Qualitative and quantitative observation of Airy pattern produced by the diffraction of laser light on circular apertures Alyssa Bendaña1, Giel Sabrine Cruz1, Stephanie Anne Fortin1 and Karl Adrian Vergara2* 1 Institute of Civil Engineering, College of Engineering 2 Department of Geodetic Engineering, College of Engineering University of the Philippines, Diliman, Quezon City 1101 Philippines *
[email protected]
Abstract Diffraction is the bending of light when it encounters an obstacle. To determine the behavior of light passing through a circular aperture, an experiment was performed involving two lasers with different wavelengths and circular apertures with dissimilar diameters. Passing light through a circular aperture will create pattern with intensified central bright spot succeeded by rings with decreasing intensities. Increasing the wavelength of the light will increase the size of the pattern as opposed to increasing the diameter of the hole which lessens the pattern size. Insignificant percent errors (0.50% - 2.50%) proved the validity of the principle of diffraction on circular apertures. Keywords: diffraction, circular aperture, airy disk, airy pattern
1. Introduction For a time, scientists were perplexed by the behavior of light as it was believed that in classical physics, that light just behaves as a wave but was then contradicted that light behaves as a particle upon the era of quantum physics. Diffraction of light has been one of the backbones of the theory that light behaves as a wave, strengthening the wave-particle duality of light upon the advent of the quantum electrodynamics. Diffraction is defined as the “way the waves spread out when they pass through a gap or round an obstacle”. [1] This diffracted light forms alternating bands of bright and dark bands that are resulted from the superposition of waves where the dark bands represent the product of destructive interference—where the waves canceled each other—and the bright bands represent the product of constructive interference—where the waves added up to each other. This paper converges on an objective of showing and investigating the principles of diffraction by sending a laser light through a circular aperture. When light is presented though a circular aperture, light impedes—forming a circular diffraction pattern with a central circular bright spot followed by alternating concentric bright and dark areas. This diffraction pattern is called the Airy pattern and central bright spot is called the Airy Disk named after Sir George Biddell Airy. With the use of the measurements that can be taken from the diffraction pattern, the wavelength of the light and the distance of the pattern to the aperture, we can compute for the diameter of the circular aperture which is given by:
𝑎=
𝑚𝜆𝐷 𝑦1
𝑚 = 1,2,3 …
(1)
where a is the diameter of the circular aperture λ is the wavelength of the incident light, m is the defined value at a certain mth minimum or maximum as shown in Table 1, y1 is the distance on the screen from the center of the pattern to the mth maximum or minimum and D is the distance from the circular aperture to screen. Table 1. Values of m on mth minimum and maximum [2] Order Minimum Maximum 1 1.220 1.635 2 2.233 2.679 3 3.238 3.690 This experiment shows that both the circular diffraction pattern vary on the wavelength λ, the screen distance D, and the diameter of circular aperture a.
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2. Methodology With the use of two diode lasers of different wavelengths, optical bench, slit disk, white paper, pencil and ruler, the principles of diffraction on circular apertures were qualitatively and quantitatively shown.
Figure 1. The Setup: Red laser diode (L1), green laser diode (L2), slit disk containing circular apertures (SD), optical bench (OB) and screen/white paper (SC)
Diode laser A was positioned first on the edge of the optical beam, while maintaining the alignment of the laser beam on the center of the circular aperture of diameter 0.02 mm (a) on the slit disk. A white paper was then placed at a certain distance (D) from the set up where the pattern created the most vivid image. Consequently, this pattern was traced on the paper, marking the positions of the mth minima and maxima in the diffraction pattern which were then measured. With the diode laser A still on its original position, the slit disk was rotated until the laser hits the circular aperture of diameter 0.04 mm. Similarly, the process of marking the positions of mth maxima was also done. Diode laser A was then replaced by diode laser B and the similar procedure was done on varying diameter of circular apertures. Using the measurements gathered, experimental wavelengths and diameters of circular apertures were calculated and compared to the theoretical values.
3. Results and Discussion Table 2. Wavelength of the red and green laser diodes Laser Red (650 nm) Green (532 nm) Circular aperture diameter a = 0.2 mm a = 0.4 mm a = 0.2 mm a = 0.4 mm Order 1st max 1st max 1st max 1st max Distance between orders 20 mm 10 mm 16 mm 8 mm Distance from center to 1st maximum, y1 10 mm 5 mm 8 mm 4 mm Calculated wavelength 661 nm 661 nm 529 nm 529 nm % difference 1.69% 1.69% 0.56% 0.56% Table 2 shows the acquired values needed for the calculation of experimental wavelength of red and green lasers. Using the equation 1, where D is equal to 1.85 m all throughout the experiment, the calculated wavelength for the red laser were 661 nm for both apertures. The same equation was also used to calculate the experimental wavelength of the green laser, which were both 529 nm for circular aperture diameters of 0.2 mm and 0.4 mm. From the equation, the diameter of the circular aperture is inversely proportional to the size of the distance from center of disk to center of subsequent circles. The calculated wavelength (λ) is inversely proportional to the diameter of the aperture. Comparing the calculated wavelengths to theoretical wavelength of the red laser which is 650 nm, a percent difference of 1.69% for both 0.2 mm and 0.4 mm slit diameter was calculated. For the green laser, which has 532 nm as its theoretical wavelength, the calculated differences were 0.56% for both 0.2 mm and 0.4 mm aperture diameters.
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Figure 2. Diffraction patterns formed with red laser diode (λ=650 nm) at 0.2 mm circular aperture diameter (left) and 0.4 mm circular aperture diameter (right). The size of the pattern decreases as the diameter of the circular aperture increases.
Figure 3. Diffraction patterns formed with green laser diode (λ=532 nm) at 0.2 mm circular aperture diameter (left) and 0.4 mm circular aperture diameter (right). The size of the pattern decreases as the diameter of the circular aperture increases.
Table 3. Data and results for the 0.2 mm circular aperture 1 2 Order Min Max Min Max Distance between orders 15 mm 20 mm 27 mm 33 mm Distance from center to mth max/min, y1 7.5 mm 10 mm 13.5 mm 16.5 mm Calculated circular aperture diameter 0.196 mm 0.197 mm 0.199 mm 0.195 mm % difference 2.00% 1.50% 0.50% 2.50% After measuring the distance between the orders, the distance between the central maximum and the mth maximum/minimum (y1) was calculated by dividing distance (y) by 2. With the use of equation 1, where D and λ were held constants with values 1.85 m and 650 nm respectively and where m is the specific value listed in Table 1, the experimental values of circular aperture diameter were calculated. Using the measurements gathered from the first order, diameter values of 0.196 mm and 0.197 mm were calculated which correspond to 2.00% and 1.50% percent difference respectively. On the other hand, 0.199 mm and 0.195 mm diameter values were computed using the 2nd order of maximum and minimum of the pattern which translate to 0.50% and 2.50% of percent difference. Table 4. Data and results for the 0.4 mm circular aperture 1 2 Order Min Max Min Max Distance between orders 7.5 mm 10 mm 13 mm 16 mm Distance from center to mth max/min, y1 3.75 mm 5 mm 6.5 mm 8 mm Calculated circular aperture diameter 0.391 mm 0.393 mm 0.410 mm 0.403 mm % difference 2.25% 1.75% 2.50% 0.75% The same process was done in computing for the values asked in Data Table 3 giving the values of calculated circular aperture diameter of 0.391 mm and 0.393 mm for the 1st order minimum and maximum of the 0.40 mm circular aperture with percent differences of 2.25% and 1.75%. Conversely, experimental diameter values of 0.410 3
mm and 0.403 mm and percent deviations of 2.50% and 0.75% came up for the 2nd order minimum and maximum of the diffraction pattern produced after passing a red diode laser through the 0.40 mm circular aperture.
4. Conclusions Two lasers of different colors were tested for diffraction. The green laser was used to confirm the result gained from the red laser. Diffraction occurs when a propagating light encounters a barrier. If the obstructing object have small openings, light tends to spread out past through the openings, forming complex patterns. In this experiment, we have seen that a light passing through circular aperture creates radial patterns.. From the figures2 and 3, we can see that the intensity of light decreases from the center going outwards. The equation 1, where a is the diameter of the aperture, is valid for this experiment. These patterns also change depending on the diameter of the circular aperture. Measuring the distances between separation of the circles of this pattern, the theoretical wavelength, theoretical diameter of the circular aperture, the experimental wavelength of the green and red laser and experimental circular aperture diameter were calculated as presented in Tables 2, 3 and 4, and compared to the theoretical values of those. We then conclude that the smaller the aperture, the larger the circular patterns at a given distance and the larger the wavelength of the light, the bigger the diffraction pattern will be created as verified and strengthened by the diminutive percent differences calculated which range from 0.50% to 2.50% only. However, due to the imperfection of the setting, unwanted light, slight carelessness on the group’s part, as in inaccuracy in gathering the measurements, improper positioning of the instruments, inconsistent handling of the green laser, poor observation and measuring skills of the members, which may have contributed to the yielded percent differences, this experiment is very open for further improvements. Aside from the elimination of the said errors, the group strongly recommends the use of a high-quality camera in gathering data, for a clearer pattern observation. This experiment may be useful to verify and elaborate the concept of diffraction in the formation of Airy Disks.
Acknowledgments The members of the group would like to express their earnest thanks to Mr. Gerold C. Pedemonte, their Physics Lab instructor who has guided them throughout the experiment and has provided the needed support and encouragement to the group. Also, this experiment would not be possible without the Physics 72.1 program of the National Institute of Physics, under the dearest University of the Philippines - Diliman. May we pronounce our gratitude to this institution.
References 1. 2.
Breithaupt, J., Key Science : Physics. 3rd ed., United Kingdom: Nelson Thornes Ltd. (2001). http://hyperphysics.phy-astr.gsu.edu/%E2%80%8Chbase/phyopt/cirapp2.html#c2
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