New phenomena in Doppleron resonances

Volume 38, number 2

OPTICS COMMUNICATIONS

15 July 1981

NEW PHENOMENA IN DOPPLERON RESONANCES R. CORBALAN, G. ORRIOLS, L. ROSO and R. VILASECA Departamento de Fisica Fondamental, UniversitatAutonoma de Barcelona, Bellaterra (Barcelona), Spain

E. ARIMONDO Istituto di Fisica Sperimentale dell'Universit~t di Napoli, Napoli, Italy and Gruppo Nazionale di Struttura della Materia, Sezione di Pisa, Pisa, Italy

Received 9 March 1981 In an inhomogeneouslyDoppler broadened three-level system we analyze the absorption of a probe laser when a strong standing-wavelaser pumps a transition sharing a level with the probe Izansition. Doppleron resonances occur for a pump field tuned off the resonant frequency. Their radiative shifts and strengths versus the pump field intensity are analyzed. The contribution of coherent and incoherent processes are separated. A large signal created by the pump field spatial dependence for the stationary molecules appears at the center of the probe absorption Doppler profile. A comparison is made with an experiment by Reid and Oka (Phys. Rev. Lett. 38 (1977) 67).

Nonlinear absorption in a three-level system is a powerful technique for high-resolution spectroscopy. Steady-state and transient regimes, optical pumping, Lamb dips, cross-over resonances, multiphoton transitions have been observed in a large variety of experiments [1 ]. A typical arrangement for the three-level investigations involve pumping with a strong laser radiation on a transition and probing with a weak laser beam on a transition sharing a common level with the first one. Travelling and standing wave irradiations produce quite different features in the experirnental observations. In the case of travelling waves for the pump and probe radiations, the steady state solution for the population and coherences is expressed in a closed analytical form. For a standing wave irradiation on a Doppler broadened line, analyses based on a variation approach [2], a numerical solution of the density matrix equations [3], a Green function technique [4] or a matrix continued fraction [5] have been presented. However several features o f the non-linear response and their dependence on the parameters of the electromagnetic field interaction have been not yet analyzed. In this letter we will study in detail the coherent and incoherent processes occurring in the case of a

standing-wave pump field tuned slightly off-resonance to an inhomogeneously broadened Doppler line. By sweeping the frequency of the probe field several resonant structures of high order in the laser pump field are observed on the absorption profile, for instance the so called "Doppleron resonances" [6]. Bunches of molecules, velocity tuned to the resonance, are saturated by the pump field and the probe laser detects the saturations through one- or multi-photon processes. These multiphoton processes have been investigated by Reid and Oka [7] on methylfluoride molecules in an infrared laser intracavity experiment. Two laser beams were used to pump and probe the transitions between rotational levels of the ground and first-excited vibrational modes. The laser electric fields were orthogonally polarized and rotational states with different M quantum numbers were involved in the pump and probe transitions. It has been shown by Skribanovitz et al. [8] that the line shape in a pumpprobe experiment on a system composed of three levels having M degeneracy is obtained by decomposing the system into a number of isolated three-level subsystems of the type (J1, M1) -~ (J0, M0) -~ (J2, M2) and summing up over the M values. In this letter we will analyze a three state configuration and show that the

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relevant features in the experiment are produced in each threedevel subsystem. Let us consider a V-folded configuration with a ground state I 0) optically connected by transitions at angular frequencies ¢o10 and 6020 respectively to the upper states I1) and [2). The generalization to other configurations is straightforward [2,5 ]. A standing wave field pumps the 10) - I1) transition and a travelling wave field probes the 10)~ 12) line. The radiation interaction hamiltonian - I t " E has the following elements different from zero in the rotating wave approximation: (it'E)01 =ha

15 July 1981

o

no 1.00

.75

.50

÷ ho

exp[i(~21t-lakz)] ,

~

ho

/a=+_l .25 I

(It" E)02 = h/3 exp [i(~22t -

kz)].

(1)

The flipping frequencies a and/3 are supposed real without loss of generality and the same wavenumber k has been introduced for the radiation fields because the 6o12 splitting is supposed small as compared to both ~01 and w02. The collisional relaxation to states external to the system isdescribed by a rate 7, independent on the 0, 1,2 state, for the populations, and rates 7i/= 7 +'r~ for the off-diagonal density matrix elements, where 3'~ is the dephasing collision rate. Spontaneous decay and cross-relaxations are neglected, but can be easily introduced in our solution [5]. At the thermal equilibrium the ground vibrational state 10) is population Idled with a maxwellian velocity distribution W(o) and the upper states I1) and 12) are empty. The ~21 laser pumping action produces hole burning in the population distribution 'of tl~e ground state 10) through absorption of one or several photons. The population in the I/') states is expressed through a spatial Fourier comP9nent A~.(u)=nOi÷ E

[n~. exp(irkz)+c.c.].

(2)

even

~r>O The z-averaged population n o is plotted in fig. I versus the velocity o along the z axis for the off-resonant pumping A 1 = ( ~ I -- c°10)/2n = 26 MHz, relaxation rates 7/2zt = 0.2 MHz, 3'~. = 0 for all i,j and Doppler width ku/27r = 50 MHz, as in the experiment of ref. [7]. Holes h 7 are produced in the n o population dis114

I

+40

J

i '

J 0

i

J

1

.410

kv (MHz)

Fig. 1. The n o population distribution versus the longitudinal Doppler shift ko. The population in absence of pump radiation is represented by the dashed line. The population was normalized to the ko = 0 value. The pump flipping a = 5 MHz was estimated from ref. [7]. Other parameters in the text. The hf Doppleron resonances with l = 0, 1, 2 appear superimposed to the Doppler profile. tribution at velocities o given by ko0/2n = +(A 1 + a2/A1) ,

kOl/27r = + [ A1 + (2/+1) a2 ) - \ 2 l + 1 l(1~ 1))X 1

t@0.

(3)

The l = 0 Bennet holes are responsible for the main contribution to the standing wave absorption of fig. 1, while (2l + 1) pump photons produce the multiDoppleron resonances. Eqs. (3) include the a 2 lowest order corrections, derived in ref. [6]. For the h~ and hi- Dopplerons, the radiative shift from the -+A1/3 position versus ot2 is represented by the dashed line of fig. 2 for the same parameters of fig. 1. The radiative corrections of higher order in the ct2 parameter, represented by the dot-dashed line of fig. 2, have been derived by our numerical solution, in agreement with the numerical results of ref. [6]. The weak ~ 2 probe detects the modified population distribution of fig. 1. The probe absorption is +oo

-2k~2,G f _00

Im[P12(v)] d°

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Shift

contributions o f ~21 multiphoton processes. Speaking in terms o f the n°(v) average population only at the numerator o f eq. (4), the hole burnings produced by the pump are detected whenever the probe is tuned into resonance with pumped molecules. By scanning the 122 frequency, the probe absorption should show up a gaussian absorption prolrde with dips at the frequencies given by

........ ::~7:.~:::_._

(MHz) 2

~,~

::::::-"-::

o~

..........

-2

s2 2 = ~O2o

-6

2 0'

0

'

4'o

'

s'o ( M H z 2)

!

a

the Fourier component o f the 0 - 2 optical coherence oscillating at the ([22t - kz) frequency. At small/3 values the/)012 coherence is expressed by the following equations, that can be easily derived from the treatment o f ref. [2] :

+/cIo

(5)

with the velocitiesvI obtained by eq. (I).However an inspection of eqs. (4) shows that the (Z 1 + Z_I ) term in the denominator introduces additionalshiftsand broadening of the Doppleron positions.Moreover the probe absorption depends on .thehigh.order spatial Fourier components n~ and ~01. This behaviour comes from the influence of the two-quantum coherence P12 on the probe absorption.As a conse-

Fig. 2. The radiative shifts for the h~ and h i Dopplerons versusa 2, the square of the pump flipping freo~ency. Radiative shifts for the Dopplerons observed on then~ population distribution are represented by the dashed and dot-dashed lines, obtained respectively from the approximate eq. (3) and the numerical solution. The continuous lines give the position of the Dopplerons observed in the velocity-integTated probe absorption. The vertical lines point out the distances 4* between the h I and h I Dopplerons, which has the same value in the population distribution and in the probe absorption.

&o,)

15 July 1981

II 1.0

=-

i.

V ;o

"

~0

": o

..

-2o

-4o

A a 1,,.)

:_

~:50 -

9'02 + i(~22 - co20 - kv) h~

=

....

b

A2 (".J

/~0

X {702 + i [ ~ 2 - 6°20 - kv - ~2(Z 1 + Z _ I ) ] } - 1 , (4) where P~2 and P~2 are the spatial Fourier components of the 1 - 2 and 0 - 2 coherences respectively. The complex Z~ continued fractions take into account the

Fig. 3. At the top the probe absorption versus the frequency detuning A2 for a weak probe # = 0.005 MHz, and parameters as in fig. 1, corresponding to the experiment of ref. [7]. The dashed line reproduces the Doppler broadened absorption profile in absence of pump field. A normalization to the A2 = 0 absorption coefficient was used. Notice the frequency scale inc~eaKng from right to left pointing out that for the given A 1 > 0 detuning the negative kv values produce absorption at positive ~2 values. In the bottom curve the central part of the probe absorption as obtained by an on-off moduiation of the pump radiation is shown. 115

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quence the Doppleron resonances examined on the velocity integrated probe absorption of fig. 3 occur at frequency detunings A 2 = (~22 -- w20)/2n different from those predicted by eqs. (3) and (5). The radiative shifts of the l = 1 Doppleron resonances detected on the weak-probe absorption are represented by continuous lines in fig. 2. It appears that for the parameters of fig. 1 the radiative shifts are quasi-linear in the ~2 intensity, and the frequency distance between h~ and h i- resonances is equal in the nO(kv) and P12(A2) profiles. While the h~ and h~ Dopplerons have the same strength in the population distribution, a line strength asymmetry appears in the probe absorption. This asymmetry arises because the travelling ~ 2 probe photons interact with ~1 copropagating and counterpropagating photons, and in a V-folded configuration copropagating photons produce a Raman velocity-compensated two-photon process between 1 and 2 states. This asymmetry in the resonances for copropagating or counterpropagating waves has been pointed out for the l = 0 resonances in ref. [2] and we produce evidence of the phenomenon for highorder Dopplerons. In fig. 3 a large signal with a dispersion derivative shape appears at the center of the integrated probe absorption. The spatial inhomogeneity of the radiative shifts for the stationary molecules (kv ~ 0) produces that features. The stationary molecules experience a pump field with a cos kz spatial dependence, and the 0 - 2 probe absorption is splitted and shifted by the presence of the pump field. In effect the dynamic Stark effect of the 0 - 1 pump transition gives rise to two resonance frequencies ~22r on the 0 - 2 transition [1] for the molecules at the z position: ~ r ( z ) = 6020 --

-- A1). The strength of the ~22 probe absorption for ko = 0 molecules depends on how many molecules along the z axis have resonant absorption at given probe frequencies, i.e. on the density of the state D(S) that produce resonance at f22:

D(S) o: (OS- /~z) -1"

(7)

The function D(S) has two maxima of equal strength at S~- = 0 and S~, and a minimum at Smi n = 21-S~. The line-shape of the signal in fig. 3, at the center of the probe absorption, closely reflects the behaviour of the D(S) function, with the resonance positions a, c and b produced respectively by the two maxima and the minimum of D(S). Fig. 4a makes evident that the D(S) function provides a faithful description of the center signal. The resgnance positions are plotted

/

j C

-r

al

-4 /

w

J

/

[ - 0

4

8

S+(z)

= 6o20 + ½A 1 + [(1A1)2 + (a cos kz)2] 1/2.

bl (6)

In presence of a A 1 detuning it is well known that the ~2r resonance frequency is nearly equal to the 6020 position of the one-photon resonance, whereas the ~2~r resonance frequency describes a two-photon process, which has a weak transition probability. Restricting our attention to the one-photon resonance, the radiative shift S - depends on the z coordinate of the stationary molecule and it is spread over a ~ frequencies between S i- = 0 and S~- = -~(X/A~ + 16a 2116

15 July 1981

z

0

4

8

a I,,Hz) Fig. 4. In the top the positions of the a, b and c points on the probe absorption versus the square of the pump intensity as obtained by the numerical solution (continuous line) and the density of states description (dashed line). In the bottom curve the strength S of the a peak versus the pump parameter ~.

Volume 38, number 2

OPTICS COMMUNICATIONS

versus the pump intensity, a continuous lines representing the results of a numerical analysis of the probe absorption, and dashed lines showing the maxima and minimum of the D(S) function. Very small differences appear in the two treatments, because the relaxation rates are not included in the density of states description and because of the contribution by/co 4:0 molecules. The a and c points in the probe absorption have radiative shifts 0 and S~- respectively and correspond to stationary molecules located respectively, in the nodal points and in the maxima of the Cos'~kz spatial dependence for the pump field intensity. Lineshapes similar to that appearing at the center of fig. 3 have been found in the treatment of a three-level homogeneously broadened system pumped by a standingwave field [9]. The central feature of the probe absorption lineshape has a large intensity and its strength, measured in units of the 0 - 2 absorption coefficient is plotted in fig. 4b versus the ct flipping frequency. The strength increases approximatively as ct3 and reaches saturation. at a ~ 8 MHz in the condition of our numerical analysis. The strength of the 3-photon Doppleron is proportional to a 6 and at low a values a large stationary-molecule signal appears while the 3-photon Doppleron is barely visible. In the'experiment by Reid and Oka [7] modulation techniques were applied and the Doppler prof'de of the probe absorption was eliminated, as we have done in fig. 3b supposing an on-off modulation of the pump field. Figs. 3a and 3b are very similar to those reported in the above experiment. There the central feature appeared with a slightly different lineshape that may be ascribed to the presence of the Mdegeneracy and a distortion introduced by the Stark modulation in the intracavity experiment. The phenomena here described are determined by the coherent interaction of the ~1 and I22 waves on the system and depend on the existence of the twoquantum coherence P12. Thus if a relaxatiortmechanism is introduced to destroy specifically the P12 coherence, the coherent phenomena disappear. This is shown in figs. 5a and 5b where the probe absorption is reported supposing 712 = 30 and 300 MHz respectively. The introduction of 7~2 destroys the coherent radiative shifts of the Dopplerons and at large 712 the Dopplerons appear at the position given by the population n o and eq. (5). The stationary-molecule signal

15 July 1981

1.

1. .75

.75

,50

20

0

-20

Fig. 5. The velocity integrated probe absorption versus the A2 detuning in presence of a relaxation rate 3,c2, ~,c2 = 30 in curve a) and ~/c2 = 300 in curve b). The normalization to the absorption coefficient of the 0-2 line has been used. is distorted or destroyed by the 742 rate, because the radiative shift S - of the dynamic Stark effect involves the two-quantum P12 coherences. In conclusion we have investigated the three-level probe absorption in a regime with a very large saturation factor, I = 4 a 2 h '2 = 100-1000 as typical of intracavity experiments. This case was not included in the otherwise detailed analysis of ref. [2]. Kyrola and Stenholm [6] have investigated the Doppleron resonances in a two-level system at large saturation factors. However they found that in a two-level system the probe absorption is originated by the simultaneous presence of coherent and incoherent processes in such a way that the resulting lineshape could not be easily interpreted. Instead we have shown that in a three-level system the contributions of the coherent and incoherent processes can be clearly analyzed. The experimental results of ref. [7] on methylfluoride molecules are well described by our numerically reconstructed lineshape. In an experiment of Keil and Toschek [10] on a three-level structure of neon atoms, the two-photon radiative shifts were studied on the probe absorption as function of the pump power, with experimental results similar to those depicted in fig. 2, and that may be analyzed in the framework of our treatment. The authors are grateful to the Centre de Calcul de 117

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la Universitat A u t o n o m a de Barcelona for the assistance in the course of this work.

References [ 11 For a review, see V.S. Letokhov and V.P. Chebotayev, Non-linear laser spectroscopy (Springer Verlag, Berlin, 1977). [2] B.J. Feldman and M.S. Feld, Phys. Rev. A5 (1972) 899; M.S. Feld, in: Fundamental and applied laser physics, eds. M.S. Feld, A. Javon and N.A. Kurnit (John Wiley, New York, 1973) p. 369.

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15 July 1981

[3] A.H. Paxton and P.W. Milloni, Optics Comm. 34 (1980) 111. [4] E. Kyrola and R. Salonma, Phys. Rev. A, in press. [5 ] R. Corbalan, G. Ortiols, L. Rose, R. Vilaseca and E. Arimondo, Book of Abstracts, XII EGAS Conf., Pisa September 1980, p. 75; and to be published. [6] E. Kyrola and S. Stenholm, Optics Comm. 22 (1977) 123; 30 (1979) 37. [7] J. Reid and T. Oka, Phys. Rev. Lett. 38 (1977) 67. [8] N. Skribanowitz, M.J. Kelly and M.S. Feld, Phys. Rev. A6 (1972) 2302. [9] E. Kyrola and R. Salomaa, Appl. Phys. 20 (1979) 339. [10] R. Kefl and P.E. Toschek, Soy. J. Quantum Electron. 8 (1978) 949.