Degenerate parametric emission in dense barium vapour

1 S October 1994

OPTICS COMMUNICATIONS

ELSEVIER Optics Communications 111 ( 1994 ) 6 13-622

Full length article

Degenerate parametric emission in dense barium vapour

Witold Chalupczak, Wojciech Gawlik, Jerzy Zachorowski Instytut Fizyki, Uniwersytet Jagielloriski, ul. Reymonta 4, 30-059 Krakdw, Poland

Received 18 April 1994

Abstract

Studies of degenerate parametric interaction in laser beam propagation through dense barium vapour are reported. Various transverse instabilities have been observed when two strong, copropagating beams intersect in the medium. Most of them are attributed to parametric interaction of the intersecting beams and their main features are consistent with the existing four-wave mixing formalism. The conditions of their generation and their mutual competition are discussed.

1. Introduction

Studies of nonlinear interaction of laser light with

gas media are one of the main topics of contempo- rary optics. These studies are concerned with such effects as changes of a probe beam intensity after pass- ing through a strongly perturbed sample, generation of new light beams via wave mixing, and changes of a transverse beam pattern, i.e., transverse instability of the transmitted beams. Most important in the first group are the three familiar absorption features that can be seen in pump-probe experiments with two- level atoms [ 11. For the pump at wL> w. (where wL, o. are the laser and resonance frequencies, respectively) the probe is amplified when its frequency op z w, + Q’ due to the three photon scattering (TPS ) where the generalised Rabi frequency a’= ( d2+ Q2)‘12, detuning J=wL-wo, and the Rabi flopping frequency O=E-D/A, E is the electric field amplitude of the light wave and D is the electric dipole mo- ment of an atom. For wp= oL-Q’ the probe is absorbed and the third, dispersion-like feature appears for op z wi_.

When in addition to the pump and probe beams a third light beam is taken into account various schemes

of four-wave couplings should be considered. Boyd et al. [2] analyzed four-wave mixing (FWM) in a strongly driven two-level system and showed that strong amplification of the probe may take place for or = wL, wL? Q’. This can be explained as a parametric process in which two oL photons are absorbed and emission of two other photons of frequencies wp and w4, such that 2oL= wp + w4, is stimulated by the probe beam. The four-wave mixing amplitude has its maxima for all three frequencies of the Mollow triplet. The actual gain of the probe beam depends on the pump laser detuning 8, on the intensities of pump and probe beams [ 3 ] and on the relaxation times. Due to

this gain any small signal (laser background, spontaneous emission) at a proper frequency can generate a well measurable signal. The first observations of non-degenerate FWM in a strongly driven system have been performed in the experiments of Plekhanov et al. [ 41 and of Chalupczak et al. [ 5 ] in which it has been shown that there is a threshold for generation of non-degenerate ( wp# wL) FWM below which the effect has to be externally initiated. In the case where collisions dominate the non-degenerate FWM is quenched and the highest gain occurs for wp% wL. One could then expect generation of para-

0030-4018/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved XWIOO30-4018(94)00394-7

614

metric signal caused by !jl quencies close to the pun noted that when the Kab rated from 0,. such a si emission, not a.ffected b;

Another important gro volve changes tof transver: behind a nonlinear sampl here: self-focusming [ 6 1, rir ical emission (CE) [ 81. may also lead to light em thereby changing the tran put light. A typical exampl’ diffraction [ 93. Generatic is another very spectacul verse instabilities. Besides of frequency w,-L?’ is In tuned (o,_> wO) light be2 light cones was seen in a single [ lo] or double ligl the beams could be co- 1 1 ing [ 13 1. The cones were gent, photorefractive crys references therein. Quite ear instabilities occur toge can be analyzed separat’el

The aim of this paper is imental studies of variou ciated with degenerate pai

copropagating laser bean tern of two-level atoms. S either seen or predi [5,9,11,12,15],buttheirl crossing angle has not bee dition to thorough recent terns and wave-coupling cones [ 15 ] and self-diffrzl vestigate in this paper t I characteristics as well as, cl their mutual competition

In Sec. 2 we describe: (I observations of various I qualitative explanation ii wave mixing is presented

W. Chaiupczak et al. /Optics Communications lll(l994) 613-622

ntaneous photons of fre- frequency. It should be

sidebands are well sepa- al is a pure parametric ‘E. 1 of nonlinear effects in- patterns of the radiation Most representative are formation [ 7 1, and con- irametric wave coupling sion into new directions ‘erse patterns of the out- s the phenomenon of self- of various cones of light manifestation of trans-

e CE where the light cone

Iced by a single, blue de- 1 a great variety of other mic vapours with either beams. In the latter case 12 ] or counter-propagat- 11~0 observed in birefrin- Is, see e.g., Ref. [ 141 and .equently several nonlin- ter, yet in some cases they

1 present results of exper- transverse patterns asso- netric coupling when two intersect in a dense sys- ne of these features were .ed in earlier works lation and dependence on discussed. Finally, in ad- tudies of the angular pat- mechanism of parametric tion spots [ 16,171, we in- :ir intensity and spectral editions of generation and

r experimental setup and msverse effects and their .erms of parametric four- t Sec. 3.

2. Experimental setup

A tunable dye laser pumped by a pulsed Nd: YAG laser was used in the described experiments. The pulse duration was about 10 ns, the energy up to 200 uJ and its spectral width was about 5 GHz. The output was split into two beams which propagate under angles *0/2 to the common axis and intersect in the centre of the interaction cell (Fig. 1 a). The intersec- tion angle 0 was varied between 4 and 100 mrad. The beams were weakly focussed with af= 80 cm lens in an oven containing barium vapour at 900°C and some lo- 15 Torr of argon as a buffer gas (the oven did not operate as a genuine heat-pipe oven). The length of the active zone was 5- 10 cm. The laser was tuned to the vicinity of the 6s’ ‘So-6s6p ‘P, (A= 553.548 nm) transition in Ba. The spatial distribution of the radiation behind the oven was recorded with a CCD camera ( 1024 x 1024 pixel) and the light intensities were measured with a photodiode.

The measurements of the absorption linewidth and its dependence on the experimental parameters (buffer gas pressure, barium density) showed that the main relaxation channel was provided by collisions. The homogeneous linewidth was about 10 GHz, i.e. an order of magnitude more than the Doppler width ( z 1 GHz ) and about three orders of magnitude more than the natural linewidth of 19 MHz.

Our experiment differs in some points from the studies already described in the literature: we reduced self-focusing and defocusing of the light beams as far as possible by keeping the light intensities and vapour pressure at a low level. At o, < o. any possible self-defocusing was balanced by a weak focusing of the laser beams. On the other side of the resonance the most interesting features appeared in such a range of detunings where self-focusing was already negligi- ble. This allows us to analyze the phase-matching condition in a most simple form related to plane waves with well defined wavevectors k, and with no need of considering light propagation in narrow, self- trapped channels (see e.g., Ref. [ 41). Another differ- ence is that we vary the angle between the two pumping beams, 8, in a range broad enough (up to 100 mrad) to establish conditions under which various transverse patterns occur either individually or jointly.

W. Chaiupczak et al. /Optics Communications 1 I1 (I 994) 6 I3-622 615

PROBE

BEAM STOP

4 W Fig. 1. (a) Experimental arrangement: dotted lines represent conical emission induced by a single, blue detuned pump beam. (b) Spatial

light distribution behind the oven registered by a CCD camera for 6= 40 GHz. The development of a crescent-like structure close to the

probe beam is observed.

3. Results and interpretation

For blue detuning of the laser light conical emission around the single, strong laser beam is observed. The second, weak beam sent under a small angle 8 to the propagation direction of the pump causes then two kinds of effects. The first is amplification of the probe beam and creation of a bright crescent around it, with the centre of curvature at the pump beam axis, Fig. lb. A similar amplification occurs also for the probe beam derived from an independent laser and tuned to the vicinity of the Rabi sideband of the pump beam frequency. When the intensities of the two laser beams are comparable and wp= w,_ the crescent structures are seen around both beams. The second effect which occurs when op x oL is creation of self- diffracted beam which yields phase-conjugate emission into the direction - ~9 with respect to the pump [ 91. For small crossing angles and comparable intensities higher order self-diffraction beams appear sep- arated from the primary beams by multiplicities of ? 8, such as observed by Heer and Griffen [ 91 and Tan-no et al. [ 17 1. This process can be observed on both sides of the resonance, but the intensities of the spots are higher for the negative (red) detunings. As

noticed above, the non-degenerate FWM can be enhanced by the ac-Stark effect and can occur for the probe beam tuned to the Rabi sideband of the pump

[2,4,51. Another transverse structure is the cone of radia-

tion spanned between the two beams which is seen as a ring on a screen placed behind the cell, Fig. 2. It can also be observed when the frequencies of both beams are close to each other, but is particularly distinct when the frequencies and intensities of both beams are equal. Frequency of the light scattered into the ring is the same as the pumping beams frequency. Such ring structures were observed by Kirilenko et al. [ 111 and later termed in Ref. [ 12 ] two-beam-excited (TBE) conical emission. The TBE ring appears for the light intensity in the beams higher than that re- quired to observe only the crescents and the FWM conjugate beam. It is most intense for laser light frequency lower than the resonance frequency (red detuning) and for angles 8= lo-30 mrad. For bigger angles and/or lower laser energy the intensity along the ring differentiates so that the sections most distant from the primary beams are the darkest. For smaller values of 8 (the beam energies being 100 pJ) additional self-diffraction spots appear on both sides

616 W. Chalupczak et al. /Optics Communications 11 I (1994) 613-622

BEAM STOPS

Fig. 2. Two-beam-excited cone produced by the interaction of two beams of the same intensity and frequency with the medium observed

on a screen behind the cell. 6= - 50 GHz, 0= 30 mrad, I,=40 uJ.

of the primary beams at multiplicities of 8. When 0~ 20 mrad we have observed creation of secondary rings between the primary beams and the self-diffraction spots, Fig. 3. For even smaller angles the ring disappears but higher order spots appear.

Fig. 4 presents integrated intensity of the light scattered off-axis (with both primary beams blocked) versus detuning of the laser light frequency from the atomic resonance registered for various energies. Fig. 4a registered for pump beam energy 60 pJ energy and probe beam energy 50 pJ shows a broad maximum of

the intensity for frequencies below w. caused by creation of the TBE cone. The cone appears for 6 in the range from -90 to 0 GHz and peaks for 6x -50 GHz. A similar maximum of the off-axis intensity for

frequencies above resonance (blue detuning ) is due to several features: conical emission, appearance of crescents and spatial broadening of the laser beams. We also observe creation of a weak TBE ring for or_> oO, shown in Fig. 5, but its intensity is much lower than for the red detuning, probably due to competition with the other processes mentioned

Fig. 3. Spatial light distribution behind the oven in the case of two beams of equal intensity IL= 40 pJ, f3= 10 mrad, demonstrates coexistence of self-diffracted beams with the emission of the two-beam-excited cones spanned between the primary beams as well as between the primary and diffracted beams.

W. Chahcpczak et al. /Optics Communications I1 I(1 994) 613-622 617

0 0 0

03~ - co, [GHz] Fig. 4. The integrated off-axis light intensity as a function of the laser detuning for various probe beam intensities. The pump beam energy is 60 pJ for all recordings and the probe beam energy is (a) 50 ).tJ, (b) 12 uJ, (c) 10 uJ, (d) 7 &I, (e) 4 pJ, and (f) 0.5 uJ, respectively. The broad maximum seen at red (negative) detunings is due to the two-beam-excited cone. The maximum for blue (posi- tive) detunings is caused by several contributions: conical emission, creation of crescents, and spatial broadening of the laser beam. Curve 1 in (f ) depicts absorption spectrum of the weak beam propagating alone across the barium cell, while curve 2 is the weak beam absorption spectrum in the presence of the strong beam of the same frequency. Barium vapour density is 5 x lOI crnm3.

above. Additional complication of the phenomenon is

brought about by another atomic transition 6s6p 3P,- 6s6d 3D1 (Lc553.586 nm) whose frequency oh lies only 38 GHz below the resonant transition being studied. To our knowledge this transition has been mentioned only in one paper [ 18 1, but without any estimation on its oscillator strength. Though no at- tempt to determine precisely this parameter has been made in this work, the magnitude of nonlinear effects observed around oh suggests that the oscillator strength is rather big, on the order of 0. l- 1. Popula- tion of the 3P1 state results from efficient excitation of the ‘Pi state by the strong laser light and subse- quent collisional transfer to the triplet state (enhanced by high vapour density, buffer gas and impurities in the cell ). Curve 2 in Fig. 4f shows absorption spectrum of a weak probe in the presence of a second strong beam, both of the same frequency. A dip in this spectrum seen at ob disappears when the strong beam is absent and does not provide sufficient pop- ulation in the ‘P, state as shown by curve 1 in Fig. 4f. Corresponding dips at wb can also be recognised in

the spectra taken with unequal beam intensities, Figs. 4b, c, d.

When the TBE ring is practically invisible, by mak- ing one beam very weak but keeping the other strong (Figs. 4e, f ), a faint crescent can be seen near the weak beam also for the frequency detuning 6~ - 10 GHz. We attribute it to the interaction with the blue wing of the 3P,-3D, transition. It should be noted that

the intensity of the crescent is determined mostly by the strong beam intensity, thus it does not change much with the increase of energy of the second, weaker beam, contrary to the behaviour of the inten-

sity of the ring which depends very strongly on the intensity of both laser beams. Fig. 6 presents the ring intensity versus the energy of one of the laser beams while the second beam energy is equal to 100 pJ. For small light levels an exponential increase followed by a gradual saturation is observed. For the ring emission the presence of both beams is crucial, it does not occur with a single beam even when its intensity ex- ceeds the sum of the intensities of two beams.

The total intensity of the light scattered off-axially for the blue detuning shows weaker dependence on

618 W Chahrpczak et al. /Optics Communications I1 l(l994) 613-622

Fig. 5. Spatial light distribution b:hind the oven registered by a

CCD camera for the interaction of two blue detuned beams of

equal intensities. Coexistence of the two-beam-excited cones and

crescents is observed, as well as a t right line between the primary

beams.

8

. . . . .

.

. I

.

.

.

Fig. 6. The integrated intensity of the TBE cone as a function of the probe beam intensity. It corresponds to the intensity of the

red frequency maxima of the recordings in Fig. 4.

the weak beam intensity, which explains the strong red-blue asymmetry in Fig. 4.

For low energy in the interacting beams and for an optically thin sample parametric mixing should have

resonant behaviour and should peak at the transition frequency. For higher energy the saturation shows up and a dip in the spectrum appears. The splitting be-

tween thus created peaks is roughly proportional to the Rabi frequency Q. The spectra of Fig. 7 show the total off-axis intensity registered for two different intensities of the equally strong incident beams. The peak for the red detuning due to parametric, TBE cone shifts off-resonance and broadens at higher energy as expected for saturation behaviour. On the other hand, the peak for the blue detuning is caused by several different processes and, therefore, exhibits less pro- nounced shift. We believe that this saturation splitting is the real cause of the disappearance of the TBE cones close to the resonance frequency and not only absorption and self-focusing and -defocusing as stated inRef. [ 121.

Another very interesting feature appears when the saturation parameter defined as the ratio of the Rabi frequency to the homogeneous width reaches higher values. We accomplished it by repeating the experiment with the same beam energies but with much lower buffer-gas pressure (0.5-0.8 Torr) and the same temperature in the oven. The cell worked then in the genuine heat-pipe regime which minimised the effect

a

0.0 I L i I -- ~__~~

-160 -60

detuni& [G$il 160

Fig. 7. The integrated off-axis light intensity as a function of the

laser detuning for different energies of the laser beams: in (a)

both beams have 60 hJ, in (b) both beams have 30 kJ. The shift of the red frequency maximum (two-beam-excited cone) off res-

onance is clearly seen while the blue frequency maximum is not

much shifted.

W Chahrpczak et al. /Optics Communications 1 I1 (1994) 613-622 619

of the rest gases and other impurities. The low energy absorption curve was then only 15 GHz broad, i.e., was much narrower than that of Fig. 4. Thus even with the comparable laser intensities we were able to attain higher saturation parameter.

Fig. 8 presents the light distribution on the screen behind the oven in such conditions. The central part of the picture is attenuated by a 2% transmission filter to avoid saturation of the camera which mask also the TBE cones. New directional emissions are clearly

visible as straight, parallel lines on both sides of the primary beams plane and perpendicular to it (Fig. Sa). For slightly smaller crossing angle (Fig. 8b) we have also observed the self-diffraction spots and another set of fainter lines, parallel to the above ones and appearing halfway between the primary and the self-diffracted spots. The lines are visible at both detunings and disappear faster than the TBE cones when the laser intensity is reduced.

Most of these phenomena are examples of degen-

Fig. 8. Spatial light distribution behind the oven for the case of low buffer-gas pressure. The central part, indicated by a dashed line, was attenuated by a 2% transmission filter. In (a) laser intensity I= 30 p.l in both beams, 6= - 15 GHz, buffer gas pressure p= 0.5 Torr,

straight, parallel lines on both sides of the primary beams plane are clearly visible. In case (b) I= 30 p.l in both beams, 6= - 10 GHz,

p=O.8 Torr; one can observe also self-diffraction spots and another set of fainter lines, parallel to the above ones and appearing between

the primary and the self-diffracted spots. Their location is indicated by the arrows.

620 W. Chahqxzaket al. /Optics Communications lll(l994) 613-622

erate wave coupling. To associate the observed features with specific mechanisms we consider a nonlinear, third-order atomic polarisation PNL:

PNL-~(3)IE12E, (1)

whereXC3’ is the third-order nonlinear susceptibility, E=EI exp(ik,.r)+Ezexp(ik2.r)+EPexp(ik,.r) de-

notes the spatial part of the incident electric field composed of two strong pump and one weak probe waves of amplitudes El,2,P and wavevectors k1,2,P, respectively. All light beams have the same frequency oL. We neglect inhomogeneous broadening which in our case is one order of magnitude smaller than the homogeneous width and two orders of magnitude smaller than the Rabi frequency being about 60 GHz.

PNL has several contributions which are responsible for modifications of propagation of each beam through the perturbed medium as well as emissions into other directions. In particular, one recognises contributions discussed by Heer [ 9,16 ] and Tan-no et al. [ 171, i.e., the terms proportional to ETET exp[i(2k,-kz).r] and E$E: exp[i(2k,-k,).r] which represent beams diffracted into ? 8 angles with respect to the primary beams. For the non-degenerate case this process can be perfectly phase matched, as indeed we were able to observe in the acStark enhanced FWM [ 51. On the other hand, with weak beam energies we observed neither conjugate nor other self-diffracted beams at op = wL. They can only be seen when both the pump and probe beam energies are sufficiently high, i.e., no less than 60 fl in our experimental conditions. Such an energy is necessary to compensate for the phase mismatch Ak= kP which is always nonzero for B# 0. Since Ak increases when 8 is increased, self-diffraction is most efficient at small angles. For 0= 4 mrad and equal energies of the intersecting beams diffraction up to the fourth order was observed in the plane defined by k, and k2. Such high orders of diffraction are associated with wavevectors of the kind mkz - (m + 1 )k, (m being an integer) and obviously require higher-order terms than in Eq. ( 1) or a nonperturbative treatment like the one in Ref. [ 171. For large 8 the phase mismatch becomes too big and self-diffraction disappears. If 8 is around 10 mrad one observes coexistence of self-diffraction with emission of the light cones spanned between the intersecting beams as well as between the diffracted and primary ones (Fig. 3 ) .

The nonlinear polarisation PNL contains also terms discussed by Kauranen et al. [ 12,15 1, in particular the one proportional to El E2E: exp[ i(k, + k2- kp) .r]. This last term is a source of a new wave with vector k4 determined by phase-matching condition k4 = k, + k2 - kp. Such wave can be generated by spontaneous emission and, in particular, as dis-

cussed in Ref. [ 16 1, can propagate into the half-angle between the two waves, i.e. with k4 = kp = $ ( kl + k2). However, the case k4 = kp is by no means the unique possibility to fulfill the phase-matching condition. More general case is when kp and k, are not coplanar with k, and k2 and can form parametric, TBE light cones as described in Refs. [ 12,15 1. The fact that there is perfect phase matching for the degenerate cone emission explains why the cone is still visible when 0 increases and does not disappear as self-diffraction does. The TBE cone emission coexists with self-diffraction for some intermediate angles but for small angles the latter prevails. This can be explained by the stimulated character of self-diffraction that is induced by one of the beams, whereas the ring emission is associated with amplification of sponta- neously emitted photons. By the same token, the di- vergence of the beams causes seeding of photons oL into directions near beams 1 and 2 which enhances the parametric coupling and is responsible for brigh- ter parts of the ring close to the primary beams than those which are most distant. The exponential dependence of the light emitted into the ring on the primary beam intensity from Fig. 6 signifies a strong parametric coupling between photons emitted into spatial directions under angle f e/2 to the bisector of the pump beams. It is thus consistent with the for- mulas given in Ref. [2] and with experimental results of Pender and Hesselink [ 10 ] and of Grigor’ev et al. [ 19 1. This situation is qualitatively different from that described in Ref. [ 5 ] where because of collisional quenching and weak pumping, i.e., low saturation parameter, the gain was small (four-wave mixing signal was found to be linearly dependent on the probe beam intensity) and the emission due to FWM had to be externally triggered by an external

probe beam. It is interesting that the emission along the bisector

direction such as reported by Tan-no [ 17 ] and discussed by Heer [ 16 ] has not been seen in our experiment (it has not been observed in an experiment by

W. Chaiupczak et al. /Optics Communications I1 1 (I 994) 613-622 621

Heer himself [ 201). Such an emission could be perfectly phase matched only for the red detuning which explains its absence in the recording of Fig. 5 ob- tained with oL>oo and for relatively big angles 8. On the other hand, a bright line is seen between the blocked primary beams in Fig. 5. Similarly as in the case of emission into the cone its intensity is higher close to the primary beams. We interpret this feature as a composition of higher-order (multiphoton) wave-coupling of the nk, - mkz kind, the modulation instability discussed in Ref. [ 151 and a non-phase- matched generation of photon pairs into the + 0’12 directions. In addition to the most efftcient, perfectly phase-matched pair emission into k 812 described above, generation of photon pairs which form cones of different apex angles 8’ is also possible. Since it is the Rayleigh scattering which is the only possible mechanism here (we do not see any other frequency component apart from oL), we limit our considera- tions to the degenerate case. For this case, however, there is a phase mismatch Ak=P2-02 which de- creases the efficiency of such generation. Thus, the emission at 19’#0 can be seen only close to the primary beams, i.e., for 8’~ 0, where considerable stim- ulation occurs.

The parallel lines seen in Fig. 8 are connected with the polarisation terms proportional to EIETEp

exp[i(k,-k,+k,).r] and E:E2EP exp[i( -k,+

k2+kp) .r] discussed by Kauranen et al. [ 15 ] under the name nonlinear Bragg diffraction. It is argued there that this term alone cannot be responsible for an amplification of a weak signal and for production of a measurable beam from noise. Our measurements seem to indicate that in fact such an amplification is possible, although its nature (e.g., the lines occurring halfway between the primary and the self-diffraction spots) is not yet clearly understood.

Similarly as in our previous work on ac-Stark effect-enhanced FWM on Rabi sidebands [ 5 1, in the present experiment with two intersecting laser beams of equal light frequency we observe a highly directional, degenerate light emission yielding the charac- teristic crescent-like pattern seen in Figs. lb, 5. The crescents appear only for the blue detunings oL > oo_ They extend symmetrically out of each primary beams if the beams have equal intensity. When one of the beams is much weaker than the other, the crescent extends only from the weak beam. The intensity of

the strong beam affects only the length and bright- ness of the crescent. The amplification profile, defined as the dependence of the integrated crescent intensity on the probe detuning from w. peaks at wL, wL? Q in which it is qualitatively consistent with the spectrum of FWM in a dressed atom [ 21. Systematic study of this effect is still to be done.

4. Conclusions

In this paper we have presented a systematic de- scription of the nonlinear effects observed when two laser beams of the same frequency oL propagate in an atomic medium and intersect under a small angle. For a nearly resonant light perturbation dramatic changes in the transverse patterns of the transmitted beams and an emission of photon pairs into a light cone has been observed and interpreted as a parametric process with an exponential gain. Most of the observations were found to be consistent with the parametric wave coupling theory but very spectacu- lar structures of the crescents and the straight, parallel lines still deserve detailed interpretation. We have also found that due to collisional transfer of the barium resonance excitation the triplet-triplet transition 6s6p 3P,-6s6d 3D1 occurs which is nearly degenerate with the resonance line and so strong that it can cause severe departures from the two-level model.

Acknowledgments

This work has been supported in part by the Polish Committee for Scientific Research (grants 201139101,2P30205205).

References

[ 1 ] See, for example, C. Cohen-Tannoudji, J. DuPont-Rot and

G. Grynberg, Atom-Photon Interactions (Wiley, New York, 1992).

[2] R.W. Boyd, M.G. Raymer, P. Narum and D.J. Hatter, Phys. Rev.A24 (1981) 411.

[ 31 H. Friedmann and A.D. Wilson-Gordon, Phys. Rev. A 38

(1988) 4087.

[ 41 AI. Plekhanov, SC. Rautian, V.P. Safanov and B.M.

Chemobrod, Zh. Eksp. Teor. Fiz. 88 ( 1985) 426 [ Sov. Phys.

JETP 61 (1985) 2491.

622 W. Chahrpczak et al. /Optics Communications 11 I (1994) 613-622

[ 51 W. Chalupczak, W. Gawlik and J. Zachorowski, Phys. Rev.

A 49 ( 1994) 4895.

[6] Y.R. Shen, Prog. Quantum I lectron. 4 (1975) I.

[7] M. LeBerre, E. Ressayre and A. Tallet, Phys. Rev. A 25

(1982) 1604.

[8]Yu.M. Kirin, S.G. Rautiai, A.E. Semenov and B.M.

Chernobrod, Pis’ma Zh. Ekip. Teor. Fiz. 11 ( 1970) 340

[JETP Lett. 11 (1970) 2261;

C.H. Skinnerand P.D. Kleiber, Phys. Rev. A 21 (1980) 151;

D.J. Hatter and R.W. Boyd, Phys. Rev. A 29 (1984) 739;

I. Golub, R. Shuker and G. Erez, Optics Comm. 57 (1986)

143;

Y. Shevy and M. Rosenbluh, J. Opt. Sot. Am. B 5 ( 1988)

116;

J.F. Valley, G. Khitrova, H.M. Gibbs, J.W. Grantham and

Xu Jianjin, Phys. Rev. Lett. 154 (1990) 2362;

W. Chaiupczak, W. Gawlik and J. Zachorowski, Phys. Rev.

A 49 (1994) 2227.

[9] C.V. Heer and N.C. Griffen, Optics Lett. 4 ( 1979) 239.

[ IO] J. Pender and L. Hesselink, J. Opt. Sot. Am. B 7 (1990)

1361.

[ 111 E.K. Kirilenko, S.A. Lesnik, V.B. Markov and AI. Khyzniak,

Optics Comm. 60 (1986) 9.

[ 12 ] M. Kauranen, J.J. Maki, A.L. Gaeta and R.W. Boyd, Optics

Lett. 16 (1991) 943.

[ 131 G. Grynberg, Optics Comm. 66 (1988) 321;

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J.R.R.

Leite, D. Bloch, S. Le Boiteux and M. Ducloy, Optics Comm.

67 (1988) 363.

[ 141 S. Odoulov, B. Sturman, L. Holtman and E. Krltzig, J. Opt.

Sot. Am. B 9 (1992) 1648;

A. Novikov, S. Odoulov R. Jungen and T. Tschudi, J. Opt.

Sot. Am. B 9 (1992) 1654.

[ 15 ] M. Kauranen, A.L. Gaeta and C.J. McKinstrie, J. Opt. Sot.

Am. B 10 (1993) 2298.

[ 161 C.V. Heer, Optics Lett. 6 ( 198 1) 549.

[ 171 N. Tan-no, K. Ohkawara and H. Inaba, Phys. Rev. Lett. 46

(1981) 1282.

[ 181 L. Jahreiss and M.C.E. Huber, Phys. Rev. A 28 ( 1983) 3382.

[ 191 I.S. Grigor’ev, V.V. Likhanskii, A.F. Semerok, V.A.

Firsovand and A.V. Chankin, Kvantovaya Electron 14

(1987) 2082 [Sov. J. Quant. Electron. 17 (1987) 13321.

[20] C. Heer, Optics Lett. 7 (1982) 593.

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Degenerate parametric emission in dense barium vapour