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Implementation of a coherent Lyman-alpha source for laser cooling and spectroscopy of antihydrogen Michan, Juan Mario 2014

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Implementation of a Coherent Lyman-αSource for Laser Cooling andSpectroscopy of AntihydrogenbyJuan Mario MichanB.ASc., The University of British Columbia, 2006M.ASc., The University of British Columbia, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2014© Juan Mario Michan 2014AbstractThis dissertation describes two related projects: the development of a co-herent Lyman-α source and the implementation of a supersonic hydrogenbeam.A two-photon resonance-enhanced four wave mixing process in kryp-ton is used to generate high power coherent radiation at ωLy−α ⇒ 121.56nm, the hydrogen Lyman-α line, to perform spectroscopy and cooling ofmagnetically trapped antihydrogen (1s − 2p transition). This is a tool todirectly test both the Einstein Equivalence Principle and Charge, Parity,and Time inversion symmetry. The former can be tested by measuring thegravity interaction of matter and antimatter. Inversion symmetry can betested by comparing the spectroscopic properties of hydrogen and antihy-drogen. Both experiments require optically cooled antihydrogen. Underthe current trapping conditions, optical cooling could be performed withnanosecond long pulses of 0.1 µJ of Lyman-α radiation at a repetition rateof 10 Hz.The process to generate Lyman-α radiation uses two wavelengths (ωR⇒202.31 nm and ωT ⇒ 602.56 nm), which are mixed in a sum-differencescheme (ωLy−α = 2ωR−ωT ) with a two-photon resonance at (4s24p55p[1/2]o← 4s24p6(1S0) ). The source implemented produces 1.2 µW at the Lyman-αline and this was confirmed by performing spectroscopy of hydrogen. Thedesign, implementation and characterization of the source are discussed inthis dissertation.In the second part of the dissertation the implementation of the hy-drogen beam and its characterization are discussed. The atomic hydrogenis generated with a thermal effusive source and it is entrained by an ex-panding noble gas. This process generates a cold beam of hydrogen atoms.iiAbstractHydrogen is separated from the noble gas with a Zeeman bender that usesthe forces generated by the Zeeman shift of low field seeking states of hy-drogen and engineered magnetic field gradients. The hydrogen beam wascharacterized with a quadrupole mass spectrometer. The seed noble gasbeam was characterized by colliding it with ultra-cold rubidium atoms ina magneto-optical trap. The trapped atoms loss rate resulting from thesecollisions can be used to measure the density of the atomic beam. Thismeasurement demonstrates the potential of using magneto-optical trapsas absolute flux monitors.iiiPrefaceAll the work presented henceforth was conducted in Dr. Takamasa Mo-mose’s laboratory with the exception of the work presented in AppendixA, which was performed at CERN on the ALPHA experimental apparatus.The material presented in Chapters 2 and 3 is being prepared for publi-cation and Dr. Momose and I are the only authors. This work has also beenpresented in two conferences: Canadian Association of Physicist Congress2013 (CAP 2013) and the 11th International Conference on Low EnergyAntiproton Physics (LEAP 2013). Portions of Chapter 3 have been submit-ted as an invited article for the Physics in Canada Magazine (due to thepresentation winning a prize during the CAP 2013 Congress). Proceedingsof LEAP 2013 have been accepted for publication in the journal HyperfineInteractions (Springer). In both publications I am the first author. I wasresponsible for all the design, implementation and assembly of the exper-imental equipment and I performed all the calculations, data collection,and data analysis presented under direct supervision of Dr. Momose.All the hardware design and implementation of Chapter 4, besides thelast section (Collision scattering experiments) was done by myself with thehelp of two undergraduate students: Andrew Wong (undergraduate sum-mer student) and Polly Yu (undergraduate thesis student). Some of the datacollection and analysis was performed by Polly Yu under my direction andis included in this dissertation (specifically the time of flight data used forFig. 4.11). The remaining data collection and analysis was done by my-self. The implementation of the pulsed valve discussed in this chapter wasdone by Pavle Djuricanin, Research Technician, and Sajjad Haidar, Elec-tronics Technician, from Technical Services in the Chemistry Departmentas noted. The aspects of the valve development presented here representsivPrefaceall of my contribution.The last section of Chapter 4 (Collision scattering experiments) is a col-laboration effort with Dr. Kirk Madison. The magneto-optical trap (opticsand light source) was mainly implemented by members of Dr. Madison’slaboratory with the help of Dr. Matthias Strebel from Freiburg University.My contribution is limited to the hardware implementation (the vacuumsystem and mechanical systems) and the integration with the supersonicbeam. All the data collection and the analysis presented was done by my-self. This work is being prepared for publication and I am the first author.Finally, all the work presented in Appendix B, regarding the carbonnanotubes was performed in collaboration with an undergraduate summerstudent, Bill Wong, who I supervised. A publication is in progress fromthis work.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation for cooling antihydrogen . . . . . . . . . . . . . . 31.1.1 A brief introduction to antimatter and antihydrogen . 41.1.2 Comparison of the properties of hydrogen and anti-hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.3 Gravitational interaction between matter and anti-matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Lyman-α radiation and hydrogen . . . . . . . . . . . . . . . . 151.3 Dissertation overview . . . . . . . . . . . . . . . . . . . . . . 172 Coherent hydrogen Lyman-α source design . . . . . . . . . . . . 192.1 Nonlinear optics . . . . . . . . . . . . . . . . . . . . . . . . . 22viTable of Contents2.1.1 Nonlinear susceptibility . . . . . . . . . . . . . . . . . 252.1.2 Quantum-mechanical expression of the nonlinear Green’sfunction . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.3 Four wave mixing solution . . . . . . . . . . . . . . . 342.1.4 Phase matching factor . . . . . . . . . . . . . . . . . . 352.2 Source design and optimization . . . . . . . . . . . . . . . . . 372.2.1 Resonance-enhancement of FWM processes . . . . . . 382.2.2 Competing and parasitic processes . . . . . . . . . . 502.2.3 Phase matching . . . . . . . . . . . . . . . . . . . . . . 532.3 Third harmonic generation . . . . . . . . . . . . . . . . . . . 593 Source implementation and characterization . . . . . . . . . . . 633.1 Source implementation . . . . . . . . . . . . . . . . . . . . . . 643.2 Lyman-α source characterization . . . . . . . . . . . . . . . . 743.2.1 Effusive hydrogen source spectroscopy . . . . . . . . 743.3 Implementation of THG Lyman-α source . . . . . . . . . . . 843.3.1 Source implementation . . . . . . . . . . . . . . . . . 853.3.2 Acetone ionization detector . . . . . . . . . . . . . . . 854 Hydrogen beam implementation . . . . . . . . . . . . . . . . . . 904.1 Supersonic beams . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Beam implementation . . . . . . . . . . . . . . . . . . . . . . 934.2.1 Source chamber . . . . . . . . . . . . . . . . . . . . . . 974.2.2 Zeeman bender . . . . . . . . . . . . . . . . . . . . . . 1044.3 Hydrogen Lyman-α detection . . . . . . . . . . . . . . . . . . 1084.4 Collision scattering experiments . . . . . . . . . . . . . . . . 1165 Current work and conclusion . . . . . . . . . . . . . . . . . . . . . 1275.1 Lyman-α Source . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2 Alternative hydrogen source . . . . . . . . . . . . . . . . . . . 1305.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137viiTable of ContentsAppendicesA Characterization of solenoids in the ALPHA trap . . . . . . . . . 154B Fabrication of carbon nanotube electron sources . . . . . . . . . 166viiiList of Tables1.1 List of particle-antiparticle property comparisons . . . . . . . 52.1 Reported coherent Lyman-α sources via nonlinear processes 212.2 Kr isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.3 Index of refraction of Kr and Ar used to calculate the phasematching mixture . . . . . . . . . . . . . . . . . . . . . . . . . 582.4 Compilation of Lyman-α sources implemented by Third Har-monic Generation . . . . . . . . . . . . . . . . . . . . . . . . . 603.1 Optimization of phase matching parameters . . . . . . . . . . 724.1 Supersonic beam characteristics for noble gases . . . . . . . . 925.1 Onset energy of electrons and cross section for hydrogen dis-sociation processes . . . . . . . . . . . . . . . . . . . . . . . . 1325.2 Summary of measured Lyman-α source parameters . . . . . . 135A.1 ALPHA coils inductances and mutual inductances . . . . . . 155A.2 ALPHA coils parasitic capacitance and series resistance . . . 155ixList of Figures1.1 Hydrogen-antihydrogen energy diagram, Lyman-α coolingtransition and CPT violation tests . . . . . . . . . . . . . . . . 112.1 Energy diagrams of sum-frequency FWM resonant processestypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2 Resonance-enhanced four wave mixing in krypton . . . . . . 412.3 Grotrian diagram of krypton and electric dipole allowed tran-sitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4 Calculated third order susceptibility for RE-FWM process . . 452.5 Calculated third order susceptibility for sum-difference andsum-frequency processes . . . . . . . . . . . . . . . . . . . . . 462.6 Two-photon resonance shape . . . . . . . . . . . . . . . . . . . 482.7 Competing and parasitic nonlinear processes . . . . . . . . . 512.8 Calculated wavevector mismatch for the RE-FWM process inKr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.9 Calculated dispersion and refractive index for Ar and Kr . . . 562.10 Grotrian diagram for argon and parasitic processes . . . . . . 572.11 Non-resonant THG process on Kr for Lyman-α generation . . 613.1 Schematic of Lyman-α source implementation . . . . . . . . . 663.2 Lyman-α mixing cell, detection and vacuum system . . . . . 693.3 Lyman-α source characterization chamber and diffraction grat-ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4 Optimal configuration of phase matching parameters . . . . 733.5 Optimal laser configuration in mixing chamber . . . . . . . . 773.6 Hot hydrogen spectroscopy signal . . . . . . . . . . . . . . . . 79xList of Figures3.7 Photomultiplier Tube Response . . . . . . . . . . . . . . . . . 813.8 Line shape of 1s − 2p transition of hot hydrogen . . . . . . . . 833.9 Acetone ionization detector . . . . . . . . . . . . . . . . . . . 873.10 THG Lyman-α source power . . . . . . . . . . . . . . . . . . . 884.1 Schematic of the hydrogen beam main components . . . . . . 954.2 CAD drawing of the complete vacuum setup . . . . . . . . . . 964.3 Pictures of hydrogen beam components . . . . . . . . . . . . 974.4 Pulsed valve components . . . . . . . . . . . . . . . . . . . . . 994.5 Pulsed valve optimization . . . . . . . . . . . . . . . . . . . . 1014.6 Gas pulse profile and discharge plates . . . . . . . . . . . . . 1024.7 Hydrogen entrainment cell . . . . . . . . . . . . . . . . . . . . 1034.8 Zeeman bender simulation . . . . . . . . . . . . . . . . . . . . 1064.9 Quadrupole implementation. . . . . . . . . . . . . . . . . . . 1074.10 QMS of filtered hydrogen . . . . . . . . . . . . . . . . . . . . . 1104.11 QMS Time of flight of Ar/H beam . . . . . . . . . . . . . . . . 1124.12 Beam spectroscopy chamber . . . . . . . . . . . . . . . . . . . 1144.13 Hydrogen beam Doppler-free LIF . . . . . . . . . . . . . . . . 1154.14 Scattered light signal of a Rb MOT during Ar beam collision 1184.15 Schematic of collision experiment and MOT . . . . . . . . . . 1204.16 Photodiode signal of Ar, Kr and He collision . . . . . . . . . . 1244.17 Beam density and velocity distribution . . . . . . . . . . . . . 1255.1 Current THG Lyman-α development . . . . . . . . . . . . . . 1295.2 Integrated QMS signal for free radical beams as function ofdischarge voltage . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3 Current THG Lyman-α development . . . . . . . . . . . . . . 134A.1 ALPHA coils parameter measurement simulation circuit . . . 156A.2 ALPHA coils parameter measurement simulation results . . . 157A.3 ALPHA coils quench of S1 circuit . . . . . . . . . . . . . . . . 159A.4 ALPHA coils quench of S1 simulation results . . . . . . . . . 160A.5 ALPHA coils quench of MA circuit . . . . . . . . . . . . . . . 162A.6 ALPHA coils quench of MA simulation . . . . . . . . . . . . 163xiList of FiguresA.7 ALPHA coils quench of S1 simulation with suggested con-figuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.1 Carbon nanotube array on silicon . . . . . . . . . . . . . . . . 168B.2 Carbon nanotube array on stainless steel . . . . . . . . . . . . 169xiiList of AbbreviationsAD Antiproton DeceleratorALPHA Antimatter Laser Physics ApparatusASE Amplified Spontaneous EmissionATHENA Antihydrogen Production and Precision ExperimentBBO β-Barium BorateCERN Conseil Europen pour la Recherche NuclaireCPT Charge, Parity and TimeCR Capacitive-resistiveCVD Chemical Vapour DepositionLIF Laser Induced FluorescenceMOT Magneto-optical TrapQED Quantum ElectrodynamicsSM Standard ModelSME Standard Model ExtensionsEEP Einstein Equivalence PrincipleEUV Extreme UltravioletFWHM Full Width at Half MaximumFWM Four Wave MixingIGBT Insulated-gate Bipolar TransistorKDP Potassium Dihydrogen PhosphateLFS Low Field SeekingMA Mirror Coil AMWNT Multi-walled NanotubesNd:YAG Neodymium-doped Yttrium Aluminum GarnetPMT Photomultiplier TubeQMS Quadrupole Mass SpectrometerxiiiList of AbbreviationsRCL Resistive-capacitive-inductiveRE-FWM Resonance-enhanced FWMRIKEN-RAL RIKEN at Rutherford Appleton LaboratorySEM Scanning Electron MicroscopeSHG Second Harmonic GenerationSWNT Single Walled NanotubesS1 Solenoid 1TEM Transmission Electron MicroscopeTEMnm Transverse Electro-magnetic Mode (nm)THG Third Harmonic GenerationTPA Two-photon AbsorptionVUV Vacuum UltravioletXUV Extreme Ultraviolet (150 nm - 30 nm)xivList of Symbolsχ(n) n Order susceptibility~P NL Nonlinear polarizationrα Transition dipole matrix elementG(n) Nonlinear Green’s functionΓ Electric transition damping constantΓBkg MOT atom loss rate constantΓR Homogeneous Lorentzian linewidth of resonant stateωLy−α Hydrogen Lyman-α line angular frequencyωR Two-photon resonant laser angular frequencyωT Tuneable laser angular frequencyF Phase matching constant or atom flux∆k Phase differencefab Oscillator strengthSab Line strengthS(ω) Plasma dispersion functionAab Transition probabilityb Confocal lengthC Vector mismatch per atomNa Density number of species aL Nonlinear medium interaction lengthf (υ) Maxwell-Boltzman distribution∆ω Doppler broadeningη Efficiency of nonlinear processxvAcknowledgementsI would like to thank Prof. Takamasa Momose for providing me with theopportunity to get involved in this project. I would also like to thank Prof.Andrea Damascelli for his mentorship and advice. Thank you to Dr. KirkMadison who has made the collision experiments possible. Finally, I wouldlike to thank Prof. Jeff Young and Dr. Roman Krems for sharing theirexpertise.xviChapter 1IntroductionHydrogen Lyman-α radiation (the 1s − 2p transition line of the hydrogenatom at 121.56698 nm or 2.466068 x 1015 Hz) is important because it al-lows for the excitation and detection of ground-state hydrogen atoms. Thetransition is the strongest resonance in hydrogen and has been mostly usedin physical cosmology to map the density and location of hydrogen clouds(absorbers) and the location of quasars (sources) in the universe. It is alsoused to calculate redshifts and to extract information about universe ex-pansion and the cosmological constant. Additionally, direct photometry ofthe spectral line has been used to study the Sun and Mars atmospheres[1].Excitation with Lyman-α radiation has also been used to measure hydro-gen density [2] and as diagnostic tool for fusion plasmas. A very importantapplication of coherent Lyman-α radiation is optical cooling of hydrogen.Laser cooling, or more precisely, Doppler cooling of magnetically trappedhydrogen was first reported by Walraven et al in 1993 [3, 4]. No other hy-drogen optical cooling experiment has been reported since then.The trapping of antihydrogen (the antimatter counterpart of atomic hy-drogen), recently reported by the ALPHA (Antihydrogen Laser PHysicsApparatus) collaboration at CERN (Conseil Europen pour la RechercheNuclaire or European Council for Nuclear Research) facility, has revivedinterest in Lyman-α sources [5]. The ALPHA collaboration is an interna-tional group of researchers (which the author has recently become partof) whose main purpose is to achieve stable trapping of antihydrogen tostudy fundamental symmetries. A number of measurements have beenperformed or planned for the trapped antihydrogen at ALPHA such as mi-crowave (hyperfine splitting), one-photon (1s−2p) and two-photon (1s−2s)spectroscopy, and gravity interferometry. The ultimate goal of these exper-1Chapter 1. Introductioniments is to test Charge, Parity, and Time (CPT) inversion symmetry (ormatter-antimatter symmetry), and antimatter-gravity interactions.One of the difficulties in performing these measurements of the mag-netically trapped antihydrogen, to the precision required for a fundamen-tal symmetry or gravity interaction test, is its relatively high kinetic energy.Energetic antihydrogen is difficult to hold in a stable position in a trap andthis leads to line broadening from the Doppler effect and from Zeemanshifts. The high kinetic energy of the antihydrogen also makes interferom-etry measurements of antimatter-matter gravity interactions difficult dueto the poor statistical significance achievable. In addition, the antihydro-gen production rate is still very low which results in very low-density sam-ples making traditional evaporative cooling impossible. For these reasonsoptical cooling of the trapped antihydrogen, using the 1s − 2p one-photon(Lyman-α) transition, is essential. The purpose of the current work is todevelop a coherent Lyman-α source capable of eventually performing op-tical cooling of the trapped antihydrogen in the ALPHA apparatus, and theinitial efforts towards this goal are described in this dissertation.In this introductory chapter I first describe the motivation for the cur-rent undertaking (i.e. the development of a source capable of optical cool-ing trapped antihydrogen) and provide a short description of the experi-ments envisioned. Some background on antimatter and the theory behindantihydrogen and fundamental symmetries research is given to substanti-ate the motivation. This is the only discussion on antimatter in this disser-tation. Secondly, hydrogen and its Lyman-α transition are discussed withemphasis on the details important for the experiment. Thirdly, the overallstructure of this dissertation is described.The remainder of the dissertation will move away from antihydrogenresearch and will focus on the design, implementation and characteriza-tion of the Lyman-α source. The Lyman-α radiation is generated througha nonlinear optical process and the theory needed to understand how thesource works is described at the beginning of Chapter 2. The rest of Chap-ter 2 and Chapter 3 will go through the design, implementation and char-acterization of the source. Chapter 4 will describe the implementation and21.1. Motivation for cooling antihydrogencharacterization of a supersonically expanded beam of hydrogen atoms. Animportant part of the beam characterization is done through elastic colli-sion scattering experiments of the beam atoms and ultra-cold rubidiumatoms on a magneto-optical trap (MOT) (used as an absolute pressure sen-sor device). These experiments are described at the end of Chapter 4 andwith this I conclude the dissertation.1.1 Motivation for cooling antihydrogenAtomic hydrogen has been one of the primary research vehicles for atomicphysics that has led to the foundations of quantum mechanics and quan-tum electrodynamics. It has been the continuous development of the hy-drogen model and metrology techniques that has made it the most pre-cisely studied system in physics. In fact, the 1s − 2s spectroscopy of hydro-gen is the only measurement with a relative precision below 5 x 10−15 with-out laser cooling [6]. Attempts to explain its spectra have triggered the de-velopment of new theories continuously changing the physics paradigms.The Balmer formula, Balmer’s attempt to explain the hydrogen emissionlines now bearing his name, provided the foundation of Rydberg’s work.The Rydberg’s formula (validated by the discovery of the hydrogen Lymanseries of emission lines) established the concepts of wave number and prin-cipal quantum number. Bohr’s model of the hydrogen atom postulatedthe foundations of quantum mechanics. Sommerfield and later Dirac in-cluded relativistic effects to the existing hydrogen models perfecting themand introducing important physics (such as the Sommerfield’s fine struc-ture constant and the antimatter solution of the Dirac equation). Lamb andRetherford measurements on hydrogen spectra triggered the developmentof quantum electrodynamics (QED) (the quantum theory of electromag-netic fields) and part of the standard model (SM) of particle physics (thecurrently accepted theory of subatomic particle interactions) leading to anexplanation of what is known as the Lamb shift of energy levels. More re-cently measurements of the 1s − 2s transition in hydrogen have been usedto test physics beyond the standard model and also what is known as the31.1. Motivation for cooling antihydrogenstandard model extensions (SME) introduced by Kostelecky´ [7]. The for-mer test set limits on the possible temporal variation of the fine structureconstant (one of the parameters that determines the hydrogen energy lev-els) [6, 8] and the latter tests (SME) are searching for Lorentz and CPTviolations using two measurements separated by half of earth solar orbitperiod [9]. The trapping of antihydrogen starts a new chapter on hydrogenresearch. With stable trapped cool antihydrogen several new experimen-tal possibilities become realizable. Two experiments that come to mindare a comparison of spectroscopic properties of hydrogen and antihydro-gen (CPT Invariance) and the measurement of the gravitational interactionbetween matter and antimatter.1.1.1 A brief introduction to antimatter and antihydrogenAntihydrogen is the bound state of an antiproton and a positron (antielec-tron). It is considered the simplest anti-atomic system and the antimatteranalog of hydrogen. However, this definition is only suitable to specialistsas it does not directly tell us anything about the system that we want tostudy. What exactly do we mean with the term antimatter? The problemdefining antimatter comes from the fact that the term matter is perhapspoorly defined in science. A starting point is defining matter as anythingcomposed of particles that have rest mass and occupy classical volume.However, quarks and leptons are considered point particles with no effec-tive volume. Nevertheless, volume can be argued to be the result of thePauli exclusion principle. In addition, a common use of the term mat-ter refers to the energy-momentum tensor that is the source of the grav-itational field in general relativity. This is a common definition in cos-mology for instance. The problem with this definition is electromagneticfields (photons) contribute to the total energy-momentum of the system(i.e. mass) and therefore add matter to it (isolated photons are generallynot considered matter as they have zero rest mass). For the sake of discus-sion let us define ordinary matter as anything composed of quarks and lep-tons. Antimatter then is anything composed of antiquarks and antileptons41.1. Motivation for cooling antihydrogenwhich are particles with the exact rest mass and spin of their correspond-ing quarks and leptons but with opposite sign charge. This is known asthe Feynman-Stueckelberg interpretation of antiparticles. These particleshave been observed and their properties have been measured to various de-grees of accuracy. Table 1.1 shows a compendium of particle-antiparticlecomparisons. A summary and discussion about these comparisons can befound in Ref. [10]. Ref. [11] provides an extensive list of comparisonsshowing no significant difference.Table 1.1: List of particle-antiparticle property comparisons and therelative precision of the measurement. The quantities measured for eachparticle pair are identical within the uncertainty of the measurement. Therelative precision is given by the difference between the measurementsof the particle-antiparticle pair normalized by one of the measurementsitself ((xparticle − xantiparticle)/xparticie). The uncertainty shown representsthe precision of a CPT test of the specific property. If the uncertainty isnot shown it was not specifically calculated in the reference.Particle Pair Property Relative Precision Referencee−e+mass (m) 8× 10−9 [12]charge (q/m) – [13]g-factor (g2) 2× 10−9 [14]ppmass (m) 2× 10−9 [15]charge (q/m) 9× 10−11 [15]magnetic moment (µ) 3× 10−3 [16]KoKomass (m) 1× 10−18 [17]decay width (τ) – [17]µ−µ+ g-factor (g2) 7× 10−8 [11]In addition to experimental confirmation, antimatter is a consequenceof quantum mechanics and special relativity. As a matter of fact, beforeantimatter particles were detected they were theoretically predicted. TheDirac equation predicted the positron [18, 19] a couple of years before An-derson reported its discovery on a cloud chamber in 1932 [20]. The an-51.1. Motivation for cooling antihydrogentiproton (composed of three antiquarks) was observed much later in 1955at the Bevatron accelerator in Berkeley [21] . The positron solution of theDirac equation (the relativistic Schro¨dinger equation) is a consequence ofspecial relativity as it contains a very important symmetry that allows theexistence of antimatter. This symmetry, strong reflection, involves the re-flection of the four coordinates (spatial and time) through the origin. Thisreflection applied on Maxwell’s equations changes the sign of the elec-tric charge [22, 23]. For instance, reversing the spatial coordinates of anelectron moving along an electric field line reverses the electric field andthe magnetic field. If time is subsequently reversed (the particle retracesits path), an observer unaware of the symmetry operations would inferthat the particle is of positive charge. In Ref. [22] Feynman shows thata positron emerges from an electron when proper time is reversed. Thequantum mechanical equivalent to strong reflection is called the CPT the-orem. In quantum theory the equations are complex and CPT inversiontakes this into account by combining strong reflection with complex con-jugation. Charge conjugation C is an operation that conjugates the com-plex wave function changing the sign of the charge in the particle. ParityP is an operation that inverts the space coordinates. Time reversal T isan operation that interchanges the forward with the backward light cone(time inversion). According to this theory antihydrogen is the CPT sym-metric reflection of hydrogen, and as such, all the physics related to hydro-gen should be identical to that of antihydrogen. More precisely, the CPTtheorem predicts that the positron and antiproton that compose the an-tihydrogen atom should have identical mass and total lifetime, but equaland opposite charge and magnetic moment as the electron and proton thatcompose the hydrogen atom. In fact, some of the initial measurements per-formed on antihydrogen such as a resonant quantum transition betweenhyperfine states confirm this theory [24].Antihydrogen was first synthesized at CERN [25] in 1995 and Fermilab[26] in 1998 through collisions of relativistic antiprotons with a nucleus. Insuch interactions, some of the antiproton’s energy could be converted to anelectron-positron pair and some of these positrons bound to an antiproton61.1. Motivation for cooling antihydrogencreating relativistic antihydrogen (i.e. moving at relativistic speeds). Thefirst non-relativistic antihydrogen atoms (with energies of the order of 1eV) were synthesized by the ATHENA experiment in 2002 at the antipro-ton decelerator (AD) at CERN [27]. These atoms were still too energeticto trap and escaped the apparatus in a few microseconds. The ALPHAcollaboration, formed as a successor to ATHENA, observed the first signsof trapped antihydrogen in 2009 [28] followed by a definitive identifica-tion in 2010 [5]. Currently ALPHA is able to trap antihydrogen for morethan 15 minutes [29]. This achievement presents a unique opportunity tofurther investigate the properties of antimatter, and to precisely comparematter and antimatter. The trapping of stable antihydrogen opens the doorto laser cooling experiments, and with it, the possibility of high precisionmeasurements becomes realizable. This is why the source implemented inthis thesis could play a vital role.1.1.2 Comparison of the properties of hydrogen andantihydrogenNone of the physical properties compared on matter/antimatter systemshave revealed significant statistical differences (see Table 1.1) and CPT in-version symmetry has been validated in every experiment. However, an-tihydrogen is the fist antimatter atom (anti-atom) ever created and thisopens the door for a range of new possible CPT inversion symmetry tests.For instance, any small difference between the properties of electrons andpositrons or protons and antiprotons would potentially result in differ-ences between electronic (or positronic) transitions of hydrogen and an-tihydrogen. This is one of the motivations for measuring the electronictransitions of antihydrogen such as the (1s − 2s) or Hyperfine transition.To measure new physics effects at short distances, colliders usually re-quire collisions with energy (Ec) of the same order of magnitude (Λsd) asthe scale of the physical phenomena studied: Ec ∼Λsd . Precision measure-ments at low energies access short distance scales by measuring energy71.1. Motivation for cooling antihydrogenshifts that typically have the form [30]:∆E ∼mn+1Λnsd, (1.1)where n depends on the effective field theory and is typically n > 0, andm is the characteristic energy of the system. CPT violations, as will beexplained below, are expected to emerge from Planck scale (very short dis-tance) physics ( Λsd ∼ 1019 GeV). As an example, if n = 1 andm ∼GeV thenthe energy shifts are in the order of 10−19 GeV and that would be the re-quired resolution to observe a CPT violation. Following this argument, forexample, ALPHA calculated the frequency precision required for (1s − 2s)spectroscopy of antihydrogen to be 103 Hz and for Hyperfine splitting to be10−4 Hz [10]. However, it can be argued that using the framework of a fieldtheory, as above, to calculate an energy shift caused by CPT violation is un-founded since there is no formal accepted field theory that includes CPTviolation. In other words, using a field theory to quantify a figure of meritfor testing a CPT violation is inconsistent with the field theory since thevalidity of the field theory relies on the validity of the CPT theorem. Theproblem on testing CPT relies on it being one of the fundamental pillars ofthe current physics paradigm. The CPT theorem of Luders and Pauli [31]states that any local (point-particle), Lorentz invariant, and causal field the-ory will be symmetric under the combined transformation C, P and T. Thetheorem is one of the foundations of quantum field theory and the standardmodel of particle physics. Nevertheless, a CPT violating theory (i.e. Kost-elecky´’s SME) could account for a quantum theory of gravity (the stringtheory which the SME is based on) and for the observed Baryon asymmetry(the observed matter antimatter asymmetry in the universe) [32].The currently accepted quantum field theory and standard model ofparticle physics incorporate the electromagnetic, strong and weak interac-tions. Gravity, on the other hand, cannot be incorporated. This has lead tothe development of grand unified theories or theories of quantum gravity.In general, these theories break Lorentz invariance (and CPT symmetry).In fact, a theory of gravity (specifically string theory) served as a motiva-81.1. Motivation for cooling antihydrogention to devise a CPT and Lorentz violating extension of the SM compatiblewith most of the SM characteristics [33–35]. String theory rejects the ideaof point-particles and replaces them with strings with dimensions of ap-proximately one Planck length (1 / mP lanck). As a result particle interactionsare non-local. By extension space-time takes a non-local character and, inthe theory, interactions in distances less than the Planck length do not ex-ist. Quantum mechanical predictions on smaller scales are ignored. Asexplained by Kostelecky´, the non-locality of string theory leads to CPT vi-olation and Lorentz symmetry breaking [34]. However, all the elementaryparticle processes currently accessible occur over distances much greaterthan the strings. The observation of these processes is equivalent to macro-scopic observations of the underlying atomic structure of matter. Macro-scopic phenomena can be approximated by physics that does not necessar-ily take into account the quantum structure of matter, but is the result ofquantum many-body interactions. Similarly, Planck-scale physics may leadto some effective continuum quantum field theory that is approximately lo-cal [36] and leads to the standard model of particle physics. Following thistrend, Kostelecky´ described various types of interactions that would revealCPT and Lorentz violations within the standard model extension [7, 37].Some of the interactions can be studied in the hydrogen/antihydrogen sys-tem and a short review can be found in the review by Holzscheiter et al.[38]. In summary, it appears that ideal candidates for Lorentz and CPTviolation tests are (see Fig. 1.1 for a graphical explanation of the tests):• Comparison of the 1s − 2s two-photon transition line of mixed spinstates in antihydrogen and hydrogen confined in a magnetic trap withan axial bias. This transition line has a very narrow intrinsic linewidth (1.3 Hz) and has been precisely measured in hydrogen (the1s − 2s transition has been measured as 2 466 061 413 187.103(46)kHz[1.9×10−14 relative uncertainty], using Doppler-free two-photon spec-troscopy [8].• Comparison of the ground state hyperfine transitions involving spin-91.1. Motivation for cooling antihydrogenflip in antihydrogen and hydrogen confined in a magnetic trap withan axial bias field. The ground-state zero field hyperfine splittinghas been measured as 1 420 405 751.7662(30) Hz [2.1×10−12 relativeuncertainty] [39].Both of these tests could potentially be performed with the antihydro-gen trapped by ALPHA. The antihydrogen in the trap has been shown tohave a lifetime longer than 15 minutes [29] and is within an axial biasfield. Also a new trap (ALPHA2) scheduled to start operating in 2014has been designed with laser and microwave access ports. However, Thetemperature of the trapped antihydrogen is still too high to perform thehigh-precision measurements required to see the Lorentz and CPT viola-tion effects. The trapped antihydrogen translational energy is less than 0.5K (× kB) and is free to roam in a volume of approximately 4 cm2 × 30 cm[29]. The non-uniformity of the magnetic field and high kinetic energy ofthe antihydrogen leads to large Zeeman broadening. This is currently themain limitation on microwave and laser spectroscopy [24, 41]. A CPT andLorentz violation test using the hyperfine transition, for instance, wouldrequire a precision of 1 / 3 mHz [38] which is currently beyond the capabil-ity of ALPHA (currently in the order of MHz) [24].This is where laser cooling of antihydrogen could play a vital role. Lasercooling would reduce the kinetic energy of the antihydrogen localizing thesample in the centre of the trap where a more uniform field is achieved.This will dramatically reduce Zeeman effects and also the second orderDoppler broadening for two photon spectroscopy. In addition, having alocalized cold sample opens the door to shelving strategies where one atomis recycled and the transition is repeated many times. This technique hasbeen showed to increase the precision of the measurement.Optical cooling (Doppler cooling 1) of magnetically trapped hydrogenwas demonstrated in 1993 by Walraven et al. [4] using a pulsed Lyman-αsource. Starting with a spin-up polarized hydrogen gas with a density n ≥1011 cm−3, at 80 mK (pre-cooled via evaporative cooling) they were able1This is in contrast to laser induced evaporation which was also used by Walraven tocool hydrogen below recoil temperatures but is not what is being referred to in this work101.1. Motivation for cooling antihydrogenCooling TransitionsMF=0MF=02S1/22P1/22P3/22S1/282 258.9191133 cm-182 258.9543992821 cm-182 259.2850014 cm-11 420 405 751.768 Hz59 169 500 Hz177 556 838.2 Hz23 651 600 HzBCPT2CPT1σ1 pi12P Ecg0.204 cm-1Hydrogen-AntihydrogenLyman-α 2 466 068 GHzHyperfine TransitionsHyperfine SplittingTwo-photon Transitionpi1 σ1F=1F=1F=1F=0F=0F=0F=1F=2Dirac+ Lamb Shift + Fine Structuren = 1n = 2Figure 1.1: Energy diagram of hydrogen and antihydrogen and the Lyman-α transition for cooling. Cooling can be accomplished with the 2s1/2−2 p3/2transitions (σ1 and pi1). The two possible CPT violation tests: CPT1 (hyper-fine splitting involving spin flip) and CPT2 (1s − 2s two-photon transitionof mixed spin states) are shown. CPT1 involves spin flip between the trap-pable states F=1 (M=0 → M=+1). It is possible to measure it using thetransitions (σ1 and pi1) to non-trappable states. The two-photon transitiontest, CPT2, is between M = +1 states and, therefore, more difficult to mea-sure. The energy levels are based on Ref. [40].111.1. Motivation for cooling antihydrogento cool the hydrogen to 8 mK in 15 min while increasing the density by afactor of 16. The Lyman-α source delivered 3×107 photons per pulse at 50Hz and 10 nsec pulses. This is equivalent to a power of 4.9 × 10−11 J at 50Hz = 2.45× 10−9 W.The linewidth of the source was not characterized but it is inferred thatit was ∼100 MHz. In this experiment, a single beam was used and the cool-ing relied on collisional mixing of the degrees of freedom. Since this work,no other optical cooling experiment of hydrogen has been reported. Anti-hydrogen could in principle be cooled using the same technique but thisis experimentally more challenging. First of all, the density of antihydro-gen in the ALPHA trap is very low (approximately one atom in the trapper experimental cycle) and the atoms have a higher kinetic energy thanthe hydrogen atoms of Walraven’s experiment. In addition, due to exper-imental requirements for the creation and trapping of antihydrogen, theALPHA trap only allows for laser access on one axis. These additional con-straints impose higher power requirements on the light source. A recentcalculation indicates that it would be possible to cool down the antihydro-gen in the ALPHA trap to ∼ 20 mK with a pulsed Lyman-α source [42].The laser cooling simulations were performed assuming a pulsed Lyman-α source with 0.1 µJ of energy per pulse, 100 MHz bandwidth, 10 nsectemporal width and 10 Hz repetition rate. This indicates that the requiredpower level of the laser is 1.0 µW (in a 2 cm diameter beam) or 6.2×1011photons / pulse. As mentioned before, the current ALPHA trap has laseraccess along one axis (longitudinal z-axis), allowing only 1D Doppler cool-ing. However, 3D cooling of antihydrogen would be possible by controllingthe dynamic coupling between the x − y and z degrees of freedom in themagnetic trap. The coupling can be enhanced by the use of non-harmonicmagnetic fields in both x − y and z directions. The current ALPHA traphas five solenoid coils along the z axis to produce the nonlinearity. Thetrap configuration and the characterization of the solenoids is described inAppendix A. In principle, 3D cooling of the magnetically trapped antihy-drogen in the ALPHA trap is possible with the current technology providedthe required laser light is available. This is the motivation for undertaking121.1. Motivation for cooling antihydrogenthe developing of a hydrogen Lyman-α source. The cooling calculation [42]is the basis of the source design discussed in this dissertation. Althoughsome of the specific laser parameters used in the simulation to accomplishthe cooling can be changed, and will most likely be refined in the future,as an initial step the source power requirement needs to be realized. Thatis the focus of our source implementation.1.1.3 Gravitational interaction between matter and antimatterAnother important experiment that requires laser cooled antihydrogen isthe measurement of the gravitational interaction of matter-antimatter sys-tems. In addition to CPT violation, quantum field gravity theories alsopredict that the gravitational acceleration of matter could be different fromthat of antimatter [43]. In string theory, for instance, the origin of the asym-metry can be tracked down to the lack of point-like interactions, which vio-late Lorentz symmetry and by extension, what is known as Einstein equiv-alence principle (EEP) [44]. EEP is the basis of general relativity. EEP isEinstein’s extension of what is known as the weak equivalence principle(WEP) and states that all the laws of special relativity hold locally in in-ertial frames. WEP states that freely falling bodies of the same mass andmass distribution should follow the same trajectory [45]. EEP stipulatesthe gravity interaction of matter-matter and matter-antimatter should beidentical. A violation of this symmetry will not necessarily be a proof ofany quantum field theory of gravity or even be a proof of CPT violation,as discussed by Nieto et al [38]. It would, however, seriously reduce ourconfidence in the theory of general relativity, and that is perhaps the mainmotivation to perform this experiment. As discussed by Roberts et al [46]the evidence of and acceleration on the expansion of the universe pointsout that either the composition of the universe or the theory describing theparticle interactions needs to be revisited. Antihydrogen, given its chargeneutrality, is a more suitable candidate for antimatter-matter gravity teststhan charged antiparticles such as antiprotons or positrons. In fact the AL-PHA collaboration has accepted a detailed proposal for gravity interferom-131.1. Motivation for cooling antihydrogenetry measurements [47]. Light-pulse matter wave interferometers split andrecombine matter waves using pulses of light. In the interim (between thepulses) the matter waves can follow different paths. During the recombina-tion, matter waves can interfere constructively or destructively dependingon their phase. This is reflected on the probability of finding the atom.The phase of the matter waves is shifted during the propagation due toits interactions with different fields. By designing the environment of thematter waves path it is possible to measure the interaction with very highaccuracy. Standing waves of nearly resonant laser light are usually used asdiffraction gratings in these interferometers. However, in the proposal forantihydrogen gravity measurement at ALPHA the authors acknowledgedthe difficulty in using these lasers. The problem is that developing a hy-drogen Lyman-α source that provides the required power is not an easytask. In fact, currently there are no cw or pulsed Lyman-α sources power-ful enough [48]. In their design they use a far-detuned high-energy pulsedlaser [47]. A requirement of this scheme is, however, that the antihydrogenis laser cooled to about 20 mK (equivalent to a thermal velocity of approx-imately 10 m/s) with a pulsed Lyman-α source. The laser cooling strategyis identical to the one discussed in the previous section (pseudo-3D coolingwith one laser beam only). Using this temperature they have estimated thatthe relative precision of the experiment, given various factors such as: thetrap magnetic field variations, retroreflective mirror vibration, sensitivityof detectors, background particle noise and number of available atoms, tobe 10−2. This estimation was performed with 250 detected atoms / month.However, upgrades at CERN are expected to increase the availability ofatoms to 3×103 atoms /month increasing the relative precision to 2×10−6.Therefore, the development of a Lyman-α source system with a power levelof 1.0 µW is key for the ongoing ALPHA experiments, which as mentionedbefore is the motivation of the present work.141.2. Lyman-α radiation and hydrogen1.2 Lyman-α radiation and hydrogenBefore moving into the details of the source design, some details of hydro-gen Lyman-α radiation and the hydrogen atom are discussed. The emissionspectrum of hydrogen is divided into spectral series given by the empiricalRydberg formula.1λ= R∞H( 1n2−1n′2)R∞H = 1.0968× 107m−1 (1.2)The hydrogen Lyman-α line is the first emission line in the series re-sulting from an electron transitioning from the excited states n′ > 1 to theground state n = 1 where n is the principal quantum number. The seriesextends from the Lyman-α line at 121.56 nm for n′ = 2 to 91.2 nm forn′ =∞. The discovery of the series by Theodore Lyman between 1906 and1916 was an important experimental validation of the predictions by theRydberg formula and Bohr’s model of the hydrogen atom. It is also marksthe beginning of the exploration of the extreme ultraviolet (EUV) band ofthe spectrumLyman-α radiation is usually considered the upper limit in wavelengthof the EUV that extends down to 10 nm, the x-ray threshold. Photons withhigher energy than Lyman-α radiation (> 10.2 eV) are considered ionizingradiation. Radiation below 200 nm is absorbed by air and propagationonly occurs in vacuum, hence this part of the spectrum down to the x-raythreshold is called vacuum ultraviolet (VUV). In addition to EUV and VUV,the part of the spectrum from about 150 nm to 30 nm is usually refereedas the XUV region and is characterized by the interaction of photons withthe chemical valence electrons of matter. Higher energy photons interactmainly with inner shell electrons and nuclei.The discovery of the VUV can be attributed to Viktor Schumann [49, 50]who made three very important contributions (some of which we still usetoday for the work in this dissertation). First, he used fluorite optics insteadof glass, which allowed him to record 182 nm. Second, he developed a spe-cial photographic plate without the commonly used gelatine [51], whichhe showed to be a strong UV absorber, and increased the detector sen-151.2. Lyman-α radiation and hydrogensitivity into the XUV. Third, he constructed vacuum spectrographs andshowed that air absorbs wavelength below 200 nm [50]. These techniquesallowed him to record the spectrum in the region from 185 nm to 123 nm,the Schumann Region. From 1904 to 1906 Theodore Lyman extended thespectrum beyond the Schumann region to 103 nm [52]. This was possibleby a significant change in instrumentation. He changed transmitting op-tics to reflective optics. Instead of using a fluorite prism he used a concaveruled grating. He also established the limit of transparency of fluorite to be126 nm[53]. Measurements of wavelength were made and checked by twomethods. One was the method of two slits, devised by Lyman, by whichthe spectrum could be displaced a known distance by utilizing a secondslit some distance from the first. The second method was by comparingthe second-order spectrum of known lines of longer wavelength with thefirst-order spectrum of the unknown ultraviolet lines. Using these meth-ods he was able to calibrate and extend the hydrogen spectrum to 95 nmwith 0.1 A resolution. By 1916 he had discovered the series of hydrogenlines now bearing his name. The light in Lymans experiment was generatedelectrically in a discharge tube of quartz provided with tungsten electrodesand the lines matched the ones predicted by Rydberg in 1890.The hydrogen Lyman-α line as first measured by Lyman was 121.6 nm(2,460,000 GHz) [53]. Since then several corrections to the theory and im-provements in experimental techniques have increased the accuracy of theline. We now know that the line actually splits into multiple lines due tothe fine and hyperfine structure. However, the fine structure doublet, withthe 2p level splitting into j = 1/2 and j = 3/2 as seen in Fig. 1.1, is whatis referred as the Lyman-α line. This separation is what the best astro-nomical observations can discern due to the line broadening caused by theparticular physical processes in solar and stellar atmospheres. The transi-tion separation is, according to the most up to date accepted experimentalmeasurements, 10,969.05 MHz [40]. It can be used to infer astronomicalprocesses and to calculate the amount of hydrogen isotopes (deuterium andtritium) on plasmas. In general the Lyman-α energy value is representedby a single number. The center-of-gravity energy is the average of the fine161.3. Dissertation overviewstructure components weighted by their statistical weights (2j + 1). For the2p level Ecg is 82,259.11089 cm−1 [40] . The wavenumber of the Lyman-αline derived from experimental fine structure intervals weighted by calcu-lated line strengths is 82,259.16(14) cm−1 or 2,466,068(3) GHz [40]. Thisis what we refer to as Lyman-α line in this work λLy−α = 121.56698 nm.1.3 Dissertation overviewThis dissertation describes the design, implementation and characteriza-tion of a coherent Lyman-α source prototype (some times refereed as laserin this dissertation) for cooling antihydrogen. The main purpose of thisintroductory Chapter, besides providing some general background, wasto describe the motivation behind this work. The key point is that lasercooling of the antihydrogen trapped in the ALPHA apparatus is essentialfor the realization of high precision experiments such as CPT violationand antimatter-matter gravity interaction measurements. The remainderof this dissertation moves away from antimatter and fundamental physicsresearch and concentrates on the project objective (i.e. making the source).The first section of Chapter 2 describes, in detail, the theory of non-linear optics. Although, there are several books and reviews on the field,the hope is that the chapter is self-sufficient. The specialist can probablyskip this section. However, the emphasis is on the theory aspects used inpractice for this specific project. The chapter’s remaining sections describethe design and optimization of the source. Chapter 3 is concerned withthe implementation and power measurements of the prototype source. Itis important to realize that this source prototype implementation is thefirst step towards a much larger and complex project. Integration of thesource with the experimental apparatus at CERN is, perhaps, more techni-cally challenging than this prototype development. It is desirable to havethe most robust and reliable source possible before initiating integration.With this in mind, a second source was implemented for comparison. Thisis briefly described in Chapter 3, as well, along with the evaluation of bothsources. Experimental confirmation that the source is resonant with the171.3. Dissertation overviewLyman-α transition of hydrogen is described in Chapters 3 and 4.Spectroscopy of a supersonically expanded beam of hydrogen was per-formed to confirm that the Lyman-α source is able to induce resonant tran-sitions in cold hydrogen. This experiment required the implementation ofthe hydrogen beam and its characterization. Chapter 4 describes this de-velopment. Some of the highlights of that chapter are the introduction ofa ”Zeeman bender” as a hydrogen filter and a rubidium-MOT as an atomicbeam density sensor. The hydrogen bender uses magnetic field gradientsand Zeeman shifts to manipulate the hydrogen beam, and the Rb-MOTuses the elastic scattering cross section of the Rb-beam atoms as an abso-lute beam flux meter.The final chapter describes the current work taking place at CERN andthe source post-prototype development. In addition, the experiment tak-ing place with the hydrogen beam and Rb-MOT is described along withsome of the attempts done to produce an alternative hydrogen source. Thischapter is finalized with a summary and conclusion.18Chapter 2Coherent hydrogen Lyman-αsource designIn the previous chapter I discussed the scientific motivation to develop ahydrogen Lyman-α source capable of optically cooling trapped antihydro-gen. The minimum requirements for such a source were given based onrealistic simulations and on the constraints imposed by a technologicallyachievable experiment in the near term. The most important requirementis power, which is estimated to be 1.0 µW (in a 2 cm diameter beam). Inaddition pulse duration, frequency and linewidth were specified for a par-ticular cooling scheme. The linewidth used in the simulation was 100 MHz.In addition to these requirements there are several technical specificationsthat need to be considered. Among these, the most important are the sizeand the stability of the source system. The system needs to fit into an ion-izing radiation protected area of the antimatter experiment and operatefor several hours without needing direct human intervention (needs to bestable). The antiproton decelerator, where the ALPHA experiment is lo-cated, receives relativistic antiproton beams that cause gamma rays duringcollisions. The area where the antihydrogen trap is located is a radiationcontrolled area and personal and equipment need to be protected. Thesource system can be placed in a protected area approximately 5 metersaway from the trap and across a concrete wall. This area, however, canonly accommodate human operators for short periods of time. Operatorsare located approximately 10 meters away from the source in a differentcompartment. This is the reason the system needs to be compact and verystable.Coherent Lyman-α radiation has been generated previously and Table19Chapter 2. Coherent hydrogen Lyman-α source design2.1 shows a list of some sources that have been reported. In addition,Lyman-α can be easily generated with a free electron laser. These devicescan produce coherent photons of tuneable energy by accelerating relativis-tic free electron beams. Furthermore, a continuous wave (cw) Lyman-αsource have been developed and is, in theory, available for an antihydro-gen experiment. However, these sources do not meet one or several ofthe experimental requirements. Free electron lasers are highly complexmachines that require large facilities called synchrotrons. Synchrotronsaround the world can currently generate a maximum flux of about ∼ 1012photons/sec at the Lyman-α spectrum. This corresponds to a power of ap-proximately 2 µW which according to simulations [42] would be sufficientfor cooling antihydrogen in the ALPHA trap. However, synchrotron radia-tion linewidths are extremely broad (∼ 10−3 eV) for laser cooling. But evenfor the sole purpose of detecting antihydrogen, for which the power avail-able from synchrotron radiation will certainly suffice and their linewidthis not a limitation, CERN is not equipped with a synchrotron light sourcefacility. There are reports of high power compact free electron laser devel-opments for medical and defence purposes but they are not available [54].The cw Lyman-α source developed by Walz et al [55, 56] is perhaps themost advanced source available for laboratory use at the moment. It hasbeen developed with the purpose of cooling antihydrogen and it has sev-eral advantages over a pulsed source. However, this system is extremelycomplex requiring several lasers. Implementing this laser in the form thatit has been reported will require more space that is currently available andit is not clear that it could be developed to operate in an unmanned form.Furthermore, there are doubts that the power reported will be sufficient forthe experiments envisioned by ALPHA. Finally, a high power source is cur-rently under development at RIKEN-RAL muon facility with the purposeof measuring the anomalous magnetic moment of the muon. The sourceuses the same nonlinear process used on a tuneable source implementedby Marangos et al [57] to produce low energy polarized muons.Besides synchrotron radiation, any other techniques for producing co-herent hydrogen Lyman-α radiation requires the use of nonlinear processes20Chapter 2. Coherent hydrogen Lyman-α source designTable 2.1: Compilation of the reported coherent Lyman-α sources vianonlinear processes in the literature.Nonlinear medium Generation Process ReferenceKr / Ar Third Harmonic Generation [58]Kr / Ar Third Harmonic Generation [59–62]Kr High Pressure Third Harmonic Generation [63]Kr / Ar (Tuneable) Four Wave Mixing [57]Hg CW Four Wave Mixing [55]Hg Four Wave Mixing [64]Be Four Wave Mixing [65]Mg Four Wave Mixing [66]in gases. In this chapter, I describe the design and optimization of a coher-ent light source based on nonlinear frequency conversion. The develop-ment of the source was approached in steps of increasing complexity. Thefirst step was to develop a source able to detect hydrogen (and antihydro-gen) and that generated, at least, the power specified for cooling. The ideais to develop a system that can be used for initial experiments and alsobe updated as to increasingly meet the experimental requirements such aslinewidth and tuneability. With regards to the actual implementation ofthe source, based on a literature review, the most common choice seemsto be a third harmonic generation (THG) process in a noble gas. Severalsources have been developed this way [59–62] and there is plenty of infor-mation about it. However, it was decided to develop a system which usesa different nonlinear process. The specific design used is very promisingwith regards to the potential power, linewidth and tuneability. The THGwas still implemented in parallel to this development with the purpose ofhaving a second source for comparison. This would allow us to evaluate thebest system to continue into the next step of integration with the ALPHAapparatus.The chapter has three sections. First, the background theory of non-212.1. Nonlinear opticslinear optics necessary to understand how the Lyman-α source works isexplained. Central to this discussion are the concepts of the nonlinear sus-ceptibility and phase matching. This is followed by a description of thedesign and optimization of the source. The source prototype implemen-tation and characterization are delayed until the next chapter where thedifferent detection techniques used are discussed. As part of the charac-terization, spectroscopy of a supersonically expanded hydrogen beam wasperformed. Details of the hydrogen beam implementation, characteriza-tion, and the spectroscopy are discussed in Chapter 4.2.1 Nonlinear opticsNonlinear optics processes such as frequency sum and difference (frequencymixing) are commonly used to extend the spectral range of lasers [67]. Thisis mostly done using crystals and visible and infrared light is commonlygenerated. Crystals, however, absorb radiation in the vacuum ultravioletspectrum (VUV) range (200 nm-100 nm). The onset of saturation, for in-stance, of β-barium borate ”BBO” crystals is 189 nm [68]. For this reason,frequency mixing in the VUV spectrum range is usually done in gases. Inthe following discussion we explore a particular process known as four-wave mixing (FWM) which is the most commonly used process for VUVlight generation.FWM arises from the interaction of four coherent optical fields throughthe third order nonlinear susceptibility. This phenomenon includes manydiverse processes such as third-harmonic generation, stimulated Ramanscattering, and Raman induced Kerr effects. Central to the discussion is thethird order nonlinear susceptibility χ(3) which is ultimately responsible forall FWM processes 2.Some of the early works, with New and Ward [69] being perhaps thefirst one on UV light generation by nonlinear processes in gases (generated2In fact, the reason FWM (χ(3)) is used to generate VUV is that gases, which are trans-parent to VUV, do not show second order nonlinear susceptibility (χ(2)) due to symmetryreasons222.1. Nonlinear optics231.4 nm by third harmonic generation), reported very low conversion ef-ficiencies. It was not until the work of Harris and Miles[70], and Young etal[71], showed that conversion efficiency could be increased by several or-ders of magnitude by using a two-component system that the technique be-came of practical interest. In this scheme one component is used to providethe overall refractive index and the second component provides the nonlin-ear susceptibility for the particular process. With two components, phasematching is possible, generating conversion efficiencies of ∼ 10−7 − 10−6[6]. The final advance that made the technique useful for practical appli-cations was the demonstrations by Bloom et al, Hogdson et al and Leunget al [72–74] of resonant enhancement of the nonlinear susceptibility. Thistechnique increases the conversion efficiency by several orders of magni-tude (≥ 10−4) [67, 68] by using a two-photon resonance transition. Mucheffort to develop new mixing schemes has been overtaken since then withthe goal of increased conversion efficiency and greater tuning range extend-ing well into the extreme ultraviolet spectrum (XUV) (100 nm -10 nm). Forinstance doubly-enhanced sum-frequency mixing have been demonstratedby Softley et al, where in addition to the two-photon resonance a secondresonance at the sum-frequency is exploited [75, 76]. There are publishedreviews of the subject with Vidals [77] being perhaps the most comprehen-sive one.A very commonly used mixing technique uses noble gases (Xe, Kr, Ar,and Ne) as the nonlinear media. Noble gases have an appropriate energy-level structure and are convenient to work with. Metal vapours such as Hgare also suitable for VUV and XUV generation but are not as convenientto work with and require heat pipes to maintain the metal in a vapourstate. An additional advantage of using atomic gases is their inherit nar-row linewidths which allow for the generation of very narrow linewidthcoherent light <1 cm−1) [78].In practical terms for radiation generated between 200 nm and 120 nmthere are windows reasonably transparent (MgF2 and LiF) and vacuum-sealed cells filled with the mixing gases can be used as the nonlinear mediumand light source. Since oxygen is the main absorber of radiation in this232.1. Nonlinear opticswavelength range, vessels filled with argon can be used as the workingmedium (for VUV processes that do not require vacuum). If shorter wave-length (≤ 120 nm) radiation is required the light source cell has to be con-nected directly to a vacuum chamber (not possible to use a LiF windowwith cut off at 105 nm) via a differential pumping scheme. In this case thenonlinear gas is usually injected into the vacuum in the source cell witha pulsed valve and nozzle system [79–82]. In most of these schemes, thefundamental laser beams are focused into the nonlinear medium in orderto increase the field strength and improve the nonlinear coupling.The most common FWM processes used to generate VUV and XUV are[83]:• Non-resonant tripling:ωvuv = 3w1.• Resonant Sum Frequency Mixing:ωvuv =ω1 +ω2 +ω3 (or 2ω1 +ω2).• Resonant Sum-Difference Frequency Mixing:ωvuv =ω1 +ω2 −ω3 (or 2ω1 −ω2).• Anti-Stokes stimulated Raman scattering:ωvuv =ω1 −ω2 +ω3.The most convenient mixing scheme for producing a widely tuneableoutput is sum-difference. The reason is there are no restrictions on thesign of the wave-vector mismatch “∆k” [84] (this point is further explainedbelow). Non-resonant tripling is convenient as only one primary beam isneeded, greatly reducing the complexity of the system. We will concentrateon these two processes.The following discussion of nonlinear optics is based on a perturbativeapproach first introduced by Bloembergen [85]. In this approach, the non-linear optical properties of the material are fully described by the nonlin-ear susceptibilities, which are obtained by applying the Fourier transformto the nonlinear Green’s function. The nonlinear Green’s function can be242.1. Nonlinear opticsobtained using a generalization of Kubos linear response theory [86]. Theperturbative theory is generally in good agreement with the experimentaldata, and comparisons have been published by Butcher [87] and Boyd [88].This approach is a semiclassical analogue of Feynman diagrams formalism[89] and diagrammatic techniques to calculate the perturbation terms ofthe density matrix have been suggested by Yee et al [90].2.1.1 Nonlinear susceptibilityThe interactions of electromagnetic waves with matter can be describedby Maxwell’s equations. At optical frequencies the materials of interestare usually non-magnetic. Also, as long as plasmas are not involved, wecan assume that the mediums are source and current free. Under theseconstraints on the Maxwell’s equations it is possible to derive the waveequation.∇2~E + 4pi~∇(~∇ · ~P )−1c2∂∂t~E =4pic2∂∂t2~P , (2.1)withρb = −(~∇ · ~P ), (2.2)and ρb is the charge density attributable to bound charges (i.e. chargesarising from the polarization of neutral atoms). The term 4pi~∇(~∇ · ~P ) isoften dropped in the literature, which is usually correct, but it should beconsidered in nanoscale, meta-materials and interface calculations.The response of the medium to an electromagnetic perturbation is fullydescribed by the above equation. The polarization term, P (~r, t), providesthe full description of the light-matter interaction and that is what we willnow derive. The polarization can be expressed by the following convolu-tion:P(T )i (~r, t) =* ∫ ∞−∞G(T )ij (~r′ , t′)Elocj (~r −~r′ , t − t′)dt′d~r ′ . (2.3)The electric field, Eloc(r, t), is the local perturbation (at ~r and t) causedby an external macroscopic field and G(r, t) is the linear Green’s functionaccounting for the response of the system. In general Eloc(~r, t) is not the252.1. Nonlinear opticsmacroscopic applied field. However, it is customary to carry on with thederivation assuming Eloc(~r, t) = E(~r, t) and modifying the final result by lo-cal field corrections [91–93].The convolution form of the polarization is particularly advantageousfor calculation as it can be reduced to a product of Fourier transforms (inspace and time).Pi(~k,w) = f{F [Pi(~r, t)]}= f{F [Gij(~r, t)]F [~Ej(~r, t)]}. = f{χij(~k,w)Ej(~k,w)}(2.4)The susceptibility, χ, is a tensor that contains all the information of the sys-tem. Additionally, in the optical frequency range the susceptibility of thesystem does not depend on the magnitude of the wave vector, which obeysthe condition |~k|d  1 where d is the characteristic order of dimensionsof the constituent elements of the material. For the atomic gases consid-ered here d is on the order of the Bohr radius. This is called the dipoleapproximation and is equivalent to dropping all the ~r dependences on theconvolution relation [87].χ(n)i,j1...jn(w) ≡ lim~k→0χ(n)i,j1...jn(~k,w). (2.5)To calculate the nonlinear terms of interest we can generalize the defi-nition of polarization. The total polarization can then be expressed as thesum of linear and nonlinear contributions at each order of nonlinearity:~P (t) = ~P (1)(t) + ~P NL(t), (2.6)~P NL(t) =∞∑n=2~P (n)(t). (2.7)And the n-order nonlinear polarization can be defined as a multiple con-volution product:P(n)i (t) =∫ ∞−∞G(n)i,j1,··· ,jn(t · · · tn)Ej1(t − t1) · · ·Ejn(t − tn)dt1 · · ·dtn. (2.8)262.1. Nonlinear opticsAs before, we compute the Fourier transform and obtain an expression forthe nonlinear susceptibility.P(n)i (ω) =∫ ∞−∞χ(n)i,j1···jnn∑l=1ωl ;ω1, · · · ,ωnEj(ω1) · · ·Ejn(ωn)× δω −n∑l=1ωldω1 · · ·dωn, (2.9)withχ(n)i,j1,··· ,jnn∑l=1ωl ;ω1, · · · ,ωn =∫ ∞−∞G(n)i,j1,··· ,jn(t1, · · · , tn)expn∑l=1ωltldt1 · · ·dtn.(2.10)The Dirac δ-function guarantees that the sum of the arguments of theFourier transforms of the electric field equals the argument of the Fouriertransform of the polarization. Physically, each ω component of the polar-ization results from the interaction of n photons mediated by the materialsusceptibility function.2.1.2 Quantum-mechanical expression of the nonlinear Green’sfunctionThe susceptibility is ultimately a function of the microscopic interactionsof the material. Therefore, we would like to derive a quantum mechani-cal expression for the nonlinear Green’s function. The quantum mechan-ical description that we follow here is based on the principle of “minimalcoupling” approximation. In general terms this refers to the coupling offields that involve only the charge distribution. Minimal coupling disre-gards higher multipole moments and magnetic moments of the particles.Under the approximation, the Schro¨dinger equation for a single particletakes the form:H =12m(−i~∇− q ~A(~ˆx, t))2+ qφ(~ˆx, t), (2.11)272.1. Nonlinear opticsandi~∂tψ(~x, t) =Hψ(~x, t), (2.12)where ~A and ψ are the vector and scalar potentials and q is the charge ofthe particle. For a many body system under the dipole approximation andusing a Legendre transformation we arrive at a total Hamiltonian:H =Ho +H1, (2.13)withHo = T +V +Hee =N∑αV (~r α) +∑α,β=1e2|~r α −~r β |, (2.14)andH1 = eN∑α~r α · ~E(t), (2.15)where the charge of the electron has been replaced q = −e. Here, all the timedependence is in the interaction part, usually referred as the Hamiltonianin the gauge of length.As we are dealing with ensembles of particles and are interested in thestatistical properties it is convenient to work with the density operator.Calculations can also be performed using the perturbed particle wavefunc-tions. However, introduction of damping due to dephasing and populationdecay is more straightforward using the density matrix approachρ =∑a,bρab |a〉〈b| . (2.16)The macroscopic polarization per unit volume can be defined as the ex-pectation value of the dipole moment per unit volume where the initialcondition is given by the Boltzmann equilibrium distribution [85–87].P(n)i (t) ≡1VT r−eN∑α=1rαi ρn(t), (2.17)282.1. Nonlinear opticswithρ(0) =∑α exp−EαKBT|a〉〈b|∑α exp−EαKBT. (2.18)The density operator obeys the Liouville differential equation (with relax-ation terms ignored).i∂tρ (t) = [H,ρ(t)] = [Ho,ρ(t)] + [HI ,ρ(t)] . (2.19)This equation can be solved using perturbation theory techniques by ex-panding the density operator in a series that leads to a system of coupleddifferential equations:i∂tρ(1)(t) =[Ho,ρ(1)(t)]+ [HI ,ρ(0)]i∂tρ(2)(t) =[Ho,ρ(2)(t)]+[HI ,ρ(1)(t)]⇓ · · · = · · · ⇓i∂tρ(n)(t) =[Ho,ρ(n)(t)]+[HI ,ρ(n−1)(t)].(2.20)It is convenient to use the interaction representation for the evolutionof the position operatorsrαi (−t) = exp[iHo(−t)~]rαi exp[−iHo(−t)~]. (2.21)Considering the n-differential equation in the system, and with the bound-ary condition ρ(n)(0) = 0, the solution of this equation can be obtained re-cursively by solving each increasing perturbation order [85, 86]. At theend ρ(n)(t) can be expressed as a function of the dipole operators and of292.1. Nonlinear opticsρ(0) only:ρ(n)(t) =( e−i~)n∫ t−∞· · ·∫ tn−1−∞Ej1(t1) · · ·Ejn(tn)∗N∑α=1rαj1(t1 − t), · · ·N∑α=1rαj1(tn − t),ρ(0)· · ·dt1 · · ·dtn. (2.22)Using the following relation:T rN∑α=1rαirαj1(−t1), · · ·N∑α=1rαjn(−tn),ρ(0)· · ·=(−1)nT rN∑α=1rαjn(−tn), · · · ,N∑α=1rαj1(−t1),N∑α=1rαi· · ·ρ(0), (2.23)and the definitions of polarization:• expectation value of the dipole operator per unit volume• convolution of the nonlinear Green’s function times n electric fieldsit is possible to derive the following relation:G(n)i,j1···jn(t1, · · · , tn) = −en+1V (−i~)n∫ ∞−∞Θ(t1) · · ·Θ(tn − tn−1)∗T rN∑α=1rαjn(−tn), · · · ,N∑α=1rα−j1(−t1),N∑α=1rαi· · ·ρ(0). (2.24)Finally, with the definition of the nonlinear Green’s function we can express302.1. Nonlinear opticsthe n-order nonlinear susceptibility of a general quantum system.χ(n)i,j1···jnn∑l=1ωl ;ω1, · · · ,ωn =∫ ∞−∞G(n)i,j1···jn(t1, · · · , tn)expn∑l=1iωltldt1 · · ·dtn =−en+1V (−i~)n∫ ∞−∞Θ(t1) · · ·Θ(tn − tn−1)∗T rN∑α=1rαjn(−tn), · · · ,N∑α=1rα−j1(−t1),N∑α=1rαi· · ·ρ(0)expn∑l=1iωltldt1 · · ·dtn.(2.25)In the preceding derivation relaxation terms were neglected but can beincluded in the equation of motion:i∂tρ (t) +( i~Γ)ρ(t) = [Ho,ρ(t)] + [HI ,ρ(t)] , (2.26)where Γ the damping constant includes the transverse (off-diagonal) andlongitudinal relaxation terms. A formal solution of the components ρ(i)(t)at frequency ±ω has been worked out by Pell and Vidal [94] in terms of theiterated integral:ρ(n)ij (t) =i~exp{−iωijt}∫ t−∞A(t′){exp{−i(ω −ωij )t′}+ exp{−i(ω+ωij )t′}}×[rij ,ρ(n−1)]dt′ , (2.27)where ωij = (Ei − Ej ) / ~ − i / Γij is the energy difference (eigenfrequencies)between level i and j. The complex term i / Γ is important in cases wherewe operate close to resonances.Using Eq. (2.25) it is possible to derive explicit expressions for the non-linear susceptibility of specific processes in media with discrete quantumlevels. This is done by performing the Fourier transform integral of the312.1. Nonlinear opticsGreen’s function and expanding the time-dependent position operators r.Of particular interest for this work is the third order nonlinear suscepti-bility of FWM processes and, specifically, non-resonant third order sus-ceptibility and resonant frequency sum-difference. Formulas for the FWMprocesses ω4 =ω1 +ω2 +ω3 which are far away from resonances have beengiven by Armstrong et al [95]. For situations where damping constantsneed to be included (resonant FWM) formulas have been worked out byBloemebergen and Shen [96].For FWM sum-difference interaction in a gas system the susceptibilityper atom is (with rij in Bohr radius units and ω in cm−1 which is customar-ily used for calculation):χ(3) (−ω4;ω1,ω1,±ω2) =13(eao)4(~c)3∑(r0irijrjkrk0)×1[ωi0 ∓ω2][ωj0 − (ω1 ±ω2)] +1[ωi0 −ω1][ωj0 − (ω1 ±ω2)]×(1ωk0 −ω4+1ω∗k0 +ω1)+1[ωi0 −ω1][ωj0 − 2ω1](1ωk0 −ω4+1ω∗k0 ±ω2)+1[ω∗i0 ±ω2] [ω∗j0 + (ω1 ±ω2)] +1[ω∗i0 +ω1] [ω∗j0 + (ω1 ±ω2)]×(1ω∗k0 +ω4+1ωk0 −ω1)+1[ω∗i0 +ω1] [ω∗j0 + 2ω1](1ω∗k0 +ω4+1ω∗k0 ±ω2). (2.28)In the gas systems that we are considering it is possible to write eachmatrix element as a coefficient times a reduced matrix element using the322.1. Nonlinear opticsWigner-Eckart theorem [97]:rab =〈αJama∣∣∣~rθ∣∣∣βJbmb〉= (−1)Ja−maJa 1 Jbma ∆mθ mb〈αJa ||r ||βJb〉. (2.29)The difference on magnetic quantum number ∆mθ in the 3j-symbol is ±1for circularly polarized light and 0 for linearly polarized. The selection ruleassociated with3j-symbol symmetry yields the following condition for thesusceptibility numerator:(r0irijrjkrk0)⇒ ∆m0i +∆mij +∆mjk +∆mk0 = 0. (2.30)There are two practical implications from these equations that are worthpointing out. First, from the matrix elements expressions it is evident thattwo right-hand and one left-hand circularly polarized waves produce a left-hand circularly polarized wave. Three right-hand or left-hand circularlypolarized waves do not produce a frequency sum [77]. Second, the selec-tion rules indicate that the states |0〉 and |j〉 have opposite parity to the|i〉 and |k〉 states. To enhance the susceptibility the resonant denominatorshould be minimized (see Eq. (2.28)). Since ωk0 and ωi0 are allowed tran-sitions, and ωj0 is not allowed, the term (ωj0 − 2ω1) should be minimized.In plain language, we wish to operate at a resonance in order to minimizethe resonant denominator in the susceptibility, and this is achieved in anoptimal way by avoiding the one-photon resonances (which would leadto absorption of the fundamental wave) and by avoiding the three-photonresonance (which would lead to absorption of the produced light. This isthe basis of two-photon resonant enhanced FWM and was demonstratedfirst by Bloom et al, Hodgson et al and Leung et al, [72–74]. This will berevisited in the next section.Finally, as mentioned at the beginning, the susceptibility derivationused Eloc(r, t) = E(r, t), and the final result needs to be modified by localfield corrections. In isotropic media such as gases the depolarization field332.1. Nonlinear opticscan be accounted for by by multiplying each field components by:L =(ω) + 23, (2.31)where (ω) is the linear dielectric constant at ω [85]. The macroscopicsusceptibility is proportional to the number of scattering centers times thelocal field corrections. In rarified gases L ≈ 1 and the macroscopic suscep-tibility is:χT =Nχa, (2.32)where N is the density number. If the density is very large, such as incryogenic crystals of noble gases, local field corrections become important.2.1.3 Four wave mixing solutionThe third order polarization describes a coupling between four waves, eachwith its own direction of propagation, polarization, and frequency. If thisnonlinear polarization is substituted into the nonlinear wave equation, Eq.(2.1), a set of four coupled wave equations may be found for the fields.As long as the variation of the field amplitude is small in an optical pe-riod the differential equations can be reduced to first order. This is calledthe slowly varying envelope approximation and is generally valid for laserradiation.The equations for the field magnitudes can be written in the fol-lowing form:∂Ej∂z+nj∂Ejc∂t=2piiωjcnjP NLj exp{ikjz}−κj2Ej , (2.33)where n is the refractive index and κ = (ωi) / (cn) is the absorption coeffi-cient. The third order nonlinear polarization is:P(3)s (ωs) =32Nχ(3)s,j1j2j3(ωs;ω1ω2ω3)E1E2E3, (2.34)342.1. Nonlinear opticswhere ωs = ω4 is the generated wave that we are trying to solve the equa-tions for.These equations cannot be solved in general and the most common ap-proach is to assume:• plane wave solutions,• the energy transferred from the input fields to the fourth field is anegligible fraction of the total energy of the fields.The solution of the intensity in the small signal approximation as firstderived by Bey et al [98] is:Isns=[24pi2ωsc2nsNLχ(3)s,j1j2j3(ωs;ω1ω2ω3)]2 Io1Io2Io3n1n2n3F(∆kL,τiτs), (2.35)with the optical depth:τj = kjL = σ(1)j (ωj )NL τs = τ1 + τ2 + τ3 (2.36)and the phase matching factor:F(∆kL,τi , τs) =exp {−τi}exp {−τsj} − 2exp{−(τi+τs)2}cos(∆kL)[ (τs−τi )2]2+ [∆kL]2, (2.37)where L is the total length of the nonlinear medium, σ is the one-photonabsorption cross section and ∆k = ks − k1 − k2 − k3.2.1.4 Phase matching factorThe phase matching factor is an expression of conservation of energy andmomentum. All the four waves must maintain a constant phase in order toavoid destructive interference. In order to optimize the FWM process onehas to minimize the three variables ∆kL, τi and τs. The limiting cases ofthe phase matching factor are [97]:352.1. Nonlinear optics• negligible absorption cross sections τi = 0, τs = 0:F(∆kL,τi = 0, τs = 0) =sin(∆kL/ 2)2(∆kL/ 2)2. (2.38)• Large absorption cross section τi  1, τs 1:F(∆kL,τi  1, τs 1) =1[(τs − τi) / 2]2 + (∆kL)2. (2.39)• Phase matched system ∆kL = 0:F(∆kL = 0, τi , τs) =[exp(−τi / 2)− exp(−τs / 2)(τs − τi) / 2]2. (2.40)Finally for the case of large absorption with τs > τi and ∆kL = 0 we haveIsns=48pi2ω2c2nsχ(3)s,j1j2j3(ωs;ω1ω2ω3)σ(1)s2Io1Io2Io3n1n2n3. (2.41)The last case is the most interesting, as it is always possible to find incidentoptical frequencies away from resonances but the absorption cross sectionof the fourth wave cannot be neglected [77]. The intensity is not dependenton the density, NL, and is limited only by the absorption cross-section andsusceptibility.The validity of the results derived using the plane wave approximationis limited, since the fields used experimentally are Gaussian. In the case ofvery short interaction length (smaller than the Rayleigh length) the phasefronts are approximately planar and if ∆kL⇒ 0 then the intensity expres-sion for Is / ns still applies. However, in general these expressions need tobe modified since Gaussian beams have a phase shift, the Gouy phase shift,of pi at the focus. Assuming a T EM00 mode, negligible cross-sections ( noabsorption), a collinear arrangement of beams, and identical confocal pa-rameter b = 2kR2 where R is the 1 / e2 radius of the intensity distribution,362.2. Source design and optimizationthe phase matching factor as calculated by Bjorklund [84] is:F(∆kL,b / L) =1L2[∫ L/2−L/2exp {−i∆kz}(1 + i2z / b)2dz]2. (2.42)The optimal conditions for FWM processes are:∆kb =−2 ωs =ω1 +ω2 +ω30 ωs =ω1 +ω2 −ω3+2 ωs =ω1 −ω2 −ω3.(2.43)Sum-difference frequency mixing with a condition ∆kb = 0 is the most for-giving and therefore the preferred process for widely tuneable lasers. Inthe case of sum frequency (or THG) the phase matching is more restricted.However, for THG the phase matching only requires two wavelengths andis tremendously simplified. Finally, excessive focusing will excite higherorder perturbations generating competing processes not included in theprevious analysis. According to Vidal [77, 97] the optimum power conver-sion in a THG phase matched system is b=L, experimentally.2.2 Source design and optimizationThe total harmonic power is obtained by integrating the power density overthe beam profile which is usually approximately Gaussian. In FWM pro-cesses the power of the generated wave can be expressed as:Ps ∝N2 · [χ(3)]2 ·P1 ·P2 ·P3 ·F. (2.44)All of these factors are related and it is not possible to optimize one ofthem without considering the others. However, the first step is to selecta medium and a process. The main criteria is usually to maximize thesusceptibility.372.2. Source design and optimization2.2.1 Resonance-enhancement of FWM processesThe susceptibility of the medium ultimately emerges from its electronicstructure. Choosing a medium with an appropriate energy structure is per-haps the most important step towards designing and optimizing a source.Important factors are the total coefficient <{χ(3)}, as it affects the powergenerated directly, and also the shape of the dispersion as it affects thephase matching factor. In general, a medium with a rapidly changing dis-persion <{χ(1)(ω)}(linear susceptibility) is not desirable as phase match-ing is very difficult. Furthermore, even if phase matching is achieved, theusable bandwidth of the input beams is very limited. The availability ofthe input lasers is, of course, important as well, and it reduces the choicesof available resonances in nonlinear medium. The susceptibility:χ(n) ∝dipole momentsresonances, (2.45)is a function of the oscillator strengths and resonances. Resonant terms inthe denominator increase the overall susceptibility. However, not all reso-nances are good. One-photon resonance occurs when the incident photonfrequency matches that of one of the allowed dipole transitions from theground state. Strong absorption is a problem and as a result the resonantdenominator cannot be made too small. Near resonance is advantageous aslong as the variation of the dispersion is not too great as to reduce the coher-ence length to the point where phase matching is impossible. Three-photonresonances, which involves combinations of three photon frequencies, arein principle the same as one-photon resonance. They involve transitionsbetween different parity states and the angular momentum restrictions de-pend on the particular coupling of the medium (L − S or J1 − J2). Thesetype of resonances are important for processes that require negative disper-sion, such as sum frequency, but they usually lead to absorption as the one-photon case. Two-photon resonances involve combinations of two photonfrequencies. They require states with the same parity and ∆J = 0,±1,±2.Two-photon absorption is generally much weaker than the one- or three-photon cases. Fig. 2.1 shows diagrams for the different cases.382.2. Source design and optimizationEOne-Photon Two-Photon Three-PhotonSum-Frequency Resonant ProcessesEigenstate of Unperturbed SystemVirtual Stateω1= ωRω2ω3ωSω1ω2ω3ω3ω2ω1ω1+ω2= ωRω1+ω2+ω3= ωRωS ωSFigure 2.1: FWM processes can be enhanced by the used of resonances.Energy diagrams of the three types of resonances available for FWM areshown for a sum-frequency resonant process. The only type that wouldnot result in absorption of one of the input beams or the generated beamis the two-photon resonant case, because the transition is electric dipoleforbidden.With regards to the medium, alkali vapours have convenient energylevels given the available input laser and K, Rb and Na have been used togenerate VUV. Hg is also very commonly used for VUV generation. The dif-ficulty of using these sources relies on the requirement of using heat pipesto maintain the vapour state of the metal. Noble gases are more attractivein this regard. In addition, noble gases have the highest ionization en-ergy of all elements; ionization is always a problem when generating VUVlight. Kr for instance has an appropriate energy level and reported highthird order nonlinear susceptibilities. Xe is also very appealing gas but thecost / mol of Xe is about two orders of magnitude higher than Kr.392.2. Source design and optimizationFig. 2.2 shows a particular resonance-enhanced sum-difference FWMprocess in Kr. The Grotrian diagram with the relevant energy levels andtransitions is shown in Fig. 2.3.In this scheme the two-photon resonance-enhanced FWM (RE-FWM)uses the resonant condition (see Eq.(2.28) of ωj0 = 2ω1with ω1 = ω2. Forthe generation of Lyman-α radiation at 121.56 nm, Kr gas is an efficientnonlinear medium since the 4s24p55s 2[1/2]1/2 J = 1 level is just∼ 1300cm−1below the Lyman-α frequency, which adds another nearly resonant condi-tion of ωkg ∼ 2ω1 − ω3 in the third order susceptibility. This process isvery promising given the extremely favourable resonance conditions (i.e.two photon resonance and nearly resonance between excited states). In ad-dition, Kr energy states conveniently suppress competing processes suchas ω1 + ω3 and one-photon resonances (from ground state). Competingprocesses, such as multiphoton absorption are important saturation mech-anisms [75, 76], and are discussed in the next section. The scheme requiresa two-photon resonant beam ωR = ω1 = ω2 = 2pi × 1.4818 × 1015 Hz or202.31 nm and a tuneable beam ωT =ω3 = 2pi×4.9753×1014 Hz or 602.56nm to adjust the output wave ωVUV = 2ωR −ωT . Both of these wavelengthare available at sufficient powers.Using Eq. (2.28) is possible to calculate the nonlinear susceptibilityof Kr for this process. The sum is over all exited states connected by al-lowed dipole transitions (see Fig. 2.3). However, the basic selection rules,which apply only to simple configurations, obey strict LS coupling. Inertgases, except helium, are more complex and the L and S levels interact.Their structure is usually described by Jj or jl coupling schemes and bothare found in the literature. Inert gases have six electrons in the outermostsubshell. These are the electronic structures of argon (argon plays an im-portant role in our scheme and is discussed later), and krypton:• Ar → Z = 18, ground state → 1s22s22p63s23p6, ionization energy15.759610 eV• Kr→ Z = 36, ground state→ 1s22s22p63s23p63d104s24p6, ionizationenergy 13.999605 eV402.2. Source design and optimization4s24p6  1S04p5(2Po3/2)5p 2[1/2]4p5(2Po1/2)5p 2[1/2]4p5(2Po1/2)5s 2[1/2]o4p5(2Po3/2)5s 2[3/2]o98855.0698 cm-194092.8626 cm-185846.7046 cm-180916.7680 cm-149427.5349 cm-182259.16 cm-116595.91 cm-1Ly-α 121.5667 nmλR 202.32 nmλT 602.56 nmλR 202.32 nmKrFigure 2.2: Nonlinear process used to generate Lyman-α light. In thisscheme the two-photon resonance-enhanced FWM (RE-FWM) uses the res-onant condition of ω1 = ω2. For the generation of Lyman-α radiation at121.56 nm, Kr gas is an efficient nonlinear medium since the 4s24p55s2[1/2]1/2 J = 1 level is just ∼ 1300 cm−1 below the Lyman-α frequency,which adds another nearly resonant condition of ωjg ∼ 2ω1 − ω3 in thethird order susceptibility.412.2. Source design and optimization4s24p6  1S0 5p 2[1/2]5s 2[1/2]o5s 2[3/2]o 98855.0698 85846.7046  80916.7680 82259.16 cm-1λT 602.56 nmλR 202.32 nmKr110290.3265cm-18d 2[1/2]o7d 2[1/2]o6d 2[1/2]o6s 2[1/2]o5d 2[3/2]o4d 2[3/2]o5d 2[1/2]o4001100711125352120<11361918122103801.7929107676.1489105146.33109342.9368105648.434104887.315112914.433 Ionization Energy Figure 2.3: Grotrian diagram of Kr. The energy levels shown contribute tothe nonlinear susceptibility for the RE-FWM process. Kr electron structureis described by jl coupling scheme an the term symbol is: 2Se+1[K]P whereP is the parity the term and ~K = ~le+ ~jp. Here ~jp is the angular momentum ofthe parent multiplet and ~le that of the excited electron. The number next toeach transition is the relative transition strength which have been compiledfrom Ref. [99, 100]. 422.2. Source design and optimizationAny of the outermost electrons can transition to excited states. In allrelevant excited states the structure remains of the form (np5n′l). A (p5)structure is equivalent to p and it reduces to (npn′l). This is equivalentto a two electron problem, one in p and the other on a s,p or d subshell.This behaviour is described by a Jj coupling where J is the total angularmomentum quantum number for the parent ion and j is for the transition-ing electron. A state in Jjcoupling notation is written as |nj1, j2JM〉, wheren is the principal quantum number of the transitioning electron, j1 is thetotal angular momentum quantum number for the parent ion and j2 forthe transitioning electron, J is the total angular momentum for the wholeconfiguration, and M the magnetic quantum number referring to J . Thedipole transition selection rules are:• ∆j1 = 0, ∆j2 = 0,±1 (but j = 0→ j = 0 is forbidden)• ∆J = 0, ±1 (but J = 0→ J = 0 is forbidden)• If M = 0 and ∆J = 0 the matrix element is zeroIn the jl scheme the electrostatic perturbation interaction produced bythe the excited electron can be approximated as ~le ·~jp. Here ~jp is the angularmomentum of the parent multiplet and ~le that of the excited electron. Thiselectrostatic perturbation is diagonal in the Hilbert space∣∣∣jpleK〉, with ~K =~le + ~jp. The coupling of the spin of the external electrons with parent ion isalso diagonal as ~je + ~jp = ~le + ~se + ~jp = ~K + ~se = ~j where ~se is the spin angularmomentum of the excited electron. The energy split by spin-orbit couplingis then characterized by a couple of values j = K ± 1 / 2 and diagonal in thevector space∣∣∣jpleKj〉. Any state can be uniquely determined by∣∣∣njpleKsej〉.The states in Fig. 2.3 are represented using the term symbols for the jlcoupling scheme, which have been used in this work for the calculations.The term symbol is: 2Se+1[K]P where P is the parity. For linear polarizationonly the z direction of the the dipole matrix elements rab is needed whichcan be obtained from tables of measured or tabulated values of oscillatorstrengths fab, line strength, S(ab) or transition probabilities Aab using the432.2. Source design and optimizationrelations[101]:|zab|2 = |〈a |z|b〉|2 = S(ab) = S(ba) (2.46)Aab =2pie2mecoλ2gagbfab (2.47)fab =23∆EgaS(ab) (2.48)where ga and gb are the statistical weights given by the degeneracy of thestate g = 2J + 1.Using Eq. (2.28), and summing over all possible combinations of al-lowed transitions under jl coupling the susceptibility for the process hasbeen calculated (see Fig. 2.4). The matrix elements were extracted fromdata tables [100, 102, 103] or tabulated from any available information us-ing the previous relations. Notice that the uncertainty of oscillator strengthmeasurements is between 20 − 50% and this is reflected in the calculatedsusceptibility.The two-photon resonance shape can be described by [104]:S(2ω1) =2√ln2∆ωDoppler√pi∫ ∞−∞exp{−x2}x − ζ, (2.49)which is a scaled Plasma Dispersion function and where ζ is:ζ = 2√ln2(2ω1 + iΓR2−ωR)/∆ωDoppler . (2.50)Here ΓR is the homogeneous Lorentzian linewidth of the resonant state(2pi × 6.4 MHz). The plasma-dispersion function is equivalent to the Fad-deeva function whose solutions are Voigt profiles (convolutions of a Lorentzianand a Gaussian profile).Since Kr has several isotopes the two-photon resonance is actually com-posed of several resonance peaks. Details of the Kr isotopes can be seen inTable 2.2. Fig. 2.6 shows the calculated two photon resonance taking into442.2. Source design and optimization2   1.5   1   0.5 0 0.5 1 1.5 2x 10400.511.522.533.54x 10−65s 2 [3/2]o5s 2 [1/2]o5d 2 [1/2]o4d 2 [3/2]o8d 2 [1/2]o7d 2 [1/2]o6d 2 [1/2]o5d 2 [3/2]o6s 2 [1/2]o|χ(3) |[(eao)4 /cm-3 ]Ly-α νT[cm-1]101.6126.8121.6112.5 91.8 84.1λVUV[nm] ωVUV=2ωR+ωTSum-Difference FWMωVUV=2ωR−ωTIonizationEnergy Figure 2.4: Calculated third order susceptibility for RE-FWM process. Us-ing Eq. (2.28), and summing over all possible combination of allowedtransitions under jl coupling, the susceptibility for the process have beencalculated (see Fig. 2.4). The matrix elements were extracted from datatables [100, 102, 103] or tabulated from any available information usingthe previous relations. Notice that the uncertainty of oscillator strengthmeasurements is between 20 − 50% and this is reflected in the calculatedsusceptibility.452.2. Source design and optimizationx 104123456789x 10-75.3481.6695Ly-α ωVUV=2ωR−ωT5s 2 [1/2]o5s 2 [3/2]o1.9 1.8 1.7 1.51.6|χ(3) |[(eao)4 /cm-3 ]νT[cm-1]5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.000.511.522.53x 103x 10-7|χ(3) |[(eao)4 /cm-3 ]ωVUV=2ωR+ωT5d 2 [3/2]o5d 2 [1/2]o6s 2 [1/2]o4d 2 [3/2]o8d 2 [1/2]o7d 2 [1/2]o6d 2 [1/2]oνT[cm-1]Figure 2.5: Same as Fig. 2.4 where the calculated third order susceptibilityfor sum-difference and sum-frequency are shown in detail.462.2. Source design and optimizationaccount all the isotopes and the relative abundance. A comparison with asingle isotope two-photon resonance indicates that the resonance strengthis only reduced by ∼ 32%.The maximum difference between the resonancepeaks of the individual isotopes is 0.0895 cm−1 ∼ 430 MHz. The linewidthof the resonance is one of the contributors to the theoretical intrinsic limiton linewidth for the generated wave in this process. There is also collisionalbroadening but the main practical limitation on the attainable linewidthare the fundamental beams. For pulsed Nd:YAG lasers, for instance, thelinewidth is on the order of ∼ 5 GHz. There are, of course, various laseroptions with narrow linewidth but they usually require amplifier stages toreach the threshold power requirement for the nonlinear process.Table 2.2: Kr stable isotopes with abundances of an earth bound mixture.The transition energy for the two-photon resonance, 5p 2[1/2] state, isshown along with the transition energy for the two lowest electric dipoleallowed transitions both around the Lyman-α energy.5p 2[1/2] 5s 2[3/2] 5s 2[1/2]Isotope Abundance 98855.0 cm−1 85846.0 cm−1 80916.0 cm−178Kr 0.00355(3) + 0.0416 + 0.6876 + 0.752880Kr 0.02286(10) + 0.0471 + 0.6892 + 0.753582Kr 0.11593(31) + 0.0528 + 0.6910 + 0.754983Kr 0.11300(19) – – –84Kr 0.56987(15) + 0.0703 + 0.7051 + 0.768586Kr 0.17279(41) + 0.1311 + 0.7624 + 0.8254As the system is perturbed by the electromagnetic fields photons areabsorbed and emitted. Some of the atoms will see two photons ωR withinτv ≈ 0.68 fsec , the lifetime of the virtual state, and will excite to the4p5(2p1/2)5p 2[1/2], J = 0 level. When this happens the populations of theeigenstates will change. If a ωT photon interacts with the atom within472.2. Source design and optimization−0.1 −0.05 0 0.05 0.1 0.15020406080100120|S(2ω1)|[cm-2 ]86Kr84Kr83Kr82Kr80Kr78KrSingle Isotope KrΔν[cm-1]Abundance 0.56987Δν = νR(center of gravity) - 2ν1Figure 2.6: Calculated two-photon resonance (5p 2[1/2]) shape vs. detun-ing from the energy centre-of-gravity of the transition (weighted averageof single isotope transition frequencies). The resonance of an ideal singleisotope 84Kr gas mixture is shown for contrast. The resonance strength isreduced by ∼ 32% and broadened due to the presence of the isotopes. Alsothe individual isotope transition frequencies and the abundance relative to84Kr are shown.482.2. Source design and optimizationτ5p = 20.8 nsec (the life time of the state) then a photon at ωs = ωLy−αthe Lyman-α line will be emitted. Otherwise, the photon will decay toanother state generating parasitic electromagnetic fields corresponding toother processes.The availability of atoms in the two photon resonant state limits the ef-fectiveness of ωT photons bringing to completion the sum-difference mix-ing process. This gives an upper limit on Io3 which, assuming the system isnot near saturation, is a function of Io1 . Notice that having more ωT pho-tons than this limit will not increase the ωLy−α photons generated. On thecontrary, the excess of photons will only increase the probability for com-peting processes to take effect such as two-photon absorption and multi-photon ionization. This is discussed in the next section. The total powergenerated can be expressed as:PLy−α ∝N2 ·[χ(3)]2·γ24·P 3R ·F, (2.51)where γ is a constant given by the requirement⌈Io3⌉= (1 / γ)Io1 . The ceilingfunction of Io3 indicates that providing more power than this maximum willnot increase the power of the generated wave and the conversion efficiencywill decrease. Based on the transition probabilities for transitions fromthe 5s states to the two-photon resonant state, then γ = 6.796. This is avery simplistic approximation but gives an indication of upper limit on theintensity Io3 . Providing any larger intensity will be counterproductive.In rarified gases the density of the system N and the phase matchingfactor F will ultimately be responsible for the maximum efficiency achievedexperimentally η = PLy−α / (PR +PT ). However, all of the factors are intri-cately related. Phase matching is perhaps the most important aspect fornonlinear laser implementation. The phase matching coefficient is a mea-sure of how efficiently the power is transferred from the input beams intothe generated beam as shown. The maximum density possible is limited byparasitic processes and by phase matching. Furthermore, through phasematching it is possible to set up the environment so that competing pro-cesses, with different susceptibilities, are not favourable. Before discussing492.2. Source design and optimizationphase matching in more detail a brief analysis of the competing processesis given.2.2.2 Competing and parasitic processesAs mentioned in the previous section some atoms will be excited to thetwo-photon resonant state. At this point they can interact with a ωT pho-ton or decay into the 4p5(2p1/2)5s 2[1/2] or 4p5(2p3/2)5s 2[3/2] states. Allof these processes compete and the total perturbation can be expressedas an infinite perturbation expansion that included not only third orderprocesses but also all the odd order processes allowed in a centrosymmet-ric medium. Furthermore, the description of power generation efficiencydoes not account for saturation effects at high powers. Factors contribut-ing to saturation in the process include the Kerr effect, power broaden-ing, stark shifts, multiphoton ionization, amplified spontaneous emission(ASE), parametric processes , hyper-Raman scattering, two photon absorp-tion, self-defocusing, and higher (5th and 7th) order nonlinear effects. Fig.2.7 shows some of the competing processes.ASE involves emission from the two-photon resonant level 4p5(2p1/2)5p2[1/2], J = 0 to the 4p5(2p1/2)5s 2[1/2], J = 1 level generating light at 557.45nm or, with much lower probability, to the 4p5(2p3/2)5s 2[3/2], J = 1 levelgenerating light at 768.74 nm. This process directly reduces the yield ofLyman-α photons. ASE presents some additional problems besides deplet-ing the two-photon resonant level. The amplified light can power broadenthe ASE transition modifying the 4p5(2p1/2)5p state and further reduce thetwo-photon resonance. In addition, decay from the 4p5(2p1/2)5s 2[1/2],J = 1 and 4p5(2p3/2)5s 2[3/2], J = 1 states to ground will produce VUVlight at 116.48 nm and 123.58 nm. This can be stimulated by the ASEcompleting a FWM process or could occur as a parametric process wherethe two photons ( i.e. ωidler = 557.45 nm and ωsignal = 123.58 nm) aregenerated. This presents a problem for detection as a bare solar blind pho-tomultiplier would not be able to discriminate between Lyman-α and theseVUV frequencies. ASE can be partially suppressed by phase matching as502.2. Source design and optimization5p 2[1/2]5s 2[1/2]o5s 2[3/2]oωRωRωRωRωRωRωRωRωR ωRωRωTωTωASEωLyαωLyαωLyαωRωLyαωIdlerωSignalωStokesHyper-Raman Multiphoton IonizationIonization EnergyASE Parametric TPAe-e-e- 5p 2[1/2]d statesFigure 2.7: Parasitic nonlinear processes that compete with the RE-FWMsum-difference process and reduces the yield of Lyman-α radiation.512.2. Source design and optimizationthe condition to generate Lyman-α radiation is different from the condi-tion for 116.48 nm. However, 123.58 nm is very close and phase matchingalone will not suppress it. Furthermore, the 123.58 nm transition is verystrong with a transition probability of 3.12×108 sec−1 [100]. The only wayto limit it is by providing enough ωT photons. There is, however, a finebalance between providing sufficient ωT photons to suppress ASE but be-low a threshold where the excess photons enhance two-photon absorptionof the combination ωLy−α +ωT .Two-photon absorption (TPA) of the combination ωLy−α +ωT into the4p5(2p1/2)5p 2[1/2] state depletes the generated VUV radiation and leads tofurther parasitic processes. These type of multiphoton absorption mecha-nisms have been shown to be an important saturation mechanism [75, 76].This absorption can be limited by controlling the intensity IoT and reducingthe interaction length (confocal length) of the tuneable gaussian beam bT .The confocal length of ωT should be shorter or equal than the one of theresonant beam bR. This condition is easily achieved with TEM00 beams.Multiphoton ionization is a serious problem for high intensity or tightlyfocused beams. The ionization threshold for Kr is ∼ 14 eV so multiphotoncombinations such as 2ωR +ωT will lead to ionization. In addition, freeelectrons accelerated by the laser fields will collide with Kr atoms and leadto further pathways of ionization (see Fig. 2.7). The contribution of theionization continuum to the refractive index may change the refractive in-dex to the point that no conversion can occur. With pulsed lasers this is aproblem that can be avoided by maintaining a loosely focused system (lowintensity). At this point it is important to clarify some of the lexicon used inthe literature when referring to tightly and loosely focused laser. In termsof the phase matching factor the tightly focused limit refers to the limitwhere b / L 1 and this definition is only concerned with the dimensionsof the terms on the phase matching integral. In terms of an optical pro-cess or photon-atom interaction it refers to some intensity threshold. Thisthreshold is different for each system and is related to the availability ofphotons within time slots given by the uncertainty principle and the life-times of eigenstates of the system. It is possible to maintain low intensity522.2. Source design and optimization(or a loosely focused system) were the effects of saturation are negligiblewithin the tightly focused limit of phase matching if the absorption is neg-ligible (L can be made very long). In our case the threshold multiphotonionization intensity limit for IoR is on the order of 1012 W/m2 based on anionization cross section of 0.2 Mb for 6 eV photons in Kr [105] and we willoperate below the limit where saturation effects become noticeable.Raman processes stimulated by the generated Lyman-α are suppressedas there are no available even parity states for the stokes transition. Hyper-Raman processes involves a two-photon transition ωR +ωLyα followed byscattering into any of the d states. This process will increase the absorp-tion of the generated Lyman-α light. However, multiphoton ionization willmost likely occur before hyper-Raman scattering.Finally, other saturation effects such as ac-stark shift and higher orderprocesses will put an upper limit on the density and intensity of opticalfields. This will result on a practical limit to the efficiency of conversion.In summary, phase matching is the principal tool to suppress parasitic pro-cesses and it is undesirable to work in the high intensity regime.2.2.3 Phase matchingPhase matching will improve the conversion efficiency and will reduce par-asitic effects by creating unfavourable conditions where energy and mo-mentum are not conserved. In the RE-FWM process under discussion theoptical depth (absorption) of the τR, τT and τLy−α are negligible. Assumingplane waves and for a phase matched system with ∆k = 0 Eq. (2.40) give usthe phase matching factor. This expression reduces to :F(∆kL = 0, τR, τT  1, τLy−α  1) ≈ 1−τR + τT + τLy−α2. (2.52)For Gaussian beams, without considering optical absorption, Eq. (2.42)gives us the phase matching factor by performing the integral. For sum-difference Bjorklund [84] has already worked out the optimum conditionwhich is ∆kb = 0 (see Eq. (2.43)). However, this is only possible if ∆k = 0,532.2. Source design and optimizationwhich is impossible to accomplish in a system with optical dispersion. Fig.2.8 is the calculated wave vector mismatch for our process in Kr which is:CN = 2pi(nR −nT −nLy−α)λLy−α, (2.53)where C is the vector mismatch per atom andN is the number density. Theindex of refraction can be calculated from the linear standard Sellmeierformula:(n− 1) =Nre2pi∑ifiλ−2i −λ−2, (2.54)and the sum is performed over all the allowed transitions i, re is the clas-sical electron radius and fi are the oscillator strengths. Fig. 2.9 shows thedispersion near the resonances of Kr and Ar and the index of refractionfor λR and λT . Also it shows the negative dispersion around the Lyman-αregion.Fortunately, the condition∆k = 0 can be accomplished in a two-componentsystem by “tuning” the effective index of refraction of the system. Twocomponent phase matching in gases was first reported by Harris et al [71,106, 107]. Effective index of refraction tuning by mixing two gases canaccomplish the phase matching condition if the gases have opposite signoptical dispersion or if the generated wave is at least near a region withnegative optical dispersion. In this case Kr has negative dispersion (or atleast the dispersion turns negative) around the Lyman-α region. Optimiza-tion is usually done by adjusting the ratio of the concentration of gases. Ifone of the components has no resonances near the optical fields appliedthen it only contributes to the linear dispersion and nonlinear effects onthis medium do not play any role. The ratio of the two components can be542.2. Source design and optimization121.1 121.2 121.3 121.4 121.5 121.6 121.700.511.522.53x 104|CN| [cm-1 ]λ[nm]Figure 2.8: Calculated wavevector mismatch for the RE-FWM process inKr where C is the vector mismatch per atom and N is the number density.This mismatch leads to destructive interference of the input beams and thegenerated Lyman-α radiation.calculated using:NaNb=ωLy−αχ(1)b (ωLy−α)−[2ωRχ(1)b (ωR) +ωTχ(1)b (ωT )][2ωRχ(1)a (ωR) +ωTχ(1)a (ωT )]−ωLy−αχ(1)a (ωLy−α). (2.55)For a second component Ar gas has a suitable index of refraction and en-ergy levels. Fig. 2.10 shows the Grotrian diagram of Ar and its ionizationthreshold [100]. In addition some of the parasitic processes that could oc-cur in it are indicated.As mentioned previously the oscillator strengths used for the suscep-tibility calculation have very large uncertainties and the calculated refrac-tive index is not very accurate. It is possible to obtain a very accurate index552.2. Source design and optimizationx 10-30.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.652.53.03.54.04.55.05.5χ(1) (λ)x 10-40.105 0.11 0.115 0.12 0.125 0.13−5−4−3−2−101234λ[μm]λ[μm]5s 2 [3/2]o5s 2 [1/2]oNegative DispersionAr KrArKrn-14s 2 [1/2]o4s 2 [3/2]oFigure 2.9: Calculated dispersion near the electric dipole resonances of Krand Ar and the calculated index of refraction for λR and λT . The negativedispersion around the Lyman-α region is important for phase matching.562.2. Source design and optimization3s23p6  1S03p5(2Po3/2)4p 2[1/2]3p5(2Po1/2)4s 2[1/2]o3p5(2Po3/2)4s 2[3/2]o98855.0698 cm-1107054.2720 cm-193750.5978 cm-149427.5349 cm-182259.16 cm-1Ly-α 121.5667 nmλR 202.32 nmλT 602.56 nmλR 202.32 nmAr95399.8276 cm-1127109.842 cm-1 Ionization Energy ωStokesAr Multiphoton IonizationAr Hyper-RamanRE-FWM in KrFigure 2.10: Argon’s Grotrian diagram and some of the possible competingnonlinear processes. The lack of energy states in the energy region usedfor the nonlinear process makes it an ideal gas for phase matching mixingwith Kr. The energy levels are based on Ref. [100]572.2. Source design and optimizationof refraction for the gases using empirical three-point Sellmeier relations[108] and this has been used to calculate nR and nT . Unfortunately, theempirical Sellmeier relation does not predict the index of refraction in theregion between the Kr resonances very well. Tabulated values for nLy−αbased on extrapolations of the linear Sellmeier relation adjusted to matchexperimental values [108, 109] have been used to calculate nLy−α and theoptimal gas ratio. The tabulated values give two regions of gas mixtureswhere it is possible to accomplish phase matching. The mixture ratiosAr:Kr are between 3-4:1 and 16-18:1. Experimentally, it was found thatthe optimal ratio for the first region was Ar:Kr ≈ 3.9:1. and for the secondregion was Ar:Kr ≈ 17.7:1. However, the mixture ratio Ar:Kr ≈ 3.9:1 pro-duced ten times more power at λLyα than at any other mixture. The indexof refraction that are consistent with this first region of gas mixture ratioare shown in Table 2.3.Table 2.3: Index of refraction from three-point Sellmeier relation of Krand Ar used to calculate the phase matching mixture.Refraction Index [n(λ)− 1]× 106Medium λ⇒ 121.56 nm 202.31 nm 602.56 nmKr 561.40 513.65 427.59Ar 488.85 320.97 281.46Now that the condition∆k = 0 can be accomplished, Eq. (2.43) indicatesan optimal ratio b / L = 1. This is identical to the plane wave case. Withoutgeometrical restrictions it is possible to use a loosely focused configuration.The loosely focused condition will assure that the intensity is below thesaturation regime. In addition plane wave approximations are effectiveas the radius of the Gaussian wave fronts RGaussian(z < b) → ∞ and theinteraction happens in a nearly plane wave region well within the Rayleigh582.3. Third harmonic generationlength.In summary, in this section a Lyman-α source based on FWM has beendesigned. The exact energy levels and two-photon resonance in kryptonto be used for the process have been identified. The third order suscepti-bility for this process has been calculated and this indicates that the pro-cess should be efficient at generating VUV. Also the two-photon resonanceshape has been calculated. An important result of this calculation is thateven with the large number of stable krypton isotopes, the resonance broad-ening caused by the isotope shift does not cause a large reduction of theresonance strength (only ∼ 32%). A limit for the power of ωT photons hasbeen calculated indicating that the power of ωT should be a maximum of∼ 15% of the power of ωR for efficient conversion. The most important par-asitic process that can potentially occur, while reducing the conversion effi-ciency, have been identified. These processes are ASE, Parametric TPA andmultiphoton ionization. Strategies to reduce their occurrence have beendiscussed. These strategies will be implemented and further discussed inthe next chapter. Finally, the wavevector mismatch has been calculated anda strategy for phase matching has been identified. Phase matching requiresthe use of a gas mixture. The second gas in the mixture has been identi-fied as argon and parasitic processes that could occur in this gas have beenstudied. The exact gas mixture ratio required for phase matching has alsobeen calculated and is Ar:Kr ≈ 3.9:1. All of this information was used toimplement the source and this is described in Chapter 3.2.3 Third harmonic generationSeveral Lyman-α sources have been designed based on THG and Table 2.4is a compilation of sources using Kr as the nonlinear medium. Based onall the information available a non-resonant THG source was implementedin parallel with the RE-FWM implementation. Fig. 2.11 shows the energydiagram of the process.Most of the optimal parameters in THG have been already worked outand the limits of conversion have been explored. The process has lower592.3. Third harmonic generationTable 2.4: Compilation of Lyman-α sources implemented by tripling 364.8nm and the reported efficiency of the conversion.Nonlinear medium Phase matching Efficiency η ReferenceKr Ar 10−5 [58]Kr Ar 10−4 [59–62]Kr 10−5 [63]conversion efficiency than RE-FMW and the phase matching parameterspace is much more restricted. However, THG is very attractive becauseit only requires one input color. This simplifies the implementation for thefollowing reasons:• Only one input beam is needed and no dichroic mirror is requiredto combine beams which is usually the mot difficult optical item toobtain.• The wavelength of the input beam is not in the UV region of the spec-trum which results in less optical losses through the optical elements.• In Kr there are less paths for parasitic processes and for instance ASEand multiphoton ionization is not a problem.• A problem in FWM processes is the aligning of the different beams inthe nonlinear medium due to dispersion and chromatic aberration ofoptical elements; this is not an issue in THG.• Since only one beam is used, the temporal alignment of pulsed laserbeams is not an issue.• The phase matching condition could be easier to achieve as only twowavelengths are involved.In addition there are some additional features that make this imple-mentation attractive to the antihydrogen experiments with ALPHA:602.3. Third harmonic generation4s24p6  1S04p5(2Po3/2)5p 2[1/2]4p5(2Po1/2)5p 2[1/2]4p5(2Po1/2)5s 2[1/2]o4p5(2Po3/2)5s 2[3/2]o98855.0698 cm-194092.8626 cm-185846.7046 cm-180916.7680 cm-126972.2560 cm-182259.16 cm-1Ly-α 121.5667 nmλT 364.68 nmKr53944.5120 cm-1Figure 2.11: Energy diagram of non-resonant THG process used for thegeneration of Lyman-α radiation in Kr. The process uses only one inputlaser at 364.68 nm.• The system requires less maintenance than a FWM based system asonly one input laser is required.• THG is a robust optical system as it does not require two collinearbeams that would misaligned over time due to vibration.• The system can be implemented with solid state lasers instead of dyelasers which require more maintenance.THG does require high intensity for the three photon process to occur.This is usually accomplished by focusing the Gaussian beam. In the limitof tight focusing, b / L 1, the limits to the phase matching integral of Eq.(2.42) can be approximated to L/ 2→±∞. The phase matching factor then612.3. Third harmonic generationreduces to:F(∆kL,b / L 1) =0 ∆kb ≥ 0pi24(bL)2(∆kb)2 exp {∆kb} ∆kb < 0,(2.56)which indicates that the wavevector difference should be negative ∆k < 0.In fact the function goes through a maximum for ∆kb = −2 and the phasematching factor becomes:F(∆kb = −2,b / l 1) =[pibL]2exp {−2}. (2.57)A two component mixture can also be used to find the phase matchingcondition in THG. A similar calculation as before predicts a gas mixtureratio of Ar:Kr⇒ of 2.8:1 for ∆k = 0. However, THG requires ∆k < 0 whichcan be reached only with a mixture ratio lower than Ar:Kr⇒ 2.8:1 and byadjusting the focus of the beam. In this implementation the optimal ratiowas found experimentally and details of this are in the next chapter wherethe laser power measurements are discussed.62Chapter 3Source implementation andcharacterizationChapter 2 addressed the design of a Lyman-α source based on a RE-FWMprocess. The resonance enhancement is used as means of increasing theconversion efficiency of the process. RE-FWM process are reported to havea conversion efficiency η on the order of (≥ 10−4) [67, 68, 110]. This is anorder of magnitude larger than typical THG processes (see Table 2.4 ). Theparticular two-photon resonance that is used in this work is particularlystrong. This resonance coupled with the quasi-resonant third photon tran-sition produces a very large third order susceptibility. We can expect a veryefficient process if the phase matching condition is met and the parasiticprocesses are reduced.As previously discussed, ASE and parametric processes also occur dur-ing the frequency mixing process generating λparametric = 123.5 nm in addi-tion to the Lyman-α radiation λLy−α = 121.56 nm. In order to measure theyield at the Lyman-α line it is necessary to separate all the spectral compo-nents. There are commercially available ”Lyman-α” filters but the trans-mission peak is at 123.0 nm. Narrow filters have a typical transmissionFWHM of 15.0 nm and a peak transmission of 19%. Furthermore, trans-mission through a narrow filter at 200.00 nm is still on the order of 1%.This means that in our RE-FWM process, assuming a maximum conver-sion efficiency of ≥ 10−4, even a 0.1% of the ωR photons surviving throughthe nonlinear process will be enough to bias a power measurement of thefiltered signal. Some of the reports on Lyman-α sources have quoted powervalues based on direct measurements with filters. These measurements areobviously case dependent, but in general have large uncertainties associ-633.1. Source implementationated with the use of the filters. In this particular case, with the detectorsavailable, filters cannot be used. Part of the problem is that solar blinddetectors, which use a Cs-I photocathode, are not characterized on the UVregion of ωR photons as these photons are supposed to be invisible. How-ever, ωR photons are not invisible. The different order of magnitudes ofωLy−α and ωR photons is such, that the detector signal for ωLy−α cannot bediscerned. The detector is blinded by the amount of ωR photons. This isthe problem and the reason this measurement is interesting. In this chap-ter the implementation of the Lyman-α source and the power measurementare discussed.During the implementation of the source the FWM process was opti-mized by changing the phase matching parameters (gas mixture, confocallength b and interaction length L) and the intensity of the input beams (viathe focusing geometry). As a metric the power of the Lyman-α radiationand light generated by the parasitic processes were monitored. These mea-surements were performed in vacuum using a diffraction grating with aphotomultiplier tube (PMT) and this is discussed in the first section alongwith the optics and vacuum system implementation. To characterize thesource, hydrogen was used as the detector and frequency discriminator.Laser Induced Fluorescence (LIF) of hydrogen was performed as means ofmeasuring the effective power at the Lyman-α source. The hydrogen forthese measurements was produced with a commercial thermal gas crackerthat is mostly used as a source for molecular beam epitaxy. The details ofthis experiment are discussed in the second section. Finally, as mentionedin the last section of the previous chapter, a THG source was developed inparallel to compare with the RE-FWM source. To measure the power of thissource a detector based on the photoionization of acetone was developedand this is discussed in the last section.3.1 Source implementationBased on the results of the previous section a two-photon RE-FWM pro-cess was used to generate high power VUV coherent light. For this pro-643.1. Source implementationcess, the nonlinear medium is Kr gas. Here, two wavelengths (ωR = 202.31and ωT = 602.56 nm) are mixed in a sum-difference scheme (ωVUV =2ωR −ωT ) with a two-photon resonance at (4s24p55p[1/2]o ← 4s24p6(1S0)), 98,855.1311 cm−1 (101.1581 nm). At this wavelength it is possible touse LiF (∼ 80% transmission) or MgF2 (∼ 50% transmission) windows andlenses. The most optimal optical layout is to have the two incoming TEM00mode beams collinear and with identical confocal parameters b. Figure 3.1shows a schematic of the optical arrangement. A Nd:YAG laser (Spectra-Physics, Quanta-Ray Lab- 130) producing light at the second harmonic,532 nm (200 ± 60 mJ in a 8-12 ns pulse with 1 cm−1 linewidth at 10 or 50Hz), was used to pump two dye lasers to generate λR ×3 = 606.93 nm ( in amixture of Rhodamine B and Rhodamine 101 dissolved in methanol) andλT = 602.56 nm (Rhodamine B in methanol) pulses. The 606.93 nm pulseswere further frequency mixed in nonlinear crystals (BBO and KDP) to pro-duce 0.2 ± 0.1 mJ of λR = 202.31 nm. The λT = 602.56 nm pulses were 0.05± 0.03 mJ. A dichroic mirror was used to combine the two pulses, and thecombined beam was focused into the gas cell with a UV fused silica lens.653.1.SourceimplementationDichroic Mirror2nd harmonic 532 nm 10-50 Hz Repetition  10-20 ns Pulse Resonant   SGH THG Telescope FocusingLens Kr,Ar Cell Vacuum  ωR ωR=Resonant LaserωT=Tunable LaserDye Lasers: Tunable   ωTNd:YAGPump LaserPellin Broka Separation Box Telescope λ/2Optical LayoutFigure 3.1: A Nd:YAG laser (Spectra-Physics, Quanta-Ray Lab- 130) producing light at the second harmonic, 532nm (200 ± 60 mJ in a 8-12 ns pulse with 1 cm−1 linewidth at 10 or 50 Hz), was used to pump two dye lasersto generate λR × 3 = 606.93 nm ( in a mixture of Rhodamine B and Rhodamine 101 dissolved in methanol) andλT = 602.56 nm (Rhodamine B in methanol) pulses. The 606.93 nm pulses were further frequency mixed innonlinear crystals (BBO and KDP) to produce 0.2 ± 0.1 mJ of λR = 202.31 nm. The λT = 602.56 nm pulses were0.05 ± 0.03 mJ. A dichroic mirror was used to combine the two pulses, and the combined beam was focused intothe gas cell with a UV fused silica lens.663.1. Source implementationThe pump laser produced a beam diameter of 10 mm with a beam di-vergence of < 0.5 mrad. The dye laser generating the λR × 3 = 606.93 nm(Narrowscan from Radiant Dyes) with a minimum linewidth of 0.04 cm−1∼ 1200 MHz and a wavelength tuneability of ∼ 0.02 nm with wavelengthstability is 0.001 nm / oC. The output was fitted with a double Pellin Brokaseparation box to extract the ASE components generated in the dye. A tele-scope is used to compensate for differences in beam divergences betweenthe beams. Collinearity is achieved by the use of the various mirrors in thepath of ωT (see Fig. 3.1). Temporal alignment of the pulses is achieved byadjusting the optical path of ωR ( by moving the laser and adjusting thepropagation length) until the pulses of the two beams coincide. The λTbeam was generated in a Lambda Physik Scanmate OPPO dye laser whichhas a minimum linewidth of 0.1 cm−1 ∼ 3 GHz and a power stability of10%. The output beams of both lasers were combined with a dichroic mir-ror. Two additional mirrors were used to align the two collinear beams,and the generated VUV beam, onto the detection system.The nonlinear medium, a mixture of Kr:Ar was injected into a vacuumtube via flow meters. The tube was pre-pumped down to 10−6 Torr with aturbo pump. The pressure of the mixture was monitored using a capacitivepressure gauge. Both sides of the tube were sealed with o-ring fitted lensterminations suitable for plano-convex lenses. The incoming side receivedthe UV fused silica lens and the output was fitted with a LiF window ora suitable MgF2 lens according to the different measurements performed.The length of the nonlinear medium and the focal length of the input lenswere optimized for the RE-FWM process.As a first step the ωR laser was calibrated to the two-photon resonanttransition using a photoionization measurement of Kr. In resonant mul-tiphoton ionization the atom absorbs photons and is excited to a quasi-stationary state (atom plus optical field) of the system. These resonancesincrease the ionization probability which is reflected on the photo-electronsignal [111]. As a first order approximation the photoionization probabil-ity peak should coincide with the two-photon resonant transition. The ωRlaser was focused between two parallel plates 0.5 cm apart with a UV fused673.1. Source implementationsilica lens of 10 cm focal length and in a chamber containing low pressureKr gas (10 Torr). A voltage was applied between the plates between 100V and 300 V. As the laser ionized Kr the field between the plates accel-erated ions and electrons toward the plates and this resulted in a currentmeasurement. The laser was tuned by maximizing the current measure-ment. The laser’s bandwidth estimated from this measurement is 0.1 cm−1∼ 3 GHz. The ωT laser was initially calibrated using a laser wavelengthmeter (Coherent WaveMaster) with an accuracy of 0.005 nm (∼ 4 GHz).Further calibration was performed by optimizing the fluorescence signalduring LIF spectroscopy of hydrogen. As shown in the previous chapter(Fig. 2.6), the two-photon resonance is broad and has a complex shape dueto the isotope shift. What this means is that all the laser ωR (with ∼ 3 GHzlinewidth) spectral energy takes part in the resonance transition. In termsof the expected linewidth of the generated Lyman-α radiation, it can be es-timated as the convolution of the ωR and ωT which are both Gaussian. Thestandard deviation of the convolution of two Gaussian functions f and gis:σf ⊗g =√σ2f + σ2g . (3.1)The FWHM ∆1/2 (linewidth) is related to σ :∆1/2 = 2√2ln2σ ≈ 2.35482σ. (3.2)With an estimated linewidth for ωT of 3 GHz, the estimated linewidth ofthe Lyman-α source is ∼ 4.2 GHz. This is only an approximation as the fullcomplexity of the two-photon resonance shape has not been considered.A vacuum system was implemented to characterize the source. Fig.3.2 show the schematic of the system which consists of a main vacuumchamber with a hydrogen source and a detector (solar blind photomulti-plier tube Hamamatsu R972). The Lyman-α chamber is attached to thelaser input port and a home-built monochromator is attached to the laseroutput port. The best vacuum achieved was ∼ 10−8 Torr with a 300 L/secturbomolecular pump. The monochromator chamber has an additional683.1. Source implementationlens, diffraction gratings and pinhole. The same photomultiplier tube wasswapped between ports for LIF experiments or source optimization withthe monochromator.Solar Blind PhotomultiplierHamamatsu R972Spectroscopy ChamberLyman-α GenerationChamberLyman-α202.32 nm602.56 nmKr/Ar HydrogenSourceDiffraction GratingFigure 3.2: Schematic of the vacuum system used to characterize thesource. It consists of a main vacuum chamber with a hydrogen source anda detector (solar blind photomultiplier tube). The Lyman-α chamber is at-tached to the laser input port and a home-built monochromator is attachedto the laser output port. The nonlinear medium, a mixture of Kr:Ar was in-jected Lyman-α chamber via flow meters. The pressure of the mixture wasmonitored using a capacitive pressure gauge. Both sides of the Lyman-αchamber were fitted with o-ring fitted lens terminations that are suitablefor plano-convex lenses.To separate the λparam = 123.5 nm component from the Lyman-α radia-tion a simple monochromator was implemented. A plane ruled diffractiongrating with 600 groves/mm × 26 mm and a maximum resolving power693.1. Source implementationλ/∆λ of 15600 was used. The grating equation:mλ = d(sinα ± sinβ), (3.3)is used to calculate the angles α (incident) and β (diffracted). In the equa-tion d is the groove spacing given by the manufacturer and m is the diffrac-tion order which in this case is negative. Fig. 3.3 shows a picture of thesetup. The angle of the grating was manipulated using a home-built tiltmanipulator which was o-ring sealed. With the o-ring sealed manipulatorit was possible to reach a vacuum of ∼ 10−6 Torr. With collimating opticsat the output of Kr cell and a pinhole 30 cm away from the grating and infront of the photomultiplier tube it was possible to discriminate betweenλparam = 123.5 and λLy−α = 121.56. In addition, the interior of the vacuumtube was covered with blackened foil to reduce reflections.This experimental setup was not suitable for quantitative measurements,but it was used to optimize the phase matching geometry (L, b and mix-ture ratio) qualitatively by comparing the power of λparam = 123.5 andλLy−α = 121.56. Two different ratios of the gas mixtures were used andthree different focusing geometries were compared. Table 3.1 shows thequalitative results.The most optimal configuration was using a nonlinear medium lengthL = 98.9 cm, an input lens focal length of 500 mm and a gas mixture ofAr:Kr of 3:1 with 100 mTorr of Kr. The gas mixture was further optimizedin this configuration to Ar:Kr 3.9:1 and 100 mTorr of Kr. TheωR laser beamhad a 1/e2 radius at the input laser of ∼ 0.75 mm. It is possible to calculatethe Gaussian beam parameters using the relations:1q=1R−iλpiW 2(3.4)andq2 =Aq1 +BCq1 +D, (3.5)where R = z(1 + z20 / z2) is the radius of curvature, W =W0√1 + z2 / z20 is the703.1. Source implementationPhotomultiplierGratingLyman-α ChamberKr,ArVacuumMgF2 LensPinholeFigure 3.3: source characterization chamber with a simple home-builtmonochromator. A plane ruled diffraction grating with 600 groves/mm× 26 mm and a maximum resolving power λ/∆λ of 15600 was used.initial beam radius and z0 = piW 20 / λ. For propagation through a lens A =1, B = 0, C= -1 / f and D = 1; for propagation through air A = 1, B = d (dis-tance), C=0 and D =1. The beam radius at the focal point is calculated thisway to be 40 µm and the confocal length b is 49.67 mm. As mentioned, thiscalculation only applies to Gaussian beams and the beams generated by thedye laser have transverse modes that are not necessarily Gaussian. Since abeam profiler was not available, the transverse modes were not character-ized. However, by visual inspection no multiple transverse modes wereidentified. The UV fused silica lens has a transmission of 93% for 602 nm713.1. Source implementationTable 3.1: Optimization of phase matching parameters. The cell lengthL, input lens focal length f and gas mixture ratio where optimized bymeasuring the power of λLy−α = 121.6 nm and the parasitic λparam = 123.5nm. The wavelength discrimination was done with a diffraction grating ina the setup shown in Fig.3.3. An optimal configuration of L= 28.9 cm, f =500 mm and gas mixture of Ar:Kr of 3.9:1 was adopted for the remainingexperiments.Relative PMT SignalL (cm) f (mm) Ar:Kr Ratio λLy−α = 121.6 nm λparam = 123.5 nm36.5 200 3:1 1 1200 16:1 – –56.9 300 3:1 8 1300 16:1 5 156.9 500 3:1 10 0.2500 16:1 1 –and < 90% for 202 nm. The average ωR intensity at the focal point with arepetition rate of 10 Hz is 3.5 ×105 W/m2. This corresponds to a peak in-tensity with a 10 nsec pulse of 3.5 ×1013 W/m2. This calculation does notconsiders the absorption of the laser before the focal point. The ωT laserbeam had a 1/e2 radius at the input laser of ∼ 2.0 mm. The beam radius atthe focal point is calculated to be 50 µm and the confocal length b is 26.06mm. Fig. 3.4 shows the configuration where the chromatic aberration ofthe input lens is not considered. Finally, similarly to the Kr ionization ex-periment, two bias plates were placed on the Lyman-α generation chamberto monitor ionization around the focal point. A very sporadic photoioniza-tion signal could be measured when the plates where biased to 100 V andmostly due to power fluctuations of the Nd;YAG pump laser. Otherwise,normal operation was below the photoionization threshold.723.1. Source implementationUV Fused Silica Lens MgF2 Lensf = 500 mmL = 989 mmb = 49.67 mmb = 26.06 mmwo = 0.050 mmwo = 0.040 mmr = 0.75 mmr = 2.0 mmwo λ = 202.38 nmλ = 602.56 nmb Ar:Kr = 3.9:1Figure 3.4: Schematic of the incoming beams’ intensity profiles in theLymanα cell. The most optimal configuration was using a nonlinearmedium length L = 98.9 cm, an input lens focal length of 500 mm anda gas mixture of Ar:Kr of 3.9:1.733.2. Lyman-α source characterization3.2 Lyman-α source characterizationIn the previous section the implementation and optimization of the sourcewere discussed. The two-photon resonant laser was calibrated by max-imizing the photoionization current generated by focusing the laser onlow pressure Kr. The phase matching parameters were optimized using ahome-built monochromator by comparing the power of the light generatedby the RE-FWM (Lyman-α) and parasitic processes in different configura-tions. However, the home-built monochromator did not provide enoughrepeatability to be used for quantitative measurements. Furthermore, withthe equipment implemented, it is not possible to do an absolute powermeasurement since there is not enough information to calculate the losses.Instead, to measure the Lyman-α power, LIF of hydrogen was performed.The Lyman-α photons excite hydrogen to the 2p state. Electrons then decayto the ground state and emit Lyman-α photons. However, the emission isomini-directional, and a detector at right angle from the laser would onlybe able to see some of the fluorescence. This is the basis of LIF power mea-surement. The hydrogen for these measurements was generated using acommercial source. In this section the LIF experiment and the absoluteLyman-α power calculated from those measurements are discussed.3.2.1 Effusive hydrogen source spectroscopyAtomic hydrogen was generated by thermal dissociation of H2 with a Man-tis MGC75 thermal gas cracker. The molecular hydrogen was flown throughthe thermal gas crackers heated capillary (tube of 2.5 mm inside diameter)at a maximum rate of 1.0 sscm to generate atomic hydrogen. Hot atomichydrogen emerging from the tip of the capillary was irradiated with theLyman-α source pulses. The fluorescence and absorption of Lyman-α lightwere measured with a solar blind photomultiplier.743.2. Lyman-α source characterizationHydrogen sourceThe gas cracker uses a biased, hot tungsten filament to generate electronsand accelerate them towards an iridium capillary. Electron bombardmentheats up the capillary to 1500 K. Molecules flown through the capillary aredissociated by collisions with the wall. The whole arrangement is encasedon a water cooled block. According to measurements provided by the man-ufacturer it is possible to achieve up to 90% dissociation of hydrogen witha gas flow of up to 1.0 sscm. A published independent characterizationof the device reported a hydrogen flux of 1.3 × 1017 atoms / (sec · cm2)impinging on a sample of highly ordered pyrolyctic graphite [112]. Themeasurement was done by characterizing the profile of chemical etchingcaused by hydrogen on the graphite. The chamber pressure was 10−4 Torrwhich, using the Hertz-Knudsen formula for incident flux F:F =P√2pimkBT, (3.6)is equivalent to an incident flux of 1.58 × 1017 atoms / (sec · cm2). Thismeasurement indicates a steady state atomic density of ∼83% hydrogen inthe chamber.Fig. 3.5 shows the experimental setup used. For LIF experiments aMgF2 lens with a focal length of 200 mm focused the laser directly in frontof the gas cracker at 2.5 cm from the capillary. The photomultiplier tubewas at right angle and assuming a point source for the fluorescence, it cap-tured a 2.2485 × 10−4 of the sphere. The hydrogen in front of the sourcecan be assumed to be thermalized at the temperature of the capillary and is,therefore, very hot at ∼ 1500 K. The velocity distribution of the hydrogenin the chamber is given by the Maxwell-Boltzman distribution:f (v) = 4pi[m2pikBT]3/2v2 exp{−mv22kBT}. (3.7)For hydrogen the mean speed v˜ =√(8kBT ) / (pim) is 5635.52 m/s. The ra-753.2. Lyman-α source characterizationdiative lifetime of the 2p state of hydrogen is:τk =∑iAki , (3.8)where Aki are the transition probabilities to all levels i lower in energy thank. For hydrogen τ2p = 2.128 nsec based on published transition probabil-ities [100]. In the chamber, an atom would have moved 12 µm betweenabsorption of the photon and the fluorescence which is not a problem forthe LIF experiment as the view area of the detector is larger. The actualspot size of the Lyman-α source is unknown. However, it is reasonable toassume that the laser has approximately the same Gaussian beam parame-ters as ωR and most of the fluorescence is produced at the focal point of thelasers. Following this assumption the Lyman-α source would have a radiusof 2 mm at the focusing lens, a waist radius less than 10 µm and confocallength b ≈ 20 µm. Given these dimensions, it is assumed that the fluo-rescence is point-like and the detector captures all of the photons withinthe solid angle viewing area (2.2485 × 10−4 × 4pi). The chamber is filledwith hydrogen and there is absorption (and fluorescence) of Lyman-α be-fore the focal point of the laser, but this is not considered. Finally, the abso-lute power measurement based on fluorescence has a large uncertainty, dueto the described assumptions. However, it is possible to measure a lowerbound for the Lyman-α power, and this bound is really what is required toevaluate the source.763.2. Lyman-α source characterizationSolar Blind PhotomultiplierHamamatsu R972MgF2 lens F = 20 cmT121.5 nm = 52%202.39 nm602.56 nmUV Fused SilicaLens F = 50 cmLyman-α Chamber98.9 cmKr 100 mTorrKr:Ar 1:3.9Thermal Gas CrackerMantis MGC75TMPH and H2MixtureMgF2 Window23.32 cmLIF SetupDiffractionGratingFigure 3.5: For LIF experiments a MgF2 lens with a focal length of 200mm focused the laser directly in front of the gas cracker at 2.5 cm fromthe capillary. The photomultiplier tube was at right angle and assuming apoint source for the fluorescence, it captured a 2.2485 × 10−4 of the sphere.The hydrogen in front of the source can be assumed to be thermalized atthe temperature of the capillary and is, therefore, very hot at ∼ 1500 K.773.2. Lyman-α source characterizationHydrogen spectroscopyTo measure fluorescence, the light scattered was recorded with the PMT. Aphotodiode (activated by the ωT ) was used to trigger the data collection.The PMT uses a MgF2 window and a Cs-I photocathode. Fig. 3.6 showsthe PMT pulses and the photodiode signal. Unfortunately, Cs-I is still ac-tivated by ωR, which is at the threshold of its spectral range (115 nm - 200nm). Most of the PMT pulse is caused by scattered ωR = 202.38 nm pho-tons. However, when the hydrogen source is turned on and off it is possibleto see a small difference caused by Lyman-α fluorescence. During the ex-periments the molecular hydrogen was continuously flowing and only thebias between the tungsten filament and the capillary was switched on andoff. The pulses of Fig. 3.6 are the result of 128 averages. The actual fluo-rescence signal was extracted by subtracting the pulses with the hydrogensource on and off. The total energy per pulse is proportional to the inte-grated fluorescence signal also shown at the bottom of Fig. 3.6. The totallength of the PMT signal pulse is 23.8 nsec, which is much longer than theexpected Lyman-α pulse length due to the detector response time.PMT’s can be modelled as capacitive-resistive (CR) circuits followed bystages of RC circuits. CR stages are differentiators and RC are integrators.The time domain output signal for a delta pulse input signal to an RC-CRcircuit is:Vout(t) =V0τ1τ1 − τ2(exp−tτ1− exp−tτ2), (3.9)where τ are the time constants and V0 is the input voltage. Fig. 3.7 showsthe time domain response for a single RC-CR circuit. For a large numberof stages the output pulse becomes Gaussian. For the Hamamatsu PMTthere are 11 stages. The rise time (time for the output signal to go from10% to 90% given a delta pulse input) is 1.6 nsec. The electron transittime is 17 nsec and the maximum gain is 105. For the linear-focused PMTs(the type used here) the expected pulse width (FWHM) of the responsefunction is 5 nsec. The standard deviation σPMT is 2.12 nsec. To calcu-late the input pulse shape, it must be de-convoluted from the output PMT783.2. Lyman-α source characterization0 20 40 60 80 100 120 140 160 180 200−300−200−1000-1.001.02.0<  23.8 nsecPMT Signal (mV)Photodiode (V)Time (nsec)PhotodiodePMT SignalH offH on14.9 mV0  PMT Pulse14.9 mV FluorescenceFLy-α =┃(Hoff- Hon)┃∫ dt F Ly-α  (mV∗nsec)160 mV∗nsecIntegrated FluorescenceFigure 3.6: To measure fluorescence the light scattered was recorded withthe PMT. A photodiode (activated by the ωT pulse) was used to trigger thedata collection. Most of the PMT pulse is caused by scattered ωR = 202.38nm photons. However, when the hydrogen source is turned on and off itis possible to see a small difference caused by Lyman-α fluorescence. Thepulses are the result of 128 averages. The actual fluorescence signal wasextracted by subtracting the pulses with the hydrogen source on and off.The total energy per pulse is proportional to the integrated fluorescencesignal shown at the bottom. The total length of the PMT signal pulse is23.8 nsec, which is much longer than the expected Lyman-α pulse lengthdue to the detector response time793.2. Lyman-α source characterizationsignal using the calculated response function of the photomultiplier. Thede-convolution of two Gaussians is another Gaussian and using Eq. (3.2)is possible to extract the standard deviation. The PMT signal can be fittedto a Gaussian with a standard deviation σsignal = 5.12 nsec and the PMTσPMT = 2.12 nsec. The laser pulse (combined ωR and Lyman-α photons)σLaser = 5.0 and the FWHM is 11.9 nsec. The Lyman-α (fluorescence) pulseis smaller with a FWHM of 9 nsec.In addition to the power measurement, LIF was also used to perform acrude measurement of the Lyman-α linewidth. To calibrate the ωT the flu-orescence signal was measured with the PMT as the wavelength was swept.The peak of the signal correlates with the power of the generated Lyman-αradiation. These measurements also provided a line shape used to estimatethe source linewidth. As mentioned before, the hydrogen in the chamberdirectly in front of the the source has a mean speed of 5635.5 m/sec andthis produces Doppler broadening:∆ω =2ω0c√2ln2kBTm. (3.10)The Doppler broadening for hydrogen is 68.42 GHz at FWHM. The naturaltransition line shapes of atoms have a Lorentzian profile and for the hydro-gen (1s − 2p) transition the linewidth of this profile is 99.7 MHz. Thermalbroadening produces an absorption line with a Gaussian profile. The netprofile is the convolution of both profiles, a Gaussian and a Lorentzian,which is called a Voigt profile. The Voigt profile is dominated by the Gaus-sian shape around the mean and by the Lorentzian in the wings (tails). Inthis case the thermal broadening linewidth is approximately three orders ofmagnitude larger than the natural linewidth and the net line shape can beassumed to be Gaussian with a FWHM of 68.43 GHz which is 2.281 cm−1.The measurements shown in Fig. 3.8 were fitted with a Gaussian that isassumed to mimic the line shape of the transition. The measurement at thepeak of the Gaussian in Fig. 3.8 corresponds to the fluorescence recordedin Fig. 3.6 with a peak fluorescence signal of 14.9 mV. The standard de-viation of the measured transition shape is σmeasured = 0.9725 cm−1 and803.2. Lyman-α source characterizationDynodes-HVRLAnodeCathodeVoltagetimePhotonRCCRRC-CRRise Time = 1.6 nsec90%10%Electron Transit Time = 17 nsecPMT Responseσ = 2.12 nsecRRC CPMT Signalσ = 5.12 nsec⊗=Photon Pulseσ = 5.0 nsecFigure 3.7: PMT’s can be modelled as capacitive-resistive (CR) circuits fol-lowed by stages of RC circuits. CR stages are differentiators and RC areintegrators. For a large number of stages the output pulse becomes Gaus-sian. To calculate the input pulse shape, it must be de-convoluted from theoutput PMT signal using the calculated response function of the photomul-tiplier. The de-convolution of two Gaussians is another Gaussian and usingEq. (3.2) is possible to extract the standard deviation. The PMT signal canbe fitted to a Gaussian with a standard deviation σsignal = 5.12 nsec and thePMT σPMT = 2.12 nsec. The laser pulse (combined ωR and Lyman-α pho-tons) σLaser = 5.0 and the FWHM is 11.9 nsec. The Lyman-α (fluorescence)pulse is smaller with a FWHM of 9 nsec.813.2. Lyman-α source characterizationthe standard deviation corresponding to a linewidth of 68.43 GHz is σH =0.968 cm−1. It is possible to de-convolute the source line shape from themeasured spectrum using the calculated line shape of the hot hydrogen.The calculated linewidth is 6.6 GHz which is 2 GHz larger than the origi-nal estimated from the ωT input laser bandwidth of ∼ 4 GHz. This is mostlikely due to the broadening of the two-photon resonance caused by col-lisions (pressure broadening), power broadening and ionization of Kr andthe linewidth of the ωR.Finally, the maximum power measured for the Lyman-α source was cal-culated from the integrated fluorescence signal as shown in Fig. 3.6. Basedon an anode minimum sensitivity at ΛLy−α of 200 A/W the power of theLyman-α radiation is:0.125 ± 0.030 µJ @ 10 Hz and 9 nsec pulses: 1.25 ± 0.30 µW.This power assumes that every Lyman-α photon is being scattered byhydrogen. The uncertainty of the PMT gain is not known and very littleinformation about it has been published. It could potentially be quite largeas the gain in region of operation is not linear. The peak power delivery perpulse 12.45 W and the area at the Lyman-α beam focus was estimated to be3.14 × 10−10 m2. The intensity is 39.63 × 109 W/m2. The conversion effi-ciency of the process is η ∼ 6×10−4 which agrees with the reported efficien-cies on the order of (≥ 10−4) [67, 68, 110]. The power generated meets therequirements set for antihydrogen cooling. Power measurements were per-formed also at 50 Hz repetition rate as it was thought that the higher rep-etition would yield higher average power. However, at 50 Hz the Nd:YAGlaser could not deliver as much energy per pulse. The maximum powermeasured for 50 Hz is:0.025 ± 0.006 µJ @ 50 Hz and 9 nsec pulses: 1.25 ± 0.30 µW.These power measurements represent the maximum obtained after care-823.2. Lyman-α source characterization16584.1 16590.1 16596.1 16602.1 16608.1051015  Fluorescence Peakmax(FLy-α (ν) =┃(Hoff- Hon)┃)Gaussian Fitσ = 0.9725μ = 16595.2FWHM = 2.290Fluorescence Signal (mV)νT (cm-1) Relative to Calibration16593.8  Figure 3.8: The net line shape of the hydrogen 1s − 2p transition can beassumed to be Gaussian with a FWHM of 68.43 GHz which is 2.281 cm−1.Fluorescence measurements performed as ωT was swept where fitted witha Gaussian that is assumed to mimic the line shape of the transition. Themeasurement at the peak of the Gaussian corresponds to the fluorescencerecorded in Fig. 3.6 with a peak fluorescence signal of 14.9 mV. The stan-dard deviation of the measured transition shape is σmeasured = 0.9725 cm−1and the standard deviation corresponding to a linewidth of 68.43 GHz isσH = 0.968 cm−1. It is possible to de-convolute the the source line shapefrom the measured spectrum using the calculated line shape of the hot hy-drogen. The calculated linewidth is 6.6 GHz which is 2 GHz larger thanthe original estimated from the ωT input laser bandwidth of ∼ 4 GHz. Thisis most likely due to the broadening of the two-photon resonance causedby collisions (pressure broadening) and ionization of Kr and the linewidthof the ωR.833.3. Implementation of THG Lyman-α sourceful optimization and capture the potential of the system. The system is notstable in the sense that it requires careful daily optimization before op-eration and changes hour-by-hour. Power fluctuations presumably due totemperature in the pump and dye lasers make these measurements chal-lenging. Also the nonlinear crystals change during the measurement run,presumably due to temperature gradients produced by the laser, and thisis reflected in the output UV light power. The density of hydrogen is high(∼ 10−4 Torr) so it is assumed all the Lyman-α photons are absorbed, whichis reasonable given the cross section for this radiation is very large. Asmentioned previously, the high temperature of the hydrogen causes largethermal broadening of 62.48 GHz. The laser linewidth was calculated tobe ∼ 6 GHz which is about a tenth of the linewidth of the hydrogen and forthat reason efficient at inducing fluorescence (all the photons can interactwith some velocity subset of atoms). It is interesting to perform the sameexperiment with cold hydrogen. However, hydrogen at 1 K already has aDoppler broadening of 1 GHz. To reduce the Doppler broadening of thehydrogen, or to “cool” the hydrogen, there are well established molecularbeam techniques. In the next chapter the implementation and characteri-zation of a supersonic beam of hydrogen is described.3.3 Implementation of THG Lyman-α sourceAs mentioned previously, the conversion efficiency of the RE-FWM pro-cess is higher than that of THG. However, implementation of THG is easiersince only one laser is required. Even though the phase matching condi-tion for THG is more restricted than other FWM processes, it is more for-giving with respect to parasitic processes. As mentioned before there areno multiphoton ionization, multiphoton absorption or ASE parasitic pro-cesses occurring during the Lyman-α generation. Power measurements arealso simpler since the input laser is easily discriminated. For instance, twoLyman-α filters in series with a transmission of < 0.1% at 364.7 nm and asolar-blind photomultiplier would be sufficient for a power measurementof the Lyman-α light generated. In this section the implementation and843.3. Implementation of THG Lyman-α sourcecharacterization of the THG source is described.3.3.1 Source implementationThe THG of Lyman-α requires an input laser λTHG = 364.68 nm. To gener-ate it an Nd:YAG laser (Spectra-Physics, Quanta-Ray Lab- 130) producinglight at the second harmonic, 532 nm (200 ± 60 mJ in a 8-12 ns pulse with1 cm−1 linewidth at 10 Hz), was used to pump a dye laser (Lambda PhysikScanmate OPPO) using Rhodamine B in methanol. This generated pulsesof 729.36 nm light which were further doubled in a BBO crystal producingλTHG = 364.68 nm pulses of ∼ 2 ± 0.5 mJ. The laser was focused into achamber with a Kr:Ar mixture using a fused silica lens with f = 300 mm.The cell length L was 56.9 cm and the 1/e2 radius of the beam at the lenswas 2 mm. The radius at the focus is calculated to be 20 µm. This is equiva-lent to a peak intensity of 1.5×1015 W/m2 which is two orders of magnitudehigher than the peak intensity of the RE-FWM process. This is required todrive the three-photon process. The optimization of the gas mixture wasdone by measuring the power of the generated VUV light. Calibration ofthe 729.36 nm light was done with the wavelength meter (Coherent Wave-Master) with an accuracy of 0.005 nm. At these power levels for λTHG itwas thought that the incoming photons could induce a three-photon res-onant process on hydrogen generating fluorescence at Lyman-α. For thisreason LIF was not used to measure the power of the THG source. Further-more, the power of the λTHG laser was beyond the damage threshold of thediffraction grating which is only a couple of centimetres squared. For thesereasons, to measure the absolute power an acetone ionization detector wasimplemented.3.3.2 Acetone ionization detectorAcetone (CH3)2CO has been previously investigated using high resolutionVUV synchrotron radiation in the energy ranges from 3.7 - 10.8 eV [113].The photo-absorption spectrum is well know and there are reliable mea-surements of the cross section. Furthermore, high purity acetone of 99.8%853.3. Implementation of THG Lyman-α sourceis readily available. The ionization threshold of acetone has been investi-gated by mass-analyzed threshold ionization and zero-kinetic energy tech-niques [114]. The ionization energy is 78299.6 cm−1 and the absorptioncross section for Lyman-α radiation is σLy−α = 38 Mb. The THG inputlaser λTHG = 364.68 nm is equivalent to an energy of 3.3998 eV (27422cm−1). These photons are well below the ionization threshold and the pho-toabsorption cross section is ∼ 0.01 Mb. For these reasons acetone is verypromising as a Lyman-α radiation power detector for the THG process.To implement a detector a vacuum chamber fitted with a LiF windowwas connected to the Lyman-α cell. The acetone chamber was fitted withtwo parallel plates 20 cm long and 2 cm wide separated by 1 cm. Thelaser propagated along the cell between the plates. The plates were biasedup to 500 V. A tube containing the liquid acetone sample is connected tothe vacuum chamber and the acetone was degassed by repeated freeze -pump - thaw cycles. The acetone was frozen by submerging the tube inliquid nitrogen. After a few cycles the acetone was frozen once more andthe chamber pumped down to ∼ 10−6 Torr. After the acetone thaws it fillsthe chamber with acetone vapour and the pressure is monitored with apressure gauge. Since acetone vapour pressure is ∼ 100 Torr at room tem-perature, the density inside the cell is high. When the laser propagatesbetween the bias plates, it is possible to measure the current generated bythe photo-electrons. The electron count is proportional to the Lyman-αpower:Γe = ΦσLy−αN, (3.11)where Γe is the electron count per time volume, Φ is the photon flux andN is the molecule number density. Lyman-α power measurements wereperformed with the acetone detector as the gas mixture was varied. Fig.3.10 shows the power measurement and the optimal gas mixture.863.3. Implementation of THG Lyman-α sourceLN2100 TorrATMPLaserCCCOHHHHHHFigure 3.9: Acetone (CH3)2CO was used to detect Lyman-α radiation.The ionization energy is 78299.6 cm−1 and the absorption cross sectionfor Lyman-α radiation is σLy−α = 38 Mb. λTHG = 364.68 nm is well belowthe ionization threshold. The acetone chamber was fitted with two parallelplates 20 cm long and 2 cm wide separated by 1 cm. The laser propa-gated along the cell between the plates. The plates were biased up to 500V. A tube containing the liquid acetone sample is connected to the vacuumchamber and the acetone was degassed by repeated freeze - pump - thawcycles. When the laser propagates between the bias plates, it is possible tomeasure the current generated by the photo-electrons. The electron countis proportional to the Lyman-α power.873.3. Implementation of THG Lyman-α source−100 0 100 200 300 400 500 600 700−0.0100.010.020.030.040.050.060.07Pressure (mBar)Energy (uJ/pulse)Lyman−Alpha Energy from THG (364.5 nm) @ 4mJKr + ArFigure 3.10: Power measurements performed in an acetone detector. Thelaser propagated along the acetone cell between the biased plates. Photo-electrons generated by the Lyman-α photons and based on the acetonecross-section of σLy−α = 38 Mb the power was calculated. The ionizationenergy of acetone is 78299.6 cm−1, well beyond the λTHG energy of 27422cm−1. During the measurement Kr was added first to the mixing cell andthen Ar. The increase in power is due to better phase matching conditions.883.3. Implementation of THG Lyman-α sourceThe maximum power generated (∼ 0.6 µW) indicates a conversion effi-ciency η ∼ 3× 10−5. Since the THG source was only implemented for com-parison with the RE-FWM source, no more measurements were performed.An advantage of the THG process is that the input power can potentiallybe increased more than in the RE-FWM case. This is due to the fact that thethreshold intensity for photoionization of Kr is much higher for the λTHGphotons than for the two-photon resonant λR photons. Also there is verylittle losses in the optics for the THG system. In Chapter 5 the two systemsare compared and the current work on the THG source is discussed.89Chapter 4Hydrogen beamimplementationIn the previous chapters the design, optimization and implementation ofa Lyman-α source were discussed in detail. Although the development ofthe source is the main focus of this dissertation, an equal or even largereffort has been put into the development of a cold hydrogen beam. In theintroductory chapter it was mentioned that in order to cool hydrogen it wasestimated that the source would require a linewidth on the order of ∼ 100MHz and a power of ∼ 1.0 µW. The hot hydrogen LIF experiment describedin Chapter 3 indicated that the source could potentially be used for the an-tihydrogen experiment as the power threshold was met. However, LIF on acold hydrogen beam would confirm that the source could detect and, sub-ject to linewidth reduction, even cool antihydrogen. It is important to pointout that the main limitations on the bandwidth of the generated laser arethe linewidth of the input lasers. The high temperature of the hydrogendissociated by the gas cracker causes large thermal broadening of 62.48GHz. The laser linewidth was calculated to be ∼ 6 GHz which is about atenth of the linewidth of the hydrogen transition and for that reason it isvery efficient at inducing fluorescence. However, hydrogen at 1 K alreadyhas a Doppler broadening of 1 GHz. To reduce the Doppler broadeningof the hydrogen transition, or to “cool” the hydrogen, we have developeda supersonic hydrogen beam. The implementation of the beam is basedon the well established techniques of supersonically expanded molecularbeams. In this chapter the implementation of the beam and its characteri-zation are discussed. The first section is a very short summary of the the-ory of supersonic beams. This is followed by a description of the hardware904.1. Supersonic beamsimplemented for the experiment. The third section describes the beamcharacterization and the LIF results. Finally, the last section describes theelastic collision experiments with a Rb MOT used to characterize the beam.4.1 Supersonic beamsA beam of cold atoms can be obtained though a supersonic jet expansionmechanism. The gas expansion from a high-pressure chamber into a vac-uum through an orifice produces internally cold, isolated gas phase atoms(or molecules). The high-pressure reservoir usually contains a dilute mix-ture of the desired molecule within a noble gas. The monoatomic noblegases have superior cooling properties due to the lack of internal degreesof freedom. If the ratio of stagnation pressure (high-pressure reservoir) tobackground pressure (vacuum) exceeds the value:P0Pb>[(γ + 1)2] γ(γ−1), (4.1)in which γ is the ratio of heat capacities γ = Cp/Cv of the mixed gas (seedgas and sample), the velocity of the exiting beam will exceed the speed ofsound in the gas mixture and undergo further expansion within the vac-uum region [115]. This expansion can be approximated as adiabatic andisentropic, so most of the work done by the differential pressure modifiesthe internal energy of the gas. In the case of monatomic species, γ = 5/3,the change of internal energy is a direct change on their kinetic energy. Theexpansion causes cooling of the expanding gas (narrowing of the Maxwell-Boltzman distribution) due to the large number of binary collisions. Theexpanding zone is insulated from the background gas by the shock wavebarrier formed on the outskirts of the expanding envelope. The coolingzone is called the zone of silence and the speed of molecules is supersonic(Mach > 1). A cold collimated beam can be extracted from this expandingcloud with the use of a skimmer (a conical or parabolic differential pump-ing orifice). With this technique, temperatures as low as ∼ 1 K are readily914.1. Supersonic beamsachieved. The speed distribution of a supersonic beam can still be approxi-mated by a Maxwell-Boltzman distribution with the central velocity shiftedby vmean:vmean =√√2γγ − 1RT0m1−[PbP0] γ−1γ, (4.2)where R is the ideal gas constant, m is the mass of the expanding atomsand T0 is the temperature of the gas reservoir. With this equation and theMaxwell-Boltzman distribution it is possible to calculate the beam charac-teristics of various monoatomic gases with a reservoir at 300 K (see Table4.1).Table 4.1: Supersonic beam characteristics for noble gases with a reservoirtemperature of 300 K.Species Mean velocity (m/sec) Velocity Spread (σv) (m/sec)He 1800 22Ne 900 11Ar 600 17Kr 450 11Xe 390 9.8With the use of pulsed valves (to limit the beam temporally and reducepumping requirements), skimmers, and differential pumping, it is possibleto make a dense collimated beam of stable species. If the desired speciesis metastable, or a free radical, the technique has to be modified so thespecies of interest is introduced at a point in the beam formation whereany reactions are reduced to a minimum. This is usually done during theexpansion.924.2. Beam implementation4.2 Beam implementationThe hydrogen beam implementation had two purposes. The first purpose,which is the main topic of this dissertation, is to confirm that the Lyman-αsource could detect cold hydrogen via Doppler-free LIF. This is the maindiscussion in this chapter. The second purpose is to perform hydrogen col-lision scattering experiments with ultra-cold Rb atoms trapped in a MOT.This particular collision experiment has not been performed by any othergroup to date. The collision experiments (H - Rb) are still ongoing andwill not be discussed in this chapter. However, collisions of a beam of no-ble gases with the Rb-MOT were performed as a way of characterizing thebeam; this is included here. There is a larger picture to this experimentthat needs to be mentioned since a lot of the experimental equipment im-plementation only makes sense in view of this larger picture.In addition to performing the LIF spectroscopy of hydrogen, this ex-periment is also setting the framework necessary for the photo-associationof the RbH molecule. Theoretical predictions indicate that RbH has anenormous dipole moment. The dipole moment calculated is the largest ofall the alkali-metal hydrides, which in turn is much larger than all the bi-alkalis. The dipole moment of RbH in the singlet state was calculated at10.46 Debye (Competing only with CsH) [116, 117]. The large dipole mo-ment of RbH can potentially lend itself to be used for electrostatic trappingand further cooling.There is a large interest in ultra-cold polar molecules with large electricdipole moments. Among their applications are: the search for a permanentelectric dipole moment of the electron [118, 119], quantum computation[120], creation of a dipolar superfluid [121] and a model system for lat-tice spin models [122]. However, the rich internal structure of molecules,which makes it difficult to implement a closed optical cycling transition,has prevented the realization of a MOT for molecules. An alternative tocooling down molecules directly is to produce them from ultra-cold atomsvia photo-association. This was proposed in 1987 [123] and the year after,ultra-cold molecular ions were produced via associative photo-ionization934.2. Beam implementation[124]. Since then, the quality of experimental spectroscopic data has beenahead of the ab-initio calculations pushing for refinement of theoreticalmodels. All of this work has only been possible in diatomic molecules.The quest now is for ultra-cold heteronuclear molecules. The first ultra-cold heteronuclear molecule was 6Li7Li [125]. Heteronuclear moleculeswith states that support dipole moments of the order of Debye were firstreported in 2004 [126].To perform the photo-association, cold hydrogen, in addition to ultra-cold Rb, is required. For these reasons the beam implementation incorpo-rate various additional features such as a Zeeman decelerator and a Zee-man bender. The purpose of the implementation is to deliver a beam ofcold hydrogen atoms (with tuneable kinetic energy) for trapping or colli-sion scattering and photo-association experiments. Fig. 4.1 shows a schematicof the hydrogen beam main components.The experimental set up consists of the following modules:• Source chamber where the hydrogen beam is created by entrainment(pick up) of hydrogen by a supersonically expanding beam of a noblegas.• Zeeman decelerator chamber where the hydrogen atoms in a LowField Seeking (LFS) state is decelerated (not discussed in this disser-tation).• Zeeman bender or filter chamber where the desired population ofLFS hydrogen atoms are separated from the ensemble of noble gasand fast hydrogen atoms.Each chamber is differentially pumped and, in that way, we can reacha vacuum of ∼10−8 Torr after the Zeeman bender, with the beam on, whilethe source chamber pressure can be as high as 10−4 Torr, during the valveopening. Fig. 4.2 is a solidworks schematic of the complete vacuum setup.The schematic also shows some additional features such as a magneticfocuser. This focuser is necessary if a well collimated beam is required after944.2. Beam implementationZeemanDeceleratorGas CrackerPulsed ValveZeemanBenderSource ChamberSkimmerAr + H Supersonic ExpansionAr + H BeamAr + H(LFS) ArH(HFS) Ar + Fast H(LFS) Slow H(LFS) ~10-4 TorrH2 Arv(z) v(r) ~10-7 Torr ~10-8 TorrLyman-α~ 580 m/sec~ 0.232 m/secEntrainment CellFigure 4.1: The experimental set up consists of the following modules:Source chamber where the hydrogen beam is created by entrainment (pickup) of hydrogen by a supersonically expanding beam of a noble gas. Zee-man decelerator chamber (not discussed in this thesis) where the hydrogenatoms in a Low Field Seeking (LFS) state are decelerated (cooled). Zee-man bender or filter chamber where the desired population of LFS hydro-gen atoms is separated from the ensemble of noble gas and fast hydrogenatoms.the bender as the hydrogen atoms exiting the bender have large transver-sal velocity and spread out. Also the Lyman-α cell can be attached to thesystem in two different positions; at the source chamber, for LIF of the hothydrogen, and at the end of the Zeeman decelerator chamber, for Doppler-free LIF of the hydrogen beam. Detectors have been integrated such as aPMT, directly above the crossing point of the Lyman-α source and hydro-gen beam, and a Quadrupole Mass Spectrometer (QMS), after the focuser.The QMS can also be placed on the Zeeman chamber instead of the PMT.Fig. 4.3 shows some pictures of each item.Regarding this dissertation, two sections of the experimental set up areimportant. The source chamber where the hydrogen beam is generated andthe Zeeman bender which is used to confirm the hydrogen in the beam.954.2. Beam implementationZeemanBender ZeemanDeceleratorLyman-α CellQuadrupole Mass SpectrometerSourceChamberPulsed ValveSkimmerGas CrackerSkimmerFocuserLyman-αFigure 4.2: CAD drawing of the complete vacuum setup. The Lyman-αcell can be placed in the source chamber for spectroscopy of hot hydrogenor in the Zeeman decelerator chamber for Doppler free spectroscopy of thebeam964.2. Beam implementationZeemanBender ZeemanDeceleratorSourceChamberFocuserFigure 4.3: Pictures of the main components: the pulsed valve, Zeemandecelerator and Zeeman bender. Th focuser at the end of the deceleratorwas not used for these experiments.4.2.1 Source chamberTo create the hydrogen beam we have used a home-built pulsed valve ac-tuated by a solenoid. The valve releases high pressure gas (seed gas) intothe vacuum creating a supersonic expansion. Hydrogen gas (atomic hydro-gen) is generated by the thermal cracker which is placed perpendicular tothe beam and directly in front of the valve. As the gas is expanding somehydrogen atoms go through the shock barrier and make it into the zone ofsilence. Here they are “picked-up” by the expanding gas and cooled dueto the large number of by binary collisions with the seed gas. The beam isextracted from the expansion with a parabolic skimmer where it goes intoa high vacuum chamber. The source chamber has a large gas load fromthe gas cracker and the pulse valve, and it is pumped down with a 1000974.2. Beam implementationL/sec turbomolecular pump. The pressure during valve opening is as highas 10−4 Torr.Pulsed valveThe design of the pulsed valve is based on the Even Lavie design [127] andwas developed by Pavle Djuricanin (mechanical) and Sajjad Haidar (elec-tronics) from Technical Services in the Chemistry Department. Fig. 4.4shows the main valve components. The valve consist of a spring loadedplunger that seals the nozzle opening. During actuation a high currentpulse drives a solenoid around the plunger. The force created by the mag-netic field pushes the plunger against the spring. The coil of the solenoiduses a Permendure shield as field concentrator. The nozzle has a conicalshape with an opening angle of 40 deg. According to the literature thisis the most optimal configuration to produce low spread (0.43 Steradians)directional beams [128]. The nozzle has a pinhole of 250 µm diameter. Themain problem with this configuration is that small misalignments betweenthe nozzle and the skimmer result in dramatic reductions of the beam in-tensity. This is very critical for this setup as the skimmer has been placedat a long distance from the valve in order to minimize the transversal ve-locity of the beam (minimizing any Doppler broadening during LIF exper-iments). The plunger and solenoid design were optimized to maximize theopening stroke force (and speed) by simulating the magnetic field and forceat different configurations.The force generated by the solenoid can be calculated with the Maxwellstress tensor:F =1µ0∮A(~B(~B · ~n)−12~B2 · ~n)dA, (4.3)where ~n is normal to the surface area A. The magnetic field was calculatedat every point for various positions of the solenoid and the most optimalposition is to have the centre of the air gap in the close position (surface Ashown in Fig. 4.5) offset from the middle of the solenoid by 750 µm. Fig.4.5 shows the result of the simulations (the best configuration). The valveopening times are as small as 20 µsec with repetition rate as high as 1 kHz984.2. Beam implementationConicalNozzleHigh Pressure GasReciprocrating PlungerPermendure ShieldReturn SpringThin Pressure VesselValve BodyGuiding FerruleSolenoid40oFigure 4.4: The design of the pulsed valve is based on Even Lavie design[127] and was developed by Technical Services in the Chemistry Depart-ment. The valve consist of a spring loaded plunger that seals the nozzleopening with a reciprocating (up-and-down) motion. During actuation ahigh current pulse drives a solenoid around the plunger. The force createdby the magnetic field pushes the plunger against the spring. The coil ofthe solenoid uses a Permendure shield as field concentrator. The nozzlehas a conical shape with an opening angle of 40 deg. According to the lit-erature this is the most optimal configuration to produce low spread (0.43Steradians) directional beams [128]. The nozzle has a pinhole is 250 µmdiameter.994.2. Beam implementationand stagnation pressures as high as 300 psi.To monitor the gas pulse in the chamber two discharge plates wereplaced in front of the valve nozzle. By applying a high voltage across thegas during the expansion it is possible to generate plasma. The ions andelectrons accelerate to the plates and a current signal proportional to thegas plasma density can be measured. Fig. 4.6 shows the profile of the gaspulse during the valve opening. Pulse widths as short as 30 µsec can begenerated. By monitoring the current it is possible to confirm the openingtime and also that the valve is not bouncing and opening multiple times.The bouncing occurs when the current supplied to the solenoid is too highand the plunger goes beyond some equilibrium point. The optimal currentis a function of the gas pressure and life time of the spring. Before per-forming experiments the discharge plates are used to optimize the currentto the valve until there is no bouncing. Also the delay time and openingtime can be read from the current profile.Hydrogen entrainmentFor the LIF spectroscopy the expanding gas used (seed gas) was Ar. Thehydrogen was sourced from the gas cracker in front of the expansion. Toincrease the density of the hydrogen (and increase the probability of hy-drogen atoms penetrating the zone of silence) an entrainment (pick up) cellwas placed in front of the expanding beam. Fig. 4.7 shows the configura-tion of the valve, cracker, entrainment cell and skimmer. Also a schematicof the pick up cell and mechanism is shown. The entrainment cell wasmade of 1/4 inch quartz tubing and connected to the gas cracker. The mainlimitation of using an entrainment cell is that recombination of rarified hy-drogen occurs predominantly on the available surfaces. Another optionto reduce the recombination is to place the hydrogen effusive source veryclose in front of the expanding beam without an entrainment cell. The twooptions were tested but no significant difference was observed. However,by optimizing the geometry of the entrainment cell and possibly coatingthe surface with Teflon™ it may be possible to get a larger percentage of1004.2. Beam implementation|B|R5.2 TSurfaceASolenoidCenter750 μmFigure 4.5: The pulsed valve stroke was optimized by maximizing theforce excerpted by the solenoid on the plunger. The force generated by thesolenoid can be calculated with the Maxwell stress tensor (Eq. 4.3. Theintegration is done on the surface A. The magnetic field calculation for thebest configuration is shown.1014.2. Beam implementationR = 100 ΩR = 0 - 10 kΩ< 1800 VSpacer PlateNozzle0.200.150.100.050.00IDTime (msec)I D [Amps]67.0 μsec53.2 μsec48.9 μsecPlunger BounceDriving Pulse50.0 μsecPlasmaFigure 4.6: To monitor the gas pulse in the chamber two discharge plateswere place in front of the valve nozzle. By applying a high voltage acrossthe gas during the expansion it is possible to ionize it and generate plasma.The ions and electrons accelerate to the plates and a current signal pro-portional to the gas plasma density can be measured. The flat top of theplasma profile indicates that the valve is fully open during the pulse. Thesecond profile indicates that the plunger is bouncing. The bouncing can beminimized by reducing the driving current1024.2. Beam implementationGas CrackerPulsed ValveSkimmerEntrainment CellHArAr + H BeamJet BoundaryFigure 4.7: To generate the hydrogen beam the hydrogen was sourcedfrom gas cracker in front of the expansion. To increase the density of thehydrogen (and increase the probability of hydrogen atoms penetrating thezone of silence) an entrainment (pick up) cell was placed in front of theexpanding beam. The expanding (seed) gas was Ar.hydrogen in the beam with this technique. Due to the large recombina-tion factor a large percentage of the hydrogen picked up by the beam ismolecular hydrogen. An advantage of using this technique in contrast tophotodissociation (a well known technique) is that the the flow of atomichydrogen is continuous. The gas cracker can continuously dissociate up to90% of 1 sccm of H2. Photodissociation usually require nanosecond longpulse lasers. With the gas cracker technique hydrogen can be entrainedinto an expanding beam that is temporarily limited by the valve and pump-ing speed of the system. Our system can potentially generate pulses upto 200 µsec. For applications such as photo-association that are not time1034.2. Beam implementationconstrained the gas cracker technique can potentially deliver much morehydrogen atoms than photoassociation.4.2.2 Zeeman benderIn order to filter the beam from unwanted atoms (such as the Ar seed gas)and molecules (H2) a filter has been implemented. The filter is a magneticquadrupole bender that uses the Zeeman effect to guide the LFS atoms tothe collision chamber while dispersing everything else. When hydrogen isplaced in a magnetic field the degeneracy of the F = 1 states is broken intothe three MF = 0, ± 1, states by the Zeeman hamiltonian (see Fig. 1.1):Hz = µeszB+µpIzB, (4.4)where µp and µe are the magnetic moment of the electron and proton, szand Iz are the spins and ~B = (0, 0, B). The energy of the M = 0 and 1 statesincreases with magnetic field. These states are called LFS’s as when thereis a field gradient they feel a force opposite to the gradient (towards thelower fields):F = −µ∇~B. (4.5)The magnetic field is created by passing a current through each leg ofthe quadrupole. A magnetic minimum is created in the middle of thequadrupole by making each leg current anti-parallel to its nearest neigh-bours. A simulation of the magnetic field shows a magnetic minimum inthe middle where the LFS atoms are guided (Fig. 4.8). The force is pro-portional to the field gradient and to the current applied. It is possibleto select the maximum speed of hydrogen atoms that would be guided bythe quadrupole by adjusting the current. The maximum current for thequadrupole is 2000 Amps and limited only by the driving electronics. Theother limitation is the transit time as the current pulse can only be sus-tained for 1.5 msec. However, at the speed of supersonically expanded Ar(the seed gas of the beam) this is not an issue. Two quadrupoles were im-plemented. The quadrupole used in this experimental set up is a 90 deg1044.2. Beam implementationturn and 110 cm radius with a leg spacing of 2.36 mm. This quadrupolecan bend hydrogen atoms at velocities of 580 m/sec when operated at 2000Amps. The second quadrupole is a 10 deg and 8 m radius with a leg spac-ing of 2.36 mm. This quadrupole can bend hydrogen at velocities of 2500m/sec when operated at 1500 Amps. Fig. 4.9 shows both quadrupoles.The quadrupole was used to confirm that the beam contained hydro-gen. As mentioned in the previous section the beam contained a large per-centage of molecular hydrogen. To characterize the beam a QMS was used.The QMS uses high energy electrons (∼ 70 eV) to ionize the species. Atthis electron energy everything is ionized. Particles with the right mass tocharge ratio are guided through the quadrupole towards the channeltron,where the current is directly measured from the anode. When molecularhydrogen is ionized 2% of it dissociatively ionizes into a H+ and is seenas mass 1 by the QMS. Since a large percentage of the beam and the back-ground gas in the chamber is molecular hydrogen, it is very difficult todiscern the hydrogen in the beam from the fraction of dissociated H2. TheZeeman bender can only extract hydrogen from the beam and it has beenused to confirm that the beam, in fact, contains hydrogen picked up duringthe expansion. This is described in the next section.1054.2. Beam implementation0.30.250.20.150.10.05Min: 1.131x10-8Magnetic Flux Density Norm |B|Copper Squares2.36 mmMax: 0.348 T-++-−5 −4 −3 −2 −1 0 1 2 3 4 5x 10-300.10.20.30.4Magnetic Field Norm on x Axisx (m)|B| (T)−1000100200Gradient on x Axis−200d|B|/dx (T/m)Figure 4.8: The Zeeman bender was implemented with a quadrupole.The magnetic field is created by passing a current through each leg ofthe quadrupole. A magnetic minimum is created in the middle of thequadrupole by making each leg current anti-parallel to its nearest neigh-bours. A simulation of the magnetic field shows a magnetic minimum inthe middle where the LFS atoms are guided. The force is proportional tothe gradient of the magnetic field and to the current applied. It is possibleto select the maximum speed of hydrogen atoms that would be guided bythe quadrupole. The maximum current for the quadrupole is 2000 Ampsand limited only by the driving electronics.1064.2. Beam implementation500 1500 2500012345678Min Radius Required at 1500 AmpH Speed (m/sec)Quadrupole Radius (m)8 m Radius, 10 deg, 2500 m/sec, at 1500 Amps0.11 m Radius, 90 deg, 580 m/sec, at 2000 AmpsFigure 4.9: Two quadrupoles where implemented. The quadrupole thatwas tested in this experimental set up is a 90 deg turn and 110 cm radiuswith a magnetic minimum size of 2 mm. This quadrupole can bend hy-drogen atoms at velocities of 580 m/sec when operated at 2000 Amps. Thesecond quadrupole is a 10 deg and 8 m radius with a magnetic minimumsize of 2 mm. This quadrupole can bend hydrogen at velocities of 2500m/sec when operated at 1500 Amps.1074.3. Hydrogen Lyman-α detection4.3 Hydrogen Lyman-α detectionThe hydrogen beam implemented is composed of Ar (the seed gas), molec-ular hydrogen H2 and atomic hydrogen. The molecular hydrogen is theresult of the recombination that occurs during the entrainment process.The Ar seed gas expansion results in a beam of approximately 600 m/sec.In order to perform Doppler free LIF spectroscopy the skimmer was placedfar from the valve to reduce the transverse velocity of the hydrogen. Witha skimmer of 2 mm diameter and a skimmer to valve distance of 25 cm thecollimation factor is 125. The transversal velocity of the hydrogen is lessthan 4.8 m/sec. This is equivalent to a transverse temperature of 0.0011K and a Doppler broadening of 58.9 MHz. The total linewidth of the hy-drogen transition can be expected to be ∼ 150 MHz. The source linewidthwas estimated to be 6 GHz so the ratio of the linewidths is 40:1. The sourcephotons are much less efficient in inducing fluorescence. The low densityof hydrogen and photons result in a very small fluorescence signal at thethreshold of detection. To be able to see any signal the PMT gain had to beincreased to the maximum (105), but this can only be done by reducing thescattered ωR photons.QMS of filtered hydrogenBefore attempting any spectroscopy the hydrogen in the beam was con-firmed by measuring the amount of hydrogen after the Zeeman bender.This was done with the QMS, which filters the charge to mass ratio se-lected, y measuring the current signal directly from the channeltron. TheQMS was initially calibrated by filling a chamber with up to ∼ 10−5 Torr ofHe and comparing the current signal with a cold cathode gauge. The samecalibration was done with molecular hydrogen. Calibration of hydrogenwas done with molecular hydrogen since 2% of it ionizes into H+ [100].With this calibration it was possible to estimate a number density after thebender. Fig. 4.10 shows the experiment with the bender. During the ex-periment the pulsed valve was running and the bender was switched onand off while the QMS was always reading the density of atomic hydrogen.1084.3. Hydrogen Lyman-α detectionSince the pressure at the QMS chamber was ∼ 108 Torr, and most of theresidual gas is H2, the QMS signal had to be averaged over long periods.Fig. 4.10 shows the QMS signal in which a window average of 1000 hasbeen applied. The valve and bender were operated at 5 Hz and the windowaverage covers 3.33 minutes of data. When the molecular hydrogen flowinto the gas cracker is decreased, the pressure in the source chamber andin the QMS chamber decreases. At a base pressure of ∼ 10−8 Torr and 60%H2 there are ∼ 1010 H2 molecules/cm3 in the chamber. When ionized 2%of them dissociate into hydrogen ions which is 108 ions/cm3 in the ionizervolume. All the measurements performed are over this background. Fromthe QMS signal it was estimated that ∼ 108 hydrogen atoms/cm3 over thebackground were added when the bender was turned on. The valve open-ing was 50 µsec and the beam diameter is 2 mm. The beam has a meanspeed of 600 m/sec so every pulse is approximately 0.94 cm3. From thiswe can estimate ∼ 108 hydrogen atoms/pulse being guided.1094.3. Hydrogen Lyman-α detection0 20 40 60 80 100 1201.81.92.02.12.22.3x 10-3Bender OnBender OffDecrease H 2 Flow0 sscm of H 2 FlowIncrease H 2 FlowBender OnBender OffBender OnBender OnBender OffBender OffBender On0 sscm of H 2 FlowTime (min)Base Pressure (Torr)QMS Signal (mV)2.54.8x 10-8Figure 4.10: The hydrogen in the beam was confirmed by measuring theamount of hydrogen after the Zeeman bender. This was done with theQMS by measuring the current signal directly from the channeltron am-plifier. The QMS was initially calibrated by filling a chamber with He andcomparing the current signal with a cold cathode gauge. The same calibra-tion was done with molecular hydrogen. Calibration of hydrogen was donewith molecular hydrogen since 2% of it ionizes into H+ [100].During theexperiment the pulsed valve was running and the bender was switched onand off while the QMS was always reading the amount of hydrogen. Sincethe pressure at the QMS chamber was ∼ 10−8 Torr, and most of the resid-ual gas is the H2 the signal, the QMS signal had to be averaged over longperiods.1104.3. Hydrogen Lyman-α detectionQMS Time of flightTo verify the beam time of flight, the QMS was placed on the PMT port(in front of the skimmer). When the QMS is running continuously, it ispossible to see the beam profile (provided there is enough signal to noiseratio). When looking for mass 1 particles the time of flight profile capturedby the QMS is mostly due to hydrogen molecules. However, a small per-centage is due to the hydrogen atoms. To discern the hydrogen atoms fromthe background hydrogen the beam was averaged 128 times and the differ-ence between a beam with the gas cracker power switched on and switchedoff was taken (the molecular hydrogen is continuously flowing). Fig. 4.11shows the time of flight profile and the difference of the beams which isattributed to hydrogen atoms. From the time of flight profile the numberof hydrogen atoms per pulse is estimated to be 2.5 × 1010 atoms/pulse.This means that the bender can bend approximately 1% of atoms. An in-teresting feature of the profile is that it appears to be two regions of higherdensity of hydrogen around the main body of the pulse. This is consistentwith reports of beam separation when the seed gas is a heavier atom [129].In this case, Ar with mass 40, appears to push the hydrogen atoms in frontand behind the pulse.1114.3. Hydrogen Lyman-α detection−5 0 5 10 15x 10-4−0.129−0.128−0.127−0.126−0.125−0.124−0.123−0.122−0.121−0.12Time of Flight  (sec)QMS Signal  (mV)  Hydrogen OnHydrogen Off04H signal (mV)Difference2.5 x 1010 H atomsFigure 4.11: To verify the beam the QMS was placed on the PMT port.When the QMS is running continuously it is possible to see the beam pro-file (provided there is enough signal to noise ratio). When looking for mass1 particles the time of flight profile capture by the QMS is mostly due tohydrogen molecules. However, a small percentage is due to the hydrogenatoms. To discern the hydrogen atoms from the background hydrogen thebeam was averaged 128 times and the difference between a beam with thegas cracker on and off was taken. From the time of flight profile the num-ber of hydrogen atoms per pulse is estimated to be 2.5 × 1010 atoms/pulse.This means that the bender can bend approximately 1% of atoms. An in-teresting feature of the profile is that it appears to be two regions of higherdensity of hydrogen around the main body of the pulse.1124.3. Hydrogen Lyman-α detectionFluorescence experimentIn Chapter 3 the density of hydrogen in the spectroscopy chamber wascalculated to be ∼ 1014 hydrogen atoms/cm3. The density of hydrogen inthe beam is estimated to be 4 orders of magnitude less. Furthermore, thelinewidth ratios of the transition and the source indicate that the photonsare at least 40 times less efficient at inducing fluorescence. To perform theLIF experiment the chamber was modified to reduce the scattered ωR pho-tons and to increase the solid angle view of the detector. Blackened baffles,to reduce the scattered light, and collimating lenses, to capture more flo-rescence photons into the PMT, where placed around the interaction area.All the lenses used were MgF2. Fig. 4.12 shows the setup. The diameterof the beam is 2 mm and the radius of the laser beam at the focal pointis less than 10 µm. The flashlight and Q-switch of the pump laser werecontrolled by a computer that also set the timing for the valve. The timingfor the valve opening was scanned and at every time step the signal wasrecorded and averaged multiple times. The timing for the valve openingwas scanned in order to find the point with the highest hydrogen densityalong the beam. The main difficulty performing this experiment was thelaser stability. The laser is not stable enough for long runs and the power ofthe Lyman-α fluctuates. Power fluctuations are due to temperature changesin the room and in the cooling water and also due to misalignment of op-tical components due to vibration. Averaging is required since the PMT isoperating very close to the detection threshold and the signal to noise ratiois very small. Fig. 4.13 shows the results of the scans. According to theQMS time of flight signal the hydrogen pulse extends from 0.5 to 1.2 msecafter the valve opening with a maximum hydrogen density at 1.2 msec. Themaximum LIF signal is just between 1.2 and 1.3 msec and this agrees withthe QMS. However, the LIF signal does not appear to reflect the densityprofile of the beam and this is troubling. The two data samples shown arethe only LIF signal that was obtained after several experiments and it wasnot possible to improve it. The signal, however, shows a clear signature ofLIF at the time that it is expected and it is always clearly distinguishable1134.3. Hydrogen Lyman-α detectionfrom noise. One potential problem is the alignment/overlap of the laserand the hydrogen beam. As currently designed adjustments of the beamand laser are very difficult to perform and for that reason it was not possi-ble to further optimize the experiment. Furthermore, since the generationof Lyman-α radiation was already confirmed (see Chapter 3) and these LIFexperiments showed some signatures of hydrogen fluorescence it was de-cided that continuing this experiment was not necessary.PMTLyman-α LaserBafflesFocusing LensMgF2f = 200 mmHydrogen BeamDiameter = 2 mmCollimating LensesMgF2Figure 4.12: To perform the LIF experiment the chamber was modifiedto reduce the scattered ωR photons and to increase the solid angle viewof the detector. Blackened baffles to reduce the scattered light and colli-mating lenses where placed around the interaction area to capture moreflorescence photons into the PMT. All the lenses used were MgF2.1144.3. Hydrogen Lyman-α detection0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.50.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7Time (msec)512 Average20 μsec Step256 Average20 μsec StepQ-Switch ScanLaser Power Decrease PMT Signal (mV)Figure 4.13: The flashlight and Q-switch of the pump laser were con-trolled by a computer that also set the timing for the valve. The laser tim-ing was scanned to find the highest density of hydrogen along the profileof the beam and every time the signal was averaged multiple times. Thedifficulty with this experiment was that the laser is not stable enough forlong runs and the power of the Lyman-α fluctuates. Stability can only beaccomplished for short amounts of time and the experiment runs need tobe short. Averaging is required since the PMT is operating very close to thedetection threshold and the signal to noise ratio is large. 1154.4. Collision scattering experiments4.4 Collision scattering experimentsIt has been proposed that the measurement of background gas collisionswith a cloud of trapped atoms can be used as a standard of the backgroundgas density [130–132]. This measuring technique (as will be shown in thissection) reduces to the calculation of a scattering cross-section. The mea-surement does not require any calibration (in contrast to ionization gaugesor other techniques) and for that reason it could be used as a measurementstandard. l In a MOT the rate of change of the number of atoms in the trapcan be given as:dNdt= R− ΓN − β∫n2MOT (r, t)d3~r. (4.6)Here R is the rate of atom capture, ΓN is the rate of loss due to collisionsof the trapped atoms with the background gas atoms and β is the rate ofloss due to collisions of two trapped atoms whose density is nMOT . If thedensity is taken to be constant then:N (t) =RΓef f(1− exp{−Γef f t}), (4.7)where Γef f = Γ + βnMOT . In a magnetic trap R is zero and if βnMOT << Γthen:N (t) =N (0)exp {−Γ t} . (4.8)In this case the rate of change of the number of atoms is only a function ofthe loss-rate constant:Γ = nBkg 〈σT B(υ)υ〉 , (4.9)where 〈σT B(υ)υ〉 denotes an average over the centre of mass speed υ ofthe colliding atoms, and σT B is the collisional cross section of the trappedatoms with the background atoms of density nBkg . If the velocity-averagedcollision cross section is known, then by measuring the rate of loss of thetrapped atoms it is possible to deduce nBkg , the density of background gas.This measurement does not rely on any other calibration method and it can1164.4. Collision scattering experimentsprovide a primary standard for density measurements. Since the densitymeasurement provided by a MOT is localized, this method can be usedto measure density differences in a volume with a resolution given by theMOT size. For instance, the density of an atom beam can be measured withthis technique.There are two main differences between a background gas density mea-surement and a beam density measurement: the type of velocity-averageperformed and the time scales involved. In the case of background gas theaverage is over a Maxwell-Boltzman distribution with a mean speed givenby the temperature of the ensemble. In the case of a beam, the distributionmean is the supersonic speed given mainly by the stagnation pressure andtemperature of the reservoir. Regarding the time scales, the rate of changeof background gas is usually slow compared with the time between colli-sion events. In fact Γ is assumed to be time independent. That is certainlynot the case in a beam density measurement where the density changes inthe order of hundreds of microseconds.In a beam density measurement the loss-rate constant has a componentdue to background collisions and a component due to the collision with thebeam; and if βnMOT << Γ then:dNdt= R− (ΓBkg + ΓBeam(t))N. (4.10)In the collision experiments that have been performed the MOT is alwaysin equilibrium, R − ΓBkgN = 0, before the collision. The time scale of thebeam collision event is less than one millisecond and the change in thenumber of atoms trapped is less than 1%. It is reasonable to assume that:dN (t)dt≈ −N (t)ΓBeam(t). (4.11)Fig. 4.14 is a plot of the scattered light from a MOT as a function of time.It shows a collision event where a beam of Ar collided with a Rb-MOT.The negligible rate of change before and after the collision support theprevious assumptions. From the scattered light signal is possible to extract1174.4. Collision scattering experimentsΓBeam(t) = nBeam(t)〈σMOT−Beam(υ)υ〉. Here υ is effectively υBeam since theMOT atoms can be assumed to be stationary compared to the supersonicspeed of the beam. In this section it will be shown that in principle, thissignal has all the information required to characterize the beam.ArAverageCollision SignalScattered Light (a.u.) Time (msec)0 2 4 6 8 10Ar Beam Rb MOTPhotodiodeFigure 4.14: Scattered light from a MOT as a function of time. It shows acollision event where a beam of Ar collided with a Rb-MOT.To perform the collision experiments, a 85Rb MOT was implementedand integrated with the supersonic beam. The trapping technique uses theD2 Rb transitions: (4p5(2P o3/2)5p2[3/2]← 4p5(2P o1/2)5s2[1/2]o), 138794.380cm−1 - 158151.666 cm−1, from the F = 3 to the F = 4’ hyperfine levels (the“pump”). To close the cycle (since some transitions occur to the F= 2 level)a “re-pump” transition is induced from the F = 2 to the F = 3’ levels. MOTs1184.4. Collision scattering experimentsrely on the momentum imparted by individual photons for trapping andcooling. To select the atoms to which momentum must be imparted, lasersare detuned from the resonance. The balance between Doppler coolingforces associated with photon absorption and the subsequent heating dueto re-emission of absorbed photons leads to the so-called Doppler cool-ing limit. The Doppler cooling limit for 85Rb is 145.57 µK. The light forthe MOT has been transported from Dr. Kirk Madison’s laboratory via apolarization maintaining fiber which carries both the pump and the per-pendicularly polarized re-pump. This particular MOT was implementedin a glass tube (square stock) bonded to Conflat™ flanges. In addition tothe anti-Helmoltz coils a couple of compensation coils have been added toallow steering of the MOT in the xy plane. The MOT chamber is pumpedwith an ion pump at the beam entrance and a turbo pump at the exit. TwoRb wire evaporation dispensers have been used: Alvasource and SAES Ad-vanced Technologies. By passing some current through the wire the Rbmetal is evaporated. Fig. 4.15 shows an schematic of the experimental setup.To integrate the MOT with the beam the Zeeman bender has been re-moved and the MOT chamber has been attached directly in front of thebeam. To perform the measurements the background density has to bestable at the MOT chamber during valve opening. This has been accom-plished by adding a second skimmer before the MOT and a turbomolecularpump behind it. With the additional differential pumping stage the pres-sure at the MOT is ∼ 10−8 - 10−9 Torr. The turbomolecular pump after theMOT captures the beam after the collisions. The skimmer, when properlyaligned with the valve, first skimmer and MOT, does not affect the beamas its opening diameter is bigger than the first skimmer. Fig. 4.15 showssome pictures of the components.1194.4.CollisionscatteringexperimentsIon Pump Ion PumpTMPTMPTMPQMSPhotodiodeValveSkimmer SkimmerRb DispenserCoilsGlass TubeBeamMOT~ 10-4 Torr ~ 10-7 Torr ~ 10-8 Torr300 psiFigure 4.15: To perform the collision experiments a MOT chamber has been attached directly in front of thebeam. To perform the measurements the background density has to be stable at the MOT chamber during valveopening. This has been accomplished by adding a second skimmer before the MOT and a turbomolecular pumpbehind it. With the additional differential pumping stage the pressure at the MOT is ∼ 10−8 - 10−9 Torr. Theturbomolecular pump after the MOT captures the beam after the collisions.1204.4. Collision scattering experimentsTo measure the MOT atom number N(t) a photodiode is used. The pho-todiode signal is proportional to the MOT number:Vph(t) = αN (t)−VN=0. (4.12)The constant α is a function of the photodiode collection efficiency andthe scattering rate of photons by the atoms in the trap. As long as thephotodiode is operated in a linear region, α can be assumed to be constant.The bias signal VN=0 is the product of the laser’s photons scattering fromthe walls of the glass tube and leaked photons from the ambient light. Tomeasure the density of the beam all that is needed is the photodiode signal:dN (t)dt≈ −N (t)ΓBeam(t) ⇒Vph(t)dt≈ −(Vph(t)−VN=0)nBeam(t)〈συBeam〉 .(4.13)The cross section σMOT−Beam(υBeam) can be calculated from first principlesunder some assumptions of the inter-atomic potential. The speed distri-bution of the beam f (υBeam) can be extracted from the normalized densitysignal:−1max(nBeam)[nBeam(t) ≈Vph(t)dt1(Vph(t)−VN=0)〈συBeam〉]. (4.14)However, there is some time delay caused by the transit time of the MOTatoms. These atoms have to move out of the detection area of the pho-todiode. This can be optimized by narrowing the photodiode view area,but the transit time represents the ultimate time resolution possible withthis measuring technique. Nevertheless, with this information the constant〈σMOT−Beam(υBeam)υBeam〉 can be calculated by proper averaging:〈σMOT−Beam(υBeam)υBeam〉 =∫ υ=∞υ=0 σMOT−Beam(υBeam)υBeamf (υBeam)dυBeam∫ υ=∞υ=0 f (υBeam)dυBeam.(4.15)Finally, the calculation of σMOT−Beam is essentially a calculation of the scat-1214.4. Collision scattering experimentstered spherical wavefunction g(υBeam,Ω), where Ω is the solid angle:σMOTBeam(υBeam) =∫|g(υBeam,Ω)|2dΩ. (4.16)The potential U (r) used in the Schro¨dinger equation to calculate the scat-tered wavefunction is assumed to be of the Lennard-Jones type (this is theassumption mentioned before):U (r) =UL−J (r) =C12r12−C6r6. (4.17)The sixth-power term in the Lennard-Jones potential arises from in-stantaneous dipole-induced dipole interactions (London dispersions forces),which represent interactions in noble gases very well. The twelfth-powerterm has no physical basis and it is chosen for convenience of calculation(it is the square of the attractive term). However, for large potential wells(or low kinetic energy), where attractive forces dominate the interatomicinteractions, the C6/r6 term dominates, and this is the most common case.In addition, according to classical calculations of gas kinetics duringa supersonic expansion the reduced collision cross section σ ∗ ( in units ofpir2m, where rm is the equilibrium potential) can be calculated as [133, 134]:σ ∗ = 1.48K−1/3 ⇒ K =µυ22, (4.18)where µ is the reduced mass and  is the potential well. This approximationcan be used as long as K < 2 and it is good for most gases where the stag-nation temperature is around, or below, room temperature. For very highreduced energies (K > 5), the cross section becomes σ ∗ = 0.74K−1/6. Thisclassical calculation gives some insight into the collision experiment andcan be used to approximate a cross section. For instance, the atom loss ratecoefficient has been calculated from these classical approximations and forthe Ar/ Rb system Γ / P (P as the pressure of Ar) is 2.3 × 107 Torr−1 sec−1[131]. This calculation assumes a 1 K trap depth and 300 K. Measurementswith similar trap depths obtained 2.3× 107 Torr−1 sec−1 [135].1224.4. Collision scattering experimentsIn summary, the term 〈σMOT−Beam(υBeam)υBeam〉 can be calculated and itis only a constant in this measurement. In fact, this term has been calcu-lated for Ar. This calculation, however, is not the author’s work and detailsare not presented in this dissertation. Regarding this discussion, it is rea-sonable to assume that the term 〈σMOT−Beam(υBeam)υBeam〉 ≈ 10−10 cm3/sec[136]. Fig. 4.16 shows the photodiode signal from measurements of col-lisions of three gases: Ar, Kr and He. The signal is quite noisy but this isnot an inherit limitation of the technique and it can be improved by hard-ware changes. The data has been smoothed using a windowed average.The second plot on the right is the averaged signal and its derivative. Thederivative is proportional to the density of the beam n(t). The QMS signalis also shown for comparison. An interesting feature is what appears tobe two different peaks in the Kr density. The QMS signal also shows thesefeature but it is less pronounced. Perhaps, the beam separated or the valvebounced, opening a second time. From the derivative of the signal the ve-locity distribution can be extracted. The absolute number of the densitycan be calculated by multiplying the constant 〈σMOT−Beam(υBeam)υBeam〉.Fig. 4.17 shows the density and absolute atom number and a fit of thevelocity distribution.1234.4. Collision scattering experiments024600.40.81.21.6−0.4−0.200.2012300.40.8−0.100.10 1 2 30240123−0.200.2−2 0 2 4 6 8 10−2−100 2 4 6 8 100.7380.6990 2 4 6 8 100.73860.69900.472 0.4720 1 2 3mVmsecArKrHedV/dtdV/dtdV/dtVphVphVphAvg(Vph)Avg(Vph)Avg(Vph)KrArHeQMS SignalPhotodiode Signal dV/dt (V/sec)Vph(mV)Time (msec)V/secVph(mV) Avg(Vph)Figure 4.16: Photodiode signal for three different experiments with Ar, Krand He. The signal has been averaged and the derivative is shown on theright. The QMS signal is also shown for comparison. A particular featureis the what appears to be two different beams in the Kr density. The QMSsignal also shows these feature but it is less pronounced.1244.4. Collision scattering experiments00.511.5200.40.81.2  0 1 2 3 4 5 6 700.10.20.30.4    ArKrHe0 1 2 3 4 5 6 7 80 1 2 3Density n(t) [x 10-10 cm-3 ]Time (msec)Speed Distribution FitTotal Atoms = ~ 1012Total Atoms = ~ 1013Total Atoms = ~ 1012Mean Speed = 580 m/sMean Speed = 400 m/sMean Speed = 1400 m/sFigure 4.17: From the derivative of the signal the velocity distributioncan be extracted. The absolute number of the density can be calculated ymultiplying the constant 〈σMOT−Beam(υBeam)υBeam〉.1254.4. Collision scattering experimentsIn conclusion, A MOT can be used as a standard to measure the abso-lute gas density, and more importantly, to measure very localized atomicdensities. The beam used to demonstrate the measurement technique hasa diameter of 2 mm. The MOT diameter is ∼ 1-2 mm diameter in size andthis is the volume resolution of the measurement. The time resolution isgiven by the time it takes the trapped atoms to escape the photodiode viewarea. In the experiment we can steer the MOT up to 1 cm in the xy planeby using compensation coils. By moving the MOT 2.5 mm off the beamthe signal disappears and this shows the potential of the technique. If thedetection can be improved, MOTs as small as a few hundred micrometerscan be used.As initially stated, the purpose of the hydrogen beam development isto eventually perform the photo-association of the RbH molecules. Hydro-gen collision experiments have also been performed to this end. However,without the Zeeman bender, the beam is a mixture of hydrogen molecules,hydrogen atoms and some seed gas. The hydrogen collision experiment wasperformed but the analysis is more difficult with the mixture of species. Infact, to perform the analysis, additional information of the species abun-dance is needed. The bender was not used because the density of the hy-drogen atoms is too low to be observed with the current system. Never-theless, the main purpose of this exposition is to show the potential of thetechnique. To measure the scattering of Rb atoms other techniques can beused such as “hot wire” (Langmuir-Taylor detector). However, the purposeof this section is to demonstrate the MOT potential as a measuring de-vice. Regarding the RbH experiments the current limitation is the hydro-gen source. The experiment requires a beam of very slow and high densityhydrogen. In order to increase the hydrogen density, different sources arebeing explored and this is discussed in the next chapter. Also the currentimplementation of the Lyman-α is discussed.126Chapter 5Current work and conclusionIn this chapter the current developments regarding the Lyman-α sourceand the hydrogen source are discussed. The final form of the Lyman-αsource is being developed. This includes the laser beam delivery systemat the CERN facility and the equipment required for optical cooling andspectroscopy of antihydrogen. Regarding the hydrogen beam and scatter-ing experiments the main focus is on increasing the density of hydrogenin the beam. Although the beam as implemented delivers sufficient hydro-gen to perform mass spectroscopy, a higher number of hydrogen atoms isrequired to observe photo-association with the Rb MOT.5.1 Lyman-α SourceThe RE-FWM process is very efficient and can potentially produce a veryhigh yield of VUV photons. However, the system is very complex and it isnot very stable. Experiments at ALPHA require a very stable and robustsystem. It was decided that the THG process would be a better candidate.The input laser required for the THG can be implemented with solid statelasers. This type of system is more robust and requires less maintenancethan a dye laser based system. Furthermore, to perform cooling, a narrowbandwidth laser of ∼ 100 MHz is required, and this is easier to accomplishwith solid state systems. Another limitation of the RE-FWM process isthe available power at 202.38 nm. The UV light is difficult to generateand there are several transmission losses in the optics. Furthermore, therequirement of collinearity of the two beams is a limitation since chromaticaberration impedes the realization of the optimal configuration.The THG system being implemented uses a diode cw laser at 729 nm.1275.1. Lyman-α SourceThis laser is amplified in a Ti:Sapphire pulse amplifier pumped by anNd:YAG pulsed laser. The generated 729 nm is further doubled in a SHGcrystal and tripled in a Kr:Ar mixing chamber. Fig.5.1 shows an schematicof the system. Before delivery into the ALPHA trap the 364 nm will befiltered with a narrow band Lyman-α filter. This system is being imple-mented and is expected to be completed during the fall of 2014. The ex-pected conversion efficiency is ∼ 10−5. Optical cooling experiments areexpected to start in the summer of 2015.1285.1. Lyman-α SourceDiode CW seed laser  729 nm (0.5W) Pulse Amp 729 nm (100 mJ)  Nd:YAG pulsed Laser 532 nm (500 mJ, 10Hz) SHG 364.7 nm 30 mJ   (121.567nm) > 0.1 μJ, 10 Hz Radiation Protected AreaLyman-αFilter Antihydrogen Trap 364.7 nm27 mJ Alignment Port Injection seeder Kr/Ar THG ChamberLyman-α Optical Transport~ 4 m Figure 5.1: Since experiments at ALPHA require a very stable and robustsystem it was decided that the THG process would be a better candidate.The input laser required for the THG can be implemented with solid statelasers. This type of system is more robust and requires less maintenancethat a dye laser based system.The THG system being implemented uses adiode cw laser at 729 nm. This laser is amplified in a Ti:Sapphire pulseamplifier pumped by an Nd:YAG pulsed laser. The generated 729 nm isfurther doubled in a SHG crystal and tripled in a Kr:Ar mixing chamber.1295.2. Alternative hydrogen source5.2 Alternative hydrogen sourceBy pulsing a high voltage across a pair of electrodes positioned in front of avalve during supersonic expansion it is possible to create a temporary dis-charge plasma. If the electron density and electron energy is sufficientlyhigh, a large number of the expanding molecules can be dissociated. Thetechnique is usually implemented with an external filament. This facili-tates the plasma creation by supplying a large number of free electrons.We have used this technique to create hydrogen by dissociating pure andseeded CH4 and H2. As a seed gas we have used Ar and Kr. We have verifiedthe dissociation of hydrogen with the QMS. Unfortunately, this dischargetechnique has been found to deplete the beam and reduce the total densityof atoms in it. A possible explanation is that the plasma generated is di-verged by the fringing electric field around the electrodes. Fig.5.2 showsthe ratio of the signal intensity for various masses as a function of the dis-charge voltage applied. Another problem with plasma expansion is thatthe discharge process heats the gas increasing the transversal speed andthe longitudinal spread of speed.Theoretically, it is possible to optimize the discharge so the beam is notdepleted. As an example we can look at dissociation of pure molecular hy-drogen. There are a few processes that can produce atomic hydrogen inplasma. In principle molecular dissociation has to compete with an ion-ization process. If the distribution of electron energies is wide, the rate atwhich any reaction proceeds depends mostly on the onset energy and thecross section immediately beyond onset. If the distribution is very narrowthen the maximum cross section can be used to increase the probabilityof a specific reaction [137]. The following table shows some of the mainhydrogen dissociation reactions ([137]:In the table reaction 2 and 3 can be neglected because of too high on-set energy. The atomic hydrogen is mainly produced by the reactions 4, 5and 6 and since the density of molecular ions is much smaller than thatof molecules the dominant process is reaction number 5. Also the otherprocesses are not very probable because there are not enough excited re-1305.2. Alternative hydrogen source 0 20.5 1 1.31.51.8Mass/Charge11216131415CH4 DissociationH+C+CH+CH2+CH3+CH4+Normalized Integrated QMS SignalDischarge Voltage (kV)Figure 5.2: Discharge plates have been used to generate hydrogen by dis-sociating pure and seeded CH4 and H2. As a seed gas we have used Arand Kr. We have verified the dissociation of hydrogen with the QMS. Un-fortunately, this discharge technique has been found to deplete the beamand reduce the total density of atoms in it. A possible explanation is thatthe plasma generated is diverged by the fringing electric field around theelectrodes.1315.2. Alternative hydrogen sourceTable 5.1: Onset energy of electrons and maximum cross section forhydrogen dissociation processes in the literature.No Process Onset (eV) σmax × 10−16 Energy at σmax(eV) (cm2) (eV)1 H2 + e→ H+2 + 2e 15.4 1.1 802 H2 + e→ H+ + H + 2e 18.0 0.005 1203 H2 + e→ H+ + H+ + 3e 46 0.005 1204 H+2 + e→ H+ + H + e 12.4 3-16 165 H2 + e→ H + H + e ≈ 8.5 0.6 126 H+2 + e→ H + H 0 100(?) –7 H + e→ H+ + 2e 13.5 0.65 408 H + e→ H∗(2p)+ e 10.2 0.7 259 H∗ + e→ H+ + 2e 3.3 15 910 H2 + e→ H∗2 + e 10.3 0.2 60action partners. Given that the cross section of reaction 5 is quite smalla large density of energetic electrons is necessary to produce a sizeableyield of hydrogen. If the voltage applied is reduced, and a large amountof monochromatic electrons is supplied to the beam, it is possible to in-crease the rate of a specific reaction without depleting the beam. This ispossible by replacing the thermionic electron emitter (filament) by a coldfield electron emitter. By selecting a material with a low work functionand a suitable geometry it is possible to extract electrons via quantum tun-neling (monochromatic electrons) with low electric field. The best mate-rial available for such application is carbon nanotubes [138, 139]. Carbonnanotubes are allotropes of carbon with a tubular structure. The carbonatoms are arranged in a honeycomb lattice that can be thought of as rolledgraphene. Nanotubes can have a very high aspect ratio. For instance, nan-otubes synthesized with length-to-diameter ratios of up to 28,000,000:1have been reported [140]. Single-walled nanotubes (SWNTs) are formedby one sheet of graphene. Multi-walled nanotubes (MWNTs) are formed by1325.2. Alternative hydrogen sourceseveral SWNTs in a coaxial arrangement (several concentric SWNTs formone MWNT). Nanotubes are formed entirely from sp2 hybrid bonds as ingraphene. This bonding structure is stronger than the sp3 bonds found indiamond and that makes them a particularly strong and stable material.These properties combined with the high conductivity found in nanotubesmake them a very good candidate for cold field-emitter sources. Due totheir high aspect ratio, an externally applied electric field is enhanced hun-dreds to thousands of times at their tip. It is thus possible to extract elec-trons with low applied voltages and, since the nanotube tip radius is onthe order of a few nanometers, it is possible to produce very localized elec-tron sources. Low-voltage operation combined with the monochromaticityof the electrons produced by quantum tunnelling makes them the perfectcandidate for our application. Fig. 5.3 shows two carbon nanotube baseddischarge valves implemented as a preliminary test of this technique. Inthe first implementation an array of MWNTs (called a forest) on a siliconsubstrate was used as the cathode of two electrodes placed across a su-personically expanding beam. In the second implementation a stainlesssteel mesh is used as anode and two carbon nanotube forests are used asthe cathode. The principle of operation is that some of the electrons emit-ted would accelerate past the mesh and dissociate molecules expanding bycollision. Since the expanding gas is in a region of no potential gradient itshould not experience any depletion. In order to implement the dischargesources the carbon nanotube forests had to be fabricated. This was donewith a chemical vapour deposition process and the development of thisproject with examples of the nanotubes grown in stainless steel and siliconsubstrates are discussed in Appendix B.1335.2. Alternative hydrogen sourcee-GasNanotube ForestElectrodesNozzleGasNanotube ForestMeshNozzleNanotube ForestFigure 5.3: Carbon nanotube based discharge valve prototypes. The firstprototype (top) used an array of MWNTs (called a forest) on a silicon sub-strate as the cathode of two electrodes placed across a supersonically ex-panding beam. In the second implementation a stainless steel mesh is usedas the anode and two carbon nanotube forests are used as the cathode. Theprinciple of operation is that some of the electrons emitted would acceler-ate past the mesh and dissociate molecules expanding by collision. Sincethe expanding gas is in a region of no potential gradient it should not ex-perience any depletion.1345.3. Conclusion5.3 ConclusionThe main topic of this dissertation was the implementation of a Lyman-αsource based on a RE-FWM process. For this work a particular strong reso-nance on krypton (4s24p55p[1/2]o← 4s24p6(1S0) ) was used. The optimiza-tion of the source was performed by changing the phase matching parame-ters such as the gas mixture, used to accomplish the condition ∆k = 0, andthe focusing geometry. In addition, the light produced by parasitic pro-cesses was monitored. The power of the source was measured indirectlyusing the LIF of hydrogen. The experimental powers measured are sum-marized in Table 5.2. These power levels are sufficient for optical cooling ofantihydrogen which is the motivation for this thesis project as described inChapter 1. The linewidth of the source was estimated to be 6.6 GHz. Whilethe power is sufficient for the described objectives, the source linewidthneeds to be narrowed for optical cooling.Table 5.2: Summary of measured Lyman-α source parameters.Energy Repetition Pulse Length Power Linewidth(µJ/pulse) (Hz) (nsec) (µW) (GHz)0.125 ± 0.030 10 9 1.25 ± 0.30 6.60.025 ± 0.006 50 9 1.25 ± 0.30 6.6Along with the RE-FWM implementation a THG source was also im-plemented. To measure the power of the THG source an acetone ionizationdetector was implemented. The maximum power generated was ∼ 0.6 µW.However, the input laser can easily be doubled and even tripled in power.Also it is possible to implement a solid state system with this process. Thisis favourable with regards to stability and maintenance in contrast witha dye laser system. These results show that THG can potentially be usedto generate Lyman-α for cooling of antihydrogen. In addition, the imple-mentation of two lasers, a RE-FWM and a THG, has provided a tangible1355.3. Conclusioncomparison of the potential of each technique. Based on the implemen-tation experience and results, it was decided that THG is a more feasiblesystem for the antihydrogen cooling.Chapter 4 described the implementation of a hydrogen beam. The hy-drogen beam was implemented by entrainment of hydrogen, producedwith a gas cracker, in a supersonic expansion of Ar. The hydrogen wasconfirmed with a QMS. 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To perform 3D cooling with one laser beam itis necessary to couple the 3 degrees of freedom. The dynamical couplingbetween the z and xy degrees of freedom can be achieved by the use ofnon-harmonic magnetic fields. The nonlinearity is produced along the zdirection by using five solenoidal coils. The ALPHA trap uses two mirrorcoils to provide axial confinement and an octupole to create radial con-finement. The antihydrogen in an LFS state is trapped at the centre in themagnetic minimum. The five solenoids are placed between the mirror coils.The 3 solenoids at the centre provide a flatter magnetic field instead of thequadratic dependence that it would have otherwise. The effective potentialhas a ∼ r6 potential along the z axis and in the radial direction in this trap.During repeated trapping of antihydrogen the coils have to be quenchedso it is necessary to understand the coupling between the coils to obtainthe desired magnetic field. In this appendix the characterization of thecoupling between the solenoids is discussed. Using PSIM, a SPICE soft-ware, the solenoids and mirrors coils were simulated during quenching.The software solves the differential equations of the RCL circuits takinginto account the couplings between the coils (mutual inductance). The cir-cuit is modelled with measured values for all the different components. Toverify how well the model represents the real device some experimentalmeasurements were recreated. The mirror coils are refereed as MA, MB,MC and MD; the solenoids as S1 and S2. Given the symmetry and the poor154Appendix A. Characterization of solenoids in the ALPHA trapcoupling between third nearest neighbours coils (for instance MA and MD)it is possible to accurately represent the real circuit by simulating only halfof the coil array. The simulation includes S1, MA, MB and MC with its re-spective RCL circuits. The inductive parts of MD and ME have also beenincluded. However, removing the MD and ME coils does not change theresults. Tables A.1, A.2 shows the values that were used in the circuit sim-ulation:Table A.1: ALPHA coils inductances and mutual inductances.(mH) S1 MA MB MC MD MES1 24.17 0.457 0.0075 0.005 0 0MA 1.032 0.075 0.00015 0.0007 0MB 1.057 0.078 0.001 0.0005MC 1.054 0.079 0.001MD 1.055 0.072ME 1.035Table A.2: ALPHA coils parasitic capacitance and series resistance.Coil Cp (µF) Rs (Ω)S1 1.116 15.4MA 21.23 1.9MB 21.22 2.1MC 21.26 2.0The source in the simulation was a 5 Vpp, 50 Hz sinusoidal. The voltageinduced was measured in the adjacent coils across a 1 MOhm resistor. Inthe real experiment when the source was applied to MA a voltage of 360mV (Vpp) and 59.2 mV (Vpp) were measured across S1 and MB respectively.155Appendix A. Characterization of solenoids in the ALPHA trapFigure A.1: ALPHA coils parameter measurement simulation. The sourcein the simulation was a 5 Vpp , 50 Hz sinusoidal. The voltage induced wasmeasured in the adjacent coils across a 1 MOhm resistor.156Appendix A. Characterization of solenoids in the ALPHA trapThe simulation reproduces the experiment very accurately (VS1 = 370 mVand VMB = 61.2 mV in the Fig. A.1 and Fig. A.2).VMAVMBVS1Time (sec) 3 0-3 0.04   0-0.04 0.2   0-0.2 0.03       0.035             0.04                  0.045   0.05Figure A.2: In the real experiment when a 5 Vpp, 50 Hz sinusoidal sourcewas applied to MA a voltage of 360 mV (Vpp) and 59.2 mV (Vpp) weremeasured across S1 and MB respectively.Resistors in parallel can be used as intrinsic protection to reduce thecurrent induced in the mirror coils during a quench. The voltage inducedacross the resistor should also be reduced as to avoid arching (and possibledamage of the insulation). To induced a quench it usually requires someactive switch and a dissipative element. The dissipative quenching resistor(RQ = 10 Ohm) was chosen so the decay time was ∼ 10 msec. The quenchwas initiated by opening the IGBT in the driving circuit. In this simulationa small series resistance of 1 mOhm was added to the circuit to account for157Appendix A. Characterization of solenoids in the ALPHA trapthe lead resistance (see Fig. A.3).Fig. A.4 shows the IGBT gate driving voltage (V1), the current decayduring the quench in S1 (IS1), and the induced current in MA (IQA) andMB (IQB) with identical protection resistors Rp = 1 ohm and with an initialcurrent IS1 = 250 Amps.158Appendix A. Characterization of solenoids in the ALPHA trapFigure A.3: The dissipative quenching resistor (RQ = 10 Ohm) was chosenso the decay time was ∼ 10 msec. The quench was initiated by opening theIGBT in the driving circuit. In this simulation a small series resistance of 1mOhm was added to the circuit to account for the lead resistance.159Appendix A. Characterization of solenoids in the ALPHA trap 0.03           0.04               0.045                       0.05 1  0 1000     0 120     0  2  0 -2IQAIQBIS1V1Time (sec)Figure A.4: IGBT gate driving voltage (V1), the current decay during thequench in S1 (IS1), and the induced current in MA (IQA) and MB (IQB) withidentical protection resistors Rp = 1 ohm and with an initial current IS1 =250 Amps.160Appendix A. Characterization of solenoids in the ALPHA trapDuring the experiment it is more likely that a quench of Mirror coils Aor E is induced. The quench of MA was simulated using a quench resis-tor (RpA = 0.5 Ohm) that induces a decay time of 10 msec. The inducedcurrent in S1 and MB were simulated using protection resistors (RpS1 andRpB) of 10 Ohms (see Fig. A.5). The capacitance, as before, did not affectthe circuit at this decays times.According to the simulation (Fig A.6) a 10 msec quench in MA or MDstarting from an initial current of 800 Amps would induce a peak current of6 Amps in S1 and 0.33 Amps in MB (with respective voltages of 60 V and3.3 V) when 10 Ohm protection resistors are used. The induced currentdecays at the same rate as the quenching current in about 10 msec. Thisis convenient in S1 as the same protection resistor could be the quench-ing resistor (quench of 10 msec in S1, R = 10 Ohm) providing an intrinsicprotection (without requiring active elements).161Appendix A. Characterization of solenoids in the ALPHA trapFigure A.5: The quench of MA was simulated using a quench resistor (RpA= 0.5 Ohm) that induces a decay time of 10 msec. The induced current inS1 and MB were simulated using protection resistors (RpS1 and RpB) of 10Ohms.162Appendix A. Characterization of solenoids in the ALPHA trapTime (sec) 0.03       0.04  0.045   0.05  0.055  0.06 1  0 800     0 8 0  0 -4V1IS1IMAIQBFigure A.6: A 10 msec quench in MA or MD starting from an initial currentof 800 Amps would induce a peak current of 6 Amps in S1 and 0.33 Ampsin MB (with respective voltages of 60 V and 3.3 V) when 10 Ohm protec-tion resistors are used. The induced current decays at the same rate as thequenching current in about 10 msec.163Appendix A. Characterization of solenoids in the ALPHA trapIf a quench were required in S1, 10 msec decay from 250 Amp wouldinduce 25 Amps in MA and 1.2 Amps in MB (with respective voltages of 25V and 1.2 V) when a 1 Ohm protection resistor is used. The quench resistorthat causes a 10 msec decay in MA is 0.5 Ohm. If intrinsic protection of MAis required while having the ability to quench S1 and MA with 10 msecdecay a possible arrangement is the following:• Resistors in parallel in S1 and S2 with total R = 10 Ohm and P = 1000W for quenching and protection from induced current during a MAor ME quench.• Resistors in parallel in MA and ME with total R = 0.5 Ohm and atleast P = 1000 W for quenching and protection from induced currentsduring a S1 or S2 quench.• Resistor in parallel in MB and MC with total R = 10 Ohm and P = 10W.Fig. A.7 is a simulation of that scenario while quenching S1 (the sameconfiguration while quenching MA can be seen in Fig. A.6).164Appendix A. Characterization of solenoids in the ALPHA trap 0.03                0.04                 0.045                             0.05Time (sec)V1IS1IQAIQB 1  0 250 150   50    0 30    0 0.3  -0.1Figure A.7: A 10 msec quench in S1 starting from an initial current of 250Amps. Resistors in parallel in S1 with total R = 10 Ohm and P = 1000 Wfor quenching and protection from induced current during a MA or MEquench. Resistors in parallel in MA with total R = 0.5 Ohm and at least P= 1000 W for quenching and protection from induced currents during a S1quench.165Appendix BFabrication of carbon nanotubeelectron sourcesIn order to implement the electron sources mentioned in Chapter 5 ar-rays of MWNT’s have been synthesized (grown) on silicon and stainlesssteel substrates. The ideal array of nanotubes consists of multiple, verticaland parallel nanotubes. Chemical Vapour Deposition (CVD) is the mostcommon method used for growing carbon nanotubes. The CVD techniquerequires nanoparticles of a metal catalyst, usually iron, nickel or cobalt,that provides a base for the nucleation of the nanotubes. The carbon isprovided in the form of a carbon containing precursor gas such as acety-lene, ethylene or methane. To grow nanotubes on silicon substrates a thinlayer of the metal catalyst ( ∼ 1 nm) is deposited using e-beam evapora-tion. In this case, iron was used as the catalyst. The silicon substrates wereplaced in the middle stage of a three stage furnace. At the reaction temper-atures, usually 600-800o C, the carbon diffuses on the surface of the metalnanoparticles and forms the carbon nanotube rings. Grows on silicon isvery reliable and Fig. B.1 shows scanning electron microscope (SEM) andtransmission electron microscope (TEM) pictures of the samples grown insilicon. The nanotubes grown this way are approximately 2 - 4 mm long.We have used these nanotubes for the prototype discharge valve shown inChapter 5. However, the electrical contact between the nanotube and thesilicon substrate is not very good since the silicon has a very thin layer ofSiO on the surface which is an insulator.in order to improve the electrical contact nanotubes were grown onstainless steel. Polished sheets of stainless steel 304 were sonicated and166Appendix B. Fabrication of carbon nanotube electron sourcesetched in 37% HCl (Sigma-Aldrich) prior to CVD. It is possible that theacid exposes the iron grains. With the heat treatment some of the ironform nanoparticles from which the nanotubes grow. Fig. B.2 shows SEMand TEM pictures of nanotubes grown on stainless steel. These nanotubesdo not form vertical arrays. It is believed this is caused by the surfaceroughness of the substrate.Finally, we are currently attempting to grow nanotubes on nickel alloysand to improve the surface of the stainless steel by electropolishing duringthe etching. All the nanotube growth was performed in Dr. TakamasaMomose lab by Bill Wong under my supervision.167Appendix B. Fabrication of carbon nanotube electron sourcesFigure B.1: SEM and TEM pictures of carbon nanotube array grown on sili-con substrates. The TEM picture indicates MWNT’s with several concentricnanotubes.168Appendix B. Fabrication of carbon nanotube electron sourcesFigure B.2: SEM and TEM pictures of carbon nanotube array grown onstainless steel substrates. The TEM picture indicates MWNT’s with abamboo-like structure.169

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