### Aim:

- To determine the refractive index of a thin glass plate.

### Apparatus:

Laser light source, Michelson interferometer kit, optical bench, glass plate, meter scale.

### Theory:

The Michelson interferometer is a historically important device which provides simple interferometric configuration for introducing basic principles. This device works under the principle of interference. Michelson Interferometers can be used to find out the refractive index of glass plates or thin films.

#### Measurement of refractive index of glass plate:

Consider a thin glass plate of thickness *t* and refractive index *n*, inserted normal to the path of one of the two interfering beams in Michelson interferometer. The optical path length of the beam through the plate is *nt*, while the optical path length through an equal thickness of air is just *t*, so the increase in optical path length caused by inserting the plate is (*n*ÃƒÂ¢Ã¢â€šÂ¬Ã¢â‚¬Å“1)*t*. The beam traverses the plate twice, so the total path difference will be 2(*n*ÃƒÂ¢Ã¢â€šÂ¬Ã¢â‚¬Å“1)*t*. If *N* is the number of fringes displaced by inserting the plate, then *NÃƒÅ½Ã‚Â»* = 2(*n*ÃƒÂ¢Ã¢â€šÂ¬Ã¢â‚¬Å“1)*t.*

After adjusting the mirrors to obtain circular fringes with a single central dark spot, the plate is introduced into the path of one of the interfering beams and the fringes are displaced. M_{1} is moved a distance *d* closer, until a single central dark spot is again obtained. The distance *d* moved is noted and the number of fringes *N* that disappear is counted. Then, since the insertion of the glass plate increased the optical path length by 2(*n*ÃƒÂ¢Ã¢â€šÂ¬Ã¢â‚¬Å“1)*t*, and the mirror motion decreased it by 2*d*, 2*d* must equal 2(*n*ÃƒÂ¢Ã¢â€šÂ¬Ã¢â‚¬Å“1)*t*, so the refractive index *n* of the plate can be calculated from *NÃƒÅ½Ã‚Â»* = 2*d* = 2(*n*ÃƒÂ¢Ã¢â€šÂ¬Ã¢â‚¬Å“1)*t*.

With a laser, the light source is more precisely monochromatic, so the measurement of *n* can be more accurate. In the Michelson interferometer, if *N* fringes are displaced when the plate is rotated through an angle *ÃƒÅ½Ã‚Â¸* from its original orientation normal to the path, the refractive index of the plate is

where *t* is the thickness of the plate and *ÃƒÅ½Ã‚Â»* is the wavelength of the laser. The last term in the numerator is often neglected. However, this causes an error that increases with *ÃƒÅ½Ã‚Â¸*, reaching ~1% at *ÃƒÅ½Ã‚Â¸* = 30Ãƒâ€šÃ‚Â°, potentially destroying the increased accuracy gained by using a laser light source.

The above equation can be approximated to get

### Applications

1. Michelson Interferometers are used in Optical Coherence Tomography (OCT), a medical imaging technique.

2. They are used in atmospheric and space applications.