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Calculating wavelength of a laser through a diffraction pattern
Дисциплина:  Физика и энергетика 

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15
Laboratory work
Calculating wavelength of a laser through a diffraction pattern
Introduction:
The aim of this experiment was to determine the wave length of a particular light source, in //our case a red laser pointer, through single slit diffraction. Manipulating the results found through measuring the interference patterns, one can easily determine the wavelength. The expected wavelength is s=
Hypothesis:
The laser emits a ray with a wavelength л. Once it passes through a slit of width b, it forms a diffraction pattern on the diffraction plate. The distance between the diffraction plate and the endpoint is marked by L. The length of the fringe, the dark areas in the diffraction pattern, is noted by x. The relationship between all of the factors can be shown in the following ratio:
The ratio between the wavelength and the diffraction slit is equal to the one between the fringes in the diffraction pattern and the distance between the aperture and the projection plate. The bigger the aperture, the smaller the ratio between the fringes and the distance between the projection plate and slit mask, so the fringes will become smaller, as the This formula can be simplified:
According to the found ratio, the wavelength can be calculated through multiplying the fringe length by the aperture length ad dividing it by the distance between the slit and the projection plate. The resulting wavelength should fit within the accepted range between 650 nm and 750nm.
Apparatus:
Stand
Slit mask
Diffraction plate
Meter stick base
Laser pen
Projection plate
Variables:
Independent
The distance between diffraction slit and diffraction plate with every trial the distance will be changed.
The aperture width with every trial, the apertures will be varied in order to see how the diffraction pattern changes.
Dependent:
The resultant diffraction pattern it will vary with every change that is made with the distance between the aperture and projection plate.
The width of the fringes will vary with every change made to distance L and aperture width.
Controlled
The wavelength the same laser pointer will be used for all the trials; the frequency and
The slit width for each particular trial; 3 slits will be used, each of them has a particular wave length.
The distance between the slit and the projection plate; for each trial, the diffraction window and projection plate will be held constant.
The position of the projection plate it is fixed at the end of the meter stick, and is not moved throughout the entire experiment.
Method:
1. Assemble a stand
2. Take the meter stick base, place the Slit mask.
3. Place the projection plate behind the slit mask
4. Place the diffraction plate window on top of the slit mask
5. Adjust the slit mask and the projection plate so that they are 0.4 meters from each other.
6. Take the laser pointer, and place it on the stand, so that the pointer is at the same level as the diffraction slit window.
7. Turn on the laser pointer
8. Adjust the position of the pointer, so that the ray passes directly through the slit mask.
9. Fix the diffraction plate window, so the light passes directly through aperture A.
10. Take a ruler and measure the average width of the fringes on the projection plate/ and measure the distance between the last and first maxima, counting the fringes.
11. Record the measurement into the data table, mentioning the uncertainty.
12. Switch to aperture B.
13. Record data in the data table.
14. Switch to aperture C and record results.
15. Move the slit mask and diffraction plate window, so that they are 0.34 meters away from each other.
16. Repeat steps 914
17. Move the slit mask and diffraction plate window so that they are 0.54 meters away from each other.
18. Repeat steps 914
Data Collection and analysis
b^{A}=0.04mm b^{B}=0.08mm b^{C}=0.16mm
The scientist can try to measure the average width of a fringe using a ruler, but that can be proven to be a wrong and simplistic approach. Through the following calculations, this method of finding the width of a fringe will be shown to be wrong.
Trial 1
Length/m  aperture/mm  fringe/mm  
0.40±0.01  0.04  2.00±0.50  
0.40±0.01  0.08  1.00±0.20  
0.40±0.01  0.16  0.50±0.10  
A:
л= 2x10^{7}m
B:
л= 2x10^{7}m
C:
л= 2x10^{7}m
Uncertainties:
Дл= 5.2x10^{8}m
Дл= 4.25x10^{8}m
Дл= 5.2x10^{8}m
The average wavelength found here is 200nm±50nm. This is clearly irrelevant towards the accepted wavelength of
Trial  L/m  Slit aperture b/m  Number of fringes Y/m  D from 2nd max to last /m  
1  0.50±0.005  A  4±1  0.031±0.001  
0.50±0.005  B  5±1  0.019±0.001  
0.50±0.005  C  6±1  0.012±0.001  
2  0.400±0.005  A  4±1  0.030±0.001  
0.400±0.005  B  6±1  0.025±0.001  
0.400±0.005  C  7±1  0.015±0.001  
3  0.300±0.005  A  5±1  0.022±0.001  
0.300±0.005  B  9±1  0.023±0.001  
0.300±0.005  C  19±1  0.024±0.001  
In order to find the wave length, a calculation involving the width of a fringe between two maxima of the diffraction pattern has to be performed:
(И being the angle between the fringe width and the distance between the aperture and the projection plate)
To complete this, the width of a fringe has to be found. This can be done by directly measuring one of the fringes but the found fringe length would be very inexact and would lead to a deceiving result. Instead of that, one can measure the distance between the first and last maxima, and divide it by the number of fringes in between those two points. So the average width of a fringe would be the distance from the first maxima to the last over the number of fringes on the particular side of measurement.
So the fringe width X, would equal to this:
As the distance between the maxima Y had to be measured by the scientist manually as well as the number of fringes, this could lead to a reasonable degree of error and uncertainty. The main uncertainty would be in measuring the various distances: the diffraction pattern of the laser could be seen clearly only in a dark environment, while the ruler with which the distances were recorded was visible in the light. This led to a serious uncertainty.
Trial 1
A
X=0.0775m
6.20x10^{7} =620nm
B
X=0.00380m
6.08x10^{7} =608nm
C
X=0.002m
6.4x10^{7} =640nm
Uncertainties
A
B
Trial 2
A
X=0.0075m
7.50x10^{7} =750nm
B
X=0.00416m
8.3x10^{7} =833nm
C
X=0.00416m
8.57x10^{7} =857nm
Uncertainties:
A
B
C
Trial 3
X=0.0044m 7.5x10^{7} =750nm
B
X=0.00255m
6.81x10^{7}m= 681 nm
C
X=0.00126m
6.73x10^{7}m= 673nm
Uncertainties
A
B
C
Calculated wavelengths
Trial  Aperture/m  Л/nm  
1  A  620±180  
B  608±158  
C  640±166  
2  A  750±188  
B  833±108  
C  857±257  
3  A  587±152  
B  681±115  
C  673±74.0  
Average    694±155  
Conclusion and evaluation
According to the hypothesis, through this experiment, one was supposed to find the wavelength of a red laser pointer. According to various sources the accepted range for the wavelength of a red light source is between 650nm and 750nm.
In order to calculate the wavelength of a wave, the width of the fringes, corresponding to the apertures and distances had to be found. There were two ways of completing this process: the scientist could find the fringe width manually by directly attempting to measure it by a ruler, or the scientist could measure the distance between the first and last maxima and then divide it by the total number of fringes, to come up with an average value for a fringe.
After obtaining the fringe width, on had to perform a simple calculation:
Where и was the angle between the total distance L, between the projection plate and the aperture, and the ray to the last maximum; as и was such a small angle the sin ratio is extremely close to
The equation could be simplified by inserting the ratio instead of the sin formula.
The scientist tried to obtain the wavelengths using the first set of wave lengths found manually, but the results were not satisfying at all.
The wavelengths found through this method of calculation are all drastically lower than the accepted wavelength of a red laser pointer or other source of light. The average wavelength found through these calculations is 200nm±50nm. Even regarding the uncertainties, the found wavelength still doesn't match to correspond to one of the red laser pointer.
This suggests that an error had been made while finding the width of the fringes. The found wavelength is roughly 500nm away.
After using the first method of retrieving the fringe widths, the scientist applies the second one, where the average width of a fringe is measured by finding the distance from the first maxima to the last and dividing it by the number of fringes.
The resulted wavelengths found through this calculation were much less precise, but were within a more reasonable range that was within the accepted wavelength spectra. The average value of the calculated wavelength was 694nm±155nm.
The average value of the accepted range for red laser pointer wave length is 700nm±50nm, so the calculated wavelength enters the range of the accepted one, totally supporting the claims of the hypothesis. Through moderately simple calculations, one can find the wavelength of a beam of light.
Even though the results of this calculation support the hypothesis and a justified to be valid, by the correlation to the world accepted data, all of the collected data wasn't completely precise and accurate.
The range between the smallest and largest wave lengths was 270 nm. It is also useful to note, that the most precise data was collected when the projection plate was further away from the diffraction slit window.
It can be noted that the base that was used for the whole construction was only 0.70m long, therefore limiting the distance that held between the diffraction window and the projection plate. On this graph one can see the variation of the distance between the second and last maxima on the diffraction pattern with the distance L of the projection plate from the diffraction window.
As the distance L grows, the range between maxima grows. This can lead to a conclusion that the further the projection plate is away from the aperture, the better and more precise the results will be.
Another limitation was that all the measurements had to be done manually, therefore this led to a vast area of uncertainty. The uncertainty for the distance between the diffraction slit window and the projection plate was estimated due to the overlapping of the plate and the slit mask. And the thickness of the both, affecting the distance, was estimated to be 0.005m. Another factor of uncertainty was the measurement of the distance between the second and last successive maxima. The diffraction pattern was more visible when in a darker environment, while the ruler o the other hand was better seen in the light. This led to a certain uncertainty within the finding the distance Y. Also, one could never be sure if the laser beam entered the aperture and slit mask through a completely right angle, therefore providing a completely correct diffraction pattern. The slit mask was of a larger width than the laser beam, and therefore gave it the freedom to pass at a small angle, which nevertheless would affect the final pattern.
Taking in account all of these limitations, now the scientist can propose several improvements. As it was noticed, the larger the separation between the diffraction window and the projection plate was, the clearer and were the diffraction patterns, and the distinction between the patterns produced by different slits. To come up with diffraction patterns, that are easier to read and process, one can increase the separation between the apertures and the projection plate. In order to decrease the inconvenience caused by the light one can observe the diffraction pattern in complete darkness, and using a writing utensil indicate marks in the places of the 2^{nd} and last maxima. When the light is turned on, one can use the taken measurements and use them to calculate the resultant wavelength. To increase the accuracy of the calculations, one can use apertures with a larger width, as they produce a diffraction pattern with narrower fringes and maxima. The laser pointer could be fixed to the base or the stand, but adjusted to have a beam of light perpendicular to the diffraction slit window, so the diffraction pattern wouldn't be affected by the angle variations.
Even though this experiment has several limitations and points of improvement, the overall results achieved through the process and calculations fully support the accepted wavelength value and the hypothesis. Therefore, one can conclude that the experiment was successful.
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