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The behavior of the map mirror with changes in temperatures as determined by double exposure holography Jackson, Miranda Sylvia 1997

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T H E B E H A V I O R OF T H E M A P M I R R O R W I T H C H A N G E S IN T E M P E R A T U R E S AS D E T E R M I N E D B Y D O U B L E E X P O S U R E H O L O G R A P H Y By Miranda Sylvia Jackson B. Sc. Hons. (Physics) University of Regina, 1994 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A July 1997 © Miranda Sylvia Jackson, 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B .C. , Canada V6T 1Z1 Date: Abstract Changes in shape with temperature, recorded using holographic interferometry, of a prototype M A P reflector are reported. The temperature of the reflector is changed between holographic exposures, so that deviations in the shape show up as fringes of interference between the light from the two images. This method can detect shape changes on the order of half a wavelength of He-Ne laser light (632.8 nm), and when analysis is performed on the image, shape changes of 0.02 microns and less are measured. A change in shape of approximately 4 microns over the entire surface is observed when the temperature of the reflector is changed by approximately 1 K from room temperature. This is insignificant compared to the figure tolerance (A/60). However, this effect may become significant when the temperature is reduced to approximately 50K in space, and therefore, the aluminum in the honeycomb material will be replaced by paper on the M A P satellite. The aluminum honeycomb structure is found to deform the surface with an amplitude of 0.008 to 0.015 microns in a regular pattern, when the temperature changes by 1 K , and this will produce sensitivity in the sidelobes of 1 0 - 7 of the main beam at 40°. This is not expected to affect the data for sources such as the earth, the moon, Jupiter, or the Milky Way, and the sun is expected to be hidden behind the instrument at all times and to never to appear at that angle from the main beam. ii Table of Contents Abstract ii Table of Contents iii List of Tables vii List of Figures viii Acknowledgments xiii 1 Introduction 1 1.1 The Cosmic Microwave Background 1 1.2 Measurements of the cosmic microwave background 1 1.2.1 Reflectors used in C M B R Experiments 1 1.2.2 Balloon Anisotropy Measurement (BAM) 2 1.2.3 The Microwave Anisotropy Probe (MAP) Satellite 2 1.2.4 The M A P mirrors and the microwave system of the M A P satellite 2 1.3 The rest of the paper 5 2 Cosmology and the C M B R 6 2.1 Big Bang Theory 6 2.1.1 Observations leading to Big Bang theory 6 2.1.2 Early Big Bang theory 6 2.1.3 Thermal history of the early Universe 7 2.2 Cosmological parameters and how they are measured 8 2.2.1 The Hubble constant and the age of the Universe 8 2.2.2 Cosmological density parameter 9 2.2.3 Using the power spectrum of the C M B R to determine the cosmo-logical parameters 10 iii 2.3 The significance on cosmology of a complete measurement of the power spectrum, such as will be provided by the M A P satellite 14 3 Diffraction estimation and control in CMBR experiments 17 3.1 Estimation of sidelobe rejection levels 17 3.2 Properties of a wavefront 19 3.3 Diffraction from a reflecting surface 19 3.4 Gain-loss and increased sidelobe sensitivity introduced by errors in a wave-front 20 3.4.1 A model of an aperture with regular sinusoidal phase errors . . . 20 3.4.2 Hatbox model of phase errors in a wavefront 21 3.4.3 A wavefront with phase errors shaped like Gaussian hats 21 3.5 Effect of correlation length and wavelength of sinusoidal patterns on the beam pattern of a microwave reflector 22 3.6 Beam size and effective area of a reflector 23 3.7 Reducing sidelobe response 23 3.7.1 Horn antennas 24 3.7.2 Curved edges on optics 24 3.7.3 The use of taper in a multiple-reflector system 24 3.7.4 Diffraction shields or ground shields 25 4 The Experiment 26 4.1 Test reflectors used in this experiment 26 4.2 First attempts at double exposure holography 27 4.3 The first test of warm-cold double exposure holography 27 4.4 The tests of the sample M A P reflector using double exposure holography 28 4.4.1 The first heating test . . 28 4.4.2 Heating the reflector with an array of resistors 29 4.4.3 Heating the reflector with a heat lamp 32 4.5 Analysis of the image to measure the deformation produced by the alu-minum honeycombs 35 iv 4.5.1 Converting the image into individual row files, containing column number and pixel intensity 36 4.5.2 Separating and analyzing the interesting parts of the image . . . . 38 4.5.3 Determining the amount of deformation produced on the reflector surface by the aluminum honeycomb 41 4.6 Final measurements of surface deformations caused by the honeycomb patterns 45 4.7 The beam pattern of the reflector, as determined from measurements of its surface deformation and antenna theory 46 4.8 Additional results not directly related to the main experiment 48 4.8.1 Cooling profile of the reflector 48 4.8.2 The warming profile of the reflector 50 5 Conclusions 52 5.1 Overall deformation of the reflector 52 5.2 Deformations caused by the honeycomb backing 52 5.3 Results of previous experiments using different methods 53 5.3.1 Photogrammetry tests done at the Goddard Space Flight Center . 53 5.4 Potential success of future similar experiments 53 5.5 Contributions to the antenna temperature by Jupiter and the Milky Way 54 Bibliography 56 A The Apparatus used in the experiment 58 A . l Basic Holography Equipment 58 A.2 The first test of warm-cold double exposure holography 62 A.3 Apparatus used to make the holograms in the large vacuum chamber . . 64 A.3.1 Elimination of vibration in the vacuum chamber 64 A.4 The configuration of the optical bench and other apparatus 65 A. 5 Measuring the temperatures at various points on the apparatus 67 A. 6 The array of resistors used to heat the reflector 69 v B The system used to cool the apparatus 71 B . l The cooling apparatus 71 B.2 Preliminary cooling tests 71 B. 3 Results of preliminary cooling tests 72 C Holography 75 C l Basic Holography Principles 75 C. 2 Coherent Light 75 C.3 The Laser 78 C.4 Models of Holography 79 C.4.1 The Zone Plate Model 79 C.4.2 Geometric Model 80 C.5 Types of holograms 80 C.5.1 In-line (Gabor) holograms 80 C.5.2 Off-axis (Leith-Upatnieks) holograms 82 C.5.3 Fresnel holograms 82 C.5.4 Frauenhofer holograms 83 C.5.5 Side band holograms 83 C.5.6 Fourier transform holograms 83 C.5.7 Image holograms 83 C.6 Holography film 83 C.7 Techniques used to optimize the quality of holograms 85 C.8 Holographic Interferometry . 87 C.8.1 Measurements of deformation using holographic interferometry . . 87 C.8.2 Types of Holographic Interferometry 87 C.9 Reproducing a holographic image as a photograph 90 vi List of Tables 4.1 Positions of diodes for the resistor-heated holograms. See also figure 4.2 for a diagram and figure A.8 for a photograph 29 4.2 Times of the exposures of holograms A and B and the locations of the resulting images and temperature profiles 32 4.3 Positions of diodes for the lamp-heated hologram. See also figure 4.8 for the diode positions 36 4.4 Positions of diodes in figures 4.18, 4.19, and 4.20 48 B . l Positions of diodes in figure B . l 74 B.2 Positions of diodes in figure B.2 74 vii List of Figures 1.1 Schematic diagram of the entire M A P satellite showing the reflectors near the top and at the sides 3 1.2 Overview of the thermal reflector system of the M A P satellite, which will be a single piece of composite material'with a paper honeycomb support. 4 2.1 The power spectrum of the C M B R 11 2.2 The results from the various C M B R experiments, plotted with error bars. The labeled result is from the B A M experiment (See section 1.2.2). The quantity Qfj a^ is defined in terms of the total power Ce at a given value of I by the relation i(£ + l)Ct = ^ Q j ^ 15 3.1 The sample beam pattern described in section 3.1 used to estimate required sidelobe rejection levels for various sources 18 3.2 Line diagram of the diffraction control flare used on the B A M experiment. Its curved edges reduce diffraction through the aperture 25 4.1 The first test of double exposure holography. 27 4.2 A diagram showing the approximate locations of the heating resistors on the rear surface of the reflector, denoted by small circles. Also shown are the positions of the measuring diodes, denoted by solid x's for those on the rear surface and by dashed x's for those on the reflecting surface. See figure A.8 for the same front view in a photograph. Concentric circles of incrementally increasing diameter are shown for reference, every 2 inches in diameter 30 4.3 The temperature profile of the reflector while hologram A was being made. The locations of the measuring diodes are listed in table 4.1 and dia-grammed in figure 4.2. The vertical lines denote the times of the first and second exposures 31 viii 4.4 Hologram A , the image produced by a double exposure hologram of the reflector when it is heated with a resistor array before the first exposure and allowed to cool for the second exposure. The temperature difference is approximately 4 degrees 32 4.5 The temperature profile of the reflector while hologram B was being made. The locations of the measuring diodes are listed in table 4.1 and dia-grammed in figure 4.2. The vertical lines denote the times of the first and second exposures 33 4.6 Hologram B, the image produced by a double exposure hologram of the reflector when it is heated with a resistor array before the first exposure and allowed to cool for the second exposure. The temperature difference is approximately 1 degree 34 4.7 The temperature profile of the reflector while hologram C was being made. The locations of the measuring diodes are listed in table 4.3 and dia-grammed in figure 4.8. The vertical lines denote the times of the first and second exposures 35 4.8 The positions of the measuring diodes, when the heat lamp is used to heat the reflector, denoted by solid x's for those on the rear surface and by dashed x's for those on the reflecting surface. The dimensions indicat-ing the distances of the diodes from the edge of the reflector are shown in inches, and concentric circles of incrementally increasing diameter are shown for reference, every 2 inches in diameter. The diodes, except diode 2, are in the same positions as they were for the resistor-heated holograms. 37 4.9 Hologram C, the image produced by a double exposure hologram of the reflector when it is heated with a heat lamp before the first exposure and allowed to cool for the second exposure. The temperature difference is approximately 3 degrees. The reflector is placed horizontally on the aluminum block 38 4.10 Using S A O I M A G E to examine the image to determine the amount of deformation produced by the honeycomb core 39 ix 4.11 The intensity profile over the image in figure 4.12, averaged over rows 500-600 40 4.12 Part of the image in figure 4.6, with the region of interest indicated in the rectangle 41 4.13 A close up of the image in figure 4.6, with the region from which the data were taken indicted in the rectangle 42 4.14 This figure shows an illustration of the analysis described in section 4.5.3. The top graph is analogous to figure 4.11 and the bottom graph shows a possible profile of the reflector surface in the region of interest 43 4.15 The discrete slope calculations of the graph shown in figure 4.11 in intensity units divided by pixels. The slope is averaged every 18 columns to produce a smooth curve 45 4.16 The deformation of the honeycomb features along the region indicated in figures 4.12 and 4.13 46 4.17 The sidelobes in the beam pattern of the reflector, produced by phase errors which occur as a result of surface deformations measured in this experiment, extrapolated to A T = 200 K , for radiation with a frequency of 90 GHz. When this is calculated for v = 60 GHz, it is found that the height of the peaks scales as and the locations of the peaks scales as A. 47 4.18 The cooling profile of the reflector 49 4.19 The warming profile of the reflector 50 4.20 The warming profile of the reflector in more detail 51 5.1 The contribution to the signal by Jupiter, when it is 0 to 90 degrees from the main beam. It is calculated using the sidelobe pattern in figure 4.17 and equation 3.5. Its contribution when it is the main beam is not included. 55 A. 2 The diagram used while calculating the divergence of a lens ball 59 A.3 The object used for the first test of warm-cold holographic interferometry. The amount of deformation has been greatly exaggerated for illustrative purposes. See section A.2 63 A.4 The setup of the apparatus for making the heated double exposure holo-grams, showing the optical bench and the computer used to record the temperatures 65 A.5 The setup of the optics for some of the heated double exposure holograms. 66 A.6 The vertical setup of the optics for some of the heated double exposure holograms. The reflector is placed on top of the aluminum block for the exposures 67 A.7 The circuit used to measure diode temperatures. The short-dashed rect-angle denotes the multiplexer switch used to measure voltage, and the long-dashed rectangle denotes the multiplexer switch used to provide cur-rent 68 A. 8 This photograph shows the back of the reflector with the heating resis-tors and measuring diodes attached. The aluminum block to which it is fastened rests on the surface of the optical bench 70 B. l The results of the first preliminary cooling test performed on the appara-tus. See section B.3 72 B. 2 The results of the second preliminary cooling test performed on the appa-ratus. See section B.3 73 C l A n example of the setup of apparatus for an in-line hologram, with a monochromatic source such as a mercury lamp with a pinhole at its en-trance to imitate a point source. This is how the first holograms were made by Gabor 81 C. 2 . A n example of the setup of apparatus for an in-line hologram, with a laser as a source and a diverging lens to produce the reference beam 81 C.3 A n example of the setup of apparatus for an off-axis hologram, with a beam splitter to separate the incoming laser beam and lenses to cause the object and reference beams to be divergent 82 xi C.4 A n example of the setup of apparatus for a Fourier transform hologram, either with or without a large converging lens between the object and film planes. A beamsplitter is used to separate the reference and object beams. The Fourier transform of the object is recorded on the film, which is parallel to the object plane 84 C.5 An illustration of masked holographic interferometry, where a number of different holographic images can be compared in various combinations on the same piece of film. See section C.8.2 89 xii Acknowledgments I would like to acknowledge the help I have received from my supervisor, Dr. Mark Halpern, over the course of my project. He has provided me with knowledge which will be applicable in my future endeavours in science, as well as with an understanding of cosmology. I would also like to thank Dr. Greg Tucker for his advice and support. It was his suggestion to use a single resistor to heat the reflector a few degrees that led to the first successful hologram with visible fringes. I would also like to show my appreciation to both of them for expecting more from me than I originally thought possible, and for showing me that the only limits on one's abilities are those that are self-imposed. As a result, I have produced a document that I believe might be useful to others making similar measurements. Others to whom I must show my appreciation include Drs. Garth Huber and Peter Bergbusch, who were my supervisors for my undergraduate work. I have found my con-tinuing correspondence with Dr. Bergbusch to be an invaluable source of encouragement. I am indebted to Dr. Paul Hickson for being so prompt in reading my thesis and providing insightful comments. My mother, Angela Jackson, proofread several versions of my thesis and provided recommendations with regard to sentence structure and English usage. xiii Chapter 1 Introduction 1.1 The Cosmic Microwave Background The cosmic microwave background was discovered by accident by Arno Penzias and Robert Wilson in 1965[1]. It was later measured to be virtually constant across the sky, with a blackbody spectrum of temperature 2.7 K[2, 3]. Anisotropics in the temperature of this radiation were first measured by the C O B E satellite in 1992[4]. Now there are numerous experiments in progress which are attempting to measure the tiny anisotropics in the C M B R , with a precision of 4p ~ 1 0 - 5 at various angular scales. See chapter 2 for a description of the cosmological principles related to the C M B R . 1.2 Measurements of the cosmic microwave background Because of the extreme precision required when making measurements of temperature differences on the sky, it is required that there be extremely low systematic errors ( A T < 5 [J,K). Since the detected temperature differences are 10~6 K , and the sun's temperature is 5 x 103 K , its contribution to the signal must be rejected at a 1 0 - 9 level. For this reason, and because the earth and the moon must also not contribute significantly to the signal, there must be a very low response in the sidelobes of the detectors. Various techniques are used to reduce sidelobe response, some of which are described in chapter 3. 1.2.1 Reflectors used in CMBR Experiments Many of the experiments designed to measure the C M B R , whether ground-based, flown on a balloon, or orbiting on a satellite, have in common large, coarse reflectors, which are used to focus the radiation onto the detectors. Since they are designed to focus microwaves, these mirrors need not be perfectly smooth by optical standards; in fact, surface irregularities large enough to be seen with the naked eye do not affect their 1 Chapter 1. Introduction 2 focusing properties significantly. Most of these reflectors are at least l m in diameter and are parabolic. 1.2.2 Balloon Anisotropy Measurement (BAM) The B A M experiment uses a cryogenic differential Fourier transform spectrometer to measure anisotropics in the C M B R in five different frequency ranges. It had its first flight[5] in July, 1995 and landed intact, except for some damage to the structure of the gondola and the pointing system. It uses off-axis optics to direct two 0.7 degree beams, 3.6 degrees apart on the sky, into the spectrometer, which measures temperature differences between the two beams. 1.2.3 The Microwave Anisotropy Probe (MAP) Satellite The M A P project is currently being developed by the Goddard Space Flight Center in partnership with Princeton University for the purpose of making a full sky map of the C M B R at high angular resolution. See figure 1.1 for a schematic diagram of the M A P satellite. The planned launch date is 2000 and it is expected to be in operation for at least 15 months after it is launched. The satellite will eventually orbit the sun at the L2 libration point. The detectors will consist of passively cooled differential microwave radiometers. The M A P instrument should determine the values of the cosmological parameters to an accuracy which will discard some cosmological theories and will perhaps produce others, and it will also enable conclusions to be drawn about large scale structure formation in the early universe. 1.2.4 The M A P mirrors and the microwave system of the MAP satellite Specifications The M A P reflectors consist of two large 1.5mx 1.3m paraboloids serving as primary reflectors and two smaller .52 m paraboloids serving as secondary reflectors. Each will have a reflecting surface made of carbon fiber and a support structure made of paper honeycomb material. The reflectors will be placed in a Gregorian configuration at either Chapter 1. Introduction 3 Figure 1.1: Schematic diagram of the entire M A P satellite showing the reflectors near the top and at the sides side of the instrument1, with a A/60 figure tolerance. The beams will be 135° apart on the sky. See figures 1.1 and 1.2. The feedhorns are designed to gather radiation from a defined cross section on the secondary reflectors, which reflect radiation from less than the entire surfaces of the primary reflectors. In this way, taper, which is described in subsection 3.7.3, is used in this system to reduce the sidelobe response. Because it is a satellite experiment, all components must be as lightweight as possible, but they must also be able to withstand the tremendous acceleration and vibration associated with the launch. In space, the reflectors will be exposed to conditions different 1In Gregorian geometry, the secondary reflector is be placed beyond the focus of the primary reflector. This is distinguishable from Cassegrain geometry, for which the secondary is placed inside the focus, thus requiring a secondary reflector with a convex hyperboloid reflecting surface. Chapter 1. Introduction 4 MAP Thermal Reflector System (TRS) Figure 1.2: Overview of the thermal reflector system of the M A P satellite, which will be a single piece of composite material with a paper honeycomb support. from those in which they were built and measured. It is important to know whether properties such as the surface shape will change, and by what amount. Also, it must be determined how the reflector material behaves when it has been cooled down, such as whether surface irregularities form which would produce errors in the wavefronts reflected from it. The errors in the wavefronts would produce a loss of optical efficiency in the instrument and increased sensitivity in the sidelobes. The experiment described in this paper measures the surface roughness and thermal stability of the proposed M A P mirrors. The relation between surface roughness and sidelobe response is calculated, and the Chapter 1. Introduction 5 sidelobe sensitivity produced by the honeycomb deformations found in this experiment is determined. Importance of Constant Temperature over the Surfaces of the Mirrors The entire instrument is shaded by the solar array, so it will cool radiatively to nearly 50K. If the surfaces do not have a uniform temperature, because of earth or moon radiation falling or reflecting onto them, the data may be affected. Solar radiation is expected to be completely shielded from hitting any part of the optical system, but the earth is expected to affect the antenna temperature by a maximum of 0.033/JK and the moon by 0.007/uK. These values are calculated using the maximum angle at which the source appears above the radiation shields as the instrument rotates and orbits the sun. The earth will not affect the data, since the temperature measurements will not have that precision [6]. 1.3 The rest of the paper Chapter 2 outlines some cosmological theory related to the experiment. The analysis of the implication of the results of this experiment on the performance of the optical system of the satellite is based on antenna theory and diffraction calculations, which are outlined in chapter 3. To measure deformations and shape changes in the mirror which form when its temperature is changed, double exposure holography, or holographic interferometry, is used. Chapter 4 outlines the experimental procedures followed, shows the data, describes how the analysis was done, and includes results not directly related to the main experiment. The conclusions reached from this analysis and comparison with previous results are found in chapter 5. Appendices A and B contain information about the apparatus used to produce the holograms and to cool the reflector. Appendix C contains information about holography and holographic interferometry, as well as the procedure followed to reproduce a holographic image onto a photograph. Chapter 2 Cosmology and the CMBR 2.1 Big Bang Theory 2.1.1 Observations leading to Big Bang theory Homogeneity On small scales, stars, galaxies, and clusters of galaxies are observed, making the Universe appear lumpy and inhomogeneous. On scales larger than 300 Mpc, however, there is no apparent structure and the Universe appears homogeneous and isotropic. Expansion In the early 20th century, Edwin Hubble discovered that objects in all directions move away at velocities proportional to their distances. The constant of proportionality is known as the Hubble constant HQ1 and the relation is written v = H0d, where v is the velocity of recession and d is the measured distance to the object. This relation was found by measuring the redshifts2 of distant galaxies. 2.1.2 Early Big Bang theory Since the constituents of the Universe are known to be receding from one another with a rate determined by the Hubble constant, the Universe is expanding and either its density is decreasing or it is continuously being supplied by new material, keeping the density constant. If the first of these assumptions is true, and the Universe was more dense and therefore hotter in the past, it would be expected that there be radiation which *The Hubble constant is also written as HQ — 100ftkm/s-Mpc, where h is thought to be 0.6 observa-tionally, but most standard cold dark matter (CDM) models employ h = 0.5. 2 The redshift is given by z where z + 1 = where A is the observed wavelength of a feature in the spectrum of a luminous object and Ao is the wavelength of the feature in the object's rest frame. 6 Chapter 2. Cosmology and the CMBR 7 decoupled from the matter at a certain epoch and which is cooling with the universe. This radiation should be observable as a homogeneous, isotropic blackbody with a temperature of approximately 5K. If the second assumption, known as the steady state theory, is true, no such radiation should be measured. When the 3K cosmic microwave background was discovered, the steady state theory was dismissed and the big bang theory was adopted as most probably correct. This simplistic picture of the history of the Universe has two fundamental flaws, known as the horizon and flatness problems. The horizon problem is explained by the fact that points on the sky more than a few degrees apart, as seen from earth, were not in causal contact when the C B R photons underwent last scattering from the matter, and thus should be observed as different temperatures. Since the radiation is known to be approximately homogeneous and isotropic over the entire sky, it is clear that something happened to change the horizon size compared to the scale factor at an early epoch. The Universe is essentially flat at present; that is, there appears to be no overall curvature of spacetime. This is an unstable state; a small deviation of the density parameter Q from 1 at one epoch will produce a greater imbalance in the future. For this reason, the balance must have been even more perfect in the past; Q must have been closer to 1. Therefore, there must have been some event which produced this balance. This is known as the flatness problem. Both of these problems are solved when a model consisting of an inflationary period at a very early epoch is considered. 2.1.3 Thermal history of the early Universe Shortly after the big bang, the scale factor of the Universe increases exponentially beyond the causal horizon. This rapid expansion, which also makes the Universe flat, solving both the horizon and flatness problems, is known as inflation. After inflation, the Universe is a dense sea of hot particles and antiparticles being created and annihilated by the high energy photons. As the Universe cools, the particles are "frozen out": the reaction rate, T, for the coupling of the photons with each particle falls below the rate of expansion. Neutrinos are among the first stable particles to freeze out and they do so when they are relativistic. The neutrino background has been cooling since then with the expansion of the Universe. As the temperature decreases, protons and neutrons freeze out and are involved in Chapter 2. Cosmology and the CMBR 8 the equilibrium reaction n + ve «-» p + e - , and neutron decay n —> p + e~ + P^. The protons and neutrons are involved in nucleosynthesis. The fractions of each atomic species produced at this time depend on initial parameters such as the baryon fraction and the relative numbers of protons and neutrons. The baryon fraction depends on reactions which do not conserve baryon number, since, according to big bang theory, there was an equal amount of matter and antimatter immediately after inflation. The baryon fraction " = rZZbTr^hotZl i s k n o w n to be approximately 1(T 9. The numbers of protons and neutrons involved in nucleosynthesis are assumed to equal the total number of baryons, and their relative numbers are determined by the rate of the above equilibrium reaction, by the rate of subsequent neutron decay before the neutrons are included in nuclei, and also by the number of light neutrino species Nv. The rates are determined by nuclear physics experiments, and Nv can be inferred from the helium mass fraction, Y, at an epoch just after nucleosynthesis, called primordial helium abundance. Eventually, the temperature decreases to a point at which neutral atoms can form. This is called recombination. At this point, scattering between photons and free electrons essentially ceases, and the matter and radiation are decoupled. Any fluctuations in the matter density on the "last scattering surface" at this moment are therefore permanently recorded in the background radiation. Before recombination, the matter has begun to dominate the energy density of the Universe, taking over from the radiation. 2.2 Cosmological parameters and how they are measured The cosmological parameters described in the following section provide a picture of the geometry, history, and future of the Universe. Information for the following section was taken from references [7, 8, 9, 10]. 2.2.1 The Hubble constant and the age of the Universe It is known that the Universe is expanding, and since there have been no known factors which would change the rate of expansion significantly, it is possible to trace the expansion back in time to the epoch of much greater density. An approximate age of the Universe can be determined by inverting the expansion rate HQ; it is determined to be 16.29 billion years, for h = .6, which is close to the various values of h found through different forms Chapter 2. Cosmology and the CMBR 9 of observation. This places an upper limit on the age of the Universe, since gravity has acted to slow the rate of expansion. The globular clusters at the edge-of the Milky Way galaxy provide a lower limit to the age of the Universe. The stars in these clusters are very metal-poor and therefore old. If the evolution tracks of the stars in a cluster are studied, the cutoff age is the approximate age of the cluster. The ages of massive white dwarfs in metal poor globular clusters also place a lower limit on the age of the universe. The temperatures of these degenerate objects are measured and compared with the expected temperature for a given cooling time. From this, an estimate can be made of the oldest white dwarfs in a cluster, since the most massive stars have very short lifetimes as compared to the cooling times. The ages of radioisotopes contained in meteors can also be measured and can place a lower limit on the age of the Universe. Supernovae in distant galaxies provide a measure of the Hubble constant. The bright-ness of a supernova in a faint galaxy makes the galaxy visible, where it was invisible or barely detectable before. The apparent brightness of an exploding star, combined with a stellar model, gives an approximate distance to the galaxy. The measured redshift of the supernova provides a velocity estimate. Given the large number of visible galaxies in the Universe and the fact that one supernova explosion occurs in a galaxy every one hundred years, a few such measurements can be taken in a night of observing. 2.2.2 Cosmological density parameter There is a critical density pc above which the Universe will stop expanding under its own gravitational pull and will recollapse, and below which the Universe will continue to expand forever. The ratio of the actual density of the Universe to the critical density is known as the cosmological density parameter Q = ^ . The density parameter is related to the curvature k of the Universe through the Friedmann-Robertson-Walker metric. If Q, > 1, k > 0 and the Universe is closed; if Q, = 1, k = 0 and the Universe is Euclidean (flat); if £2 < 1, k < 0 and the Universe is open. The density parameter for baryonic matter only (Q,R) has been determined to be approximately 0.1 by measuring rotation of galaxies. The total density parameter is Chapter 2. Cosmology and the CMBR 10 assumed to be approximately 1, and therefore there is matter in the Universe not yet measured. This is known as dark matter, since it cannot be seen. Cosmological models have been based on cold (nonrelativistic) dark matter (CDM) and hot (relativistic) dark matter (HDM). Pure H D M models have been ruled out by observation, but further measurements of the C M B R are necessary to determine which of the C D M models best fit observation. 2.2.3 Using the power spectrum of the CMBR to determine the cosmological parameters As better measurements are made in the various regions of the power spectrum, estimates can be made of the cosmological parameters based on the many models. In this way, in-correct cosmological models can be eliminated and perhaps new models can be conceived. A complete map of anisotropics on the whole sky at all angular scales, for which the M A P satellite is designed, will place constraints on the geometry of the Universe and therefore the present density parameter Q0, the cosmological constant A, the baryonic density n B , the present Hubble constant Hn, the epoch of first structure formation, and indirectly, the age of the Universe. Figure 2.1 shows the power spectrum computed from a number of different models, courtesy of [11]. The multipole number £ is inversely proportional to the angular scale, 9, over which the temperatures are compared, and C( is the total power at that scale[12]. Figure 2.2 contains results from various C M B R experiments. The following sections outline the effects of various conditions in the early Universe on the power spectrum, some of which are illustrated in figure 2.1. Recombination At the time when the C B R photons decouple from the baryons (z « 1000), an imprint of the last scattering surface is made on the radiation. Anisotropies in the radiation arise at this time from gravitational redshifts which occur as some photons climb out of potential wells. This is known as the Sachs-Wolfe effect. The peculiar velocities of the matter at the last scattering surface causes Doppler redshifts in the radiation. Overdense regions are intrinsically hotter, thus causing the radiation which scatters from these to have a higher temperature. This reduces the effect of gravitational redshifts. On large Chapter 2. Cosmology and the CMBR 11 Figure 2.1: The power spectrum of the C M B R , plotted with various values for the cos-mological parameters. O is the cosmological density parameter, ClR is the contribu-tion to the density parameter by baryons, h is related to the Hubble constant, H0 by H0 = 100/zkm/s • Mpc, r is the optical depth, and A is the cosmological constant. Chapter 2. Cosmology and the CMBR 12 scales (a few degrees on the sky), these two terms reduce to a constant, producing the almost flat region at low I of the power spectrum. On smaller scales, the fluctuations in the potential </>, the peculiar velocities, and the overdensities 6 = ^ are able to undergo acoustic oscillations before recombination, and those oscillations which have already entered the horizon produce the Doppler peaks at high I in the power spectrum. At this point the rate of expansion has an effect on the oscillations. The dashed, long dash-dot, and the long dash-short dash lines in figure 2.1 show the effect of three different present Hubble constants, holding all other parameters constant. The epoch of equality of the energy densities of matter and radiation can be made earlier by increasing the value of h, thus deepening the potential wells at recombination. The Sachs-Wolfe effect will overcompensate for the increase in the anisotropics in the radiation, thus decreasing the amplitudes of the Doppler peaks. If the pressure is lowered by increasing fi^/i2, the Sachs-Wolfe effect will not be able to balance the adiabatic growth of perturbations, increasing the height of the Doppler peaks. This condition is valid for £2B > 0.05. Comparing the solid line to the dashed line in the same figure shows how the baryonic density affects the amplitude of oscillations. The pressure is increased by reducing £2R, increasing the impact of the Sachs-Wolfe effect and lowering the Doppler peaks. The non-baryonic matter does not participate in the oscillations and therefore it has no effect on the Doppler peaks. The last scattering surface has a non-infinitesimal thickness, due to the non-instanta-neous nature of the decoupling of the baryons and photons. Photons are able to random walk from regions of high density to regions of lower density during this time, damping out small perturbations. This is known as Silk damping and can be seen in figure 2.1 as a reduction in the height of the peaks toward high £. Another effect which causes a reduction in the heights of the peaks on small scales is the destructive interference of anisotropies of wavelengths smaller than the thickness when they pass through the last scattering surface. The position of the first (leftmost) Doppler peak in figure 2.1 provides an estimate of Q. The surface of last scattering is expected to be at a redshift z & 1000 in standard C D M models, independent of cosmological parameters. The angle the horizon at the LSS subtends on the present sky ( « 01/,2Gfe<?rees[13]) is dependent only on the geometry Chapter 2. Cosmology and the CMBR 13 of the Universe. Thus, the position of the first Doppler peak is virtually independent of n, ClR, A, and the history of the Universe since recombination, and depends only on the total density parameter Cl. This will perhaps be used to find the density parameter when the position of the first Doppler peak can be accurately determined. Immediately after recombination After recombination, only gravity connects the matter to the radiation, and before large structures begin to form, the gravitational potential 4> is assumed to be virtually constant. The photon contribution to the density of the universe changes the potential cj), however, for a short time after recombination. This effect is known as the early integrated Sachs-Wolfe (ISW) effect and has more of an effect as the total density Cl is reduced. It has a peak in the power spectrum slightly to the left of the first Doppler peak. Reionization If a region of space becomes so dense and hot that it becomes ionized, the radiation in that region is again coupled to the matter through collisions with the electrons. This happens in hot clusters of galaxies, and the mechanism through which it occurs is known as the Sunayev-Zeldovich effect. Thomson scattering between photons and free electrons in a cluster moving towards an observer produces a blueshift in the direction of that cluster, and the photons will tend to be given a higher energy by the hot electrons, distorting the Planck spectrum toward the high energy side. These effects are used to measure properties of cluster cores and are not expected to have a significant impact on the power spectrum of the C M B R . If the entire Universe became reionized and remained ionized until the present, this would have a significant impact on the power spectrum at small scales. The magnitude of the effect depends on the optical depth r of the ionized Universe and the redshift at which reionization occurs. At £ 3> 10, the power is suppressed by a factor e~2T, but at larger scales the power is unaffected. This is illustrated with the long-dashed line for r = 1 and the dash-dot line for r = 2 in figure 2.1. Compare this with the solid line, for which all parameters are identical to these except the optical depth, which is 0. Chapter 2. Cosmology and the CMBR 14 Low redshift effects If there is a cosmological constant A, it will contribute to 0, and if k / 0, the curvature will change 0, and these effects will both contribute to the power spectrum at low z. This is known as the late integrated Sachs-Wolfe (ISW) effect. The first condition is illustrated with the dotted line in figure 2.1, where there is a non-zero cosmological constant. This effect decreases the power at low £, and when the curve is normalized to the C O B E results, it appears to raise the curve at the high i side. As overdensities eventually collapse, radiation passing through them undergoes a net blueshift. This is known as the Rees-Sciama effect and is not expected to have a significant effect on the power spectrum of the background radiation in standard C D M models. A small amount of gravitational lensing may occur in or near regions of high density, such as clusters of galaxies, slightly washing out the Doppler peaks. This may be detectable in future C M B R experiments, as models and detection strategies become more precise. 2.3 The significance on cosmology of a complete measurement of the power spectrum, such as will be provided by the MAP satellite At present, only small portions of the power spectrum have been measured by the various C M B R experiments, and they have large uncertainties. See the error bars on figure 2.2 for an illustration of this. A clear determination of the positions and relative heights of the Doppler peaks is thus not yet possible. If a complete measurement of the sky is made at various frequencies, the temperature difference between any two points will be known, and therefore it will be possible to graph the power spectrum at all angular scales. The positions and heights of the-Doppler peaks at the right side of the graph in figure 2.1, which at present are not known with great precision, will provide a better measure of the cosmological parameters. Because M A P is a satellite experiment, it has much more time to take data than ground-based, balloon-borne, or rocket-flown C M B R experiments. Also, errors caused by the transmission of radiation through the atmosphere are not a concern. For these reasons, the systematic errors are expected to be much smaller than for all previous Chapter 2. Cosmology and the CMBR 15 O o CO O ^ o O* o CO o i—r 1 — I I i i i M I 3 5 . 9 ! ' " M K T o u — L 4 f JLU Ji=* T 1 I I JLU J l_jfc fc=U_ 10 100 I 1000 Figure 2.2: The results from the various C M B R experiments, plotted with error bars. The labeled result is from the B A M experiment (See section 1.2.2). The quantity Qfiat i s defined in terms of the total power Ce at a given value of I by the relation 5 ^ f l a f 1(1 + l)Ce = ^Q} Chapter 2. Cosmology and the CMBR 16 experiments. To make a complete measurement of the power spectrum of the C M B R with low systematic error is the purpose of the M A P experiment. Chapter 3 Diffraction estimation and control in CMBR experiments One important error to be overcome while measuring the C M B R is diffraction effects caused by the various apertures in the optical system. Errors are introduced in the wavefronts of the beams because of flaws or irregularities in the reflecting surfaces, and these contribute to the diffraction, reducing the gain in the main beam and increasing sensitivity in the sidelobes. There are many ways to overcome these effects, and each has drawbacks and advantages. The method used depends on the purpose, sensitivity, and angular resolution of the experiment. 3.1 Estimation of sidelobe rejection levels If an instrument used to measure temperatures at a particular frequency has an angular response G(0 ,0) , a noncosmological source with a temperature distribution T(9,fa) will produce an increase in the antenna temperature T 4 of This is valid for the regime where ^ < 1, where intensity is proportional to temperature in the blackbody spectrum. The M A P instrument has a beam size of approximately | ° . A simple model of the beam is therefore This is illustrated in figure 3.1. It is desirable to know the approximate value of the background, x, acceptable for various sources at reasonable angles from the main beam. The sources expected to have a significant effect are the sun, the earth, the moon, Jupiter, and the Milky Way. The instrument will point directly away from the sun at all times and will be shielded by a solar (3.1) (3.2) 17 Chapter 3. Diffraction estimation and control in CMBR experiments 18 A G(e,<|>) x - - o 4 10 10 0 4> Figure 3.1: The sample beam pattern described in section 3.1 used to estimate required sidelobe rejection levels for various sources. array. The radiation will be diffracted by the edges of the shields. The earth' subtends approximately 0.2° on the sky and is nearly directly behind the instrument. It, and the moon, which will always be more than 25° from the main beam, are approximately SOOK. The contribution of the Milky Way to the measured signal can be estimated as follows. Model the galaxy as a stripe about 10° wide and 30° long on the sky, at a temperature of 2mK. In this simple model, there is no contribution from the galaxy outside of this area. Using equation 3.1 and the above model for the beam given in equation 3.2, with a solid angle of Clbeam, the contribution, TA, from a source of average temperature T at an angle </> from the main beam, with a solid angle Ctsource, is given by source- (3.3) With G(0) « 1, Clbeam ~ ^ square degrees, G(4>) = x, T & 2mK, Clsource « 300 square Chapter 3. Diffraction estimation and control in CMBR experiments 19 degrees, and an acceptable antenna temperature AT A of 2pK, X « 1.3 x 10 , - 7 (3.4) Thus it is necessary for the sidelobe response, averaged over the extent of the Milky Way, to be less than the above value. Jupiter is approximately 250 K and subtends 1 x 1 0 - 4 radians on the sky. It will be used to calibrate the detectors. If its signal is to be rejected when it is not in the main beam, using the above argument (equation 3.3), the value of x must be less than 8 x 1 0 - 5 . Therefore, Jupiter requires less sidelobe rejection by about an order of magnitude when compared with the rejection required by the Milky Way. Equation 3.1 can also be expressed as the equations for an extended source, such as the Milky Way galaxy. The function G(6, (f>) is calculated directly from the surface roughness measured in this experiment, using equation 3.8. This is done in section 4.7. 3.2 Properties of a wavefront In the Huygens formulation, it is postulated that any wavefront is composed of a spher-ical wavefront source at each point. These wavelets interfere, reproducing the original wavefront if no disruptions are present. If the wavefront passes through an aperture which blocks some of the wavelets, the transmitted Huygens wavelets passing through the aperture interfere with each other and produce a diffraction pattern. The resulting beam pattern is a Fourier transform of the aperture shape. 3.3 Diffraction from a reflecting surface (3.5) for a pointlike source such as the sun, earth, or moon, and (3.6) In antenna theory, a reflector is considered equivalent to a lens of the same diameter, and will diffract as if it were an aperture of the same size. With a radially symmetric Chapter 3. Diffraction estimation and control in CMBR experiments 20 circular reflector, the image is broadened due to diffraction by an angle 8 ~ \/D, where A is the wavelength of the radiation and D is the diameter of the reflector. 3.4 Gain-loss and increased sidelobe sensitivity introduced by errors in a wavefront If an aperture introduces phase errors to an incoming wavefront, a loss in gain of the antenna is experienced [14]. For small phase errors and shallow reflectors, the fractional loss is given by the weighted mean squared error in the phase. 3.4.1 A model of an aperture with regular sinusoidal phase errors An approximation can be made of the sidelobe response of a reflector with a regular sinusoidal deformation across its surface. This is done by approximating the surface deformation of the reflector by a sine function of x, where x and y are in the plane of the aperture (the reflector), and by approximating the aperture shape by a square of side a. Since, for this experiment, only an order of magnitude estimate is needed of the sensitivity produced as a result of the phase error in the aperture, such an approximation is valid. If 9 is the angle from the axis of the system to a point on an axis parallel to the y-axis on the plane of observation, 4> corresponds in the same way to the x-axis, A is the wavelength under consideration, k = 2jL is the wavenumber (not to be confused with the curvature, k, described in the previous chapter), i ? 0 is the perpendicular distance from the aperture plane to the plane of observation, B is the amplitude of the surface deformation, and d is the wavelength of the deformations, then the equation describing the electric field at a point on the observation plane, in the far field (Frauenhofer) approximation, is: iA ra/2 iBsin(^r), , /W 2 , , , E = ——e-lkRo e~— x(l + - ^ d i e*ydy (3.7) XR0 J-a/2 A J-a/2 When the integral is computed, the fraction of the Electric field produced by the phase error, E$, at an angle 6, normalized by the no-error electric field in the main beam, E0(0), is W) _ iB ( f coa(*l*)8in(t) - \cos{^)sin{^)) E0(0) irXa & ~ & ( j Chapter 3. Diffraction estimation and control in CMBR experiments 21 Since the measured quantity is the intensity, and the above expression is out of phase with the component of the main beam, whose central height is 1, the square of the above gives the beam producedby the phase errors caused by the reflector deformation. 3.4.2 Hatbox model of phase errors in a wavefront If the surface deformations are irregular on the surface of the reflector, the analysis of the beam pattern is somewhat more complicated. For large phase errors, a model is devised of a wavefront consisting of a large number of discrete phase errors over an aperture, shaped like cylindrical hatboxes. With this model, the resulting radiation pattern is found to contain a component of a diffraction pattern which is reduced by a factor given by the exponential of the RMS phase error, in addition to a scattered field, whose angle of scattering is inversely proportional to the size of the correlated regions, where 2c is the size of the correlated regions, as given by the first and second terms, respectively, in the following equation: G(9,<j>) = G0(9, <f>)e^ + (~f(l - e - » ) K x ( ^ ) (3.9) where G0(9, 4>) is the no-error radiation diagram whose axial value is n ( ^ ) 2 , D is the diameter of the aperture, r\ is the aperture efficiency, A is the wavelength of the radiation, u is sin9, and A i is the lambda function. The equation arises from the Fourier transform of a beam which has had phase errors introduced into it which are proportional to the heights of the hatboxes. Thus, the uncorrelated beam will have a greater beamwidth with a rough surface (smaller c), and a narrower beam with a smoother surface. 3.4.3 A wavefront with phase errors shaped like Gaussian hats If the hatboxes are replaced by more realistic Gaussian shaped surfaces, a similar but more complicated result is obtained, as given by the following equation: G(9,4>) = G 0 ( M ) e + ( — ) 2 e 6 £ -e » ? ? ( — ) ¥ A ) , (3.10) A n = l n ' n l A where S2 is the average of the squared phase errors, e is the rms surface error on a shallow reflector which produces such phase errors, and the rest of the symbols have the same Chapter 3. Diffraction estimation and control in CMBR experiments 22 meaning as in equation 3.9. The summation in the second term is equivalent to the function 5(m, x), defined as: oo n 2 S(m, x) = e~x (3-11) n = l 7 1 ' n -and has been evaluated by Ruze [14] at various values of m and x. If the correlation lengths of these surfaces are small compared to the diameter of the reflector, and if there are small losses due to surface roughness, the scattered signal is diffuse enough to be ignored in a calculation of gain, and only the diffracted pattern, whose gain has been reduced by the exponential factor, remains. The expression for the gain is then G = G0e-s* = r ^ V - ^ 2 (3.12) A where Go is the gain of the original radiation pattern, 82 is the R M S phase error produced by the reflector, r\ is the aperture efficiency, D is the diameter of the reflector, e is the effective reflector tolerance, and A is the wavelength of the radiation being considered. If this relation is differentiated with respect to A and set to zero to find the wavelength of maximum gain, the result is that the maximum gain occurs for A m = 47re and is GmaX « ^ ( f ) 2 - (3-13) Y is known as the precision of manufacture. For a shallow reflector, the effective reflector tolerance e is the RMS surface error which will produce a phase front variance 82. Since small reflectors, on the average, can be manufactured with more accurate surfaces,- the wavelength of radiation for which a reflector is best suited depends strongly on the size of the reflector. 3.5 Effect of correlation length and wavelength of sinusoidal patterns on the beam pattern of a microwave reflector From the model of an aperture described above, it can be concluded that the beamwidth of the scattered portion of the radiation field will be much greater with surface defor-mations of small correlation lengths (c) and sinusoidal wavelengths (d). Therefore, since a narrow beam is desirable to reduce radiation from non-cosmological sources, a large Chapter 3. Diffraction estimation and control in CMBR experiments 23 reflector which experiences deformation over its entire surface is preferable to one which experiences many small deformations. The beam pattern of a reflector or aperture which produces phase errors contains two components: a component of the original beam pattern multiplied by a factor depending on the aperture efficiency, and a broad, scattered pattern produced by surface rough-ness. The correlation length c describes the distance along the surface of a reflector an irregularity extends. If this quantity is a small fraction of the reflector's diameter D, the scattered beam will subtend a larger angle than for greater correlation lengths. It is necessary to have a narrow, well-defined beam for C M B R experiments, because they measure such small temperature differences that radiation from a hot source, such as the sun, or an extended source such as the Milky Way, contained in the scattered beam pat-tern, could affect the antenna temperature. The sidelobe rejection levels for two sample sources are calculated in section 3.1. 3.6 Beam size and effective area of a reflector The total solid angle a beam subtends on the sky is given by the integral of the normalized power pattern over the entire sky. The main beam subtends a solid angle which is given by the same integral, with the position of the first nulls of the diffraction pattern on either side of the axis as its limits. The main beam will therefore subtend a solid angle less than the total solid angle; the ratio of these values is known as the main beam efficiency. If the power pattern of the main beam is approximated to be a Gaussian, the main beam solid angle is given by Qm = jj^Q2^, where #!/2 is the angular full width at half maximum (FWHM)of the main beam. Since B^ii ~ A / D , &mA oc A 2 , and from thermodynamics, the constant of proportionality can be found and the effective area Ae is related to the total solid angle and the wavelength by QAe = A 2 . This value is always less then the actual area of the reflector. 3.7 Reducing sidelobe response In experiments designed to measure low temperature phenomena on the sky, a contri-bution from a hot source such as the earth, the sun, or the moon in the main beam or Chapter 3. Diffraction estimation and control in CMBR experiments 24 the sidelobes could drown out the desired signal, and could perhaps damage the sensitive detector, if the sensitivity in that part of the beam is sufficient. To prevent this, measures are taken to reduce the sensitivity in the sidelobes as much as possible. Optical systems are designed to allow only radiation from within a certain angle to reach the detector, to reflect away radiation from the sides and bottom of an instrument, and/or to eliminate or reduce diffraction effects at some apertures. 3.7.1 Horn antennas Horn antennas are used without reflectors to reduce the number of diffracting apertures. They are used when it is very important to reduce sidelobe response but when angular resolution is not important. The only diffracting aperture on a horn antenna is the opening. In order to decrease radiation from the edges of the horn flowing along the inside walls, the horn is flared at the opening or corrugations of a certain size are machined inside it. The corrugations must be A/4 in size so this method is good only for a narrow bandwidth. Both modifications decrease the effective aperture and thus increase Clm, and they also make the beam more symmetric. 3.7.2 Curved edges on optics A large amount of the diffraction produced by an optical system exists as a result of radiation passing by sharp edges in that system. If some or all of these edges are reduced in sharpness, the sensitivity in the sidelobes of the optical system becomes smaller. This concept is applied to the B A M experiment in the form of a diffraction control flare, which has a smooth, curved surface, and is used to direct radiation towards the detectors, without producing diffraction. This is illustrated in figure 3.2 as a line diagram of the flare[15]. 3.7.3 The use of taper in a multiple-reflector system In a dual-reflector system, for maximum signal-gathering efficiency, the secondary reflec-tor is of a size and shape such that it will reflect the radiation from the entire primary reflector onto the detector. The secondary reflector can be tapered so that it reflects less Chapter 3. Diffraction estimation and control in CMBR experiments 25 Diffraction Control Flare Figure 3.2: Line diagram of the diffraction control flare used on the B A M experiment. Its curved edges reduce diffraction through the aperture. of the signal from the edges of the primary reflector onto the detector, or, most com-monly, the feed geometry can be constructed such that it gathers radiation from only the center of the reflector and not its edges. This is especially important in ground-based experiments where radiation from the hot ground is diffracted by the edge of the pri-mary reflector onto the secondary reflector. This method reduces the effective area of the primary reflector. 3.7.4 Diffraction shields or ground shields Most flying C M B R experiments have shields to protect the instrument from signals from large, high temperature sources, such as the earth, the sun, and the moon. For long wavelength experiments, these can be made of sheet metal or even wire mesh at the lowest frequencies. The shields keep the instrument cool and replace the emission from the ground with the less intense radiation from the sky, so that if this radiation is picked up in the sidelobes, it will not dominate the signal. Chapter 4 The Experiment This chapter describes the test reflectors and outlines the various procedures followed while producing holograms and heating and cooling the reflector, and describes the results and analysis. The procedures and their outcomes are placed in the order in which they were performed during the course of the experiment. 4.1 Test reflectors used in this experiment A sample reflector, manufactured by PCI (Programmed Composites Incorporated, Ana-heim, California) of the type to be used on the satellite has been tested. It is an on-axis paraboloid with a carbon fiber and epoxy composite reflecting surface. It has a 14" di-ameter, with a 2.5" hole in the center and a height of 2.5." It has a core of aluminum honeycomb which is sandwiched between two layers of composite. It has no support structure. Another sample reflector, made by COI (Composite Optics Incorporated) is available for tests. It has a 24" diameter and is 2.5" high, with a tab protruding from one edge at its focus. Is has a square composite box beam support and a very thin (0.50") layer of composite as the reflecting surface. This reflector is not tested using holography in this experiment because it is not part of M A P , but the cooling and warming profiles at the end of the chapter refer to this reflector. The test reflector is dark in colour, which increases the exposure time required to produce a hologram. It has a small visible pattern on its reflecting surface, however, which makes it easier to see whether a successful hologram has been made. Its small size makes it possible to illuminate its entire surface when the holographic exposures are made. A thin layer of chalk dust is spread on the surface of the reflector to make it lighter in colour so it produces a more pronounced image in the hologram. 26 Chapter 4. The Experiment 27 Figure 4.1: The first test of double exposure holography. The cardboard box deformed when a light aluminum weight was removed. The weight is seen as the rectangular object between the letters. See section 4.2. 4.2 First attempts at double exposure holography A small cardboard box which originally held unexposed holographic film is placed on the table with a weight on its top surface, and an exposure is taken from above. The weight, which is a thin piece of aluminum, is removed from the box, causing it to change shape. A second exposure is made. The resulting image is shown in figure 4.1. As can be seen, the fringes are not symmetric about the weight, as would be expected. The surface deforms in a somewhat complex way. The deformation appears to be centered at a point near to the lower right-hand corner of the box. 4.3 The first test of warm-cold double exposure holography The original purpose of the experiment was to measure shape changes in the reflector as it was cooled 200K or more below room temperature. To test the usefulness of the cooling and optical apparatus for this purpose, a double-exposure hologram is recorded of the object described in section A.2. The object is suspended into a container made of sheet copper soldered along the sides, with a lid made of clear Plexiglas which has been sealed to the container. The optical equipment is set up in such a way that it Chapter 4. The Experiment 28 produces the hologram from above. Two exposures are made without changing anything in between, except that for one exposure the container has liquid nitrogen boiling in it. The first warm-cold double exposure hologram was not successful because of the moving cold nitrogen gas enveloping the object. However, a dim hologram was obtained as the object warmed up after the nitrogen had boiled off. At this time, it became apparent that a system in which there is no gas or vapour (i.e. a vacuum) was necessary for cooling the object down to liquid nitrogen temperature. This led to the idea of using a vacuum chamber with a Plexiglas window through which a hologram could be taken, as described in appendix B. The object is cooled by allowing liquid nitrogen to flow through the coil. The same principle is applied, making one exposure when the apparatus is cool and another when it is warm. It was discovered over the course of the experiment that the copper table deforms when it is cooled so that a double exposure hologram is not possible with the cooling apparatus. Therefore, the reflector is heated instead. 4.4 The tests of the sample M A P reflector using double exposure holography The intended experiment, the purpose of which was to measure deformations in the reflector surface when it is cooled to less than 100 K , could not be performed because the reflector deforms significantly when its temperature is changed by a small amount. Temperature changes of more than approximately 10°C produce fringes so numerous and small that they are not visible. Instead of cooling the reflector to measure its deformations, it is heated by a small amount in air. 4.4.1 The first heating test A small portion of the reflector is heated with a single resistor to produce a temperature difference between the front and back surfaces and a gradient along those surfaces. The resistor is attached to the back surface of the reflector with aluminum tape. The reflector is mounted vertically, firmly attached to a heavy block of aluminum with a bolt and some washers. A potential of approximately 4 volts, which is just sufficient to heat the reflector without fear of damaging its surface, is applied to the single resistor, while the array of diodes, described in appendix A measures the temperatures at various points on the front and back surfaces of the reflector. An exposure is taken when the temperature of the Chapter 4. The Experiment 29 reflector at the resistor has been increased by a few degrees. The power supply to the resistor is switched off and the reflector is allowed to cool before taking another exposure. Four large, vertical fringes can be seen across half of the reflector centered at the point of heating, indicating that the reflector surface moves approximately 2 wavelengths or 1 micron at this point. Due to the difficulty in reproducing holograms, a photograph was not made of this image. 4.4.2 Heating the reflector with an array of resistors Twenty resistors, fastened to the back of the reflector with aluminum tape, as shown in figures A.8 and 4.2 and described in section A.6, are used to heat the reflector. A constant potential of 4 volts is applied and the reflector is heated to a temperature a few degrees above room temperature. After the power supply was switched off, the reflector achieved uniform temperature as it cooled. An exposure was taken at approximately 296K and another was taken as it neared its original temperature of 293K. Diode # Position 1 Back surface, 1" from edge 2 Attached to the aluminum block at the center of the hole in the reflector 3 3" from edge, front surface 4 2" from edge, back surface 5 2" from edge, back surface 6 3" from edge, front surface 7 4" from edge, front surface 8 1" from edge, front surface Table 4.1: Positions of diodes for the resistor-heated holograms. See also figure 4.2 for a diagram and figure A.8 for a photograph. For the first hologram (Hologram A) taken this way, shown in figure 4.4, the power supply for the resistors was left on for the first exposure so the surfaces of the reflector were not allowed to achieve a uniform temperature. The temperature profile of the reflector for the time during which the holograms (Holograms A and B) were made is shown in figure 4.3 and the positions of the diodes are listed in table 4.1. The image in figure 4.4 is reconstructed from a hologram for which exposures were taken at the two times labelled A on table 4.2, corresponding to the temperatures on Chapter 4. The Experiment 30 Figure 4.2: A diagram showing the approximate locations of the heating resistors on the rear surface of the reflector, denoted by small circles. Also shown are the positions of the measuring diodes, denoted by solid x's for those on the rear surface and by dashed x's for those on the reflecting surface. See figure A.8 for the same front view in a photograph. Concentric circles of incrementally increasing diameter are shown for reference, every 2 inches in diameter. figure 4.3. It can be seen that there are many small fringes, some distributed around the locations of the resistors, and some indicating an overall deformation of the reflector. There are approximately 50 parallel fringes along the surface of the reflector, and those in the middle are nearly stationary when they are observed from varying viewpoints. This indicates either that the reflector has bent slightly at the center or that it has tilted. The local deformations at the resistors are approximately 5-10 fringes deep. These deformations are due to the fact that for the first exposure of this hologram, the reflector was not at a uniform temperature because the resistors were still being provided with a potential at this time. The temperature difference between the back and front surfaces at these points causes the reflector to expand at one surface and not the other, producing g g g I i i i ' I i i i i U i i J i I i i ' i I i i i i I 0 500 1000 1500 2000 2500 T ime (s) Figure 4.3: The temperature profile of the reflector while hologram A was being made. The locations of the measuring diodes are listed in table 4.1 and diagrammed in figure 4.2. The vertical lines denote the times of the first and second exposures. the deformations. The image in figure 4.6 is reconstructed from a hologram for which exposures were taken at the two times labelled B on table 4.2, corresponding to the temperatures on figure 4.5. It shows only the horizontal parallel fringes and no obvious local deformations, except the honeycomb shapes produced by the support structure, visible near the dark fringes. The analysis to determine the size of the deformations is described in section 4.5. Chapter 4. The Experiment 32 First exposure Second exposure Figure Temp. Profile Hologram A 970 1320 4.4 4.3 Hologram B 3790 3900 4.6 4.5 Table 4.2: Times of the exposures of holograms A and B and the locations of the resulting images and temperature profiles. Figure 4.4: Hologram A, the image produced by a double exposure hologram of the reflector when it is heated with a resistor array before the first exposure and allowed to cool for the second exposure. The temperature difference is approximately 4 degrees. 4.4.3 Heating the reflector with a heat lamp Another method of heating the reflector is with the use of a heat lamp The lamp is held approximately a foot away from the reflector and is directed towards its front surface for a minute. The lamp is turned off and the reflector becomes a nearly uniform temperature as it cools, while the film is loaded into the holder. The temperatures of various parts of the surface of the reflector are measured by diodes, the positions of which are listed in table 4.3 and illustrated in figure 4.8. An exposure was taken when the temperature Chapter 4. The Experiment 298 33 296 SH Si V O H 294 292 3000 3500 4000 Time (s) Figure 4.5: The temperature profile of the reflector while hologram B was being made. The locations of the measuring diodes are listed in table 4.1 and diagrammed in figure 4.2. The vertical lines denote the times of the first and second exposures. was approximately even at 294K over the entire reflector, a couple of degrees higher than its original temperature. After a few minutes, the reflector cooled to nearly room temperature (291.5K) and another exposure was taken (Hologram C). Using the heat lamp instead of the resistors has the advantage that the surface is heated more evenly. The image of the reflector when it is placed horizontally onto the aluminum block, shown in figure 4.9, which is a reconstruction of a hologram taken when the reflector had been heated by a heat lamp, shows a deformation of the entire reflector in one direction Chapter 4. The Experiment 34 Figure 4.6: Hologram B, the image produced by a double exposure hologram of the reflector when it is heated with a resistor array before the first exposure and allowed to cool for the second exposure. The temperature difference is approximately 1 degree. only, similar to when the reflector was rigidly attached to the aluminum block. This is an unexpected result; it would be assumed that any deformation of the total reflector would be symmetric about its center hole, producing concentric rings in the holographic image, but this seems not to be so. There is also a slight deviation from completely parallel, straight fringes near one edge of the reflector, barely visible in the center right of figure 4.9. These parallel fringes could also indicate that the reflector has tilted. However, it is unlikely that the reflector would tilt by the same amount when it is heated by different methods on either its front or its back surface. Also, the fringes appear to be oriented the same way when the reflector is vertically oriented as when it is lying on its back surface and not attached to the aluminum block by a bolt. Chapter 4. The Experiment 35 2go i i i i L i i i I I i i i I i I i i 1 ' ' i I 0 200 400 600 BOO 1000 Time (s) Figure 4.7: The temperature profile of the reflector while hologram C was being made. The locations of the measuring diodes are listed in table 4.3 and diagrammed in figure 4.8. The vertical lines denote the times of the first and second exposures. 4.5 Analysis of the image to measure the deformation produced by the alu-minum honeycombs The steps used in the analysis of the image in figure 4.6, which shows most clearly the hexagonal deformations over part of its surface, are outlined in this section. Chapter 4. The Experiment 36 Diode # Position 1 2 3 4 5 6 7 8 Back surface, 1" from edge Taped to optical bench 3" from edge, front surface 2" from edge, back surface 2" from edge, back surface 3" from edge, front surface 4" from edge, front surface 1" from edge, front surface Table 4.3: Positions of diodes for the lamp-heated hologram. See also figure 4.8 for the diode positions. 4.5.1 Converting the image into individual row files, containing column num-ber and pixel intensity Figure 4.10 shows how S A O I M A G E and I R A F are used to view and manipulate the image with pronounced honeycomb-shaped deformation (figure 4.6). 1. The image file is converted from TIFF to FITS format using XView. S A O I M A G E can display FITS files. 2. The FITS image is converted to I R A F format using IRAF's rfits command. This is necessary so that I R A F can be used to manipulate the image. 3. Using IRAF's imslice command, the 2-dimensional image is sliced into 1-dimen-sional rows, which are stored in image files. Files containing the columns are also separated in this way. 4. S A O I M A G E is used to determine the row and column numbers of interest. 5. The image files with those row and column numbers are saved as ASCII files using IRAF's wtextfile command. 6. A simple C program is used to convert these rows and columns of pixel values into files with two sets of data, the column or row numbers (starting from the first column or row in the file) and the pixel (intensity) values. 7. Supermongo is used to plot these files individually or in combination. Chapter 4. The Experiment 3 7 V — 2 — 0 " T V V 14 . Figure 4.8: The positions of the measuring diodes, when the heat lamp is used to heat the reflector, denoted by solid x's for those on the rear surface and by dashed x's for those on the reflecting surface. The dimensions indicating the distances of the diodes from the edge of the reflector are shown in inches, and concentric circles of incrementally increasing diameter are shown for reference, every 2 inches in diameter. The diodes, except diode 2, are in the same positions as they were for the resistor-heated holograms. Three distinct features are seen in the data for each row: the overall illumination pattern, visible as a broad curve; the large fringes produced by the deformation; and the smaller, less distinguishable patterns due to the honeycomb shapes, visible as bumps on the fringe pattern curves. There is also a moderate amount of pixel noise present in each individual row, so the above features are best viewed when the data have been averaged over many rows, such as in the graph for which rows 500-600 are averaged, shown in figure 4.11. The honeycomb shapes are not as visible in this graph as in the data for the individual rows, since the light and dark parts of the honeycomb pattern are averaged together over this Chapter 4. The Experiment 38 Figure 4.9: Hologram C, the image produced by a double exposure hologram of the reflector when it is heated with a heat lamp before the first exposure and allowed to cool for the second exposure. The temperature difference is approximately 3 degrees. The reflector is placed horizontally on the aluminum block. area of the image. 4.5.2 Separating and analyzing the interesting parts of the image 1. The image shows the pronounced honeycomb pattern adjacent to the fringes in the lower right-hand corner of the image, so this portion is saved as a separate file in FITS format, using the XView program, and converted into an I R A F file as described in subsection 4.5.1. 2. From the close-up image, shown in figure 4.10, it is determined that rows 500-600 are suitable to use for the analysis. The criteria for this are that a sufficient number of rows must be averaged together to produce a standard pattern independent of the honeycomb deformations, which are approximately 15-20 columns wide, and Chapter 4. The Experiment 39 Figure 4.10: Using S A O I M A G E to examine the image to determine the amount of de-formation produced by the honeycomb core. that the effects of the brightness variation and the fringe pattern must be nearly constant over the range of rows. The average intensity profile of the selected rows is shown in figure 4.11. 3. If it is assumed that the reflector's surface is deformed evenly in the area between the two most distinguishable fringes, the functional form of the deformation with respect to the column number is trivially determined by finding the column numbers at the approximate centres of the fringes. These are columns 432 and 657, meaning that, over the area of interest, the surface is deformed by half a wavelength or Chapter 4. The Experiment 40 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 Column Number Figure 4.11: The intensity profile over the image in figure 4.12, averaged over rows 500-600 approximately 0.3 microns in 225 pixels along the rows of the image. Thus, the conversion factor between columns (x direction) and deformation in the z direction (perpendicular to the surface) is 225 pixels for 0.3 microns or 0.00133 microns/pixel or 0.00133 microns/column. Figures 4.12 and 4.13 show the section of the image from which the data were taken. Chapter 4. The Experiment 41 Figure 4.12: Part of the image in figure 4.6, with the region of interest indicated in the rectangle. 4.5.3 Determining the amount of deformation produced on the reflector sur-face by the aluminum honeycomb See figure 4.14 for an illustration of the analysis. To determine the amount of deformation indicated by a certain variation of intensity from the average, in a given set of pixels, a relationship between intensity deviation, A 7 , and distance along the z-axis (surface deformation), Az, is needed. There is a clear relation between average z-position and column number x, i f it assumed that the reflector has tilted or deformed in a linear fashion over the region of interest, which is illustrated in figure 4.6 as approximately equidistant, parallel fringes: z = mx + b, (4.1) where m is 0.00133 microns/column, from subsection 4.5.2. Therefore, only a means of inferring a distance in columns, Ax, that a given location on the image deviates from the average intensity, shown as the curve of figure 4.11, which has the functional form (4.2) Chapter 4. The Experiment 42 Figure 4.13: A close up of the image in figure 4.6, with the region from which the data were taken indicted in the rectangle. is necessary. If the derivative of equation 4.2 is taken at each value of x, as in A/(x ) = f'(x)Ax, (4.3) the surface deformation Az at each position (x, y) on an image can be determined, because, from equation 4.1, Az = mAx (4.4) and, from equation 4.3, the distance across the image, in the x, that a given location on the image deviates from the average intensity, is given by so that the desired relation is Al(x.y) , x A z = m m - (4-6» This gives the deformation Az at each location (x, y) on the image, and depends on the slope m in equation 4.1, the deviation AI(x) in the intensity of a pixel in the position i ^ x Figure 4.14: This figure shows an illustration of the analysis described in section 4.5.3. The top graph is analogous to figure 4.11 and the bottom graph shows a possible profile of the reflector surface in the region of interest. from the average intensity for that column (x), and the derivative of the graph given in figure 4.11 at that column.It is not possible using this technique to determine the sign of A z ; it is not clear whether the honeycomb pattern produces convex deformations or concave deformaations. This does not affect the resulting beam pattern, since it is calculated from phase errors. . If a set of adjacent columns in an image is considered, the deformation at each location is calculated, then the average over each rva lue in the column are calculated. 1. To find the value of f'(x) for each column x to be used in the analysis, the slope of the graph in figure 4.11 at each point is determined. This gives a derivative of the Chapter 4. The Experiment 44 average intensity with respect to column number and this is shown in figure 4.15. The original data are in the form of two sets of numbers, one of which is the column number x which runs from 1 to 990, and the other of which is the intensity I[x] at each of those columns, averaged over the rows of interest. The slope at each column number can be found by subtracting the intensity in the previous column from the intensity in the present column, as in ~[x] = I[x] - I[x - 1}. (4.7) This produces a very noisy result, which contains multiple passes through zero. To make it smoother, instead of subtracting single intensity values, the slope of the graph at a column number is given by the difference between the average intensity of the pixels three columns after it and the previous three pixels, divided by four, because that is the average "run" which corresponds to the "rise." The slope at a given value of x is given by the above as T\x+l]+r\x+2]+r\x+3] _ T\x-3]+T\x-2]+r\x-l} x] = 3 2 _ ( 4 8 ) 2. To further smooth the result, every eighteen adjacent pixels are set to their average intensity value. This is to reduce the effect of divisions by zero or very small num-bers caused in part by poor statistics. The resulting graph is shown in figure 4.15. Once the above calculations have been made, the difference between the above graph of average intensities, and a set of intensities averaged over a strategically chosen range of rows, can be given meaning. Since the graph of average intensities shown in figure 4.11 is not monotonic, the analysis must be done carefully, avoiding regions where the slope approaches zero. 3. If an intensity deviation from the average in a region (AI(x, y)) is found, it can be converted into a measure of local deformation by multiplying the value by the previously obtained 0.00133 microns/pixel (m = Az/Ax, where x is the column number and the x-axis of the graph), and dividing by the slope of the intensity graph at that point (AI/Ax). Chapter 4. The Experiment 45 •5 o -2 h T I 1 1 , p Rows 500-600 L a 4 ^ J l 200 400 600 Column Number 800 1000 Figure 4.15: The discrete slope calculations of the graph shown in figure 4.11 in intensity units divided by pixels. The slope is averaged every 18 columns to produce a smooth curve. 4.6 Final measurements of surface deformations caused by the honeycomb patterns As can be seen from figure 4.16, the deformation caused by the honeycomb pattern is 0.008-0.015 microns at the most. It was difficult to find useful sections of the image to analyze, since the slope of the intensity vs. pixel number graph passes through zero many times. Near these locations, the slope is very tiny and thus has a very large relative uncertainty, producing a large uncertainty in the result. For this reason, these areas must be eliminated from those to be analyzed. Since an order of magnitude estimate of the Chapter 4. The Experiment 46 Columns 605-620 i _ i i i ' i i I i i i I i i i__ I 500 520 540 560 580 600 Row Number Figure 4.16: The deformation of the honeycomb features along the region indicated in figures 4.12 and 4.13. deformation produced by the honeycomb backing is the goal, only one region containing a number of honeycomb-shaped deformations, shown in figures 4.12 and 4.13, was analyzed. 4.7 The beam pattern of the reflector, as determined from measurements of its surface deformation and antenna theory Since the measurements in this experiment determine that, when it is cooled, the reflector attains regular hexagonal surface deformations produced by the aluminum honeycomb backing, the analysis described in subsection 3.4.1 is most appropriate for determining the sidelobe pattern. The normalized pattern is given by the square of the following Chapter 4. The Experiment 47 S ( d e g r e e s ) Figure 4.17: The sidelobes in the beam pattern of the reflector, produced by phase errors which occur as a result of surface deformations measured in this experiment, extrapolated to A T = 200 K , for radiation with a frequency of 90 GHz. When this is calculated for v = 60 GHz, it is found that the height of the peaks scales as and the locations of the peaks scales as A. expression: B (jco8(>to)8in(>?) - ±co8(>f)8in(*to)) *A« (I)* - ( f ) 3 ( 4 - 9 j where a is the size of the aperture (in this case the size of the proposed beam pattern, 0.65 m); for the expected worst case, A = 0.33 cm, so that value is used; and d is the distance between consecutive honeycomb shapes, approximately 0.2"; and B is the ampli-tude of the surface deformation, determined to be 0.008-0.015 microns, where the average Chapter 4. The Experiment 48 value of 0.012 microns is used, multiplied by 200, since the reflector will be cooled by 200 degrees, instead of just one degree. The last item represents a worst-case scenario; it is unlikely that the deformation in the reflector is linear in temperature. The resulting sidelobe response is shown in figure 4.17. The peaks in this pattern at 9 ~ 40° reach a fraction of 1 0 - 7 of the main beam. 4.8 Additional results not directly related to the main experiment The COI reflector is cooled in the vacuum chamber, and then allowed to return to room temperature, while the temperatures at various points on its surface are measured at reg-ular intervals. The following profiles can be used to determine the thermal conductivity and thermal time constant of the reflector material. 4.8.1 Cooling profile of the reflector Diode # Position 1 Not working 2 Center bottom of reflector near copper plate 3 Approx 2" radially outward from #2 4 Approx 2" radially outward from #3 5 Approx 2" radially outward from #4, just inside support 6 Underside of edge 7 Halfway up copper wall 8 At the edge on the top of the tab Table 4.4: Positions of diodes in figures 4.18, 4.19, and 4.20. The reflector is placed on the copper table in a vacuum chamber, both of which are described in appendix B , and cooled in a vacuum with liquid nitrogen flowing through a copper coil which is attached to the table. Diodes are used to measure temperatures at various points on the reflector, and the positions of these are recorded in table 4.4. As can be seen in figure 4.18, the various parts of the reflector have similar shaped cooling profiles. As expected, the parts of the reflector closer to the copper base cool faster and reach a lower temperature than those farther from it. When the long-dashed curve of diode 4 and the dash-dot curve of diode 5 are compared, it can be seen that diode 5 Chapter 4. The Experiment 49 Time (min) Figure 4.18: The temperature profile of the reflector, as it is cooled in a vacuum envi-ronment by liquid nitrogen flowing through a copper coil. The reflector is situated on top of a copper table to which the coil is attached, inside a large vacuum chamber. The positions of the diodes used for measuring temperatures are listed in table 4.4. reaches and maintains a lower temperature, even though it is farther away from the base of the reflector. This is because diode 5 is closer to the support, which conducts the heat from that area towards the cold copper plate. Diode 8 (dark, solid curve) reaches a lower temperature than diode 6 (long dash-dot curve) because it is closer to the copper wall. The bump at the beginning of the curves is due to the diodes offgassing or responding in some other way to the sudden decrease in pressure. Perhaps the moisture which is on them condenses and then is removed by the pump. The temperature readings seem to Chapter 4. The Experiment 50 be reliable after this point, however. 4.8.2 The warming profile of the reflector i i i i i i i i i i i i i i i i_ i i i i i 200 300 400 500 600 Time (min) Figure 4.19: The temperature profile of the reflector, as it is allowed to return to room temperature in a vacuum environment after having been cooled by liquid nitrogen. The positions of the diodes used to measure temperatures are listed in table 4.4. From figure 4.19, it can be seen that the reflector warms uniformly when the flow of liquid nitrogen is stopped at 201 minutes. A closer look at this process in figure 4.20 reveals that diode 8 (dark solid curve), which is most closely coupled to the ceiling of the chamber, warms with a slightly greater slope than the other diodes, which are closer to Chapter 4. The Experiment 200 51 T 5 T 6 " T 7 T8 i i i i i i i i i i I i i i i I ' i i i ' I 200 210 220 230 240 Time (min) Figure 4.20: The data shown in figure 4.19, plotted in more detail. the copper plate. The fact that there is no delay between when the copper base begins warming and when the other parts of the reflector begin warming, indicates that the reflector material has a reasonably high thermal conductivity. This is also illustrated by the fact that the reflector rapidly assumes a uniform temperature after parts of it have been heated with resistors or with a heat lamp, illustrated in the temperature profiles in chapter 4, though the air surrounding the reflector in that case may contribute in a significant way to this phenomenon. Chapter 5 Conclusions 5.1 Overall deformation of the reflector With a temperature change of approximately 4 degrees, the shape is seen to change by about 50 fringes or 50 x 0.3 /x = 15 p. If one assumes a linear relationship between temperature change and shape change, the expected shape change for a temperature change of approximately 150 degrees can be estimated. This is the temperature difference between the photogrammetric measurements done at the Goddard Space Flight Center, and the result can be used for comparison of the two measurements. The shape change will be: Az = 15 a x n 600 u (5.1) 4 K This shape change is over the entire reflector and is therefore not as worrisome as the small, localized deformations, which occur as a result of the aluminum honeycomb back-ing. The deformation appears to be in one direction along the surface of the reflector and not circularly symmetric about its center. 5.2 Deformations caused by the honeycomb backing The reflector is found to deform by 0.016 to 0.030 microns, peak to peak, in many locations, each approximately 2 mm across, when the reflector is heated by 1 degree. Because the wavelength of these deformations is so small, approximately 5 mm across, this will produce an increase in sensitivity in the sidelobes of 1 0 - 7 of the main beam at approximately 40° from the center of the beam, as seen in figure 4.17. If radiation from a source above a temperature of approximately 100 K , such as the earth, the moon, or Jupiter, reaches the reflectors at such an angle, then that radiation is not expected to affect the measured temperature differences, which are approximately 10~6 K . The sun is not expected to shine at such an angle, because it will be behind the instrument, but if it 52 Chapter 5. Conclusions 53 appeared at that angle from the main beam, it would produce an antenna temperature of 5 x 1 0 - 4 K at the most. Since it is desired than the rejection of non-cosmological sources be at better than the 10~6 level, the aluminum honeycomb is acceptable, and the paper honeycomb design will also be acceptable if it is found to deform by approximately the same amount as the aluminum honeycomb, though it is expected to deform less. 5.3 Results of previous experiments using different methods 5.3.1 Photogrammetry tests done at the Goddard Space Flight Center A number of reflective stickers are placed at various points of the reflector surface. Pho-tographs are taken of the surface from various angles, and then the reflector is cooled to approximately 150 K. The photographs are then analyzed with a complex computer program. The surface deforms by a few hundred microns in a circularly symmetric way using this method[16]. From the results of the holography experiment described in this paper, it appears that the measurement is confirmed to an order of magnitude, although, according to the two experiments, the reflector appears to have deformed differently. Be-cause of the overall shape change, the aluminum honeycomb will be replaced by paper in the backing of the M A P mirrors. 5.4 Potential success of future similar experiments Although the complete intended experiment could not be performed, a similar method, coupled with a system to cool an object while keeping it stationary, can produce a precise measurement of deformations of its entire surface, provided the deformations are not greater than the amount which can be measured in this way. It seems very rare to find an object that will not deform significantly when it is cooled by such an amount, since the reflector material was designed not to change shape under such conditions, and does so by a measurable amount when the temperature is only slightly changed. The shape change caused by moisture absorption can also be measured in this way. A holographic exposure is made when an object has been sealed in a vacuum chamber immediately after the chamber reaches its lowest pressure. Over time, while the object is being pumped on, the moisture which it has absorbed from the air will be removed by the Chapter 5. Conclusions 54 pump and another exposure can be taken. While this process was not actually measured in this experiment, the same apparatus can likely be used for such an endeavour. It is likely that the test reflector with a paper honeycomb core, which will be produced towards the end of 1997, will be tested in the same way. It is expected and hoped that the honeycombs will not produce as much deformation as do the aluminum honeycombs. If this is the case, a greater temperature difference may be required to see the deformations. 5.5 Contributions to the antenna temperature by Jupiter and the Milky Way The contribution to the antenna temperature by Jupiter, when it is at various angles from the main beam is shown in figure 5.1. The peak near 40° will not affect the experiment, because it is much smaller than 2 pK. With the same approximation of the Milky Way as is given in section 3.1, the galaxy is found to contribute a maximum of ATA = 2 x 1 0 - 1 0 K in the sidelobes, using equation 3.6. Its contribution is small because the average sensitivity in the sidelobes is much smaller than the value at the peak, and the average over the entire source must be taken into account when the antenna temperature is estimated. Figure 5.1: The contribution to the signal by Jupiter, when it is 0 to 90 degrees from the main beam. It is calculated using the sidelobe pattern in figure 4.17 and equation 3.5. Its contribution when it is the main beam is not included. Bibliography [1] A . A . Penzias and R.W. Wilson. ApJ, 142:419, 1965. [2] H.P. Gush, M . Halpern, and E .H. Wishnow. Phys. Rev. Lett, 65:537, 1990. [3] J.C. Mather et al. ApJ, 354:L37, 1990. [4] A . Kogut et al. ApJ, 401:1, 1992. [5] G.S. Tucker, H.P. Gush, M . Halpern, I. Shinkoda, and W. Towlson. Anisotropy in the microwave sky: Results from the first flight of the balloon-borne anisotropy measurement (bam). ApJ, 475:L73-L76, 1997. [6] L . Page. Sun-, earth-, and moonshine on reflectors. Internal M A P document, August 1996. [7] R .B . Partridge. 3K: The Cosmic Microwave Background Radiation. Cambridge University Press, Cambridge, 1995. [8] M . Tegmark. Doppler peaks and all that: Cmb anisotropics and what they can tell us. preprint astro-ph/9511148, November 1995. To appear in Proc. Enrico Fermi, Course C X X X I I , Varenna, 1995. [9] W. Hu. In L . M . Krauss, editor, CMB Anisotropics two years after COBE: Obser-vations, Theory, and the Future. World Scientific, Singapore, 1994. [10] N . Sugiyama. In L . M . Krauss, editor, CMB Anisotropics two years after COBE: Observations, Theory, and the Future. World Scientific, Singapore, 1994. [11] N . Sugiyama. ApJ Supp., 100:281, 1995. [12] D. Scott and M . White. Echoes of gravity. Address: Center for Particle Astrophysics and Department of Astronomy, University of California, Berkeley, C A 94720-7304. [13] M . Kamionkowski. In L . M . Krauss, editor, CMB Anisotropics two years after COBE: Observations, Theory, and the Future. World Scientific, Singapore, 1994. [14] John Ruze. Antenna tolerance theory - a review. In Proc. IEEE, volume 54, pages 633-640. IEEE, Apri l 1966. [15] http://cmbr.physics.ubc.ca/experimental.html. 56 Bibliography 57 [16] L . Page. The effects of cooled composite optics on the map beams. Internal M A P document, March 1997. [17] R. Guenther. Modern Optics. John Wiley & Sons, Toronto, 1990. [18] G.W. Stroke. An Introduction to Coherent Optics and Holography, 2nd Ed. Aca-demic Press, New York, 1969. [19] P. Hariharan. Optical Holography. Cambridge University Press, New York, 1984. [20] N . Abramson. The Making and Evaluation of Holograms. Academic Press, New York, 1981. [21] L . M . Soroko. Holography and Coherent Optics. Plenum Press, New York, 1980. [22] R . R . A . Syms. Practical Volume Holography. Clarendon Press, Oxford, 1990. [23] J.E. Kasper and S. Feller. The Complete Book of Holograms. John Wiley & Sons, New York, 1987. [24] G.O. DeVelis, J.B.and Reynolds. Theory and Applications of Holography. Addison-Wesley, Reading, 1967. [25] J. Zhang et al. Optics Communications, 87:263, 1992. [26] T . Y . Chen. Experimental Techniques, 17(1):30, 1993. Appendix A The Apparatus used in the experiment A . l Basic Holography Equipment To produce the transmission holograms in this experiment, the following equipment is needed: optical bench, lenses, mirrors, laser, beamsplitter, and holographic film. Optical bench The optical bench has a grid of closely spaced threaded holes so that optical components can be easily positioned and adjusted. This surface must be free of vibration for successful holograms to be obtained. The bench is usually weighted and placed on a vibration-blocking layer commonly comprised of partially compressed tennis balls which in turn rest on a slab of concrete in a basement. This is standard for most holography experiments. Laser Laser light is used because of its coherence and monochromaticity. See appendix C for a description of the properties of laser light. The lasers used to make and reproduce the holograms in this experiment are 0.5 mW red (632.8 nm) He-Ne lasers. The beam width at the entrance of these lasers is approximately 1.9mm. Beamsplitter The beamsplitter splits the single laser beam into two beams: the refer-ence beam and object beam. The reflectance of the beamsplitter must be chosen so that the film is illuminated approximately equally by the reference beam and the light reflected from the object. For a large object, this requires that the inten-sity of the illumination beam be much greater than the intensity of the reference beam. For the main part of this experiment, a microscope slide is used because most of the laser light is required to illuminate the large surface of the reflector so that a large enough proportion of the light reaches the film from the object. A simple beamsplitter of any desired trarismittance can be produced by depositing aluminum or another metal onto a flat piece of glass, such as a microscope slide; 58 Appendix A. The Apparatus used in the experiment 59 the transmittance depends on the thickness of the metal deposited. A beamsplitter with varying transmittance along its length can be made by holding one end closer to the molten metal while the deposition is taking place. Flat mirrors The mirrors are used to direct the light in the desired directions. The mirror mounts are held onto the optical bench with mounting brackets held with \ - 20 bolts. Lenses The lenses are used to spread out the reference beam and the beam used to illuminate the object. Lenses of an unusually small focal length are required to illuminate the surface of the reflector, which is a small distance away from the lens. The amount of divergence from a transmitting lens ball can be calculated by using the index of refraction of the material (n), its diameter (d), and the incident laser beam width (b). See figure A.2. outgoing divergent beam Figure A.2: The diagram used while calculating the divergence of a lens ball Snell's law: sinO = rising (A.2) where n is the index of refraction in the lens (n=1.77). The index of refraction of the incident medium is that of air (n=l). From this relation and the isosceles triangle formed by the beam and the two radial lines, it can be seen that the outgoing beam Appendix A. The Apparatus used in the experiment 60 is at an angle 6 from a line which starts at the center and ends at the point on the surface of the lens where the beam emerges. It can be seen from the diagram that r? + 0 = 0 (A.3) and therefore T = n-4) + 0 = 26-2(j) (A.4) and from equation A.2 it can be seen that 0 = s m - 1 ( — ) (A.5) n and since sin9 = - (A.6) r we have, when this is used in equation A.4: r = 2 s m _ 1 ( - ) - 2sin-1(—) (A.7) r rn If 6 <C r, equation A.7 reduces to r = 2 ( - ) - 2 ( - ) = 2 6 ( — ) (A.8) r rn rn but with the laser used in this experiment, which had a beamwidth of approximately 2mm, and the 2.5mm diameter lens ball, this approximation is not valid, and the divergence is T = 2stn-1(—- - 2s%n~l — - r r — r r = 52.5° A.9 v1.25mm y v(1.25mm)(1.77)' Similarly, the divergence of the 3 mm lens ball is 39.4°. The lens balls have other advantages over microscope objectives, such as the simplicity of cleaning their sur-faces. Any dust can be blown off with an aerosol blower. This is important, because any dust or other small particles on the surface of the lenses produce patterns in their beams. It is desirable to eliminate this effect as much as possible, so any inter-ference patterns of experimental interest are not obscured by, or confused with,these patterns. Their mounts, which are simply pieces of appropriately sized aluminum Appendix A. The Apparatus used in the experiment 61 angle, are also compact and easy to incorporate into the optical system. Unfortu-nately, the lens balls with small diameters similar to the beamwidth of the incoming laser beam produce interference fringes on the objects which are visible across the holograms. This is due to the multiple path lengths of the light diverged by the lens. This effect is much more visible in a holographic image produced with such lenses than on the illuminated object itself. Because of this effect, a microscope objective is used to illuminate the reflector, with a large, flat mirror which increases the distance the divergent beam travels, maximizing its spread. See figures A.4, A.5, and A.6. Spher ical m i r ro r A special mirror is used to produce a divergent reference beam. A small spherical lens, onto which has been deposited a film of reflective aluminum, is placed near the object, which is illuminated by a divergent beam. When a laser beam is directed towards the convex aluminized surface, it is reflected as a divergent beam of approximately the correct size and intensity to illuminate the film and interfere with the light reflected from the object at the film plane. A small ball bearing is also used for the same purpose. The divergence of a spherical mirror is T = sm _ 1 (^ ) , and therefore, a 1 cm diameter spherical mirror will produce a beam which diverges by 5.7° when a 2mm beam is directed at it. A steel ball bearing, about 0.8 cm in diameter, is used to spread out the reference beam further than the 1cm mirror, which is necessary because otherwise, the reference beam would be brighter on the film than would the light from the object. A second diverging lens is usually used to spread out the reference beam. Holography f i lm The holography film (fine grained photographic film) used in this experiment is Kodak SO-253 high speed holographic film. A disadvantage of this type of film is that it has an antihalation coating which does not allow light to penetrate to the emulsion from the back of the plate, so that the object and reference beams must be on the same side of the plate. After exposing the film, it is placed in a lightproof plastic canister with the emulsion side inwards and attached with a rubber bracket which holds it in place. The method of developing such film is as follows: Appendix A. The Apparatus used in the experiment 62 1. D-19 developer is poured into the canister and the canister is rolled on a special machine for 5 minutes, then emptied with a funnel into the developer bottle. 2. The film is rinsed with tap water for 30 seconds and then the water is poured out. 3. Fixer is poured into the canister which is rolled for 2 minutes then emptied into the fixer bottle. 4. The film is taken out and washed in a tap water bath for 1 minute or more. 5. If necessary, the film is placed in bleach until it is clear, then rinsed thoroughly. 6. Photo-flo can be used to prevent water marks. 7. The film is squeeged then placed in an air dryer or hung to dry. The water used during this process should be as close to the developer temperature as possible, which should be near 69°F. A.2 The first test of warm-cold double exposure holography For the initial test using liquid nitrogen, the apparatus is set up as shown in figure A.3. The object is a piece of h-beam aluminum, of which half has been removed with a milling machine, and to which is attached some brass threaded rod. This is done so a double exposure hologram yields a calculable number of fringes when it is cooled to low temperature. This number is found by using the thermal expansion -y^ of the two metals. See figure A.3 dBrass = d ^ ~ ((-^")Brass,300K ~ (-^-)Brass,80K)) ( A - 1 0 ) dA\ = d& - «17)A1,300K ~ (77)^°*)) (A.11) $ = ^Brass ~ d>A] I d (A.12) (A.13) Q x = r(l — cos—) (A.14) 2sm(f) 6 '"2' Appendix A. The Apparatus used in the experiment 63 ' u n — f i Figure A.3: The object used for the first test of warm-cold holographic interferometry. The amount of deformation has been greatly exaggerated for illustrative purposes. See section A.2. and with - 5 (AL/L0A1.300K = 4 3 1 x 1 0 ( A L / L , ) A l j 8 0 K = 24 x 10 (AL/L0Brass,300K = 3 9 7 * 10 (AL/LOBrass^OK = 34 x 10 d = 10 cm - 5 / = 5 cm Appendix A. The Apparatus used in the experiment 64 the distance that the middle of the surface moves, x, can be calculated: ^Brass = 1 0 c m ( l - (397 - 34) x 1(T 5) = 9.9600 cm (A.15) d'M = 10cm(l - (431 - 24) x 10~5) = 9.9593 cm (A.16) . 9.9600cm - 9.9593cm . 0 = _ = 1.4 x 10~4 (A.17) 5cm 10 cm R = 0 . ^ 1 - 4 x 1 0 - 4 = 7.14 x 104cm (A.18) 2sin 2 1 4 x 1 0 - 4 x = 7.14 x 10 4cm(l - cos-— ) = 1.74 x 10~4cm (A.19) This can be expressed as a number of fringes by dividing by a half wavelength (in this case, A = 6.328 x 10 _ 5cra): Number of fringes = ^ ^ x i o - g r i n " = 5 f r i n g e s (A.20) 2 This is an acceptable number of fringes, since they will be large and therefore easily visible. See section 4.3 for the results of this first test. A.3 Apparatus used to make the holograms in the large vacuum chamber The apparatus used to take the double exposure holograms of the reflectors consists of all the items previously mentioned, as well as a large vacuum chamber, and a copper table to which has been soldered a coil of copper tubing. See appendix B. A.3.1 Elimination of vibration in the vacuum chamber As with all holography experiments, reduction of vibration to a minimum is essential. This is illustrated by the wild movement of the fringes when any part of a Michelson interferometer is disturbed. In holography, such movement during an exposure would extinguish the holographic effect and an image would not be visible in the resulting hologram. A type of holographic interferometry known as time average holography (see subsection C.8.2), is used to find the modes of vibration of an object. To isolate the apparatus, 16 tennis balls have been placed underneath the vacuum chamber in a ring around the bottom flange through which the coils are supplied with liquid nitrogen. Appendix A. The Apparatus used in the experiment A.4 The configuration of the optical bench and other apparatus 65 computer • circuit used for measuring temperatures heat lamp optical table reflector m wm. atiddiodex Figure A.4: The setup of the apparatus for making the heated double exposure holograms, showing the optical bench and the computer used to record the temperatures. Figures A.4 and A.5 show the configuration of the apparatus when some of the holo-grams were produced. The optics equipment is placed so that nearly the entire reflector is illuminated evenly. This is done by directing the light from a microscope objective lens onto a large, flat mirror, which is placed at the opposite end of the bench as the reflector. The maximum beam spread and the most efficient use of the available laser light occurs when the beam from the microscope objective nearly fills the flat mirror. The flat mirror can be rotated or tilted to direct the light toward the reflector. Appendix A. The Apparatus used in the experiment 66 Figure A . 5 : The setup of the optics for some of the heated double exposure holograms. The reference beam is produced by directing the light reflected from the beam splitter towards a shiny steel ball bearing perched in the socket of a \ — 20 bolt which is partially threaded into the optical bench. The ball bearing is slightly smaller than the previously mentioned aluminized lens ball so it produces a more greatly fanned out beam. The ball bearing is placed in line with the reflector but separated from it from the point of view of the film. This is so that the image of the reflector is not near the laser spot when the hologram is being viewed or reproduced. Appendix A. The Apparatus used in the experiment 67 Figure A.6: The vertical setup of the optics for some of the heated double exposure holograms. The reflector is placed on top of the aluminum block for the exposures. A.5 Measuring the temperatures at various points on the apparatus An array of diodes provides reasonably accurate measurements of temperatures primarily on the surface of the reflector. The precision of the temperature measurements is at least 0.01 K and the accuracy depends on the voltage-temperature calibration and the uniformity of the supplied current. Variations in the voltage supplied to the current source may change the temperature calibration slightly, but not more than 0.1 K . The diodes are attached with plastic tape, as shown in the photograph of figure A.8, and since they were intended to make measurements of temperatures much lower than room Appendix A. The Apparatus used in the experiment 68 }To digital I/O card in the PC (4 leads for each multiplexer switch) Voltmeter 50 |xA current source Harris 508-5 CMOS 8-channel analog multiplexers Figure A. 7: The circuit used to measure diode temperatures. The short-dashed rectangle denotes the multiplexer switch used to measure voltage, and the long-dashed rectangle denotes the multiplexer switch used to provide current. temperature, manganin wire is used to connect them electrically to the circuit, which is shown in figure A.7. Manganin is used because of its low thermal conductivity. The wire is provided with a nonconductive coating, which must be stripped off at the ends to enable it to be soldered. Strip-X can be used for this purpose, and it can take 15 minutes or more for the stripping process to be completed.1 The multiplexer switches used in this experiment accept 8 analog inputs and choose one of them based on a second input from a digital i /o card installed in a P C . These chips are powered by common + and —15 V D C power supplies. A program enables •"This is in disagreement with the instructions on the bottle, which instruct the user to wait only 5 minutes. Appendix A. The Apparatus used in the experiment 69 the 8 diodes to be provided with current and measured in sequence at regular intervals, determined when the program is started. A digital multimeter, also connected to and controlled by the P C , is used to measure the analog output from the voltage switch. It communicates with the computer via a GPIB connection. A constant 50/iA current source, powered by a 15V D C power supply, is used to supply current to the circuit. A .6 The array of resistors used to heat the reflector Twenty 49.90 resistors are attached to the surfaces of the reflector with aluminum tape, as illustrated in figure 4.8. They are connected in parallel in two groups and two D C power supplies are needed to produce a potential of 4V across each resistor. This provides adequate power to increase the temperature of the entire reflector by 4°C. Appendix A. The Apparatus used in the experiment 70 Figure A .8 : This photograph shows the back of the reflector with the heating resistors and measuring diodes attached. The aluminum block to which it is fastened rests on the surface of the optical bench. Appendix B The system used to cool the apparatus B . l The cooling apparatus To cool the object of which the hologram is being taken, liquid nitrogen is forced to flow through a coil of copper tubing which has been soldered to the bottom of a large ^ r " copper plate. This table has fiberglass legs and a vertical copper wall attached at its edge. Fiberglass was chosen because of its low thermal expansion coefficient. It is desired that the object on the table move vertically as little as possible. Superinsulation is wrapped around the table and wall to block radiation from the warm walls of the vacuum chamber, except at a hole in the top through which the holograms are taken. It is desired that the mirror come to a uniform temperature near to liquid nitrogen temperature. This is confirmed with a system of 8 diodes attached to various parts of the mirror and at various places on the inside of the chamber. A program was written to measure and record these temperatures in sequence at a given time interval. The cold hologram is taken when the temperature is reasonably uniform across the mirror. B.2 Preliminary cooling tests To perform these tests, a small aluminum plate on an aluminum leg is placed on the copper table, which is inside the vacuum chamber. Measuring diodes are attached with tape onto the object and in various places on the copper table and walls. Care is taken to touch only the grounded side of each diode with the aluminum tape, to prevent short circuits. The vacuum chamber is sealed and pumped down. When it has reached a low pressure, liquid nitrogen' is allowed to flow from a slightly pressurized {PgaUge > 10 psi) dewar into the chamber and through the coil on the copper plate. The various parts of the apparatus inside the chamber eventually assume a quasi-constant temperature. 71 Appendix B. The system used to cool the apparatus B.3 Results of preliminary cooling tests 72 Tima (min) Figure B . l : The results of the first preliminary cooling test performed on the apparatus. See section B.3. The temperatures read by the various diodes in the first and second tests are shown in figures B . l and B.2 respectively. The positions of the diodes are noted in tables B . l and B.2. As can be seen from figures B . l and B.2, the copper plate reaches a low temperature relatively rapidly. A l l parts of the copper reach a temperature and then maintain a constant temperature until the supply of liquid nitrogen is terminated. The temperature of the copper has a strong correlation with the height above the copper plate. Some interesting effects are noted when the pumping valve is closed and the pressure in the chamber is allowed to increase. This is illustrated in figure B.2 for diodes 3, 4, and 6 in particular, and for diodes 1 and 8 to a lesser extent. The following table shows the times in minutes when pumping was stopped and resumed. The pressure in the chamber is allowed to reach approximately 400 pm Hg before pumping is resumed. Appendix B. The system used to cool the apparatus 73 Time (min) Figure B.2: The results of the second preliminary cooling test performed on the appara-tus. See section B.3. Stop pumping Start pumping 177 219 222 264 268 307 310 345 347 Appendix B. The system used to cool the apparatus 74 Diode # Position 1 Not working 2 Center top of the copper plate 3 Halfway up the vertical wall -4 Hanging above the copper plate 5 Near edge of copper plate 6 Near top of vertical wall , 7 Near bottom of vertical wall 8 Not working Table B . l : Positions of diodes in figure B . l . Diode # Position 1 Bottom of the aluminum plate 2 Center top of the copper plate 3 Halfway up the vertical wall 4 Hanging above the copper plate 5 Near edge of copper plate 6 Near top of vertical wall 7 On opposite edge to diode #5 8 Side of Al/brass test piece (section A.2) Table B.2: Positions of diodes in figure B.2. Appendix C Holography C l Basic Holography Principles Photographic images are made by recording the intensity of light falling on a piece of film at each part of a focussed image, while a holographic image is a recording of both amplitude information and phase information. This permits the reconstruction of the entire original wavefront, and allows for the 3 dimensional quality of holographic images. The viewer has the ability to look around the image to see parts that initially appear hidden by others. There is no detector which can record phase information directly, so such a complete measurement of an incoming wavefront as is required in holography can only be done with light itself. The interference pattern of two wavefronts is recorded on very fine-grained photographic film, or other recording device, and one wavefront can be used to reproduce the other when it is directed toward the processed film. Since interference of light is important in holography, a discussion of the various properties of coherent light is included. C.2 Coherent Light Interference is a manifestation of light's wavelike quality. To produce visible interference patterns, the light used must be very nearly coherent. The relation describing wave propagation of light [17] is E(x, t) = E0cos(ujt - kx + 4>) ( C l ) where E(x,t) is the electric field measured at position and time (x,t), EQ is the amplitude, OJ is the frequency, and <j> is the phase. The following discussion concerns the coherence of light from a single monochromatic source. If such light is permitted to interfere with 75 Appendix C. Holography 76 itself, such as with a Michelson interferometer, the path length difference between the light in the two arms is important. The light emitted at a given time interferes with light which was emitted at a previous time and the time difference or retardation time corresponds to the path length difference Ad by the relation Ad = CTd. If the frequency of light emitted by the source changes in this time, the interference is described by the addition of two waves of different frequencies u\ and u>2-Ei(x, t) = E0cos(uit - k xx + fa) (C2) E2(x, t) = Eocos(co2t - k 2x + fa) (C.3) When these are added, the resulting wave is described in one dimension1 by the equation E{x, t) = 2E0cos(cot - kx)cos(Aut - Akx) (C.4) where Au, = — — , u, = - T -2 ' 2 Au> is the beat frequency of the interference. The interference pattern gives a measure of the time averaged intensity, which is determined from equation C.4 above to be I(x) 2 -T4 = 2 + — — [cos[Auj(2t0 + T) - 2Akx]sin{AuT)) (C.5) IQ Aui where the quantity I0 normalizes the total intensity, t0 is the moment of initial intensity measurement, and T is the duration of measurement. When the above is compared with the resulting intensity from the vector addition of two waves of identical frequency, ^ = 2 + 2cos8 (C.6) where 5 is a quantity related to the differences between the phases, fa and fa, and the wavevectors, ki and k2 of the two waves, it is clear that the second term in equa-tion C.5 is the interference term. This interference term clearly becomes negligible if 1 Waves interfere in each direction of polarization separately so the interference can be described in this way without loss of generality. Appendix C. Holography 77 AooT » 1, when, for a reasonable integration time, the beat frequency exceeds an amount corresponding to a wavelength deviation A A of 1 0 - 4 nm or less. If AooT < 1, the approximation sin(AooT) « AooT becomes valid and the equation becomes ^ - = 2 + 2cos[Aoj(2t0 + T)~2Akx) (C.7) If the beat frequency Aoo is small, the above equation reduces to equation C.6, and the waves interfere like those of identical frequency. There are three different types of coherence[18]: Spectral coherence This refers to a condition where the waves from a source are regu-lar. It is also called short term temporal coherence. This means that there can only be one wavelength; otherwise, waves of different wavelengths would be emitted at different times. This requires monochromaticity. This type of coherence refers to a condition in which there is a single value of oo in the above argument and the total intensity when such light interferes is given by equation C.6 above. Spatial Coherence At an instant in time, the waves at points equidistant from the source are all in phase. The only true source of spatially coherent light is a point source. A wave is spatially coherent if at all points along a wavefront, the wavevec-tor k is identical. Temporal or longitudinal coherence There are two types of temporal coherence: Constant frequency over time This requires that the source emit constant en-ergy. This requirement is satisfied if two waves from a single source separated by a retardation time with an optical system differ in frequency by an amount less that that which would make the interference term in equation C.5 negli-gible. Constant phase over time This type of coherence requires that a source emit waves of constant phase. A source will satisfy this condition over its own coherence time r c . The coherence time is measured by determining the path length distance between two interfering rays from a single source for which the fringe visibility vanishes. Appendix C. Holography 78 The fringe visibility V is related to the intensities of the bright (Imax) and dark (Imin) fringes in an interference pattern by Imax Irrnn - m u x -ram -t-rn.ar. I •im.i'n lm x "" I m A quantity related to the coherence time is the coherence length (ie), which is evaluated with it = cr c. The coherence length associated with frequency stability is These coherence conditions are by no means limited to electromagnetic waves. Water waves can be produced which exhibit these qualities, especially spatial coherence, since a small disturbance on the surface of water produces a number of concentric wavefronts indicative of a point source. Sound waves can also be coherent. Some waves are coherent in some ways but not in others. Using the concept of spatial coherence, interferometry can be used to measure dis-tances to stars and galaxies, because they are almost point sources,and only plane waves reach the Earth. Therefore, if two detectors are placed a known distance apart,an anal-ysis of the interference between the two sets of data from the source will yield distance estimates. This is the principle on which very long baseline interferometry (VLBI) is based. It is used both to measure distances to radio sources and to produce images of nearby sources. C.3 The Laser Lasers (Light Amplification by Stimulated Emission of Radiation) are very nearly com-pletely coherent, due to the process by which they emit light. They work on the basis of electrons falling from a specific atomic energy level to another, so that they are very nearly monochromatic. The atoms in a laser's gas tube exhibit a cascade reaction whereby one atom's emission causes other atoms to drop from a metastable energy state to a lower state in phase. This produces waves which are very spatially coherent. Long term tem-poral coherence is evaluated by the coherence length of the laser, which is important in interferometry and therefore holography, and affects how the apparatus is assembled. Appendix C. Holography 79 The coherence length depends to a large degree upon the length of the gas tube of the laser. If it is very long, it will be bright, because there are more atoms to emit light, but there will be more possible modes, and the laser will switch between those modes while it is in operation, abruptly changing the phase of the emitted light. The average time between such mode-changes is called the coherence time r c and is discussed above in section C.2. The coherence length of a laser can be evaluated by assembling a Michelson interfer-ometer with one arm much longer than the other. The level of coherence is determined by measuring or estimating the fringe visibility of the interference pattern. The coherence length of the laser can be estimated by multiplying V, the fringe visibility, by the current pathlength difference. The coherence length of the laser used in this experiment has been determined by this method to be at least 100 meters. C.4 Models of Holography A piece of film on which is recorded a hologram contains the interference patterns pro-duced by the reference beam and the light reflected from each point on the object. Therefore, a hologram of any object beyond a speck of dust is very complicated and a model is needed to describe the process. The zone plate model is a good approximation for holograms recorded on thin film and the geometric model is applicable to holograms recorded on thick film, or volume holograms, but is not a good approximation for those recorded on thin film. The information about models and types of holography were obtained from references [19, 20, 21, 22, 23, 24]. C.4.1 The Zone Plate Model This model works on the principle of the interference between the spherical wavefront produced by reflection from each point on the object and the plane wavefront of the reference beam, whose source is comparatively distant. In an on-axis system, the two wavefronts will produce ; a pattern of concentric circles when a cross sectional slice is made. A plane hologram of an object is thus the superposition of these "zone plates" from each point on the object. In an off-axis system, the cross sectional slices are made at an angle to the axis of the spherical waves and are thus ellipses. Appendix C. Holography 80 When a plane wavefront is directed towards the hologram, each zone plate diffracts and reproduces the original spherical wave from each point, thus creating an image of the original object. By the nature of this process, a real and a virtual image are produced. The transmissivity profiles of the zone plates do not have square edges, with either maximum or minimum opacity; these are known as classical or Fresnel zone plates and would produce an infinite number of images. Instead, the profiles are sinusoidal. They are also called Gabor zone plates, after the man who first conceived of them, and who invented holography. This model is valid when the object is close enough to the plate to make spherical waves and the source of the reference beam is distant enough that plane waves reach the plate. C.4.2 Geometric Model When two spherical waves are allowed to interfere, a system of fringes in the shape of hyperboloids of rotation is produced. If this can be recorded by a light sensitive material with a thickness great enough to accommodate a number of these patterns, a volume hologram is produced. The dark fringes in the recording medium perform sequential Bragg reflection of incoming light, reproducing the original wavefront which reflected from the object. The wavelength of light which is reflected most efficiently in this manner depends on the original wavelength and the angle at which the hologram is subsequently viewed. Therefore, this type of hologram is visible in white light. Both transmission and reflection holograms are made in this way. C.5 Types of holograms C.5.1 In-line (Gabor) holograms This type of hologram is recorded on a thin layer of light sensitive material with an on-axis optical system. It works under the principle of the previously mentioned zone plate model. The object is a flat piece of transparent material with opaque patterns on it, and a single coherent beam serves as both the reference and object illumination beams. With this configuration, one real and one virtual image are produced when the plate is illuminated. See figures C l and C.2. Appendix C. Holography 81 Figure C l : A n example of the setup of apparatus for an in-line hologram, with a monochromatic source such as a mercury lamp with a pinhole at its entrance to imi-tate a point source. This is how the first holograms were made by Gabor. Figure C.2: A n example of the setup of apparatus for an in-line hologram, with a laser as a source and a diverging lens to produce the reference beam. Appendix C. Holography 82 Figure C.3: An example of the setup of apparatus for an off-axis hologram, with a beam splitter to separate the incoming laser beam and lenses to cause the object and reference beams to be divergent. C.5.2 Off-axis (Leith-Upatnieks) holograms If the reference beam is offset from the beam used to illuminate the object, a transparent object is not required. The object can be solid and opaque and is placed close to the photographic plate. This eliminates unwanted interference between the images found in on-axis systems. See figure C.3. C.5.3 Fresnel holograms This refers to holograms made in either of the above ways with the interference between the near-field (Fresnel) diffraction from each point on the object and the reference beam recorded on the plate. Appendix C. Holography 8 3 C.5.4 Frauenhofer holograms Holograms of this type are recordings of the interference between the far-field (Frauen-hofer) diffraction pattern and the reference beam. Since the object is sufficiently small and far away from the plate (d >^ (g2+i/2); where d is the distance to the plate and x and y are the dimensions of the object), its images will not interfere and an on-axis configuration can be used. C.5.5 Side band holograms Side-band Fresnel and Frauenhofer holograms are holograms recorded in the above way with an off-axis system. C.5.6 Fourier transform holograms This type of hologram records the Fourier transform of the object beam. This requires a flat, transparent object to be placed in the same plane as the point source of the reference beam and the recording plate to be parallel to this plane. Sometimes a lens is placed between the two planes; otherwise, the hologram is called a lensless Fourier transform hologram. See figure C.4. C.5.7 Image holograms A hologram can be taken of a real image of an object in the same way as of the object itself. This image can be located with its center at the plate, creating a dramatic effect when reproduced. Lenses and/or mirrors are used to position the image in the desired location. C.6 Holography film To capture the tiny interference patterns associated with holography, the film must have very small grain size. There are many different types of holographic film. Some of these are: Appendix C. Holography 84 Transparent Diverging lenses Figure C.4: A n example of the setup of apparatus for a Fourier transform hologram, either with or without a large converging lens between the object and film planes. A beamsplitter is used to separate the reference and object beams. The Fourier transform of the object is recorded on the film, which is parallel to the object plane. Appendix C. Holography 85 Thin transmission film This is the most readily available type of holographic film. It has an antihalation coating which prevents light reaching the emulsion from the back of the film, reducing unwanted exposure to stray laser light. The film is called thin if it is not thick enough to store a significant amount of information along its thickness, and produces a virtual image as well as a real image. Thin reflection film This type of holography film lacks the antihalation coating found on transmission film and therefore can be illuminated by the object and the refer-ence beam on opposite sides, and can be viewed with the reference beam on the same side of the film as the viewer. It can also be viewed in the same way that a transmission hologram can, and, depending on the developing method used, can be viewed in white light. Thick holographic film This type of film is used to make volume holograms, which have the interference pattern recorded along the thickness, as well as the length and width. Rainbow holograms, visible in white light, are commonly produced with this type of film. The recording medium can be composed of dichromated gelatin, photothermoplastics, or photorefractive crystals. A holography film is described as thick if it is thick enough to record many Bragg reflecting surfaces. C.7 Techniques used to optimize the quality of holograms The sharpness of holographic images can be improved by following these guidelines: 1. The holography film should be illuminated by the same amount by the reference beam and the light reflected from the object. The extra light from either source over the other will merely add to the noise. 2. A n even divergent beam can be produced by first directing the beam without the diverging lens to the center of the target, then aligning the lens so that the beam passes through the middle of it and is not obscured by the edges of the lens holder. 3. Laser light which is not part of the reference beam and which is not reflected from the object should be prevented from reaching the film, as it will cloud the hologram Appendix C. Holography 86 and will reduce the signal to noise ratio. For the same reason, an attempt must be made to eliminate unrelated light, but some types of holographic film are insensitive to green light, for example, so a green safety light can be used when handling undeveloped holograms. 4. The total path lengths of light rays in the object beam, and in the reference beam, from the laser to the film must be as close as possible. Otherwise, the coherence length of the laser will become very important. The coherence length of the laser used in this experiment is sufficient that this condition need not be satisfied pre-cisely. 5. The directions from which the two beams arrive at the film must be as close as possible to take advantage of the fine grain of the film. Otherwise, the interference pattern of which the hologram is composed will tend to fall along the thickness of the film, instead of its length, and the pattern will be lost. 6. The correct exposure is obtained when the film shows a grayish pattern but is not completely opaque. If the film is underexposed, the signal to noise ratio is reduced, and if it is overexposed, the interference pattern is washed out. It is sometimes possible to bleach an overexposed transmission hologram to make the image easier to view, since it transmits more light. This produces a phase hologram, which, instead of containing opaque regions which transmit very little light, contains regions of varying index of refraction, which have the same effect when producing a holographic image. 7. Transmission holograms, as a general rule, should transmit more light than reflec-tion holograms, since the image is made up of the light that is allowed to pass through the film. Bleaching can be used to make the emulsion transparent, while still retaining its ability to produce a holographic image. A hologram prepared in this way is often called a phase hologram, since it modifies the phase of the incoming wavefront, and not its amplitude. 8. To prevent vibration, all components of the apparatus with which the hologram is taken should be bolted or clamped to a vibration-free surface. This is not always Appendix C. Holography 87 possible, such as when a hologram is being taken of a delicate or awkward object which cannot be clamped. It is especially important in this case to set the apparatus up i n such a way as to make the necessary exposure time as short as possible. C.8 Holographic Interferometry Holographic interferometry is used to measure tiny movements and deformations wi th a precision better than half a wavelength of the light used in the process. See references [25, 26] for experiments. In al l cases, fringes are visible as a result of interference between the wavefronts from a holographic image and either another hologram or the original object which has been slightly deformed by a source of stress, such as force, pressure, or temperature difference. If the interference is between two holograms, they have commonly been exposed on the same piece of film with the same reference beam. C.8.1 Measurements of deformation using holographic interferometry A s wi th a l l forms of interferometry, fringes are produced by differences in the path length of light between two beams. In this case, at least one of these beams has been recorded holographically and a reference beam is used to reproduce it. The distance a point on the surface of the object has traveled since the first exposure is given by the number of fringes visible between that point and a point which has not moved, multiplied by a half wavelength of the light used. Therefore, the precision of measurement is determined by the wavelength of the laser light. Deformations of a fraction of a wavelength of light can be measured wi th sufficient analysis of a photograph of the holographic image, as is done in this experiment. C.8.2 Types of Holographic Interferometry Time average holographic interferometry This type of holography is used to measure the vibration modes of an object. Since the position of a vibrat ing object or part of an object is, on the average, close to the extremes for most of the time, the images of the object at these two positions interfere and the resulting image is a map of the parts of the object which vibrated the most and the least, Appendix C. Holography 88 and a measure of the amplitude of vibration. In this process, the object is subjected to vibration, such as from a speaker, and a hologram is taken. This can be done at various frequencies of vibration to determine the effect on the object. Real time holographic interferometry Real time holographic interferometry allows changes in an object over time to be observed and measured. A single exposure hologram is made, developed, and returned to the exact position at which it was exposed. The illuminating beam is kept on the object and it and the reference beam are adjusted in intensity so that both the object and its holographic image look equally illuminated. The shape change is then initiated by changing the temperature, or by adding some other stress to the material, and as the light from the surface of the stressed object interferes with the light produced by the hologram, the fringes can be seen to evolve over time as the shape change takes place. This can be photographed, or a movie can be taken, and these data can be analyzed later. This procedure has been used to measure shape changes over time or with various stressors. One significant drawback of this method is that the holographic plate must be returned to precisely the same position as when it was exposed, or else the fringe pattern will not be visible. In addition, the image may be distorted slightly by the developing process. Double Exposure Holography Double exposure holography solves the problem associated with real time holography by capturing the object in both unstressed and stressed conditions on the same hologram, without moving the film between exposures. The developed film has merely to be exposed to a beam of coherent light to view both images and the interference between the light from the object in two different states, instead of having to be returned to a precise location. Any distortion caused by processing affects both images in the same manner. There are various methods of double exposure holography, the chosen method depending on the purpose for which the hologram is intended. Masked Multiple Exposure Holography This process is based on the same concept as double exposure holography, except interference between any two of many images Appendix C. Holography 89 First exposure exposed region masked region 9 Second exposure P 1&2 2&3 Regions in which combinations of holographic images interfere Third exposure Figure C.5: A n illustration of masked holographic interferometry, where a number of different holographic images can be compared in various combinations on the same piece of film. See section C.8.2. can be viewed by modifying the way in which the exposures are taken. By covering, or masking, strategically chosen parts of the plate for each of many exposures, the holographic patterns for only two different exposures will occur on each part of the film, and viewing the holographic image through a given part will produce interference between the corresponding two states. See figure C.5 for an example of this process. D y n a m i c Interferometric Holography This type of double exposure holography is concerned with measuring the effects of high-speed phenomena, such as the turbu-lence formed in air when a bullet passes through it. One exposure is taken of a flat background; then a second exposure is triggered by a given occurrence, such as an instant after a gun is fired when the bullet will be in the field of view. This obvi-ously requires a powerful laser and/or very high speed holographic film, because a short exposure is needed to capture the action. A pulsed ruby laser, which has a power of 1 0 1 2 W / m 2 for 15 ns, is commonly used for this type of interferometry. Appendix C. Holography 90 Contour Interferometry This type of double exposure holography measures the shape of an object, rather than a change in shape. Two similar wavelengths of light are used for the exposures, and when the hologram is viewed with one or other of the wavelengths, the patterns interfere with each other, and shape contours can be seen. This type of holography is limited in its use by the unavailability of suitable lasers, which include the dye laser and lasers which are tuned to emit two very close wavelengths, both of which are very expensive and complex instruments. C.9 Reproducing a holographic image as a photograph It is possible to capture the real image produced by a hologram on a photographic plate by positioning the hologram in a stationary position, shining a laser beam through the exposed part, and positioning a piece of photographic film in the real image. This is most easily done with Polaroid film on a film holder. The image will of course be a mirror image of the original object so it is best to place the holographic film in backwards (the emulsion side facing away from the laser). The typical exposure time for this method is a few seconds with a 5 mW laser. This will produce a marginal photograph because the laser beam has a width great enough that the images from more than one perspective are captured on the film in the same place. A better, yet more involved, method is to shine a divergent beam through the holo-gram, preferably using a spatial filter, and to photograph the virtual image behind the hologram (on the same side as the laser). If a small aperture is used, a lens, such as one found in a Polaroid camera, can be used to focus the light from the image onto the film inside the camera. This requires much patience, since it is difficult to find the image with the camera. The exposure time for this method is much longer, but the image quality is probably the best that is obtainable. The image is first focussed on the frosted glass at the back of the camera, and then the film holder is inserted. The camera must not be moved as the film is inserted or the image might be lost. The best way to prefocus the image is by focusing the portion of the diverged laser beam which enters the camera to one small spot.Then the image must be found and it will be in approximate focus. This is the method used in this experiment to capture most of the holographic images. The exposure time used for ISO 100 Polaroid film was 80 seconds with an aperture of f/16, Appendix C. Holography 91 but it is likely that holograms produced differently will require more or less exposure time. Exposure testing is required to find the best exposure time. A 35mm SLR camera (or, in fact, any SLR camera) can also be used to capture the image, but, again, it is very difficult to find and focus the image, since only a small proportion of the original laser light enters the camera and a small aperture must be used. 

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