**Optics Series, lecture – XI**

This lecture ( 1.5 hrs ) was delivered on 16 – 02 – 2017 to honors students

( all optics series lectures ) go to other available optics lectures

**“Fresnel’s Bi-prism: measurement of wavelength of light”**

Today we will discuss another interesting interference set – up, now that we have discussed the Young’s double slit experiment, in lecture – IX. A few words about the general mechanism behind interference. There are two kinds of interference basically that we will be discussing in our lectures. We discussed the Young’s double slit interference pattern based on our understandings of intensity or irradiance patterns that we discussed in lecture – VII.

Interference is sustained and visible if the corresponding sources of light are coherent among themselves, that is, if the sources have phase differences that are not arbitrarily or abruptly changing, as a consequence we can safely assume the phase differences are constant and therefore predictable. Incoherent light makes this impossible.

Incoherent light is that light source whose production itself is arbitrary and abrupt and unpredictable, hence nothing can be definitively said on its phase, as a result the coherence is only short lived. If two light sources are so generated that their respective coherence time (or coherence length) are well within each others span, they are said to be coherent light.

Recall the idea of temporal and spatial coherence here that we discussed: here, when we discussed Young’s DS interference. We observe one basic thing about interference here. The two coherent sources S_{1} and S_{2} that we considered give rise to two different wave-fronts that meet up after traversing their respective optical paths. When they meet they produce interference. For this reason such type of interference are called as wave-front splitting interference.

Young’s DS experiment is an example of wave-front splitting interference. The Fresnel bi-prism that we will discuss is also an example of wave-front splitting interference. But there is yet another type of interference mechanism. Its called amplitude-splitting interference, examples of which are colors of thin films and Newton’s ring phenomena which we will study soon enough in future lectures.

In an amplitude splitting interference what happens is there is only one wave (or its wave-fronts) which splits into different components such as reflected or transmitted (refracted) parts according to the respective coefficients for these processes. So the amplitude has a fraction which is reflected and another which is refracted.

Naturally the question of coherence does not deter the production of interference effects. There always is inherent coherence in the amplitude splitting processes. When these different components meet up later, they produce interference.

### Fresnel’s Bi-prism

Fresnel’s bi-prism is a combination of two prisms with very small base angles, typically of ~20′, when monochromatic light from a slit falls on it. It refracts and produces two virtual images which act like coherent sources. An interference pattern is produced which can be seen with an eyepiece. Relevant parameters can be measured by a micrometer attached to the eyepiece. Let us make a diagram suitably representing our situation.

Let us briefly explain the set up. The refracted light rays from S and virtually from S_{1} and S_{2} which act like coherent sources now, go through a convex lens placed in between the Bi-prism and the eyepiece. If the eyepiece is fixed in location there are two positions L_{1} and L_{2} at which images of S_{1} and S_{2 }can be seen at the eyepiece.

As should be clear by now to the attentive learner the arrangement of Fresnel’s bi-prism must produce the same interference pattern that we discussed for Young’s Double Slit arrangement. We are concerned about measuring the wavelength of light of the source we are using — eg the monochromatic Sodium light source that we use usually. There are two ways to measure the wavelength.

#### Method — I

If n represents the refractive index of the bi-prism material and α the base angle of the prism, then the angular deviation is given by: δ = (n-1) α. From geometrical considerations S_{1}S_{2} ≅ 2a (n-1) α where a is the distance of S to the base of the bi-prism.

Typically n = 1.5 — glass, α = 20′ = 5.8 × 10^{-3} rad, a ≅ 2 cm which gives d = 0.012 cm. We have seen in the Young’s DS arrangement the fringe spacing β is related to the wave-length λ as follows: λ = dβ/D. We measure β through the micrometer attached to the eye piece. Thus we can determine the wavelength: λ.

#### Method — II

d and D can be measured by placing a convex lens between bi-prism and eye-piece. Let d_{1} be the distance between two images when lens is placed at L_{1}, which is at a distance b_{1} from eyepiece.

Similarly let d_{1} be the distance between two images when lens is at L_{2} at a distance b_{2} from eyepiece. Then d is the geometric mean of d_{1} and d_{2}, that is: d = square root (d_{1}d_{2}) and D = b_{1}+b_{2}. By using the formula λ = dβ/D again we can have the wavelength of a monochromatic light source.

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