Alex Trinh Senior Physics Report 18 April 2014

1

The Michelson Interferometer A. Trinh

School of Physics, University of Sydney

The Michelson Interferometer is a historically important piece of equipment that is

classified as an amplitude splitting interferometer and has a variety of applications. Using this

interferometer, we explore concepts of fringe localisation and fringe visibility using light

sources such as a sodium vapour lamp, mercury lamp and an incandescent lamp to determine

the d=0 position of the interferometer (the white light position with the incandescent lamp). We

also used this position to determine the group refractive index of a piece of glass, found to be

1.554. The value of d=0 position ranged from 11.60 mm to 12.03 mm. Discrepancies were due

to background sources which affected the contrast of the fringes.

1. INTRODUCTION In 1887, Michelson and Morley performed a famous

experiment in which they attempted to determine the

relative velocity of the Earth through the luminiferous

aether, providing evidence against the theory that light

required a medium to travel through. Throughout their

experiment, they used a notable and one of the most

historically important pieces of equipment, known as the

Michelson Interferometer.

There are many uses of the Michelson

Interferometer. Besides the Michelson-Morley

Experiment, Michelson went on to measure the length of

the standard metre through the use of precisely known

wavelengths of atomic spectral lines. The interferometer

is also used in Fourier transform spectroscopy,

refractive index determination and precision

spectroscopy [1].

Interferometers can be classified into two classes,

wavefront-splitting and amplitude splitting [2].

Wavefront-splitting interferometers involve splitting

two coherent portions of a wavefront and having them

interfere. This was explored by Dr. Thomas Young in

his slit experiments. The second class of interferometers,

amplitude-splitting, involves light being incident on a

partially reflective surface. Parts of the wave will be

transmitted, while the rest will be reflected. The

amplitudes of these waves will be smaller than the

incident wave. If the waves recombined, interference

would occur as long as the coherence length isn’t

changed. This is how the Michelson Interferometer

functions.

This report details the methods used in investigating

fringe localisation and fringe visibility using the

Michelson Interferometer. Attention will be centred on

determining the d=0 position of the interferometer using

light sources such as a Sodium lamp, Mercury lamp, and

an incandescent light source. This position was used in

finding the group refractive index of glass.

2. THEORY A problem with producing interference patterns is

that the light source used must be coherent. We classify

coherence into two categories, spatial and temporal [3].

Spatial coherence describes the correlation between

wavefronts at different points in space. Temporal

coherence describes the correlation between the values

of a wave at different moments in time. The Michelson fringe visibility is given by

𝑉 𝑥 =𝐼𝑚𝑎𝑥 −𝐼𝑚𝑖𝑛

𝐼𝑚𝑎𝑥 +𝐼𝑚𝑖𝑛 (1)

For a monochromatic source, if the optical path

difference, 𝑥=2d, is zero then 𝑉=1. For real light

sources, increasing |𝑥| decreases the visibility. The

coherence length of the light source is the full width at

half maximum (FWHM) of 𝑉(𝑥) and is given by

ℓ ≈ 𝛿𝜎−1 ≈𝜆0

2

𝛿𝜆 (2)

Alex Trinh Senior Physics Report 18 April 2014

2

If a glass plate with group refractive index 𝑛𝑔 and

thickness 𝑡 is placed into one arm of the interferometer

then the optical path difference will be

2 𝑛𝑔 − 1 𝑡 (3)

This leads to a change in d of [4]

𝑛𝑔 − 1 𝑡 (4)

3. EXPERIMENTAL PROCEDURE

3.1 Setup and Calibration

The interferometer was set up in a fashion shown in

Fig. 1. In order to calibrate the interferometer, a

translucent screen with a cross a cross drawn on the

centre was placed at the input in order to centre the

image. This was done by illuminating the screen with a

sodium vapour lamp and adjusting the tilt of M1 until

the images of the cross were superimposed. Once the

images were superimposed, fringes became evident

across the field of view as a result. A telescope focused

at infinity was used to estimate the d=0 position by

adjusting the micrometer on the interferometer until the

field of view was completely illuminated.

3.2 The Sodium and Mercury Spectrum Starting from the d=0 position, the micrometer was

adjusted in a clockwise direction to change the fringe

visibility. The position where the fringe visibility was

zero was noted as d1. d2 was found by adjusting the

micrometer in the opposite direction until the visibility

was zero. The fringe visibility should be symmetrical

around the d=0 position. Using these values, the optical

path difference was determined using

𝑋 = 2(𝑑2 − 𝑑1) (5)

The Na spectrum consisted of a sodium doublet centred

on the mean wavelength 589.3 nm. The wavelength

difference between the two lines in the Na doublet is

given by

∆𝜆 =𝜆2

𝑋 (6)

The Na lamp was replaced with a Hg lamp. The most

prominent lines in the Hg spectrum are at 435.8 nm

(blue), 546.1 nm (green) and 578.0 nm (yellow). A

green filter was used to isolate the green line. Returning

to the d=0 position, the micrometer was adjusted again

until the visibility was zero to determine the d1 and d2

positions, hence another estimate of the d=0 position

was found.

3.3 Refractive Index of Glass

The Hg lamp was used to calibrate the interferometer

for an incandescent light source. The tilt fringes were

adjusted such that they were as broad as possible.

Returning to the d=0 position and using a

spectroscope, the micrometer was adjusted until channel

fringes could be seen on a continuous spectrum. At this

point, coloured fringes could be seen at the output. The

central fringe is known as the white light fringe (WLF)

and will be used to determine the group refractive index

of glass. A glass plate was placed at one arm of the

interferometer, and using an estimated value for the

refractive index lead to an estimated adjustment of d

given in Eqn. 4 as the WLF disappeared. The

micrometer was adjusted in baby steps until the fringes

reappeared on the glass and using Eqn. 4 again, the

group refractive index of the glass was found.

4. RESULTS AND ANALYSIS Table-1 shows the measured results for d=0 using the

Fig. 1: Schematic of apparatus. Input is positioned at the

source and output is positioned at the observer side. C and

B on the diagram represent the compensator and

beamsplitter respectively.

Alex Trinh Senior Physics Report 18 April 2014

3

telescope focused at infinity. The average value for the

position was (11.60 ± 0.09) mm with the error

calculated by taking the standard deviation of the

repeated attempts. All measurements were performed

qualitatively, hence finding the position where the field

of view was completely illuminated was difficult due to

light coming from other sources. To resolve this issue, a

black cloth was placed over the interferometer to block

the background light. Also, the contrast of the fringes

could be increased by adjusting the tilt of M1 so that it

could be easily viewed. The visibility of the fringes was

improved by placing a slit at the input. Non-localised

fringes are produced but are faint because the light

source was uncollimated. Placing the slit at the input

collimates the light and allows it to propagate parallel to

the optical axis. If the virtual mirror M1’ is tilted, the slit

focuses the light onto the wedge formed by M1 and the

virtual mirror and will act as though d has decreased.

Table-2a shows the measured values for the optical

path difference found by finding where the visibility

goes to zero with the Na lamp. The d=0 position was 11.68 ± 0.11 mm, found by taking the average

between d1 and d2. The error in d was found using the

standard deviations of d1 and d2. The value of X was

found to be (0.58 ± 0.04) mm and was found by taking

the average of all attempts to find X. Again the error in

X was found by calculating the standard deviation of all

the X’s. Using this optical path difference, the

separation in the Na doublet was calculated to be

5.99 ± 0.41 × 10−10 m.

There is a large discrepancy in row 4 of Table-2a,

where d2=11.61 and d1=11.30, as it is noticeably smaller

compared to the other results. The second and last

calculation of X, both X=0.54, is noticeably smaller than

the other results for X. These values may have affected

the final calculation for X and the d=0 position and may

be a result of bad lighting, which can be rectified by

using a black cloth to block background light. Though

these discrepancies existed, the new d=0 position falls

within the uncertainty found in our first measurement.

The theoretical wavelength difference for the Na

doublet is 5.97 × 10−10 m [5]. Comparing this value to

the calculated value, our calculation is quite consistent

with theoretical values as the theoretical value falls

within the uncertainty of the calculated value. This

means that the discrepancies found when calculating the

optical path difference and the d=0 position can be

considered as minor as it did not affect the final

calculation by a large amount and also means that we

can be certain about our measurements.

Table-2b shows the measurements for the Hg lamp.

The coherence length of the green line was found to be 3.56 ± 0.26 mm and the linewidth was found to be

(8.38 ± 0.61) × 10−11 m. Error was calculated using

the same method as in the measurements of the Na

lamp. We see variations in the values for X which are a

result of a large coherence length of the green line. The

FWHM was determined by eye which is highly

Table-2a: Optical path difference measurements for Na lamp

obtained by taking positions where minima occurred. Table-

2b: Coherence length measurements for Hg lamp obtained

using the method used in Na lamp measurements

(b)

(a)

Table-1: Repeated measurements of the d=0 position using

telescope. Measurements are in mm

Alex Trinh Senior Physics Report 18 April 2014

4

inaccurate as it is very objective, as it differs from

person to person. It was also difficult because of the

particularly large coherence length of the green line,

making it harder to determine where half intensity was

located. A more accurate method for determining the

FWHM is necessary for future investigations.

Photomultipliers may be used in place of the naked eye

such that we can determine the half intensity of the light

at the output as a function of d.

Table-3 shows the measurements for the WLF

position with and without the glass. The WLF position

was found to be at (12.032 ± 0.004) mm, and the WLF

position with the glass plate at one arm of the

interferometer was found to be at (11.465 ± 0.005). Taking the difference between the two values, the shift

in d, and using Eqn. 4, the group refractive index was

found, having a value of 1.554 ± 0.008. The errors of

the WLF positions were found by calculating the

standard deviation of the different WLF positions found

by repeating the experiment, while the error on the

group refractive index was found by considering the

relative errors of the WLF positions. Though it was

difficult to locate the channel fringes using the

spectrometer, due to background light which made the

fringes appear faint, it was enough to give an estimate of

the positions, with errors having values of 0.03% and

0.04% respectively. This may be a result of high

contrast fringes, allowing us to determine the positions

more easily compared to previous measurements.

Overall, most discrepancies were due to background

light and low contrast fringes, which can be easily

rectified by using a black cloth to cover the

interferometer and adjusting the tilt on M1 to limit

the number of fringes in the field of view.

The group refractive index of common optical glass

is 𝑛𝑔 ≈ 1.55. This falls within the error of our measured

result so we can be fairly certain that our measurements

were correct.

5. CONCLUSION

This experiment demonstrated the functions of the

Michelson interferometer in viewing and manipulating

interference fringes. It examined the concept of fringe

visibility, allowing us to measure the d=0 position of the

interferometer, which was found to be within the range

of 11.60 mm and 12.03 mm, which in turn allowed us to

determine the coherence length of the green line of a

mercury lamp, measured to be 3.56 mm, and the

refractive index of glass by examining the white light

fringe, measured to be 1.554. Estimates of the d=0

position could have been improved by reducing the

background light to improve the contrast of the fringes.

6. REFERENCES [1] University of Manitoba, nd, Michelson

Interferometer, viewed 13th April 2014,

http://www.physics.umanitoba.ca/undergraduate/

phys2260/Lectures/Intro%20Optics%20- %20PPT%20v2part%2004.pdf

[2] Hecht, E, Optics, 4th ed., Addison-Wesley, 2001

[3] Duke University, nd, Spatial and Temporal

Coherence, viewed 13th April 2014,

http://dukemil.bme.duke.edu/Ultrasound/k-

space/node8.html

[4] Experiment 24. The Michelson Interferometer

Experiment notes

[5] HyperPhysics, nd, Sodium Doublet on Fabry-Perot,

viewed 14th April 2014, http://hyperphysics.phy-

astr.gsu.edu/hbase/phyopt/fabry3.html

Table-3: White Light Fringe positions with and without the

glass at an arm of the interferometer