A Kivimäki
email: kivimaki@tasc.infm.it

J Alvarez-Ruiz

J Wasowicz

C Callegari

M De Simone

M Alagia

R Richter

M Coreno

T J Wasowicz

O 1s excitation and ionization processes in the CO2 molecule studied via detection of low-energy fluorescence emission

Keywords: numbers: 33, 50, Dq,33, 80, Eh

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Introduction

Several kinds of electronic transitions take place at and above core-level thresholds of (simple) molecules. A core electron can be excited to an empty valence orbital or to a Rydberg orbital and/or the molecule can be ionized. Furthermore, these basic transitions can be accompanied by the simultaneous excitation or removal of a valence electron, leading to double excitations, shake-up and shake-off processes. All the resulting core-hole states decay very fast, in the femtosecond timescale, by emitting Auger electrons or, more rarely, soft x-ray photons, or in some cases by ultrafast dissociation. In order to help the reader to follow the present study, we depict in figure 1 different core excitation and ionization processes as well as their main decay channels through electron emission. (Radiative decay and ultrafast dissociation will not be discussed in this paper.)

Information on core-hole processes has traditionally been obtained by detecting the energy taken by the molecule (photoabsorption and electron energy loss spectroscopy) or by observing any primary particle created in de-excitation events (electron, ion, and soft x-ray emission spectroscopies). Ultraviolet-visible (UV-Vis) fluorescence spectroscopy does not detect any of the primary particles, hence it does not appear particularly suitable for studies of core electron phenomena. However, the final states of the Auger decay and soft x-ray emission process are usually excited ones, and they may directly emit low-energy photons, or they may dissociate into excited fragments that fluoresce. Fluorescence in the VUV, UV, visible and near infrared wavelengths generally constitutes the last step in the de-excitation pathway of core-hole states. The question is whether monitoring such emission can tell something about the process that created the initial core hole. The best chances for that occur when fluorescence immediately follows resonant Auger decay, in which case it can be used as a high-resolution technique to study core-hole decay. For instance, this condition is fulfilled in the N 2 molecule by resonant Auger transitions from the N 1s -1 π * state to the three lowest excited states of the N + 2 ion which all can decay via fluorescence emission to the ground state of N + 2 , see e.g. [1,2]. The situation is less promising for emission from fragments, as the same excited dissociation products can be created starting from different kinds of core-hole states. Using N 2 again as an example, the excited N + ion emitting at 500 nm is created at the excitations of the N 1s electron to the π * and Rydberg orbitals, but also in the N 1s ionization continuum at the positions of the double excitations and the shape resonance [3,4]. On the other hand, the creation of excited neutral N atoms may be limited only to the de-excitation of neutral core hole states, including doubly excited states [4]. Therefore the applicability of UV-Vis emission as a probe of core electron processes greatly depends on which emission is measured as a function of photon energy.

Particularly informative has been the detection of Balmer emission due to excited H atoms, that is H(n > 2) → H(n = 2) transitions. It has been observed that Balmer emission is increased at the core-to-Rydberg excitations as compared to the core-tovalence excitations in H 2 O, H 2 S, NH 3 and CH 4 [5,6,7]. The different members of the Figure 1. A simplified presentation of core excitation, core ionization and Auger decay processes. Following abbreviations are used: CO = core orbital, VO = valence orbital, UO = unoccupied orbital, IP = ionization potential, SR = shape resonance. In the designated configurations, K = core hole, V = valence hole and e = electron in an unoccupied orbital.

Balmer series were found to have their intensity maxima at different photon energies: the higher the principal quantum number n was, the closer to the core ionization potential (IP) peaked the intensity of the H(n ) → H(n = 2) transition. This behavior was explained by the spectator resonant Auger decay of the core-excited states to two-hole one-electron (2h-1e) final states (these states are designated as VVe in figure 1) whose subsequent dissociation yields excited H atoms. The excited electron may shake to another orbital of the same symmetry during resonant Auger decay, but in general 2h-1e final states with an electron on a high-n Rydberg orbital are expected when the initial core excitation promotes a core electron to a high-n Rydberg orbital. If dissociation of the molecular ion in a 2h-1e Rydberg state is adiabatic, one of the fragments gets an excited electron in a Rydberg orbital with the same principal and orbital quantum numbers [5]. In that way highly excited H atoms can be produced by exciting molecules just below their core IPs. In the H 2 O, NH 3 and CH 4 molecules also core valence double excitations, which are located above the core IPs, appear more prominently in the excitation functions of the Lyman and Balmer emission than in the total ion yield [8,9]. The emission remained at an elevated level even at higher photon energies, which was attributed to fluorescence emission following the de-excitation of shake-up states to which doubly excited states converge. It was concluded that emission from excited H atoms is enhanced at any core electron transitions that involve the population of a Rydberg orbital [8].

Coincidence experiments incorporating photon detection can bring additional information on the characteristics of fluorescence emission and on pathways leading to it. They can also help identify weak emission that can not be resolved in conventional dispersed fluorescence measurements. Photoelectron-fluorescence (PEFCO), threshold photoelectron-fluorescence (TPEFCO) and photoion-fluorescence coincidence (PIFCO) techniques were introduced already in the 1970s [10,11,12]. They can be used to correlate fluorescence emission to the discrete excited state of the molecular ion and to determine the radiative lifetimes and fluorescence quantum yields of those excited states. In PEFCO spectroscopy, the electronic state of the ion is selected by the measurement of photoelectrons in the kinetic energy range of interest. TPEFCO is a variation of PEFCO where the kinetic energy is restricted to be very close to zero, only few meV. The ionic state is instead chosen by scanning the incident photon energy. It is possible to achieve vibrational resolution with this technique. In PIFCO spectroscopy, the detection of an ion (with a time-of-flight (TOF) spectrometer) fixes the ionic species, and the specific excited state of that ion can be selected by narrowing the bandpass of the detected fluoresence. The first coincidence studies [10,11,12] used photomultiplier tubes (PMT) for photon detection in the UV and visible range. The application of the PIFCO technique was extended to the studies of inner-shell phenomena by Meyer et al [13]. They found that UV/visible photon -photoion coincidence spectra at the N 1s edge of N 2 and N 2 O as well as above the I 4d edge in ICN are dominated by fragment ions, while fluorescence in the parent ion is far less important. Quite recently, Taylor and Eland [14] have combined the detection of electrons, ions and photons in a single experimental set-up. By measuring ion spectra relative to electrons, but in coincidence with photons (using a PMT with 200-500 nm sensitivity range), they could improve signal to noise ratio and observe emission in fragment and parent ions of several diatomic and triatomic molecules. The same authors have later developed an electron-electronphoton coincidence setup, where a magnetic bottle electron TOF spectrometer was used to observe an electron pair resulting from double photoionization and the abovementioned PMT collected photons emitted by the N 2 O 2+ or CS 2+ 2 dication [15,16]. The PIFCO technique has also been exploited to study dissociative photoionization [with] excitation, i.e., photon-induced ionization processes where the parent molecular ion dissociates and fluorescence emission occurs in an excited neutral fragment [17,18].

In those studies, undispersed VUV photons were detected with a channeltron electron multiplier [17] or with a microchannel plate (MCP) detector [18]. Moreover, coincidences between two VUV photons, e.g., in the H 2 molecule have been observed with an experimental setup that consists of two MCP detectors facing each other [19]. The events were attributed to neutral dissociation of doubly excited states, upon which two excited H atoms are created. Also soft x-rays have been exploited in coincidence experiments: the detection of threshold electrons in coincidence with soft x-ray emission allows one to study core photoionization at the very threshold without the post collision interaction (PCI) effect that occurs when the core hole decays by Auger electron emission [20].

We were interested to investigate the sensitivity of fluorescence emission as a probe of core excitations in molecules that do not contain H atoms. The O 1s edge of the CO 2 molecule is a good choice for such a study because it shows a rich collection of various core excitation phenomena. The pre-edge O 1s photoabsorption spectrum contains excitations to both valence and Rydberg orbitals, while the above-edge part reveals double excitations and shape resonances, see e.g., [21,22,23,24,25,26]. Furthermore, fragmentation of the CO 2 molecule after core excitation and core ionization has been studied extensively using different electron -ion and ion -ion coincidence spectroscopies, see e.g., [27,28,29,30,31], which can provide complementary information to the present investigation. We can observe the production of excited O and C atoms as well as of excited O + and C + ions, as these species have strong emission lines within the operation range (190-1000 nm) of our fluorescence spectrograph. We followed the intensity of some selected emission lines across all the core excitations up to about 40 eV above the O 1s ionization potential (IP). Preliminary results of a neutral oxygen emission line at 845 nm were reported in Ref. [32]. We also recorded a non-dispersed fluorescence yield using a photomultiplier tube (PMT). In an independent experiment, photoions were measured in coincidence with undispersed VUV photons, which gives information on the production of ions in highly excited states.

Experimental details

The experiments were performed at the Gas Phase Photoemission beamline [33] of the Elettra synchrotron radiation laboratory (Trieste, Italy). The beamline uses radiation from an undulator in the photon energy range of 14-900 eV. Synchrotron radiation was monochomatized by a spherical grating monochromator, equipped with a planar premirror, and sent to the experimental station. The photon energy scale was calibrated against the O 1s → π * resonance of CO 2 at 535.4 eV [21].

In the dispersed and non-dispersed fluorescence experiments, a hypodermic needle was used to introduce the CO 2 gas just below the incident photon beam. The light emitted from the interaction region was collimated by a spherical Al mirror. The collimated beam exited the vacuum chamber through a quartz window and was focused with a lens onto the entrance slit of a Minuteman 305MV fluorescence spectrograph. Fluorescence light was dispersed with a 1200 lines/mm grating and detected with a liquid-nitrogen cooled CCD detector (Princeton Instruments 10:100B). The detection sensitivity curve of the experimental system was recently determined [34] and is used here. The scheme of the experimental set-up was also presented in [34]. In the measurements, the incident photon energy was scanned in small energy steps in the excitation region of interest and the fluorescence spectrum was collected at each energy point typically for five minutes. The intensity of the emission line was obtained by integrating the counts over the peak. The total ion yield was measured simultaneously using a microsphere plate detector operating in current mode. The photodiode at the exit of the vacuum chamber monitored the photon flux for normalization purposes. An undispersed fluorescence yield of the CO 2 molecule was measured using a photomultiplier tube Hamamatsu R6095 whose operation range is 300-650 nm, as given by the manufacturer. It was installed after the same exit window that was used for dispersed fluorescence experiments.

The ambient gas pressure in the experimental chamber was 8•10 -5 mbar during fluorescence measurements, and it is estimated to be 10-50 times higher in the interaction region. The gas pressure was few times higher than in our previous experiments (e.g., [39]) because in the present study we aimed to probe processes of rather low cross sections. We cannot completely rule out the possibility that ejected photoelectrons and Auger electrons may have caused collision-induced fluorescence emission. Such contamination would affect the intensity ratios between the fragment and CO + 2 emissions at a given photon energy as well as the intensity behaviour of a given emission line or band between the regions of high and low photoabsorption cross section. No such effects are evident in our results, but we limit the discussion of relative intensities in the fluorescence excitation functions to a qualitative level.

During another experimental session at the Gas Phase Photoemission beamline, VUV photon -photoion coincidences were measured at the O 1s edge of the CO 2 molecule. The scheme of the experimental set-up is presented in figure 2. Fluorescence photons were detected with a microchannel plate (MCP) detector. The MCP used can detect photons with wavelenghts below about 150 nm. The MCP was wired for ions, hence electrons were prevented from being detected by the high voltage at the front side of the MCP. A mesh (90 % transmission) was installed in front of the MCP and set at a positive voltage (+300 V) to repel positive ions. The MCP had a diameter of 40 mm and it was located ∼17 mm from the interaction region in the direction of the electric vector (E) of the linearly polarized incident light. Consequently, VUV photons were collected along the E vector within the cone whose aperture was about 100 • . An ion time-of-flight (TOF) spectrometer, with a drift tube length of 105 mm, was located on the opposite side of the interaction region, also along the light polarization vector. It was operated using constant voltages that fulfilled the Wiley-McLaren space focusing condition and that allowed to collect with 4π solid angle ions whose kinetic energies were less than ∼12 eV. The signals from the detectors were fed into a Time-to-Digital Converter system (TMD-GPX from ACAM [35]) in order to record arrival times. The photon signal provided the start signal, while the arrival of photoions generated the stop signal. The Figure 2. The scheme of the experimental set-up used to measure VUV photonphotoion coincidences. The incident photon beam comes from the back side of the drawing and crosses the effusive gas jet in the interaction region, which is shown by a dot above the needle. The potentials applied to a few surfaces are given.

acquisition system was configured to operate with 81-ps time resolution. Typical count rates for photoions and VUV photons were 40000 and 200 per second, respectively. The gas pressure in the experimental chamber was about 2•10 -6 mbar during the coincidence measurements. VUV-photoion coincidence spectra were collected usually for 45 min at each selected photon energy. Conversion of the coincidence spectra from the time scale to the mass/charge scale was achieved by the measurement of the mass spectrum of Ar with identical voltages in the ion TOF spectrometer.

Results and discussion

Dispersed and undispersed fluorescence experiments

The ground-state electronic configuration of the CO 2 molecule is 1σ

2 g 1σ 2 u 2σ 2 g 3σ 2 g 2σ 2 u 4σ 2 g 3σ 2 u 1π 4 u 1π 4 g ( 1 Σ + g )
, where the 1σ g and 1σ u molecular orbitals (MO) correspond to the atomic O 1s orbitals. The molecule has three virtual or unoccupied valence MOs: 2π u (often denoted as π * ), 4σ u and 5σ g . The O 1s photoabsorption spectrum of CO 2 shows an intense resonance due to the 1σ g → 2π u excitation at 535.4 eV [21]. The O 1s excitations to various Rydberg orbitals are also located below the O 1s edge. Their symmetries were nicely revealed by the angle-resolved ion-yield measurements of Okada et al [26]. Watanabe et al [23] applied the same technique in the O 1s continuum, showing clearly that the σ * u and σ * g shape resonances due to the 1σ g → 4σ u and 1σ u → 5σ g excitations appear at about 559 eV and 542 eV, respectively. These assignments have been confirmed by relaxed-core Hartree-Fock calculations of Saito et al [36]. A structure at ∼554-eV excitation energy in the Σ → Π transition channel was also observed in [23]; it has been assigned to corevalence double excitations. Our total ion yield spectrum, shown as a solid line in figure 3, displays these features. Figure 3 also contains the excitation function of the 845-nm emission line, which is due to the 1s 2 2s 2 2p 3 3p 1 ( 3 P) → 1s 2 2s 2 2p 3 3s 1 ( 3 S) transitions in neutral oxygen (a dispersed fluorescence spectrum was reported in [32]), and the partial fluorescence yield measured with the PMT that is sensitive to the light at wavelengths 300-650 nm.

Whereas the non-dispersed fluorescence yield mostly mimics the TIY curve, the excitation function of the neutral oxygen line has quite a different shape. The Rydberg excitations appear much stronger in the oxygen emission than in the TIY. It was shown in Ref. [32] that the enhancement in 845-nm emission follows resonant Auger decay of the core-excited states. The emission weakens gradually just above the O 1s IP. A similar behavior of the same oxygen emission line at the O 1s excitations in the water molecule was previously correlated with recapture processes induced by the PCI effect [39]. The effect is general and is by no means limited to oxygen atoms. Indeed, the recent observation of metastable H atoms just above the O 1s IP of the water molecule was explained using the same mechanism [40]. Briefly, just above the core IP, a fast Auger electron can overtake the slow photoelectron and push it back towards the ion where it ends up in an empty Rydberg orbital. Thus 2h-1e states can be created both below and above the O 1s threshold, and their dissociation can lead to the production of the same excited fragments. The probability of the PCI recapture decreases fast with the excess energy, which is the difference between the used photon energy and core ionization energy.

The σ * g shape resonance around 542 eV can not be seen in the oxygen excitation function at all, but it is visible in the TIY and in the partial fluorescence yield. The excitation function of the oxygen emission line displays a clear feature at ∼549 eV with a shoulder at ∼547 eV, whereas TIY looks quite flat in the same energy range. A resonance around these energies has been observed in the yield of O -ions [25], and it was attributed to double excitations. We concur with that interpretation. The energies of these features (547-549 eV) agree very well with those estimated for the core valence double excited states, obtained by adding the energy of the O 1s → π * excitation (535.4 eV) and the shake-up energies of 11.5-13.8 eV for the O 1s photoelectron satellites S 1 -S 3 , as labeled in [37] and in figure 3. In other words, the same valence-to-virtual orbital excitation is suggested to accompany both the O 1s → π * transition in core valence double excitation and O 1s photoionization in a shake-up process; the 1π u → 2π u monopole excitation is likely involved here, similarly to the case of the C 1s shake-up transitions [22]. The better-known double excitation feature around 554 eV also appears with high intensity in the excitation functions of the 845-nm oxygen line.

The next feature in the excitation function of the oxygen line peaks at a lower photon energy and ends earlier than the σ * u shape resonance in the TIY. The differences are so clear that this feature in the excitation function of the 845-nm line may not be caused by the σ * u shape resonance, but more likely by other kind of electronic transitions. Maier et al [37] observed that the photoionization cross sections of the two unresolved O 1s shake-up satellites S 1 and S 2 experience a huge intensity increase around 6-eV kinetic energy. The shake-up energies of these satellites are about 12 eV (see figure 3). Taking into account the O 1s binding energy of 541.254 eV [24], the increased population of shake-up satellite states occurs around 559-eV photon energy, i.e., close to the maximum of the feature in the excitation function of the 845-nm oxygen line. This process would explain the differences in the resonant position and width between the excitation function of the oxygen line and TIY. If a shake-up satellite experiences a resonance, it implies that KVe type of states, where K stands for an O 1s hole, are created more abundantly (see row 3 in figure 1). Because of the electron in a virtual orbital after shake-up photoionization, subsequent Auger decay produces final states that have spectator electrons. When such final states dissociate, excited fragments may be expected. In that way resonances in shake-up satellites cross sections may increase the number of excited fragments and, consequently, enhance fluorescence. Indeed, we suggest that the feature about 559 eV in the fluorescence yield originates at least partly from O 1s shake-up photoionization.

It appears that in CO 2 fluorescence emission from the excited oxygen atoms is less probable at the shape resonances than at double excitations. This observation can be rationalized as follows. The shape resonances decay via tunneling to the (O 1s) -1 state which further decays via normal Auger transitions (see row 5 in figure 1). The final states of normal Auger decay generally have two valence holes (and no excited electrons). On the other hand, core-valence doubly excited states can decay by resonant Auger transitions to a multitude of final states ( see row 6 in figure 1). The Auger spectra of CO 2 measured at photon energies around 554 eV, corresponding to the core valence double excitations, gave experimental evidence of transitions to VVe or VVVee final states [37]. Calculations for double excited states above the 4d ionization limit of the Xe atom predict that such decay channels are the most probable in which both the excited electrons remain as spectators [41]. Spectator electrons are thus expected to feature with high probability in the final states after Auger decay of core valence double excitations, but much less so at the shape resonances. As discussed above, the presence of electrons in virtual orbitals (= spectator electrons) after Auger decay leads to an efficient creation of excited fragments. In the specific case of CO 2 , some lowest-energy doubly charged states are quasi-stable [42], as is also indicated by the resolved vibrational structure in the recently published high-resolution Auger spectrum after C 1s ionization [43]. To the best of our knowledge, fluorescence emission has not been observed between the states of CO 2+ 2 . Thus Auger decay to quasi-stable final states probably further reduces the fluorescence emission yield at the shape resonances of CO 2 . Note that the shape resonances are also absent or very weak in the partial ion yields of O -and C -measured above the O 1s IP of CO 2 [25].

We have also remeasured with high photon energy resolution the excitation function of the 845-nm oxygen line in order to search for a possible correlation between fluorescence emission and the vibrational energy content of the core-to-Rydberg excited states. In the previous low-resolution study (∆hν ∼ 0.8 eV) [32], the yield of this oxygen line seemed to be systematically enhanced at photon energies slightly higher than in the total ion yield. In the new measurements, the incident photon energy resolution was improved to about 0.15 eV and the spectral resolution for fluorescence photons was slightly relaxed to ∼2.5 nm. Despite somewhat low statistics, the new results clearly show (figure 4) that the TIY and the excitation function of the oxygen line are aligned also at the Rydberg resonances around 539 eV. Thus vibrational excitations that accompany a core-to-Rydberg electronic transition is not found to increase the probability of subsequent fluorescence emission in the oxygen atom.

The partial fluorescence yield (wavelength range 300-650 nm) from all core-to-Rydberg excitations appears reduced as compared to the TIY. This behavior is just opposite to that of the excitation function of the 845-nm oxygen line. The most likely reason for the enhanced partial fluorescence yield at the O 1s → π * excitation is participator decay that populates the single-hole states of the CO + 2 ion [44,45], followed by fluorescence emission to lower ionic states of the parent ion. In particular, the A 2 Π u → X 2 Π g emission system of CO + 2 occurs at wavelengths between 300 and 475 nm [46] and is detected by our PMT. Participator decay from core-to-Rydberg excited states usually has a very low probability due to negligible interaction between the ionic molecular core and the electron on the diffuse Rydberg orbital. The intensity of the partial fluorescence yield at the Rydberg excitations and above the O 1s edge should mostly come from excited O + and C + ions that have numerous emission lines between 300 and 650 nm [START_REF] Yu | NIST Atomic Spectra Database (ver.4.0.1)[END_REF]. Some of them could be identified in the dispersed fluorescence spectrum (figure 5) that was measured at the O 1s → π * excitation.

The intensity behavior of the emission lines in the window 422-462 nm was followed in the photon energy range 522-588 eV with 0.4-eV energy steps by measuring the emission spectrum with 2.5-nm resolution for fluorescence photons. The excitation functions obtained are shown in the upper panel of figure 6 together with TIY. All the curves have been scaled to have the same amplitude at the O 1s → π * excitation. The two O + peaks at 441 and 459 nm displayed similar trends, hence only their sum is reported. All the excitation functions show the double excitation feature at 554 eV and the shape resonance around 559 eV. The excitation function of the measured C + line also resonates just above the O 1s IP, which may be due to the σ * g shape resonance. These findings imply that the production of excited ions is possible after shape resonance excitation, its subsequent decay and dissociation. At photon energies above 570 eV, one can observe that all the excitation functions of the ionic emitters gain strength relative to TIY. This is more clearly visible in the lower panel of figure 6, where we report the intensity ratios of the emission lines and TIY. Note that the possible photon energy dependence of the photodiode current is cancelled out in these ratios. The intensity ratios show wide maxima around 559 eV, which may be due to the σ * u shape resonance in the single-hole photoionization cross section or the above-mentioned resonance in the shake-up satellite cross sections, and increasing trends towards the end of the studied photon energy region. These results indicate that O 1s photoionization has not reached a stable situation by ∼588-eV photon energy, but relative strengths between different photoionization channels still vary. In particular, shake-up and shakeoff ionization channels may still evolve. Maier et al [37] identified several shake-up satellites with binding energies in the range 552-566 eV, i.e. their thresholds overlap the double excitation feature around 554 eV and the σ * u shape resonance around 559 eV (see figure 3). The lowest threshold for O 1s shake-off ionization occurs at about 567 eV [38]; this channel directly populates KV states (see row 4 in figure 1). As the cross sections of the O 1s shake-up satellites mostly decrease with the increase of the photon energy [37], we attribute to shake-off photoionization the relative enhancement of the ionic emission toward higher photon energies.

VUV photon -photoion coincidence measurements

We studied ionic emission of CO 2 at the O 1s edge also via VUV photon -photoion coincidence experiments. Some of the measured spectra are shown in figure 7, after counting the number of arrivals in 6-ns wide windows and conversion of the time scale to mass/charge scale. Most observed coincidences originate from VUV emission in the C + and O + ions. The small peak at m/z = 6 is attributed to emission from the C 2+ ions. There are some hints of O 2+ emission around m/z ≈ 8, but the spectra also have some background intensity at the same position. The source of that background structure, and also of the one with m/z between 23 and 29, could not be identified, but it was clearly connected to photoionization of the sample molecules. The CO + 2 (m/z = 44) and CO 2+ 2 (m/z = 22) ions are not observed in coincidence with VUV photons in the individual spectra within the statistical noise of the measurements. These ions are observed abundantly in the partial ion yields at the O 1s edge [25], but they do not emit in the VUV. A small CO + 2 peak does appear if all our photon -photoion coincidence spectra are added together, corresponding to a total measuring time of about 11 hours. That peak is attributed to soft x-ray emission from the core-ionized states, which directly populates the bound single-hole valence states of CO + 2 [START_REF] Gunnelin | [END_REF]. The CO + (m/z = 28) ion does not emit either in the VUV, but it might be expected to appear in the VUV photon -coincidence spectra, as CO + could be produced together with an VUV-emitting O + (or O) when CO 2+ 2 (or CO + 2 ) fragments. That only trace amounts of CO + may be present in the VUV -ion coincidence spectra could indicate that VUV emission in O + is mostly correlated with complete fragmentation of the parent molecule. Spectrum (a) in figure 7 was measured at the photon energy of 525 eV. The VUV photon -photoion coincidences predominantly originate from C 1s single-hole, shake-up and shake-off photoionization. The contribution of direct valence ionization is deemed negligible, as practically no intensity was observed in the coincidence spectrum measured at the photon energy of 90 eV and the cross section of such processes is even smaller at hν = 525 eV. At that photon energy, the O + peak has two components which result from ions whose initial velocities are directed toward and away from the ion TOF spectrometer, indicating that these ions have received kinetic energy from the photodissociation reaction. The shape of the O + peak changes completely at the O 1s → π * excitation (spectrum b in figure 7): the doublet peak structure of O + is replaced by a single, rather wide peak. This change can be explained by the observation that at this resonance most ions are ejected in the direction perpendicular to the electric vector of the linearly polarized light [23]. The underlying reason is that a Σ → Π excitation in a linear molecule occurs if the molecular axis is oriented perpendicularly to the electric vector, and this orientation is maintained during resonant Auger decay of the core-excited state and the subsequent dissociation, which are both much faster than molecular rotation. The resulting ions and/or neutrals are then pushed apart along the molecular axis in a linear molecule. However, this so-called axial recoil approximation breaks down in polyatomic linear molecules if a transition to a core-excited state induces bending vibrations in the molecule, which is the case for the core-to-π * excited states of CO 2 [49,50]. A partial preferentiality of the ionic expulsion along the perpendicular direction is nevertheless kept at the O 1s → π * excitation [23], and this is also reflected in our experiment by the appearance of a single peak in the VUV -O + coincidence spectrum. In spectrum (b), the C + ion peak appears wider than in the other spectra, which implies that excited C + ions receive most kinetic energy from dissociation at the O 1s → π * excitation. Flight-time simulations (not shown) reproduce the experimental line shapes reasonably well when the C + and O + ions have the kinetic energies of 10 eV and 12 eV, respectively. The latter value is rather the lower energy limit and some fast O + ions probably escape detection. The simulation also indicated that the angular anisotropy parameter β for both ions is negative at the O 1s → π * excitation. Our result for C + clearly differs from the value β ≈ 0.9 found for fast C + ions (kinetic energy = 3-8 eV) in non-coincidence measurements [51]. The different β value is probably due to the fact that our experiment only observes a small fraction of all the events, namely the events that are correlated with VUV emission from the ions.

Slightly above the O 1s IP (spectrum (c), hν = 544.3 eV), the O + peak is very clearly split in two components, which indicates the dominance of the Σ -Σ transitions, possibly caused by the influence of the σ * g resonance. This attribution agrees with the calculations of Saito et al [36] according to which the σ * g resonance extends to several eVs above the maximum position at ∼542 eV. The peak shape of the O + ion can be simulated satisfactorily by an O + kinetic energy of about 10 eV and β ≈ 1. The width of the C + peak implies the ion kinetic energies in the range 2-4 eV, but statistics are too low to extract the β parameter. At the core-valence double excitation feature at ∼554 eV (spectrum d), the two components of the O + peak partly merge. We ascribe this change to the excitation of core-valence doubly excited states of π symmetry [23], which enhances the ion ejection along the perpendicular direction, similarly to the O 1s → π * excitation. At the energies of the double excitations, however, direct O 1s photoionization is also strong, and the observed peak shape is a composition of at least these two contributions. Spectrum (e) in figure 7 was measured at the σ * u resonance. The O + peak is again split into two components, which evidences the dominance of the Σ -Σ transitions, in agreement with [36], and molecular dissociation along the E vector of the incident light. We thus see that the line shapes in VUV photon -photoion coincidence spectra change when different core holes states are created by soft x-ray absorption. This also indicates that all these core-hole processes (i.e., core excitations, shape resonances and double excitations) contribute to the VUV -O + signal.

Figure 8 shows the intensities of the C + and O + ions at the O 1s edge, as obtained from the fits of the VUV -photoion coincidence spectra, and the total ion yield measured with the ion TOF spectrometer. Both the C + and O + peaks were fitted with two Gaussian profiles in order to reproduce better the experimental data. The coincidence data points for the C + and O + ions display a similar behavior and follow the TIY, except perhaps for the region just above the O 1s IP, where VUV emission from the ions appears slightly reduced. No conclusive effects on the ratio of excited C + and O + ions can be observed at the double excitations (around 554 eV) or at the σ * u resonance (around 559 eV). In fact, the C + :O + intensity ratio appears to be rather constant at 0.83±0.06 in all the coincidence spectra measured above the O 1s IP. The intensity of the C 2+ emission is too weak to draw detailed conclusions on its behavior in the O 1s continuum. However, the branching ratio of C 2+ emission to all ionic VUV emission was smallest at the O 1s → π * excitation (spectrum (b) in figure 7) and largest at hν = 525 eV (spectrum (a) in figure 7), i.e. following C 1s ionization processes. We briefly consider de-excitation processes that can lead to VUV emission in ionic fragments. Following single-hole core ionization (both O 1s and C 1s) in CO 2 , normal Auger transitions also populate such dicationic final states that have one or two holes in the inner-valence molecular orbitals [52]. The inner-valence photoelectron spectrum of CO 2 is very complicated and can not be described using single-electron configurations [53]. Nevertheless, in the simplest molecular orbital approximation, the inner-valence MOs of CO 2 have significant contributions from the O 2s and/or C 2s atomic orbitals. Consequently, when doubly charged states with inner-valence vacancies dissociate, one can expect that C + and O + ions with 2s holes are produced. The NIST tables [START_REF] Yu | NIST Atomic Spectra Database (ver.4.0.1)[END_REF] list numerous transitions in the VUV region for the O + ion with the upper state electron configuration 2s 1 2p 4 (for instance, 2s 1 2p 4 ( 4 P) → 2s 2 2p 3 ( 4 S) transitions around 83 nm) and for the C + ion with the upper state electron configration of 2s 1 2p 2 (for instance, 2s 1 2p 2 ( 2 P) → 2s 2 2p 1 ( 2 P) transitions around 90 nm). VUV emission in the C + and O + ions is thus expected to follow normal Auger decay of the core-ionized states. Corevalence doubly excited states probably de-excite to final states having inner-valence vacancies with about the same probability as the core-ionized states do. Subsequent dissociation pathways are more complicated following core-valence double excitation, but some of them will surely result in VUV emission from the C + and O + ions having a hole in the 2s subshell. Following core-valence double excitations, VUV emission may also be expected from ionic states that have electrons on Rydberg orbitals. The O + ion, in particular, has several states with electron configurations such as 2s 2 2p 2 3d 1 and 2s 2 2p 2 3s 1 that emit in the VUV region [START_REF] Yu | NIST Atomic Spectra Database (ver.4.0.1)[END_REF]. However, the population of such ionic states may not be significant after dissociation, as the 2π u (π * ) orbital, which is most likely involved in these core-valence doubly excited states, is built from the atomic C 2p and O 2p orbitals [START_REF] Harris | Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy[END_REF]. It appears therefore plausible that also in the case of corevalence double excitations the main decay channel resulting in VUV emission from the the C + and O + ions is the one that leaves inner-valence holes in the Auger final states. This reasoning may explain the similar normalized intensities at the shape resonances and core-valence double excitations when VUV emission from the C + and O + ions is concerned.

Conclusions

We have studied excitation and ionization processes at the O 1s edge of the CO 2 molecule by monitoring subsequent dissociation pathways that involve the emission of low-energy photons. The emission line at ∼845 nm due to a neutral oxygen atom is most enhanced at core-to-Rydberg and at core valence double excitations, but appears much reduced or even negligible at the σ * g and σ * u shape resonances around 542 and 559 eV, respectively. The oxygen emission is suggested to gain intensity again when O 1s shake-up ionization channels open. In general, the observed neutral O emission likely follows when core hole states decay via (resonant) Auger transitions to final states that also have electrons on virtual orbitals. Instead, less excited neutral fragments are expected after the decay of the shape resonant states, as it mainly involves the population of two-hole final states (without electrons on virtual orbitals). Additionally, we have measured the partial fluorescence yield in the wavelength window 300-650 nm as well as the excitation functions of some C + and O + ionic states emitting in the visible region. They display all the O 1s absorption features, hence ionic emission in the visible range appears less sensitive to different core excitation processes than neutral emission. However, shakeoff photoionization seems to produce much more excited ionic fragments than neutral fragments, which is deduced from the relative increase of ionic emission toward the end of the studied photon energy range, 588 eV.

The VUV photon -photoion coincidence spectra measured in this work reveal that C + and O + are the most important ionic emitters of VUV radiation at the O 1s edge of the CO 2 molecule. Their yields follow rather well the total ion yield. In other words, the partial VUV yields of different ions do not show different sensitivies to the core-valence double excitations or shape resonances. However, the line shapes in the coincidence spectra themselves, and particularly that of the VUV -O + coincidence peak, reflect to some extent the symmetry of the final state of the initial core excitation process that eventually leads to VUV emission. VUV photon -C + /O + coincidences are suggested to originate mostly from such (resonant) Auger decay transitions that populate final states having inner-valence holes.

Figure 3 .

 3 Figure 3. The excitation function of the 845-nm neutral oxygen emission line (open circles), the partial fluorescence yield in the wavelength window 300-650 nm (dots) and the total ion yield (TIY, solid line) measured across the O 1s edge of the CO 2 molecule. The curves have been scaled at the same height at the O 1s → π * excitation (535.4 eV). The main spectral features are labeled. Various ionization thresholds[24,37,38] are also indicated. See text for details.

Figure 4 .

 4 Figure 4. The excitation function of the 845-nm neutral oxygen emission line at the O 1s → Rydberg excitations in comparison with the total ion yield measured simultaneously. The assignments are according to Ref. [26].

Figure 5 .

 5 Figure5. The fluorescence spectrum of CO 2 measured at the O 1s → π * resonance after correction for relative detection efficiency. The spectrum is a composition of three measurements taken with different wavelength windows. The emission lines have been assigned using the NIST database[START_REF] Yu | NIST Atomic Spectra Database (ver.4.0.1)[END_REF].

Figure 6 .

 6 Figure 6. (upper panel) Total ion yield (TIY) and the excitation functions of some emission lines observed in figure 3. (lower panel) The ratios of the excitation functions and TIY. The solid curves show the data smoothed over 7 adjacent points.

Figure 7 .

 7 Figure 7. VUV photon -photoion coincidence spectra measured at the O 1s edge of the CO 2 molecule. Spectrum (a) was measured below the O 1s excitations (photon energy 525 eV), (b) at the O 1s → π * excitation (535.4 eV), (c) slightly above the O 1s IP (544.3 eV), (d) at the core-valence double excitation resonance (553.9 eV) and (e) at the σ * u shape resonance (560.3 eV). The spectra have been scaled to have approximately the same heights and their base levels have been shifted for clarity. The observed ions are labelled and the expected positions of the CO + 2 , CO + and CO 2+ 2 ions are indicated.

Figure 8 .

 8 Figure 8. The intensity of VUV -C + and VUV -O + coincidences at the O 1s edge of the CO 2 molecule. Total ion yield is shown for comparison.

Figure 1 Figure 5

 15 Figure 1 (CO2newFig1.EPS)

Figure 8

 8 Figure 8 (CO2newFig8.EPS)

Acknowledgments

We are grateful to the staff of Sincrotrone Trieste and to Andrea Martin of the CNR-IOM electronic workshop for technical assistance during the beamtime. The research leading to these results has received funding from the European Communitys Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 226716 and from the Italian National Research Council under projects MD.P06.016.002 (CNR-IOM) and MD.P03.026.001 (CNR-IMIP).


A Armyras

D Pentaris
email: dion.pentaris@gmail.com

T Efthimiopoulos

N Merlemis

A Lyras

Saturation and population transfer of a two-photon excited four-level potassium atom

Keywords: Wave mixing process, destructive quantum interference, saturation and population transfer PACS: 42.50.Hz, 42.50.-p, 42.65.-k

A theoretical study of the nonlinearity reduction in a four-level system of potassium, atom in the presence of a strong nanosecond laser field, which excites the transition

emission, without population inversion, in a cascade scheme.

4S

6S ↔ with two photons, is presented. It is shown, that the destructive quantum interference between the laser field and the internally generated radiations results in a linear response of the atomic path-1

( 1/2 1/ 2 3/2 1/2 5P 4S 6S 4S ↔ ↔ ↔
) emitted parametric fields.

For sufficiently high laser intensities and/or atomic densities the path-1 emitted fields are driven into saturation, a regime accompanied by substantial population redistribution among the states. A reasonable agreement between earlier experimental results and the theoretical ones is obtained. It is also shown, that upon saturation of path-1, for low atomic density, the path-2

( 1/2 1/2 3/2 1/2 4P 4S 6S 4S ↔ ↔ ↔
), is activated. A subtle interplay between laser intensity and atomic density may determine the activation of path-2. Also, it is shown that the path-2 emission 1/2 3/2 4P 6S ↔ is an amplified spontaneous emission process which induces a

Introduction

The laser-atoms interaction induced by a moderate intensity laser beam tuned near an allowed two-photon transition, results in the generation of new radiation fields from nonlinear processes, such as parametric four-wave mixing (PFWM), stimulated hyper Raman scattering (SHRS) and amplified spontaneous emission (ASE) [1,2] and six-wave mixing (conical third harmonic generation-THG) [3]. New effects can occur because of the constructive or destructive quantum interference (QI) between transitions along multiple excitation pathways in a given set of optically coupled states, where one of the pathways involves internally generated fields [4]. Under destructive interference some resonant enhancements can completely canceled (suppressed) [5]. This suppression effect can take place with odd or even photon excitation schemes [1][2][3][4][5][6][7][8]. As a consequence, laser depletion, saturation of the nonlinear susceptibility and population transfer are significantly modified. Population transfer leads to significant changes of the refraction index and thus limits the conversion efficiency of the wave mixing process. Nonlinear radiation trapping at high optical depths is also shown to drastically change the spatial distribution and temporal evolution of the excited state population and of the fluorescence light trapped by the nonlinear medium [9]. The response of the nonlinear medium can be also considerably modified by the ac Stark shifts and by the power broadening induced by the generated fields [1], when a sum or difference wave mixing occurs. These phenomena and the suppression effects observed when quantum interference is present can cause a linear response of a nonlinear medium [4-8 and 10].

In our previous work, we have shown experimentally [6] and theoretically [8] in the case of potassium atom that even photon QI resulting from two different excitation paths to the 1/2 6S excited state, one by the two photon pumping laser and another by the generated fields, can lead to a linear dependence of the emitted radiation field on the excitation intensity. In this case, it was estimated that the ratio of the frequencies of the two generated photons is equal to the ratio of the corresponding intensities. Specifically, in the case of the potassium atom, it was shown that the ratio int n of the peak intensities of the PFWM generated fields close to the ( )

1/2 3/2 3/2 1/2 int 6S 5P 5P 4S max n I I - - = = 1/2 3/2 3/2 1/2 6S 5P 5P 4S - - ω ω ≈0.
11 [8]. It was also shown that for sodium [1,4] and potassium [6,8] atoms, the pumping laser intensity and the atom number density are crucial factors that determine whether the destructive QI effect is insignificant (nonlinear region) or dominant (linear region). In order to elucidate the relative importance and complicated interplay of the various parameters of the driving laser and the atomic system, we have set out to model the dynamics of the laser-driven four-level system. In an early attempt [7], we had employed a simplified formulation of the self-consistent evolution of the internally generated field intensities in the context of a density matrix description of the laser-atom dynamics. The essential features of the emitted radiation along both deexcitation paths, as identified in earlier experiments [6], were recovered. However, various temporal and spectral details of the emitted pulses were not reproduced in a fully satisfactory way. In order to improve our understanding of the dynamics of the system, we have proceeded with a detailed formulation of the self-consistent propagation of all internally generated fields. A preliminary account of our results was presented in a conference paper [8] and a fuller, yet still partial and incomplete, account in a subsequent conference presentation [START_REF] Armyras | AIP Conf. Proc[END_REF].

Therefore, this study investigates in considerable detail the temporal dynamics of the different emission channels induced by two-photon excitation of the potassium atoms in conjunction with the QI effect and population transfer. We focus on the description of the rather complicated dependence of the emission intensity on the pumping laser intensity which shows a nonlinear, a linear and a saturation region. The study employs a simplified but adequate theoretical model for the simulation of the temporal profiles of the internally generated fields, for several laser intensities and atomic density values, as well as the population distribution between the states involved. The theoretical results are also compared with earlier experimental results obtained under similar conditions.

Theoretical modelling

In our model we assume that the excitation field does not propagate because its envelope is spatially more extended than the atomic medium.

4S

to the state

( ) ( ) ( ) 4 ij ij ij ij ij 1 E z, t z, t exp[ i v t k z ] 2 c.c = ⎡ ⎤ = ε - - + ⎣ ⎦ ∑
, where the complex amplitudes ( ) ij z, t ε are regarded as slowly varying in amplitude and phase within a period and a wavelength of the carrier wave (SVEA) [8].

Solving the von Neumann equation

( ) I i H , Ι ⎡ ⎤ ρ = - ρ ⎣ ⎦ & h
i σ = Ω σ -Ω σ + Ω σ -Ω σ +Ω σ -Ω σ + Γ σ + Γ σ + Γ σ & , (1) ( ) (
i σ = Ω σ -Ω σ + Ω σ -Ω σ +Ω σ -Ω σ -Γ + Γ + Γ σ & , (2)
( )

33
i σ = Ω σ -Ω σ +Ω σ -Ω σ -Γ σ +Γ σ & , (3) (
i σ = Ω σ -Ω σ +Ω σ -Ω σ -Γ σ +Γ σ & , (4) ( ) ( )
i i i i σ = Δ + γ + γ σ + Ω σ -σ + Ω σ + Ω σ -Ω σ & , (5) ( ) ( )
i i i i σ = Δ + γ + γ σ + Ω σ -σ + Ω σ + Ω σ -Ω σ & , (6) ( ) ( )
i i i i σ = Δ + γ + γ σ + Ω σ -σ + Ω σ +Ω σ -Ω σ -Ω σ & , (7) ( ) ( ) (
i i i σ = -Δ -Δ -γ + γ σ + Ω σ + Ω σ -Ω σ -Ω σ & , (8) ( ) ( ) ( ) (
i i i ( ) i σ = -Δ -Δ -γ + γ σ + Ω σ -σ + Ω σ + Ω σ -Ω σ & , (9) ( ) ( ) ( ) (
i i i ( ) i σ = -Δ -Δ -γ + γ σ + Ω σ -σ + Ω σ + Ω σ -Ω σ & , (10)
c.c + .

We added phenomenologically the coherence decay rates of the system which are defined according to:

( ) 5S , through which the atom decays from state 2 to 1 .

Additionally, we took into account the contribution of collision dephasing rate (or collision phase relaxation) col col 1 γ = τ by appropriately including it in the dephasing term for the non-diagonal density matrix elements. In the above equations ( 5)-( 10) the collision dephasing term was added in the first brackets on the right-hand side of the equations according to the convention ( )

ij col i γ + γ .
Experimental studies of the collision broadening and energy shift of potassium resonance lines by noble gasses, supplemented by semiclassical and quantum mechanical calculations have been performed in many cases [13,14]. Specifically, it was shown [13] that the impact of the above phenomena on potassium states, for a variety of noble gasses (e.g. helium, neon and argon), is significant for relatively high densities ( 18 3 10 cm -), considerably higher than the ones relevant for our study ( 133 10 cm -). In previous experiments, we used helium as a buffer gas to keep the potassium away from the cell windows for average densities of 15 3 10 cm - [5]. Therefore, in order to take into account potassium-noble gas collisions, we used the value col γ = 0.017 1 cm -which is appropriate for elastic dephasing collisions in the system Na-He, (for pressures less than 20 Torr) as estimated in reference [15], a system quite similar to the one (K-He) studied in this work for average potassium densities of 14

( ) ( ) ( ) ( ) ij ij 0 ji ij , i k 4 p , ∂ ∂ζ Ω ζ τ = ε μ ζ τ h , where
( ) ( ) ij p , NTr ζ τ =
μρ is the quantum mechanical polarization, N the atomic density, 0 ε the permittivity of free space, ij k the wave-number for each transition and ji μ the corresponding matrix element of the electric dipole operator. From reference [16] we find that the matrix elements of the transitions of interest are: 24 μ = 10.7 a.u., 41 μ = -0.453 a.u., 23 μ =1.07 a.u.

and 31 μ = -5.13 a.u. respectively. Also, the two-photon matrix element of the pumping transition is ( )

2 12
μ = -950 a.u. More specifically, the propagation equations for the internally generated Rabi frequencies obtain the following form:

( ) ( ) ( ) 2 24 24 0 42 24 , iN k 2 ∂ ∂ζ Ω ζ τ = ε μ σ h , (11) ( ) ( ) (
)

2 41 41 0 14 41 , iN k 2 ∂ ∂ζ Ω ζ τ = ε μ σ h , (12) ( ) ( ) (
)

2 23 23 0 32 23 , iN k 2 ∂ ∂ζ Ω ζ τ = ε μ σ h , (13) ( ) ( ) (
)

2 31 31 0 13 31 , iN k 2 ∂ ∂ζ Ω ζ τ = ε μ σ h . ( 14
)
The coupled system of equations (1)(2)(3)(4)(5)(6)(7)(8)(9)[START_REF] Armyras | AIP Conf. Proc[END_REF][START_REF] Hart | [END_REF](12)(13)(14) is solved numerically in order to obtain the unknown Rabi frequencies ij Ω and density matrix elements ij σ . We use a fourth-order Runge-Kutta method of constant step size. Initial conditions are such that the atoms are in the ground state

for each ζ at τ = 0. The boundary conditions for the internally generated Rabi frequencies at ζ = 0 are defined by quantum noise [15] i.e. we assume that the initial values of the fields that induce one-photon transitions of interest result from quantum fluctuations.

Results and discussion

Excitation of the transitions show a linear dependence on the laser intensity max I , a feature we call "linearization" of path-1. In the following sections we present our results concerning: the path-1 emissions, the population transfer among various states and the comparison of the simulation results with the experimental results (for low atomic density). Finally, we present some results for the path-2 emissions for high atomic density at which they start to become significant.

Intensity of the path-1 emissions as a function of the excitation max I

(a) The pumping laser intensity effect. The three regions The transition between the three regions becomes less sharply defined as the atomic density is increased. This may indicate the emergence of a fourth region, in between the "nonlinear" and "linear" ones, for still higher atomic densities, a feature we have not further investigated. However, the transition from the "linear" to the "saturation" regions does not seem to support a similar tentative conclusion, at least for the range of atomic density values we employed in our calculations. It is a typical transition from a power-law intensity dependence to a saturated one.

(b) The QI effect

The calculation of the ratio ( ) transitions for the linear region, results in an average value of 0.14. This value is close to 0.11 (the ratio of the frequencies involved) as expected for a two-photon parametric process. This result agrees with the experimental results for sodium [1] and potassium atoms [6]. It was shown in [1,6] that the mechanism causing the linear response of the atomic medium is the destructive QI between the twophoton laser excitation and the excitation by the two photons 24 ω and 41 ω generated by the parametric process, which is 180 0 out of phase. We should mention that by parametric process we mean either the simultaneous emission of two photons corresponding to the

1/2 3/2 3/
transitions 2 4 1 ↔ ↔
or the wave mixing of two laser photons with a photon which came from the SHRS process connecting the states 1 and 4 . These two processes produce coherent emissions but with slightly different wavelengths due to phase matching constrains.

In the nonlinear region, the intensities of the two fields at 24 ω and 41 ω have not grown enough in the atomic medium in order to compete effectively with the laser excitation. Other possible emission mechanisms (such as weak ASE from the state 2 or SHRS that transfers population to the state 4 ), are not yet fully developed to provide a strong contribution to a four-wave mixing process, that could counteract the laser excitation. In short the QI effect in the nonlinear region is small at low laser intensity until the parametric emissions become strong enough to counteract the laser.

(c) The atomic density effect

In figure 2, the effect of the atomic density on the peak intensity of the radiation emitted at the 4 1 ↔ transition is demonstrated. In the nonlinear region, the rate of change of the emitted peak intensity as a function of the laser intensity is reduced as the density increases.

Moreover, the transition to the linear region, a manifestation of QI, occurs at lower laser intensities as the atomic density is increased. Finally, the saturation of the emitted radiation is predicted to occur at higher laser intensities as the atomic density is increased. The results in the nonlinear and the saturation regions can be consistently understood as a result of the diminishing number of available driving photons per atom as the atomic density is increased.

The onset of the linear region at progressively lower laser intensities as the atomic density is increased can be understood as a result of the enhanced emission from a larger number of available atoms.

We should mention that identical response is also found for the other emission of the path-1 at the frequency 24 ω of the 2 4 ↔ transition, which indicates their connection to the same generation process according to our model. In conclusion, we have presented the main features of the path-1 emission peak intensities as a function of the laser intensity and atomic density. The effect of the same parameters on the temporal profile of the internally generated fields is discussed in the following.

Temporal profiles of the generated fields of path-1

Figure 3 presents the temporal profiles of the generated fields along path-1 for a low laser intensity corresponding to the nonlinear region ( 2 max I 0.08MW/cm = ) and a low atomic N 4 10 cm - = × , for ζ = 5 cm. We notice that the fields are weak, narrower than the laser pulse and reach a maximum value a few ns after the laser maximum. The peak intensity ratio of the two emitted fields at 24 ω and 41 ω is equal to 1 which is typical for the nonlinear region. Additionally, the radiation at 41 ω has a similar temporal shape with that of 24 ω and their time delay, with respect to the laser pulse maximum, coincides with the increase of the 44 σ population by a stimulated process (SHRS) directly from the ground state (inset in figure 3(b)). The temporal shape of the pulses, their delay and the delay of the 44 σ population seem to indicate a four-wave mixing process of one SHRS photon and two laser photons. Figure 3(b) shows also, the population of all the atomic states for this particular low laser intensity and low atomic density. The population rise of the level 4 is delayed with respect to the level 2 population and rises faster than that of level 2 indicating a SHRS, that populates directly level 4 , while the level 2 population is due to the direct twophoton absorption of the laser photons.

Figure 4 shows a similar graph for the linear region, (

2 max I 0.4MW/cm =
), where the generated fields become stronger and they are maximized close to the maximum laser intensity ( 0 τ ≈ ). In the linear region the emissions at 24 ω and 41 ω have a temporal structure similar to the laser pulse but with an asymmetry in the second half of the pulse which seems to correlate to the ground state population reduction (a topic discussed below). The emitted intensity ratio of the two components is well below 1 and equal to 0.14, a feature discussed in detail earlier.

For the saturation region (

2 max I 4MW/cm =
), as shown in figure 5, the generated fields are broader, activate earlier and maximize before the laser maximum. The temporal profile of the two emissions at 24 ω and 41 ω is similar. The emission widths at 24 ω and 41 ω in the saturation region, are even shorter than the ones in the linear region, start earlier and finish more suddenly. These features are well correlated with the increased population transfer shown in the inset of figure 5 where even a transient population inversion of the excited state 4 with respect to the ground state is observed. In short the emitted fields along path-1 show a time shift of their maxima, a temporal profile change, and emitted intensity ratios which depend on the excitation laser intensity. These changes are well correlated with the population redistribution, which also depends on the laser intensity.

Population transfer

In this section we investigate the population distribution among the atomic states, as a function of the laser intensity, covering the three regions of interest, for a low atomic density of potassium. The internally generated emissions and the processes involved are directly related to the population transfer. In our simplified model the ionization, the non-elastic collisions, the backward emissions of ASE or SHRS and the conical emission are not included yet. In the following we investigate the population redistribution in the three regions of interest for the low atomic density case: a) In the nonlinear region, of excitation the population remains mainly in the ground state ( 11 1 σ ≈ ) while a very small percentage of the population is transferred to the state 2 by laser absorption and to the state 4 by SHRS mainly, as shown in figure 6(a) in detail.

The 22 σ population follows the envelope of the laser pulse and the 44 σ population starts 5 ns later in a very abrupt way indicating the presence of a stimulating process with population oscillations between the states 2 and 4 . There is very small population in the state 3 .

The ratio of the emitted intensities of the path-1 is close to 1, which indicates that the SHRS process has a strong contribution to the emission from the

I

40MW/cm = respectively. We notice a population inversion between the states 1 and 4 . The population of state 2 is equal to the population of state 1 for this onresonance case and its relaxation to state 3 is rather clear, providing a maximum population 28 ns after the laser pulse and initiating the emissions of path-2, as shown in figure 7(a). In figure 7(b) and 7(c), the population of state 3 evolves monotonically with the laser intensity increase. This is due to the response of path-2 generated fields, as we present in the following. Significant population transfer to the state 4 (over 53%) is also clearly shown in figure 7(c) for a laser intensity of 2 max I 40MW/cm = . From figure 7, it is evident that on two-photon resonance, the population of state 2 , tagged to that of the state 1 , slightly decreases with increasing laser intensity while the population of state 4 increases accordingly.

Another important observation from figure 7 is that maximum population is achieved for the state 2 at shorter times with increasing excitation intensity, which is reasonable since higher excitation intensity leads to rapid growth of the fields ( 24 ω and 41 ω ) which results in a shorter period of time for the QI effect to develop and suppress the population of 2 . Additionally, higher population of the state 4 compared to that of the state 2 is an indication of a SHRS process which transfers population from the state 1 immediately to the state 4 . This process is a kind of stimulated hyper Raman adiabatic passage (STIRAP)

for high laser intensity and detuning from 2 . The four-wave mixing process (involving two laser photons and one SHRS photon) can still take place to generate a field at frequency 41 ω in the saturation region, but the efficiency of the process is reduced. The slow decay rate from 4 to 1 represents a bottleneck for any further growth of the emitted field, despite the sizable increase in the population of state 4 . The large transfer of the population of state 1 to different levels also affects the emission processes through the modification of the index of refraction and the phase matching because the resonances involved are populated.

Comparison with experimental results

In order to compare the results of the simulation with relevant experimental results, we use the work of N. Merlemis et al. [6]. In figure 8 a plot of ( )

41

Emissions of the path-2

The path-2 emissions refer to the radiation corresponding to the transitions

1 2 3 1 ↔ ↔ ↔ .
Experimental results [6,8,15] and the theoretical simulation both show that path-2 is activated only when the laser intensity and the atomic density have values such that path-1 is driven into saturation, for a low atomic density and in the linear region, for a high atomic density.

In figure 9 we present the temporal profiles of the generated fields near the transitions one. We think that the difference between the experimental and theoretically predicted values could be in part due to some important simplification of the model, such as: i) the exclusion of state in the experiment and ii) the exclusion of any conical divergence of both emissions [17].

In figure 10 we present the population distribution for an atomic density of N ~10 cm -, path-2 is activated while path-1 is in the linear region, as presented in figures 9 and 10.

Conclusions

In this work we studied the two-photon interaction of a laser field with potassium atoms and compared the simulation results with relevant experimental observations. We have found that there is significant interplay between two different paths of de-excitation, with the activation of the second path (path-2) closely related to the linearity of response or saturation of the first path (path-1). In particular: i) There is a nonlinear region for the generated fields of path-1, where the parametric fields are relatively weak to counteract the laser excitation and thus the destructive QI is small. There is insignificant population transfer in this region.

ii) In the linear region, the generation process for path-1 seems to be mainly parametric, with the destructive QI effect to be strong. There is few population transfer and the atomic system remains mainly in the ground state.

iii) Approaching the saturation region, there is significant population transfer from the state 1 to the state 4 , indicating the presence of a SHRS process. The path-2 is activated.

iv) The excitation energy is directed to path-2 when path-1 is close to saturation, for a low atomic density ( 133 N ~10 cm -). For higher values of atomic density, the path-2 is activated at lower laser intensity, where the path-1 is still in the linear response region, because the gain of the emissions is density dependent. This indicates an interplay between laser intensity and atomic density in determining the transition between regions, a feature to be further explored in subsequent publications.

v) The theoretical simulation results concerning the nonlinear and the linear regions are in reasonable agreement with the experimental results.

vi) The emissions of the path-2, in high atomic density, seem to be of a cascade type of emission without population inversion and show a considerable time delay which depends on the laser intensity. The fields corresponding to the transitions

  The coupled system of Maxwell-Bloch equations is solved numerically in one spatial dimension z allowing for the generation of new fields. The potassium atom is modelled by the four-level configuration shown in figure 1. An external field with FWHM c τ = 9 ns and waveform in order to resemble experimental laser pulses), excites the atom from the state 1/2

3 10

 3 cm -. The generation and propagation of the internally generated fields are described by the Maxwell equations for the field amplitudes, under the transformation to the retarded time frame, for axially-forward propagating waves with t z c τ =and z = ζ . The Maxwell equation for the Rabi frequency becomes:

Figure 2

 2 Figure2shows the calculated emitted peak intensity at 41 ω as a function of the laser intensity for three values of the atomic density,

7

 7 four-wave mixing (with two laser photons and one SHRS one) to generate the photons at the 4 1 ↔ transition. The relaxation of the population of states 2 and 4 starts about 15 ns after the laser pulse maximum and after the laps of the oscillations. b) In the linear region, figure 6(b), where we have 11 σ ≈ 0.80-0.85 and int n ≈ 0.11, the oscillations between 22 σ and 44 σ are more intense (because of the higher intensity of the SHRS 2 4 ↔ emission) and of higher frequency. The oscillations are, in this case, shifted in time earlier than the laser pulse maximum and the two populations, ( 22 σ and 44 σ ), reach a maximum after the end of the oscillations. The state 3 population is still very small. c) Finally, in the saturation region the picture completely changes as shown in figure

2 MW

 2 max I as a function of the laser intensity max I is shown for two values of the atomic density ( 3 10 cm - = ×). The width (FWHM) of the laser pulse is 25 ns, the cell length is17 cm ζ =and the two-photon detuning is 12 0 Δ = . There is reasonable agreement between the experiment and the simulation. The slopes of the nonlinear region in the Log-Log diagram for the theoretical and the experimental results are: 8(b)). Reasonable agreement also exists in the case of the linear region, where, for the atomic density of predicted region (saturation) is absent in the experimental plot presented, since the laser used in the experiments[6] could not provide enough emitted intensity. In addition, we observe a difference for the critical point NL L I -(at which the system passes from the nonlinear to the linear region). In particular: b), correspondingly. This laser intensity difference can be mainly attributed to the following reasons: The linewidth of the laser used in the experiments was So, when tuning on two-photon resonance, a significant amount of the laser intensity is practically off-resonance from state2 . This effect causes a shift of the critical driving intensity to larger values, in the experimental case, compared with the theoretical model in which the Fourier bandwidth ( Δτ ∝ ⋅τ ) of the laser pulse of 25 ns used, is 0.0013 1 cm -. The vertical separation of the plots is due to the different units used ( / cm in the simulation and arbitrary units in the experiment). In conclusion, the results of the simplified model agree to a large extent with the experimental results and the difference in the critical point values are justified by the laser linewidth difference between theory and experiment.

3 ↔ 1 ↔

 31 photon parametric emission, in which case we would expect the same number of photons in each field. This result was also experimentally observed by M. Katharakis et.al.[17] for an atomic density of

3 N 3 N

 33 that the population transfer to the excited states is significant and the activation of the emissions of path-2 approximately starts, when the population of ground state is at its minimum value. It is also clear that there is no population inversion between the states 2 and 3 for the particular parameters used. In the previous sections concerning the path-1 emission, figures2 and 8 show that the critical point NL L I -(which indicates the transition from the nonlinear to the linear region) is shifted to lower laser intensity values with increasing atomic density, while the critical point L SAT I - (which indicates the transition from the linear to the saturation region), is shifted to higher laser intensity values, respectively. The activation of path-2, for the low atomic density case ( 13 ~10 cm -) is connected with the linear or the saturation response of path-1. When the atomic density increases to 14 ~10 cm -or 15 3

Figure 1 .(

 1 Figure 1.

Figure 2 .

 2 Figure 2. Peak intensity of the generated radiation near the 4 1 ↔ transition as a

Figure 3 .

 3 Figure 3.

Figure 4 .

 4 Figure 4.

Figure 5 .

 5 Figure 5.

Figure 6 .

 6 Figure 6.

Figure 7 .

 7 Figure 7.

Figure 8 .

 8 Figure 8. Peak intensity of the emission, near the 4 1 ↔ transition, as a function of

Figure 9 . 3 ↔ transition and a weak one near 3 1 ↔

 931 Figure 9.

Figure 10 . 3 N

 103 Figure 10.

  )

	22	21 12	12 21	24 42	42 24	23 32	32 23	23	24	2R	22

  )

	44	41 14	14 41	42 24	24 42	41 44	24 22

  )

	34	14	13	34	col	34	31 14	32 24	14 31	24 32

  )

	23	12	13	23	col	23	23	33	22	21 13	24 43	13 21

  )

	24	12	14	24	col	24	24	44	22	21 14	23 34	14 21