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Theory of an optically pumped gas laser
J
Quonr.
Speclrosc.
Radiar
THEORY
Transfer.
Vol.
13. pp. 235-254
Pergamon
Press 1973 Pnnled
OF AN OPTICALLY JAMES J. HEALY and
in Great Britain
PUMPED
GAS
LASER
T. F. MORSE
Division of Engineering and Center for Fluid Dynamics, Brown University. Providence, R.I. 02912, U.S.A. (Received 5 Junv 1972)
Abstract-A theoretical description of the steady-state operation of an optically-pumped gas laser is presented. The three-level gas is described by a suitably modified form of the Boltzmann equation. This formulation is intermediate between the rate equation and semi-classical descriptions of laser behavior. Allowing coupling between gas and radiation, solutions are obtained for population inversion, gain coefficient, radiative intensity and output power, in terms of system parameters and spatial variables. The role of Doppler (inhomogeneous) broadening as compared with collisional (homogeneous) broadening is clearly displayed. Furthermore, it has been possible to include non-linear saturation effects due to the lasing radiation in a simple, quantitative fashion. The theoretical predictions are compared with the limited experimental data available, and agreement is good. Although the quantitative results are applicable only to optically pumped systems, the qualitative predictions should be valid for any type of gas laser.
INTRODUCTION DESPITE the great amount of work on gas lasers in recent years, the problem of the opticallypumped gas laser appears so far to have escaped detailed theoretical investigation. This is perhaps because of the considerable experimental difficulties associated with this type of laser, and the relatively low output power developed by the systems hitherto investigated. Because of the desirability of strong coincidence between absorber and pump lines, few systems are capable of efficient optical pumping. Only cesium vapor, pumped by a resonance helium line has been studied in detail. (l-3) Broad band pumping of the CO, system with solar radiation has also been suggested as a possible laser system.‘4’ Certain aspects of the optically pumped system make it attractive from the theoretical viewpoint, however. Since the excitation mechanism is optical rather than collisional, a great simplification in the equations governing the problem is effected, as compared with collisionally pumped gas lasers. In addition, the low operating pressure renders the dynamics of the gas tractable from the viewpoint of nearly-free-molecule kinetic theory. Thus we are led to consider formulation of a Boltzmann equation for a gas possessing internal energy states, incorporating appropriate radiative source and sink terms. (5) In addition, we can treat the radiation at the pumping and lasing frequencies by means of the equation of radiative transfer, where the absorption and emission coefficients are dependent on the gas particle distribution functions. We then find a set of coupled kinetic and radiative equations which, in principle, can be solved to display population number densities, gain coefficient, and radiative intensity, as functions of the spatial variables and the system parameters. This approach has several advantages over more conventional treatments employing rate equations, among which are : 235
JAMES J. HEAL-Y and T. t:. MORSF
‘76
(1 I The random motion of the gas particles is specifically accounted for. This is an important consideration in radiative transfer problems. since the macroscopic emission and absorption profiles are intimately related to the gas velocity distribution functions. In particular, one would not expect that the type of formulation developed by BIRERMAN and HOLSTEIN(~) would be adequate for the present problem, since the key assumption there is that all velocity distributions are Maxwellian. This would not be true in a laser, where the intense, unidirectional. monochromatic lasing beam will severely distort the velocity distributions of the relevant excited atomic levels. Thus. no a priori restrictions on the excited-level distribution functions should be made. and the structure of the distribution functions must be part of the solution of the problem. (2) The interaction of the pumping process with the depopulation due to lasing action is clearly displayed, whereas normally the pumping process must be treated as a parameter of the system. (3) The effect of spatial variations and the system geometry can be explicitly treated. (4) The quenching effect of wall and interparticle collisions can be studied. (5) Although optically pumped systems were chosen as models for this theory. the theory itself is capable of treating the more complex situations of inhomogeneous broadening in collisionally pumped gases. The disadvantage of this approach. at least in its present form, as compared with more complete quantum mechanical approaches, (‘w is that by treating the radiation by means of the equation of radiative transfer, all information regarding photon coherence properties is lost. Thus, we can hope to obtain detailed predictions for the population inversion. gain coetficient. radiative intensity and output power, but no information connected with coherence phenomena can be extracted. The physical problem considered in the remainder of this work consists of a box of low-pressure monatomic gas, which is being irradiated at one of the gas resonance frequencies. See Fig. 1. Two of the walls of the box are plane, partially transmitting mirrors, set at an appropriate spacing so that they form a Fabry-P&rot cavity, which is resonant at the gas lasing frequency. Thus, the population of excited gas atoms at steady state represents a balance between the rate of excitation due to the pumping radiation, and deexcitation due to spontaneous and stimulated emission, and collisions. Similarly. the intensity of the lasing radiation represents a balance between losses in the system, and stimulated emission by the gas. Throughout this work. the emphasis will be on obtaining manageable approximations to the solutions of the complex governing equations, which provide insight into the operation of the laser, and make quantitatively useful predictions of the relevant parameters.
I. THE
ATOMIC
MODEL
AND
GOVERNING
EQUATIONS
Consider a FabryyPkrot cavity formed by two plane mirrors of depth 20 and length 2L, which are infinite in the y-direction, as shown schematically in Fig. 1. The lasing gas is contained between the two mirrors by means of transparent walls. and pumping radiation at approximately the resonance frequency I’~~ is supplied to the laser through these walls. between the semi-transparent mirrors. The lasing radiation at frequency v z 1 propagates and the cavity is assumed to be tuned so that the spacing 2L corresponds exactly to a
231
Theory of an optically pumped gas laser PUMPING
PUMPING
RADIATION
RADIATION
AT
V20
AT vzo
FIG. 1. Geometry of laser.
resonance condition at vZ1. It is assumed that only a single cavity mode is excited above threshold. The latter two restrictions will be removed in a subsequent paper.“’ In the atomic model adopted here, a ground level and two excited levels are present, and for simplicity, all degeneracies are assumed to be unity. All of the levels are allowed to have a collisional half width, sO, sr, and e2. For purposes of this work, these widths correspond to the homogeneous broadening of the energy levels by any process other than Doppler shifting. For an optically pumped laser, it is required that the atomic transition very closely with the frequency of the pumping radiation exfrequency v10 correspond ternally supplied, and, in addition, the atomic frequency vZ1 must. correspond to the resonance frequency of the optical cavity formed by the two mirrors. Figure 2 summarizes the details of the atomic model. The general problem of the interaction between a gas and line radiation has been treated in detail through the use of kinetic theory and the equation of radiative transfer derived by these authors to the by CIPOLLA and MORSE.(‘) In applying the equations optically pumped gas laser oscillator, only one modification is required; the collisional broadening, assumed zero in Ref. (5) must be explicitly taken into account. It will be shown later that if the collisional linewidth of the lasing transition were zero, then lasing action would be virtually impossible. The relationships among the various linewidths pertinent
COLLISIONAL
BROADENING 2nd
2E2
Is.1
EXCITED
EXCITED
LEVEL
LEVEL
/ PUMPING TRANSll Y 10
-&A
FIG.
TERMINAL DEPOPULATING TRANSITION
GROUND LEVEL
2. Description of atomic energy-level model.
JAMES J. HEALY and T. F. MORSE
238
to the present problem may be summarized as follows: bandwidth of pumping radiation >> Doppler width >> collisional width at vZO; collisional width of lasing transition >> bandwidth of lasing radiation. The collisional linewidth of the lasing transition may in general be less than, equal to, or larger than the Doppler width, and all of these possibilities will be treated. As discussed in Ref. (5) the governing kinetic equations can now be written as
The velocity
in the y-direction
has been integrated
where n,(x, z) is the number-density per unit volume nondimensionalizations have been used in equations x=F
for
-l
z=i
for
-1lxll
L T
-
l - (2RT,)“2;
out, and the normalization
of atoms (1) :
on .fi is
in level “i”. The following
D
(2)
T2 = (2RT,)“’
Here 5 is the dimensional atomic velocity vector as measured in the laboratory reference frame, and 7” is the temperature of the laser walls, assumed uniform. z, and z2 are essentially the mean times of flight for an atom to traverse the laser in the x and z directions. The quantities Ji denote the source and sink terms for atoms in level “i” due to collisions,““’ and the A,, denote the Einstein coefficients for spontaneous emission. Thus the terms .4,,Q,,f, give the rate of stimulated emission by atoms in level “m”, per unit of fourdimensional phase space. For a system of atoms interacting with line radiation, the steady-state equation of radiative transfer may be written asC5’
1.2=c c hvB,,[.~~,(f,)+.f,.;//,,(f, -1,)l n “
m
Theory of an optically pumped gas laser
239
where f, is the photon distribution function.‘5’ The quantity &,,(f,) in equation (3) is defined as the total number of atoms per unit volume in level “m” which are capable of emitting a photon of frequency v,, into the element of solid angle dQ about 1.Since we wish to retain the effect of broadened energy levels in the present problem, the assumption is now made that the collisional lineshape q,,(p) shown in Fig. 3a, can be approximated as in Fig. 3b. Physically, this amounts to saying that all frequencies “p” which fall within the cross-hatched region of Fig. 2, will be absorbed with equal probability, but frequencies outside these limits will be unable to interact with the atom. Mathematically stated, this becomes q
(p)
=
~[~v,,+&"+&,}-~l.~[~-{v",-&"--E,}l
nm
2(&n + &J
where H(x) denotes the Heaviside unit step function, and equation (4) satisfies the normalization
I
qmh4dp = 1.
Y nm P’ FIG. 3a. True collisional lineshape.
u nm FIG. 3b. Modelled collisional lineshape.
(5)
JAMES J. HEALY and
‘30
T. I-. MORSE
Now, a photon of frequency p in the atomic reference frame will appear to have a frequency 18in the laboratory rest frame, where v is given by the Doppler formula as C’= speed of light
since ~1
“II”, Substituting
from equation
(6) into equation
(51, we find
where dimensionless ratio of collisional linewidth to Doppler width for 17+ 111transition dimensionless The total number which are capable
frequency
difference.
of atoms in level “II”, per unit volume, per unit solid angle about 1, of absorbing a photon in the line centered on \I,,,~is then given by”’
We now turn to the development of expressions for the terms Q,,,,, which were introduced in the kinetic equations (1). The rate of stimulated emission from level “nr” to “II”. per unit solid angle, per unit of phase space volume, is given by the product of the Einstein coefficient for stimulated emission, multiplied by the number of photons per unit volume in the element d0 about 1, multiplied by the number of particles per unit of phase space volume capable of emitting a photon of frequency I’~,,,.Thus, the substantial derivative of ,Jm(q) is given by
= The quantity
Q,,(f,)
is now defined
-7
Thus Q,,,, gives the number capable of being absorbed
A,nJ;,,(~)Q,,,,( f;). by
equations
(9)
(7) and (9) as
in of photons per unit volume in the n-m resonance by a particle in a given region of phase space.
line which arc
Theory
of an optically
pumped
241
gas laser
To complete the mathematical formulation of the problem, boundary conditions need to be specified on equations (1) and (3). For the kinetic equations, the simple assumption will be made that particles leaving any solid surface after a collision will have a MaxwellBoltzmann velocity distribution : ,f;(x,r~,E,) = N,Q-‘(2nRT,))‘exp-
{ $+vI’+--~ -x z ;j
This boundary condition is motivated by the fact that interactions _ generally tend to destroy an inversion. The boundary condition on radiation centered at v2,, is : j;,,,(z = *l) I0 is a dimensionless
measure
= :a(1 Jr
of the pumping
FI=)exp(-/PC:, intensity
(11) with a boundary
1
at z = f 1. It can be shown”
will
(14 ” that (13)
where qi, is the flux of pumping energy supplied at either transparent wall, and fiml is defined as the bandwidth of the pumping radiation, divided by the Doppler width of the lasing gas. For radiation at the lasing frequency v2 1, the condition for stable operation is that the single-pass amplification at line center must balance the system losses.“3’ In addition, it is reasonable to assume that the lasing frequency must be confined to propagation directions very close to I, = k 1; otherwise the photon will be reflected out of the cavity, and will not participate in the feedback process. No boundary condition is available at x = + 1 for either the amplitude or spectral distribution of the lasing radiation. This difficulty will be circumvented later by using the fact that the laser bandwidth is many orders of magnitude smaller than all other bandwidths in the problem, and the amplitude will be fixed by means of a self-consistency requirement. The above discussion leads to an important simplification of the equation of radiative transfer (3). We are dealing here with two spectrally disjoint radiation packets, which are propagating in different directions. See Fig. 4. The absorption of radiation at the pumping frequency will be controlled by (f, -_&) ‘v fo, since fi will always be several orders of
FIG. 4. Qualitative
dependence
off, on frequency.
JAMES J. HEALY and
242
T. F. MORSE
magnitude smaller than .f, provided hv,, >> kT,. Similarly, radiation at v2i will interact only with (f2--fi), SO that for all practical purposes, the two frequencies interact with separate atomic populations. Thus the two frequencies may be uncoupled and equation (3) may be replaced by the two simpler equations +
z
=
(14a) Dh~,20BZ0[.~20(.1;)_~PLD.~~O(.fO_.f2)l
=
Lhv,,B,,[,~~,(J;)+f;,L,X,,(f;-f;)l. (14b)
and +
The set of kinetic equations (1) can also be simplified considerably. Since the laser cavity is not resonant at via, the terminal depopulating transition, we can assume that stimulated emission at this frequency is negligible. Thus Qro = 0 in equation (1). A further simplification is motivated by the fact that the ratio D/L as defined in Fig. 1 is normally small, values of lo-’ being typical for gas lasers. It is intuitively clear that as D/L 4 0, the derivative with respect to x in the kinetic equations may be neglected, when x is not close to the mirrors. Thus the solution to the kinetic equations (1) in a cavity where D/L cc I can be found to a high degree of accuracy over almost the entire length of the laser by retaining only the derivative with respect to z.(i4) The role of collisions is expected to be of secondary importance in an optically pumped nonequilibrium system, since an appreciable number of collisions would drive the system toward equilibrium. However, to retain the physical effect of collisions, we adopt the following simplified form of a relaxation model. J, 1 (M,, -.f&ri. Here gi is an inelastic
collision
frequency,
gi = C
defined as(lo,l I)
fj,ll-‘lslI~~(l’l-~sl,~)dodrls.
jkl ss
The simplified
“-2 z2
v* af1
Ty’ay
kinetic
equations
now read
2: (M,o-f,)ao+4~(A,of,+A,of,)-A20Q20.1b
N (M,1-f,)o,+4~(A2,f2-A,~f,)+A2,Q21(f2-f,)
(15)
The sets of equations (15) and (14), together with the definitions (8) and (10) and the boundary conditions now comprise a set of five coupled linear integro-differential equations of for the five unknowns fo, fi, f2, f,,, and f,,, , and we now turn to the construction approximate solutions for these quantities.
243
Theory of an optically pumped gas laser
2. ITERATIVE
SOLUTION
TO
AND POPULATION
THE
GOVERNING
NUMBER
EQUATIONS
DENSITIES
The analytic solution of the coupled radiative and kinetic equations, even with the simplifications so far discussed, is prohibitively difficult. However, CWOLLA and MORSE”) have shown that an iterative approach to this type of problem can be expected to converge, and to yield physically satisfactory results after one iteration. The iterative strategy to be adopted here is as follows; we first obtain a solution to the radiative equation (14a), on the assumption that the kinetic distribution functions are Maxwell-Boltzmann everywhere. The solution for f,,, so obtained should be valid when the gas is optically excited, since even then, there will be only a very small percentage change in fo, provided I0 is not too large. The solution for f,,, can then be substituted into equation (10) to obtain Q$?, where the superscript (0) denotes a zeroth-order quantity. To generate a zeroth-order expression for Q2i, the assumption is made that we can write
(16) Thus, it is assumed that the lasing radiation is a delta-function in frequency and direction. Lo is an undetermined constant, which is related to the quantity of energy stored in the lasing mode. Now, with f. given by its Maxwell-Boltzmann value, one finds from equations (7) and (8) cc
co
%:“d(f,) = N
dqXexP(-“) s
Jn
-m
j. dr],
_m
H[a-(i,o-rl,)lH[a+(i~~-vl,)l 260+ 4 .
exp(-13
Jn
(17) where no N N has been used, and a = (a0 +EJc/v~~(~RT,,,)~‘~. The parameter “a” will normally be much less than unity, and for the cesium system to be discussed later, a - O(10m3). For a -+ 0, equation (17) becomes
~%fo) = Now, using equation
(18) in equation
af1’5:
*aZ=with the boundary
condition
JRv20&y2 .exp( - Go),
(18)
(14a) we find :
~hv20B20Nc exp(-ii,) &cv,o(2RTW)“2
(12) this has the solution
: (19)
The + superscripts refer to radiation propagating The quantity S in equation (19) is defined as
in the + z and -z directions
Nt2c3 A,, s = 2J(7c)v:o .
respectively.
JAMES J. HEALY and T. I-. MOKSI
243
The dimensionless photon mean free path, for photons in the &2 resonance line can be defined”“’ as /I,, = D exp(ci,)/S. which merits discussion. The contribution Equation (19) for .f;,,, reflects an assumption to ,f;.,, from spontaneous emission by the lasing gas has been neglected. The results of Van Trigt,“” which refer to a strictly two-level gas, optically pumped by resonance radiation. show that the spontaneous emission contribution is always important in an optically thick gas. However, these results assume that there are no intermediate levels in the gas between the excited level and the ground level. Thus spontaneous emission is always in the same If, however, the gas has a frequency band as the pumping radiation, and is “trapped”. large number of intermediate levels between the excited level and the ground level. then the spontaneous emission will be at a number of frequencies, only one of which will be trapped. Thus the contribution to .fl,,) by spontaneous emission in the gas is controlled by the branching ratio for level “2”. If the branching ratio is such that (Azo/xnX2 A,,) - 1.0. then the spontaneous emission term will be important. If on the other hand (A2O/x,1 ~ L .A2,,) CC 1.0 then one can conclude that most of the spontaneous emission is not at the trapped frequency \n2(), and the contribution to ,fiJO is negligible. The latter situation is the relevant case for a laser. This can be seen by considering the terminal lasing level (level “I” in Fig. 2). If there are not a number of levels between the terminal level and the ground state, then spontaneous emission from “1” will also be trapped, and the effective value of Al,, will be reduced to _ A 1,/SJ(ln S).‘6’ Th us it is expedient to provide “escape routes” for spontaneous emission in the form of intermediate levels, if a bottleneck in the terminal level is to be avoided. We conclude then that the effect of spontaneous emission at \I~,~ will normally be of secondary importance in an optically pumped laser, and the solution as given by equation (19) should be valid. into the expression defining Q2,,. and one The solution for ,f,,,, is now substituted finds, for n + 0,““’ Q’,“d= 4J(7r)Z” cosh[(Sz) exp( - qf)] Similarly,
using equation
(16) in equation
exp[ - (S) exp( - qz)]
(20)
(10) we find
(21) where h = (e, +EJc/\~~~(~RT,)“~, the dimensionless collisional linewidth for the lasing transition normalized by the Doppler width. The physical interpretation of equation (21) is that atoms whose dimensionless velocity components in the x-direction lie between h and -h can interact with the lasing radiation, even if the lasing radiation has essentially into equation (15) zero bandwidth. Having calculated Q’pd and Q’p:, these are substituted which can then be solved by standard methods. The full solution for equation (15) is discussed in detail in Ref. (14) and the dominant portion of the solution is given by 4J(71)7 .l““(--“Z)
4 20A4,,[47-~.4~~+0~ +A,,Q$‘/]
e-P cosh(pz)
(22)
= j[471(~~~0+~21)+~2][4~~~0+~1]+t2,Q~o~[4~(~~0+~20)+t~+~2])
P,“(Z ; VZ) =
4J(Z)L”AZ0M
,,0[47tA,1 +A,,Q;‘j]
ee@ cosh(pz)
~~471(~~~+~~,)+~2][471A10+~,]+~A21Q!o~[471(~~O+A20)+~I+~~])
where p = S exp( -r&
(23)
Theory
of an optically
pumped
gas laser
245
Terms of order M,, and MWLhave been dropped, so that these solutions are valid only when I0 is so large that fi and fr are both much larger than their Maxwell-Boltzmann values. In addition, the validity is restricted to the case where (S/t) 5 1 -_IzI << 1, where t E r,[4n(A,,+A,,)+a,+ A21Q\o~] = time of flight/excited lifetime. This means that there are two narrow “boundary layers” adjacent to the transparent walls where equations (22) and (23) are invalid, and the full solutions as given in Ref. (14) must be used. Note also that equations (22) and (23) are the solutions one would obtain by simply neglecting the derivatives in equation (15), and solving the resulting set of simultaneous linear algebraic equations. It is clear then that the particle streaming terms, represented by the derivatives, are important only inside the wall boundary layers, and the thickness of these layers is sO(S/t). For the experimental systems discussed below, S/t < 10-r. The population inversion is found by integration over all velocity space, and we have
where the saturation
parameter
XE
x is defined as
27cA,,LO b
[47+4,0+~2o)+~,
+a21
‘[4~(~20+~2,)+~21[4~~~0+~~1’
The importance of an efficient means of depopulating the terminal lasing level is clearly shown by the fact that the inversion is proportional to the term (47~4,~ + or -47~4,~). ThusA,, >> A,,, if a “bottleneck” in the terminal level is to be avoided. In a real laser, the requirement would be more correctly stated as c, A,, >> A,, where “n” includes all levels between the terminal lasing level and the ground state. The population inversion is also predicted to be inversely proportional to the plate spacing D, which in practice would mean that large diameter optically-pumped lasers would not operate unless 1’ could be increased appropriately. Finally, equation (24) shows the nonlinear effect of the lasing field, which is parameterized by x. Clearly, the maximum inversion occurs for x = 0, and we can write:
(25) The importance of collisional linewidth in a situation involving large values of the lasing radiation can be clearly demonstrated by taking the limit b + 0 in equation (24). Then we find lim(ny)(z+n~l~(z))
b+O
=
1
1671 e- 1~0v~oW7~w)“2 COW) JWC3J(ln S)
JAMESJ. HEALY and T. F. MORSE
246
and equation (26) is identical to equation (25). The physical interpretation of this is that when b = 0, atoms whose x-component of velocity is nonzero will be Doppler-shifted so that they cannot interact with the lasing field, regardless of its intensity, and thus all saturation effects due to the lasing field will be lost in an analysis which fails to consider collisional linewidth. 3. GAIN
COEFFICIENT. AND
INTENSITY POWER
OF LASTING OUTPUT
RADIATION
From equation (14b), we can define the first order gain coefficient lasing frequency as
G”‘K,
1,
2) =
for radiation
at the
Lhv,,B,,.X,,(f:“-f:“) A,,LC3
(f:“-f:“)H[b-(i,,
= 4b~;,(2RT,)“~
-q.l)l.dv,drlz
-I1.I)]H[b+(i,,
(27)
Setting 1 = I, = 1, we now evaluate the gain for radiation propagating in the +x direction. using equations (22) and (23) in equation (27). From symmetry, the gain for radiation propagating in the -x direction must be identical. Also, since the gain must be symmetric with respect to the dimensionless frequency, equation (27) will only be evaluated for V &21 2 0. The details of the evaluation are contained in Ref. (14). and the result is
G’1)((21, z) =
2nexp(-
1)1°A2,[4~(A,,-
A,,)+a,]
~~~~~~~~[~~~~20+~2,~+~,l~~~~,o+~,l J{ERF(i’2,+b)-ERF([2,
where equation
cash(z)
-b)}
(28)
(28) is valid for i2, 2 2b. For 0 < iZ, 5 2b we find
G”‘((,,
271exp(-
. z) = -~
l)l”A2,[4n(A,,
- A,,)+a,]
cash(z)
b~(2~nS)[4~(A2,+A2,)+a2][4~A,,+o,] ERF(l,
I + b)- ERF(b) +
ERF(b) - ERF(i, I -b)\ >. ~~--~~~~~~~~~~~ (I +.x) I
(29)
The physical content of equations (28) and (29) is made more evident by examining two limiting cases. First consider the situation in which the Doppler width of the lasing transition is much larger than the collisional width. In terms of the dimensionless parameters. this means 2b cc 2J(ln 2). By expanding the error functions in this limit, we find that equation (28) becomes
G”‘(i,,
, z) =
8J(x) exp( - l)I’A,, 2J(ln
S)
cash(z) ~‘~
L D
[‘WA,o-A2,)+~,1 ---.exp( ‘[4~(A2o+AAZ1)+~21[4~A,o+a,l
-[z,)
for <2, 2 2h. and b cc 1. (30)
Theory
Similarly,
from equation G’ ’ I((’
of an optically
pumped
247
gas laser
(29) we find 4J(70 exp( - l)Z’A,,
cosh(.z)
21’z)=
for c2, CCb and b cc 1. From equations (30) and (31) the qualitative behavior of the gain as a function of frequency is shown in Fig. 5. The gain is a Gaussian, with an approximately triangular “hole” in the region -2b I (Yzl I 2b. The depth of the hole is determined by the intensity of the lasing radiation Lo, and in the limit Lo + 0, the hole disappears and equation (28) becomes valid for all frequencies, and is then the “small signal” gain of the laser. From the viewpoint of this work, the “hole-burning” phenomenon’7~8~‘3’ can be interpreted as the result of the preferential depopulation by the lasing radiation of atoms with a small x-component of velocity. Those with larger components in the x-direction are adversely Doppler shifted, and do not interact with the lasing radiation. However, large velocities are less probable than small, so that the gain eventually drops, since fewer and fewer atoms are available to participate in stimulated emission. The precise geometry of the “hole” in Fig. 5 will not be accurately given by the present analysis, since it will clearly depend on the detailed collisional line-shape, which has been approximated here by two Heaviside distributions.
FIG. 5. Normalized
2sA,, Lo gain for b = 0.2; x = ___ b
4n(A,,+A,,)+a,
+o,
[~~(Az~+A,,)+~,I[~~A,,+u,I
We now investigate the second limiting case of equation (29) namely 2b >> 2J(ln 2) or collisional line width much larger than the Doppler width. In this case equation (29) becomes 4~ exp( - l)Z’A,, cash(z) L G”‘(i,, , z) = bJ(2lnS) ‘0’ (32) and equation (32) is valid for b >> 1 and 0 I Czl CCb. Now the gain close to line center is seen to be independent of frequency, and the magnitude of the gain is fixed by the value of Lo. From equations (30) and (32) we sketch the behavior of the gain in this limit in Fig. 6.
JAMES J. HEALY
and ‘I. F. MORSE
z
z cO8-
2 _g SF3
mn\
06-
10
2 5:04m, si? =,02-
-8.0 -60
-40
-20
0 2.0 EPI-
40
60
80
Flti.6.
The limitations on x and z for which the above expressions are valid are similar to those on the number-density formulae of the last section. To this point, the parameter Lo has been treated as an undetermined constant, related to the energy stored in the lasing mode. For steady state operation. this quantity can be determined in a self-consistent fashion by applying the condition that single-pass gain must equal the sum of single-pass losses at line center. Let exp(-f‘) be defined as the fraction of the photons which survive one trip along the laser axis in one direction. in the absence of gain. Then the condition for steady state operation with gain may be written as (33)
f‘ = 2G’1’([zI = 0).
We evaluate Lo for z = 0, and assume that the value so found will be a good approximation for all values of z outside the wall “boundary layers”. From equations (29) and (33) at z = 0 and i2, = 0 we find j’
4nexp( - l)I’A,,
ERF(b)
L
1 'm-p, (34) [4~(A,,+A,,)+azl[4~A,+a,l (lf-~o) [47+4,o-~,,)+~,l
Setting Lo = 0 here, we obtain the threshold at which lasing action can commence. Thus
pumping
intensity,
which is the value of I”
,hreaho,d ~ I* _ 1‘J!21nS)[4~(A,o+A,,)+o,l[47cA,o+a,l
(IO)
871exp(-l)A,,[4n(A,,-A,,)+a,l
The threshold pumping requirement is seen to increase with the system losses, the cube of the ratio of lasing to pumping frequencies, and the ratio of the laser “diameter” lI to its length. I* also increases slowly with total number density N, which appears in J(2 In S) and in c1 and rsz. The dependence of I* on the collisional linewidth is interesting. For increases linearly with h. b K 1, I* is independent of b, but as h increases. I* eventually The physical interpretation is as follows ; as b increases from values CC1, with the Doppler width held constant, the number of excited atoms capable of stimulated emission increases,
249
Theory of an optically pumped gas laser
since atoms with a larger range of velocities can interact with the lasing frequency. However, the amount of stimulated emission from a given number of excited atoms which can “see” the lasing frequency decreases with b, since an increase in b is equivalent to weakening the gas-radiation coupling at a given frequency. Thus for small values of b, the two effects cancel, and I* is independent of b. For b >> 1, however, all the excited atoms are capable of interacting with the lasing frequency. If b is then increased, there is nothing to offset the reduction in gas-radiation coupling, and I* must increase. Equation (34) is linear in Lo, and the solution may be written as (36) The self-consistent value of Lo can now be substituted back into the earlier expressions for population inversion and gain, to obtain the steadystate values of these quantities. In this way we incorporate the nonlinear effect of the lasing field in a self-consistent, quantitative manner.” 6,1 ‘) An expression for the power output at each of the partially transmitting mirrors can be obtained by considering the intensity of the lasing radiation at x = f 1, and multiplying by the fractional transmission of the mirrors, denoted by “T”. For simplicity, it has been assumed that the transmission at each of the two mirrors is equal. We then find m
qT OUL -- T
ss dn.
di,,
-cc
Wl v21(2~T!JP2 C2
. ~“a21)w
c
*LJ
(37)
27r
where q,‘,, is the coherent power radiated at each mirror per unit of mirror area, and the superscripts refer to directions of propagation along the negative and positive x-axes respectively. Evaluating the integrals we have
42 =
2b~v~,(2~T,)“2T[4~(A20+A21)+a2][4nAlo+al]
(38)
~21C3[4~(~20+~lo)+~l +a,]
The power output is predicted to increase linearly with the intensity of the pumping radiation I’, as long as we meet the condition that even for weak signal gain, the population of the upper lasing level will be much less than the ground state population. For a given laser, with fixed pumping intensity and Doppler width, there is clearly an optimum value of b which will maximize the power output, and this is found by setting aL’/ab = 0 in equation (36). The optimum value of b is reached when exp(b2). ERF(b) b
I0
2 (39)
= F Jrl
or b(opt) =
16J(n)exp(In
f&In
l)Z”A2,[47r(A,,-A,,)+a,]
S)[4~(~,o+~,,)+~,l[4~~,o+~,l
L D
(40)
Figure 7 describes the variation of bcoptjas (2/Jrc) x (IO/Z*) is varied. This shows that for situations where the laser is very far above threshold, it is advantageous to have a homogeneously broadened lasing transition, but for operation fairly close to threshold, b should
250
JAMES J. HEALY and T. F. MORSE
40-
30-
bow 2 O _
IO-
0
IO’
IO2
FIG. 7. Optimum
103
IO4
value of h vs
IO6
2 I” -,-. I;
\’ 171
not be larger than roughly the Doppler width. This conclusion should not be restricted to optically pumped systems. and should be valid for any gas laser. Examination of equation (36) for IO/I* >> 1 shows that Lo is essentially unaffected by b. provided that h is sufficiently large so that ERF(b) = 1. The important practical conclusion here is that maximum power output for a laser operating far above threshold occurs when homogeneous broadening dominates. The power output can also be maximized with respect to the output coupling “T".This has already been done by RIGROD.“~,” and an essentially identical result can be derived from the present analysis.
4. COMPARISON
OF
THEORY
WITH
EXPERIMENTAL
DATA
The only published information currently available on optically pumped steady-state gas lasers appears to be contained in Refs. (I-3). and all of these refer to the situation in which the lasing gas is monatomic cesium vapor. and the pump is a helium discharge lamp. The helium lamp emits strongly at 0.3888 p. which corresponds closely to the 6S,/, --t 8P, ,z transition in cesium. We first compare the number density predictions, using the experimental data of Ref. (2). The experimental system consisted of a long cylindrical tube of cesium vapor. with transparent windows at each end. The amplifier gain was measured by comparing the intensity ofan input signal to that at the output window. The pertinent data. in the notation of the present work. are as follows: Amplifier length = 2L = 92 cm Internal diameter = 20 = 1 cm Wall temperature = T, = 440°K Total output power = 10-j pW/cm’
Theory
of an optically
pumped
gas laser
251
Doppler width of pump radiation N 0.06 A (Ref. (3b)) Measured gain coefficient N 0.04 Vapor pressure of cesium ‘v 0.025 torr Total number density N 5 x 1014 atoms/cm3 Lasing frequency = vZ1 = 4.3 x 1013 cycle/set (8P,!, -+ 8S,,J Pump frequency = v2,, = 7.5 x 1014 cycle/set (6S,,, -+ 8PIiZ) Einstein coefficient A,, = 4.5 x lo4 set- ‘f12- ’ Einstein coefficient A,, = 6.7 x lo4 set- ‘R- ’ Ref. (15) Einstein coefficient A r0 = C Ai, = 7.7 x 10” set- ‘0-l 1 Measured population of upier lasing level z 10’ atoms/cm3 Measured population of lower lasing level 2 10’ atomsjcm3. The Doppler width of the lasing gas was 0.004 A, and therefore /I N (0.004/0.06) = 0.06 << 1 as assumed above. From the very low value of the output power, it is clear that Lo N 0, and saturation effects can be neglected. The published information is insufficient to determine I’. However, we can use equation (30) and the measured gain to compute the value of I’. This can then be used in the integrated forms of equations (22) and (23) to obtain predicted values of n2 and n,. We then find I0 N 1.1 x 10-s n2c,heoryj2. 1.5 x lo8 atoms/cm3 ; n2,exp,J = 10’ atoms/cm3 nlcexptj 2 10’ atoms/cm3 n 1 (theory) N 1.0 x 10’ atoms/cm3; Reference (3) contains data on two laser oscillators. lasing at different frequencies, under conditions where saturation effects were significant and these can be used to check the predictions of nonlinear gain and power output. The appropriate values for b can be estimated on the assumption that the important broadening mechanisms, other than Doppler broadening, are radiation damping and resonance broadening.‘19-21’ For the the data from Ref. (2) can be summarized laser operating on the 8P1,2 + 8S,,, transition, as: Cavity length = 2L = 92 cm Laser tube i.d. = 20 = 1 cm Wall temperature = T, = 440°K Single pass loss factor = f 21 0.04 Output coupling through mirrors = T = 0.005 Total power output = 50 x lop6 W Cesium vapor pressure N 0.025 torr Pump frequency = vlo = 7.7 x 1Ol4 c/set (8P,,, + 6S,,J Lasing frequency = vZ1 = 4.3 x 1013 c/set (8P,,, + 8S,,J A,, = 4.5 x lo4 see-‘0-l A,, = 6.7 x lo4 set-‘0-l Ref. (15) A,, = 7.7 x lo5 set-‘R-l I Length of pump lamp = 15 cm Diameter of pump length = 0.4 cm Power output from pump lamp at 0.388 p = 8 x 10e3 W/cm2-Q Calculated value of b 2: 13. The value of I0 can be calculated from equation (13). The effective value must be reduced by a factor of about 3 to allow for various experimental imperfections.‘3’ Thus the effective
JAMES J. HEALY and T.
252
F. MORSE
value of I0 is estimated as I0 = 4 x IO->. while the threshold requirement is found from equation (36) as I* r~. 5 x 10mh. The value of Lo is now found from equation (37) as Lo z 3 x 102, and this in turn is used in equation (32) to find G”‘(~,, = 0) 1.9 x IO- ‘. This is to be compared to the experimental value of f/2. or 2 x 10m2. From equation (39) the total power output can be calculated as = 70 pW/cm’. which compares favorably with the experimental value of 64 pW/cm2. Details on the oscillator operating on the (8P, ,* - 6D,,,) transition are found in Ref. (3b). and we have : Cavity length = 2L = 50cm Tube i.d. = 20 = 1 cm Wall temperature N 440°K Single pass loss factor 2 .1‘N 0.18 [estimated value’14’] Mirror output coupling = T = 0.007 Total power output = 51 pW/cm2 Cesium vapor pressure = 0.02 torr Pumping frequency = \‘20 = 7.7 x 10’” c/set (6Srj2 + 8P,,,) Lasing frequency = v2, = 9.7 x 1013 c/set (8P,:, -+ 6D,,,) A 2. = 4.5 x lo4 set- ‘R- ’ A 21 = 5.9 x lo4 set- ‘R-l Ref. (15) = 1.2 x lo6 set-‘R-i A I Pum$ng lamp = same as before. In this case the value of h was estimated at 0.4, so that Doppler effects were significant. Proceeding as before. we calculate I0 = 7 x 1O-i. I* = 3.5 x lop4 and Lo 1 1.3. The predicted value of the saturated gain from equation (29) is G”‘([,, = 0) = 0.1, and the estimated experimental value is 0.09. The predicted power output from equation (39) is 24 pW/cm’, as compared to the experimental value of 50 pW/cm2. In all of the above comparisons, the largest sources of error are likely to be in the calculated values of 1’ and b. In each case. errors of several hundred percent are possible. In addition, the effects of cylindrical rather than plane geometry and curved rather than flat mirrors will introduce smaller errors. Thus the experimental values can only be expected to agree with the theoretical predictions in an order of magnitude sense.
5
DISCUSSION
AND
CONCLUSIONS
The kinetic and radiative equations formulated by Cipolla and Morse to treat the interaction of line radiation with a gas. have been modified to include the effect of natural linewidth. and applied to the problem of a three-level optically pumped laser. Quantitatively useful expressions for population inversion, gain, and power output have been developed, which display the roles of system geometry, radiative parameters and collisional effects in a conceptually simple fashion. It has been shown that when the intensity of stimulated emission is low, the population inversion may be correctly calculated by ignoring all sources of line-broadening, other than the Doppler effect. However, the neglect of collisional linewidth in calculating the small signal gain is correct only in the limit that the Doppler width is much larger than the collisional width. It is never correct to neglect collisional linewidth when a high lasing intensity is present, since the nonlinear saturation
Theory
of an optically
pumped
gas laser
253
process has been shown to be intimately connected with collisional linewidth. Although these conclusions have been arrived at for an optically pumped laser, they should be basically valid for any gas laser. As far as laser geometry is concerned, the most significant result is that gain is proportional to the ratio of the laser length to diameter. The dependence of gain on diameter is similar to the behavior of conventional collisionally-excited gas lasers. in which the gain is normally observed to decay as diameter is increased. The physical explanation is obviously quite different in the case of the optically pumped laser, however. An important consequence is that it rules out the possibility of significantly increasing the output power by enlarging the tube diameter, since, for a given pumping intensity, the gain will decrease as the diameter is increased until the pump is no longer able to maintain the system above threshold. Thus, the only means of achieving higher output power is through the use of longer lasers, more powerful pumping sources, and optimization of the output coupling for a given system. In arriving at this conclusion, the implicit assumption has been made that the conclusions from the analysis of the plane slab are applicable to cylindrical geometry. Also, the discussion earlier regarding the role of spontaneous emission and trapping needs to be kept in mind. If a laser could be made to operate under conditions where the branching ratio were close to unity, then the gain would no longer be inversely proportional to plate spacing. It might be possible to construct a laser where the effect of spontaneous emission would be neither negligible (the case considered here) or dominant. Then the gain might be expected to be proportional to D-” with 0 I c( i 1. In that case, large diameter systems might be of practical interest. With these qualifications in mind, the optically pumped gas laser is not likely to become attractive for technological application, unless much more intense pumping sources become available. In addition, the overall efficiency of the optically pumped system is several orders of magnitude lower than for a comparable collisionally pumped system. The limiting factor on power output is clearly the low-intensity pumping devices currently available, and not the small number density typical of optically pumped systems. Number density will become a limiting factor only when the pump removes an appreciable fraction of the atoms from the ground state, and “bleaching” becomes important. This is clearly not the case for the cesium systems discussed here. The role of collisions has been treated by means of a largely phenomenological model of the collision integral, which recognizes the tendency of inelastic collisions to restore the distribution functions for each level to the MaxwellLBoltzmann value. In particular the model used does not treat the crossrelaxation effect of elastic collisions, which would tend to counteract the hole-burning caused by the coherent radiation. (22) A much more refined quantum-mechanical treatment of the role of collisions, valid for low levels of radiative intensity and Doppler width >> collisional width has been developed by BERMAN and LAMB. Finally the connection between the present work and that of GORDON et~1."~) should be mentioned. The latter paper dealt with a Maser amplifier (as opposed to an oscillator) using a formulation very similar to that developed here. Although a more realistic model of the collisional lineshape was used, the form of the distribution function in velocity space was assumed a priori. The present work shows the essential connection between the distribution function in velocity space, and the behavior of gain in frequency space. Thus a “hole” in a plot of gain vs. frequency implies a “hole” in the distribution function when plotted against velocity. The a priori assumption of a Maxwellian velocity distribution is
254
JAMES J. HEALY and T. F. MORSE
therefore invalid when h CC 1 and Lo >> 1. For b >> 1 or Lo CC I the assumption wellian should not produce significant error.
of a Max-
.‘I C./ttlo~l~kvI,~~~t~IPt2~J A helpful discussion with Professor Y. P. PAO on this problem is gratefully noted The authors are indebted to Professors M. SIRULKIN and H. J. GERRITS~N for reading and improving the manuscript. This work was supported by the Mechanics Division. Office of Aerospace Research, under contract F44620-71. (‘-0030.
REFERENCES
I. t-1.Z CURRINS. I. AH~.LLA. 0. S. HI:AVENS, N. KNA~I.I. and (‘. TOWN% ,~~~.N~Icc\in L)uc~?I/z/~)IE/~,~./rorl~~s. p. 12. Columbia University Press. New York (1961). ’ S. JACOBS. G. Gour.~~ and P. RAI~NOWI~~‘. P~II:v. Rw. Lcrt. II, 415 (1961 ). <: (a) P. RABINOWIIL. S. JACOUS and G. GOUID: App/. Op/ I, 513 (1962); (b) P. RAHINOWI~I and S. JAC OHS. QU~UI~I~ E/cvrronic.c. p. 489. (Edited by GRIV~~ and B[.OMBI:RGEN).Columbia Ilniversity Press f lY64). 4 Prof. W. CHRISTIANSI‘N.University of Washington. Private Communication. 5. J. W. CIPOI.I.A, JR. and T. F. MORSE, PIzj,.\. p/uidy 14. 9 (lY71 ). 6. (a) T. HOI.STHN. Ph,,.c. RN. 72. 1212 (1947): 83, II59 (I951 ): (b) I,. M. BIBERMAN..%I,. Pln,v Jk’7’P 19. 5X4 (lY4Y). 7 W. R. B~NN~I I. JK.. l,cr.scr.v.- .4 C’ol/ec~/r~~,?CJ/ Rcxprirrr.t wit/~ C’r~tnn~m~trr~~ Vol. 10~. (Edited by J. WI I+.K) Gordon and Breach (1968). X. W. E. LAMR. JR., Ph~,v. RN,. 134, Al429 (1964). 9. J. J. H~AI Y and T. 6 MORSE. Phv~ Ret,. .4. 6, 2457 (1977). IO. C. S. WANG &ANG and G. E. U~~I~NI%FC’K.Engineering Rebearch Institute. Univ of Michigan Report (‘M6XI. I I .I. W. CIIYN I A. JR . Ph.D. Thesis. Brown University (1970). I2 C‘ VAU TKK;I. P/I,L\. Rr/rlSl, 97 (1969); Al, 1298 (1970): A4, 1303 (lY7ll. Ii W. R. BENNETT. JR. Pll,s. Rw 126, 580 (1962). 14 J J. HIXLY. Ph.D. Thesis. Brown llniversity (1072). 15. 0. S. HI:AVTNS. J. opf. SM. Am. 51, 1058 ( I96 I ). 16. W W RIGKOD. J. Appl. Phy.c. 34, 2602 ( 1963). 17. W. W. RIGROI). J Appl. P/IK\. 36, 2487 ( 1965) IX. P. DA~IIIOVI-~S, App/. Ph.vs. Lctr. 5, IS (1964). 19 I+. R. GRIMM, Plrrsmrr Sp~~c~rmc~p.,~.Ch. 4. McGraw-Hill (1964). 20. R. G. BKEENE,JR.. Thr Shifr und Shtrpr of Sp~ctrul Lines Ch. 5. Pergamon Press (196 I). 2 I. J. 7‘. JI:I;FRIFS, .Spw!ruI Line Fwmrrinn. Blaisdell (1968). 22. A. Y, CABEZAS and R. P. TREAT. J. Appl. Phy.v. 37,3556 (1966). 23. I’. R. BI:RMAN~~~ W. E. LAMB, JR.. Phys Rrr. A. 2, 2435 (1970);4, 319 (1971). 24 E. I GORDON. A. D. WHITE and J. D. RIGIIFN. ,S~~mpotium on Oprid Mtr.crr.r, Polytech. I n\t of Brook tyn. April lY63.
Quonr.
Speclrosc.
Radiar
THEORY
Transfer.
Vol.
13. pp. 235-254
Pergamon
Press 1973 Pnnled
OF AN OPTICALLY JAMES J. HEALY and
in Great Britain
PUMPED
GAS
LASER
T. F. MORSE
Division of Engineering and Center for Fluid Dynamics, Brown University. Providence, R.I. 02912, U.S.A. (Received 5 Junv 1972)
Abstract-A theoretical description of the steady-state operation of an optically-pumped gas laser is presented. The three-level gas is described by a suitably modified form of the Boltzmann equation. This formulation is intermediate between the rate equation and semi-classical descriptions of laser behavior. Allowing coupling between gas and radiation, solutions are obtained for population inversion, gain coefficient, radiative intensity and output power, in terms of system parameters and spatial variables. The role of Doppler (inhomogeneous) broadening as compared with collisional (homogeneous) broadening is clearly displayed. Furthermore, it has been possible to include non-linear saturation effects due to the lasing radiation in a simple, quantitative fashion. The theoretical predictions are compared with the limited experimental data available, and agreement is good. Although the quantitative results are applicable only to optically pumped systems, the qualitative predictions should be valid for any type of gas laser.
INTRODUCTION DESPITE the great amount of work on gas lasers in recent years, the problem of the opticallypumped gas laser appears so far to have escaped detailed theoretical investigation. This is perhaps because of the considerable experimental difficulties associated with this type of laser, and the relatively low output power developed by the systems hitherto investigated. Because of the desirability of strong coincidence between absorber and pump lines, few systems are capable of efficient optical pumping. Only cesium vapor, pumped by a resonance helium line has been studied in detail. (l-3) Broad band pumping of the CO, system with solar radiation has also been suggested as a possible laser system.‘4’ Certain aspects of the optically pumped system make it attractive from the theoretical viewpoint, however. Since the excitation mechanism is optical rather than collisional, a great simplification in the equations governing the problem is effected, as compared with collisionally pumped gas lasers. In addition, the low operating pressure renders the dynamics of the gas tractable from the viewpoint of nearly-free-molecule kinetic theory. Thus we are led to consider formulation of a Boltzmann equation for a gas possessing internal energy states, incorporating appropriate radiative source and sink terms. (5) In addition, we can treat the radiation at the pumping and lasing frequencies by means of the equation of radiative transfer, where the absorption and emission coefficients are dependent on the gas particle distribution functions. We then find a set of coupled kinetic and radiative equations which, in principle, can be solved to display population number densities, gain coefficient, and radiative intensity, as functions of the spatial variables and the system parameters. This approach has several advantages over more conventional treatments employing rate equations, among which are : 235
JAMES J. HEAL-Y and T. t:. MORSF
‘76
(1 I The random motion of the gas particles is specifically accounted for. This is an important consideration in radiative transfer problems. since the macroscopic emission and absorption profiles are intimately related to the gas velocity distribution functions. In particular, one would not expect that the type of formulation developed by BIRERMAN and HOLSTEIN(~) would be adequate for the present problem, since the key assumption there is that all velocity distributions are Maxwellian. This would not be true in a laser, where the intense, unidirectional. monochromatic lasing beam will severely distort the velocity distributions of the relevant excited atomic levels. Thus. no a priori restrictions on the excited-level distribution functions should be made. and the structure of the distribution functions must be part of the solution of the problem. (2) The interaction of the pumping process with the depopulation due to lasing action is clearly displayed, whereas normally the pumping process must be treated as a parameter of the system. (3) The effect of spatial variations and the system geometry can be explicitly treated. (4) The quenching effect of wall and interparticle collisions can be studied. (5) Although optically pumped systems were chosen as models for this theory. the theory itself is capable of treating the more complex situations of inhomogeneous broadening in collisionally pumped gases. The disadvantage of this approach. at least in its present form, as compared with more complete quantum mechanical approaches, (‘w is that by treating the radiation by means of the equation of radiative transfer, all information regarding photon coherence properties is lost. Thus, we can hope to obtain detailed predictions for the population inversion. gain coetficient. radiative intensity and output power, but no information connected with coherence phenomena can be extracted. The physical problem considered in the remainder of this work consists of a box of low-pressure monatomic gas, which is being irradiated at one of the gas resonance frequencies. See Fig. 1. Two of the walls of the box are plane, partially transmitting mirrors, set at an appropriate spacing so that they form a Fabry-P&rot cavity, which is resonant at the gas lasing frequency. Thus, the population of excited gas atoms at steady state represents a balance between the rate of excitation due to the pumping radiation, and deexcitation due to spontaneous and stimulated emission, and collisions. Similarly. the intensity of the lasing radiation represents a balance between losses in the system, and stimulated emission by the gas. Throughout this work. the emphasis will be on obtaining manageable approximations to the solutions of the complex governing equations, which provide insight into the operation of the laser, and make quantitatively useful predictions of the relevant parameters.
I. THE
ATOMIC
MODEL
AND
GOVERNING
EQUATIONS
Consider a FabryyPkrot cavity formed by two plane mirrors of depth 20 and length 2L, which are infinite in the y-direction, as shown schematically in Fig. 1. The lasing gas is contained between the two mirrors by means of transparent walls. and pumping radiation at approximately the resonance frequency I’~~ is supplied to the laser through these walls. between the semi-transparent mirrors. The lasing radiation at frequency v z 1 propagates and the cavity is assumed to be tuned so that the spacing 2L corresponds exactly to a
231
Theory of an optically pumped gas laser PUMPING
PUMPING
RADIATION
RADIATION
AT
V20
AT vzo
FIG. 1. Geometry of laser.
resonance condition at vZ1. It is assumed that only a single cavity mode is excited above threshold. The latter two restrictions will be removed in a subsequent paper.“’ In the atomic model adopted here, a ground level and two excited levels are present, and for simplicity, all degeneracies are assumed to be unity. All of the levels are allowed to have a collisional half width, sO, sr, and e2. For purposes of this work, these widths correspond to the homogeneous broadening of the energy levels by any process other than Doppler shifting. For an optically pumped laser, it is required that the atomic transition very closely with the frequency of the pumping radiation exfrequency v10 correspond ternally supplied, and, in addition, the atomic frequency vZ1 must. correspond to the resonance frequency of the optical cavity formed by the two mirrors. Figure 2 summarizes the details of the atomic model. The general problem of the interaction between a gas and line radiation has been treated in detail through the use of kinetic theory and the equation of radiative transfer derived by these authors to the by CIPOLLA and MORSE.(‘) In applying the equations optically pumped gas laser oscillator, only one modification is required; the collisional broadening, assumed zero in Ref. (5) must be explicitly taken into account. It will be shown later that if the collisional linewidth of the lasing transition were zero, then lasing action would be virtually impossible. The relationships among the various linewidths pertinent
COLLISIONAL
BROADENING 2nd
2E2
Is.1
EXCITED
EXCITED
LEVEL
LEVEL
/ PUMPING TRANSll Y 10
-&A
FIG.
TERMINAL DEPOPULATING TRANSITION
GROUND LEVEL
2. Description of atomic energy-level model.
JAMES J. HEALY and T. F. MORSE
238
to the present problem may be summarized as follows: bandwidth of pumping radiation >> Doppler width >> collisional width at vZO; collisional width of lasing transition >> bandwidth of lasing radiation. The collisional linewidth of the lasing transition may in general be less than, equal to, or larger than the Doppler width, and all of these possibilities will be treated. As discussed in Ref. (5) the governing kinetic equations can now be written as
The velocity
in the y-direction
has been integrated
where n,(x, z) is the number-density per unit volume nondimensionalizations have been used in equations x=F
for
-l
z=i
for
-1lxll
L T
-
l - (2RT,)“2;
out, and the normalization
of atoms (1) :
on .fi is
in level “i”. The following
D
(2)
T2 = (2RT,)“’
Here 5 is the dimensional atomic velocity vector as measured in the laboratory reference frame, and 7” is the temperature of the laser walls, assumed uniform. z, and z2 are essentially the mean times of flight for an atom to traverse the laser in the x and z directions. The quantities Ji denote the source and sink terms for atoms in level “i” due to collisions,““’ and the A,, denote the Einstein coefficients for spontaneous emission. Thus the terms .4,,Q,,f, give the rate of stimulated emission by atoms in level “m”, per unit of fourdimensional phase space. For a system of atoms interacting with line radiation, the steady-state equation of radiative transfer may be written asC5’
1.2=c c hvB,,[.~~,(f,)+.f,.;//,,(f, -1,)l n “
m
Theory of an optically pumped gas laser
239
where f, is the photon distribution function.‘5’ The quantity &,,(f,) in equation (3) is defined as the total number of atoms per unit volume in level “m” which are capable of emitting a photon of frequency v,, into the element of solid angle dQ about 1.Since we wish to retain the effect of broadened energy levels in the present problem, the assumption is now made that the collisional lineshape q,,(p) shown in Fig. 3a, can be approximated as in Fig. 3b. Physically, this amounts to saying that all frequencies “p” which fall within the cross-hatched region of Fig. 2, will be absorbed with equal probability, but frequencies outside these limits will be unable to interact with the atom. Mathematically stated, this becomes q
(p)
=
~[~v,,+&"+&,}-~l.~[~-{v",-&"--E,}l
nm
2(&n + &J
where H(x) denotes the Heaviside unit step function, and equation (4) satisfies the normalization
I
qmh4dp = 1.
Y nm P’ FIG. 3a. True collisional lineshape.
u nm FIG. 3b. Modelled collisional lineshape.
(5)
JAMES J. HEALY and
‘30
T. I-. MORSE
Now, a photon of frequency p in the atomic reference frame will appear to have a frequency 18in the laboratory rest frame, where v is given by the Doppler formula as C’= speed of light
since ~1
“II”, Substituting
from equation
(6) into equation
(51, we find
where dimensionless ratio of collisional linewidth to Doppler width for 17+ 111transition dimensionless The total number which are capable
frequency
difference.
of atoms in level “II”, per unit volume, per unit solid angle about 1, of absorbing a photon in the line centered on \I,,,~is then given by”’
We now turn to the development of expressions for the terms Q,,,,, which were introduced in the kinetic equations (1). The rate of stimulated emission from level “nr” to “II”. per unit solid angle, per unit of phase space volume, is given by the product of the Einstein coefficient for stimulated emission, multiplied by the number of photons per unit volume in the element d0 about 1, multiplied by the number of particles per unit of phase space volume capable of emitting a photon of frequency I’~,,,.Thus, the substantial derivative of ,Jm(q) is given by
= The quantity
Q,,(f,)
is now defined
-7
Thus Q,,,, gives the number capable of being absorbed
A,nJ;,,(~)Q,,,,( f;). by
equations
(9)
(7) and (9) as
in of photons per unit volume in the n-m resonance by a particle in a given region of phase space.
line which arc
Theory
of an optically
pumped
241
gas laser
To complete the mathematical formulation of the problem, boundary conditions need to be specified on equations (1) and (3). For the kinetic equations, the simple assumption will be made that particles leaving any solid surface after a collision will have a MaxwellBoltzmann velocity distribution : ,f;(x,r~,E,) = N,Q-‘(2nRT,))‘exp-
{ $+vI’+--~ -x z ;j
This boundary condition is motivated by the fact that interactions _ generally tend to destroy an inversion. The boundary condition on radiation centered at v2,, is : j;,,,(z = *l) I0 is a dimensionless
measure
= :a(1 Jr
of the pumping
FI=)exp(-/PC:, intensity
(11) with a boundary
1
at z = f 1. It can be shown”
will
(14 ” that (13)
where qi, is the flux of pumping energy supplied at either transparent wall, and fiml is defined as the bandwidth of the pumping radiation, divided by the Doppler width of the lasing gas. For radiation at the lasing frequency v2 1, the condition for stable operation is that the single-pass amplification at line center must balance the system losses.“3’ In addition, it is reasonable to assume that the lasing frequency must be confined to propagation directions very close to I, = k 1; otherwise the photon will be reflected out of the cavity, and will not participate in the feedback process. No boundary condition is available at x = + 1 for either the amplitude or spectral distribution of the lasing radiation. This difficulty will be circumvented later by using the fact that the laser bandwidth is many orders of magnitude smaller than all other bandwidths in the problem, and the amplitude will be fixed by means of a self-consistency requirement. The above discussion leads to an important simplification of the equation of radiative transfer (3). We are dealing here with two spectrally disjoint radiation packets, which are propagating in different directions. See Fig. 4. The absorption of radiation at the pumping frequency will be controlled by (f, -_&) ‘v fo, since fi will always be several orders of
FIG. 4. Qualitative
dependence
off, on frequency.
JAMES J. HEALY and
242
T. F. MORSE
magnitude smaller than .f, provided hv,, >> kT,. Similarly, radiation at v2i will interact only with (f2--fi), SO that for all practical purposes, the two frequencies interact with separate atomic populations. Thus the two frequencies may be uncoupled and equation (3) may be replaced by the two simpler equations +
z
=
(14a) Dh~,20BZ0[.~20(.1;)_~PLD.~~O(.fO_.f2)l
=
Lhv,,B,,[,~~,(J;)+f;,L,X,,(f;-f;)l. (14b)
and +
The set of kinetic equations (1) can also be simplified considerably. Since the laser cavity is not resonant at via, the terminal depopulating transition, we can assume that stimulated emission at this frequency is negligible. Thus Qro = 0 in equation (1). A further simplification is motivated by the fact that the ratio D/L as defined in Fig. 1 is normally small, values of lo-’ being typical for gas lasers. It is intuitively clear that as D/L 4 0, the derivative with respect to x in the kinetic equations may be neglected, when x is not close to the mirrors. Thus the solution to the kinetic equations (1) in a cavity where D/L cc I can be found to a high degree of accuracy over almost the entire length of the laser by retaining only the derivative with respect to z.(i4) The role of collisions is expected to be of secondary importance in an optically pumped nonequilibrium system, since an appreciable number of collisions would drive the system toward equilibrium. However, to retain the physical effect of collisions, we adopt the following simplified form of a relaxation model. J, 1 (M,, -.f&ri. Here gi is an inelastic
collision
frequency,
gi = C
defined as(lo,l I)
fj,ll-‘lslI~~(l’l-~sl,~)dodrls.
jkl ss
The simplified
“-2 z2
v* af1
Ty’ay
kinetic
equations
now read
2: (M,o-f,)ao+4~(A,of,+A,of,)-A20Q20.1b
N (M,1-f,)o,+4~(A2,f2-A,~f,)+A2,Q21(f2-f,)
(15)
The sets of equations (15) and (14), together with the definitions (8) and (10) and the boundary conditions now comprise a set of five coupled linear integro-differential equations of for the five unknowns fo, fi, f2, f,,, and f,,, , and we now turn to the construction approximate solutions for these quantities.
243
Theory of an optically pumped gas laser
2. ITERATIVE
SOLUTION
TO
AND POPULATION
THE
GOVERNING
NUMBER
EQUATIONS
DENSITIES
The analytic solution of the coupled radiative and kinetic equations, even with the simplifications so far discussed, is prohibitively difficult. However, CWOLLA and MORSE”) have shown that an iterative approach to this type of problem can be expected to converge, and to yield physically satisfactory results after one iteration. The iterative strategy to be adopted here is as follows; we first obtain a solution to the radiative equation (14a), on the assumption that the kinetic distribution functions are Maxwell-Boltzmann everywhere. The solution for f,,, so obtained should be valid when the gas is optically excited, since even then, there will be only a very small percentage change in fo, provided I0 is not too large. The solution for f,,, can then be substituted into equation (10) to obtain Q$?, where the superscript (0) denotes a zeroth-order quantity. To generate a zeroth-order expression for Q2i, the assumption is made that we can write
(16) Thus, it is assumed that the lasing radiation is a delta-function in frequency and direction. Lo is an undetermined constant, which is related to the quantity of energy stored in the lasing mode. Now, with f. given by its Maxwell-Boltzmann value, one finds from equations (7) and (8) cc
co
%:“d(f,) = N
dqXexP(-“) s
Jn
-m
j. dr],
_m
H[a-(i,o-rl,)lH[a+(i~~-vl,)l 260+ 4 .
exp(-13
Jn
(17) where no N N has been used, and a = (a0 +EJc/v~~(~RT,,,)~‘~. The parameter “a” will normally be much less than unity, and for the cesium system to be discussed later, a - O(10m3). For a -+ 0, equation (17) becomes
~%fo) = Now, using equation
(18) in equation
af1’5:
*aZ=with the boundary
condition
JRv20&y2 .exp( - Go),
(18)
(14a) we find :
~hv20B20Nc exp(-ii,) &cv,o(2RTW)“2
(12) this has the solution
: (19)
The + superscripts refer to radiation propagating The quantity S in equation (19) is defined as
in the + z and -z directions
Nt2c3 A,, s = 2J(7c)v:o .
respectively.
JAMES J. HEALY and T. I-. MOKSI
243
The dimensionless photon mean free path, for photons in the &2 resonance line can be defined”“’ as /I,, = D exp(ci,)/S. which merits discussion. The contribution Equation (19) for .f;,,, reflects an assumption to ,f;.,, from spontaneous emission by the lasing gas has been neglected. The results of Van Trigt,“” which refer to a strictly two-level gas, optically pumped by resonance radiation. show that the spontaneous emission contribution is always important in an optically thick gas. However, these results assume that there are no intermediate levels in the gas between the excited level and the ground level. Thus spontaneous emission is always in the same If, however, the gas has a frequency band as the pumping radiation, and is “trapped”. large number of intermediate levels between the excited level and the ground level. then the spontaneous emission will be at a number of frequencies, only one of which will be trapped. Thus the contribution to .fl,,) by spontaneous emission in the gas is controlled by the branching ratio for level “2”. If the branching ratio is such that (Azo/xnX2 A,,) - 1.0. then the spontaneous emission term will be important. If on the other hand (A2O/x,1 ~ L .A2,,) CC 1.0 then one can conclude that most of the spontaneous emission is not at the trapped frequency \n2(), and the contribution to ,fiJO is negligible. The latter situation is the relevant case for a laser. This can be seen by considering the terminal lasing level (level “I” in Fig. 2). If there are not a number of levels between the terminal level and the ground state, then spontaneous emission from “1” will also be trapped, and the effective value of Al,, will be reduced to _ A 1,/SJ(ln S).‘6’ Th us it is expedient to provide “escape routes” for spontaneous emission in the form of intermediate levels, if a bottleneck in the terminal level is to be avoided. We conclude then that the effect of spontaneous emission at \I~,~ will normally be of secondary importance in an optically pumped laser, and the solution as given by equation (19) should be valid. into the expression defining Q2,,. and one The solution for ,f,,,, is now substituted finds, for n + 0,““’ Q’,“d= 4J(7r)Z” cosh[(Sz) exp( - qf)] Similarly,
using equation
(16) in equation
exp[ - (S) exp( - qz)]
(20)
(10) we find
(21) where h = (e, +EJc/\~~~(~RT,)“~, the dimensionless collisional linewidth for the lasing transition normalized by the Doppler width. The physical interpretation of equation (21) is that atoms whose dimensionless velocity components in the x-direction lie between h and -h can interact with the lasing radiation, even if the lasing radiation has essentially into equation (15) zero bandwidth. Having calculated Q’pd and Q’p:, these are substituted which can then be solved by standard methods. The full solution for equation (15) is discussed in detail in Ref. (14) and the dominant portion of the solution is given by 4J(71)7 .l““(--“Z)
4 20A4,,[47-~.4~~+0~ +A,,Q$‘/]
e-P cosh(pz)
(22)
= j[471(~~~0+~21)+~2][4~~~0+~1]+t2,Q~o~[4~(~~0+~20)+t~+~2])
P,“(Z ; VZ) =
4J(Z)L”AZ0M
,,0[47tA,1 +A,,Q;‘j]
ee@ cosh(pz)
~~471(~~~+~~,)+~2][471A10+~,]+~A21Q!o~[471(~~O+A20)+~I+~~])
where p = S exp( -r&
(23)
Theory
of an optically
pumped
gas laser
245
Terms of order M,, and MWLhave been dropped, so that these solutions are valid only when I0 is so large that fi and fr are both much larger than their Maxwell-Boltzmann values. In addition, the validity is restricted to the case where (S/t) 5 1 -_IzI << 1, where t E r,[4n(A,,+A,,)+a,+ A21Q\o~] = time of flight/excited lifetime. This means that there are two narrow “boundary layers” adjacent to the transparent walls where equations (22) and (23) are invalid, and the full solutions as given in Ref. (14) must be used. Note also that equations (22) and (23) are the solutions one would obtain by simply neglecting the derivatives in equation (15), and solving the resulting set of simultaneous linear algebraic equations. It is clear then that the particle streaming terms, represented by the derivatives, are important only inside the wall boundary layers, and the thickness of these layers is sO(S/t). For the experimental systems discussed below, S/t < 10-r. The population inversion is found by integration over all velocity space, and we have
where the saturation
parameter
XE
x is defined as
27cA,,LO b
[47+4,0+~2o)+~,
+a21
‘[4~(~20+~2,)+~21[4~~~0+~~1’
The importance of an efficient means of depopulating the terminal lasing level is clearly shown by the fact that the inversion is proportional to the term (47~4,~ + or -47~4,~). ThusA,, >> A,,, if a “bottleneck” in the terminal level is to be avoided. In a real laser, the requirement would be more correctly stated as c, A,, >> A,, where “n” includes all levels between the terminal lasing level and the ground state. The population inversion is also predicted to be inversely proportional to the plate spacing D, which in practice would mean that large diameter optically-pumped lasers would not operate unless 1’ could be increased appropriately. Finally, equation (24) shows the nonlinear effect of the lasing field, which is parameterized by x. Clearly, the maximum inversion occurs for x = 0, and we can write:
(25) The importance of collisional linewidth in a situation involving large values of the lasing radiation can be clearly demonstrated by taking the limit b + 0 in equation (24). Then we find lim(ny)(z+n~l~(z))
b+O
=
1
1671 e- 1~0v~oW7~w)“2 COW) JWC3J(ln S)
JAMESJ. HEALY and T. F. MORSE
246
and equation (26) is identical to equation (25). The physical interpretation of this is that when b = 0, atoms whose x-component of velocity is nonzero will be Doppler-shifted so that they cannot interact with the lasing field, regardless of its intensity, and thus all saturation effects due to the lasing field will be lost in an analysis which fails to consider collisional linewidth. 3. GAIN
COEFFICIENT. AND
INTENSITY POWER
OF LASTING OUTPUT
RADIATION
From equation (14b), we can define the first order gain coefficient lasing frequency as
G”‘K,
1,
2) =
for radiation
at the
Lhv,,B,,.X,,(f:“-f:“) A,,LC3
(f:“-f:“)H[b-(i,,
= 4b~;,(2RT,)“~
-q.l)l.dv,drlz
-I1.I)]H[b+(i,,
(27)
Setting 1 = I, = 1, we now evaluate the gain for radiation propagating in the +x direction. using equations (22) and (23) in equation (27). From symmetry, the gain for radiation propagating in the -x direction must be identical. Also, since the gain must be symmetric with respect to the dimensionless frequency, equation (27) will only be evaluated for V &21 2 0. The details of the evaluation are contained in Ref. (14). and the result is
G’1)((21, z) =
2nexp(-
1)1°A2,[4~(A,,-
A,,)+a,]
~~~~~~~~[~~~~20+~2,~+~,l~~~~,o+~,l J{ERF(i’2,+b)-ERF([2,
where equation
cash(z)
-b)}
(28)
(28) is valid for i2, 2 2b. For 0 < iZ, 5 2b we find
G”‘((,,
271exp(-
. z) = -~
l)l”A2,[4n(A,,
- A,,)+a,]
cash(z)
b~(2~nS)[4~(A2,+A2,)+a2][4~A,,+o,] ERF(l,
I + b)- ERF(b) +
ERF(b) - ERF(i, I -b)\ >. ~~--~~~~~~~~~~~ (I +.x) I
(29)
The physical content of equations (28) and (29) is made more evident by examining two limiting cases. First consider the situation in which the Doppler width of the lasing transition is much larger than the collisional width. In terms of the dimensionless parameters. this means 2b cc 2J(ln 2). By expanding the error functions in this limit, we find that equation (28) becomes
G”‘(i,,
, z) =
8J(x) exp( - l)I’A,, 2J(ln
S)
cash(z) ~‘~
L D
[‘WA,o-A2,)+~,1 ---.exp( ‘[4~(A2o+AAZ1)+~21[4~A,o+a,l
-[z,)
for <2, 2 2h. and b cc 1. (30)
Theory
Similarly,
from equation G’ ’ I((’
of an optically
pumped
247
gas laser
(29) we find 4J(70 exp( - l)Z’A,,
cosh(.z)
21’z)=
for c2, CCb and b cc 1. From equations (30) and (31) the qualitative behavior of the gain as a function of frequency is shown in Fig. 5. The gain is a Gaussian, with an approximately triangular “hole” in the region -2b I (Yzl I 2b. The depth of the hole is determined by the intensity of the lasing radiation Lo, and in the limit Lo + 0, the hole disappears and equation (28) becomes valid for all frequencies, and is then the “small signal” gain of the laser. From the viewpoint of this work, the “hole-burning” phenomenon’7~8~‘3’ can be interpreted as the result of the preferential depopulation by the lasing radiation of atoms with a small x-component of velocity. Those with larger components in the x-direction are adversely Doppler shifted, and do not interact with the lasing radiation. However, large velocities are less probable than small, so that the gain eventually drops, since fewer and fewer atoms are available to participate in stimulated emission. The precise geometry of the “hole” in Fig. 5 will not be accurately given by the present analysis, since it will clearly depend on the detailed collisional line-shape, which has been approximated here by two Heaviside distributions.
FIG. 5. Normalized
2sA,, Lo gain for b = 0.2; x = ___ b
4n(A,,+A,,)+a,
+o,
[~~(Az~+A,,)+~,I[~~A,,+u,I
We now investigate the second limiting case of equation (29) namely 2b >> 2J(ln 2) or collisional line width much larger than the Doppler width. In this case equation (29) becomes 4~ exp( - l)Z’A,, cash(z) L G”‘(i,, , z) = bJ(2lnS) ‘0’ (32) and equation (32) is valid for b >> 1 and 0 I Czl CCb. Now the gain close to line center is seen to be independent of frequency, and the magnitude of the gain is fixed by the value of Lo. From equations (30) and (32) we sketch the behavior of the gain in this limit in Fig. 6.
JAMES J. HEALY
and ‘I. F. MORSE
z
z cO8-
2 _g SF3
mn\
06-
10
2 5:04m, si? =,02-
-8.0 -60
-40
-20
0 2.0 EPI-
40
60
80
Flti.6.
The limitations on x and z for which the above expressions are valid are similar to those on the number-density formulae of the last section. To this point, the parameter Lo has been treated as an undetermined constant, related to the energy stored in the lasing mode. For steady state operation. this quantity can be determined in a self-consistent fashion by applying the condition that single-pass gain must equal the sum of single-pass losses at line center. Let exp(-f‘) be defined as the fraction of the photons which survive one trip along the laser axis in one direction. in the absence of gain. Then the condition for steady state operation with gain may be written as (33)
f‘ = 2G’1’([zI = 0).
We evaluate Lo for z = 0, and assume that the value so found will be a good approximation for all values of z outside the wall “boundary layers”. From equations (29) and (33) at z = 0 and i2, = 0 we find j’
4nexp( - l)I’A,,
ERF(b)
L
1 'm-p, (34) [4~(A,,+A,,)+azl[4~A,+a,l (lf-~o) [47+4,o-~,,)+~,l
Setting Lo = 0 here, we obtain the threshold at which lasing action can commence. Thus
pumping
intensity,
which is the value of I”
,hreaho,d ~ I* _ 1‘J!21nS)[4~(A,o+A,,)+o,l[47cA,o+a,l
(IO)
871exp(-l)A,,[4n(A,,-A,,)+a,l
The threshold pumping requirement is seen to increase with the system losses, the cube of the ratio of lasing to pumping frequencies, and the ratio of the laser “diameter” lI to its length. I* also increases slowly with total number density N, which appears in J(2 In S) and in c1 and rsz. The dependence of I* on the collisional linewidth is interesting. For increases linearly with h. b K 1, I* is independent of b, but as h increases. I* eventually The physical interpretation is as follows ; as b increases from values CC1, with the Doppler width held constant, the number of excited atoms capable of stimulated emission increases,
249
Theory of an optically pumped gas laser
since atoms with a larger range of velocities can interact with the lasing frequency. However, the amount of stimulated emission from a given number of excited atoms which can “see” the lasing frequency decreases with b, since an increase in b is equivalent to weakening the gas-radiation coupling at a given frequency. Thus for small values of b, the two effects cancel, and I* is independent of b. For b >> 1, however, all the excited atoms are capable of interacting with the lasing frequency. If b is then increased, there is nothing to offset the reduction in gas-radiation coupling, and I* must increase. Equation (34) is linear in Lo, and the solution may be written as (36) The self-consistent value of Lo can now be substituted back into the earlier expressions for population inversion and gain, to obtain the steadystate values of these quantities. In this way we incorporate the nonlinear effect of the lasing field in a self-consistent, quantitative manner.” 6,1 ‘) An expression for the power output at each of the partially transmitting mirrors can be obtained by considering the intensity of the lasing radiation at x = f 1, and multiplying by the fractional transmission of the mirrors, denoted by “T”. For simplicity, it has been assumed that the transmission at each of the two mirrors is equal. We then find m
qT OUL -- T
ss dn.
di,,
-cc
Wl v21(2~T!JP2 C2
. ~“a21)w
c
*LJ
(37)
27r
where q,‘,, is the coherent power radiated at each mirror per unit of mirror area, and the superscripts refer to directions of propagation along the negative and positive x-axes respectively. Evaluating the integrals we have
42 =
2b~v~,(2~T,)“2T[4~(A20+A21)+a2][4nAlo+al]
(38)
~21C3[4~(~20+~lo)+~l +a,]
The power output is predicted to increase linearly with the intensity of the pumping radiation I’, as long as we meet the condition that even for weak signal gain, the population of the upper lasing level will be much less than the ground state population. For a given laser, with fixed pumping intensity and Doppler width, there is clearly an optimum value of b which will maximize the power output, and this is found by setting aL’/ab = 0 in equation (36). The optimum value of b is reached when exp(b2). ERF(b) b
I0
2 (39)
= F Jrl
or b(opt) =
16J(n)exp(In
f&In
l)Z”A2,[47r(A,,-A,,)+a,]
S)[4~(~,o+~,,)+~,l[4~~,o+~,l
L D
(40)
Figure 7 describes the variation of bcoptjas (2/Jrc) x (IO/Z*) is varied. This shows that for situations where the laser is very far above threshold, it is advantageous to have a homogeneously broadened lasing transition, but for operation fairly close to threshold, b should
250
JAMES J. HEALY and T. F. MORSE
40-
30-
bow 2 O _
IO-
0
IO’
IO2
FIG. 7. Optimum
103
IO4
value of h vs
IO6
2 I” -,-. I;
\’ 171
not be larger than roughly the Doppler width. This conclusion should not be restricted to optically pumped systems. and should be valid for any gas laser. Examination of equation (36) for IO/I* >> 1 shows that Lo is essentially unaffected by b. provided that h is sufficiently large so that ERF(b) = 1. The important practical conclusion here is that maximum power output for a laser operating far above threshold occurs when homogeneous broadening dominates. The power output can also be maximized with respect to the output coupling “T".This has already been done by RIGROD.“~,” and an essentially identical result can be derived from the present analysis.
4. COMPARISON
OF
THEORY
WITH
EXPERIMENTAL
DATA
The only published information currently available on optically pumped steady-state gas lasers appears to be contained in Refs. (I-3). and all of these refer to the situation in which the lasing gas is monatomic cesium vapor. and the pump is a helium discharge lamp. The helium lamp emits strongly at 0.3888 p. which corresponds closely to the 6S,/, --t 8P, ,z transition in cesium. We first compare the number density predictions, using the experimental data of Ref. (2). The experimental system consisted of a long cylindrical tube of cesium vapor. with transparent windows at each end. The amplifier gain was measured by comparing the intensity ofan input signal to that at the output window. The pertinent data. in the notation of the present work. are as follows: Amplifier length = 2L = 92 cm Internal diameter = 20 = 1 cm Wall temperature = T, = 440°K Total output power = 10-j pW/cm’
Theory
of an optically
pumped
gas laser
251
Doppler width of pump radiation N 0.06 A (Ref. (3b)) Measured gain coefficient N 0.04 Vapor pressure of cesium ‘v 0.025 torr Total number density N 5 x 1014 atoms/cm3 Lasing frequency = vZ1 = 4.3 x 1013 cycle/set (8P,!, -+ 8S,,J Pump frequency = v2,, = 7.5 x 1014 cycle/set (6S,,, -+ 8PIiZ) Einstein coefficient A,, = 4.5 x lo4 set- ‘f12- ’ Einstein coefficient A,, = 6.7 x lo4 set- ‘R- ’ Ref. (15) Einstein coefficient A r0 = C Ai, = 7.7 x 10” set- ‘0-l 1 Measured population of upier lasing level z 10’ atoms/cm3 Measured population of lower lasing level 2 10’ atomsjcm3. The Doppler width of the lasing gas was 0.004 A, and therefore /I N (0.004/0.06) = 0.06 << 1 as assumed above. From the very low value of the output power, it is clear that Lo N 0, and saturation effects can be neglected. The published information is insufficient to determine I’. However, we can use equation (30) and the measured gain to compute the value of I’. This can then be used in the integrated forms of equations (22) and (23) to obtain predicted values of n2 and n,. We then find I0 N 1.1 x 10-s n2c,heoryj2. 1.5 x lo8 atoms/cm3 ; n2,exp,J = 10’ atoms/cm3 nlcexptj 2 10’ atoms/cm3 n 1 (theory) N 1.0 x 10’ atoms/cm3; Reference (3) contains data on two laser oscillators. lasing at different frequencies, under conditions where saturation effects were significant and these can be used to check the predictions of nonlinear gain and power output. The appropriate values for b can be estimated on the assumption that the important broadening mechanisms, other than Doppler broadening, are radiation damping and resonance broadening.‘19-21’ For the the data from Ref. (2) can be summarized laser operating on the 8P1,2 + 8S,,, transition, as: Cavity length = 2L = 92 cm Laser tube i.d. = 20 = 1 cm Wall temperature = T, = 440°K Single pass loss factor = f 21 0.04 Output coupling through mirrors = T = 0.005 Total power output = 50 x lop6 W Cesium vapor pressure N 0.025 torr Pump frequency = vlo = 7.7 x 1Ol4 c/set (8P,,, + 6S,,J Lasing frequency = vZ1 = 4.3 x 1013 c/set (8P,,, + 8S,,J A,, = 4.5 x lo4 see-‘0-l A,, = 6.7 x lo4 set-‘0-l Ref. (15) A,, = 7.7 x lo5 set-‘R-l I Length of pump lamp = 15 cm Diameter of pump length = 0.4 cm Power output from pump lamp at 0.388 p = 8 x 10e3 W/cm2-Q Calculated value of b 2: 13. The value of I0 can be calculated from equation (13). The effective value must be reduced by a factor of about 3 to allow for various experimental imperfections.‘3’ Thus the effective
JAMES J. HEALY and T.
252
F. MORSE
value of I0 is estimated as I0 = 4 x IO->. while the threshold requirement is found from equation (36) as I* r~. 5 x 10mh. The value of Lo is now found from equation (37) as Lo z 3 x 102, and this in turn is used in equation (32) to find G”‘(~,, = 0) 1.9 x IO- ‘. This is to be compared to the experimental value of f/2. or 2 x 10m2. From equation (39) the total power output can be calculated as = 70 pW/cm’. which compares favorably with the experimental value of 64 pW/cm2. Details on the oscillator operating on the (8P, ,* - 6D,,,) transition are found in Ref. (3b). and we have : Cavity length = 2L = 50cm Tube i.d. = 20 = 1 cm Wall temperature N 440°K Single pass loss factor 2 .1‘N 0.18 [estimated value’14’] Mirror output coupling = T = 0.007 Total power output = 51 pW/cm2 Cesium vapor pressure = 0.02 torr Pumping frequency = \‘20 = 7.7 x 10’” c/set (6Srj2 + 8P,,,) Lasing frequency = v2, = 9.7 x 1013 c/set (8P,:, -+ 6D,,,) A 2. = 4.5 x lo4 set- ‘R- ’ A 21 = 5.9 x lo4 set- ‘R-l Ref. (15) = 1.2 x lo6 set-‘R-i A I Pum$ng lamp = same as before. In this case the value of h was estimated at 0.4, so that Doppler effects were significant. Proceeding as before. we calculate I0 = 7 x 1O-i. I* = 3.5 x lop4 and Lo 1 1.3. The predicted value of the saturated gain from equation (29) is G”‘([,, = 0) = 0.1, and the estimated experimental value is 0.09. The predicted power output from equation (39) is 24 pW/cm’, as compared to the experimental value of 50 pW/cm2. In all of the above comparisons, the largest sources of error are likely to be in the calculated values of 1’ and b. In each case. errors of several hundred percent are possible. In addition, the effects of cylindrical rather than plane geometry and curved rather than flat mirrors will introduce smaller errors. Thus the experimental values can only be expected to agree with the theoretical predictions in an order of magnitude sense.
5
DISCUSSION
AND
CONCLUSIONS
The kinetic and radiative equations formulated by Cipolla and Morse to treat the interaction of line radiation with a gas. have been modified to include the effect of natural linewidth. and applied to the problem of a three-level optically pumped laser. Quantitatively useful expressions for population inversion, gain, and power output have been developed, which display the roles of system geometry, radiative parameters and collisional effects in a conceptually simple fashion. It has been shown that when the intensity of stimulated emission is low, the population inversion may be correctly calculated by ignoring all sources of line-broadening, other than the Doppler effect. However, the neglect of collisional linewidth in calculating the small signal gain is correct only in the limit that the Doppler width is much larger than the collisional width. It is never correct to neglect collisional linewidth when a high lasing intensity is present, since the nonlinear saturation
Theory
of an optically
pumped
gas laser
253
process has been shown to be intimately connected with collisional linewidth. Although these conclusions have been arrived at for an optically pumped laser, they should be basically valid for any gas laser. As far as laser geometry is concerned, the most significant result is that gain is proportional to the ratio of the laser length to diameter. The dependence of gain on diameter is similar to the behavior of conventional collisionally-excited gas lasers. in which the gain is normally observed to decay as diameter is increased. The physical explanation is obviously quite different in the case of the optically pumped laser, however. An important consequence is that it rules out the possibility of significantly increasing the output power by enlarging the tube diameter, since, for a given pumping intensity, the gain will decrease as the diameter is increased until the pump is no longer able to maintain the system above threshold. Thus, the only means of achieving higher output power is through the use of longer lasers, more powerful pumping sources, and optimization of the output coupling for a given system. In arriving at this conclusion, the implicit assumption has been made that the conclusions from the analysis of the plane slab are applicable to cylindrical geometry. Also, the discussion earlier regarding the role of spontaneous emission and trapping needs to be kept in mind. If a laser could be made to operate under conditions where the branching ratio were close to unity, then the gain would no longer be inversely proportional to plate spacing. It might be possible to construct a laser where the effect of spontaneous emission would be neither negligible (the case considered here) or dominant. Then the gain might be expected to be proportional to D-” with 0 I c( i 1. In that case, large diameter systems might be of practical interest. With these qualifications in mind, the optically pumped gas laser is not likely to become attractive for technological application, unless much more intense pumping sources become available. In addition, the overall efficiency of the optically pumped system is several orders of magnitude lower than for a comparable collisionally pumped system. The limiting factor on power output is clearly the low-intensity pumping devices currently available, and not the small number density typical of optically pumped systems. Number density will become a limiting factor only when the pump removes an appreciable fraction of the atoms from the ground state, and “bleaching” becomes important. This is clearly not the case for the cesium systems discussed here. The role of collisions has been treated by means of a largely phenomenological model of the collision integral, which recognizes the tendency of inelastic collisions to restore the distribution functions for each level to the MaxwellLBoltzmann value. In particular the model used does not treat the crossrelaxation effect of elastic collisions, which would tend to counteract the hole-burning caused by the coherent radiation. (22) A much more refined quantum-mechanical treatment of the role of collisions, valid for low levels of radiative intensity and Doppler width >> collisional width has been developed by BERMAN and LAMB. Finally the connection between the present work and that of GORDON et~1."~) should be mentioned. The latter paper dealt with a Maser amplifier (as opposed to an oscillator) using a formulation very similar to that developed here. Although a more realistic model of the collisional lineshape was used, the form of the distribution function in velocity space was assumed a priori. The present work shows the essential connection between the distribution function in velocity space, and the behavior of gain in frequency space. Thus a “hole” in a plot of gain vs. frequency implies a “hole” in the distribution function when plotted against velocity. The a priori assumption of a Maxwellian velocity distribution is
254
JAMES J. HEALY and T. F. MORSE
therefore invalid when h CC 1 and Lo >> 1. For b >> 1 or Lo CC I the assumption wellian should not produce significant error.
of a Max-
.‘I C./ttlo~l~kvI,~~~t~IPt2~J A helpful discussion with Professor Y. P. PAO on this problem is gratefully noted The authors are indebted to Professors M. SIRULKIN and H. J. GERRITS~N for reading and improving the manuscript. This work was supported by the Mechanics Division. Office of Aerospace Research, under contract F44620-71. (‘-0030.
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