Effects of Slit Width and Separation in Single-Slit and Double-Slit Diffraction Patterns of Monochromatic Light John Paul B. de Blas1, Kira Andrea C. Chan2, Renz Joshua B. Labay3, Marion P. Ordillano4* Department of Mining, Metallurgical and Materials Engineering, University of the Philippines, Diliman, Quezon City 2 Department of Industrial Engineering, University of the Philippines, Diliman, Quezon City 3 Department of Mechanical Engineering, University of the Philippines, Diliman, Quezon City 4 Department of Geodetic Engineering, University of the Philippines, Diliman, Quezon City *Corresponding author:
[email protected]
1
Abstract The experiment studied light as a wave using the concepts of interference and diffraction. Laser was used to hit a screen and produced different patterns and widths depending on variables such as slit-to-screen distance and use of double or single slits. Boundaries of each bright fringe were marked and measured. After collecting data, calculations were made. It was observed that an inverse relationship between slit width and the diffraction envelope exists. The calculated wavelength of the laser diode was both
7 14 .29 nm
for the 0.02
mm and 0.04mm slit width which yielded to a percent difference of 9.89% from the theoretical value of 650 nm.
1. Introduction Light can behave as a particle or wave, depending on how it is manipulated. In this experiment, light is studied as an electromagnetic wave using the concept of interference and diffraction. Interference is a phenomena of waves. It is observed when light passes through a narrow slit. Diffraction is a phenomena when many sources of waves are present. These occur due to the principle of superposition. When waves encounter one another, they either add up (constructive) or cancel out (destructive). When light passes through a slit with finite width, it makes a diffraction pattern of alternating bright (constructive) and dark (destructive) fringes. The widest bright fringe is called the central maximum. The intensity of brightness is inversely proportional to the distance between a fringe and the central fringe. Thus, brightness in fringes closer to the central fringe is greater than the ones farther or having larger distance from the central one. When light passes through two identical slits with finite width, light will diffract at both slits, but at the same time, interferes with one another. This forms a pattern of both double-slit interference and single-slit diffraction. Laser beam is used in this experiment as the source of monochromatic light. Laser will be passed through slits and distance between the center to side (y) will be measured. The following equations are used to solve for the wavelength and slit width in the double-slit interference and single-slit diffraction: ym,diff=
mλL a
(m=±1,±2,±3,⋯)
ym,int=
mλL d
(m=±1,±2,±3,⋯)
(1)
(2) where λ is the wavelength of the light source, L is slit-to-screen distance, a is slit width, and d is the slit separation. Equation (1) tells position of intensity minima in a single-slit setup, while equation (2) tells the intensity maxima in a double-slit setup. The experiment aims to observe behavior as variables are changed by differentiating patterns produced by single-slit and double-slit, and to quantitatively relate the pattern to slid width.
2. Methodology
A. Single-Slit diffraction The experiment was set up by placing a 650 nm laser diode on the end of the optical bench and the single slit disk together with its holder 3 cm away from the laser. Then, a white sheet of paper was fixed to a vertical surface and was placed on the other end of the optical bench making sure that the beam would hit it. The 0.04-mm slit width single slit was selected by rotating the disk until it was centered in the slit holder. The laser beam was then aligned on the slit by adjusting the knob horizontally and vertically until the beam was centered. Utmost precaution was taken to avoid direct eye contact with the laser beam. Figure 3 shows this experimental setup. The lights were turned off before the experiment started to clearly observe diffraction patterns in the paper background. The boundaries of dark fringes were marked using a pencil and the intensity minima were also identified. The distance between the first order minima and second order minima were measured and recorded. The distances between side orders were divided by two to get the distances from the center of the pattern to the first and second order minima and were recorded as y m. The diffraction pattern displayed on the sheet of paper was subsequently sketched. The steps were repeated for slit widths 0.02 mm and 0.08 mm. Then, the wavelength and the percent difference were calculated. B. Double-Slit diffraction The 650 nm laser diode was placed on one end of the optical bench and the multiple slit disk together with its holder 3 cm away from the laser. Then, a white sheet of paper was fixed to a vertical surface and was placed on the other end of the optical bench making sure that the beam would hit it. A qualitative observation on a variable double slit with a width of 0.04 mm and a slit separation of 0.125 mm to 0.75 mm was done. The interference fringes and diffraction envelopes were observed as the slit width is kept constant and the slit separation was varied. The variable double slit was replaced by double slit with a slit width of 0.04 mm and a slit separation of 0.25 mm. The distance between the slit and screen (paper) were measured and recorded. The group made sure that the slit and the pattern were at the same level vertically before measuring the distance. Then, the diffraction envelope and the interference fringes were observed. The dark fringes boundaries were then marked and the intensity minima were located using a ruler. The distances between the first order minima and second order minima were measured and divided into two to get the distance from the center. Using the obtained data, the slit width and percent difference were calculated. The double-slit pattern was projected onto the white paper fixed on a vertical surface. The position of the screen was fixed all throughout the experiment. The interference fringes located inside the central maximum were counted and recorded. Also the width of central maximum was measured and divided by the number of interference fringes this gave an approximate value for the width of each interference fringe. The double-slit diffraction pattern was then sketched to scale.
3. Results and Discussion The Interference and Diffraction Experiment aims to find the difference between single-slit and double-slit diffraction through varying slit widths and separation. Table W1. Wavelength of the laser diode
α = 0.02 mm, m = 1 Distance between side orders,
Distance from center to side, Calculated wavelength Percent difference
Δ y1 y1
α = 0.04 mm, m = 1
2
1
1
0.5
7.1428571 x 10−7 m 9.89%
7.1428571 x 10−7 m 9.89%
Average wavelength (
λ )
7.1428571 x 10−7 m
Slit-to-screen distance (L)
28
Table W1 shows the experimental wavelength of the laser diode using different slit widths. The wavelength was determined using the equation λ = ya/mL (3) where y is the distance from center to side, a is the width of the slit, the intensity minima and L is the slit-to screen-distance. The calculated average wavelength of the laser diode is 714.29 nm yielding a 9.89% percent difference from the theoretical wavelength, 650 nm.
Figure W1. Sketches (to scale) of diffraction pattern for various slit widths and fixed slit-to-screen distance
Table W2. Data and results for the
Distance between side orders,
Distance from center to side,
α = 0.04 mm, d = 0.25 mm double slit
Δ ym ym
Calculated slit width Percent difference Slit-to-screen distance (L)
m=1
m=2
1
2
0.5
1
0.00364
0.00364
9%
9% 28
Table W2 shows the distances between side orders for double-slit with slit width 0.04mm and slit separation of 0.25 mm. Given the mth intensity minima and the measurement of the central maximum width, the experimental slit width was calculated using Equation 4, a = mλL/y m,diff
(4)
Since actual slit width is a = 0.04 mm, calculated percent difference for slit width on m=1 and m = 2 is both 9%. Table W3. Data and results for double-slit interference II
α = 0.04 mm
α = 0.04 mm
α = 0.08 mm
α = 0.08 mm
d = 0.25 mm
d = 0.50 mm
d = 0.25 mm
d = 0.50 mm
Number of fringes
10
22
6
10
Width of central maximum
2.5
2.5
1.75
1.25
Fringe width
.25
.11363636
.29166667
.125
Table W3 shows pertinent measurements of the diffraction pattern and the fringes on a double-slit using different slit widths and separations which can be calculated using Equation 5. Fringe width = width of central maximum/no. of fringes
(5)
Figure W2. Sketches (to scale) of interference pattern for various slit widths and slit separations
Based on these observations, the increase in slit width results into the decrease in diffraction length, and vice versa. On the other hand, the increase in slit separation length results into the increase in the number of fringes. This means that the slit separation is directly proportional to the number of fringes. However, this also means that the width of each fringe is inversely proportional to the slit separation. Based on the table, the fringe width, decreases as the slit separation increases, and increases as the latter decreases. When the slit width is being increased, interference in waves occurs, thus, more wave antinodes are created which are represented by the dark bands observed in the diffraction pattern and as a result, bright bands are more separated from one another.. The resulting fringes are caused by the interference that arises from secondary waves formed from the two slits. Increasing slit separation results to a higher number of fringes but a decrease in fringe width. The reason here lies on the fact that increasing slit separation also increases the difference in distance travelled by light waves emerging from the slits to reach the same point in the screen. On the other hand, slit width affects diffraction pattern specifically on the intensity of emergent diffraction envelopes. Furthermore, decreased slit width increases the width of the central maximum and diffraction envelope due to wider wave diffraction as a result of increased secondary wave curvature. Also, a smaller slit width results to increased number of fringes and therefore, smaller fringe width.
4. Conclusion In this experiment of interference and diffraction of monochromatic light, the effect of the slit width and slit separation on diffraction patterns of laser beam was observed. As the slit width increases, the diffraction envelopes decreases and vice versa. With this, it could be said that there is an inverse relationship between slit width and the
diffraction envelope. On the other hand, the slit separation was observed to have no significant effect on diffraction envelopes. In the case of double-slit diffraction, the change in slit width had no significant changes on the fringe width while the change in slit separation does affect the width of the fringes. The experimental wavelength of the laser diode was also calculated to be both
714.29 nm
for the 0.02mm and 0.04mm slit width respectively.
These calculated value yield to a percent difference of 9.89% from the theoretical value of 650 nm. For better results, it is recommended to conduct this experiment in a darker room so that the fringes would be observed more sharply and minimize the percent difference
5. Acknowledgement We would like to express our gratitude to our Physics 72.1 professor, Mr. Denny Lane Sombillo, for his guidance and for providing careful intructions throughout the whole experiment. Likewise, we would like to extend our gratitude to the National Institute of Physics for providing us with the facilities needed to complete this study.
6. References 1. 2.
Young, H. D. and Freedman, R. A. (2012). University physics with modern physics (13th ed.). San Francisco, California: Pearson Education, Inc. Kamal, A. A. (2010). 1000 Solved Problems in Modern Physics. New York: Springer-Verlag Berlin Heidelberg
