Progress Report to National Aeronautics and Space Administration (NASA) 

Greenbelt, MD 2007 1 

NAG 5-526 

Theoretical Studies of Resonance Enhanced Stimulated Raman 

Scattering (RESRS) of Frequency-Doubled Alexandrite Laser Wavelength 

in Cesium Vapor 

Principal Investigator 

Nabil M. Lawandy 

Associate Professor of Engineering and Physics 

Brown University 

Providence, RI 02912 

January-June, 1987 

(NASA-CR-192870) THEORETICAL STUDIES OF N88-22133 

RESONANCE ENHANCED STIflULATED RAHAH 

SCATTERING [RESRS) OF FREQUENCY-DOUBLED 

ALEXANDRITE LASER WAVELENGTH IN CESIUH VAPOR 

Proqress Report, Jan. - Jua. 1987 (Broun G3/25 0142329 

Unclas 
r 

Propress Report 

The work during this period was aimed at elucidating the effects of a velocity 

distribution on the RERS process. The particular case of Cs vapor was investigated. The 

results of this work are best summarized in the publication preprint which follows. This 

paper was submitted to the Journal of Applied Physics B. 
MULTIPLE RELAXATION AND INHOMOGENEOUS BROADENING IN 

RESONANCE ENHANCED RAMAN SCATTERING: APPLICATION 

TO TUNABLE INFRARED GENERATION ._ 

John C. Ryan* and Nabil M. Lawandyt 

*Department of Physics 

tDivision of Engineering and Department of Physics 

Brown University 

Providence, Rhode Island 029 12 

Abstract 

The solutions for the imaginary susceptibility of the Raman field transition 

with arbitrary relaxation rates and field strengths are examined for three different 

-sets of relaxation rates. These rates correspond to (1) Far infrared (FIR) Raman 

lasers in the diabatic collision regime without consideration of coupled population 

decay in a closed system, (2) Raman FIR lasers in the diabatic collision regime with 

coupled population conserving decay, and (3) IR Raman gain in cesium vapor. The 

model is further expanded to include Doppler broadening and used to predict the 

peak gain as a function of detuning for a frequency doubled alexandrite 

- 

laser-pumped cesium vapor gain cell. 

This work was supported by an Alfred P. Sloan Fellowship and a grant from the 

National Aeronautics and Space Administration (NAG-5-526). 
-2- 

Introduction 

In this paper we consider the gain in a coherently pumped three level 

system by examining solutions of the density matrix calculation with arbitrary 

relaxation rates, field strengths, and detunings) ‘1 

The problem of a three level laser was first examined quantum mechanically 

by JavanJ2] However, his solution is valid only for a weak stimulated field, or for 

equal pump and stimulated field detuning in the strong field case. Many other 

 author^[^,^,^,^] have examined the three level system with various approximations, 

most notably Panock and Temkin16] whose solution is applicable for arbitrarily strong 

pump and stimulated fields, but assumes all relaxation rates to be equal. These 

solutions are necessary for the description of resonance enhanced Raman techniques 

for generating tunable radiation. Tunable radiation has been obtained using tunable 

pump lasers and various gases in the far infrared and near infrared regions of the 

spectrum. The far infrared systems utilized CO, TEA lasers as tunable pumps and 

molecular gases such as CH,F. The infrared systems were based around tunable dye 

laser pumps and atomic vapors such as the alkali metals. The recent development of 

the tunable alexandrite laser and efficient frequency doubling and tripling crystals 

has made possible solid state laser pumped alkali metal vapor infrared systems. 

In Section I we present the model used to approximate the gain in a 

coherently pumped three level system and the basis of the calculation of the 

susceptibilities. In Section I1 we examine the effects of different relaxation rates on 

the homogeneous line shape including how the various contributing terms in the 

imaginary stimulated susceptibility saturate with stimulated field intensity. 

In Section I11 the model is expanded to include Doppler broadening. We then 

briefly examine some of the properties of the Doppler broadened line shape and show 

that although the results at low pump power and detuning within the Doppler width 

agree with the previous cal~ulation,[’~~] the line shape for large pump fields and 
-3- 

detunings outside of the Doppler width docs not reduce to the homogeneous case. We 

then apply the inhomogeneous model to a doubled alexandrite pumped cesium vapor 

laser in order to study stimulated infrared Raman emission. Much of the present 

theory of stimulated Raman emission in non-linear optics is based on calculations of 

x@) from second order perturbation the~ryI'>~'*~ '1 This approach neglects many 

'.: 

important effects of the complicated population dynamics associated with the 

relaxation processes of any multi-level quantum system. It is also shown that the 

Doppler broadened prediction of stimulated gain with tuning differs significantly 

from the homogeneous prediction. 
L 

-4- 

I. Theory 

We have modeled coherently pumped atomic and molecular lasers of a 

three-level system in an "inverted V" configuration, shown in Fig. 1. The complex 

susceptibilities were obtained['] using the density matrix formalism while neglecting 

all non-resonant terms in the interaction picture. This treatment differs from 

 other^[^*^,^,^,^] in that both arbitrary longitudinal and transverse rates as well as 

,. ._ 

arbitrary field strengths are allowed. The system is assumed to be closed so that 

there is no net loss of probability. 

The zero field rate of decay of the average probability in any state will 

depend upon the configuration of the entire system and not only upon the value of 

the probability in that particular state. For instance, if the system in Figure 1 was 

prepared in a configuration in which the ensemble averaged probability of finding 

the system in state 12' was equal to its thermal equilibrium value while that of state 

13' were greater than its thermal equilibrium value, state 12' will not remain in 

equilibrium, since one channel of decay for state 13' is through state 12'. Therefore, 

there will be an increase in the probability of finding the system in state 12'. It is 

clear then that any one state will1 retain its equilibrium probability only if the 

cntire system is in equilibrium. Rates of decrease or increase of probability in the 

zero field case can only be defined between level pairs so that when fields 3re 

present the time dependence of the diagonal elements are governed by 

, 
I 

-5- 

where the fij's are equilibrium occupation ratios (ratio of ith to jth equilibrium 

population), and T,, T,, and Ts are the spectroscopically measured relaxation rates of 

the specified level pairs. Notice that trp = 0 is satisfied identically, showing 

probability conservation. 

Rewriting Eqs. (1-3) in terms of the population differences A,, = pll - p,, 

and AS2 = pss -pZ2 and Trp = 1, we have 

-1 2 * 

'1s TIm + - ti Im[p,,a,,E , ]

where uij are the density matrix elements in the interaction picture. E, and E, are 

the complex amplitudes of the signal and pump fields and the constants k,, k,, k3, 

k,,A!2 and A!, are defined in terms of T,, T,, T, and fij by the following relations. 

1 1 [ 1 - f,, 2 + 4f,, 1 + 2f,, 

k. * - + '+ 

- 1 + 2f32 2 - 2f,, 2 + f,, 1 

+ 

T2 T3: 

k2 = 

T, 

1 ks=i [ + + 2 - 2f3, 1 + 2f3, 1 + fZ1 

3 =1 T2 T3 

(7) 
-6- 

The off diagonal elements, on the other h 

2 + f21 

=3 

nd, give the average ampli 

(9) 

ide mixing 

between level pairs. The zero field decay of these elements can be approximated to 

first order by a simple exponential decay without contradiction.[ 12] 

The resulting susceptibilities are given by: 

IBp'2 ccL ] (13) 

(6; + Tz-2) 

] + A13 

h(6: + (6: + X' = 

I Bs I 2yl 

XI:= (6; + T2-2) 

I BPI 2y3 

XI:' (6,' + Tl-2) 
-7 - 

where 6, and 6, are the signal and pump detunings; the 7’s are shown in Fig. 1, A A 

-b 

PlS - Ep 

2fL Bp= :,E* 2fL 

the a’s, y’s and A’s are defined in Ref. [l], and B, = 

To explore the effects of different relaxation rates, we compare the 

susceptibilities of three different sets of rates at various detunings, pump intensities, 

and signal intensities. The first set of rates correspond to previous results first 

obtained by Panock and Temkid6] in which the decay of each state is uncoupled 

from the other states and all rates are assumed equal. The next set also assumes 

equal rates. However, since the rates are the transition rates between level pairs, 

the pump and signal inversion decays become coupled and the effective decay rates 

are no longer equal. This set of rates is meant to approximate a homogeneously 

broadened molecular gas laser in the infrared or far infrared where energy spacings 

are small compared with the collision energy. In this diabatic collision regime the 

level pair and polarization decays are collision dominated, and thus, all decay rates 

are equal to the collision rate. 

The third system to be modeled is a cesium vapor Raman laser which is 

pumped from the 6S1/, level to the 8P1/, level and emission is to the 8S,/, level. 

The vapor is at low pressure so collisional effects are negligible and decay is 

dominated by spontaneous emission. 

Since the Doppler width of cesium at temperatures above the vaporization 

temperature is much greater than the homogeneous line width, the susceptibilities 

must be Doppler integrated to obtain the actual gain of the system. However, it is 

instructive to first examine the effects of the different relaxation rates on the 

- 

homogeneous line shape, before including inhomogeneous effects to more accurately 

predict the tuning curve of a cesium vapor Raman cell as a tunable infrared source. 

The presence of the 6P1/,, 7S1/, and 7P,/, levels complicate matters slightly since the 

system actually has six levels. However, neglecting the spontaneous emission fields 

\ 
and assuming the only fields present are the fields nearly resonant with the pump 

and laser transitions and that all non-resonant contributions of these fields are 

negligible, then there are no field dependent perturbations connected to the 6S1/2, 

7Sl12, or 7Pl12 levels. The energy levels of the cesium system are shown in Fig. 2. 

The equations of motion of the diagonal elements are as follows: 

?_ 

>_ 

-1 -1 -1 

p44 = ‘887~ p22 - [‘7p7a + 7p6s]p44 

-1 -1 -1 

p56 = ‘8~7s p33 + t7p7s p44 - 786p PI6 

-1 -1 -1 

p66 = bpb p33 + t7r6p pS6 6p6s p66 

Where the labels 4, 5, and 6 denote the 7P1/,, 7S1,,, and 6P1/, labels, respectively, and 

the zero field equilibrium occupation of the upper levels are negligible. In steady 

state, Eqs. (16-21) can be rewritten, using trp = 1, as two equations in terms of A,, 

and A32 exactly as in the three-level case except the effective rates k,, k2, k,, and k, 

will be different. The dephasing rates in this radiative limit will be taken to be the 

sum of the spontaneous emission rates of each decay channel for both levels 

constituting the level pair in question. The dephasing rates are given by 
-9- 

-1 -1 -1 -1 -1 -1 -1 

'1 = '32 = 8~8s' 8~78' 8~6s' 8s7p' 8s6p 

-1 -1 -1 -1 

'3 = '21 = 887~' 8e6p 

The spontaneous emission rates are calculated using 

1 2 4jw3 

- 3 

with the dipole moments obtained from Ref. [13]. The spontaneous emission rates, as 

well as the effective transverse and longitudinal relaxation rates are given in 

Table 1. 

11. Homogeneous Broadening Results 

In this section we compare the imaginary part of the susceptibility at the 

Raman field frequency, which is responsible for gain or attenuation, for d fferent 

sets of rates. It will be shown that many of the qualitative features of the previous 

results, corresponding to the first set of rates, will be changed by the more involved 

relaxation process occurring in the second two systems. The small signal or 

unsaturated case was first examined for equal uncoupled decay rates by Javan12] and 

later Panock and Temkin[6] as a special case of their work. It was shown that for 

off resonance pumping (6,>Av), the susceptibility exhibited two peaks of equal height, 

one near = 0 and the other near 68 = 6,, but displaccd by the Rsbi shift of the 

energy levels. When the laser field is turned on the symmetry is broken and the 68 

2 0 peak saturates faster than the Sa 2: 6p peak until there is only the peak which 

shifts towards the center (Fig. 3(a-d)). Physically, this is because the quantum 
t 

-10- 

efficiency of a double quantum transition is one. Or in semiclassical language, it is 

because the single photon, 6, 2: 0, contribution is proportional to pll - pZ2, while the 

two photon, 6, 2 6, is proportional to pll - pZz. The effect of the saturating field 

to first order is to induce transition from pss to p22, which reduces the single photon 

inversion, but for any reasonable pump intensities the ground state will have a much 

greater population than either excited state so that the two photon inversion saturates 

much slower than the single photon. 

When the susceptibility was decomposed into single and double photon 

contributions, it was seen that the one photon part was absorptive at the gain peak 

of the two photon part, and similarly, the two photon part was absorptive at the one 

photon peak. The single photon contribution had a much higher peak than the two 

photon contribution, but the two photon absorption at the one photon peak was much 

greater than the one photon absorption at the two photon peak, such that when 

plotted together the 6, 2: 0 and the 6, It 

can also be seen from the decomposed susceptibility that the decrease in gain with 

increasing laser field was due not only to saturation but also to an increasingly more 

effective competition between gain and absorption. It can be seen in Fig. 5 that 

initially only the single photon peak saturates until it is approximately equal to the 

two photon absorption. At this point, the one photon absorption begins to grow 

while the two photon gain then begins to saturate (Fig. 6). 

6, peaks were of equal height (Fig. 4). 

In addition, the one and two photon contributions nearly cancel, leaving a 

single peak in the total susceptibility between the original 6a 1 0 and 6, 6, peaks, 

approximately an order of magnitude less than ,the peak of the single photon 

contribution. The absorptive part of the two photon contribution remains 

approximately the same value as its = 0 value and only now begins to saturate as 

both contributions saturate uniformly and the resultant peak in the susceptibility 

shifts towards 6, = 0 (Fig. 7). 
-11- 

The on resonance case is harder to interpret in terms of a photon picture, 

especially at powers at which the Rabi splitting becomes significant. At low pump 

powers the one and two photon contributions both have peaks at 6, = 0, with the 

two photon contribution having absorptive side bands (Fig. 8). At higher powers, the 

single photon term surprisingly becomes a split absorption peak at 6, = 0, with gain 

on the side bands, while the double photon term has a split gain peak with 

absorption on the side bands (Fig. 9a). The sum of these parts give the familiar 

Rabi split line shape (Fig. 9b). With increasing signal field both parts saturate and 

the splitting and sidebands disappear and the one and two photon contributions are 

respectively single absorption and gain peaks at 6, = 0. This causes the total 

susceptibility to decrease in height and the splitting is lost (Fig. 10). The scenario 

described above applies only to the equal uncoupled rate case. 

.. 

When the rates are unequal, the resonance behavior is qualitatively similar 

but there are significant differences in the off resonance behavior. This is to be 

expected, since the denominators of the population differences A,, and A3, contain the 

population relaxation rates k,, k,, k,, and k, , as well as T,, T,, and T, in the 

different parts, which alter what might be called the effective saturation intensities 

of the different parts. Even for B, = 0 there will be differences because of a 

bottlenecking effect which occurs when one of the population relaxation rates is less 

than the others and thus cannot relax fast enough to keep the occupation of that 

level down to the valve it would have in the equal rate case. As a result the 

balance of absorption and gain required to make the 6a 0 and the 6a Y 6p peaks 

equal at B, = 0 will not be present. 

In the FIR system with coupled rates, the gain of the one photon term is 

less than the absorption of the two photon term, and therefore, the resulting 

susceptibility has net absorption not gain at 6, 1 0 (Fig. 11). Then, as the signal 

field is turned on, the peak of the one photon term incrcases to cancel the two 
-12- 

photon absorption instead of saturating, as would be expected (Fig. 12). 

In the cesium system the peak of the two photon term is much greater than 

the one photon peak or the two photon absorption, which makes the 6, 6, peak in 

the susceptibility about an order of magnitude larger than the 6, 2 0 (Fig. 13) peak. 

For increasing values of B,, the two photon peak saturates very rapidly, since the 

single photon peak is approximately the same height as the two photon absorption. 

The one photon term also has a small gain peak as well as an absorption peak at the 

two photon gain peak (Fig. 14). 

Another difference between the cesium system and the previous two is the 

narrowness of the two photon gain peak. This is because in the cesium set of rates 

the polarization decay rates are due to spontaneous emission and not collisions. Since 

all of the rates in the plots are normalized to T~ and the width is roughly 

determined by T~/T~, which is the relative rate of two-photon coherence decay. In 

the first two systems this number is unity, while in the cesium system it is 

T~/T~ = 0.063. Fig. 15(a) shows how the 61 2 6, peak narrows as the two photon 

contribution becomes more dominant with increasing pump field detuning, while Figs. 

15(b), (c) shows how the width of the the peaks in the other two rate cases remains 

about the same. 

Figs. 15(a-c) also show that the absolute maximum of the susceptibility 

initially increases with pump detuning at constant pump field in both the cesium and 

the coupled rate FIR system, while there is a monotonic decrease in the equal rate 

case. Figs. 16-18 show the value of the absolute maximum of the susceptibilities as a 

function of increasing pump detuning for pump field Rabi frequencies of two, five, 

and ten in units of T;'. In the equal rate case there is always a rapid monotonic 

decrease, and although the coupled rate FIR system rises to a peak, it then decreases 

rapidly with detuning. The cesium system has its maximum gain at a greater 

detuning than the FIR system and falls off much slower with detuning. This 
-13- 

suggests that the cesium may be used more effectively as a tunable laser medium 

than would be expected from looking at the equal rate case, since there is an 

appreciable gain peak over a significantly larger range of detunings. 

Fig. 19 shows the position of the peak signal frequency detuning as a 

function of pump detuning and the deviation from 6, = 6, as 6, - 6, is plotted vs. 

13,. The peak location is the same for the three sets of rates, illustrating that for 

this range of pump power, the peak location depends only on B, and 6,. At 6, = 0 

the peak is located at Only at very low powers where 

the Rabi splitting is not simply equal to the Rabi frequency do the rates affect the 

position of the peaks (Fig. 20). Fig. 21 shows the peak position vs. pump field 

Rabi frequency B, at 6, = 0 and 6, = 10. For 6, = 0, the peak is at 6, = 0 for 

low power, then splits at different powers for the three different rate cases, and 

eventually approaches 6, = B, at higher powers. For 6, = 10, the peak is at 6, = 6, 

and then as pump power is increased the peak shifts towards 6, = Bp. 

= B, the Rabi frequency. 
-14- 

111. Inhomogeneous Modcl 

In this section we will examine the effects of Doppler broadening in the 

cesium system for different values of the pump field detunings and Rabi frequencies. 

We will first see how the gain curves evolve as the detuning approaches and exceeds 

the Doppler width. Then, in the second half of this section, we will compare the 

predicted tuning curves for the Doppler broadened system vs. the homogeneous system 

for pump detunings and pump Rabi frequencies much greater than the Doppler 

width, as would be present in the doubled alexandrite pumped cesium vapor Raman 

laser. 

The Doppler broadened gain is given by the homogeneous response of a 

specific velocity group integrated over the Boltzman velocity distribution, which can 

be written as 

where E = k,/k,, k, and k, are, respectively, the pump and signal field wave vectors, 

w is the angular frequency shift of the pump, and AwDop is the full width at half 

max of the pump transition given by 

AwDOp = 2kp 

2kT ln2 

M 

The - and + are, respectively, the co- and counter-propagating bcams. 

For cesium, the transition energies in wavc numbers are 25709 cm-' for thc 

6S1/$P1/, transition and 1572 cm-' for the 8S1/,8P,/, transition. Using thcse valucs 

and a temperature of 1000K[14] (the cesium vaporization tcmpcraturc is 93?K), AwDop 
-15- 

and the wave vector ratio were calculated to be 

The integral was evaluated numerically. 

For small pump field and detunings['] within the Doppler width, the line 

shape exhibits a co- and counter-propagating asymmetry with the co-propagating peak 

located at cgp and the counter-propagating peak located at -E 6,. The co-propagating 

peak is slightly higher and narrower than the counter-propagating peak by 

approximately 1 + E (Fig. 22), and as the pump Rabi frcquency is increased above 

B, = 7/5 Ti', the co-propagating peak splits, although the magnitude of the splitting 

is less than in the homogeneous case (Fig. 23). 

As the detuning is increased, the co-propagating split disappears, and a wide 

peak begins to appear at the Rabi shifted two photon resonance. The origin of this 

can be seen by approximating the homogeneous line as 3 sum of delta function 

responses at the one and two photon resonances. 

X,, = VI 6(6,- A,) + V,6( 6,- 6, - A2) (29) 

where VI and V, are the areas of the two peaks, and AI and A2 are the Rabi light 

shifts. Although the homogeneous line will only look likc delta functions when 

(k,/kp) AWDop >>Ti, k, this approximation shows some important features of the fully 

integrated line shape. 
-16- 

where the weak dependence of A1 and A2 on 6, was neglected. The first term is just 

a gaussian near line center with the width of the laser transition eAwDop. The second 

term near 6# 6, is what the Raman peak becomes in the presence of Doppler 

broadening. It is a gaussian with the width of the pump transition i the width of 

the laser transition, depending on co- or counter-propagating beams. Also, the peak 

value of the homogeneous two photon transition is diminished by 1/( 1 ie)AwDop, while 

the one photon peak is only diminished by l/eAwDop. Thus, the 6, = A, peak will be 

higher and narrower than the 6, = 6, + A2 peak. Fig. 24 shows the fully integrated 

co- and counter-propagating gain curve for 6, = 2007;' and B, = 7;'. It is 

interesting to contrast the Doppler integrated gain where the one photon peak is 

significantly higher than the two photon peak, with the homogeneous gain curve, in 

which the single photon peak is almost negligible in comparison to the two photon 

peak (Fig. 25). 

For higher pump powers the on resonance counter-propagating gain peak 

begins to split (Fig. 26), and as the pump Rabi frequency, B,, approaches the half 

width of the Doppler distribution, both the co- and counter-propagating gain curves 

split into two distinct peaks located at *B,. The evolution of the curves with 

increasing detuning proceeds much the same as the low pump power except that the 

co- and counter-propagating asymmetry persists on the single photon peak. 

We will now compare the predicted gain tuning curves of a doubled 

alexandrite pumped cesium vapor Raman $laser to examine the effects of Doppler 

broadening. The alexandrite laser is a solid state laser, tunable from 7000A to 8OOOK 

We will assume a pulsed laser with an energy of 150 mJ/pulse and 3 pulse width of 

T = 300 ns, focused to a spot size of 100 pm radius, which gives a peak Rabi 

frequency of B, Y 3.6 x 1057;'. The tuning range which will be examined is from 

25709 cm" to 27200 cm", keeping in mind that the higher wave numbers will begin 

to pump the 9P1/2 level, and thus will bc most accurate closer to resonance. When 
-17- 

tuned to resonance at peak power, the susceptibility has two peaks Rabi split to 

approximately *275 cm" wave number detuning from the laser transition line center. 

The peak position is the same for both the homogeneous and inhomogeneous 

susceptibilities, although the peak gain value is much lower in the Doppler case than 

it would be if the system were not inhomogeneously broadened. As the pump is 

detuned the peak closest to the pump frequency begins to increase and move away 

from the center, while the other peak decreases and moves towards line center. Fig. 

27 shows the resonantly pumped susceptibility and the value of the peak as the 

detuning is increased. In Fig. 28 the position of the peak is given for varying pump 

detunings, with the diagonal line denoting the 6, = 6,, line. The dotted line indicates 

the peak closer to line center, while the solid line indicates the "two photon" peak, 

which approaches 6, = 6, as the detuning becomes large compared with the Rabi 

frequency. 

When inhomogeneous broadening is taken into account, there is a drastic 

deviation from the homogeneous results. Although the peak position remains the 

same as in the homogeneous line, the peak which approaches 6, = 6,,, widens and 

decreases in height from its 6, = 0 value while the other peak narrows and increases 

sharply, even though the detuning and Rabi frequency are both huge in comparison 

to the Doppler width. Figure 29 shows how the peak changes with detuning from 

the 6, = 0 value. 

This illustrates the necessity of including Doppler broadening in the gain 

calculation for such a system. The essence of the lower gain on the two photon 

transition is because of the near proportionality ,of peak position, 6$, to pump 

detuning, 6,. In contrast to the homogeneous calculations, including the calculation 

of x ('1 [9,10,111 in which each atom radiates at the same frequcncy, the Doppler 

broadened calculation includes a frequency shifted peak for each vclocity group. This 

results in a two photon gain which is spread out with the width approximately equal 
-18- 

to the Doppler width of the pump transition and height proportionately diminished, 

as shown in Eq. (30). In addition, if the vapor cell is used as a one or two pass 

Raman amplifier, the line width of the output radiation will be that of the pump 

Doppler width, and not the homogeneous or power broadened linewidth. 
-19- 

IV. Conclusion 

We have examined the imaginary susceptibility on the laser transition 

(proportional to the gain or attenuation) of a three-level system, with arbitrary 

relaxation rates, detunings, and field strengths interacting with two laser fields, 

obtained analytically in the density matrix formalism. The solution assumed 

negligible spatial and temporal variation and monochromatic laser fields (laser line 

width much smaller than the homogeneous width). It was found that differing 

relaxation rates cause important qualitative differences, and some case, gain in one 

system may become absorption in another. 

The results were then applied to a cesium vapor laser pumped by a tunable 

pulsed doubled alexandrite laser. Since the pulse width is just slightly larger than 

the homogeneous line width, the system will be just on the limit of the adiabatic 

following regime. Therefore, the steady state results can only be loosely applied and 

must serve as a rough model for the position and relative strengths of the gain 

peaks. Another complication due to the pulsed pump arose because of the sensitivity 

of the gain peak position on pump field strength. This will result in "chirped" 

output pulses. However, since the peak gain for fixed detuning and pulsed pump 

field strength occurs at the peak of the pump pulse, the dominant contribution to the 

output gain will come from the peak field strength, resulting in a relatively narrow 

output pulse. 

The system was then Doppler intcgrated, since the Doppler width is many 

times the homogeneous width. It was found that even though the detuning and Rabi 

frequency were much greater than the Doppler width, it caused the gain to shift 

from the predominantly two photon (6s 6J hornogcneous susceptibility result to 3 

predominantly single photon contribution (6, - 0). 
-20- 

Acknowledgment 

The authors are grate 11 to Dr. J. Degnan of the Goddard Space Flight 

Center for his useful discussions concerning the directions of this work. We are also 

grateful to Mr. Jan Nordgren for his help with the numerical analysis. 
-21- 

Rcf ercnces 

1. ADD^. Phvs,, Vol. 40, pp. 3370-3374, 1969. 

J. C. Ryan and N. M. Lawandy, "Density Matrix Solutions for the 

Susceptibilities of a Three-Level System with Arbitrary Relaxation Rates and 

Field Strengths," IEEE JQE, Vol. QE22( 1 l), 2075-2078, 1986. 

A. Javan, "Theory of a Three Level"Maser," Phvs. Rev., Vol. 107, pp. 1579-1589, 

1957. 

I 

A. M. Clogston, "Susceptibility of the Three Level Maser," J. Phvs. Chem. 

Solids, Vol. 4, pp. 271-277, 1958. 

S. Yatsiv, "Role of Double Quantum Transitions in Masers," Phvs. Rev., Vol. 113, 

pp. 1538-1544, 1959. 

R. G. Brewer and E. L. Hahn, "Coherent Two Photon Processes: Transient and 

Steady State Cases," Phvs. Rev. A, Vol. 11, pp. 1641-1649, 1975. 

R. L. Panock and R. J. Temkin, "Interaction of Two Laser Fields with a Three 

Level Molecular System," IEEE JQE, Vol. QE-13, pp. 425-434, 1977. 

M. S. Feld and A. Javan, "Laser-Induced Line-Narrowing Effects in Coupled 

Doppler Broadened Transitions," Phvs. Rev., Vol. 177, pp. 540-562, 1969. 

D. Seligson, Martial Ducay, J. R. R. Leite, A. Sanchez, and M. S. Feld, 

"Quantum Mechanical Features of Optically Pumped CW FIR Lasers," IEEE JQE, 

Vol. QE-13, pp. 468-472, 1977. 

J. J. Wynne and P. P. Sorokin, "Optical Mixing in Atomic Vapors," in Nonlinear 

Infrared Generation, Y.-R. Shen, Ed., Berlin-Heidelberg: Springer-Verlag, 1977. 

D. Cotter, D. C. Hanna, P. A. KZrkkCinen, R. Wyatt, "Stimulated Electronic 

Raman Scattering as a Tunable Infrared Source," ODt. Commun., Vol. 15, pp. 

143-146, October 1975. 

D. Cotter, D. C. Hanna, "Stimulated Electronic Raman Scattering in Cs Vapor: 

A Simple Tunable Laser System for the 2.7 to 3.5 pm Region," ODticfil and 

Ouantum Electronics, Vol. 9, pp. 509-518, 1977. 

N. M. Lawandy, Ali Fahmy, and Kayee Lee, "Excitation Dependent Dephasing 

and the Stability of the Lorenz Model for a Homogeneously Broadened 

Unidirectional Ring Laser," International Meeting on Instabilities and Dynamics 

of Lasers and Nonlinear Optical Systems," Rochester, NY, 1985. 

Richard B. Miles and Stephen E. Harris, "Optical Third Harmonic Generation in 

Alkali Metal Vapors," IEEE JQE, Vol. QE9, pp. 470-484, 1973. 

C. R. Vidal and J. Cooper, "Heat Pipe Oven for Spectroscopic Measurements," 
-22- 

Figure Captions 

Figure 1: Schematic of 3-level system. 

Figure 2: Energy levels of atomic cesium. 

Figure 3: % fr/T,p2co is plotted versus 6,T, with decreasing amplitude corresponding 

to increasing stimulated fields of B,T~ = 0, 1, 2, 5 and (a) B T1 = 1, 

6,T1 = 5, (b) BpTl = 5, 6,T1 = 5, (C) BpTl = 1, 6,T, = 10, (d) EPT1 = 5, 

6-T. = 10. 

Figure 4: (a) One and two photon terms with two photon term absorption near 

6,Tl = 0 of % fr/T,p2c, and (b) % fr/T,p2co are plotted versus 6,T1 for 

SpTl = 10, BpT1 = 2, BETl = 0. 

Figure 5: Same as Fig. 4 except for 6,T, = 10, BpTl = 2, BETl = 1. 

Figure 6: Same as Fig. 4 except for 6,T1 = 10, BpT1 = 2, BaT1 = 5. 

Figure 7: Same as Fig. 4 except for 6,T, = 10, BpTl = 2, BETl = lo. 

Figure 8: (a) One photon term has higher amplitude of % h/T1b2~, and (b) 

% h/T,p2co are plotted versus 6,T1 for 6pT1 = 0, B,T, = 0.2, BaT1 = 0. 

Figure 9: (a) Single photon term is absorptive at 6,= 0 and (b) are 

plotted versus 6,T1 for 6,,T1 = 0, B,T, = 5, BET1 = 0. 

Figure 10: Same as Fig. 9 except for 6,T, = 0, B,T, = 5, B8Tl = 8. 

Figure 11: Same as Fig. 4 except with coupled FIR ratcs for 6pT1 = 10, BpT1 = 2, 

BaT1 = 0. 

Figure 12: Same as Fig. 11 except for 6pT, = 10, DpT, = 2, BaT1 = 2. 

Figure 13: Same as Fig. 11 except with cesium ratcs for 6,T, = 10, BpT1 = 2, 

BITl = 0. 
-23- 

Figure 14: Same as Fig. 13 except for 6,T1 = 10, BpTl = 2, BETl = 1. 

Figure 15: x fi/T1p2e, is plotted versus fjET1 at values of 6,T1 = 2, 7, lo, 15 for (a) 

cesium rates and (b) equal coupled rates. 

Figure 16: Peak value of x h/~,p~~, is plotted versus 6ET1 at BpTl = 2, BET1 = 0 for 

(a) cesium rates, (b) equal coupled FIR rates, and (c) equal uncoupled 

rates. - 

Figure 17: Peak value of fi/T1p2€, is plotted versus 6,T1 at B,Tl = 5, BET1 = 0 for 

(a) cesium rates, (b) equal coupled FIR rates, and (c) equal uncoupled 

rates. 

Figure 18: Peak value of fi/~,p~€, is plotted versus 6 T1 at BpTl = 10, BET1 = 0 

for (a) cesium rates, (b) equal coupled SIR rates, and (c) equal 

uncoupled rates. 

Figure 19: (a) peak position (6ET1) peak is plotted versus SpT, at B,T, = 10, BETl = 0 

and (b) deviation from 6E = 6, is plotted versus 6,Tr 

Figure 20: (a) peak position (6ET1) peak is plotted versus B T1 at 6pT1 = 0, BsTl = 0 

for (a) cesium rates, (b) equal coupled FIK rates, and (c), equal 

uncoupled rates. 

Figure 21: (a) peak position ("T1) peak is plotted versus BpTl at SpT, = 10, 

BITl = 0. 

Figure 22: (a) Co-propagating and (b) counter-propagating values of ~/T,$E, 

versus 6,T1 Doppler integrated at Bp~l = 0.5, Bsfl = 0, 6 T = 10. Pl 

Figure 23: (a) Co-propagating and (b) counter-propagating values of fi/~,p'~, 

versus 6,Tl Doppler integrated at BpT1 = 1.0, DsTl = 0, SpTl = 10. 

Figure' 24: (a) Co-propagating and (b) counter-propagating values of xz ~/T,c('E, 

versus 68T1 Doppler integrated at 6, = 1.0, BETl= 0, SpTl = 200. 

Figure 25: Homogeneous value of fi/T,j~~~~ versus sET1 for 0, = 1.0. Bs = 0, 

6pT1 = 200. 
-24- 

Figure 26: (a) Co-propagating and (b) counter-propagating >cd fi/T,fi2co versus 6sT1 

Doppler integrated at S,?, = 20, BsTl = 0, spT1 = 0. 

Figure 27: Vertical lines are a plot of the on resonant fi/T,fi2co at 

13, = 3.6 x 105~,, and the curves are a plot of the maximum value of 

the respective homogeneous peaks versus 6s in wave numbers 3s pump 

detuning is decreased below zero. 

Figure 28: A plot of the values of the two peaks of >cd fi/?,p2co at 8, = 3.6 x 105T, 

versus 6, in wave numbers. 

Figure 29: Vertical lines are a plot of the on resonant Doppler integrated 

fi/?,p2c,, at S, = 3.6 x IO5?,, and the curves are a plot of the 

maximum value of the respective Doppler integrated peaks versus 6s in 

wave numbers as pump detuning is decreased below zero. 
Table 1 

Dipole Matrix Element (C-rn) 

2.8 10-29 

3.1 x 10-30 

I .4 x 10-30 

2.6 10-29 

5.7 10-30 

6.1 x 

6.6 x 

5.8 I 0-29 

1 .o x 10-28 

Effective Rates (sec-I) 

Spontaneous Lifetime (Sec) 

1.09 x 10-8 

1.17 x 

3.59 10-7 

4.54 x 10-8 

1.68 x 1 0'7 

9.41 x 10-8 

7.30 x 

2.67 x 

2.77 x 

k, = 2.62 x IO7 

k, = 3.06 x IO7 

k, = 1.24 x IO7 

71-1 = 1.53 x IO8 

~~-1 = 1.43 x IO8 

T3-l = 9.71 x IO6 

k, = 2.60 x 107 
'1 3 

12 > 

Fig. 1 
27632 9p312 

27637 9~112 

9s 112 

2691 1 

8s 112 Os c 

241 37 

7s 112 

18536 

6s 112 

25792 ap 312 

25709 8p 112 

21947 7~312 

21766 7p 112 

11732 6p 312 

11178 6p 112 

Fig. 2 
-20 -1 0 0 10 20 

-20 -1 0 0 10 20 

vi 

Fig. 3 
0 

w 

(Y 

=L 

\ 

e." 

0.06 - 

0.04 - 

A (a> 

0.04 I 

-20 -1 0 0 10 20 

VI 

Fig. 4 
0 

w 

N = 

e- 

\ 

\F 

-x 

-.L 

-0.02 -j 

0.02 -j 

-0.04 1 I I I I 

-20 -1 0 0 10 20 

0.03 

0.02 

0.01 

0.00 

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65.5 

Fig. 5 
0 

u 

cy 

=L 

0.04 

0.02 

0.00 

-0.02 

-0.04 ! I I I 

-20 -1 0 0 10 20 

621 

-20 -1 0 0 

Vi 

10 

Fig. 6 

20 
0.03 

0.02 

0.01 

0 .oo 

-0 .o 1 

-0.02 

-0.03 I 1 I 

-20 -1 0 0 10 20 

vi 

0.001 2 

0.001 0 

0.0008 

0.0006 

0.0004 

0.0002 

-0.0000 I I I I 

-20 -1 0 0 10 20 

Fig. 7 
0.08 

0.06 - 

0.04 - 

0.02 - 

0.00 

-0.02 

-20 -1 0 0 10 

(a> 

- 

I I I 

0.1 0 

0 .08 

0.06 

0.04 

0.02 

0 .oo 

20 

-20 -1 0 0 10 20 

65'1 

Fig. 8 
'I 

-1 f 1 I I 

-20 -1 0 0 10 20 

-20 -1 0 0 10 20 

621 

Fig. 9 
.. 

0.1 

0 

u 

e4 = 

e- 0 .o 

-0.1 I I I 

-20 -1 0 0 10 20 

0.003 

(b) 

0.000 I I I I 

-20 -1 0 0 

621 

10 20 

Fig. 10 
-0.2 I I 1 I 

-20 -1 0 0 10 20 

-0.1 f I 1 I 

-20 -1 0 10 20 

Fig. 1 1 
0.04 

Oeo2L 0.00 z i -0.02 

-0.04 I I 

V I 

-20 -1 0 0 10 20 

0.03 

(b) 

0.02 - 

0.01 - 

0.00 

-0.01 f 1 I I 

-20 -1 0 0 10 

VI 

Fig. 12 

20 
-20 -1 0 0 10 20 

VI 

-20 -1 0 

Fig. 13 

10 20 
0.03 

(a> 

-0.03 I I I 

-20 -1 0 0 10 20 

621 

0.008 I 

(b) 1 

0.006 

0.004 1 i 0.002 

-0.000 ! I I I 

-20 -1 0 0 10 20 

VI 

Fig. 14 
1 

0.2 

2 8 14 20 

0 

0 

e4 

a 

\ I e0.1 \F -* -vl 

0 .o 

u 

s 

\ 

e4 

e- 

\F 

-% 

-vl 

u 

s 

\ 

e4 

e- 

\F 

-% 

-vl 

2 14 20 

0.3 

(C> 

2 8 14 20 

vi 

Fig. 15 
0 

u 

N 

5 

\ 

n 

Y 

W 

a 

e" 

\F 

n 

-* W -* 

0.3 '1 

0.2 - 

0.1 - 

0 .o 

0.3 

.(a> 

0.2 - 

0.1 - 

0 .o I I I I 

(C> 

I I I I 

0 20 40 60 80 100 

VI 

0 

w 

N 

e- 

\ 

n 

Y 

(0 

a 

\F 0.1 

n 

-4 W -* 

0 .o 

0 

0 

N 

5 

\ 

n 

Y 

a 

a 

-* v 

e- 

\F 

n 

-* 

0 20 40 60 80 100 

wt 

Fig. 16 
0 

u 

cy 

a 

\ 

n 

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a 

Q) 

e." 

\F 

n 

0.3 i 1 

0.2 

0.1 

0.0 I I I I I I 

0 20 40 60 80 100 

0 

u 

cy 

=L 

\ 

n 

Y 

a 

Q) 

-m W 

e." 

s 

n 

-% 

0.2 

0.1 

0 .o 

0 20 40 60 80 100 

6pTI 

0 20 40 60 80 100 

Fig. 17 
0 20 40 60 80 100 

n7. 

0.0 I I I I I 

- 

0 20 40 60 80 100 

sp'r1 

0.3 

0.2 

0.1 

0 .o 

0 20 40 60 80 100 

6pTl 

Fig.1 8 
o! I I I I 

0 20 40 60 80 100 

10 

8 

2 

I I I I 

0 20 40 60 80 too 

Fig. 19 
, 

0 1 2 3 4 5 

0 1 2 3 4 5 

PP'l 

0 1 4 5 

Fig. 20 
20 

15 

cg e: 10 

5 

0 

10 

PP5 

Fig. 2 1 

15 
0 .oo 

0.00 

0.oou 

-20 -15 -10 -5 0 5 10 15 20 

621 

0.003 

0.002 

0 .oo 1 

0 .ooo 

-20 -15 -10 -5 0 5 10 15 20 

V1 

Fig. 22 
0.004 

0.003 0 

u 

cy 

=L 

\ 

Pd 0.002 

s 

-% 

-% 

0.001 

0 .ooo 

-20 -15 -10 -5 0 5 10 15 20 

65% 

0.004 

0.003 

0.002 

0.001 

0.000 

-20 -15 -10 -5 0 5 10 15 20 

621 , 

Fig. 23 
4 .Oe-6 

3.0e-6 - 

2.0e-6 - 

1 .Oe-6 - 

O.Oe+O 

-20 20 60 100 140 180 220 

vi 

4.0e-6 

0 3.0e-6 w 

(Y 

e" 2.0e-6 \ 

\F 

-?e 

-cL 

1 .Oe-6 

O.Oe+O 

-20 20 60 100 140 180 220 

65% 

Fig. 24 
0 

u 

e4 

=L 

\ 

e- 

\F 

-% 

5.0e-4 . 

4 .Oe-4 

3.0e-4 

2 .Oe-4 

1 .Oe-4 

O.Oe+O 

-20 20 60 100 140 180 

Ul 

Fig. 25 
u.uo 

0.00 0 

u 

(Y 

=L 

e- 0.00 

\ 

\F 

-% 

-m 

0.00 

0 .oo 

-60 -40 -20 0 20 40 60 

6511 

0.oooI . , I I , . , I I 

-60 -40 -20 0 20 40 6 3 

Fig. 26 
-500 0 500 1000 1500 

Laser Field Detuning (1 /cm) 

Fig. 27 
n 

E 

9 

c 

v 

m 

e 

e 

¶ 

01 

P 

01 

*c 

Y 

3! 

z 

a 

L 

1500 

1000 

500 

0 

-500 

-1 000 

-1 500 

-1500 -1000 -500 0 500 1000 1500 

Lasern Field Detuning (1 /cm) 

Fig. 28 
0.01 2 

0.01 0 - 

0.008 - 

0.006 - 

0.004 - 

0.002 - 

0.000 

-500 0 500 1000 1500 

Laser Field Detuning (1 /cm) 

1 

I 

Fig. 29