Principle of superposition:

When two or more waves overlap, the resultant displacement at any point and at any instant is equal to the vector sum of instantaneous displacements that would be produced at the point by the individual waves if each wave is present alone.

Interference of light waves (Definition):

The modification in the intensity of light (redistribution of light energy) produced by the superposition of two or more light waves is called interference of light. A typical interference pattern consists of bright and dark bands that are formed in the region of interference pattern. The light energy, which disappears at the position of minima (dark band), appears at the position of maxima (bright band).

Interference of light waves (Description):

When two light waves of same wavelength, say wave 1 (of amplitude A1) and wave 2 (of amplitude A2), meet at a point then interference results. If these waves meet such that crest of wave 1 meets with crest of wave 2 then addition of amplitudes takes place, the resultant wave has amplitude A1 + A2, and this is called constructive interference. If these waves meet such that crest of wave 1 meets with trough of wave 2 then subtraction of amplitudes takes place, the resultant wave has amplitude A1 – A2, and this is called destructive interference. Intensity of light is maximum where constructive inteference occurs and intensity of light is minimum where destructive interference occurs.

Examples of interference in everyday life:

Examples of interference in everyday life are as follows:

(a) Beautiful colors are observed in a light reflected from a thin film of oil floating on water surface,

(b) Beautiful colors are observed in a light reflected from soap water bubble.

Constructive interference:

If two waves are meeting at a point such that crest and trough of one wave falls on crest and trough of other wave respectively, then constructive interference takes place and amplitude of resultant wave is maximum.

Destructive interference:

If two waves are meeting at a point such that crest of one wave falls on trough of other wave and trough of one wave falls on crest of other wave then destructive interference takes place and amplitude of resultant wave is minimum.

Interference and Fringes:

Let two coherent sources of light 1 and 2 are emitting the light. Let these sources emit the light rays, say, ray 1 and ray 2. These light rays – ray 1 and ray 2 – meet at a point. Now depending upon the path difference between these two light rays, we get bright fringe or dark fringe.

If path difference PD satisfy the following relation:

PD = n λ

then the interfering rays undergo constructive interference and generate bright fringe. Here, PD = path difference between the two rays meeting at a point on the screen, n = 0, 1, 2, 3, ....., and λ = wavelength of light.

If path difference PD satisfy the following relation:

PD = (n - 1/2) λ

then the interfering rays undergo destructive interference and generate dark fringe. Here, PD = path difference between the two rays meeting at a point on the screen, n = 1, 2, 3, ....., and λ = wavelength of light.

Condition of constructive interference:

If two waves starting from the points A and B with no phase difference reach at point P. Then path difference between these waves reaching at point P is BP – AP. The point P will be bright if there is constructive interference. For constructive interference to occur path difference and phase difference between the waves reaching at point P should be as follows:

Path difference = 0, 2λ/2, 4λ/2, 6λ/2, .........

Phase difference = 0, 2π, 4π, 6π, ..........

Condition of destructive interference:

If two waves starting from the points A and B with no phase difference reach at point P. Then path difference between these waves reaching at point P is BP – AP. The point P will be dark if there is destructive interference. For destructive interference to occur path difference and phase difference between the waves reaching at point P should be as follows:

Path difference = λ/2, 3λ/2, 5λ/2, 7λ/2, .........

Phase difference = π, 3π, 5π, 7π,..........

Coherent sources:

The sources of light which emit: (a) light waves of equal frequency, and (b) light waves with no phase difference or constant phase difference, are called coherent sources of light.

Monochromatic source:

The source of light which emits light of single frequency is called monochromatic source of light.

Conditions for producing interference pattern:

Conditions for producing interference pattern are as follows:

(a) There should be two sources of light and these sources should be coherent.

(b) The two sources of light should be monochromatic.

(c) The light waves coming from the two sources of light should be in the same state of polarization.

(d) The two sources of light should emit light waves of equal amplitude or intensity.

(e) The sources of light should be narrow.

(f) The separation between the two light sources should be as small as possible.

(g) The distance of the screen from the two sources of light should be large.

(h) The two interfering waves should travel in the same direction.

Band width or fringe width:

The distance between the centers of two adjecent bright bands (or two adjecent dark bands) is called band width or fringe width.

Young’s double slit experiment:

Thomas Young in 1801, allowed to fall light from a monochromatic source on a single slit S. The light coming from the slit S was allowed fall on two similar slits S1 and S2 (hence the name “double slit”). Thus S1 and S2 become coherent sources of light. The light coming from S1 and S2 was allowed to fall on the screen and interference pattern consisting of bright and dark bands was then displayed on the screen. Young’s experiment confirmed the wave nature of light very first time. In this experiment, the expression for band width or fringe width is as follows:

β = λ D / d

where β = band width or fringe width, λ = wavelength of light, D = distance between slits and screen, d = separation between the slits.

Formula for double slit interference, bright fringes:

In case of double slit interference for bright fringes, the distance between centre of screen and m^th bright fringe, y, is given by:

y = m λ D / d

where D = distance between slits and screen, d = spacing between slits, λ = wavelength of light, m = order of bright fringe and y = distance between centre of screen and m^th bright fringe.

Formula for double slit interference, dark fringes:

In case of double slit interference for dark fringes, the distance between centre of screen and m^th dark fringe, y, is given by:

y = (m – 1/2) λ D / d

where D = distance between slits and screen, d = spacing between slits, λ = wavelength of light, m = order of dark fringe and y = distance between centre of screen and m^th dark fringe.

Formula for double slit interference, fringe width (β):

In case of double slit interference, the fringe width β is given by:

β = λ D / d

where β = fringe width, D = distance between slits and screen, d = spacing between slits, λ = wavelength of light.

Michelson inteferometer formula:

The basic formula of Michelson interferometer is given below:

d = m λ / 2

where, d = distance through which mirror is displaced, m = number of fringes shifted, λ = wavelength of light.

Newton's rings:

When a convex lens is placed on a plane glass surface, concentric circular fringes are observed around the point of contact due to interference of light. These concentric circular fringes are called Newton's rings. These rings were discovered by Robert Hooke in 1665. However, these rings were studied in detail by Newton and hence named after him.

Diffraction:

The bending of light near the edges of an obstacle or slit and spreading into the region of geometrical shadow is known as diffraction of light.

Everyday examples of diffraction:

Everyday examples of diffraction are as follows:

(a) glittering of the edge of mountain when Sun is behind the mountain, after sunset or before sunset,

(b) glittering of the edge of cloud when Sun is behind the cloud,

(c) hologram, and

(d) beautiful colors observed on the shiny side of CD or DVD.

Types of diffraction:

There are two types of diffractions : (a) Fraunhofer diffraction, and (b) Fresnel diffraction.

Fraunhofer diffraction:

In Fraunhofer diffraction, the source of light and the screen on which diffraction pattern is obtained are effectively at infinite distances from the diffracting system. In this case plane wavefront is considered and diffraction pattern is obtained by using convex lens. Fraunhofer diffraction was discovered by Joseph Ritter von Fraunhofer, German physicist, who lived from 6 March 1787 to 7 June 1826.

Fresnel diffraction:

In Fresnel diffraction, the source of light and screen are kept at finite distance from the diffracting system. In this case cylindrical or spherical wavefronts are considered. Fresnel diffraction was discovered by Augustin-Jean Fresnel, French physicist, who lived from 10 May 1788 to 14 July 1827.

Comparison of Fresnel diffraction and Fraunhofer diffraction:

Fraunhofer diffraction due to a single slit:

In this case the diffraction pattern consists of a central bright maximum followed by alternate secondary minima and maxima on its both sides. Various aspects of this diffraction pattern are considered below:

a) Position of secondary minimum are given by the following formula:

sin θ_n = n λ / a

where n = ±1, ±2, ±3, ±4, ........ ; θ_n = angle that gives the direction of the n^th minimum, a = width of slit, λ = wavelength of light.

b) Position of secondary maximum are given by the following formula:

sin θ_n = (2n + 1) λ / a

where n = ±1, ±2, ±3, ±4, ........ ; θ_n = angle that gives the direction of the n^th minimum, a = width of slit, λ = wavelength of light.

c) Width of central maximum is given by the following formula:

w = 2 λ f / a

where w = width of slit, λ = wavelength of light, f = focal length of lens, a = width of slit. Here, it is assumed that lens is placed close to the slit and hence the focal length ‘f’ of the lens is approximately equal to the distance ‘D’ between slit and screen.

Airy’s disc:

When the beam of a light from a point object passes through the objective of an optical instrument, with lens acting like a circular aperture then it produces diffraction pattern instead of a point image. The diffraction pattern is a bright disc surrounded by alternate dark and bright rings, whose intensity goes on decreasing. This is known as Airy’s disc.

Limit of resolution:

The smallest angular or linear separation between the two point objects at which they appear to be just resoloved is called limit of resolution of an optical instrument.

Resolving power:

The reciprocal of the limit of resolution is called its resolving power.

Rayleigh’s criterion of resolution of images:

According to Rayleigh’s criterion,

(a) the images of two point objects close to each other are regarded as just resolved (i.e., just separated) if the central maximum of one falls on the first minimum of the other. In other words, when central bright image of one falls on the first dark ring of the other, the two images are said to be just resolved (i.e., just separated),

(b) the images of two point objects close to each other are regarded well resolved, if the separation between the central maximum of the two images is greater than the distance between the central maximum and first minimum of any of the two images,

(c) the images of two point objects close to each other are regarded unresolved, if the separation between the central maximum of the two images is less than the distance between the central maximum and first minimum of any of the two images.

Limit of resolution of microscope:

The minimum distance by which two point objects are separated from each other so that their images as produced by microscope are just seen resolved or separate is called limit of resolution of microscope. It is given by the following expression :

d = λ / (2 sin α)

In case of oil immersed objective (in which space between the objective and the object is filled with oil in order to avoid the loss of light by reflection at first lens surface and to improve the resolving power of microscope), the limit of resolution of microscope is given by the following expression:

d = λ / (2 μ sin α)

where d = distance between two point objects = limit of resolution, λ = wavelength of light, 2 α = angle subtended by an object at the aperture of objective, μ = refractive index of an oil.

Numerical aperture:

The term ‘μ sinα’ in the above given expression for ‘d’ is called numerical aperture of the objective of the microscope.

Resolving power of microscope:

The reciprocal of limit of resolution is the resolving power of a microscope. It is given by the following expression:

R.P. of microscope = 1/d = 2 μ sin α / λ

where d = distance between two point objects = limit of resolution, λ = wavelength of light, 2α = angle subtended by an object at the aperture of objective, μ = refractive index of an oil.

Resolution of Telescope According to Rayleigh:

According to Rayleigh the limit of angular resolution of telescope is given by:

θ = 1.220 λ / D

Where θ = angular resolution (in radians), λ = wavelength of light used, and D = diameter of the lens' aperture.

Resolving Power of Telescope According to Rayleigh:

According to Rayleigh the resolving power of telescope, RP, is given by:

RP = 1 / θ = D / (1.220 λ)

Where θ = angular resolution (in radians), λ = wavelength of light used, and D = diameter of the lens' aperture.

Difference between interference and diffraction:

Differences between interference and diffraction are as follows:

(a) Interference pattern is obtained as the result of interaction of light coming from two different wavefronts from two coherent sources while diffraction pattern is obtained as the result of interaction of light coming from different parts of the same wavefront.

(b) Interference fringes are of the same width while diffraction fringes are not of the same width.

(c) In interference pattern all bright bands are of same intensity while in diffraction pattern all bright bands are not of same intensity. Intensity is maximum for central maximum in diffraction pattern.

(d) The dark fringes are perfectly dark in interference pattern but in diffraction pattern they are not perfectly dark.

Diffraction grating:

Diffraction grating is nothing but large number of eqally spaced parallel slits. Despite its name, diffraction grating is analyzed using the "interference formulae" and not the "diffraction formuale." In order to create diffraction grating, large number of parallel, thin lines of black ink are drawn on glass plate. The gap between two consecutive black lines serve as slit. In market, diffraction gratings with 10,000 lines per centimeter are available. There are two types of diffraction gratings, namely, (a) transmission grating, and (b) reflection grating. The grating described above is "transmission grating." In reflection grating, instead of black ink, silver ink is used which reflects the incident light.

Formula for m^th order maxima in diffraction grating:

Formula for m^th order maxima in diffraction grating is given below:

d sinθ_m = m λ

where d = (1/N) meter = distance between the slits, N = number of lines per meter on grating, θ_m = angle of m^th order maximum, λ = wavelength of light, m = ....., -2, -1, 0, 1, 2, .....