Fraunhofer Diffraction Experiment

INTRODUCTION
Diffraction is one of the most important topics in optics, it refers to a spectacle which occurs when a wave encounters an obstacle or slit in its path. The wave will then bend around the edges or corners of the obstacle or aperture, into the region of a geometrical shadow of the obstacle. The Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the object, in the near field region, is given by the Fresnel diffraction equation.

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If the shadow of an object cast on a screen by a small source of light is examined, it is found that the boundary of the shadow is not sharp. The light is not propagated strictly in straight lines, and peculiar patterns are produced near the edges of the shadow, which depend on the size and shape of the object. This breaking up of the light, which occurs as it passes the object, is known as diffraction and the patterns observed are called diffraction patterns. The phenomena arise because of the natural wave nature of light. Apertures and objects produce a similar effect.
In Fraunhofer diffraction, a parallel beam of light passes the diffracting object in question and the effects are observed in the focal plane of a lens placed behind it.
From the diagram in FIG 1, AB represents a slit whose length is perpendicular to the plane of the paper given by the distance $d$, and which parallel beam of light passes through from left to right. Per Huygens’s principle, each point in the slit must be considered as a source of secondary wavelets that spread out in all directions. Now the wavelets travelling straight forward along AC, BD, and so on, will arrive at the lens in phase and will produce strong constructive interference at point O. Secondary wavelets spreading out in a direction such as AE, BF, and so on will arrive at the lens with a phase difference between successive wavelets, and the effect at P will depend on whether this phase difference causes destructive interference or not.
It will be noticed that there will always be a bright fringe at the centre of the diffraction pattern. The separation of the diffraction bands increases as the width of the slit is reduced; with a wide slit the bands are so close together that they are not readily noticeable. The separation also depends on the wavelength of light, being greater for longer wavelengths.
In the case of the slit shown in the diagram, the first dark line at P is in a direction $theta$ such that BG is one wavelength, $lambda$. If d is the width of the slit, then $theta = lambda/d$. This is assuming the angle is so small then $sin(theta) approx theta$.
EXPERIMENTS
In these sets of experiments a low power (0.5 mW) Helium-neon laser is used as the source of light. The laser light produced by the laser used is coherent and parallel, but for these sets of experiments the beams diameter is far too small. To get around this problem a beam expander arrangement is set up in front of the laser source to expand the beam to a larger width before hitting the object being examined.
From FIG 2 it can be seen that the biconcave lens A causes the beam to diverge, and appear to emerge from the point X in the focal plane of the lens A. If a second lens B with focal length $f_B$ and place it $f_B$ away from X as shown, the outputted laser light will be parallel again, but it will have a large width.
The output of this beam is used to examine Fraunhofer diffraction patterns produced under various circumstances, viewing the resulting patterns on a white screen or with the use of a photodetector to detect beam intensity at varying locations.
A good bit of time is spend aligning the laser to be as close to the center of the lenses as possible and therefore careful note is taken for where each position of the lenses stands are set, this will help with consistency between different days and if the apparatus is tampered with. The distance from the object being examined to the photodetector was kept at a constant $(0.53pm 0.01)m$ throughout all experiments carried out.
SINGLE SLIT
The first object to be examined is the simple single slit. Setting up a variable slit in the object path the slit width can be adjusted allowing investigation of slit width and intensity to be measured. The intensity distribution on the screen is given by the equation,
The resulting laser beam from the beam expander passes through the single slit then through another lens to focus on a detector screen. Placing a white sheet of paper on this screen the maxima’s can easily be seen by eye allowing simple marks to be placed where they are. These marks then can be easily measured with a set of digital callipers, which have a measurement uncertainty of $pm$0.02mm for measurements less than 100mm and $pm$0.02mm for less than 200mmcite{digitalcalipers}. It is seen that for a varying single slit the separation of the diffraction bands increases as the width of the slit is reduced; with a wide slit the bands are so close together that they are not readily noticeable. This is as expected from the predicted theory.
Using a single non-variable slit as the object, the resulting slit separation can be calculated. This is done by taking the measurements from the central maximum and plotting them against their order. This relation is given by Young’s equation, where $y_m$ is the distance from the central maxima for the m’th order fringe, $lambda$ is the wavelength of laser light used, $D$ is the distance from the object to the screen and $a$ is the slit width.

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Plotting the values of $y_m$ versus the corresponding order value $m$ the resulting line of best fit is the value of $frac{lambda D}{a}$, with the use of the known constant the value of $a$ can be determined. This calculation is easily done with MATLAB which would give a more accurate result than hand drawing a graph, using the function $nlinfit$ the error in the line of best fit can be obtained and thus the uncertainty in the measurement of the slit width. Each value for $y_m$ is taken multiple times to reduce reading uncertainty and also the marking of maxima on the paper is repeated to further reduce reading uncertainty.
From measurements taken the calculated value for the slit width was found to be $(7.31pm 0.39)cdot10^{-5}m$, this agrees with typical values for a single slit which are in the order of Nano meters.
At this point it was found that the photodetector didn’t function properly. Trying to measure intensity it was seen that the measured value was negative. It was also not notable to see second and third maxima’s, just the central maxima could be clearly detectable. Many attempts were made to correct this, re alignment of the laser had very little effect. Ensuring the room was constantly dark to try to eliminate the background light was also tested, but again no improvement in the reading. It was decided to stop taking any measurements of the intensities for the remaining experiments.
MULTIPLE SLITS
An arrangement consisting of many parallel slits, of the same width and separated by equal distance is known as a Diffraction grating. When the spacing between the lines is of the order of the wavelength of light, then a noticeable deviation of the light is produced.
The intensity of light can be adapted from one single slit to a generalisation for N number of slits, the distribution for N number of slits is given by,
The $sin^2beta/beta^2$ term is describing the diffraction from each individual slit. While the $(sin^2(NY))/(sin^2(Y))$ describes the interference for the N slits, and so this gives a maximum and minimum where,
Each diffraction grating was placed in the source holder one by one and the outputted diffraction patterns on the detector screen were observed. It was found to be that the second maxima were weaker as the number of slits on the source was increased and the central maxima became sharper. Grating with 6 slits was found to be the sharpest central image while the slit with only 2 was the weakest.
ONE AND TWO DIMENSIONAL
One dimensional gratings can now be used to examine the difference in slit width and to examine the difference in diffraction patterns observed, for this part there were three unknown one dimensional gratings to be examined. The gratings were loaded in one by one and marking the central maximum and other maximum observed on the screen the distances can be measured allowing slit width to be calculated. It was observed that the different gratings gave a different spread of maxima on the screen. For a one dimensional grating the measurements were repeated 3 times for three different gratings. The same method is used to calculate the slit distance as in the single slit experiment. The measurements for the gratings widths were found to be, $(6.90pm 0.51)cdot10^{-5}m$, $(2.37pm 0.46)cdot10^{-5}m$ and $(1.49pm 0.14)cdot10^{-5}m$. All these values lie within the expected range for a slit to diffract light.
To measure the output of the two-dimensional grating we can model it as two one dimensional problems. Measuring the maxima in one direction then again in the other direction, these two can be compared and should be with in similar value is the grating is equally spaced in both directions. Results were found to be $(5.84pm 2.62)cdot10^{-5}m$ and $(5.24pm 2.62
CONCLUSION
All parts of the experiments were carried out effectively and for all parts of the experiment data was collected and analysed. For a single slit of unknown width the calculated value for it was found to be $(7.31pm 0.39)cdot10^{-5}m$, which is in the right order of magnitude for a single slit resulting in light diffracting. Also observing multiple slits on a source was found to show that the second maxima were weaker as the number of slits on the source was increased and the central maxima became sharper. Finally, a one and two-dimensional grating was analysed to calculate wire separation. It was found for the one dimension samples the separation width was $(6.90pm 0.51)cdot10^{-5}m$, $(2.37pm 0.46)cdot10^{-5}m$ and $(1.49pm 0.14)cdot10^{-5}m$ and for the two dimensional it was found that in each directions the width was $(5.84pm 2.62)cdot10^{-5}m$ and $(5.24pm 2.62)cdot10^{-5}m$.
Unfortunately, the photodetector did not work accordingly. The values obtained from one measurement did not match with values obtained later or on different days. Attempts were made to try and improve readings; keeping room constantly pitch black and realignment of the mirrors. It was decided to stop taking detector measurements.