Coherent Microwave Scattering from Laser-induced Plasma Zhili Zhang A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Mechanical and Aerospace Engineering September 2008 Thesis Advisor: Richard B. Miles 3324313 3324313 2008 c Copyright by Zhili Zhang, 2008. ° All Rights Reserved Abstract This thesis demonstrates a novel diagnostic technique: coherent microwave Rayleigh scattering from a small plasma region generated by resonance enhanced multiphoton ionization (REMPI), Radar REMPI. The technique combines the high sensitivity of microwave detection and the extreme selectivity of nonlinear optical processes. Time accurate, high precision and non-intrusive spectroscopy is achieved. Properties of microwave scattering from the small laser plasma are first explored to verify its coherent nature. Experimental results show that microwave scattering is in the Rayleigh regime when both the microwave wavelength and skin layer thickness are greater than the size of the plasma. A general solution of Mie scattering is needed when either assumption is not satisfied. Then the concept of Radar REMPI is demonstrated both experimentally and theoretically in inert gases. Theoretical plasma dynamic and gas dynamic models are developed. The computational results match the experimental data satisfactorily. REMPI spectra obtained by microwave scattering are verified by comparison to the classical electron collection method. The highly time accurate, in-situ and non-intrusive nature of Radar REMPI is expected to attract numerous applications in fluid dynamics, combustion, chemistry and semiconductor physics. The applications of Radar REMPI are illustrated here by trace species detection and weakly ionized plasma amplification by REMPI enhanced avalanche ionization. Trace species detection is shown by detecting below 1 ppm (parts per million) nitric oxide in nitrogen. The nanosecond response of the excitation and detection makes Radar REMPI robust to quenching and useful for the reacting environment. The time accuracy and quick response of Radar REMPI is finally exploited in the plasma amplification. Sequential ionization by REMPI of argon and avalanche ionization of xenon in the argon and xenon mixture is observed experimentally and verified theoretically. iii Acknowledgements During my years in Princeton, I have been fortunate to find support, guidance and generous encouragement from many people. Without them, this thesis would not be possible. It is my pleasure here to express my gratitude to all those who helped me during this process. First I would like to express my most sincere thanks to my advisor Professor Richard Miles. He taught me about not only how to do science, but also how to be a scientist and gain individual insight. His deep understanding of physics, contagious enthusiasm and excellent leadership set up an ideal figure for me. Second I want to thank Dr. Mikhail Shneider with whom I have been working closely on a daily basis. He spent countless hours with me sharing his deep insights and bold ideas, and discussing various aspect of physical modeling and experimental results. The support from Dr. Sergey Macheret and Dr. Sohail Zaidi is also essential. Dr. Macheret taught me how to gain physical insight to a complex system with simple estimation and Dr. Zaidi showed me how to be a good experimentalist. I also feel indebted to many fellow students and friends at Princeton. I still clearly remember Xingguo Pan showed me around the laboratory when I first arrived at Princeton. Lipeng Qian spent a lot of time teaching me how to align the big Ti: Sapphire laser. I want to thank Manny Stockman, Sasha Likhanskii, Bruce Alderman, Demitry Opaits and CJ for many stimulating discussions. Finally I want to thank my family. My father Fumin Zhang and my mother Xianqin Meng give me endless support. My wife Meng Zhao gives boundless love and support. My son Haoming always greets me with his smile. It is you that make my work and life meaningful and colorful. This dissertation carries the number 3188T of in the records of the Department of Mechanical and Aerospace Engineering. iv Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 1 Introduction and Background 1 1.1 Optical diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Resonance Enhanced Multi-Photon Ionization (REMPI) . . . . . . . 3 1.3 Rayleigh scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Theory of coherent microwave scattering 2.1 2.2 8 Coherent microwave scattering . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Microwave scattering by a small-volume plasma . . . . . . . . 9 Rayleigh scattering and Mie scattering . . . . . . . . . . . . . . . . . 13 3 Microwave scattering from a laser spark in air 18 3.1 Laser spark in air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Avalanche ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Coherent microwave scattering experiments . . . . . . . . . . . . . . . 24 3.4 Rayleigh and Mie regime . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Coherent Microwave Rayleigh scattering of Resonance Enhanced v Multiphoton ionization in inert gas 34 4.1 The Concept of Radar REMPI . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.1 REMPI in argon . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.2 Plasma dynamic model of REMPI plasma at low pressure . . 46 4.2.3 Plasma dynamic model of REMPI plasma at high pressure . . 50 Experimental setup and results . . . . . . . . . . . . . . . . . . . . . 53 4.3 5 Radar REMPI spectroscopy for trace species detection 61 5.1 Radar REMPI spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Demonstration in neutral argon . . . . . . . . . . . . . . . . . . . . . 62 5.3 Xenon experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.4 Nitric oxide and sub-ppm detection . . . . . . . . . . . . . . . . . . . 75 6 Plasma amplification by REMPI and avalanche ionization 80 6.1 Idea of plasma amplification . . . . . . . . . . . . . . . . . . . . . . . 80 6.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.3 Plasma dynamic models . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7 Conclusions and future research 94 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Appendix 97 A Resonance enhanced multiphoton ionization 98 A.1 Coupling schemes of electrons in complex atoms . . . . . . . . . . . . 98 A.1.1 Pair coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 A.1.2 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 vi A.1.3 Energy level notations . . . . . . . . . . . . . . . . . . . . . . 102 A.1.4 The 3n-j symbols . . . . . . . . . . . . . . . . . . . . . . . . . 102 A.2 The Interaction Picture for Quantum Mechanics . . . . . . . . . . . . 104 A.3 Perturbation theory for multiphoton ionization . . . . . . . . . . . . . 107 A.3.1 Single Atom without losses . . . . . . . . . . . . . . . . . . . . 107 A.3.2 Continuum states . . . . . . . . . . . . . . . . . . . . . . . . . 112 A.3.3 Damping and Level shifts . . . . . . . . . . . . . . . . . . . . 113 A.4 Evaluation of Nth-order Matrix elements . . . . . . . . . . . . . . . . 114 A.5 Resonance Enhanced Multi-photon ionization . . . . . . . . . . . . . 114 A.6 Arbitrary polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.7 Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.8 Evaluation of the matrix . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.9 Keldysh theory of multiphoton ionization and tunneling ionization . . 121 vii Chapter 1 Introduction and Background This dissertation presents an experimental and theoretical study of a new diagnostic technique: coherent microwave scattering from laser-induced plasma. This technique combines highly selective capability of laser diagnostics with extremely sensitive microwave detection methods. These features combine to offer a high-precision ionization spectroscopy, which can do sensitive trace species detection, flow temperature measurement and possibly remote sensing. In this approach, two physical processes happen. First a frequency tunable laser beam is focused to generate a small-volume plasma by Resonance Enhanced MultiPhoton Ionization (REMPI) of some sample material. In REMPI, first m photons are simultaneously absorbed by an atom or molecule to bring it to an excited state. Another n photons are absorbed afterwards to generate an electron and ion pair. The so-called m+n REMPI is a nonlinear optical process which can only happen within the focus of the laser beam. Then a small-volume plasma is formed near the laser focal region. Second, under illumination of microwaves (Radar), electrons inside the plasma oscillate with the electric field of the microwave. The re-radiation from the electrons forms coherent scattering. When the skin depth at the microwave frequency is larger than the size of the plasma, in the far-field approximation, the microwave scatter- 1 ing from the small volume plasma lies in the Rayleigh range. The most important feature of the scattering is that the electric field of the scattered microwave directly reflects the total electron number inside the plasma. By combining the selectivity of REMPI and the sensitivity of microwave detection, a new diagnostic technique, Radar REMPI can achieve time-accurate, highly selective, extremely sensitive and non-intrusive measurements. 1.1 Optical diagnostics Optics has a long history of being a tool for people to understand the nature [1]. From ancient Europe to modern laboratories, optical diagnostics serves as an irreplaceable tool. By using optical telescopes, Nicolaus Copernicus discovered the heliocentric cosmology, which is regarded as the start of modern science. Niels Bohr’s theory of hydrogen spectra and Albert Einstein’s theory of the photoelectric effect opened the door of modern quantum physics. The invention of the laser (an acronym for Light Amplification by Stimulated Emission of Radiation) is one of the biggest achievements in the 20th century. The laser has inspired enormous theories and applications. Optical diagnostics based on the laser provides another important tool for studying complex physical-chemical system, fluid dynamics, combustion, semiconductor and biology etc. The important properties of laser-aided diagnostic techniques are their active nature, high spatial and temporal resolution and non-intrusiveness. The object to be measured is illuminated by the laser light and the response to the light is detected. The type of response differs, such as intensity, phase, frequency, polarization and direction of propagation, but the response is absent without the illumination of the laser. The highly coherent laser beam provides a unique characteristic from other background noise. With the aid of ultrashort (femtosecond) pulse and laser interfer- 2 ometry, equipment at a sub-angstrom and sub-femteosecond resolution is commercially available. And the most important feature of all is that the measurement is non-intrusive. Absorption spectroscopy is a common technique for optical diagnostics. If a laser beam passes through a gaseous molecular sample, molecules in the lower state may absorb radiant power at their eigenfrequencies, which are thus missing in the transmitted power. The difference in the spectral distributions of incident minus transmitted power is the absorption spectrum of the sample. Nowadays diode laser absorption spectroscopy has been gaining popularity in a wide array of practical measurement applications ranging from trace-gas detection[2], and combustion diagnostics [3] to numerous biomedical applications. Of course absorption spectroscopy is limited by the availability of diode laser source and the signal is path-integrated. Nonlinear optical spectroscopy, including Coherent Anti-Stokes Scattering (CARS) [4, 5] and its variations femtosecond adaptive spectroscopic techniques for coherent anti-Stokes Raman spectroscopy (FAST CARS)[6] etc, has been developed. When the molecules are put into coherent oscillation by a pair of preparation pulses and a third pulse is scattered off of this coherent molecular vibration, a strong anti-Stokes signal is generated. It has been used in the trace species detection [6], diagnostics in combustion environment[5] and a lot of biomedical applications[7]. The measurement is single-point and usually requires several lasers. 1.2 Resonance Enhanced Multi-Photon Ionization (REMPI) Among these optical diagnostics techniques, multi-photon ionization (MPI) is a nonlinear optical process. The ionization of a quantum system (atoms, molecules, ions) is called nonlinear if the condition h̄ω < Ei , where h̄ω is the photon energy of radiation 3 and Ei is the binding energy of the outermost electron in this system. It contradicts Einstein’s relation for the photo effect, which is the inequality h̄ω > Ew , where Ew is the work function of the system. However the multiphoton ionization, Kh̄ω > Ei (K is an integer) is in agreement of Einstein’s relation.[8] Multiphoton effect has been investigated since the beginning of the 20th century. Surprisingly two-photon absorption was predicted by Maria Goeppert-Mayer in 1929 [9] long before the invention of the laser. Bohr’s postulation states that an atom can only be populated when the photon energy matches the energy gap between two energy levels. The possibility of an electronic transition which violates Bohr’s postulation results from the uncertainty principle for energy and time, ∆E∆t ≥ h̄, where ∆E is the energy defect between a transition and ∆t is the lifetime of the transition. One interpretation is that the transition in a quantum system between initial and final states does occur with the energy defect ∆E, however, an electron can be found in the final state during the time ∆t only. Such transition and states are called virtual, unlike real transitions to states with long lifetimes. A simple estimation can be made. The inverse of the transition energy, typically is 10−15 second. Consequently the absorption of photons through virtual states must occur within a period of < 10−15 second. Therefore the photon flux has to be strong enough for there to be a large number of photons within < 10−15 second. This was only possible after the laser was invented. Since its invention, quantitative measurement of MPI is the concern of much research. Time of flight (TOF) spectroscopy and REMPI probe are most widely used techniques for detection of the ionization. TOF spectroscopy measures the time that it takes for a particle, object or stream to reach a detector while traveling over a known distance. For the ionization detection, ions are accelerated by an electrical field of known strength, with the velocity of the ion depending on the mass-to-charge ratio. Thus the time-of-flight can be used to determine the mass-to-charge ratio. The 4 modern TOF spectroscopy was improved by Wiley [10] in 1950s. Now applications of TOF spectroscopy are in physics, chemistry and biology. The Chemistry Noble Prize was awarded to the application of the TOF mass spectroscopy to biological macromolecules in 2002. As an alternative to the measurement of ions, a probe [11] can be used to quantitatively collect the electrons generated by MPI. A positive voltage is applied to the probe, electrons are collected and the current is measured. Both the TOF spectroscopy and the probe methods are intrusive and have limited applications in high pressure environment ∼ 1 atmosphere because ions or electrons may be lost through collisions during the drift to the collecting electrodes. 1.3 Rayleigh scattering Rayleigh scattering is named for Lord Rayleigh, who successfully explained the color of the sky in 1871 [1]. Rayleigh scattering can be explained classically based on the electric dipole approximation. Electrons in atoms, molecules or small particles are forced to oscillate by an applied electromagnetic field. The acceleration and deceleration of the electrons generate radiation. When the wavelength of the electromagnetic field is much larger than the size of the radiator, the dipole approximation of the radiation is valid. If the scattering sources are stationary, then this secondary radiation is phase locked to the driving electromagnetic field. In this case, each scatterer is driven by a field which is the coherent sum of the applied field and the fields from all the other scatterers. If the density of sources is high enough that they may be treated as a continuum, and if they are uniformly distributed, then the radiation cancels in all but the forward direction, where it adds coherently, leading to a change in the index of refraction of the propagating wave as shown by Ewald and Oseen [1]. In a gas, the motion of the molecules leads to microscopic density fluctuations that randomize the phases and 5 cause the scattering to be incoherent in all but the forward direction. Away from the forward direction, very rapidly changing interferences occur which average to remove coherent effects and make the scattering intensity just proportional to the number of scatterers. In the forward direction, coherence is maintained since there is no momentum transfer. The phases add together, independent of the motion, and the Ewald and Oseen solution remains valid [12]. For this reason, Rayleigh scattering is called coherent. This coherent nature of the interaction means that Rayleigh scattering is sensitive to long range coherent density fluctuations associated with acoustic waves and turbulent phenomena [13]. An analog of Rayleigh scattering can be extended to the microwave scattering. The electric field of the microwave produces charge separation inside a scatterer, which could be atoms, molecules or a plasma. If the wavelength of the microwave is much larger than the size of the scatterer, the dipole approximation is valid. Without the complication of the multiple scatters, coherent Rayleigh scattering of a single scatter is expected. The coherent nature of the scattering is very important because it limits the detection system to a signal with a fixed phase shift. This makes it possible to implement heterodyne, homodyne and IQ (In-phase and Quadrature) detection of the microwave at an extremely low level, even below the thermal noise floor at the fundamental microwave frequency. Background noise from thermal and other fluctuations only adds incoherent background. 1.4 Outline of this Thesis In this thesis, in chapter 2, a theory of coherent microwave scattering from a smallvolume plasma is presented. When both microwave wavelength and skin layer thickness at the microwave frequency are much greater than the size of the plasma and the detector is far away from the plasma, the scattering lies within the Rayleigh regime. In 6 this case the dipole approximation remains valid. A discussion of Rayleigh scattering and Mie scattering is also given. In chapter 3, microwave scattering from a laser spark in air is presented. It demonstrates the coherence nature of microwave scattering. At the initial phase of laser spark, i.e., avalanche ionization, microwave scattering lies in the Rayleigh range as seen by the isotropic nature of the microwave scattering. It is also shown that at later time, the dipole approximation is invalid: an anisotropic microwave scattering is observed. In chapter 4, Radar REMPI is demonstrated by combining coherent microwave scattering with resonance enhanced multiphoton ionization (REMPI) in an inert gas. It is shown that Radar REMPI is a time-accurate, sensitive, selective and nonintrusive detection method. A simplified model of plasma evolution is given. A comparison between experimental and theoretical results gives good agreement. In chapter 5, the application of Radar REMPI for trace species detection is presented. A nitric oxide REMPI spectrum near 226 nm was obtained using (1+1) Radar REMPI. In addition 160 parts per billion (ppb) of nitric oxide inside nitrogen was detected near 380 nm by using (2+1) REMPI. In chapter 6, plasma amplification by combining REMPI and avalanche ionization is demonstrated. A mixture of argon and xenon is ionized by a laser beam which is tuned to 261.27nm in three photon resonance with the argon (3d, J = 3) level. A combination of REMPI of argon and avalanche ionization of xenon is shown both experimentally and theoretically. In chapter 7, conclusions are given and future work is discussed. In the appendix A, a general theory of resonance enhanced multiphoton ionization is given. Based on time dependent perturbation theory, a general formula of multiphoton absorption is derived. The other type of ionization, tunneling ionization is also discussed using Keldysh theory. 7 Chapter 2 Theory of coherent microwave scattering This chapter presents the theory of coherent microwave scattering. The microwave scattering from a small volume and low conductivity plasma is isotropic in space and lies within the Rayleigh regime. The new diagnostic method can directly measure the total electron number inside a small-volume plasma when it lies in the Rayleigh regime. In this chapter, first a general formula of coherent microwave scattering is derived. A discussion of coherent and incoherent scattering is given. Rayleigh and Mie scattering patterns are calculated using the general Mie scattering solution of Maxwell equations. 2.1 Coherent microwave scattering Plasma diagnostics with microwaves has been investigated for more than 40 years [14, 15]. Typical experimental and theoretical research has focused on plasmas whose sizes are much larger than the microwave wavelength. Microwave transmission, reflection, interferometry, incoherent scattering and microwave radiation from plasma were examined. However when the size of a plasma is much smaller than microwave 8 wavelength, the usual microwave measurement methods do not apply. 2.1.1 Microwave scattering by a small-volume plasma A small-volume, non-stationary plasma can be produced by a focused, pulsed laser beam through avalanche ionization or REMPI. The plasma located near the focal point of the laser beam is generally on the scale of tens of microns in diameter and less than a few millimeters in length. When it is illuminated by microwave whose wavelength is on the order of centimeters, the charge separation can be modulated, √ creating oscillating induced dipoles inside. If the skin layer thickness, δ = 2/ 2µ0 σωm at microwave frequency ωm and with plasma conductivity σ , is greater than the characteristic size of the plasma, all the electrons in the plasma oscillate in the same phase as the microwave. So, in the far field, the plasma can be regarded as one induced point dipole radiation source of microwaves, and the scattering falls into the Rayleigh scattering approximation. In that case, the scattered microwave power is directly proportional to the square of the number of electrons. On the other hand, if the skin layer thickness δ is small, then some of the electrons are shielded. In that case, the scattering falls into the Mie regime and no longer has a dipole character. The Rayleigh range is particularly interesting since in that regime, the microwave scattering intensity is directly related to the total number of electrons and can be used for quantitative measurement of avalanche ionization and REMPI processes. The physical process can be described as the following. An incident microwave of frequency fm = ωm /2π illuminate a small-volume plasma. The microwave penetrates and interacts with the quasi-neutral weakly ionized plasma column. Inside the plasma, electrons oscillate with the microwave while much heavier ions can be regarded as immobile. Mainly three forces act on the electrons: electron-neutral collisions, electron-ion attractions and microwave forces. The volume of the plasma is considered to be small if the wavelength of the microwave is much larger than the size 9 z, E I0 x, k L y, H R Figure 2.1: Scheme of coherent microwave scattering by a small-volume plasma. of the plasma. The field strength or intensity of the microwave can then be regarded as uniform across the spatial scale of the plasma. Within the skin layer δ at the microwave frequency ωm , a partial differential equation for electron movement ∆z can be written [16, 17] ¨ + υen ∆z ˙ + ωp2 ∆z = e E0,i cosωt ∆z me (2.1) Where e is electron charge, me is the mass of electron, E0,i is the electrical field of the microwave, ωp = q e2 m/ε0 m is the electron plasma frequency and υen is collisional frequency between electrons and neutrals. The microwave radiation can be of any polarization, but the strongest effect will be when the microwave radiation is linearly polarized with its electric field E along the plasma column. Solving the equation of 2.1 gives, ∆z(t) = ∆z0 cos(ωm t + ϕ) (2.2) q 2 2 )2 + (ν ω )2 ). ) and ∆z0 = eE0,i /(m (ωp2 − ωm Where tan ϕ = −νen ωm /(ωp2 − ωm en m 10 Plasma S kin la ye r Microwave E k B Figure 2.2: Principles of microwave scattering by a laser-induced small-volume plasma. Within the skin layer(unshaded region), plasma is transparent to the microwave. Outside the skin layer, microwave is shielded. In the far field, where the distance between the receiver and the plasma is much greater than the microwave wavelength λ, which, in turn, is much greater than the scale of the plasma, L, the instantaneous dipole radiation power is Θ= d¨2 6πυ0 c3 (2.3) Where Θ is the intensity of scattered microwave radiation, d = ene ∆zLS is the dipole moment, S is the cross-sectional area of the plasma, and ne is the electron number density in plasma. The total intensity of coherently scattered microwave radiation averaged over a microwave cycle can be found as hΘi = = 4 Im0 V 2 ωp4 ωm 2 )2 + (ν ω )2 ] 6πc4 [(ωp2 − ωm en m 4 Im0 ωm e4 N 2 2 )2 + (ν ω )2 ] ε2 m2 6πc4 [(ωp2 − ωm en m 0 e (2.4) where h·i means average over a microwave cycle, c is light speed in vacuum, ε0 is the dielectric constant in vacuum, V ≈ LS is the volume of the plasma, Im0 is the 11 incident microwave intensity, N ' V ne is the total electron number in the plasma. In the collision dominated regime where νen >> ωm , ωp , the effective total ”Rayleigh” scattering cross section σR , which can be written as [17] σR = 2 hΘi e 4 ωm N2 = 2 4 2 2 Im0 6πε0 c me υen (2.5) The effective differential cross section is ∂σR 3 = σR sin2 φ ∂Ω 8π (2.6) where φ is the angle between polarization of the microwave and the direction of scattered signal, as shown in figure 2.1. θ is the angle in the plane normal to the polarization vector and θ = 0 is defined to be along the y axis, orthogonal to the propagation direction of the incident microwave beam. Note that for the experiments, the laser propagates in the direction of the microwave polarization vector. This selection helps to assure that the induced polarization vector lies in the same direction as the polarization of the incident microwave radiation and assures cylindrical symmetry. At νen >> ωM W >> ωp , the equation 2.4 can be approximately simplified as 2 hΘi ≈ Ii N 2 ωm (2.7) The averaged power of microwave scattering is approximately proportional to the intensity of the incident microwave, the square of the total electron number inside the plasma and the square of the microwave frequency. Of course, the averaged electric field of the microwave scattering is the square root of the averaged power. So the averaged electric field of the microwave scattering is approximately proportional to the field of the incident microwave, the total number of the electrons inside the plasma and the frequency of the microwave. 12 2.2 Rayleigh scattering and Mie scattering In 1871, Lord Rayleigh successfully explained the blue color of the sky which is caused by light scattering of sun light by air molecules with diameters much less than the light wavelength λ . Using Maxwell’s equations, Rayleigh calculated the scattering cross section of the dielectric particle and explained the polarization of the sky light. In 1908, G. Mie published his famous paper about a rigorous solution for the diffraction of a plane monochromatic wave by a homogeneous sphere of any diameter and of any composition situated in a homogeneous medium [1]. Mathematically, Mie’s theory is a solution of Maxwell’s equations which describe the field arising from a plane monochromatic wave incident upon a spherical surface, across which the properties of the medium change abruptly. So Mie’s solution gives a general formula of scattering of electromagnetic wave. Rayleigh scattering is an approximation of Mie’s formula when the wavelength of electromagnetic wave is much larger than the size of the scatterer. The diffraction of a plane, linearly polarized, monochromatic wave with a frequency ω by a sphere with a radius of a, dielectric constant εII in a homogeneous, isotropic medium with dielectric constant εI , is shown in figure 2.3. The medium around the sphere is nonconducting and both the medium and the sphere are nonmagnetic. The Maxwell’s equations are − → − → ∇ × H = −k1 E − → − → ∇ × E = k2 H 13 (2.8) x E P r k z σ εI εII Θ a Figure 2.3: Diffraction by a conducting sphere. Where iω 4πσ (ε + i ) c ω iω = c k1 = k2 (2.9) The wave number k is real outside and complex inside the sphere. Boundary conditions are that the tangential components of E and H should be continuous across the surface of the sphere, (I) (II) (I) (II) Etang = Etang Htang = Htang (2.10) The Mie’s solution is written as the superposition of the electric wave and the magnetic waves in spherical coordinates [1]. 1 cos φ ∞ (1) (I)(cos θ) Σl=1 l(l + 1)e Bl ζl (k (I) r)Pl 2 r 1 cos φ ∞ e 1 (1)0 (I)0 (cos θ) (I) (I) = − (I) sin θ − im Bl ζl (k (I) r)Pl (cos θ) Σl=1 [ Bl ζl (k (I) r)Pl ] k r sin θ 1 sinφ ∞ e (1)0 (I)0 (cos θ) 1 (I) (I 0 ) Σl=1 [ Bl ζl (k (I) r)Pl − im Bl ζl (k (I) r)Pl (cos θ) sin θ] = − (I) k r sin θ Er(s) = (s) Eθ (s) Eφ k (I)2 14 i sinφ ∞ (I) (I) Σl=1 l(l + 1)m Bl ζl (k (I) r)Pl (cos θ) 2 r 1 sinφ ∞ e 1 (I) (I) (I)0 (I)0 = − (I) Σl=1 [ Bl ζl (k (I) r)Pl (cos θ) + im Bl ζl (k (I) r)Pl (cos θ) sin θ] r sin θ k2 1 cos φ = (I) × r k2 1 (I) (I) (I)0 (I)0 (I) e Σ∞ (cos θ)sin θ + im Bl ζl (k (I) r)Pl (cos θ) ] (2.11) l=1 [ Bl ζl (k r)Pl sin θ Hr(s) = (s) Hθ (s) Hφ (II) k (I) k2 where e Bl = il+1 2l + 1 × l(l + 1) (II) (I) k2 k (II) ψl0 (k (I) a)ψl (k (II) a) − k2 k (I) ψl0 (k (II) a)ψl (k (I) a) (I) m Bl (I)0 (II) (I) k2 k (II) ζ1 (k (I) a)ψl (k (II) a) − k2 k (I) ψl0 (k (II) a)ζl (k (I) a) 2l + 1 = il+1 × l(l + 1) (I) (II) k2 k (II) ψl (k (I) a)ψl0 (k (II) a) − k2 k (I) ψl0 (k (I) a)ψl (k (II) a) (I) (I) (I)0 (II) k2 k (II) ζ1 (k (I) a)ψl0 (k (II) a) − k2 k (I) ψl (k (II) a)ζl and ψl (ρ) = q (k (I) a) (2.12) q πρ/2Jl+1/2 (ρ) is Bessel function of the first kind, χl (ρ) = − πρ/2Nl+1/2 (ρ) (1) (m) is Bessel function of the second kind, ζl (ρ) = ψl (ρ) − iχl (ρ), Pl (cosθ) is Legen(I) dre function and the addition of a prime to the function ψl , χl and Pl denotes √ differentiation with respect to their argument. k (I) = 2π/λ0 ε(I) = 2π/λ(I) is the (I) wavenumber in the surrounding medium and k2 = i2π/λ0 is the imaginary part of the wavenumber. From these expressions,it is seen that isotropic scattering occurs for spheres with small diameters and low conductivity and anisotropic scattering occurs for spheres with larger diameters and higher conductivity. In figure 2.4, scattering of microwave at frequency of 12 GHz from spheres with different radii of 0.5 mm, 1 mm and 2mm and conductivity of 50 mho/cm is shown. In figure 2.5, scattering of microwave at frequency of 12 GHz from spheres with a radius of 1 mm and conductivities of 10, 100 and 1000 mho/cm is shown. 15 r = 0.5 mm, σ = 50 /Ω cm r = 1 mm, σ = 50 /Ω cm r = 2 mm, σ = 50 /Ω cm 90 90 90 60 120 30 150 180 180 0 180 0 210 210 330 240 0 210 330 240 30 150 30 150 330 300 300 60 120 60 120 240 300 270 270 270 Figure 2.4: Mie scattering pattern in x-z plane by spheres with radius of 0.5mm, 1mm and 2mm and conductivity of 50 mho/cm. r = 1 mm, σ = 100 /Ω cm r = 1 mm, σ = 10 /Ω cm 90 120 150 120 30 180 330 210 300 120 60 150 0 270 90 90 60 240 r = 1 mm, σ = 1000 /Ω cm 150 30 180 0 330 210 300 240 60 30 180 0 330 210 300 240 270 270 Figure 2.5: Mie scattering pattern in x-z plane by spheres with diameters of 1mm and conductivity of 10, 100, and 1000 mho/cm. 16 Figures of 2.4 and 2.5 clearly show that isotropic microwave scattering is obtained for spheres with small diameters and low conductivity. The rigorous Mie’s solution here serves as a criterion of Rayleigh and Mie scattering. Rayleigh scattering is isotropic while Mie scattering is anisotropic. Since the size and conductivity of the small plasma is hard to measure accurately, and the skin layer thickness at the microwave frequency is almost impossible to obtained accurately, anisotropic Mie scattering is difficult to quantify. So an isotropic Rayleigh scattering is desirable for quantitative measurement of the number of electrons in the plasma by microwave scattering method. A measured anisotropic scattering pattern is an indication that the plasma is not in the Rayleigh range. This will be used in the next chapter to show scattering from the Rayleigh range to the Mie range with the temporal evolution of the plasma. 17 Chapter 3 Microwave scattering from a laser spark in air In this chapter, microwave scattering experiments from a laser spark in air are shown to verify the applicability of microwave Rayleigh and Mie scattering. Microwave scattering from avalanche ionization which happens on the order of tens of nanoseconds is isotropic in space and lies within the Rayleigh regime. While the scattering from the after-spark plasma expansion and evolution is non-isotropic and lies in the Mie regime. The experimental results verify the coherent microwave scattering theory shown in chapter 2. It also shows that the time accuracy of the detection method is fast enough to capture the dynamics of the avalanche ionization in air. 3.1 Laser spark in air Almost as soon as the invention of the giant pulse laser in the 1960’s, people found that a tight focus of the laser beam can generate laser spark or laser breakdown. For decades, research has been carried on the mechanisms leading to breakdown, threshold power and other influences of various parameters such as wavelength, pressure, pulse length and material [18, 19, 20]. Different methods have been applied 18 to diagnose the laser spark, for example, fast cameras [21], X-ray spectrometry [19], Mach-Zehnder interferometry [22], Rayleigh scattering [23] and beam deflection techniques [24] etc. Here a microwave scattering method is used to diagnose the laser spark. The traditional microwave diagnostic methods, such as microwave absorption and interferometry [14] are extremely difficult to apply for laser spark studies. The volume of the laser spark is small relative to the microwave wavelength, so diffractive effects dominate, and the phase of the microwave is not changed significantly. However, microwave scattering from a laser spark, analogous to induced dipole scattering of light from an atom or molecule [13], can be applied for a plasma region whose scale is much less than the wavelength of the microwave radiation. Not only is microwave scattering a non-intrusive method of diagnostics, but it is also time accurate so it can follow the dynamics of the plasma formation and subsequent evolution with precision of nanosecond or even less, depending the configuration of the microwave system. Moreover, microwave scattering has a detection limit which can be as low as the thermal noise level, and heterodyne and homodyne detection can be employed for very high noise rejection. Here a microwave scattering method is used to measure the avalanche ionization phase in a laser spark to test the coherent nature of the method [17]. Avalanche ionization gives a quantitative understanding of the laser spark formation. The detection of the transition following avalanche ionization can remove the uncertainty associated with the beginning time of the after-spark mechanisms in the theory of spark evolution [25, 24, 26, 27]. Due to the high sensitivity of detection and its direct relationship to the density of charges rather than to optical emission, the whole life cycle of the laser spark can be shown, from the initial stages of ionization during the nanosecond scale laser pulse to about 50 microseconds of spark evolution following the laser pulse, including spark growth, recombination and attachment stages. This means that the microwave scattering method can achieve a continuous measurement 19 of the laser spark. As such, it provides a powerful tool for studying the physics of spark mechanisms, especially the early stage where few techniques exist. Of particular interest here is the rising edge of the precursor pulse which occurs during and shortly after the laser pulse and corresponds to the avalanche ionization process. According to the electron-impact avalanche ionization theory, the rise time of the precursor is directly related to the initial electron number density. 3.2 Avalanche ionization Avalanche ionization can be explained as the following [18, 19]: If an electron is accelerated in an electric field without collisions, the movement of the electron is pure oscillation. The electric field does no work to the electron on the average. When collisions between the electron and neutrals happen, they disturb the purely harmonic course of the electron’s oscillation. A sharp change in the direction of motion after scattering stops the electron from achieving the full range of displacement; the electron starts oscillating anew after each collision, with a new phase and new angle relative to the instantaneous direction of velocity. At this early stage of laser-induced breakdown, quasi-neutral plasma has not been formed yet. Theoretically the growth of the electron density with time should be analyzed by solving the Boltzmann equation for the electron distribution function in the velocity space. Here for simplicity, only a simplified theory of optical breakdown will be presented. So the equation for the electron’s movement can be written as m dv = −eEm sin(ωt) − mvνm dt (3.1) where e and m are the electric charge and mass of the electron, v is the velocity, Em is the electric field, ω is the frequency of the electric field, and νm is the collision frequency between electrons and neutrals. 20 Solving the equation 3.1 gives " eE cos ωt v = Re me (iω − νm ) # (3.2) The power gained from the field is obtained by taking the time average of −eE · v, Q= e2 E02 νm 2) 2me (ω 2 + νm (3.3) The ionization rate under the applied laser field EL = Re[E0 exp(−iωL t)] can be expressed as the ratio of the power absorbed by a free electron to the ionization energy [18] e2 E02 νen 1 × 2 2 2me (ωL + νen ) ξion e2 IL νen 1 = × 2 2 )ε c me (ωL + νen ξion 0 νi (t) = (3.4) where νi (t) is the ionization frequency, ξion is the ionization threshold of neutrals and IL (t) is the laser intensity, IL = 21 E02 ε0 c. Here it is assumed that the ionization during the avalanche phase is mainly due to the collisions between electrons and neutrals and that there is no energy loss for electrons other than ionization. These assumptions are valid for the initial stages of laser-induced breakdown, when the ionization rate is much greater than the losses due to recombination and diffusion, however it is obviously invalid at the late stage of breakdown where electron-electron collisions and Coulomb interaction dominate [28, 19]. This method underestimates the effective ionization threshold by excluding other losses. For room air (N2 : O2 ≈ 4 : 1) the ionization threshold can be approximated, 21 ξion ≈ ξion,O2 ξion,N2 /(0.8ξion,O2 + 0.2ξion,N2 ) ≈ 14.77ev (3.5) where ξion,O2 and ξion,N2 ionization threshold of O2 and N2 , respectively. At the initial stage of the breakdown, the plasma volume is approximately the same as the laser focus size. Electrons mainly collide with neutrals, and the electron number density approximately follows the relationship dne = νi ne dt (3.6) So the electron number density can be expressed as ne = ne0 exp µZ t 0 ¶ νi dt 0 (3.7) Where ne0 is the initial electron number density in the plasma region before ionization by the laser. One can get the expression for the microwave scattering signal intensity in terms of the parameters of the laser and the microwave source during the precursor rise by combining equation 2.7 and 3.6. The scattered signal intensity becomes [17] µ Z t 2 hΘi ∝ Im0 ωm V 2 n2e0 exp 2 2 ∼ Im0 ωm V 2 σω2 0 ¶ νi dt0 (3.8) 2 2 ) is the high-frequency conductivity of the + ωm where σω (t) = e2 ne (t)νen /m(νen weakly ionized plasma [18] during the initial avalanche phase of a laser breakdown. In this phase, electrons have temperature Te ∼ 1eV which is estimated from the 22 Q=50 mJ 35 1.0 25 1012 10 ne(t)/ne,0 0.8 Laser pulse 10 0.6 108 106 0.4 104 0.2 50 35 25 102 IL (Arbitrary units) rL=7.5 µm 1014 b 0 10 0 1 2 3 4 5 6 7 8 9 0.0 10 t, ns Figure 3.1: Relative electron number densities at avalanche breakdown, computed with equations 3.4 and 3.5. Arrows show the time delays when the breakdown criterion (the dotted horizontal line) is realized. experimental value [16] of the electron mobility and the collision frequency is νen ≈ 3.5×109 ·p·300/T , where p and T are the gas pressure in Torr and the gas temperature in Kelvin [16]. When only avalanche ionization is included within tens of nanoseconds, the plasma volume V0 , the incident microwave intensity, the microwave frequency, and the initial number density of electrons can be considered constant [22]. Taking into account 1/3 the condition of plasma transparency, δ > V0 , from equation 3.8 it follows that ³ R t the scattered signal intensity is Is ∝ hΘi ∝ exp 2 ´ 0 , where νi (IL , t0 ) is the 0 νi dt ionization rate, which is a function of laser intensity IL as in equation 3.4, i.e. νi ∝ IL . A characteristic of the observed microwave signal is the delay time between the laser pulse and the microwave scattering. This delay occurs because of the time required for the electron number density to build up from the ambient level to a level high enough to be observed. According to Ref. [18, 16], nanosecond (and 23 longer) laser pulses creating breakdown in air at pressures above several tenth of one atmosphere always produce avalanche ionization, and the breakdown condition corresponds to about 1013 avalanche electrons per one initial electron (non-stationary breakdown criterion). Therefore, we can suppose that ne (t)/ne,0 = 1013 is a criterion for a time delay corresponding to the precursor peak, observable by the microwave scattering. To produce a realistic estimation of the delay, a typical laser pulse shape from our frequency doubled, Q switched Nd:YAG laser was used in Equation 3.8. From equations 3.4 and 3.5, for a measured effective laser beam radius, rL = 7.5µm, we predict ne (t)/ne,0 for different energies per pulse. The build up rates for 50mJ, 35mJ and 25mJ pulses and the laser pulse shape assumed are shown in Figure 3.1. The predicted delay times for this experimental pulse shape and for the assumed criterion, ne (t)/ne,0 = 1013 are also shown in the figure 3.1. 3.3 Coherent microwave scattering experiments The experimental setup is shown in figure 3.2. A Q-switched, frequency-doubled, Nd: YAG laser (Continuum YG661-10, pulse width 8 nanoseconds) was used to generate a breakdown of the room air. One Glan-Thomson prism was used to vary the power of the laser beam without changing its time characteristics. The beam was focused in the room air by a lens of 10cm focal length. The laser focal region in the air had dimensions of about 1mm in length and about less than 15 microns in diameter. A tunable Gunn-diode microwave source (West Divident, operating frequency: 12.6GHz, wavelength: 2.36cm, power: about 15mW ) was beamed into the breakdown area through a microwave horn (WR75). The polarization of the microwave radiation was along the propagation direction of the laser beam. The source itself was covered by a metal box to minimize the background signal. The distance between the laser spark and the source microwave horn (WR75) was 30cm. The microwave scattering 24 Glan polarizer Q-switch, Nd:YAG laser power meter microwave absorber x z y 12.6 GHz microwave source Microwave receivers power meter Figure 3.2: Experimental setup of microwave scattering from laser spark 25 0.6 Arbitrary Units Microwave intensity (Arbitrary Units) 0.4 0.2 0.0 0 20 40 60 80 100 120 140 160 180 200 Time(ns) µs Figure 3.3: Sample measurement signal of microwave scattering from a laser spark in room air. The inset window shows zoomed in view of the precursor pulse. was detected with a second horn (WR75). The distance between the laser spark and the receiving horn was between 15 ∼ 35cm, which is much greater than the 2.38cm wavelength of the microwave and very much greater than the scale of the plasma, so that the far-field scattering approximation is valid. The received microwave signal was amplified by an amplifier with a gain factor 30dB and then rectified by a diode (HP8427A). Care was taken to make sure the whole breakdown area was surrounded by microwave absorbers (> 20dB) to minimize electronic disturbances and spurious scattering that would otherwise affect the microwave scattering measurement. Figure 3.3 shows a sample microwave measurement from the laser induced breakdown. For this shot, the laser beam had an energy of 55mJ. The signal typically is composed of two features. In time sequence, the first peak occurs over an approxi- 26 mately 20ns time scale and is called the ”precursor” pulse. The second peak which occurs at 7.5 microseconds is the after-spark evolution of the plasma. The rising edge of the precursor pulse reveals information on the avalanche phase of the laser breakdown. The avalanche phase of the optical breakdown is only valid until thermal ionization becomes dominant, which occurs when the temperature of the plasma reaches greater than 104 K [18]. During the precursor pulse, the scale of the plasma remains small, and for the whole ionization process, the plasma region is expected to remain transparent to the microwave radiation because the skin layer thickness is projected to be greater than plasma scale. For example, even for a fully ionized laser induced air plasma, with a conductivity σ ∼ 104 /Ohm · m , at the microwave frequency used in these experiments ω ≈ 2π · 12.6 × 109 rad/sec the skin layer thickness (δ ≈ 4 × 10−5 m) is greater than the plasma scale (r0 ≈ 10−5 ) m. When the plasma is close to a thermal, fully ionized plasma, the conductivity depends primarily on temperature σ ∝ T 3/2 [16]. From equation 3.8, the microwave scattering intensity Is ∼ hΘi ∝ σ 2 V02 ∝ T 3 in that regime. Therefore, the scattered signal intensity increases during the laser pulse both from the avalanche ionization and the rapid temperature increase. After the laser pulse ends, cooling due to the radiation losses causes the plasma temperature and, therefore the conductivity to go down, leading to the observed decrease of the scattered precursor intensity. This explains the precursor temporal shape. We can approximately regard the lowest point in the curve as the transition from the precursor to the after-spark evolution. It may remove an assumption of the start time in many theories of after-spark evolution. The second broad peak is related to the after spark dynamics of the plasma, including the growth of the plasma region and expansion cooling. There are many papers [25, 24, 26, 27]that describe the post discharge plasma evolution after the laser spark [24, 28] as well as regular pulsed arcs [25, 26, 27]. 27 3.4 Rayleigh and Mie regime The first experimental task was to establish the point dipole nature of the precursor scattering. If this can be shown to be the case, it substantiates the Rayleigh rather than Mie nature of the scattering process and confirms that the skin layer effects can be neglected. As discussed in the section of 2.2, a Rayleigh-like dipole field is cylindrically symmetric surrounding the axis of the dipole and the scattering intensity goes to zero for scattering along the axis, as indicated in equation 2.6. If the scatterer falls into the Mie regime, then the scattering is no longer cylindrically symmetric, and the pattern of the scattering can be related to the size, shape, conductivity and dielectric constant of the scatterer. A non dipole scatterer may also depolarize the scattering, leading to a non zero signal along the polarization axis. In the absence of background interference, in the far field the measured signal intensity from a dipole scatterer will drop as the distance squared. Both the spatial distribution and the scattering fall off of the signal with distance of the signal scattered from the precursor were measured. Figure 3.4 shows the precursor scattering at different positions of the microwave receiver relative to the microwave induced polarization direction. For a point dipole, it is assumed that the induced polarization lies along the polarization direction of the applied microwave field. The data in Figure 3.4 indicate that the scattered microwave has a dipole radiation pattern, with a maximum at φ = 900 and 0 for φ = 00 . Scattering signals from receivers at φ = 450 and 1350 positions show signals that are lower than the ones at 900 , as expected. Figure 3.5 shows the same precursor but in the φ = 900 plane through the dipole center and normal to the polarization axis. On the right are the data from the full time evolution of the laser induced discharge, and on the left is an expanded view of the scattering from the precursor portion of that time interval. The precursor signals measured with the receiver located at θ = −450 , θ = 00 and θ = +450 are almost identical, showing that during the precursor period, the scatterer 28 0 45 0 0 5 13 0 90 0 ø 1.0 0 135 90 0 45 0 00 Microwave Intensity (Arbitrary Units) 0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 50 60 Time(ns) Figure 3.4: Precursor part of microwave scattering signal at various locations around the laser breakdown region relative to the microwave induced polarization direction. The receiver angles are 00 (dashed dot dot (green) line), 450 (dashed (red) line), 900 (solid (black) line), and 1350 (dotted (blue) line). Each curve represents an average of 64 pulses. For all curves, θ = 00 29 5 0 +4 θ 0 0 5 -4 1.2 1.0 0 Forward 45 0 Backward 45 Microwave Intensity (Arbitrary Units) 0 Microwave Intensity (Arbitrary Units) 0 Forward 45 0 Backward 45 1.0 0 0.8 0.6 0.4 0.2 0 0.8 0 0.6 0.4 0.2 0.0 0.0 -0.2 -10 0 10 20 30 Time(ns) 40 50 60 0 10 20 30 40 50 Time(µs) Figure 3.5: Short time (left) and long time (right) microwave scattering at φ = 900 and at θ = ±450 and 00 . Note that the precursor scattering is independent of θ, indicating scattering in the Rayleigh range, whereas the long time scattering is in the Mie regime. is acting like an ideal dipole and the scattering is Rayleigh in nature. In the longer time regime, however, the plasma grows and moves into the Mie regime, where it is no longer symmetric about the polarization axis. Figure 3.6 shows the maximum microwave scattering intensity at different distances of the receiver from the dipole precursor scatterer at φ = 900 . The black line is a fit to the data. It shows microwave scattering intensity has a square dependence of distance in the far-field and also, that the scattering from the background is small as compare with the scattering from the laser spark. If that were not the case, then interference effects with the background would cause the dependency to be oscillatory and less than the squared dependence measured. All results in figures 3.4, 3.5, 3.6 demonstrate that the received microwave signal from the precursor is consistent with microwave scattering that has a dipole radiation pattern. In that case, one can be confident that the precursor microwave scattering signal is a direct measure of the electron number. 30 -2.4241 y = 0.0129x Microwave Intensity (Arbitrary Units) 1 0.8 y = kx-1 0.6 0.4 y = kx-2 0.2 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Distance (m) Figure 3.6: Dependence of peak microwave scattering signal on distance of the receiver. The (black) curve is a fit to the maximum scattering intensity at different distances. For comparison, (green) curve y = k/x and (blue) curve y = k/x2 are also shown in the figure. 31 Laser pulse microwave scattering Q=50 mJ Q=35 mJ Q=25 mJ 1.0 IL, Is, (rel.units) 0.8 0.6 0.4 0.2 35 25 50 0.0 0 5 10 15 20 25 30 t, ns Figure 3.7: A zoomed precursor part of normalized microwave scattering signals at different laser pulse intensities Arrows show the time moments when the accepted breakdown criterion ne (t)/ne,0 = 1013 (the horizontal line as in figure 3.1) is realized. If one supposes that the scattered microwave signal level during in the rising portion of the precursor pulse represents well defined number of electrons, then the delay time between the laser pulse and the observation of that particular signal strength can be quantitatively related to the laser power, the focal volume, the gas density, the laser pulse shape, and the initial electron number density. Under the same laser pulse shape and focusing conditions and with the same initial gas density, the rise time becomes a unique function of the laser power. That functional behavior was shown in figure 3.1. Current experimental results for the scattered microwave signal versus time during the precursor pulse, obtained at different laser pulse powers, are shown in figure 3.7. The dotted line indicates the normalized laser pulse amplitude. The microwave scattering signals are normalized relative to the 50mJ case. Note that the laser shots at sequentially lower power are sequentially more delayed. The small arrows at the bottom are taken from figure 3.1 and indicate good agreement with predicted 32 delays. Measuring of the scattering delay time for reaching of a given scattered signal intensity can give information about the initial bulk plasma density preceding the optical breakdown [29]. 3.5 Conclusions This chapter presents theoretical estimations and experimental results of microwave scattering from laser spark in air. It verifies the coherent nature of microwave scattering from a small-volume plasma in the chapter 2: the scattering from a small-volume, low conductivity plasma lies within the Rayleigh regime while the scattering from a larger and high conductivity plasma switches to the Mie regime. Microwave scattering from the early stage of laser breakdown, avalanche ionization, accounts for the precursor part of the scattering signal, which is isotropic in space and has a dipole radiation pattern. The delay between the laser pulse and the microwave scattering signal decreases as the energy of the laser beam increases. Microwave scattering from after spark evolution is anisotropic in space. 33 Chapter 4 Coherent Microwave Rayleigh scattering of Resonance Enhanced Multiphoton ionization in inert gas This chapter shows the coherent microwave Rayleigh scattering for the measurement of resonance enhanced multiphoton ionization in inert gas. It fully demonstrates the concept of Radar REMPI by measuring microwave scattering of a small volume plasma generated by resonance enhanced multiphoton ionization in argon. First, three photons are absorbed to excite an argon atom from ground state to an excited state, then another photon ionizes the excited atom. Microwave is scattered by the plasma generated by the ionization. For argon at low pressure, a simplified plasma dynamic model for argon is also presented and is shown to be in good agreement with the experimental results. For argon at high pressure, a fully developed plasma dynamic model coupled with the gas dynamic equations is discussed and agreement with the experimental results is shown. 34 4.1 The Concept of Radar REMPI Although the first paper about multiphoton absorption appeared as early as the 1930s [9], multiphoton absorption and multiphoton ionization (MPI) were intensively studied only after the invention of the giant pulsed laser because not enough photons were available within a short period time to generate significant multiphoton absorption or ionization for a common light source. At the focus of coherent laser light, breakdown can be generated with shock waves and flashes. MPI and avalanche ionization are recognized as two important mechanisms in the formation of the plasma. MPI provides a few seed electrons and avalanche ionization sequentially multiplies them. The breakdown process is like explosion and it does not have high selectivity over different species. Pure multiphoton ionization is greatly enhanced if an intermediate state is single or multiphoton resonant with the laser frequency. By using these resonances, it can become an accurate spectroscopic method for measuring trace species or pollutants [11, 8], which is called Resonance Enhanced MultiPhoton Ionization (REMPI). Common detection methods for the MPI or REMPI signal are based on collection of electrons by electrical probes/electrodes [11, 8] or collection of ions by Time Of Flight (TOF) mass spectroscopy [8]. Both methods require that the electrons or ions be extracted from the ionization region, so neither method directly measures all the free electrons that are generated and neither can follow the electron recombination process with high temporal accuracy. Radar REMPI, coherent microwave scattering from a REMPI produced plasma provides a new means for the direct, time accurate observation of the free electrons and thus a new method for high sensitivity REMPI spectroscopy of a gas and a new method for the measurement of electron formation and loss processes. The REMPI plasma acts as a coherent microwave scatterer, with the scattering electric field amplitude proportional to the number of electrons. Since the size of the REMPI 35 plasma is small compared to the microwave wavelength, the scattering falls into the Rayleigh regime [17, 29]. No local probes are required and the electrons do not have to be extracted from the REMPI region to be detected. The absence of a probe or mass spectrometer allows this approach to be used for non local measurements, such as in combusting environments or at long distances for pollution and threat gas detection. Measurements can also be made at high pressure, since even though the lifetime of the free electrons may be short, the high bandwidth capability of microwave detection allows them to be measured. Common microwave diagnostics for plasmas, including microwave absorption [30, 14] and microwave interferometry [14], are not suitable for the measurement of these REMPI plasmas because the volumes are small compared to the microwave wavelength. Furthermore, the use of scattering rather than transmission avoids the need for a detector located on the opposite side of the sample volume. 4.2 4.2.1 Theory REMPI in argon Argon is a simple atom to calculate the transition probability of multiphoton process. Argon can be excited by three photon at ∼ 260nm from ground state 3p6 1S0 , J = 0 to an intermediate state 3p5 3d, J = 3 by both linearly and circularly polarized light. selection rule discussion Since excited states of argon are best described by Jc K coupling, one must apply Jc K selection rules which are described in the equation of A.3, ∆jc = 0, 36 ∆K = 0, ±1, ∆le = ±1 (4.1) together with ∆J = 0, ±1 and parity change[31]. If argon is in its ground state, 3p6 , J = 0, MJ = 0, from table A.1, the excited state must have M = 0 for linearly polarized light along z axis, M = ±3 for circularly polarized light within x and y plane. For linearly polarized light, electric dipole momentum can be written as, hαLSJMJ |R|α0 L0 SJ 0 MJ 0 i = (−1)J−M hαJ|R|α0 J 0 i J 1 0 J −M q M 0 (4.2) From table A.1, q = 0. From ground state 3p6 , J = 0, MJ = 0, and selection rule M 0 = M = 0, then Three-J symbol would be J 0 1 J 0 0 0 (4.3) From the property of the Three-J symbol, J − J 0 = ±1 only. So J = J 0 transition is forbidden. If one considers a linearly polarized photon as a sum of two opposite circularly polarized photons. Then the probability of transition to the level of M = ±1 will be the same. The final state is an admixture of these two states. It is a coherent superposition of waves. So statistically only M = 0 state is possible to populate in linearly polarized light. For multiphoton transitions, the probability of occupying a level is the square of the sum of probability amplitudes transiting to the certain level. Statistically the probability of transition to J = even, M = even for n = odd or J = odd, M = odd 37 J = 0, M = 0 ( 0 0 ( 11 -11 10 +( -11 1 1 1 0 ( { ( 11 -11 00 =( 1-1 11 ≠0 J=1 M = +1 M=0 M = -1 M=0 J=1 ( =0 =( 1-1 11 20 =( 1-1 -11 22 ≠0 ( J=1 M = +1 M=0 M = -1 M = -2 M = -1 J=2 1 1 1 1 ( ( J=1 M = +1 M = +1 M=0 M=0 1 -1 1 1 2 0 2 -2 ( ( M = +2 M=0 ( ( M = +1 ≠0 M = -1 M = -1 J=1 1 1 1 -1 ( ( 00 -11 11 =( 00 ≠0 ( J = 0, M = 0 Figure 4.1: Two photon selection rules for linearly polarized light. One linearly polarized photon = left circularly polarized photon (spin -1)+ right circularly polarized photon (spin 1). for n = even would be zero. Arbitrary Polarizations To begin with arbitrary polarization, one must go over the derivation of the selection rules. A general discussion can be found in the references [32, 33]. From section A.1, one knows the rules of different coupling schemes. Now those schemes will be used to simplify the dipole momentum matrix and get the selection rules. Before the derivation, it is important to know two identities. One is famous Wigner-Echart theorem. It says the transition from |α0 j 0 m0 i to |αjmi can be simplified as 38 0 0 hα0 j 0 m0 |T (k, q)|αjmi = (−1)j −m j 0 k j −m0 q m 0 0 k hα j |T |αji (4.4) Basically it simplifies the dependence of m, m0 and q from the dipole momentum matrix. The other one is called uncoupling formulae for reduced matrix elements. If the operator T (K) operates only on |α1 j1 m1 i then hα1 j1 α2 j2 j|T (K) |α10 j10 α20 j20 j 0 i = 0 j1 δα2 j2 ,α02 j20 (−1)j1 +j2 +j +k [j, j 0 ]1/2 j0 j2 j k j10 hα1 j1 |T k |α10 j10 i (4.5) Now for argon or any Jc K coupling atoms, one has [34] n−1 0 0 0 ~ h(nl)n−1 Lc Sc Jc , ne le (K), 1/2, JmJ |~r · E|(nl) Lc Sc Jc , n0e le0 (K 0 ), 1/2, J 0 m0J i 0 0 ~ 0i = (−1)J−mJ +K+me +J +1+Jc +le +K+l+le [J, J 0 , K, K 0 ]1/2 hn|~r · E|n × le 0 1 le0 0 −mJ 0 where [J]1/2 = √ J 0 K 1 J q J0 mJ J K 0 le 1/2 K Jc K0 le0 1 1 (4.6) 2J + 1, q is the polarization term, 0 is the linear, 1 is right circular and -1 is the left circular and J −mJ +K+me +J 0 +1+Jc +le0 +K+l+le = −mJ +J 0 +le0 . If one defines +~y as the direction of propagation of the wave. The plane wave field can be expressed as [35, 36] ~ r, t) = E0~²exp[−i(ωt − βy)] + c.c. E(~ 39 (4.7) The unit vector ~² = ²1~x + ²3~z describes the polarization of the field. ~ into the matrix elements, one can calculate any polarization, By including ~r · E i.e., hn|²1~x + ²3~z|mi. For bound-bound transition of argon(Jc k coupling), the transition probability amplitude can be expressed as [37, 32], 0 0 0 0 0 0 0 ~ DJK = h[(...α1 L1 S1 )J1 , l2 ]K, s2 J|R|[(...α 1 L1 S1 )J1 , l2 ]K , s2 J i 0 0 q = (−1)s+J −J1 −l2 (2J + 1)(2J 0 + 1)(2K + 1)(2K 0 + 1) K s J0 1 1 K0 J1 J l2 K K l20 0 (1) R l2 l 0 (4.8) 2 For bound-free transition, we use partial wave expansion for the continuum state wave function [38, 37],i.e., the total wave function is the sum of the plane wave of the electrons and converging spherical wave. |~ki = ∞ X 0 l X 0 4πil exp(−iξl0 )Gl0 (r)Pl (~k · ~r/kr)χs l0 =0 ml0 =−l0 = 0 ∞ X l X l0 =0 ml0 =−l0 0 4πil exp(−iξl0 )Gl0 (r)Yl∗0 ml0 (Θ, Φ)Yl0 ml0 (θ, φ)χs Here we have used the equation Pl (n~0 · ~n) = 4π/(2l + 1) Pl m=−l (4.9) Ylm (n~0 )Ylm (~n). In this equation, Θ,Φ represent the direction of the the momentum of the photoelectron, while r,θ,φ indicate the magnitude and direction of the electronic spatial position. Gl0 (r) is the radial part of the continuum function. ξl0 is the phase factor of the continuum wave function which will give the angular distribution of the photoelectrons but be irrelevant for the calculation of the total ionization cross section. Now if we write the program to calculate the cross section for any polarization with the selection rule implemented, we just put hn|²1~x + ²3~z|mi into the equation 4.8. While any radial dipole momentum for bound-bound transition can be calculated 40 from Einstein coefficient shown in next section; One for bound-free transition can be calculated from [39]. Transition lines and Einstein coefficients From the web site of NIST (National Institute of Standards and Technology), we can get all available transition lines, energy levels and Einstein coefficient of transitions. From the time dependent theory of quantum mechanics, we can derive a formula connecting Einstein coefficient and electric dipole moment [32, 37]. In the case of an electric dipole transition (λ = 2πc/ω À dimension of the system ) between the states |γJM i and |γ 0 J 0 M 0 i, the probability of spontaneous emission is given by dWρ ω3 = |ekρ · Dab |dO 2πc3 h̄ ω3 |ekρ hγJM |D|γ 0 J 0 M 0 i|2 dO = 2πc3 h̄ (4.10) In order to simplify the notation we shall hence forth omit the index k from ekρ . We shall transform 4.10 by using the addition theorem for spherical harmonics X Pl (cosω) = m X = m l∗ l Cm (θ1 , ϕ1 )Cm (theta2 , ϕ2 ) l l (−1)m Cm (θ1 , ϕ1 )Cm (theta2 , ϕ2 ) (4.11) so that, eρ · D = DcosθeD = D 0 0 0 eρ hγJM |D|γ J M i = X q X q ∗ (θe , ϕe )C1q (θD , ϕD ) = C1q ∗ (θe , ϕe )hγJM |Dq |γ 0 J 0 M 0 i C1q 41 X q e∗q Dq = X q e∗q hγJM |Dq |γ 0 J 0 M 0 i (4.12) Here eq and Dq are the spherical components of the vectors eρ and D. According to the general formula J hγJM |Dq |γ 0 J 0 M 0 i = (−1)J−M hγJ|D|γ 0 J 0 i 1 0 J −M q M 0 (4.13) From the properties of 3j symbols, it follows that the selection rules, ∆J = J − J 0 = 0, ±1; J + J 0 ≥ 1 ∆M = M − M 0 = q = 0, ±1. (4.14) (4.15) And these rules must include a parity change. For each of the three possible transitions ∆M = 0, ±1, only one term is nonzero in the sum 4.12. Thus for ∆M = 0, X = e0 hγJM |D0 |γ 0 J 0 M i = ez hγJM |Dz |γ 0 J 0 M i (4.16) q For ∆M = +1, X q = e∗1 hγJM |D1 |γ 0 J 0 M − 1i = 1/2(ex − iey )hγJM |Dx + iDy |γ 0 J 0 M − 1i and for ∆M = −1, 42 (4.17) X q = e∗−1 hγJM |D−1 |γ 0 J 0 M − 1i = 1/2(ex + iey )hγJM |Dx − iDy |γ 0 J 0 M + 1i (4.18) The angular distribution of radiation for each of the transitions ∆M = 0, ±1 is ∗ determined by the factor |C1q (θe , ϕe )|2 ; q = 0, ±1, in which the angles θe and ϕe describing the direction of the vector eρ have to be expressed in terms of θk = θ and ϕk = ϕ. Consider the transition ∆M = 0. In this case the polarization vectors can be chosen so that cosθe1 = sinθ and cosθe2 = 0. Therefore one obtains dW1 (γJM ; γ 0 J 0 M ) = ω3 |hγJM |Dz |γ 0 J 0 M i|2 sin2 θdO 2πh̄c3 dW2 (γJM ; γ 0 J 0 M ) = 0 (4.19) Summing over ρ = 1, 2 and integrating over all angles, one obtains W (γJM ; γ 0 J 0 M ) = 4ω 3 |hγJM |Dz |γ 0 J 0 M i|2 3h̄c3 (4.20) If all directions in space are equivalent, then the atom can be in any of the M states with equal probability. Thus the probability of a transition γJ → γ 0 J 0 can be obtained by summing 4.10 with respect to M 0 and averaging over M X 1 ω3 dW (γJM ; γ J M ) = |hγJM |Dz |γ 0 J 0 M 0 i|2 dO 3 3πh̄c 2J + 1 M M 0 0 0 (4.21) One know the summations X |eρ hγJM |D|γ 0 J 0 M 0 i|2 = 1/3|hγJ|D|γ 0 J 0 i|2 (4.22) MM0 Now we can get the formula for Einstein coefficient in terms of transition matrices 43 and electromagnetic field variables. From experimental value of Einstein coefficient, we can deduce the dipole momentum matrix of A.66 by applying linearly polarized light 3j symbol to the equation 4.21. For different polarizations, we multiply different 3j symbols. Note: J in equation 4.21 is always the total angular momentum of the upper level. Based on the above derivation, a FORTRAN program was developed to calculate the REMPI spectrum of argon. In this program, all known Einstein coefficients and transition wavelengths in the NIST database were taken into account. The broadening effects were considered too. When the laser wavelength is exactly at the (3+1) multiphoton resonance of argon. The first step of the REMPI process is three-photon excitation to an excited state (3p5 3d[5/2], J = 3) by three circularly polarized 261.27nm photons. The second step is one-photon ionization to the continuum state. Including the laser linewidth, natural and pressure broadening effects, the three-photon excitation rate to the intermediate state (in s−1 ) can be written as we derive in the appendix A [40, 41] (3) Wf,g = (2παF ω)3 | XX (ω2,g − 2ωL )2 + γ22 /4 |2i |1i h1|r|gi q h2|r|1i hf |r|2i q (ω1,g − ωL )2 + γ12 /4 |2 G(ωL ) (4.23) where g and f denote the ground state (3p6 1S) and the excited state (3p5 3d[5/2], J = 3) of the argon atom, respectively. The circular polarization of the photons leads to ∆J = ±3, ∆M = ±3 selection rules, which uniquely select the 3p5 3d[5/2], J = 3 state from among the various states that fall within the tuning range of the laser. The summations are over dipole allowed off resonant transitions to intermediate states, ω2g is the frequency difference between states |2i and |gi, ω1g is the frequency difference between states |1i and |gi, α is the fine-structure constant (1/137.036), F (r, t) = 44 I(r, t)/h̄ωL is the total photon flux measured in number of photons per unit area per second, I(r, t) = I0 (t)exp(−r2 /rb2 ) is the laser intensity which has a Gaussian profile, where rb , is the radius of laser focus, ωL is the angular frequency of the exciting incident laser, h|r|i are the dipole transition matrix elements, connecting the ground state, intermediate states and excited state g, 1, 2 and f. γi is the linewidth of the ith intermediate states. G = √ √ 1 2ln2 √ ∆2L +δ 2 +D2 (ω3,g −3ωL )2 +γ32 /4 exp (3ωL − ωD )2 /2(3∆2 + δ 2 + D2 ) is a generalized lineshape scale factor, which is the convolution of level Lorentz broadening, Doppler broadening and laser broadening. D is the Doppler linewidth, ∆L is the laser linewidth, δ is natural linewidth and ωD is the peak frequency of the ensemble of argon atoms. G is strongly peaked at ω3,g = ωL , the three photon resonance condition. The general three-photon excitation cross section is defined as (3) Wf,g = σ(3) F 3 . Equation 4.23 gives the peak three photon excitation cross section σ(3) ≈ 2.81 × 10−93 m6 · s2 . A quantum-defect approximation is used for the one-photon ionization from the excited bound state of an atom, which is exact for hydrogenic atom and quite good for Rydberg states of rare gases [42]. σP i = 8 × 10−22 [m2 ] Z(UI /R)1/2 (h̄ωL /UI )3 (4.24) where Z is net charge on the ion, UI is the ionization potential for the atom at the excited state, R is Rydberg constant and h̄ωL is the photon energy. This gives σP i = 7.77 × 10−23 m2 . Note that the kinetic energy of the free electrons generated by the ionization process is εph = h̄ωL − UI ≈ 3.2eV . As a comparison, the direct multiphoton ionization rate using Keldysh theory [43] is about 3 orders of magnitude lower than that of the resonance enhanced multiphoton ionization. Microwave scattering from the plasma generated by REMPI can be calculated by following [17, 29] 45 Epeak ∝ Em0 N (4.25) where Epeak is the time dependent electric field of the scattering signal, Em0 is the incident microwave electric field, and N is the time varying number of electrons inside the plasma generated by REMPI. Note that the microwave scattering field strength is proportional to the total electron number inside the plasma when skin layer thickness is greater than the size of the plasma. By measuring the peak electric field of the microwave scattering as a function of the laser wavelength, the REMPI spectrum can be obtained. 4.2.2 Plasma dynamic model of REMPI plasma at low pressure1 At low pressure, gas temperature and pressure are assumed unchanged during the REMPI process. Molecular ions Ar2+ are neglected. Avalanche ionization is not considered either. The rate equation for the local density of argon atoms in the excited state (3p5 3d[5/2], J = 3) can be written as Ã 1 ∂ ∂N ∗ ∂N ∗ = rD∗ ∂t r ∂r ∂r ! + Ng σ(3) F 3 − N ∗ σP i F − (1/τ + k1 Ng + k3 Ng2 + ke ne )N ∗ (4.26) with the initial condition, N ∗ (t = 0) = 0. 1 The plasma dynamic models shown here and the next section for argon at high pressure were developed by Dr. Mikhail N. Shneider. The results were published in [44] and [45] respectively. They were presented here for completeness of the thesis. 46 Ng = Ng (r, t) = Ng (0) − N ∗ (r, t) − n+ (r, t), N ∗ and n+ are number densities of the ground state argon atom, of the excited state and of the atomic ions (Ar+ ). D∗ is the diffusion coefficient of the argon atom in the excited state. τ ≈ 6.66 × 10−8 s is the spontaneous relaxation time of the excited state. k1 , k3 and ke are the quenching collision coefficient of two-body and three-body collisions between ground state atoms and excited atoms, and the quenching collision coefficient between excited atoms and electrons, respectively. Due to the long relaxation time of the electrons, the plasma is non-equilibrium, Tg = T0 = T+ 6= Te . Since the plasma is generated in the laser focal volume which has an extended elliptical shape, it is assumed that the plasma region has cylindrical shape. This symmetry allows the plasma to be modeled with time dependent one dimensional codes. Electrons and ions remain near the laser focal region, so the rate equations for the electron number density ne and the ion number density n+ can then be written as ∂ne 1 ∂rΓe + = N ∗ σP i F − βne n+ ∂t r ∂r (4.27) ∂n+ 1 ∂rΓ+ + = N ∗ σP i F − βne n+ ∂t r ∂r (4.28) where β is the effective recombination rate including the three-body and radiative recombination rates between electrons and atomic ions [46]. The electron and ion fluxes are Γe = −µe ne E − De ne ∂Te ∂ne − De ∂r Te ∂r (4.29) ∂n+ ∂r (4.30) Γ+ = µ+ n+ E − D+ 47 Figure 4.2: Computed and measured microwave scattering signal. Laser has a wavelength of 261.27 nm and an energy of 3.5mJ/pulse, 3mJ/pulse and 2.5mJ/pulse. The experimental microwave scattering signal has been averaged over 64 laser shots. where the mobility of the electron is µe = e/m(νen + νc ). νc and νen are Coulomb collision frequency and electron-neutral collision frequency, and the diffusion coefficient is De = µe Te . For the low pressure we have neglected the conversion Ar + Ar+ ⇒ Ar2+ . The electrical potential is calculated by Poisson equation, 1 ∂ ∂φ e (r ) = − (n+ − ne ) r ∂r ∂r ε0 (4.31) with boundary conditions ∂φ/∂r|r=0 = 0 and φ(∞) = 0, ε0 is the permittivity of free space. The field is calculated as E=− ∂φ ∂r (4.32) One supposes the geometry of the temperature field is spherical because electron 48 n 50 100 150 200 250 μm Figure 4.3: Plasma composition of 5 Torr argon at 25 ns after the laser pulse starts. The laser pulse energy is 3.5mJ, radius is 6.5 µm in this calculation. heat conductivity is very high. The electron temperature in the plasma thus obeys [16], ∂ 3 1 ∂ 2 ∂Te ( ne kTe ) + (r (5/2Γe kTe − λe )) ∂t 2 r ∂r ∂r = JL + JM W − 3/2ne k(Te − T0 )(νen + νc )δ + N ∗ σP i F εph + ke ne N ∗ ε∗ (4.33) where JL = e2 ne I(r, t)(νc +νen )/ε0 cme (ωL2 +(νc +νen )2 ) and JM W = e2 ne IM W (νc + 2 2 νen )/ε0 cme (ωM W + (νc + νen ) ) are heating by the laser pulse and the microwave respectively. The heating by the microwave is considered negligible. k is Boltzmann constant, λ = 5/2kne De is the electron heat conductivity, δ = 2me /M is the collision constant of electrons with neutrals or ions, me and M are the mass of electrons and neutrals or ions, ε∗ = 14.099eV is the energy of the excited state. 49 By solving equations 4.26 - 4.33 where constants are found in the literature [47], we can get the time evolution of the excited argon atoms, electrons and ions. If the plasma dimension is small relative to the microwave wavelength and the electron density is low enough so that skin effects can be neglected, the electric field of microwave scattering is proportional to the total electron number inside the plasma region by equation 4.25, where N (t) ∼ 2π R∞ 0 ne (r, t)rdr. For the conditions here, the skin layer thickness is about 110 microns at the peak electron number density, and then increases as electron number density decreases, so it is always larger than the physical extent of the plasma. Figure 4.2 shows computed curves of the time dependent microwave scattering from the plasma generated by REMPI in neutral argon (dotted lines). The rising part is due to the increasing electron number inside the plasma during the ionization by the laser pulse. The decreasing part is due to the recombination losses of electrons. Plasma expansion is followed by ambipolar diffusion. Figure 4.3 shows an example of the plasma structure at 25 ns after the laser pulse starts. 4.2.3 Plasma dynamic model of REMPI plasma at high pressure At higher pressure, plasma evolution and molecular ion formation must be considered, together with a complete set of gas dynamic equations. The plasma dynamic model must include the dynamics of molecular ions. The heating due to friction can not be neglected any more. The friction force between the charged particles and neutral gas in a process of ambipolar plasma expansion can be comparable to the pressure gradient force or even exceed it [45]. Generation and loss of the excited atom N ∗ are same, so the rate equation stays unchanged as 4.26. The rate equations for electron number density ne , atomic ion number density n+ and molecular ion number density n2+ are 50 ∂ne 1 ∂rΓe + = N ∗ σP i F + νi ne − βef f ne n+ − βd ne n2+ ∂t r ∂r (4.34) The electron is generated by one photon ionization from the excited state and avalanche ionization by the laser field and lost by recombination with atomic and molecular ions. ∂n+ 1 ∂rΓ+ + = N ∗ σP i F + νi ne − βef f ne n+ − kconv n+ Ng2 ∂t r ∂r (4.35) The atomic ion is generated by one photon ionization from the excited state and avalanche ionization by the laser field and lost by recombination with the electron and conversion to molecular ion by Ar+ + Ar + Ar ⇒ Ar2+ + Ar. ∂n2+ 1 ∂rΓ2+ + = kconv n+ Ng2 − βd ne n2+ ∂t r ∂r (4.36) The molecular ion is generated by conversion from atomic ion by Ar+ + Ar + Ar ⇒ Ar2+ + Ar and lost by recombination with the electron. The fluxes of electrons, atomic ions and molecular ions are Γe = ne ue (4.37) Γ+ = n+ u+ (4.38) Γ2+ = n2+ u2+ (4.39) where ue , u+ , and u2+ are the corresponding drift velocities in the laboratory reference frame ;νi is an approximate rate of classical avalanche ionization by the bulk electrons oscillating in the laser radiation field, 51 νi ≈ e2 IL [νen + νe,+ + νe,2+ ] m[ωL2 + (νen + νe,+ + νe,2+ )2 ]ε0 cξi (4.40) and ξi = 15.76eV is the ionization potential for argon atom. The instantaneous potential distribution is calculated by the Poisson equation, Ã ∂φ 1 ∂ r r ∂r ∂r ! =− e (n2+ + n+ − ne ) ε0 (4.41) with the boundary condition ∂φ/∂r|r=0 = 0 and φ(∞) = 0. The radial field is E=− ∂φ ∂r (4.42) The electron temperature in the plasma obeys a hydrodynamic equation, ∂ 3 1 ∂ ∂Te ( ne kTe ) + (r(5/2Γe kTe − λe )) ∂t 2 r ∂r ∂r = JL + JM W − 3/2ne k(Te − T0 )(νen + νe,+ + νe,2+ )δ + N ∗ σP i F εph + ke ne N ∗ ε∗ − νi ne ξi (4.43) where JL = e2 ne I(r, t)(νen + νe,+ + νe,2+ )/ε0 cme (ωL2 + (νen + νe,+ + νe,2+ )2 ) and 2 2 JM W = e2 ne IM W (νen + νe,+ + νe,2+ )/ε0 cme (ωM W + (νen + νe,+ + νe,2+ ) ) are heating by the laser pulse and the microwave respectively. The heating by the microwave is considered negligible. k is Boltzmann constant, λ = 5/2kne De is the electron heat conductivity, δ = 2me /M is the collision constant of electrons with neutrals or ions, me and M are the mass of electrons and neutrals or ions, ε∗ = 14.099eV is the energy of the excited state. To complete the problem, a gas dynamic model is included for modeling of gas temperature and pressure. 52 ∂ρ 1 ∂rρu + = qρ ∂t r ∂r Ã (4.44) Ã ! ∂u 1 ∂rρu2 ∂p 1 ∂ ∂u u2 + =− +η r − ∂t r ∂r ∂r r ∂r ∂r r Ã ! ! + Fd Ã ∂ρ(ε) + u2 /2 1 ∂{r[ρ(ε + u2 /2) + p]u} 1 ∂ ∂T ∂u + = Q+ rλ +2η ∂t r ∂r r ∂r ∂r ∂r (4.45) !2 u2 + 2 +Qd r (4.46) By solving the equations from 4.34 to 4.46, the time evolution of the excited argon atoms, electrons, atomic ions Ar+ , and molecular ions Ar2+ , together with the neutral gas parameters of pressure, density, temperature, and radial velocity. 4.3 Experimental setup and results The experimental setup is shown schematically in figure 4.7. A custom built frequency tunable, frequency tripled, Q switched Ti: Sapphire laser, shown in the figure 4.6, was used to generate the ionization. The laser was pumped with two Q-switched N d : Y V O4 lasers. The N d : Y V O4 oscillator output was split and amplified. The 1053 nm laser output was frequency doubled by two temperature controlled BBO crystals. The 527 nm laser beams were sent into four sides of two Brewster-angle cut Ti:Saphhire crystals. The Ti:Sapphire laser cavity is 1 meter long. The laser frequency was roughly selected by four prisms inside the cavity. After separation by a prism, the third harmonic of the laser passed through a quarter wave plate and a UV polarization beam splitter to purify the polarization. An advantage of using the combination of the quarter wave plate and polarization beam splitter is that the energy of the REMPI pulse can be changed without changing the laser pumping or up-conversion optics. A second quarter wave plate was used to produce circular 53 Figure 4.4: Comparison between experimental and theoretical results of microwave scattering from plasma produced by REMPI in argon at different pressures. Laser wavelength is 261.27nm, linewidth = 0.1nm, laser focal radius is rb=7.5 m; energy per pulse E=2.1 mJ. All results, presented here correspond to the same laser beam parameters 54 Figure 4.5: Plasma composition of 5 Torr argon at 25 ns after the laser pulse starts. The laser pulse energy is 3.5mJ, radius is 6.5 µm in this calculation. polarization. Then laser beam was focused by a UV fused silica lens (f = 7.5 cm) into the test chamber. The maximum energy of the beam was approximately 4mJ/pulse, with a pulse length of about 25 ns. Note that the laser pulse length is short enough and the pulse energy is low enough to minimize direct avalanche ionization at low pressure. The linewidth of this laser is measured to be less than approximately 0.2nm by using a monochromator (Acton, PI500Max). The monochromator has a resolution of 0.03nm and the wavelength was calibrated by using a commercial mercury lamp line at 253.65nm. The laser wavelength was monitored by the monochromator during the experiment. A 10 mW microwave source at a frequency 12.6 GHz (wavelength 2.3 cm)illuminates the ionization volume from the upper side of the chamber through a microwave horn (WR 75). A microwave receiver was placed perpendicular to the propagation directions of both the laser and the microwave sources. The receiver is located 30cm 55 Nd: YVO4 Oscillator BBO 1053nm am plifier 527 nm R etro-reflector dispersion prism s beam stops P iezo-driven m irror KDP BBO Ti:S apphire Ti:S apphire Figure 4.6: (Color) Schematic of the custom built frequency tunable, frequency tripled, Q switched Ti:Sapphire laser. 56 Microwave Source Beam stopper Quarter waveplate m is Pr THG z,E Glan x,k spectroscopy Tu Ti na :S bl ap e, ph fre ire qu la enc se y r tri pl ed Beam polarizer stopper vacuum system z, E I0 x, k L y, H R Figure 4.7: (Color) Schematic of the microwave scattering experiment used to measure multi-photon ionization in neutral argon: THG, third harmonic generation 100 100 10-1 10-2 10-2 1.0 1.5 0.8 10 1.0 10-5 10-6 10-7 0.5 Calculation Arbitrary Units 10-3 -4 Experiments Arbitrary Units Calculation Arbitrary Units 10-3 10-4 0.6 10-5 10-6 0.4 10-7 10-8 10-8 10-9 Experiments Arbitrary Units 2.0 10-1 0.2 10-9 0.0 10-10 260.2 260.4 260.6 260.8 261.0 261.2 261.4 261.6 261.8 262.0 262.2 0.0 10-10 260.2 260.4 260.6 260.8 261.0 261.2 261.4 261.6 261.8 262.0 262.2 wavelength(nm) wavelength(nm) (a) (b) Figure 4.8: (Color) REMPI spectrum of neutral argon at 5 Torr.(a) was obtained by microwave scattering method and (b) is by traditional electron-collection electrodes. Both are obtained by using the circularly polarized light. Solid lines are theoretical REMPI calculations by equations 4.23 and 4.24. At the wings off the resonance direct multiphoton ionization is about three orders of magnitude smaller. Circle marks are experimental results. 57 away from the plasma. A microwave homodyne receiving system was used for detection and the scattered microwave signal was amplified by about 90dB. The vacuum chamber was made of microwave transparent PVC plastic. The inner diameter of the chamber is 8 cm. The cell was purged three times before filling with argon gas. The microwave source was placed in a metal box, and the experiment was covered by microwave absorber to minimize background. Figure 4.2 shows comparisons between the model predictions based on plasma dynamic model at low pressure and experimental results at 5 Torr. It shows that good agreement with the predicted electron recombination rate has been achieved for three values of laser pulse energy. The residual discrepancies are likely due to the limitations of the two dimensional model and noise in the detection system. Figure 4.4 shows comparisons between the model predictions based on plasma dynamic model at high pressure and experimental results at higher pressures. It also gives good agreement between models and experiments. By measuring the peak intensity of the microwave scattering signal at different wavelengths, REMPI spectra were obtained as shown in 4.8 (a). The curve in the figure represents the predicted spectrum using the model described in the appendix A and this chapter with convolution of 0.2 nm laser linewidth. The points are experimentally measured values. Because of the very low off resonant signal strength, the measured values shown for off resonant points are representative of the background noise in the microwave detection apparatus. The REMPI peak at 261.27nm matches the empirical value for the circularly polarized light(left and right-handed circularly polarized light are equivalent). To validate the microwave scattering experimental results, a traditional electron collection method was conducted as shown in the figure 4.9. The argon sample cell consists of two quartz Brewster-angle windows, a 28 mm inner diameter Pyrex body with 4 pairs of electrodes and two vacuum valves. The focal length of lens into the cell 58 High Voltage Beam stopper Quarter waveplate P THG Glan Circuit rip le d Beam polarizer stopper C Tu Ti na :S bl ap e, ph fre ire qu la enc se y r t R Ground R + - spectroscopy m ris Figure 4.9: Schematic of the classical electrode experiment used to measure multiphoton ionization in neutral argon: THG, third harmonic generation was 15 cm. The ionization current was monitored through a load resistance of 10kΩ. A capacitor of 0.01µF separates the DC current from the ionization current. Finally the measured current pulse was amplified and integrated. The REMPI spectrum taken with circularly polarized light by this method is shown in figure 4.8 (b). The spectrum is in good agreement with the one obtained by microwave scattering. In conclusion, coherent microwave scattering to detect resonance enhanced multiphoton ionization of neutral argon and to follow the recombination of the free electrons is demonstrated both experimentally and theoretically. The time evolution of the recombination process from the ellipsoidal laser produced plasma is modeled with a cylindrical geometry for the electrons, positive ions and excited state argon atoms, and a spherical geometry for the long range electron temperature. Predictions of the time dependent one dimensional model at low pressures give good agreement with the measured recombination rates. This non-intrusive Radar REMPI detection method provides a method for the time accurate measurement of free electron generation and loss processes and for a high sensitivity detection of the REMPI plasma. For argon at higher pressures, a complete model of plasma evolution and gas dynamic equations are solved numerically. The corresponding theoretical prediction 59 of microwave scattering signal gives good agreement with the experimental results at higher pressure. This model includes the generation and evolution of molecular ions, the friction heat by ambipolar diffusion and joule heating by laser pulse and microwave. 60 Chapter 5 Radar REMPI spectroscopy for trace species detection In this chapter, Radar REMPI spectroscopy is demonstrated both in argon and in the mixture of nitric oxide and nitrogen. Accurate REMPI spectra of neutral argon and nitric oxide were obtained within the limit of the laser linewidth. Trace species concentration of nitric oxide in the nitrogen at a sub-ppm level was detected. 5.1 Radar REMPI spectroscopy Trace species detection has become a field of growing importance for fluid dynamics, combustion, as well as environmental, health and security applications. Various spectroscopic techniques have been explored and used, such as laser induced fluorescence (LIF), cavity ring-down spectroscopy (CRDS), photo acoustic spectroscopy and REMPI (Resonance Enhanced Multi-Photon Ionization). REMPI [11, 8, 40] is very attractive because it requires relatively little alignment and has a high signal to noise ratio compared to other methods. Because of its nonlinearity, REMPI occurs most strongly near the focal point, where the photon flux is highest. A three dimensional localization can thus be achieved. Common detection methods for the REMPI 61 signal are based on electron or ion collection, such as electrical probes/electrodes and Time Of Flight (TOF) mass spectroscopy. Electrical probes with high voltages are intrusive, while TOF spectroscopy is only possible for measurement at relatively low pressure. Both methods are limited to local applications. The Radar REMPI concept demonstrated in the chapter 4 provides another possibility to do the trace species measurement in-situ and non-intrusively. 5.2 Demonstration in neutral argon To demonstrate Radar REMPI ’s ability to do accurate spectroscopy, a narrow linewidth laser together with microwave scattering system and automatic data acquisition was set up, shown in figures 5.3, 5.4 and 5.5. An injection-seeded, cavity-locked, frequency-tunable, frequency-tripled, Ti: Sapphire laser [48] was used to generate a small volume plasma by the REMPI process in neutral argon. The Ti: Sapphire laser is made of three lasers. A multi-mode argon-ion laser (Lexel 85) at 514nm was used to pump a continuous wave (CW) ring cavity Ti: Sapphire laser. As shown in the figure the CW laser has multiple frequency controlling components inside the cavity, including a photo diode, a Bi-refringent (BRF) plate and a voltage-controlled etalon. The photo diode eliminates the bi-directional oscillation of the laser, allowing unidirectional lasing, so that the spatial hole burning is avoided. The BRF plate roughly limits the oscillating frequency of the laser. The voltage-controlled etalon selects the single frequency lasing. The frequency of the CW laser can be tuned either by manually tuning BRF orientation for a rough scan ( 0.5nm per step) or by scanning voltage applied to the etalon for more accurate wavelength scan (∼ 200 MHz per step). Then the CW laser was used to injection seed a pulsed, tunable, frequency-tripled Ti: Sapphire laser, which was used in the broadband experiments. The two Faraday isolators were used in the optical path of the injection seeding to 62 Figure 5.1: Detail setup of CW ring cavity Ti:Sapphire laser. eliminate the feedback to the CW laser. The pulsed laser has an active cavity control system (Ramp and Lock) to match the laser frequency shift due to the wavelength scan of the CW seeder laser and other disturbances. The injected seeded laser has a linewidth of ∼ 20 MHz at the fundamental frequency, instead of several hundred GHz in the broadband experiment. The comparison of laser linewidths for the broad and narrow linewidth laser is measured by a 0.5 meter monochromator (Acton PI500), shown in figure 5.6. The microwave setup is shown in figure 5.4. A 10 mW, 12.6 GHz Gunn diode microwave source is pre-amplified to 50mW and illuminates the ionization point through a microwave horn. Microwave scattering from the plasma was collected by another microwave horn and amplified sequentially by one preamplifier at 12.6 GHz at a factor of 30dB. After the frequency was converted down by the microwave mixer, two other amplifiers with a bandwidth of DC-1GHz amplified the scattering signal by a factor of 60dB. The output time-accurate signal was monitored by an oscilloscope. The polarization of the microwave lies along the propagation direction of the laser and thus the elongated axis of the plasma ellipsoid to maximize the scattering signal. 63 Figure 5.2: Detail setup of injection-seeded Ti:Sapphire laser. Figure 5.3: Experimental setup of microwave scattering from REMPI using a narrow bandwidth laser. PD: Photo diode, M1-M4 are high reflective mirrors. 64 Figure 5.4: Homodyne microwave detection system, Microwave splitter, preamplifiers and transmitting and receiving horns (red in the plot) work at 12.6 GHz, the amplifier (blue) after mixer works from DC − 1 GHz. 65 Photo Diode microwave system Labview Interface BoxCar integrator computer Q-switch Trigger Oscilliscope laser scan control Figure 5.5: Automatic Data acquisition system, Photo diode signal is from PD in figure 5.3. Q-switch from the pulsed Ti: Sapphire laser trigger the boxcar integrator, oscilloscope and Labview. Laser scan control signal out of the Labview controls the CW laser. The microwave scattering signal follows the electron number density in nanoseconds scale in this setup. It is limited by the bandwidth of amplification arm after the microwave frequency is mixed down. A higher bandwidth microwave amplifier will give a more accurate time response. An automatic data acquisition system, as shown in figure 5.5, was developed to collect microwave scattering signals. The output of the microwave system was input into one channel of a boxcar integrator. The laser pulse monitored by a photo diode at the entrance of the gas cell was input into another channel of the boxcar. A LabView program was developed to collect the data. The laser scan control to the CW laser is also achieved by the LabView program. The whole system is synchronized by the Q switch of the pulsed Ti: Sapphire laser. A sample of time accurate microwave scattering signal from REMPI generated plasma in argon at 97.4 Torr using the narrow linewidth laser at different laser pulse energies is shown in figure 5.7. Argon REMPI spectra for linear and circular po66 Figure 5.6: Comparison between seeded and unseeded laser profile measured by 0.5 meter monochramator. The upper picture is a seeded laser beam used in narrow linewidth laser experiment, while the lower one is an unseeded beam is used in the broadband laser experiment. larizations near 261.27nm were obtained by picking up the peaks of the microwave scattering signal, shown in the figure 5.8. At this wavelength both circular and linear polarizations produce REMPI signals, but the circular polarization is significantly more effective due to stronger coupling to the three photon resonance [8, 40]. The experiments on microwave scattering from the plasma generated by (3+1) REMPI successfully demonstrate that Radar REMPI can give a time accurate (in but not limited to 1 nanosecond scale), localized (about 100 microns) and nonintrusive measurement of a weakly ionized plasma. The response time is short enough to provide a method for measuring fast evolving plasma dynamics, which otherwise limits our understanding of fundamental process in plasma. The time dependent information of electron temperature, number density of atomic and molecular ions and electric fields can be inferred from the model after which is calibrated by the experimental results. The experiments using the narrow linewidth laser and automatic data acquisition system further confirm the resonance nature of the process and give an accurate 3+1 REMPI spectrum of argon at 261.27 nm for both linearly and circularly polarized 67 Figure 5.7: Time accurate microwave scattering signal from REMPI generated plasma in neutral argon at 97.4 Torr using the narrow linewidth laser at different energies. 68 Figure 5.8: REMPI spectra for circularly and linearly polarized light of argon at 30.3 Torr obtained by the automatic data acquisition system shown in the figure 5.3 and 5.5 with a scanning step of ∼ 200 MHz. The asymmetry in the spectra may be due to collisional broadening of Ar and Ar2+ . 69 be am stope r M1 M6 Nd: YAG laser 2 26 nm L1 M9 be am stop 5 32 nm M3 M7 be am stop 1 06 4 n m M2 m ixing crysta l Dye laser (Rhodeman 6G) 2 87 nm M8 M4 M5 Figure 5.9: Experimental setup of Nitric oxide. light. That the circularly polarized light gives higher signal than the linearly polarized one qualitatively validates the time dependent theory of quantum mechanics as described in the chapter of 4 and the appendix A. 5.3 Xenon experiments The experimental setup is shown in figure 5.9. A broad band frequency-doubled Nd: YAG laser was used to pump a dye (Rhodeman 6G) laser (Sirah precision). The frequency-doubled dye laser beam near 287nm was mixed with the fundamental frequency of Nd: YAG laser at 1064nm to generate a laser beam near 226nm (pulse energy = 1mJ/pulse, pulse length = 10ns, linewidth was roughly estimated to 2cm−1 ). The beam was used to generate a small-volume plasma by the (2+1) REMPI process in xenon. The microwave illuminates the ionization point from the upper arm of the vacuum chamber. The microwave scattering signal is collected perpendicular to the plane of the paper as in the experiments in argon, shown in 5.4. The automatic data collection system, as shown in figure 5.5, was also used in the experiments. 70 The accurate wavelength and linewidth of the Nd: YAG, the dye laser and mixing output were all unknown. It generates uncertainties for the experimental results, especially for some species having rich spectra, such as nitric oxide. Radar REMPI in xenon was used here to calibrate the system. Xenon is an inert gas. It has well defined spectral characteristics near 226nm. The ground state of xenon is 5p6 (1S), J = 0. Based on the selection rules of twophoton transition [31], the state of (5p5 (2 P 3/2)7p, J = 2) E = 88351.681cm−1 , is a resonance level of the two-photon transition (transition of a p state electron is p state to p state via a virtual transition s state p-s-p, ∆J = 2. Details about selection rules in multiphoton absorption or ionization could be found in the chapter 4and the appendix A ). Other atomic energy levels are far away from the resonance level, so it provides a good calibration tool for the laser wavelength. The xenon line is also very useful for the measurement of the linewidth of REMPI spectrum. Because of the broad bandwidth of the Nd: YAG laser which is about 1cm−1 , the mixing output of the laser beam should be roughly 1 ∼ 2cm−1 . Since the spectrum of REMPI is dependent of different broadening mechanisms, laser linewidth, pressure broadening, Doppler broadening and natural broadening, the measurement of REMPI spectrum of xenon provides an accurate measurement of the dominate broadening effect. The natural broadening of xenon is proportional to the Einstein coefficients of the excited state, which is on the order of 105 s−1 , and much smaller than the laser broadening. The Doppler broadening for xenon could be estimated from equation 5.1 as 1.43GHz at 300 K, which is also much smaller than the laser linewidth. (1cm−1 = 30GHz) s ∆ωDoppler ω0 kT =2 2ln2 c m0 (5.1) where ∆ωDoppler is the linewidth of the Doppler effect, ω0 is center frequency of the transition, k is Boltzmann constant, T is the temperature, m0 is the mass of the 71 Figure 5.10: 2+1 REMPI spectra at different pressures. xenon atom and c is the speed of light. To confirm the estimation, RADAR REMPI experiments of xenon at different pressures were done to get an accurate 2+1 REMPI spectrum of xenon. The experimental results are shown in the figure 5.10. Based on the experimental data of xenon at 10 Torr, a Gaussian fit has been done as shown in figure 5.11. We get the laser linewidth at F W HM = 1.4 ± 0.02cm−1 ≈ 42GHz. As expected, the laser linewidth dominates all other broadening effects. For higher pressure case, the REMPI spectrum fits Gaussian profile well at the high frequency side, fits the Lorentz profile at the low frequency side as shown in the figure 5.12. This may be due to the laser induced collisional effects. Since the pressure broadening has a Lorentz profile, it may suggest the possibility of pressure broadening of Xe and Xe+ 2. 72 Figure 5.11: Comparison of experimental 2+1 REMPI spectrum of xenon at 10 Torr and Gaussian fit, which gives laser linewidth ∼ 42GHz 73 Experimental data at 200 Torr Gaussian Fit Lorentz Fit 1.0 Arbitrary Units 0.8 0.6 0.4 0.2 0.0 -0.2 44168 44172 44176 44180 44184 44188 -1 W avelength(cm ) Figure 5.12: Comparison of experimental 2+1 REMPI spectrum of xenon at 200 Torr with Gaussian fit and Lorentz fit. The asymmetry of the spectrum may suggest the pressure broadening. 74 5.4 Nitric oxide and sub-ppm detection The demonstration of Radar REMPI spectroscopy in molecular gases was undertaken after the experiments in inert gases. Nitric oxide was selected as an example because of its relative simple structure, its well-known spectrum and its importance for combustion pollution and hazard detection. The experimental setup for Radar REMPI spectroscopy of nitric oxide is same as for xenon, as shown in figure 5.9. A laser linewidth limited (42GHz) spectrum of nitric oxide at 1.91 Torr has been obtained by this method. With the calibrated laser wavelength, the nitric oxide REMPI spectrum is shown in figure 5.13 and shows good agreement with the reference spectrum [49]. The transition involved in the REMPI process is from the ground state X 2 Π to the excited state A2 Σ+ and then to the ionization state. It is so called 1+1 REMPI. The experiments in the nitric oxide demonstrate the possibility of using RADAR REMPI to obtain accurate spectral measurement in diatomic gases. To demonstrate trace species detection, 2+1 REMPI was conducted where the two-photon resonance was to the nitric oxide gamma manifold in the region of 190 nm. The laser is a Ti: Sapphire laser, operated in the region of 760 nm and frequencydoubled to the UV around 380 nm . This laser has a broad bandwidth and is hand tuned, so it is not particularly useful for spectroscopy, but it can be used to identify the presence of nitric oxide by tuning on and off the nitric oxide two-photon resonance. Air is not transparent at 190 nm, but it is at 380nm, so this detection utilized the subharmonic capability of REMPI. The experimentally measured spectrum of 720 and 160 parts per billion of nitric oxide in 200 Torr of nitrogen is shown in Fig 5.14, together with the 2+1 classical REMPI spectrum of NO from the literature [11]. This mixture was prepared by diluting a 20% NO plus 80% nitrogen mixture several times. For this experiment the laser pulse energy was approximately 20 mJ/pulse and it was focused into the cell with a 10 cm focal length lens. The microwave power was 50 milliwatts and the receiver was located 30 cm away from the plasma region. The 75 1+1 Radar REMPI signal ) Figure 5.13: 1+1 Radar REMPI spectrum of nitric oxide at 1.91 Torr (Bold lines in the upper figure). For comparison, nitric oxide REMPI spectrum from the reference [49] is given. 76 Figure 5.14: Comparison of experimental 2+1 REMPI spectrum of nitric oxide in nitrogen at different concentration levels with the one from the reference [11] shown as solid line in the plot. figure shows the spectrum as the laser is hand-tuned across the NO resonance. Each point represents an integration of 32 laser pulses. The time dependence of the signal is also of interest. Figure 5.15 shows the time evolution of the 2+1 Radar REMPI microwave scattering signal from nitric oxide at 15 ppm, 720 ppb and 160 ppb during and after the laser pulse. The ionization takes place in two steps: REMPI ionization of NO, followed by avalanche ionization of the nitrogen. The avalanche process is dominated by the electrons that were generated in the REMPI process. These two mechanisms may be directly observable in the rise 77 Figure 5.15: Microwave scattering signal from nitric oxide in nitrogen at different concentration levels, (red) line with solid dots is for 15 ppm NO in N2 , (blue) line with triangles is for 0.72pm NO in N2 and (green) line with circles is for 0.16ppm NO in N2 . Dotted line is 1/t line. time of the microwave scattering, where the shoulder of the ionization indicated by the arrow corresponds to the REMPI process, and the subsequent rise in the signal is due to the avalanche process. Shorter laser pulses ( 100 psec) are expected to suppress the avalanche component and will provide for better quantitative measurements, however, the avalanche process may be of interest for achieving higher sensitivity for trace species detection. The decay rates of the signals are directly related to the loss rate of electrons. As shown by the dotted line in the figure, the decay rate of electrons in the NO and nitrogen mixture has the 1/t dependence expected for dissociative recombination. In this chapter, the demonstration of Radar REMPI spectroscopy is given in argon, xenon and nitric oxide. By using different lasers and by matching different resonance 78 levels of these species, high-precision REMPI spectra of argon, xenon and nitric oxide are obtained. Sub-ppm nitric oxide in the nitrogen is detected by the non-intrusive method. The time accuracy of the method is about 1 nanosecond and it can show a distinction between the immediate REMPI and the delayed avalanche ionization processes. 79 Chapter 6 Plasma amplification by REMPI and avalanche ionization Plasma amplification is achieved by combining REMPI with avalanche ionization. The REMPI process can greatly enhance the MPI process and produce orders-ofmagnitude more electrons when the laser wavelength is on the resonance of one species. Avalanche ionization can then take advantage of the existing electrons and amplify the plasma generation. 6.1 Idea of plasma amplification The generation of laser spark or laser induced breakdown is well understood. Experimentally a Q-switched giant pulse laser beam is focused to a small volume. A light flash and shock waves are observed. Theoretically, the first multiple photons are absorbed simultaneously and generate some electrons. Normally the multiphoton absorption cross section is very small. Only a tiny amount of electrons is produced. Then avalanche or cascade ionization multiplies the electrons. The sequential plasma generation by multiphoton ionization and avalanche ionization produces the violent optical breakdown and can be regarded as a good example of plasma amplification. 80 The idea here is to amplify the REMPI plasma by avalanche ionization. Instead of direct multiphoton ionization, which can only happen within the limit of the uncertainty principle such that the cross section is very small, the REMPI process has much higher cross section and well-defined property of species selectivity. When the laser frequency is tuned to a resonant level of the species, the species can be ionized by the laser beam with much higher probability. Since the number of electrons can be controlled by the laser frequency, REMPI can provide the seeding source for the cascade process. A nanosecond laser pulse is long enough to generate REMPI and avalanche ionization at the same time since REMPI usually can happen within 1 nanosecond [50]. To prove this concept, microwave scattering experiments were done to measure the multiphoton ionization and electron avalanche ionization simultaneously in a gas mixture. As an example, a simple mixture of two inert gases: argon and xenon, was considered. When the laser wavelength is scanned around the resonance level of argon, ∼ 261.27 nm, electrons are mainly generated from the (3+1) resonance multiphoton ionization process of argon. When the laser pulse length is long enough and/or the laser pulse energy is high, the additional avalanche ionization of argon as well as xenon will happen, which results in a strong amplification of the initial plasma. Since the microwave scattering signal in the Rayleigh regime is determined by the total number of electrons, a stronger microwave scattering signal is expected. 6.2 Experiments The schematic diagram of experimental setup was presented in the previous chapters. Briefly, an injection-seeded, cavity-locked, frequency-tunable, frequency-tripled Ti: Sapphire laser tuned to ∼ 261.27 nm was used to do the experiments in the mixture of argon and xenon. After passing through the cell the laser beam was monitored 81 Figure 6.1: Measured microwave scattering signal by homodyne detection system for different Ar:Xe partial pressures. Laser wavelength is 261.27 nm, linewidth 200 MHz; energy per pulse ∼ 1.0 mJ. by a monochromator (Princeton Instruments, PI-500A) to measure the wavelength within an accuracy of 0.03 nm. The gas cell is made of microwave transparent PVC plastics. The microwave source is located just outside of the cell. The microwave receiver was positioned (> 30 cm) above the cell. A microwave homodyne system was used. A 12.6 GHz Gunn diode was split by a 50/50 microwave splitter. One part was sent to the plasma region and the other was sent to one arm of a mixer in the receiving part. The receiving horn sent the microwave through a pre-amplifier to the other arm of the mixer. The output of the mixer was low-pass filtered and amplified by an amplifier. Finally the signal was monitored by an oscilloscope. In the figure 6.2, a set of microwave scattering signals from plasmas initiated in Ar:Xe mixtures at T0 ≈ 290K and partial pressure of argon at ∼ 5.4 Torr and different partial pressures of Xe are shown. All experiments are done at the wavelength of 261.27 nm which is resonant for (3+1) REMPI of argon, a linewidth of 200 MHz 82 and energy per pulse of ∼ 1.0mJ. It is clearly seen that with increasing of the density of xenon, the scattered microwave signal increases. This is related to avalanche ionization in xenon. A small amount of electrons are generated by REMPI of argon at low pressure. Additional avalanche ionization then develops on the basis of the REMPI electrons. To verify the non-resonance nature of xenon, an experiment in pure xenon was done where the laser wavelength stays at 261.27nm and energy per pulse is ∼ 1.0mJ . No microwave scattering signal is detected for xenon up to 250 Torr. It shows that 261.27 nm is far from the resonance of xenon and a low energy laser pulse cannot initiate an avalanche ionization without the aid of seed electrons from REMPI of argon. To further confirm the resonance nature of argon, another experiment was done in the mixture of argon and xenon. The same laser was tuned to 261.5 nm, which is resonance of neither argon nor xenon. No microwave scattering signal is detected for the mixture of xenon at 250 Torr and argon at 10 Torr when the laser has 1.0 mJ/pulse. This confirms that the amplification process does not happen when the seed electrons are not generated by REMPI of argon. 6.3 Plasma dynamic models1 From the experiments, it is clear that the REMPI process happens only in argon. The model presented in previous chapters covers the physics of REMPI. The introduction + of xenon into argon has several effects: first the molecular ions such as Xe+ 2 , Ar2 and ArXe+ have to be taken into account in the gas mixture at high pressure. This means the reactions of production and recombination of these molecular ions have to be included in the model. Second the Penning ionization of xenon by the excited 1 The plasma dynamic model shown here was developed by Dr. Mikhail N. Shneider. It was presented here for completeness of the thesis. 83 argon atom Ar∗ + Xe ⇒ Ar + Xe+ + e must be taken into account because the energy of the excited argon atom at 3d level is 14.265 eV and the ionization threshold of xenon is only 12.139 eV. For production of the molecular ions, selected reactions are the following, by neglecting Ar4+ and Xe+ 4 , [51] Ar+ + Ar + Ar ⇒ Ar2+ + Ar (6.1) Ar2+ + Xe ⇒ Xe+ + 2Ar (6.2) Ar2+ + Xe ⇒ ArXe+ + Ar (6.3) ArXe+ + Xe ⇒ Ar + Xe+ 2 (6.4) ArXe+ + Ar ⇒ Xe+ + 2Ar (6.5) Xe+ + Ar + Ar ⇒ ArXe+ + Ar (6.6) Xe+ + Xe + Xe ⇒ Xe+ 2 + Xe (6.7) Recombination process can happen in several ways. βef f is the effective recombination rate between electrons and atomic ions (including the 3-body collisional-radiative recombination and photo-recombination) βef f = 8.75 · 10−39 Te−4.5 ne + 2.7 · 10−19 Te−0.75 [m3 /s] 84 (6.8) where Te is in eV and ne in m−3 . Dissociative recombination can be described as the following, Ar2+ + e ⇒ Ar + Ar∗ ∗ ∗ Xe+ 2 + e ⇒ Xe + Xe βd,3 = 0.91 · 10−12 (300/Te[K] )0.61 [m3 /s] (6.9) βd,4 = 2.3 · 10−12 (300/Te[K] )0.6 [m3 /s] (6.10) βd,5 = 1 · 10−12 (300/Te[K] )0.5 [m3 /s] (6.11) ArXe+ + e ⇒ Ar + Xe∗ The frequency of a Penning ionization can be estimated with the simplest gaskinetic approximation π νp = NXe v (d1 + d2 ) 4 where v = (6.12) q 8kT /πµ12 is an averaged relative velocity; hereinafter the symbols µij = Mi Mj /(Mi + Mj ) are the reduced masses for collisions of Mi with Mj species and d1 , d2 are effective diameters of the argon and xenon atoms. In a one dimensional cylindrical model, the plasma evolution and radial expansion for electrons and ions can be described ∂ne 1 ∂rΓe ∗ + = qR,Ar + ne (νi,Ar + νi,Xe ) + νp NAr ∂t r ∂r −βef f (nAr+ + nXe+ ) − ne (βd,3 n+,3 + βd,4 n+,4 + βd,5 n+,5 ) 85 (6.13) ∂nAr+ 1 rΓAr+ + = qR,Ar + ne νi + kArXe nArXe NAr ∂t r ∂r 2 −kconvAr+ nAr+ NAr − βe f f ne nAr+ 2 (6.14) ∂nXe+ 1 ∂(rΓXe+ ) + = ne νi,1 + νP N ∗ + kArXe nAr2+ NXe ∂t r ∂r 2 2 −kconvXe+2 nXe+ NXe − kconvArXe nAr+ NAr − βef f ne nXe+ ∂nAr+ 2 ∂t + 1 ∂(rΓAr2+ ) 2 = kconvAr+ nAr+ NAr − (kArXe,1 2 r ∂r +kArXe,2 )nAr2+ NXe+ − βd,3 ne nAr2+ ∂nXe+ 2 ∂t + (6.15) (6.16) 1 ∂(rΓXe+2 ) 2 = kconvXe+ nXe+ NXe + kArXe,3 nArXe+ NXe − βd,4 ne nXe+ (6.17) 2 2 r ∂r ∂nArXe+ 1 ∂(rΓArXe+ ) 2 + = kArXe,2 nAr+ NXe + kArXe,5 nXe+ NAr 2 ∂t r ∂r −kArXe,3 nArXe+ NXe − kArXe,4 nArXe+ NAr − βd,5 ne nArXe+ (6.18) For excited argon atom, it obeys the following equation, ¶ µ 1 ∂N ∗ 1 ∂ + (rΓ∗ ) = N1 σ(3) F 3 − N ∗ σP i F + + k1 N1 + k3 N12 + ke ne + νP (6.19) ∂t r ∂r τ 86 The Poisson equation for the electric potential is Ã 1 ∂ ∂φ r r ∂r ∂r ! =− e (n+,1 + n+,2 + n+,3 + n+,4 + n+,5 − ne ) ε0 (6.20) The electron temperature obeys µ ¶ " Ã !# ∂ 3 1 ∂ 5 ∂Te ne kTe + r Γe kTe − λe ∂t 2 r ∂r 2 ∂r 3 = JL + JM W − ne k (Te − T ) [(νe,N1 + νe,n+,1 ) δ1 + (νe,N2 + νe,n+,2 ) δ2 ] + N ∗ σP i F εph 2 + ke N ∗ ne I ∗ − ne (νi,1 Ii,1 + νi,2 Ii,2 ) + Nm νP εP (6.21) where 2 JM W = e2 ne IM W (r, t)(νe,+ + νe,2+ + νe,N1 + νe,N2 )/ε0 cm[ωM W +(νe,+ + νe,2+ + νe,N1 + νe,N2 )2 ] << JL (6.22) and JL = e2 ne IL (r, t)(νe,+ + νe,2+ + νe,N1 + νe,N2 )/ε0 cm[ωL2 +(νe,+ + νe,2+ + νe,N1 + νe,N2 )2 ] (6.23) are, respectively, the Joule heating by the microwave and laser pulse. Again the gas dynamic effects are taken into account by adding the Navier-Stokes equations, ∂ρ 1 ∂ (rρu) + = qρ ≈ 0 ∂t r ∂r 87 (6.24) Figure 6.2: Computational results of rate of ionization of argon (5 Torr) and xenon (5 Torr) by the laser pulse of 1 mJ. Ã Ã ! ∂ρu 1 ∂ (rρu2 ) 1 ∂ ∂u u2 ∂p r − + =− +η ∂t r ∂r ∂r r ∂r ∂r r ! + Fd Ã ∂ρ(ε + u2 /2) 1 ∂ (r[ρ(ε + u2 /2) + p]u) 1 ∂ ∂T + =Q+ rλ ∂t r ∂r r ∂r ∂r (6.25) ! Ã !2 2 ∂u u +2η + 2 + Qd ∂r r (6.26) By solving equations 6.13 - 6.26, the time evolution and spatial distribution of the Ar:Xe partially ionized plasma components, together with the neutral gas parameters of pressure, density, temperature, and radial velocity can be obtained [51]. 88 Figure 6.3: Computational results of rate of ionization of argon (5 Torr) and xenon (150 Torr) by the laser pulse of 1 mJ. 89 Figure 6.4: Computational results of rate of ionization of argon (5 Torr) and xenon (250 Torr) by the laser pulse of 1 mJ. Figures 6.3 - 6.3 show the relative magnitudes of the ionization rates at the plasma axis different initial compositions of Ar:Xe mixture, 5:5, 5:150, 5:250 and 5:300 Torr. REMPI is denoted by qREM P I , and the electron avalanche ionization by the laser field is denoted by qav . It is clearly seen that when partial pressure of xenon is low, REMPI remains the main mechanism of plasma generation during the whole laser pulse. But with the increasing of xenon initial partial pressure, the role of avalanche breakdown becomes more important and, even dominant at 300 Torr. It follows from this result that for the long laser pulses, the breakdown which starts from the REMPI phase develops into a classical avalanche on the later times. Such a combination of the REMPI with avalanche ionization can reduce substantially the laser breakdown threshold. This can be important for gas mixtures when one component can be ionized in a REMPI process with subsequent avalanche ionization of other mixture components. It provides an amplification mechanism for mixtures. 90 Figure 6.5: Computational results of rate of ionization of argon (5 Torr) and xenon (300 Torr) by the laser pulse of 1 mJ. 91 A buffer gas which is easily ionized can be seeded into the sample. After it is ionized by REMPI, the triggered avalanche ionization will be detected. In the experiment, when the laser frequency lies exactly on the resonance level, an avalanche breakdown can be observed at 200 Torr or higher. While the frequency is tuned off the resonance, only the REMPI part can be observed by microwave scattering. 6.4 Discussions Plasma amplification by the combination of REMPI and avalanche ionization is demonstrated both experimentally and theoretically. Coherent microwave Rayleigh scattering was used to measure REMPI and avalanche ionization in Ar:Xe mixtures. It was shown that REMPI ionization of a relatively small density component (Ar) can catalyze the avalanche ionization process in a buffer gas (Xe) by the use of a laser beam at very low intensity, which itself is not enough for initiation of the avalanche ionization of the buffer gas. Theoretical plasma dynamic model verifies the finding. Based on the presented results, several important applications are possible. First, it can improve the detection sensitivity of Radar REMPI. Complete ionization of a gas mixture can be made by combining REMPI with an additional avalanche ionization of bulk gas, acting as an amplifier of the weakly ionized REMPI plasma. It can significantly increase the scattered microwave signal. Second, it suggests that plasma generation can be achieved at reduced gas densities. For many applications, it is very important, but usually difficult, to generate weakly ionized plasma at different gas pressures and temperatures with heating of the bulk gas up to arbitrary temperature. Here we can suggest the following two methods: Long laser pulse: The pulse front generates REMPI and subsequent pulse initiates avalanche ionization and joule heating. 92 Two subsequent laser pulses: A short laser pulse tuned on a REMPI of mixture component generates weakly ionized REMPI plasma and a long off-resonant laser pulse for the avalanche ionization and Joule heating. A good example is atmospheric air. 1% Argon in air can be ionized by REMPI and other species can act as amplifier buffer gases. In this case the breakdown initiation and gas heating might be achieved by laser beams at relatively low intensity. 93 Chapter 7 Conclusions and future research 7.1 Conclusions In this thesis, a novel diagnostic method, coherent microwave scattering from laser induced small volume plasma, is demonstrated for the first time. The physics of microwave scattering and REMPI is first studied. The applications of Radar REMPI are then explored. This new diagnostic technique combines the highly sensitive microwave detection and extremely selective laser material interaction. A time accurate, non-intrusive and localized measurement is achieved. The underlying physics of microwave scattering at short time is developed and validated. When microwave wavelength is much larger than the size of plasma, the plasma can be regarded as a point. When the skin layer thickness is also greater than the size of the plasma, all electrons inside the plasma oscillate according to the field of the microwave. The acceleration and deceleration of the electrons are all in phase and generate coherent emission or scattering, because all are approximately at the same location of the microwave field. The scattering has a simple interpretation Rayleigh scattering. And more important, the power of the scattering is proportional to the square of the total electron number inside the plasma. The theoretical and 94 experimental verification is conducted. A general Mie scattering formula is derived, in which microwave wavelength, the size of the plasma and conductivity are parameters. It verifies the above conclusions, i.e., when size of the plasma is much smaller than the microwave wavelength and smaller than the skin layer thickness, the scattering lies within the Rayleigh range. The experimental demonstration is done with the microwave scattering from a laser spark in air. In the avalanche phase of laser induced breakdown, the size of the plasma is approximately same as the size of the laser focus. The conductivity is relatively low so that the skin layer thickness is larger than the plasma size. The microwave scattering is shown to have an isotropic scattering pattern in the plane perpendicular to the polarization of the microwave and have a classic dipole radiation pattern in the plane parallel to the polarization. In the after spark evolution, plasma evolves and the size becomes larger so that the approximation for the point dipole is invalid. The experimental results show an anisotropic Mie scattering pattern. The REMPI process is then discussed and demonstrated. Off resonance multiphoton absorption is a nonlinear optical process which only happens when there is enough photon flux to make the transition within femtoseconds according to the uncertainty principle. It is greatly enhanced when energy of one or multiple photons matches the gap of different energy levels. When laser wavelength is scanning across a resonance level, the total electron number inside the plasma is changing accordingly and the spectrum of the gas can be recorded. A plasma dynamic model is used to predict the evolution of this plasma and an experimental demonstration in argon is successfully conducted. Good agreement on the REMPI spectrum obtained by Radar REMPI, classical electron collection and quantum mechanical predictions is achieved. A good match between the time evolution of the experimental microwave signals and plasma dynamic model prediction is obtained. Second applications of the novel diagnostic method are explored. First time 95 resolved, high precision REMPI spectroscopy is demonstrated in different atomic and molecular species. An injection-seeded, frequency tunable, frequency doubled or tripled Ti: Sapphire laser is used to generate REMPI in nitric oxide or argon and spectra are observed. The time accuracy of the measurement is about 1 nanosecond. Second trace species detection is demonstrated. 160 ppb (parts per billion) nitric oxide inside nitrogen is detected using Radar REMPI. Third, plasma amplification is demonstrated. In the classical optical breakdown, it is believed that multiphoton ionization generates a few seed electrons and avalanche ionization multiplies them. An experiment is done in a mixture of argon and xenon to show the amplification of argon REMPI plasma by avalanche ionization in xenon. When the laser is tuned to the resonance of argon, microwave scattering signal increases with the increasing pressure of xenon. While the laser is off the resonance of argon, laser is not powerful enough to generate any microwave scattering signal. A theoretical plasma dynamic model of mixture of argon and xenon is developed. Qualitative agreement with the experimental data is achieved and suggests a new way to heat the gas to an arbitrary temperature without a classical optical breakdown. 7.2 Future work The new diagnostic method has several advantages over other methods. First it provides a time-resolved signal. The non-intrusive Radar REMPI measurement can give detailed view of the ionization and recombination processes. Second the use of combination of high power laser and pulsed microwave can extend the measurement to the far field. So possible remote sensing scheme may be possible [52]. The Radar REMPI technique could also be used to measure rotational temperature of molecular species [53]. Accurate optical measurements of temperature are often required in combusting or high-speed flows for which high temperatures or 96 pressures, high velocity, or highly reactive environments preclude the use of physical probes. The possibility of measuring the vibrational and rotational temperatures of an individual specie is also of interest for reacting environments, supersonic flows, surface interactions and for tracking the evolution of species generated by photo induced fragmentation. REMPI has proved to be a useful method for measuring trace species and it can be used to determine vibrational and rotational temperature [8, 11, 40]. Radar REMPI has been demonstrated to have the capability to do a non-intrusive remote sensing, to achieve high-spatial resolution measurement, and to determine accurate concentration profiles without the use of probes or electrodes. With a narrow linewidth laser, an accurate rotational REMPI spectrum can be obtained, which can be used to determine rotational temperature of the observed species. Velocity measurement is clearly another important application. When the plasma is moving with the flow toward or away from the receiver, Doppler shift of the microwave will occur. By measuring the frequency shift, the velocity of the flow can be deduced. This may best be detected using quadrature techniques. 97 Appendix A Resonance enhanced multiphoton ionization This chapter we will derive a general formula of resonance enhanced multiphoton absorption in an atom. First some coupling schemes used in the complex atoms and corresponding selection rules will be discussed. In the interaction picture of quantum mechanics, time dependent perturbation theory is then used to get the general formula for multiphoton absorption. Discussions for different polarization of the light and different broadening mechanisms are also given. Finally direct multiphoton ionization based on Keldysh theory is discussed. A.1 Coupling schemes of electrons in complex atoms For the experiments, argon was used to demonstrate the concept of Radar REMPI because of its relative simple structure, well-known spectrum and reaction constants for plasma dynamic models. To fully understand the theory of REMPI, different coupling schemes will be introduced here. Because in argon the 3p5 4p, 3p5 4d and the lower np5 n0 s states are LS coupled and other states are jc K coupled. There are some approximations to make the coupled Hamiltonian of the multi98 electron configuration of the complex atoms tractable.[32] 1. LS coupling 2. jj coupling 3. pair coupling It is very common to find descriptions about LS and jj coupling. Here only the pair coupling will be introduced. A.1.1 Pair coupling More common than jj coupling, though less well known, are coupling conditions under which the energy levels tend to appear in pairs. These conditions occur for excited configurations in which the energy depends only slightly on the spin of the excited electron; the level pairs correspond to the two possible values of J or Jc that are obtained when s is added to the resultant, K, of all other angular momenta. Pair-coupling conditions occur mainly when the excited electron has large angular momentum because such an electron tends not to penetrate the core and thus experiences only a small spin-dependent Coulomb interaction, and its spin-orbit interaction is likewise small. jc K coupling The most common limiting type of pair coupling, jc K coupling, occurs when the strongest interaction is the spin-orbit interaction of the more tightly bound electron, and the next strongest interaction is the spin-independent portion of the Coulomb interaction between the two electrons. The corresponding angular-momentum coupling scheme is 99 lc + sc = jc j c + le = K K + se = J (A.1) This type of coupling occurs particularly in excited configurations of the noble gases (Ne, Ar, kr, Xe, Rn) and of the carbon-group elements, but also in many other cases. In this case the eigenstate of the atom can be expressed as |[(nl)n−1 [lc sc ]Jc , nle ]K, 1/2, JmJ i (A.2) 0 For example argon first excited state is |3p5 (2 P3/2 )4s[K = 3/2]J = 2i. LK coupling L-K coupling conditions occur rather infrequently, notably in the configurations N II 2p4f and PII 3p4f. A.1.2 Selection rules jc K coupling Selection rules for an N-photon transition are most easily derived by successive applications of the selection rules for one-photon processes. Both LS and jc K coupling occurs when the ”electrostatic interaction is weak compared to the spin-orbit interaction of the parent ion, but is strong compared to the spin coupling of external electron”[32]. The selection rules for one-photon processes are 100 ∆jc = 0, ∆K = 0, ±1, ∆le = ±1 (A.3) together with ∆J = 0, ±1 and parity change. LS coupling Just for illustration, LS coupling selection rules for a dipole approximation are listed. For linearly polarized light and single photon transition, if the perturbation is much less than the LS coupling, the selection rules for one photon transitions are ∆J = 0, ±1 ∆L = 0, ±1 ∆S = 0 ∆M = 0 J + J0 ≥ 1 L + L0 ≥ 1 (A.4) For circularly polarized light, the selection rules for one-photon transition become ∆J = 0, ±1 ∆M = ±1 101 (A.5) A.1.3 Energy level notations Notations are determined by the coupling schemes. For a common LS coupling, the notation is 2S+1 Lj , where 2S + 1 is the total spin quantum number, j is total angular momentum number. While in a jc K coupling scheme, the level is labeled as nl[K]J . For example, the 4s1 P1 state of neon is written in JK coupling as 4s[1/2]1 . Levels belonging to the 2 P1/2 core are indicated with a primed l value, the unprimed belonging to the 2 P3/2 core. For example, 3p5 nl[K]J and 3p5 nl0 [K]J correspond to jc = 3/2 and jc = 1/2 ionic states respectively. A.1.4 The 3n-j symbols 3n-j symbols are created to calculate Clebsch-Gordon coefficients or other representation transition. First we consider the 3-j symbol. It is an algebraic function of six arguments that may be defined by the expression j1 j2 j3 m1 m2 m3 (A.6) In quantum mechanics, angular momentum is discrete. If we want to couple two angular momentums, we must find a common complete set to describe them. One complete set is |j1 m1 , j2 m2 i = |j1 m1 i|j2 m2 i, j12 |j1 m1 , j2 m2 i = j1 (j1 + 1)|j1 m1 , j2 m2 i 2 j1z |j1 m1 , j2 m2 i = j1 (j1 + 1)|j1 m1 , j2 m2 i j22 |j1 m1 , j2 m2 i = j2 (j2 + 1)|j1 m1 , j2 m2 i 102 2 j2z |j1 m1 , j2 m2 i = m2 |j1 m1 , j2 m2 i (A.7) the other complete set is the |jmi, j12 |jmi = j1 (j1 + 1)|jmi j22 |jmi = j2 (j2 + 1)|jmi j 2 |jmi = j(j + 1)|jmi jz |jmi = m|jmi (A.8) Then we must find the transformation to connect two complete sets. It is related to the 3j symbol. |jmi = X C(j1 j2 j; m1 m2 m)|j1 m1 , j2 m2 i (A.9) m1 ,m2 where C(j1 j2 j; m1 m2 m) = hj1 m1 , j2 m2 |j3 m3 i is the Clebsch-Gordon coefficient. It is related to 3j symbol by j1 j2 j3 m1 m2 m3 j1 −j2 −m3 (2j3 + 1)−1/2 hj1 m1 , j2 m2 |j3 − m3 i = (−1) (A.10) Physically Clebsch-Gordon coefficient represents the probability amplitude that the coupled state |jmi will be found having its component parts j1 and j2 making the projection m1 and m2 for j1z and j2z respectively. The corresponding 3j symbol is the probability amplitude divided by the factor (2j + 1)1/2 . The 3j symbol function can be calculated by the built-in function T hreeJSymbol in Mathematica 6.0. The similar case holds for the 6-j symbol. It is created to couple 3 angular momen103 tums. Of course if 4 angular momentums are coupled, we will have 9j symbol. The function can be calculated by the function SixJSymbol in Mathematica. General algorithms of calculating 3n-j symbols can be found in the reference [33]. A.2 The Interaction Picture for Quantum Mechanics There are three main pictures for description of quantum mechanics, Schrodinger picture, Heisenberg picture and interaction picture [38]. The one used here is ”Interaction picture”. All quantities for interaction picture will be marked with subscripts ”I” .[54] A common Schrodinger description of a quantum system is, ih̄ d |φ(t)i = Hs |φ(t)i dt (A.11) where Hs = Hs0 + Hs1 (t) is the total Hamiltonian for the system. Hs0 is constant with time, while Hs1 (t) is a function of time. If we define a propagator Us (t, t0 ) by |φs (t)i = Us (t, t0 )|φs (t0 )i (A.12) it follows that ih̄ dUs = H s Us dt (A.13) Since Us (t, t0 ) is a unitary operator, which is the generalization of the rotation operator to complex space, we may describe the time evolution of the state vectors as ”rotations” in Hilbert space. The rotation is generated by Us (t, t0 ) or equivalently 104 by Hs (t) = Hs0 + Hs1 (t). If Hs1 = 0, the equation A.13 becomes ih̄ dUs0 = Hs0 Us0 dt (A.14) the formal solution to which is Us0 (t, t0 ) = exp(−iHs0 (t − t0 )/h̄) (A.15) If Hs1 6= 0 , both Hs0 and Hs1 jointly produce the rotation Us . The interaction picture states that the stationary frame is the one that rotates at a rate that Us0 or Hs0 generates by itself. In this frame, the state vector moves because Hs1 6= 0. So the state vector in the interaction frame is related to the one in the Schrodinger frame by |φI (t)i = [Us0 (t, t0 )]† |φs (t)i (A.16) The time evolution of |φI (t)i is as followed: † ih̄ d dU 0 d|φsi † |φI (t)i = ih̄ s |φsi + Us0 ih̄ dt dt dt † = Us0 Hs1 |φs i † † = Us0 Hs1 Us0 Us0 |φs i † = Us0 Hs1 Us0 |φI (t)i (A.17) Now we can define HI1 (t) = Us0 † Hs1 (t)Us0 105 (A.18) as the perturbing Hamiltonian seen in the rotating frame. So we can write ih̄ d |φI (t)i = HI1 (t)|φI (t)i dt (A.19) This is the Schrodinger equation in the interaction picture. Now we can define a propagator UI (t, t0 ) in the interaction picture, |φI (t)i = UI (t, t0 )|φ(t0 )i (A.20) which obeys ih̄ dUI = HI1 UI dt (A.21) Since HI1 depends on time, the solution to equation A.21 is not UI = exp(−iHI1 (t− t0 )/h̄). A formal solution, with the initial condition, is UI (t, t0 ) = I − iZt HI (t0 )UI (t0 , t0 )dt0 h̄ t0 (A.22) This is not a solution, but an integral equation. The nice thing for this is that we can iterate the solution by substituting UI (t, t0 ) into the right hand side to get a desired accuracy. To first order, we can keep only powers of HI1 . So we use the zeroth-order value for UI iZt 1 0 0 UI (t, t0 ) = I − H (t )dt + O(HI2 ) h̄ t0 I (A.23) To the higher orders, we can repeat feeding into the right hand side. Z t Z t0 i 1 0 0 2 UI (t, t0 ) = I − HI (t )dt + (−i/h̄) HI1 (t0 )HI1 (t00 )dt0 dt00 + ... h̄ t0 t0 (A.24) Premultiplying by Us0 (t, t0 ) and expressing HI1 in terms of Hs1 , we get the Schrodinger 106 picture propagator Us (t, t0 ) = iZt 0 † − Us (t, t0 )Us0 (t0 , t0 ) Hs1 Us0 (t0 , t0 )dt0 + ... h̄ t0 iZt 0 = Us0 (t, t0 ) − U (t, t0 )Hs1 Us0 (t0 , t0 )dt0 + ... h̄ t0 s Us0 (t, t0 ) (A.25) By using equation A.15, we can calculate the transition from |i0 i to |f 0 i. The second term says that between t0 and t0 the eigenstate |i0 i picks up just a phase(i.e. responds to Hs0 alone). At t0 it meets the perturbation, which has an amplitude hf 0 |H 1 |i0 i of converting it to the state |f 0 i. Thereafter it evolves as the eigenstate |f 0 i until time t. The total amplitude to end up in |f 0 i is found by integrating over the times at which conversion could have taken place. Thus the first order transition corresponds to a one-step process i → f . At the second order, we see a sum over a complete set of states |n0 i. It means the system can go from |i0 i to |f 0 i via any intermediate or virtual state |n0 i that H 1 can knock |i0 i into. Thus the second order amplitude describes a two-step process, i → n → f . Higher order amplitudes have a similar interpretation. It is exactly how multiphoton process can happen. A.3 Perturbation theory for multiphoton ionization A.3.1 Single Atom without losses The Hamiltonian for a bound electron interaction with a radiation field can be written as[31] H = H0 + HIN T 107 (A.26) where H0 = Ha + Hr is the total Hamiltonian of the unperturbed system. Ha = p2 /2m + V (r) is the unperturbed Hamiltonian of the atom. Hr = P λ aλ † aλ h̄ωλ is the Hamiltonian of the radiation field. aλ and aλ † are the photon annihilation and creation operators at the frequency of λ. HIN T = −e/me ε · r is the Hamiltonian of the interaction between the atom and radiation field. Note that HIN T is independent of time. In the interaction picture, the time evolution of the system comes entirely from the interaction part of the Hamiltonian HIN T . Suppose that at time t0 = 0, the system is in one of the eigenstates |gi of the unperturbed Hamiltonian, H0 . Then after time t, the state of the system is ψ(t)i. The propagator UI (t) is such that |ψ(t)i = UI (t)|gi. The probability amplitude bn that the system is in some other eigenstate of H0 , say |ni, is clearly given by the projection of |ψ(t)i to |ni, bn = hn|ψ(t)i = hn|UI (t)|gi (A.27) If |ψ(t)i is represented in terms of the complete set of orthonormal eigenstates of H0 , then bn is just one of the expansion coefficients of |ψ(t)i = X bm (t)|mi (A.28) m Thus, by definition, the probability that the system has undergone a transition from |gi to |ni in time t is Wn,g (t) = |hn|UI (t)|gi|2 = |bn |2 (A.29) From equation A.22, the propagator UI (t) can be written by iterating right hand side as 108 UI (t) = 1 + ∞ X (N ) UI (t) (A.30) N =1 where HI0 (t) = e(iH0 (t−t0 )/h̄) HIN T e−iH0 (t−t0 ) (N ) UI (t) −n = (ih̄) Z t dtn 0 −n = (ih̄) Z t 0 Z tn 0 dtn−1... Z t2 0 (A.31) dt1 HI0 (tn )HI0 (tn−1 )...HI0 (t1 ) dn tHI0 (tn )HI0 (tn−1 )...HI0 (t1 ) (A.32) To calculate the Nth-order contribution to the transition rate between, the ground (N ) state |gi and some final state |f i, we must determine the matrix element of UI (N ) From the definition given by equation A.32 for UI (N ) hf |UI |gi −N = (ih̄) −N = (ih̄) Z t 0 Z t 0 (t). (t), we have dN thf |HI0 (tN )HI0 (tN −1 )...HI0 (t1 )|gi dN t X X ... X hf |HI0 (tN )|mN −1 i (A.33) mN −1 mN −2 m1 0 ×hmN −1 |HI (tN −1 )|mN −2 i...hm1 |HI0 (t1 )|gi The sums over the mν are extended over the complete set of states of H0 , discrete plus continuum. Now for one sum, we can get Z t X 0 mN −1 hf |HI0 (tN )|mN −1 idt = = Z t X 0 mN −1 Z t X 0 mN −1 hf |eiH0 tN /h̄ HIN T (tN )e−iH0 tN /h̄ |mN −1 idtN hf |eiEf tN /h̄ HIN T (tN )e−iEmN −1 tN /h̄ |mN −1 idtN 109 X ei(Ef −EmN −1 )t/h̄ − 1 = mN −1 i(Ef − EmN −1 )/h̄ hf |HIN T (tN )|mN −1 i (A.34) If we define ωi,j = (Ei − Ej )/h̄, then we integrate all integrals. Next we integrate over time. To eliminate rapidly oscillating terms, the lower limits on the intermediate integrals are taken as t0 → ∞. The lower limit on the final integral is taken at t=0 to ensure the final state was not occupied before the perturbation was on. Thus, (N ) hf |UI (t)|gi = (−h̄)−N eiωf g t − 1 (N ) Mf,g ωf g (A.35) where (N ) Mf,g = X X ... mN −1 mN −2 X hf |HIN T |mN −1 i m1 hmN −1 |HIN T |mN −2 i ωmN −1 ,mN −2 hmN −2 |HIN T |mN −3 i hm1 |HIN T |gi ... ωmN −2 ,mN −3 ωm1 ,g (A.36) For N-photon ionization process, the lowest order contribution to photoionization cannot involve the creation of photons. Therefore, the only part of the field operator ε that requires consideration is ε = i/c X ωλ aλ Aλ (A.37) λ where aλ is the photon annihilation operator defined as aλ |λi = [h̄/2ωλ nλ ]1/2 |nλ − 1i. Aλ is given as Aλ = (4πc2 )1/2 eikλ ·rε HI = −eε · r = −ie/cωλ aA · r (A.38) Now we can define R = eik·r which operates only on atomic states. The matrix elements of HI can be immediately evaluated. 110 Let the state |gi be denoted by |ag ; ni, where ag specifies the atomic quantum numbers and n specifies occupation number of the field. Similarly the state |mυ i are represented by |aυ ; n0 i. Using the definition of the annihilation operator a and recalling that for each successive matrix element, the radiation field is depleted by (N ) one photon, the N th order matrix element Mf,g becomes (N ) Mf,g = [−ie(2πh̄ω)1/2 ]N {[n − (N − 1)] × [n − (N − 2)] × ... (A.39) ) × [n − 1] × n}1/2 Ka(N f ,ag (A.40) where ) Ka(N = f ,ag X X aN −1 aN −2 ... X haf |R|aN −1 i × a1 ha1 |R|ag i haN −1 |R|aN −2 i × ... × (A.41) ωaN −1 ,ag − (N − 1)ω ωa1 ,ag − ω ωm1 ,g = ωm1 − ωg = [ωa1 + (n − 1)ω] − [ωag + nω] = ωa1 ,ag − ω (A.42) Since for multiphoton ionization, photon number n is on the order of 1020 and number of quantum states N is at most 100. We can neglect the depletion of the photon field, i.e., {[n − (N − 1)] × [n − (N − 2)] × ... × [n − 1] × n} ≈ nN . Finally, we get the transition probability (N ) (N ) Wf,g (t) = |hf |UI (t)|gi|2 (N ) = h̄−2N f (ωf,g , t)|Mf,g |2 ) 2 = [2π(e2 /h̄)ωn]N f (ωf,g , t)|Ka(N | f ,ag 111 (A.43) where f (ωf,g , t) = |(eiωf,g t − 1)/ωf,g |2 We also can replace number of photons n by F/c where F is photon flux (number of photons cm−2 sec−1 ). When time is long compared to 1/ωf,g , we can replace f (ω, t) by its asymptotic value, 2πtδ(ω), and α = e2 /h̄c. The transition rate per atom can be written (N ) dWf,g (t) ) 2 = 2π(2παF ω)N |Ka(N | δ(ωf,g ) f ,ag dt A.3.2 (A.44) Continuum states When transitions are to the continuum, we must treat a group of neighboring states within a small energy range d²f,g rather than a single state. From equation A.44, we have upon the integration over ²f,g , (N ) ) 2 Wf,g = 2πh̄(2παF ω)N |Ka(N | ρ(0) f ,ag (A.45) The density of final states, ρ, depends on the normalization of the continuum state function. Here we use ρ((k)) = (2π)−3 . Then the density of states with respect to energy becomes, ρ(²) = (m/h̄2 )k dΩk (2π)3 (A.46) where k is the wave number of the electron and energy is defined by ²f,g = ²I − N h̄ω + h̄2 k 2 /2m = 0 (A.47) where ²I is the ionization energy of the atom, h̄ω is the photon energy, and the last term is the kinetic energy of the electron. 112 Now we put equations A.46 and A.47 into the equation A.45, (N ) wf,g (θk , φk ) = m/h̄ ) 2 (2παF ω)N |Ka(N |k f ,ag 2 (2π) (A.48) (N ) Here,wf,g (θk , φk ) denotes the differential transition rate corresponding to the probability for emission of an electron in the direction of (θk , φk ) within the solid angle dΩk . The total transition rate per atom is (N ) Z wf,g = A.3.3 (N ) dΩk wf,g (θk , ωk ) (A.49) Damping and Level shifts For the damping and level shifts, a rigorous treatment is difficult even for ”onephoton” problems. Here we assume the light field is small compared to the field of the atom. So the level shift is omitted. But we certainly can add a phenomenological damping term to the initial and final state wave functions with time dependence |ψa (t)i = e−iωa t−γa t/2 |ai (A.50) The principal consequence is an addition of a 1/2iγ to every energy denominator appearing in our equations, where γ is the combined width of initial and final states. If we use a quantum mechanical treatment, we can get the spontaneous decay rate γ= 4ωmn |Rnm |2 3c3 (A.51) With the damping terms considered, the N th-order transition operator can be expressed as 113 ) Ka(N = f ,ag X aN −1 X haf |R|aN −1 i × ... ωaN −1 ,ag − (N − 1)ω + iγ/2 aN −2 (A.52) (N ) Then we can correspondingly get the expression for Mf g . A.4 Evaluation of Nth-order Matrix elements The main difficulty in calculating the transition probability for N-photon ionization ) is the evaluation of the Nth-order matrix element Ka(N in the equation of A.41. f ,ag The most conspicuous feature is the appearance of many infinite summations over electronic eigenstates. There are several approaches for the evaluation of those infinite summations. One is the truncated summation method [31]. The method start with a given atomic state, and the matrix elements decrease as one goes to higher states. At the same time, the energy denominators increase after a certain state. As a result, the contribution of the higher terms decreases. Therefore, keeping a finite number of terms will in many cases give a good approximation. The other commonly used method is the Green’s function method. Using the phenomenological Green’s function method, one has a way of performing the infinite summation without truncation. Since the technique is based on quantum defect theory (QDT), it is usually stated that it should be as accurate as QDT. In this thesis, I used the truncated summation method and intermediate levels are limited to the ones found in NIST physics reference website. A.5 Resonance Enhanced Multi-photon ionization If the energy of K 0 photons lies in resonance with some intermediate level, then ) in the equation of A.41 vanishes, making one of the energy denominators in Ka(N f ,ag 114 the expression inapplicable. In this case, the ionization rate has to be described by formulas that are more complex. The process is the so-called process of resonance ionization. The ionization rate, when there is an intermediate resonance, is always much greater than the rate of direct multiphoton ionization. In fact, the appearance of an intermediate resonance corresponds to a decrease of the detuning so that it is on the order of the reduced transition width [55]. Also in the case when spontaneous relaxation of the resonance state dominates the induced transitions from this state, the resonance ionization is a cascade process. Then the probability of ionization of the initial state i during 1 second can be obtained by multiplying the excitation probability Wik of the resonance state k by the ionization rate wkE of the resonance state. In any field, the probability of finding the electron in a given state is a smooth function of the frequency. The Lorentzian line shape for a spontaneous decay is a well-known example of such a function. Hence there are no strict limits on the resonance frequency of the external field. As the detuning from resonance increases, the non-resonant interaction of these states with other states, bound or free, starts to compete with the resonant interaction. First, the criterion for the resonance enhanced ionization can be defined as ∆(F ) = |Ek (F ) − Ei (F ) − K 0 h̄ω| ≤ Γki (F ) (A.53) Here ∆(F ) is the resonance detuning; Ei (F ) and Ek (F ) are the energies of the initial and resonance states, respectively, taking into account of their perturbation by the radiation field with field strength F; Γki (F ) is the reduced width of the resonance transition. This is for monochromatic radiation and a single motionless atom. In the more realistic case of an ensemble of atoms with a given temperature and quasi monochromatic radiation, Γki (F ) in the equation A.53 must be replaced by the maximum width for the given ensemble and laser mode distributions. In addition to 115 the field width Γki (F ), the spectrum width ∆ωK 0 reduced for the K 0 − photon process should be taken into account, the Doppler width ΓD for the target gas or beam and the space-time distribution of the radiation intensity. By following Breit-Wigner procedure [38], the rate of resonance enhanced ionization in a weak field is simplified by factorization of the compound matrix element ) Ka(N = f ,ag X X aN −1 aN −2 ... X haf |R|aN −1 i × a1 ... haN −1 |R|aN −2 i × ωaN −1 ,ag − (N − 1)ω + iγN −1 /2 haK |R|aK−1 i ha1 |R|ag i ... × (A.54) (ωak ,ag − Kω) + iγK /2 ωa1 ,ag − ω + iγ1 /2 where γk is the natural linewidth of the resonance level k. The K-photon resonance transition happens when ωak ,ag − Kω < ∆ωL , i.e. difference is less than the laser linewidth, the term ωak ,ag − Kω must be approximated by ∆ωL . So the transition probability from equation A.43 is (N ) ) 2 Wf,g (t) = h̄−2N |Ma(N | f ,ag eiωf,g t − 1 2 (N ) 2 | |Kaf ,ag | ωf,g eiωf,g t − 1 2 X X X = (2παF ω)N | |{ ... ... haf |R|aN −1 i ωf,g aN −1 aK a1 = [2π(e2 /h̄)ωn]N | haN −1 |R|aN −2 i × q 2 (ωaN −1 ,ag − (N − 1)ω)2 + γN −1 /4 haK |R|aK−1 i ... q 2 (ωaK ,ag − Kω)2 + γK /4 ha1 |R|ag i ... × q (ωa1 ,ag − ω)2 + γ12 /4 × }2 (A.55) Now all the theory is only applicable to the single motionless atom in a not too 116 strong and strictly monochromatic light field. A.6 Arbitrary polarization If we define +~y as the direction of propagation of the wave, the plane wave field can be expressed as [35, 36] ~ r, t) = E0~²exp[−i(ωt − βy)] + c.c. E(~ (A.56) The unit vector ~² = ²1~x + ²3~z describes the polarization of the field. ~ into the matrix elements, we can calculate any polarization, By including ~r · E i.e., hn|²1~x + ²3~z|mi. A.7 Broadening Besides the damping due to the emission of the radiation itself, there are several other causes which actually broaden a line: Temperature Doppler broadening, collisional broadening and Stark broadening etc. For simplicity, we assume that each atom has a frequency response which is Gaussian [56]. Here collisional broadening which has a Lorentz lineshape is not included. The Doppler broadening of atoms can be expressed as 1 ω − ωa 2 )] Pn (ω) = Wn exp[− ( 2 δ (A.57) where Wn is the total absorption probability for unit laser intensity, ωa is the peak response frequency of the atom, δ is the spread of the transition, which is equal to √ half maximum of the transition divided by 2 2ln2. In general δ is determined by the radiative width of the final atomic state. 117 The atomic frequencies, ωa , of the atoms are also distributed normally, due to the Doppler shift, so that the ensemble response is give by N0 1 ωa − ωD 2 exp[− ( N (ωa ) = √ )] 2 D 2πD (A.58) where N0 is the number density of the absorbers, ωD is the peak frequency of the ensemble and D is the Doppler or ensemble spread. The laser frequency also has a Gaussian profile, I(ωa ) = √ 1 ω − ω Li 2 1 exp[− ( )] 2 ∆i 2π∆i (A.59) where Ii is the total cycle-averaged intensity of the ith laser photon, ωLi is the peak frequency of laser i and ∆ is the spread of laser i. Now corresponding to n photons picked from n laser frequency distributions, the new distribution is also Gaussian I (n) (ω) = √ 1 P 2π( n ∆2i )1/2 exp (−1/2 (ω − P ωLi )2 ) 2 n ∆i P n (A.60) The rate of absorption of n photons by one atom is given by R(ωa ) = Z ∞ 0 I (n) (ω)Wn (ω)dω P Z ∞ Wn (ω − n ωLi )2 (ω − ωa )2 = exp (−1/2 − 1/2 )dω P P 2 (2π)1/2 ( n ∆2i )1/2 δ2 0 n ∆i P ( n ωLi − ωa )2 Wn (δ) exp(−1/2 P 2 = √ P 2 ) (A.61) 2 2( n ∆i + δ 2 )1/2 n ∆i + δ The weighted response for the ensemble can be given by 118 Rn (ωD ) = Z ∞ 0 Rn (ωa )N (ωa )dωa P Wn δN0 ( n ωLi − ωD )2 = exp(−1/2 ) P P 2 2 2 2( n ∆2i + δ 2 + D2 )1/2 n ∆i + δ + D (A.62) This equation A.62 includes all the linewidth parameters involved. Finally we can write the probability of the resonance enhanced N-photon ionization for an ensemble(N0 atoms), (N ) Wf,g N0 δ (nωL − ωD )2 exp(−1/2 ) 2(n∆2 + δ 2 + D2 )1/2 n∆2 + δ 2 + D2 N0 γ(2ln2)1/2 (nωL − ωD )2 (N ) exp(−1/2 ) = Wf,g (∆2L + δ 2 + D2 )1/2 n∆2 + δ 2 + D2 (N ) = Wf,g (A.63) Here we assume that all photons have same macroscopic linewidth ∆L and same frequency ωL . Now we can define a lineshape function G(ωL ) √ G(ωL ) = 2ln2 1 (nωL − ωD )2 exp[ ] (∆2L + δ 2 + D2 )1/2 2 n∆2 + δ 2 + D2 So the final expression for the transition probability is (N ) Wf,g ) 2 | ∗ G(ωL )γ = h̄−2N |Ma(N f ,ag eiωf,g t − 1 2 (N ) 2 | |Kaf ,ag | G(ωL )γ ωf,g eiωf,g t − 1 2 X X X = (2παF ω)N | |{ ... ... haf |R|aN −1 i ωf,g aN −1 aK a1 = [2π(e2 /h̄)ωn]N | 119 (A.64) haN −1 |R|aN −2 i × q (ωaN −1 ,ag − (N − 1)ω)2 + γ 2 /4 haK |R|aK−1 i ... q (ωaN −1 ,ag − (N − 1)ω)2 + γ 2 /4 ... × q ha1 |R|ag i (ωa1 ,ag − ω)2 + γ 2 /4 × }2 ∗ G(ωL )γ (A.65) when there is a resonance, the resonance term will vanish and one of the γs in the denominator will be eliminated. This result is identical to the result obtained by using another method [57]. A.8 Evaluation of the matrix If we assume that in the case of linear polarization only an increase of orbital momentum is possible, then the K-photon matrix elements of absorption of circularly and linearly polarized radiation differ from each other only in the Clebsch-Gordon coefficients. The radial matrix elements and energy denominators are the same in these compound matrix elements. From [32], we can get the first order dipole matrix element, |hαLSJMJ |R|αL0 SJ 0 MJ0 i|2 = (2J 0 + 1) 2 J 1 −MJ q J 0 A(s) 3 σ 2 (0.66702) ∆E (A.66) MJ0 where the dipole matrix element is in units of Bohr radius (5.2917 × 1011 m), A(s) is the spontaneous coefficient in s−1 , σ is transition wavelength in wavenumber, ∆E is the energy gap between the two states non-dimensionalized by Rydberg constant (1Ry = 109737.315cm−1 ). For different polarization, M , M 0 and q have different values according to the table A.1. 120 Table A.1: Polarization of Electric Dipole Radiation ∆M q polarization M = M0 0 linear, parallel to z axis M = M 0 + 1 +1 circular, clockwise in (x,y) plane 0 M = M − 1 -1 circular, counterclockwise in (x,y) plane A.9 Keldysh theory of multiphoton ionization and tunneling ionization In 1964, Keldysh [43] published his famous paper about direct multiphoton ionization and tunneling ionization of one-electron atom. The general formula for the ionization probability can be written as µ I0 P = Aω h̄ω Ã ¶3/2 Ã γ (1 + γ 2 )1/2 !5/2 Ã Ie0 S γ, h̄ω 2Ie0 (1 + γ 2 )1/2 × exp − sinh−1 γ − γ h̄ω 1 + 2γ 2 ! ! (A.67) where A is numerical coefficient of order of unity, ω is the frequency of the electromagnetic wave, I0 is the ionization potential of the atom, Ie0 = I0 + e2 F 2 /4mω 2 is the effective ionization potential, F is electric field intensity of the electromagnetic wave, γ is defined as γ = ω/ωt = ω(2mI0 )1/2 /eF (A.68) The parameter γ is now called Keldysh parameter. The Keldysh parameter is the ratio of the characteristic time that the electron takes to pass through the barrier formed by the electric field and static atomic potential to the cycle time of the oscillating electric field. ³ ´ The function S γ, Ie0 /h̄ω varies slowly compared to the exponential function and is defined as 121 ( S(γ, x) = Σ∞ 0 exp −2[< x + 1 > −x + n] sinh " −1 γ γ−√ 1 + γ2 #1/2 2γ Φ0 √ (< x + 1 > −x + n) 1 + γ2 ) (A.69) and the symbol < x > denotes the integer part of the number x. In the limit of γ >> 1, the characteristic time to pass through the barrier is much greater than the cycle time. 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