a model of temperature effects in pulsed and continuous wave co2 lasers and optimization using...

8/13/2019 A MODEL OF TEMPERATURE EFFECTS IN PULSED AND CONTINUOUS WAVE CO2 LASERS AND OPTIMIZATION USING

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D TIC MODEL OF TEMPERATURE EFFECTS INELECTE PULSED AND CONTINUOUS WAVE CO2 LASERSDEC 2' 989 AND OPTIMIZATION USINGRESPONSE SURFACE METHODOLOGY

-c -THESIS

Thomas B. Melancon, B.S.'Captain, USAF

AFIT/GSO/E?'P/ENS/89D- 5

D , . . .C u h releaeG,e

DEPARTMENT OF THE AIR FORCEAIR UNJIVERSITYAIR FORCE INSTITUTE OF TECHNOLOGY

Wright-Patterson Air Force Base, Ohio

S9 12 26 159


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AFIT/GSO/ENP/ENS/89D-5

A MODEL OF TEMPERATURE EFFECTS IN PULSED ANDCONTINUOUS WAVE CO2 LASERS AND

OPTIMIZATION USING RESPONSE SURFACE METHODOLOGY

THESIS

Presented to the Faculty of the School of Engineeringof the Air rorce Institute of Technology

Air UniversityIn Partial Fulfillment of th e

Requirements for the Degree ofMaster of Science in Space Operations

Thomas B. Melancon, B.S.Captain, USAF

December 1989

Approved for public release; distr ibution unlimited


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PrefaceThe first purpose of this study is to improve th e

pulsed CO2 laser model built by Major Stone, AFIT/ENP, byincluding the effects of temperature change throughout th elaser pulse. The second purpose is to build a continuouswave (CW) CO2 laser model. The two models will complementeach other for the study of trends in the lasers, for thesimulation of laser systems, and as instructional aids. Thelast purpose of this study is to explore optimizatien usingresponse surface methodology (,13M).

Throughout the model building process Major Stone, myadvisor, gave me invaluable help, guidance, and the wisdomof experience. I am also indebted Major Kelso, my co-advisor, for help and feedback during the writing process,and to Major Bauer for guidance with RSM.

My greatest thanks, however, is due my wife, Angela, whoexpected nothing but the best from my efforts, was mygreatest encourager, and endured my emotional absence formany weeks. She gave me peace and freedom to work byexpertly managing the household, raising the children,taking them to the doctor, teaching them diligently andexpertly, and introducing them to the Lord.

Thomas B. Melancon

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Table of ContentsPage

Preface ................................................ iiList of Figures..................... viiList of Tables .......................................... xiAbstract ........ .............. ............ xii

T. INTRODUCTION ............ ................. IBackground ................... ..................... 1Statement of the Problem ........... ...............Objectives of the Research ......... ... ............. 2Scope and Limitations-. . . . . . . . . . . . . 3

Assumptions. ............... .................. 3

II. LITERATURE REVIEW ............ .................. 6Overview ................. ....................... 6The Lasing Process ................ ................. 6Pumping the Laser ........... ... ............... 10Description of a Pulsed TEA CO2 Laser .. ....... 11Gilbert 's Model of a CO2 Pulsed Laser ....... 12Description of a Four-Level Point Model . . .. 12Gilbert 's Assumptions ....... ............. 13Gilbert 's Rate Equations ........... i5Stone's Mo,_ of a Pulsed CO 2 Laser .. ........ 16Stone's 4odifications ....... ............. 16Calculation of the Laser Output ....... ........ 18Temperature Effects in the Laser ... .......... 19Reduced Gain ............ ................. 19

Power Limitation .......... ................ 21Reduced Optical Quality ....... ............ 22

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PageTemperature Dependent Parameters ... .......... 23Thermal Populations of the Upper and LowerVibrational Levels ....... ............ 23

Thermal Population of the Rotational Levels 24Relaxation Rates .......... ............... 26Line Width ............. .................. 27Gas Density ............. .................. 29Heat Capacity ........... ................. 29

Temperature Rise During Lasing ..... ........... 30Energy Balance Equations ..... ........... 31

III. PULSED CO2 LASER MODEL, METHODOLOGY and ANALYSIS 33Overview ................. ........................Developing the Model ........... ................ 33

Rate Equations ........... ................ 33Energy Balance to Calculate the New TemperatureRecalculation of the Temperature DependentParameters ........... ................ 39Summary of Model Development ... ......... 41Verification and Analysis ....... ............. 42Verifying the Temperature rise .. ........ 42Verifying the Effect of Temperature on Peak Gain. 43Verifying the Effect of Temperature on Peak Pow.er

and Pulse Shape ....... ............. 45Verifying the Effect of Temperature on Energy 53Verifying the Effect of the Thermal EquilibriumPopulat ions of the Excited States . . .. 54Verifying the Effect of the New Equation for nb 55Verifying the Overall Computer Code ...... 56Validat] ,n and Analysis ....... .............. 57

Valiating the Power and Energy vs Temperature 57Validat.ing the Temperature Rise ... ........ 59Validating the Pulse Shape ... .......... 59Summary of Validation ....... ............. 69

Limits of the Model ............ ................ 69Four-level Model .......... ............... 70Point Model ............. .................. .... 71Fast Vibrational and Rotational Relaxation 72Sparsely Populated Excited States ....... .. 74iv


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PagePractical Limits Found During Validation . 76Limits Imposed by Equations and Data ..... 76Summary of Model Limitations ... ......... 77

IV. CONTIOUOUS WAVE CO2 MODEL, METHODOLOGY AND ANALYSIS 79Overview ................. ...................... 79Description of a CW Laser ........... ............. 79Steady-State Conditions ........... ............ 80Survey of Previous Models of CO2 CW Lasers ..... 80Developing the CW Laser Model ............ ........... 82

Assumptions ................. .................. 83Calculations ............ ................. 84

Verification and Analysis ........... ............. 93Validation and Analysis .......... ............... 97Validation Against Experimental CW Lasers . 97Validat ion Against Fowler's CO 2 CW Model . 104Summary of Validation ...... ............. .. 116Limits to the CW Model ......... ............... 117

Limits Imposed by Assumptions ... ......... 11 7Practical Limits Found During Validation 119Limits Imposed by Valid Range of Equations 120Summary of Limits to Model ...... .......... 120V. OPTIMIZING THE CW MODEL USING RSM ..... .......... 122

Overview ................. ...................... 122Response Surface Methodology (RSM) ... ......... 122Developing the CW RSM Model ...... ............ 124Validation ............... ..................... 130Optimization ............... .................... 130Results and Analysis ......... ................ 132Conclusions ................ .................... 137VI. CONCLUSIONS ............. .. .................... 140

Strengths of the Pulsed CO2 Model ....... . 140Limits of the Pulsed CO2 Model ..... ........... 14]Strengths of the Continuous Wave CO2 Model ..... 141v


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PageLimits of the Continuous Wave Model .... ........ 142Strengths of the RSM Model ....... ............. 143Limits of the RSM Model ........ .............. 143

VII. RECOMMENDATION....... ................... 145Pulsed CO2 Model ............. .................. 14 5Continuous Wave CO2 Model ........ ............. 14 6RSM Modeling and Optimizatizn ...... ........... 14 8

Appendix A: Definition of Terms .............. ............ 149Appendix B: RSM ANOVA Tables ......... .............. 1543ibliography .................. ...................... 158Vita ........................ .......................... 161

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List of FiguresPage

Figure 1. CO 2 Laser Vibrat ional Energy Levels . . . . 7Figure 2. Rotational Distribution as a Function of

Temperature .......... ................ 25Figure 3. Comparison of 7a Cdlculated by Wittemanand Wutzke, as a Function ofTemperature .......... ................. 27Figure 4. Peak Gain vs Initial Temperature ...... 43Ficrlire 5. Peak Power as a Function of Initial

Temperature .......... ................ 46Figure 6. Lctt Pulse Shape vs Initial Temperatureat Constant Pressure of 1 atm .. ....... 47Figure 7. Laser Pulse Shape vs Initial Temperatureat Constant Number Density ......... 48Figure 8. Laser Pulse Shape vs Initial Temperatureat Low Pressure (100 torr) ... ......... 49Ficure 9. Laser Pulse Shape vs Initial Temperatureat High Pressure (4 atm) ... .......... 49Figure 10. Calculated Temperature Rise During theLaser Pulse for the Rising TemperaturePulse (Pulse 1) in Figure 11. ...... ........ 50Figure 11. Comparison of Laser Pulse Shapes, RisingTemp (Pulse 1), Const Temp (Pulse 2) . . . . 51Figure 12. Populations of the Upper VibrationalLevel, na, and the Lower VibrationalLevel, nb, for the Rising Temp (Pulse 1)and the Const Temp Pulses (Pulse 2) inFigure 11.. .............. ................. 52Figure 13. Total Energy in the Laser Pulse vs

Initial Temperature ...... ............ 53Figure 14. Upper and Lower Vibrational Populations,na and nb, at 600 0 K ........... ............. 54

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PageFigure 15. Upper and Lower Vib Pops, na and nb,Calc by Old Eqn for nb, arid New Eqn fo rnb . . . . . . . . . . . . . . . . . . . . . 56Figure 16. Peak Power and Total Pulse Energy vzInitial Temperature ....... ............ 58Figure 17. Pulse Shape vs Pump Energy from Manes'Study ...................................... 61Figure 18. Calculated Pulse Shape as a Function ofPump Energy ............... ................ 62Figure 19. Pulse Shape vs Pump Energy from Manes'Study .................. 63Figure 20. Calculated Pulse Shape as a Function ofPump Energy . . . . . . . . . . . . . ..4Figure 21. Pulse Shape vs Gas Mix from Manes'Study............ . . . . . ...Figure 22. Calculated Pulse Shlape as a Function ofGas Mix . ..Figure 23. Experimental Pump Pulse and Lazer Pulse . . . 67Figure 24. Calculated Laser Pulse and Gain fromWitteman's Pive-Level Model, Duplicating

the Conditions of the Experimental Laserin Fiqure 23 ......... ................. ,.Figure 25. Calculated Z.dser Pulse and Gain usingFour- Level Model, Duplicating theConditions of Figure 23 .... .............Figure 26. Ca2cu].ted Vibrational Temperature fromWitteman's Five-Level M-Iodel ... . 71Figure 27. Calculated and Experimental VibrationalTemperatures of the Upper Vibrational

Level (CO2 001) as a Function of PumpEnergy . . . . . . . . . . . . . . . . . . .Figure 28. Calculated Vibrational Temperatures ofCO2 001 vs Pump Energy .......... 5

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PageFigure 29. Comparison of* c at 500'K vs Pressure.Calculated using Doppler, Pressure, andQuasi-Voigt Line Shapes .... .......... 88Figure 30. Laser Pulse from the Quasi-CW code .. ..... 94Figure 31. CW Power vs Reflectivity ....... .......... 97Figure 32. Comparison of Beverly's ExperimentalOutput and Author's Calculated Output vsPumD Power . . . . . . . . . . . . . . . . IFigure 33. CW Laser Power vs Temperature andPressure from Mitsuhiro ..........Figure 34. Calculated CW Laser Power vs Temperature

and Pressure . . . . . . . . . . . . . . . .Figure 35. Small-Signal Gain vs Temperature andElectron Number Density from Fowler . ..Figure 36. Small-Signal Gain vs Temperature andPump Power . . . . . . . . . . . . . . . . .Figure 37. Saturation Flux Calculated by Fowler andAuthor's Model . . . . . . . . . . . .. . .Figure 38. Calculated Saturation Flux from

Douglas-Hamilton and Author's Mce1 vsTemperature . . . . . . . . . . . . . . . .

Figure 39. Optical Power Density, Is*Go, vsTemperature and Electron Number Densityfrom Fowler ............ ...................

Figure 40. Calculated Optical Power Density, Is*Go,vs Temperature and Pump Power ... ....... 11 0Figure 41. Small-Signal Gain and Satuiation Flux vsTotal Gas Pressure from Fowler .. ........ I.1Figure 42. Calculated Small-Signai Gain andSaturation Flux vs Total Gas Pressure 112Figure 43. Small-Signal Gain and Optical PowerDensity vs Gas Mix from Fowler .. ....... .. 114

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PageFigure 4, Calculated Small-Signal Gain vs Gas Mix.Same conditions as Figure 43 ... ........ 114F:gure 45. Calculated Optical Power Density vs GasMix ................ .................... 115Figure 46. Residual Plot from the Second CW RSMExperimental Design, Second-Order Model . . 128Figu3re 47. Residual Plot from the Fourth CW RSMExperimental Design, hird- Order

High-Resolution Model . . . . . . . . . . . ....Figure 48. CW Output Power as a Function of % N-and Temperature, and % N2 andReflectivity. 10 torr and 4% CO2 ......

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List of Tables

Table PageI. Gain vs Temperature ........................... 45

II. Experimental and Calculated CO2 CW LaserOutput .............. .................... 102III. Summary of CW RSM Models ..... .......... 13 4

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AFIT/GSO/ENP/ENS/89D-5

Abstract

The purpose of unis study is to develop models of pulsedand continuous wave (CW) CO. lasers on a personal comrnuter,and to explore optimization of the computed output powerusing response surfa-e methodloqy (RSM).

The first program is based on i four-level point modelof a pulsed CO 2 laser that predicts the temperature rise andthe effects of temperature on the power, energy, and gain ofthe pulsed laser. The second progran, derived from tfefirst, is based on a four-level point model of a cont inuouswave CO 2 laser and calculates the power, gain, small-signalgain, and saturation flux of the CW laser. Both mode.s wereverified and extensively val idated against published datawith good results.

Finally, this study explored the optimization of the CWlaser model using RSM and found an operat ing point fo rmaximum power at 10 torr and 4% CO 2.

Several recomyrendations are included for upgrading th eCO 2 laser models and cont inuing the optimi;-ation.

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A MODEL OF TEMPERATURE EFFECTS IN PULSED ANDCONTINUOUS WAVE CO 2 LASERS AND

OPTIMIZATION USING RESPONSE SURFACE METHODOLOGY

I. INTRODUCTION

BackgroundSince the invention of the carbon dioxide (C0 2 ) laser in

1964, CO2 lasers have proven superior in many applicationsbecause of their high efficiency and output. They arecurrently being considered for a space-borne LADAR (laserradar) system as an integral part of the Strategic DefenseInitiative. Both continuous wave (CW) and pulsed C02 laserswould be part of such a system.

Many researchers have worked to optimize both the CW andpulsed C02 laser, and several have built computer models.Major Stone, AFIT/EN, built a model of a pulsed C02 laser ona personal computer to investigate its trends, model a CO,LADAR system, and optimize the computed output.

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Statement of the ProblemWhile Major Stone's current pulsed laser computer model

is good, it has l imitations. It cannot simulate a CW laser,is difficult to optimize, and accounts for only two of sixpossible temperature dependent parameters. In his currentmodel, temperature is accounted for only by the change indensity of the active medium and the change in the linewidth. Temperature needs to be accounted for by the changein the rotational levels of the CO2 molecules, by the changein their relaxation rates, by the change in the thermalpopulation of the vibrational levels, and by the change inthe heat capacity of CO2. Integrating these additions intothe Runge-Kutta integration algorithm which calculates th elaser output would allow a more accurate simulation of thepulsed C02 laser.

In addition, the computer model could be expanded tosimulate a C02 CW laser. A CW model would complement th epulsed model, could simulate a more complete LADAR system,and might be more easily optimized.

Oblectives of the ResearchThis thesis will expand the current computer model of

the C02 laser through three objectives.The first objective of the research is to write an

algorithm to be incorporated into the current computeL model

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of a pulsed CO2 laser to model the temperature change of th eactive medium and to i teratively calculate the temperaturedependent parameters.

The second objective is to build a simple, user-friendly, fast-executing computer model of a CW CO2 laser.

The third objective is to explore using response surfacemethodology (RSM) to optimize the computed output power ofthe CW laser model and to evaluate its use for optimizingthe pulsed laser model.

Scope and Limitations. This thesis effort will onlyconsider temperature efrects on the gas kinetics and photoninteraction in the active medium of the pulsed CO2 laser.The temperature effect on laser structure, like the mirrors,will not be considered. In addition, the temperature effecton the optical quali ty or frequency of the laser beam willnot be considered.

The CW model will also only consider the reactionkinetics of the active medium that produce the laser beam.This study will not include fluid dynamics (flowingsystems), heat transfer, resonator modes, or the physics ofelectron beam pumping.

Assumptions. Both the pulsed and CW models in thisstudy rest on the assumption that Gilbert 's four-level pointmodel is adequate. Gilbert assumes that the vibrational androtational modes reach equilibrium much faster than other

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processes and the excited states of CO 2 and N2 are sparselypopulated. The following processes are assumed by Gilbertto be slow or have little effect and are icnored: th edissociation of C0 2 , the relaxation of the CO2 001vibrational level directly to the ground state, and th eradiative relaxation of the excited states. Thus, allrelaxation rates are collisional. Finally, the rate ofenergy exchange between excited N2 and CO2 is assumed to bemuch faster than the relaxation of either N, or CO-(11:2524-26). G ilbert 's assumptions are further enumeratedin the literature review.

This study also adopted some of Stone's assumptions.He assumes that only one fourth of the CO2 molecules thatrelax from the upper vibrational level (001) add to thepopulation of the lower vibrational level (100). Stone alsoassumes that the laser operates in only one resonator modeand lases on the center of the P(20) line shape.

This study added the assumption that only one fourth ofthe CO 2 molecules that leave the upper laser level bystimulated emission also add to the population of the lowervibrational level.

This study made three additional assumptions whilebuilding on G ilbert 's and Stone's models. The model of thetemperature effects in the pulsed laser assumes constant gasdensity throughout the laser pulse forming process. To

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calculate gas heating, any energy added by the electron beampump that does not appear as molccular vibrational energy orlaser energy is assumed to be thermal energy of the gas. Inaddition, only the heat capacity of CO2 is assumed to changesignificantly with temperature over the operating range oftne laser.

The CW laser model ircluded all the above assumptionsand is modeled as a four-level point model at steady state.The excited populations, photon density, and temperature areassumed to be constant throughout the laser cavity. Nospecific accounting of the heat remoial system is included;it is assumed that the cooling system is able to maintain aspecified bulk temperature in the laser. In addition, thequasi-Voigt l ine shape is assumed to be adequate.

Finally, for the RSM study, this study assumes that athird-order polynomial is adequate to fit the computed powerfrom the CW laser model over the region of interest.

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II. LITERATURE REVIEW

OverviewThis literature review describes the las ing process, a

pulsed CO 2 laser, and analytical models of the pulsed CO 2laser. A review of the effects of temperature on the lascrfollows, and then equat ions for the temperature dependentparameters are described. Last, the energy balance equationsrequired to calculate the temperature rise of a laser duringoperation are presented.

The Lasing ProcessThe foundation of the CO 2 laser is the vibrational and

rotational energy levels of the CO 2 molecule and theirinteraction with the laser light. The following simplifiedillustration (Figure 1) of how the energy levels of the CO 2molecule produce laser light will provide the background forunderstanding the effects of temperature on the lasingprocess. The 000 level is the ground vibrational mode, the100 energy level is the first symmetric stretch vibrationalmode, the 010 and 020 energy levels are the bendingvibrational modes, and the 001 energy level is th easymmetric stretch vibrational mode.

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/C02 N 2i- 001 v=l

CO CO iO. 6um 2 C-S0202 100 photon

" pump pump pump010 1

Energy 000 ground stateFigure 1. CO Laser Vibrational Energy Levels (>icollisional refaxation rates.)

At room temperature, nearly all of the CO2 molecules arein the ground vibrational state (000). An electron beam,normally produced by a cathode and anode at high voltage, isused to excite the molecules to the different vibrationalenergy levels. This process is called pumping. Nitrogen isalmost always mixed with the CO2 in the active medium (thegas mixture in the laser) because N2 is excited moreefficiently than CO 2 and very readily transfers itsvibrational energy to the CO2, pumping it up to the 001level.

The 001 level, called the upper vibrational level, willemit a photon with a wavelength of 10.6 microns (in the farinfrared spectrum) and leave the molecule in the 100 level,called the lower vibrational level. If this photoninteracts with another CO2 molecule in the 001 level, it can

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stimulate the second molecule to emit a photon in the samedirection and in phase with the first. This is calleds t im ulated emission. However, in a process calledst im ulated absorption, if the photon encounters a CO 2molecule in the 100 level , the molecule can absorb thephoton and be raised to the 001 level .

A complete description of the CO 2 moiecule 's energystate must include the rotational energy levels, which alsoparticipate in the las ing process. Rotational eneraylevels, described by a "j" number, are spaced much moreclosely than vibrational levels, and each vibrational levelcontains many rotational levels. For the purposes of thisstudy, a single CO 2 molecule can have only one vibrat ionaland one rotational level. Thus, the complete description ofthe upper laser level is 001, j = 19, and the lower laserlevel is 100, j = 20. This is called the P(20) transition,and is the normal lasing transition for the CO 2 laser. TheCO 2 laser normally operates on the P(20) line shape(produced by the P(20) transition) because j :19 is the mostpopul ted rotational level at room temperature.

If the active medium of the laser is to produce a netpositive output of photons for the laser beam, there must bemore molecules in the upper laser level (001, j=19) than inthe lower laser level (100, j=20). This unnatural state of


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affairs is called a population inversion and is absolutelynecessary for lasing.

After a molecule is excited, it can either lase or relaxcollisionally by transferring its vibrational energy toanother molecule or to the walls of the laser cavity. Itcan also relax radiatively by emitting a photon that carriesaway the vibrational and rotational energy the moleculeloses.

The quantum efficiency ot the CO2 laser can also bededuced from the energy level diagram. It is the ratio ofthe energy -- r led away by the emitted photon to the energystored in the CO2 molecule at the 001 level. This isapproximately 38% and is the limiting effiuiency for th elaser.

In summary, the electron beam pumps some of the CO 2 upto the 001 level (either directly or through the N2 ) toproduce a population inversion. The excited CO2 moleculelases on the P(20) transition, producing a 10.6 micronphoton and leaving the CO, molecule in the 100, j = 20state. This molecule then relaxes through the 020 and 010levels to the ground state, giving up the excess vibrationalenergy as heat. At the ground state, it can be pumped up tolase again. This process occurs in both CW and pulsedlase,- .

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Pumping the Laser. As mentioned previously, the laseris pumped by applying a high voltage across the anode andcathode in the laser cavity. The pruduct of the voltage andthe resulting current is equal to pump power, Pin = E ICIwhere Pin is the pump power, E is the voltage, and IC is th ecurrent. The current is proportional to the electron numberdensity in the active medium, Net and is often held fairlyconstant. The voltage, however, is often varied with th etotal molecular number density, No, so as to keep E/N-constant at approximately 2.OxlO'1 6 V-cm2 (24:159). Thisproduces an average electron energy of about 2 eV, which isthe most efficient for pumping N2 (24:67). According toWitteman, electrons with less than 0.3 eV of energy heat th egas instead of exciting the upper vibrational states of CO,and N2 . Electrons with more than 3 eV excite electronicstates of the molecules, resulting in lower efficiency(24:67).

Since the best pump efficiency is obtained by keepingE/N., and thus the average electron energy, constant, thenthe pump power is proportional to the total number densityand electron density:

Pin = (E/No) No Ic a (E/No) No Ne (1)

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If the electron number density is not constant but is alsoproportional to the molecular number density (to keep th eratio of electrons to molecules equal), then the pump powervaries as the square of the molecular number density.

Description of a Pulsed TEA CO2 LaserA Transversely Excited Atmospheric (TEA) pulsed CO 2

laser is one of the possible CO 2 laser systems thatcapitalize on the lasing process described above. It canoperate at high pressure (up to several atmospheres) and iscapable of very high energy output. In one possibleconfiguration of a TEA laser, the laser cavity consists of along cavity (which contains the active medium), a fullyreflective mirror at one end, and an output mirror at th eother end. The cathode and anode are positioned parallel tjthe cavity so that the electrons flow through the activemedium t ransversely to the axis of the cavity.

During operation, the active medium is pumped with apulse of electrons (the pump pulse) which flow from thecathode to the anode. This pulse )f electrons excites anumber of CO2 molecules, which begin to lase. The number ofphotons in the cavity continues to build exponentially untilthe gain ,creases below the threshold gain, and the laserpulse stops building. This short pulse of laser radiation

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begins to exit the laser through the output mirror while itis forming and is completely gone in a few microseconds.

Gilbert's Model of a CO 2 Pulsed LaserStone's current computer model of a pulsed TEA laser is

based on the work of Gilbert, et al. (11:2523-35), andunderstanding Gilbert's model is basic to understandingStone's model. Gilbert's work was particularly suited toadaptation to a personal computer by Stone because Gilbertused a four-level point model and several simplifyingassumptions that reduced the computational complexity.

Description of a Four-Level Point Model. Gilbert'sfour-level point model assumes that both a four-level modeland a point model are adequate A point model assumes thepoculation inversion and the radiation intensity are equolat every point throughout the laser cavity. This assumptionlimits the rate at which the photons can leave the cavity,and it also limits the round trip gain - length pnoduct ofthe laser to less than one. Both l imits mean a point modelapplies only to a short laser or low gain lasers.

A four-level point model keeps track of the populationsin four energy levels. Gilbert chose the four levels to be :

1. CO2 (001), the upper vibrational level;2. CO2 (100 + 020 + 010), the lower vibrational level

plus levels it is in equilibrium with;

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3. N2 (v=l), the first excited state of N2;4. and the ground states of CO2 and Nd.The foundational assumption for a four-level model is

that the relaxation rate between the 100 and 020 vibrationallevels of CO2 is much faster than the other relaxationrates, and so the 100 and 020 levels are always roughly inequilibrium. Using Witteman's rate equations, the rate forexchange between the i00 and 020 levels is 1.2x1C3 per sec,-, is 2.05x10 6 , and a is 8.6x10 4 per sec at 3iLC an(24:164).

Other authors, however, have used five-level thzseparate CO2 (100) from CO2 (020 + 010). For a comrar inof my four-level model (derived from Gilbert 's rodelWitteman's five-level model, see page 70.

Gilbert 's Assumptions. In addition to the assc-that a four-level point model is adequate, Gilbert -a

three other major simplifying assumptions. The firstassumption is that the vibrational and rotational ojJesreach equilibrium much faster than the other processes Ir.the laser. The second assumption is that the excited statesof CO2 are sparsely populated. The third assumption is th3tall reaction rates except those specifically accounted fcrare negligible. These assumptions were confirmed by Gilbertin the literature or through his experimentation with anactual laser (11:2523-27).

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Fast Vibrational and Rotational Equilibrium.Gilbert 's first simplifying assumption is that th evibrational and rotational modes reach equilibrium fasterthan other laser processes. The vibrational and rotationalmodes reach equilibrium very quickly because the CO 2molecule can exchange vibrational or rotational energy everyfew collisions, and the time between coll i- ions is verysmall compared to other processes in the laser. This -.asconfirmed by Witteman, who reported the lifetime of the CO2001 level to be 10'3 seconds, while the time betweencollisions is only 10s seconds at 10 tort (24:63). Atnigher pressures, the collision time can be 10-1 seconds,while the pulse forming process takes approximately 10-seconds. Each vibrational and rotational distr ibution isthus maintained in equilibrium by frequent collisionsthroughout the pulse forming process, which allows th ecomplex rotational distr ibution to be ignored for a four-level model (11:2523-27).

Spars-ely Populated Excited States. Gilbert 'ssecond simplifying assumption is that the excited states aresparsely populated. Assuming the excited states of C02 aresparsely populated led Gilbert to assert that the relaxationrates are independent of the concentration of the excitedstates, the ground state population stays essentiallyconstant, and the pumping efficiency of C02 does not vary

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with the concentration of N2 (11:2523-27). See page 74 foran analysis of this assumption.

This assumption of sparsely populated excited statesalso means that the lowest excited levels above ground statecontain almost all the excited molecules. This modelassumes that all the CO 2 excited by the electron beam ispumped into the 100, 010, 020, and 001 levels, and all theN2 excited can be found in the first N2 vibrational level(v=1).

Negligible Reaction Rates. Gilbert 's thirdsimplifying assumption is that all the reaction rates exceptthose specifically accounted for are negligible. Thefollowing processes are assumed by Gilbert to be slow orhave little effect and are ignored: the dissociation of C02,the relaxation of the 001 level directly to the groundstate, and the radiative relaxation of the excited states.In addition, the rate of energy exchange between excited N2and CO2 is assumed to be much faster than the relaxation ofeither N2 or CO 2.

All of the above assumptions and simplifications allowedGilbert to model the very complex CO2 laser as a four-levelsystem, described by four coupled rate equations (11:2523-27) .

Gilbert 's Rate Equations. The heart of the Gilbert 'smodel consists of four differentlai rate equations which

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describe the rate of change of the upper vibrational level,the lower vibrational level, the excited N2 , and the photonsin the cavity as follows:

dna /dt = I c c(nb, - na) - 7a na +-I,(n - n.) + w, (2)dnbl /dt = I ac (n. - nbl) + -a na - 7b nb- + wb (3)dnc /dt = yc(na - nc) - 7co nc + w (4)dI/dt = -7o I + I a c(na - nrb,) + na ws (5)

where n, is the number density of the ith excited level, Iis the photon flux, -y is the relaxation rate from the ithlevel, and wi is the pump rate into the ith level. Thesubscripts are defined, in general, as follows: a refers toC02 (001), b' refers to CO2 (100 + 010 + 020), and c refersto N2 (v=l) (11:2525). All the terms are specificallydefined in appendix A.

Stone's Model of a Pulsed CO2 LaserStone took Gilbert 's model, modified it, and wrote th e

model into Quick BASIC computer code to fit on a personalcomputer. This fast executing code was designed todemonstrate the trends of a pulsed CO2 laser.

Stone's M odifications. Stone's modifications toGilbert 's model included adding one change and two major

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assumptions, changing the rate equations, and usingWitteman's l ine width and relaxation rate equations.

The first change is that the lower vibrational levelconsists of only CO 2 (100), the lower excited state thatactually interacts with the laser photon, as opposed toG ilbert 's lower vibrational level, which consists of threeexcited levels, CO2 (100 + 020 + 010). The first assumptionis that the 020 and 010 levels are always in equilibriumwith the CO 2 (100) level and equal to it.

Stone changed Gilbert's rate equat ions by incorporat ingthe above assumptions, adding the degeneracy ratio, andseparating the rates between CO 2 (001) and N2 (v=l) . ThusStone s model is still a simple four-level model, but th erate equat ions are changed as follows:

dna /dt = I a c(gu/gl n. - na) - 7'a na + Yco2 nc-- Yc,2 na + wa (6)

dnb /dt = I c c(na - gu/gl nb) + -y n,/4- 7b nb + Wb (7)dnc /dt = 7cn2 na - 7co2 nc + wc (8)dI/dt = I a c(na - gu/gl nb) - I/rcav + n, w. (9)

where nb is the population of CO2 (100) level, gu/gl is th edegeneracy ratio, 7co2 is the energy transfer rate from 12 toC02, 7cn 2 is the rate from C02 to N2, and TC., is the photonl ifetim e in the cavity. The divisor of 4 in the equation

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for nb accounts for the assumption that only one fourth ofthe CO 2 (001) molecules that relax to the lower vibrationallevel enter the 100 level: the other three fourths enter th edouble 020 levels and the 010 level, which are roughly inequilibrium with the 100 level. The term for directrelaxation of N2, 7co nC, was dropped from Equation 4 becauseStone assumed that all the N2 relaxes through the CO2vibrational levels.

The second assumption added by Stone is that the laseralways operates in one laser cavity mode and on the centerof the P(20) l ine shape. Thus, only the rotational levelsj=19 and j=20 participate in lasing. Although Gilbertmentioned that the point model does not limit the number ofcavity modes that can exist inside the cavity (11:2523-27),Stone assumes one longitudinal cavity mode only.

Calculation of the Laser Output. The coupled

differential rate equations presented by Gilbert andmodified by Stone (Equations 6 to 9) are integrated inStone's computer model with a fourth-order Runge-Kuttaintegration algorithm to calculate the laser output overtime.

Stone's Accounting of Temperature. In solving thecoupled rate equations, temperature is used in Stone's modelto calculate two parameters: the l ine width -- used tocalculate the gain -- and the density of the gas mixture in

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the laser. The l ine width, Av, is related to temperature byAV = Avo*(300/T) , where Lv is the line width attemperature T, 6vo is the l ine width at 300 0 K, and T is th etemperature in "K, assuming constant pressure.

The density of the gas is calculated using the ideal gaslaw at constant volume, No = P/kT. The density and mix ofthe gas determines the number of CO2 molecules available tolase.

Temperature Effects in the LaserAlthou-h Stone's computer model accounted for two

temperature dependent parameters, it did not account for allthe effects of temperature. The temperature of the gaseousmedium greatly affects the power output of a CW laser andthe total energy and shape of the pulse from a TEA laser.The greatest effects are reducing the gain, l imiting th epotential power, and reducing the optical quality of thelaser output. The following paragraphs discuss the effectsof temperature on a C02 laser.

Reduced Gain. The first effect of increased temperatureis reduced gain. Temperature reduces the gain through threemechanisms: reducing the population inversion, decreasingthe radiative cross section, and reducing the number densityof the gas. The gain equation

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G = (na Cu - nb CL gu/gl) Z a (n. - nb) (10)

illustrates each of these effects. Increasing temperature

will reduce the populat ion inversion, (na - nrb , byincreasing the population of the lower vibrational level,nrb, and increasing the relaxation rate of the uppervibrational level, na, (24:3, 71). Both effects reduce th epopulation inversion and thus reduce the gain of the laser.

Increasing temperature will also increase the line widthand decrease the population of rotational level j = 19, bothof which decrease the radiative cross section, an,: thusreduce the gain. The last effect of temperature on gain isto reduce the number density in the gas (at constantpressure), and thus reduce the number of CO2 moleculesavailable to be excited and lase.

Fowler found that the small-signal gain drops off tonearly zero at temperatures of 500 to 700

0K. His conclusionis, Low gas temperature is seen to be one of the most

important properties that a CO2 laser plasma can have"(10:3484-86).

One exception to the rule, however, was highlighted byDeutsch. He found that "the more effective rotational crossrelaxation at higher temperatures can result in an increaseof the saturation parameter and can partially compensate forthe decrease of gain with temperature" (6:947). In spite of

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this partial compensation, the CW laser Deutsch studieddecreased in power from 43 watts to only 28 watts as th etemperature of the cooling jacket was raised from -60 0C to+90'C (6:974).

Power Limitation. The second effect of increasedtemperature is limited power. Witteman found, like Deutsch,that the power decreases with increasing temperature.According to Witteman, since the quantum efficiency is 18%,at least 62% of the power introduced by the pump iseventually converted to heat. This heat reduces the gain,which limits the power output. At about 150 0C (423 0 K) , th epower begins to decrease. Without active cooling to carryaway the generated heat, the maximum pump energy is 300joules/liter (J/l) and the maximum output is 40 J/l (24:3).

To overcome this l imitation, Witteman discussed twotypes of active cooling that have been designed into manylasers. In the first type, the gaseous medium is cooled byconduction through the walls of the laser cavity to a water(or other coolant) jacket. The second cooling technique isto replace the hot gas with new, cool gas by flowing the gasthrough the laser cavity (24:3). Taking this technique onestep further, Mitsuhiro studied a fast flowing CW CO2 laseroperated below room temperature and found a 50% increase inpower at 200 0 K. He also found the pump efficiency increasedfrom 15% to 20% (13:680). Drobyazko found a similar 50%

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increase in energy for a cooled, pulsed CO2 laser operatedat 200 0 K. He found the heat created during the pulseformation at room temperature increased the lowervibrational level population to the point of terminating thepulse before the upper vibrational level was depleted(8:974).

Reduced Optical Quality. The last effect of hightemperature in the laser medium is to reduce the opticalquality of the laser. Poor optical quali ty of the laserbeam is the result of inhomogeneities in the laser medium.*rhese inhomogeneities arise when a temperature change causesthe density and refractive index of the gas to change. BothCW and pulsed TEA lasers are vulnerable to temperaturevariations. Witteman calculated the temperature profile inthe medium of a CW laser and found the temperature variedfrom 290 0 K at the cavity wall to over 800 0K at the centerfor an input energy of 500 watts/meter (24:91). Theresulting density gradient acts like a diverging lens tospread the beam and distort its intensity profile.

The heat introduced to the medium of a pulsed laser alsochanges its density and optical properties. Milonnicalculated a temperature rise from 300 to 550 0K in 25microseconds during the pulse forming process of a high-power laser (15:3595). Fedorov calculated approximately th esame temperature rise in 15 microseconds for an input energy

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of 250 J/l. This rapid rise in temperature creates a localpressure wave wYeh non-uniform density which travels throughthe laser cavity at the local speed of sound. The transittime of the pressure wave, however, is longer than laserpulse time, so the effect on optical quali ty is not severe(9:629-30). This model does not calculate the effect oftemperature on optical quali ty.

Temperature Dependent ParametersOf the three major effects of temperature, two --

reduced gain and power l imitation -- are mathematicallydescribed by six tempe-ature dependent parameters. The sixparameters are the thermal populations of the vibrationallevels, the thermal population of the rotational energylevels (j levels) of the C02 molecule, the relaxation ratesof the vibrational enerqy levels of the molecules, the linewidth, the gas density, and the heat capacity of CO2.

Thermal Populations of the Upner and Lower VibrationalLevels. The first temperature dependent parameter is th ethermal population of the vibrational levels (CO2 001 and100). The thermal population of a vibrational level isdescribed by the Boltzmann equation:

Ni = No exp(-Ei/k T) (11)

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where N, is the number of molecules in the ith vibrationallevel, No is the number of molecules in the ground level,and E, is the energy difference between the grc-..id and th eith vibrational level.

The thermal population of the vibrational levels isdistributed exponentially. At low temperatures th eexponential term is very small and the population of th elower vibrational level is small. At higher temperaturesthe population can be substantial -- 2.0% at 500 0K and 6.2%at 700 0K for CO2 (100).

Thermal Population of the Rotational Levels. The secondtemperature dependent parameter is the thermal population ofthe rotational levels. The fraction of molecules in arotational level is also given by the Boltzmann equation,presented by Witteman as:

fvi = rvj /n, = C (2j+l) exp(-F(j) h/k T) (12)

whereC = 2 h B , k T, a normalization factorF(j) = B, j(j + 1) + D, j (j+l)' (Hz)Bv,Dv = rotational constants, different for each

vibrational level (Hz) (24:16)

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and where f. is the fraction of molecules in vibrationallevel v that are in rotational level j, n, J is the number ofmolecules in rotational level j and vibrational level v, andn = number of molecules in vibration level v.

This equation describes the population of the rotationallevel in the upper laser level, j=19 (fu), and the lowerlaser level, j=20 (fL)' for the P(20) transition attemperature T, and is used in the calculation of th eradiative cross sections, ao and aL. Figure 2 shows th edistribution of the rotational levels are not exponential as

-S FuCtc~0 f 7e~me'r- _-

LJ

2j~(A.

0 IQ 20 30 43

Figure 2. Rotational Distribution as a Function ofTemperature. Calculated from Equation 12 .

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expected, but peak at different values of j depending on th etemperature.

Relaxation Rates. The third set of temperaturedependent parameters are the relaxation rates (-y).According to Witteman, the time constant T (1/-y) is afunction of pressure, the number density and type ofcoll ision partners, and the temperature. For example, thetime constant for the relaxation to ground state from theupper vibrational level (001) is approximated by:

73 = r 3o T exp[h(V 3 - V2 - 1) V T - ki (1 )

where T3o = (P(Z Vi Ki))I, the time constant at the baselinetemperature for the particular pressure and gas mixinvolved, and K. is the rate constant for gas component i.Finally, v, is the frequency of emission for the CO2 100,010, and 001 levels, respectively.

Other authors, however, have proposed different modelsfor the temperature dependence of the time constants.Douglas-Hamilton used the relation r = l/(2(Q kj)), whereP is the fraction of a component and k is the rate forthat component. The rate constant, k is then a function ofinverse of the cube root of the temperature, k, = (T)and is displayed in graphical form (7:7-10). Wutzke, on th eother hand, used a fitted curve of the form:

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-y sum K, N1 ) (14)Ki = A'x 2 exp(-B'x + C'x 2 + D'x 3) (15)

where r is the relaxation rate, Ki are fitted rate constantsfor the gas components, N. is the number density of gascomponent i, x = T / 3, and A'-D' are fitted constants(2 5: 100-115) .

Figure 3 shows a comparison between the relaxation ratescalculated by Witteman, Equation 13, and Wutzke, Equations14 and 15.

; 3ax-i~C Ptea~

iU

ch o------

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different equations, depending upon the pressure. Atrelatively high pressure (greater than 50 torr), the lineshape is Lorentzian (pressure-broadened), but at less than10 torr the l ine width is Doppler-broadened.

Lorentzian Line Width. The pressure-broadenedLorentzian line width has been measured by severalresearchers. Brimacombe measured the half width at halfmaximum l ine widths (cm ) at the P(16) and P(32) lines andfound the following relations for pressure broadening:

At constant number density:L = P/760[Z(Vi a) ](T/300) n16)

At constant pressure:1-n

LL = P/760[E (0 a) ](300/T) (17)

where a, are linear functions of the rotational number andthe exponential term, n, equals 0.42+ 0.06

The value of n compares favorably with the valuemeasured by Pack and quoted by Brimacombe as 0.38 for heliumbroadening (4:1671-72). In addition, Gross measured theself-broadened line width of CO2 (CO2 colliding with C02 ) andfound n to be 0.7 (12:2253).

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Witteman gave the pressure-broadened line width (Hz) as(24:61):

ALvp = 3.925xI06 (CO 2 + 0.730N2 + 0.640He)P(300/T) (18)

Doppler-Broadened Line Width. The second form ofl ine broadening is Doppler broadening. For the CW laser atlow pressure, where pressure broadening is not a factor,Witteman gives the Doppler l ine width (Hz) as (24:60):

ALd = v[(in2) 2 k T/M c2 ] (19)

where v is the laser frequency at line center and M is th emolecular weight of the lasing molecule (C0 2).

The l ine widths are used to calculate the radiativecross sections, au and aL, where a is inversely proportionalto the l ine width.

Gas Density. The fifth temperature dependent parameteris gas density. The number density of the gas is calculatedby the ideal gas law, No = P/k T. Brimacombe stated that apulsed laser operates at constant volume, so the density ofthe gas does not change during the laser pulse (4:1671).

Heat Capacity. The sixth and last temperature dependentparameter is the heat capacity of CO2. The heat capacity ofpoly-atomic molecules increases with temperature because

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molecular vibrational levels can become populated, absorbingenergy that would otherwise increase the temperature of th egas. Of all the gases used in the CO2 laser, only the CO2(010) vibrational level is significantly populated at thelaser's operational temperatures (300 - 700 0 K). Therefore,only the heat capacity of CO2 varies significantly withtemperature and is described by Vincenti as

Cv =R (5/2 + [ hk / ifhh jJ 2)(0

where Cv is the constant volume heat capacity of CO2(23:136). The next paragraphs will discuss the calculationof the temperature rise in the laser medium duringoperation.

Temperature Rise During LasingSince the temperature of the lasing medium has such a

large effect on the laser output, "it is thereforeimperative to determine the kinetic gas temperature"(25:116). The gas temperature may be determined byaccounting for all the energy that is introduced into thegaseous medium by the electron beam pump. The electron beamenergy is divided into exciting the CO2 , the N2 , anddirectly heating the gas. Some of the excited CO2 and N2molecules contribute their energy to the laser output, but

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the rest relax to the ground state, giving their energy upto heat. Anywhere from 5% to 20% of the electron beamenergy directly heats the gas (7:28), and no more than 38%(quantum efficiency) of the rest can be converted into laseroutput. Milonni calculated the temperature rise to beapproximately constant over the time span of 25 microseconds(15:3595). A well-designed CO2 laser can have an overallenergy efficiency of 10% - 20%.

Energy Balance Equations. An energy balance equationcan be used to calculate the amount of energy that isintroduced as heat and raises the temperature of the gas.An equation that accounts for nearly all the energyintroduced by the electron beam was presented by Milonni:

q = (W - Re) - Rst + Rvt + R v (21)

whereq = rate thermal energy is added to the gasW = rate total energy is added to the medium by pumpRe = pump rate into all vibrational levelsRst= rate energy is lost to st imulated radiationR,, = rate energy is converted to heat from

vibrational relaxation to the ground stateR,, = rate energy is converted to heat from

vibrational energy exchanges between molecules

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The first term on the right, (W - Re), accounts fo renergy from the pump that is converted directly into heat(15:3596).

Each of the terms presented above has been calculatedexplici t ly by Wutzke (25:49-50, 116-119) for each of th emajor vt and vv reactions. Wutzke estimated the 80% of allthe heat added to the gas comes from the vt reactions. Inaddition, he expands q to be q = (dT/dt)[Z(Oi C,)] wheredT/dt is the change in gas temperature with time, ip is thefraction of the various gas components, and C, is the heatcapacity of each gas component (25:50).

The only energy not accounted for in the energy balanceequations is the thermal energy lost to the laser cavitywalls. In a CW laser at steady-state operation, the thermalenergy transported through the cavity walls must equal th ethermal energy created, and a steady-state temperature isreached. For a pulsed laser, however, the laser pulse isformed and gone so quickly that none of the heat isconducted away during the pulse; it all stays in the gas toraise its temperature.

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III. PULSED CO 2 LASER MODEL. METHODOLOGY and ANALYSIS

OverviewIn this chapter, the development of the temperature

dependent pulsed CO 2 laser model, henceforth referred to as"the author 's model" or "this model", is discussed indetail . In addition, the verif ication (internal tests) andthe validation (external tests) of the model are presentedand analyzed. Finally, the limits of the model, boththeoretical and practical, are discussed.

Developing the ModelThis model incorporates four major changes to Stone's

pulsed CO 2 model: thermal populations of the excitedvibrational states, thermal populations of the upper andlower rotational levels, another divisor of four in the rateequation for the population of the lower laser level nb),an energy balance to calculate the new temperature, and arecalculation of the temperature dependent parameters.

Rate Equations. The rate equations incorporate thefirst three major changes. The thermal populations of theupper and lower vibrational levels, nae and nbe, are includedin the rate equations for na and nb. The fraction of CO 2

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molecules in the upper and lower rotational levels areincorporated into the radiative cross sections cu and oL .These temperature dependent parameters are calculated hyEquations 11 and 12. The extra divisor of four in the rateequa iun f,, ;b.-- disc_'S=",, e o

d(na)/dt = I c(gu/gl CL nb - 0U n.) + 7a(nae - na)- cn2 na + 7co2 nc + wa (22)

d(nb)/dt I c(ou na - gu/gl oL nb)/4+ ya(na - hae)/4 + yb(nbe - nb) + Wb (23)

d(nc)/dt = cn2 na 7co2 nc + wc (24)d(I)/dt = I c(au na - gu/gl CL nb) - I/Tcay + na ws (25)

where nie are thermal equilibrium number densities of CO2 001and 100 levels, and au and cL are the effective radiativecross sections of the upper and lower laser levels.

The thermal population of N2 was not included because nodirect relaxation term (similar to 1a) was found.

Rate Equation for nb. The extra divisor of four inthe rate for the population of the lower vibrational level,nb, is based on the assumption that only one fourth of th eCO 2 molecules that leave the upper vibrational level bystimulated emission enter the 100 level to increase nb.

Stone assumed that only one fourth of the CO 2 moleculesthat relax from the upper vibrational level (001) enter th e

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100 level. These molecules enter one of the two 020 le-vels,and then very quickly thermalize to the other threevibrational levels in equilibrium with it (100, the other020, and 010).

St-r- -ontin'ipd to assume, however, that All of the CO 2molecules that leave the upper vibrational level bystimulated emission enter the 100 level. This caused aproblem in the energy balance, so this study assumes thatonly one fourth of the CO 2 molecules that leave the uppervibrational level by stimulated emission enter the 100 levelto increase nb. These CO2 molecules from stimulatedemission, l ike those from relaxation, are thermalized veryquickly to equally fill all four levels (the 100, two 020,and 010 levels). The new rate equation for nb is Equation23 above.

Energy Balance to Calculate the New Temperature. Thethird major change to Stone's model is the set of energybalance equations to calculate the energy that goes intoheating the gas, and the corresponding temperature rise.As explained in the literature review (pages 19), hightemperature is detrimental to the laser, and lasing causesthe temperature of the active medium to increase due to th etransfer of vibrational energy to translational energy(heat) as the lower vibrational level (100) relaxes toground state. The energy balance equations required to

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calculate the increasing temperature are derived using theconservation of energy principle. The principle ofconservation of energy maintains that the total enervy inputto the laser (pump energy) is equal to the sum of the energyAcrumimited in the laser (vibrational and translationalenergy) and the energy leaving the laser (photon energy).This study accounted for energy in six different "bins" andbalances them to keep the total input equal to totalaccumulated plus total out. The six "bins" are pump erinrgy;vibrational energies of na, nb, no; photon energy; andtranslational (heat) energy.

Pump Energy. The first enorgy bin is pump energy.In the pulsed CO 2 laser model the pump is defined by th enumber of molecules pumped into excited vibrational states.In addition, a variable fraction of the pump energy alsodirectly heats the gas and is accounted for. The pumpenergy is modeled by the following equat ions:

W = ((Wc + Wa)h V3 + 4 Wb h i) / (1 - fh) (26)TotPumpE(j) = TotPumpE(j-l) + W At (27)

where W is the rate energy is pumped into the laser medium,fh is the fraction of pump energy that directly heats th egas, TotPumpE(j) is the cumulat ive pump energy at step t imestep j, and At is the t ime step size in units of laser

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cavity lifetime. The pump energy in-lit to the laser isincremented at each time step until the pump is turned off.

Vibrational Energies. The next three energy binsare the vibrational energies. At each time step the totalvibrational energy of the excited populations over and abovethe thermally excited populations of C02 and N2 were alsocalculated as follows:

NaEng = (na - nae) h V3, for C02 (001) (28)NbEng = 4(nb - nb) h Ll for CO 2 (1on (29)N2Eng = (nc - nce) h -3, for N2 (v=1) (C'

where NiEng is the energy density of the correspondingvibrational level added by the pump.

The factor of 4 in NbEng accounts for C02 vibrationallevels 100, 010, and the two levels of 020. Assuming thatthe 100, two 020, and 010 levels are in equilibrium andequally populated, then the sum of the vibrational energystored in each is equal to four times the energy stored inthe 100 level.

Photon Energy. The fifth energy bin is photonenergy. The total cumulative photon energy is calculate by

Energy(j) = Energy(j- ) + Power(j) nt (31)

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where Energy(j) is the cumulative photon energy (J/m 3 ) atthe jth time step, Power(j) is the current laser l ight power(W/m 3), and 6t is the time step. This is a rectangularapproximation to integration using Simpson's rule (17:108)

Thermal energy. The sixth and last energy bin isthermal (translational) energy. The pump energy that doesnot appear as photons or vibrational energy appears as heatin the laser medium and is calculated as follows:

TotHeatE(j) = TotPumpE(j) - (N2Eng(j) + NaEng(j)+ NbEng(j) + Energy(j)) (32)

The new temperature is calculated from the thcr-malenerg;; ,ising an equation derived from Wutzke:

T = Tinit + TotHeatE(j) / (N0 k Z(i C-, )) (33)

where T is the new temperature of the laser medium, Tini isthe initial temperature at time 0, TotHeatE(j) is th ecumulative thermal energy added to medium since time 0, andCvi is the constant volume heat capacity of constituent i(25:49).

Recalculation of the Temperature Dependent Parameters.The fourth major change to Stone's model is th erecalculation of the temperature dependent parameters. Once

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a new temperature is calculated, the temperature dependentparameters (described in Chapter II) can be recalculated.These temperature dependent equations were placed inside th eintegration loop of the computer code to be recalculatedeach time step. A review of the five sets of temperature'iependent equations used in the computer code follows.

Thermal Population of Excited Vibrational Levels.The first set of temperature dependent equations is th ethermal population of the excited vibrational states. Thethermal equilibrium population of an excited vibrationallevel at temperature T is described by Equation 11.

Thermal Populations of Rotational Levels. Thesecond set of temperature dependent equations that arerecalculated are the populations of rotational levels j = 19and j = 20. Equation 12 is used to calculate f. th efraction of CO2 molecules in rotational level 19 for th eupper vibrational level (001) , and fL, the fraction inrotational level 20 for the lower vibrational level (100).Because the model assumed lasing in one cavity mode and online center only, these two fractions are the only sets ofCO 2 molecules that actually participate in lasing.

Relaxation Rates. The third set of temperaturedependent equations are the vibrational relaxation rates.The temperature dependent relaxation rates were calculated

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using the reciprocal of the time constants found inWitteman's work:

-Y a- ( T )-f exp[-h (L 3 W2 - V/1)t o (34)7o=-b' (T/300) exP[-h dl - oj (35)

7rcn2 = 7cn 2 , (T/300) 3/2 367co2 = 7co2l (T/300)3/2 (37

where -yi is the temperature dependent relaxation rate ofvibrational level i and -y, is the relaxation rate at 30'JD(24:165-6).

As mentioned in the l i terature review (page 14), therotational relaxation rates are assumed to be infinite.

Line Width. The fourth temperature dependentparameter is the line width. Witteman's equation for linewidth (Equation 18) was modified to change it to a constantnumber density line width as opposed to Witteman's constantpressure line width equation. The modification ,.as derivedusing the ideal gas law, P = NokT . Since the numberdensity is assumed to remain constant, the pressure isproportional to temperature. Substituting for P in Equation18 changed the (300/T) term to (T/300) as follows:

v = 7.85(OC0 2 + 0.73 ON2 + 0.64 OHe)P(T/300)" (38)

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where P is now the initial pressure. The form of thisequat ion matches Brimacombe's constant number densi ty linewidth (4:1672). This study choose W itteman's line widthover Brimacombe's because Witteman was published four yearslater, and there is very little difference between the twoequat ions.

Heat Capacity. The fifth and last temperaturedependent parameter is the constant vclume heat capacity ofC0 2, calculated using Equation 20.

Summary of Model Development. Tn summary, the majorchanges to Stone s model operate in the fol lowing way.Beginning with the initial condit ions, the model calculatesthe total energy supplied by the pump for one t ime step. Itthen calculates which bins that energy has been transferredto, and by subtraction, calculates the fraction of pump

energy that heats the gas. Using the constant volume heatcapacities of the gaseous components, the model thencalculates the new temperature of the laser medium. At th ebeginning of the next t ime step, the model recalculates th etemperature dependent parameters using the new temperature,and then begins the energy balance calculations again tofind the next new temperature for the next t ime step.

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Verification and AnalysisVerification is a set of internal tests to confirm the

model is functioning as expected. Since G ilbert 's andStone's models were previously verified, this study verifiedonly the three major changes to Stone's model and that thosechanges interacted correctly with the overall model. Thusverif ication occurred in the following five steps:

1. First, this study verified that the temperature didrise monotonically throughout the laser pulse.

2. Next, the adverse effects of the temperature riseon gain, power, and total energy in the pulse wereverified.

3. Third, this study verified the effect of th ethermal equilibrium populations.

4. Fourth, this study verified that changing the rateequation for nb did change the pulse shape and theexcited populations.

5. Last, this study verified that the overall computercode functioned correctly with the new changes.

In the following paragraphs, each verif ication step ispresented and then analyzed.

Verifyina the Temperature rise. The study firstverified that the temperature did rise during the pump pulseand the laser pulse. The energy balance equations

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calculated a temperature rise (Figure 10, page 56) that ispropor t ional the pump energy, as expected.

Verifying the Effect of Temperature on Peak Gain. Next,the study verified the effect of temperature on peak gain(Figure 4). As expected, the peak gain decreased an orderof magnitude as the initial temperature increased from 200to 700 0 K. Although the temperature increases above theinitial temperature throughout the laser pulse, the increasedoes not play a significant role because most of th e

Pea< Gain vs lnKt Te m

14w I

Co't t P1

21 1

ConSt No

40

100 200 300 400 500 F _Initial T rriperture, K

Figure 4. Peak Gain vs Initial Temperature. Const P = 1atm; Const No = 2.44xi0 24 ; Mix = 10% C0 2, 10% N2 , 80% He;Pump Eff = 0.2; Length = 1 m; Refl = 71.6%; Pump Pulse200 nsec, square; Losses = 0.

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temperature rise occurs after the gain and power peak.Figure 4 shows the gain does not decrease monotonically, butpeaks at about 3000K.

This local maxima in the gain curves may be due to threecompeting factors that influence the gain: the populationsna and nb; their relative relaxation rates, -a and t, andthe radiation cross secticn, c. Two of these parameters,the populations and the radiative cross section, decreasethe gain with increasing temperature, but the effect of therelaxation rates tends to help the gain at highertemperatures.

These competing effects may be seen in the gainequation, Equation 10, where the gain is directlyproportional to the population difference and a, th eradiative cross section. As the temperature increases above200 0 K, the relaxation rates of both na and n. increase, but-y increases faster than 7Ya, possibly increasing th epopulation difference and thus the gain and power. However,as the temperature increases above 300 0 K, both populationsna and nb decrease through relaxation, decreasing th epotential population difference. In addition, the radiativecross section decreases, further decicasing the gain. Thefollowing table illustrates the change in ratio ofrelaxation rates, a, and gain with temperature at 1 atmpressure and a gas mix of 10% C0 2 , 10% N2, 80% He.

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Table I. Gain vs TemperatureTemperature -Y/ a a Gain (%/m)

2000K 12 11.5 x10"24 119250 0K 18 10.2 x10"24 122300 0K 24 8.9 x10"24 122500 0K 42 5.5 x10"24 67.5700 0K 54 3.7 x10-24 20.0

The difference between the peak gain at constantpressure and constant number density at low temperatures(shown in Figure 4) is probably a function of the totalnumber density. At high temperatures the effects ofrelaxation rates and radiative cross section overpower th eeffect of total number density.

Verifying the Effect of Temperature on Peak Pcwer andPulse Shape. The study then verified the effect oftemperature on peak power and pulse shape. Figure 5 showsthe peak power, at constant pressure and constant numberdensity, which also reach a maximum at an intermediatetemperature (about 350 0 K). This behavior was not expectedat first, and could not be confirmed in the literature.Since power is a function of gain, however, and observingthe behavior of the gain, it is reasonable to expect th epeak power to occur at an intermediate temperature. With

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Peak Power vs Init Temp120

100

80

60

20

100 200 300 400 5C3 E Linrti~l Te~r~prture, K

Figure 5. Peak Power as a Function of InitialTemperature. Same conditions as Figure 4.the effect of number density removed (constant numberdensity curve, Figure 5), the temperature has a much greatereffect on the peak power. At temperatures below 300 0 K, th enumber density of CC2 is higher for the constant pressurecase, and the power is higher. At temperatures above 300 0 K,the pressure for the constant number density case is higherthan 1 atm, so the relaxation rates and the number densityare higher. These higher relaxation rates may enhance thepeak power up to the point where the temperature kills th egain.

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Figures 6 and 7 show the effect of initial temperatureon the shape and timing of the laser pulse over th etemperature range of 200 to 600 0 K for both constant pressureand constant number density.

Pulse Shape vs TemperatufreConstant Pressure

?0 400 K60

50 2U0 KN 500 K

40L 300

20600 K

10,01i I III0 200 400 600 800 1000 ,2'D .

iv in narcsonC3

Figure 6. Laser Pulse Shape vs Initial Temperature atConstant Pressure of 1 atm. Mix = 10% C02, 10% N2, 80% He;Pump Eff = 0.2; Length = 1.0 m; Refl = 71.6%; Pump Pulse200 nsec, square; Losses = 0%.47


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Pulse Snape vs Init Temp at 1o00 orr35

400K350K I i503 0

_ 300K

C 200Ki1 600K1

05

0 200 400 600 800 1000 1200 1400 1600 18C0 2C0Tin[ narosecor*

Figure 8. Laser Pulse Shape vs Initial Temperature at LowPressure (100 torr). Other conditions same as Figure 6.

Pulse Shape vs Init Temp a: 4300

300K

S30000 430

20C

100

0C0 100 200 300 400 500 600 100 KC- C3

T, f nanoseconrjsFigure 9. Laser Pulse Shape vs Initial Temperature at HighPressure (4 atm). Other conditions same as Figure 6.

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Pu1se STipe ,e Tp Ps20C

Pulse 2, 177 X -cjst Terip

150

100

Pulseso

0 200 400 600 800 -1000 1200 140C IEi' 1 302S7 i f- I~ rnnonosec on

Figure 11. Comparison of Laser Pulse Shapes, Rising Temp(Pulse 1), Const Temp (Pulse 2). P = 1 atm; T = 300"K; Mix= 17% C02 1 33% N2, 50% He; Pump Eff = 0.4; Length = 1.0 m;Refl = 71.6%; Pump Pulse = 200 nsec, square; Losses = 0%.

The rising temperature in pulse 1 increases th erelaxation rates, which drains the energy out of the uppervibrational level, na, and reduces the height and length ofthe tail portion of the laser pulse. Figure 12 shows th eeffect of the temperature rise on the excited CO2populations, na and nb.

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ExciteO Populations, Na & t,

Pulse 250-401C0 30-

0 200 400 600 Boo 1000 1200 1.40 1E C E1E. ZZTinf in mnosecorns

Figure 12. Populations of the Upper Vibrational Level, na ,and the Lower Vibrational Level, nb, for the Rising Temp(Pulse 1) and the Const Temp Pulses (Pulse 2) in Figure 11.

Although the absolute populations of the upper and lowervibrational levels, na and nb? are affected immediatelyafter the laser spike, the gain is not. The gain in boththe constant temperature and the rising temperature pulsesare nearly identical out to about 2000 nanoseconds, wherethe increasing temperature begins to shut down the gain ofthe rising tempe:ature pulse. The increased temperatureshuts down the gain by rapidly draining n. and increasingthe thermal population of nb-

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Verifying the Effect of Temperature on Energy. The lastlaser parameter to be affected by temperature is the energyof the laser pulse. Unlike the power and gain, the totalenergy in the laser pulse does not peak at an intermediatetemperature, but decreases steadily with temperature asshown in Figure 13. This was expected, and occurs because

Total Puise Energy vsemr>_e1811 Cor t P14

- ~12SCC~

864:

150 200 250 300 350 400 450 5C0 5o 6CC 5

Terprature InOerees K

Figure 13. Total Energy in the Laser Pulse vs InitialTemperature. Const P = 1 atm, Const No = 2.44x10 25/m 3 ; Mix= 10% CO2 , 10% N2, 80% He; Pump Eff= 0.2; Length = 1.0 m;Refl = 71.6%; Pump Pulse = 200 nsec, square; Losses = 0%.

the number of CO2 molecules available to lase decreases withtemperature at constant pressure. Even when the numberdensity of CO2 is held constant, the total laser energystill decreases with increasing initial temperature as shown

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in Figure 13. The energy available in the upper vibrationallevel drains out faster through relaxation at highertemperatures and is unavailable for lasing.

Verifyinq the Effect of the Thermal EquilibriumPopulations of the Excited States. The third major changethat this study verified is the effect of the thermalequilibrium populations, nee and nb, on the excitedpopulations during lasing. This effect is best seen at hightemperature. Figure 14 shows na and nb for a pulse at600 K.

Excited PoPuLLt ions, E :at 5SO K

a +

12

0 50 050 ISW 2000 2SC0 z3

Figure 14. Upper and Lower Vibrational Populations, naand nb, at 6000K. P = 1 atm; Mix = 10% CO2 , 10% N2, 80% He;Pump Eff = 0.2; Length = 1.0 m; Refl = 71.6%; Pump Pulse= 200 nsec, squ-re; Losses = 0%.

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At time 0, nb starts at its thermal population for 600K(5xI016/cc), and is pumped up from there. After the pumppulse, it quickly drains back down to its thermal population(6xi016/cc) for the current temperature in the laser (about630 0 K). The laser- spike barely effects nb and it remains atits thermal population (6xi0' 6/cc) for the currenttemperature for the rest of the pulse. na, on the otherhand, is pumped far above its thermal population and lases.It eventually relaxes down to its thermal population ofabout 0.7x10 6/cc at the end of the run.

Verifying the Effect of the New Equation for n,. Thefourth major change to Stoie's model that this studyverified is adding the divisor of four to the photon fluxterm in the differential equation for n,. This equati--will be referred to as the new rate equation for n.(Equation 23). As expected, the additional divisor of fourin the rate equatioin for nb had a significant effect on th epeak power and the populations of na and nb after the oowerpeaks. It was not expected, however, that there would be noeffect on the gain. Figure 15 shows the populations n. andnb throughout the laser pulse for both the old and newequation for nb-

The populations na and nb are much lower after the pulsespike using the new equation. This occurs because only onefourth of the CO 2 molecules that transit ion from 001 due to

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Excitea Populations, Na & voD25.

02

0 503 1000 1500 2000 2520 7-

T~T in nanos~conrFigure 15. Upper and Lower Vib Pops, n8 and n., Calc byOld Eqn for nrb, and New Eqn for nb. P 1 atm; T = 300K;Mix 10% C0 2, 10% N2, 80% He; Pump Eff = 0.2; L = 1.0 m;Refl = 71.6%; P.P. = 200 nsec, sqr; No Loss.

stimulated emission enter 100. Thus, nb grows to only onefourth it's original value during the las.- pulse spike, butn, is driven down to meet nb. The extra molecules in 001that respond to stimulated emission add to the laser spikeand increase the peak power 77%.

The gain, however, is nearly identical in each case.The gains for the old and new equation are nearly identicalbecause the gain is proportional to the populationdifference, which stays approximately the same (Figure 15).

Verifyina the Overall Computer Code. Finally, thisstudy verified that the overall computer code functionedcorrectly with the new changes. This was done by varying

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all ten input parameters and observing the resulting laserpulse, temperature change, and the upper and lower laserpopulations, na and nb. In each case, the results wereconsistent with the input parameters.

Validation and AnalysisValidation involves a set of external tests to compare

the calculated output with actual laser output and withother CO2 laser models. This study validated three areas ofthe model: power and energy vs temperature, the temperaturerise, and the pulse shape.

Validating the Power and Energy vs Temperature. Thisstudy first validated the relationship of power and energyvs temperature (Figure 16). Witteman stated that the pulsedlaser's output begins to decrease above about 450 0 K (24:3).in studying the CW laser, Fowler found that the po.er dropsoff sharply at higher temperatures (500 - 700 0 K). Since th epulsed laser depends on the same mechanisms to produce laserpower, its power and total energy should also drop off5 narply at higher temperatures. Figure 16 shows both th epower and total energy of this pulsed model decrease sharplytoward zero at 600+ K.

The power and energy of the pulse decrease withtemperat- re because the number density decreases, th e

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Peak Power and Total Energy

CC

S I I200 250 30D 30 400 450 500 550 6CO 5510 7

initial Teffl erat~re,K

Figure 16. Peak Power and Total Pulse Energy vs InitialTemperature. P = 1 atm; Mix = 10% C02, 10% N2, 80% He;Pump Eff = 0.2; Length = 1.0 m; Refl = 71.6%; Pump Pulse= 200 nsec, square; Losses = 0%.

relaxation rates increase, and the gain substantiallydecreases. When the gain drops below the threshold gain,the laser shuts down.

At low temperatures (200 0 K) , the power should increase20 - 50% according to Mitsuhiro for the CW laser (13:68).Drobyazko showed that at 200 0 K, the total pulse energyincreased 50% for a pulsed laser (8:29). Figure 13 on page53 shows my model calculated a 62% increase in total pulseenergy at 2000K at constant pressure, and a 9% increase atconstant number density. The increase in number density atconstant pressure and low temperature accounts for most ofthe increase in the total pulse energy. At constant aumbe-

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density, the reduced relaxation rates at low temperatureprobably account for the small increased total energy.

Validating the Temperature Rise. This study nextvalidated the temperature rise. Milonni's pulsed C02 lasermodel included an energy balance and he calculated a 'ineartemperature rise from 300 0K to 5000K in 25 microsecondsduring the pump pulse (15:3695). Fedorov also calculatedthe same linear temperature rise during the pump pulse(9:630). This study duplicated Fedorov's conditions (a 304sec pump pulse of 250 J/l and mix of 3:2:1 He:N2:CO2) andobtained a linear temperature rise from 300 to 550)K duringthe pump pulse.

Validating the Pulse Shape. Last, this study validatedthe pulse shape against data supplied by Manes. Manes useda five-level model of a TEA pulsed C02 laser to calculatepulse shape for different initial conditions. He made threesets of three runs each. In the first set he set the cavitylength to 0.48 meter, and varied the pump energy. In th esecond set, he set the cavity length to 2.5 meters andvaried the pump energy. For the last set of runs, he keptthe pump energy and cavity length constant and varied th egas mix. He compared his model with the output from anexperimentai puised CO2 laser and found "good agreement"(14:5077). He did not include the rise in temperature inthe laser medium because he found the rise to be only about

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500 K and to have little effect (14:5075). In duplicatinghis conditions, this study found the largest temperaturerise to be 125 0 K, and the average to be 70 0 K.

Throughout the nine test runs duplicatirg Manes'initial conditions, the change in peak power and delay timealways agreed with the trend in Manes' calculated output.In addition, the relative size and structure of the tailportion of the pulse agreed with Manes' calculations(Figures 17 to 22).

In each of the Figures showing the calculated pulseshapes for comparison against Manes' pulse shapes, P.E. isthe pump energy for that pulse, in J/l.

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Peak Power =36-013[ (C) 166 MW/I

I[V LMAX Ne 296.10)2 c,-3erg

Cm2 see

0 2 3 4

Peak Power120 MW/i2,6-10'3 (d)l

MAX Ne:222."O12 cM-3Iv

ergscm2 see L

03 4

Peak Power =o 1J1 (6) 46 MW/iMAXNe:133-1O12cm-3Iv

cm2 see0 2 3 4

in 'Usec

Figure 17. Pulse Shape vs Pump Energy from Manes' Study.Length = 0.48 m; Pressure = 1 atm; Temperature = 3000 K; Mix= 10% CO 2 , 10% N2 , 80% He; Ref = 74%; Pump Pulse = 1 4sec,sin shape; Losses = 0%. Reprinted from (14:5074)

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alculated Pulse ShapeResonator Length 0.48 M

120P.E. 0.4

100

80PRE 0 17'5

06

02

0 500 1000 5Time~ in nanoseconds

Figure 18. Calculated Pulse Shape as a Function of PumpEnergy. Resonator Length = 0.18 mn Same conditions asFigure 17.

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401013 Peak Power =22 MW/iRON L MAX Ne] 63-10 2 cmn-3was Iv

500 *sec/d . 0 1 2 3 4

2.0-10"13 Peak Power =11 MW/i

1V MAX No-074 l0 2crmi

500S soc/d., 0 1 2 3

09-1013

Peak Power :_______.________,M-X 5 MWi 1AX.cmZsoc

(C)200 -c/di, 0 I 2 3 4

in '.sec

Figure 19. Pulse Shape vs Pump Energy from Manes' Study.Length = 2.5 m; Pressure = 1 atm; Temperature = 300 K; Mix= 10% C0 2, 10% N2 , 80% He; Refl = 70%; Pump Pulse = 1

a model of temperature effects in pulsed and continuous wave co2 lasers and optimization using...

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