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Diagnostic experiments and modeling of the 118 µm CH3OH laser
J.-M. Lourtioz, R. Adde
To cite this version:
J.-M. Lourtioz, R. Adde. Diagnostic experiments and modeling of the 118µm CH3OH laser. Journal de Physique, 1980, 41 (3), pp.251-258. �10.1051/jphys:01980004103025100�. �jpa-00209240�
Diagnostic experiments and modeling of the 118 03BCm CH3OH laser (*)
J.-M. Lourtioz and R. Adde
Institut d’Electronique Fondamentale (**), Bâtiment 220, Université Paris-XI, 91405 Orsay, France (ReCu le 29 aout 1978, révisé le 17 octobre 1979, accepté le 2 novembre 1979)
Résumé. 2014 Une analyse quantitative du rendement en puissance du laser CH3OH à 118 03BCm est présentée. Un
modèle à taux de population a été étendu dans ce but à cette transition à courte longueur d’onde dont l’élargisse-
ment est Doppler ou mixte. Des expériences de diagnostics sur cette raie laser permettent la mesure de certains des termes qui déterminent le rendement du laser, ainsi que l’analyse de la puissance de sortie. Elles montrent aussi le degré appréciable de saturation lointain infrarouge pour un large domaine des valeurs des paramètres.
Abstract. - A quantitative investigation of the 118 03BCm CH3OH laser power efficiency is performed. A rate equation
model is extended to this short wavelength transition (Doppler or mixed broadened) giving a close formulation of the laser power. Diagnostic experiments of the 118 03BCm line allow the measurement of some of the terms which determine the laser efficiency and give an overall fit of the FIR power. They also indicate an appreciable degree
of FIR saturation in a wide range of parameters.
Classilication Physics Ahslracls 42.60B
1. Introduction. - The rate equation (R.E.) models have been up to now the main quantitative method
used to. evaluate the power efficiency of CW opti- cally pumped FIR lasers [1-6]. They omit quantum effects which have been analyzed recently for short wavelengths transitions (Doppler or mixed broadened) by small signal gain measurements [7]. However
there is not yet a quantum model of such systems
interacting with two intense laser fields although
laser operation of powerful FIR lines occurs in a wide
range of conditions under significant FIR saturation (see section 3). Therefore we have followed the fruitful approach of Danielewicz and de Temple [1 ] in the study of the 496 ym CH3F laser to extend the R.E.
models at the 118 03BCm CH 30H laser line. We have analyzed our diagnostic experiments of this line in this framework. We present briefly in section 2 the assumptions of the R.E. model and give the expres- sion of the FIR output power for on resonance IR
_pumping at short wavelengths. The diagnostic experi-
ments of the 118 ym line described in section 3 allow
the measurement of some of the terms which determine the laser efficiency and their comparison with the
calculated values. They also lead to an overall fit of the FIR power as a function of the different laser parameters.
2. R.E. modeling of the 118 pm Iine. - 2.1
FIR LASER GAIN. - The 118 pm CH30H line is
mixed broadened in a wide range of operating condi-
tions (gas pressure p and pump power PIR). Then
besides the common assumptions to R.E. models [1-7],
the molecular velocities denoted by v must be intro-
duced in the FIR stimulated emission cross-
section [1-4] and in the pump source term [1] which
must be fully treated to allow the R.E. model extension at short FIR wavelengths. However simplifications
occur since the P(36) C02 pump emission frequency
may be tuned at the center of the CH 30H gas absorp-
tion. In this case the expression of the FIR laser gain
at the center emission frequency vFIR deduced from the R.E. equations (Appendix I) may be written :
The first term in the bracket of Eq. (1) represents the contribution of the active molecules to the FIR gain
(*) This work was supported in part by the Groupement de Recherches Coordonn6es du C.N.R.S. no 11 (Greco Micro-ondes).
(**) Laboratoire associ6 au C.N.R.S. n> 22.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004103025100
252
whereas the second term is related to the excited state FIR absorption [2]. The parameters in Eq. (1) are
listed below :
- 92’ 93 are the respective degeneracy degrees
of the FIR lasing levels 2 and 3../2, f3 are the popu- lation equilibrium fractions of these levels within the
upper vibrational state M..I:’ is the equilibrium
fraction of the upper vibrational state referred to the total number of molecules per unit volume No.
- VFIR is the FIR central emission frequency.
- The rotational relaxation rate is denoted by
7r = n AvH, where AvH is the homogeneous linewidth
of both the IR and FIR pumped gas transitions.
r v synthetizes the vibrational relaxation to the ground
state manifold.
- The forward and backward FIR travelling waves
are assumed to have the same intensity, FIR,* which simply neglects all the FIR cavity asymmetries (see
section 2.2).
- The FIR saturation intensity ISAT is defined by
where the quantities h, c, T, . Pont. have their usual
meanings.
- Co(v, VAR) is the normalized Lorentzian function of width 2 bvo = AIIR AVH which enters in the FIR emission cross-section
Then
is the power broadened Lorentzian of width
- f (v) denotes the maxwellian velocity distribution of width 2 Ov = 2 y’2 kB Tim. 4v is related to the
Doppler widths of the IR and FIR transitions
through :
- S(v) = S + {v) + S - (v) is the pump source term, i.e. S ± (v) dv represent the number of molecules in the velocity range (v, v + dv) which are pumped per unit volume and unit time by the two IR travelling
waves in the upper vibrational manifold. If I K are
the IR travelling wave intensities inside the FIR cavity
and Io is the saturation intensity of the absorption transition, the rate equations associated with the IR absorption transition give :
where ao is the unsaturated IR absorption coefficient at the line center and the velocity distribution width
corresponding to the hole burning [8] is
+ o,,
In other
words - 00
(S :!:{v). hVIR/Il) dv representsJ-oo
the saturated IR absorption coefficient Lx± for the forward (backward) travelling pump wave and takes the usual form :
2 . Z CIRCULATING INTENSITIES IN THE CAVITY, FIR
OSCILLATION CAVITY, FIR OSCILLATION CONDITION. -
The pump beam is usually focused into the FIR laser
through a small input coupling hole and suffers
multiple specular reflections on the waveguide walls (our present experiments, see section 3). For this
reason, the radial variations of the pump are averaged
inside the FIR cavity and interference effects at the pump frequency are also neglected (see section 3).
If there is moderate pumped gas absorption (low
pressure or weak gas absorption) besides the above condition, it is reasonable to consider besides the above assumption, that the two propagating pump
waves have the same intensities which may be
expressed as [1] :
where aL and f3IR are the IR losses per pass related to the pumped gas absorption and to the walls of
the FIR waveguide cavity (Length L and cross-
section S), PIR is the injected pump power. Averaging
the radial distributions of the FIR intensities in the
cavity and assuming a symmetrical FIR cavity; Eq. (7)
leads to the equality of the forward and backward FIR intensities I’ F R and IFIR. Then the threshold FIR
oscillation condition may simply be written as :
where t is the FIR coupling loss and a represents all
other losses per single pass. The FIR propagation
losses are often negligible for low order modes in
waveguide conditions [9, 10] and a is essentially
related to the absorption loss at the mirrors of the
cavity. The total FIR power which is coupled out
is given by :
- On the other side, important longitudinal varia-
tions of I,’ and IIR occur at high pumped gas pressure and (or) with a strong absorber. Then situations of directional pumping (li£ » (g rr 0) may be reached (11]. We have performed a Rigrod’s treatment
to evaluate the influence of directional pumping
on the FIR gain and the FIR output power [12].
This was done in the framework of a R.E. model, assuming again uniform radial pump and FIR inten-
sity distributions. The calculation including the longi-
tudinal variations [13] shows that Eqs. (7-9) may still be applied with a good approximation provided
that averaged quantities 7,R, a, 7f;IR, 9FIR are introduced.
Therefore, a possible directional pumping does not
restrict the validity of the R.E. analytical treatment
presented in the paper. The slow dependence of the
saturated IR absorption coefficient with the IR inten-
sity is the main cause of this result.
2. 3 ANALYTICAL EXPRESSION OF PFIR. - The expres- sion of P F1R is calculated from Eqs. (1-9), when the
FIR cavity is tuned near the center of the FIR emission
line. Assuming 6’v small compared to w i.e. moderate IR saturation conditions (see experiments in section 3)
we obtain :
Ae is related to the FIR excited state absorption,
+ x,
where
p(c5v/LBv) = -a;
,f’(v) . £(v, VFlJ dv characterizes the mixed-broadened regime and requires generallya numerical integration. At low pressure and low FIR saturation it reduces to the analytical expression
n 1/2 6v/Av.
The implicit Eq. (10) of PFIR may be written in a close form [14, 15]
qt h I + 92/93 I
th g 2 g 3VF;IR VIR
vIR is the quantum efficiency-
F - aL
abs. is the fractional absorption ofaL -I- pIR p
the pump power
- F t ran s.
= t (I - A e) . -1 I.. hIS
the fractional trans-t+a 1 + h s
mission loss of the FIR radiation. Ae is related to the excited state absorption and h to the hole burning.
3. Experiments and quantitative analysis of the 118 PM CH30H line. - The behavior of
for an intense FIR transition such as the 118 gm
line (strong dipole moment i.e. weak FIR saturation
intensity ISAT) is governed essentially by the terms figuring in FabS. and F,,,a.,;.. Therefore we describe
first the experimental evaluation of FabS. which we
compare to the calculated value. Then we discuss the nature of the excited state absorption and finally
we correlate the model presented in section 2 with
the experimental curves PFIR = .1’(p, PR). The experi-
mental set-up has been described recently [17] and
is shown in figure l. The CO 2 power available on
P(36) of the 9 J.1m band is about 20 W. The CO2
laser frequency is stabilized by maximizing the FIR amplitude of a reference FIR laser (’). The 1.80 m long and 25 mm ID pyrex waveguide FIR cavity
Fig. I. - Schematic diagram of experiments.
has hole coupling mirrors at both ends (2 mm injection
hole and 4 mm output hole). The CH30H gas pressure is measured with a Pirani gauge adequate for the operating range 10-300 mtorr and previously cali-
brated against an absolute capacitive gauge. The
(’ ) A simpler frequency stabilization loop uses now the absorption signal detected with a spectrophone cell.
254
input pump power which may be varied continuously
with an FIR gas attenuator is measured on detector I
(Moll thermopile) and calibrated against the effective
power P1R injected in the FIR cavity. Measurements ol’ the FIR cavity losses at the pump frequency leading
to fJlR and aL are done with detector 2 (Moll thermo- pile) which measures the power Plk of the IR beam propagating back through the injection hole. This experimental arrangement allows to monitor conti- nuously either PIIR (detector 3-Moll thermopile) or PI*R as function of FIR gas pressure, FIR cavity tuning and injected pump power P1R.
3.1 SPECTROSCOPIC DATA. - The 118 pm CH30H
line is well identified as a pure rotational transition in the torsional ground state of the first excited CO- stretch vibration level [16]
with n, i, K, J = 0, 1, 8, 16. The main CH30H spectro- scopic parameters of interest are listed in table I with their origin. We have added for completeness
the measured waveguide cavity parameters. A good
coincidence between the P(36) CO2 pump and the
absorption transition [17] may be obtained since the line centers are distant of - 25 MHz as measured
by saturated absorption spectroscopy (2). Then it is
doubtless that FIR optimization occurs when FIR cavity is tuned very near the FIR center emission frequency and Eq. (10) is to be employed.
3.2 DETERMINATION OF F - The procedure
used to measure Fab4, is the following :
- measurement of the empty FIR cavity losses ( - #IR) at the pump frequency,
- measurement of the gas filled FIR cavity
losses - (PIR + aL) at the pump frequency versus CH30H pressure p and pump power PIR - compa- rison with theoretical evaluations and determination of Fabs. = aL/(aL T PIR).
The figure 2 shows in the insert the IR power
coupled out of the empty cavity (detector 2) and
measured as a function of the FIR cavity length.
The low level of these variations (= 15 %) indicates
that interference effects at the CO2 frequency are
small. This results from the multiple specular reflec-
Table I.
(2) We have measured this frequency distance through the interferogram of the FIR cavity at very low pressure (p > 30 mtorr), when
the C02 pump laser is locked at the maximum of its tuning curve. Then each FIR mode is splitted and the peak separation
ðVFIR = 2 ).VIR.À1R/ÀFIR confirms the frequency offset of the P(36) pump line center [17]. -
(") Although there arc multiple refiections of thc pump from the walls, the observation of the F1R beam reveals a strong linear pola-
rization perpendicular to the pump. Therefore the I R radiation must keep in the FIR cavity a large part of its original polarization and we
have used in table I the value of the dipole moment corresponding to a polarized FIR gain.
Fig. 2. - Outcoupled IR power PIR (detector 2) - 1 /(§L + PIR)
as a function of the injected pump power P1R, with pressure as a parameter. Empty cavity (0), p = 30 mtorr (1), p = 53 mtorr (2)
p = 73 mtorr (3), p = 100 mtorr (4), n = 200 mtorr (5). Full lines (experiments), dotted lines (calculated curves). PIR is represented
in the insert as a function of the FIR cavity length. The weak modulation rate of the outcoupled IR power indicates that inter- lerence etiects at the pump frequency are blured due to the multiple specular reflections of the IR beam in the cavity.
tions off the waveguide walls since the injected pump beam is uncollimated. It is reasonable to assume that the C02 power inside the cavity is radially averaged.
A Rigrod’s treatment allows to write the IR power
(detector 2) coupled back from the empty FIR cavity
-where t in is the input mirror transmission coefficient.
If we replace the input cavity mirror by a zero reflecting diaphragm, the above expression becomes .
In this way, we have obtained PIR - 0.17, a value quite independent of the output mirror coupling
hole (1-5 mm). Therefore the IR propagation losses
on the waveguide walls are dominant in our system.
The IR power (detector 2) coupled out of the gas filled FIR cavity is monitored as a function of pres-.
sure p. This measurement gives
represented on figure 2 (full lines). The corresponding
theoretical curves in the same figure (dashed lines)
are obtained through the relations (6) and (7) where
the averaged intensity fIR is calculated using Weiss
Fig. 3. - Fractional absorption of the pump power as a function
of the CH30H pressure, PUt being a parameter. PIR = 14.3 W (1), PIR = 1 I W (2), PIR = 7.8 W (3), PIR = 4.23 W (4), P,R = 1.9 W (5).
The dotted curve represents the evolution of F abs. at the optimum
pressure.
data [18] (saturation intensity and absorption coeffi- cient). The agreement between the two sets of curves is very good. The fractional IR absorption coeffi-
cient Fabs. extracted from the above results is plotted
as a function of pressure on figure 3, PIR being a parameter. The corresponding values of the optimum
FIR operating pressure have been also indicated and it must be already emphasized that the values F abs. (PoPt) = f(P1R) all lie in a very narrow range.
Figure 4 represents an important laser pump para- meter, the saturation degree of the IR absorption
Fig. 4. - Saturation degree of the IR absorption transition
(IR/ 10) 1/2 as a function of the CH30H pressure, PIR being a para- meter ,PIR = 14.3 W (I), PiR = I I W (2), PIR = 7.8 W (3) PIR. = 4.23 W (4), PIR = 1.9 W (5). The dotted curve represents
the evolution of (I,R,Io)1/2 at the optimum pressure.
256
transition. It must be pointed out that this parameter keeps a moderate value in the range of optimum
pressure although it is responsible of the slow increase of Fabs. with P opt.. On the other side strong IR satu- ration is reached at low pressure and it is in this domain that quantum effects may manifest essen-
tially.
3. 3 FIR EXCITED STATE ABSORPTION IN THE 118 gm
LINE. - The FIR excited state absorption which is
the consequence of the velocity selective pumping
may be amplified by the presence of the vibrational bottleneck. This latter effect has been demonstrated to be dominant in CH3F [1, 3]. However it appears to be small for the 118 pm line [7]. Our diagnostics
at the cut-off pressure which are described in this section confirm this behavior.
If the vibrational bottleneck is neglected 1/ r v disappears in Eq. (11) and at the cut-off pressure
(JHR = 0) A e may be expressed as :
On the other side Ae may be measured from the cut-off condition which gives (see Eq. (12)) ;
The cut-off pressure as a function of pump power PIR
is obtained with precision from figure 5 and the cor- responding value of Fabs. is deduced from figure 3.
Fig. 5. - Measurement of the threshold conditions of the 118 03BCm CH30H laser : (1)p = 33 mtorr, (2) p = 42 mtorr, (3)p = 52 mtorr, (4) p = 77 mtorr, (5) p = 95 mtorr, (6) p = 160 mtorr, (7) p = 230 mtorr, (8) n = 270 mtorr.
ISAT is calculated using Eq. (2) and the data available in the litterature. The small value
is associated to the strong dipole moment of the FIR
transition (/123 # 2.2 x 10-30 C . m) and the mode-
rate rotation relaxation rate
The hole burning term expresses simply at the threshold
condition,
and may be calculated from figure 4.
In our laser waveguide configuration the FIR cavity losses are essentially located at the mirrors [10]
and we have evaluated the loss coefficient per pass
a = 0.03. The transmission coefficient per single
pass t has been evaluated by iteration using a parti-
Fig. 6. - Dependence of y(p) (bracket in Eq. (13)) with CH30H
gas pressure.
cular operating point of the curves PFIR = f ( p, P1R)
which is indicated as on figure 7 and we have obtained
t = 0.025. This value lies fairly well in the middle
of the range of a numerical estimation.
We have investigated Ae through the p2 depen-
dent expression y(p) (bracket in Eq. (13)) which
is the term of interest. Therefore we have plotted
in figure 6 the theoretical and measured values of
AeFabs. P1R/P(AvH/AvD,FIR) obtained from Eqs. (13-14).
Figure 6 shows that the experimental curve (dots) approaches the theoretical one reasonably well,
and lies always beneath the latter one. This is a
confirmation a posteriori that the vibrational bottle- neck may be neglected.
Therefore Ae will be calculated in the expression
of Pmg within this approximation. Besides we will
use the values of y(p) extrapolated from the experi-
mental curve of figure 6.
3.4 PRESSURE AND IR POWER DEPENDENCES OF
PFIR- - The FIR output power measured at detector 3 and monitored continuously versus pressure is drawn in full line on figure 7 with PIR as a parameter. The theoretical curves (dashed line) have been plotted
from Eq. (10) and we have introduced a multiplicative
factor to take into account the laser input coupling
hole (t;n = 0.25 tout) and the transmission of the output quartz window ( 70 %). Eq. (10) is an implicit expression of PAR through IFlR/IsAT which
enters in the terms Ae and h. Consequently we have
Fig. 7. - Ilg um CH30H laser FIR power PFIR monitored continuously as a function of gas pressure p, P,R being a controlled parameter. The different values of PIR are : 1) 14.3 W, 2) 1 W, 3) 7.8 W, 4) 4.25 W, 5) 1.92 W. Full lines (experiments), dotted lines (calculated lines).
Fig. 8. - Saturation degree of the 118 um CH30H FIR transition
as a function of pressure, with PIR as a parameter (same as Fig. 7).
The dotted curve represents the evolution of (IFIR/lo)1/2 at the optimum pressure.
used an iterative method to calculate PFIR starting
from PFIR/2 tS (experimental) as the initial value of JFlR. The theoretical curves of figure 7 which
have been fitted using only the threshold points (determination of y(p)) and one point at PÃRx (deter-
mination of t) are in excellent agreement with the
experimental curves throughout most of the pressure
Fig. O. - Variation of 11(1 + h) related to the hole burning (9a) and of (1 I - Ae) related to the excited state absorption (9b) as a
function of pressure, PIR being a parameter : I ) 14.3 W, 2) 7.8 W, 3) 1.92 W. These factors enter in F,ran,. the fractional transmission loss (Eq. (12)). The dotted curve (Fig. 9h) represents the evolution of (I - Ae) at the optimum pressure.
