Title: Computer simulation of an optically pumped methyl fluoride laser
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Title: Computer simulation of an optically pumped methyl fluoride laser
Physical Description: v, 110 leaves : ill. ; 28cm.
Language: English
Creator: Schau, Harvey Charles, 1949-
Copyright Date: 1975
 Subjects
Subject: Gas lasers   ( lcsh )
Chemical lasers   ( lcsh )
Engineering Sciences thesis Ph. D
Dissertations, Academic -- Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Statement of Responsibility: by Harvey Charles Schau III.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 106-109.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00098935
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000170955
oclc - 02940210
notis - AAT7376

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COMPUTER SIMULATION OF AN OPTICALLY
PUMPED METHYL FLUORIDE LASER










By

HARVEY CHARLES SCHAU III


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA


1975























THIS WORK IS DEDICATED TO THE

SCIENTISTS AND CREW OF THE R.V. GULFSTREAM
















ACKNOWLEDGEMENTS


The author would like to thank his graduate committee

and his department chairman, Dr. Knox Millsaps, for

helpful discussions during the past year. The author

would like to give particular thanks to his advisor

and friend Dr. Dennis R. Keefer for suggesting and

discussing the problem. Special thanks go to

Dr. Willis B. Person of the Department of Chemistry

for his encouragement and interest in the problem.

The author wishes to thank his parents Mr. and

Mrs. Harvey C. Schau for moral and financial support

and Miss Judith Van Der Walt for her excellent typing of

the manuscript.

Lastly the author wants to thank his wife Sharron

for her understanding and endurance of his ravings

about methyl fluoride lasers for the past year, and

his friends Falmouth, Monroe, Pete, and Fred for their

welcome diversions.



















TABLE OF CONTENTS



ACKNOWLEDGEMENTS . . .

ABSTRACT .

CHAPTER


page

iii


v


I. INTRODUCTION 1

II. ENERGY TRANSFER IN MOLECULES 29

III. METHYL FLUORIDE 39

IV. MODEL 53

V. RESULTS 62

VI. CONCLUSION 97

APPENDIX A 99

APPENDIX B 101

APPENDIX C 104

BIBLIOGRAPHY 106

BIOGRAPHICAL SKETCH 110

















Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



COMPUTER SIMULATION OF AN OPTICALLY
PUMPED METHYL FLUORIDE LASER

By

Harvey Charles Schau III

August, 1975

Chairman: Dennis R. Keefer
Major Department: Engineering Sciences

Employing a semi-classical model, numerical solutions

were obtained for a ten level model of methyl fluoride.

Laser emission from methyl fluoride at approximately

16 microns was found to be possible by optically

pumping with a carbon-dioxide laser. The methyl fluoride

laser was found to have gain of approximately .51% per

cm. and energy storage of approximately 31 mJL-ltorr-.















CHAPTER I INTRODUCTION


Laser (La'zar) n. lighth) a(mplification by)

s(timulated) e ( mission of ) radiationn] ; a device,

containing a crystal, gas, or other suitable substance, in

which atoms, when stimulated by focused light waves,

amplify and concentrate these waves, then emit them in a

narrow, very intense beam, optical maser.1

Thus Webster defines the laser. Webster's definition

exemplifies the importance and rapid growth of this new

light source. Prior to the summer of 1960 however the

word laser had an entirely different meaning; "Laser; the

juice of the laser tree, laserpitium latifolium, also

called silphium, greatly esteemed by the ancients as an

antispasmodic deobstrucent and diuretic."2 Although

the word has an ancient heritage, today it brings to mind

a much more modern meaning. The laser is an example of

the same idea occurring to different people in different

parts of the world at about the same time. The first

lasers were actually masers (microwaveamplifiers) which

were jointly developed in 1951 by C. H. Townes of Columbia

University and 1952 by N. G. Basov and A. M. Prokhorov of

the USSR. For their work on coherent microwave amplifiers

Townes, Basov, and Prokhorov were jointly awarded the Nobel

prize in Physics in 1964. Following the development of the









maser, A. L. Schawlow of Bell Telephone Laboratories and

Townes outlined the theory of an optical maser in 1958.

After publication of this theory, T. H. Maiman in 1960 was

the first to obtain actual laser action in the visible

spectrum. Maiman employed a crystal of ruby as a lasing

medium. This was surprising since at that time scientists

were working on primarily gas lasers, and felt that

probability of success was highest for gaseous devices.

Time has proven their premonitions correct since most high

power laser development in recent years has been from

gases. Solid state lasers have,however,been found to be

sources of tunable radiation important to ultra-high

resolution spectroscopy, and recently glass lasers have

been used in controlled fusion studies. Outside of these

two areas new high power laser applications are being

developed primarily around gas lasers.

Prior to the late sixties the laser was largely a

scientific toy. It was at that time that molecular gas

lasers were developed. These lasers,particularly the CO2,

have exhibited several important qualities. First is

their efficiency (30-40%). This is extremely high when

compared with efficiencies of less than 1% for most

conventional lasers. Second, and perhaps more important,

is their ability to be scaled up to industrial size.

This gave great impetus toward developing commercially

available high power C02 systems. In recent years CO2

lasers have been moving out of the scientific laboratory









and into industry and the production line.3,4,5

Continuous wave (CW) lasing at 10.6 microns from CO2

was first reported by C. N. K. Patel in 1964.6 Since that

date the CO2 laser has undergone rapid improvement.7

Attention is called to the review of CO2 laser development

by Robinson.8

Laser action from CO2 proceeds via population inversion

of vibrational levels. In electric discharge systems

nitrogen is excited by electron collisions, which in turn

excites the 0001antisymmetric stretch mode (see figure 1)

of CO2. Lasing is achieved from this level to either the

1000 symmetric stretch at 10.6 microns or the 0200 second

bending mode at 9.6 microns (figure 2).9 In order to

maintain a population inversion the lower (0110) level must

be depleted. Normally helium is added to collisionally

depopulate this level. In addition to undergoing ir active

vibrational energy transitions, CO2 may change its rotational

energy by one quantum. This leads to a vibration-rotation

spectrum of many-frequency laser lines in each the 10.6

micron band (figure 3). Each term labeled by J represents

a rotational angular momentum state of CO2 and thus a

splitting of energy levels.10 The terms R(J) and P(J)

represent a transition of AJ = -1 and +1 respectively,

following ir selection rules for CO2. Figure 4 illustrates

the wavelength difference for P,Q, and R bands (Q;AJ = 0).

Notice that odd J terms are missing. This is due to

nuclear spin statistics for zero nuclear spin molecules



























3000-




2000


-E
00 020
1000--
S-- 010




CO2 N2

Figure 1. Energy levels of C02
and N2 laser levels.



























T
-0 C


Oc
6


0- sym. str.

0-' asym. str.


bend


Figure 2. Laser vibrations of
CO2.




























001-020 band CO2
J R(J) P(J)
4 1067.50 cm- 1061.61
6 1068.89 1059.04
8 1070.43 1057.30
10 1071.87 1055.58
12 1073.28 1053.91
14 1074.63 1052.13
16 1076.00 1050.47
18 1077.30 1048.66
20 1078.57 1046.85
22 1079.85 1045.04
24 1081.08 1043.19
26 1082.29 1041.29
28 1083.48 1039.34
30 1084.63 1037.40
32 1085.74 1035.46
34 1086.84 1033.48
36 1087.90 1031.56
38 1088.97 1029.44
40 1090.04 1027.38
42 1090.99 1025.27
44 1092.00 1023.17
46 1093.01 1021.03
48 1093.85 1018.85
50 1094.81 1016.67
52 1095.71 1014.46
54 1012.25
56 1010.00
58 1007.76
60 1005.38

Figure 3. V-R spectrum of C02'





























001-100 band CO2

J R(J) P(J)
4 964.74 957.76
6 966.18 956.16
8 967.73 954.52
10 969.09 952.88
12 970.50 951.16
14 971.91 949.44
16 973.24 947.73
18 974.61 945.94
20 975.90 944.15
22 977.18 942.37
24 978.47 940.51
26 979.67 938.66
28 980.87 936.77
30 982.08 934.88
32 983.19 932.92
34 984.35 930.97
36 985.42 928.94
38 986.49 926.96
40 987.56 924.90
42 988.63 922.85
44 989.61 920.77
46 990.54 -918.65
48 991.47 916.51
50 992.46 914.41
52 993.34 912.16
54 994.18 909.92
56 907.73

Figure 3. V-R spectrum of CO2.




























001 j






100
^p J'


Figure 4. Diagram of V-R selection rules.









(for further discussion see Herzberg).

The theory of laser oscillation is easily understood

if presented within the framework of amplifier dynamics.

The laser as a device consists usually of a pair of

parallel mirrors between which is placed the material which

is to act as the amplifying medium over a limited frequency

range (figure 5). L is the length of the active material

and the optical distance will be denoted L'. If mirrors

are placed on the ends of the cavity containing the

amplifying material L'= nL where n is the index of

refraction. For gas lasers n=1 however the mirrors are

usually located outside the laser cavity so that L'> L.

At best one of the reflectors is partially transmitting so

that its reflection coefficient (reflectivity) is less

than one. The reflection coefficient r is defined as the

fraction of light intensity reflected. Thus at each

reflection, (l-r) of the intensity is transmitted as output.

If the reflectivities of the two mirrors are rl and r2

respectively, the energy of the wave is diminished by a

factor rlr2 in one transit. If the fraction of intensity

remaining after one round trip through the laser is denoted

by 62y 7 is seen to be the loss coefficient per length.

It is seen that

y = -1/2 Log rlr2

if all other losses other than reflection are neglected.

In gas lasers, diffraction losses are sometimes quite

significant, but they will be neglected here.





















M
L M

tive materialoutput
active material


Figure 5. Diagram of gas laser.







11

Oscillation may be sustained if the amount of amplifi-

cation per passage is equal to or greater than the losses.

If the amplification per unit length is denoted by a,,

the threshold condition is seen to be

(1) amL = y.

This equation also defines the amplification since the laser

will always operate at threshold, as in electrical circuits.

The amplification coefficient (negative of the absorption

coefficient) may be written as

(2) a(v) = k(v)N

where v = frequency

N = relative population difference between emitting

and absorbing levels

N = 1/N0 (g1/g2 N2- N)

g,, g2 are the quantum degeneracies of levels 1 and 2

respectively. N and N2 are the populations of levels 1

and 2 respectively. It can be shown that

(3) k(U) = A21 /8Rin22 g2/g 1 G(vv) = KG(vv)

where A2 = Einstein spontaneous emmission coefficient

(to be discussed later)

A21 = t2 the lifetime of atoms in level 2.

r = index of refraction

G(vo) = lineshape around line center v.o

Combining equations (1), (2), and (3) we find

(4) g,/g, N- N = y/L
where we assume emmision is at line center.









The time for a photon to make a single laser

passage is T = L /C so that P0 photons travelling

through the laser after m passages will be reduced to

P = Poe-Yt/

thus the average photon lifetime may be defined as

(5) tp= T/y = L'/cy.

Combining this with equation (4)

(6) EL N N= rT22 02 L t.
g, 2 1 C G(0) g2 L tp

Equation (6) is known as the Schawlow-Townes condition,

and gives the minimum population inversion necessary

for lasing.

Once a population inversion has been established the

principle of stimulated emission may be simply stated as:

A photon will interact with a molecule in an excited state

to produce two coherent photons travelling the same direct-

ion as the first, plus a molecule in the ground state.

The initiation and maintenance of lasing action is most

easily explained by use of the Einstein coefficients.

These two coefficients are

spontaneous A12 [sec-'] and

stimulated B12 [cm'J- sec-2].

An excited state has probability Pmn of decaying from

state m to state n defined as

Pmn = Amn + UvBmn

where U is the radiative density at the frequency

satisfied by the Bohr condition. Note that Amn and Bmnhave









different units. Amn is a measure of the natural life-

time of state m before spontaneous radiative decay to state

n, analogous to noise in an electrical amplifier. Just as

in the electrical analogy, the spontaneously emitted photons

start the oscillations which lead to a radiation density

thus stimulating more photons. Once the oscillations have

reached threshold, the stiumlated term will continue to

act as an amplifier as long as a population inversion is

maintained. It is common in lasers for population inversion

to be established and oscillation to start, where the rate

of depletion of the upper level is faster than it can be

refilled. In this case g1/g2 Ni- N2 < 0 within a short

time, and laser oscillations cease. This type of laser is

called a pulsed laser, as opposed to a CW laser. CO2 may

be used as either and may be constructed in several

different configurations. The earliest and simplest was by

Patel (see figure 6)." It is interesting to note that the

CO2 was not placed directly in the discharge. Nitrogen is

excited in the discharge and being homonuclear, is not ir

active. CO2 is then collisionally excited by the

nitrogen to achieve a population inversion. Today commer-

cial lasers place a 1-1-8 mixture of N2- CO2- He in the

discharge to obtain pulsed or CW lasing.

An important advancement in CO2 laser technology was

the development of the TEA (Transverse Electric Atmospheric)

laser." As the acronym suggests this laser operates at

atmospheric pressures thus eliminating the need for vacuum






































Pump



Figure 6. Early CO2 laser by Patel.










systems and increasing power output. The pulse length is

largely determined by the relaxation times of the gas

molecules. As these become shorter at higher pressures,

the pulse length decreases proportionally. Typical

pulse length of a TEA laser is a few nanoseconds

(1 nsec = 10l-sec). It is difficult to achieve uniform

excitation of CO2 at atmospheric pressure, since at about

200 Torr the normal glow discharge constricts to an arc,

heating the gas and destroying laser action. Several

techniques have been developed to overcome this.

Beaulieu originally used a very short 17 KV pulse to

excite a transverse discharge between a row of pin

cathodes and a round bar anode.12 Today numerous TEA

lasers use a helix geometry transverse discharge. The

helix is designed to improve the uniformity of the discharge.

The radial radiation distribution in a cylindrical

laser is very close to the eigenfunctions of a cylinder

(see figure 7). Because of diffraction losses, the TEM,,

mode propagates best, and to prevent mode switching and

competition most lasers are operated by placing a stop with-

in the optical cavity so that the laser operates in TEMo .

It was indicated earlier that gas lasers radiate on

many vibrational-rotational transitions. Because cavity

quality factor is narrow compared with the Doppler or

pressure broadened half width, the laser has longitudinal

modes within the natural half width.13 The condition for re-

inforcement is that nX=2Ln or v/c=n/2Ln, defining the

optical distance














TEM



t @ $


00 10 20






01 11 21

Figure 7.Radial distribution of radiation for
cavity with circular mirrors. TEM (Transverse
Electric and Magnetic) modes are indicated by
arrows indicating magnitude of Electric field.










L
L' = If n(z) dz

the condition for reinforcement becomes

v/c = n/2L'.

Thus the spacing between consecutive axial modes is

Vn+i Vn = Av = c/2L'.

Axial mode spacing is compared with natural line width in

figure 8, notice that the cavity spacing is narrow with

respect to the natural line. Only those cavity modes above

threshold will lase, and in most gas lasers several axial

modes will operate simultaneously giving multimode operation.

There may possibly be from several to several thousand modes

excited during laser operation.

It is of interest to consider the type of output that

would result if a large number of modes, equidistant in

frequency, could be excited with approximately equal

amplitude, and, if these modes could somehow be locked

together in a constant phase relationship. Each oscillating

mode gives rise to a wave with an amplitude described by

the real part of eiw(t-x/c) where m = Wo + kAn and Ah = 2nAv

(Av = half width of mode). The addition of 2n + 1 of such

waves with frequency centered around m0 all having equal

amplitude leads to the expression

Zn ei(wo + kAo)t
k=-n


which isequaltoe 0ot times the amplitude function

F(t) = sin(n+1)At. The result of this synthesis is an
sin -Awt

amplitude-modulated wave of frequency i0, whose intensity

























natural line


threshold


Figure 8.Axial mode spacing.









varies at the rate F(t)2. Figure 9 illustrates a typical

example. The peak intensity reaches (2n+l)2 and the first

zero of F(t) is separated from the peak by a time interval

T = T/2n+l where T is the period equal to 2r/Aw. For a

laser of optical length L' = 60cm, T = 4x10-9sec. If 100

adjacent axial modes are locked together the resulting peak

intensity is 104 times that of the individual modes, that

is, 100 times the sum of all intensities. The peaks

would repeat at the rate of T-' = 250MHz and peak pulses

would have a half width of 4x10-n sec.

Experimentally, phase-locking of the axial modes was

first realized by means of an acoustic modulator incorporated

in a He Ne laser. Further experimentation with mode-

locking revealed that it could be accomplished without the

use of an externally driven modulator. Incorporation of a

suitable bleachable absorber cell produces self-locking of

the longitudinal modes. Figure 10 illustrates schematically

the use of a bleachable absorbing cell as a mode locker.

The complete description of this process requires a detailed

mathematical analysis although several things can be said

qualitatively.14 The modes are all locked in phase when the

dye cell is short and is located near one of the mirrors.

Mode locking is accomplished by the nonlinear interaction

of radiation of differing frequencies within the dye cell.

The nonlinearity of the interaction is the consequence of

the fact that the dye cell is operated in a region of

intensity where the transitions are nearly saturated.





































Figure 9. Example of mode locked pulses.




















4 0 aser

M dye
cell


Figure 10. Mode locked laser.









The need for saturable absorbers was one of the early

motivations for finding gases which efficiently absorbed

laser radiation. In the case of CO2, several gases were

found such as SF6, NH3, PF5, and CH3F.1 It was soon

discovered however that laser absorption provided a means

of preparing a molecule in an excited state in order to

study energy transfer processes and chemical reactions. The

criterion for good absorption of laser radiation is that the

gas must have an energy level resonant with the laser

emission. For infrared lasers, this corresponds to

molecular vibrational frequencies, and it was soon dis-

covered that the laser could play an important role in the

study of a process mediated through vibrations such as

chemical reaction.16-23 Lasers have become an important tool

in fluorescence studies, ultra high resolution spectroscopy,

and molecular energy transfer, as well as developing passive

mode lockers for other lasers.24 Figure 11 shows a typical

experimental set-up for a quasi- CW fluorescence experiment.

Infrared fluorescence may be monitored from a gas which

absorbs CO2 radiation directly, or by the addition of a

low concentration of sensitizer such as SF6, the fluorescence

from the collisionally excited molecule may be observed.

Laser induced fluorescence spectroscopy is also a valuable

technique for following the course of laser induced chemical

reactions.25,26 Molecules can be excited by the absorption

of CO2 laser radiation, and the kinetics of the subsequent

reactions can be monitored by ir fluorescence spectroscopy.






















C02 s
laser


M
reference
detector





recorder


Figure 11.
study.


Experimental set-up of laser fluorescence


Power
meter









In general the reactions are monitored by observing the

fluorescence from a given vibrational level as a function

of time. Detailed energy transfer and evidence of non-

Boltzmann chemical reactions have been observed in CH3F,

CH3Cl' and CH2C12.52

If,as postulated, chemical processes can be affected

by selectively exciting vibrational levels in one of the

reactants producing non-equilibrium distributions,

preferential reaction channels or increased reaction

rates may be obtained. Possible applications of this

technique include accelerated catalysis, efficient

fractionation of hydrocarbons, and isotope separation.

To date the possibility of stiumlating chemical reactions

with lasers has been shown to be possible; however as

yet there has been no large scale applications of non-

Boltzmann chemical reactions.28

The possibility of separating isotopes with lasers

has received much attention in recent years. The central

idea is selective excitation of the isotopic species by

a narrow band ir laser. This occurs because of the

isotope shift,which is in the infrared portion of the

spectrum, so that the laser leaves only a single isotope

vibrationally excited. Photoionization may proceed

from the excited molecule by a visible laser which

would not contain enough energy to ionize the non-

excited molecule. The ionized molecule may then be

removed electrostatically or chemically.23









One major problem that must be overcome is the

exchange of charge or vibrational energy between isotope

and non-isotope before the separation takes place. This

would mix the molecules enough so that no substantial

increase in isotope concentration would result. Lasers

may be the solution since a laser induced reaction might

proceed rapidly.

Although the idea of laser isotope separation

appears sound, in practice, few isotopes have been

separated by either the photoionization, photon recoil,

or induced chemical reaction method. One successful

isotope separation has been with the isotope of clorine.

The reason for this success is the fact that BC13 has

an energy level resonant with CO2 laser radiation.

Not surprisingly,boron isotopes have also been separated

by this process. There are however many isotopes which

do not absorb resonantly with conventional lasers. It

is the separation of these isotopes, which include

uranium,that have stimulated interest in developing

middle infrared lasers of high (several percent)

efficiency.

The most important isotope separation problem

today is the U235 U238 pair for obvious reasons. A

glance at the spectrum of UF, reveals that the absorp-

tion at 16 microns is by far more intense than other

wavelengths and the isotope shift is largest there also.29

A 16 micron gas laser does not currently exist and its







26


development has stimulated interest in several laboratories.

A new development that appears promising in genera-

tion of ir lasers is optical pumping with a conventional

laser source. This eliminates the need for operation in

an electric discharge or flowing gas chemical system

which usually requires a homonuclear or metastable

collision partner such as N, and H2. The fundamental

idea is absorption of laser radiation of one wavelength

and regeneration at another. The quantum efficiency of

such a process would be the ratio of the two wave-

lengths or about 62% for CO2 absorption and 16 micron

emission, although quantum efficiency is seldom approached.

CO2 lasers commonly operate with 20% efficiency which

would yield a 16 micron laser with 1-2% efficiency.

Although middle infrared lasers have not yet been

generated in this manner, nearly 200 different wave-

lengths lasers in the far infrared have and it is

hoped that this method will eventually provide lasers

at nearly any desirable wavelength. 332 The application

of optical pumping for generation of middle infrared

lasers to be used in the study of isotope separation,

laser induced chemistry, and ultra high resolution

spectroscopy should deserve more and more attention in

the future.

The author believes an important problem which

may be solved by optical pumping is the development

of a laser at approximately 16 microns for the separation







27

of uranium isotopes. By using a conventional laser

source such as CO2, a molecule may be excited and made

to regenerate at around 16 microns. Figure 12 shows a

diagramatic representation of the energy levels required

for successful absorption of CO2 radiation and regenera-

tion at 16 microns.

The purpose of this paper is to demonstrate the

feasibility of generation of middle infrared lasers

by optical pumping. The particular wavelengths used

will be absorption of CO2 radiation and emission at

16 microns, although the ideas should be equally

applicable to other needs.























948 or
1046


Figure 12. Expected energy levels leading
to a 16micron laser.
















CHAPTER II ENERGY TRANSFER IN MOLECULES


Molecular energy transfer is important to both laser

absorption and laser emission processes. For radiation

in the infrared, energy transfer is largely in vibra-

tional and rotational modes. Schrodinger's equation is

written by assuming the Born-Oppenheimer approximation

(4) Y(r,R) = Te(r)Xn(R)

where Ye(r) is the election wavefunction and Xn is that

for the nucleus. Xn includes all nuclear terms such as

vibrational and rotational motion. The energy is the

sum of the individual contributions viz,

Etot = Etans + Erot + Evb + Eel+ Espin-orbit


Normally the contributions from infrared processes are

those of vibrational and rotational motions. The

electronic configuration is usually considered constant

for infrared processes. Eigenfunctions for vibrational

motion are usually found by first assuming a harmonic

potential and adding anharmonic perturbations. The

eigenfunctions for a harmonic potential are harmonic

oscillator wavefunctions which lead to an evenly spaced

spectrum with selection rules Av = 1 (see figure 13).

The addition of anharmonic terms perturbs the spectrum






30










3


S plllrim
2 --:s I I I I I I


spectrum


Figure 13. Spectrum of harmonic oscillator.









qualitatively as shown by the dashed lines.

As indicated earlier, rotational energy splits each

vibrational energy level into many sublevels, the

eigenfunctions of which are found by assuming a rigid

rotor Hamiltonian and are proportional to spherical

harmonic functions. This leads to selection rules

AJ = 0, 1, for a totally symmetric electronic wave-

function.33 Figure 14 shows the energy levels of a

rigid rotor and indicates dipole-allowed transitions.

Emission or absorption of light is usually explained

within the framework of time dependent perturbation

theory.34 After separating the unperturbed Hamiltonian

into vibrational, rotational,and electronic motion,

solutions are found in the absence of any electromagnetic

radiation. The oscillating field is then treated as a

perturbation and first order quantities such as probabil-

ities of transition from one stationary state to another,

and energy corrections are calculated. If one writes

the perturbing hamiltonian as

H'(Rt) = H'(R)eimt

then the probability of the system being in state m at

time t (after starting in state k) for a dipole trans-

ition is

(7) jCm(t) 2 = 41Hk 1 sin[(Ek- Em ho)t/2h]/(Ek-Emhw)2

where Em, Ek are the energies of the m and k states.

The case where Em= Ek+h is called resonance absorption

or emission (depending on the sign), and for this case


































Figure 14. Energy levels of
rigid rotor.










the transition probability depends linearly on time for

short times. (Recall that this is only a perturbation

approach and the interaction must still be small or occur

for a short time.) The matrix elements coupling the

states may be written

Hk = < mlH'(R) k > = <, H (R) I k>

where the wave functions are those calculated from the

unperturbed Hamiltonian. A first order calculation

results in

Hm = Ep (mk)

where p(mk) is the transition dipole moment

p(mk) = < m!l|k >

and V is the permanent dipole of the molecule.

The probability of a level changing from state k to

m in a radiation field with a finite spectrum rather than

simply a monochromatic field may be found by integrating

(7) over the frequency domain and averaging over all

spatial directions. The result may be written in terms

of the Einstein stimulated emission coefficient

Pk-m = UvBmk

where U is the radiation density at frequency corres-

ponding to energy difference of levels m and k, and Bmk

is the Einstein stimulated emission coefficient

Bka = SWT/3h2|lp(mk) i

Transitions which are forbidden; those for which the

transition dipole (or dipole moment) are identically zero

are found by symmetry operations and it is usually









relatively easy to find which states are not infrared

active. 33

The mechanism by which laser action initiates and

subsequently amplifies is most easily explained with use

of the Einstein coefficients. In the early part of this

century it was shown by A. Einstein that a resonant photon

travelling in one direction could react with an excited

molecule and produce two coherent photons traveling in

phase, plus an unexcited molecule. Once laser action is

started, the laser medium acts like an amplifier. The

probability of an excited system undergoing such a

transition in a radiation field of energy density U, has

already been given as

P21i = UVB12
and similarly for absorption. This does not,however,

explain how any radiation gets started in the cavity since

initially for zero radiation there is zero probability of

initiation. Actually,atoms or molecules in excited

states remain so only for limited periods of time due to

the Heisenberg uncertainty principle. The uncertainty

may be expressed

AE At = h or

Av At = 1

so that for an ensemble of molecules of lineshape with

half width Av the excited state can be expected to decay

spontaneously within At. Therefore we must add a term to

our probability for a molecule to go from state 2 to










state 1: P21 = U B21 + A21.

A,1 is the Einstein spontaneous emission coefficient (At-).

Thus we see that it is through the natural decay that

noise is able to start the laser amplifier.

Vibrational or rotational energy can be transferred

collisionally as well as through radiative transition

(for infrared we do not consider electronic transitions).

Collisional transitions occur when one molecule collides

with a second and transfers some, or all,of its energy to

the different degrees of freedom of the second molecule.

This energy may go into rotational, vibrational, or

translationalenergy of the second molecule. The time a

molecule remains in an excited state before the energy

redistributes back to equilibrium conditions is called

the relaxation time. In optical pumping, energy transfer

can proceed three ways; transfer from one excited

vibrational state to another (V-V),transfer from an

excited vibrational state to higher rotational energy

(V-R),and transfer from an excited vibrational state to

the translational motion of the molecule as a whole (V-T).

It is commonly believed that T _v< TR_< TVT where 7

is the relaxation time for that particular process and

we assume that transfer within one vibrational mode is

faster than between two different modes.20'23'35

Theoretical accounts of collision processes have only

moderate success in predicting probability of energy

transfer. The reader is referred to an excellent review








36

of the theoretical treatment of collisional energy transfer

by Bailey and Cruickshank.36 The most popular theory to

compare with experimental results is a quantum mechanical

V-T theory developed by Schwartz, Slawsky, and Herzfeld

(SSH theory).37 SSH theory, as well as most others, does

not succeed in providing absolute probabilities. However

it does predict that resonant processes (those for which

the second of the colliding pair has an energy level

resonant with the excited level of the first) have the

largest relative probability,and that pairs with a

smaller reduced mass will have higher probability of

energy transfer than others.

Although there are no rigorous selection rules for

molecular collisions, there appear to be some collisional

transfer processes which occur with much higher proba-

bility than others. This is based on the fact that

symmetric states do not combine with antisymmetric for

any kinds of transitions including collisions. It is for

this reason that the two modifications of symmetric top

molecules such as NH3, CHC1, CH F (e,a) are transferred

into each other only extremely slowly, just as are ortho-

and para- hydrogen.38

Recently Oka has given some approximate selection

rules observed by microwave studies of NH, and the

qualitative interpretation of them.39 To summarize these

he finds










(i) Collision-induced transitions with dipole selection

rules (AJ = 0, 1, parity + *--) are "preferred".

(ii) The AJ = 0 dipole-type transitions (same selection

rules as dipole transitions) have much greater proba-

bility than the AJ = 1 dipole-type transitions for levels

with JK but they have probabilities of equal order of

magnitude for levels with J>>K.

(iii) The AJ>I transitions have much smaller probabilities

than the AJ = 1 transitions.

(iv) The AK $ 0 transitions have much smaller proba-

bilities than the AK = 0 transitions.

(v) It is suggested that AJ = 1 quadrupole-type trans-

itions (parity + +-+) are the same order of magnitude as

those of the corresponding dipole transitions.

These "selection" rules are very important since

most experiments measure the rate of transfer of

vibrational energy between different modes of a molecule

or different molecules, although they don't give information

as to the particular rotational levels involved.

We have seen in this chapter that there are two basic

energy transfer mechanisms: radiative and collisional.

Both are important to lasers, sometimes beneficial some-

times detrimental. Radiative transitions are obviously

most important when optically pumping a gas with a laser

such as C02; however if laser action is expected from

any level other than that pumped, one must rely on

collisions to transfer energy selectively into that level.






38


Transitions due to the natural lifetimes of the state

must initiate stimulated emission before collisions

populate the lower state, thus depleting the population

inversion. Both spontaneous and stimulated coefficients

must be known as well as the collisional rates and their

selection rules if one is to successfully predict how

a collection of molecules will react after optical pumping.

The regeneration of laser radiation at a different

frequency will be a function of all three processes.
















CHAPTER III METHYL FLUORIDE


The success of producing an optically pumped laser is

largely dependent on the gas to be used as a lasing

medium. It must absorb CO2 laser radiation strongly

and be able to achieve a population inversion between two

levels that would result in 16 micron emission. When

modeling a process such as this, a good deal of informa-

tion is required to accurately predict the results.

The first area that must be understood is the absorp-

tion process. It must be known how strongly and on what

CO2 line the gas absorbs. A high resolution spectrum is

helpful to see to what degree the gas and pumping line-

shapes overlap. Normally, one is not given this informa-

tion, rather an experimental absorption coefficient is

given for a particular CO2 line. The overlap integral

may still be estimated if the Einstein coefficients can

be found from vibrational band intensity measurements

which have been carried out for many molecules. In

general the absorption coefficient will depend on the

particular rotational levels involved, line broadening,

degree of resonance, and pressure. One advantage of

modeling a process such as this is that these parameters

may be changed and the absorption coefficient still known

39









if a measurement has been carried out for any one partic-

ular set of parameters.

The second area of importance is the collision

kinetics of the molecule. This is important to the

problem of rotational bottlenecking. Often during laser

operation, a particular rotational state will be filled

faster than the collisional rotational relation can relax

it. When this occurs, the population inversion cannot

be maintained and lasing stops. The bottleneck effect

is so termed because it is normally the process that

limits laser efficiency. Within this area also,is the

V-V energy transfer process. If laser action is expected

from any level other than the upper state of the pumping

process, energy will have to be transferred via V-V

collisions. The time scale and efficiency of these

collisions will be very important to the success of the

laser.

The gas to be used in generation of a 16 micron laser

by optically pumping with CO2 must strongly absorb CO0

radiation, and must have two levels separated by approx-

imately 624 cmT' The lower level must be far enough

above ground, that there is no appreciable thermal

population. Although several molecules fulfill the

above requirements, only one has had enough experimental

work done on it to make it attractive. The molecule,

methyl fluoride (CH,F),had been used as a saturable

absorber in CO2 mode-locking studies and far infrared









laser generation, 32 a source of photon echoes in phase

coherence studies," and recently the collisional kinetics

have been studied by following laser induced fluoresence.25

Figure 15 displays the energy level diagram for methyl

fluoride. The ground state to V3= 1 transition is at

approximately 9.55 microns which is within the 9.6 band

of CO,.'4 The second excited state of V, is reported at

2081 cm-1 and the lower pair of the V25 doublet hybrid

band is reported approximately at 1460 cm-'.42

Methyl fluoride is a symmetric top molecule with Clv

symmetry.38 The rigid rotor term values are given by

F(J,K) = AJ(J+1)+(A-B)K2

where J is the total angular momentum and K is the

component of J on the internuclear axis. The constants

A and B are defined as

A = h2/2cI, B = h2/2cIb

where I is the moment of inertia about that particular

axis. Selection rules yield radiative transitions (rigid

rotor, harmonic oscillator) AJ=0,l,AK=0,l which lead

to the following term symbols, labeled

KJ(J ,K
K lower Klower
QQ(J,K)=VO

QP(J,K)=vo+2AJ

QR(J,K)=vo-2A(J+l)

FQ(J,K)=v-(A-B) (2K+1)

PP(J,K)=v,+2AJ-(A-B)(2K+l)

PR(J,K)=vo-2A(J+1)-(A-B) (2K+1)





































Figure 15.Energy levels of CH3F.









RQ(J,K)=v0+(A-B) (2K+1)

Rp(J,K)=V,+2AJ+(A-B) (2K+1)

RR(J,K)=vo-2A(J+1)+(A-B) (2K+1)

Actually the rotational constants change slightly from

one vibrational level to the next and centrifugal

distortion terms may be included so that the spectrum is

slightly more complicated than is indicated here.

Thus it may be seen that the spectrum of methyl

fluoride is quite complex. Parallel bands, those for

which the dipole changes parallel to the applied field

have selection rules AK=0, while perpendicular bands

have AK=l. It is customary to make the following

vibrational assignments for methyl fluoride.43 Actually

the V2 and V5 vibrations must be considered degenerate

thus creating a V25 and V, hybrid pair. This band is

composed of both a perpendicular and a parallel band.

The transition from V3=2 to V2, to be considered should

therefore have several rotational lines on which lasing

could occur around 16 microns.

Vibration Freuency(c) Species Type

VI-CH3 s-stretch 2930 a parallel

V2-CH3 s-deform 1464 e parallel

V3-CF stretch 1049 e parallel

V- CH, d-stretch 3006 e perpendicular

V,-CH, d-deform 1467 e perpendicular

V6-CH, rock 1182 e perpendicular









Methyl fluoride has been used as an absorber of

9.6 micron CO, radiation and absorption on the P(20)

and P(32) CO2 lines has been reported. The P(20)

absorption is believed to be the QQ(12,1) and QQ(12,2)

meythl fluoride transition while the P(32) absorption is due

to another isotopic species of methyl fluoride." 32 The

lack of absorption of other CO2 wavelength is somewhat

surprising since a medium resolution spectrum exhibits a

large Q branch centered around the P(18), 9.6 CO2 band."4

Closer inspection reveals the K splitting of this branch

may account for the possibility that the extremely

narrow laser line can fit between two methyl fluoride

transitions. The laser used for this experiment was a

Q-switched mode-stabilize CO2 and the possibility of

hole burning exists. The effect of a TEA laser pulse or

a mode stabilized TEA laser can be modeled as described

in Chapter I. This can be written into any simulation

scheme and results checked to observe the effect of

different forms of pumping radiation.

Recently the collisional kinetics and energy transfer

have been studied by monitoring fluorescence rise times

from various levels after pumping the V3=l with a CO2

laser.25'35'4548 This technique has been fruitful in

observing collisional energy transfer from a particular

vibrational level to a diluent gas also.49 Figure 16

indicates the relative speed of V-V transfer after

pumping the V3=l level in methyl fluoride. The results




























(1) Excitation of V3 by 9.6 P(20) C02
(2) 2CH3F(V3) CH3F1O) + CH3F(2V3) + 10 cr
CH3F (2V3) + CH3F (V3)- CH3F (0) + CH3F (3
10 collisions
(3) CH3F(3V3) + CH3F(0)7 CH3F(0) + CH3F(V]
CH3F (3V3) + CH3F (0) CH3F (0) + CH3F (V4
70 collisions
(4) CH3F(VI) + CH3F (0)Z CH3F (0) + CH3F (2V
CH3F (V1) + CH3F (0): CH3F(0) + CH3F(2
CH3F (V4) + CH3F (0) CH3F(0) + CH3F (2
CH3F(V4) + CH3F(0)- CH3F (0) + CH3F(2+
10 collisions
-1
(5) CH3F(2V2) + CH3F (0)- 2CH3F (V2)-0 cm
CH3F(2V5) + CH3F (0)- 2CH3F(V5)-10 cm-1
50 collisions
(6) CH3F(V3) + CH3F (0) C CH3F (0) + CH3F(V)
40 collisions
(7) CH3F(V-T/R) 15,000 collisions


-1
[1 -1
V3) + 20 cm
-1
) + 120 cm-1
) + 100 cm

V2)
/5)
2)
5)


-133
)-133 cm


Figure 16. V-V,V-T/R rates for methyl fluoride.









were obtained by monitoring the fluoresence risetime in

an experiment similar to one described in Chapter I.

The processes are step-wise in that they follow in the

order indicated. Comparison with SSH breathing sphere

amplitudes reveal the relative collision rates are in

good agreement although as is usually true, absolute

probabilities are not.4'47

The very long V-T time is due to the fact that the

lowest level of methyl fluoride is still over 1000 cm-'

above ground. By using a double resonance experimental

setup, where a particular rotational level is populated

into the upper vibrational state, and microwave absorption

monitored as a function of time for AJ=1, AK=0 in C3 H,F

the rotational relaxation was found to be

T = (10.56)sec/mtorr, which is long as compared with the

V-V rates.50

As can be seen from the preceding discussion, a

great deal is known about methyl fluoride. Figure 17

indicates the match up between known CO2 laser lines and

the theoretical methyl fluoride spectrum around the

reported P(20) absorption.4'"'5 Other CO2 lines have close

coincidence, although we will primarily discuss the P(20).

This figure clearly indicates that even high resolution

is not enough to guarantee good absorption of CO2.

Due to the number and narrowness of methyl fluoride

absorption lines, one might expect little absorption of

a CO2 line that chanced to fall between two methyl








47
























J K qQ term value
12 0 1046.837 Reported CO2 line 1046.85
12 1 1046.827 P(20) absorption
12 2 1046.826 reported.
12 3 1046.824
12 4 1046.821
12 5 1046.817
12 6 1046.813
12 7 1046.808
12 8 1046.802
12 9 1046.796
12 10 1046.788
12 11 1046.780
12 12 1046.771

Half width CH3F; 2.21 x 10-3cm-1 at 1 torr.
Half width CO2; 2-32 x 10-3cm-1 at 1-40 torr.
Cavity width CO2; 10-5cm-1.


Figure 17. Theoretical spectrum of CH3F









fluoride lines. The question of how pressure broadening

of both CO, and methyl fluoride affects absorption has

not been answered. It is probably true that the QQ(12)

transition is involved instead of the next closest QP(1)

since the thermal population peak is around J = 11.31 32

Lasers which operate on a single axial mode, tunable over

their doppler width have recently been made commercially

available and will no doubt play an important role in

studying and maximizing absorption in molecules such as

methyl fluoride.

Of course the real interest of this study is the

generation of a 16 micron laser. Figure 18 shows the

approximate expected spectrum for V3= 2 to V2 transition.

The rotational constants are not accurately known for

either level so the ground state to V,=l rotational

constants were used.32'43 These may be in error by severalwave

numbers; however it serves to give an idea of the relative

differences among transitions.

Although this paper is concerned with the computer

simulation of laser construction, Figure 19 illustrates

a typical laboratory construction of an optically pumped

laser. The KBr prism acts as a dispersing element thus

enabling reflection of the CO2 and 16 micron radiation

to physically take place in different regions. The

grating is needed to operate on the 9.6 band since gain

is normally higher in the 10.6 region. Mirrors M, and M2

should be gold or dielectric coated to reflect at 16 microns.







49























J'=12, K'=2 J'=12, K'=l

F(JK) 'J K F(J,K) JI K"
619.23 12 2 619.23 12 1
qR 641.07 11 2 599.07 13 1
qp 599.07 13 2 641.07 11 1
rQ 631.57 12 2 622.78 12 0
PQ 596.37 12 3 605.17 12 2
rR 651.73 11 1 642.93 11 0
PR 616.52 11 3 625.32 11 2
r 607.73 13 1 600.93 13 0
PP 674.53 13 3 583.33 13 2


Figure 18. 2V3-V25 approximate spectrum.

































































Ccy


E-

$4
0
r-4
4-4


_0










4J
0







r.
0
-4

r4
in








44


al-
in


sl


w









If a line other than that at 16 microns has a lower

threshold, it will lase first. In this event, mirror M2

will have to be replaced with a grating so that the

cavity becomes tuned to 16 microns.

It can be seen in figure 19 that there are many

different external conditions under which lasing may be

attempted. These include gain length, pressure, mirror

reflectivities, pump power and duration, and type of CO2

pumping. The purpose of this simulation is to check the

effect of different parameters and narrow the conditions

under which one should expect laser action. This will

eliminate much trial and error experimental work and

should help in the planning and construction of the laser.

A second area where computer simulation is expedient

is the determination of how accurately a particular

molecular constant is needed to be known for accurate

prediction of laser generation. This will help plan

additional experiments which need to be performed prior

to laser construction. In the case of methyl fluoride

there are two areas of importance that may need further

study. The first is the absolute frequency of the

V3= 2 to V25 transition. This will have to be measured

if it is deemed important. The second area of importance

is the life-time of the V,= 2 state in methyl fluoride.

This is important because a long lifetime (determined by

the Einstein spontaneous emission coefficient) will allow

the level to collisionally populate, thus depleting the










population inversion. The spontaneous emission acts as

an initiator of laser action and must initiate rapidly

enough to allow the energy to transfer radiatively from

V3= 2 to V25 = 1 rather than collisionally. Thus the

competition between collisional and radiative lifetimes

may be extremely important. An important aspect of

modeling as we have described, is that the collision

rates are known. This should enable us to predict for

what range of values of the Einstein coefficient to expect

lasing. It is conceivable that a rough estimate within

a factor of 100 is all that is needed to assure lasing,

or an accurate measurement may have to be made if the

results are very sensitive to the actual numerical value.

We believe that the Einstein coefficient for the V3= 2

state may be estimated fairly accurately from the Einstein

coefficients for the V = 1 level. This coefficient

may be determined by integrated intensity measurements

using the relation

r(cm2 mole') = 2.505 x 10EP2
52
substituting the measured value for F yields.

r = 9055.6 10%

B 2 = 6.8 x 1023 3 J1sec

It is expected that the Einstein coefficient for the V3= 2

to V25 = 1 will be substantially smaller than B12 since the

transition is forbidden by harmonic oscillator selection

rules (AV / 2). Thus by estimating the coefficient for

this transition we may determine roughly to what accuracy

it must be known. See Appendix B.

















CHAPTER IV MODEL


The model which we employ is a standard kinetic

rate equation used by many authors. The two types of

terms are those for radiative52 and collisional35

transitions. Figure 20 shows the interaction of

different levels in this model. This figure should be

compared with the energy level diagram for methyl

fluoride in figure 14. We assume that collisionally

AK = 0, AJ = 0 selection rules predominate so that

relatively few rotation levels need to be represented.

The V2, state is represented by two rotational levels,

one for collisional population and one for radiative

population since it is desired to see what effect the

different symmetry combinations have on lasing.

Rotational levels of methyl fluoride are of two types,

the K = 0,3,6,...are symmetric (type a) while the

K = 1,2,4,5,...levels are asymmetric (type e). Levels

with overall species e are doubly degenerate, thus

direct product tables show that the V, level has rotational

levels a and e, while the others have only the a

rotational level; these yield the overall e state

(a x e = e) while the e rotational level gives an

a + a + e(e x e = a + a + e).38 Molecules of methyl

53



















lot II


2v3 -, ;7

16yJ
jlot JOIr 'K
[10% 10 =
'6


0


collision
laser


0.b UU2


laser

1K
Figure 20. Schematic representation of model.
(large numbers indicate number of collisions
and arrows, the direction for that rate.)


^









fluoride in the ground state are either in rotational

type a or e so that even collisionally, little mixing is

expected. The possibility of changing rotational levels

by AK = 0,1l exists, since V2 is actually a hybrid

(parallel and perpendicular allowed transitions) and

thus the lower laser level could possibly be a different

symmetry than the upper level which is filled collisionally.

The equations when non dimensionalized in time with

respect to collision frequency, number density with

respect to density of the ground rotational level, and

laser output with respect to the input CO2 power assume

the form


dT = QW(N 3-- NL ) + P
t- gV25V2


dN3 = r (NN3 e0/kT N ) + r4 (NN33 e20/kT N 3N3)
dt

+ r12 (NNV6 e133/kT No NV3) + Z 0 (NO- N3 )


dt r,(elkT N3 NN0 N2) + r (NoNv3- e20/kT N2vN

Z23 (N2v3 9 1 NL )
V25 V25


d = r (e20/kT N23Nv N N v3) + r(NONV1 el10/kT NoN33)
dt


d_ = r (e"l/kT NN 3- N N ) + r, (NoN2V25
dt v3 DNv41


- NoNV41 )










dN5= (N N N N2V25 + r (N25 e /kT No N 2v25)
dt = rV(NoNv-2 NvN 7 0

+ r (Nv2 eo/kT N N2v5)


dN' r (e10/kT NCN) +, (NO N N N2
dt NoN2V25 V25 9 N N52 NV25
ri NoNv25+r1 N Nv6 TN25+ rF Z23
L c
(N23 g2y3 N5 )- U(Nv25
V25


d = 23 Z (N3- g2V3 N )L TNvL + rF[r(e10/kT NN2v2S
21 2V3 V25 V25 7 2V25
92V 3
Nc2 ) + r, (No Nv N N25 ) r,1(No NV25
V25

+ r1No0 N6 ]- U(N 25)


2 c
d = r8 (eo0/kT No,2 NNc2 + r (N9 N 25 N0 v
dt
ro NoN52 + r11 NNv YN2



dNv r,1 (el3/kT NNv N Nv,) 2r,, No Nv
dt
+ rO N (NV25 + NV52 ) XNv


No = K [N3 + N2V3 + N3V3 + N225 + N2
+ NL + v + ]
V 2 5 NV5 2 N6
where
Y = 16 micron laser

D = input CO2 laser

N.= number density of particular energy level.
1









where


9
K => N.(t=0)
i=o

and we define the following constants

Q = hvoNB23 W
V

P = C In rI r,
2Lv

Zll = Q V B12V


Z3 = k B__' W
vuI CV

r. kN/


(Collisional reaction rate ki is defined by

dN = kilNiN kiNN
dt i2Nk

for a process such as

CHF(Ni) + CHF(Nk) CH3F(Ni) + CH3F(Nj) + TV

from the equilibrium condition

Kil [CH3F(Ni)CH3F(Nj)] fv/kT
Ki2 [CH3F(Ni)CH3F(Nk)]

T = r/v rotational relaxation

RF = % relaxation between two V2, rotational levels.

If levels are of different species RF = 0.

U,Y,X = collisional depopulation with diluent gas for

that particular level.

Initial conditions on the vibrational level popula-

tions are given by Boltzmann statistics; while the initial

condition on the 16 micron laser is given by the









spontaneous emission. The following parameters are used

in the above definition:

h = Plancks constant = 6.626 x 10-27 erg sec

v = frequency of 16 micron Laser

vo = frequency of CO2 Laser

B12 = Einstein stimulated emission coefficient for

ground to V3 transition.

B23 = Einstein stimulated emission coefficient for 2V3

to V25 transition.

A23 = Einstein spontaneous emission coefficient for

2V3 to V2s transition.
NO = Population of the ground state rotational level

which is optically pumped.

V = collisional frequency

W = methyl fluoride lineshape at 16 microns

V = overlap between pumping lineshape and absorbing

lineshape.

Q = Max power per unit area for CO2 pumping laser

c = velocity of light in vacuum 3 x 1010 cm sec-1

rr2 = mirror reflectivities

1 = laser length (cm)

The effect of tuning the laser around the methyl fluoride

line is contained within the constant V. From the quoted

absorption coefficient the overlap internal may be

evaluated numerically.3032 Using equation Al, (Appendix A)

we see this gives a distance between the CO2 and

absorbing methyl fluoride line of approximately









1.61 x 10-2 cm-1. This agrees favorably with the

predicted distance from the theoretical spectrum of

figure 17. The laser used in this experiment was a Q-

switched, mode-stablized laser. By changing the frequency

difference between CO2 and methyl fluoride lines, the

effect of tuning the transition to exact resonance may

be simulated. Figure 21 shows the predicted position

for a common Q-switched laser relative to methyl fluoride

(Appendix A). This situation may be changed however by

changing (-,). An experimental condition such as

figure 22 may be simulated by bringing the CO2 laser line

very close to the methyl fluoride transition, or moving

it across the methyl fluoride transition in time.

The Einstein coefficients for the V3= 2 to V25= 1

transition may be varied to see how sensitive laser

operation is to them. As a first guess (see Appendix B)

we choose B 2= 4.58 x 1019 cm3J-'sec2. The collisional

rates are given in figure 19 and diluent collision rates

will either be taken from experiment or assumed.5'49

The rotational relaxation rate of methyl fluoride,

as previously stated, has been measured to be on the order

of 103 collisions. We will use 1000 collisions. All

calculations are carried out at 300 degrees Kelvin. Only

these parameters which may be easily met in the laboratory

will be considered thus eliminating exotic experimental

set-ups to check the computer predictions.


























CO2 doppler
Profile


CH3F


axial
mode


Figure 21. CO2 and methyl fluoride lineshapes.





























Pressure broadened
l C02


K: 01 2 3 4 5


J;12


Figure 22. Broadened
methyl fluoride.


CO2 in resonance with

















CHAPTER V RESULTS


Numerical solutions of the equations described in

the last chapter were carried out at the University of

Florida on an I.B.M. 360-75 computer. The numerical

technique employed was Hamming's modification of the Milne

predictor -corrector method. A fourth order Runge-Kutta

method is used to generate the first time-increment

solution since the predictor -corrector method is not

self-starting. Hamming's method is a stable fourth

order integration procedure which has the advantage of

a variable step size. This saves computing time without

sacrificing accuracy.

The accuracy for all the results was kept between one

part in 103 to one part in 10 This is better than

actually required; however it facilitates faster overall

integration since solutions were not allowed to start

diverging at any point. The time step was between .5

and .01 measured in units of time where one unit was the

time for one collision. The predictor corrector method

was able to bisect the time step up to 10 times if

required to obtain the specified accuracy.

The first question which needed to be answered was

what parameters to start the model with since there are

62







63

a large number for which threshold and subsequent lasing

would not occur. This depends on the Einstein coefficient

as was discussed in Chapter IV. Figures 23 and 24 show

the reflectivity needed at a particular pressure to

satisfy the Schawlow-Townes condition. The population

inversion may be estimated from the thermal population

of the lower laser level since little additional

population is expected during the first several collisions.

The three curves are drawn for the Einstein coefficients:

(L = 100cm)

A; B12 = 4.58 x 1019 cm3J-f sec-2

B; B12 = 4.58 x 1020 cm3J-1 sec-2

C; B12 = 4.58 x 1018 cm3J- sec-2.

All numerical work was performed for case A although if

the actual B12 is anywhere in the above range, the

equations will take the same form by picking a point on

the appropriate curve. The laser output however will

decrease as the reflectivity increases so that for

reflectivity above .99 little output will be expected.

It can be seen that case C is more or less a limiting

case for practical application.

Unless otherwise stated the conditions and parameters

were as follows:

Pressure = 10 torr

Pump power = 10 watts/cm2

Pump duration = CW

Reflectivities = 100%, 98%































.98



.95



.94-



.92


1 10 100
Pressure (torr)

Figure 23. Schalow-Townes condition for inversion of
5 x 10-3 ( L= 100cm ).





























.96



: .S4



.92


B

.1 1 10 100
Pressure (torr)

Figure 24. Schalow-Townes condition for inversion of
1x 10-2, ( L= 100cm ).










Laser length = 200 cm

Pump J,K; J'= 12 K'= 2

Laser transitions (lower) J,K; J"1= 12, K"= 2

Distance between CO2 and CH3F linecenters

= 1.62 x 10-2 cm-1

Rotational relaxation = 103 collisions

Diluent relaxation = none.

Cases of departure from these conditions will be dealt

with as necessary. Figure 25 shows the laser pulse as

it first starts to develop for 250 nsec after being pumped

by a kilowatt CO2 laser pulse for 50 nsec. Notice that

the laser does not immediately start to amplify after

threshold is reached, but waits several hundred nanoseconds

before the pulse starts to curve upwards again. This

is due to the relatively small Einstein coefficient

(~10 J-' cm3 sec-2) for the 2V, to V2s transition. In

studying the populations of each level in the model during

the first few collisions several things were apparent.

The relatively strong absorption coefficient (.018cm-ltorr-

for P(20)) caused the ground to V3 transition to saturate

within a few collisions. This strongly limited the

amount of energy in the CO2 pulse which was utilized,

since after saturation nearly all the laser pulse

propagates through the laser medium as a bleaching wave.

Increasing the power of the pump CO2 laser for this type

of situation was found to have little effect on output

or populations as might be expected.






67























41

E input

=2




2 4 6 8 10
collisions
Figure 25. CH3F pulse following threshold.







68

In the figures presented for energy level populations,

the following symbols are used.

SYMBOL ENERGY LEVEL

2 V3

3 2V,

4 3V3

5 V 4

6 2V25

7 V's collisionally filled level
L
8 V2s laser level

9 Vs2

0 V,

Figures26a through 261 show the results of pumping

methyl fluoride with a megawatt 100nsecCO, laser. The

resulting 16 micron laser output has a peak intensity of

approximately 5 watts/cm2 in 10-6 sec. Notice that only

3.4 microseconds elapse between pump and laser output.

The maximum population inversion occurs in only 20

collisions and decays nearly linearly for the next 150

collisions. The population of V3=l level saturates

immediately and decays via V-V collisions. The populations

of the V3= 1,2,3 levels all peak with a phase delay

corresponding to the V-V equilibrium rate. Notice that

the maximum population of each level is less than the

level previous thus indicating the collisional equilibrium

described earlier. This is not true for the V2s level

which is also populated by radiative transitions. It is









interesting to note that although the V2 level is not

populated by radiative transitions, it is always in

equlibrium with the V25 level. This is due to the fact

that both levels form a hybrid pair and are different

linear combinations of the same two levels. These levels

are degenerate and the resulting pair are in Fermi

resonance and are thus split by only 2cm-. In contrast

to these figures consider figures 27a through 271. This

set show response to a 5 watt CW CO2 laser. Notice that

the 16 micron laser takes nearly 2.5 times longer to

form its pulse, and the pulse is nearly 7 watts. The

long phase delay is understood if it is recalled that

lasing occurs from the V3= 2 level. By pumping with only

5 watts it takes nearly 100 collisions to saturate the

ground to V, transition and thus population in V,= 2

does not peak until 180 collisions after CO2 pumping

is initiated. The V2s level is being collisionally

populated during this time; however the figure clearly

shows that the rate for collisional population is much

smaller than that for radiative.

This explains why the population inversion has such

a slow rise time and rapid decline, as compared with the

previous figure. An important difference to note between

CW and short pulsed pumping is that in the CW case the

upper vibrational levels are collisionally populated

much more than the pulsed case. This is because of the

long phase delay of the output laser thus requiring levels


















PRESSL:E TORP
tO.uo0
PUMP J.K LOWFP LASER LEVEL J,K GO/iVI, GV3/GV?5
12 2 12 2 !.OOC0 1.UOC
NU4MER DENSITY Nc-CM-3
0.q558E 16

EINSTIEN CoEFS CU3 J-1 SEC-?2 55,a25,P12
n.45AOF ?0 0.9q2PE 03 0.680OE 2~
COLLISION FRr 'ENCY SEC-1
0.390OE n8
nVEPLAP INTfERAL FrP T,,.' LroP IT7r1.s RVDc '0 L S'"F
LINE CF !TEP nTqTACE-HAL T IT"' E-E'TYL FL' CIPIt A2'D C02SEC-1
O.4V50E 09 ".6000E 08 U.3000E 06
OVEPLAP INTERNAL L-L SFC
0.40CbE-10
CO? LASER IPIlT WATTS/Cv2 OURATIOu IN CnLLISIONS
0.1000E 07 4.n00
711....Z23 23...
.?2044F L 0.5706E 0? 0.2065F 01 0.1846E-06
DILUENT CCLLISTOl.AL rEPqPL,.IATIln 0O0AT InhAL RFLAXATTON
NL2 -N52-N6-Rn-T RELAXtvCCL LTSIC S-1
0.o (. 0.0 0.001
%RELAXATION BETWEEN kL?5-"L5?.A F: OLF J,K.
1.0000
MIPRIOPR EFLECTIVITIF-S, LASEo CAvITY LENT Ch
1.000 0.9pO 200.000

P= "00345


Figure 26a. Parameters.


















16 MICRON LASER OUTPUT.ATT/CU2
20.000 ---------.---------+----- ------.-----
SI T I I
I T T I
I I I I I
I I I I I
I I T I I
I I T I I
I T T I I
I T T I I
15.000 +---------+-------+---------------------
I I I I I
I I I I I
I I I I I
I I I I I
I 1I I I
T I T I
I I 1 I I
I I I I


L I I 1 I1
I T I I T
10.000 +---..---.---------.--------- +--------- .



a I I I
SI T T I I

0 1 T T I I
II I I I I I
S5.000 +---------+ ..-----+.------+-------+
6 I I I I
NI I T I


U I I I I
M T T I I

I I ** I I
SI I T* T I
S I I ** I I
T I I ** iI I
E I *I I ***** T I
0 o.0 +*****A**--------------------****
0.0 90.250 180.500 270.750 361.000
time



Figure 26b. Methyl fluoride laser.
















PnPULATTION 7VERSICM'
O.PCO +--.------+---------+ .-----------------
I I I I I
I I I I I
I I I I I
I I I I I
I I I I I
I I T
ST T I I
I I I T I
I I T I
.125 +-------------------.........--.........----------+.........---
I* I T I I
I T T I I


I I *I I II
I I T I I
V T T I T T
E 0.050 +---------+ ----.---+-.--------+---------+
N I I I
I i T I I T
0 I T I T
N I T *** I I t
/ 1 T ************* T
G I I T T*********I
p T T I I I
N I I I T I
0 -1.P25 +--------- .-------- .------- ---.. ....
I T T I
P I I I I I
n I T I I
T I I I I
I I I I I
L I I I I I
E I I T I I
V I I I T I
E I I I I I
L -0.100 +---------.. -------+--------+--------.+
0.0 90.250 1P0.50C 270.750 361.000
time


Figure 26c. Population inversion.







73









ENERGY LEVEL POPULATIOnS
".500 +---------+---------+--------*---------+
2 I T I I
I I I I I
I I I I I
I I I I I
I I T I
I I I I I
I I T I I
I I I I I
I I T I I
0.375 +-------- ---------.---------+---------+
I I I I T
12 T I I I
I I I I I
I I I I
I I T I I
I I 7 TI
I I T I I
I2 I I I I
I T T I I
0.250 +--------+---------+.-------+--+..--..--
I 2 I I
I 22 I T I T
N I 222 I I I I
2 I 222 T I I

G T I 22P22222222222222222221
R I I I T
N I T 7 T I
O 0.125 .....-----.. ------------ ----.... ---.......
I I I I I
I I I I I
I I
I I I I I
L I T I T I
E T T T I I
V I I T I I
E I I I I
L (.0 +--.------+--+----------------+-----
0.0 90.250 10.500 27U,750 361.000
time


Figure 26d. Population of V3.







74











ENERGY LEVEL 3OPULATI1NS
0.250 +--.----------------+---- --------+-------
SI T I
I I I T 1
I I I I I
I I I I I
I I I I I
I I I I I
I I I T I
I I T I I
0.188 +--------+-------------+--------+ -----+*
I I T I
I I T I I
T I I I I
I T T I I
I I I I I
II I I I
I I I I
I I I I
T 3 I T I I
0.125 +--3-------------------+-- .---+-----
13 1 I I I I
I 3 I T I
T I 3 I T I
SI T I
I 3 T I
Q I 13 T I I
N I 13 T I I
D 9.063 +---------+--3t33---+----------+--------+
I I 33313333333333313 I
R I I T I 3333331
T I
O I I I
I T I I I
L 3 I I T I
E I I T I I
V I I I I I
E I I I I I
L U.0 +------+------------+--------+--------
0.0 90.250 IA0.500 270.750 361.000
time


Figure 26e. Population of 2V3.


















ENERGY LEVEL POPULATIn q
0.125 +--.------+---------+---------+--------+
I I I I I
I I T I I
I T
I I I I I
I I I I I
I I T 1 I
I I I I I
I I I I I
I I I I I
0.094 +--.------+*--------+---------+---------
I I T I I
II I
I T I I I
I I I I I


0.063 - ----. ... .. +.... ....- .. .... .... ...
SI T T I
I T I I I
I I I I I

I I I I I

SI I T I 1
0.063 +--------+--------------------- -

I I I i I
R 4 I T i I


S1 4 I I


I I I T aaaouou
L I I I I I
E I I I I I



v T I T I
N I I I I



I I I T I I
F 4I I LL 1 1 I
r I I aldaIL I I
I I I Incagageson I
I T I I 44444J41
L I I I I I
E I I I I
V I I I I I
F I I I I I
L 0.0 ..--------+---- .---------------+----------
0.0 90.250 190.500 270.750 361.000
time


Figure 26f. Population of 3V3.
















ENERGY LEVEL POPULATIONS
0.100 +-------+---------+--------+--------
T T I I I
I I I I
I I
I I I I X
I r I
IT I T I
I I I I T
I I T I I
I I T I I
11.075 +I~----~--------------------.----+----+--

I I I I
I I I I I
I I I I I
I I T I I
I I I T
I I T I
I I
0.050 +---------+---------+---------------
I T I I
T I
r I T r I
T I I I
f. i I T
3 I I I I
0 .025 ...----------------.+-------- -+------ --
I 1 I I I
RI T I T
n I 559c5595 T I
T I 5551 55 55555555 555 T
T 55 I T 555551
L I I I I I
E T 5 I T T T
V TS5 I I I
V T5 1 17 I
E I I I T
L -U.00 55-.-------+---------*--------4.----------*-
0.0 90.'25 140.500 270.7q0 361.000
time


Figure 26g. Population of V41.

















ENEPGV LEVEL POPULATIr',l
0.050 +---------------------------+---- ----+
I T I I I
I I T T I
I I I I
I I I I I
I I I II
I I I T I
I T I I
I I I r I
0.038 .-------- -------------+- -------
I I T I I
I I I I I
I I T T I
I I I T I
I I I I I
I I I
I T T I
I I I T I
.m25 +-.---.---..--.-----. --------...
SI I i f
N I I I I I
G I IT I I
SI I I T I
G I I I I I
G I I T I

D 0.013 +--.-- ...+-66h6ft^^6h6-h^6^6.-........
T 66 T (6 6^b666dl
R I TI I I I

N I I I I I
D o.013 +---------+-6666666A6,666666---+--------+
1 66 1 666b6b66bbbr

SI 6 1 I I
T 1 6 I I I
TI 6 I I I
L I I I I I
E T 6 T T I
E I 66 I T I
L 0.n0O0 bb --------------+-----------------+
0.0 90.250 l0,.500 270.750 361.000
time


Figure 26h. Population of 2V25.



















ENERGY LEVEL POPULATIONS
!,.125 +-----+---------+---------+--------+
I I T I I
I I I I
I I T I I
I I T I I
I I T I I
I I T T I
I I I I I
I I T I
0.f94 *--.------+---------+-------.-+--------+
I T 7 I I
I I T I I
I I I I
I I T I I
I T T I I
I I T I I
I I T I I
I I T I i
.063 +--------+---------+---------+-----------
I I I T I
I T I T T
N I I 777777777777777 I
SI I 77777 I 777777!
I T 7 I I I
T 1 7 I I I
G I I I I I
I T 7 I I I
N I I I T I
D O.n31 +--------+-------+--------+-- --------+
I 17 I T I
I I I I I
SI 7
T I I t
I 71 T I I
L I 7 I I I
E I 77 1 1 I I
V I 77 I T I I
E I 77 I i I I
L 0.0 77------------------+---------+--------+
0.0 90.250 180.500 270.750 3h1.000
time


Figure 26i. Population of Vc5.







79












ENERGY LEVEL POPULATIrNS
0.125 +..-----.------------*------ ---+---------
I I I I I
I I I I I
I I T I I
I 1
I I T I
I I I I I
I I I I I
I T I I I
O.094 +-- ------+-----------------+----------+
I T T I I
I I T I I
I T I I I
T I T I I
I I I I I
I I T I I
I I I I I
I I I I I
I T I T I
0.063 +--------+----------+---------+---------

N T I BPPPFPs8PAoPaTTaggg8P8I
0 I I 8 !8 I T I
T T I I I
/ I I T I
SI I I I
TT I I I
D 0.031 +----- 8---8-------------------------+
T I I I I
R I I T I I
0 I I
I 81 I T I
E I 8 T T T I
E I 88 I I I
L 0.0 88--------+---------- -----------------
0.0 90.250 1i0.50C 27u.750 361.000
time


1
Figure 26j. Population of V25.
25'







80








ENERGY LEVEL POPULATIr',S
,.125 +-----------+----------+---------.------
I I I T I
I I I I I
SI I T I
I I I I I
I I I I I
I I T I I
I I I I I
I I I I I
I I T I I
0.09La *-------------------- -----------+-----+-----------+
I I I I I
I I I I I
I I T I I
I I I I I
I I I T I
I I I I I

I I T I I

I I I 909!
0.063 *--.---------------------+--------"----------------
I I T T I


T I T 09 T T
I I I I

& I I I I
P I T 9 I I
N I 1 9 I T I
0 U.031 +--.-----+-------- --------------*-------+
I I9 I I I
P I I T T I
n I T I I
SI 9 9 7 1 I
I 91 T T
L I 91 i I
I 99 I 7
V I 99 T I

L 0.0 q-------------------+-----------------------
0.0 90.290 IA0.50C 270,7S0 361.000
time


Figure 26k. Population of V52.

















ENERGY LEVEL POPULATIONS
0.250 +--------+---------+.---------+---------+
I T T I I
I I I I I
I I i I I
I I I I
I I I
I I I I I
I I T I
I T I I
0.188 +------------------+------------------+
1 1 I I
I I I I I
T I I I
I I I I I
I I I I I
T T T I I
I I I I I
I T I T I
0.125 +--.-----------+ -------+--------........



I 0 I I
/ I I I I I
R I T I I I
N I I T I T
D o.063 +-0. ----------------.--.------.-+
I 0I I I
R I I I I I
0 10 I T I. I
T T I I I
I I I I I
L I T I I I
E I I I I I
V I T I T I
E 0 1 T I I
L 0.0 +-----------------------.-+--- -----+
0.0 90.2=0 1R0.500 270.750 361.000
time


Figure 261. Population of V6.



















PRESSURE TnRp
t10o .'J"
PUUP J.K LC'OrF LASER LEVEL I,W G0I/V3, GV3/GVP5
12 2 12 2? 1.000 1.0000
NUMBER DE.SITY NP-CM-3
O.qS58E 16

EINSTIEN CnEF_ CV3 J-1 SFC-2 ;.15,A25;,12
0.0580F ?0 0.92a81E 03 .6800E ?4
COLLISTON cEP'IFy Y SEC-1
O.'390e 18
rVFRLAP IpTErp &L F R T'r) L"'? CTZrF' nRPO 2D''0 1 I I'JE
LIlE CF IEQ )TqTAn CE- ALF '-I3)TS AETY'!. L'O -IDET AND Cfl2SEC-1
0.49S1E 09 '.63%00 %9 0.3000E T0
OVERLAP TITEGRPL l-L SEC

C02 LASER IMPUjT IATT'/Cm? DiURATIO TN CILLISIn4S
0.5000E 11 3t1.00o
711.".. 23 Q?3.: -
0.1owqE-Ai A.2"'PE-03 I.PDJAc 11 0.3692E-01
D~IUENT COU LTTONAL 'EfPluLATJI"% oOTAT:L'NAL RELAXATION
MLP5-"52-, nT, RELiX!tLr-.LISTOCYS-1
0.. 0.) 0.0 0.001
RELAXATIONN 9FTEEN *L25- C2 .& FN C .!,<.
1.0000
MIRRIOP REFLFTIVITIES, LASER CAVITY LE'.GTw CM
1.000 0.P 0O 210.090

P= -0.0315




Figure 27a. Parameters.


----------------







83










16 MICRON LAER nOUTOJT.WATT/Cm?
2'.000 +---------+---------+---------+--------+
I I T I I
SI T T I
I T T I I
I I I I
I T I T T
T T T T I
I I T I I
I I T
I T T I I
15.600 +---------+----- ------------+..------+
I I T I I
I I T I I
T T T I T
I T T t I
I I I I
I T T I T
SI I I
I T I


A I 1 I I
E I I I I I
E I I I ** I I
T I T I I
0 I I T *I I


U I I I I I
T 5.000 +---------+---------*.--------*+---------*

1 T TT II
6 T I T I I

L I I I .*I
M I I I I I
I I I T *** I
L I I I**I
A T T I I I
S I I 7 I I
E I I *T I T
R 0.0 *****t****k****.**-.+--------.+-.-------
0.0 9C.250 1R0.500 270.710 361.000
time


Figure 27b. Methyl fluoride laser.

















PnPULATIU! INVERSION
u.200 *+---------+---------+-----------------+
SI T T I
I I T I
T T T I I
I I T I I
I I T I I
I I I I
II I I I
T I I I
I T I I
0.125 +--.------+------+----- ------ --------
I I T I I
I I I I
I T I I
I I T T 1
I I *** I I I
I I *** *T I I
I I 1** I I
N I ** I I
V I I I
E '.050 +------*--+---- --- +k+------ .+---------
P 7 { 1 T
S I T I I
1 I I T I I
0 I T I I
N I I I ***&*****+****T*I
/ I I T I
G ** T T I
P I I f T I
T I T I T
D -0.025 + ---------+---------+-------+.--------+
SI I T I
S1 I TI T I
0 I I 1 j I
0 I
I I T I
L II I I I
E I I I I
V I i I
E T T I I
L -0.100 +-----------+---------+------- --.---------
0.0 9G0250 1O.500 270.750 361.000
time


Figure 27c. Population inversion.

















EIEDGV LEVEI. OUPULATInr S
'.500 --------- -------------------- --------
I I I
I I I I

I I I I I
I I T T !
I T T I I
0.375 +..---.---------------+---- -+--------
I T T 7 I
I I I I I
I I T I I
I I I I
I I I I I
I I I I I
I I I I I
I I I I I
u.250 +--------+---------+-----------------+
I T222 I I
I 2222 2?222? I i
N I 2 I iT2 I I
SI 2 I 22 1 1
I 2 I 222 I I
I I I 222P222?2 I
G I 2 T T I 221
R I I I I I
N I T I I
0 0.125 +-.-.-------+-------------+- ---+------
I I II I
SI I I I I
nI I I I I
T 12 1 I I I
L I T I I
E I I T I T
v I I I I
E 2 T I I
L I,0 I +----------+--------'---------+---------
0.0 90.250 1A0.500 270.750 361.000
time


Figure 27d. Population of V3.


















ENERGY LEVEL rOPULATIr'JS
0.250 +---------------------+---------+--------

I T I T I
I I I I I



II

I T I I I
I I I I I
I I I I I
I I I I I
I I I I I
I 1 T I I

qI I I I I
I I I I I
I I I I I



D~2 n.6 ---------------------+--------------
I I TT t
SI I T I I
T T 3 T I


SI 3 I I I
R I 31 I 1 I
v I T 31 I


ET I 3 I T
I I 3 1 I I


t 0.063 +---.---+------------.--------+---------

L I I I I I
V I 7 I T I
E I T T T I I
L '.0 33--------+---------+---------+-- -**
0.0 90.250 1A0.50) 270.750 361.000
time


Figure 27e. Population of 2V3.


















ENERGY LEVEL POPULATIn,':
n.125 +-------+---------+---------+--------- +
I T I I
T T T I I
I I I T I
I I I I I
I I I I
I I I
I I T I I

O.-90 +--.--- ----- ---+---.---.---..------
I I T7 I
I T I I I
I I T I T
I I T T I
T I I I I
I I T I

'.063 +.--..----+--.-------.. -------.---- 444--
I 1 T aaUa T
t U i an 44i a T t
I I L47 T I
SI I a I T
I T 4 I T I
T I o I I I
I T L T T I
F I T a 7 I I
I I I T I
S".031 *---------+--------+---------+ ........
1 1 IT I
I 4I t T I
T u T I I I
I T I T I
I 4 I T I
T 4 I T I I
SI I I I 1
TI a T I I T
,p 8u44u -------- ------- --- ---- ---------+
0.0 90.250 1P.500 270.7C0 361.000
time


Figure 27f. Population of 3V3.
















ENERGY LEVEL POPULATIn S
1?.100 +--------------------------------.-----+
I r I I
i T T i
I I I I I
I r r I I
I I T I
I i T I
T r T I
T I T I T
I T i I i

I I 7 T !
I I T T I
I I I
T I T I I
T I I I I
I TI I I
I I I I I
I T T I I
0.050 +.-------+---------+-------------------+
I I 1 T I
T 7 7 I I
N I I T I
SI 555
TI I I 5555
G I T I T 55 I
R I I I 55 I
N I I I 595 I
D n.025 +--.------+--------+----55---+---------+
I I 535 T T
R T I 55 I I
0 I I 557 I I
T I I 55 I T I
I T 5 T I T
L I T 5S I I I
E I T59 I T I
I 55 I I I
E 555 I T I
L -0.000 c55c55 ----+---------+-------- --------- +
0.0 90.2c0 IA0.500 270.750 361.00
time


Figure 27g. Ppoulation of V41


















ENERGY LEVEL POPULATInrI
(1.050 +--------+---------+---------+---------

I T T 1 I
I I I I I
I I I I I

I I I I I
I I T r !
T T T T I
I T I I I
0.-38 +--.------+----------- ------+-----------
I I I I I
I T I I I
I T I I I
I I T T 66!
I I I T 6 I
I 7 T I 6 I
T I I 66 I
I T I 6 I
I I I I 6 I
0.025 +--------+--------+------+-- -----6--+
T T I 66
I I T 6
N T I 1 6 1 I
0 I I T 6 7
I I I 6 1 I
TI T 6 I
S1 T6 T1
R I T 6 T I

I I 67 I I
R I T 6 TT I
n I T 66 1 I I
T I 1 6 I I I
T I 6 T I I
L I I 66 T T I
E I T6 I I I
V I 66 T I I
E I 666 I I T I
L 0.000 666666 ----+---------+---------. --------*
0.0 90.250 IP0.500 27C.7=0 361.0n0
time


Figure 27h. Population of 2V25.






90









ENERGY LEVEL PODULATIr1
).125 +-......- -+--------------.........-------
I I I I I
I I I I I
I I I I I
I T I I I


I J T I I
I I 1 I
0.091 +--------+---------+---------- +--------+
I I T I I
T I T I I
I T I 17777777771
I 7 I I
I I T 77 T I
I I I I I
T T T 7 I I
SI T I !
0.063 +---...------ 7 -------- ----------------
I T I I I

7 i i 77 i
. I 1 T I

R I I 7 I
N I T 7 I I
0 0.131 +---------+--------+---------+.---------+
T I 71 I I
R I 1 7 1 1 I
0 T T 77 T I I
T I I 77 T T I
SI I77 I I I
L I 77 T T I
EI 777 I
v 77 T T
E 777 I T I I
L 0.0 777--------.------------+----+---------+
0.0 90.32S 10C.500 270.750 361.0n0
time





Figure 27i. Population of V25.

















ENERGY LEVEL POPULATIQ'1';
0.125 +---------+---------+---------+----------+
T I I
I I I I I
IT I I
I I T I
SI T I I
I I T I I
T I I
I I I I
O.094 +--.------* ---------+---------+--------
SI T T I
I I T I 818I
I T T 8P8888 I
I I I R8I I
I T I 9 I T
I I T I I
I I T T I
IT I I
I T i I I
0, 3 +,--------+------------+------------------
I T T T I
I I I I I
N I T T I
0 I I I I I
N I 1 I i
GI T I I
O I T I I
N I I T I I
0 Q.031 +--------------------------+----------
I I T I
0 I 1 I I I
T I I a9 T T I
I T T I I
L I 88 I T T
E I 881 T T I
V I 88 I 7 I I
E I B8 TB T 1T
L 0.0 889-------+---------+---------------
0.0 90.250 1;0.5)0 270.750 361.000
time


Figure 27j. Population of V25.
















ENERGY LEVEL POlPULATINFr
0.125 +---------+--------+---.-----+-------- +
I T T T I
I I I I
I I I I I
I I T I I
I T I I
I I T I
I T I I
I I T T I
I I 7 I I
~.ns +--- t ------------,,------------------
I i I
T I T
I T T T 99991
I T I 599 I
I I T 999 I
I T T 9I T
I T T 9 I I
I I T q T I
0.063 +--.----+---------..---- 7.--.+-----..-
I I T 9 I I
I I T T
N I T T 9 I T
0 T T I
T T 0 I I
/ I T I I
R I T T I T
SI T T9 I T
D 1.031 +---------+------------------+--------+
1 T 1) I I
9 I I 19T I I
n I T q9 I I
T I T q9q I
I I C) I I I
L I q9 T I I
E I 997 I I
V T 99 1 T T I
E I 9q T y I
L . 99q.------+----.-_------------------ ---
0.0 90.25n 1i0.500 270,750 361.000
time


Figure 27k. Population of V52.


















ENERGY LEVEL POIPUJLATI ';
0.250 +-------*--------.--+ -----------------+

I I I I

I I I I I
I I T I I
I I T I I
I I T T I
0.18 +--.,--.+-------------+--------+---------
I I T I I
I I I I I
1 1 1
T I I T
I I T I I
T T i I
I T T I I
0.125 +- -----+----0000000 000000

00IO 1
I O0 I T I
o I 0T I I I

0 IT I
I 00 T I I
N I 0 0 I I I
0I T I I
R I 0 I I I

SI o I T T I
SI 0 I T I
0 0.063 +-4_,----+--4------+4-------+---------+
I 0 1 T I I
R I T I I I
f I I T 7 I
T I T I T I 1
I I T I I
L I 0 I T I I
EI 7 I
V 10 I T I
E 0 T I I
L 0.0 +-....---+------------ ------.--- --------
0.0 90.251 180.50C 270.7r0 361.000
time


Figure 271. Population of V6.









to remain in equilibrium. Thus during CW operation the

lower laser level (V2s) is populated much higher before

lasing thus reducing the population inversion. The

explanation of why the CW case has higher output and

lower population inversion is revealed by comparing the

two figures for each case (figures 26c, 27c). Although

the inversion for the pulsed input reaches a higher

value, it also takes nearly three times as long to

release its energy. The power output of the CW pump is

then understandably higher since the time rate of change

of energy released is higher.

Of the 22 different parameters sets run numerically,

figures 26 and 27 are representative for all. As might

be expected there was no magic combination of parameters

which greatly increased laser output, although all 22

combinations yielded a 16 micron laser of at least

several watts/cm2. The figures for all 20 other combina-

tions will not be shown since they resemble so strongly

those of 26 and 27 that little additional information

would be gained. (See Appendix C).

Figure 28 indicates the efficiency for different

types of carbon-dioxide lasers. The parameter I is a

measure of energy (I = .023mJ). The first thing one

notices about this figure is that all points fall more

or less along a straight line which peaks around log(I)=3

and sharply drops off below that. This represents the




















CW
cw




0
Ccw



- CW





Stie
-1



0O


-3
0

3 4 5 6 7

log(I) I, Watt x time in collisions
Figure 28. Effect of CO2 pumping on 16 micron laser
efficiency.




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