## Light wave properties

### Shape of fringes

1) The interference fringes are usually hyperbolic I in shape, because for given value of n , locus of path difference between light waves from two slits is a hyperbola.

2) When the screen is at 90˚ to the line joining foci of the hyperbola, the fringes are circular.
3) When distance of screen is very large compared to the distance between the slits, the fringes are straight. Note that the fringe obtained at the centre of the screen is called fringe of zeroth order.

### Circular Wave front

We know that when we through a piece of stone on the surface of water, we get circular wave fronts. As the wave fronts expand, the heights of crests and depths of troughs go on decreasing. This is because same energy is distributed over circular wave fronts of larger and radii. If r is radius of circular wave fronts, its perimeter = 2πr.

### Coloring of thin films

We know that oil spreads over the surface of water and forms an extremely thin layer. When this film is seen in sunlight, we observe beautiful rainbow colures. It can be explained in terms of interference of light. We can show that condition for maxima and minima in the reflected system are complementary to those in the transmitted system. It means a film that appears bright in the reflected system shall appear dark in the transmitted system and vice-versa.

Suppose t= thickness of film
µ = refractive index of material of the film
r = angle of refraction
λ = wavelength of light used.
In the reflected system,
Condition for maxima is      2 µ t cos r = (2n-1) λ/2.
And condition for minima is     2 µ t cos r= n λ
In the transmitted system,
Condition for maxima is      2 µ t cos r = n λ.
And condition for minima is     2 µ t cos r = (2n-1) λ/2.

### Fresnel’s Biprism

It is equipment, which is used for obtaining two coherent sources of light from a single source by refraction. Further, it demonstrates experimentally, the phenomenon of interference of light and is also used for the determination of wavelength of light.
The interference pattern consists of concentric dark and bright rings. The centre of the pattern is bright.
For bright fringe, path diff. = n λ, and
Far a dark fringe, path diff. = (2n-1) λ/2
Again, fringe width, β = λD/d.

### LIyod’s Single Mirror

It is equipment which is used for obtaining coherent sources of light by reflection from a plane mirror. Only half of the interference pattern is available. The central fringe is dark. As the rays suffer a phase change of p on account of reflection from a denser medium, therefore, conditions for bright and dark fringes are just reversed. For bright fringe, path diff. = n λ
Far a dark fringe, path diff. = (2n-1) λ/2

### Diffraction of light

It is the phenomenon of bending of light around corners of an aperture in the path of light. On account of this bending, light penetrates into the geometrical shadow of an obstacle. The diffraction pattern due to a single slit consist of a central bright brand having alternate dark and weak bright bands of decreasing intensity on both sides.

The condition for nth secondary minimum is that

Path difference = a sin ѳn = n λ, where n= 1,2,3…And the condition for nth secondary maximum is that

Path difference = a sin ѳn = n λ, where n= 1,2,3…And the condition for nth secondary maximum is that path difference = a sin ѳn= (2n+1) λ/2 where n= 1, 2, 3, …..
Width of central maximum is 2x = 2 D λ/a =2 f λ/a

Here, a is width of slit and D is distance of screen from the slit; f is focal length of lens for diffracted light. Diffraction is supposed to be due to interference of secondary wavelets from the exposed portion of wave front from the slit. Whereas in interference, all bright fringes have same intensity, in diffraction, bright bands are of decreasing intensity.

### Diffraction Grating

A diffraction grating is an optical device, which is used for studying the spectra of sources of light, and in the determination of wavelength of light. To prepare a diffraction grating, we take an optically plane glass plate. A number of equidistant parallel lines are ruled on the plate using a fine diamond point. The region where the line is drawn becomes opaque to light, but the space between every two lines is transparent. If a is width of each transparency and b is width of each opacity, then (a + b) is called grating element. If N is number of lines per inch of the grating plate, then

a + b = 1 inch/N=2.54 cm/N

The condition for principal maximum in the grating spectra is (a + b) sin ѳ = n λ, Where n is the order of the spectrum. When the light used is not monochromatic, the grating spectra are colored. However, the central maximum is white.

### Doppler’s effect in light

According to Doppler’s effect in light, whenever there is a relative motion between a source of light and observer, the apparent frequency of light received by observer is different from the true frequency of light emitted from the source of light. The apparent frequency v’ is given by

v’= v(1±v/c)

Where v is velocity of star and c is velocity of light in vacuum, v being the actual frequency of light emitted from the source + sign is used when source moves towards the observer and vice-versa.
We can, show that

Δv=±uv/c and Δλv/c

When a star approaching earth, v’>v, Δvis +.Accordingly, λ’< λ and Δλ is negative.

The spectrum of the star shifts towards violet end (lower wavelength side). Similarly, when a star moves away from earth, the spectrum of the star shift towards red end,  (higher wavelength side).
Doppler’s effect in light is used in measuring speed of a star/galaxy, satellites, aero planes, submarines etc.
Note that width of spectral line = 2Δλ.

### Polarisation of light

It is the phenomenon of restricting the vibration of light (electric vector) in a particular direction; on passing ordinary light (unpolarised) through certain crystals like tourmaline crystal. This crystal acts as a polarizer. When un polarised light is seen through a single crystal (Polaroid), intensity of transmitted light decreases to 50% on account of polarization. However, on rotating the crystal about the direction of propagation as axis, intensity of polarized light does not change. On the contrary, when polarized light is passed through another crystal called analyser and the analyser is rotated, the transmitted fraction of light changes from maximum to zero. This is how we detect polarized light.

### Law of Malus

According to law of Malus, when a beam of completely plane polarized light is incident on an analyser, the resultant intensity of light (I) transmitted from the analyser varies directly as the square of the cosine of the angle (θ) between plane of transmission of analyser and polarizer i.e. I ∝ cos2 θ

### Brewster’s law

When unpolarised light is incident at polarizing angle (ip) on an interface separating a rarer medium from a denser medium of refractive index μ, such that μ = tan ip , then light reflected in the rare medium is completely polarized. The reflected and refracted waves, in this case are perpendicular to each other. Obviously polarizing angle depends on nature of media in contact, and on the colour of light.