# Michelson Interferometer- Definition, Principle, Construction and Working, Applications.

An interferometer is an instrument that uses interference phenomenon in the measurement of the wavelength of light in terms of standard of length or the measurement of distance in terms of the known wavelength of the light.

# Michelson Interferometer

## What is Michelson Interferometer?

An optical instrument which is used to measure the physical properties of light, such as wavelength, refractive index, and interference patterns is called Michelson Interferometer.

The Michelson Interferometer was first developed by the American physicist Albert Michelson in the late 19th century and has since become an essential tool in scientific research, especially in the field of optics.

## Michelson Interferometer Theory

The Michelson Interferometer is based on the idea of interference from light waves. When two light waves with similar wavelength are combined they may either enhance each one another (constructive interference) or cancel one another off (destructive interference) and result in a pattern consisting of light and dark fringes.

The Michelson Interferometer consists of a beam splitter, two mirrors and an instrument. It is reflective mirror which splits up the laser beam of light into two pathways. One path is called the reference path, while another is that of the test path. Both paths are reflected by mirrors, and they merge when they reach the beam splitter in which they conflict with one another.

Mathematically Michelson Interferometer can be described by the following equation:

I = I1 + I2 + 2√(I1 I2) cos(δ)

where I is the intensity of the interference pattern, I1 and I2 are the intensities of the two light beams, and δ is the phase difference between the two beams.

The phase difference is determined by the path length difference between the two beams. If the path lengths are equal, there is no phase difference and the interference is constructive, producing a bright fringe. If the path lengths differ by half a wavelength, the interference is destructive, producing a dark fringe.

## Michelson Interferometer Construction and Working

### Michelson Interferometer Working Principle

The Michelson Interferometer works based on the principle of interference of light waves.

In this instrument, light from an extended source is divided into two parts by partial reflection and transmission. These two beams are sent at right angles to each other in the two directions. They get reflected from the mirror and form interference fringes which are observed and investigated.

### Construction of Michelson Interferometer

It consists of a semi silver plate P  placed at an angle of 45 degrees to the horizontal. M1 and M2 are two highly polished silver mirrors. These Mirrors are provided with a labeling screw at the back.

Mirror M1 can be moved towards or away from plate P with the help of micrometer screws. P1 is another glass plate whose thickness is exactly equal to the thickness of plate P.

It is transparent and parallel to P.  S  is a monochromatic source of light, say sodium lamp. Lens  L is a Converging lens whose function is made to the source of light as an extended one.

### Michelson Interferometer Working

Light from an extended monochromatic source S rendered parallel by the lens L is made to fall on semi-silvered plate P. Here, the light is divided into a reflected and transmitted beam of equal intensity.

The reflected beam which falls on mirror M1 and the transmitted beam that falls on mirror M2 is incident normally. When the Apparatus is in normal adjustment.

After reflection from mirror M1 and M2, a part of the amplitude of the wave from M1 is transmitted along with AE. While a part of the amplitude of the wave from M2 is reflected along with AE.

As these two waves entering the telescope are derived from the same source S, hence these waves are coherent waves.

Coherent waves interfere with each other and interference fringes formed and seen through the telescope. The ray which gets reflected from M1 crosses the plate P twice while the ray which from M2 Travels only in the air.

This means that the ray which travels along the mirror M1 covers an additional part 2 (n-1) t. Where t is the thickness of the plate P and n  is its refractive index.

Which does not get any difficulty if we are using monochromatic light but if white light is used. Then it does create a problem because of the variation of n with wavelength.

To overcome this difficulty another plate P1 is introduced between P and mirror M2. This plate P1 is called a compensating plate.

The plate P and P1 are of equal thickness being cut from a single optical plane parallel plate to ensure the equality of thickness and the nature of the material.

The function of the compensating plate is that the Ray of light traveling towards M1 and M2 must travel equally pass through the glass plate.

The phase changes on reflection at mirror M1 and M2 are smaller, the phase changes due to reflection in air and class are also similar, is equal to π. The two rays reaching the telescope interfere constructively or destructively depending upon the path difference.

The path difference between the two rays reaching the telescope is mλ. Where M is the integer and λ  is the wavelength of light used.  Then constructive interference takes place.

On the other hand, if the path difference is (2m+1) λ/2, Then destructive interference occurs. The path difference between the two beams AM1A and AM2A can be altered by moving mirror M1.

## Formation of Fringes

When we see through the telescope, the mirror M1 is seen directly and a virtual image of M2 is also seen in M1. If the distance of M1 and M2 from plate P are not exactly equal, the image of M2 is formed slightly ahead or at the back of M1.

Thus the system reduces to the equivalent of an air film and enclosed between the mirror M1 and the virtual image of M2.

Now to see whether both the mirrors are at right angles to each other or not, a pin is placed between the source as and the plate P. Looking along EA, two images of the pin are seen.

The Adjustment of mirror M2 is made with the help of three screws such that both the images of the pin coincide. Which means both the mirrors are at right angles to each other.

The pin is now removed. Now if the distance AM1A and AM2A  are exactly equal the field of view shall be totally dark. A slight movement of mirror M1 forward or backward shall produce concentric circular fringes in the field of view.

## Types of fringes in Michelson Interferometer

There are two types of fringes generated by Michelson Interferometer:

• Circular Fringes.
• Localized Fringes.

The shape of the fringes formed in the Michelson interferometer depends on the inclination between mirror M1 and M2. Let M2 be the virtual image of mirror M1 as S’ be the image of source  S.

Let D be the separation between mirror M1 and the virtual image M2’  of the mirror M2. Therefore, the interference pattern formed will be due to air film enclosed between M1 and M2’. Let S1’ and S2’ be the images of the Saras M1 and M2 respectively.

It means, we have two coherent sources S1’ and  S2’ to obtain due to the process of the division of amplitude.

### Circular Fringes in Michelson’s interferometer

If the mirror M1 and M2 are exactly at right angles, then the mirror ember and the image M2’ two of the mirror M2 are parallel to each other.

Then circular fringes are formed. Let D be the distance of mirror M1 and mirror M2 from the plate P then the distance between M1 and M2’ who is also d. It means, the thickness of air film enclosed between M1 and M2’ is d  as shown in the figure.

However, the distance between two coherent sources  S1’ and S2’ is 2d. The path difference between the rays coming from S1’ and S2’=2d cosθ. Where θ  is the angle of inclination of the ray Falling On The Mirror.

### Localized fringes in Michelson’s interferometer

When mirrors M1 and M2’ are not exactly parallel or not equally distant, a wedge-shaped film is formed between them. The path of two reflected rays, originating from the same incident ray why reflection from M1 and M2’ you are no longer parallel.

They intersect near M1 and hence the fringes are formed near M1. The fringes are called localized fringes and to see them the eye must be focused on the vicinity of M1.

These fringes are curved with their convex side toward the thin edge of the wedge as shown in the figure. The thin edge of the wedge is to the left and therefore the observer fringes are convex toward the left.

As we go on decreasing the separation between M1 and M2’, the fringes move across the field of view away from the thin as of the wedge and at the same time gradually become straight. When M1 and M2’ intersect, the lines are perfectly straight as shown in the figure.

We have to wedges opposing each other so the line should appear curved on both sides of the intersection but for a small field of view, they appear straight. When M1 is still moved such that the mirror M1 and the virtual image M2’ of mirror M2 get a position as shown in the figure.

The fringes are again curved but with their convex side towards the right. Localized fringes become invisible for large path differences of the order of several millimeters.

### White Light Fringes

If monochromatic light is replaced by white light, its constituent wavelength gives rise to its own set of fringes two different widths. The zero-order fringes corresponding to each wavelength will coincide and hence we get a dark central fringe.

But if the path difference between the interfering rays is considerable. The central fringe will be surrounded by a few colored fringes. And then there will be so much overlapping that the pattern appears to be white.

The importance of these white light fringes is that the position of zero-order fringe which is dark can be located very easily in the Michelson interferometer.

## Applications of Michelson Interferometer

Michelson Interferometer is used to determine:

• Wavelength of monochromatic light.
• The refractive index of a thin film.
• Resolution of spectral lines.
• The evolution of meters in terms of the wavelength of light.
• The angular diameter of stars.
•  Presence of ether.
•  The accuracy of the surface of the prism and lens.

These are some advantages of using a Michelson Interferometer:

1. High precision: The Michelson Interferometer can measure displacements as small as a fraction of a nanometer, making it a highly precise instrument for measuring physical quantities.
2. Versatility: The Michelson Interferometer can be used to measure various physical parameters, including distance, refractive index, and wavelength of light. It is also used in Fourier transform spectroscopy and laser stabilization.
3. Non-destructive: The Michelson Interferometer is a non-destructive method of measurement, making it ideal for studying biological samples and other delicate materials.
4. Interference-based measurement: The Michelson Interferometer is based on the interference of light waves, which makes it a highly accurate and reliable method of measurement.
5. Easy to use: The Michelson Interferometer is easy to set up and use, and does not require specialized knowledge or skills to operate.
6. Low cost: The Michelson Interferometer is relatively inexpensive compared to other precision measurement instruments, making it accessible to a wide range of users.
7. Wide range of applications: The Michelson Interferometer has applications in various fields, including physics, engineering, astronomy, and biology.

• Susceptible to vibration.
• Limited range.
• Alignment issues.
• Limited applications.
• Limited resolution.

## Michelson Interferometer Short Notes

Here are the some short notes on the Michelson Interferometer:

• The Michelson Interferometer is an optical instrument used to measure small displacements, changes in refractive index, and other physical parameters using interference of light waves.
• It was invented by Albert Michelson in 1881 and is still widely used today in various fields, such as physics, engineering, and biology.
• The Michelson Interferometer consists of a beam splitter, two mirrors, and a detector. A light source is directed towards the beam splitter, which splits the light into two beams that travel along different paths.
• The two beams are reflected by mirrors and recombine at the beam splitter, producing an interference pattern that is detected by a photodetector.
• The interference pattern consists of a series of bright and dark fringes, which are produced by constructive and destructive interference of the two light beams.
• The Michelson Interferometer can be used to measure the distance between the mirrors, changes in refractive index of a material, and the wavelength of light.
• The Michelson Interferometer is also used in Fourier transform spectroscopy, laser stabilization, and gravitational wave detection.
• The Michelson Interferometer is a highly precise instrument and can be used to measure displacements as small as a fraction of a nanometer.
• The Michelson Interferometer is widely used in research and industry for its versatility, accuracy, and reliability.

These are just a few key points about the Michelson Interferometer.

## FAQ on Michelson Interferometer

### How many types of fringes in Michelson Interferometer.

1. Circular Fringes.
2. Localised Friges.

### How does interferometer work?

In this instrument, light from an extended source is divided into two parts by partial reflection and transmission. These two beams are sent at right angles to each other in the two directions. They get reflected from the mirror and form interference fringes which are observed and investigated.

### What is the principle of Michelson interferometer?

The Michelson Interferometer works based on the principle of interference of light waves.