#### Interferometry, Physics tutorial

Introduction:

Interferometry is an instrument mainly designed to exploit the interference of light and the fringe patterns which result from the optical path differences, in any of a variety of ways, is termed as an optical interferometer.

In order to attain interference between the two coherent beams of light, an interferometer splits an initial beam into two or more divisions which travel diverse optical paths and then superpose to generate an interference pattern. One of the criterions for generally categorizing interferometers differentiates the way in which the initial beam is separated. Wavefront division interferometers sample parts of the similar wavefront of a coherent beam of light, as in the case of the Young's double slit, Lloyd's mirror or Fresnel's Biprism arrangement. The Amplitude-division interferometers, rather, make use of several kind of beamsplitter which splits the initial beam into two portions.

The Michelson interferometer is of this kind. Generally the beam splitting is managed through a semi-reflecting metallic film. In this interferometer, the two interfering beams are broadly separated, and the path difference among them can be varied at will by moving the mirror or via introducing a refracting material in one of the beams.

Michelson Interferometer:

This kind of interferometer was originally designed by A. Michelson for the precise spectral analysis of the light sources. For illustration, by using the instrument he was capable to represent that the red Hα Balmer line of the hydrogen spectrum was mainly composed of two components separated through just 0.014 nm. He was as well capable to compute the natural width of the cadmium red line. These days this instrument is generally employed for the accurate determination of wavelengths, refractive indices of gases and transparent materials and minute changes of length.

The central features of the interferometer are represented schematically in figure below:

M1 and M2 is the two plane mirrors silvered on the front surfaces. They are mounted vertically on two rigid holders positioned at the sides of a flat metal stand. Screws are provided in front of the holders, adjusting of that lets M1 and M2 to be tilted. M1 can as well be moved horizontally through a micrometer joined to the M1 holder. 'G' that is, the beam splitter, is a plane glass plate slightly silvered on one side. 'C' is the compensator plane glass plate of the identical thickness as 'G'. Both are mounted vertically and at an angle 45° to the direction of the incident light.

Whenever light from the extended light source is permitted to fall on 'G', one part, calling it beam 'A', is transmitted via 'G' and 'C' to M2 and the other, calling it beam 'B' is reflected through 'G' to M1. Beam 'A', returning from M2, is reflected at the back of 'G' to the eye positioned at 'E' and beam 'B', after reflected from the M1 passes via 'G' to reach the eye 'E'.

We will now observe that how the combination of such two parts of light that have began as one single beam and gone via various optical paths give mount to interference fringes. To do this we refer to figure shown below:

The mirror M2 is here substituted through its virtual image M2' formed via reflection in 'G'. Whenever M1 and M2 have been set vertical and perpendicular to one other, then M2' is parallel to M1. Owing to the mirror action of 'G', we might now assume that the light source as being the observer and as forming two virtual images L1 and L2 in M1 and M2' correspondingly. If 'd' is the separation of M1 and M2', the virtual sources L1 and L2 will be separated via 2d.

For the rays of light reflected normal to the mirrors, the phase difference due to the path difference is 4πd/λ. As well, an extra phase shift of 'π' is introduced as beam 'A' is reflected off the outer side whereas beam B is reflected off the inner side of the beam splitter. Thus, the total or net phase difference is:

δ = 4πd/λ - π

When the light rays satisfy the conditions:

δ = 2mπ

Or 2d = (m + 1/2) λ

Here, 'm' is an integer and 'λ' the wavelength of incident light, they will be in phase. Though, parallel rays of light reflected at an angle are usually not in phase. They will be in phase just for those angles 'θ' which satisfy the relation:

2dcos θ = (m + 1/2) λ

On entering the eye that has been adjusted to get or obtain parallel rays of light, such rays will reinforce one other to generate constructive interference resultant in maxima of light intensity at their focal points on the eye retina.

For such reflected rays of light satisfying:

2dcos θ = m λ

destructive interference will take place resultant in the minima of light intensity at their focal points on the eye retina.

As for fixed 'd' and 'λ' and various values of 'm', a system of dark and bright circle concentric fringes each corresponding to a constant 'θ' will be observed, such interference fringes are thus termed as fringes of equivalent inclination. They are localized at infinity.

Whenever the two mirrors are not precisely vertical and perpendicular to one other, then the space between M1 and M2' will be in the shape of a wedge, the fringes noticed in this case will be straight and parallel. These fringes are termed as fringes of equivalent thickness. They are localized at the finite distance.

Fringe Shapes:

The accurate form the fringes depends on whether mirrors M1 and M2 are precisely perpendicular to one other and whether the optical lengths of the two arms is similar or not. Suppose that M1 is the movable mirror.

1) Circular fringes:

The angle θ is necessarily similar for the two rays if M1 is parallel to M2' in such a way that the rays are parallel. Therefore if the eye is focused to obtain parallel rays one can view circular fringes that are localized at infinity if the two arm lengths are not equivalent. In this case, the fringes are stated to be localized at infinity. As in any single circular fringe the rays come parallel to one other to make fringe and all the rays enclose equivalent angle by the optic axis these fringes are termed as Haidinger's fringes.

2) Straight fringes:

Whenever the two arm lengths are equivalent and the two mirrors M1 and M2' are not parallel to one other the rays that form interference fringes are not parallel and appear to come from a point closer to the virtual mirrors. In this condition, the fringes observed are termed as the Fizeau fringes and they are localized close to the mirror. The localized fringes are practically straight as the variation of the path difference across the field of view is now due mainly to the variation of the thickness of the 'air film' between the mirrors. By the wedge-shaped film, the locus of points of equivalent thickness is a straight line parallel to the edge of the wedge.

3) Curved fringes:

Whenever the two arm lengths are not equivalent and the two mirrors are arranged in a manner explained in case 2 we get curved fringes, which are the portion of circular fringes however the fringes are stated to encompass both the components of Haidinger's fringes and Fizeau's fringes.  The fringe localization plane in this condition is between the infinity and mirror.

Fabry-Perot Interferometer:

This is mainly based on the principle of multiple beam interference. This is a high resolving power instrument that makes use of the 'fringes of constant inclination' generated by the transmitted light after multiple reflections among the two parallel and highly-reflecting glass plates.

It comprises of two optically-plane glass plates A and B as shown in figure above, with plane surfaces. The inner surfaces are coated by means of partly transparent films of high reflectivity and positioned accurately parallel to one other. Screws are provided to secure the parallelism if disturbed. The two uncoated surfaces of each and every plate are made to encompass a slight angle between them in order to avoid unwanted fringes formed due to the multiple reflections in the plate itself.

One of the two plates is kept fixed, whereas the other can be moved to differ the separation of the two plates. In this configuration, the instrument is termed as a Fabry-Perot interferometer. At times both the plates are at a fixed separation by the help of spacers. The system having fixed spacing is termed as Fabry-Perot etalon. The Fabry-Perot interferometer (or etalon) is employed to find out the wavelengths precisely, to compare two wavelengths, to calibrate the standard metre in terms of wavelength and so on.

S1 is a wide source of monochromatic light and L1 a convex lens that makes the beam more collimated. The incident ray suffers a huge number of internal reflections successively at the two silvered surfaces, as represented above. At each and every reflection a small fractional part of the light is as well transmitted. Therefore, each and every incident ray generates a group of coherent and parallel transmitted rays by a constant path difference among any two successive rays. A second convex lens, L2, brings these rays altogether at a point P in its focal plane, where they interfere. Therefore, the rays from all points of the source generate an interference pattern on the screen S2 positioned in the focal plane of L2.

Formation of the Fringes: Let 'd' be the separation among the two silvered surfaces, and 'θ' the inclination of a specific ray with the normal to the plates. Then the path difference among any two successive transmitted rays corresponding to the incident ray is 2 d cos θ. The medium among the two silvered surfaces is generally air. As phase 'π' changes takes place on both of these (air-to-glass) surfaces, therefore, the condition:

2 d cos θ = nλ

holds for the maximum intensity.

Here, 'n' is an integer, termed as the order of interference, and 'λ' the wavelength of light. The locus of points in the source that give rays of a constant inclination 'θ' is a circle. Therefore, with an extended source, the interference pattern comprises of a system of bright concentric rings on a dark background, each and every ring corresponding to a specific value of θ.

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