Theory of Michelson's interferometer and Types of fringes

INTERFEROMETRY

The phenomenon of interference has been used to test the flatness of surfaces and also used to reduce the reflecting power of the lens and prisms. Instruments based on the principle of interference of light are known as interferometers. Michelson designed an interferometer to determine the wavelength of the light, resolution of the spectral line and the thickness of the thin transparent materials.

THEORY OF MICHELSON'S INTERFEROMETER

Principle

The amplitude of light beam from a source is divided into two parts of equal intensities by partial reflection and transmission. These beams are then sent in two directions at right angles and are brought together after they suffer reflection from plane mirrors to produce interference fringes.

Construction

The Schematic diagram of a Michelson interferometer is as shown in Fig. 8.9. It consists of two highly polished plane mirrors M₁ and M₂ which are at right angles to each other. There are two optically flat glass plates G₁ and G₂ of same thickness and made up of same material placed parallel to each other. These plates are inclined at an angle of 45° with the mirrors M₁ and M₂.

The plate G₁ is half silvered at the back so that the incident beam is divided into two beams viz., reflected beam and transmitted beam of equal intensity. The mirrors M₁ and M₂ are provided with screws on their backs, so that they can be adjusted exactly perpendicular to each other.

The mirror M₁ is mounted on a carriage which can be moved forward and backward using the handle 'H'. The distance at which the M₁ is moved can be read with the help of the scale as shown in Fig. 8.9. The interference fringes can be observed in the field of view of the telescope 'T'.

Working

Light from a monochromatic source S is made parallel with the help of collimating lens L. The light beam is allowed to fall on the semisilvered glass plate G₁. It is partly reflected at the back surface of G₁ and travel towards M₁ i.e. along (AC) and partly transmitted towards M₂ (i.e. along AB). These two rays travel along two mutually perpendicular paths and are reflected back by the mirror M₁ and M₂. These two rays again meet at glass plate G₁ and enter a short focus telescope T.

The two rays which enter the telescope are originally derived from the same single beam, hence they cause the interference fringes in the field to view of the telescope. Hence a path difference can be introduced between the two reflected rays by moving the mirror M₁.

It is clear from the Fig.8.9 that a ray PC passes twice through the glass plate G₁ i.e., 1st through PQ and 2nd through QP, after reflection from the mirror M₁, whereas the ray PB does not even passes once through G₁, even after reflection from the mirror M₂. Thus in the absence of the glass plate G₂ the path traced by the beam between G₁M₁ and G₁M₂ are not equal.

To equalise the path difference, a glass plate G₂ of same thickness and material as that of G₁ is introduced between G₁ and M₂. So that the ray PB will also pass twice, i.e., 1^st through RO in glass plate G₂ and 2^nd through OR in glass plate G₂, after reflection from the mirror M₂.

Since the glass plate G₂ is used to compensate the path difference between the two rays, it is called as a compensation plate. Thus the path of the two rays viz., PB and PC are made equal.

Adjustments

(i) The distance of the mirrors M₁ and M₂ are adjusted to be nearly equal from G₁.

(ii) In order to make the incident beam parallel, a tin sheet with a small hole is placed in front of source S.

(iii) The hole and the light from the source are adjusted in line with the centre of glass plates G₁ and G₂ and mirror M₂.

(iv) A lens is then placed between G₁ and tin sheet and a plane mirror is placed between G₁ and lens normally.

(v) The lens position is adjusted till the image of the hole falls back on the tin plate very close to the hole, hence the light is made as a parallel beam when it leaves the lens.

(vi) If the plane mirror is removed, and if the beam is seen in the direction of AT. We can see four images of the hole. The mirrors are adjusted till the images coincide two by two.

(vii) At this stage, if the tin sheet is removed, then the two paths of light are exactly parallel to each other giving rise to circular fringes in the field of view. By tilting mirror M₂ slightly, the fringes can be made straight.

TYPES OF FRINGES

(i) Circular fringes

When mirror M₂ is exactly perpendicular to mirror M₁ and the virtual mirror M₂' (which is image of M₂) an air film of constant thickness is enclosed between them as shown in Fig. 8.10.

The airfilm gives reflected beam to interface. The path difference depends upon the following factors.

(i) The separation between M₁ and M₂'

(ii) The angle subtended on the eye

(iii) The inclination between M₁ and M₂'

It can be seen from the Fig. 8.10 that the path differences are different for different values of θ. The source S is an extended source and S₁ and S₂ are the virtual source due to M₁ and M₂'. If the distance between M₁ and M₂' is d and the distance between S₁ and S₂ is 2d, then

The path difference between the two rays = 2dcosθ

Since the light reflected from mirror M₂ is again reflected by the semi‒silvered glass plate G₁, an additional path difference of λ/2 has to be introduced.

Therefore, the total path difference between the two rays = 2dcosθ + λ/2

If the total path difference is = nλ, we get bright fringes and

If the total path difference is = (2n+1) λ/2, we get dark fringes.

Case i) When M₁ and M₂' coincides, the path difference is zero and therefore the field of view is perfectly dark, as shown in Fig. 8.11.

Case ii) When M₁ is moved either way parallel to itself, widely spaced circular fringes are produced. The width of the fringes depends upon the path difference between the rays. (Fig 8.12)

(ii) Localised fringes

When the mirror M₂ is not exactly perpendicular to M₁ (or) if M₁ and the virtual mirror M₂' are inclined, then the airfilm enclosed between them is wedge shaped.

For small path difference, the fringes are nearly straight but for large path difference the fringes are generally curved and are always covex towards the thin edge of the wedge. When the two mirrors M₁ and M₂' middle, straight line fringes are observed as shown in Fig. 8.13. When the two mirrors are inclined, curved fringes are observed as shown in Fig. 8.14 and 8.15.

iii) Localised white light fringes

With white light, the fringes are observed only when the path difference is very small. The fringes of zero thickness is dark (i.e) central fringe and the other fringes are coloured due to overlapping of various colours. If the film is thick, uniform illumination is observed. White light fringes are useful to find the zero path difference, especially in the standardisation of metre.