Manual Diffraction. Manual remote experiment Project e-xperimenteren+ J. Snellenburg, J.M.Mulder

Transcription

1 Manual remote experiment Project e-xperimenteren+ J. Snellenburg, J.M.Mulder

2 Colofon Manual diffraction Manual remote experiment Project e-xperimenteren+ Stichting Digitale Universiteit Oudenoord 340, 3513 EX Utrecht Postbus 18, 3500 AD Utrecht Phone Fax Author(s) J. Snellenburg, J.M.Mulder Copyright Stichting Digitale Universiteit De Creative Commons Naamsvermelding-GeenAfgeleideWerken-NietCommercieel-licentie is van toepassing op dit werk. Ga naar om deze licentie te bekijken. Date page of 1

3 Table of contents 1 Introduction 4 Theory 4.1 Interference 4. Diffraction 5.3 Fresnel diffraction 5.4 Diffraction patterns of various objects 7 3 Setup Schematic view 9 3. Pictures 10 4 Remote interface Settings Measure Analyse Journal 18 5 Constants and parameters 19 6 References 19 7 Appendix A: system requirements Hardware 0 7. Software 0 page 3 of 1

4 1 Introduction This manual explains the working of a remotely controlled diffraction experiment. The experiment is set up so that a diffraction pattern can be measured with several kinds of objects. The distance of the source to the object can also be varied. The goal of the experiment is to learn how to calculate the dimensions of the object by analyzing the diffraction pattern. The experimental setup is located at the Vrije Universiteit Amsterdam and is maintained by the Physics student laboratory. The web-experiment can be accessed from practically any location in the world, provided the user s system meets the minimum system requirements as listed in Appendix A. Before starting the experiment it is recommended to formulate a research plan to work as productively as possible. To this end, besides reading this manual, it may help to study the background theory available on the website to find out what the possibilities and limitations are of online experimenting. Theory.1 Interference The wave nature of light causes effects that cannot be explained with geometrical optics. Interference is an example of such an effect and does not only occur with light, but with other wave like phenomena, like sound, as well. Interference can be understood with the principle of superposition. According to this principle the result of two interfering waves is just the addition of those two waves. Whenever two waves are in phase they interfere constructively, increasing the amplitude of the total wave, and when two waves are out of phase they interfere destructively, reducing the amplitude of the total wave, possible even to zero. An example of interference of two wave sources is given in Figure.1. Figure.1: Interference of two waves. Left: the dotted and marked lines represent the minimum and maximum values respectively. Right: two waves are in phase when the difference in optical wavelength is a multiple of the wavelength λ. In this figure, plane waves with wavelength (λ) are incident on a surface with two slits separated by a distance D. In accordance to Huygens' Principle each of these slits will act as a secondary, coherent wave source and will produce cylindrical waves (spherical in the case of holes) that interfere with each other. At the points where the distance of the wave coming from the two thin slits differ by a multiple (n) of a wavelength the two waves interfere constructively. When the difference in distance to both sources is n+½ times the wavelength the two waves interfere page 4 of 1

5 destructively. When the interference pattern is projected on a screen set at a distance R, the distance between points of maximum intensity along the x-axis (see figure.1) is found to be dependent on the wavelength (λ), the distance to the screen (R) and the space between the slits (D). In first order approximation (for small angles θ) the distance from the central maximum to the n-th order maximum is given by the formula: dn ( ) nrλ D (1). Diffraction When dealing with the superposition of many waves the term diffraction is used rather than interference. Because of the short wavelength of visible light it is difficult to observe the effect of diffraction. But when a laser bundle is used to illuminate a small slit the effect can be made visible. Figure. is an example of the diffraction pattern of a single slit. Figure.: single slit diffraction pattern Diffraction can be described by Fresnel or near field diffraction theory. This is the general theory taking into account the curve of the wave fronts, but is rather complex. When distances from source-object and object-screen are so large that the wave fronts can be approximated by plane waves one can use a simplified theory called Fraunhofer or far field diffraction theory. In this experiment we will be dealing with relatively small distances and therefore we have to use the Fresnel theory..3 Fresnel diffraction The general idea of Fresnel diffraction can be well illustrated by the example of a single slit. For a more detailed treatment of the Fresnel theory see text books on optics ref. [1,]. In this case light generated by a monochromatic source in a point S, illuminates the slit, located at a distance p from the source. Behind the slit a screen is placed at a distance q from the slit (see figure.3). Figure.3: construction of the diffraction pattern of a single slit with width B page 5 of 1

6 First of all we will calculate the light intensity in point P at the center of the screen. According to Huygens Principle all points in the slit will contribute to the intensity in point P of the screen. The important parameter in the calculation is the phase difference between the different paths that the light travels for the different points in the slit. Light that goes straight from S to P travels a distance p+q. To calculate the phase one needs the optical path length, which is (p+q)/ λ in case of a wavelength λ. Light that follows a path S-O-P has an optical path length of ( (p +z )+ (q +z ))/ λ. As z is much smaller than p and q the phase difference Δϕ between these two paths is given by: Δϕ ( z) πz = ( p + q) pqλ () Thinking of the slit divided in infinitely small strips with width dz the amplitude of the wave front in P is given by: + B / A( P) = A0 exp( iδϕ ( z)) dz (3) B / With A 0 depends on the intensity of the light source in S. It is common to introduce a new variable ν to transform equation 3 such that it does not depend directly on the geometrical parameters. 1/ ( p + q) ν = z (4) pqλ The intensity in point P is equal to A(P) and is after the transformation given by: +Δ ν / +Δ ν / πν πν I ( P) = I 0 cos dν + sin dν (5) Δ / ν Δν / With 1/ ( p + q) Δν = B and I 0 depends like A 0 on the intensity of the light source. (6) pqλ To calculate the intensity at a point R that is not located at the centre of the screen (see Figure.4). Figure.4: construction of diffraction pattern of a single slit with width B page 6 of 1

7 one only has to change the boundaries of integration in equation 5 ν ν πν πν I ( ρ) = I0 cos dν + sin dν (7) ν ν with 1 Δv Δv p v1 = ρ, v =+ ρ and ρ = δ qλ( p+ q) 1 The integrals ν πν ' C( ν ) = cos d ν ' 0 And ν πν ' S( ν ) = sin d ν ' 0 are the so called Fresnel integrals. (8) (9) [ ] Using these Fresnel integrals equation 7 can now be written in a very simple form: I ( ρ) = I [ ( ) ( )] [ ( ) ( )] 0 C ν C ν1 + S ν S ν1 (10) The Fresnel integrals cannot be solved analytically, but the results can be computed numerically. Usually the integrals are tabulated for different values of ν to avoid computing the integral each time again..4 Diffraction patterns of various objects The same methods to calculate the diffraction pattern of a slit can also be applied to other objects like a wire or a half plane. Because of the different shape of the diffraction object other points on the wave front give a contribution to the intensity pattern on the screen. Mathematically this amounts to adjusted integration intervals. Wire The derivation for the diffraction pattern of a wire with thickness B is almost identical to that of the derivation of a single slit. The difference is that not the points on the interval B/<z<+B/ (points within the slit) should be taken into account, but the points outside that interval extending to infinity. This and the fact that + + πν ' πν ' cos ' = 1 d ν and sin (11) ' = 1 d ν results in: [ 1 ( C( ν ) C( ν ))] + [ 1 ( S( ν ) S( ) ] ] I ( ρ) ν Half-plane = I ) (1) In the same way an equation for a half plane can be derived. Now the boundaries of integration change to page 7 of 1

8 v 1 = ρ, v = so that we can write: I (13) 1 1 ( ρ) = I0 C( ρ ) + S( ρ) Other objects It is possible to calculate the diffraction pattern for a wide variety of other objects. A grating for example can be seen as a collection of many slits with the same width and positioned at equal distances. The contribution of each slit has to be taken into account to get the diffraction pattern for the whole grating. page 8 of 1

9 3 Setup 3.1 Schematic view Figure 3.1: schematic view of the experimental setup The experimental setup of the diffraction experiment consists of several components. He-Ne laser Provides a constant-intensity laser bundle with a wavelength of 63,8 nm. Spatial filter Consists of an ocular and a pinhole. Converts the laser beam into a point light source. Diffraction object Located in a carousel this object can be either a grating, a wire, one or two slits or a half plane, each with its own diffraction pattern. Detector A photo diode connected to a multi meter. The value of the multi meter is read out by the LabVIEW program. The detector is located at approximately 1000 mm from the source. It can be displaced in a direction perpendicular to the laser beam to measure the diffraction pattern. page 9 of 1

10 3. Pictures Below some pictures of the experimental setup are shown. Legend: A B C D E F laser spatial filter carrousel with objects web cam detector power supply detector A B C D F Side view of the experimental setup. page 10 of 1

11 C A B D A side view with spatial filer and object holder. C A B D F A side view with spatial filter and laser page 11 of 1

12 E Detector with 0. mm slit. E D D C B A Top view experimental setup. page 1 of 1

13 4 Remote interface Below is a description of the software interface of the experiment. The panel is organized into several tabs: 1. Settings. Measure 3. Analyse 4. Journal Each tab is described below: 4.1 Settings On this tab one can adjust two experimental parameters. First, the distance from source to object. Second, and most importantly, the object itself which determines the characteristics of the diffraction pattern. The difference in patterns of different objects can be large so it s important to make good estimations of the dimensions and magnitudes of the various parameters involved when analyzing the patterns. Figure 4.1: screenshot of the SETTINGS tab in the remote panel overview. Choose object: with this popup button an object can be selected: a wire (two thicknesses available), a single slit, a double slit, a grid, a half plane, a screen and empty. Refresh rate: sets the update rate of the web cam images. Camera 1: shows the distance between the source and object. page 13 of 1

14 Camera : shows the experimental setup of the laser and the object carrousel. Camera 3: shows a close-up of the detector. Small image: when selected the color will change to blue and a small web cam image is displayed. Large image: when selected the color will change to blue and a large web cam image is displayed. Distance source-object: Here the distance between the source and object is set by dragging the black object with holder to a new position. The actual change is done by clicking the Do it button. 4. Measure On this page the diffraction pattern can be measured as a function of the detector position along the diffraction pattern. The detector will move from the initial position (current) to the destination and measures the intensity of the incident light at a number of points along the way. The number of points can be controlled by the step size. Once the measurement is done the pattern can be saved for later analysis. It is also exported to the ANALYSE tab so that it can be analyzed instantly. Figure 4.: screenshot of the MEASURE tab in the remote panel overview. Detector position: the red pointer in the horizontal slider is the current detector position. The yellow pointer is the destination position. Change the destination by dragging the pointer or changing the yellow value in the numeric control above the slider. The actual change has to be confirmed by the Move to destination button or by starting the measurement. page 14 of 1

15 Current position: The current position of the detector. Intensity: The light intensity that illuminates the detector in a.u. Step size: The resolution of the measurement. (The step that the detector makes.) Points to measure: this indicator shows the number of measuring points calculated from the current position, the destination position and the step size. Start measurement: Starts the measurement. The detector moves from start to destination position with steps size. Is only visible when the measurement is stopped. Stop measurement: Stops the measurement. Is only visible when the measurement is running. Save data: If the measurement is stopped the experimental data can be saved. Graph: Displays the measured light intensity as a function of the detector position. page 15 of 1

16 4.3 Analyse The analyse tab automatically imports the data from the measure page but it s also possible to read in older data via the Read Data button. Once a diffraction pattern is loaded a pattern can be fitted to it by adjusting the various parameters at the right side of the page. The distance between the detector and the source is an experimental constant and fixed at 1000 mm. The distance between the object and the source can be read out using camera 1 or the indicator at the settings tab. Note that the unit of this quantity on the settings scale is in centimeters whereas on the analyse tab it should be converted to millimeters. Figure 4.3: screenshot of the ANALYSE tab in the remote panel overview diffraction pattern: Displays in blue the measured signal and in black the theoretical signal calculated with the parameters on the right hand side. Vertical offset: adds an offset value to the calculated value, thus moving the theoretical curve upwards. Horizontal offset: moves the theoretical signal to the left. Log fitting parameters: Stores all the fitting parameters in the log file. This can be read back on the journal tab. Fitting parameters Type: here the object type: slit, wire or half plane is set. Size object [mm]: this value is the thickness of the wire or the width of the slit. page 16 of 1

17 Number of objects: set when for example a slit is chosen (single or double). Distance between objects: the distance between two slits. Distance source-object [mm]: the distance between the source and the object. This can be read from a web cam image position object. Distance source-detector: the distance between the object and the detector. This is fixed at 1000 mm. Pattern width: the width of the pattern. This is determined by the start and stop position of the detector. Vertical scale factor: this value is multiplied with the calculated values. Wavelength source [nm]: the wavelength of the used laser light (He-Ne laser: nm). Calculate chi squared: clicking this button will calculate the chi squared with the given standard deviation and the calculated chi squared values are displayed in the upper graph. Standard deviation: the standard deviation of the measured values. Read data: previously saved data can be read back into the graph. page 17 of 1

18 4.4 Journal In this tab you can enter items to your lab journal. This is done by selecting a Journal Item and Journal Status from the lists at the top of this tab and entering a text in the field Journal Text. Optionally you can add a reference to an earlier journal entry or log entry. This is done by clicking the entry in the list box at the bottom of the tab or entering its number in the Reference field. By clicking the button Submit, the journal entry is added to the log. Note that some actions are automatically logged. Figure 4.4: screenshot of the JOURNAL tab in the remote panel overview Journal Item: here it is possible to choose a category for your journal entry. Journal Status: here you can choose a qualification of your journal entry. Journal Text: here the actual text of the journal entry can be typed. Reference: You can refer to an earlier journal entry or experiment log entry by clicking in the table of entries or choosing a number in the Reference field. Submit: by clicking the button Submit the journal entry is added to the experiment log. page 18 of 1

19 5 Constants and parameters Name Description Value / range, error, unit He-Ne Laser beam Wavelength 63.8 nm Source Detector Distance 1000 mm 6 References 1. F.A. Jenkins and H.E. White, Fundamental of Optics; chapter 18. E, Hecht and A. Zajac, Optics; chapter 10 page 19 of 1

20 7 Appendix A: system requirements 7.1 Hardware Processor Screen resolution Internet connection Free disk space PIII - 1Ghz or equivalent 104 x 768 pixels 56k but broadband (>51k) recommended 100MB 7. Software For e-xperimenteren+ you need the following software (free), required programs are marked by an asterisk. * LabVIEW runtime engine 7.1 ftp://ftp.ni.com/support/labview/runtime/ * Java Runtime Environment 5.0 (or higher) Adobe Reader (free pdf reader) Foxit Reader (free pdf reader windows only) page 0 of 1

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