Comment on the paper Transit-time effects in saturated absorption resonances by A. Titov, Yu. Malyshev and Yu. Rastorguyev

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Comment on the paper Transit-time effects in saturated absorption resonances by A. Titov, Yu. Malyshev and

Yu. Rastorguyev

E.V. Baklanov, B. Ya. Dubetsky

To cite this version:

E.V. Baklanov, B. Ya. Dubetsky. Comment on the paper Transit-time effects in saturated absorption

resonances by A. Titov, Yu. Malyshev and Yu. Rastorguyev. Journal de Physique, 1988, 49 (7),

pp.1307-1313. �10.1051/jphys:019880049070130700�. �jpa-00210811�

Comment on the paper Transit-time effects in saturated absorption

resonances by ^A. Titov, ^Yu. Malyshev ^{and Yu.} Rastorguyev [1]

E. V. Baklanov and B. Ya. Dubetsky

Institute of Thermophysics, Siberian Branch of the U.S.S.R., Academy ^{of Sciences, 630090} Novosibirsk, U.S.S.R.

(Requ le 2 novembre 1987, révisé le 17 f6vrier 1988, accepté ^{le 23} février 1988)

Résumé.

Ce commentaire traite d’un important problème ^de spectroscopie d’absorption saturée relatif à la

largeur de la raie d’absorption ^non linéaire dans un gaz dilué ^et dans le champ d’une onde stationnaire. Il montre le caractère erroné des conclusions des références [1-3] ^{tendant à montrer} que, à la limite des faibles saturations, ^la largeur de la résonance est à un facteur constant près, égale à l’inverse du temps de transit d’un atome de vitesse thermique ^{à travers} le faisceau lumineux. Un traitement détaillé de ce problème, ^{donné en} appendice, ^{est en accord avec des travaux} plus ^anciens [5-9].

Abstract.

The Comment discusses an important problem of saturated absorption spectroscopy ^{on the width} of nonlinear absorption ^{resonance in} low-pressure gases in the standing-wave field. The conclusions of references [1-3] ^are erroneous, according ^to which, with small saturations, ^{the resonance width} is, ^{within a}

constant factor, the inverse of the transit time of an atom with thermal velocity through ^the light ^{beam. A}

careful treatment of this problem given ⁱⁿ supplement agrees with earlier works [5-9].

Classification

Physics ^Abstracts

33.70J

42.65G

Recently the Journal de Physique published ^a paper

by ^A. Titov, ^Yu. Malyshev ^{and Yu.} Rastorguyev [1]

devoted to the important problem ⁱⁿ precision spectroscopy ^and frequency standards which is the

lineshape of saturated absorption resonances in a

low-pressure gas. As well ^as the previous ^{works of}

one of the authors [2, 3] ^this paper presents ^{a result} that qualitatively ^differs from the conventional

one [4-9]. This has incited us to turn to this problem

once more. Here we shall try ^to show that the results of [1-3] ^are wrong and ^{are not} related to the critical remarks given ⁱⁿ [1-3] against ^{the works} [4-9].

Since the notations and the methods used in [4-9]

are different, ^we have written a supplement equival-

ent to [4-9] ^{with a} result similar to that obtained in

[4-9]. ^{We shall} therefore compare this supplement

with references [1-3].

Let us now consider the difference between the results of [1-3] ^and [4-9]. ^The problem treated is the interaction of a standing ^{wave that has a} finite beam dimension with a gas. A width of the saturated

absorption ^resonance is found in the limit of a very low saturating field and low pressures, when the

parameter TTo ^{is small} (T ^{is the} collisional linewidth, To

a/vo is the average transit time of an atom through ^the light beam, a ^{is the} light ^beam radius, vo is the average thermal velocity). According

to [5-9] ^{the resonance width is} (r /TO)1/2 ^{and tends to}

zero at low pressures (1). ^In references [1-3] ^{at these}

pressures the ^resonance width is 1 / To, ^{i. e. tends to a}

constant (the ^{resonance widths are} given ^{within a}

numerical factor of the order of unity).

First we shall quote the critical remarks from [1]

(p. 2036) : « First of all, ^we would like to compare the main conclusions of the presented theory ^with

the results of the previous papers [4-8]. ^{We shall}

remind that according ^to these theories the linewidth

at the HWHM level is equal (2) ^{to 1.51} T/To ^and

(1) The results of reference [4] ^{have been obtained with}

the restrictions that do not permit ^{their use to} analyse

saturated absorption resonances in the transit-time con-

ditions (r To « 1 ).

(2) ^In [5] the coefficient 1.51 is obtained by ^numerical integration, ⁱⁿ [6] ^{it is} expressed through the Euler and Catalan constants.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070130700

1308

the width of the discriminant curve is 1.44 r [9] ⁱⁿ

the transit-time regime. ^{In our} opinion, these results

are the mere consequence of the fact that these authors have not taken into consideration the flux of molecules arriving ^at the interaction zone through

the boundary surface of the beam. » However, ^as

shown in supplement, equation (S3) derived within the frame of the approach ^of [4-9] after transfor- mations takes the form of (S5) that takes into account the particles arising ^{inside the beam} (the

first term) ^{and the} particles arriving ^{at the beam} through ^the boundary ^surface (the ^second term).

Moreover, ^the expression ^{for the} absorption ^coef-

ficient (4) ^from [1] ^{and the formula} (S5) ^{of the} supplement coincide when obvious typographical

errors made in [1] have been removed. This means

that the approaches ^of [1] and of the supplement ^are identical, i.e. the critical remarks in [1] ^{are not}

related to [4-9].

The coincidence of the expressions assumed for the absorption coefficient means that the mathemati- cal enunciations made in [1] and in the supplement

are identical. Now the point is how the solution has been found. In the supplement ^the probability ^of

transition in the standing-wave field has been found

by solving ^kinetic equations ^{for an atom} density matrix, ^i.e. by ^the conventional method in the literature. Then after necessary integrations ⁱⁿ (S3)

the expression 2.0 (T/ 70 )1/2 has been obtained for the resonance width in the transit-time domain

(F To -r- 1 ).

In reference [1] the formula (6) ^{for the} standing

wave absorption coefficient has not been derived but written with reference to unpublished ^work (see

Ref. [18] ⁱⁿ [1]). ^{The same} formula numbered (8) ^is given ⁱⁿ [3], where the authors wrote it in analogy

with the one-dimensional theory: ^{« In} approxi-

mation of rate equations ^the expression ^for standing

wave, in analogy with the one-dimensional theory,

can be written in the form... ». This analogy ^cannot

be used in the transit-time case. By analogy ^the

solution is fulfilled in the case where assumption equations ^coincide (e.g. equations ^{for a mechanical}

oscillator and an electric vibrational contour). ^In

case of references [1-3] ^{there is no} such coincidence.

Calculation of the saturated absorption ^{resonance in}

the transit-time regime ^differs from the one-dimen- sional case by the fact that the assumed equations

should allow for a field variation along ^{the atom} trajectory. ^In equation (S8) ^{of the} supplement ^{this is}

allowed for by the time restriction of the field

(cp (t ) ⁱⁿ (S9) depends ^on time). In the one-dimen- .sional theory ^{this is not allowed for as} the time of atom-field interaction ^T is assumed to be much greater ^{than the atom lifetime} r-1 (T T > 1 ).

It should be emphasized that in the case where the

equations ^{can be} solved, ^{there is no need to use an} analogy. The solution of equations ^{for the} density

matrix in the rate approximation under the transit- time conditions in a strong ^{field is} given ⁱⁿ [10].

The use of the analogy with the one-dimensional

case in [1, 2] ^has given unreasonable results. Evi-

dently, ^the absorption coefficient in a gas may be

presented in the form :

where a (v ) ^d3v is the contribution of atoms moving

with velocity ^v within the velocity ^interval d3v to the

absorption coefficient. The absorption coefficient of

a standing ^{wave in} [1] (formula (6)) is of the form :

where y

kv, ^{v is the atom} velocity projection ^onto

the direction of wave propagation. ^The remaining quantities ^are defined in [1]. ^{It is} important ^{for us}

that R and Ll ^are expressed through the function cp ^that contains the integration ^{over a transverse} velocity ^{u. As a} result, ^the integration ^{over trans-}

verse velocities turned out to be in the denominator of the integrand (2). This contradicts (1) and, hence,

the obvious physical conception ^{that atoms with}

different transverse velocities provide ^{an additive}

contribution to absorption.

As a matter of fact, ^{use of} analogy ^has resulted in the fact that the authors of [1] averaged the Bennett

dip ^over transverse velocities and used some effective

Bennett dip ^{to find an} absorption coefficient of a

standing ^{wave. This} practice ^is erroneous. An ab-

sorption coefficient is the product of the Bennett dip

and the transition probability ^that depends ^{on the}

time of atom-field interaction, ^{i.e. on the transverse} velocity. Averaging ^over transverse velocities should therefore be made only in the final expression ^{for an} absorption coefficient.

As far as experimental ^{results are concerned} [1], ^it

is not understandable why they ^coincide quantitat- ively with the results of the incorrect theory. ^There

are fitting parameters (light ^{beam radius,} dipole

moment of transition and so on) ^{in the} theory [1].

Choice of these parameters ^can provide satisfactory

agreement.

The absence of an effect of slow atoms in exper- iment [1] ^{is due to} the fact that in [1] the condition of field weakness V /T ¹ (V ^{is the Rabi} frequency)

has not been realized. The experiment ^{was oriented}

to the theory [1], where the condition of field weakness is given wrongly ^as V To -- 1. In the transit- time regime rTo « 1, ^so much smaller fields are

needed for achieving ^the region ^{of slow atoms} V /T ^1. The saturated absorption ^{resonance in the}

transit-time regime ^was studied in [10], ^{where an}

effect of slow atoms has been detected (resonance

width was - (r / TO )112) ^{and a} good agreement ^with theory has been achieved [5] (see Fig. 1).,

Fig. ^1.

Dependence ^{of the resonance halfwidth} r 1/2 in methane as a function of the transit parameter

TTO (a

^0.25 cm, To

4.5 x 10-6 s). The continuous

curve is the theoretical value in agreement ^with [5-6].

In [1] ^it is stated that the second-order Doppler

effect can be measured by using ^{the data on broaden-} ing and shifts of saturated absorption resonances in the transit-time domain. It should be noted here that in [1] this effect is measured indirectly, ^{so an} analysis

of experimental ^data requires ^{a correct} theory. ^Since

the theory ^of [1] ^is erroneous, such a statement in [1]

about shift measurement related to the second-order

Doppler ^{effect cannot be} trusted. The theory ^of broadening and shift of the saturated absorption

resonance because of the second-order Doppler

effect based on the solution of kinetic equations ⁱⁿ

the transit-time conditions is given ⁱⁿ references [11- 13] ^{which are} considered to be wrong in [1] ^without justification.

Acknowledgments.

The authors thank Profs. S. N. Bagayev, ^{V. P.}

Chebotayev ^{and E. A.} Titov for valuable dis- cussions.

Supplement.

1. Let us consider the interaction of an atom with a resonance wave propagating along ^{the z} axis in the

cylindrical region ^of radius a and length ^{Az. In the}

rest system ^{of an atom} the field is

where E and co are the amplitude ^{and the} frequency

of the field. If ^u and v are the transverse and

longitudinal components ^{of the atom} velocity, ^then

the time of interaction with the field is

T

2 a2 - y2/u, where the y axis is perpendicular

to the z axis and the atom trajectory. The function

g (t ) # ^{0 with 0 t T,} g (t )

exp (i kvt ) ^corres- ponds ^{to a} traveling wave, g (t )

^{2 cos} (at) ^{to a} standing one, where k is the wave vector. The

equations ^{for the atom} density ^{matrix are of the}

form

where f2 = úJ - úJ 21 is the field frequency detuning

with respect ^{to the} frequency ^{of the atomic transition}

w21 between an upper level 2 and ^a lower level 1, V (t )

Vg (t ), ^the quantity ^V

Ed21/1í ^{is assumed}

for simplicity ^{to be a real one,} d21 ^{is the} dipole ^matrix

element of the transition, ^r is the relaxation constant that is, ^for simplicity, ^{taken identical} for all Pik-

An average dipole ^{moment of an atom is}

the energy ^{an atom} absorbs during ^{time T} (when flying through ^a light beam) ^is

Now we shall consider atoms crossing ^an infinitely

small area dy dz with velocities within the intervals

[u, u ⁺ du ], [v, ^{v +} dv ]. ^The contribution of these atoms to the absorbed power dN is equal ^{to an} energy E(7-) ^absorbed by ^{each atom} per number of atoms arriving per unit time dn

dJ dy dz, ^where dJ = no u f (u, v ) du dv is the flux of considered atoms, no is the gas density,

is the Maxwellian velocity distribution function.

Summing the contributions from all atoms and all

points of arrival (3) ^we shall obtain for the power absorbed by the gas (4) :

(3) ^The impression ^{can be} given ^that in this way ^we do not take into account the atoms excited inside the beam. It is not the case. The right-hand ^{part of} (Sl) contains the

pumping ^{to the lower level, which} automatically ^includes

the contribution of these ^atoms in the calculation of

£ ( T ).

(4) Earlier this approach ^to the calculation of absorption

was used in [14].

1310

The absorption coefficient « (,fZ ) is ^{the ratio of the}

power absorbed _{per unit} length N /Az ^{to the total}

field power that in ^case of a standing ^{wave is} W = cE2 a2 :

Now we shall show that this formula is equivalent

to the appropriate ^formula (4) ^{in reference} [1]. ^For

this purpose, by integrating ^{the first} equation (Sl)

from 0 tao ? we obtain

As the condition Pn(0) ^{= 1 is} fulfilled, p 22 (o ) _ P 12 (o )

^{0, then} P 22 (t )

P (t ) is the transition

probability. ^Then

In the first term instead of y and t ^we introduce the polar coordinates r, cp :

in the second we make substitution _y = a sin cp’. ^Then

Thus using (S3) ^we represent ^the absorption coefficient as a sum of two summands. The first summand

corresponds ^to absorption ^{of atoms} arising ^{in the beam,} the second to the contribution of atoms arriving through ^the boundary surface of the beam. Each of these summands coincides to the appropriate ^{term in the}

formula (4) from reference [1] ^{with the obvious errors in} [1] ^{to be} corrected. Therefore we consider the

equivalency ^{of our} assumption ^formula (S3) ^{and the} assumption ^{formula in} [1] ^{to be} proved.

Now we shall derive a formula in quadratures for linear and nonlinear additions to the absorption

coefficient. Let an atom interact with a standing ^wave. The solution of (Sl) ^is sought in the form :

in the right-hand part ^of equations ^for populations ^{the terms} proportional ^to exp (± ² i kvt ) being omitted,

i.e. we use the rate approximation that in the infinite Doppler width limit :

gives ^{the correct} result for weak fields. Then we obtain

where p ^{= ± 1}

Substituting (S6) ^into (S2) ^and neglecting ^the terms - exp (± 2 ikvt ) ^we obtain for an absorbed _energy

The equations (S8) ^{are solved} by iterating ^{over V. We obtain}

where pP ^and ph ^{are the terms of the V S order} in the field expansion ^of non-diagonal ^and diagonal ^elements

of the density matrix, respectively.

The field expansion ^{of the} absorption coefficient is of the form :

where as(n) is the contribution into the absorption coefficient - V’. From (S3, S5, S10, Sll) one can see

that a, (f2 ) ^can be obtained by substituting ^either pP + 1 ^{in (S3) or the} populations ph + 2 in ^{(S5). We shall use}

the first choice. Then for the linear absorption coefficient we obtain

Here t - t, - 7 > (kvo)- 1. Therefore the main contribution to the integrals ^{over v is} provided by ^the regions

v -- -:t f2 /k. Integration ^{over v} gives ^{the term} exp [- (a / kvO)2] (4 ^7T /k ) ⁸ (t - tl ). ^The argument ^{of the 5-} function becomes zero at the boundary ^{of the} region ^of integration ^over tl. ^{A more strict} analysis ^{with use of}

a probability integral ^{from the} complex argument ^{shows that it} should be assumed that

dt2 8 (t - tl ) F (t1 ) = i 1/2 F (t). ^After integration ^over tl, t, ^u, y ^we obtain

Thus, ^{with our} approach by allowing for transit effects we have obtained the known Doppler ^broadened

contour of linear absorption.

Now we shall turn to the calculation of the Lamb dip, the nonlinear correction a2(03A9) ^to (S13). ^{Let us}

consider the region ^{near the line} centre 03A9[ -% max (F, T - 1) K vo. ^{It is} known that atoms with small

velocity projection ^{onto the z} axis I v I :$ I n /K ^{vo are} responsible ^for the formation of a Lamb dip.

Therefore in the expression ^for a2(n) one can assume that exp (- V2/V2) _ ^{1. Then from} (Sll) ^{we obtain}

Since t - tl > ⁰ > t3 - t2, ^{the first summand in the} curly brackets in (S14) may be neglected. ^Then

1312

Let us introduce new variables x

(t - r/2) ^u, 03BE1

(tl - r/2) ^u, 03BE2

t2 - T 2 ) ^{u. In} these variables the Lamb dip ^is

where

for for

2. Now we shall prove that the effect of slow atoms follows from (S15) : narrowing ^{of the Lamb} dip in the transit regime ^{down to the width}

(r / TO)ll2.

Let us assume that for the atoms with thermal transverse velocities the life-time T-1 is _greater than the time of flight through ^the light ^{beam :}

Let us consider a range of detunings such that :

Below we shall prove that this is sufficient for

determining the halfwidth.

Turning ^{to a} dimensionless velocity ^u

u/vo ^we

can see from (S16) ^{that the} following integral ^should

be considered

where p is the complex variable and lp I - (r2 + ⁿ 2)1/2 ^{To, i.e. from} (Sl8) ^and (Sl9) it follows that

For this integral ^reference [15] gives the formula

where C

0.577... is the Euler constant. We shall

now prove (S22). Let 0 1 and 0 2 be contributions to

(S22) ^{from the} integrations regions [0, 8 J ^and [6, oo J, where p ^{6 « 1.} Assuming exp (- Ù2) = 1, passing ^{to the variable z}

plii ^and integrating by parts ^{we obtain} for ~ 1 :

oo

where we used the formula C

Jo ^{dz e- z In z} (Ref. [15], p. 587). Assuming exp (- p/u ) = 1, turning ^to

o

the integration ^{variable z}

^{ii2 and after} integration by parts ^{we have for} 0 2 :

Adding 0 ¹ and 0 2 ^we obtain the formula (S22). Using this formula for the integration ^over

u in (S15) ^we shall obtain the approximate expression for the Lamb dip in the range defined by (S19) :

The Lamb dip ^{width is} determined as a root of the equation :

From a more general ^formula (S16) it follows that a2(oo)

a2(0)/2. The solution of (S24) ^{should be} sought ^{in the} region ^{r «} r 1/2 TO 1. Then ^{we have}

It is seen that the dependence TII2 ’- (F/TO)1/2 in the transit region in the weak field is fulfilled for a field of

arbitrary configuration. ^{For the field} configuration (S17) chosen here and in [1] ^{we have} finally

We should note that the numerical value 2.0(F/TO)112 ^{has been obtained in} [10].

References

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