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Selvi Afstudeerverslag Eindhoven University of Technology MASTER On the reservoir model for CO2-laser amplification Selvi, S.C. Award date: 2018 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain On the reservoir model for CO2-laser amplification S.C. Selvi November 15, 2018 2 Abstract In many technological applications, such as extreme ultraviolet lithography, high-intensity CO2- laser pulses are used, which are created in a CO2-laser amplification system. Research concerned with CO2-laser amplification tries to describe the dynamics of such amplifications systems to be able to make predictions about its characteristics and performance. The system under consider- ation is a single plasma cell of the amplification system having a pre-mixture of CO2-CO-N2-He and infra-red radiation with a wavelength of 10:58 µm and 10:206 µm. This thesis provides an in-depth treatment of laser physics together with a derivation of the reservoir model for CO2- laser amplification describing the evolution the infra-red CO2-laser radiation through the system under consideration. Furthermore, a numerical implementation of the model in the PLASIMO code is used to simulate the evolution of the radiation through various cases, thereby showing its verification and the capabilities of the model. Keywords: CO2, laser, PLASIMO, reservoir model. 3 4 Preface This thesis is written for my physics master performed in the Elementary Processes in Gas Dis- charges (EPG) group at the Physics Department of the Eindhoven University of Technology under the supervision of Jan van Dijk and Diana Mihailova (Plasma Matters B.V.). The modeling re- search of EPG involves the use of the PLASIMO code and directly involves Plasma Matters B.V. under the supervision of Diana Mihailova. The research at EPG and Plasma Matters B.V. is, among many other subjects, concerned with CO2-laser amplification. This research is performed in close relation with partner ASML, which uses it in and investigates it for its application in extreme ultraviolet lithography. The research tries to describe the dynamics of a CO2-laser am- plification system to be able to make predictions about its characteristics and performance. The modeling research involves theoretical derivations and development of the model, together with the investigation and verification of the simulation results using its numerical implementation in PLASIMO. These parts of the research go hand in hand, and the people, next to myself, working on these aspects of the research, whom I worked with, are Diana Mihailova, Jan van Dijk, and Marc van Dort. This thesis provides an extensive and in-depth treatment of laser physics for the theoretical derivation of the reservoir model for CO2-laser amplification from first principles. Furthermore, the evolution of CO2-laser radiation pulses through a CO2-N2-He-CO laser plasma is investigated using its numerical implementation in the PLASIMO code, thereby showing its verification and the capabilities of the model. 5 6 Contents 1 Introduction 9 1.1 The development of CO2-laser amplification . .9 1.2 Present work . 17 2 Background theory 21 2.1 Dynamics of a general molecule . 21 2.1.1 Structure . 21 2.1.2 Rotation . 27 2.1.3 Vibration . 30 2.1.4 Interaction of rotation and vibration and finer details of spectra . 33 2.2 Population distribution in a laser medium . 36 2.2.1 Boltzmann distribution . 36 2.2.2 Partition function . 38 2.3 Light-matter interaction in lasers . 42 2.3.1 Atom-field interaction . 42 2.3.2 Einstein coefficients . 46 3 Reservoir model and CO2-laser amplification 51 3.1 CO2 molecule . 51 3.1.1 Structure of CO2 ................................. 51 3.1.2 Rotation of CO2 ................................. 52 3.1.3 Vibration of CO2 ................................. 52 3.1.4 Molecular state of CO2 ............................. 53 3.1.5 Energy levels of CO2 ............................... 53 3.1.6 Fermi resonance of CO2 ............................. 55 3.1.7 Coefficients and wave-numbers of CO2 ..................... 56 3.2 Reservoir model . 59 3.2.1 Species densities . 59 3.2.2 From species densities to energy densities . 60 3.2.3 Energy densities . 60 3.2.4 Stoichiometry . 62 3.2.5 Ro-translational reservoir . 63 3.2.6 Vibrational temperature . 63 3.2.7 Populations . 64 3.3 Reservoir model for a CO2-N2-He-CO gas mixture . 65 3.3.1 Chemistry . 65 3.3.2 Energy density reservoir equations . 68 3.3.3 VT-transitions . 70 3.3.4 VV-transitions . 74 3.3.5 VVV-transitions . 80 3.3.6 VV transition between symmetric and bending mode of CO2 ........ 82 3.3.7 Coupling of VV-transition power sources . 83 3.3.8 Gas flow . 83 3.3.9 Electron excitation . 84 7 8 CONTENTS 3.3.10 Reservoir temperatures . 84 3.3.11 Reservoir model equations . 84 3.4 Reservoir model for CO2-laser amplification . 88 3.4.1 Evolution of the electromagnetic field . 88 3.4.2 Population inversion and rotational relaxation . 92 3.4.3 Summary of the reservoir model for CO2-laser amplification . 96 3.5 Numerical approach using PLASIMO . 98 4 Results: CO2-laser amplification 99 4.1 Setting up the system . 99 4.2 Creating the laser plasma . 102 4.3 Convergence . 105 4.4 Forward pulse . 108 4.5 Rotational relaxation . 110 4.6 Long forward pulse and pulse train . 113 4.7 Backward pulse . 115 4.8 Forward and backward pulse train . 118 5 Summary, discussion, and outlook 121 5.1 Summary . 121 5.2 Discussion . 122 5.3 Outlook . 123 Appendices 125 A Born-Oppenheimer approximation . 125 B Boltzmann distribution in statistical physics . 126 C Derivation of the Boltzmann distribution . 130 D Derivation of entropy in terms of partition function . 133 E Derivation of internal energy of a harmonic oscillator . 133 F Time-dependent perturbation theory . 134 G Separation of variables of the time-dependent Schr¨odingerequation . 136 H Derivation of the set of expansion coefficient differential equations . 137 0 0 I Evaluation of the matrix elements H and H ................... 138 f1i f2i (1) J Determining the first order coefficient cf ....................... 139 K Continuity equation . 140 L Amplified spontaneous emission . 141 Bibliography 149 Introduction "Waltzing Matilda, waltzing Matilda. You'll go waltzing Matilda with me." Rod Stewart 1.1 The development of CO2-laser amplification Lasers Lasers were invented in the early 1960's and are now common devices in commercial and scientific environments. A laser produces light, which is different in both intensity and quality than any other source. It has, therefore, a vast variety of applications among a large part of the technological platform. Laser light is used as the carrier wave in long distance communication, while in the manufacturing industry it is mainly used for cutting and welding. In medicine lasers are used for eye surgery to the sealing of dental enamel, while in biology they are applied to holographic microscopy and pattern recognition. Within this wide range of applications an important role is taken on by the carbon dioxide laser (CO2-laser), which emits infra-red light. Using CO2-lasers most nonmetallic materials can be cut, thereby for example increasing the cutting speed of garments. For laser surgery, potential advantages are the ability to cauterize and cut with minimum damage to adjacent tissue. The control of power and energy distribution makes a CO2-laser preferable over ruby and neodymium. Vibrational relaxation The behavior of a volume of gas in a sound field or gas flow can be described by the standard equations consisting of the conservation of mass, the conservation of momentum, the conservation of energy, and the equation of state. However, in many physical processes relaxation processes occur, such as in dielectric polarization, para-magnetism but also in molecular rotation and vibration. These relaxation processes are characterized by the change of a physical quantity followed by a slower process of equilibration of other physical quantities. The characteristic time for such a process is the relaxation time τ, giving an indication of the time in which stationary conditions are reached after the change of a physical quantity. The process of vibrational relaxation can be illustrated as follows. If energy is supplied to a gas, the translational degrees of freedom (dofs) will take up this energy and the temperature of the gas will increase. This increased translational energy is not in equilibrium with the internal dofs, being the rotational and vibrational dofs. As a result, energy flows from the translational dofs to the internal dofs continuing until all dofs are in thermal equilibrium. This energy transfer between the different dofs occurs during molecular collisions. In the study of vibrational excitation, by means of inelastic molecular collisions, the main difficulty lies in determining a solution for the system concerning the relative motion of two colliding molecules.
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