MAGNETIC MOMENT IMAGING By NICKOLAS A. PTSCHELINZEW A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2015
Â© 2015 Nickolas A. Ptschelinzew
To my friends, family, colleagues , and mentors who supported me as well as a special thank you to M ajor Donaldson, Dr. Williams , and Mr. Sade.
4 ACKNOWLEDGMENTS Coming this far in a project of this scope is like life : it The help , support, and guidance of my advisors, Dr. Mark Davidson and Dr. Paul Holloway , has been absolutely invaluable in navigating the waters of my research as as his boundless enthusiasm has made every day an oppor tunity to learn more. Dr. Holloway has contributed more than I could have thought , both in terms of his knowledge and insight on the concepts of this project but also his sage like advice on how to proceed when I saw no way forward and his no nonsense app roach to science and life; I felt like I was being advised by Tony Stewart git r be overstated , and this project would never have begun without his understanding of a tomic interaction and his ability to explain that understanding to others. The work of Dr. Yunmei Chen and her group made the simulations and the image recovery possible , and I am extremely appreciative of the patience they showed the house. I would also like to thank my committee members , select faculty at the department of Materials Scie nce and Engineering, and past advisers who have guided me to this path as well as continually mentoring me over the years : Dr. Rolf Hummel, Dr . Jon Dobson, Dr . John Forder, Dr . Gerald Bourne, Dr . Michele Manuel, Tiza Garland and Dr. Don Swieter . Thank you to Chuck Rowland for all the help with building and machining, lab upkeep and safety, the common sense, and everyday advice of actual implementation
5 of what I learned in a classroom. Juan Fernandez was invaluable for his programing and electronics experien ce as well as keeping me f ro m being alone in the building for the last year of the project. He completed in days what would have taken me weeks. Deepti Pant needs to be thanked for not only dealing with me and my constant questioning but also for her early work and assistance with programing during initial testing and simulation. I would also like to acknowled ge the administrative staff at Microfabritech and MSE : Ludie Harmon, Joni Nattiel, and Flo Orlik are always so positive and helpful that the y made an arduous task pleasant and made sure I was able to stay ahead of the bureaucratic weight graduate school brings to bear on a person. Next, I need to thank my lab mate, , for keeping me up to date in the world, reminding me that I am o ld, and making sure I was still having fun . My dear friend Athena Buell not only let me rant for the last few years at her but spent literally hours helping me with English and grammar on this dissertation as well as many other papers I have written. She a lmost makes me wish English was my first language. I would like to thank my family for supporting, encouraging, and harassing me into pursuing my academic achievements my mother S ally, father Bernard, my adoring wife Lourdes, my beautiful child Helios, a nd all of my extended family without whom I could not have done this or really anything else : Alex , Amanda, Brandon , Broham, Chris, Lainie, and Rob. I should also thank the family of those friends whom I lean on so heavily and take so much of their time, s o thank you Emily, Jen, Grady, Eve, Jasmin e , and AJ.
6 Finally I must thank my Cutie Booty; you know what you did, and I thank you for all the late nights, the patience, and the generally being you while letting me be me.
7 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ............................ 4 LIST OF FIGURES ................................ ................................ ................................ .... 9 LIST OF ABBREVIATIONS ................................ ................................ ..................... 11 ABSTRACT ................................ ................................ ................................ ............. 15 CHAPTER 1 INTRODUCTION ................................ ................................ .............................. 17 2 LITERATURE REVIEW ................................ ................................ .................... 25 Historic ................................ ................................ ................................ .............. 25 Contemporary ................................ ................................ ................................ ... 26 3 HYPOTHESIS ................................ ................................ ................................ ... 28 Zeeman Splitting ................................ ................................ ............................... 28 Optical Atomic Magnetometers ................................ ................................ ......... 33 Anisotropic Detection ................................ ................................ ........................ 42 4 SIMU LATIONS ................................ ................................ ................................ . 46 Computer Modeling ................................ ................................ .......................... 46 Mathematical Model ................................ ................................ .......................... 47 Modeling R esults ................................ ................................ .............................. 51 5 EXPERIMENTAL PROCEDURE ................................ ................................ ...... 61 Introduction ................................ ................................ ................................ ....... 61 Experimental Planning ................................ ................................ ...................... 61 Materi al Selection ................................ ................................ ............................. 62 Magnetic Environment Testing ................................ ................................ ......... 63 Rubidium Cell ................................ ................................ ................................ ... 70 Optics ................................ ................................ ................................ ................ 72 Coils ................................ ................................ ................................ .................. 73 Signal Processing ................................ ................................ ............................. 74 Experimental Configuration ................................ ................................ ............... 76 Free Induction Decay ................................ ................................ ................. 76 Rubidium MX Magnetometer Configurations ................................ .............. 79 Square Wave Configuration ................................ ................................ ....... 79 DC S weep Configuration ................................ ................................ ............ 81
8 AC Sweep Configuration ................................ ................................ ............ 83 6 RESULTS ................................ ................................ ................................ ......... 86 Goals ................................ ................................ ................................ ................ 86 Free Induction Decay Results ................................ ................................ ........... 86 AC Sweep Results ................................ ................................ ............................ 87 DC Sweep Results ................................ ................................ ............................ 87 Square Wave Results ................................ ................................ ....................... 90 Conclusions ................................ ................................ ................................ ...... 94 7 F UTURE WORK ................................ ................................ ............................... 9 5 APPENDIX A: LIST OF VARIABLES ................................ ................................ ....................... 98 B: LINEARIZATION EQUATION ................................ ................................ ......... 101 LIST OF REFERENCES ................................ ................................ ....................... 103 BIOGRAPHICAL SKETCH ................................ ................................ .................... 108
9 LIST OF FIGURES Figure page 1 1 T1 vs T2 scan. ................................ ................................ .............................. 18 1 2 Isotropic detector map. ................................ ................................ ................. 19 1 3 Anisotropic detector map. ................................ ................................ ............. 20 1 4 Simplified MRI T1 concept. ................................ ................................ ........... 22 1 5 Simplified MMI concept. ................................ ................................ ............... 24 3 1 Fine energy splitting in rubidium D 1 transition. ................................ .............. 29 3 2 Hyperfine energy states of the D 1 transition in rubidium 87 . ........................... 31 3 3 Excited hyperfine transition in a magnetic field. ................................ ............ 32 3 4 Ground state hyperfine transition in a magnetic field. ................................ ... 33 3 5 Bell Bloom Magnetometer. ................................ ................................ ........... 37 3 6 MX magnetometer. ................................ ................................ ....................... 38 3 7 Modified MX magnetometer. ................................ ................................ ......... 40 3 8 Diagram of experimental hemispherical array simulator.. ............................. 41 3 9 Orientation of dipole and detector. ................................ ................................ 43 3 10 Vector Orientation.. ................................ ................................ ....................... 45 4 1 The simulated hemisphere of detectors.. ................................ ...................... 51 4 2 Geometric target. ................................ ................................ .......................... 52 4 3 Central slice of simulated target. ................................ ................................ .. 53 4 4 Recovered image. ................................ ................................ ........................ 54 4 5 Direct Least Square Image recovery. ................................ ........................... 54 4 6 Shepp Logan reconstruction with 133 detectors. ................................ .......... 56 4 7 Shepp Logan reconstruction with 333 detectors. ................................ .......... 57 4 8 High resolution 2D target reconstruction with 134 detectors. ....................... 58
10 4 9 High resolution 2D target reconstruction with 333 detectors. ....................... 59 4 10 3D low resolution target reconstruction.. ................................ ...................... 60 5 1 Un shielded magnetic environment. ................................ ............................. 64 5 2 Early magnetic shielding.. ................................ ................................ ............. 65 5 3 Thin walled vs thick walled magnetic shielding. ................................ ........... 68 5 4 Detector apparatus. ................................ ................................ ...................... 69 5 5 Electronics diagram. ................................ ................................ ..................... 75 5 6 Free induction decay diagrams. ................................ ................................ .... 78 5 7 Square wave detection diagrams. ................................ ................................ 81 5 8 DC sweep electronics diagram. ................................ ................................ .... 83 5 9 AC sweep electronics diagram diagrams.. ................................ ................... 85 6 1 Recovered signal from square wave test.. ................................ .................... 93 7 1 Phantom.. ................................ ................................ ................................ ..... 95
11 LIST OF ABBREVIATIONS AC Alternating Current, used to refer to the flow of electricity and the a lternating amplitude of electro magnetic fields. ADMM Alternating Direction Method of M ultipliers , is a class of algorithms used to solve complex optimization problems by breaking them into smaller pieces, each of which are then easier to handle. AWG Arbitrary Wave form Generator, a device used to set a frequency and often drive an experiment by providing a voltage at the desired frequency and amplitude. BB Barzilai/Borwein , a method used to solve steep gradient problems especially for large scale optimization. D.I. Water De Ionized W ater, water that has had any H+ , OH , or other ions removed fro m it prior to use, leaving onl y largely nonconductive, charge neutral water remaining. DA C Digital to A nalog Converter , a device that converts an analog signal into a digital one. It is also used as a hub to split signals up or collect st r eams of data f ro m multiple sources into one piece of software. DC Direct Current, used to refer to the flow of ele ctric ity as well as some steady state electro magnetic fields (fields that do not have a modulating amplitude) . EEG Electroencephalography, is a technique used to monitor the electrical activity in the brain. EKG Electrocardiography, is a technique used to monitor the electrical activity in the heart, also abbreviated as ECG. EM Electro M magnetic radiation, refers to any form of radiation but in the case of this project refers to the magnetic and corresponding field from a few hundred to a few million Hz. ESD Electrostatic Discharge, the sudden flow of electricity between two points. Although cau sed by a wide variety of circumstances and actions, in this paper it is usually used to refer to a very small voltage discharge caused by induction in a line that may damage sensitive electronic components.
12 FDA Food and Drug Administration, the U.S. government office in charge of testing and regulating any and inspecting food items, as well as electromagnetic radiating devices. GPIB General Purpose Interface Bus, IEEE 488 is an 8 bit communication cable used com monly before USB and other forms of connections . GPS Global Positioning System, a series of communication devices based in orbit and terrain that allow for precise measurements of position based on the location of the satellite relative to the area in que stion. Positional accuracy is heavily based on the precision of the time measurement monitored at points in the system. JCPDS Joint Committee on Powder Diffraction Standards, this is the organization that maintains the largest database on known crystallographic scatter patterns to be used in material identification with XRD. Mag Wire Enamel co ated copper wire, usually of a thin gauge and used for , and not hardened enamel, is well bond ed to the surface allowing it to be bent and turne d without risk of conduction fro m wire to wire. MCG Magnetocardiography, is a technique similar to an EKG, used to monitor the electrical activity in the heart. In an MCG a magnetometer monitors the magn etic component of the EM fields with SQUID magnetometers. MEG Magnetoencephalography, a technique similar to an EEG, used to monitor the electrical activity in the brain. In an MEG a magnetometer monitors the magnetic component of the EM fields SQUID magnetometers. MMI Magnetic Moment Imaging or more accurately Nuclear Magnetic Moment Imaging , a method of usin g the atomic magnetic moment of a sample to create a compositional image via computer tomography. MRI Properly , N uclear Magnetic Resonance Imaging or NMRI , a variant of NMR in which the magnetic resonance information of an atom is used to visually differe ntiate it from nearby atoms of a different type or atoms in a different bonded configuration.
13 NIST The National Institute of Standards and Technology, a department Promote U.S. innovation and industrial competitiveness by advancing measurement science, standards, and technology in ways that NMR Nuclear Magnetic Resonance, a technique by which the magnetic resonance of an atom gives d etailed information about the identity of that atom as well as the environment around it with respect to how it is bonded and to what. PTFE Polytetrafluoroe thylen e , most commonly known by as Teflon, a trademarked formula based on PTFE. This polymer has a high heat tolerance and is extremely chemical resistant. In addition , it is ridged and easily workable into many shapes. RF Radio Frequency, the frequency band from 3 KHz to 300 GHz, named so because most commercial communications fall in this bandwidth. In this paper the term may be used loosely as some experiments may be performed technically below this bandwidth. SBI Split Bregman I teration , is a technique for solving a variety of L1 regularized optimization problems, and is particularly effective for problems involving total variation regularization. This method is primarily used when the number of parameters exceeds the amount of data and regression breakdowns. SEMPA Scanning Electron Microscopy with Polarization Analysis, a spectroscopy technique employing the electron bean beam to pick up magnetic information from the target by examining the spin of secondary el ectrons emitted by the target. SQUID Superconducting Quantum Interference Device, a high sensitivity magnetometer. T 1 Spin lattice relaxation time constant, this refers to a type of MRI scan in which the relaxation of the induced magnetic moment is , the lattice in which it is in. To create a T1 weighted image magnetization is allowed to recover before measuring the MR signal by changing the repetition time. T 2 Spin Spin relaxation time constant, us ually refers to an MRI scan in which the more rapid spin spin relaxation is looked for. This is the exponential decay towards nuclear magnetic equilibrium within the atom .
14 TEC Thermal Electric Cooler, usually employing the Peltier effect, to maintain a te mperature for a laser diode and therefor e a laser wave length. TEM Transmission Electron Microscopy, a spectroscopy technique that utilized the interaction of stream of electrons as it passes through a thin sample to gather information. TV Total Variation, a mathematic concept used to help relate changes in a function to the codomain. In Imaging specifically it is used to help isolate an image from noise during the digital processing of the signal by means of a function with x >1 real variable. U SB Universal Serial Bus, the now ubiquitous , data connection protocol cable and mass storage device s . In the context of this paper it refers to the cable and usually has a data transfer rate of 480 M M bit/ s or slower. XRD X Ray Diffraction, a method of identifying crystalline materials by the scattering of X rays; however, results usually need to be compared to a known standard.
15 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MAGNETIC MOMENT IMAGING By Nickolas A. Ptschelinzew December 2015 Chair: Paul Holloway Major: Materials Science and Engineering Magnetic moment imaging is a method of generating tomographic images using the induced magnetic moment response to a magnetic field in t he area of the target. Although small and decaying rapidly, the induced magnetic moment is dete ctable if the sensor is sensitive and fast enough. Optical atomic magnetic detectors represent such a possibility. By developing very small, very sensitive detectors it is possible to form an array that would be able to give a composition image of a target magnetic responses. These responses can be detected in many ways, often generating images based on analyzation of the magnetic field strength or resonance; however , this detection system will use decay time to reconstruct contrast maps . U nlike tradition al magnetic imaging, this can be done without a complex gradient f ield applied over the target during testing if the magnetometer in question is anisotropic . In this way it was thought a portable imaging d evice providing MRI like images could be fabricated. By anisotropic it is meant that the magnetometer distinguish es different magnetic responses not only by magnitude and distance, but also by spatial positioning regardless of actual displacement f ro m the detector. Simulations completed early in this
16 project have shown the number of detectors needed for a reasonable resolution for medical purposes , and other groups have been working towards miniaturization as well as improvements in efficiency. As such , th is project centered on using our simulated data to test the hypothesis governing the detection mapping in order to further refine the simulations and move toward fabricating an imaging device.
17 CHAPTER 1 INTRODUCTION Magnetic Moment Imaging (MMI) is a novel way to use the magnetic properties inherent in all objects to gain a detailed internal view without the use of ionizing radiation of other harmful methods. In addition to being non destructive this technique may off er an alternate to Transmission Electron Microscopy ( TEM ) , Scanning Electron Microscopy with Polarization Analysis ( SEMPA ) , and Lorentz microscopy [ 1] . All materials have a magnetic response, be it diamagnetic, paramagnetic, ferromagnetic, or another atom ic or electron based response and thus all will respond to a magnetic field. However all of these responses have differing time constants and it is the atomic induced magnetic moment that is the focus of the project . Not only are these times long enough to measure , being on the order of 100s to 1000s of milliseconds [ 2 ], but they are highly charac teristic of specific materials as this relaxation time is deter mined by the gyromagnetic ratio of the individual atoms and the mobility of the lattice [ 3 ] . By exposing a target to a magnetic field of even a few mT, the response can be detected as a change in the magnetic moment of the target. The rate at which this induced magnetic moment decays can be monitored by the equilibrium change in the MMI detector . Additionally , if the system is fast enough, this decay can be viewed as a direct current ( DC ) magnetic field on a small enough time scale and thus can be identified by its Larmor frequency, although this is not the primary focus of this project. With this information it is possible to differentiate between areas of different atomic composition , by differences in their decay time . To form an image, a way to determine spatial resolution is also necessary. In Magnetic Resonance Imaging (MRI) this is most commo nly done by applying a gradient to the magnetic fields applied over the target [ 4 ] .
18 By using a n anisotropic detector , no external variation is needed as the positon of the detector it s elf give s spatial information on the target being examined. It is for t his reason that atomic optical magnetometers have been utilized in this experiment. The detection of these devices is suspected to be anisotropic , unlike a typical Hall Effect detector , SQUIDS, or even a simple coil of wire , but instead follows a mathematical equation developed during this project. Using this information and existing computer tomography technologies as well as algorithms newly developed for this project , it should be possible to recreate a visual representation of the internal stru cture of even a complex system such as a fruit undergoing FDA inspection or a human brain being examined for signs of trauma. The results are be comparable to an MRI T 1 (spin lattice relaxation) image. Figure 1 1 . T1 vs T2 scan . T his image shows the diff erence in the contrast between a spin lattice (T1) image and a spin spin (T2) image. The difference has to do with the relaxation mechanism of the induced magnetic response as well as the timing of the scan. Photo credit http://www.nvvn.org/patienteninfo/img mri scan.php
19 This anisotrop y had not been previously explored but is key to the operation of this imager. Gradient fields are commonly used in MRI to determine positional informati on as the receiving coils are an isotropic detector , I.e. all magnetic field lines of equal magnitude and emanating from an equal distance would be read as the same regardless of spatial positioning rel ative to the detector. However , if the detector used was anisotropic , these gradient fields, and the cause for much of the characteristic MRI noise, would be eliminated. Figure 1 2. I sotropic detector map . A n isotropic detector (a) interprets the same signal from the magnetic charges of (b) and (d) as they ar e of the same strength and distance from (a). Magnetic charge (c) is weaker but closer so it is also detected as the same intensity as (b) and (d).
20 Figure 1 3. Anisotropic detector map. The anisotropic detector (a) detects different signals from the ma gnetic charges of (b) and (d) although they are of the same magnitude and distance from (a), they are not the same position in X Y Z space. Magnetic charge (c) is weaker and closer but also at a different position so although it has the same magnitude as ( a) it is detected as a different signal intensity. The suspected detection mapping for (a), represented as red dashed lines (e), influences how magnetic signals are interpreted. At the same time , the increased sensitivity of atomic magnetometers over con ventional magnetic detectors [ 5 ] is considerable, with this style of magnetometer detecting fields as small as 10 f T/Hz [ 6] . This leads to an overall reduction in the strength of the magnetic coils needed to produce some kind of internal imaging. This reduction in strength of the magnetic coils would also lead to a decreasing need for magnetically shielded rooms and safety concerns that are produced by large magnetic fields in the medical environment [ 4 ]. Some shielding would , of course , still be n eeded
21 but it would be to prevent stray fields f ro m impacting the imaging device or washing out the desired signal, not to prevent injury or damage to the apparatus being used. coils cool enough to operate in the superconducting region and removing them allows furthe r reductions of support structure . In this way not only has the system been simplified by removing the need for large magnetic gradient fields to be constantly gener ated, the system gains a considerable reduction in cost and weight, thus allowing portability. Lastly , advances in size reduction of optical atomic magnetometers mean s that these type s of detectors can now be fabricated on the chip scale with volumes on t he order of 25 mm 3  . Along with the small size the power requirements are also considerably reduced, requiring less than 200 mW to power the entire apparatus, heating, laser, alternating current ( AC ) and DC magnetic fields. This does not include the com puter used to collect and analyze the data but reductions in computer size and power consumption compared to computation power are outside the scope of this project. It is this combination of high sensitivity, small size, and rapid processing that will allow the decay time of even extremely small fields to be detected and used to create a contrast map. To be clear, although the aim is to produce images that would provide similar data as an MRI T1 scan would , it is different principles that are being exploited to do so. To clarify these differences an example of how both sy stems would gain information fro m a target follows.
22 In an MRI the target of examination is exp osed to a DC field which aligns the magnetic moments of the target. To produce the T1 weighted image a perpendicular field is used to knock the atomic moments 90 Â° to the longitudinal DC field, this perpendicular field is an AC field that matches the resona nce of the atomic precession. As the atomic moments relax back into the longitudin al DC filed they will release a Radio Frequency (RF) pulse equal to any energy they absorbed once they are in the lowest energy configuration and by reading this emitted sign al the MRI determines the T1 time (note this is something of a simpl ification but generally covers the principle ) . Figure 1 4. Simplified MRI T1 concept. A) The target , which typically has zero net magnetic moment. It is exposed to a DC gradient field th at aligned the net magnetic moment, although individual atoms will have a magnetic moment that precession about the axis of the DC field. An RF pulse that is matched to this atomic precession frequency is directed perpendicular to the DC field taking the n et magnetic moment of the affected atoms 90 Â° out of alignment with the DC field and as a result into a higher energy state. B) The net magnetic moment before the RF pulse and C) the net magnetic moment during. The time it takes for the atoms to relax back into alignment with the DC field is the T1 relaxation time. When the RF field is active there is a net field perpendicular to the DC field but this net field decays quite rapidly as the individual spins rapidly decouple far faster than the atoms can relax back into the DC field and this is the T2 relaxation time. These fields are measures by receiver coils around the target.
23 Compare this to an MMI where the target is not exposed to a consider magnetic field but instead only a short pulse. The MMI detector has two magnetic fields in it, the first is a DC filed aligning the magnetic moment of the internal gas atoms; the second is an AC field that runs perpendicular to the DC field and is matched to the atomic precession of the gas atoms. The precession rate i s determined by the DC field and when the system operates in this way it is in an equilibrium state as read photos absorption with in the cell. When the target of examination is given a magnetic pulse by a coil with a sharp drop, the target will develop an induced magnetic moment. This magnetic moment will change the effective DC field of the detector taking it out of equilibrium and thus disturbing the photo absorption rate. As this induced magnetic moment decays back to its pre pulse state so too will the detector return to its equilibrium state. The length of this perturbation can be used to measure the decay rate of the target, which is a material dependent property and used as a contrast mechanism for imaging.
24 Figure 1 5. Simplified MMI concept. A) T he target which typically has zero net magnetic moment. It is exposed to a DC Pulse that aligned the net magnetic moment of the target, although individual atoms will have a magnetic moment that precession about the axis of the DC pulse. B) The net magnet ic moment before the DC pulse and C) the net magnetic moment during. As the pulse has forced the atoms into a higher energy state they will decay back to the lower energy , randomized, and thus zero net magnetic moment , configuration as fast as diffusion wi ll allow. However between the time of the DC pulse ending and the zero magnetic moment equilibrium state being reached the decaying field produced by the target will disrupt the MMI detector and the timing of this disruption will be used to measures the de cay time that can be used for image contrast.
25 CHAPTER 2 LITERATURE REVIEW Historic Since the late 1950 s, with the development of optical magnetometers [ 8 ], the improvement and variation in optical magnetometer s have been in the background of the field. Used primar il y as an offshoot o f atomic clock miniaturization, chip scale magnetometers have been developed and deployed as components for positioning and communication devices [ 9 ] . At the same time , magnetic imag ing has advanced f ro m early NMRI images to modern open medic al MRI, high and low field MRI , and magnetic echocardio grams [ 10 ] . Despite these advances , Ha ll E ffect and SQUID magnetometers dominate the magnetometer usage , and ionizing radiations are still t he domina nt form of imaging internal structures [ 4 ] [ 11 ]. The primary reason for this is the expense and size of NMRI type imaging versus the more damaging, but highly acceptable, ionizing radia tion techniques such a s X ray or CAT , with most MRIs costing 1.4 million US dollars an average , a life time maintenance cost equally to the approximate purchase cost. Spec ialty models such as high field MRI may cost in excess of 10 million US dollars [ 12 ] . With no mobility , high cost, and complex shielding requireme nts , it is not surprising to see why X rays are the primary imaging technique despite the risks of ion izing radiation . T he high sensitivity of the atomic optical magnetometers is highly desirable for some applications but for many other s unnecessary . Add to this their poor commercial availability and the cost of replacing established technology with new technology and it becomes clear why optical magneto scopy is still in its infancy after more than fifty years.
26 At the same time , the magnetic imaging system, MRI, has changed in terms of sensitivity, size, and capabilities over the last few decades , but they are still room sized imaging apparatus. First developed in the early 1970s [ 13 ] upon the princip le s of NMR, the NMRI has beco me one of the most useful medical imaging devices [ 4 ]. A lthough secondary to X ray in cost and quantity of use, it is vastly superior both in patient safety [ 14 ] [ 15 ], utility and diversity [ 16 ]. Only now are optical magnetometers beginning to be looked at as a way of developing portable MRI or MRI like devices [ 17 ] [ 18 ]. Contemporary Atomic magnetometers were first developed by William Bell and Arnold Bloom [ 8 ] [ 19 ] based on some early work of K a stler [ 20 ] but did not really make any advancement until an effort was made to miniaturize atomic clocks; the similarities both in pri ncipl e and design have led to a jump in the development of chip scale atomic magnetometers. Both the optical atomic clock and the optical atomic magnetometer have at their core a laser tuned to the frequency of a transition state of the atom with which the laser is interacting . From this interaction , frequency information can be used , in an a tomic clock , to provided high ly accurate time keeping, on the order of 1 secon d for billion s of years [ 21 ]. In an atomic magnetometer this frequency information gives details about the magnetic environment around the atom [ 22 ]. With this shared development, there has been a renewed interest in chip scale optical atomic clocks and th eir applications. NIST , among other , has been instrumental in pushing the boundaries of both the size [ 6] [ 7 ] of this type of magnetometer and also the limits of sensitivity [ 23 ] [ 5 ]. Other groups have focused on application such as EKG (MCG) and ( MEG) [ 24 ] or on using 25 ]. Others have attempted to apply atomic
27 magnetometers as ways of examining polymer products that may be too delicate for use with other methods [ 26 ]. With the current developments in atomic magnetomete rs the use of them in magnetic imaging is inevitable but upon closer examination it may be possible that they will not be used to improve MRI type imaging but to be utilized for an entirely new form of imaging.
28 CHAPTER 3 HYPOTHESIS Zeeman Splitting Before discussing the operation of an optical atomic magnetometer , it is necessary to understand Zeeman splitting. Electrons bound to an atom are described in terms of their quantum numbers : the principal quantum number , n , describing the shell ; the angular quan tum number , l , (sometimes called Azimuthal) describing subshell as well as the orbital angular momentum ; the magnetic quantum number , m , which details the specific orbital and the projection of angular momentum ; and finally , the s pin quantum number , s , which describes the intrinsic angular momentum for that electron along an axis within that orbital. There is a set of spectral lines associated with each atom based on the energy levels described by these quantum numbers. Under most circumstances, the m and s do not play a major role in these spectral lines , but when the atoms are exposed to a magnetic field, the changes in m do have an impact and actually split this spectral line. This may be easier to conceptualize in terms of energy levels rather than spectral lines.
29 Figure 3 1 . Fine energy splitting in rubidium D 1 transition . This is one part of the D doublet, not to scale . Note that the values given for the un spilt level assume zero net magnetic field strength F ntum , see Equation 3 3 . Energy level data f ro m [ 27 ] [ 28 ] Rubidium has a single val e nce electron and , when ignoring relativistic effects and spin coupling, the two energy state model is an accurate depiction of the ground and excited s t ates for the electron. However, when electron spin and relativistic corrections are taken into account, the degeneracy of the energy levels is broken into the fine energy structure seen in Figure 3 1 . The transition is one of the components of a f ine structure doublet D 1 transition , and each of these energy states additionally has a hyperfine structure. The fine structure is a result of the coupling between the orbital angular momentum , l , of the outer electron and its spin angular momentum , s . T he total electron angular momentum vector is then given by Equation 3 1 [ 29 ] :
30 Where is the vector of the orbital angular m omentum and s is the vector of spin angular momentum. Both of these are based on the quantum numbers l and s , respectively. This new quantum number, , has a range found by Equation 3 2 : Thus , in the D 1 transition is only Â½. This is sometimes noted as [ 30 ] with the superscript on the orbital shell number indicating 2 s +1 and the subscript on the a ngular quantum shell number is . This notation is particularly helpful when discussing hyperfine transitions . Hyperfine transitions take place within a magnetic field due to the interaction between and , the total nuclear angular momentum. , the total atomic angular momentum, seen earlier in Figure 3 1 when discussing fine splitting of energy levels , is defined as : A n example of these hyperfine levels can be seen in Figure 3 2 below.
31 Figure 3 2 . H yperfine energy states of the D 1 transition in rubidium 87 . The F1, m1 ground state is known as a dark ground state as electrons falling into this state have no lower energy to move into and no higher energy state that can be excited by a photon of 794. 76 nm. alkali metal vapor as discussed in the next subchapter. Each of these hyperfine states has magnetic levels can be near degenerate in a weak magnetic field but quite divergent as the magnetic field strength the atom is subjected to increases as seen in Figure 3 3 and Figure 3 4 . For the purpose of these experiments, the magnetic fields were kept in the weak field / anomalous Zeeman Effec t region and not the higher fields Paschen Back Effect region [ 3 1 ] . This is not a hard number but based on the effects produced by various fields. The generally or anomalous Zeeman Effect has been discussed here and occurs at relatively weak field s , usually less than 0.25 T for rubidium 87 . In this regime total
32 angular momentum, from Equation 3 1, is conserved. In higher fields the coupling between and breaks down and is beyond the parameters of this project. With this understanding of fin e and hyperfine energy levels a discussion on the operation of optical atomic magnetometers and atomic clocks f ro m which the technology developed can proceed. Figure 3 3. Excited hyperfine transition in a magnetic field. This is the energy levels of the excited D 1 transition, , in rubidium 87 with increasing magnetic field. The weak field, or Zeeman Effect region is on the left hand side of this graph and the stronger field, Pashen Back effect region is on the right hand side. Image and data from [ 27 ] [ 28 ]
33 Figure 3 4 . Ground state hyperfine transition in a magnetic field. This is the energy levels of the grunt state D 1 transition, , in rubidium 87 with increasing magnetic field. Note the change in scale of the X axis (magnetic field strength). The weak field, or Zeeman Effect region is on the left hand side of this graph and the stronger field, Pashen Back effect region is on the right hand side. Image and data from [ 27 ] [ 28 ] Optical Atomic Magnetometers The deve lopment of optical magnetometers followed closely the development of atomic clocks as previously discussed. In this section an examination of the mechanism of the atomic clock will be made. Following the atomic clock the optical atomic magnetometer will be discussed, with the more traditional methods of and moving into the novel appr oach of this project. The basic princip le of atomic clocks is that the frequency of an energy transition, usually in cesium or rubidium, at a given temperature are used as a standard for the passage of time. This energy transition, or hyperfine transition depending on the clock, is the feedback t c clock [ 3 2 ].
34 Obviously , as advances were made in the ability to discern and monitor energy transitions in alkali metals, this same technology could be applied to noting changes in this freque ncy due to changes in the magnetic fields to which the atoms are exposed. As improvements in time keeping improves fields f ro m navigation via GPS to computer processing [ 32 ], to fundamental sciences whose measurements often depend on the precision of time keeping , the design of atomic clocks has been an ongoing endeavor . Th e expansive application s of this technology have driven further development in atomic clocks that ha s benefited the development of atomic magnetometers. The most common methods of atomi c magnetometers are the Bell Bloom [ 8 ] and MX [ 33 ] methods. They share many similarities and operate on the same princip les but both will be briefly outline d for comparison purposes , before examining the modified MX method employed in this project . In the case of the Bell Bloom configuration, a pump laser is oriented along the axis of the steady state magnetic field. Then a probe laser is oriented orthogonally with respect to the pump laser. T his system requires two magnetic fields, one in a DC steady state and one in an AC or RF that is set perpendicular to the DC field. The DC fi el d i s the stronger of the two and is used to align the magnetic moments of the gas atoms in the cell. Although the DC >> AC with respect to field strength it must still fall with in the anomalous Zeeman Effect region so below 0.25 T for the DC field and roughly two order s of magnitude lower for the AC field [ 34 ]. In this project the typical DC field was on the order of 7 mT, based on the amperage limitation of the wire in the solenoid ; this can be seen in Equation 5 3. The weaker RF fi el d is used to align the precession of the atomic magnetic moments. The frequency of which is set by the strength of the DC field by :
35 Where is known as the Larmor frequency which is based on the gyro magnetic ratio , and the magnetic field to which the atom is exposed. This Larmor frequency describe s the precession of the magnetic dipole about the direction of the large DC magnetic field which it is exposed to. Units for are Hz over t esla and B is in t esla. The gyromagnetic ratio is a combination of the mass of the precessing system , , the g factor, denoted . As proportionality ratio relating the intrinsic magnetic moment to the angular moment of the nucleus in question , is a unitless value. For rubidium 87 a value of = 7 Hz/nT is used [ 35 ] [ 36 ]. This gives a Larmor frequency range of about 49 MHz to 35 kHz for this project. This does not take into account changes in the Larmor frequency due to an induced magnetic moment occurring outside of the detector as the focus will be on timing the disruption , not tracking the decaying field strength. With these four basic systems in place , the Bell Bloom magnetometer works as follows. The atoms are exposed to the DC magnetic field and the pump laser, which has been circularly polarized. This induces a magnetic moment in the gas atoms as well as align s them all along the direct ion of the fi eld and laser. However , at this point
36 precession of the individual atomic dipoles are out of phase with the other atomic precession s of the system state that is, one of the hyperfine ground state s that are not excited by the photons of the wavelength used in the pump beam. Referring back to Figure 3 4 , it can be seen that the same energy gap that exists in the absence of a magnetic field ( or a very small magnetic field) will not be the same for t he hyper fine states. Due to this fact , the pump laser is able to excite electrons from some ground states but not from others , lea d ing to the eventual situation of all val e nce electrons being in the dark ground states. Thus the perpendicular photons , which are at the same wavelength as the pump beam, will not be absorbed . However , if the AC magnetic field running perpendicular to the DC field is out of phase with the precession the system will absorb some energy and this is enough to allow some of the elect rons to move to a non dark ground state and once again become excitable. In this way the intensity and frequency of the probe beam intensity at the photo diode is a result of the frequency difference between the Larmor frequency and that of the RF magnetic field. The data f ro m the probe photo diode can be used as part of a feedback loop to the RF coil. Once the RF field is in resonance with the Larmor frequency Equation 3 4 can be solved to find the magnitude of the DC magnetic field to which the alkali met als are being exposed . See Figure 3 5 .
37 Figure 3 5 . Bell Bloom Magnetometer. (a) The pump laser of circularly polarized photons goes through the cell (b) aligning the magnetic moments in the direction of the DC magnetic field (c), which is the field to be measured. Once the p ump laser absorption has become stable, the probe laser (d) is used in the direction of t he RF field (e) which is perpendicular to the axis of the pump laser and the DC magnetic field This probe laser photo diode (f) picks up the modulation difference between the Larmor frequency and the RF field , which in turn is used as feedback (g) for the RF field generator (h). Once the RF field is at the Larmor frequency the strength of DC magnetic field can be determined. The MX method is similar ; however , the DC magnetic field and pump laser are oriented at a 45 Â° angle to each other and the RF field at a right angle to the D C field. In this configuration the projection of the precession polarization onto the photons leads to a periodic modulation of the optical absorption which can be read and used to make adjustments to the RF coil until the Larmor fr equency can be extrapolated. The 45 Â° angle is sometimes debated as the angle give s the best optimization [ 33 ].
38 Figure 3 6 . MX magnetometer. Unlike the Bell Bloom type, only one laser is used in the MX method. This combined pump/probe laser (a) runs through the alkali gas cell (b) at an angle to the direction of the DC magnetic field (c) which , again, is the field to be measured. Usually this is a 45 Â° angle and often in a plane perpendicular to the R F field (d). When the RF field if off the Larmor frequency the energy it adds to the system can move electrons to the non dark ground state allowing them to absorb energy from the pump/probe laser a nd this can be read at the photo diode (e). This information is used to adjust (f) the AC field generator until the system is in resonance and the strength of the DC magnetic field has been determined. There are numerus variations of both of these method s developed to optimize different aspects of the magnetometer. The Davidson group developed a variation of the MX method developed to be simple to fabricate and miniaturize as well as optimized for image reconstruction by producing a n in line detector str ucture while maintaining the anisotrop y of the de tection mapping. The specific configuration used will be reviewed in
39 detail in the experimental procedure chapter , but a summary of the method will be given here. A combined pump / probe laser tuned to the D 1 transition energy of 1.560 eV, which is a wavelength of 794.6 nm, aligned down the axis of the cell contain ing the rubidium gas as well as being the axis of the steady state magnetic field is the core configuration. A coil is set up to create an AC field perpendicular to the DC field , the magnitude of which will vary slightly for different experimental configurations but the DC field is typically 6 mT and the AC field usually in the 10s to 100s of Âµ T . Unlike most optical magnetometers , this project proposed to measure decaying fields and to use that decay as a way to determine the intensity of the magnetic signal. This requires the target of the measurement to not have a magne tic signal under normal circumstances, but have a decaying signal after being exposed to a magnetic pulse. It is this decaying induced magnetic response that will be the magnetic signal in t he early experiment s to test the validity of the system, often usi ng a controlled or steady state phantom to mimic the magnetic response of a target , and in later experiments this decaying field will disrupt the system , and the time length of that disruption will be used to provide a contrast signal . B efore the target emits a signal, the detector will have both DC and AC fields on as well as the laser. If the AC field is in reson ance with the Larmor frequency , then the majority of electrons in rubidium val e nce shell will be in a dark state and the photons will not be absorbed. On c e this state is reached the targ et can be pulsed and will there for e emit a magnetic signal that will change the Larmor frequency and take the system out of resonance. Once this happens , the energy of the AC field will be enough
40 to allow some electrons to move to the pumpable non dark ground state and be absorbed by the laser. This in turn will be read as a change in photon absorption , and the time it takes for the system to return to equilibrium can be used to determine decay time as well as g iving a qualitative magnetic signal intensity. Figure 3 7 . Modified MX magnetometer. In the magnetometer developed by the Davidson group a single pump/probe laser (a) is orientated along the axis of a rubidium cell (b) and the D C magnetic field (c). An AC magnetic field (d) orientated perpendicular to this , if not in resonance, causes the variation in the laser that is de tected by the photo diode (e). A t this point there are two ways to operate : 1) a target (f) is pulsed and the i nduced magnetic signal (g) disrupts the system by effectively changing the DC magnetic environment of the rubidium cell but as this signal is decaying and the DC magnetic field was known the decay time can be used to identify or at least differentiate this target ; or , 2) as a more traditional MX detector where the photo diode can provide a feedback loop (h) for the AC coil generator (i) until it is again in resonance and thus the total DC magnetic environment of the rubidium cell determined. This does , how ever , require a very quick feedback and response loop from not only the AC magnetic coils but also the electronics package analyzing and controlling the feedback loop.
41 Furthermore , this configuration allows for easy replication of a final device without t he need for multiple detectors. As discussed in the computer modeling chapters, an improvement in image reconstruction is achieved with an increasing number of detectors. As the original concepts, and the computer modeling, called for a hemispherical arra y of detectors, an in line approach to bui lding this detection made it easier to fabricate it in such a way as to mimic a hemispherical array. By placing the target to be examined on a rotating stage and placing the detector of an arm that moved through a n arc over the center point of the rotating base plate , all positions of a hemisphere could be simulated experimentally. The spacing and maximum number of detectors used for computer modeling were established using the plans for this apparatus. Figure 3 8 . Diagram of experimental hemispherical array simulator. The detector can be moved through an arc (x) about the object that is on a rotating stage giving bot h the t a and phi motion and there for e simulating the effect of a full array of detectors in a he misphere about the object. The large Helmholtz coils are to provide a magnetic pulse in order to induce a magnetic moment i n the object. These coils will be able to produce a fast pulse that has a magnetic field which would collapse on a much shorter timef rame than the induced magnetic moment decay of an organic target.
42 A nisotropic Detection Dr . the following equations that may dictate the anisotropic nature of these kind s of at omic detectors, based on the orbitals of the atoms in question. In order to develop an equation of detection , a magnetic dipole was considered first . This is a magnetic source with negligible volume, analogous to an electric mono pole (point charge) . The magnetic field, B, generated by such a dipole is [ 37 ] : Where is the strength of the dipole, is the permeability of fr ee space ; is a unit vector giving the orientation of the dipole (assumed to be aligned with external field ); is the distance from th e dipole to the detector ; and is a unit vector pointing from the sou rce to the detector . At the detector, the magnetic field lines from the dipole will point in a specific direction due to the curve of the field lines; this will be denoted by , which is the unit vector of the magnetic field line as it enters the detector and can be seen in t he equation and graphic below.
43 Figure 3 9 . Orientation of dipole and detector. Diagram of the orientation between the magnetic dipole and the atomic optical detector along with a view of one of the field lines . Note that although in reality there are many field lines emanating from the dipole, many of which would intercept the detector, the assumption that the detector i s very small means that for a first approximation a single field line can be used. The magnetic moment of the dipole should not be aligned with the axis of the laser as that would lead to the detector being struck by two equal and opposite magnetic field lines , resulting in a net magnetic strength of zero being detected. The signal generated from the detectors will be a function of the strength of the magnetic dipole, the distance from the dipole, and the angle of the laser, giving the spatial component. This is seen in E quation 3 8 , where i s a unit vector along the .
44 Expanding this to the signal received by a number of detectors around a volume of many dipoles , a total signal from each detector can be explained as : If there are K sources distributed across a volume, for the i th of which has an amplitude (magnetic strength of the dipole at its origin ) of and an orientation unit vector of ( i , K ) , there would also need to be N detectors, for the j th of which will have an internal laser orientation unit vector of ( j N ) , and in which is the unit vector from the i th source to the j th detector. The signal from the j th detector can be seen in E quation 3 9 . In E quation 3 7 and 3 8 these changes can be made, for example is the distance from the i th source to the j th detector. With those additions , Equation 3 7 now looks like this : and Equation 3 8 becomes
45 Figure 3 10 . Vector O rientation . This diagram depicts the vectors of the detector and the magnetic diploe. In the next chapter this information has been modeled and presented graphically in the third section, Modeling Results.
46 CHAPTER 4 SIMULATIONS Computer Modeling varying magnetic strength due to the anisotrop y of the detector interpretation of fields was created by the Davidson and Chen groups . This simulation was done not only to check viability but to help determine the kinds of experiments needed to prove the hypothesis by dictating the quantity, arrangement, and distances at which an image could be formed. The simulated target was also varied to help determine initial testin g targets. Although the end goal was to use the timing of the decaying response time to reconstruct images it was important to check the viability of the system by making sure a magnetic signal external to the apparatus can be detected. The base equations , Equation 3 6, Equation 3 10, and Equation 3 11, which dictate the intensity of detecti on signal versus spatial positions, were assumed to be true for all of the following simulations. Initial simulations all used a geometric shape that could be easily cr eated in the lab to test against simulated data. All magnetic sources were given a value of or . A two dimensional target was first simulated , but a three dimensional target and array of detectors was found to be more suitable and was quickly adopted. The i nitial target was a simple geometric shape that was both easy to describe digitally and suitable for 3D printing. The target design was copied over to a printable format and was to be printing out with a high porosity such that it could be so aked in a gar gel, which has similar magnetic propert ies to water with regard to induced magnetic moment response [ 38 ] . Around this the simulation assumed a hemispherical array of detectors.
47 The number, position, and distance from the target to the detectors was varied for the actual simulation based on the expected physical limits of the system, with an approximate detector volume of 10 mm 3 [ 39 ] and a distance to the target of 2 .5 cm . These simulations used algorithms that allowed for greater resol ution th an the number of detectors normally allow , as detailed below . This is achieved through specific projections of lines and an algorithm that identifies edges to prevent blurring or removal of areas of distinction between magnetic regions. Below can be seen the mathematic and imaging techniques used as well as the results of the modeling demonstrating a higher than expected number of detectors but clearly recognizing images. Mathematical Model The starting point of the mathmat ical modeling was the ma gnetic dipole, . The area of int e rest was defined as i is a sub set of . R ecall that each i represents a magnetic source in a specific location ( e.g., x i , y i , z i ). In this model was the only unknown, as the location and orientation of the detector and target would be known, and as all magnetic moments in the target are aligned by an external field th a t too also be known. With the knowl edge of E quation 3 1 0 could be solved for , which leaves as the only unknown variable in E quation 3 11 . It has already been established that the signal from the i th detector was given in E quation 3 9 . Integrating this over the area of interest gives :
48 Which can be written as : Where is : In the case of these experiments would be known due to the fact that the position of the laser and the driving magnetic field are a controlled parame te r, and thus and are known, leaving only to be solved for. The goal of reconstruct ing the image (magnetic strength) , , for using the measurements from the detectors requires solving the following optimization problem [ 40 ]: Note that the first term in E quation 4 4 is a special intergral for TV (Total Variation) and does not require a or a as the case may be. Where is a parameter that must be greater than zero and is the gradient of , the TV regularization smooths only along its iso intensity contours, and never across them. Hence, the smoothing does not smear out the boundaries between
49 h etero geneous regions (e.g. boundaries of two different types of contrast from varying magnetic response or decay time ), and in this way maintains hard boundaries . The second term in Equation 4 4 is a data fidelity term that forces the reconstructed image of to satisfy Equation 4 2 . Th is model will allow to be reconstruct ed even if there are fewer detectors ( j ) than the number of pixels in the image domain. The fidelity term alone i n Equation 4 4 could be an under determined system , but with r egularization term the freedom of the pixel intensities , , are much reduced. As a result of this TV regularization, within a target cannot have much oscillation. This has led to concern about doing initial testing on organic targets and thus an electronic phantom with a steady, non oscillating , magnetic signal was to be used in cases where this became problematic. To solve Equation 4 4 efficiently , the idea developed in [ 41 ] by Dr . Chen was adopt ed . A variable splitting algorithm w as used to decouple the original problem into more easily solved sub problems. A combination of alternating direction method of multipliers (ADMM) [ 42 ] , or Split Bregman I teration (SBI) [ 43 ] , with a Barzilai/Borwein (BB) approximation to the second order square matrix [ 44 ] was used to solve Equation 4 4 . More precisely, if the image domain , earlier referred to as the area of interest , , consist of D pixels in the image domain, , the k th of which is numbered . Additionally , a 3D positioning system will be assigned as detectors N located at , magnetic dipoles K located at , and D pixels located at . In this regime , subscript s 1 , 2 , and 3 will refer to the X , Y, and Z in the Cartesian
50 coordinate system or in unit vector coordinate system. For discrete total variation regularization, is the discrete gradient (finite differences along the coordinate directions) of at the k th pixel , and the TV of is written as : W here is the Euclidean norm in . To this end Equation 4 4 can be discretized as : W here . To simplify the notations, we write i t in the following vector form: where ; ; is a matrix ; ; and . By introducing an auxiliary variable and adding a constraint , Equation 4 7 is equivalent to the following constrained problem:
51 S ubject to V= M , w here , and . The augmented Lagrangian associated with Equation 4 8 is : Modeling Results The success of the computer modeling of the imaging process gave not only a n idea of the expected results but also of a clear range of detectors needed to achieve the desired image quality and the viability of our lab built targets. The computer simulation assumed a hemispherical array of detectors around the target , but the numbe r and position of the detectors was varied to optimize image quality. D ue to the small size and relatively low cost of chip scale optical magnetometers , this was a viable approach as well as having practical applications in this configuration. Figure 4 1 . The simulated hemisphere of detectors. The blue sphere represents the potential area of interest, and the red dots represent potential detector locations. This particular image shows 4547 detectors, represented as points, evenly distributed across t he top hemisphere. This was the maximal count of The scale is in m eters .
52 The number of detectors used was varied from as few as 134 up to just over 4500. The location s of positions are generated by subdividing an icosahedron on the sphere, thus they were nearly evenly distributed. Early simulations used a 19x19x19 grid w here denotes 6859 voxels. The simulated target was chosen to be a simple geometric shape as it was easy to render digitally as well as to fabricate in the laboratory for experimental confirmation. Figure 4 2 . Geometric target. This shape was selected as it had hard edges for the simulation to test contrast differences as well as its ease in real wor ld fabrication. This shape could easily be printed in a desktop 3D printer using a porous structure and then soaked in agar gel to make what is effectively a block of water in that shape orga nic targets with respect to atomic magnetic response. The scale is in cm. This was then modeled with a varying number of detectors as well as being simulated with a more common minimizing method the Direct L east S quare method  of solving equations of the type similar to Equation 4 7. With the high number of
53 detectors , the image recover y was impressive, matching almost exactly the ground truth, while minimizing the least square s produced poor results. Part of the poor performance of least s quare methods is dealing with the large numbers needed to resolve this many voxels f ro m a large number of detectors. The matrix has a size of 4547x6859 in this configuration leading to a pproximately 10 15 elements to be solved. This can be seen in the fi gures below. Figure 4 3 . Central slice of simulated target. A layer by layer approach was used to create the image but the results are most easily com pared when looking at a single slice, such as this taken transversely through the center .
54 Figure 4 4 . Recovered i mage . As can be seen , the recovered central transverse slice is very similar to the ground truth central transverse slice. The colors represent strength of the magnetic dipole at each point, white being a maximal of and black being a minimal of . Image provided by the Y. Chen group. Figure 4 5 . Direct Least Square Image recovery. The large size of matrix A means that this type of image recover y will have issues resulting from errors in the numerical process , such as the computer round off error. The colors represent strength of the magnetic dipole at each point, white being a maximal of and black being a minimal of . Image provided by the Y. Chen group.
55 For these simulations, no noise was added. The target area was given a value of for each representing the entirety of the geometric target and a value of for each that did not have anything simulated in that space, effectively the remainder of the area of interest. As this was done as a steady state signal without decay of any kind , this would represent a single moment in time as the detector readings in an actual target will decay over time. To solve Equation 4 8 for this simulation an ADMM that was iteratively solved with the following argument was used : Further , led to a highly efficient form that only requires a shrinkage operator and a Fourier transform. This reduced form can be found in Appendix B . From this the next step of simulation was to model a real world target. This would not be within the scope of this project experimentally , but would be a necessary step to demonstrating commercial viability as well as using a more standard target for comparison to other developing and established technologies . A Shepp Logan phantom [ 4 6 ] was chosen as the new simulated target as this is the standard used in many image reconstruction algorithms. As above, this model response is based on solving Equation
56 4 7 , and many of the same methods applied to the geometric target were also applied to the 2D simulation. Due to the curved nature of the contrast changes in this target, as opposed to the clear lines provide by the geometric target, the Shepp Logan target is more difficult to model accurately . Initially for this round of simulations , a re duced number of detectors were wanted to get an idea of how many were actually needed. To this end a low resolution, 36x36 pixels, Shepp Logan image was used and tested with 134 and 333 detectors arranged in a hemisphere , a D irect Least S quare model was al so completed for the ground truth to provide a basis of comparison. Results can be seen below : Figure 4 6. Shepp Logan reconstruction with 133 detectors. A) T he total variation regularized reconstructed image, and although distorted the basic shape of th e ground truth ( B), is clear. C) T he Direct Least Square method reconstruction and is a complete failure. The colors represent strength of the magnetic dipole at each point, white being a maximal of and black being a minimal of . Image provi ded by the Y. Chen group.
57 Figure 4 7 . S he p p Logan reconstruction with 333 detectors. The colors represent strength of the magnetic dipole at each point, white being a maximal of and black being a minimal of . A) Is once again the total variation regularized reconstructed image using our algorithms. More definition can be seen and better color contrast through the internal grey shaded area of the image but more aliasing can be seen on the exterior where no image should have been rep orted. B) The low definition ground truth. C) The Direct L east Square method reconstruction which once again fails to even get the basic shape of the target. Image provided by the Y. Chen group. Although the shape is close , the interior detail is bar e ly r esolvable, even with 333 detectors. The shading shows that the intensities being read were not as accurate as they were with a binary intensity geometric target , Figure 4 4 . That being the case the Sheep Logan image is still recognizable and this shows a b asic viability as the signal could be detecting and therefor e decay time identified. This is viewed as a success as this is only the initial modeling , and with further testing and experimental data it was hoped it could be refined. Having completed a low resolution test, a high resolution Sheep Logan target was built as a ground truth with 256x256 pixels. As the algorithms were not changed and the computational time allows for only a low resolution 36x36 recon struction, this round of modeling was expected to have little significants and generally poor results .
58 However, even with a low resolution target and low number of detectors, general shapes were still recognizable as well as decent contrast/intensity rec onstruction. Although not be suitable for many ap plications, especially in the medical field, it does prove the viability of this form of detector and open s up possible deployment where coarse resolution is sufficient. Figure 4 8. High resolution 2D target reconstruction with 13 4 detectors. A) A 36x36 total variation regularized recons truction of the target taken fro m 134 simulated detectors. B) The High resolution , 256x256, 2D target ground truth . C) For comparison purposes, the Direct Least Square method. The colors represent strength of the magnetic dipole at each point, white being a maximal of and black being a minimal of . Image provided by the Y. Chen group. Although the smaller number of detectors produced more image aliasing , the basic shape is still recognizable and the comparison to the D irect L east S quare still shows a dramatic improvement over the baseline algorithm.
59 Figure 4 9. High resolution 2D target reconstruction with 333 detectors. A) A 36x36 total variation regularized recons truction of the target taken fro m 333 simulated detectors. B) The High resolution , 256x256, 2D target ground truth . C) For comparison purposes, the Direct Least Square method. The colors represent strength of the magnetic dipole at each point, white being a maximal of and black being a minimal of . Image provided by the Y. Chen group. As was the case in Figure 4 8, the D irect L east S quare method utterly fails to reconstruct the image. Our method does not produce anything close to a complete reconstruction but does provide a good approximation, with basic shape and contrast / intensity both being of the approximately correct values. L astly , another 3D simulation was render ed with a simplified 13 block geometric shape in a 10x10x10 pixel low resolution ground truth. 500 detectors were used in this simulation in order to provide more detail than the 2D tests but not require the computati onal power that the 4547 detectors needed when resolving the high definition 3D geometric target as a ground truth from Figure 4 4. It is believed that the small resolution size led to the smallest blocks in the ground truth not being recognized. It is wor th stating at this point that the goals of the project included not only a portable but fast imaging system. By reducing the resolution of the reconstructed images , the processing power of the computers was reduced . In this way it was hoped an overall
60 redu ction in the size, weight, and expense of the computer processing unit on any portable imaging device could be achieved. This also aided in reducing simulation time needed to complete these studies. Figure 4 10. 3D low resolution target reconstruction. A) The total variation regularized reconstruction of the target taken from 500 simulated detectors. B) The center transverse slice of the ground truth. C) For comparison purposes, the Direct Least Square method. The colors represent strength of the magneti c dipole at each point, white being a maximal of and black being a minimal of . Image provided by the Y. Chen group. In conclusion , the simulation of atomic optical magnetometers showed the promise this project has. With a simple array of detectors or even a single detector moved through a hemispherical pattern on a mobile arm ( see Figure 3 6 ) a viable magnetic imager could be realize d. Furthermore , by showing how well the array of detectors could perceive the magnetic response f ro m a target it is clear this would have a noticeable impact on the system when run in equilibrium, and thus the interruption caused by the induced atomic magn etic moment could be monitored and timed. This timing data could be used to reconstruct the image based not on magnetic strength as presented above , but on decay time.
61 CHAPTER 5 EXPERIMENTAL PROCEDURE Introduction In this chapter the development of the experiments carried out in order to test the anisotrop y as well as verify the simulations discussed earlier, will be discussed . Starting with the planning and preparation phases, an experimental goal and ambient magnetic conditions were established. The design of the optical magnetometer was then considered, from the requirements of the cell to the vaporization of the alkali metal , the lasers used for the pumping and probing of the cell, and finally to the magnetic coils needed to control the atomic magnetic moment. After this, the electronics package will be discussed as well as configuration changes for specific experiments. Exp erimental Planning In this section the planning and process of the experimental part of this project is discussed. A computer controlled system was used to set the AC frequency and DC field intensity as well as automate data collection . Large cells were us ed throughout this project. Future work would be accomplished using microcells. The driving laser and photon detector array was next assembled, followed by the magnetic coils, and finally signal processing equipment as discussed in the section of the same name. Finally , a stage was designed that was to consist of an arm that moved through an arc used to simulate an array of detectors as well as phantoms and test targets , as presented in Chapter 3. The idea was to design a series of experiments that would test the basic functionality of the lab built optical magnetometer. It was decided that by testing it in a
62 more typical manner, that of a Larmor frequency detector, the equipment could be tested so that it was not only functioning correctly but that it wa s also able to come to an equilibrium state that would be disrupted if other fields were present. In addition to establish ing working parameters many of these experiments calibrate components such as the magnetic coils or heating system. Material Selectio n The process of atomic optical magneticspectro s copy comes from the advances and development of atomic clocks and can be done with any of the alkali metals (or titanium ) , hypothet ically. Of the six candidates rubidium, cesium , and francium have a low enough heat of vaporization to easily generate the needed partial pressure at near room t emperature [ 4 7 ] . Of those three , francium is incredibly rare and short lived. Cesium is considered more toxic and is generally more expensive than rubidiu m , thus rubidium was selected as the alkali metal gas for the optical magnetometer in our experiments. Looking at construction material choices for the cell , all metals were rejected for anything other than fluid dynamic testing where the easy machinabili ty of metals allows for easy design modification. Even nonferrous metals such as brass could not be used due to electrical Foucault currents, also known as eddy currents, that could interfere with testing [ 4 8 ] . Due to the heat requirements this left few op tions and a high temperature polymer was decided upon as well as glass cell s and windows. The wavelength of the D 1 transition in rubidium 87 , 796.4 nm, is not absorbed by boron silicate glass and it is easier to modify than quartz . These options are discussed in further detail in the cell subsection.
63 Magnetic Environment Testing Using the Lakesure 3 axis magnetometer, the locations of minimal DC and AC om laboratory equipment or power lines were characterized to ensure minimal environmental interference . AC fields from 10 to 400 Hz could be detected with this magnetometer as well as DC fields of as little as 0.1 ÂµT . Once the location with minimal magneti c field fluctuations was determined, the magnetometer was set up in a more stable manner , affixed to a mount, and calibrated before each test with a zero field box provided by the manufacturer. The data in Figure 5 1 show a typical result from changes in t he measurement s of the magnetic field strengths in Cartesian coordinate X Y Z orientation versus time. Although there is a fluctuation with time it is slow enough to be ignored for these experiments. The downward slope of the filed in the Z direction (alon g the axis of the experiment) is from a saturation effect in the magnetometer . After each re zeroing, the field lines return to approximately their starting positions and decay again, so it was surmised after this pattern was repeatedly observed that the a pproximate initial positi ons of the field lines are an accurate descriptor of the magnetic environment in which the experiment was conducted.
64 Figure 5 1. Un s hielded magnetic environment. This is a typical example of the DC magnetic environment in the experimental area over a one hour period. The X direction is taken on a horizontal transverse of the work area (south to north), the Y direction is taken vertically through the work area (up t o down) and the Z direction is taken in the horizontal/longitudinal direction (east to west). All results are in m T and are on the low side of the average earth magnetic field of 0.025 0.065 m T [ 25 ] . Some early procedures, such as the Free Induction Decay experiments, were able to be conducted in this environment unshielded; however, due to the desirability to have a smaller magnetic field in some experiments as well as equipment limitations, magnetic shielding was require d . Initially , thin strips of a high permeability metal, such as mu in a five laser and photo diode were exposed. However , due to the need for heating pipes, sensor cables, and mounting poles it was determined that not only was this shielding -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0 500 1000 1500 2000 2500 3000 3500 X Y Z No shield Time in Seconds Field in mT
65 less than optimal it was also cumbersome to work with due to frequen t tear downs and the brittleness of the metal strips. Figure 5 2 . Early magnetic shielding. Initial passive shielding was composed of a fiberglass shell with several layers of a thin high magnetic permeability metal separated by blue paper tape. The (pink) fiberglass thermal insulation can be seen surrounding the cell as well as the w iring lines for the internal coils and heating fluid lines coming into the apparatus. A removable face cover was fabricated but only used during an experimental run, as they were difficult and time consuming to attach and remove. Photo c o urtesy of the auth or. Active shielding was considered and after constructing a frame used to house a He lmholtz C age and attaching the H all E ffect sensors it was found that the demands of the controlling software as well as the control of the power supplies necessary to cre ate a functional active shielding unit was beyond the scope of the project. Subsequent
66 magnetic environmental testing showed satisfactory results for low sensitivity tests due to the reasonably stable environment within the laboratory and the slow changes with respect to time. An outer shell type shield was devised; however, due to the loss of efficiency of magnetic shielding with increasing size [ 4 9 ], an extended tube shield was construct ed using the ratios that optimize efficiency. This was also a five layer, thin metal shield that conformed to dimensions to maximize shielding and a calculated shielding ratio of 0.034 in the transverse direction and 0.008 in the longitudinal direction according to Equation 5 1 and Equation 5 2 : Where S is the shielding factor, a ratio of the external field to the shielded or internal field ; n is the number of layers in the shield ; is a discrete shield layer ; and D is the diameter . When tested this shielding proved more effective th a n calcula ted ratios would indicate despite being open on each end, even on magnetic field lines along the open axis ( Z direction). This is likely due to a breakdown of these equations as shield diameter increases and becomes a fraction. Once exper imentation was performed inside the shielding, results were inconsistent and upon further research, the
67 possibility that the thin shield was quickly becoming saturated due to the magnetic coils in close proximity to it was realized. Thicker shield walls of a high permeability metal or frequent degaussing are the common solutions to saturation problems [ 50 ] [ 5 1 ] [ 5 2 ] . The fabrication of a thick walled shield was beyond the capabilities of the facilities the group had access to ; however , a s a result a new shield was ordered from a manufacturer that would incorporate both . A thick walled , three layer shield with an integrated degaussing coil was acquired from Magnetic Shield Corp. This proved to be an improvement by an order of magnitude over the previous s hielding, and the integrated degaussing coil could be used periodically as needed. The thick walled shield also had a permanently affixed cap at one end and a removable cap at the other. Due to the number of cables and heating pipes needed to enter and exi t the shielding, the removable cap was left off. The shield was aligned such that the environmental magnetic field lines along the Z axis would impact on the affixed cap, which did have a small hole through which the laser was projected. The following grap h shows a typical reading of the lab built thin walled shield and the Magnetic Shield Corp. thick walled shield, again with the slop ing field lines due to saturation of the Lakesure detector.
68 A) B) Figure 5 3 . Thin walled vs thick walled magnetic shielding. Measurements taking in the longitudinal ( Z axis) as well as the transverse horizontal (X axis) and vertical (Y axis) directions. A) Thin walled shield over a one hour period . B) Thick walled shield over the same time period, a noticeable improv ement in both intensity and stability. -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0 1000 2000 3000 X Y Z Time in Seconds Field in mT Thin Walled Shield
69 Althoug h device application still requires improvements in shielding in order to make the device portable, it was hoped that smaller magnetic shielding, either integrated into a fixed form factor or via some kind of woven high magnetic permeability material , could be constructed . However , the complete apparatus at this point weighs only 38.7 kg and could be moved wit h little difficulty, assuming a 1.4 m 2 area could be cleared and sufficient power and outlets provided. Figure 5 4. Detector apparatus. This is the current state of the detector apparatus including the heating unit and laser driver. Note that the heating unit has additional magnetic shielding around it to prevent stray EM fields. Not seen are the electronics stack (located to the right, o ut of frame) including arbitrary waveform generator ( AWG ) , oscilloscope, lock in amplifiers, AC and DC power supplies, in line amplifiers, and in line attenuators. Photo courtesy of the author.
70 Rubidium Cell Most of the recent research and development on optical magnetometers centers on small or even chip scale rubidium cells [ 53 ]. There are many advantages to decreasing the size of the rubidium cell : the smaller cell is easier to heat and control temperature ; the smaller form dictates a smaller volume of shielded area and magnetic shielding efficiency goes up as shielded volume goes down [ 49 ] . T he shorter the path of the laser the less sources of noise can be introduced as well as papers pointing to smaller cells having higher sensitivity . However , fab rication is not one of the advantages. Starting with a larger cell was an expedient way to begin working on magnetometer designs while exploring the capacity to fabricate smaller cells in the lab or acquire them from other sources. This large cell had a volume of 38.611 cm 3 and a length of 7.62 cm, and was backfilled with nitrogen as a buffer gas to 3 atmospheres pressure. An unknown quantity of rubidium 87 was filled into the cell; the manufacturer Early designs focused on a way to heat the cell without causing magnetic field noise. Rubidium sublimes at 100 Â° C but a significant partial pressure can be sustained at lower temperatures , with a calculated vapor pressure of 0.042 Pa based on Equation 5 3 [ 54 ] [ 47 ] . A and B are constants with a value for Rubidium of 4.857 and 4215 respectively, T in temperature in degrees K, and P is the vapor pressure in Pa.
71 This was verified experimental by monitoring the photon absorption as the cell was heated to the limit of heat sen sitive shell, when a minimal absorption was stable it was determined to indicate sufficient rubidium partial pressure. In this project an Silicon heating fluid was initial used followed by heated air flow in latter experiments to avoid the EM interference form resistive heating. Other groups have tried [ 7 ], hot air ovens [ 5 ] [ 55 ], or try to adjust the heater duty cycle [ 56 ] to avoid this problem. All of these methods have their drawbacks and advantages . The first pr ototype s for this project were built around a fluid heating system using a silicon heating fluid to control cell temp erature . Although this was a very efficient method of controlling temp erature without any interference from the heater on the magnetometer, it was very bulky and made working on the cell very difficult and time consuming, whether it was a clamshell style immersion bath with exposed faces or a polytetrafluoroethylene tube coiled around the cell. Time lost to the heating system became more prob lematic resulting in the project moving to the less efficient but easer to maintain air heating system. In this model hot air is pumped into a shell built around the cell. Temperature was monitored qualitatively by the amount of visible rubidium metal in t he cell and quantitatively by a K type thermal couple attached to the surface of insulation layer to help determine the heat loss through that layer. Due to heating loss o f the long tubes necessary to separate the heater unit f ro m the magnetometer and the poor efficiency of the air heating unit a typical routine had the heater set to 150 Â° C for 45 minutes before stepping down to 1 15 Â° C in order to maintain the cell at 102 Â° C.
72 During the course of the project a condensed layer of rubidium became visible on the face of the cell that, even when directly exposed to sustained temperatures of 120 Â° C , does not dissipate but also does not seem to impact testing. No anti reflective coating was applied to the cell, so the long axis (Z axis) was tilted slightly with respect to the path of the laser and most interior surfaces within the magnetic shielding were coated with a non reflective paint to reduce internal reflect ions . Prior to any test run, the laser and photo diode were turned on and measurements taken through the warm up cycle starting with a c old cell of about 22 Â° C, to ensure proper absorption of photons by the rubidium once at operating temperature . Optics Through the single mode laser diode tuned to 794.6 nm, or 3.773x10 14 Hz, via an external TEC was used to achieve a monochromatic photon source. This provided photons with an energy of 1.5603 eV per photon [ 57 ]. Both the laser driver and the TEC are analog controlled with potentiometers. The laser di ode was first calibrated with an Ocean Optics USB 2000 USB spectrometer and then further refined by monitoring the absorption through the rubidium cell; on ce the cell is at operating temperature of 102 Â° C, the rubidium in the cell will absorb photons at the desired wavelength, and thus by finding the TEC setting that produces the minimal transmitted light, the laser wavelength is set to the D 1 transition [ 58 ]. This laser was used to excite the rubidium electrons, thus acting as the pump laser , while at the same time subtle changes in absorption due to resonates with th e perpendicular magnetic field will have been able to be detected as well, making it also a probe laser.
73 Coils There are two primary coil systems used in the optic magnetometer apparatus: the solenoid wrapped around the cell to provide magnetic field lines running along the Z axis, or longitudinal axis, of the cell along the path of the laser, and a coil mounted outside of the thermal insulation that is orientated to form a Helmholtz coil pair along the X axis, which is horizontally perpendicular to the longitudinal axis of the cell. The solenoid is referred to as the DC coil and the Helmholtz pair mounted across the cell is called the AC coil. Although some papers with large cell s report better results with a coil mounted at a 45 Â° angle to the path of the pump laser [ 59 ], the constraints of the insulating layer made a more traditional approach necessary. Both coils use AWG 28 mag wire. The solenoid coil has 178 turns of radius 12.7 mm and a length of 76.2 mm with a small defect in the wrapping to accommodate the fill stem on the cell approximately mid length on the cell. As a magnetometer probe cannot be placed in the sealed cell, a mock up solenoid was fabricated and tested; measured values were within 5% of calculated values for DC fields. The perpendicular coil set has 200 turns with a diameter of 152.4 mm, separated by a distance of 76.2 mm. The use of the coils will be discussed further in the configuration chapter. These coils have independent power supplies and amplifiers as needed. Originally controlled by Microsoft Visual Studio software run over GPID, latter experiments used manual s ettings and integrated step or sweep functions. This was due to a late stage software change to MATLAB , but time did not allow for code to be written that would control most of the hardware over GPID. A third coil, configured as a large Helmholtz pair, wa s also fabricated. This coil of 32.39 cm diameter and 1600 turns was to be placed across a target in order to expose it
74 to a strong magnetic field that would , in turn , induce an atomic magnetic moment in the target for the optical magnetometer to sample. T he free induction decay experiment These coils were powered by a Keithley 237 high voltage power supply for discreet DC currents from 100 fA to 100 mA or an Agilent E36 34A for AC current. A Hewlett Packard 467 A amplifier was used when these power supplies could not provide the necessary current to generate the desired field. Signal Processing The signal from the photo diode was delivered to a digital to analog converte r ( DA C ) and there split up to be sent to the lock in amplifier as well as to the computer for raw data collection. The signal moved through an attenuator before going to the lock in if necessary. For signal processing a Stanford Research SR530 Lock in Ampl ifier was used. The post lock in signal was collected from the computer via a coaxial cable returning to the DA C . It is of note that for the majority of this experiment, the GPID connections were used to control the lock in amplifier and collect data from it. The reference source for the lock in amplifier was set according to the experimental configurations and target Larmor frequency.
75 Figure 5 5 . Electronics diagram. Photons collected by the photo diode (a) are converted into a voltage signal and sent t o a DA C (b) along coaxial cable (black dashed line) where the raw signal is sent to the computer (orange dotted line) (c) for data collection as well as to the lock in amplifier (d) for processing. The output from the lock in amplifier can be sent to the c omputer via the DA C or by a general purpose interface bus ( GPIB ) cable (blue dashed line) depending on the software used. The AWG (e) is often used as the driving source for the AC coil as well as providing the reference signal to the lock in amplifier. In other cases the lock in amplifier reference is provided by the lock in e wave signal generator. The Helmholtz coils (f), placed perpendicular to the path of the laser, are often used as RF coils. Finally , a DC power supply (h) with an optional amplifier (g) was used to drive DC fields, usually in the solenoid (i) wrapped around the cell
76 Experimental Configuration Within this magnetic shielding was placed the detector. The core of the detector is a rubidium cell held in a PTFE case into which hot air was pumped through a PTFE tube. The front and rear faces of the PTFE case are open but have Kapton tape sealing off t he majority of this opening, leaving enough space for the laser beam to pass though unimpeded. External of the magnetic shielding there is the optic, heating, and signal processing equipment. The openings at both ends of the magnetic shielding as well as the optic equipment are covered with black velvet, and all overhead lights in the room are off during experimental runs . Free Induction Decay Based on the classic experiment [ 60 ] [ 61 ] the free induction decay experiment sets up an induced atomic magnetic moment in a target and the n monitors the decay in the magnetic field produced. A slightly modified Robi n son oscillator was used to account for modern electronic available and the unavailability of some components due to their age [ 62 ] [ 63 ]. The free induct ion decay experiment was performed to test some apparatus as well as hardware for a preliminary evaluation. From this the larger pulsing coils that are to be used for later experiments would be calibrated, the thin gauge wire solenoids made in the lab coul d be tested, and the viability of the lock in amplifier to detect the decaying signal on the time frame of milliseconds would be confirmed. The experiments were carried out in an unshielded environment with an average DC magnetic ambient field of 0.02 mT from south to north ( Z direction), 0.03 mT from east to west ( X direction) and 0.001 mT in the vertical axis ( Y direction) of the experiment. No AC magnetic fields were detected in the 10 to 400 Hz range. A small volume of de ionized water (D.I. water) , 2 ml, was placed in an NMR tube ( thin walled,
77 high purity borosilicate glass tube of high uniformity wall thickness), that had a coil of AWG 28 enamel coated wire wrapped around the circumference of the tube. This solenoid was 40 turns covering the bottom 1.28 cm on the tube. This apparatus was then suspended so that the coil and D.I. water were held in the center of a pair of Helmholtz coils ( two coils of enameled copper wire placed apart at a distance equal to their inner radius, thus providing a uniform and stable magnetic field ) . The Helmholtz coil pair had a DC power source with an AC offset, a 73 Hz wiggle signal added in via a small transformer. The solenoid around the tube was connected to an AC power supply going through a modified Robinson oscillator and then a lock in amplifier for signal processing. See the diagram below:
78 Figure 5 6 . Fre e induction decay diagrams. A) The electronics set up for the FID experiment. An AC sine wave from the lock through the Robinson oscillator (b) to the solenoid (c) wrapped around the base of the NMR tube holding the D.I . water sample. This sample is saturated in a magnetic field produced by the large Helmholtz coils (d) straddling the NMR tube. The Magnetic field is driven by the DC power supply (f) and added via a transformer (g). This AC offset is used as the reference as well and is what the lock in is looking to detect. The Robinson oscillator forms a tank or resonant circuit with the smaller solenoid at the given frequency range. The DC cu rrent was then adjusted slowly until the lock in amplifier detected the wiggle signal, indicating that the AC signal is at the Larmor frequency induced by the larger DC field. The fields are turn ed off and the decay of the atomic magnetic moment is monitor ed via the lock in amplifier .
79 Rubidium MX Magnetometer Configurations All systems are turned on for their full warm up period, 30 minutes for most configurations, and then photo diode dark current is recorded. Photo current is monitored with a cold cell a t 22 Â° C and recorded as the cell comes to temperature of 102 Â° C over a period of an hour or more . To avoid overloading the lock in input, the photo diode signal must be kept under 1 volt. Typically the photo voltage is adjusted via neutral density filters to be just above 1 volt when the cell is cold. At temperature, a 30% decrease in transmitted light is normal with this set up. Several configurations have been evaluated to gather information on various aspects of the optical magnetometer properties . Squ are Wave Configuration Th e square wave configuration is used to determine the change in intensity in optical absorption with magnetic field of different strengths. The AC coils are not active during this experiment. A square wave, offset to have no negati ve current, was fed into the DC coil. Due to the limitations of the lock in amplifiers and the 28 AWG wire used in the coil, a minimum voltage of 0 and maximum voltage of 2 was used; this translates as a current 0.5128 A and a peak magnetic field of 1.426 mT.
80 is the generated magnetic field (in t esla), Âµ 0 is the magnetic permittivity of free space, is the number of turns in the solenoid, I is the current, L is the length of the solenoid, is the outer radius of the coil, and is the inner radius of the coil. This square wave signal was also fed into the lock in amplifier as a reference signal. The period was initially set very low, 1 H z (the lock in lower limit for a square wave is 0.5 H z ), but it was suggested that despite th e listed specifications, lock in amplifiers experience difficulty determining such low signals. This was confirmed by an aliasing issue where the lock in amplifier identified signals that were not present at very low frequencies. Although the low frequenc y was not the sole cause of the artificial signal, it was greatly exacerbated by it. Once a signal was found, the experiment could be repeated with a higher or lower amplitude square wave. As the AC coil was not active, we had no expectation of detecting a Larmor frequency but just an absorption intensity change that followed the squar e wave. This configuration yield s data that could be used to determine limits of intensity change in the optic system to avoid saturation or too few photons. This information could also be used to compare intensities from the phantom target in order to calibrate the signal generated there versus the DC coils (calibrated in the DC sweep configuration).
81 Figure 5 7 . Square wave detection diagrams. An offset square wave is gener ated (a) and sent to the solenoid around the cell as well a reference signal (b) to the fluctuate with this changing magnetic field. The absorption is recorded by the photo di ode (e), passed onto the lock in amplifier via the DA C , and be proceed there to find any signal buried in the noise. Finally the signal is sent to a PC for to recoding and analysis. DC Sweep Configuration In this configuration the AC coil is keep at a static frequency sine wave, signal and current provided from an arbitrary wave form generator. The same signal is sent to the lock in amplifier as a reference signal. This frequency is based on the expected La rmor frequency. The DC coil power supply is set to step up the current through a range with the target current being the midpoint of this range. The current to the DC
82 current will produce a magnetic field that will saturate the rubidium, dictating the orie ntation of the induced magnetic moment and the Larmor pr e cession rate , Equation 3 4 . When combined with the solenoid E quation 5 3 W here is the resulting Larmor frequency and is the gyromagne tic ratio, 7 Hz/nT for rubidium 87 . This magnetic field is used in turn to calculate the Larmor frequency used as the AC set point. Step size and hold time between steps are adjusted to optimize lock in probability, approximately 0.001 A step with 0.1s hold time or the equivalent of 2.7 ÂµT at 10 Hz , although depending on the lock in amplifier refer ence frequency and software package, longer hold times of up to 2 s have been used. In this configuration one would expect the lock in amplifier to identify the signal that matches the reference signal only when the reference frequency is in resonance wit h the Larmor frequency due to changes in the quantity of transmitted photos due to dark ground states becoming more stable. This is a direct effect of the perpendicular AC magnetic field not imparting more energy into the system once resonance is reached. This test would be used to calibrate the DC coil system as well as to establish the detection limits of the cell. This experiment was conducted using GPID connections and controlled by Microsoft Visual Studio software based on a C++ platform.
83 Figure 5 8 . DC sweep electronics diagram. Lock in amplifier (a) sends a set AC sine wave to the RF coil (b), through an amplifier(c), as well as using this signal as a reference. The computer (d), via a GPIB interface, controls the DC power supple (e) and thus the solenoid (f) current and magnetic field. The phot diode (g) collects the transmitted photons and sends the converted signal, as volts, to the lock in amplifier. AC Sweep Configuration A more difficult configuration when compared to the DC sweep configurat ion but a useful configuration when developing a portable imaging system due to the unknown magnitude of the decaying atomic magnetic moment in real world samples and the desire to test the system on changing fields such as an induced magnetic dec ay . The DC coil is held static at a known current and thus known DC magnetic field and Larmor frequency , in this case 0.01 A with a resulting DC B field of 28.57 ÂµT and a
84 Larmor frequency of 0.19 MHz . The AC coil frequency is swept through a range with the target Larmor frequency being the midpoint of the range. The step size of the frequency could vary greatly as the ideal lock in amplifier would produce a stronger signal as the reference frequency approaches the Larmor frequency; steps of 1 HZ to 0.01 H z were used. The hold time unfortunately was considerably greater than expected due to the issues lock in amplifiers have when forced to re acquire a changing reference signal; hold times of 5 s or more were often used. An additional issue was discovered w hen some software packages used to control the sweep of the AC coil caused a data collection rate much lower than the reported collection rate. This issue is discussed further in computer software. The data from this configuration would establish the need ed response time of supporting equipment as well the viability of matching the Larmor frequency to a decaying field with any degree of accuracy. Although this configuration was attempted, it became apparent tha t the equipment in the lab is not be able to process signals fast enough for this configuration to yield any meaningful results.
85 Figure 5 9 . AC sweep electronics diagram diagrams. PC (a) controlled AWG (b) steps frequency sent to AC coils (c) as well as to the lock in amplifier (d) as a re ference. A steady DC static is sent to the solenoid (e) to establish a magnetic environment and Larmor frequency. Any transmitted photons are detected by the photo diode (f) and sent to the lock in amplifier as a voltage signal. After processing , this sign al is sent to the PC for data recording.
86 CHAPTER 6 RESULTS Goals The primary goal of this project, to create a magnet moment imaging system that could be made portable, depended of testing the hypothesis that rubidium optical magnetometers are anisotropic . At this stage this hypothesis has not be en conclusively proven or disproved. Simulation data suggest that imaging should be possible with a sufficient number of detectors in an array. The national labs have produced magnetometers of sufficient size and sensitivity to form these arrays. But the key element, the anisotrop y of the detector, is still undergoing testing . Additionally , the proof of concept method of identifying a target by decay time instead of relying on resonance with the Larmor frequency was simulated but was not able to be replicated in the laboratory although the square wave test showed that it is possible . Free Induction Decay R esults After the experiment it was concluded that the lock in amplifier was able to correctly extrac t signal from the decay ing field and to correctly identify the time constant within a standard deviation for the field used. This would prove to not be the case later when it could not be replicated at high Larmor frequencies or when more advanced software coding was used to control more complex experiments and caused a lag in the system to produce faulty data sampling rates. With this confirmation an offset and stability tolerance was established for the pulsing coils to be used in later experiments. Thi s was done by repeating the experiment several times and noting the difference between calculated values and the values necessary to produce the detected Larmor rate and thus magnetic fields .
87 AC Sweep Results With the success of the Free Induction Decay test it was assumed the equipment was sufficient to move on to more advanced testing stages. Some early electric shielding issues led to false signals that appeared real but upon detailed analysis were proven to be faulty wire configuration and then faulty electrical insulation. Once the wiring issues were addressed no signal was able to be recovered. The lock in amplifier required a long hold time, on the order of 5 or more seconds, in order to re acquire the reference signal after every step. Efforts were made to overcome this, such as using a dual interface lock in amplifier so that one input was acquiring the next step of the AC signal while the other recorded data f ro m the previous step. This required an independent reference source from the AC sine wav e feed into the experimental apparatus. It was hoped that by having both s ine waves initiated by the same software they would be synchronized. However, due to the limitations of computer communication there was a consistent but variable difference in the s tart point for each sine wave leading to phase differences . Further testing showed that even when the reference source was set up independently to each lock in amplifier and phantom signal, the retrieved lock in data was not able to be correlated due to th e incoherent phase . Slight calibration issues on the channels of the dual lock in amplifier were the suspected cause of the data mismatch. It was decided at this junction to run an experiment with less parameters to account for, the DC sweep configuration, due to the lock in reference acquisition rate issues . DC Sweep Results This experiment has been performed in several groups and a lthough not no vel in and of itself, it prove d the viability of the lab built magne tometer apparatus and that will
88 allow testing of the anisotrop y of the rubidium detectors as well as testing the decay rate identification method. When no viable signal could be obtained from this configuration a dummy signal was introduced to help identif y the fault by running an AC component matching the target Larmor frequency of 0.7 MHz . This eventually led to the discovery of some damaged electronics within the AWG leading to the generation of a signal that was not only inconsistent but also not accura tely displayed or reported. After all hardware was functioning correctly the DC sweep was again attempted unsuccessfully. At this time it was suspected that the problem could be excessive noise or some unknown factor causing a greater variation f ro m calcul ated results than expected. To examine the latter, DC range was increased to 0.05 A at the top of the range and hold time was increased, this caused test runs that took nearly 2 days . Step size was adjusted and sample rate was reduced. The data collected h ad to be analyzed in batches as the limits of the available calculating software were exceeded by the volume of data collected. It was believed that even at the reduced data collection rate the lock in amplifier would be able to isolate a signal f ro m the b ackground noise. Hoping to isolate the issues, the probable noise sources were examined and atte mpts were made to minimize them. Among the identified possible sources was the laser diode and the supporting TEC and driver, the photo diode, environmental fac tors impacting the photons due to the length of the laser path, and lastly an out of calibration magnetometer giving poor verification of other testing parameters. To address these issues the laser driver was monitored for power output to both the diode, TEC and monitor diode, all of which were within tolerance of the driver. The
89 TEC power was monitored to within the ability to do so while the laser diode was active. Additionally , th e TE C was disassembled and Lasorb brand electrostatic discharge ( ESD ) prot ection was added to help mitigate minor power fluctuations or static charges. The laser diodes were tested with the spectrometers available and were found to be accurate within the limits of the detect ors . Laser output was not tested as no means of doing so with any degree of accuracy was available at the time within the facility. It was determined that the laser diode and TEC would be changed out to attempt to verify any errors; the replacement components provided the same test and experimental results. D uring this period it was noted that one of the manual potentiometers used to adjust TEC settings on the laser driver was faulty and although it was replaced a replacement laser driver was also installed. Similar testing of the phot o d iode was carried out and the phot o diode was also replaced, although all combinations of photo diode, laser diode, laser driver, and TEC were tested to check for any device failures . The magnetometer was tested to within the limits of lab personnel and after support from the m anufacturer was determined to be operating within expected parameters. To address environmental issues all sources of environmental impact that could be eliminated were, even if they were not suspected to have had an impact on the testing environment. The air conditioning was turn ed off for the duration of all test runs and doors were not used. External air handlers, water circulators, and air compressors that were in proximity to the area of the lab where testing was taken place were also suspended during testing. As a safety note, no other experiments were active during this time period and no systems depended on the mechanical systems that were shut down during this time. Additionally , cardboard tubes were set up within the magnetic
90 shielding to further protect the laser path without interfering with any of the EM fields within the testing area . The optic array was minimized in terms of focus mirrors, expanders, and any other components that could be eliminated. It was noted during the installation of the cardboard tubes that the rubidium that had condensed on the face of the cell seem ed to be larger than it had on pervious tear downs. The apparatus was torn down and the heating systems were rebuilt in an attempt to optimize heating efficiency and ensure even heat distribution. The resulting changes led to a longer warm up period but more even distribution of heat ; however , the condensed material on the face plate was unable to be cleared up. The rubidium cell was removed and baked as per the manufacture r s suggestion and allowed to cool with change the size of the deposit. Direct heat, f ro m a hot air source, was applied to the face. After 20 minutes of air heated to 120 Â° C directed at the face from less than 5 cm away it was determined that the deposit would not be sublimed and to replace the cell. Further test ing on the replacement cell showed a similar, although smaller and less opaque, condensation deposit. After i nstallation it was determined that it was not a major factor. Despite all of these efforts the DC sweep experiment was still unsuccessful . As of these writing s many of the solution s are in place and physical failings are no longer suspected to be the caus e, but a failing of the electronics software package being used for the majority of this project as covered in the following section. Square Wave Results After encountering some trouble getting viable or repeatable results from the most current detector a pparatus, a fundamental experiment was devised to ensure that
91 the Zeeman type splitting could be detected. Conceptually there should be a difference in photo n such as the lab ambient field, see Figure 5 3 , and one existing in a stronger magnetic field , such as when the solenoid provides a field of 1.38 mT . Initially direct photo diode readings were taken but there was too much noise in the system to discern the change in absorption. Using an AC square wave of a few hundred Hz with an offset so the current ranged from 0 to 2 V on the solenoid wrapped around the cell the magnetic field could be modulated between an on or off position. When the field is off the r ubidium absorbing photons will remain constant and consistent with the readings taken on the cell at operational temperature. However, when the magneti c field is engaged the energy levels split and the ground levels become divided between pumpable and un pumpable, or dark, states as described in Chapter 3 . Electrons in this dark state are not excited by the photo n s and absorption decreases. The lock in a mplifier, having this same AC sq uare wave as a reference, will be able to detect this pattern only if the photon absorption is being modulated as expected. This experiment would not provide any information with regard to field strength via Larmor frequenc y detection. That information is only discernable with a transverse alternating field and is the only way to monitor field intensity with this kind of optical the basic vi ability and functionality of the detector working with the decay time method of magnetic field tomography . Early experimental runs had no success in isolating the signal from the background noise, either with direct PD readings or with the l ock in amplifie r.
92 Suspected sources of noise have been addressed and adjustments have been made where possible to minimize this. The large optical table on which the apparatus is situated does not have a functional air lift system providing hydraulic vibration damping. T his is due to both the building air compressors being non operational and missing regulators on the table. Photo diodes, laser diodes, TEC, and laser controllers were all tested and calibrated to the limit s of our ability in the lab. After these avenues we re exhausted and the signal could still not be separated from the noise the use of Fourier transforms were employed. Th is was also unsuccessful but le d to the discovery that our data collection rate was significantly lower tha n reported by the equipment. A lthough our data collection rate was set between 100 and 10 , 000 samples a second, depending on the length on the test run, our actual rate was found to be never higher than 80 samples a second and often lower , with false data being generated from the syste m to backfill these gaps . With further investigation we found the controlling software, Microsoft Visual Studio 2010, when coupled with GPIB IEEE 488 interface bus , was the source of the slow down. The GPIB is limited to 1 Mb per second but the slowest dev ice sets the speed for all devices on the GPID control. Add to t his the way in which Microsoft V isual S tudio handles the command threads, data reporting threads, communication between devices, and data recording was leading to a high overhead in terms of p rocessing time, from ms to tenths of seconds. At this point alternate software was looked at, initially MATLAB was rejected as a software package due to some team members unfamiliar ity with it and concerns with the interface between some of the older equi pment. To date the software has not worked well with the GPID interface or some of the hardware (arbitrary wave form generator and DC power supply , primarily) but has performed
93 beyond expectations with the recording data f ro m both the lock in amplifier and the photo diode. With data acquisition rates in the 40 kHz range a signal was able to be isolated f ro m the noise with high reproducibility. Results were not clearly defined until very small segments, 500 or so data points, thus it would have been impossib le to achieve results with the combination of the software and hardware that was employed prior to this discovery . Work has begun to proceed using the higher data collection rates and previous experiments are being revisited as soon as the code can be comp leted to control the sweeps and frequency changes in an automated fashion . However , that data that was collected shows that the system can track changes at least as fast as 1 ms and this falls well within the range of most organic targets for induced magnetic moment decay rate . Figure 6 1. Recovered signal from square wave test . A n 800 Hz square wave was sent to the solenoid, which caused that frequency to be imparted on the laser as it passed through the cell due to the changes in the rubidium valence electrons moving f ro m dark to non dark states. The X axis is in data points but the step between each represents 0.25 ms.
94 Conclusions The computer modeling of the detector showed the viability of using an array of detectors without any kind of gra dient field to successfully reconstruct a contrast image using atomic optical magnetometers. Image recovery was far better than the Direct Least Square method in 2D, 3D, high and low definition simulations. The TV algorithms developed produced images with pixels far in excess of the number of detectors reducing the need for large arrays and adding in moving to a portable system. The experimental square wave test showed that the system could detect changes in magnetic fields that impact the cell on the orde r of ms, as was the goal of this project. The failure of software and signal processing equipment should have been rectified sooner but the focus on seeking a solution within the physics of the project eliminated many future sources of probable issues as w ell as determining that, even on a limited budget, a magnetic imager could be fabricated. All of this points to the viability of a magnetic imager based on measuring the decay of an induced magnetic moment. This could also be used as a materials identific ation tool in a manner similar to XRD by comparing it to a database like the JCPDS but of known decay time s instead of crystallographic information.
95 CHAPTER 7 FUTURE WORK Beyond what was accomplished there are objectives that could still be expanded upon. Testing was done via simulation and with some AWG but a phantom signal generator could be fabricated that would more closely mimic a target with a decaying magnetic momen t. During the course of this project one was constructed with a variable potentiometer as part of the attenuation circuit and a variable resistor to control the initial moment and the decay rate. Although it was never tested it is simple to create and woul d enable the testing of the MMI with a controlled decaying field, as well as perform some initial positional testing as the phantom emitter head is relatively small and separated from the electronics package that controls it by a flexible cable. Figure 7 1. Phantom. An image of the phantom created for this project. Shielded wire leads to a small exposed loop that emits an EM signal designed to mimic a decaying magnetic moment. Magnetic strength and decay rate are set by manual potentiometers on the phan tom circuit board and monitored via an oscilloscope. An alternate emitter head is shown as well with a glass capsule that could hold a liquid used to further replicate the decay of real world targets. Photo courtesy of the author.
96 The next reasonable step would be to work on miniaturization of the detector. Numerous companies now produce chip scale atomic clocks that have the potential to be modified into magnetometers of the relatively low sensitivity required for this application. However, even with the large rubidium cell the apparatus could be greatly reduced in size by modifying the thermal and magnetic insulation. Currently the magnetic shielding is 1 m in length and 0.3 m in diameter; this could be reduced at the cost of shielding efficiency. When t he stability, in terms of rapid changing fields, of the laboratory is considered this may not have as high an impact as was initially thought. Alternatively, the shielding could be expanded to encapsulate the entire experiment instead of just the detector . This would reduce overall portability but would allow for a more controlled experimental environment. Exchanging the heating tubes for a high temperature resistant rubber would also help reduce size as the rigid PTFE pipe and insulation are bulky, as can be seen in Figure 5 4. Once the detector has been miniaturized the array can be simulated (Figure 3 8). This could be accomplished by mounting the target on a rotation stage while mounting d over the center of rotation of the platform the target is mounted on. In this way a hemispherical array of any number of detectors in any configuration could be simulated with a single detector. Hemispherical diameter is determined by detector position on the arm. This is most useful if the magnetic shielding is removed from the detector and the entire experiment, pulsing coils included, are e ncased in a magnetic shield. This would introduce the possibility of the stage and arm interacting with the mag netic fields and careful consideration of the magnetic properties of the materials will
97 need to be considered before construction. For example, the rotating stage and the mobile arm should be constructed on a non metallic/non conducting material to avoid the possibility of eddy currents. The polymers or ceramics used should be carefully examined for induced magnetic responses as well as magnetic decay rates. Finally, once the computer modeled data and equations match up with the experimental data, actual real world targets could be used with the pulsing coils. Starting with 3D printed porous polymers in geometric configurations soaked in agar gel, such as represented in Figure 4 2, and moving onto more complex targets such as bell peppers or other shapes that have complex internal arrangements. This information would be directly applied towards the design and fabrication of an actual detector array. In addition to any experimental configurations the limits of how fast a magnetic decay could be detected or how much ambient interference could be added before the image was compromised could also be examined and compared to computer simulations.
98 APPENDIX A LIST OF VARIABLES n T he principal quantum number , describing the shell , written an a number l T he angular quantum number (sometimes called Azimuthal) describing subshell , given as a letter S, P, D, or F , generally m T he magnetic quantum number, which details the specific orbital s The spin quantum number, which describes the intrinsic angular momentum The total electron angular momentum , a vector T he vector of the orbital angular momentum s T he vector of spin angular momentum Non vector form of total electron angular momentum T he total atomic angular momentum L armor frequency , describes the precession of a rotating body G yromagnetic ratio , ratio of magnetic dipole moment to angular momentum Mag netic field , also called magnetic flux M ass of the particle, in this case the nucleus of the rubidium atom G factor , unitless proportionality constant Elementary charge, coulombs T he strength of the magnetic dipole, measured in tesla T he magnetic permeability of free space , U nit vector giving the orientation of the dipole (assumed to be aligned with external field), subscript indicates which source T he distance from the dipole to the detector
99 U nit vector pointing from the source to the detector, subscripts indicate starting and ending point U nit vector of the magnetic field line as it enters the detector, the subscript denotes the originating source and the detector to which it is interacting U nit vector along t indicates the identity of the detector Signal output of the detector, subscript indicates detector identity i The identity of magnetic sources in the area on interest , numbered from 1 to K j The identity of detectors in the system , numbered from 1 to N K Number of magnetic sources in the area of interest N Number of detectors in the system The area of interest, composed of both magnetic sources and empty space An isolated section of Equation 4 1 Special TV parameter that must be greater than 0 D Number of pixels in the reconstructed image The identity of pixels in the reconstructed image , numbered from 1 to D Ratio of magnetic strength inside the shielded volume compared to the strength outside the shielded volume in the transvers e direction of a cylindrical shield Ratio of magnetic strength inside the shielded volume compared to the strength outside the shielded volume in the longitudinal di rection of a cylindrical shield Diame ter of magnetic shielding layer , where refers to the discreet shieldi ng layer
100 Number of turns in a solenoid I Current in amperage L L ength of the solenoid R Radius of the solenoid, subscript indicated inner or outer measurement P Absolute vapor pressure of the substance T Temperature in K
101 APPENDIX B LINEARIZATION EQUATION With reference to Equation 4 solution to further reduce the problem : Since is not invertible in general, solving the sub problem was achi e ved by the following line a rization of at by where the step size is chosen using BB method, that is Then, by a computation involving complete squares, Equation 4 10 was reduced to where was determined in Equation A 2 , and The optimal condition of the M subproblem in Equation A 3 le d to The operator is a block circulant matrix ; it was diagonalized by discrete Fourier transform. Then, an explicit expression for was achi e ved . The solution for the V subproblem in Equation A 3 was obtained by the shrinkage operator:
102 In this manner a highly efficient formulation of a complex optimization problem was achieved with successful results.
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10 8 BIOGRAPHICAL SKETCH Nickolas Ptschelinzew is a naturalized citizen of the United States of America, being originally from Australia. Raised in south Florida, he enlisted in the United Sates Air Force and d id one tour of duty before being honorably discharged. He then returned to Miami i n order to pursue a degree. It was while attending classes at Florida International University that Nickolas found his passion for science. He moved to Gainesville with the intent to enroll at the University of Florid a . After complet ing an mmunity College he was accepted into the D epartment of Materials Science and Engineering at UF. After attaining a Bachelor of Science he joined Dr . Bourn e molding of bulk metallic glasses, before joining the Davidson gr oup to work on his PhD. Nickolas plans to work in industry for several years before eventually returning to academia .