Theoretical investigation of superfluorescent emission from an optically pumped CF4gas columnSucharita Sinha Citation: Journal of Applied Physics 64, 4293 (1988); doi: 10.1063/1.341303 View online: http://dx.doi.org/10.1063/1.341303 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/64/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Blue light emission from an organic nonlinear optical crystal of 4-aminobenzophenone pumped by a laser diode Appl. Phys. Lett. 70, 562 (1997); 10.1063/1.118208 Effect of superfluorescent emission on the operation of an optically pumped CF4 laser J. Appl. Phys. 70, 1172 (1991); 10.1063/1.349595 Optically pumped superfluorescence S2 molecular laser Appl. Phys. Lett. 36, 509 (1980); 10.1063/1.91587 Optically pumped superfluorescent Na2 molecular laser J. Appl. Phys. 47, 1515 (1976); 10.1063/1.322817 Superfluorescent laser emission from electronbeampumped Ar–N2 mixtures Appl. Phys. Lett. 25, 735 (1974); 10.1063/1.1655381

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Theoretical investigation of superfluorescent emission from an optically pumped CF4 gas column

Sucharita Sinha Laser Division, Bhabha Atomic Research Centre, Bombay-400085, India

(Received 24 February 1988; accepted for pubHcation 5 July 1988)

A theoretical study of the superfluorescent emission from an optically pumped CF4 gas column is reported. We have estimated the dependence of the superftuorescent pulse energy on the various operating parameters, such as temperature, pressure, pump rate, and the extent of frequency detuning between the pump radiation and the pumped CF4 transition. The calculated temperature dependence of the superfiuorescent pulse energy confirms that at any given pressure, the highest output energy would be achieved at the lowest possible temperature determined by the vapor pressure of the CF4 gas. Our estimate for the dependence of the superfluorescent pulse energy on the temporal profile of the CO2 pump pulse indicates the significance of the narrow gain switched spike over the fiat tail of the CO2 pump pulse for efficient pumping of the CF4 molecules. We have also estimated the superfiuorescent pulse energy in the double-pass configuration and compared it with that for the single-pass configuration.

I. INTRODUCTION

The requirement ofa coherent radiation source in the 16 j.tm wavelength region for laser isotope enrichment ofuran­ium has led to the rapid development of the CF4 laser. Since the first report on the CO2 taser-pumped CF4 iaser1

,2 exten­sive research has resulted in the incorporation of various design improvements l and pulse energies of over 100 m] and energy conversion efficiencies up to 10% have been achieved.4 Laser action on several lines in the frequency re­gion covering 605-655 em -. l have been reported.5 The 9-Pill band C02 1aser pumps the V 2 + V 4 combination band of the CF4 molecule leading to stimulated emission in the 16 j.tm

region from transitions to the V z vibrational level. The high collisional deactivation rate of the combination level and the presence of the absorption band vo--'>v,,1imit the operating pressure range for the CF 4 laser. Consequently, long CF4 gas columns are used for efficient absorption of the pump ener­gy. Long gain lengths, however, result in enhanced super­fluorescent emission from such CF4 1aser systems. Thus, op­tically pumped CF4 gas can be operated in both laser as wen as the superftuorescent mode. Theoretical investigation of the continuous wave CF4 laser has been reported earlier us­ing a rate equation model. {; In this case it suffices to consider the space-averaged steady-state solutions of the rate equa­tions describing the population densities in the various inter­active states. However, in order to study the superfiuores­cent emission from a CO2 laser-pumped CF4 gas column it is necessary to incorporate both time and space dependence in the rate equation modeL Such a calculation is reported in this paper.

We have estimated the dependence of the supertluores­cent pulse energy on the various operating parameters such as operating pressure, temperature, pumping rate, and the extent of detuning between the pump radiation and the pumped CF4 transition using our model. Earlier observa­tions 7 on the temperature dependence of the output energy from a CO2 laser-pumped CF4 laser reported an initial in-

crease in the output energy with decreasing temperature fol­lowed by a saturation of the output energy for temperatures in the region of 150--100 K. However, later investigations8

,9

reported a steady increase in the output energy as the tem­perature was lowered down to 120 K. Our calculation indi­cates a steady increase in the superfiuorescent pulse energy with decreasing temperature, confirming that, at any given pressure, the highest output energy would be achieved at the lowest possible temperature determined by the vapor pres­sure of the CF4 gas. The temporal profile of the CO2 pump pulse plays a significant role in the efficient operation of a CF4 1aser,9 Using our theoretical model we have studied the effect of the temporal shape of the CO2 pump pulse on the superfluorescem output energy. Our results indicate the im­portance of the spike over the tail of the CO2 pump puise for efficient pumping of the CF4 gas. We have also estimated the superfiuorescent pulse energy in the double-pass configura­tion and compared it with that for the single-pass configura­tion.

II. RATE EQUATION MODEL

The kinetic scheme for the COzlaser-pumped CF4 gas is shown in Fig. 1. The 9R ( 12} rotovibrational line of CO2

laser at 1073.278 cm- 1 pumps the R + (29) transition in the CF4 molecule from the Va vibrational state to the (Vz + v 4 )

combination state. Stimulated emission occurs on the p + (31) transition from (v2 + V 4) to (V2 ) vibrational leveL The ( + ) sign denotes the CorioUs sublevel involved in the respective transitions. The 9 R (12) pump laser radiation is 19 MHz to the red side of the pumped transition line cen­ter. 10 The relevant energy-level diagram is shown in Fig. 2.

Our model is based on a set of rate equations describing the dynamics of the CF4 molecules belonging to the six rel­evant groups represented by no i = 1-6, in Fig. 1. The popu­lation densities in the rotational-vibrational levels interact­ing directly with the fields are denoted by ni , i = 1,2,3, while fli' i = 4, 5, 6 represent the population densities in the

4293 J, Appl. Phys. 64 (9), 1 November 1988 0021 -8979/88/214293-08$02.40 @ 1988 American Institute of Physics 4293

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FIG. 1. Kinetic scheme for the CO2 laser pumped CF4 gas. T" is the rota­tional relaxation time. 'T,,, is the vibrational-vibrational relaxation time, 7',,1'

is the vibrational-translational relaxation time; OJp and IUSF denote the fre­quencies corresponding to the pumped and superfluorescent emission tran­sitions, respectively.

ground vibrational state (vo) and the two vibrationally ex­cited states (v2 + v4) and (v2 ), respectively. Each of the states n" i = 1,2,3, is coupled to a rotational reservoir through rotational relaxation, and these rotational reser­voirs interact with each other through vibrational-vibration­al relaxation as wen as vibrational-translational relaxation.

In the absence of stimulated emission, the time evolu­tion of the population densities in these six levels are given by the fonowing equations:

h l = -SuQ(n,-gln3/g3)+K4In4-KI4nl' (I)

il2 = K62i16 - K 26n2 , (2)

it) = SuQ(n j - gln3/g1) - K15113 + K'DnS' (3)

n4=KI4ni-K41114' (4)

hs = K35113 - KS3nS - K""n5, (5)

n6 = K 26nZ - K62f16 + Ko"ns - K uTn6 • (6)

The pump photon density is represented by Q, while g, denotes the degeneracy of the ith level. The first term in Eqs. ( 1) and (3) denotes the net optical pumping rate, where S13 = CO'n represents the stimulated pump transition rate per pump photon number density, c is the velocity of light, and 0'13 is the absorption cross section for the pump transi­tion. Rotational relaxation rate between the ith and the jth levels is denoted by Ki/ while K"u = 1/T"v denotes the vibra­tional-to-vibrational relaxation rate [(0000) + (0101) J =? (0100) + (0001) J, i.e., [( V 2 + v4 ) + (vo)] =? [( v 2 ) + ('V4 )], andKuT = 1/'01' denotes the vibrational­to-translational relaxation rate, the numerical values of

..l~·"23~O~[~==;=====~== V2 + Uk" 0 I i + I 1-_ p+ (31) EMISSION AT 61S em' \ j Jdl J!2

R~(29) 1_- PUMPED BY , 9R(12) 01' C02

1

Va JQ~ 29

FIG. 2. Relevant energy-level diagram for superftuoresccnt emission from optically pumped CF4 molecules. The total angular momentum quantum numbers ill the ground, Vz -+- )'" and V2 vibrational state are denoted by.l", J', and .I, respectively. The two arrows indicate the pumped transition R + (29) tothe Coriolissublevel ( -+- ) and thesuperfluorescentemission on the P + (31) transition, respectively.

4294 J. AppL Phys., Vol. 64, No.9, 1 November 19B8

which have been taken from Ref. 6. These collisional relaxa­tion processes are indicated in Fig, 1.

The above set of rate equations describe the temporal evolution of the population inversion (n~ - g"n 2Ig2 ) and that of the small-signal gain /3 for optically pumped CF4 gas molecules,

/3 = 0'32(11:3 - g3n2/ g2) ,

where (]'12 is the emission cross section for the transition. Propagation and growth of superf'iuorescent emission

from optically pumped CF4 gas column for a homogeneous­ly broadened transition is given by the equation

dI;z(Z) = ±IJI:±-(z)/{l+[l+(z)+r-(z)]IIsa!}' (7)

where z denotes the axial direction, i.e., the direction of prop­agation of the pump beam and the superscripts ( ± ) denote the two superfiuorescent beams propagating in opposite di­rection through the gain medium, [i,e., copropagating ( + ) and counterpropagating ( - ) with respect to the pump beam]. The equations for the beam intensities, It (z) and I -- (z), are coupled since they share the same group of mole­cules in the gain medium. I sat denotes the saturation intensi­ty, which is a function of the dipole moment for the transi­tion and the collisional lifetime of the upper excited level.

Estimates for the various parameters of the output su­perfluorescent emission are obtained on solving these equa­tions numerically, The foHowing simplifying assumptions have been made in our model:

0) The rotational rate constants Kij and Kji are related by the principle of detailed balance. When considering a par­ticular rotational level an the remaining rotational levels as­sociated with the same vibrational level are represented by the rotational reservoir. The equilibrium relation for the ro­tational relaxation rates given by K!j = Kjf//; is assumed to hold well for the case of optically pumped CF4 molecules also, where/; andJ; denote the thermal fractional population in the ith and the jth levels, respectively. Further, a single rotational relaxation time is used for an the rotational levels irrespective of the vibrational state to which they belong.

(ii) Spatial distribution of the pump beam in the trans­verse plane and therefore that of the small-signal gain of the optical1y pumped CF4 gas has been neglected. Spatial depen­dence in the axial direction, i.e., direction of propagation of the pump beam has been included.

(iii) The effect of pumping with a polarized CO2 beam has not been incorporated explicitly, although the use of the pump rate as an adjustable parameter in our calculation does account for this to some extent.

(iv) Fundamental band absorption corresponding to the Vo'" V4 transition of the superfluorescent emission has been neglected.

The fractional popuiation at thermal equilibrium in the relevant molecular groups were evaluated using the rota­tional partition function QR for a CF4 spherical top mole­cule ll

;

QR = [(2Iy + 1)4/12]V1T(BhcIKT)-3f2 exp(Bhc/4KT) ,

where I y is the nuclear spin of the fluorine atoms ( = ~), B is the rotational constant = 0.191 688 em -- 1,11 and the re-

Sucharita Sinha 4294

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maining symbols have their usual meaning. The vibrational partition function is given by

Qv = [1 - exp( - hUJ Ic!K7,) ]-1

X [1 - exp( - hW2c/ KT} ]-2

X [1 - exp( - hw3c/KT)]-3

X [1 - exp( --- hUJ4c/KT)]-1

where {Vi' (v2' W 3, and W 4 are the wave numbers for the four fundamental modes of vibration. II The number density N of CF4 molecules at pressure p and temperature T is related to No, the molecular density at standard temperature (To = 273.15 K) and pressure (Po = 760 Torr) by the usual relation l'

N = No(pTo/PoT},

with

No = 2.687X 101'lmol/cm3•

The dipole moments corresponding to the pump and superfiuorescent emission transition were obtained from Ref. 11, while the pressure broadening coefficients for these two transitions were taken to be 10.8 MHz/Torr. 12 The satu­ration intensity I'Rt was evaluated as in Ref. 6 using the expression

I,at = h 2c/(81TlfL~n!2r),

where 1t~12 denotes the dipole moment corresponding to the superfiuorescent emission and 7 is the relaxation time of the upper excited level.

The shape of the CO2 pump pulse has been approximat­ed by a Gaussian distribution in time, i.e., the functional form of Q, the pump pulse photon density has been taken to be of the form

Q = Qo exp( - (t - to)2a],

where a = 41n 2/t;, and tp is the full width at half maxi­mum (FWHM) of the Gaussian profile and has been taken to be 100 n8 for a typical CO2 pump beam. Qo represents the peak pump photon density and is the only adjustable param­eter in our calculation. For our calculations a time span cov­ering eight times the FWHM on either side of the Gaussian peak has been considered, beyond which the pump rates be­come negligibly small.

nI. SOLUTION AND ALGORITHM

Based on the set of rate equations describing the popula­tion densities in the six relevant energy levels, the smaH­signal gain at 16 pm has been calculated at time intervals of 0.1 ns for a 3-m-long CF4 gas column similar to the experi­mental setup.8,9 Time intervals smaller than this resulted in no significant change in the observable parameters such as output energy density, intensity, population densities in the various levels, and output pulse shape. In order to study the propagation and growth of the superf'luorescent emission at 615 cm- 1 the entire CF4 gas column was divided into 100 segments along the axial direction, and the gain associated with each segment was assumed to be uniform over that seg­ment. The rate equations were solved at each point in time using the Runge-Kutta method of integration. Using the

4295 J. AppL Phys., Vol. 64, No, 9, i November 1988

value of the instantaneous small-signal gain in each segment the equations describing the growth of I + (z) and J - (z)

were solved. The intensity at the output of a given axial seg­ment was used as the input condition for the next segment of the CF4 gain length. The initial value of superfluorescent intensity at t = 0 was taken to be the background noise level and a value of 10 14 W /crn 2 was assumed. Our calculation showed that the final output pulse energy of superfluores­cent radiation does not depend strongly on the initialleveI of noise taken, and a value of 10- u W /cm 2 results in an output pulse energy which is about 10% larger than that corre­sponding to an initial noise level of 10- 14 W/cm 2 for the same pump rate. At every instant of time the intensity ofthe superfluorescent radiation emerging from the last axial seg­ment was taken as the output and the entire process was repeated over the total axial gain length of the CF4 gas col­umn over a total time interval of2.5I1S. The temporal evolu­tion of the superfiuorescent intensity when integrated over time gave estimates for the total energy density contained in the pulse. Estimates for the total pulse energy were obtained assuming a normalized cross-sectional area of 1 cm2 for the superfiuorcscent radiation beam.

IV. RESULTS

A. Pressure dependence of superfluorescent energy

Output energy from optically pumped CF4 gas in both laser and superfiuorescent mode of operation exhibits a strong pressure dependence. 7

•H There exists an optimum

pressure as well as a maximum operating pressure for the CF4 gas.

We have used our rate equation model to estimate the pressure dependence of superfiuorescent output energy at various operating temperatures. In Fig. 3 the experimenta18

and computed 16-ltlTI single-pass output energies are plotted as a function of the CF4 gas pressure at the two operating temperatures of 117 and 145 K, respectively. The solid curve depicts the computed resuit while the circles denote the ex­perimental observations. A peak pump rate of 5.2 X lOR pho­tons/8 which corresponds to an input pump energy density of70 mJ/cm2 resulted in the best agreement. The agreement between the experimental data and our calculation is satis­factory. In addition, the estimates for the optimum and max­imum operating pressures at the two temperatures are also in good agreement with the experimental observations.

The optimum performance of the optically pumped CF <\

gas would correspond to the condition for maximum gain. For low operating pressure (Doppler-broadened pumped and laser transition), the gain increases with increasing pres­sure because of increased population inversion. However, for operation in the pressure broadened regime the output energy is determined by the pressure dependence of the laser transition linewidth and of the population inversion. With increasing pressure the number of molecules pumped to the upper laser level becomes Ilcarly a constant. However, in­creased population in the lower laser level as well as the increased linewidth of the laser transition result in a reduc­tion in the net gain beyond an optimum pressure,

The effect of pumping rate on the optimum operating

Sucharita Sinha 4295

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E

>

" a: w z w

16f 14

12

10 '

o ___ 1

4

CF4 PRESSURE (m bar)

pressure was studied theoretically using the rate equation modeL In Fig. 4 we show the calculated pressure dependence of output energy for peak pump rates varying from 1,5 to 5.2 X 108 photons/s at a constant operating temperature of 117 K. As is evident from Figs. 3 and 4, the optimum pres­sure increases with increased pump rates and reduced tem­peratures. With increasing pressure, the increased transition linewidth tends to reduce the net gain. However, reduction in temperature and increased pump rate enable the CF4 gain medium to counteract this effect, thus leading to higher opti­mum pressures. The assumption of a homogeneously broad­ened emission profile for the 16 f-lm superfluorescent emis­sion is justified for pressures greater than 3 Torr at the

'l I

16

1

141 ~

E 12 ~ >-<!l a: '" z w 10

.... J z w u oJ1 w cr

J 0 3 "-a: w

·f a. ::J

""

zr

L 0

_L ___ .l

4 6 ~'---

B

CF4 PRESSURE (m bar)

4296 J. Appl. Phys., Vol. 64, No.9. 1 November i9B8

.l 24

FIG. 3. Comparison of the experimental and theoretk:al dependence of supertluorescent pulse energy on the CF4 gas pressure at tbe two operating temperatures of 117 and 145 K, re­spectively. The solid line depicts the theoretical dependence while 0 (117 K) and. (145 K) denote the expcrimental observations.

highest operating temperature of220 K, which has been con­sidered in our calculation.

80 Temperature dependence of superfluorescent energy

The lower laser level being very close to the ground state (L\.v~435 cm-- 1

), the superfluorcscent output from CF4 is expected to show a strong temperature dependence. Green7

had reported a temperature dependence which indicated an initial increase followed by a saturation of the output as the temperature was lowered in the region of 150-100 K. Cool­ing of the CF4 gas has two effects. First, it reduces the ther-

1

20 22

FIG. 4, Theoretical curves for the dependence of super­fluorescent pulse energy on the CF4 gas prl'Ssure at a con­stant operating temperature of 117 K for different values of the pumping rate. Peak pumping rates varied from 1.5 to 5.2 X lOx photons/so

Sucharita Sinha 4296

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FIG. 5. §uperfluorescent pulse energy as a function afCF'. gas temperature for an operating pressure of 6 Torr. The solid line denotes the theoretical dependence while open circles denote the experimental data.

mal population of the lower laser level (V2' J) and second, it changes the initial thermal population of the source pumped level (va' Jo). Later reports,8,9 however, showed a steady increase of the output energy as the operating temperature was lowered down to 120 K.

Using the rate equation model we have obtained the temperatl're dependence of the superfluorescent output en­ergy and the calculated and the experimenta18 observations are in satisfactory agreement. In Fig. 5 the experimental data is denoted by the open circles and the solid line denotes the computed dependence for a peak pump rate of 5. 2 X 108 pho­tons/so OUf calculations confirm that fOf any given pressure, the highest output energy would occur at the lowest possible temperature determined by the vapor pressure of the CF4

gas.

C. Dependence on pump rate

The calculated dependence of the superfluorescent pulse energy density on the pumping rate is shown in Fig. 6 at an operating temperature of I 17 K and at the optimum pressure of operation for the CF4 gas. Assuming a resonant pumping condition for the CF4 gas the output energy density is seen to increase linearly as the peak pump rate is increased from 5 X 106 to 5 X 109 photons/so Our calculations indicate that pulse energy density of the order of 180 mJ/cm2 for the single-pass superfl.uorescent configuration can be achieved with pump pulse energy density of the order of 700 mJ/cm2

from a 3-m-long CF4 gas column operated at the optimum pressure and at a temperature of 117 K.

Do Effect of detuning of the pump laser

The large fluctuations in the output energy as is evident from the error bars associated with the experimental data in

4297 J. Appl. Phys., Vol. 64, No.9, 1 November 1988

................... ;.;.;.;.; •• -.; ••••• ,. •••••••• ,' ••• ' •• ;0 ••••••••••••• -.~.:.:.;.; •••••••••• '......... • •••••••••• ~.:.; ••••••••••••• ,. •••••••••••••••••• ~ •• - ••• - ••••••• -.... ",' •• ~......... ,'.-." •• , •••• '.' .~ ••••• .,. ••••••• "' •• -.-.~

"'g ,~ot

E 1201

2°1 'L-__ --L ___ .1 _____ L... __ --1 ___ l.. .. __ . ...-J

o 10 20 30 40 50 60

FIG. 6. Theoretical dependence ofsuperfllloresccnt pulse energy density 011

the pumping rate at an operating temperature of 117 K and at the corre­sponding optimum pressures. Peak pump rates varied from 5 X 10" to 5 X 109 photons/so

Figs. 3 and 5 is essentially due to the frequency fluctuation of the pump CO2 laser pulse. Although the presence oftlle low­pressure gain cell in the CO2 hybrid oscillator confines the CO2 output frequency within the narrow gain bandwidth characteristic of the low-pressure gain ceH, the frequency fluctuation, as a result of the fluctuating resonator length of the CO2 oscillator, affects the CF4 superfluorescent output energy dramatically. This leads to a large shot-to-shot vari­ation in the pump energy absorbed by the CF4 molecule and is reflected in its turn in the output superfluorescent energy which is rather erratic, falling to zero at times. Using the rate equation model for supertluorescent emission we have esti­m-ated the dependence of output energy at 16 l"m on the detuning of the pump frequency from the CF4 absorption band line center. In our calculation the separation of 19 MHz between the line centers of the 9R (12) CO2 pump radi­ation and the R -+ (29) CF4-pumped transition has been tak­en into account and a drift in the CO2 axial modes over a gain bandwidth - 50 MHz, corresponding to the gain profile of the low-pressure gain cell of the hybrid CO2 oscillator, has been considered. Figure 7 shows the calculated dependence of the puise energy on the frequency detuning. The strong dependence indicates the necessity offrequency locking and st;bilization of the pump pulse corresponding to a maximum energy absorption in the CF4 gas.

It is important to note that the pump energy absorbed by the CF4 molecules also depends on the Rabi width of the pumped transition in addition to the frequency detuning and the transition linewidth. However, this effect has been ne­glected since the linewidth ofthe pumped transition is much greater than the Rabi width under the typical operating con­ditions considered in our calculation. The Rabi width is

Sucharita Sinha 4297

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about 40 MHz in this case, while the transition Iinewidth is about 80 MHz. Even for higher pump powers, the linewidth of the pumped transition continues to be greater than the Rabi width since the optimum operating pressure increases with increased pump powers.

Eo Effect of temporal shape of the CO2 pump pulse

Earlier workers had suggested that in CO2 laser­pumped molecular gas lasers strong emission occurs only after the completion of the pumping by the gain-switched spike of the CO2 pump pulse. Therefore, the tail of the CO2

laser pulse which is relatively fiat in time was expected to play a more significant role in the optical pumping of the moleCUlar gas lasers.13 In contrast, Bonanni, Castiglione, and Salvetti'} reported that in the case of the CO2 laser­pumped CF4 laser the spike rather than the tail of the CO2

pump pulse was observed to affect the performance of the laser more strongly. They observed that the CF4 pulse ener­gy dropped by a factor of nearly 6 on attenuating the pump pulse energy uniformly by 60%, whereas the drop was by a factor of only 2.4 when the pump pulse energy was reduced by 60% by reducing the energy contained in the tail of the CO2 pump pulse, thus implying the importance of the energy contained in the spike of the CO2 laser pulse. We have used our rate equation model to investigate the effect of the tem­poral shape of the CO2 pump pulse on the 16 pm super­fluorescent energy by simulating a CO2 pump pulse shape which is a superposItIon of a narrow Gaussian (FWHM = 100 ns) and a broader half Gaussian (FWHM = 1 fls) distribution in time. On reducing the total pump pulse energy by 60% by uniform attenuation of the pulse, the superfiuorescent energy dropped by a factor of about 5. On the other hand, the superfiuorescent output en­ergy fell by a factor of only 2. i when the 60% reduction in the pump pulse energy was effected by removal of energy

4298 J. Appl. Phys., Vol. 64, No.9, 1 November 1988

FIG. 7. Calculated dependence ofsuper!luorescent pulse energy on the extent of frequency detuning (~v) between the pump radiation and the CF.­pumped transition.

from just the tail of the CO2 pump pulse. This is in close agreement with the reported observations.9 In Fig. 8 we show the calculated dependence of the 1611m superftuores­cent pulse energy on the fractional energy content of the tail of the CO2 pump pulse. Our calculation shows that the su­perfiuorescent pulse energy falls as the fractional energy contained in the tail rises, indicating the importance of the spike rather than the tail of the CO2 pump pulse in the case of optically pumped CF4 gas. This can be understood by com­paring the decay rate of the upper excited level of the CF4

molecule and the pump pulse duration. Since the decay time of the upper excited level is of the order of 5 ns at the CF4

J I ,+

0:; lJ E I

~ I ~ tOt § ! ~ 81 ~ 6~ '5 I

~ 4r

z I 1_._, _._, __ "'--_.-L _ -L_.---'- _ -'-_ .. -L-. __

o ~ w ~ ~ ~ M ro ~

~RAC'fIONAl ENERGV IN C02 PULSE TAil (Of.)

FIG. 8. Calculated dependence of the supertluorescent pulse energy on the fractional energy content of the tail of the CO2 pump pulse.

Sucharita Sinha 4298

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\

o L_ -~-- - -~---372

_t __________ >

1246

'<Il

i I

I ~ I : 002S1 ~ \

I I

TIME (n&i'\csec 1

o l ________ _ ____ .J-.:: ~ ____ "_. __ ._" ____ .J

372 1245 2121

TIME (nano!i€'c)

FIG. 9~ Temporal evolution of (a) a typical 16 /lm superfluoresceni pulse computed using the rate equation model, (b) corresponding CO2 laser pump pulse.

operating pressure of6 Torr, which is much shorter than the pump pulse duration consisting of a 100 ns spike and a 1 f1s tail, the energy contained in the long duration tail is not expected to playa significant role in the efficient pumping of the CF4 molecule.

Fo Tempora! evolution of the superfluorescent pulse

Figure 9 (a) shows the temporal evolution of a typical 16 .urn superfiuorescent pulse computed using the rate equation model. The narrow spike has a FWHM of about 60 ns. For an operating pressure of 6 Torr and a temperature of 117 K, a peak intensity of 0.14 MW /cm2 is calculated for a peak pump rate of 5.2 X 108 photons/so Figure 9 (b) depicts the temporal evolution of the corresponding pump CO2 laser pulse assumed to be a Gaussian of FWHM = 100 ns.

\10 DOUBLE~PASS SUPERFLUORESCENT EMISSION

Double-pass superfiuorescent output essentially in­volves the configuration as shown in Fig. 10. The CO2 pump beam which is linearly polarized is transmitted into the CF4

gas cell through a brewster-oriented germanium plate. The radiation emitted by the CF4 molecules in the reverse direc­tion with respect to the propagating CO2 beam has both par­allel (p) and perpendicular (s) components of polarization. Of these, the radiation component polarized parallel to the

4299 J. Appl. Phys., Vol. 64, No.9, 1 November 1988

. -z ~-~Ip

Ij;--­-------5'1" I; I;~

GAIN MEDIUM

FIG. W. Schematic diagram of the double-pass superftuorescent configura­tion Ge: Germanium Brewster plate; M, reflecting mirror.

pump CO2 beam is wholly transmitted by the brewster ger­manium plate; the perpendicularly polarized (s) beam is partly reflected by the germanium plate on to a gold mirror which reflects it back into the CF4 cell, thus ensuring a dou­ble-pass for this component of the beam through the active medium.

In order to study the emission characteristics from the double-pass configuration, the rate equation model has been extended to account for the presence of both p- and s-polar­ized emission traveling both paranel and antiparallel with respect to the pump beam. For a homogeneously broadened gain profile the component beams having the same state of polarization but traveling in opposite directions will interact with the same group of molecules, thus sharing the same population inversion. The corresponding equation describ­ing the growth and propagation through the optically pumped CF4 gas column is given by

df /' (z)

dz

dl t (z)

dz

± Pll (z)/{ 1 + [I p-t (z) + f p- (z)]I I,aJ,

(8)

± .821/ (z)/{l + [I;' (z) + I s- (z)]IIsat },

(9)

where, as before, z denotes the direction of propagation of the CO2 pump beam. The subscripts p and s denote the state of polarization, while the superscripts ( ± ) indicate the di­rection of propagation as before.

Optical pumping of CF4 gas molecules using a linearly polarized CO2 laser beam results in an orientational anisot­ropy in the stimulated emission cross section for the optical­ly pumped molecules, In the case of CF4 molecules pumped with a linearly polarized CO2 beam, the effective transition probabilities for emission polarized parallel and perpendicu­lar to the pump beam has been given by the ratio 4:3 in the case when both pumping and lasing are on the R or P transi­tion, 14 Thus, the small-signal gains for the p- and s-polarized components denoted by /31 and P2 are in the ratio 4:3.

As in the previous case, the algorithm is based on a series of axial segments along the gain length of the CF4 gas col­umn, the only distinguishing feature being that at each in­stant the sets of four equations are solved simultaneously and values for 1.;1 (z), I; (z), 1.,* (z), and J,~ (z) obtained in each segment are taken as the input condition for the next axial segment. Further, the component l,- on reflection at the germanium brewster plate and the gold mirror is once again fed back into the CF4 cell in this configuration, thus contributing to the input signal for 1/ in this case. Propaga­tion of I s- through free space, together with reflection at the

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germanium and gold surfaces, has been simulated by subdi­viding the intermediate free space into several axial zones where propagation has been assumed to be lossless. While the superftuorescent emission in the reverse direction having p polarization is lost, the remaining three components con­tribute to the total superfluorescent pUlse energy in the dou­ble-pass configuration.

Proceeding as before, the set of rate equations describing the population inversion in the active medium is solved and the smaU-signal gain of the pumped CF4 molecules is calcu­lated at time intervals of 0.1 liS. Using the value ofthe instan­taneous gain the four equations describing the dynamics of the superfiuorescent intensity for the four components are solved simultaneously in each axial segment. Incorporating the contribution of the Is emission on reflection at the ger­manium brewster plate and the gold mirror to the input for 1.-' emission, the energy density of the p- and s-polarized components, as wen as the total energy density in the pulse, was evaluated for various temperatures of operation. OUf

results showed an increase in the double-pass superfiuores­cent pulse energy over that in the single-pass configuration by a factor of the order of 2. 5 for an operating temperature of 117 K and pressure of 6 Torr. Experimental observations8

indicate an improvement factor of 1.5 for the above operat­ing condition. Some possible causes of this discrepancy are suggested in the next section.

A reflectivity of 99% for the gold mirror and 78% for the brewster-oriented germanium plate for the s-polarized component has been taken. Depolarization of the laser gain through collisional reorientation of molecules has been ne­glected.

VI. DiSCUSSION

Although our theoretical investigation has been carried out on the superfluorescent emission from CF4 molecules at 615 cm-- 1

, the rate equation model is applicable to emission on other transitions as well, provided the CO2 pump radi­ation matches spectrally with an absorption profile of the CF4 molecule.

In our approach of theoretical modeling of the optically pumped CF4 gas column, the depletion in population inver­sion which occurs at the end of each time step has been ne­glected. As a result, the small-signal gain has been overesti­mated. In order to estimate the effect this would have on the superftuorescent radiation, we considered the numerical so­lution of the rate equations describing the molecular density in the various energy levels in the presence of a nonzero photon fiux. Thus the set of equations describing the popula­tion density and the photon flux were solved simultaneously in each of the axial segments. The computer time required for a single run in this approach was nearly eight times larger than that required in the first approach. However, the out­put pulse energy densities differed by about 10% only. Com­paring the small-signal gain and the gain in the presence of a nonzero photon fiux, we find that these two differ marginal­ly in the first half of the CF4 gas column, implying that satu­ration effects begin to affect the gain mainly in the second half of the CF4 gain length. This is due to the low level of photon fiux in the first half length of the gain medium.

4300 J. Appl. Phys., Vol. 64, No.9, 1 November 1988

Therefore, it is only half of the total CF4 gain length in which the difference in these two approaches becomes important. Keeping in view the computer times involved, the first ap­proach has been fonowed in the rest of our study.

However, in the case of the double-pass configuration, since the ccunterpropagating s-polarized beam is reflected and fed back into the CF4 gas column, it is more likely that the small-signal gain and the gain in the presence of photon flux would differ over a larger fraction of the total gain length, i.e., more than just half the total gain length. This would therefore result in larger discrepancies between the final result as obtained from the two approaches. This, to­gether with the fact that all intracavity losses have been ne­glected, would lead to an overestimate for the improvement factor when comparing the single- and double-pass configu­rations.

VII. CONCLUSIONS

We have made a numerical study of superf'iuorescent emission from an optically pumped CF4 gas column using a rate equation model. Estimates for the dependence of super­fluorescent emission on the various operating parameters, such as temperature, pressure, pumping rate, and the detun­ing between the pump radiation and the pumped CF4 transi­tion have been obtained. OUf results confirm that, at any given pressure, the highest pulse energy would be achieved at the lowest possible temperature determined by the vapor pressure of the CF4 gas. The effect of the temporal shape of the CO2 pump pulse on the superfiuorescent emission energy was estimated which shows the significance of the gain­switched spike over the relatively flat tail of the CO2 pump pulse for efficient performance of the system. Estimates for the superfiuorescent emission energy in the double-pass con­figuration were also made using our rate equation model and compared with that for the single-pass configuration.

ACKNOWLEDGMENTS

The author would like to thank Dr. S. C. Mehendale, Dr. K. C. Rustagi, and Dr. L. G. Nair for some helpful discussions and a critical reading of the manuscript.

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