ABSTRACT Title of Dissertation : INFRARED OPTICAL SENSOR FOR COMBUSTION DIAGNOSTICS USING WAVELENGTH MODULATION SPECTROSCOPY Mohammadreza Gharavi, Doctor of Philosophy, 2004 Dissertation Directed By: Assistant Professor Steven G. Buckley Department of Mechanical Engineering In this dissertation, an optical sensor for combustion diagnostics, basedon near- infrared distributed feedback (DFB) tunable diode lasers, is developed. The sensor is implemented to perform simultaneous species and temperature measurements in a combustion system. The use of optical sensing allows non-intrusive measurements, which are useful particularly in systems sensitive to perturbations caused by external probes. The tunable diode laser sensor is based on absorption spectroscopy, where absorption of laser light by a particular rotational-vibrational absorption feature of a molecule is related to the molecular concentration. In addition, based on absorption ratio of two particular absorption features of a molecule, temperature is determined. A TDL sensor was designed, built, and tested in various conditions in the flame and a static cell for CH4 and H2O concentration and for temperature measurement. For high sensitivity, Wavelength Modulation Spectroscopy (WMS) was used. In general, quantification of WMS, particularly in combustion systems, requires detailed spectroscopic information of all absorption features probed by the sensor. Typical absorption features are often overlapped, and therefore measurement of spectroscopic information for individual transitions may be very difficult, or sometimes impossible. The lack of detailed spectroscopic information for the near-infrared transitions of interest, combined with the complexity of WMS technique itself, are the main issue for quantification WMS in combustion systems. In this dissertation, these problems and their solutions are discussed. Following a detailed theoretical study of WMS, new approaches and techniques for concentration and temperature measurement in combustion systems are developed. INFRARED OPTICAL SENSOR FOR COMBUSTION DIAGNOSTICS USING WAVELENGTH MODULATION SPECTROSCOPY By Mohammadreza Gharavi Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park, in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2004 Advisory Committee: Assistant Professor Steven G. Buckley, Chairman/Advisor Professor James M. Wallace Professor Christopher C. Davis Associate Professor Gregory S. Jackson Assistant Professor Bao Yang ? Copyright by Mohammadreza Gharavi 2004 ii DEDICATION To my wife Simin iii ACKNOWLEDGEMENTS I would like to thank my advisor Professor Steven Buckley for his great support and guidance and his patience during this research work. I would also like to thank the members of my committee for agreeing to be on my committee and for their diligence in reading this thesis. I would like to acknowledge the loving support of my parents. Finally, I would like to thank my wife, Simin, for her understanding, love, and encouragement. iv TABLE OF CONTENTENTS List of Tables ???????????..???????????????... ix List of Figures ??..??????????????????.?????.? xi List of Symbols .?.?????????????????????...??.. xxii 1 Introduction ??????????????????????????.....1 1.1 Motivation for use of IR diode-laser diagnostics ??????????... 2 1.2 Background ?????????????????????????. 3 1.3 Outline of the dissertation ???????????????????... 4 2 Theory of Absorption of Electromagnetic Waves by Molecules ?????... 6 2.1 Postulates of quantum mechanics ????????????????... 6 2.2 Important Stationary Energy States in Molecular Spectroscopy ????? 9 2.2.1 Translational states ??...?????????????.???. 10 2.2.2 Electronic states ????..????????????.???.. 11 2.2.2.1 Electronic states in atoms ??????????????. 11 2.2.2.2 Electronic states in Molecules ????????????.. 19 2.2.3 Rotational states ????????????????????.. 21 2.2.3.1 Rotational states of a diatomic molecule ????????.. 21 2.2.3.2 Rotational states of a polyatomic molecule ???????.. 24 2.2.4 Vibrational states ???????????????????? 26 2.2.4.1 Vibrational states for a diatomic molecule ???????... 26 v 2.2.4.2 Vibrational states for a polyatomic molecule ??????... 30 2.2.5 Total molecular energy state ???????????????.. 33 2.3 Interaction of weak electromagnetic radiation by a molecule ?????.. 33 2.3.1 Attenuation of radiation beam due to absorption and stimulated emission ????????????????????...??... 38 2.3.2 Pressure broadening ??????????????????? 40 2.3.3 Temperature broadening ?????????????????. 42 2.3.4 Voigt line shape function ????????????????? 43 2.4 The Selection rules ?????????????????????... 44 2.4.1 Selection rules for rotational and vibrational transitions ????? 44 2.4.2 Selection rule for rotational- vibrational absorption ??????... 45 2.5 Vibrational-Rotational Spectrum ????????????????.. 47 2.6 Beer-Lambert Law ?????????????????????? 50 3 Techniques of Tunable Diode Laser Absorption Spectroscopy ?????... 52 3.1 Direct absorption ??????????????????????.. 52 3.2 Wavelength Modulation ???????????????????... 56 3.3 Frequency Modulation ????????????????????.. 58 3.4 Two-tone Frequency Modulation Spectroscopy ??????????... 66 4 Theory of Wavelength-modulation Spectroscopy ?????...??..???71 4.1 Theory of wavelength-modulation spectroscopy ??????????.. 71 4.2 Properties of the in-phase signal at the nth harmonic component, )( , pnfI .. 81 vi 4.2.1 Properties of )( , pnfI in the absence of intensity modulation ??... 81 4.2.2 Properties of )( , pnfI in the presence of intensity modulation ?.?. 88 4.3 Methods for concentration and temperature measurement ??????... 94 4.3.1 Method for concentration measurement ???????????. 94 4.3.2 Method for temperature measurement ???????????? 95 4.4 Issues and solutions for concentration and temperature measurement in a combustion system using WMS ????????????????... 98 4.5 Information required for quantification of WMS for combustion diagnostics ????????????????????????... 100 5 Measurement of spectroscopic parameters of H2O and CH4 ?????... 102 5.1 Theory of measuring spectroscopic parameters ??????????. 103 5.2 Measurement of CH4 spectroscopic parameters ??????????. 105 5.2.1 Experimental details ??????????????????. 106 5.2.2 Line strength at room temperature ????????????... 109 5.2.3 Line strength at elevated temperature ???????????... 116 5.2.4 Pressure broadening calculation ?????????????... 118 5.3 Measurement of H2O spectroscopic parameters ??????????. 125 5.3.1 Experimental details ??????????????????. 126 5.3.2 Line strength measurement and temperature sensitivity of the transitions ??????????????????????. 128 5.3.3 Pressure broadening calculations ?????????????. 135 vii 5.4 Approximate pressure broadening calculation for some quenching gases 142 6 Temperature and concentration measurements using the WMS sensor ? 145 6.1 A Model for simultaneous temperature, CH4 concentration, and H2O concentration measurement in combustion systems ????????... 145 6.1.1 Calculation of TI I OH f vs.)(? 22  and TTR III vs.)(, correlations at optimum conditions ??????????????????.. 151 6.1.2 Calculation of TI CHf vs.)(? 042  correlation at optimum conditions .... 159 6.2 Comparison of the modeling and experiments ??????????... 160 6.3 Sensor architecture ?????????????????????.. 167 6.4 In-flame measurements ???????????????????... 169 7 Conclusion ??????????????????????????.. 177 Appendix A1: A program for simulation of the second harmonic spectra of a Voigt profile when Doppler and Collisional half-width are known 180 Appendix A2: A program for simulation of the second harmonic spectra of the selected H2O transitions ????????????????... 182 Appendix A3: A program for simulation of the second harmonic spectra of the R(4) manifold of the 23 band of CH4 ???????????.. 188 Appendix B1: Some spectroscopic parameters of R(3) and R(4) manifolds of the 23 band of CH4 ??????????????????.. 192 viii Appendix B2: Some spectroscopic parameters of the selected H2O transitions ?.. 193 Appendix C: Design and building the Ring Interferometer ???????.?? 194 Appendix D: Glass Cell and its accessories ???????????????. 196 Appendix E: Absolute error analysis for linestrength ST and pressure broadening coefficient  ?????????????????????? 197 Appendix F: Measurement of amplitude of modulation and intensity-frequency Parameters ??????????????????????... 201 Appendix G: Temperature correction of the thermocouple measurements for radiation loss ?????????????????????. 205 References ???????????????????????????.. 207 ix LIST OF TABLES Table 5.1: Individual line strengths of R(3) at T=296 K ??????????. 107 Table 5.2: Individual line strengths of R(4) at T=296 K ??????????. 108 Table 5.3: Variation of Stot in R(3) with temperature ???????????.. 117 Table 5.4: Variation of Stot in R(4) with temperature ???????????.. 117 Table 5.5: Variation of collisional half-width with temperature in R(3) manifold 122 Table 5.6: Variation of collisional half-width with temperature in R(4) manifold 122 Table 5.7: Temperature dependence of collisional broadening line width parameter n for different colliding gases. Uncertainty represents the standard deviation in the data ???????????????? 124 Table 5.8. Frequency spacing and line strengths at T=977 K (comparison between results of HITRAN 96 and measurement) ???????? 130 Table 5.9: Variation of collisional half-width with temperature in transition 1 and 2 ?????????????????????????? 140 Table 5.10: Variation of collisional half-width with temperature in transition 3 ? 140 Table 5.11: Temperature dependence of collisional broadening line width parameter n for different quenching gases ??????????... 142 Table B1.1: Vibrational quantum numbers of the 23 band of R(3) manifold of CH4 absorption transitions (Extracted from HITRAN database) ? 191 Table B1.2: Vibrational quantum number of the 23 band of R(4) manifold of CH4 absorption transitions (Extracted from HITRAN database) ? 191 x Table B2.1: Global and local quanta index of the selected H2O absorption transitions (Extracted from HITRAN database) ????????.. 192 xi LIST OF FIGURES Figure 2.1: A particle confined in a box ????????????????? 10 Figure 2.2: The energy levels for a particle in a cube ???????????? 11 Figure 2.3: Quantization of angular momentum L (left) and spin angular momentum S (right) along the direction of an electric field ???....... 16 Figure 2.4: The first few electronic states of atomic hydrogen ????????. 18 Figure 2.5: Definition of the distances involved in the Hamiltonian operator for +2H ?????????????????????????. 20 Figure 2.6a: Variation of energy, E, versus inter-nuclear distance, R, at ground electronic states for H2+ ???????????.??????? 21 Figure 2.6a: A potential energy diagram of O2 ??????????????.. 21 Figure 2.7: Two mass m1 and m2 shown rotating about their center of mass ??? 22 Figure 2.8: The rotation of a single particle about a fixed point ???????? 23 Figure 2.9a: Two masses connected to a spring, which is a model used to describe the vibrational motion of a diatomic molecule ?????????. 27 Figure 2.9b: A mass connected to a wall by a spring ???????????? 27 Figure 2.10: The energy states of a harmonic oscillator (dashed line) and anharmonic oscillator superimposed on a harmonic-oscillator potential and a more realistic internuclear potential ???????. 29 Figure 2.11: Normal (vibrational) modes of H2O ?????????????. 32 Figure 2.12: Two-level system ????????????????????.. 34 xii Figure 2.13: The function 212122 )/(]2/)[(sin)2()(  hhh == EEtEEFF which represents the probability of making a 21 transition in the time interval 0 to t, plotted against frequency  ???????. 36 Figure 2.14: A system with dimension 1 m ? 1 m ? l m that contains N0 and N1 molecules per cubic meter with energies E0 and E1, respectively ??. 38 Figure 2.15: The phase of an oscillating dipole moment randomly interrupted by Collision ???????????????????????? 41 Figure 2.16: Interaction of a lane electromagnetic wave with a moving atom ??.. 42 Figure 2.17: Normal modes of CO2 ??????????????????... 45 Figure 2.18: The ro-vibrational spectrum of the 10  vibrational transition of HBr(g) ???????????????????????? 46 Figure 2.19: One of the Parallel band (left trace) and perpendicular band (right trace) in CO2 ????????????????????. 46 Figure 2.20: A typical ro-vibrational transition ??????????????. 49 Figure 2.21: The left graph shows the variation of population density in lower state energy with 0> J in a diatomic molecule based on rigid rotor model. Right graph shows the spectrum of a diatomic molecule based on rigid rotor model (the transitions for all possible J are considered) ??????????????????????... 49 Figure 2.22: A typical spectrum of a real diatomic molecule ????????? 50 Figure 3.1: Typical experimental setup used direct absorption ????????. 53 Figure 3.2: Typical data processing used for quantification of direct absorption technique ????????????????????????. 55 xiii Figure 3.3. A typical experimental setup for wavelength-modulation spectroscopy ???????????????????????. 56 Figure 3.4: Typical power spectrum of the modulated laser in FMS ?????? 62 Figure 3.5: The schematic of an experimental setup for FMS ????????... 63 Figure 3.6: Output signal in frequency-modulation spectroscopy of Fabry-Perot resonance ????????????????????????. 64 Figure 3.7: Spectral distribution of the laser field in two-tone frequency modulated diode laser at 1 and 2 with  =1.0 and M =0 ??????????. 67 Figure 3.8: Typical experimental arrangement for two-tone frequency modulation 70 Figure 4.1: Comparison power spectrum between FMS and WMS ??????. 72 Figure 4.2: Amplitude of absorption and dispersion of the first harmonic signal as a function of normalized modulation frequency line/  = fxm when FM index 1/ = = fm ???????????????. 72 Figure 4.3: Intensity variation of a typical diode laser as a function of optical frequency ????????????????????????.. 76 Figure 4.4: The functional block diagram of the Stanford Research SR830 lock-in amplifier which was used in this research ???????????... 78 Figure 4.5: The effect of variation of modulation index m, on the spectra of )( ,2 pfI for a case of a Voigt line-shape function with 05.= D cm-1, 05.= L cm-1, 0== , and no intensity modulation ( 021 === fFF sss ) ????????????????????... 82 xiv Figure 4.6: In-phase signal, )( , pnfI for the first four harmonic components for the case of a Voigt line-shape function with 01.)( 0 = , 05.= D cm-1, 05.= L cm-1, m=1, 0== , 1= K , and no intensity modulation ( 021 === fFF sss ) ???????????????... 83 Figure 4.7: The effect of detection phase on pnfI , ?????????????... 85 Figure 4.8: The effect of detection phase on in-phase signal )( , pnfI for the case of a Voigt line-shape function with 01.)( 0 = , 05.= D cm-1, 05.= L cm-1, m=1, 0= , 1= K , and no intensity modulation ( 021 === fFF sss ) ???????????????????? 86 Figure 4.9: The effect of modulation index, m, on the maximum of the in-phase signal )( ,2 pfI for the case of a Voigt line-shape function with 01.)( 0 = , 100= K , and no intensity modulation ( 021 === fFF sss ), with the optimum detection phase ??????. 87 Figure 4.10: The effect of modulation index, m, on the maximum of the in-phase signal )( ,2 pfI assuming a Voigt line-shape function with 01.)( 0 = , 100= K , and no intensity modulation ( 021 === fFF sss ), when the detection phase is not optimum ??... 88 Figure 4.11: The effect of the two different mechanisms of intensity modulation on the spectra of )( ,2 pfI ?????????????????... 90 xv Figure 4.12: In-phase signal, )( , pnfI for the first four harmonic components calculated with a Voigt line-shape function with 01.)( 0 = , 05.= D cm-1 and 05.= L cm-1, m=1, 0== , 1= K , and intensity modulation parameters of ( ][1/cm2,0],[1/cm1 -12-11 === fFF sss ) ??????????. 91 Figure 4.13: The effect of detection phase on the )( ,2 pfI spectra. The calculated results in this figure assume a Voigt line-shape function with 01.)( 0 = , 1= K , m=1, deg20= , deg50min == , 02 =Fs ... 92 Figure 4.14: The effect of detection phase on in-phase signal )( , pnfI for a signal with a Voigt line-shape function with 01.)( 0 = , 05.= D cm-1 and 05.= L cm-1, m=1, 0= , 1= K , and linear intensity modulation ( 2,0,1 21 === fFF sss ) ???????????. 93 Figure 4.15: Second harmonic signal variation as a function of optical frequency for three some arbitrary overlapped absorption transitions ????... 98 Figure 5.1: Line positions and strengths of the 23 band of CH4 ???????. 105 Figure 5.2: Schematic diagram of the experiment ????????????... 107 Figure 5.3: From top to bottom: Ring interferometer signal, absorption signal for R(3), and absorption signal for R(4) ?????????????.. 112 Figure 5.4: Measured room temperature (296 K) line strengths for each of the R(3) lines of CH4, and total line strength at 296 K ?..??????.. 113 xvi Figure 5.5: Measured room temperature (296 K) line strengths for each of the R(4) lines of CH4, and total line strength at 296 K ???????? 114 Figure 5.6: Comparison between the experimental and fitted model results for R(3) transitions (upper trace) and R(4) transitions (lower trace) at room temperature (T=296 K) ????????????????. 115 Figure 5.7: Comparison between the experimental and fitted model results for R(3) transitions (upper trace) and R(4) transitions (lower trace) at high temperature (T= 908 K) ????????????????. 119 Figure 5.8: Collisional broadening half-width of CH4 by N2 for R(3) transitions as a function of pressure. Upper trace at room temperature (296 K), and lower trace at high temperature (860 K) ??????????. 120 Figure 5.9: Collisional broadening half-width of CH4 by CO2 for R(3) transitions as a function of pressure. Upper trace at room temperature (296 K), and lower trace at high temperature (908 K) ??????????. 120 Figure 5.10: Collisional broadening half-width of CH4 by N2 for R(4) transitions as a function of pressure. Upper trace at room temperature (296 K), and lower trace at high temperature (952 K) ?????????... 121 Figure 5.11: Collisional broadening half-width of CH4 by CO2 for R(4) transitions as a function of pressure. Upper trace at room temperature (296 K), and lower trace at high temperature (952 K) ?????????... 121 Figure 5.12: The temperature variation in broadening coefficient half-width of CH4 by N2 (top trace) and CO2 (bottom race) for R(3) transitions ?.. 123 xvii Figure 5.13: The temperature variation in broadening coefficient half-width of CH4 by N2 (top trace) and CO2 (bottom race) for R(4) transitions ?.. 124 Figure 5.14: Schematic diagram of the experimental set up used to measure line strengths of H2O ????????????????????... 128 Figure 5.15: Transmitted signal (I) and RI signal at different temperature and pressure conditions. The top figure was at T= 468 K, PH2O=24.6 torr and the bottom was at T= 977 K, PH2O=27.6 torr ???????... 129 Figure 5.16: Measured and fitted results for H2O absorbance corresponding to the data shown in Figure 2. The top figure was at T= 468 K, PH2O=24.6 torr and the bottom was at T= 977 K, PH2O=27.6 torr ?? 131 Figure 5.17: Comparison between the measured line strengths and HITRAN results. Top trace shows (S1+S2) and the bottom trace illustrates S3 ... 132 Figure 5.18: Comparison between the measured absorption ratio (S1+S2)/S3, and HITRAN results ??????????????????? 135 Figure 5.19: Measured and fitted results for H2O absorbance at different pressure for calculating pressure broadening coefficient due to N2 at T=483 K ???????????????????????? 136 Figure 5.20: Measured and fitted results for H2O absorbance at different pressure for calculating pressure broadening coefficient due to N2 at T=999 K ???????????????????????? 137 Figure 5.21: Collisional broadening half-width of H2O transitions by N2 as a function of pressure. Upper trace at room temperature (483 K), and lower trace at high temperature (999 K) ???????????.. 138 xviii Figure 5.22: Collisional broadening half-width of H2O transitions by CO2 as a function of pressure. Upper trace at room temperature (483 K), and lower trace at high temperature (860 K) ???????????.. 139 Figure 5.23: The temperature variation in broadening coefficient half-width of H2O by N2 for transitions 1 and 2 (top trace) and transition 3 (bottom trace) ????????????????????....... 141 Figure 5.24: The temperature variation in broadening coefficient half-width of H2O by CO2 for transitions 1 and 2 (top trace) and transition 3 (bottom trace) ?????????????????????... 141 Figure 6.1: Flow diagram of the main steps performed for quantification of the sensor for temperature and species measurement in combustion system with a multiplexed WMS sensor ???????????? 146 Figure 6.2: Typical second harmonic signal of H2O related to the selected H2O transitions ?????????????????????... 149 Figure 6.3: Typical second harmonic signal of CH4 related to the selected CH4 transitions ???????????????????????.. 149 Figure 6.4: Variation in TR III vs., curve at different modulation depths and optimum detection phase o155= for CH4/air flame with =1 ??? 152 Figure 6.5: Calculated spectra of )(? 22 IOHfI   as a function of modulation depth, ,m for CH4/air flame with =1 at condition: T=500 K, o155= and optical path of L=10 cm ????????????? 153 xix Figure 6.6: Calculated spectra of )(? 22 IOHfI   as a function of modulation depth, ,m for CH4/air flame with =1 at condition: T=2000 K, o155= and optical path of L=10 cm ????????????? 154 Figure 6.7: The modeled effect of the 4th transition in )(? 22 IOHfI   on the third transition for a CH4/air flame with =1 at -1cm0.09= m o155( = , and optical path L=10 cm) ??????????...?? 157 Figure 6.8: The modeled effect of the 4th transition in )(? 22 IOHfI   on the third transition for a CH4/air flame with =1 at -1cm0.16= m o155( = , and optical path L=10 cm) ????????????... 157 Figure 6.9: Variation of )(? 22 IOHfI  with temperature at two different modulation depths for a CH4/air flame with =1, at detection phase of o155= and optical path of L=10 cm ????????????????.. 158 Figure 6.10: Variation of )(? 042 CHfI with temperature at three different modulation depths for a CH4/air flame with =1, at detection phase of o140= and an optical path of L=10 cm ??????????????... 159 Figure 6.11: Calculated spectra of )(? 042  CHfI for R(4) transitions at T=500 and 2000 K for a CH4/air flame with =1, at condition: o140= and optical path of L=10 cm ???????????????? 160 Figure 6.12: Comparison between measured and calculated 2f spectra of the selected H2O absorption transitions for a mixture of N2-H2O in the gas cell ??????????????????????? 161 xx Figure 6.13: Comparison between measured and calculated 2f spectra of the selected H2O absorption transitions for a mixture of CO2-H2O in the gas cell ??????????????????????... 162 Figure 6.14: Comparison between measured and calculated 2f spectra of the selected H2O absorption transitions for pure H2O vapor in the gas cell ????????????????????????. 163 Figure 5.15: Comparison between a set of (T , RI,II) measured by experiment in the gas cell and corresponding values calculated by the modeling results ????????????????????... 164 Figure 6.16: Comparison between measured and calculated 2f spectra of the selected CH4 absorption transitions, R(4), for a known CH4-N2 mixture in the gas cell ??????????????????.. 166 Figure 6:17: Schematic of the sensor architecture ????????????... 168 Figure: 6.18: Simplified Figure of the setup of the in-flame measurement ??.? 170 Figure 6.19: Pre-calculated correlations, TI I OH f vs.)(? 22  , TI CHf vs.)(? 042  , and TR III vs., , for the CH4/air flame with =1.70. The 2f signals are normalized by the 1f signal at the central absorption frequencies ?... 172 Figure 6.20: Calculation of K by comparing the measured second harmonic signal at know temperature and concentration in the gas cell and calculated second harmonic signal with unit gain ???????... 173 Figure 6.21: Temperature variation at different height above the burner surface using the optical sensor ?????????????????... 174 xxi Figure 6.21: Variation of the H2O and CH4 concentrations at different heights above the burner surface using the optical sensor ???????.. 175 Figure C.1: Schematic of the ring interferometer ????????????? 193 Figure D.1: From top to bottom: Glass Cell and heater assembly, heater dimensions, and Glass Cell and its accessories dimensions ????. 195 Figure F.1: Ring interferometer signal of the H2O laser when it is modulated around II=6771.47 cm-1 at f=10 kHz ????????????? 201 Figure F.2: Ring interferometer signal of the H2O laser when it is modulated around II=6771.47 cm-1 at F=10 Hz ?????????..??????... 202 xxii LIST OF SYMBOLS )(rr Time-independent Wave function )(* rr Complex conjugate of )(rr H? Hamiltonian operator h Plank?s constant V(x,y,z) Potential energy at location (x,y,z) m Mass of a particle A? Operator n Eigenfunctions of A? an Eigenvalues of A? En Eigenvalues of Hamiltonian operator a Average value of the observable corresponding to A? ),( trr Time-dependent wave function t Time nx, ny, nz Quantum numbers of a particle confined in a box zyx nnn State functions of a particle confined in a box zyx nnn E Ttranslational energy states of a particle confined in a box Z Number of protons in a nucleolus N Number of electrons in a atom/molecule, number of atoms in a molecule me Mass of electron xxiii e Electrical charge of electron jr r Position of electron j in a molecule ),,,( 21 zj rrrG rKrr Potential energy of electron j due to presence of electron 1 at position 1r r , electron 2 at position 2r r ,? and electron z at position zr r n Priciple quantum number l Angular quantum number ml Magnetic quantum number ms Spin quantum number L Magnitude of the orbital angular momentum of electron, length, optical path S Magnitude of the spin angular momentum of electron E Energy of electron  Spin variable, absorption cross section () Spin eigenfunctions corresponding to ms= 1/2 () Spin eigenfunctions corresponding to ms= -1/2 ),( rr Spin orbital wavefunctions  Angular velocity I Moment of inertia, radiation intensity ? Reduced mass J Rotational quantum number g Degeneracy in energy level CBA ~,~,~ Rotational constants D~ Centrifugal distortion constant xxiv k Spring constant  Fundamental vibrational frequency, optical frequency  Vibrational quantum number ex ~ Anharmonicity constant ?r Net dipole moment of a moleucule 01P Probability of absorption which causes transition from state 0 to state 1 10? Transition dipole moment between states 0 and 1 )( Radiation density 01B Einstein coefficient for absorption 01B Einstein coefficient of stimulated emission c Speed of light in vaccum Nj Population density at state j F Neutron flux, ramp (saw tooth) frequency )( 0 g Line-shape function M r Dipole moment of a system T2 Average time between collisions L Collisional (Lorentzian) half-width v r Velocity k r Wave vector )( 0 Lg Lorentzian line-shape function (related to collisional effect) )( 0 Dg Gaussian line-shape function (related to Doppler effect) xxv T Temperature M Molecular weight, AM index D Doppler half-width )( 0 Vg Voigt line-shape function Q(T) Partition function at temperature T kB Boltzmann constant k() Absorption coefficient Pabs Partial pressure of absorbing species )( 0  Line-shape function S(T) Line strength NL Loschmidt?s number E Lower-state energy level f Oscillator strength, modulation frequency Inf nth harmonic signal H2( ? 0) Second harmonic component of the modulated normalized line- shape (with a maximum of 1.0) 0 Maximum absorbance (absorbance at the central absorption frequency) I2f,max Peak of the second harmonic signal m Modulation angular frequency fm Modulation frequency 0 Laser carrier angular frequency  FM index xxvi  Phase difference between intensity/amplitude and frequency modulation E(t) Amplitude of electric field Tn Complex transmission function n Refractive index at 0 + nm n Absorbance at 0 + nm n Amplitude attenuation at 0 + nm n Optical phase shift at 0 + nm  Intermediate frequency i(t) Diode laser injection current  Laser center frequency 1Fs , 2Fs Intensity-frequency parameters at saw tooth frequency F fs Intensity-frequency parameter at modulation frequency f i Amplitude of current modulation I Amplitude of intensity modulation I Average intensity m Amplitude of frequency of modulation K Opto-electrical gain )(DI Photodetector signal Detection phase )( , pnfI nth harmonic of the in?phase signal i Pressure broadening coefficient xxvii yi Mole fraction of species i fI 2? Normalized second harmonic signal IIIR , Ratio of two absorption peaks of the H2O selected transition xxviii 1 1 Introduction In this dissertation, an optical sensor based on near-infrared distributed feedback (DFB) tunable diode lasers for combustion diagnostics is developed. The sensor is implemented to perform simultaneous species and also temperature measurements in a combustion system. The use of optical sensing allows for a non- intrusive measurement, which is very important for systems sensitive to perturbations caused by external probes. The sensor is based on absorption spectroscopy, where absorption of laser light by a particular rotational-vibrational absorption feature of a molecule is related to the molecular concentration. In addition, based on absorption ratio of two particular absorption features of a molecule, temperature is calculated. In this dissertation, a sensor based on absorption spectroscopy was designed, built, and tested in various conditions in the flame and a static cell for CH4 and H2O concentration and for temperature measurement. For high sensitivity, Wavelength Modulation Spectroscopy (WMS) was used. In general, quantification of WMS technique, particularly in combustion systems, requires detailed spectroscopic information of all absorption features probed by the sensor. Usually many absorption features are overlapped, and therefore measuring spectroscopic information of individual transitions may be very difficult, or sometimes impossible. The lack of detailed spectroscopic information for the near-infrared transitions of interest, combined with the complexity of WMS technique itself are the main issue for quantification WMS in combustion systems. In this dissertation, the problems and 2 also their solutions will be discussed. Following a detailed theoretical study of WMS technique explained in this dissertation, new approaches and techniques for concentration and temperature measurement in combustion systems developed herein are presented. For temperature and H2O concentration measurement, a set of H2O transitions with significant variation in line strength ratio with temperature, accessible by a single tunable diode laser within the telecommunication S band operating around 1477 nm, has been identified. The main spectroscopic properties of H2O absorption lines of the selected transitions, including absorption line strengths and pressure broadening parameters, with respect to the most important combustion species such as CO2, O2, N2, CH4, and H2O, were measured. For CH4 measurement a distributed feedback laser operating around 1650 nm suitable for measuring R(3) and R(4) transitions of 23 rotational-vibrational band of CH4 was used. For these transitions also all important spectroscopic properties required for WMS quantification, including line strengths and broadening coefficients at different temperatures were measured. The measured properties then used in a high-temperature model for flame measurement. 1.1 Motivation for use of IR diode-laser diagnostics The need for real-time and in-situ measurements for process control in combustion systems is primary motivated for this work. Also, for emission control and theoretical study of combustion physics the need for fast and sensitive methods of combustion diagnostics is well recognized. Non-intrusive and non-perturbing 3 methodologies may enhance both the speed and accuracy of a diagnostic method. In addition, since combustion processes generally deal with harsh environments, the capability of remote control is another parameter which important for sensor selection. And finally, cost may be a significant factor in the choice of a sensor. All the aforementioned requirements can be realized in IR diode-laser sensors. DFB lasers and related components, which are used in telecommunication systems, are among the best options for combustion diagnostics sensors. High speed current modulation, for absorption measurements on many species using strong ro-vibrational transitions in the infrared, yields great potential for fast and sensitive measurement. Also, the availability of fiber pigtailed lasers and related optical fiber components allows us to use them for remote sensing and control. In addition, because of massive investment by the telecommunication industry (Steele 1997) DFB lasers, IR detectors, and all related products, for optical design and signal processing, are relatively accessible and inexpensive. As will be discussed later, there are different techniques of modulations and analysis for Tunable Diode Laser Absorption Spectroscopy (TDLAS), which can be used for combustion diagnostics. Among these techniques, wavelength modulation is considered to be the best method in which a simple hardware design and the capability of fast measurements are combined with highly sensitive measurements. 1.2 Background The use of diode laser for species measurement began shortly after the demonstration of direct current injection semiconductor lasers in the late 1960s. The 4 first atmospheric measurement was realized in 1975 by employing a long open path (Ku and Hinkley 1975; Henkley 1976) and this technique is still used today. In 1977, diode lasers found application to in-situ combustion diagnostics (Hanson, Kuntx et al. 1977). Since that time much research has been conducted to apply diode laser spectroscopy for combustion measurements. Today, diode lasers are common measurement tools, used for sensitive laboratory-based spectroscopic measurements in the post flame region and for remote sensing and atmospheric measurements (CHou, Baer et al. 1997; Milhalcea, Baer et al. 1997; Nelson 1998). Advances in detection methods (e.g. wavelength modulation spectroscopy, one- and two-tone frequency modulation spectroscopy) over the past decade have greatly increased the sensitivity of diode laser absorption spectroscopy measurements (Reid and Labrie 1981; Lenth, Ortiz et al. 1982; Pokrowsky, Zapka et al. 1983; Cooper and Gallagher 1985; Janik, Carlise et al. 1985; Whittaker, Shum et al. 1988; Bomse, Silver et al. 1992; Feher and Martin 1995; Avetisov and Kauranen 1996). 1.3 Outline of the dissertation In the second chapter, the theory of absorption of electromagnetic waves by molecules is discussed. In this chapter at the beginning, different electronic states of a molecule based on quantum mechanics are explained. Then absorption of electromagnetic radiation by molecules, which causes molecules to go to different energy states, based on simple semi-classical theory is explained. Following that, Beers?s law, which is the main governing equation in absorption spectroscopy, will be derived. In this chapter, the physical meaning of spectroscopic properties of a 5 molecule including, rotational-vibrational energy states and spectra, absorption line strength, line-shape function, etc. has been outlined. Third chapter reviews different techniques of absorption spectroscopy using tunable diode laser; measurement methods including direct absorption, wavelength modulation, and also frequency modulation are briefly explained. In the fourth chapter, detailed theory of a model developed for this work, based on Wavelength Modulation Spectroscopy (WMS) for temperature and concentration measurement suitable for combustion diagnostics is explained. The measurement of spectroscopic parameters required for quantification of the sensor data, for combustion measurements, is discussed in the fifth chapter. In the sixth chapter, the sensor architecture, experimental results and also the data processing models developed in this research are presented and discussed. And finally the seventh chapter summarizes all important results and new findings in this dissertation. 6 2 Theory of Absorption of Electromagnetic Waves by Molecules In this chapter the physical phenomena of absorption of electromagnetic waves by gas-phase molecules (particularly in the gas) will be discussed. The materials covered in this chapter, are mainly extracted from (McQuarrie and Simon 1997) and (Bernath 1995) and are presented such that the physics behind the absorption phenomena is better understood. At the beginning of this chapter, I will review some aspects of quantum-mechanics that form the basis for absorption theory. For example, different quantized modes of energy storage leading to observed spectroscopic transitions of molecules, including translational, electronic, vibrational, and rotational transitions will be briefly explained. Following this, the process of absorption of electromagnetic waves by molecules which leads to the so-called ?selection rules? governing transitions will be reviewed. Finally, based on these development, the Beer-Lambert Law, which is the key equation for absorption spectroscopy, will be discussed. The material covered in this chapter will enable a basic understanding of some important spectroscopic properties, such as the transition line strength, line shape, absorption spectra, partition function, etc. 2.1 Postulates of quantum mechanics (McQuarrie and Simon 1997) Based on the first postulate of quantum mechanics, the quantum-mechanical state of a closed system, meaning an isolated system or a system which does not interact with an external field, is completely specified by a wave function )(rr , 7 which depends on the coordinate of the particle in the system. As in classical mechanics, the state of a system at particular time is determined by the position of a particle, and its velocity or momentum at that time. Based on the first postulate, the wave function has the important property that dxdydzrr )()( * rr is the probability that the particle lies in the volume element dxdydz at position rr , where )(* rr is the complex conjugate of )(rr . According to the second postulate, to every observable (a measurable dynamic variable such as energy, momentum, and etc.) in classical mechanics there corresponds a linear operator in quantum mechanics. For instance, the quantum- mechanical operator for the energy, which is called Hamiltonian operator H? , is defined as: ),,( 2 ),,( 2 ? 2 2 2 2 2 2 2 22 zyxV m zyxV zyxm H ++      +  +  hh (2.1) where m is the mass of the particle and 2/h=h , in which h is plank?s constant, and V(x,y,z) is the potential energy of the particle in the system. According to the third postulate, in any measurement of the observable associated with the operator A? , the only values that will ever be observed are the eigenvalues an, which satisfy the following equation, nnn aA =? (2.2) For example, the quantum-mechanical operator corresponding to the energy is the Hamiltonian operator H? , which is defined in (2.1). Therefore, based on the third 8 postulate, the only measurable energies of a particle with mass m and potential energy of V(x,y,z) will be the eigenvalues of the following equation: EzyxV m =+ ),,( 2 2 2h (2.3) Equation (2.3) represents the time-independent Schr?dinger equation. The eigenvalues of this equation, En, will be the only possible energies that we can experimentally measure. One of the properties of Hamiltonian operator is that if it can be written as a sum of independent terms, the total wave function will be a product of individual wave functions and the energy will be a sum of independent energies. Also, based on the fourth postulate of quantum mechanics, if a system is in a state described by a wave function , then the average value of the observable corresponding to A? is given by dxdydzAa spaceall = ?* (2.4) The fifth postulate of quantum mechanics specifies the variation of wave function of a system with time. It states, the wave function, or state function, of a system evolves in time according to time-dependent Schr?dinger equation t tritrH   = ),(),(? r hr (2.5) if the operator H? dos not contain time explicitly, we can write the wave function ),( trr as )()(),( tfrtr rr = (2.6) in which )(rr is the spatial wave function obtained from (2.3). Doing the separation of variables as in (2.6) we can directly calculate f(t) and therefore relate ),( trr and )(rr by 9 hrr /)(),( iEtertr = (2.7) It should be stressed here Equation (2.7) is only valid for closed, stationary system, where the operator H? dose not depend on time. The time-independent Shr?dinger Equation (2.3) can be used for calculating the stationary states in a closed system such as atoms or molecules; however, Equation (2.5) is applied for a system undergoing external interaction. For example, the interaction of an external electromagnetic field with molecules could be studied using the time-independent Schr?dinger equation. 2.2 Important Stationary Energy States in Molecular Spectroscopy (McQuarrie and Simon 1997) As was mentioned above, the stationary states of a closed system can be obtained from the time-independent Schr?dinger Equation (2.3). Based on the geometry, potential energy, and the boundary conditions of a given system, Equation (2.3) can (in principle) be solved. The solution of Equation (2.3) will be a set of eigenfunctions n, corresponding to eigenvalues En. Based on the third postulate of quantum-mechanics, the only observable energy for the system will be En. Using the forth postulate, one can also show that the variance of the energy, En, will be zero, indicating that the only measurable value is En. In the following, the energy levels En of some important systems in molecular spectroscopy are explained. 10 2.2.1 Translational states If Equation (2.3) is applied to a molecule (or any other particle) with mass m confined in a rigid cubic box (container) with the dimensions of a (Figure 2.1), translational energy states zyx nnn E , and state functions zyx nnn , which are the eigenvalues and eigenfunctions of (2.3) respectively, will be calculated by a zn a yn a xn a nnnnnn ma hE zyx nnn zyxzyxnnn zyx zyx  sinsinsin8 ,2,1,,)( 8 2/1 3 222 2 2    = =++= K (2.8) a a a Figure 2.1: A particle confined in a box. The combinations of nx, ny, and nz corresponding to allowed translational energy states of a molecule in the box, based on Equation (2.8) are shown in Figure 2.2. For each set of nx, ny, and nz, there is a corresponding wave function zyx nnn , which specifies the translational state of the molecule in the box. Different states (different state functions) which have the same energy are called degenerate states. 11 Figure 2.2: The energy levels for a particle in a cube. The degeneracy of each level is also indicated from (McQuarrie and Simon 1997). 2.2.2 Electronic states The electronic states of a molecule are based on the electronic states of the constituent atoms, hence the electronic states of atoms are discussed as a prelude to discussion of molecular electronic states. 2.2.2.1 Electronic states in atoms Assume an atom with Z protons in a fixed nucleus (zero kinetic energy) contains Z electrons which are around the nucleus. Using the time-independent Schr?dinger equation, the energy of the atom can be obtained from 12 [ ] ),,,( ),,,(),,,(),,,(),,,( ),,,( 444 ),,,( 222 21 2121212211 21 0 2 20 2 10 2 21 2 2 2 2 2 2 1 2 z zzzzz z z zz eee rrrE rrrrrrGrrrGrrrG rrr r Ze r Ze r Ze rrr mmm rKrr rKrrrKrrLrKrrrKrr rKrrL rKrrhLhh """ = +++ +     +     (2.9) In Equation (2.9), me is the mass of electron and 2j is the Laplacian operator with respect to the position of the electron j coordinates. The term Ze2/4 0rj represents potential energy of electron j with respect to the nucleus, and ),,,( 21 zj rrrG rKrr is the potential energy of electron j due to presence of electron 1 at position 1r r , electron 2 at position 2r r ,? and electron z at position zr r (inter-electronic repulsion). By solving equation (9) and finding the eigenvalues, one finds the all possible electronic energy states of the atom. Except for hydrogen-like atoms (atoms with Z protons in nucleus and one electron in orbital), this equation can not be solved analytically. However, if the effect of the interelectronic repulsion ),,,( 21 zj rrrG rKrr , is ignored, Equation (2.9) can be simplified as )()()(),,,( 2121 zz rrrrrr rKrrrKrr = (2.10) where, )( jr r is wave function of electron j in the hydrogen-like atom. In fact, )( jrr is the eigenfunction of )()( 4 )( 2 0 2 2 2 jjj j jj e rEr r Ze r m rrrh " =+ (2.11) 13 Solving Equation (2.11) yields the energy levels and also wavefunctions of all possible states that one electron in a hydrogen-like atom can have. In fact, there is an analytical solution for Equation (2.11). The eigenfunctions of this equation, which specifies the state functions (wave functions) of the electron in hydrogen-like atom, depend upon three different quantum numbers (similar to those quantum numbers for translational states of the particle in the box). These quantum numbers are principal quantum number n, angular quantum number l, and magnetic quantum number ml. Based on the solution of Equation (2.11), these quantum numbers can only have certain values as shown in (2.12), which completely specify the state function of the electron for a particular energy state. llllm nl n l ,1,,1,0,,1, 1,,2,1,0 ,2,1 += = = KK K K (2.12) Here, n represents the orbits in which the electron can be found around the atom nucleus. l is used to calculate L, the magnitude of the orbital angular momentum of the electron, given by )1( += llL h (2.13) The Z component of the orbital angular momentum of the electron is determined by ml through Equation (2.14). hlz mL = (2.14) where Z is the direction of the electric field the axis. Based on the solution of equation (2.11), the electron in hydrogen-like atom can have energies given only by 14 n emZ E e 22 0 2 42 32 h"= (2.15) which shows that the energy of the electron for the hydrogen-like atom, only depends on principal quantum number n. This means that all states with same n but different l and ml are degenerate. Briefly, the solution of Equation (2.11) gives us some important information about the property of the electron in the hydrogen-like atom, indicating that the electron can have only certain energies and orbital angular momenta while it is confined in the atom. In other words, the magnitude of the energy and orbital angular momentum of the electron are quantized. Also the direction of the orbital angular momentum of the electron is quantized. Equation (2.15) agrees very well with experiments for hydrogen-like atoms. As discussed above, for multi-electronic atoms, if the inter-electronic repulsion terms ),,,( 21 zj rrrG rKrr , in Equation (2.9) were negligible, we could use Equation (2.10) to calculate wave functions )( jr r and energy states Ej of each electron. If that was the case, the energy states of each electron in a multi-electron atom could be found from Equation (2.15), and also the total electronic energy states of the atom could be found from: n emZZEE e Z j j 22 0 2 42 1 32 h"== #= (2.16) It turns out that Equation (2.16) does not agree with experiment. For example, the electronic ground state energy (n=1) for helium atoms is overestimated by 38% using Equation (2.16), indicating that the inter-electronic repulsion term ),,,( 21 zj rrrG rKrr in 15 Equation (2.9) makes a significant contribution in the total potential energy and cannot be neglected. This means that the simplified form of the state function suggested by equation (2.10) is not valid for multi-electron atoms. Hence, to solve Equation (2.9), we must resort to approximate methods. Fortunately, different approximation methods (e.g. perturbation theory and variational method, etc.) have been used and these methods can yield extremely good results. Although the Schr?dinger equation is quite successful in predicting or explaining the results of the most experiments, it can not explain a few phenomena, such as doublet yellow line in the atomic spectrum of sodium. In this regard, in addition to the three quantum numbers obtained directly from the solution for hydrogen-like atoms, it was suggested by Goudsmit and Uhlenbeck that an electron behaves like a spinning top, contributing its own z components of spin angular momentum. This electronic angular momentum is quantized into two (?up? or ?down?) values of 2/h? and motivate the fourth quantum number ms, which is called spin quantum number and can only be 2/1? for each electron. Thus the spin quantum number ms, is used to calculate the z component of the spin angular momentum Sz by 2/1,2/1, == ssz mmS h (2.17) Therefore the magnitude of spin angular momentum S, similar to orbital angular momentum, is 2/1, 2 3)1( ==+= sssS h (2.18) 16 Figure (2.3) shows examples of the z components of the orbital and spin angular momenta. Figure 2.3: Quantization of angular momentum L (left) and spin angular momentum S (right) along the direction of an electric field. Based on the spin angular momentum, the one-electron wave function also includes the spin function and it is postulated that the spatial and spin parts of its wave function are independent and therefore the wave function is defined by )()(),()()(),(    rrorrr rrrr == (2.19) in which  is spin variable, () and () are (orthogonal) spin eigenfunctions corresponding to ms= 1/2 and -1/2 respectively. The spatial and also the complete form of one-electron wave functions )(rr and ),( rr are called orbital and spin orbital wavefunctions, respectively. As explained above, the spin orbital wavefunction for each electron in an atom depends on four quantum numbers n, l, ml, and ms. Based on Pauli Exclusion 17 Principle, no two electrons in an atom can have the same values of all four quantum numbers. Using this postulate and also the fact that electrons are indistinguishable, the wave function of an atom containing N electrons is usually specified by )()()( )2()2()2( )1()1()1( ! 1),,2,1( 21 21 21 NNN N N N N N %%% %%% %%% L MMMM L L K = (2.20) where ?s in Equation (2.20) are spin orbitals. Based on Pauli Exclusion Principle, all ?s are different; otherwise, the wave function of the atom will be zero. Each i in Equation (2.20) is dependent on a set of quantum numbers n, l, ml, and ms and also depends on the inter-electronic repulsions. As a result, for each electron the wave function can be written as a function of the quantum numbers as ),,,,,,,( 21 iiiii psliii mmln %% K= (2.21) where the arbitrary parameters iii p  ,,, 21 K are used to account for inter-electronic repulsions affecting the selected electron. Considering the effect of inter-electronic repulsions for each electron the orbital energy i is defined by iiiiF %"% =? (2.22) where the energy operator iF? includes all inter-electronic repulsions due to other electrons and can be found from Equation (2.20) when the other spin orbitals , ,are known. Using a trial and error procedure (called self-consistent method), the spin orbital i (or atomic orbital i), and the orbital energy i, can be found where every atomic orbital and spin orbital are related by 18 2/1if,)( 2/1if,)( == == sii sii m m  %  % (2.23) Briefly, we can see from (2.20), for an atom with N electrons, each electron can take a set of quantum numbers given by Equations (2.12) and (2.17), and based on this configuration the atom will have particular wave function (state function) and therefore particular energy state (?electronic state?) corresponding to the quantum numbers of its electrons. For example the electronic energy states of the hydrogen atom are shown in Figure 2.4. Figure 2.4: The first few electronic states of atomic hydrogen from (McQuarrie and Simon 1997). 19 2.2.2.2 Electronic states in Molecules The electronic energy states of molecules can be explained with the same concepts developed for atoms. The wave function for a closed-shell molecule with N electrons (N must be an even number) is given by )()()()()()()()( )2()1()2()2()2()2()2()2( )1()1()1()1()1()1()1()1( ! 1),,2,1( 2/2/11 2/2/11 2/2/11 NNNNNNNN N N NN NN NN       L MMMM L L K = (2.24) where the individual entries are product of (one-electron) molecular orbitals s,i and spin functions  and . The same as atomic orbitals ?s, molecular orbitals s,i specify the spatial probability density of each electron around the molecule. Usually, the ith molecular orbital is obtained from linear combination of atomic orbitals by # = = M j jjii c 1 % (2.25) in which M is the number of atomic orbitals used to construct a molecular orbital. The atomic orbitals for each atom are obtained from the procedure outlined in the previous section. Again using a trial and error procedure, the constant cji and therefore the (one-electron) molecular orbital s,i are calculated. It should be stressed that each molecule contains multiple atoms. Based on the spacing between atoms in a molecule, from Equation (2.25), it is clear that the molecular orbital s,i will be also a function of atomic spacing. Knowing that for each set of molecular orbitals 2/21 ,,, N K , the molecule has a certain electronic energy, we realize that the electronic energy of a polyatomic molecules also depends on its atomic spacing. 20 To clarify this more, consider the molecular orbitals of H2+ which are the simplest orbitals among all polyatomic molecules. Generally, it is a good approximation that the nuclear motion of atoms in molecules can be neglected. Using this approximation for H2+, the time-independent Schr?dinger equation for this system (one electron molecule H2+ as shown in Figure 5) can be written as """ ER e r e r e m BAe =++ ) 444 ( 2 0 2 10 2 10 2 2 2h (2.26) Figure 2.5: Definition of the distances involved in the Hamiltonian operator for +2H from (McQuarrie and Simon 1997). It is obvious from Equation (2.26) for each state j, the molecular orbital j and therefore energy Ej, eigenfunction and eigenvalue of (2.26), will depend on atomic distance R. The inter-nuclear distance between the atoms at particular electronic state where the energy of molecules is minimum, is called bond length. As an example, the variation of energy, E, versus inter-nuclear distance, R, at ground electronic states for H2+, is shown in Figure 2.6a. Also, in figure 2.6b some electronic states of the oxygen molecule O2 are illustrated. Hence, just as atoms, molecules have many different electronic states, which are derived from equation (2.25). 21 2.2.3 Rotational states Rotational states of molecules are most easily understood through a discussion of the simpler states of diatomic molecules, after which the discussion can be easily extended to polyatomic molecules. 2.2.3.1 Rotational states of a diatomic molecule Assume a diatomic molecule contains two atoms with mass m1 and m2 at fixed distances r1 and r2 from their center of mass (the bond length r is assumed to be constant, Figure 2.7) rotates at angular velocity . For such rigid system, the total kinetic energy will be 222 11 2 11 2 22 2 11 2 1)( 2 1 2 1 2 1  IrmrmvmvmK =+=+= (2.27) where I is called the moment of inertia, given by Figure 2.6a: Variation of energy, E, versus inter-nuclear distance, R, at ground electronic states for H2+ from (McQuarrie and Simon 1997). Figure 2.6a: A potential energy diagram of O2 from (McQuarrie and Simon 1997). 22 2 22 2 11 rmrmI += (2.28) Figure 2.7: Two mass m1 and m2 shown rotating about their center of mass, from (McQuarrie and Simon 1997). The two-body system can be transformed to a single-body system consisting of an effective mass ? rotating at a distance r from a fixed center (which is the center of mass of the two-body system given by 222211 rmrm = ). Figure 2.8. In this new, simpler system, the moment of inertia is given by 2rI ?= (2.29) in which ? is reduced mass defined by 21 21 mm mm + =? (2.30) This model is called the rigid rotor model for a molecular system. Writing the time- independent Schr?dinger equation for the transformed one-body system, the total rotational kinetic energy of the molecule will be Kh 2,1,0,)1( 2 2 =+= JJJ I EJ (2.31) 23 Figure 2.8: The rotation of a single particle about a fixed point, from (McQuarrie and Simon 1997). Equation (2.31) says that a diatomic molecule with a fixed bond length r has only a set of discrete allowed energies and can not have any energy. Also, based on the solution of the Schr?dinger equation for such system, each allowed energy level has a degeneracy gJ given by 12 += Jg J (2.32) Usually, energy in molecular spectroscopy is expressed in units of cm-1, related to energy in Joules as: hc JoulesE cmE )()( 1 = (2.33) Using Equation (2.33), the rotational energy of the diatomic molecule in cm-1 is written by Kh 2,1,0,)1( 8 )1( 2 2 2 =+=+= JJJ cI hJJ Ihc EJ  (2.34) 24 Per the rigid rotor model assumptions, Equation (2.34) is only valid when the bond length r is constant. It turns out as molecule rotates more energetically (increasing J), the centrifugal force causes the bond to stretch slightly. If this small variation in r is treated with perturbation theory, the rotational energy (in cm-1) becomes K2,1,0,)1(~)1(~ 22 =++= JJJDJJBEJ (2.35) where cIhB 28/~ = and D~ (the centrifugal distortion constant) for each molecule are usually obtained by fitting Equation (2.35) to the experimental data. 2.2.3.2 Rotational states of a polyatomic molecule For simplicity, we first assume a polyatomic molecule can be considered as a rigid network of N atoms. For such a system, the moment of inertia Ixx with respect to any chosen Cartesian axes can be defined as ])()[( 2 1 2 cmj N j cmjjxx zzyymI += # = (2.36) where mj s the mass of the jth atom situated at point xj, yj, zj and xcm, ycm, zcm are the coordinates of the center of mass of the molecules. Similar definations are applied to two other moments of inertia Iyy and Izz. Also for this system the product of inertia Ixy is defined by )()( 1 cmj N j cmjjxy yyxxmI = # = (2.37) and as in Equation (2.37), Ixz and Iyz are defined similarly. 25 There is a theorem saying that for such a system there will always be a particular set of Cartesian axes X, Y, Z, called the principal axes, passing through the center of mass of the system such that the products of inertia vanish. The moments of inertia with respect to the principal axes are called principal moments of inertia. The principal moments of inertia of the polyatomic molecules are usually denoted by IA, IB, IC such that IA  IB IC. The principle axes of molecules with some degree of symmetry are more easily found. Usually an axis of symmetry of a polyatomic molecule is a principal axis. Principal moments of inertia are usually expressed in terms of rotational constants in units of cm-1, defined by CBA cI hC cI hB cI hA 222 8 ~ and, 8 ~ , 8 ~  === (2.38) Since CBA III '' , the rotational constants always satisfy CBA ~~~ (( . Basically, the characteristics of a prototypical rigid polyatomic molecule are determined by its three principal moments. If all principal moments are equal, the molecule is called spherical top; if only two of them are equal, it is called a symmetric top; and if all three are different, the molecule is called an asymmetric top. For example, CH4 and SF6 are spherical tops: NH3 and C6H6 are symmetric tops; and H2O is an asymmetric top. The rotational energy states for a spherical top and symmetric top can be obtained analytically from the time-independent Shr?dinger equation, however, for a asymmetric top molecule, the rotational energy sates are very complicated and there is no analytical expression. The rotational energy states for a spherical top molecules 26 ( CBA ~~~ == ) are exactly the same as for a diatomic molecule given by Equation (2.34). Polyatomic atomic molecules are usually less rigid than diatomic molecules and therefore the centrifugal distortion effect is more important. For this reason, as for a diatomic molecule, the effect of centrifugal force in rotational energy states of a polyatomic is treated by perturbation theory. Again, for spherical top and symmetric top molecules there are analytical solutions for rotational energy states where centrifugal distortion are included. For example, the rotational energy states of a spherical top polyatomic molecule are given by Equation (2.35), which again is exactly the same as for diatomic molecules. 2.2.4 Vibrational states In addition to translational, electronic, and rotational energy states, a molecule can also have vibrational energy states. As for rotational energy states, the vibrational energy states for the simplest molecules (diatomic molecules) are discussed first, and then the results are extended to polyatomic molecules. 2.2.4.1 Vibrational states for a diatomic molecule We will consider the so-called ?harmonic oscillator? model of a diatomic molecule with two atoms with mass m1 and m2, located at x1 and x2 respectively, Figure 2.9a. The basis of the harmonic oscillator model is that the internuclear (repulsion and attraction) forces between the two atoms are symmetric and are proportional to the 27 displacement from equilibrium ( xkF = ), as in a simple spring system. If this is true, then the equations of motion of the atoms are given by )( )( 122 2 2 2 122 1 2 1 rxxk dt xd m rxxk dt xd m = = (2.39) Since the motion of the two atoms in harmonic oscillator model only depends on the relative separation between the atoms 12 xxx = , it is better to transform the above two-body system to a one-body system illustrated in Figure 2.9b. The equation of motion (2.39) becomes 02 2 =+ kx dt xd? (2.40) where ? is reduced mass of the molecule given by equation (2.30). The vibrational energy states for a diatomic molecule predicted from the harmonic oscillator model can be obtained from the time-independent Schr?dinger Figure 2.9b: A mass connected to a wall by a spring, from (McQuarrie and Simon 1997). Figure 2.9a: Two masses connected to a spring, which is a model used to describe the vibrational motion of a diatomic molecule, from (McQuarrie and Simon 1997). 28 equation, when the potential energy function V(x) is known. The potential energy for this system can be easily obtained form  ==+= 0)0(Assuming,2)()( 2 VxkcdxxFxV (2.41) If this potential is put in the time-independent Schr?dinger equation, for the aforementioned one-body system, the eigenvalues, or vibrational energy states, will be (below it should say ?where EJ is in Joules?) 1in ishere w2,1,0, 2 1 ~ Joulein ishere w2,1,0, 2 1  =   += =   += cmEE EhE J J K K     (2.42) in which E is the vibrational energy state, an allowable value for vibrational energy E =p2/2?+kx2/2), and , or~ , is the fundamental vibrational frequency defined by 2/12/1 2 1 ~or 2 1    =   = ?? k c k (2.43) The harmonic oscillator model embodied by equation (2.42) predicts that vibrational energy states for a diatomic molecule are equally separated. If the vibrational energy states are obtained precisely by experiment, it is observed that they are not equally separated. To calculate vibrational energy states for a diatomic molecule more precisely, the potential energy in the time-independent Schr?dinger equation must be the real potential energy of a diatomic molecule, incorporating nonlinearities not in the Hookian spring model. The real inter-nuclear potential energy obtained from electronic state energy diagram (such as Figure 2.6a and 2.6b) illustrates how the actual potential energy of a diatomic molecule varies as the inter-nuclear separation 29 changes. It turns out the potential energy diagram of a molecule in a particular electronic state is not a simple parabola. To obtain a more accurate vibrational energy, the higher orders of relative displacement x is considered in the potential energy equation such as L+++= 423122)( xcxcx k xV (2.43) If the anharmonic terms in (2.43) are treated separately by applying perturbation theory the time-independent Schr?dinger equation, then the modified vibrational energy states will be given by KL 2,1,0,) 2 1(~~) 2 1(~ 2 =+++=  eee xE (2.44) where ex~ is called the anharmonicity constant. The energy levels determined by Equation (2.44) are not equally separated, and in fact match with experiment. Figure 2.10 shows the vibrational energy states for a diatomic molecule using harmonic oscillator model, equation (2.42) and more precise model, given by (2.44). Figure 2.10: The energy states of a harmonic oscillator (dashed line) and anharmonic oscillator superimposed on a harmonic-oscillator potential and a more realistic internuclear potential, from (McQuarrie and Simon 1997). 30 It should be stressed that in Equation (2.44) different electronic states have different potential energy diagrams, and therefore the constants e~ and ex~ are unique to each electronic state. For each molecule, these constants are usually obtained by fitting the results of measured vibrational states with Equation (2.44). 2.2.4.2 Vibrational states for a polyatomic molecule As for diatomic molecules, the key factor in determining vibrational states of a polyatomic molecules is the potential energy of the molecule. In this section, the vibrational energy of a polyatomic molecule based on harmonic oscillator approximation is discussed. A complete specification of an N-atom molecule in space requires 3N coordinates, each atom requires three coordinates. In this regard, the N-atom molecule has 3N degrees of freedom. Three of these coordinates are used to specify the center of mass of the molecule. Because motion along these coordinates corresponds to translational motion, we can say the molecule has three translational degrees of freedom. Two coordinates are required to specify the orientation of a linear molecule about its center of mass, and three coordinates to specify the orientation of a nonlinear molecule about its center of mass. Since motion along these coordinates corresponds to rotational motion, we can say linear and nonlinear molecules have two and three rotational degrees of freedom, respectively. The remaining coordinates (3N- 5 for a linear molecule and 3N-6 for a nonlinear molecule) specify the relative position of the N nuclei. Since the vibrational motion depends on the relative position 31 of the nuclei in a polyatomic molecule, we say that a linear and nonlinear molecule have 3N-5 and 3N-6 vibrational degrees of freedom. As explained in Section 2.2.2.2, the potential energy of a polyatomic molecule, in the absence of external fields, only depends upon the relative position of the nuclei. Therefore, the potential energy will be only a function of 3N-5 or 3N-6 vibrational coordinates. If we let the displacements about the equilibrium values of these coordinates be denoted by vibN qqq ,,, 21 K , where vibN is the number of vibrational degrees of freedom, then the potential energy for the molecule is given by ## ## = = = = += +      += vib vib vib vib N i ji N j ij N i ji N j ji vib qqf qq qq VVqqqV 1 1 1 1 2 21 2 1 2 1)0,,0,0(),,,( L LKK (2.45) where Equation (2.45) is a generalized form of Equation (2.41) for a multidimensional case. As we can see, the anharmonic terms are not considered here and therefore Equation (2.45) represents potential energy for the N-atom molecule with the harmonic-oscillator approximation. The presence of cross terms in Equation (2.45) makes the solution of the corresponding Schr?dinger equation very difficult to obtain. However, by using a particular transformation (mapping {qi} coordinates to {Qj} coordinates), the potential energy is related to new coordinates by # = = vib vib N j jjN QFQQQV 1 2 21 2 1),,,( K (2.46) These new coordinates are called normal coordinates and corresponding 3N-5 or 3N- 6 vibrations are referred to as normal modes of vibration. In normal modes of vibration, the nuclei move in phase, and the motions of nuclei in a normal mode are 32 such that the center of mass dose not move and the molecule as a whole does not rotate. Sometimes, several modes have identical vibrational frequencies and are referred as degenerate modes. The Hamiltonian operator H? for the vibrational energy of the N-atom molecule with the harmonic-oscillator approximation in terms of normal coordinates can be written as ## == =    += vibvib N j jvib N j jj jj vib HQFdQ dH 1 , 1 2 2 22 ? 2 1 2 ? ? h (2.47) According to the property of Hamiltonian operator, mentioned in Section 2.1, the vibrational energy states of the molecule can be written as # = =+= vibN j jjjvib hE 1 2,1,0,each ) 2 1( K (2.48) This means that under the harmonic oscillator approximation, the vibrational motion of a polyatomic molecule appears as Nvib independent harmonic oscillators, each vibrating with a fundamental frequency j . For example the normal modes of H2O are shown in Figure 2.11. Figure 2.11: Normal (vibrational) modes of H2O, from (McQuarrie and Simon 1997). 33 As mentioned above, Equation (2.48) only considers the quadratic term in the potential energy Hamiltonian. If the anharmonic terms (cubic, quartic,? terms) in the potential energy are considered, then modified vibrational energy states are obtained: L+++++= # ## = = ( ) 2 1() 2 1(~) 2 1(~ 1 1 ,, k N j N j N jk jjkejjevib vib vib vib xE  (2.49) As explained for diatomic molecules, the anharmonicity coefficients je,~ and jkex ,~ are obtained by fitting the experimental results to Equation (2.49). 2.2.5 Total molecular energy state The total energy of a molecule, in general, is the sum of translational, electronic, vibrational, and rotational energy. The translational energy states in the molecules are very closely spaced and can not be probed using molecular spectroscopy; hence, they are neglected here. Based on the discussion above, the total energy of a polyatomic molecule is given by rotvibelect EEEE ++= (2.50) 2.3 Interaction of weak electromagnetic radiation by a molecule (Bernath 1995) In this section a semi-classical model for the interaction of a weak electromagnetic wave with a molecule, leading to absorption of electromagnetic energy is discussed. In this model, the energy levels of the molecules are obtained from the time-independent Schr?dinger equation, as explained in the proceeding sections, and the electromagnetic wave is treated classically. Consider a molecule with only two states, with lower and upper state function 0 and 1 and corresponding energy levels E0 and E1, as shown in Figure 2.12. Such 34 a two level system is a good approximation for weak electromagnetic wave where the electric field strengths are low enough such that there is a negligible buildup of population in the exited state{Bernath, 1995 #59}. According to the Bohr condition, an electromagnetic wave with frequency  such that hEE = 01 can be absorbed by a molecule and induce a transition from E0 to E1. Figure 2.12: Two-level system, from (Bernath 1995). The nature of such transition is a time-dependent phenomenon and therefore this transition probability is obtained from the time-dependent Shr?dinger equation by [ ] ),(??),()(?),( )1()0( trHHtrtH t tri rr r h +==  (2.51) where )0(?H is Hamiltonian operator for an isolated system, given in Equation (2.3), which we used to obtain different stationary states in a molecule. )1(?H is a time- dependent term of )(? tH in equation (2.51), due to the interaction of the electromagnetic wave with molecule, representing the time variation of potential energy in the system due to electromagnetic radiation and obtained by )2cos(E?)2-K.rcos(E.?(t)E.?? 00)1( ttH  === rrrr (2.52) Equation (2.52) is valid if the oscillating electric field and the net dipole moment are in the same direction and the wavelength of electromagnetic wave is much bigger than the dimension of the system, such that there is an equal electric field strength at 35 every point in the molecule. In Equation (2.52), E0 and  represent the amplitude and frequency of the electromagnetic radiation, and ? is the net dipole moment of the system (nuclei and electrons), calculated by #= ii rq? rr (2.53) where rr is the coordinate of the particle i (nuclei and electrons) relative to the center of mass of the molecule and iq is the charge of the particle. Assume that the system is initially in state 0 and the interaction of electromagnetic radiation with the molecule occurs at t=0. Based on time-dependent perturbation theory the state function of the system at any time t will be a linear combination of state function of stationary states 0 and 1, ),(0 tr r and ),(1 trr . Therefore one can calculate the state function of the system from ),()(),()(),( 1100 trtatrtatr rrr += (2.54) where a0(t) and a1(t) are to be determined. In such a linear combination ai(t) ai(t)* is the probability that the molecule is in state i, where the asterisk denotes the complex conjugate. Using Equation (2.7) in (2.54) and then solving Equation (2.51) with initial condition of a0(t) = 1 and a1(t) = 0 and also assuming the electromagnetic field vector and dipole moment vector are in the same direction, solution is possible for a1(t), and therefore a1(t) a1(t)*, which is the probability of absorption or intensity of absorption. The final results will be ( )[ ] ( )201 01 2 2 0 2 10 * 1101 2/sin E?)()(   hEE thEE tataP   == h (2.55) where 10? is called transition dipole moment between states 0 and 1 and is defined by 36 + d0*110 ?? = (2.56) From (2.55), it is observed that the maximum probability of absorption, in the time interval 0 to t, occurs when 01 EEh = . A plot of a1(t) a1(t)* based on (2.55) is shown in Figure 2.13. Figure 2.13: The function 212122 )/(]2/)[(sin)2()(  hhh == EEtEEFF which represents the probability of making a 21 transition in the time interval 0 to t, plotted against frequency . This function peaks when  hEE == h12 , from (Bernath 1995). Equation (2.55) is somewhat inconsistent because it assumes monochromatic radiation and short interaction time, which are incompatible with one another according to Heisenberg uncertainty principle: h( tE or  2 1( t (2.57) Equation (2.57) says if monochromatic radiation is applied to the system for a time t , then the system sees radiation of bandwidth t1/2 =  in frequency space, which is certainly not monochromatic. 37 Considering broadband radiation with radiation density 2/)( 20E" = , where )( is assumed to be slowly varying near hEE /)( 010 = , the total transition probability over all radiation frequencies can be calculated as ( )[ ] ( ) ( )[ ] ( ) t d hEE thEE d hEE thEE P )( 2 ? 2/sin)(2? 2/sin)(2 02 0 2 10 2 01 01 2 0 0 2 10 2 01 01 2 0 2 10 10 "  "  " ? h h h =   =   =   (2.58) Equation (2.58) has been derived using plane wave radiation traveling in a particular direction parallel to the dipole moment of the molecule. In general, for radiation oriented at some arbitrary direction with respect to the molecule, only the component of the radiation aligned with the molecular axis will be absorbed, and therefore in general case we should divide Equation (2.58) by 3. Therefore the absorption rate per molecule with a population density of N0 and N1 (as number of molecules per unit volume in states E0 and E1 respectively) can be calculated as 00010 2 1003 0 2 101 )()( 3 2 NBN hcdt dN dt dP ?"    === (2.59) in which 01B is Einstein coefficient for absorption. In addition to absorption, there is another important interaction of light with molecules, suggested by Einstein in 1917, called stimulated emission. In this process, a photon of energy h corresponding to the transition E1  E0 induces an atom or molecule in the upper E2 state to relax to the lower state by emitting a second photon of energy h, in phase with the incident photon. This is the process by which a laser achieves gain, through stimulated 38 emission in the lasing medium. The change of population in state E1 due to stimulated emission is given by )( 01011 NBdt dN = (2.60) It can be shown that 01B , Einstein coefficient of stimulated emission, is equal to 01B and therefore can be obtained from (2.59). Using (2.59) and (2.60), the total change rate of the population in state E1 due to stimulated emission and absorption can be obtained by )( 3 2)( )()( 0 2 103 0 2 10 00110 1 ?"   hc NN BNN dt dN = =  (2.61) 2.3.1 Attenuation of radiation beam due to absorption and stimulated emission Figure 2.14: A system with dimension 1 m ? 1 m ? l m that contains N0 and N1 molecules per cubic meter with energies E0 and E1, respectively, from (Bernath 1995). 39 Consider a system containing N0 molecules per cubic meter in the ground state (with energy E0) and N1 molecules per cubic meter in the excited state (with energy E1), as shown in Figure 2.14. If a flux of photons with  hIhcF //)( 00 == (photons/m2s) is incident upon the system from the left, the photons will be either absorbed as they travel through the system or will stimulate emission. In this case, the rate of change of population at state E1 can be obtained by )()( 3 2 1010 0 2 10 2 1 NNFNNF hcdt dN == " ? (2.62) The parameter , called the absorption cross section with has units of m2, is measure of the effective area that a molecule presents to absorb photons. The absorption of photons (which is always accompanied with stimulated emission) by molecules causes an attenuation of flux by dF when it passes through element with thickness dx. Therefore the change in flux caused by traveling through element with thickness of dx is dxNNIdIdxNNFdF )(or )( 1010 ==  (2.63) By taking the integral with respect to x, the transmitted (attenuated) intensity I due to absorption after travel of L (m) through the absorbing media is given by LNN eII )(0 10  =  (2.64) in which I0 is the incident light intensity. Typically N0 >>N1, and therefore N , N0+N1 and the effect of stimulated emission in absorption spectroscopy, particularly at low temperatures, is negligible. In (2.61), many phenomena, such as collisions and also the effect of molecular motions have been ignored. When these phenomena are considered, the molecular 40 absorption line shape changes from an infinitely sharp and narrow Dirac delta function )( 0-  to a real molecular line shape function )( 0 g . [Note that Figure 2.13 does not show a delta function ?] Based on Equation (2.61), a realistic absorption cross section can be defined as )( 3 2)( 0 0 2 10 2 " ? == g hc (2.65) It is this cross section that should be considered in equations (2.62) to (2.64) for applications to real systems. 2.3.2 Pressure broadening Consider a two-level system as described in Section 2.3. In the absence of radiation the state of the system is given by hh / 11 / 00 1100 10 tiEtiE eaea aa  += += (2.66) In such system, a0 and a1 are constants because no electromagnetic radiation is present in the system. It can be shown that the dipole moment of the system M r naturally oscillates at the Bohr frequency, hEE /)( 010 = , in the expression )2cos(M )2cos(?2?M 00 01010 * t taad  + r r = ==  (2.67) The variation of the dipole moment of the system with time specifies the line shape function, which can be easily obtained by a Fourier transform of the dipole moment of the system. In the absence of collisions, the dipole moment oscillates exactly at the 41 Bohr frequency 0, and therefore the line-shape function would be )( 0-  since the Fourier transform of the infinite cosine wave (2.67) yields a frequency of exactly 0. However, in the case of intermolecular collisions, the phase of the oscillating dipole moment is altered in a random manner. If the average time between collisions is T2 , as shown in Figure 2.15, then the infinite cosine wave is broken into pieces of average length of T2. The Fourier transform of such oscillating dipole moment will be a Lorentzian function with a full width at half maximum (FWHM) given by 2T 12FWHM  = = L (2.68) and therefore the line-shape function will be: 2 0 20 )()( /)(   + = L L Lg (2.69) Since the average time between collisions is proportional to the reciprocal of the pressure p, therefore bpL = = 2FWHM (2.70) where b is called pressure broadening coefficient and can be calculated through the experiment. Figure 2.15: The phase of an oscillating dipole moment randomly interrupted by collision, from (Bernath 1995). 42 2.3.3 Temperature broadening Consider further an atom with velocity vr interacting with a plane wave with frequency  and a wave vectorkr , as in Figure 2.16. In this case the atom sees a Doppler-shifted frequency v/c)1( ?=  depending on if the atoms and plane wave are moving in the same direction (-) or the opposite direction (+). In general when the two vectors v and k are not parallel, the Doppler shifted frequency can be calculated by     = kc v.k1 (2.71) Figure 2.16: Interaction of a lane electromagnetic wave with a moving atom, from (Bernath 1995). In the frame of the moving molecule, the frequency of the absorbed or emitted electromagnetic wave is unshifted at 0, but the in the laboratory frame the resonance frequency (of the atom moving at velocity v) has been shifted to the new value of v/c1 0 0 ? =  (2.72) At a given temperature T, the molecules in a gaseous system at equilibrium have Maxwell-Boltzmann velocity distribution given by 43 v T2 vv)( T)2/()v( 2/1 2 de k mdp km   =  (2.73) in which m is the mass of a molecule and k is the Boltzmann constant. Using 00 )/(cv  = dd from (2.72), the velocity distribution function of molecules given in (2.73), yields a normalized frequency distribution for a fixed resonance frequency 0 in the laboratory frame of )T2/(])(c[ 2/12 0 0 2 0 2 0 2 T2 c1)(g  km D ek m     = (2.74) As shown in (74), the Doppler effect causes a Gaussian line-shape. It can be easily shown that the FWHM in (74) is M T101.7 c )2ln(T22 0720  ?== m k D (2.75) where, T is in K, M in amu, 0 in cm-1 and 0 is in cm-1. Using (2.75), the Gaussian line-shape function in (2.74) can be simplified as 22 0 ]/)[(2ln4 0 )2ln(2)(g De D D    = (2.76) 2.3.4 Voigt line shape function In many systems both Doppler broadening and collisional broadening are important, and therefore the line-shape function is affected by both of these broadening mechanisms. Consider a system containing molecules with resonance frequency 0 related to a particular absorption transition. As the result of the Doppler effect, the resonance frequency 0 will be shifted to 0 . Therefore probability that a 44 system has a resonance frequency in the interval between 0 to 00  + d is 000 )(   dgD . Including the effect of collisions, the shifted frequency 0 will itself become broadened. In this case, the probability of a frequency shift from 0 to the interval between  and  d+ is given by  +/= /=  = 0 0 00000 )()()(    ddggdg LDV (2.77) where the line-shape function )( 0 Vg , which is a convolution of a Lorentzian and a Gaussian function is called the Voigt function. This can be written as: )()()( 0000 +/ /  =  dggg LDV (2.78) The Voigt function is a very common line-shape function used in spectroscopy. 2.4 The Selection rules (Bernath 1995; McQuarrie and Simon 1997) One of the most important results from equation (2.55) is the fact that molecular transitions will occur upon interaction of a molecule with electromagnetic radiation only if the transition dipole moment between the lower and upper state energies, 10? , is not zero. General rules, incorporating quantum mechanics, indicating transitions for which 10?  0 are given by selection rules. Below selection rules for some important transitions are explained. 2.4.1 Selection rules for rotational and vibrational transitions Using the rigid-rotor model as explained in section 2.3, upon calculating a state function, the transition dipole moment can be determined. In this case, for the 45 transition dipole moments 10? to have a non-zero value requires 1?= J , where J is difference between rotational quantum number of upper and lower states. Also, the molecule must have a permanent dipole moment, 0? 0 . These are the selection rules corresponding to a harmonic oscillator. A selection rule for vibrational absorption is that the dipole moment of the molecule must vary during the normal mode motion. When this happens, the normal mode is said to be infrared active. Otherwise, it is infrared inactive. For example, CO2 has four normal modes, shown in Figure 2.17. The completely symmetric motion, which does not result in a change in the dipole moment, is infrared active. Figure 2.17: Normal modes of CO2, from (McQuarrie and Simon 1997). 2.4.2 Selection rule for rotational-vibrational absorption The selection rules for rotational-vibrational absorption mainly depend on how the dipole moment oscillates with respect to the molecular axis. If the dipole moment oscillates parallel to the molecular axis, the selection rule is given by 1+=  and 1?= J (2.79) in this case the rotational-vibrational spectrum consists of a R branch (rotational- vibrational transition with 1+= J ) and a P branch (rotational-vibrational transition 46 1.0 0.8 0.6 0.4 0.2 0.0 Ab so rp tin Sp ec tru m 23802360234023202300 Frequency (cm-1) R branch P brach 1.0 0.8 0.6 0.4 0.2 0.0 Ab so rp tio n Sp ec tru m 700680660640 Frequency (cm-1) P branch R branch Q branch with 1= J ). Such absorption band is called parallel band. It should be mentioned that all diatomic molecules fall in this category. Figure 2.18 shows R and P branch of 10  vibrational spectrum of HBr. Figure 2.18: The ro-vibrational spectrum of the 10  vibrational transition of HBr(g). The R branch and P branch are indicated in the figure, from (McQuarrie and Simon 1997). If the dipole moment oscillates perpendicular to the molecular axis, the selection rule is 1+=  and 1,0 ?= J (2.80) in this case in addition to R branch and P branch, the absorption spectrum consists a Q branch (rotational-vibrational transition with 0= J ) and the absorption band is called a perpendicular band. Figure 2.19 shows a parallel and a perpendicular band of CO2. Figure 2.19: One of the Parallel band (left trace) and perpendicular band (right trace) in CO2. 47 2.5 Vibrational-Rotational Spectrum (MIT(website); Bernath 1995; Nagali, Chou et al. 1996; McQuarrie and Simon 1997) As explained in Section 2.3.2, based on Equation (2.64), the attenuation of a radiation beam due to absorption depends on absorption cross section, the population difference, 10 NNN = , and the optical path length. The integral of the cross section as a function of frequency represents the oscillator strength and is a fixed quantity for a particular transition. The oscillator strength depends primarily on the upper and lower state functions and the dipole moment(s) of the molecule. However, the relative population between upper and lower states depends on temperature, and this controls the shape of an absorption spectra looks like. Here I try to draw a simple picture of vibrational-rotational spectra for a very simple case (a diatomic molecule based on harmonic oscillator and rigid rotor). As a definition, when a molecule absorbs infrared radiation, the vibrational transition is accompanied by a rotational transition. This kind of transition is called vibrational-rotational or a ro-vibrational transition. (Figure 2.20). Consider a system comprised of N particular molecules having different electronic, vibrational and rotational energy levels at temperature T. Based on a Boltzmann distribution, at thermal equilibrium, the number of molecules at particular energy state Ei can be calculated by )/exp()( TkEgTQ NN Biii = (2.81) 48 where, Bi kg and are degeneracy of the i th state and Boltzmann constant respectively. Q(T) is the partition function of the molecule and is defined by # = i Bii TkEgTQ )/exp()( (2.82) Now, lets consider a particular ro-vibrational transition, which takes molecules from state Ei to Ef. Suppose Ei and Ef are energy of states with quantum numbers of ),( Jv and ),( Jv respectively. Using the fact that, in many cases there is no appreciable thermal population in upper state ),( Jv , and also molecules are usually at their vibrational ground state, only transitions initiating from ground vibrational state shown in Figure 2.20 are assumed to occur, and the corresponding change in population will be: ]/)1(~exp[)12( )/exp()( ),(),( TkJJBhcJ TkEJg JNJNN B BJ + + =  1  =  (2.83) where the rotational degeneracy and rotational for a diatomic molecule based on rigid rotor from Section 2.2.3.1 are used. Therefore the absorption spectrum, which is proportional to the change in population, can be obtained from (2.83) and the results would appear as in Figure 2.21. In Figure 2.21, the vibrational-rotational spectrum is comprised of equally spaced lines and is symmetric. In fact, as discussed in Section 2.2.3, the vibrational and rotational transitions are not independent and a real spectrum looks like figure 2.22. 49 Figure 2.20: A typical ro-vibrational transition, from (MIT(website)). Figure 2.21: The left graph shows the variation of population density in lower state energy with 0> J in a diatomic molecule based on rigid rotor model. Right graph shows the spectrum of a diatomic molecule based on rigid rotor model (the transitions for all possible J are considered (MIT(website)). 50 Figure 2.22: A typical spectrum of a real diatomic molecule (MIT(website)). 2.6 Beer-Lambert Law Based on discussions provided in previous sections, the attenuation of a radiation beam through a region can be calculated from equation (2.64). This equation may be written as LkeII )(0  = (2.84) where )()()( 10  NNk = (2.85) is called absorption coefficient. For a ro-vibrational transition, as explained in Section 2.4, if the effect of stimulated emission is neglected, then the population change will be proportional to NTkEhc TQ gJNJNJNN BJ )/exp()()(),(),( 010 = ,  =  (2.86) where, ETQgJ and,)(, are lower-state degeneracy, molecular partition function at T and lower-state energy. If the effect of stimulated emission also is considered, then the right hand side of Equation (2.86) must be multiplied by )/exp(1 0 Tkhc B . Considering above assumptions, and writing the absorption coefficient based on the partial pressure of absorbing gas Pabs (instead of molecular number density N), Equation (2.84) can be written as 51 LPTS abseII )()(0 0   = (2.87) Equation (2.87) is called the Beer-Lambert Law. In this equation S(T) is called absorption line strength, which is purely a temperature dependent parameter and may be expressed by 23 456 7              = Tk hcf Tk Ehc TQ g cm e T NTS BB J e L 0 2 2 exp1exp)( 273)(  (2.88) where NL (cm-3), e (e.u.), and me (g) are Loschmidt?s number, the electron charge, and electron mass, respectively; c (cm s-1) is the speed of light; Q(T) is the molecular partition function; h (J s) is Planck?s constant; kB (J K-1) is Boltzmann?s constant; and Jg , E , and f are the lower-state degeneracy, the lower-state energy level, and the oscillator strength for the transition, respectively. )( 0  in Equation (2.87) is line- shape function and in general, as explained in previous sections, it depends on temperature and pressure and usually expressed by a Voigt function. 52 3 Techniques of Tunable Diode Laser Absorption Spectroscopy Absorption spectroscopy using tunable diode lasers (TDLs) is based on tuning of the laser optical frequency across a selected absorption transition of a particular species. In general, tunable diode lasers have very narrow line widths and their optical frequencies can be tuned by changing the laser temperature or laser injection drive current. Several techniques for laser modulation and signal interpretation exist; these are the subject of this chapter. In the implementation of all of the techniques, the laser frequency is kept around the center of the transition frequency by adjusting the laser temperature. Then, the frequency of laser light is tuned by changing the injection current. As we will explain in next sections, based on the type of current modulation, different methods of absorption spectroscopy are realized. 3.1 Direct absorption In this technique (Ku and Hinkley 1975; Hanson, Kuntx et al. 1977; Nagali, Chou et al. 1996; Mihalcea, Webber et al. 1998; Durry and Megie 1999) the wavelength of the tunable diode laser is tuned across the selected transition by ramp- modulating the injection current. A simple experimental set up is shown in Figure 3.1. Typically the emitted laser light is split into three beams by use of two optical beam splitters and one mirror. According to Figure 3.1, the first beam is directed through a probe region and the attenuated beam is detected by the first photodiode detector. The second beam is sent to an interferometer (e.g. ring interferometer) and is detected by the second detector. The signal from the second detector is used to track the 53 Absorption region BS BS wavelength tuning of the laser with time, to enable the transfer of time domain information to the frequency domain. Finally, the third beam is passed through the ambient and is detected by a third detector. The output signal from the third detector provides relevant information about the background signal. Figure 3.1: Typical experimental setup used direct absorption. TDL controller Diode laser Function generator Photo detector Photo detector Photo detector I0 (t) time i(t) time v(t) time time IF (t) time I(t) Interferometer M 54 As shown in Figure (3.2), for quantification of the measurements (after background correction), first the absorbance of laser light at time t, )/ln( 0 II , is determined from the measured transmitted intensity I(t), normalized by the un- attenuated signal I0(t). By comparison of the attenuated signal I(t) and interferometer signal IF(t), the absorption of the light as the laser is scanned is obtained. Using the results from step one and two, the measured absorbance as a function of time is converted to absorbance as a function of frequency. Upon obtaining absorbance v.s optical frequency, the measured absorbance is fit with suitable line shape (e.g. Voigt function) that can describe the line-broadening processes. Using the fact that the integral of line-shape function is unity,  = 1)( 0  dg , the integral of the fitted curve is used to calculate the absorber concentration (Equation. 2.6). =  d I I LTS Pabs )ln()( 1 0 (3.1) where line strength S(T) is only a temperature dependent parameter and is obtained from a database (e.g. HITRAN) or from experiment. It should be mentioned that if the signal to noise ratio (S/N) of measured absorbance is large enough, (S/N > 100), the absorbance integral can be calculated directly from measured absorbance and therefore fitting is no longer required. 55 Figure 3.2: Typical data processing used for quantification of direct absorption technique A direct absorption technique is thus very simple and the quantification of the measurement is straightforward but there are two main drawbacks. First, it requires integration over the whole spectra, which is not always possible, either due to interference between transitions or spectral broadening. Second, the primary disadvantage is the low sensitivity of direct absorption. In addition, quantification of this technique is very sensitive to the measured reference signal and therefore any miscalculation of reference signal causes an error in the concentration calculation. time I(t) time IF (t) I0 (t) time time  time Ln (I0/I)  Ln (I0/I) time I(t) 56 Absorption region 3.2 Wavelength Modulation In Wavelength Modulation Spectroscopy (WMS), (Reid and Labrie 1981; Bomse, Silver et al. 1992; Philippe and Hanson 1993), the injection current is usually swept at sweep frequency (50-200 Hz) and a small sinusoidal modulation (10-50 kHz) or ?dither? (Uehara and Tai 1992; Jin, Xu et al. 1997) is superimposed upon the sweep signal. Figure 3.3. shows a typical setup for wavelength-modulation spectroscopy. Traditional WMS, as discussed in the literature, is confined to the regime in which the modulation frequency is much smaller than the frequency of the absorption half-width. Figure 3.3. A typical experimental setup for wavelength-modulation spectroscopy. time Mixer Function Generator TDL Laser Current Source and TEC Drive current Photo Detector Lock-in amplifier 2f signal Modulation Signal F I2f I2f, max time i(t) 0 f Sweep Signal 57 This modulation allows increased sensitivity compared with direct absorption, through the rejection of lower frequency noise by employing a relatively high dither frequency (decreasing 1/f laser noise), and also by decreasing the detection bandwidth by use of lock-in amplifier. It has been demonstrated that WMS, may improve sensitivities up to three orders of magnitude over conventional direct absorption method (Reid and Labrie 1981). As shown in Figure 3.3, the absorption signal is detected by a photodector and the voltage signal from the detector, I, is locked at twice (or for times, etc.) of fundamental frequency, f, and corresponding harmonic signal (I2f , I4f , etc.) is captured by a lock-in amplifier. In a simple picture, for small absorption, 1.0)()( 0 <= absPLgS  , the output signal from lock-in amplifier, I2f, is related to absorber pressure by 002 00022 )( )()()(   = = HK PLgSHKI absf (3.2) where H2( ? 0) is the second harmonic component of the modulated normalized line-shape (with a maximum of 1.0) and K is electro-optical gain. In this equation, absPLgS )( 000  = represents the maximum absorbance in the transmitted light. Parameters H2( ? 0) and 0 both depend on the line-shape function and therefore depend upon temperature and gas species concentrations. H2( ? 0) also depends on the modulation index. In WMS, the peak of second harmonic signal I2f,max is related to the absorbing specie concentration by absf PLgSHKI )( )( 0002max,2  = (3.3) 58 where, )( 02 H is the second harmonic of modulated normalized line-shape function at the center of absorption feature. Equation (3.3), only requires information at the center of absorption line for concentration calculation. If this information is available, there is no need to transform temporally-acquired signal to optical frequency, and hence typically in WMS an interferometer is not required. In general, the second harmonic signal, I2f, is more complicated than as expressed Iin Equation (3.2), but this equation shows the main properties of the second harmonic signal. For example, Equation (3.2) illustrates that for quantification of one species, the line-shape function and therefore concentration of all species in the system and temperature must be known. This is an enormous challenge and represents a large hurdle for quantification of WMS, particularly in combustion systems where the concentrations of all species are generally unknown. In the next chapter main issues and solutions for quantification of WMS will be addressed, along with a more advance treatment of WMS theory. 3.3 Frequency Modulation Frequency Modulation Spectroscopy (FMS) is an offshoot of WMS, in which the modulation frequency is larger than the absorption half-width frequency (Silver 1992). The rationale for this approach is that, at such frequencies (typically at least several hundred megahertz) the laser source (excess) noise term is negligible (Lenth 1983; Gehrtz, Lenth et al. 1986). This translates to potential sensitivities of 10-7-10-8 absorbance with a 1-Hz detection bandwidth (Silver 1992). In contrast to the theory 59 of WMS, which is based on the perturbation in the intensity of laser light, FMS theory is based on how a laser electric fields being transmitted through an absorptive (and dispersive) medium. This approach retains phase information, unlike WMS theory, which uses an intensity rather than electric field. In addition, FMS theory includes residual amplitude modulation (AM) effects. Thus the FMS derivation provides a richer picture of the results and these results are applicable to the WMS regime as well (Silver 1992; Avetisov and Kauranen 1997). When laser current of a diode laser is modulated at frequency m=2 fm, simultaneous AM and FM of electric field occurs such that (Lenth 1983): [ ] [ ])sin(exp)sin(1)( 0 tititMEtE mmo    +++= (3.4) in which, 0, M, and  are the laser carrier frequency, and the AM and FM index respectively. Also the phase difference between AM and FM is denoted  and has been found to have a value of /2 in number of experiments for frequency modulation fm < 750 MHz (Lenth 1984; Gehrtz, Lenth et al. 1986). The instantaneous optical frequency is )cos()( 0 tt mm   += . Since the intensity is proportional to the square of electric field, the total light intensity is (Silver 1992): [ ])sin(21)( 0 tMItI m+= (3.5) where I0 is detected intensity at 0 and the AM index can be obtained from the following equation: 0 max0 2I II M  = (3.6) 60 The phase modulation component can be expanded in terms of Bessel functions )( nJ as (Abramowitz and Stegun 1972) #+/= /= = n n mnm tinJti )exp()()]sin(exp[   (3.7) By expanding the AM term in (3.4) in terms of exponentials, then E(t) can be written as #+/= /= = n n mn tinrtEtE )exp()exp()( 00  (3.8) where (Cooper and Warren 1987) # =  + = += 1 1 11 )(. )]()exp()()[exp( 2 )(),,( k knk nnnn Ja JiJi i MJMr (3.9) with ,10 =a )exp(21 ii M a ?9?=? (3.10) Based on Equation (3.8), as result of modulation, the radiation field will no longer contain only the single optical frequency 0 , but it includes other frequency components with frequency 0 mn + and amplitude of nr given in (3.9) (with ),2,1,0n K??= . The frequency 0 is carrier frequency and the other frequencies are called sideband frequencies. While the carrier frequency has the maximum amplitude, the amplitude of sidebands decreases by increasing the absolute of n. 61 For a better understanding of the absorption phenomena in FMS, consider a modulation with small M and . Then the radiation field given in (3.8) as first described by(Lenth 1983) will be simplified as: 23 456 7 +++= + tiitiiti mm ee i M ee i M eEtE )()(0 000 )22()22()(    (3.11) Therefore for small modulation index, the radiation field is described by a strong carrier frequency at 0 and two weak sidebands at m ?0 . If = 0 or , then: 23 456 7 +++= + tititi mm e i M e i M eEtE )()(0 000 )22()22()(  (3.12) If this radiation is detected by a square-law wave photodetector, in which the photodetector generates output signal proportional to the input optical power or square of amplitude of electric field, then the two side bands each beats with the central component to produce signals at m. These two beat signals are 180o out of phase, and since the side bands are of equal intensity the beat signals cancel exactly and giving no signal component at m. However; if , n0 the detected radiation will have always a component at m. The schematic of a typical power spectrum of the modulated laser (when M and  are small) is shown in Figure 3.4. 62 Figure 3.4: Typical power spectrum of the modulated laser in FMS If the modulated laser light passes through absorptive medium, each of the sidebands is attenuated differently and the transmitted radiation field will be: 23 456 7 +++= + tiitiiti mm eei MTee i MTeTEtE )(1 )( 100 000 ) 22 () 22 ()(    (3.13) where, Tn is the complex transmission function and is defined as: 1,0,1==  neT nn in - (3.14) in which, n and n are amplitude attenuation and optical phase shift at 0 + nm and are calculated as: 2 n n - = , 1,0,1)( 0 =+= n C nL mn n : (3.15) in which n and n are absorbance and refractive index of the probe region at frequency 0 + nm respectively. L is length of probe region and C represent light velocity in vacuum. The transmitted light intensity IT(t) impinging onto the photodetector is proportional to )(2 tET . Under the further assumption that the absorption losses and phase shifts experienced by the two sidebands and the carrier  = 0,  0+ m0- m 0 0+ m0- m 0   0,  63 Absorption region are small, i.e. 110 ;; ?-- , and 110 ;; ? the transmitted intensity is readily obtained as (Lenth 1983) )]sin()2()sin()22( )cos()(cos)(1[)( 011110 11110 ttM tMtItI mm mmT  --- -- ++++ +++1   (3.16) Using a radiofrequency (rf) mixer, one can select any of the terms in (3.16) by adjusting the phase at the rf mixer. A typical experimental setup to detect these components is schematically illustrated in Figure 3.5. Figure 3.5: The schematic of an experimental setup for FMS Based on the experimental setup shown in Figure 3.5, a single mode diode laser is used as a light source. The diode drive current consists of a DC bias current, a relatively slow saw-tooth signal, of a few hundred Hz up to a few hundred KHz, Addition Function Generator TDL Drive current Photo Detector RF- Oscillator In-phase signal RF Mixer Phase Adjuster 64 which allows the frequency tuning of laser along absorption transition, and also, a sinusoidal rf signal, in the range of a few hundred MHz to a few GHz. The modulated laser light is directed to absorbing medium and is detected by a fast photodetector. The output of the detector is phase-sensitive detected by a double balanced mixer, a passive circuit that forms the product of two analog waveforms, and can be controlled by phase adjuster. This adjustment allows us to pick either absorption (signal containing n terms) or dispersion signal (signal containing n terma) from the output of the photodetector (Lenth 1983; Gehrtz, Lenth et al. 1986). And finally, the resulting signal is displayed on a digital oscilloscope or computer for signal processing. Typical results of FMS are shown in Figure 3.6. Regarding the detection of the photodetector signal using rf mixer, the component of the photodetector signal which is exactly in-phase with the modulation signal is called in-phase signal, and the component with 90 phase difference is called quadrature signal. Figure 3.6: Output signal in frequency-modulation spectroscopy of Fabry-Perot resonance: left trace shows in-phase component, right trace shows quadrature component (Lenth 1983). 65 For quantification, based on Equation (3.16), one can relate the in-phase component (terms with tmcos ) to the concentration. As an example, in a condition when 2/ = , where FWHMm >> of the absorption feature, each absorption spectra (of Figure 3.6), would purely represent the absorption of each sideband (say m +0 ) and therefore the other frequency components (carrier frequency and the other sideband frequency) would not contribute to the in-phase signal (because 010 == -- ). Under this condition, the in-phase component can be given by )()(21)](2[)(1 111phase-in - - - MMMI ++=+1 (3.17) in which, using Equation (3.15), 1- is related to partial pressure of the absorbing species by LPgTSv abs)()(2 1)( 2 1)( 111 - == (3.18) In Equation (3.17), and M are laser properties, and depends on the amplitude and the frequency of the modulation current to laser, and can be obtained experimentally. Usually the frequency modulation index is much larger than the intensity modulation index and a typical value of /M, 20 is reported in (Lenth 1983). For trace gas detection for improving the detection limit, larger values of are usually used (e.g. ~ 1.0), and this results in the presence of higher order sidebands 2,1,0,1,-2,-, LL=n . Also, the FMS condition that m is much greater than the absorption line width is not always met because of the difficulty of achieving the necessary detection bandwidth. In this regard the theory of Two-Tone Frequency 66 Modulation Spectroscopy (TTFMS) was introduced, which is explained briefly in the following section. 3.4 Two-tone Frequency Modulation Spectroscopy In discussions of TTFMS (Janik, Carlisle et al. 1986; Cooper and Warren 1987; Avetisov and Kauranen 1996), the conventional FMS is termed as Single-Tone Frequency Modulation Spectroscopy (STFMS). In TTFMS, a diode laser is two-tone frequency modulated by imposing two closely spaced RF signals 2/1 <+= m and 2/2 <= m directly to the diode laser drive current. In general, the generator frequency m and /2 are chosen such that m is comparable with or larger than the absorption half width (a few GHz) to obtain optimum sensitivity, and the intermediate frequency 21  =< that is detected is small in comparison with the modulation frequency but large enough to avoid low frequency (1/f) noise, i.e., typically 5-20 MHz. Similar to discussion provided for single tone FMS in Section 3.3, injection current modulation of a diode laser at two closely spaced frequencies 2/1 <+= m and 2/2 <= m produces the electric field [ ][ ] [ ] [ ])sin(exp)sin(exp )sin(1)sin(1)( 220110 222111 titititi tMtMEtE o      ++? ++++= (3.19) where 0 is laser carrier frequency. As the modulation frequencies, 1 and 2, are generally selected very close to each other, the phase difference , and the FM and AM indices of the diode laser at the two modulation frequencies are nearly the same 67 and therefore the following approximation is adopted in Equation (3.19), as in (Avetisov and Kauranen 1996) 212121 ,, === MM (3.20) Following the same procedure explained for single tone FMS in section 3.3, and using approximation given in (3.20), the electric field in (3.19) can be simplified by )](exp[)exp()( 2 , 100  mnirrtiEtE mn mn += #+/ /= (3.21) where rn and rm can be obtained from (3.9). A schematic of the spectral distribution of the two-tone frequency-modulated laser field for =1.0 and M=0 is shown in Figure 3.7. In this figure, spacing between different side bands is greatly exaggerated for illustration. The central component in (n = m = 0) is the laser carrier frequency,  2/00 = , and the sidebands are 210  mn ++ . Figure 3.7: Spectral distribution of the laser field in two-tone frequency modulated diode laser at 1 and 2 with  =1.0 and M =0. 0+ 21- 2 0+ 2 0+ 1 0 68 As result of the interaction of the electric field given in Equation (3.21), with a sample containing absorbing molecules, the electric field is perturbed according to (Avetisov and Kauranen 1996) )2/1exp()](exp[)exp()( ,,2 , 100 mnmn mn mn imnirrtiEtE  += #+/ /= (3.22) where )( and )( are defined as c mnLmn mn LmnPTSmn mn abcmn )()()( )g()()( 210210 210, 210210, :  ++++ =++= ++=++= (3.23) The intensity detected by a photodetector 2/)( *0EEctI "= is given by (Avetisov and Kauranen 1996) )]()(2/1exp[ ])()(exp[ 2 )( ,,,, 21 , ,, **2 0 0 mnmnmnmn mn mn mnmn i tmmitnnirrrrE c tI +? + = #  " (3.24) where c and 0" are light velocity and permeability in air respectively. The component of time varying intensity given in (3.24), arising from the heterodyning (mixing the detected and modulation signals to obtain particular frequency component of the detected signal at modulation frequency) of adjacent frequency sidebands at 700 K the opposite behavior is observed. The modulation depth should be chosen such that it results in a stronger 2f signal, while minimizing interferences. Based on these factors and the results shown in Figure 6.10, modulation depth of -1cm057.0= m is chosen for the CH4 measurement. Using the spectra such as those shown in this Figure, correlation )(? 042 CHfI 160 between )(? 042 CHfI and temperature can be obtained at any equivalence ratio and a fixed modulation depth.The spectra of the normalized 2f signal of CH4, ),(? 042 CHfI at two different temperatures are shown in Figure 6.11. As we can see in this figure, the overall spectra of the R(4) transitions changes by in creasing temperature. Figure 6.11: Calculated spectra of )(? 042  CHfI for R(4) transitions at T=500 and 2000 K for a CH4/air flame with =1, at condition: o140= and optical path of L=10 cm. 6.2 Comparison of the modeling and experiments In this section the results of the model are compared with experiments performed in the gas cell at known concentration and temperature. The experimental setup for this part of the measurement was the same as that for the direct absorption 161 measurement in Chapter 5; except in addition to the sweep signal (F=10 Hz), a modulation signal, with f=10 kHz and f=12 kHz for H2O and CH4 measurement, respectively, was added to the injection current. Also, the detector signal is sent to a lock-in amplifier and the second harmonic component of the photodetector signal is captured. More details of the setup for modulation is explained in the next section. Figure 6.12: Comparison between measured and calculated 2f spectra of the selected H2O absorption transitions for a mixture of N2-H2O in the gas cell. 162 Since the 2f signal of H2O is used for temperature measurement, and for accurate measurement the model has to predict the spectra precisely to avoid significant error in temperature which would propagate to error in concentrations, many experiments were performed to compare the measured spectra of H2O signal and the corresponding spectra computed (predicted) by the model for mixtures of H2O -N2, H2O -CO2, and pure H2O at different temperatures (483-999 K). Figure 6.13: Comparison between measured and calculated 2f spectra of the selected H2O absorption transitions for a mixture of CO2-H2O in the gas cell. 163 Figure 6.14: Comparison between measured and calculated 2f spectra of the selected H2O absorption transitions for pure H2O vapor in the gas cell. The bottom trace shows the contribution of individual transitions to the total spectra according to results of the modeling. 164 Figures 6.12 and 6.13 (on the previous pages) compare measured and calculated 2f spectra of the H2O absorption transitions at temperatures of 483 K and 999 K for H2O-N2 for H2O-CO2 mixtures respectively. Also the comparison between the calculated and measured second harmonic spectra of pure H2O vapor in the gas cell at T=483 and 999 K for modulation depth -1 , cm09.0= Im is shown in Figure 6.14. In this figure (bottom trace) the spectra of calculated individual transitions (1 through 3) related to the middle trace (T=999 K) are shown. Figure 5.15: Comparison between a set of (T , RI,II) measured by experiment in the gas cell and corresponding values calculated by the modeling results. . 165 As noted above, many experiments were performed for evaluation of the accuracy of the modeling in the quantification of the second harmonic signal before using the sensor for in-flame measurement. The results shown in Figures 6.12 through 6.14 illustrate the accuracy of the mathematical model for concentration measurement, but the accuracy of the model for temperature measurements still requires assessment. For this reason, for known temperature and concentrations in the gas cell, the measured peak ratio RI,II, at temperature recorded by thermocouple, was compared to corresponding calculated values from the modeling. The results of such comparison are shown in Figure 6.15 (on the previous page). The results in Figure 6.15 show that the difference between the temperature measurement using the sensor (based on peak ratio) and the thermocouple used in ?experiment? is about 30 ?C while the uncertainty in the thermocouple measurement is ?15 ?c (not considering a ?20 ?C uncertainty due to temperature variation along the optical path inside the gas cell). In these experiments, the measured uncertainty in the peak ratio RI,II, based on three measurements, is about 1% (the error bar associated with RI,II can not be seen in the graph). These results illustrate that the optical measurement for temperature gives a very good estimate of the gas temperature, within ?30 ?C of the thermocouple measurement for the temperature range shown in Figure 6.15). Comparison of the experimental results in the gas cell and quantification of the sensor measurement using the model show that the quantification of the sensor measurement for temperature and concentration measurement of H2O has an accuracy of about 3% and 5% for temperature and concentration measurements respectively. 166 The same model is applied for quantification of CH4 measurement. Since the model for quantification is already validated, there is no need to repeat all of the experiments performed for H2O absorption measurement for CH4. Figure 2.16 compares the measured and calculated spectra of R(4) transitions for a mixture of CH4 and N2 at T=719 K. Also in this Figure the contribution of each of the individual R( 4) transitions to the total profile is shown. See Appendix A3 for the C++ program. Figure 6.16: Comparison between measured and calculated 2f spectra of the selected CH4 absorption transitions, R(4), for a known CH4-N2 mixture in the gas cell. The bottom trace shows the contribution of individual transitions to the total spectra according to results of the modeling. 167 The accuracy of quantification CH4 concentration measurement strongly depends on the measured temperature. As we can see in Figure 6.10, the normalized signal (PCH4=1 torr) decreases exponentially with increasing temperature, and therefore any error in temperature measurements would result in a considerable error in CH4 concentration after quantification of the CH4 signal. This behavior is mainly due to the fact that the line strength of the selected CH4 transitions decreases exponentially with temperature. 6.3 Sensor architecture The schematic of the sensor architecture for simultaneous temperature and CH4 and H2O concentration measurements, is shown in Figure 6.17. In this schematic, a ramp (or sawtooth) signal from the function generator at frequency F=10 Hz is used for tuning the optical frequency of the H2O and CH4 lasers. The sinusoidal signals taken from the lock-in amplifiers at frequency f1=10 kHz and f2=12 kHz, which are used as modulation signals, are added to the ramp signal separately using two summing circuits that have the ability to adjust the amplitude of the input signals. The output voltage from each summing circuit is sent to the Tunable Diode Laser (TDL) controller, and through that the corresponding injection current is sent to the laser. The TDL controllers also control the temperature of the lasers using thermo-electric circuits. The laser beam emitted from each laser are combined using a pellicle beam splitter and then the multiplexed light is directed through the flame region where the absorption of laser light in the reaction zone takes place. The attenuated light, as a 168 result of absorption in the reaction zone, is detected by a main Photo Detector (MPD) which generates a voltage signal proportional to the impinging laser intensity at the Figure 6:17: Schematic of the sensor architecture. Lock-in amplifier Lock-in amplifier Function Generator A/D CH4 laser H2O laser TDL Controller TDL Controller MPD RPD Modulation signal at f2 Modulation signal at f1 Ramp signal at F TTL signal Modulation plus ramp Modulation plus ramp 2f1 signal, H2O 2f2 signal, CH4 + + - Fiber collimator Beam splitter SM Fiber 1X2 Fiber splitter Collimating lens Reference photo- detector Main photo- detector Ar Ar Flame region 169 surface of its photodiode. Due to some nonlinearity in the optical signal of the H2O laser, which results in a nonzero second harmonic signal at zero absorption, 10% of the laser light is separated from the main H2O beam. This laser beam is first detected by the Reference Photo Detector (RPD), then the RPD signal is subtracted from the MPD using a subtracting circuit. In the subtracting circuit the RPD signal is adjusted so as to exactly cancel the MPD signal related to H2O. In this manner, the second harmonic signal related to the nonlinearity in intensity of the H2O laser is removed. The subtracted signal is then sent to two lock-in amplifiers, where demodulation of PDS is performed. From the lock-in amplifiers, the second harmonic signals related to H2O and CH4 are obtained. These signals, with references based on the triggering signal of the function generator (TTL signal), are digitized using an A/D converter. The data are either stored in the computer (for post processing) or processed in real- time for simultaneous temperature and concentration measurement. As explained in Chapter 5, to eliminate the background H2O absorption related to moisture in the air, the optical path of the H2O laser outside the reaction zone is purged either by Argon or dry air as shown in the Figure. 6.4 In-flame measurements In this section measurements of temperature and measurements of H2O and CH4 concentrations in a flame, based on the setup shown in Figure 6.17, are presented. The first set of experiments performed involves measurements at different heights in the flame by changing the height of the burner using a transition stage with 0.025 mm resolution, while keeping all optics fixed. 170 For these measurements an adjustable slot burner with 2 mm width and length of 10.0 cm generated a premixed CH4/air flame. The air and fuel flow rate of 9.96 and 1.78 LPM respectively were measured by two calibrated rotameters. When the burner was running, the multiplexed laser beams were directed through a very uniform flam. In addition, two type R thermocouples with diameter of 0.008 inches were used to measure the flame and post flame temperature (see Figure 6.18). As result of considerable heat conduction in thermocouple wires in the pre-flame zone, the thermocouple could not be used in these regions for temperature measurement. Therefore, thermocouple measurements were limited to flam or post-flame zones where the temperature gradients in the thermocouple wires were negligible. For the non-intrusive optical sensor, measurement of temperature at different zones (along the vertical direction) was possible. Figure: 6.18: Simplified Figure of the setup of the in-flame measurement 171 As discussed, TI I OH f vs.)(? 22  , TI CHf vs.)(? 042  , and also RI,II vs. T correlations must be available for the mixture. These correlations are determined as shown in Figure 6.19 for this experiment. During the measurement, the peak ratio, RI,II, can be calculated from the measured 2f signal of H2O. Using this ratio, the temperature along the optical path in the combustion zone is calculated as Tm (see the top trace in Figure 6.19). This temperature is used to calculate the normalized second harmonic signals of CH4 and H2O at that condition based on ?complete combustion approximation?. The graphs in the middle and bottom are pre-calculated normalized 2f signal of H2O and CH4 respectively. The amplitude of the second harmonic signals in these graphs are normalized to an absorbing species concentration of one torr and by intensity. For normalization with respect to optical power (intensity), they are divided by the amplitude of 1f signal at the transition frequency as discussed in Chapter 4. The normalized signals obtained from the previous step are used to calculate the concentrations using Equation 6.2, in which K in general depends on both electrical gain and intensity. If the measured 2f signal is divided by the 1f signal at the central absorption frequency, then K in Equation (6.2), K will become independent of intensity. This allows us to use a value of K from measurement in the gas cell where the concentration and temperature are known. Based on this procedure, the constant K is determined for both H2O and CH4 species as 1/23.0 and 1/1.9 respectively (see Figure 6.20). 172 Figure 6.19: Pre-calculated correlations, TI I OH f vs.)(? 22  , TI CHf vs.)(? 042  , and TR III vs., , for the CH4/air flame with =1.70. The 2f signals are normalized by the 1f signal at the central absorption frequencies. )(? 042 CHfI )(? 22 IOHfI  173 In this experiment a large diameter photodiode diameter is used to detect the absorption signal to minimize effects of beam steering. For all flame experiments, the intensity of the laser impinging on the photodiode was essentially constant, with the fluctuation due to beam steering less than 2%. Therefore, the amplitude of the first harmonic signals (1f) could be predetermined, using the two lock-in amplifiers available for this research, before the flame measurement. This helped us to simplify the set up and no further component needed to monitor the laser intensity during the experiment. Figure 6.20: Calculation of K by comparing the measured second harmonic signal at know temperature and concentration in the gas cell and calculated second harmonic signal with unit gain. 174 Using the information explained above, the temperature and the concentrations of CH4 and H2O at 5 different heights were measured. The results of these measurements are shown in Figures 6.21 and 6.22. The uncertainty in the measured in temperature in Figure 6.21, is as result of signal fluctuation, which is attributed mainly to acoustical vibration. However for the concentrations in Figure 6.22, uncertainty is related to the both fluctuation of the signals and uncertainty in quantification as result of temperature fluctuation. The temperature uncertainty is especially dominant in CH4 results in Figure 6.22. The bottom graph of Figure 6.19 shows that the normalized 2f signal of CH4 decreases exponentially with temperature, and hence even a small uncertainty in temperature causes a large uncertainty in the concentration. Figure 6.21: Temperature variation at different height above the burner surface using the optical sensor. 175 Figure 6.21: Variation of the H2O and CH4 concentrations at different heights above the burner surface using the optical sensor. All the results (each point in the graphs) are the average of 4 measurements and each measurement is the average of 10 points with frequency of 10 Hz. Therefore each measurement represents the average values over 1 second and each point in the graphs indicates the average over 4 seconds. The thermocouple values presented in Figure 6.20 are the average of 40 points over 4 seconds and are corrected for radiation loss. The method of radiation correction is presented in Appendix G. According to the results presented in Figure 6.20, the difference in measured temperature by thermocouple and optical sensor in the post- flame zone is 135 K while the uncertainty in the thermocouple measurement is ?100 K. This shows, as we expected, the optical sensor gives very a good estimate for 176 temperature measurement in a region where the complete combustion assumption is valid. Also, in Figure 6.21, the mole fractions measured by the optical sensor seem reasonable. In the pos- flame zone, according to the measurement, H2O concentration is 0.184 and based on complete combustion, the mole fraction at this region is expected to be 0.178. If the reaction is assumed to be complete, the optical sensor overestimates the concentration by 3%. For CH4 also very reasonable trend is observed. At a distance about 5.5 mm above the burner surface, where the reaction rate is small, the measured mole fraction at this point is 0.155 while the mole fraction of CH4 in the reactant is 26 %. Also close to the flame-zone, from Figure 6.21, it is concluded that CH4 concentration is not zero and this is what we expect because there has been excess fuel in the reactants. According to this discussion, it is concluded, as for the measurement inside the gas cell, the developed optical sensor gives good estimates for temperature and concentration of CH4 and H2O. 177 7 Conclusion In this dissertation, a sensor to measure temperature and concentrations of CH4 and H2O based on absorption spectroscopy was designed, built, and tested in a static cell under various conditions and in a flame. For high sensitivity, Wavelength Modulation Spectroscopy (WMS) was used to quantify the absorption of the tunable diode laser beam. In general, quantification of WMS, particularly in combustion systems, requires detailed spectroscopic information for all of the absorption features probed by the sensor. At the same time, many absorption features are usually overlapped, and therefore measurement of spectroscopic information for individual transitions may be very difficult or sometimes impossible. The lack of detailed high- temperature spectroscopic information for the near-infrared transitions of interest, combined with the complexity of WMS technique itself, have in the past been barriers for quantification of WMS in combustion systems. In this research, a few strong absorption transitions suitable for CH4 and H2O measurement have been identified. In addition to having fairly strong line strength, the transitions are selected such that, they are located in the range of standard telecommunication lasers, which are readily available and inexpensive. The selected H2O transitions are unique as they are close enough to be within the tuning of a single diode laser. Spectroscopic properties of the selected transitions, which are overlapped under most conditions of temperature and pressure, are determined using a new technique 178 developed in this research, whereby individual spectroscopic properties of the transitions can be extracted from experiment. This spectroscopic information for each of the strongly overlapped transitions allows the prediction of the behavior of these transitions as a function of temperature and pressure. Using this technique, spectroscopic properties of CH4 and H2O for the selected transitions were determined. For quantification of WMS in combustion systems, a mathematical model based on laser and absorption transition properties has been developed. Using the model, detailed properties of WMS when both intensity and frequency are modulated are studied. The model implements a Voigt function, and therefore is also applicable for Lorentzian and also Gaussian line-shape profiles. Using the measured spectroscopic properties of the selected H2O and CH4 transitions in the WMS model allowed us to validate the accuracy of the WMS model through experiments in a static quartz cell at known temperatures and concentrations. The results of these experiments proved the accuracy of the model. Finally, the optical sensor is applied to simultaneous in-flame measurements of temperature and concentration of CH4 and H2O. A primary application of the sensor described in this work might be for emission or process controls in high temperature processes including gas turbines, boilers, engine exhaust, semiconductor manufacturing, or optical fiber manufacture, among many applications. For such applications, the response time of the sensor could be improved dramatically (up to a few kHz) by centerline-locking technique. Here, instead of tuning the laser frequency across an absorption feature, through a feedback control the laser frequency is kept at the center of absorption feature and 179 only a modulation current is applied to the laser. Demodulation of absorption signal with such modulation scheme provides absorption information at the same rate of modulation frequency (tens of kHz). In future work, quantification of the WMS model, which here was based on a complete combustion approximation, could be extended to the case where the reaction has progressed partially. In this case accurate quantification might be possible through an iterative method in which the absorption spectrum is modified in each iteration after calculating H2O concentration. This particular WMS quantification could be useful for applications in which a detailed study of combustion process is needed. 180 Appendix A1: A program for simulation of the second harmonic spectra of a Voigt profile when Doppler and Collisional half-width are known //===================================================================== // This Program simulates the Inf,p (in-phase signal) when the // absorption spectra has a Voigt line-shape function for the // case when intensity modulation is taken into account. //===================================================================== #include #include int main() { // Tuning range = 1.6 cm-1 centered at nuhat=0 cm-1 int i,j,k; const int nn=2000; const int mm=1000; double gamal,gamad,nu0,gamav,x,sigmanu0,a,stepwt,sum1,sum2,sum3,sum4,pi,sf,sF,sai,phi,gain; double snu,alpha,cos1,cos2,cos3,cos4,y,term,m,I1f[mm+1],I2f[mm+1],I3f[mm+1],I4f[mm+1],nuhat[mm+1]; pi=atan(1.0)*4.0; gamal=0.05; // Lorentzian half-width in [cm-1] gamad=0.05; // Doppler half-width in [cm-1] sF=-1.0; // Laser intensity-frequency parameters (assumed linear variation) sf=-2.0; sai=(-20.0/180.0)*pi; // is phase difference b/w IM and FM in (rad) phi=(50.0/180.0)*pi; // is detection phase in (rad) m=1.0; // Ratio of FM amplitude to the absorption profile half-width (Voigt half-width) nu0=0.0; // central absorption frequency char* out="IMFMn.out"; gamav=.5346*gamal+pow(.2166*pow(gamal,2.0)+pow(gamad,2.0),0.5); x=gamal/gamav; sigmanu0=.01; gain=1000.0; stepwt=2*pi/(nn); nuhat[1]=-.80; for(k=2;k<=mm;k++) nuhat[k]=nuhat[k-1]+1.6/mm; a=m*gamav; 181 for (i=1;i<=mm;i++) { sum1=0.0; sum2=0.0; sum3=0.0; sum4=0.0; for(j=1;j<=nn;j++) { y=fabs((nuhat[i]-nu0)/gamav-m*cos(-pi+(j-1)*stepwt+sai)); alpha=sigmanu0*((1.0-x)*exp(-.693*pow(y,2.0))+x/(1.0+pow(y,2.0))+.016*(1.0-x)*x*(exp(- .0841*pow(y,2.25))-1.0/(1.0+.0210*pow(y,2.25)))); snu=sF*nuhat[i]-sf*a*cos(-pi+(j-1)*stepwt); cos1=cos(1.0*(-pi+(j-1)*stepwt)+phi); cos2=cos(2.0*(-pi+(j-1)*stepwt)+phi); cos3=cos(3.0*(-pi+(j-1)*stepwt)+phi); cos4=cos(4.0*(-pi+(j-1)*stepwt)+phi); term=(1.0-alpha)*(1.0+snu); sum1=sum1+term*cos1; sum2=sum2+term*cos2; sum3=sum3+term*cos3; sum4=sum4+term*cos4; } I1f[i]=gain*sum1*stepwt/pi; // First four harmonics I2f[i]=gain*sum2*stepwt/pi; I3f[i]=gain*sum3*stepwt/pi; I4f[i]=gain*sum4*stepwt/pi; } // -------------------- Making output file (START) FILE *fpr1; fpr1=fopen(out,"w"); for(k=1;k<=mm;k++) fprintf(fpr1,"%f %f %f %f %f\n", nuhat[k],I1f[k],I2f[k],I3f[k],I4f[k]); fclose(fpr1); // -------------------- Making output file (END) return 0; } 182 Appendix A2: A program for simulation of the second harmonic spectra of the selected H2O transitions //========================================================================= // This Program simulates the I2f signal when the absorption // profile has a Voigt profile for the case when intensity // modulation is taken into account. In this program the effect // of two types intensity modulation sF, sf, and also phase shift // between intensity and frequency modulation, sai, are considered. // In addition the detection phase, phi, is also included // For Selected H2O transitions //========================================================================= #include #include int main() { // Tuning range = 1.6 cm-1 centered at nuhat=0 cm-1 int i,j,k; const int nn=2000; const int mm=400; double a0,a12,a3,a4,aav,sum,snu,cos2,term,alpha,s1F,s2F,sf,T,Tb,Av,R,stepwt,L,Lb,sai,phi; double XCO2,XN2,XO2,XH2O,pCO2,pN2,pO2,pH2O,pH2Oc,pH2Ob; double gamaL12H2O,gamaL3H2O,gamaL0H2O,gamaL4H2O,gamaL12O2,gamaL3O2,gamaL0O2, gamaL4O2,gamaL12CO2,gamaL3CO2,gamaL0CO2,gamaL4CO2,gamaL12N2,gamaL3N2, gamaL0N2,gamaL4N2; double n0[7]={-1.134302397663504e-042,9.676714398274705e-039,-3.352457574161414e- 035,5.918478065251312e-032,-5.229449580648281e-029,1.382024888890897e-026, 8.736570371216040e-024}; double n1[7]={1.182880452915844e-040,-9.730660245129210e-037,3.208326620720904e-033,- 5.276879397285108e-030,4.238665213413604e-027,-1.240711482449527e-024, 1.060758405615869e-022}; double n3[7]={-6.705028656148200e-042,1.016906199980311e-037,-5.613008939092201e- 034,1.541989562850704e-030,-2.263749771221360e-027,1.612103724409066e-024, -2.888611866904023e-022}; double n4[7]={1.885184012772765e-041,-1.447567686064963e-037,4.293572378735591e-034,- 5.824645724200227e-031,2.757247450868865e-028,9.006868603789042e-026, -4.092023902523880e-023}; double nuhat[mm+1],I2f[mm+1],I2f1[mm+1],I2f2[mm+1],I2f3[mm+1],I2f4[mm+1],I2f0[mm+1]; double I2fb[mm+1],I2f1b[mm+1],I2f2b[mm+1],I2f3b[mm+1],I2f4b[mm+1],I2f0b[mm+1]; double S0,S1,S2,S3,S4,S0b,S1b,S2b,S3b,S4b; double HWHML0,HWHML12,HWHML3,HWHML4,HWHMD0,HWHMD1,HWHMD2,HWHMD3, HWHMD4,HWHMV0,HWHMV1,HWHMV2,HWHMV3,HWHMV4; double HWHML0b,HWHML12b,HWHML3b,HWHML4b,HWHMD0b,HWHMD1b,HWHMD2b, HWHMD3b,HWHMD4b,HWHMV0b,HWHMV1b,HWHMV2b,HWHMV3b,HWHMV4b; double m0,m1,m2,m3,m4,nu00,nu01,nu02,nu03,nu04,xx0,xx1,xx2,xx3,xx4,y0,y1,y2,y3,y4,pi; double m0b,m1b,m2b,m3b,m4b,xx0b,xx1b,xx2b,xx3b,xx4b,y0b,y1b,y2b,y3b,y4b; double sigma00,sigma01,sigma02,sigma03,sigma04,alpha0,alpha1,alpha2,alpha3,alpha4; double sigma00b,sigma01b,sigma02b,sigma03b,sigma04b,alpha0b,alpha1b,alpha2b,alpha3b,alpha4b; double sum1,sum2,sum3,sum4,sum0,term1,term2,term3,term4,term0,snu1,snu2,snu3,snu4,snu0; 183 double sum1b,sum2b,sum3b,sum4b,sum0b,term1b,term2b,term3b,term4b,term0b; char* out="I2fnw.out"; // FSR=.022694; Free Spectral Rang of RI pi=atan(1.0)*4.0; T=2000; // in (K) Tb=23+273; // Ambient temperature (in K): for the case the background H2O absorption needs // is not zero L=10.0; // in (cm) Lb=0.0; // in (cm) pH2Ob=10.0/760.0; a0=0.08; // Differnt FM amplitude at differnt region a12=0.090; // in (cm-1) for transition 1 and 2 a3=0.095; // in (cm-1) for the transition 3 a4=0.10; // a0 and a4 are FM amplitude of adjucent transition to the main transitions s1F=-1.1928; // laser intensity-frequency parameters s2F= -0.3985; sf=-2.3; sai=(-15/180.0)*pi; // phase differnce b/w FM and IM in (rad) phi=(155/180.0)*pi; // detection phase in (rad) aav=(a12+a3)/2; // average FM amplitude over the main transitions Av=6.02e23; R=82.08; XCO2=1/(1+2+2*3.76)*100; // methane/air combustion: mole fraction in complete //combustion approximation. XN2=2*3.76/(1+2+2*3.76)*100; // for Equivalence ratio=1.0 XO2=0; XH2O=2/(1+2+2*3.76)*100; pN2=XN2*1e-2; // partial pressures in (atm) pO2=XO2*1e-2; pCO2=XCO2*1e-2; pH2O=XH2O*1e-2; pH2Oc=1.0; // Normalized H2O pressure in atm (however the results are divided by 760.0 to // give the normalization based on 1 torr H2O // Calculation of line strength at the specified temperature S0=n0[0]*pow(T,6.0)+n0[1]*pow(T,5.0)+n0[2]*pow(T,4.0)+n0[3]*pow(T,3.0)+n0[4]*pow(T,2.0)+ n0[5]*T+n0[6]; S1=n1[0]*pow(T,6.0)+n1[1]*pow(T,5.0)+n1[2]*pow(T,4.0)+n1[3]*pow(T,3.0)+n1[4]*pow(T,2.0)+ n1[5]*T+n1[6]; S3=n3[0]*pow(T,6.0)+n3[1]*pow(T,5.0)+n3[2]*pow(T,4.0)+n3[3]*pow(T,3.0)+n3[4]*pow(T,2.0)+ n3[5]*T+n3[6]; S4=n4[0]*pow(T,6.0)+n4[1]*pow(T,5.0)+n4[2]*pow(T,4.0)+n4[3]*pow(T,3.0)+n4[4]*pow(T,2.0)+ n4[5]*T+n4[6]; S4b=3.980E-24; S3b=3.910E-23; S2b=1.880E-24; 184 S1b=5.470E-24; S0b=8.450E-24; S0=Av/(R*T)*S0; S1=Av/(R*T)*S1; // Conversion of Si to atm,cm system units S2=S1/2.91; S3=Av/(R*T)*S3; S4=Av/(R*T)*S4; S0b=Av/(R*Tb)*S0b; // Background information S1b=Av/(R*Tb)*S1b; // Conversion of Si to atm,cm system units S2b=S1b/2.91; S3b=Av/(R*Tb)*S3b; S4b=Av/(R*Tb)*S4b; // Calculation of broadening coefficient for different quenching species at the specified temperature gamaL12H2O=3.0116/pow(T,0.475); // Self H2O broadening coefficient of transition 1 and 2 gamaL3H2O=26.759/pow(T,0.766); // Self H2O broadening coefficient of transition 3 gamaL0H2O=0.4389*pow(T/296,-0.68); // Self H2O broadening coefficient of transition 0 gamaL4H2O=0.4170*pow(T/296,-0.68); // Self H2O broadening coefficient of transition 4 gamaL12N2=1.0053/pow(T,0.552); // H2O-N2 broadening coefficient of transition 1 and 2 gamaL3N2=6.5581/pow(T,0.768); // H2O-N2 broadening coefficient of transition 3 gamaL0N2=0.0828*pow(T/296,-0.68); // H2O-N2 broadening coefficient of transition 0 gamaL4N2=0.0967*pow(T/296,-0.68); // H2O-N2 broadening coefficient of transition 4 gamaL12O2=.27008/pow(T,0.467); // H2O-O2 broadening coefficient of transition 1 and 2 gamaL3O2=2.4758/pow(T,0.751); // H2O-O2 broadening coefficient of transition 3 gamaL0O2=0.0828*pow(T/296,-0.68); // H2O-O2 broadening coefficient of transition 0 gamaL4O2=0.0967*pow(T/296,-0.68); // H2O-O2 broadening coefficient of transition 4 gamaL12CO2=1.4911/pow(T,0.563); // H2O-CO2 broadening coefficient of transition 1 and 2 gamaL3CO2=1.6644/pow(T,0.505); // H2O-CO2 broadening coefficient of transition 3 gamaL0CO2=0.0828*pow(T/296,-0.68); // H2O-CO2 broadening coefficient of transition 0 gamaL4CO2=0.0967*pow(T/296,-0.68); // H2O-CO2 broadening coefficient of transition 4 // Calculation of effective (total) collisional halfwidths HWHML12=gamaL12H2O*pH2O+gamaL12N2*pN2+gamaL12O2*pO2+gamaL12CO2*pCO2; HWHML3=gamaL3H2O*pH2O+gamaL3N2*pN2+gamaL3O2*pO2+gamaL3CO2*pCO2; HWHML0=gamaL0H2O*pH2O+gamaL0N2*pN2+gamaL0O2*pO2+gamaL0CO2*pCO2; HWHML4=gamaL4H2O*pH2O+gamaL4N2*pN2+gamaL4O2*pO2+gamaL4CO2*pCO2; // Calculation of effective (total) collisional halfwidths HWHML12=gamaL12H2O*pH2O+gamaL12N2*pN2+gamaL12O2*pO2+gamaL12CO2*pCO2; HWHML3=gamaL3H2O*pH2O+gamaL3N2*pN2+gamaL3O2*pO2+gamaL3CO2*pCO2; HWHML0=gamaL0H2O*pH2O+gamaL0N2*pN2+gamaL0O2*pO2+gamaL0CO2*pCO2; HWHML4=gamaL4H2O*pH2O+gamaL4N2*pN2+gamaL4O2*pO2+gamaL4CO2*pCO2; HWHML12b=.3853/pow(Tb,0.400)*(3.76/4.76)*1+.1456/pow(Tb,0.371)*(1/4.76)*1; HWHML3b=15.101/pow(Tb,0.903)*(3.76/4.76)*1+2.1717/pow(Tb,0.723)*(1/4.76)*1; HWHML4b=0.0828*pow(Tb/296,-0.68)*(3.76/4.76)*1+0.0828*pow(Tb/296,-0.68)*(1/4.76)*1; HWHML0b=0.0967*pow(Tb/296,-0.68)*(3.76/4.76)*1+0.0967*pow(Tb/296,-0.68)*(1/4.76)*1; HWHMD1 =.5*(7.1623e-7*6771.747*pow(T/18.0,0.5)); HWHMD2 =.5*(7.1623e-7*6771.710*pow(T/18.0,0.5)); HWHMD3 =.5*(7.1623e-7*6771.475*pow(T/18.0,0.5)); HWHMD4 =.5*(7.1623e-7*6771.1974*pow(T/18.0,0.5)); HWHMD0 =.5*(7.1623e-7*6771.96150*pow(T/18.0,0.5)); HWHMD1b =.5*(7.1623e-7*6771.747*pow(Tb/18.0,0.5)); 185 HWHMD2b =.5*(7.1623e-7*6771.710*pow(Tb/18.0,0.5)); HWHMD3b =.5*(7.1623e-7*6771.475*pow(Tb/18.0,0.5)); HWHMD4b =.5*(7.1623e-7*6771.1974*pow(Tb/18.0,0.5)); HWHMD0b =.5*(7.1623e-7*6771.96150*pow(Tb/18.0,0.5)); HWHMV1=.5346*HWHML12+pow(.2166*pow(HWHML12,2.0)+ pow(HWHMD1,2.0),0.5); HWHMV2=.5346*HWHML12+pow(.2166*pow(HWHML12,2.0)+ pow(HWHMD2,2.0),0.5); HWHMV3=.5346*HWHML3+pow(.2166*pow(HWHML3,2.0)+ pow(HWHMD3,2.0),0.5); HWHMV4=.5346*HWHML4+pow(.2166*pow(HWHML4,2.0)+ pow(HWHMD4,2.0),0.5); HWHMV0=.5346*HWHML0+pow(.2166*pow(HWHML0,2.0)+ pow(HWHMD0,2.0),0.5); HWHMV1b=.5346*HWHML12b+pow(.2166*pow(HWHML12b,2.0)+ pow(HWHMD1b,2.0),0.5); HWHMV2b=.5346*HWHML12b+pow(.2166*pow(HWHML12b,2.0)+ pow(HWHMD2b,2.0),0.5); HWHMV3b=.5346*HWHML3b+pow(.2166*pow(HWHML3b,2.0)+ pow(HWHMD3b,2.0),0.5); HWHMV4b=.5346*HWHML4b+pow(.2166*pow(HWHML4b,2.0)+ pow(HWHMD4b,2.0),0.5); HWHMV0b=.5346*HWHML0b+pow(.2166*pow(HWHML0b,2.0)+ pow(HWHMD0b,2.0),0.5); xx1=HWHML12/HWHMV1; xx2=HWHML12/HWHMV2; xx3=HWHML3/HWHMV3; xx4=HWHML4/HWHMV4; xx0=HWHML0/HWHMV0; xx1b=HWHML12b/HWHMV1b; xx2b=HWHML12b/HWHMV2b; xx3b=HWHML3b/HWHMV3b; xx4b=HWHML4b/HWHMV4b; xx0b=HWHML0b/HWHMV0b; nu01=0; nu02=-.03724; nu03=-.27199; nu04=-.5499; nu00=+0.2142; sigma01=S1*pH2Oc*L/(2*HWHMV1*(1.065+.447*xx1+.058*pow(xx1,2.0))); sigma02=S2*pH2Oc*L/(2*HWHMV2*(1.065+.447*xx2+.058*pow(xx2,2.0))); sigma03=S3*pH2Oc*L/(2*HWHMV3*(1.065+.447*xx3+.058*pow(xx3,2.0))); sigma04=S4*pH2Oc*L/(2*HWHMV4*(1.065+.447*xx4+.058*pow(xx4,2.0))); sigma00=S0*pH2Oc*L/(2*HWHMV0*(1.065+.447*xx0+.058*pow(xx0,2.0))); sigma01b=S1b*pH2Ob*Lb/(2*HWHMV1b*(1.065+.447*xx1b+.058*pow(xx1b,2.0))); sigma02b=S2b*pH2Ob*Lb/(2*HWHMV2b*(1.065+.447*xx2b+.058*pow(xx2b,2.0))); sigma03b=S3b*pH2Ob*Lb/(2*HWHMV3b*(1.065+.447*xx3b+.058*pow(xx3b,2.0))); sigma04b=S4b*pH2Ob*Lb/(2*HWHMV4b*(1.065+.447*xx4b+.058*pow(xx4b,2.0))); sigma00b=S0b*pH2Ob*Lb/(2*HWHMV0b*(1.065+.447*xx0b+.058*pow(xx0b,2.0))); m1=a12/HWHMV1; m2=a12/HWHMV2; m3=a3/HWHMV3; m4=a4/HWHMV4; m0=a0/HWHMV0; m1b=a12/HWHMV1b; m2b=a12/HWHMV2b; m3b=a3/HWHMV3b; m4b=a4/HWHMV4b; m0b=a0/HWHMV0b; 186 nuhat[1]=-1.3; for(k=2;k<=mm;k++) nuhat[k]=nuhat[k-1]+1.9/mm; stepwt=2*pi/nn; for (i=1;i<=mm;i++) { sum1=0.0;sum2=0.0;sum3=0.0;sum4=0.0;sum0=0.0; sum1b=0.0;sum2b=0.0;sum3b=0.0;sum4b=0.0;sum0b=0.0; for(j=1;j<=nn;j++) { y1=fabs((nuhat[i]-nu01)/HWHMV1-m1*cos(-pi+(j-1)*stepwt+sai)); y2=fabs((nuhat[i]-nu02)/HWHMV2-m2*cos(-pi+(j-1)*stepwt+sai)); y3=fabs((nuhat[i]-nu03)/HWHMV3-m3*cos(-pi+(j-1)*stepwt+sai)); y4=fabs((nuhat[i]-nu04)/HWHMV4-m4*cos(-pi+(j-1)*stepwt+sai)); y0=fabs((nuhat[i]-nu00)/HWHMV0-m0*cos(-pi+(j-1)*stepwt+sai)); y1b=fabs((nuhat[i]-nu01)/HWHMV1b-m1b*cos(-pi+(j-1)*stepwt+sai)); y2b=fabs((nuhat[i]-nu02)/HWHMV2b-m2b*cos(-pi+(j-1)*stepwt+sai)); y3b=fabs((nuhat[i]-nu03)/HWHMV3b-m3b*cos(-pi+(j-1)*stepwt+sai)); y4b=fabs((nuhat[i]-nu04)/HWHMV4b-m4b*cos(-pi+(j-1)*stepwt+sai)); y0b=fabs((nuhat[i]-nu00)/HWHMV0b-m0b*cos(-pi+(j-1)*stepwt+sai)); alpha1=sigma01*((1.0-xx1)*exp(-.693*pow(y1,2.0))+xx1/(1.0+pow(y1,2.0))+ .016*(1.0-xx1)*xx1*(exp(-.0841*pow(y1,2.25))-1.0/(1.0+.0210*pow(y1,2.25)))); alpha2=sigma02*((1.0-xx2)*exp(-.693*pow(y2,2.0))+xx2/(1.0+pow(y2,2.0))+ .016*(1.0-xx2)*xx2*(exp(-.0841*pow(y2,2.25))-1.0/(1.0+.0210*pow(y2,2.25)))); alpha3=sigma03*((1.0-xx3)*exp(-.693*pow(y3,2.0))+xx3/(1.0+pow(y3,2.0))+ .016*(1.0-xx3)*xx3*(exp(-.0841*pow(y3,2.25))-1.0/(1.0+.0210*pow(y3,2.25)))); alpha4=sigma04*((1.0-xx4)*exp(-.693*pow(y4,2.0))+xx4/(1.0+pow(y4,2.0))+ .016*(1.0-xx4)*xx4*(exp(-.0841*pow(y4,2.25))-1.0/(1.0+.0210*pow(y4,2.25)))); alpha0=sigma00*((1.0-xx0)*exp(-.693*pow(y0,2.0))+xx0/(1.0+pow(y0,2.0))+ .016*(1.0-xx0)*xx0*(exp(-.0841*pow(y0,2.25))-1.0/(1.0+.0210*pow(y0,2.25)))); alpha1b=sigma01b*((1.0-xx1b)*exp(-.693*pow(y1b,2.0))+xx1b/(1.0+pow(y1b,2.0))+ .016*(1.0-xx1b)*xx1b*(exp(-.0841*pow(y1b,2.25))-1.0/(1.0+.0210*pow(y1b,2.25)))); alpha2b=sigma02b*((1.0-xx2b)*exp(-.693*pow(y2b,2.0))+xx2b/(1.0+pow(y2b,2.0))+ .016*(1.0-xx2b)*xx2b*(exp(-.0841*pow(y2b,2.25))-1.0/(1.0+.0210*pow(y2b,2.25)))); alpha3b=sigma03b*((1.0-xx3b)*exp(-.693*pow(y3b,2.0))+xx3b/(1.0+pow(y3b,2.0))+ .016*(1.0-xx3b)*xx3b*(exp(-.0841*pow(y3b,2.25))-1.0/(1.0+.0210*pow(y3b,2.25)))); alpha4b=sigma04b*((1.0-xx4b)*exp(-.693*pow(y4b,2.0))+xx4b/(1.0+pow(y4b,2.0))+ .016*(1.0-xx4b)*xx4b*(exp(-.0841*pow(y4b,2.25))-1.0/(1.0+.0210*pow(y4b,2.25)))); alpha0b=sigma00b*((1.0-xx0b)*exp(-.693*pow(y0b,2.0))+xx0b/(1.0+pow(y0b,2.0))+ .016*(1.0-xx0b)*xx0b*(exp(-.0841*pow(y0b,2.25))-1.0/(1.0+.0210*pow(y0b,2.25)))); snu1=s1F*nuhat[i]+s2F*pow(nuhat[i],2.0)-sf*a12*cos(-pi+(j-1)*stepwt); snu2=snu1; snu3=s1F*nuhat[i]+s2F*pow(nuhat[i],2.0)-sf*a3*cos(-pi+(j-1)*stepwt); snu4=s1F*nuhat[i]+s2F*pow(nuhat[i],2.0)-sf*a4*cos(-pi+(j-1)*stepwt); snu0=s1F*nuhat[i]+s2F*pow(nuhat[i],2.0)-sf*a0*cos(-pi+(j-1)*stepwt); cos2=cos(1.0 *(-pi+(j-1)*stepwt)+phi); term1=(1.0-alpha1)*(1.0+snu1); term2=(1.0-alpha2)*(1.0+snu2); term3=(1.0-alpha3)*(1.0+snu3); term4=(1.0-alpha4)*(1.0+snu4); term0=(1.0-alpha0)*(1.0+snu0); term1b=(1.0-alpha1b)*(1.0+snu1); 187 term2b=(1.0-alpha2b)*(1.0+snu2); term3b=(1.0-alpha3b)*(1.0+snu3); term4b=(1.0-alpha4b)*(1.0+snu4); term0b=(1.0-alpha0b)*(1.0+snu0); sum1=sum1+term1*cos2; sum2=sum2+term2*cos2; sum3=sum3+term3*cos2; sum4=sum4+term4*cos2; sum0=sum0+term0*cos2; sum1b=sum1b+term1b*cos2; sum2b=sum2b+term2b*cos2; sum3b=sum3b+term3b*cos2; sum4b=sum4b+term4b*cos2; sum0b=sum0b+term0b*cos2; } I2f1[i]=sum1*stepwt/pi; I2f2[i]=sum2*stepwt/pi; I2f3[i]=sum3*stepwt/pi; I2f4[i]=sum4*stepwt/pi; I2f0[i]=sum0*stepwt/pi; I2f1b[i]=sum1b*stepwt/pi; I2f2b[i]=sum2b*stepwt/pi; I2f3b[i]=sum3b*stepwt/pi; I2f4b[i]=sum4b*stepwt/pi; I2f0b[i]=sum0b*stepwt/pi; I2f[i]=I2f1[i]+I2f2[i]+I2f3[i]+I2f4[i]+I2f0[i]; I2fb[i]=I2f1b[i]+I2f2b[i]+I2f3b[i]+I2f4b[i]+I2f0b[i]; } // -------------------- Making output file (START) FILE *fpr1; fpr1=fopen(out,"w"); for(k=1;k<=mm;k++) fprintf(fpr1,"%f %f %f\n", nuhat[k],I2f[k]); fclose(fpr1); // -------------------- Making output file (END) return 0; } 188 Appendix A3: A program for simulation of the second harmonic spectra of the R(4) manifold of the 23 band of CH4 //========================================================================= // This Program simulates the I2f signal when the absorption // profile has a Voigt profile for the case when intensity // modulation is taken into account. In this program the effect // of two types intensity modulation sf, sF, and also phase shift // between intensity and frequency modulation, sai, are considered. // In addition the detection phase, phi, is also included. // *** R(4) transition of CH4 *** //========================================================================= #include #include int main() { // Tuning range for simulation= 1.6 cm-1 centered at nuhat=0 cm-1 int i,j,k; const int nn=2000; const int mm=400; double cc,kk,bb,hh,ssigma,Qnucl,Qrot,Qvib,QT0,QT,nu1,nu2,nu3,nu4; double am,cos2,s1F,s2F,sf,T,Av,R,stepwt,L,sai,phi; double XCO2,XN2,XO2,XH2O,XCH4,pCO2,pN2,pO2,pH2O,pCH4,pCH4c; double gamaL1H2O,gamaL2H2O,gamaL3H2O,gamaL4H2O,gamaL1O2,gamaL2O2,gamaL3O2,gamaL4O2, gamaL1CO2,gamaL2CO2,gamaL3CO2,gamaL4CO2,gamaL1N2,gamaL2N2,gamaL3N2,gamaL4N2, gamaL1CH4,gamaL2CH4,gamaL3CH4,gamaL4CH4; double nuhat[mm+1],I2f[mm+1],I2f1[mm+1],I2f2[mm+1],I2f3[mm+1],I2f4[mm+1]; double S1,S2,S3,S4; double HWHML1,HWHML2,HWHML3,HWHML4,HWHMD1,HWHMD2,HWHMD3,HWHMD4, HWHMV1,HWHMV2,HWHMV3,HWHMV4; double m1,m2,m3,m4,nu01,nu02,nu03,nu04,xx1,xx2,xx3,xx4,y1,y2,y3,y4,pi; double sigma01,sigma02,sigma03,sigma04,alpha1,alpha2,alpha3,alpha4; double sum1,sum2,sum3,sum4,term1,term2,term3,term4,snu1,snu2,snu3,snu4; char* out="I2CH4R4.out"; pi=atan(1.0)*4.0; T=2000; // in (K) L=10.0; // in (cm) pCH4c=1.0; // in (atm), however in the graphs in the dissertation it is devided by 760 // for normalization per 1 torr am=0.057; // amplitude of FM (cm-1) 189 s1F=-1.869; // Laser intensity-frequency parameters s2F=-.900; sf=-3.5; sai=(0.0/180.0)*pi; // phase difference b/w FM and IM in (rad) phi=(140.0/180.0)*pi; // detection phase in (rad) Av=6.02e23; R=82.08; XCO2=1/(1+2+2*3.76)*100; // methane/air combustion phi=1.0 XN2=2*3.76/(1+2+2*3.76)*100; XO2=0; XH2O=2/(1+2+2*3.76)*100; XCH4=0.0; pN2=XN2*1e-2; // partial pressure of species in (atm) pO2=XO2*1e-2; pCO2=XCO2*1e-2; pH2O=XH2O*1e-2; pCH4=XCH4*1e-2; // Calculation of line strength at a given temeprature based on SHM Qnucl=16.0; QT0=585.132; bb=5.24; ssigma=12.0; cc=2.998e+10; kk=1.38066e-23; hh=6.62608e-34; nu1=2914.2; nu2=1526.0; nu3=3020.3; nu4=1306.2; Qrot=(1/ssigma)*pow( (pi/pow(bb,3.0)) * pow((kk*T/(hh*cc)),3.0),0.5 ); Qvib=pow(1- ( exp(-hh*cc*nu1/(kk*T)) ) ,-1.0) * pow(1- ( exp(-hh*cc*nu2/(kk*T)) ) ,-2.0) * pow(1- ( exp(-hh*cc*nu3/(kk*T)) ) ,-3.0) * pow(1- ( exp(-hh*cc*nu4/(kk*T)) ) ,-3.0); QT=Qnucl*Qrot*Qvib; S1=(0.021854*QT0*296.0)/(QT*T) * exp( -hh*cc*104.7800/kk*(1.0/T-1.0/296.0) ) * ( 1- ( exp(-hh*cc*6057.127/(kk*T)) ) ) * 1.0/ ( 1- ( exp(-hh*cc*6057.127/(kk*296.0)) ) ); S2=(0.022598*QT0*296.0)/(QT*T) * exp( -hh*cc*104.7747/kk*(1.0/T-1.0/296.0) ) * ( 1- ( exp(-hh*cc*6057.100/(kk*T)) ) ) * 1.0/ ( 1- ( exp(-hh*cc*6057.100/(kk*296.0)) ) ); S3=(0.031716*QT0*296.0)/(QT*T) * exp( -hh*cc*104.7728/kk*(1.0/T-1.0/296.0) ) * ( 1- ( exp(-hh*cc*6057.086/(kk*T)) ) ) * 1.0/ ( 1- ( exp(-hh*cc*6057.086/(kk*296.0)) ) ); S4=(0.020293*QT0*296.0)/(QT*T) * exp( -hh*cc*104.7760/kk*(1.0/T-1.0/296.0) ) * ( 1- ( exp(-hh*cc*6057.078/(kk*T)) ) ) * 1.0/ ( 1- ( exp(-hh*cc*6057.078/(kk*296.0)) ) ); // Calculation of broadening coefficient for different quenching species at the specified temperature gamaL1H2O=14.098/pow(T,0.907); // CH4-H2O broadening coefficient of transition 1 gamaL2H2O=gamaL1H2O; // CH4-H2O broadening coefficient of transition 2 gamaL3H2O=gamaL1H2O; // CH4-H2O broadening coefficient of transition 3 gamaL4H2O=gamaL1H2O; // CH4-H2O broadening coefficient of transition 4 gamaL1N2=10.984/pow(T,0.928); // CH4-N2 broadening coefficient of transition 1 gamaL2N2=gamaL1N2; // CH4-N2 broadening coefficient of transition 2 gamaL3N2=gamaL1N2; // CH4-N2 broadening coefficient of transition 3 gamaL4N2=gamaL1N2; // CH4-N2 broadening coefficient of transition 4 gamaL1O2=12.021/pow(T,0.916); // CH4-O2 broadening coefficient of transition 1 190 gamaL2O2=gamaL1O2; // CH4-O2 broadening coefficient of transition 2 gamaL3O2=gamaL1O2; // CH4-O2 broadening coefficient of transition 3 gamaL4O2=gamaL1O2; // CH4-O2 broadening coefficient of transition 4 gamaL1CO2=16.213/pow(T,0.950); // CH4-CO2 broadening coefficient of transition 1 gamaL2CO2=gamaL1CO2; // CH4-CO2 broadening coefficient of transition 2 gamaL3CO2=gamaL1CO2; // CH4-CO2 broadening coefficient of transition 3 gamaL4CO2=gamaL1CO2; // CH4-CO2 broadening coefficient of transition 4 gamaL1CH4=16.466/pow(T,0.952); // CH4-CH4 broadening coefficient of transition 1 gamaL2CH4=gamaL1CH4; // CH4-CH4 broadening coefficient of transition 2 gamaL3CH4=gamaL1CH4; // CH4-CH4 broadening coefficient of transition 3 gamaL4CH4=gamaL1CH4; // CH4-CH4 broadening coefficient of transition 4 // Calculation of effective (total) collisional halfwidths HWHML1=gamaL1H2O*pH2O+gamaL1N2*pN2+gamaL1O2*pO2+gamaL1CO2*pCO2+ gamaL1CH4*pCH4; HWHML2=gamaL2H2O*pH2O+gamaL2N2*pN2+gamaL2O2*pO2+gamaL2CO2*pCO2+ gamaL2CH4*pCH4; HWHML3=gamaL3H2O*pH2O+gamaL3N2*pN2+gamaL3O2*pO2+gamaL3CO2*pCO2+ gamaL3CH4*pCH4; HWHML4=gamaL4H2O*pH2O+gamaL4N2*pN2+gamaL4O2*pO2+gamaL4CO2*pCO2+ gamaL4CH4*pCH4; // Calculation of Doppler halfwidths HWHMD1 =.5*(7.1623e-7*6057.127*pow(T/16.0,0.5)); HWHMD2 =.5*(7.1623e-7*6057.100*pow(T/16.0,0.5)); HWHMD3 =.5*(7.1623e-7*6057.086*pow(T/16.0,0.5)); HWHMD4 =.5*(7.1623e-7*6057.078*pow(T/16.0,0.5)); HWHMV1=.5346*HWHML1+pow(.2166*pow(HWHML1,2.0)+ pow(HWHMD1,2.0),0.5); HWHMV2=.5346*HWHML2+pow(.2166*pow(HWHML2,2.0)+ pow(HWHMD2,2.0),0.5); HWHMV3=.5346*HWHML3+pow(.2166*pow(HWHML3,2.0)+ pow(HWHMD3,2.0),0.5); HWHMV4=.5346*HWHML4+pow(.2166*pow(HWHML4,2.0)+ pow(HWHMD4,2.0),0.5); xx1=HWHML1/HWHMV1; xx2=HWHML1/HWHMV2; xx3=HWHML3/HWHMV3; xx4=HWHML4/HWHMV4; nu01=.0412; nu02=0.0137; nu03=0.0; nu04=-.0083; sigma01=S1*pCH4c*L/(2*HWHMV1*(1.065+.447*xx1+.058*pow(xx1,2.0))); sigma02=S2*pCH4c*L/(2*HWHMV2*(1.065+.447*xx2+.058*pow(xx2,2.0))); sigma03=S3*pCH4c*L/(2*HWHMV3*(1.065+.447*xx3+.058*pow(xx3,2.0))); sigma04=S4*pCH4c*L/(2*HWHMV4*(1.065+.447*xx4+.058*pow(xx4,2.0))); m1=am/HWHMV1; m2=am/HWHMV2; m3=am/HWHMV3; m4=am/HWHMV4; nuhat[1]=-0.7; for(k=2;k<=mm;k++) nuhat[k]=nuhat[k-1]+1.4/mm; 191 stepwt=2*pi/nn; for (i=1;i<=mm;i++) { sum1=0.0;sum2=0.0;sum3=0.0;sum4=0.0; for(j=1;j<=nn;j++) { y1=fabs((nuhat[i]-nu01)/HWHMV1-m1*cos(-pi+(j-1)*stepwt+sai)); y2=fabs((nuhat[i]-nu02)/HWHMV2-m2*cos(-pi+(j-1)*stepwt+sai)); y3=fabs((nuhat[i]-nu03)/HWHMV3-m3*cos(-pi+(j-1)*stepwt+sai)); y4=fabs((nuhat[i]-nu04)/HWHMV4-m4*cos(-pi+(j-1)*stepwt+sai)); alpha1=sigma01*((1.0-xx1)*exp(-.693*pow(y1,2.0))+xx1/(1.0+pow(y1,2.0))+ .016*(1.0-xx1)*xx1*(exp(-.0841*pow(y1,2.25))-1.0/(1.0+.0210*pow(y1,2.25)))); alpha2=sigma02*((1.0-xx2)*exp(-.693*pow(y2,2.0))+xx2/(1.0+pow(y2,2.0))+ .016*(1.0-xx2)*xx2*(exp(-.0841*pow(y2,2.25))-1.0/(1.0+.0210*pow(y2,2.25)))); alpha3=sigma03*((1.0-xx3)*exp(-.693*pow(y3,2.0))+xx3/(1.0+pow(y3,2.0))+ .016*(1.0-xx3)*xx3*(exp(-.0841*pow(y3,2.25))-1.0/(1.0+.0210*pow(y3,2.25)))); alpha4=sigma04*((1.0-xx4)*exp(-.693*pow(y4,2.0))+xx4/(1.0+pow(y4,2.0))+ .016*(1.0-xx4)*xx4*(exp(-.0841*pow(y4,2.25))-1.0/(1.0+.0210*pow(y4,2.25)))); snu1=s1F*nuhat[i]+s2F*pow(nuhat[i],2.0)-sf*am*cos(-pi+(j-1)*stepwt); snu2=snu1; snu3=snu1; snu4=snu1; cos2=cos(2.0 *(-pi+(j-1)*stepwt)+phi); term1=(1.0-alpha1)*(1.0+snu1); term2=(1.0-alpha2)*(1.0+snu2); term3=(1.0-alpha3)*(1.0+snu3); term4=(1.0-alpha4)*(1.0+snu4); sum1=sum1+term1*cos2; sum2=sum2+term2*cos2; sum3=sum3+term3*cos2; sum4=sum4+term4*cos2; } I2f1[i]=sum1*stepwt/pi; I2f2[i]=sum2*stepwt/pi; I2f3[i]=sum3*stepwt/pi; I2f4[i]=sum4*stepwt/pi; I2f[i]=I2f1[i]+I2f2[i]+I2f3[i]+I2f4[i]; } // -------------------- Making output file (START) FILE *fpr1; fpr1=fopen(out,"w"); fprintf(fpr1,"x=[\n"); for(k=1;k<=mm;k++) fprintf(fpr1,"%f %f\n", nuhat[k],I2f[k]); fclose(fpr1); // -------------------- Making output file (END) return 0; } 192 Appendix B1: Some spectroscopic parameters of R(3) and R(4) manifolds of the 23 band of CH4 The vibrational quantum numbers of R(3) and R(4) manifolds of the 23 band of CH4 transitions are listed in Tables B1.1 and B1.2, respectively. Table B1.1: Vibrational quantum numbers of the 23 band of R(3) manifold of CH4 absorption transitions (Extracted from HITRAN database).  E'' v1' v2' v3' v4' n/C' V1'' v2'' v3'' v4'' n/C'' 6046.9420 62.8768 0 0 2 0 1F2 <- 0 0 0 0 1A1 6046.9527 62.8758 0 0 2 0 1F2 <- 0 0 0 0 1A1 6046.9647 62.8782 0 0 2 0 1F2 <- 0 0 0 0 1A1 Table B1.2: Vibrational quantum number of the 23 band of R(4) manifold of CH4 absorption transitions (Extracted from HITRAN database).  E'' v1' V2' v3' v4' n/C' v1'' v2'' v3'' v4'' n/C'' 6057.0778 104.776 0 0 2 0 1F2 <- 0 0 0 0 1A1 6057.0861 104.7728 0 0 2 0 1F2 <- 0 0 0 0 1A1 6057.0998 104.7747 0 0 2 0 1F2 <- 0 0 0 0 1A1 6057.1273 104.78 0 0 2 0 1F2 <- 0 0 0 0 1A1 More information about the spectroscopic parameters of CH4 can be found in the paper by Brown et al. (Brown, Benner et al. 2003) 193 Appendix B2: Some spectroscopic parameters of the selected H2O transitions Global and local quanta indexes, which are used to describe a particular transition, for selected absorption transitions are listed in Table B2.1. Table B2.1: Global and local quanta index of the selected H2O absorption transitions (Extracted from HITRAN database).  E'' v1' v2' v3' v1'' v2'' v3'' J' K'a K'c F' J'' K''a K''c F'' 6771.4753 931.237 0 2 1 0 0 0 6 4 2 7 4 3 562 6 6771.7101 1590.691 2 0 0 0 0 0 7 6 2 8 7 1 462 6 6771.7473 1590.69 2 0 0 0 0 0 7 6 1 8 7 2 462 6 More information about HITRAN transition parameters can be found in the paper by Rothman et al. (Rothman, Barbe et al. 2003) 194 Appendix C: Design and building the Ring Interferometer The ring interferometer is designed based on information presented in paper by Stokes (Stokes, Chodorow et al. 1982) Based on information presented in this paper, a 95/5 single mode directional coupler is used for building the interferometer. One of the input/out ports of the coupler is first cut in such a way that by splicing the ends of these ports together, a ring with an approximate length of 30 cm has been made. See Figure C.1. Figure C.1: Schematic of the ring interferometer The Free Spectral Range (FSR) of such interferometer will be 1/nL cm-1, where n is refractive index of waveguide and L is the length of the ring in cm. Since the waveguide of the coupler is made of fused silica, and fused silica has an approximate refractive index of 1.5 for infrared light, the FSR of the designed ring interferometer is approximately 1/(1.5?30)=0.022 cm-1. Since during building the ring interferometer there was some uncertainty in measuring the length of the ring, it is decided to calculate FSR more precisely. For measuring FSR, the spectra of the ring interferometer signal, as shown in Figure 5.15, was compared to absorption spectra of two H2O transitions, shown in Figure 5.15, where their frequency spacing is well known from HITRAN database. By comparison of these spectra, the FSR for the H2O 2?2 coupler 195 laser (1477 nm) is obtained as 0.0022694 ? 0.0000020cm-1. Also for CH4 laser, the FSR is calculated by 4622.1 4634.1 FSR FSR H2O CH4 CH4 H2O == n n (C.1) Therefore for CH4 laser, the FSR is calculated as 0.0022675 ? 0.0000020cm-1. 196 Appendix D: Glass Cell and its accessories Figure D.1: From top to bottom: Glass Cell and heater assembly, heater dimensions, and Glass Cell and its accessories dimensions 197 Appendix E: Absolute error analysis for linestrength ST and pressure broadening coefficient  In the following the uncertainty in the linestrength and also pressure broadening coefficients of the selected CH4 transitions are explained in detail and for H2O transitions only the results are presented. E.1 Uncertainty in LineStrength ST of the CH4 transitions From measurements of linestrength ST can be determined from partial pressure of the absorbing species Pabs, the path length, and the wavelength-integrated absorbance over the line, as     9= dI I LP S abs T 0ln1 E.1 From this expression the estimated uncertainties ( P/P), ( L/L), and ( /), where  is the value of the integral, can be obtained in the usual way 222    +   +   =   L L P P S S T T E.2 with the uncertainty ( P/P)=1%, due to pressure gauge, and estimating ( L/L)=0.3% (1 mm in 32.1 cm), and ( /)=2% from the uncertainty in the fit, we arrive from Equation (E.2) at a combined uncertainty estimate for ST of 2.3%. 198 E.2 Uncertainty in the pressure broadening coefficients of the CH4 transitions As p L = , one can calculate the relative error in pressure broadening coefficient as 22     +    = p p L L -  -  - (E.3) During the fitting processes, the uncertainty in Voigt half-width  and the Doppler half-width D , is propagated to the Lorentzian half-width L , by 2 2 2 2 )()( D D L V V L L - - -       +       = (E.4) Using Whitting?s expression, 5.022 )2166.0(5346.0 DLLV  + + = (E.5) one can calculate the derivative terms in Eq. (E.4) as shown in Eqs. (E.6) 2)/(3195.01 7257.67254.7 VDV L   +=   (E.6) 2)/(3195.0 1488.2 DVD L   +=   In these experiments, the maximum fractional error in pressure broadening  - occurs at the highest temperature T=952 K, for the N2 measurement which exhibits the maximum error in L L  - . At this condition 0167.0= D cm-1, 199 0057.00018.0 ' ' L cm-1, and 020.00177.0 ' ' V cm-1, therefore 95.083.0 ' ' V D   . Using this in Equation (E.6) we get 80.164.1 '  ' V L   , and 14.294.1 '  ' D L   . (E.7) For this condition the maximum error in Voigt half-width, which is due to the limited number of points in each sweep, is estimated as 00013.0020.0*)149/1(max == V- cm-1 (E.8) in which 1/149 represents the relative error in the Voigt half-width corresponding 1 data point error in the 149 points of each collected sweep over the line. The maximum error in Doppler half-width based on 2% uncertainty in temperature can be calculated as: 00017.0)95202.0( 95216 11.60571079.111079.1 70 7 =?? ? ???=?=  T TMD -- cm-1 (E.9) Using Eq. (E.7), Eq. (E.8), and Eq. (E.9) in Eq. (E.4), the maximum uncertainty in Lorentzian half-width is estimated as ( ) ( ) 00043.0)00017.0(14.2)00013.0(8.1 2222 =+= L- cm-1 (E.10) For the worst case, the pressure broadening coefficient is calculated based on the maximum absolute error of 0.00043 cm-1 over six different pressures. For the rang of pressures studied, the Lorentzian half-width is in the range of 0057.00018.0 ' ' L . This means that in the worst case the pressure broadening 200 coefficient is calculated based on six values for Lorentzian half-widths each having a relative error in the range of 24.0075.0 ' ' L L  - . Taking the maximum relative error in the measured Lorentzian half-width due to temperature and Voigt half-width uncertainty to be the average of range: 16.0max = L L  - (E.11) From this, the maximum absolute error in pressure broadening coefficient due to 1% uncertainty in pressure, 2% uncertainty in temperature, and 1/149=0.7% in Voigt half-width for the worst case measurement would be: ( ) ( ) 16.001.16.0max 22 =+= - For the lower temperature measurements the absolute error becomes much smaller. Using the same procedure explained above, the uncertainty in linestrength and pressure broadening coefficients of the selected H2O transitions are estimated as 7% and 15%, respectively. 201 Appendix F: Measurement of amplitude of modulation and intensity- frequency parameters F.1 Calculation of sF1, and sF2 These parameters are directly calculated from the second-order least square fitting of the intensity variation of the ring interferometer signal versus frequency. Since these parameters are normalized with respect to intensity at one of the absorption transitions (for H2O they are normalized at the absorption peak located between transition 1 and 2, while for CH4 they are normalized at the absorption peak of the manifold), the absorbance must be calculated in order to determining the location (frequency) where the parameters should be normalized. Typical values of these parameters for H2O and CH4 lasers can be found in Appendix A2 and A3 in the C++ program written for modeling the WMS signal of the selected transitions. F.2 Calculation of amplitude of modulation m Amplitude of modulation varies as the laser frequency is tuned over absorption transitions. At a given optical frequency, the amplitude of modulation can be obtained by detecting the ring interferometer signal when the central frequency of laser is fixed, at the given frequency, and only modulation current is applied to the laser. Based on the number of fringes during half cycle of the modulation and also Free Spectral Rang (FSR) of the ring interferometer, the amplitude of modulation m 202 can be calculated. Figure F.1 shows the frequency modulation of the H2O laser around optical frequency of II=6771.47 cm-1. Figure F.1: Ring interferometer signal of the H2O laser when it is modulated around II=6771.47 cm-1 at f=10 kHz. According to this figure, m is approximately equivalent to 4FSR but more accurate value of m can be obtained by comparing the calculated 2f signal with experiments at known condition. In this case, by only adjusting the m in the modeling, and comparing the results with measured 2f signal at known condition, m can be calculated more precisely. F.3 Calculation of sf For calculating sf, it is needed the ring interferometer signal at a given optical frequency is measured at the ramp frequency (see Figure F.2). Comparing this signal with corresponding signal at modulation frequency (such as the signal shown in Figure F.1), and also the calculated value for sF1, as explained in section F.1, this parameter can be calculated. 203 Figure F.2: Ring interferometer signal of the H2O laser when it is modulated around II=6771.47 cm-1 at F=10 Hz. Then sf, can be calculated by F f fm Fm Ff I I Ss 9 =   1 (F.1) where, subscripts f and F represent calculated parameters when the modulation frequency is f and F, respectively. I represents amplitude of intensity modulation (amplitude of intensity as shown in Figures F.1 and F.2 without considering fringes). Calculated values of sf can be found in the written C++ program for modeling 2f signal presented in Appendixes A2 and A3. F.4 Calculation of phase between frequency and intensity modulation  Based on Equation (4.19), the detection phase  is adjusted such that the magnitude of the measured 2f signal at the center of a particular transition frequency becomes maximum. At this condition, based on the measured 2,max, using Equation (4.19) one can calculate . 204 The measured values of  for H2O and CH4 lasers at the selected modulation frequencies (10 kHz for H2O laser and 12 kHz for CH4 laser) are -15? and 140?, respectively. 205 Appendix G: Temperature correction of the thermocouple measurements for radiation loss The in-flame temperature measurement using thermocouple must be corrected for radiation loss. The method of temperature correction used in this dissertation was based on a information presented in a paper by Weissweiler (Weissweiler 1994). According to this method, the corrected temperature Tc (in K) is calculated by ATdTT ddc 99 999 += H " 2 4 G.1 where Td is measured (displayed) temperature in K,  (in W/mK) is thermal conductivity of the thermocouple wire, d is bead diameter (in m), is the emissivity, and  is Boltzmann constant ( = 5.67?10-8 W/m2K4) and A is the view factor. The thermal conductivity of the type R thermocouple at temperature Td is calculated from (1994) 2853 104547.1101225.8106942.4 dd TT  ??+?=H G.2 For the in-flame measurement, explained in Chapter 6, type-R thermocouples with a thinness of 0.008 in. were used. In that experiment, the view factor of A=1 is considered as the surrounding temperature around the bid was almost the same as room temperature. The emissivity of type-R thermocouple is approximately 0.2. The bid diameter based on the manufacturer specification sheet is 2.5 of the wire diameter. 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