coherent light sources of light examples and conditions class 12 why are needed for interference

know coherent light sources of light examples and conditions class 12 why are needed for interference ?

Wave Nature of Light

Light is an electromagnetic wave which does not require a material medium for propagation. The electric and magnetic fields vary in space and time resulting in the propagation of an electromagnetic wave even in free space. The electric field varies in space and time as

E = A sin (wt – kx)

which represents a wave travelling along the + x direction. A = amplitude, ω = 2πv (w is angular frequency in rad s^-1 and v is frequency in Hz) and k = 2π/λ ;

λ = wavelength. Also

where v is the wave velocity.

Phase : The phase Φ of a wave at a point x and at time t is given by the argument of the harmonic function (sine or cosine) representing the wave, i.e.

Φ = wt – kx

Phase Difference

Suppose two waves meeting at a point P are represented by

where x1 and x2 are paths of the waves up to point P where they meet. The phase difference between them is

1. If the two waves have different frequencies, i.e., v1 ≠ v2 then λ1 ≠ λ2 and 𐤃Φ depends on time t.
2. If v1 = v2, then λ1 = λ2. In this case

or Phase difference = 2π/λ  x (path difference)

i.e., the phase difference is independent of time and depends only on the path difference (x2 – x1). This holds only if the two sources of wave are ‘coherent’, i.e., they have a constant fixed phase relationship. Intensity The intensity of a wave at any point in its path is proportional to the square of its amplitude at that point.

2. Reflection and Refraction of Light

When a light wave falls on a reflecting surface, it is reflected obeying the usual laws of reflection. When a wave travels from one medium into another, its velocity and wavelength undergo a change and the wave is said to suffer refraction. The frequency of the wave does not undergo any change in refraction (and reflection). If v1 is the velocity of the wave in the medium in which the incident wave propagates and v2 is the velocity of the wave in the medium in which the refracted wave propagates, then 1 m2, the reflective index of the second medium with respect to the first, is defined as

where λ1 and λ2 are the wavelengths of the same wave in the two media. The frequency of the refracted wave remains the same as that of the incident wave. When a wave, travelling in a rarer medium, is reflected at the boundary of a denser medium, the reflected wave suffers a phase change of 180° (or π radians) in relation to that of the incident wave. No phase change occurs if a wave, travelling in a denser medium, is reflected at the boundary of a rarer medium. The refracted wave, in both cases, does not undergo any phase change.

Interference of Light

When two or more light waves travelling in the same direction meet (or superpose) at a point in a medium, the electric field of the resultant wave can be obtained by using the principle of superposition which states that the resultant electric field is given by the algebraic sum of the individual electric fields, at that point, due to the individual waves, i.e., E = E1 + E2 + ………

resulting in a change in amplitude (and hence in intensity) at that point. The phenomenon in which the intensity of light at a point is modified by the superposition of two or more waves is known as interference.

If two waves of intensities I1 and I2, differing in phase by Φ, superpose, the resultant intensity is given by

Constructive Interference The resultant intensity I is maximum if cos Φ = + 1, i.e

where 𐤃 is the path difference between the interfering waves. Then

The interference is said to be constructive. If the two interfering waves have equal intensities I₁ = I₂ = I₀, then

I_max = 4 I₀

Destructive Interference The resultant intensity I is minimum if cos Φ = – 1, i.e

At maxima, the waves reinforce each other and at minima they cancel out each other. These maxima and minima constitute the bright and dark fringes.

Coherent Light Sources

The resultant intensity of light at a point on the screen depends on the phase difference (Φ) between the two interfering waves. This phase difference depends upon two factors—

(1) the initial phase difference between the waves emitted by the two sources and

(2) the phase difference resulting from the path difference for that point. The initial phase difference depends upon the time and remains constant only for about 10^–8 to 10^–10 second. Thus the resultant intensity changes so rapidly with time that, due to persistence of vision, we are unable to see the interference pattern. Thus, non-coherent sources cannot produce sustained interference effects. We conclude that, for a steady interference pattern, the following two conditions must be satisfied.

1. The sources must be coherent.

2. The wavelengths of the interfering waves must be the same. Thus, only monochromatic coherent light sources produce observable interference pattern.

Young’s Double Slit Experiment

Monochromatic light from a source slit S illuminates two slits S1 and S2 which are very close together and equidistant from S

Secondary waves from S1 and S2 interfere giving rise to bright and dark fringes on the screen. There is bright fringe at centre P₀ of the screen.

(i) The distance of the nth bright fringe from the centre of the fringe system is

where l = wavelength of light used, λ = seperation between slits S1 and S2 and D = distance between the screen and the plane of the two slits.

(ii) The distance of the n th dark fringe from the centre of the fringe system is

(iii) The separation between two consecutive bright or dark fringes is called fringe width (β) which is given

(iv) Angular separation between nth bright fringe and the central fringe is

(v) Angular separation between nth dark fringe and the central fringe is

Displacement of Fringes

If a transparent plate of thickness t and refractive index µ is introduced in the path of one of the interfering waves, the entire fringe pattern is shifted by a distance

Diffraction at a Slit

When a parallel beam of monochromatic light falls normally on a narrow slit, the diffraction pattern on a screen has a bright central maximum bordered on both sides by secondary maxima of rapidly decreasing intensity. If λ is the wavelength of light and a is the width of the slit, then

f = focal length of the convex lens placed close to the slit.