Michelson Interferometer

Michelson Interferometer by Sameer Saini

The aim of this experiment is to set up a laser according to the Michelson interferometer experiment, using this set-up to measure the wavelength of the laser, the refractive index of air and of glass. The experiment resulted with a couple of experimental values which did not seem reliable or accurate for the refractive indices, but the wavelength of the laser was accurately found.

Introduction Background Theory Michelson Interferometer Interference The interferometer divides an incident beam of light into two parts via a partially reflecting mirror called a beam-splitter. The two resulting beams are coherent and travel two separate, but nearly equal, paths. Both beams are reflected by mirrors in such a manner that they recombine (see Figure 1). If the mirrors are perfectly parallel, the interference pattern, when viewed on a screen, will be a set of circular rings called fringes. One of the mirrors can be moved in very small increments using a micrometer driven device attached to the mirror's stage. If the path lengths differ by λ/2+nλ, one would see destructive interference, that is the two beams add up to give 0. and if they differ by only nλ, they would add constructively. Effect of a change in pressure In the interferometer system the characteristics of the fringe pattern depend on the phase relationship between two interfering beams. One way to affect the beams is to change medium through which the beam passes, or its pressure. For a light of specific frequency the wavelength varies as such, =

0 n

where λ0 is the wavelength of light through a vacuum and n is the refractive index for the material. For low pressures the refractive index varies linearly for a gas with its pressure. Effect of Glass plate in the system The optical path length of one of the light paths will change if a glass plate is inserted into it. As the plate is rotated, the length of glass in the path will increase and therefore the number of wavelengths in that path will increase. This will, of course, change the interference pattern. The index of refraction of the glass plate can be calculated from the number of interference fringes shifted during the rotation of the glass plate through some angle θ.

Aim To set up a laser according to the Michelson interferometer experiment, using this set-up to measure the wavelength of the laser, the refractive index of air and of glass

Method The equipment was set up as according to figure 1.0, it was aligned in such a manner that the beams from both arms would rejoin perfectly on the splitter again and pass on to the camera to produce an interference pattern. For the case of calculating the wavelength of the laser, the moveable mirror was put on a motor and starting from 0, was moved by some set distance, at the same time the number of fringes that passed on a set point on the computer screen was measured. This was done several times in order to get a reliable set of results. The a similar set-up was used to measure the refractive index of air, instead this time having a moveable mirror, that was set to be stationary, and in between that mirror and the beam splitter a gas chamber, through which the light could pass through. This time several runs were done in a similar fashion as to before but instead the pressure of the gas chamber was changed to move the fringes. For the final section of the experiment, measuring the refractive index of glass, the gas chamber was replaced by a rotatable glass slide, the glass slides angle started at 0, which was set as the glass side being perpendicular to the laser light. The angle of the slide was then slowly changed and the corresponding fringes passing by on the screen measured, this was then repeated several times. Figure 1.0

Results The full set of results can be found at Appendix 1.0

Part 1: Wavelength of laser light Distance(m) versus Fringes

For result 1, the

experimental wavelength was found to be 624.7±3.9nm, for result 2, the wavelength was 637.6±4.6nm. Therefore the average wavelength from the results was 630.5±6.0nm. The wavelength of the laser given by the manufacturer was stated to be 630nm, which seems to fall quite close to the experimental result. The calculations for this result are as follows; Average Wavelength=

Average Wavelength=

Result 1 Result 2 2

624.7nm 637.6nm 2

Average Wavelength=630.5nm Total Error =   Error 12 Error 22× Average Wavelength



Total Error = 

3.9 2 4.6 2    ×630.5nm 624.7 637.6

Total Error =6.0nm

Part 2: Refractive Index of Air ∇Path Length(m) versus Fringes

Therefore extrapolating the data, at room temperature, the change in path length for the laser light through the gas chamber would be, 0.0000527671±1.76769*10-7m. Using the prior information, plus the fact that the gas chamber was 10.7±0.5cm in length (only regarding the section that contained the increased pressure), it is possible to now calculate the refractive index of air; Refractive Index=

∇ Path length 1 Distance

Refractive Index=

0.000052761 1 10.7×10−2

Refractive Index=1.00049



Refractive Index Error= 

0.5×10−2 2 1.76769×10−7 2    ×0.000493 −2 0.0000527671 10.7×10

Refractive Index Error=2.310×10

−5

Therefore the refractive index of air was experimentally found to be 1.00049±2.310*10-5, the actual value for the refractive index of air according to reference 1, is 1.000292, which falls outside the range error for experimental calculations.

Part 3: Refractive Index of Glass We know that N=

2n a d a 2 n g d g  0

Where N is the number of fringes and λ0 the wavelength of the light, from this equation we can then simplify it to something more useful. 2t−N 01−cos  N= 2t 1−cos −N 0  where t is the thickness of the glass plate. From the results

Thus we can conclude that the experimental value for the refractive index of the glass was found to be 1.28±0.06. According to reference 2 the theoretical value for the refractive index of glass is 1.52, which falls does not fall within the range of error for the experimental value, there are several reasons for why this may have occurred, which can be found in the discussion.

Fringes Versus Rotation

(Another method for calculating the Refractive Index for glass, not sure how correct this method is) The data from the experiment can be approximated into a parabola, with this information, it can be seen that when the laser beam is perpendicular to the glass (when the rotation is 0) the ratio for the distance travelled between the arm is 1.84±1.34, thus it can be said using Snell's law; n2 =1.84 n1 n 1≈1 due to it being air therefore n 2=1.84 ± 1.34 According to reference 2 the theoretical value for the refractive index of glass is 1.52, which falls within the range of error for the experimental value. It should be noted though, the range of error for this result is quite large, making it very inaccurate.

Discussion For part 1 of this experiment, that is calculating the wavelength of the laser, even though the experimental value contained the theoretical value produced by the manufacturer within its error, there were several important factors when completing this experiment that made it largely error prone. Firstly, there was big chance that there would be random miscounts when measuring the number of fringes passing by, second of all the motor was slightly faulty and would sometimes jump which would cause miscounts and a misreading of the first fringe after restarting the motor. Finally there was also a problem with stopping the motor at half a fringe, where we just rounded to the closest fringe. The experimental value calculated for the refractive index of air, did not seem to agree with the theoretical value as proposed by reference 1. There are several reasons as to why this result may have occurred. First of all, it was assumed that the room pressure on the day was at 1 atmosphere, which may or may not true. Secondly and more importantly, the theoretical value for the refractive index assumes that the temperature in the room would have been 0C, while in reality it was definitely not the case, and this factor was not taken into account when conducting the measurements. For part 3, the experimental results did not contain the theoretical value, which was most likely caused by systematic error, as the method for determining how many fringes passed as the glass was rotated was all based on human accuracy, which in our case was very poor.

Conclusion To conclude all tasks were successfully completed, but in many cases the results proved to be invalid and/or inaccurate.

Appendix 2.0 References 1. http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_7.html 2. http://hyperphysics.phy-astr.gsu.edu/Hbase/tables/indrf.html 3. www.phy.duke.edu/courses/143/labs/michelson.pdf 4. www.physics.iitm.ac.in/courses_files/...odd/Michelson.htm