TIME-RESOLVED LASER SATURATION SPECTROSCOPY

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TIME-RESOLVED LASER SATURATION SPECTROSCOPY

M. Ducloy

To cite this version:

M. Ducloy. TIME-RESOLVED LASER SATURATION SPECTROSCOPY. Journal de Physique

Colloques, 1979, 40 (C1), pp.C1-63-C1-70. �10.1051/jphyscol:1979115�. �jpa-00218394�

JOURNAL DE PHYSIQUE Colloque C1, supplkment au n o 2, Tome 40, fkvrier 1979, page (21-63

TIME-RESOLVED LASER SATURATION SPECTROSCOPY M. DUCLOY

Laboratoire de Physique des Lasers, Associd au C.N.R.S., Universitb Paris-Nord, 93430 VILLETANEUSE

Rdsumb

-

Cet article est consacrb 1 une revue des transitoires cohbrentes. AprZs un bref rappel des transitoires P une onde (prbcession libre, nutation optique, etc

...

^{) ,} on discute plus en ddtail les transitoires P deux ondes, c.a.d. la spectroscopie de saturation en temps rbsolu. On montre que cette technique combinant spectroscopies laser r6solues en temps et en frdquence permet d'dtudier les divers processus quantiques responsables des formes de raies, en particulier ( 1 ) l'interaction entre le laser et le milieu irradib (2) la dynamique de la relaxation du systsme dtudi6. On montre les applications 1 l'btude des transitions de type Raman dans les systsmes P trois niveaux, de 1' blargissement radiatif et de la diffusion dans l'espace des vitesses.

Abstract

-

A review of coherent transients is presented in this article. After a brief discussion of single-wave transients (free induction decay, optical nutation, etc

...),

two-wave transients, i.e.

time-resolved saturation spectroscopy, are treated in more detail. It is shown that this technique of combining both frequency-and-time-resolution in laser spectroscopy allows one to study the va- rious quantum-mechanical processes contributing to the lineshape, in particular (i) the interac- tion between the laser field and the atomic (or molecular) system, and (ii) the dynamics of the re- laxation of the system. Applications to the analysis of Raman-type processes in three-level systems, power-broadening and velocity diffusion are given.

INTRODUCTION

With the development of laser sources, time- resolved spectroscopy has become a very useful tool for exploring laser interaction and relaxation pro- cesses. These techniques can be divided in two ma- jor parts : time-resolved fluorescence following pulse excitation and coherent transients (see e . g . the various contributions in

111

^[2]). The purpose of this article is to review the field of coherent transients techniques, and more particularly, time- resolved saturation spectroscopy.

application of accelerating or decelerating fields over short distances. This technique is potential- ly very promising for the study of coherent tran- sients. The remaining part of this article will be devoted to coherent transients in gases since most of the experiments have been performed with cells.

SINGLE WAVE COHERENT TRANSIENTS

These transients are observed in the transmis- sion of a single monochromatic wave (amplitude E, frequency Q) propagating through a sample cell, when either E or Q undergoes a sudden change. For For the major part, coherent transients have example, when the field is suddenly switched off been studied in gas samples. However one has to at time to, the dipoles coherently excited before point out that time-resolved experiments have been to freely precess and decay. These phenomena may performed on beams with cross-excitation by C.W. be observed on the re-radiated field (Free Induc- lasers. For instance, Ramsey fringes have been ob- tion Decay, FID). On the other hand, if the laser served in a saturated-absorptiop experiment using field is suddenly turned on, it forces atoms to three distinct interaction zones on a neon beam undergo nutations between ground and excited sta-

[3J. Also the free decay of dip~les coherently ex- tes (Optisal Nutation, ON).

cited in a molecular iodine beam has beeit. recently

A powerful method to study these transients analyzed [4J. On the other hand, the Villeurbanne'

-

the "Stark switching'' technique

-

has been intro- s group recently introduced the "in-flight" Lamb-

duced by Brewer and collaborators [6] [7]. The dip technique

[sJ

using a super-imposed beam geome-

principle is to induce a sudden change in the mole-

-

try (ion beam add laser beam propagating parallel

cular resonance frequency by applying an electro- to each other). The ions are brought into or out of

static field which suddenly Stark shifts the ener- resonance by Doppler tuning. The mean beam velocity

gy levels. Suppose, for instance, that for t < o , may be altered abruptly along the ion beam pass by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979115

JOURNAL D E PHYSIQUE

the molecular center frequency is w,. The laser then induces a macroscopic dipole whose axial velocity is centered at v, = (R

-

^w,)/k (k, wave vector). If at t = o, the molecular frequency is suddenly shifted to a new value w:, these dipoles no longer interact with the laser and freely precess at the Doppler-

shifted frequency, w: + kvo = w:

-

^w, + R. The re- radiated field then interferes with the incident field to give a heterodyne beat signal at frequency w:

-

w,, decaying in a time of the order of the transverse relaxation time. Optical nutation and photon echoes may also be observed with this tech- nique. For more detail on the experiments and the information which is obtained in this way (relaxa- tion times, etc

...),

the reader is referred to re- cent reviews [7].

The above experiments are essentially devoted to the time response of the irradiated medium. As is usual in gases the frequency behaviour is very broad, of the order of the Doppler linewidth. Howe- ver, by Fourier-transforming the transient response

[7], it is possible to get some of the characteris-

tic frequencies of the system with a resolution li- mited by the natural linewidth, if these frequencies are lower than a few GHz. To analyze the frequency response over a broader range with the same resolu- tion, it is necessary to use more than one running wave : this is the goal of saturation spectroscopy.

SATURI.TION SPECTROSCOPY

Saturation spectroscopy is performed with two e.m. fields : the first one

-

the "saturating"

field

-

alters the axial velocity distribution of the energy levels over a narrow interval (limited by the natural linewidth of the resonant transi- tion), and this change induces a sharp resonance in the frequency behaviour of the gain for another tu- nable monochromatic field

-

the "probe" field. This may be carried out with either two counter-running waves of the same frequency (saturated absorption, or Lamb-dip spectroscopy), or two co-running (or counter-running) waves of different frequencies, in two

-

^{or three}

-

level systems.

The resonance position yields spectroscopic information

,

while its lineshape provides insight in the system's relaxation. If the relaxation may be characterized with a single decay constant, then

the lineshape is Lorentzian. But several processes often contribute to the signal which becomes more complicated. For instance, in three-level spectros- copy, both Raman-type two-photon transitions and po- pulation saturation contribute [8]. Also velocity- changing processes may broden the resonance, in ad- dition to the phase-interrupting processes which are usually responsible for line-broadenings. In ge- neral, it is quite difficult to separate the diffe- rent contributions in the steady-state lineshape, but, as is shown below, this may be carried out by observing their time-evolution in a transient regi- me.

TIME-RESOLVED SATURATION SPECTROSCOPY (TWO-WAVE TRANSIENTS)

Time-resolved saturation spectroscopy [9] com- bines the methods of saturation spectroscopy and co- herent transients : it consists in observing the transient response of the probe wave to a sudden change in the saturating field amplitude. As an example, consider a conventional saturated absorp- tion experiment (Fig. 1) in which the saturating field is suddenly switched off and the lineshape of the narrow resonance induced in the probe trans- mission is analyzed a fixed interval of time later.

Fig. 1 - Time-delayed Lamb dip

-

As the time delay is increased the change signal will become smaller, corresponding to the decay of the saturated molecules and their return to equili- brium. Furthermore as shown below the shupe of the resonance may evolve from its steady state form.

This information can be combined to form a surface in a coordinate system having axes _: probe intensi- ty (z-axis), frequency detuning (x-axis), time delay (y-axis) (Fig. 2). Sections parallel to the x-axis give the resonance lineshape at delayed times. Simi- larly, sections parallel to the y-axis give the free decay of the system at various values of frequency detuning

.

Notice that the time-delayed lineshapes may bear no relationship to the corresponding decay ti-

Fig. 2

-

Three dimensional representation of the time-delayed change signals.

mes. In fact, different portions of the lineshape may decay at different rates, resulting in a defor- mation of the overall lineshape. Lineshape evolu- tion of this type becomes particularly important when :

(i) Various processes contribute to the inte- raction between laser field and irradiated me- dium, and decay at different characteristic times : Raman-type processes, dynamic Stark splitting, coherence effect, etc...

(ii) The system's relaxation is not simple and includes such different contributions as phase- changing collisions, M-level- changing colli- sions, thermalization by elastic collisions, radiation trapping, etc...

These points are elaborated on in the following sec- tions by analyzing the free decay of saturation re- sonances. However the wide range of techniques to which time-resolved saturation spectroscopy can be applied, also includes optical nutation [lo] and photon echoes in two and three level systems (both cascade and folded), for probe and saturating waves either co

-

or counter

-

propagating.

RELAXATION : DIFFUSION IN THE VELOCITY SPACE Since saturation spectroscopy lies in the se- lection of a particular velocity group by the satu- rating field, every process which randomly changes

the absorber's velocity without perturbing its in- ternal state, will broaden the resonance. This effkct is well-known in Lamb-dip spectroscopy. In general, it is quite difficult in line broadening to distin- guish phase-interrupting processes (responsible for the optical dipole relaxation) from velocity-chan- ging processes which tend to thermalize the velocity distribution. Only strong-collision effects, in which the mean velocity jump is large compared to the natural width, are easily accessible from steady state experiments [I 11. However, in the transient regime, the effect of elastic processes accumulates as time evolves, contrary to the dipole relaxation which is time-independent. Then one can consider the following ideal experiment

-

for a zero natural width

-

^: selection of one well-defined axial velo- city v, by a fixed-frequency saturating field at t = o and subsequent analysis of the return to the equilibrium distribution (time-dependent shift and broadening of the vo- class) by means of the tuna- ble probe field. This kind of experiment in excited neon is in progress in our laboratory.

A time-delayed saturated absorption experiment has been performed on Na by Hznsch et aZ. [12],

using one pulsed dye laser to produce counter-propa- gating beams. A variable delay was introduced on the probe pulse. When Ar buffer gas was added into the Na vapor cell, the authors observed the increase of the resonance width with delay time, showing the

gradual return to the Maxwell velocity distribu- tion in the ground state of sodium. This experiment yielded the average collisional velocity jump, assu- ming the Na-Ar elastic scattering cross-section to be known from earlier atomic-beam studies. It should be noted that pulsed lasers are not ideal for such studies (due to the limited time

-

and frequency

-

resolution), and the use of a single laser beam al- lows in general the excitation of zero-axial-veloci- ty groups only. Similar experiments (but without the frequency tunability) have also been performed by Cahuzac et aZ. [I 33

.

We now turn to the analysis of the laser interaction it-self in the following sec- tions.

RAMAN-TYPE PROCESSES IN THREE LEVEL SPECTROSCOPY In this section, one analyzes the time evolu- tion of Raman-type contribution in three level free

C1-66 JOURNAL DE PHYSIQUE

decay [g]

[lo]

[14t. Consider a three level system, 2-0-1 (Fig. 3). The saturating field (amplitude E2, frequency Q2) is resonant with the Doppler broadened 0-2 transition and alters the populations of levels 0 and 2 over a narrow range of axial velocities cen- tered about v2, satisfying the resonance condition

n2 -

^{k v} _{2 2} = w2 (u2 molecular frequency, k 2 = Q /c) ₂

.

Fig. 3

-

Energy level diagram

This resonant change in the velocity distribution manifests itself in the spectral profile of a Dop- pler-broadened transition, 0-1, sharing a common le- vel with 0-2. If a weak monochromatic field, El(Ql), colinear with the intense field, probes the 0-1 transition, center frequency w l , a sharp change in transmission will occur when Q is tuned into reso-

1

nance with the molecules of velocity v

-

2 '

(E = + I and - I ?or co-propagating and counter-propa- gating waves, respectively). This effect has been the subject of numerous theoretical and experimental investigations devoted to studying the steady state

where L 1 = yO1 + i(Rl

-

^w1

-

^klv) ; n. is the ther- J

ma1 (or background) population of level j ; 6 and a are the Rabi frequencies ( 6 = p2E2/.k and a = pIEl/t', with 11 the 0 - j dipole moment matrix element). As

j

can be seen from Eq. (2), the polarization at fre- quency Q1 is driven by both the population of the common level O(as influenced by E ) and Raman-type

2

contributions (az1 is the macroscopic 1-2 coherence induced by Raman-type two-quanta transitions).

The solution of (2) in the transient regime (6 = 0, t > 0) is given by (if we neglect the time- independent linear polarization proportional to n,

-

^no)

with yo the decay rate of level 0. uO1(0) and uO0(0) are the initial values (t = 0) of the saturated po- larization and population respectively. The last term in (3) describes the coupling of oO1 with uO0.

Since 6 = 0 for t

.

0, transient Raman processes do not occur (as they would in three level optical nutation). But their influence is contained in the initial polarization, along with that of the satura- ted population of level 0 :

change signals at the probe transition. It is now

The resulting probe lineshapes, obtained by integra- well known that the change signal lineshape cannot

ting (3) over velocity, depend on the relative direc- be accounted for in terms of population considera-

tion of the two waves (E = f 1). The signal for weak tions alone, and that coherent Raman-type processes

saturating fields (6 << yij) and close transition such as two-photon transitions play an important ro-

frequencies (wl

-

^{w )} is given by the real part of

le [8]. Thus, for example, the widths of the change 2

-(I'+i&) t signals in the forward (E = + I ) and backward S(E) = (- 1 + -.-)-- I+E I e +

yo 2 y12+i6 r+16.

(E = -1) directions can differ considerablv.

Our purpose is to analyze the transient beha- vior of the change signal lineshape when the satura- ting field is suddenly terminated at time t = 0. The system is described within the density matrix forma- lism. Single relaxation rates, yij, are associated with each i- j density matrix element, ui j . (Veloci- ty-changing processes are no longer taken into ac- count). In the thin sample approximation the change signal is givenlythe velocity-integrated value of Im (u ) since uO1 is proportional to the amplitude

0 1

of the op'tical polarization induced by the probe field. In the slowly-varying envelope approximation uO, obeys an oscillator equation of the type [I41

where I' = y 01 + yO2 and 6 = ill

-

^{Q (E)} is the fre- 1

quency detuning [14]

.

(i) The first term in (5) describes the decay of the initial polarization. Its t = 0 value gives the steady state lineshape in which a directional anisotropy is induced by the Raman contribution (yT2 term). Its decay rate, T, consists of two terms, the ordinary relaxa- tion rate, yO1, and a "Doppler dephasing"

contribution, yO2. This latter term is due to the velocity spread Av2 = y /k of the ini-

02 2

tially excited molecules, which gives rise to a corresponding spread in the reradiated fre- quencies and leads to a destructive interfe-

-

1

rence in a time t yo2. Notice that this term exhibits power-broadening for large saturating intensities (see next section).

(ii) The second term in (5) comes from the cou- pling of the polarization with the saturated le- vel population, and decays at characteristic rate y

0'

Notice that S simplifies noticeably in the im- portant case of a large d i p o l e relaxation

( r

>> Yo, y12) :

The first term describes a broad resonance (of width equal to the natural linewidth), induced by population saturation, which decays at the popula- tion decay rate, yo. This is the only contribution for counter-running waves (E = -1). The second term, which exists for co-running waves only (c = + I), is a narrow resonance of width yI2, induced by Raman- type coherent processes, which decays at the much faster rate

r

= y 01 + yO2, characteristic of the decay of the initial polarization. This leads to the remarkable result that after a time = l/T the narrow

contribution decays away and the forward change si- gnal evolves to a broad Lorentzian identical to that of the backward signal. This behaviour is exampli- fied in Fig. 4.

Experimental checks were performed on NH3 [15]. The v2 asQ (8,7) transition of NH was saturated and

3

probed using two C.W. N 0 lasers oscillating on the 2

P(13) line ( A = 10.78 um), which falls within the NH Doppler profile (Fig. 5).

3

/-

^*

- 4

^{Probe N20 Loser} PZT

I

I

I Soturot ing N20 Loser

I

I I ^{NoCl No Cl}

I 1 . -

- ^t --- -- I

I

I ^{/0 N"3'} 1 \

\---/

/ ^Detector CU-Ge$

Fig. 5 - Experimental set-up

The two laser beams were linearly polarized at right angles, and Brewster angle NaCl beam splitters were used to overlap the beams before the cell and separate them afterwards. The probe beam was monito- red using a He-cooled Cu-Ge detector.

The transient change signals were observed by turning on and off the saturating beam with an exter- nal electro-optic modulator, a GaAs crystal to which high voltage square pulses were applied (rise-time 30ns, duration lous, repetition rate IkHz), thus in- ducing a fast rotation of the polarization of the

C.W. saturating beam. A subsequent analyzer, a Brewster angle silicon plate, yielded the square pulses of the saturating beam. Probe signals analy- zed with a Boxcar integrator are shown in Fig. 6, for various delay times.

Since the NH transition is degenerate (J = 3

8+8) and the two beams have perpendicular linear po- larizations, the saturating field can be considered to induce AM = 0 transitions, and the weak field then probes AM ==+ltransitions. In the absence of Fig. 4

-

Backward (a) and forward (b) change

M-changing collisions the system decomposes into two signals for strong phase-changing collisions

groups of coupled three-level systems having the and weak saturation (wl = w2, yo = y2 = y 1 2 = y,

common level in the ground (g) and excited ( e ) sta- yo, = yO2 = 5y). Time delays : t, = 0, t2 = 0.5/y0,,

tes, respectively. The Raman coherence responsible t3 = 1/yO1, t4 = 2/yO1 and t5 = 4 / ~ ~ ~ .

for the forward-backward asymmetry of the t = 0 si-

C 1-68 JOURNAL DE PHYSIQUE

Fig. 6

-

Time-delayed lineshapes observed in for- ward (a) and backward (b) directions ; p = 30m Torr. The dashed curves give the theoretical fit.

gnals of Fig. 6 is thus the coherence between adja- cent M-Sublevels (AM =

+

^{1 ) .} As predicted by Eq.

(6) and Fig. 4, the Raman-type contribution of Fig.

6 decays away rapidly, leaving a slowly decaying population-induced component which is the same for both propagation directions. Figure 6 gives direct experimental evidence for the two distinct physical processes which contribute to the narrow resonances of laser-induced line narrowing.

However a quantitative analysis of the experi- mental results must take into account the occurrence of M-changing collisions in NH3. As is well known from the symmetry properties of collisional relaxa- tion processes in gases 1161, allowance must then be made for different relaxation rates of the various multi~ole moments of each level. Using the tensorial formalism [16] to account for these features, the following expression for the change signal can be derived [17] :

where L = 2y1 _{e g} + i6 and L: =

yz

^{+ i6. y:g} is the re- laxation rate of the optical dipole polarization and y, the decay rate of the kth order multipole moment k in level a : k = 0, population ; 1, orientation ; 2, alignment. Eq. (7) shows that the width of the narrow Raman-type contribution of the forward signal is primarily determined by alignment relaxation pro-

cesses (ya). A detailed analysis of the experiments 2 [9] [15] shows that the lifetime of the NH3 excited state is much longer than the one of the ground sta- te. The best fit of the experimental lineshapes to Eq. (7) gives the following set of relaxation rates:

These values show that, in NH3,

[la

(i) Raman-type contributions come essentially from the excited state alignment (linewidth ye) and decay at rate 2y1 2 eg (48MHz/T).

(ii) The excited state population provides the dominant contribution to the broad resonance (width 2y 1 ) which then decays at a characte-

eg

ristic rate yz(3.5 MHz/T).

Finally notice that these experiments yield the first measurement of the alignment relaxation rate in the ground electronic state of a molecule.

POWER-BROADENING OF THE SATURATION RESONANCES In this section, one presents a second example of application of time-resolved saturation spectros- copy to the analysis of laser interaction processes, namely the various contributions in power broadening and their origin.

First considerer three level free decay for large saturating intensities. In the case wl = W2 and yii = y, Eq. (3) may be integrated over veloci- ty to give, instead of Eq. (51, the following re- sult valid independently of the saturating intensi- ty ~ 1 4 1 ,

where Q is the saturation parameter,

and y(a = f 1) is the steady-state linewidth (t=O),

~ ( € 1

= $1 t 2 Q

-

^{EQ) (10)}

As in Eq. (5), the first term of Eq. (8) describes the decay of the initial palarization. Notice that, in its decay rate, the "Doppler dephasing" contri- bution, yQ, is now power-broadened. This comes from the power-broadening of the velocity group excited during the preparative stage (the velocity spread is Av2 = y Q/k ) . This broadening of the velocity

02 2

group induces a corresponding power-broadening in

the natural linewidth which directly appears in the denominator of the second term of Eq. (8) describing the transient coupling with the population of the common level, 0. However, in the permanent regime, there is another contribution to the power-broade- ning coming from the dynamic Stark splitting [18]. The strong saturating field mixes the wavefunctions of levels0 and 2, thus causing the transition fre- quency to be split in two frequencies whose posi- tions depend on 6 and on the velocity-dependent frequency detuning,

F -,

In general, this splitting is washed out by the Dop- pler effect and, after velocity integration, brings a new contribution to the power-broadening which may add or subtract to the population contribution, de- pending on the sign of E

[lq.

Its net effect is to narrow the steady-state linewidth for co-propagating beams Cy(+) = y(l+Q)/2] and to broaden it for coun- ter-propagating beams

Ey(-)

^{= y (1+3Q)}

127.

As shown by Eq. (8), the contribution of the high-frequency Stark effect disappears as the ini- tial polarization decays away (in a time of order y-l(l+Q)-l = 6-I for B >> y), and then leaves the population saturation contribution alone (width yQ).

As the backward signal decays from t = 0, its ampli- tude increases and its linewidth narrows, while the forward signal decreases and broadens. This behavior is clearly seen in Fig. 7.

Similar effects also occur in two level sys- tems. For instance, the time-delayed lineshapes in saturated absorption free decay are given by the real part of [ZOJ

r;( 1+5Q)+i61e-E'( '+Q)+~~I

with 6 = ill + i12

-

2w (w, transition frequency). No- tice the close similarity of Eq. (12) with three le- vel free decay

Bq.

^{(8)]. When 6 >>} y, the high-fre- quency Stark splitting contribution decays in a time of order 6-', and the linewidth narrows from about

1.6 6 to 6 [2a.

Narrowing of this type has been reported by Shahin et aZ. [211 in a time-delayed Lamb dip experi- ment on sodium, using a pulse dye laser ( ~ i ~ : 8). As the authors point out, this narrowing (of about

Fig. 7

-

Backward (a) and forward (b) change si- gnals for strong saturation (wl = w2, 6 = 10y).

Time delays : tl = 0, t2 = 0.2/y, t3 = 0.5/y, and t4 = I/y.

23 MHz) must be attributed to the disappearance of the contribution from the dynamic Stark effect. Ho- wever a careful analysis of the experimental condi-

PROBE DELAY (nsec)

Fig. 8

-

Average linewidth in the saturation spec- trum of the Na D2 line versus delay time between

saturating and probe pulse (from Ref. C211).

cl-70 JOURNAL D E PHYSIQUE

tions shows that the power-broadened decay time (= 8-I) is about 3ns, much shorter than the pulse duration, 4411s. This means that, as compared to the effective lifetime of the cltical dipole, the satu- rating field is turned off adiabatically rather than suddenly, and the narrowing then occurs in a delay time of the order of the pulse duration, as observed in Ref. [21] (Fig. 8).

CONCLUSION

In this review, one has presented the princi- ple of time-resolved saturation spectroscopy, and its application to the analysis of laser interac- tion processes and relaxation dynamics in gas phase, by means of the free decay of saturation resonances.

This technique combining frequency

-

^{and time}

-

^reso-

lution is very general, and can be applied to optical nutation, photon echo, Ramsey fringes, etc., in two -or multi- level systems. This technique is also ap- plicable to solids, or beams, as pointed out in the introduction. For example, the "in flight" Lamb-dip technique [5], in which an ion beam is optically pum- ped and probed by means of two successive zones of interaction with a super-imposed laser beam, is quite similar to the forward free decay of saturation reso- nances analyzed in this article. The principal diffe- rence is that ions are observed through their fluo- rescence [22], while gas cells are studied in gene- ral through the probe absorption.

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[I]

Laser Spectroscopy III, Springer Series in Opti- cal Sciences, Vol. 7, ed. J.L. Hall and J.L.

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[7] See the review articles by Brewer e t aZ. in Refs.

[I]

p. 220 and 121 p. 341. A different tzchnique

-

"laser frequency switchingr1

-

^has

a..so be used.

[8] See, for example, M. S. Feld in Fundamental and Ap?Zied Laser Physics, edited by M.S. Feld, N.A. Kurnit and A. Javan (Wiley, New York,

1973), pp. 369-420.

191 M. Ducloy and M.S. Feld, Ref. [I], pp. 243-257.

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^{1465 and}

1469 ( 1977).

[16] For a general review of tensorial formalism and multipole moments see A. Omont in Progress i n Quantwn EZectronics, Vol. 5, p. 69 (Pergamon Press, Oxford, 1977). For their application in laser spectroscopy see B. Decomps, M. Dumont and M. Ducloy in Laser Spectroscopy o f Atoms and MoZecuZes p. 283 (Topics in Applied Phy- sics, Vol. 2, Springer-Verlag, 1976).

[lq

M. Ducloy and M. Gorlicki, to be published.

[IS] S.H. Autler and C.H. Tomes, Phys. Rev.

100,

703 ( 1955)

.

[I~J

However, in the particular case, E = + 1 and w2 > w l , the dynamic Stark splitting is not washed out by Doppler averaging, and the satu- ration resonance is composed of two distinct peaks whose frequency separation is proportio- nal to 13. One shows in Ref.

[lq

that the decay of each peak exhibits a novel type of oscilla- tory behavior.

[ZOJ S. Kumar and M. Ducloy, to be published, and communication to the S.F.P. Colloquium "Spec- troscopie RIsoZue dans Ze Tempsrr, Villeneuve dlAscq, (France), 1978.

[

2]I I. S. Shahin and T.W. ~snsch, Opt. Corn.

8,

^{3 12}

( 1973)

.

[22] Time-delayed %luorescence signals in three le- vel free induction decay have been analyzed in Ducloy et a l . , Ref. [14], pp. 639-641

.