Transit-time effects in saturated absorption resonances
A. Titov, Yu. Malyshev and Yu. Rastorguev
USSR State Committee for Standards, 117049 Moscow, U.S.S.R.
(Requ le 3 fevrier 1986, revise le 7 juillet, accepte le 26 août 1986)
Résumé.
2014Les caractéristiques principales des résonances d’absorption saturée (leur forme, leur amplitude, l’élargissement de puissance et de collision) sont étudiées
endétail expérimentalement et théoriquement pour des niveaux de saturation très petits et relativement grands. A la différence des travaux précédents
onmontre
que la largeur de la résonance dans le régime de vol libre n’est déterminée que par le temps typique de transit
des molécules à travers le faisceau lumineux, tandis que la largeur de la première dérivée est proportionnelle à
la racine carrée du produit des largeurs de collision et de transit. Pour la première fois
on adétecté et étudié
l’influence des molécules lentes
surla forme de la résonance et
surle décalage à
causede l’effet Doppler de
second ordre. Nous
avonsatteint pour la première fois
uneprécision de 3
x10-13 pour les standards laser de
fréquence.
Abstract.
2014The main characteristics of the saturated absorption
resonance(its line shape, its amplitude,
power and collisional broadening)
arestudied in detail, experimentally and theoretically, covering both the
domain of very small and relatively large saturation levels. In contrast with
numerousprevious papers, it is shown that the width of the
resonancein the free-flight regime is governed only by the typical transit time of molecules
acrossthe beam, while the width of the first derivative is equal to the square root of the product of
the collisional and transit-time linewidths. The effect of slow molecules
onthe shape of the
resonanceand the
second-order Doppler shift has been detected and studied experimentally for the first time. The unprecedented
accuracy of about 3 parts in 1013 is reported for the laser frequency standard.
Classification
Physics Abstracts
76.03X
-06.63
It is well known that the saturated absorption
method is still widely used for optical frequency
standard applications and for spectroscopic studies
of extremely high resolution. Naturally, long-living
molecules are of primary importance in these
cases [1] and the operation at very low gas pressures is a common feature. The main goal of this paper is to show that many concepts, often being used in
current publications and corresponding to the main
characteristics of molecular resonances, should be
significantly changed in the transit-time regime.
Here detailed experimental and theoretical studies of saturated absorption resonances are presented for
a wide range of gas pressures and field intensities.
In the first part of the paper, the line shapes of
methane resonances together with their derivatives
(discriminant curves) are reported in the transit-time domain for very low saturation levels, for the first time. The effect of .slow molecules on the second- order Doppler shift and on the line shape has been
discovered experimentally and studied in detail. The
resonance lineshapes in the transit-time regime and
the deviations from Lorentzians are presented both
for very low and relatively high saturation intensities,.
The width of the resonance at the half-intensity level
for very low gas pressures and saturation fields (in
contrast with a widely spread opinion) is found to depend only on the ratio of the most probable
thermal velocity, vo, to the beam radius, a (as in
molecular beam experiments). Meanwhile, the
width of the discriminant curve approaches zero as
the square root of the product of the collisional, r,
and the transit-time linewidth for r approaching
zero.
In the second part, we present a theory of the
saturated absorption resonance for spatially-boun-
ded plane laser beams which properly describes the main experimental features and is valid up to moderate field intensities and gas pressure levels. In contrast with the previous theories, but in accor-
dance with the requirements of the thermal equili- brium, it takes into account the dependence of the
molecular free path on the thermal velocity of the particle. As it gives the appropriate line shapes and
linewidths for arbitrary collisional and transit-time
frequencies, for very weak and moderate saturation fields, and
asits predictions of the second-order
Doppler shift are in reasonable quantitative agree-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470120202500
ment with the experimental data for a wide range of the relaxation parameter r a / vo, this theory may be
used to find the unperturbed molecular transition
frequency
-the final goal of all frequency standard applications.
Comparisons with the previous publications is performed mainly in the third part of this paper.
1. Experiment.
The experiments were performed using He-Ne laser
frequency stabilized by the transition, without a magnetic hyperfine structure (the E-component of
the v3 P (7) transition in methane). The laser
spectrometer was based on the idea of the offset
locking, first proposed by R. L. Barger and
J. L. Hall [1], and consisted of two lasers with intra-
cavity absorption cells and a local laser oscillator.
The latter was locked by a relatively fast (wideband) loop to the frequency stabilized He-Ne/CH, laser.
Using another frequency/phase lock loop, the perfect
short-term and long-term stabilities of the reference laser were transferred to the other He-Ne CH4 laser
under investigation. So, we were able (by changing
the tuning of the frequency synthesizer in the first PL loop) to study in detail any portion of the
molecular resonance.
fThe most important design features of our laser have been described already in our previous
papers [2, 3], so we shall note the latest improve-
ments and some of the parameters which are crucial for the operation of the device as an optical standard
and spectrometer.
First of all, the single line in the absorption spectrum of a 12CH4 molecule, which is free from
magnetic hyperfine structure, was used in our experi-
mental studies, thus enabling to perform quantitative comparisons with the theoretical predictions. Then,
the fine adjustment and an accurate stabilization of the transverse magnetic field, applied to the dis- charge tube, gave us an opportunity to achieve exact
coincidence between the molecular resonance and the maximum output power of the laser, operating in
a single frequency regime, for each operating point.
So, precise frequency measurements were possible
even for very low saturation fields when the ampli-
tude of the inverted Lamb dip was very small. The
performance of such measurements is of crucial
importance for the detection of the second order
Doppler shift and for the studies of the effect of slow molecules.
But the key feature of our stabilized laser is a very small radial variation of the gain across the discharge
tube (on the mode size scale), which is simulta-
neously achieved with the operation in a single mode regime without the use of any limiting aperture.
Under these conditions, the output power of the laser does not depend significantly on the mode size,
and the frequency shift due to the saturation of the refractive index of the absorption medium is drasti-
cally reduced [3]. Applying the necessary corrections to this effect, in accordance with’the method descri-
bed in [3], and extrapolating the observed frequency dependences to zero intensity (to exclude the shift
resulting from the distortion of the mode curvature due to saturation of the gain and of the absorption media), we obtained the reproducibility and the
agreement between two independent standards
within (1-2) parts in 1013. The observed RMS varia- tion of the stabilized frequency (not the Allan variance) was only 2 parts in 1014 for the integration
time of 100 s and for half an hour observation time.
The other features of our lasers worth mentioning
are as follows : The absorption cell, of about 1.3 m
long, was installed in such a way that its centre coincided with the position of the beam waist of the
cavity. The transparency of the mirror near the cell
was about 1.3 %, so the frequency shift due to the
curvature of the mode resulting from laser mirrors
was very small (1). The absorption cell was connected
to a gas handling system, and the methane inside it
was always free from any contaminations. The gas pressure level in our studies was usually from 0.2 to
3 mTorr. In order to perform measurements at
higher pressures, this cell was exchanged to a shorter
one, of about 30 cm in length, and the extension to 20 mTorr methane pressures was achieved. All the measurements were performed either at room or liquid nitrogen temperatures. The diameter of the Gaussian mode waist was about 1.43 mm. The transmittance of the mirror near the gain tube was
3.4 %. In our experiments InSb nitrogen cooled photo-detectors were used with a sensitive area
much greater than the mode spot size on it. The amplitude noise in the servo-system was primarily
due to the current through the photo-resistor.
Using such lasers we were able, for the first time,
to study and record the shape of a molecular
resonance and its first derivative (a discriminant
curve) in the transit-time domain for very low saturation fields. As predicted by theories [4-6], the typical
«transit-time
»shape was observed (curve 1
in Fig. la) with an important sharpening of the curve
for the normalized detunings X 1; with a slower
decrease of the intensity relative to the Lorentzian
lineshapes with the same half-widths for intermediate
detunings 1 X 3; again with a faster decrease in the wing of the curve for X > 5. This specific sharpening of the top of the curve, which according
to theories [4-6] is the manifestation of the effect of slow molecules, giving an excessive contribution
(1) In accordance with estimates based
on ourexperi-
mental studies, the frequency shift due to the wavefront curvature resulting from the laser mirrors is within 6
x10-14 for the uncertainty of the location of the cell of 5
cmrelative to the beam waist for
amethane pressure of 1 mTorr at the liquid nitrogen temperature. The shift due
to the beam diffraction, the finite length of the cell and the
gas absorption is about 7
x10-14 for the
sameconditions,
and the appropriate correction may be applied, if
neces-sary, in this
case.The asymmetry of the line, typically
afew parts in 103,
canbe detected only by precise frequency
measurements.
Fig. 1.
-Experimental line shapes and Lorentz fits for
amethane pressure of 0.6 mTorr and
anoutput power
Pout of 8 JLW (curve No.1 in Fig. la), Pout
=38 pW (Fig.1b), Pout = 160 RW (Fig.1c) and for
amethane
pressure of 3.2 mTorr (curve No. 2 in Fig. 1a)., The
Lorentz fits
aredepicted in dashed lines.
near the transition frequency, is observed only for
very small saturation fields and very low gas pressure levels. Namely, the curve 1 in figure la corresponds
to the relaxation parameter Fo ro
=0.12 (where ro
is the collisional linewidth and ro is the typical transit
time equal to the ratio of the beam radius, a, to the most probable gas velocity, vo). The saturation parameter, R, for each of the running waves in the cavity was estimated from the amplitude characteris- tics of the resonance and was found to be equal to
0.02-0.04 for the laser output power of about 8 row
(2),
.It is very important that a relatively small increase
of the laser power or methane pressure effectively
suppresses the effect of slow molecules. The cur-
(2) The intracavity circulating power may be calculated
using the transmittance of the corresponding mirror of
3.4 % and the beam waist Wo = 1.43
mm.Unfortunately,
a
closed form expression for R is not available in the
general case,
asaveraging
overthe transverse velocity
component must be performed. But for small saturation
levels the estimates
canbe reliably performed
asthe Lamb
dip amplitude is simply equal to R/2 in this
case.ves a, b, c in figure 1, corresponding to the gas pressure of 0.6 mTorr and the output power of 8, 38
and 160 RW, illustrate the change of the shape and a
power broadening effect. It is clearly seen from figure lc that for relatively high saturation levels,
when the dip amplitude is large enough to perform
any kind of exact measurements and when the saturation parameter R is about 1, the resonance line
form decreases already more slowly for small detu-
nings and decreases faster for large detunings,
relative to the Lorentzian with the same halfwidth at half-maximum. One can also find, that from the
presented set of curves it follows that the saturation parameter for the 8 )JLBV power level is indeed very
small, as the power broadening for the curve la is only a few per cent.
Another important result which follows from
figure 1 is that the line shape in the limit of small saturation and high pressure level is practically a
pure Lorentzian. For example, the deviations from the Lorentzian line form for the curve 2 in figure 1, corresponding to a methane pressure of 3.2 mTorr and a power level of 9 RW, are quite negligible. We
also note here that a line shape analogous to the
curve 1c has been previously observed by J. L. Hall
in [7], but it is difficult to say whether it is
aneffect of the unresolved hyperfine structure [8], or the field
effect as in curve 1c.
The effect of sharpening of the top of the curve, which is observed in the transit-time regime for very low saturation fields and which may seem not to be of any significance, still plays a crucial role in many
applications. Due to this effect, the width of the discriminant curve is greatly reduced relative to the width of the Lorentzian’s first derivative with the
same half-width. This effect is clearly observed in
figure 2, where the curve 1 corresponds to the mole-
cular resonance depicted by curve 1 in figure 1.
Naturally, the increase of the saturation field sup- presses the contribution of slow molecules and the difference between the discriminant curve and the function x / (1 + X2) 2 is gradually diminishing, as
illustrated by the curve 2.
Fig. 2.
-Discriminant
curvesfor
amethane pressure of 0.6 mTorr and
anoutput power level of 8 pW (curve No.1) and 160 JLW (curve No. 2), with corresponding
Lorentzian derivative functions (la and 2a).
Performing the measurements for each pressure level at different saturating field intensities and
removing the field broadening from these data, we
were able to find the dependences of the widths of the resonance and of the discriminant curve versus
the methane pressure both for room and nitrogen temperatures, which are presented in figure 3.
The new experimental results of crucial impor-
tance are the following : the width of the saturated
absorption resonance in the limit of very low gas pressures and field intensities is determined only by
the typical time of flight of molecules across the beam (as in molecular beam experiments and in
contrast with theories [5, 6, 9-11]). The deviations from the linear dependence of the linewidth versus
methane pressure are relatively small in the transit- time regime. On the contrary, the width of the discriminant curve is strongly affected by the slow
molecules and approaches zero as the square root of the product of the collisional and transit-time line-
widths, for the collisional width approaching zero.
The pressure (3) broadening at liquid nitrogen tem- perature is substantially greater (24.5 kHz/mTorr ±
10 %) than at room temperature (15.8 kHz/m
Torr ± 6 %). But as the width of the resonance in the transit time regime is determined mainly by the
transit-time linewidth, the observed resonance at low pressures, at low temperatures are significantly
narrower than at room temperature. We note here
Fig. 3.
-Experimental (1, 2, 5) and calculated (3, 4) dependences of the width of the
resonance(HWHM) and
of the discriminant
curve(2, 4)
onthe methane pressure for
room(1, 2) and liquid nitrogen (5) temperatures.
(3) The pressure
wasmeasured in the section of the
absorption cell kept at
roomtemperature.
that according to all the previous theories [5, 6, 9-11]
gas cooling to liquid nitrogen would not lead to any
crucial narrowing at the fixed methane pressure, as the decrease of the transit-time linewidth vo Wo is
practically compensated by the increase of the collisional linewidth (or in other words, by more
than a 3-fold increase of the relaxation parameter
rTO).
In figure 4 the experimental dependence of the
molecular resonance width with the methane pres-
sure is presented for a wider range of pressures. It is obvious that the deviation from the linear depen-
dence for 0.3-20 mTorr pressure domain is quite negligible. This result is in agreement with previous experimental studies of the v3 P ( 7 ) transition in methane [1, 12], and the pressure broadening coeffi-
cient is practically the same for the E- and
F 2 2-components of this transition.
As mentioned above, the sharpening of the top of the molecular resonance, which is observed at very low intensities and gas pressures, is attributed to the effect of slow molecules. We have tested this conclu- sion experimentally by measuring the frequency shift
of the resonance due to the second-order Doppler
effect by cooling the methane gas from room to the
liquid nitrogen temperatures. The observed shift,
after applying the necessary corrections to the gas lens effect [3], was found to be substantially less than
the value k ATI (mC2) , which corresponds to the frequency shift due to molecules with the most
probable velocity vo
=J2 k åT/m. (Here k is the
Boltzmann constant, m is the mass of molecules, AT
is the change of gas temperature). The results of this
unique experiment will be presented in detail elsewhere, but we want to note here that by measu- ring the resonance frequency shift, we were able to
detect such important features of this effect as the
Fig. 4.
-Experimental dependence of the width of the
resonance on
the methane pressure at
roomtemperature for
anextended pressure range.
change of the contribution of slow molecules when
changing the methane gas pressure, or the laser
output power, or the amplitude of the frequency modulation, necessary for the operation of the servo system. The latter effect has been previously detec-
ted only in the best primary caesium standards used for the generation of the national time scales [13].
As the measurements of the shift are very sensitive
and accurate, there is a possibility to detect the effect of slow molecules even for higher pressures and saturation fields in comparison with the studies of the resonance line shape (Fig. 1). For example, at
1 mTorr pressure level and for a saturation parame- ter R
=0.07 the frequency shift for the cooling of
the methane from room to nitrogen temperatures is measured to be 65 ± 15 Hz for low frequency sine
wave modulation with a deviation of 50 kHz [14] and only 45 ± 10 Hz for the methane pressure of 0.9 mTorr and a deviation of 23 kHz. These values should be compared with the quantity of 110 Hz,
which follows from the estimate performed for the
most probable velocity.
Now we should like to make a few remarks on the saturation parameter in the transit-time regime.
Here, the experimental results of primary impor-
tance have been obtained by J. L. Hall in [7], where
it has been shown that for very low gas pressures the level of the saturation of the transition is characteri- zed by a saturation power, which does not depend on
the beam diameter and linearly depends on the absorption gas temperature. In other words, the
saturation intensity is determined by the square of the typical transit-time frequency volga, but not the
collisional linewidth n In the intermediate pressure range, according to [7], the saturation depends linearly on r and this function converts into a
parabolic dependence in the high pressure limit. The
experimental plots of the molecular resonance ampli-
tudes versus laser output power obtained for diffe-
rent methane pressures are depicted in figure 5 and present another confirmation of these results.
Indeed, for low field intensities, the amplitude of the
saturation resonance is proportional to the saturation parameter of the transition and the gas pressure level in the cell (which determines the unsaturated
absorption in the gas). The nearly constant value of
the resonance, observed in the intermediate pressure domain for 0.2 TTo , 2 for low field intensities, clearly indicates that the saturation parameter is, roughly, inversely proportional to r, and the quadra-
tic dependence on it in the high pressure domain
immediately results in the decrease of the resonance
amplitude (curve 5 in Fig. 5). If one moves further to
the transit-time regime, then the amplitude of the
resonance will begin to decrease, and in the free-
flight regime the linear dependence will be observed,
as the unsaturated absorption linearly depends on
the gas pressure and the saturation parameter is
determined by the transit time of molecules across
the beam. Such a relatively slow decrease of the
amplitude in the free-flight regime is a manifestation
of the fact that, for the spatially-bounded laser
beam, the zone of creation of molecules with the
- --
B of . %-, ,
Fig. 5.
-Experimental (5a) and calculated (5b) molecular
resonanceamplitudes
versuslaser output power for methane pressures of 0.7 (1), 1.8 (2), 3.15 (3), 4.2 (4) and 16 (5) mTorr.
necessary v, component interacting with the field is
rapidly increasing with the decrease of gas pressure
[4]. For a relatively large free path, A, it greately
exceeds the beam radius.
Naturally, if one compares the amplitudes at
different output power levels, for example, for the
intensities corresponding to the maximum of the Lamb dip, then the results will be quite different. In the very low gas pressure domain the amplitude of
the resonance is a linear function of pressure. For the intermediate pressure range, where the contribu- tion to the saturation parameter of the collisional
broadening is already important relative to the
transit-time one, but the dependence of the satura-
tion parameter is practically a linear function of pressure, the dependence of the amplitude on the
pressure will be quadratic. In the high pressure domain the dependence of the resonance amplitude, corresponding to the maximum dip, will be a cubic
one, as the saturation parameter is a quadratic
function of the gas pressure in this case.
The linear dependence of the saturation parameter
on the pressure level for r TO 1 and the quadratic dependence for higher gas pressures have been
previously obtained by us in the studies of the molecular resonance frequency-pulling effect [15].
In contrast with the studies just described, that experiment was performed at relatively high satura-
tion fields, corresponding to the maximum pulling
effect.
We note that all these experimental results are in
obvious disagreement with the assumptions of the
authors of [9-11], who considered that the saturation parameter in the transit-time regime was the same as
in the case of the infinite plane wave [16], in particular, the transit time did not affect its value.
Naturally, the estimates of the saturation parameter according to the formulae p E. 1 1 h / r 1’a 1’b + 1 [9-
11] ] and P E 2 T 2 [7] in the free-flight regime may
h / 0 g g Y
differ by several orders. In this respect it should be reminded, that the maximum Lamb dip amplitude in
all cases (it does not matter whether it is a transit- time or collisional regime) is observed at the satura- tion parameter R for each travelling wave only slightly exceeding one [4].
2. Theory.
Now we shall proceed to a theory of the inverted Lamb dip for a spatially-bounded plane laser beam.
An improved version of [4] will be presented, which
takes into account the second-order Doppler effect
and the dependence of the molecular free path on
the velocity of the particle. In order to simplify the calculations, it is assumed that a plane wave E ( r ) cos ( f t - Kz ) of frequency, f, and wave
vector, K, is propagating in the, z, direction within a
molecular gas. The field amplitude is assumed to be
equal to a constant value of Eo inside the cylindrical region of radius, a, and is supposed to be zero
outside it. We also consider that the time interaction with the field of a long-living particle is governed by
molecular collisions, and each collision terminates or starts such interaction, as r « Kvo. So, if the mole-
cule, after a long series of collisions, is created with the necessary v z component and is in resonance with
the laser field, it will be, for sure, in the ground state
and is eligible for absorption of energy. Then, under
these approximations for the two-level system with
equal decay constants ya and 1’b’ the exact solution
fot the transition probability of a molecule with U
and v, velocity components, subjected to the spatial-
ly-bounded plane wave of arbitrary intensity, is [4]
Here,
and p is the dipole moment, w is the transition frequency, Eo is the field amplitude, h is the Planck constant, T is the collisional width, L is the maximum path of molecules inside the beam for the specified
direction and the creation position.
Naturally, this solution in the free-flight regime
converts into the Rabi formula
with the saturation level determined by the quantity
V2 L 2/ U2 for Xl
=0, and into the Lorentzian
V 2 V 2 + Xf + r 2) with the saturation parameter
v2/r2 in the collisional limit. In contrast to [4], here
the explicit dependences of the molecular free path, A, and the collisional width on the velocity of the particle have been taken into account. Namely, the approximate relation
is used, where no is the density of molecules, ao is the collision cross-section, v
=v vo is the normalized
velocity and To
=no ao vo. It is important that this
relation gives exact values of the molecular free path
A both for slow molecules v vo
very fast molecules v > v Y / k (v) = no 1 ao 00
=k 0,
and differs in magnitude from the appropriate ex- pression of the gas kinetic theory [17] by only a few
per cent for all other intermediate molecular vel- ocities. The corresponding distribution function for the number of particles being created per unit time in unit volume with the specified velocity v is
It follows from (2) and ergodic theorem, which in particular says that for stationary processes the ensemble and the time averages coincide. This
means that the velocity (Maxwellian) distribution function for the molecules in gas may be obtained as
a product of a time interval between the adjacent collisions, which for the specified velocity group is 1/r ( v ), and the normalized creation rate function
F 1 ( v). So, the expression (3) describes exactly
both the slow and the very fast molecules, and the
difference from the thermal equilibrium distribution function is a few per cent for intermediate velocities.
This seems to be quite sufficient for the most of
applications, but in case of necessity the relation (5)
from [4] may be used.
In accordance with [4], integrating (1) with the
distribution function (3) and with the free path given by the relation
over the whole space, and dividing the result by the
flux of energy V2/ ’IT’a2, one obtains the following expression for the absorption coefficient, K r, for the
running spatially-bounded plane wave :
The first term in (4) corresponds to the absorption by the molecules which are created inside the beam and it gives the main contribution in the collisional
regime when J1 a. For this term the length of the path, L, inside the beam for the molecules, which
are created at the position specified by the vector
r(originated from the
zaxis of the system), is
L
= rcos cP + J a2 - ,2 sin2 ’P, where cp is the angle
between
rand the tranverse velocity of the molecule U.
The second term in (4) describes the absorption by
the particles created outside the beam and arriving
at the beam zone due to the thermal motion. It plays
the main role in the transit-time limit, as the ratio of contributions of the second and of the first terms in
(4) for large saturation levels is simply equal to the
ratio Vo / r 0 a = À 0/ a. This result is in agreement
with the classical gas kinetic theory, as the number
of molecules arriving through the boundary surface
of a sphere of radius, a, per unit time to the number of particles created by collisions inside that sphere is
determined by the ratio A/a. The corresponding length of the path of molecules inside the beam for this term is equal to 2 a cos cp’, where cp’ is the angle
between the vector U and the normal to the beam’s
boundary surface at the point of crossing with the particle.
In the second term in (4) the integration over the
space, which is outside the beam, has been perfor-
med and has resulted in the distribution function for
the flux of molecules across the unit surface in a gas :
F2 ( v ) = V3exp(-v2) cos cp’ (5)
which is valid for all the molecular velocities in a gas under the thermal equilibrium conditions [17]. But
as the contribution to the Lamb dip is mainly due to
the molecules with extremely small values of the
vz components, for these molecules v
=V U2 + vi
U, the distribution functions (3), (5) in the cylin-
drical coordinate system convert to the forms used in
(4).
If the integration over vz is not performed, then
the function 0 (x + y, V ) determines the form of the Bennett hole lineshape Li and the saturation
parameter R for a spatially-bounded running wave.
After obtaining these results, one can repeat all the considerations of [18] and find in the rate equation approximation (REA) of [18] the following expres- sion for the absorption coefficient of a standing
wave :
and for the Lamb dip line form
We note here, that REA is not practically a limita- tion, as according to [18] the REA solution begins to
differ from a more accurate one only for the
saturation parameters of 10 or 15, while the maxi-
mum Lamb dip is observed at a saturation parameter less than 3, and the operation beyond that value is not usually reasonable either for frequency standard
or spectroscopic applications of this method.
The calculated inverted Lamb dip amplitudes as a,
function of the standing field intensity W2
=2 V 2,
which is measured in Rabi frequency, for different collisional linewidths r 0’ are presented in figure 6.
These curves correspond to methane molecules at room temperature and to a beam radius of 1.3 mm.
Comparing the intensities corresponding to the maxi-
mum dip values, one will find the dependence of the
saturation intensity on the collisional linewidth
depicted in figure 7. The other curve in figure 7, corresponding to a larger beam radius of 2.1 mm,
clearly shows that in the free-flight regime the
saturation parameter is, indeed, inversely proportio-
nal to the square of the ratio a vo [4]. The calculated
dependences of figures 6 and 7 are in relatively good quantitative agreement with the experimental data,
and has enabled us to calculate the resonance
amplitude versus the laser output power and methane pressure, using only one scaling parameter
(Fig. 5b).
The other very important result, which we would
like to emphasize, is the line shape dependence on
the relaxation parameter Fo T0* Several line forms,
which have been computed using the relations (6), (7), are presented in figure 8, corresponding to very
low saturation fields. As in the experiment pre-
viously discussed, for very low gas pressure levels the typical transit-time line shape is observed
(Fig. 10a), with an obvious sharpening of the top, with a slower decrease of the amplitude of tbe dip
for intermediate detunings, and a faster decrease in the wing, in comparison to the Lorentzian with the
same half-width at the half-maximum basis. Again in
accordance with the experimental data, for relatively
Fig. 6.
-Computed dependences of the Lamb dip amplitudes
versussaturation intensity W2 = 2 V 2 for
To=1kHz (1), To = 5 kHz ( 2 ) , To =10 kHz ( 3 )
and To
=20 kHz (4) and
abeam radius of 1.31
mm.Fig. 7.
-Theoretical dependence of the saturation
intensity W2, corresponding to the maximum dip ampli- tude,
versusthe collisional linewidth 1’o for
abeam radius of 1.31 and 2.1
mm.high pressure levels and very small saturation fields the deviation of the line form from the Lorentzian is
negligible. The curve 5 in figure 8 corresponding to To = 40 kHz (or methane pressure of about 3.2 mTorr) examplifies this case. We also note here,
that for low saturation fields the width of the dip (HWHM) in the free-flight regime (as in molecular beam systems) is determined by the typical transit-
time of molecules across the beam To, in agreement with the experimental data presented in figure 3. For
the adopted model, the calculated width for extre-
mely small collisional rates approaches the value of 0.105 vola at the half-maximum level, when measu-
red in Hz. This width is about 3 times less than the width of the Rabi resonances in molecular beam systems at very low saturation levels. This difference
stems from the fact that the distribution function for molecules effectively interacting with the field is
112 exp ( - 112) for the saturation absorption
Fig. 8.
-Calculated lineshapes at very small saturation levels for r 0 = 1 kHz (1), fo=5kHz(2), r 0 =
10 kHz (3), ro
=20 kHz ( 4 ) , ro = 40 kHz ( 5 ) and
a
beam radius of 2.1
mm.Fig. 9.
-Shapes of the discriminant
curveand the derivatives of the Lorentzians, with the
samewidths
asthe
molecular resonances, for
for
abeam radius of 2.1
mm.method and !? exp (- ’V2) for the molecular beams,
with a substantial deficit of slow molecules in the
latter case.
Fig. 10.
-Change of the shape of the
resonancein the free-flight regimen = 1 kHz, a = 1.31 mum) with the
increase of the saturating field :
The Lorentzians with the
samehalf-width
aredepicted in
dashed lines.
The conclusion that the Lamb dip width is gover- ned by the transit time only in the free-flight domain
is relatively obvious, as the main contribution to the
resonance is due to the molecules arriving through
the boundary beam surface. This contribution is described by the second term in (4), where Pq is
simply a Rabi type solution. As the transition
probability and the distribution function both do not
depend on the collisional linewidth, but the expres- sion (4) does converge, so the widths of the Bennett holes and the Lamb dip are not equal to zero and can only depend on the parameter vda. The correspon-
ding coefficient has been obtained by computation.
The effect of slow molecules is rather weak at the
half-intensity level and the deviations from the
straight line on the dependence of the dip’s width
versus the collisional rate are not very large. The corresponding plot (curve 3 in Fig. 3), computed for
the beam radius of 2.1 mm, is in fairly good agree- ment with the experimental data. The comparison performed between the experimental and the calcu-
lated curves for relatively high pressures, where the increment is found to be equal to 1.27 Fo, gave us
the opportunity to obtain the relation between
Fo and methane pressures. For room temperature, this dependence is found to be 12.4 kHz/mTorr,
while for liquid nitrogen temperature it is 19.2 kHz/mTorr.
As mentioned above, the shape of the discriminant
curve is much more sensitive to the effect of slow
molecules,, as their contribution rapidly increases
with a decrease of the detuning from the transition
frequency. A spectacular confirmation of this effect is presented by the plots in figure 9, corresponding
to very low saturation fields, from which it is clear that for small relaxation parameters the discriminant
curve becomes much narrower than the first deriva- tive of the Lorentzian, with the same half-width at half-maximum as the molecular resonance. The theoretical curve n° 4 in figure 3 shows that in the transit-time regime the width of the discriminant
curve (in contrast with molecular beam systems) is
approximately equal to the square root of the
product of the collisional and the transit-time line- widths.
The suppression of the effect of slow molecules and the change of the shape of the resonance with
the increase of saturating field intensity is illustrated
by the plots in figure 10. As in the experimental studies, the transit-time lineshape is observed at very low intensities, with a typical sharpening of the peak
of the resonance and a relatively large, nearly linear portion of the curve. With the increase of the field slow molecules are rapidly saturated, and it results in
a fast broadening of the top of the line and a shift of
a linear portion of the curve towards larger detu- nings. For relatively high saturation fields, which provide the dips with amplitudes near their maxi-
mum values, the intensity of the resonance in the wing falls faster than the intensity of the correspon-
ding Lorentzian, again in agreement with the experi-
mental data (Fig. 1).
We note, that the domain of field intensities, corresponding to the curves 2 and 3 in figure 10 is
not accessible to the previous theories [5, 6, 9-11], as
the normalized quantity V2/ r2 is much larger than
one. On the other hand, the saturation parameter R is less than one (for example, for the curve 3 it is equal to 0.17), as it may be inferred from the
amplitude values of the resonances. Thus, the field
limitations of the previous theories [5, 6, 9-11] in the
transit-time domain would restrict experimentalists
to extremely small resonance values, for which the
precise frequency measurements could hardly be performed.
A reliable estimate of the saturation parameter for the transition, as usually, may be obtained from the field broadening experiments. For this purpose, the
computed dependence of the dip width on the field intensity for methane molecules at room temperature and for a beam radius of 2.1 mm is presented in figure 11, together with the experimental data deri-
ved from figure 1. It follows from these curves that for relaxation parameters Fo To
=0.12 and 0.24 the relative field broadening is about 1.5 and 1.52,
Fig. 11.
-Calculated field broadening for To
=5 kHz ( 1 ) and To = 10 kHz ( 2 ) and experimen-
tal line (3) for
amethane pressure of 0.6 mTorr.
correspondingly, for the saturation level
( R = 1.2-1.26) providing the maximum dip amp- litude. We emphasize that the curved portions of the plots which are observed at very low field intensities will become practically straight lines, when plotted
versus the field strength V. This is, indeed, the
manifestation of the transit-time effects, which are quite similar to the effects in molecular beam
systems [17].
The amplitude of the resonance in the output power, A, of a laser with an intracavity cell may be
reliably estimated in the transit-time regime, using
the expression :
where P and G are the unsaturated absorption and
the gain in the cells, correspondingly ; d is the
calculated Lamb dip amplitudes is a scaling parameter which gives the relation between the saturation parameter for the gain medium Jo (on the
beam axis) and the saturation intensity W2 = 2 V 2
for the absorption medium, W2
= Uj 0; dJo dF 0 is the
derivative of the saturation function of the gain medium, which for a homogenius broadening and
Gaussian beams is [3]
In order to compare the calculated amplitudes with
the experimental data, we measured the gain of the
tube and the output power of the laser with the evacuated absorption cell versus the discharge cur-
rent, and in this way the parameter M, giving the
relation between the output power and the saturation parameter of the gain tube Jo, was found ( M
=833 tkW . Then measuring the contrast of
the resonance for a specified methane pressure for very small saturation levels, we were able to find the scaling parameter 4 ( pL = 11 200 kHz2 ) using the
equations (8) and (9), and the calculated values of the Lamb dip presented in figure 6. After that, we
were able to find the dependences of the resonance amplitudes on the laser output power and the methane pressure, which are presented in figure 5b.
Comparing them with the experimental data (Fig. 5a) and finding again a fairly good agreement,
we acquired the confidence that the presented theory would give reliable values for the second order Doppler effect frequency shifts, as the lines- hape, the power and the collisional broadening, and
the amplitudes of the dips predicted by the theory
were in agreement with the experiment.
The analysis of such shifts for very low gas pressures will be presented elsewhere [18], so we
shall note only the main results. First of all, for high
saturation levels, when the transit-time effects
become unimportant and the transition probability
in accordance with (1) is practically equal to its asymptotic value of 0.5, the frequency shift is equal
to - 3/2 kT / mC 2, as predicted earlier by [19]. This
value, naturally, differs from the quantity
2 kT/mC2, corresponding to the molecular beam systems for very high saturations, as the distribution functions for the interacting particles are different (Eqs. (6) and (4)). For the saturation levels corres-
ponding to the maximum dip amplitude, the depen-
dence of the second-order Doppler shift on the
modulation amplitude becomes already important.
For example, for the frequency deviation equal to
0.7 of the half-width of the molecular resonance, the calculated shift is approximately equal to
-
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