Measuring the Wavelength of Hg and Na Light using a Michelson
Interferometer
Abstract: A Michelson interferometer has been used to determine
the wavelength of the green emission line from a mercury lamp. The
measured wavelength was 549.0(9)nm. Using the known
wavelength of the same line, 546.074nm, the reduction lever of the
interferometer was found to reduce the micrometer travel by a ratio
of 5.026(8):1, rather than the quoted 5:1. Using this calibrated
reduction ratio, the average wavelength of the yellow sodium
doublet was determined to be 589.3(10)nm and the splitting between
the lines forming the doublet was 0.5971(5)nm. These are in
excellent agreement with the literature values of 589.294nm and
0.5974nm, respectively.
The literature value of the same line is 546.074nm, and this was used
to calibrate the ratio of the reduction arm. A plot of micrometer
reading versus Nl/2, where l=546.074nm should give a straight line
with a gradient equal to the gearing ratio.
3.16
Leverarm corrected micrometer reading versus fringes
from Hg light (gradient is lambda/2)
3.14
5:1 reduction
micromter reading (/5 mm)
Micrometer
Mirror 1
Ground
Glass
Mirror 2
3.12
3.1
3.08
3.06
3.04
3.02
Light
Source
3
0
50
100
150
200
250
300
Fringes
350
400
450
500
Compensator
Filter
Beamsplitter
0.0012
The data are shown in Figure 3. The error bars are smaller than the
symbols used to plot the points and have been omitted. A leastsquares fit to the data using LINEST gave a gradient of 5.026(8). A
plot of the residuals of the fit showed no systematic deviations, but
again suggested that the estimated uncertainty of ±0.002mm in the
micrometer readings is probably overestimated.
Residual versus Fringes
Micrometer Reading (mm)
(all ±0.002)
Corrected for Gearing (mm)
(all ±0.0004)
15.735
15.530
15.391
15.186
15.049
3.1470
3.1060
3.0782
3.0372
3.0098
A plot of micrometer reading versus the number of fringes passed
should be linear with a gradient l/2. The data are shown in Figure 2a.
The error bars are smaller than the symbols used to plot the points and
have been omitted. A least-squares fit to the data using LINEST gave
a gradient of 274.5(5)nm/fringe. The wavelength of the green
emission line is therefore 549.0(9)nm. A plot of the residuals (Figure
2b) shows no systematic deviations, but suggests that the estimated
uncertainty of ±0.002mm in the micrometer readings is probably an
overestimate.
Residual (mm)
0
-0.0004
15.8
Micrometer reading versus N x lambda/2
(gradient is (–) the gear ratio)
Micrometer reading (mm)
15.7
-0.0008
-0.0012
15.6
0
100
200
300
400
500
Fringes
15.5
15.4
15.3
15.2
15.1
15
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
N x lambda/2 (mm)
Figure 2: (Top) Gearing corrected micrometer
reading versus fringe number for the green mercury
emission line. The dashed line through the data
points is the best-fitting straight line from the leastsquares fit. (Bottom) The residuals from the leastsquares fit. The distribution of the points suggests
that the uncertainties are probably overestimated.
0.005
0.006
Residual versus N x lambda/2
Micrometer reading vs n x (lambda^2)/2 for Na doublet
0.003
0.005
0.001
-0.001
-0.003
0
150
250
400
500
Finally, the interferometer and the calibrated lever arm ratio were
used to determine both the average wavelength of the two emission
lines that make up yellow sodium-doublet, and their wavelength
splitting.
0.0004
micromter reading/gearing (m)
Fringes
Figure 1: Schematic diagram showing the layout of
the Michelson interferometer.
0.004
0.003
0.002
0.001
-0.005
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0
0
N x lambda/2 (mm)
5E-13
1E-12
1.5E-12
2E-12
The average wavelength of the sodium doublet was obtained by
counting fringes, as for the 549nm Hg line. From a plot of gearingcorrected micrometer reading versus fringes (using the
experimentally determined gearing ratio), a least-squares fit to the
data using LINEST gave a gradient of 294.6(5)nm/fringe. The
average wavelength of the sodium doublet is therefore 589.2(9)nm,
in excellent agreement with the literature value of 589.294nm.
Contrast
Minimum (n)
0
1
3
4
7
8
13
14
2
n l /2 (m)
0
–13
1.73634´10
–13
5.20901´10
–13
6.94535´10
–12
1.21544´10
–12
1.38907´10
–12
2.25724´10
–12
2.43087´10
Micrometer reading (mm)
(all ±0.01)
24.245
22.715
19.730
18.260
13.870
12.415
5.100
3.630
2.5E-12
n x lambda/2 (m)
Figure 3: (Top) Micrometer reading versus N ´l/2,
with l= 546.074nm. The dashed line through the data
points is the best-fitting straight line from the leastsquares fit. (Bottom) The residuals from the leastsquares fit. The distribution of the points suggests
that the uncertainties are probably overestimated.
3.00E-05
Residual vs n x (lambda^2)/2 for sodium doublet
2.50E-05
2.00E-05
Residual (m)
Using a mercury lamp with a green filter to transmit only the
546.074nm green emission line, the number of localised fringes that
passed the centre of the field of view when mirror M1 was translated
was determined.
0.0008
Residual (mm)
In a Michelson interferometer (Figure 1), monochromatic light from
an extended source is divided into two equal-amplitude beams by a
beam splitter. One beam travels towards a fixed mirror (M2) and is
reflected back to the beam splitter. The other beam travels to a
moveable mirror (M1), and is reflected back to the beam splitter. The
two beams then recombine and are then detected showing
interference fringes. If the effective pathlength of one of the optical
path lengths is changed, then any given point on the interference
patterns shifts from light to dark, or vice-versa, for each halfwavelength of path length change. Thus if N fringes shift across the
field of view when M1 is translated, then the distance x moved by M1
is Nl/2.
1.50E-05
1.00E-05
5.00E-06
0.00E+00
-5.00E-06
-1.00E-05
0
5E-13
1E-12
1.5E-12
2E-12
2.5E-12
n x (lambda^2)/2 (m)
Figure 4: (Top) Micrometer reading versus Nl2/2,
with l= 546.074nm. The dashed line through the
data points is the best-fitting straight line from the
least-squares fit. (Bottom) The residuals from the
least-squares fit show a clear systematic deviation.
To determine the wavelength splitting of the doublet, the distance
traveled by mirror M1 between successive contrast minima in the
fringe pattern was determined. The data for 14 such minima are
shown above. A linear plot of the gearing corrected micrometer
(using the calibrated ratio of 5.026(8)) versus Nl2/2 is shown in
Figure 4a. The error bars are again smaller than the symbols used to
plot the data. A linear least-squares fit to all the data using LINEST
gave a wavelength splitting of 0.5951(9)nm. However, a plot of the
residuals (Figure 4b) clearly shows a systematic deviation from the
straight line. Refitting the data omitting the two worst fitting data
points (n=0 & 1) gave a wavelength splitting of 0.5970(6)nm, in