Purpose: The aim of doing this experiment was to examine diffraction and interference effects of light passing through various apertures, and use the diffraction patterns obtained by single and double slit apertures to find the wavelength of the light source used. Theory: We know that light can be described by two theories, namely the particle theory and the wave theory of light, each having its own experimental proofs. In this experiment, we examine the interference and diffraction phenomena of light, both of which can be described by the wave theory of light. While interference is just the superposition of waves, diffraction is also any deviation from geometrical optics that results from the obstruction of a wavefront of light. In other words, diffraction is considering the double-slit experiment by taking into account the width of the slit openings, too. Another way of distinguishing between interference and diffraction is to consider the interfering beams in diffraction phenomena as originating from a continuous distribution of sources, whereas the interfering beams in interference phenomena as originating from a discrete number of sources.

This way of treatment of interference and diffraction is a result of Huygens principle which states that every point of a given wavefront of light can be considered a source of secondary spherical wavelets. Hence, superposition occurs between these secondary waves emitted from different parts of the wavefront, taking into account both their amplitudes and phases. Diffraction effects can also be classified according to the mathematical approximations used in calculations. In the case of the light source and the observation screen being very far from the slit, relative to the slit width, the incident and diffracted waves are assumed to be plane and the diffraction type is called Fraunhofer, or far-field diffraction.

In this case, as the viewing screen is moved relative to the aperture, the size of the diffraction pattern changes, but not the shape. We are going to use this kind of approximation in this experiment. We should keep in mind that the Huygens principle used to find the diffraction relations is itself an approximation. When calculating the single-slit Fraunhofer diffraction a rectangular aperture with a length much larger than its width is considered. In this case the intensity of the light reaching the screen at point P, at an angle Î¸ is given by: Is=I0(sin2Î±Î±2)

where

Î±=12kasinÎ¸=Ï€asinÎ¸Î»

In the above relations I0 is the intensity at the middle of the central maxima and a is the slit width. Hence, by taking the limit as ï„ƒâ†’0, we observe that this pattern attains its maximum at Î¸=0. Similarly, equating ï„ƒ=mÏ€, we obtain the minima of the pattern and we get the following relation for this case: nÎ»=asinÎ¸

where n=1,2,3,¦ For small angles we can make the sinÎ¸=tanÎ¸ approximation and, calling L the distance between the slit and the screen, we can get y=LsinÎ¸, where y is the distance from the central maximum to the observation point. For this case, we conclude that on the screen, the irradiance is a maximum at Î¸=0, hence y=0, and it drops to zero at values of y such that y=Î»La . Therefore, we can find Î» using this relation. (Here, y is the average distance between adjacent minima).

When we regard the double-slit diffraction we see that we have to do with two different terms, one of which belongs to the interference pattern, and the other to the diffraction pattern. If we ignore the effect of the slit widths, we get the intensity of the pattern given by only the interference term as I=4I0cos2Î², where Î²=(Ï€bÎ»)sinÎ¸. Here, Î¸ is the angle of observation and b is the slit separation. Nevertheless, since the intensity from a single slit depends on the angle Î¸ through diffraction, we should take into account the diffraction pattern, too. Now, the intensity is given by: I=4I0(sin2Î±Î±2)cos2Î²

In this case ï„ƒ is again Î±=12kasinÎ¸=Ï€asinÎ¸Î». Hence, we conclude that in double slit diffraction the intensity is the product of the interference and diffraction patterns. By analyzing the intensity relation, we observe that an interference minimum occurs whenever Î²=(n+1/2)Ï€ for n=0,1,2,3,¦, and an interference maxima occurs whenever Î²=nÏ€, again for n=0,1,2,¦ Using the approximation sinÎ¸=tanÎ¸, we obtain y=LsinÎ¸, and y=Î»Lb, where y is the average distance between either adjacent maxima or minima.

Data and Results: Part A: Single Slit

Pattern| A| B| C|

Width of the slit, a| 4—10-5m| 8—10-5m| 16—10-5m|

Distance slit-screen, L| 1m| 1m| 1m|

Average dist btw minima, y| 1.67 cm| 0.75 cm| 0.45 cm|

Î»=ay/L| 668 nm| 600 nm| 720 nm|

Error âˆ†y on y| 0.08173 cm| 0.138 cm| 0.0548 cm|

Error âˆ†Î» on Î»=aâˆ†y/L| 32.7 nm| 110 nm| 87.7 nm|

Î»=Î»±âˆ†Î»| 635.5 nm| 710 nm| 632.3 nm|

| y1| y2| y3| y4| y5| y6|

A| 1.8| 1.6| 1.7| 1.7| 1.6| 1.6|

B| 0.5| 0.8| 0.8| 0.8| 0.9| 0.7|

C| 0.5| 0.5| 0.5| 0.4| 0.4| 0.4|

The error on y is found using the relation below:

âˆ†y=i=1N(yi-y)N-1

Part B: Double Slit

Pattern| D| E| F|

Width of the slit, a| 8—10-5m| 8—10-5m| 4—10-5m|

Slit separation, b| 5—10-4m| 2.5—10-4m| 2.5—10-4m|

Distance slit-screen, L| 1m| 1m| 1m|

Average dist btw minima, y| 0.00160 m| 0.00300 m| 0.00155 m| Î»=by/L| 800 nm| 750 nm| 387.5 nm|

Error âˆ†y on y| 0.000342m| 0.000524m| 0.000342m|

Error âˆ†Î» on Î»=bâˆ†y/L| 171 nm| 131 nm| 85.5 nm|

Î»=Î»±âˆ†Î»| 629 nm| 619 nm| 473 nm|

y| D| E| F|

1| 0.138| 0.110| 0.053|

2| 0.141| 0.106| 0.051|

3| 0.143| 0.101| 0.048|

4| 0.146| 0.095| 0.045|

5| 0.148| 0.090| 0.043|

6| 0.151| 0.086| 0.040|

7| 0.154| | 0.038|

8| 0.156| | 0.035|

9| | | 0.033|

We calculated the difference between each successive data to obtain the displacement. Then, we multiplied each displacement value with a factor of (21.5/34.5) because the scale of the linear translator and the interface were not equal. Having done this we calculated the average distance. The error on y is found again by using the relation

âˆ†y=i=1N(yi-y)N-1

Discussion and Conclusion: In part A we considered interference and diffraction pattern of a single slit opening for three different slits. We measured the distance between the source and the slit to be 1m and we used the relations found in the theory part in order to find the wavelength of the light source used. We found the average distance between minima to be 1.67 cm for slit A, 0.75 cm for slit B and 0.45 cm for slit C. Hence, we found the wavelength of the light source to have values of 668 nm for slit A, 600nm for slit B and 720nm for slit C. However, after calculating the error in the average distance and using this error, the wavelengths turned out to be 635.5nm for slit A, 710nm for slit B and 632.3nm for slit C. We know that theoretically the wavelength is expected to be 650±10nm. Our experimental values, despite the fact they are close to, do not fit totally to the expected theoretical ones.

Hence, we argue that any discrepancy in the values found is a result of the imprecise equipment used, especially the light sensor. Furthermore, we claim that these discrepancies are also a result of the fact that we had to move the linear translator with our hand slowly enough so that the detector could detect the intensity peak and the other maxima. Hence, it is very much likely that we could not carry this process out precisely enough as it is required in order to have correct data, since we are human beings and it is impossible for us to achieve such a thing. We also think that the light coming from the surrounding might have had a negative effect on our results since the room where the experiment was carried out was not evacuated well enough. Moreover, we point out that the relations between wavelength, distance between minima and slit width used to find the wavelength and the Huygens principle itself are all approximations, since as it was stated in the theory part, we used far field mathematical approximations in order to obtain these relations.

In part B, we used a double slit opening in order to observe the interference and diffraction pattern. In this case both the slit width and the slit separation have an effect when finding the intensity at a certain point. However, in the relations used to find the wavelength we considered only the slit separation b. In this part, after calculating the error in displacement and using this in Î», we found the wavelength values to be of 629nm for slit D, 619nm for slit E and 473nm for slit F. We observe that, except for slit F, these values of Î» agree with the values found in part A. We claim that the discrepancies in this part are a result of the same reasons causing the discrepancies in part A. As for the case of slit F where Î» turned out to be 473nm (much smaller than the theoretical value) we think that the main reason for such a result is the change in width of the slit, which in this case, unlike the other two cases, is 0.04mm. This leads us to conclude that, as expected theoretically, the width of the slit also affects the intensity pattern, and in these cases more precise relations should be used in order to obtain correct data.

Applications: Interference and diffraction phenomena of light have found a quite large application in science and technology. Understanding these phenomena has led to understanding the world around us and being able to use it in a better way in order to fulfill our needs. Among the most important applications of diffraction for example, is the fact that it is used to obtain accurate information about the atomic scale structure of the matter around us. Since the number of atoms or molecules inside a crystal is arranged in such a way that it resembles a grating with very thin spacing,

diffraction phenomena leads to understanding the insights of each crystal structure.

Diffraction phenomena was also used to learn that the sodium and chloride ions are bonded in a lattice fashion and not molecules, to distinguish between different cubic lattice, to analyze all kinds of materials, even biological samples, etc. Using diffraction interesting things such as hair thickness can also be measured .The interference phenomenon, on the other hand, is used to make highly-wavelength specific mirrors for lasers. Furthermore, interference is the reason why soap bubbles appear colorful. Many other optical coatings owe their optical properties to the interference phenomena. An example of this is the antireflection coatings on lenses that we use everyday. Another application of interference is holography, which is a way of reconstructing three dimensional images with laser light.

Perhaps the most fascinating application of interference is to create holograms. This is done by reflecting a coherent light source, such as a laser, off of an object onto a special film. The interference patterns created by the reflected light are what result in the holographic image, which can be viewed when it is again placed in the right sort of lighting. Moreover, diffraction and interference can be observed when an atom passes through a standard light wave and its position is localized. In this case, the localization can be thought of as the creation of virtual slits leading to the above mentioned phenomena. Diffraction is also used to understand the insights of the ionosphere. All in all, by doing this experiment we learned the importance of the phenomenon of interference and diffraction in our lives.

References:

http://online.physics.uiuc.edu/courses/phys214/spring09/Lectures/Lect04.pdf http://bigbro.biophys.cornell.edu/~toombes/Science_Education/Laser_Diffraction/Diffraction_Lesson.pdf http://answers.yahoo.com/question/index?qid=20080509124425AAyW8bl http://physics.about.com/od/mathematicsofwaves/a/interference.htm URL: http://link.aps.org/doi/10.1103/PhysRevLett.68.472

This way of treatment of interference and diffraction is a result of Huygens principle which states that every point of a given wavefront of light can be considered a source of secondary spherical wavelets. Hence, superposition occurs between these secondary waves emitted from different parts of the wavefront, taking into account both their amplitudes and phases. Diffraction effects can also be classified according to the mathematical approximations used in calculations. In the case of the light source and the observation screen being very far from the slit, relative to the slit width, the incident and diffracted waves are assumed to be plane and the diffraction type is called Fraunhofer, or far-field diffraction.

In this case, as the viewing screen is moved relative to the aperture, the size of the diffraction pattern changes, but not the shape. We are going to use this kind of approximation in this experiment. We should keep in mind that the Huygens principle used to find the diffraction relations is itself an approximation. When calculating the single-slit Fraunhofer diffraction a rectangular aperture with a length much larger than its width is considered. In this case the intensity of the light reaching the screen at point P, at an angle Î¸ is given by: Is=I0(sin2Î±Î±2)

where

Î±=12kasinÎ¸=Ï€asinÎ¸Î»

In the above relations I0 is the intensity at the middle of the central maxima and a is the slit width. Hence, by taking the limit as ï„ƒâ†’0, we observe that this pattern attains its maximum at Î¸=0. Similarly, equating ï„ƒ=mÏ€, we obtain the minima of the pattern and we get the following relation for this case: nÎ»=asinÎ¸

where n=1,2,3,¦ For small angles we can make the sinÎ¸=tanÎ¸ approximation and, calling L the distance between the slit and the screen, we can get y=LsinÎ¸, where y is the distance from the central maximum to the observation point. For this case, we conclude that on the screen, the irradiance is a maximum at Î¸=0, hence y=0, and it drops to zero at values of y such that y=Î»La . Therefore, we can find Î» using this relation. (Here, y is the average distance between adjacent minima).

When we regard the double-slit diffraction we see that we have to do with two different terms, one of which belongs to the interference pattern, and the other to the diffraction pattern. If we ignore the effect of the slit widths, we get the intensity of the pattern given by only the interference term as I=4I0cos2Î², where Î²=(Ï€bÎ»)sinÎ¸. Here, Î¸ is the angle of observation and b is the slit separation. Nevertheless, since the intensity from a single slit depends on the angle Î¸ through diffraction, we should take into account the diffraction pattern, too. Now, the intensity is given by: I=4I0(sin2Î±Î±2)cos2Î²

In this case ï„ƒ is again Î±=12kasinÎ¸=Ï€asinÎ¸Î». Hence, we conclude that in double slit diffraction the intensity is the product of the interference and diffraction patterns. By analyzing the intensity relation, we observe that an interference minimum occurs whenever Î²=(n+1/2)Ï€ for n=0,1,2,3,¦, and an interference maxima occurs whenever Î²=nÏ€, again for n=0,1,2,¦ Using the approximation sinÎ¸=tanÎ¸, we obtain y=LsinÎ¸, and y=Î»Lb, where y is the average distance between either adjacent maxima or minima.

Data and Results: Part A: Single Slit

Pattern| A| B| C|

Width of the slit, a| 4—10-5m| 8—10-5m| 16—10-5m|

Distance slit-screen, L| 1m| 1m| 1m|

Average dist btw minima, y| 1.67 cm| 0.75 cm| 0.45 cm|

Î»=ay/L| 668 nm| 600 nm| 720 nm|

Error âˆ†y on y| 0.08173 cm| 0.138 cm| 0.0548 cm|

Error âˆ†Î» on Î»=aâˆ†y/L| 32.7 nm| 110 nm| 87.7 nm|

Î»=Î»±âˆ†Î»| 635.5 nm| 710 nm| 632.3 nm|

| y1| y2| y3| y4| y5| y6|

A| 1.8| 1.6| 1.7| 1.7| 1.6| 1.6|

B| 0.5| 0.8| 0.8| 0.8| 0.9| 0.7|

C| 0.5| 0.5| 0.5| 0.4| 0.4| 0.4|

The error on y is found using the relation below:

âˆ†y=i=1N(yi-y)N-1

Part B: Double Slit

Pattern| D| E| F|

Width of the slit, a| 8—10-5m| 8—10-5m| 4—10-5m|

Slit separation, b| 5—10-4m| 2.5—10-4m| 2.5—10-4m|

Distance slit-screen, L| 1m| 1m| 1m|

Average dist btw minima, y| 0.00160 m| 0.00300 m| 0.00155 m| Î»=by/L| 800 nm| 750 nm| 387.5 nm|

Error âˆ†y on y| 0.000342m| 0.000524m| 0.000342m|

Error âˆ†Î» on Î»=bâˆ†y/L| 171 nm| 131 nm| 85.5 nm|

Î»=Î»±âˆ†Î»| 629 nm| 619 nm| 473 nm|

y| D| E| F|

1| 0.138| 0.110| 0.053|

2| 0.141| 0.106| 0.051|

3| 0.143| 0.101| 0.048|

4| 0.146| 0.095| 0.045|

5| 0.148| 0.090| 0.043|

6| 0.151| 0.086| 0.040|

7| 0.154| | 0.038|

8| 0.156| | 0.035|

9| | | 0.033|

We calculated the difference between each successive data to obtain the displacement. Then, we multiplied each displacement value with a factor of (21.5/34.5) because the scale of the linear translator and the interface were not equal. Having done this we calculated the average distance. The error on y is found again by using the relation

âˆ†y=i=1N(yi-y)N-1

Discussion and Conclusion: In part A we considered interference and diffraction pattern of a single slit opening for three different slits. We measured the distance between the source and the slit to be 1m and we used the relations found in the theory part in order to find the wavelength of the light source used. We found the average distance between minima to be 1.67 cm for slit A, 0.75 cm for slit B and 0.45 cm for slit C. Hence, we found the wavelength of the light source to have values of 668 nm for slit A, 600nm for slit B and 720nm for slit C. However, after calculating the error in the average distance and using this error, the wavelengths turned out to be 635.5nm for slit A, 710nm for slit B and 632.3nm for slit C. We know that theoretically the wavelength is expected to be 650±10nm. Our experimental values, despite the fact they are close to, do not fit totally to the expected theoretical ones.

Hence, we argue that any discrepancy in the values found is a result of the imprecise equipment used, especially the light sensor. Furthermore, we claim that these discrepancies are also a result of the fact that we had to move the linear translator with our hand slowly enough so that the detector could detect the intensity peak and the other maxima. Hence, it is very much likely that we could not carry this process out precisely enough as it is required in order to have correct data, since we are human beings and it is impossible for us to achieve such a thing. We also think that the light coming from the surrounding might have had a negative effect on our results since the room where the experiment was carried out was not evacuated well enough. Moreover, we point out that the relations between wavelength, distance between minima and slit width used to find the wavelength and the Huygens principle itself are all approximations, since as it was stated in the theory part, we used far field mathematical approximations in order to obtain these relations.

In part B, we used a double slit opening in order to observe the interference and diffraction pattern. In this case both the slit width and the slit separation have an effect when finding the intensity at a certain point. However, in the relations used to find the wavelength we considered only the slit separation b. In this part, after calculating the error in displacement and using this in Î», we found the wavelength values to be of 629nm for slit D, 619nm for slit E and 473nm for slit F. We observe that, except for slit F, these values of Î» agree with the values found in part A. We claim that the discrepancies in this part are a result of the same reasons causing the discrepancies in part A. As for the case of slit F where Î» turned out to be 473nm (much smaller than the theoretical value) we think that the main reason for such a result is the change in width of the slit, which in this case, unlike the other two cases, is 0.04mm. This leads us to conclude that, as expected theoretically, the width of the slit also affects the intensity pattern, and in these cases more precise relations should be used in order to obtain correct data.

Applications: Interference and diffraction phenomena of light have found a quite large application in science and technology. Understanding these phenomena has led to understanding the world around us and being able to use it in a better way in order to fulfill our needs. Among the most important applications of diffraction for example, is the fact that it is used to obtain accurate information about the atomic scale structure of the matter around us. Since the number of atoms or molecules inside a crystal is arranged in such a way that it resembles a grating with very thin spacing,

diffraction phenomena leads to understanding the insights of each crystal structure.

Diffraction phenomena was also used to learn that the sodium and chloride ions are bonded in a lattice fashion and not molecules, to distinguish between different cubic lattice, to analyze all kinds of materials, even biological samples, etc. Using diffraction interesting things such as hair thickness can also be measured .The interference phenomenon, on the other hand, is used to make highly-wavelength specific mirrors for lasers. Furthermore, interference is the reason why soap bubbles appear colorful. Many other optical coatings owe their optical properties to the interference phenomena. An example of this is the antireflection coatings on lenses that we use everyday. Another application of interference is holography, which is a way of reconstructing three dimensional images with laser light.

Perhaps the most fascinating application of interference is to create holograms. This is done by reflecting a coherent light source, such as a laser, off of an object onto a special film. The interference patterns created by the reflected light are what result in the holographic image, which can be viewed when it is again placed in the right sort of lighting. Moreover, diffraction and interference can be observed when an atom passes through a standard light wave and its position is localized. In this case, the localization can be thought of as the creation of virtual slits leading to the above mentioned phenomena. Diffraction is also used to understand the insights of the ionosphere. All in all, by doing this experiment we learned the importance of the phenomenon of interference and diffraction in our lives.

References:

http://online.physics.uiuc.edu/courses/phys214/spring09/Lectures/Lect04.pdf http://bigbro.biophys.cornell.edu/~toombes/Science_Education/Laser_Diffraction/Diffraction_Lesson.pdf http://answers.yahoo.com/question/index?qid=20080509124425AAyW8bl http://physics.about.com/od/mathematicsofwaves/a/interference.htm URL: http://link.aps.org/doi/10.1103/PhysRevLett.68.472