment known as Michelson Morley experiment was also carried out using this instrument.