Measuring the Wavelength Seperation of Sodium D Doublet
Lines and the Thickness of Mylar
G.S. Sagoo
With Laboratory Partner: Nilesh Agnihotri
University College London
18th November 2009
Abstract: The wavelength separation of the Sodium D Doublet Lines measured using a
Michelson Interferometer was found to be (6  0.1) 10 10 m .The accepted value of
5.97  10 10 m [1] is within our experimental uncertainty suggesting that the methodology
was carried out precisely and accurately. The thickness of Mylar was found to be
(1.2  0.3)  10 5 m by using differences in optical path difference using the same
equipment.
1
Introduction
The primary objective for this experiment was to find the wavelength separation of the
Sodium D Doublet using a Michelson Interferometer. The secondary objective was to
find the thickness of a sample of Mylar once the wavelength separation distance was
found.
Figure 1 shows a diagram of a Michelson Interferometer. It works by taking a beam of
light from a source and splitting it into 2, each beam of equal intensity. A path difference
of 2d is introduced to one beam, then both beams are recombined at the beam splitter.
The recombined beam is then reflected to an observer and an interference pattern is
observed.
Figure 1: Schematic of Michelson Interferometer [2]
For two monochromatic light sources, much like the sodium lamp used in this experiment
dark fringes will occur when the optical path difference, d, is:
(1)
2d  m11  m2 2
However this only applies when wavelength 1  2 and where m1 and m2 are integers.
From equation (1) the difference in wavelength can be obtained where:
m
2d
Where λm is the mean wavelength of λ1, λ2 and Δd is the mirror shit.
 
2
(2)
By adding a material of refractive index μ, the optical path length increases by μt where t
is the thickness of the material. Thus upon knowing the refractive index of the sample,
cab be used to calculate its thickness by:
(3)
d  (  1)t
Equation 3 will only be valid if before adding the material the interferometer is adjusted
for zero path difference (i.e d=0).
2
Method
Figure 2 shows a schematic diagram of the apparatus used in the setup. The apparatus
were first calibrated for mirror movement to determine the correction factor [2]. The
correction factor is a constant relating micrometer movement to mirror shift and needs to
deduced so that the micrometer reading accurately reflects the mirror M1’s location.
Once the apparatus was calibrated a sodium discharge lamp was replaced as the source.
Using the micrometer, mirror M1 was then shifted so that fringe visibility varied with
movement, and its location noted. 8 changes in minima were observed and the final
location of the micrometer noted. This was because recording one change in minima
resulted in a large error. Reading 8 minima and dividing the result by 8 was a far better
way of getting an accurate result of one change in minima. From the two micrometer
readings the wave separation of the sodium d doublet lines can be deduced.
Figure 2: Diagram showing setup of apparatus [2].
After recording this data, M1 was adjusted so that the centre fringe filled the field of view
as shown on Figure 3.1. M2 was then moved so that 7 fringes were in view, then M1 was
then re-adjusted so the curvature of the fringes were straight, almost so that they were
parallel to each other as shown in Figure 3.2. The apparatus is now calibrated for zero
path difference, so the sodium discharge lamp was replaced with a white incandescent
lamp. Adding a crosshair to the incoming beam before it is observed and by slowly
changing the position of M1, coloured fringes were observed and the location of the
central dark fringe was noted when it was lined with the crosshair on the micrometer. The
sample of Mylar was placed in arm one of the Interferometer and M1 shifted so that the
central dark fringe in line with the crosshair and its corresponding micrometer reading
noted. This was repeated for arm two.
Figure 3.1: Diagram showing
centre of circular fringes
Figure 3.2: Diagram showing
straight fringes
3
Results and Analysis
Given that the correction factor is a constant relating mirror shift with micrometer
movement, the mirror shift must therefore be:
(4)
d  fL
Where f is the correction factor and ΔL is the micrometer movement.
Also since the mirror shift is essentially the optical path difference introduced so that
fringes are observed, the mirror shift must be equal to:

(5)
R
2
Where R is the number of fringes observed when the micrometer is moved through a
distance of ΔL. Since the fringe spacing was very small, when turning the micrometer,
some fringes passed through the field of view when returning the hand from maximum
rotation, thus leading to the number of fringes observed to be miscounted. With the error
associated with a micrometer reading of 0.005m, equating equations (4) and (5);
combining these errors gave an error in the correction factor was given by:
d 
 R 
 eL 
(6)
f  
 
  f
 R 
 L 
Where eΔL is the error associated with the difference in micrometer readings.
Using equations (5) and (6) with the data shown in Figure 4, a correction factor value of
0.192  0.004 was obtained
2
2
Figure 4: Table of Results Showing Data used in Calculating f , the Correction Factor
Initial Micrometer Final Micrometer
Reading (mm)
Reading (mm)
.
17.000
9.650
10.000
9.875
9.575
Wavelength of Filter (given):
16.690
9.320
9.710
9.575
9.305
Difference in
Error in
Error in
Number of
Micrometer
Micrometer
Number of
Fringes
Readings (mm) Difference (mm)
Fringes
0.310
0.00707
211
10
0.330
0.00707
220
15
0.290
0.00707
204
10
0.300
0.00707
204
10
0.270
0.00707
201
5
5.46E-07 m
Mirror Shift
(mm)
0.05761
0.06007
0.05570
0.05570
0.05488
Error in
Mirror Shift
(mm)
0.00273
0.00410
0.00273
0.00273
0.00137
Correction
Factor
0.18584
0.18202
0.19207
0.18566
0.20326
Weighted Mean of Correction Factor, f :
0.19244
Uncertainty on Weighted Mean of Correction Factor, f : 0.00431222
Figure 5 shows the data used in calculating the wavelength separation. Due to time
constraints the procedure set in collecting these results was only carried out once.
Initial Micrometer Reading:
Final Micrometer Reading:
Error In Micrometer Reading:
Number of successive minmums recorded:
ΔL
Correction Factor:
fΔL
λm
Δλ
1.35E-02
1.50E-03
5.00E-06
8
0.00149
0.19244
0.00029
5.89E-07
6.0403E-10
m
m
m
Error in ΔL
Error in ΔL for one fringe
Error in Correction Factor
Error in fΔL
Error in Δλ
7.1E-06
8.8E-07
4.3E-03
6.4E-06
1.4E-11
m
m
m
m
m
m
m
m
Figure 5: Data showing results gathered for the calculation un
the difference of the Sodium D Doublet Lines.
4
Error in
Correction
Factor
0.0097746
0.0130091
0.010515
0.0100986
0.0073418
Substituting equation (4) into (2), the wavelength separation becomes:
 
m
2 fL
(7)
Given the mean wavelength of the Sodium D Doublet to be 589.3 nm [2], the errors
associated with the correction factor and micrometer readings needed to be propagated
for the wavelength separation to have a correct error associated with it. This is given by:
2
 f   eL 
e     
  
 f   L 
Where eΔλ is the error associated with the wavelength separation.
By applying equations (7) and (8), the wavelength separation was found to be:
  (6  0.1)  10 10 m
2
(8)
The accepted value for the wavelength separation of the Sodium D Doublet Lines is
5.97  10 10 m [1], which is within our experimental uncertainty suggesting that the
acquisition of the correction factor and experimental methodology was carried out
precisely and accurately considering the equipment used. However it is felt that the
micrometer movement sensitivity needed to be decreased. Because of the high sensitivity
fringe counts were not counted properly, leading to increased error in the correction
factor. By using a less sensitive device, when returning the hand from maximum rotation
the micrometer should not move, increasing precision.
Upon determining the separation in wavelengths, the thickness of a sample of Mylar
needed to determined. Figure 6 shows the data collecting during the experiment which
was used to determine the thickness of Mylar. Given the refractive index of Mylar to be
1.64 [2], substituting equation (4) into (3), it can re-arranged so that:
t
Location of Mylar
Arm
Arm
Arm
Arm
Arm
Arm
1
1
1
2
2
2
f L
 1
Position of Position of
The
dark fringe dark fringe
difference in
without
with
position/mm
Mylar/mm Mylar/mm
6.755
6.720
0.035
6.755
6.720
0.035
6.755
6.720
0.035
6.755
6.800
0.045
6.755
6.800
0.045
6.755
6.800
0.045
Average
0.040
(9)
Combined
Error on the
micrometer
reading/mm
0.007
0.007
0.007
0.007
0.007
0.007
0.001
Figure 6: Table of results showing difference in micrometer readings with
and without the sample of Mylar in place.
As stated above there were errors associated with both f and ΔL. By using the same
method for calculating the error in wavelength separation, the error in the thickness of
Mylar can be determined by:
5
2
 f   eL 
  
t  
(10)
 t
 f   L 
Combining equations (9) and (10) the thickness of the sample of Mylar was found to be:
t  (1.2  0.3)  10 5 m
2
Although there is no accepted value for the thickness of Mylar, our experimental value of
(1.2  0.3)  10 5 m is of the right order of magnitude and seems sensible. Our peers
achieved a value of 1.4  10 5 m which lies within our experimental uncertainty
suggesting the values achieved are precise, but may not be accurate. The main problem
that faced us in determining the thickness of Mylar was that the sample was not tight in
the holder, meaning that when viewing the sample through the interferometer, the straight
fringes without the Mylar, refracted at random angles, leading to these fringes to become
wobbly. Since the sample of Mylar was binded into the holder, it was impossible to
remove and make tight, but in future the sample must be tight so ‘random’ refraction can
occur.
Conclusion
In conclusion the wavelength separation of Sodium D Doublet Lines measured using a
Michelson Interferometer was found to be   (6  0.1)  10 10 m , in which the accepted
value of 5.97  10 10 m [1] lies within, suggesting that the methodology of the experiment
was correct and executed precisely and accurately. The thickness of Mylar was measured
to be t  (1.2  0.3)  10 5 m using the same equipment. Our peers measured the Mylar to
be t  1.4  10 5 m suggesting that both our methodologies were correct. Their value lies
within our experimental uncertainty, but without knowing our peers uncertainty it is false
to say for sure that their value is correct, as it may be accurate, but not precise.
References
1. Spectrum Lines, Zaidel (1976) .
2. Experiment O4 Lab Script, Dept Physics and Astronomy, UCL, course PHAS2440, (2003).
6