PHYS 351
Augustana College
Winter 2010-2011
Observation and Determination of the Wavelengths of a Diode Laser and Sodium D lines
Derick Peterson, Augustana College, Rock Island, Illinois, USA
ABSTRACT
This paper presents a utilization of the Michelson Interferometer in determining the
wavelengths of a diode laser and sodium D lines. The distance between a repeat in fringe patterns (d)
was measured for a diode laser and sodium D-lines, and this was used to calculate the wavelengths of
each light source (667.5 +/- 6.79%nm and 633.8 +/- 8.39%, respectively). The distance between adjacent
visibility maxima in the sodium source was also recorded and was used to calculate the wavelengths of
each individual wavelength of the 2 components of the sodium D-lines (633.4 +/- 8.40% and 634.2 +/8.39%).
INTRODUCTION
The interferometer is a device which can measure changes in length at great accuracy by
observation of interference fringes. In 1887 Michelson and Morley used a highly-accurate
interferometer with 11-meter paths to test for the existence of the luminiferous ether, and again in
1907, when Michelson was able to show conclusively that the standard meter is equal to 1553163.5
wavelengths of a certain monochromatic red light. In this experiment we will be using the
interferometer to find the wavelengths of certain kinds of light.
The basic Michelson Interferometer is shown in figure 1. The incoming light beam hits the beam splitter
and half of the light is transmitted through to M1 through the compensator plate and the other half
reflected toward M2. M1 and M2 reflect the light back to the beam splitter, and half of each beam
PHYS 351
Augustana College
Winter 2010-2011
reaches the detector while the other half is pointed at the source and lost. The position of M2 can be
altered using a precision micrometer, and in this way the path length of that light beam can be altered.
By placing a lens in front of the laser source an interference pattern. This interference is due to the
phase difference between the two highly coorelated beams, which is due to the different optical paths
each beam travels. Since the beam has to travel to and from each mirror, moving M2 by a distance d
would increase the path length D by D = 2d. If the fringes change in a way that is sinusoidal, the fringes
should repeat their pattern when the path length changes by λ, so using these two relationships we can
get λ = 2d.
The sodium D lines are most easily seen as a single wavelength, but in truth they are made up of
two individual wavelengths spaced close together. When both of the lines are in phase then the fringes
are most clear, and if you scroll through them you can find other visibility maxima. Over the distance
between adjacent visibility maxima (D), each wavelength of light should give rise to an integer number
of fringes since they both repeat their behavior, so we can write
D = m1λ = m2λ
(1)
The shorter wavelength should contain more wavelengths since the distance is equal for both,
specifically 1 more wavelength (m2 = m1 + 1). Usuing this and equation 1 we can therefore find that the
difference between the 2 sodium D-line wavelengths is
λ2– λ1=( λ2λ1)/2D
We can use this relationship to calculate λ2, λ1, and Δλ.
(2)
PHYS 351
Augustana College
Winter 2010-2011
METHOD
In order to properly utilize the interferometer, it first had to be aligned. The PASCO scientific OS9255A Precision Interferometer was setup as shown below in figure 1. The laser beam was adjusted so
that it was parallel with the base in such a way that it struck the center of the movable mirror (M1) and
reflected back into the laser aperture. The beam-splitter was placed as shown so that the beam
reflected to the center of the adjustable mirror (M2). The compensator plate was positioned as seen at a
45o angle, so that the beam hit the plate near the center. The viewing screen was not used, so two sets
of bright red dots were seen on the far wall of the room, where they could be easily seen. The beam
splitter was adjusted so that the two dots were relatively close, then the two thumbscrews on the back
of M2 were used to fine-tune the dot’s position until they overlapped.
Figure 1. The Michelson Interferometer
To obtain the wavelength of the diode laser, a converging lens was placed as shown above in
figure 1, and the position of M2 (the only mirror adjusted after this point) was adjusted until a point was
found where circular fringes in the diverging beam could be clearly seen. The mirror was then further
adjusted in order to determine the average distance that you move it in order to get a repeat in fringes.
Rather than observing the fringes on the viewing screen, we projected the fringes onto the far wall in
order to get a more accurate reading by comparing the relatively larger fringes with markings made on
PHYS 351
Augustana College
Winter 2010-2011
the wall. The mirror was adjusted so that ten repeating fringes passed, and the distance moved, d, was
recorded. This distance was then used to calculate the wavelength of the light of the diode laser.
To obtain the wavelength of the sodium D lines, the same setup as that of the diode laser was
used, but the converging lens was removed. Instead, the converging lens of the human eye was used by
looking directly into the instrument to observe the fringes. In a similar fashion as with the diode laser,
the mirror was adjusted so as to get a repeat in the pattern of fringes. The distance, d, between ten
fringes was recorded, and this was used to calculate the wavelength of the sodium line, which is the
average value of the two sodium D lines.
To obtain the wavelength difference between the two close lines that comprise the sodium D
line, we used the average wavelength and observations of the visibility of the fringes. The same setup as
in finding the wavelength of the sodium D lines was used. The mirror was adjusted across a very wide
range, and the changes in the fringe patterns from sharply visible to ‘washed out’ were observed. It was
noted roughly where two visibility maxima were, and the distance between these points was recorded.
This value was used to calculate the difference between the two sodium D line wavelengths, and each
individual wavelength.
RESULTS
Table 1 lists the experimentally determined average values for the wavelengths of the light from
the diode laser and the Sodium D lines. The values for each wavelength were computed using the
formula
λ = 2d
(3)
where d is the distance that the mirror was moved to get a repeat in the pattern of fringes. The smallest
reading on the precision Interferometer is 1 µm, so we assigned an uncertainty to these measurements
PHYS 351
Augustana College
Winter 2010-2011
of .5 µm. These wavelength data were compared against their standard values, and the percent
difference was computed. The error in each experimental value is given as a percentage, and is
computed using the standard propagation of error technique.
Diode laser (λD)
Sodium D lines (λA)
λ (nm)
667.5 +/- 6.79%
633.8 +/- 8.39%
Standard val. (nm)
650
559.3
% diff
2.69%
7.56%
Table 1. Experimental and Standard wavelengths for Diode laser and Sodium D lines.
Table 2 shows the experimentally determined average values for the wavelengths of the light from each
individual Sodium D line (λ1, λ2), as well as that of the difference between the two (Δλ). The wavelengths
were computed using the formulas
λ2 =( -(4D-2 λA ) + sqrt ( (4D-2 λA)2 – 4(-4λAD) ) )/2
(4)
λ1 = λA - λ 2
(5)
Δλ = λ2 - λ1
(6)
where D is the distance between the visibility maxima and λA is the average wavelength of the sodium D
lines, seen in table 1. These data were compared against their standard values, and the percent
difference was computed. The error in each experimental value is given as a percentage, and is
computed using the standard propagation of error technique.
Experimental value
Standard value
% difference
λ1 (nm)
633.4 +/- 8.40%
589.00
7.55%
λ2 (nm)
634.2 +/- 8.39%
589.6
7.57%
Δλ (nm)
0.8153 +/- 14.52%
0.5974
36.48%
Table 1. Experimental and Standard wavelengths for Diode laser and Sodium D lines.
PHYS 351
Augustana College
Winter 2010-2011
DISCUSSION
The percent difference between our experimentally determined values of the wavelengths of
light and their standard values were 2.69% for the diode laser (λD) and 7.56% for the average of the
Sodium D lines (λD), compared to the propagated error of 6.79% and 8.39%, respectively. Thus, both of
these data fell within computed error of the standard. Additionally, the percent difference between the
experimental values of λ1, λ1, and Δλ and their standards were 7.55%, 7.57%, and 36.48% compared to
the propagated error of 8.40%, 8.39% and 36.48%, respectively. Therefore λ1 and λ1, fall within
computed error of the standard, but Δλ did not. This larger percent error is likely due to the fact that Δλ
is a very small value (<1nm) and our interferometer has a measurement uncertainty of .5µm, so even
taking using very large measurements to calculate λ2 and thus Δλ as we did (D was equal to 286.383µm)
will not eliminate this source of error. It is likely that a large part of the error in Δλ arose from human
error in observing the sodium fringes with the naked eye. The sodium fringes proved extremely difficult
to focus on, and this may have resulted in miss-readings. We could improve the accuracy of this
experiment by using a higher-precision interferometer and using a computerized device to observe the
fringes of the sodium band, thus minimizing human error.
CONCLUSION
We have determined the wavelengths of a diode laser, the average of the sodium D lines and
the wavelength of each individual line in the sodium D lines.
References:
1. An Introduction to Error Analysis, John R. Taylor, 2nd Ed. (1997).