The physics of diffraction

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The Physics of Diffraction

Dilen Sachania, Matthew Spear and Christopher Strange

Project Supervisor: Christoph Hombach

Group: 21

April 2013


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Abstract

This report outlines a series of experiment on the topic of diffraction; webelieve that through conducting different experiments we achieved our aimof gaining a deeper understanding of the phenomenon that is diffraction.Specifically we focused on light wave diffraction through the use of a laser.We test some of our own ideas about the diffraction of light such as thevector line equations of beam paths after mesh diffraction. We believe weachieved our aim of developing our experimental skills, by learning frommany mistakes, the most important being that experiments should be fo-cused on reducing error. We also developed many transferable skills such ascommunication skills, advanced excel functions and statistics.


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Contents

I Introduction 4

II Experiments 7

1 Diffraction Gratings 81.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Single Slit Diffraction 13

2.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Double Slit Diffraction 183.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Aperture Diffraction 224.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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4.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Wire Diffraction 265.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Mesh Diffraction 296.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.4 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Angled Gratings 347.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Angled (x axis) diffraction gratings 36

8.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

III Conclusion 39

IV References 41

V Appendix 44

9 Derivation of vector line equations 459.1 Single slit/double slit/grating diffraction . . . . . . . . . . . . 45

9.1.1 Notation + assumptions . . . . . . . . . . . . . . . . . 459.1.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 46

9.2 Mesh diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 489.2.1 Mesh definition . . . . . . . . . . . . . . . . . . . . . . 48

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9.2.2 Notation + assumptions . . . . . . . . . . . . . . . . . 48

9.2.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 509.3 Overlaid diffraction gratings at an angle . . . . . . . . . . . . 51

9.3.1 Overlaid diffraction gratings definition . . . . . . . . . 519.3.2 Notation + assumptions . . . . . . . . . . . . . . . . . 51

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Part I

Introduction

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In our project we undertook the topic of Diffraction in physics. Our aims

for the project were to gain a deeper understanding of the subject area ofdiffraction, furthermore we wanted to learn new experimental techniquesand methods to develop our knowledge of physics. We planned to do thisby carrying out a series of experiments.

Diffraction can be seen as the consequence when light waves encounter an ob-ject, it is observed in light waves bending around an obstacle or the spreadingout of waves past small slits. Diffraction occurs with all types of waves in-cluding water, sound and electromagnetic waves such as visible light. Whilediffraction occurs whenever waves encounter an object its effects are muchmore pronounced whenever the wavelength is approximately the size of the

object. When a wave propagates past the object or slits a complex pat-tern of dark and coloured areas can be produced, this effect was named byThomas Young as interference.

The effects of diffraction were first observed in 1665 by Francesco MariaGrimaldi, who named the effect of light breaking up into different directionsas diffraction. Isaac Newton also investigated diffraction and attributedthese effects to inflection of light rays. Thomas Young was celebrated forhis experiments on diffraction in 1801; he conducted an experiment knownas the double slit experiment which included two closely spaced slits with amonochromatic light source. Young explained the effects of his experiments

with interference. He deduced from his experiment that light must propa-gate as a wave.

The Huygens-Fresnel principle is a method of analysis applied to problemsof wave propagation. In 1678 Huygens proposed that every point to which aluminous disturbance reaches becomes a source of a spherical wave, the sumof these secondary waves determines the form of the wave at any subsequenttime. He assumed that the secondary wave travelled only in the forwarddirection; but this is not explained in his theory. Huygens was thus ableto provide a qualitative explanation of linear wave propagation. HoweverHuygens failed to explain the effects that occur when light encounters edges,

apertures. Also Huygens theory of light traveling as a wave depended on ainvisible medium called the ether.

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Later in 1816, Fresnel, showed that his own theory of interference, together

with Huygens theory of light could explain the propagation of light as a waveand the effects of light diffraction. However to obtain agreement with hisresults he had to include additional arbitrary assumptions about the phaseand amplitude of the secondary waves. Fresnel concluded that interferencebetween beams of light could only be obtained if they were plane polarised.In other words polarised light waves having their oscillations directions ori-entated parallel to each other can combine to create interference, whereasthose that are perpendicular to each other cannot interfere.

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Part II

Experiments

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Chapter 1

Diffraction Gratings

1.1 Aim

The aim was to find the wavelength of the laser using a diffraction grating.

1.2 Background

Diffraction gratings split and diffract light into several beams travelling indifferent directions; the direction depends on the spaces between the grat-ing and the wavelength. Gratings are most commonly used in monochro-

mators and spectrometers. Diffraction gratings were discovered by JamesGregory, about a year after Newtons prism experiment, the phenomenon wasstill highly controversial. Gregory discovered diffraction gratings by passingsunlight through a bird feather and observing the diffraction patterns thatwere produced. In particular he observed the splitting of sunlight into thecolours which comprise it. The first practical diffraction gratings were madeby Joseph von Fraunhofer in 1820. He stretched fine parallel wires betweentwo parallel rods. Diffraction gratings are often used in monochromators,spectrometers, lasers, wavelength division multiplexing devices, optical pulsecompressing devices, and many other optical instruments. Diffraction grat-ings are most commonly used in everyday CDs and DVDs and can also be

used to demonstrate diffraction by reflecting sunlight off them onto a whitewall. Diffraction gratings are also used to distribute evenly the front lightof e-readers such as the Nook Simple Touch with GlowLight.

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1.3 Equipment

Green (532nm) class 2 laser

Diffraction grating slides (100, 300 and 600 lines/mm)

Optical bench/range and rail

Screen

Graph paper

Metre rule and 30cm ruler both with millimetre markings

Set square

Blutack and tape

1.4 Method

The experiment consisted of three sets of data, with each set based on oneof the three diffraction gratings (100, 300 and 600 lines/mm). Within eachset multiple measurements of grating-to-screen distance were taken rangingfrom 20 - 50 cm.

Figure 1.1: Ruler and set square.

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The first step was to set up the chosen diffraction grating at the initial

grating-to-screen distance of 20 cm, using a ruler attached to the bench anda set square to ensure the slide was in the correct position.

Once the slide was in the correct position the distance between the two out-ermost visible maxima (on the screen) was measured. This was followed bycounting and recording the number of visible maxima.

Once all the data had been recorded the experiment was repeated incre-mentally at 1 cm intervals from 20 - 50 cm grating-to-screen distance. Thismethod was repeated again for each of the other diffraction gratings (100,300 and 600 lines/mm).

1.5 Equation

n = d sin (1.1)

Therefore:

=d sin

n=

d opp

n hyp(1.2)

However we used:

=d sin

1

2(MaxM 1)

=d opp

1

2 (MaxM 1) hyp

(1.3)

A value of n was not measured, however a value for the number of maximavisible on the screen was recorded, subtracting the central maximum and

then halving would give us a theoretical maxima number (even if it was anon-integer) for that particular environment. Where n = maxima number, = wavelength, d = slit separation, = angle of laser beam, opp = distancefrom central maximum to relevant maxima, adj = slit to screen distance,hyp = slit to relevant maxima distance and MaxM = number of maximaon screen.

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1.6 Analysis

We obtained a reasonably large sample size of 93 readings, but the originalexperiment suffered from widespread systematic errors as detailed below.It was incorrectly assumed from a preliminary experiment that all maximawere the same distance away from their closest neighbour. This skewed theresults, leading to an overestimate of the bright fringe separation. The brightfringes appear at certain angles away from each other, which geometricallymeans that fringes tend to have larger separations between themselves thefurther away they are from the central maxima.

Originally assuming that the fringes were all the same distance from each

other led us to measure only the outermost fringes, thinking that it waspossible to increase accuracy this way. However a solution was found; wecounted the number of maxima on the board in the experiment, which im-plied the maxima number of a hypothetical fringe with a distance half ofthat measured. For example If there are nine fringes then one of them isthe central maximum so minus one then divide by two to get the maximanumber, in this case four. This fix led to a huge increase in precision andaccuracy.

Number ofdata sets

Totalsample size

Weightedmean

Standard errorof mean

Uncertainty

3 93 533 nm 0.1 nm 1.41 %Table 1.1: Basic descriptive statistics about the diffraction gratings experi-ment.

The actual labelled value given by the laser for the wavelength was 532 nm,0.27 % lower than the calculated value, which is accurate. The skew of thedata was 1.60 which shows that more values were larger than the mean thanwere smaller than it.

Confidence intervals

90% 1.2899% 2.01

Table 1.2: Confidence Intervals for diffraction grating experiment.

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There was quite a small standard deviation of 7.52 nm. There was an un-

expected cluster, four consecutive points, of data that were between threeand four standard deviations away from the mean, all skewed in the samedirection, which implies some kind of unaccounted for systematic error.

It should however be noted that later data sets were considerably less accu-rate that sets recorded first because of the higher fractional error due to alower number of visible maximas observed.

Figure 1.2: Data from the diffraction gratings experiment. Calculated wave-lengths of laser displayed alongside the labelled value, with error bars.

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Chapter 2

Single Slit Diffraction

2.1 Aim

The aim was to find the slit width of the single slit using its diffractionpattern.

2.2 Background

The difficulty confronting Young when conducting the single slit experimentwas the light sources available at the time, such as candles and lanterns.

These light sources were not suitable as they could not serve as a coherentlight source. Instead Young devised a method which involved using sunlightwhich entered the room through a pinhole in a window shutter, and then amirror was used to direct light across the room. To obtain two sources oflight he used a small paper card to break the light into two separate beams.Since two beams came from the same source they could be considered com-ing from two coherent light sources. Young predicted, the light waves fromthese two sources would interfere. This pattern could then be projected ontoa screen where measurements could be taken to determine the wavelengthof the light source.

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2.3 Equipment


Single slit


Screen

Graph paper


Set square

Blutack and tape

2.4 Method

The experiment produced 4 sets of data at fixed distances from the screen(35, 40, 45 and 50 cm). Each set contained measurements of the distancebetween the pairs of maxima.

To begin with the single slit was placed and aligned on the laser rail with

the slit-to-screen distance starting at 35 cm. A piece of paper was attachedonto the screen and the central maxima was marked on with an X. The restof the maxima (that fitted on the paper) were marked with a +. The paperwas then removed and taken into another room with better light conditionsfor measuring, allowing the markings to be seen more clearly. From week2 onwards we followed this convention of recording the measurements fromthe paper in well lit conditions, in an attempt to reduce human errors. Nextmeasurements were taken of the distance between each pair of maxima ofthe same order. The order of the measurements were taken working out-wards (with the first pair after the central maxima defined as of the firstorder m = 1).

The experiment was then repeated for further sets of data at the new slit-to-screen distances (35, 40, 45 and 50 cm).

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2.5 Equation

(n +1

2) = d sin (2.1)

Therefore:

d =(n + 1

2)

sin= (n +

1

2)

hyp

opp(2.2)

Where n = maxima number, = wavelength, d = slit width, = angle oflaser beam, opp = distance from central maximum to relevant maxima, adj= slit to screen distance, hyp = slit to relevant maxima distance.

2.6 Analysis

A very small sample size of 21 readings was obtained, but as we were plottingresults as we went along, we saw good agreement between results and so

justified not taking more readings. However we were plotting points usingan equation that did not apply to single slit diffraction. After correcting for

this during processing more data should have been collected (as always).

Number ofdata sets

Totalsample size

Weightedmean


Uncertainty

4 21 62.7 m 0.1 m 12.52 %

Table 2.1: Basic descriptive statistics about the single slit experiment.

It was wrongly assumed that the equation for single and double slit diffrac-tion was the same. A fix of sorts was implemented by assuming that maximaoccur directly between minima. It should be noted that there is no simple,accurate way to predict maxima positions for a single slit. However the

assumption that maxima occur directly between minima is a close enoughapproximation for our purposes.


90% 2.82

99% 4.41

Table 2.2: Confidence Intervals for single slit experiment.

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The standard deviation was 7.85m, rather high as is shown in the uncer-

tainty. This was down to the large fractional errors for small input data,which combined in quadrature. The slit to screen distance should have beenlarger to combat this. The skew was 2.95, caused by the high error firstmaxima measurements, one of which was more than three standard devia-tions away from the weighted mean, quite odd for such a small sample size.

Figure 2.1: Data from the single slit experiment. Calculated slit widths ofsingle slit displayed alongside the weighted mean, with error bars.

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As can be seen in Figure ??, predictions of maxima positions using the

weighted mean closely agree with the positions observed. It should again bestressed that there is no simple model that predicts maxima positions forsingle slit diffraction. Hypothesis testing failed to reject the null hypoth-esis that the average difference between measured and predicted maximapositions was zero. The 2 test implied a very bad fit between values, butas visually they dont seem too bad, this suggests that some error was notaccounted for.

Figure 2.2: Data from the single slit experiment. Calculated slit widths ofsingle slit displayed alongside the weighted mean, with error bars.

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Chapter 3

Double Slit Diffraction

3.1 Aim

The aim of the experiment was to find the slit separation of the double slit.

3.2 Background

The Double Slit experiment was first conducted by Thomas Young in 1801;he was the first person to provide strong evidence which supported the wavemodel of light. Youngs experiment was based upon the hypothesis that if

light were wave like in nature then it should behave like ripples in a pond.When two waves are in step, they should combine to make a larger wave,however when two waves are out of step they should cancel out creating aflat surface.

The Double slit experiment consisted of a screen with two slits cut into it,with a monochromatic light source. The purpose of this was to measurethe resulting impacts on the screen behind the two slits. The results of thisexperiment showed the patterns of interference which could only occur ifthe wave patterns were involved. As Young changed the distances betweenthe slits to the screen, and the space between the silts, he observed that the

light passing through the slits produced distinct bands of colour separatedby dark regions.

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At the time, Youngs experiment seemed to provided conclusive proof that

light travelled in waves, which would mean a revitalisation of Huygens theoryof light, which said that light waves propagated through an invisible medium,which at the time was coined the ether. However Youngs theory was notaccepted at first by his peers. Other events such as phenomena like therainbow colours and Newtons rings, although explained by his work werenot immediately obvious to many scientists at the time who firmly believedthat light propagates as a stream of particles. Other experiments were laterdevised which demonstrated the wave like nature of light and interferenceeffects.

3.3 Equipment Green (532nm) class 2 laser

Double slit


Screen

Graph paper


Set square

3.4 Method

The experiment consisted of 4 sets of data at varying distances from thescreen. The distance between the pairs of maxima was measured and usedto determine the separation between the double slit.

The first step was to stick graph paper up onto the screen and the pointwhere the laser hit the graph paper was marked with an X. Next the double

slit was put into the holder and then placed onto the laser rail. Due to theshort range of pattern clarity, it was decided to pick random data pointsbetween the range of 75 - 85 cm slit-to-screen distance. So next the slidewas placed onto the rail, the slit-to-screen distance was set and the maximamarked on the paper with horizontal lines. In recording the measurementsof the distance between the pairs of maxima it is important ensure that themaxima are of the same order. In between each measurement the slide wastaken off the rail to ensure that the values were chosen randomly.

This experiment was then repeated at other random distances.

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3.5 Equation

n = d sin (3.1)

d =n

sin= n

hyp

opp(3.2)

Where n = maxima number, = wavelength, d = slit separation, = angleof laser beam, opp = distance from central maximum to relevant maxima,

adj = slit to screen distance, hyp = slit to relevant maxima distance.

3.6 Analysis

A very large amount of data was collected for this experiment, 110 read-ings. There were no systematic errors, although there was suspiciously littleagreement between the data.

Number ofdata sets

Totalsample size

Weightedmean


Uncertainty

4 110 200 m 0.07 m 9.00 %

Table 3.1: Basic descriptive statistics about the double slit experiment.

It is noteworthy that other physics groups reported calculating similar val-ues for the slit separation.


90% 2.82

99% 4.42

Table 3.2: Confidence Intervals for double slit experiment.

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There was some agreement between experimental data and theoretical pre-

dictions assuming that the separation was the average, mostly in sets twoand three. Sets one and four did not look like they fit the prediction muchat all. See chart five (in appendix, too big to fit into this section).

Figure 3.1: Data from the double slit experiment. Calculated slit seperationof double slit displayed alongside the weighted mean, with error bars.

Hypothesis testing failed to reject the null hypothesis that the average dif-ference between the measured and theorised values was zero. The 2 test

again suggested a very bad fit, which as does not visually look like the casewas likely down to underestimating errors.

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Chapter 4

Aperture Diffraction

4.1 Aim

The aim of the experiment was to find the diameter of an aperture using itsdiffraction pattern.

4.2 Background

When Light waves pass through a small circular hole and onto a screen, itcan be seen that light passing through the hole interferes with itself. On

the screen a circular diffraction pattern can be observed. This diffractionpattern is called the Airy Disc, it is the central bright spot, surrounded bya series of concentric rings of alternate dark and bright fringes.

This type of diffraction has many applications as the eye and many opticalinstruments have circular apertures. if the smearing of the image of thepoint source is larger than that produced by the inadequacies of the system,the image processing is said to be diffraction limited.

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4.3 Equipment


Circular aperture


Screen

Graph paper


Blutack and tape

4.4 Method

The experiment consisted of 5 sets of data. Each set of data uses the sameaperture at different distances to the screen.

The first step was to insert the aperture slide into the holder and fix thegraph paper into position on the screen so that the laser hit the centre ofthe paper. The point at which the laser hits the screen was then markedon to the graph paper with an X. Next the holder was placed onto the rail

and the aperture-to-screen distance was set and fixed in position. We pickedvalues at random between the range of 96.6 - 102.2 cm aperture to screendistance which was selected based on the clarity of the diffraction pattern.Once the aperture was in position the diffraction pattern was recorded bymarking every other maxima from the centre (e.g. 1st, 3rd, 5th etc . . . ) inboth the vertical and horizontal directions either side of the central max-ima. The first ring from the central maxima is defined to be of the first order.

Once the diffraction pattern had been recorded, both the vertical and hor-izontal distances between the pairs of maxima were measured. The reasonfor recording both the horizontal and vertical distances was so that the data

could be averaged to increase accuracy.

This method was repeated until an adequate amount data had been col-lected, which was after 4 different aperture-to-screen distances.

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4.5 Equation

(n + k +1

2) = d sin (4.1)

d =(n + k + 1

2)

sin= (n + k +

1

2)

hyp

opp(4.2)

Where k = 0, 1, 2, 3 . . .

Where n = maxima number (recorded), = wavelength, d = aperture diam-eter, /theta = angle of laser beam, opp = distance from central maximumto relevant maxima, adj = slit to screen distance, hyp = slit to relevantmaxima distance and k = an arbitrary maxima modifier.

4.6 Analysis

A small sample size of 32 readings was obtained, however each reading tooktwice as long as other previous experiments, more could have been recordedin the allocated time. Every second maxima was recorded to maximise

spread of collected data.

It was assumed that the apertures bright ring diameter could be modelledas twice the relevant maxima position for a single slit. It was also initiallyassumed that the first visible bright ring was maxima one and so on, howeverwe suspected that the bright central maximum may have overlapped withother bright areas meaning that the visible bright disk we recorded may nothave been maxima one but maxima two or three or even four.

This was accounted for by calculating the diameter four times, first assum-ing that what the maxima recorded really were true, then assuming that

they were one off, then two off and then three off. This was further com-plicated by the fact that each data set could have been recorded differently,for example set one could have recorded the true maxima, and set two couldbe one off.

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Therefore weighted averages (and other relevant statistics) had to be cal-

culated separately for each data set. These separate weighted averages andtheir standard errors could then also be taken as a data set, the weightedaverage of these weighted averages could then be calculated along with otheruseful information.

The separate weighted averages were then used to model where the brightrings would occur, and the best models corresponding value for the diameterwas used for each data set. Then the weighted mean of the values obtainedthrough the models was used as the final value.

Number ofdata sets Totalsample size Weightedmean Standard errorof mean Uncertainty

5 32 657 m 1 m 6.75 %

Table 4.1: Basic descriptive statistics about the aperture diffraction experi-ment.

The 2 test was applied to each of the models and informed the choice ofwhich model was the best for each particular data set. A few of the data setshad models that were convincingly accurate, but others seemed to indicatethat there was some factor which was missed.

Figure 4.1: Data from the aperture diffraction experiment. Calculated di-ameters of aperture displayed alongside the weighted mean, with error bars.

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Chapter 5

Wire Diffraction

5.1 Aim

The aim of the experiment was to find the thickness of the wire in the slideby its diffraction pattern.

5.2 Equipment


Wire slide


Screen

Graph paper


Blutack and tape

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5.3 Method

The experiment consisted of 11 sets of data taken for varying slide-to-screendistances. Each set of data contains 10 or more individual readings for eachorder of minima measured either side of the central maxima.

To start with the slide containing the wire was placed in a holder with thewire parallel to the bench so that it would produce a vertical diffractionpattern on the screen. Next the graph paper was attached to the screen,the point the laser hit the screen was marked with an X and orientation ofthe paper marked on. The holder was then placed on the rail and randomslide-to-screen distances were selected. Again random values were used as

the clarity varied depending on the slide-to-screen distance, the total rangewas 66.4 - 90.6 cm. The minima were marked with a horizontal line oneither side of the central maxima.

The pairs of minima were numbered on the paper for easy reference whilstmeasuring. The distance from the central maxima to an order of minimawas recorded down separately for both the left and right sides of the diffrac-tion pattern. Since the wire was turned through 90 and was parallel tothe bench creating a vertical diffraction pattern, the right measurement wasdefined as towards the top of the graph paper.

This method was then repeated at different slide-to-screen distances untilan adequate amount of readings had been taken.

5.4 Equation

n = d sin (5.1)

d =n

sin= n

hyp

opp(5.2)

Where n = maxima number, = wavelength, d = wire width, = angle oflaser beam, opp = distance from central maximum to relevant maxima, adj= slit to screen distance, hyp = slit to relevant maxima distance.

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5.5 Analysis

A large amount of data was collected for this experiment, especially consid-ering that each data point required two measurements.

Number ofdata sets

Totalsample size

Weightedmean


Uncertainty

10 103 81.9 m 0.02 m 17.57 %

Table 5.1: Basic descriptive statistics about the wire diffraction experiment.

Minima were measured instead of maxima under the assumption that max-ima were directly in between minima, in order to decrease error, leading to

a marginal increase in precision but a decrease in accuracy.


90% 2.33

99% 3.65

Table 5.2: Confidence intervals for wire diffraction experiment.

The results had a standard deviation of 14.4 m and a skew of 1.86. Apartfrom the first reading of every data set, the results seemed to be very regular.

Hypothesis testing and the 2

test suggest deep incompatibilities betweenmeasured and theorised x coordinate position. Maybe another model needsto be explored. See below.

Figure 5.1: Data from the wire diffraction experiment. Calculated wirewidths of the wire displayed alongside the weighted mean, with error bars.

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Chapter 6

Mesh Diffraction

6.1 Aim

The aim of the experiment was to test the theory of diffraction patternsproduced by a mesh at different distances.

6.2 Equipment


Mesh (two overlaid diffraction gratings)


Screen

Graph paper


Blutack and tape

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6.3 Method

This experiment consisted of 2 sets of data. Each set of data was the co-ordinates of the maxima on the screen taken with varying mesh-to-screendistances.

The first grating was attached into the holder and the second fixed per-pendicular to the first. The initial laser point was marked onto the graphpaper with an X. Next the grating-to-screen distance was set. Since therewere two gratings forming the mesh, the mesh-to-screen distance was takento the grating closest to the laser. The diffraction pattern produced was agrid of maxima which were marked on the graph paper with a +. Other

important information like the orientation of the paper, the distance fromscreen and grating selection were recorded on the paper.

Once the diffraction pattern had been recorded, we defined a coordinatesystem. As mentioned before the central maxima is marked with an X.The central maximas composite order is defined as [0, 0] with all other co-ordinates in reference to this point in terms of x and y. The points werelabeled in relation to the centre (an example would be the composite orderfor the maxima to the left of the central maxima, it would be [1, 0]). Mea-surements were taken (in centimetres) from the centre in both the x andy directions to mark position. To keep consistency in the order of writing

down the coordinate we followed an anti-clockwise spiral system in sortingthe coordinates.

This experiment was repeated at different mesh-to-screen distances in orderto model and predict the outcome.

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6.4 Equation

If maxima with composite order (nh, nv)Let:

h = tan(arcsin(nhd1

)) (6.1)

v = tan(arcsin(nvd2

)) (6.2)

x = z h (6.3)

y = z v (6.4)

Where adj is z coordinate and opp is x and y coordinate respectively.

(x,y ,z) = z (h, v, 1) (6.5)

For a more detailed explanation see appendix.

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6.5 Analysis

There was only 2 data sets measured with a sample size of 43. There was asource of systematic error present; the slit to screen distance was measuredas the distance from the screen to the grating closest to the laser, but themodel assumes that the distance between gratings is negligible, the gratingclosest to the screen would have given the best representation of the model.

This was accounted for by assuming that the error on the slit to screenmeasurement was half a centimetre, which was justified by measuring thedifference in length from the first and second slits, which was roughly a cen-timetre.

Another systematic error was the uneven distribution of blu-tack on thegrating closest to the screen, which would have been at a slight angle rel-ative to the first slit. As shown in the final experiment and the diagrambelow, a grating at an angle relative to the laser beam can cause a curvatureto form in the line of maxima.

Furthermore the distance between the gratings was not negligibly smalland probably caused a similar effect as described above as the first grat-ing diffracts the laser, the beam splits, the beams which hit the edge of thesecond slit would be at a non-negligible angle, therefore also adding error.

The effect of this error would be much larger than that of the previouslydescribed error.

With the assumption of additional error on the slit to screen distance, the 2

test indicated a very good compatibility between theory and data for boththe x and y axis predictions. Hypothesis testing however did not agree. Thenull hypothesis that the average difference (for the x axis and then sepa-rately for the y axis) between measured and theoretical values was zero, wasbarely not rejected for the x axis (0.05 significance for both x/y axis) andwas rejected for the y axis.

Considering the 2

test and the average difference and the multiple non-negligible systematic errors outlined above, the hypothesis test seems to beindicating a positively skewed systematic error(s) for both x and y coordi-nates. See below.

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Figure 6.1: Data from the mesh diffraction experiment sets 1 and 2.

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Chapter 7

Angled Gratings

7.1 Aim

To determine the relationship between a pair of angled gratings and thediffraction pattern they produce.

7.2 Equipment


two overlaid diffraction gratings at varying angles


Screen

Graph paper


7.3 Method

This experiment consists of 6 sets of data. Each set of data is at the same

grating-to-screen distance with a change in either angle or grating combina-tion.

The slides were set up with the first grating placed in the holder with theslits perpendicular to the laser. The second grating was then fixed onto theholder on the side closer to the screen with a 90 (measured angle) definedas the second grating being parallel to the bench. A plumb bob was usedto ensure that the distribution of bluetack was even and the mesh was per-pendicular to the laser.

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The angle between the two gratings was defined as looking towards the grat-

ings from the view point of the board, with the angle being measured fromthe horizontal. The measured angle was later used to calculate the anglefrom the vertical. It was also defined that clockwise was a positive angleand anti-clockwise a negative angle.

Next the paper was stuck up onto the screen and the laser point was markedonto the screen with an X. The initial distance between the first and secondgratings was also taken. The slide holder and gratings were then place ontothe rail.

The remaining points were marked onto the graph paper with a +. When

the diffraction pattern was recorded the data kept with the conventions ofthe previous experiment with the use of same coordinate system and usedthe spiral ordering system with adjustments made for the skew of the gridchanging.

This experiment was repeated for a series of different angles and gratingcombinations.

7.4 Equation

See Appendix for derivation.

7.5 Analysis

A large amount of data (86) was collected for this experiment with consid-erably more sets (6) than the previous experiment. The same systematicerrors that befell the above experiment also apply to this experiment withthe exception of the uneven distribution of blu-tack which was correctedfor using a plumb bob. Additional error due to not knowing the distancebetween the slits was accounted for similarly to the previously mentioned ex-

periment. The 2

test for both the x and y coordinate predictions indicateda good fit between measured and predicted values. Hypothesis testing failedto reject the null hypothesis that the average difference between measuredand predicted values is zero.

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Chapter 8

Angled (x axis) diffraction

gratings

8.1 Aim

To find a relationship between angle of a diffraction grating and the bendingof the diffraction pattern.

8.2 Equipment


Diffraction grating slides (100, 300 and 600 lines/mm)


Screen

Graph paper


Blutack and tape

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8.3 Method

This experiment contained 6 sets of data. Each set of data consisted of thegrating at a different angle of skew.

To start with the graph paper was attached to the screen and the laser pointwas marked on with an X. Next a single diffraction grating was placed at90 into the slide holder (with slits parallel to the bench), this allows forthe angle of the slide to change on another axis to our previous experiment.

Figure 8.1: Example of curved diffraction pattern of angled grating.

In order to change the angle, the grating starts parallel to the screen. Thedistance from the screen to each corner of the slide holder was measured. Inorder to change the angle we increased the distance to the far side by 1cmwhich in turn reduced the near side by the same amount. The measurement

is then taken by marking the remaining maxima with a +.

The experiment was repeated for a variety of angles and grating types.

8.4 Equation

No equation used/found.

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8.5 Analysis

There were not many conclusions that could be drawn from this small adata set (44) with such a small spread. It was found that over the rangewe covered the x coordinates were approximately the same as if there waszero rotation. Whilst rotated through a certain angle the change in x coor-dinates from when there was zero rotation appeared to be proportional to n.

A theory that n was proportional to the difference between the x coordinateat zero rotation and at an angle, was tested and appears not to be the caseas the covariance (covar. for short) was -0.10 and the correlation coefficient(correl. for short) was -0.29. It was also found that average y coordinates

change with a particular theta, although the exact nature of the relationshiphas not been determined. The average y coordinate position for each angleincreases as the angle increases. A direct proportionality may be possible ascorrel. is 0.778 but covar. is 0.065.

Generally y is larger for a large absolute value of either the order or thex coordinate, implying the quantities are directly proportional. For n vsy covar. = 0.28 and correl. = 0.36, which is not promising. For x vs ycovar. = 1.02 and correl. = 0.441, implying some kind of more complicatedrelationship than a simple probability.

Furthermore a smaller slit distance produced a larger average y coordinateposition, implying an inverse proportionality, but its only speculation as wechanged slit distance only once.

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Part III

Conclusion

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Our aims for the project were to gain a deeper understanding of the sub-

ject area of diffraction, furthermore we wanted to learn new experimentaltechniques and methods to develop our knowledge of physics. We plannedto do this by carrying out a series of experiments. We believe that throughconducting different experiments we achieved the aim of gaining a deeperunderstanding of the phenomenon that is diffraction. Specifically we focusedon light wave diffraction through the use of a laser. However this was notpart of the original plan as we wanted to look into all aspects of diffractionincluding particle diffraction; this allowed us to test some of our own ideasabout the diffraction of light such as the vector line equations of beam pathsafter mesh diffraction. We also believe we achieved our aim of developingour experimental skills, by learning from many mistakes, the most impor-

tant being that experiments should be focused on reducing error. We alsodeveloped many transferable skills such as communication skills, advancedexcel functions and statistics. Another important skill which will help us inlater years is the use of the LATEX typesetting system. Moreover we usedother technologies to help with communications to enhance group work suchas google drive. this was beneficial as we could all edit a ocument at thesame time without the need to meet up in the same location. Also usingvideo conferencing tools such as Skype allowed us to communicate and workon the project during the Easter break. There were many improvementsthat could have been made throughout the project. The most importantprogression we could have achieved in the project was to start off with the

aim of focusing on reducing the errors in each experiment. Also anotherimprovement we could have implemented was to increase efficiency by plan-ning in more detail then we had been doing. To conclude we feel we havemet our aims and developed many skills.

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Part IV

References

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Introduction:

Authorship: Dr Rod NaveTitle: DiffractionAvailable at: Hyper physicswww.hyperphysics.phy-astr.gsu.edu/hbase/phyopt/diffracon.html

Authorship: Andrew Zimmerman JonesAvailable at: About.com Education Physicshttp://physics.about.com/od/mathematicsofwaves/a/huygrnspriciple

Diffraction Gratings:

Authorship: Dr Rod NaveTitle: Diffraction gratingsAvailable at: Hyper physicswww.hyperphysics.phy-astr.gsu.edu/hbase/phyopt/diffracon.html

Single Slit :

Authorship: Michael W. Davidson [of Florida State University]Title: Thomas Youngs Double Slit experimentAvailable at:

www.micro.magnet.fsu.edu/primer/java/interference/doubleslit/

Double Slit Experiment:

Authorship: Michael W. Davidson [of Florida State University]Title: Thomas Youngs Double Slit experimentAvailable at:www.micro.magnet.fsu.edu/primer/java/interference/doubleslit/

Authorship: AnonTitle: Lesson 58: Youngs Double Slit Experiment

Available at :www.studyphysics.ca/newnotes/20/unit04light/chp1719light/lesson58.htm

Aperture:

Authorship: AnonTitle: Diffraction of a circular apertureAvailable at :www.physics.nus.edu.sg/.../experiments/diffraction

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Wire Diffraction

Authorship: AnonTitle: Diffraction with Hair or wireAvailable at:www.physicsed.buffalostate.edu/pubs/StudentIndepStudy/EURP09/Young/Young.html

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Part V

Appendix

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Chapter 9

Derivation of vector line

equations

9.1 Single slit/double slit/grating diffraction

9.1.1 Notation + assumptions

Using the Cartesian coordinate system in three dimensions, define the pointwhere the laser strikes the slit(s) as (0, 0, 0).

The unit vector i acts in the horizontal plane perpendicular to the beam of

the laser striking the slit(s) and is positive to the right of the direction oftravel of the beam. The unit vector j acts in the vertical plane perpendicularto the beam of the laser striking the slit(s) and is positive upwards. Thevector k acts and is positive in the direction of the laser beam. Let maximamean points where constructive interference occurs. Let n denote the orderof the maxima, where n = 0 is defined as the central maximum (CM),n = 1 is defined as the first maximum observed in the i direction after theCM, n = 2 is the second and so on. The first maximum to the left of theCM would mean n = 1, the second n = 2 etcetera. Let denote thewavelength of the laser light. Let d denote slit separation. Let denotethe angle at which maxima occurs, positive in i direction. opp = distance of

maxima from the CM positive in the i direction (shown as y in diagram) = xcoordinate of maxima. adj = slit to screen distance (shown as L in diagram)= z coordinate of maxima. The equation n = d sin is assumed to betrue, which implies a finite number of maxima. For single slit equivalentreplace n with (n + 1

2).

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9.1.2 Derivation

n = d sin (9.1)

Therefore:

n

d= sin (9.2)

To find x coordinate of maxima in terms of slit to screen distance.

opp

adj= tan (9.3)

Where:

= arcsin(n

d) (9.4)

opp

adj= tan(arcsin(

n

d)) (9.5)

Therefore:

opp = adj tan(arcsin(n

d)) (9.6)

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Where opp = x, adj = z, and y coordinate is always 0.

x = z tan(arcsin(n

d)) (9.7)

Therefore the 3d vector line equations of the laser beams where maxima willoccur is:

M(tan(arcsin(n

d))i + k) (9.8)

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9.2 Mesh diffraction

9.2.1 Mesh definition

Here we shall define a mesh as two overlaid diffraction gratings. The gratingclosest to the laser being fixed horizontally and the grating closest to thescreen fixed vertically. The distance between gratings must be negligiblysmall so as to reduce diffraction effects between gratings.


Use the same definition of (0, 0, 0) in the Cartesian coordinate system andfor as in the previous derivation.

It is assumed that the diffraction pattern will be the ordinary horizontaldiffraction pattern for which each horizontal maxima is diffracted verticallyby the second vertically positioned grating, to produce a diffraction array.

Assuming the diffraction array model described above is true then the vec-tor line equations of the laser beam after passing through the mesh can beusing the above derivation, for the first slit, then applying it again to thoselines but vertically, using details of the second slit as appropriate.

Let d1 denote the slit separation of the grating closest to the laser and d2

denote the slit separation of the slit closest to the screen. Let nh of a max-imum denote the order of the maxima that corresponds to the first slit. Itcan be thought of as the horizontal order.

Conversely Let nv of a maximum denote the order of the maxima that cor-responds to the second slit. It can be thought of as the vertical order. Thei direction is positive for the horizontal order and the j direction is posi-tive for the vertical order. Therefore each bright fringe in the array can beassigned a composite order in a coordinate form (horizontal order, verticalorder). See fig below.

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For example the central maxima shown highlighted with a red ring would

have composite order (0, 0), the other highlighted maxima would have com-posite orders as follows: White (2, 1), purple (1, 2), pink (4, 0), blue(2,2), yellow (0,3).

Figure 9.1: Mesh diffraction pattern with marking.

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9.2.3 Derivation

If maxima with composite order (nh, nv). Let:

h = tan(arcsin(nhd1

)) (9.9)

v = tan(arcsin(nvd2

)) (9.10)

Therefore:

opp

adj= tan(arcsin(

n

d)) (9.11)

opp = adj tan(arcsin(n

d)) (9.12)

Therefore:

x = z h (9.13)

y = z v (9.14)

Where adj is z coordinate and opp is x and y coordinate respectively.

(x,y ,z) = z (h, v, 1) (9.15)

Therefore the 3d vector line equations of the laser beams where maxima willoccur is:

M(hi + vj + k) (9.16)

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9.3 Overlaid diffraction gratings at an angle

9.3.1 Overlaid diffraction gratings definition

Define an overlaid diffraction grating as a mesh, where the second gratingis placed at an angle from the vertical.


Use the same definition of (0, 0, 0) in the Cartesian coordinate system andfor as in the previous two derivations.

It is assumed that the diffraction pattern will be similar to the mesh. The

pattern formed by the first slit, the horizontal one, will remain the same,each of those maxima will then be diffracted at whatever angle the anglebetween the two slits. The resulting array can be imagined as the same asthe mesh pattern but with each of the vertical diffraction patterns rotatedaround the beam that it is diffracting.

Let d1 denote the slit separation of the grating closest to the laser and d2 de-note the slit separation of the slit closest to the screen. Let nh of a maximumdenote the order of the maxima that corresponds to the first grating. It canbe thought of as the horizontal order. Conversely Let nv of a maximumdenote the order of the maxima that corresponds to the second grating. It

can be thought of as the vertical order rotated through an angle, from thevertical, . The i direction is positive for the horizontal order and the jdirection is positive for the vertical order before rotation.

Therefore again similarly to the mesh each bright fringe in the array canbe assigned a composite order in a coordinate form (horizontal order, verti-cal order). The difference being that the coordinate system applies beforerotation. If a particular maxima has a composite order (a, b) in one anglethen although its position will be different at a different angle, its compositeorder will be the same. See fig.

The anticlockwise direction will be defined as positive so the diagram in figwill be a negative angle.

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Figure 9.2:

If maxima with composite order (nh, nv).Let:

h = tan(arcsin(nhd1

)) (9.17)

v = tan(arcsin(nvd2

)) (9.18)

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Start off imagining the mesh situation, if the mesh is rotated a certain angle

through a circle, then if turning anticlockwise is a positive angle and left toright is the positive x axis then the new x position is h h sin. Itsthe original the change caused by rotation. The new y position is easier tounderstand its simply h cos, the original multiplied by cos of the angle.The z coordinate always stays the same.

(x,y ,z) = z (h hsin,hcos, 1)

M (h hsini, hcosj, k)

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