Measuring the Change in the Refractive Index of Water with a Michelson Interferometer

measuring the change in the refractive index of water with a michelson interferometermeasuring the change in the refractive index of water with a michelson interferometer
andy tan
introduction
when hot water is poured into a glass containing cold water, there appears to be “ripples” within the glass. it seems like two immiscible liquids have been put together and two liquid phases can almost be distinguished. this is due to the difference in the refractive index of water at different temperatures.
a michelson interferometer (figure 1) can be used to measure changes in the index of refraction by inserting a water-containing cell in one leg of the interferometer and letting the laser beam pass through the water. as the laser beam interferes with another beam from a second leg of the interferometer, an image consisting of a pattern of stripes (light and dark lines) appears on a screen placed in front of the laser. if the temperature of the water changes, then so does the refractive index of the water, which affects the manner in which laser beams from each leg of the interferometer interfere. this is seen as movement of the pattern of light and dark stripes on the screen.
the object of this paper is to measure the change in the refractive index of water by counting the number of dark-light-dark fringe shifts that appeared per degree celsius.
the refractive index n of a substance varies as a function of temperature. a water-containing cell was placed in one leg of a michelson interferometer, producing fringe shifts (laser interference patterns) on a screen as the water cooled. by counting the fringe shifts for each change in temperature of 1°c, a set of values for n could be created. these values agreed with values of n found in two separate papers to within 2 x 10-5.
figure 1. diagram of the michelson interferometer. the distance between the beam splitter to the screen and each mirror is 15 cm, the distance between the beam splitter and the laser is 30 cm.
diagram of michelson interferometer
direction of movement
fringes
screen
mirror
lens
mirror
cell
he-ne laser
beam splitter
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methods
a michelson interferometer was first constructed from a 5mw 632.8 nm he-ne laser, a beam splitter, two mirrors, a converging lens and a white screen (figure 1). the mirrors and beam splitter were adjusted as necessary to observe an interference pattern of light and dark fringes on the display screen. a lens was placed before the screen to widen the fringe pattern and facilitate observation.
a rectangular optical cell was then constructed out of three flat pieces of wood and two microscope slides (figure 2) and placed in one leg of the michelson interferometer (figure 1). the cell was filled with warm, deionized water at about 70°c and allowed to cool to room temperature. a thermometer supported by a stand was suspended in the cell as close to the laser beam as possible without disturbing it.
as the water cooled, the refractive index of the water changed, affecting the way light from one leg of the interferometer interfered with light from the other leg (e.g. from constructive, to destructive, to constructive again) causing fringe shifts, which move in a direction perpendicular to the stripes of light, to appear on the screen. at any particular point on the screen, one fringe shift is said to have occurred if a dark stripe passed by that point, followed by a light stripe, followed by a dark stripe again. the number of fringe shifts passing a marked point on the screen was recorded for every droping of a degree in temperature. one cycle of constructive-destructive-constructive interference (one fringe shift) occurs as the number of wavelengths of light contained in the cell changes by 1.
the total path length of the laser as it passes through the cell twice is 2l where l is the interior length of the cell. the wavelength of the laser in the cell ? is given by
????= ???????????????????? (1) where n is the index of refraction of the cell s contents. then the number of wavelengths m in the cell is
????= 2????????= 2?????? ???????????????? (2)
as the number of wavelengths changes by 1, one fringe shift is seen and so the number of fringe shifts seen is equal to the difference in the number of wavelengths at temperatures t1 and t2. the difference in the number of wavelengths at two temperatures t1 and t2 is given by ?????? = ????2? ????1 (3)
figure 2. the base and the two long vertical sides were made by joining the pieces of wood (pine) with a framing machine, while the front and back faces of the cell were made by gluing one microscope slide on each face. the top of the cell was left open.
diagram of the glass cell
9 cm
2 cm
4 cm
laser beam
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but since n changes with temperature, m2 and m1 are given by
????2=2????????2???????????????? and ????1=2????????1????????????????. (4), (5) combining equations (3), (4), (5) and rearranging for n2 - n1 results in the following expression ????2?????1=?????? ????????????????2???? (6) the change in the index of refraction can be written as ?n = n2 - n1. making this substitution and dividing both sides of (6) by ?t = t2 - t1 yields the final expression
?????? ????????=?????? ????????????????2?????? (7)
?????? , the number of fringe shifts seen for each change in temperature of 1°c was observed and recorded. ?????? ???????? for a particular temperature could then be calculated. this process was repeated about 5 times for each temperature from 28°c to 72°c and the average ?????? ???????? found for each temperature was plotted (figure 3).
results & discussion
table 1 contains the errors that arose when conducting the measurements.
source of error
summary of possible errors
error contribution
fringe miscount at t ? 60°c
6.7% 1 miscount in 15 fringe shifts
fringe miscount at 45°c ? t < 60°c
2.5% 1 miscount in 40 fringe shifts
fringe miscount at t < 45°c
1% 1 miscount in 100 fringe shifts
cell length measurement
0.1%
cell length thermal expansion
0.0225%
table 1. the error rate is greater at higher temperatures due to the greater rate of cooling, and hence a greater speed at which fringe shifts occurred. the cell length was measured by a digital caliper with an accuracy of 0.02 mm. the cell length expansion for pine is 5 parts per million per change in °c measured along the grain[1]. with a cell length of 2 cm and maximum temperature difference of 45°c, the change in length of the cell is at most 0.0225%.
the length of the cell was measured prior to each trial to minimize the effect of the swelling of wood due to water. the fringe miscounts were chiefly due to the necessity of monitoring the thermometer while counting fringes. there were other factors that limited the quality of the data such as the precision of the thermometer, the warming of the air by the faces of the cell, and the exact wavelength of the laser output. their contribution to error was difficult to quantify.
it was found that as the temperature increased, the rate of change of the refractive index of water grew increasingly negative (figure 3). as ?????? ???????? appears to be nonlinear in temperature, n is likely nonlinear in temperature as well. there appears to be anomalies in the temperature ranges of 33°c to 39°c and 50°c to 57°c which deviate from the general decreasing trend. there were fewer trials conducted in the lower temperature range of 40°c and below, which may have lead to rougher data in comparison to temperatures above 40°c. at temperatures 50°c to 57°c, the data is again irregular as this was the range where the refractive index of water had changed sufficiently compared to at 72°c that the beams from each leg of the interferometer strayed from alignment and no longer interfered with each other (perfectly positioning the mirrors right at the start of the trial to avoid this problem was difficult for practical reasons). a readjustment of the mirrors became necessary to continue getting interference of the laser.
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