Angelica Moise

Michele Alagia

Lorenzo Avaldi

Vitaliy Feyer

Kevin C Prince

Robert Richter
email: robert.richter@elettra.trieste.it

Photoionization of laser-excited caesium atoms above the 4d ionization threshold

Keywords: core ionization, atomic spectroscopy, two-color experiments, laser and synchrotron radiation, PACS: 82.53.Hn, 42.62.Fi, 32.30.-r, 32.80.Dz

Introduction:

Inner shell photoemission spectroscopy is an ideal tool to investigate the electronic structure of atoms and to test theoretical methods. The possibility to examine how certain parameters or couplings change with the size of the atom and atomic number allows detailed comparison with theory. From the experimental point of view gas phase studies are often complicated by the fact that atoms different from rare gases need to be prepared in situ. Alkali metals, being easy to evaporate, are a relatively convenient target for studies and were investigated over the years. The series from Lithium to Cesium has been studied extensively [1,2,3] using inner-shell spectroscopy techniques. Alkali metals are however very reactive, thus the target densities have to be kept relatively low in order to avoid contamination of the experimental setup. High resolution studies were started only in the last 15 years or so with the advent of third generation light sources which provide a sufficiently high flux to offset the low target concentration [4,5,6,7]. Theoretically alkali metals are appealing, because of their hydrogen-like electronic structure with only one electron in an s-orbital outside a noble gas closed shell configuration. In addition, for the heavier atoms relativistic effects become very important, and the change of coupling from LS to jj [8] can be investigated moving from one atom to the next.

Two color experiments add a further degree of control, as the target can be prepared in a well-defined quantum state by the absorption of the first photon, and then studied using the second. This type of investigation can give very detailed insights into the electronic structure and ionization dynamics, as, depending on the characteristics of the two light sources, very high experimental energy resolution, time resolution, or population of specific sublevels (by polarization control) can be achieved. Thus the application of laser-synchrotron pump-probe techniques is an important tool in the study of the electronic structure [9]. Alkali atoms are also a particularly convenient target for such experiments. All of them have very strong resonance lines, which are located in the tuning range of commercial lasers, so that a substantial fraction of the ground state population can be transferred into an excited state. The coupling of lasers and synchrotrons to investigate alkali atoms was started already in the eighties at second generation light sources [10]. Those pioneering studies were very important to show the potential of such experiments. With the development of third generation light sources the available photon flux and resolution have improved, but the main experimental difficulties -low target density and the risk of contamination due to a highly reactive sample -have remained unchanged [3,4,5,9]. Investigations of laser-excited atoms using high resolution have been essential to unravel the complexity of the satellite structure in the np valence ionization of Rubidium [11] and Cesium [12]. The selective excitation gave access to final states which are otherwise only populated via shake-up processes, significantly simplifying the spectra. Thus theoretical predictions could be tested in detail. Also, the excitation of the outermost ns electron leaves the atom in a state with an electron less tightly bound. The study of photoelectron and shake-up spectra, in comparison to their ground state counterparts, permits the evaluation of electron correlation and relativistic effects in theoretical models [2].

Here we describe experiments carried out using a tunable mode-locked Ti:Sapphire oscillator and synchrotron light pulses in the EUV range applied, to the 4d ionization of Cesium.

The spectrum of Cesium in the vicinity of the 4d ionization threshold is very complex. Below threshold the spectrum is dominated by several sharp features assigned to excitations of 4d electrons to empty np orbitals, although it has been shown that a single configuration approach is not sufficient for a correct description of the spectrum [13,14,15]. Above threshold Cesium exhibits a giant resonance due to transitions into the εf continuum, where the cross section increases to ~35 Mb [16].

The spectrum of ground state atoms has been studied by absorption [13], ionization [14,15], photoelectron [4,17] and Auger [18,19] spectroscopy with always increasing resolution, and has also been the subject of theoretical investigations [4].

There are fewer reports on laser-excited Cesium atoms, but Schulz et al [12] have studied 6p excited atoms near the 5p ionization threshold. More recently, the absorption spectrum below the 4d ionization threshold has been investigated both experimentally and theoretically [20]. Here we present results on the 4d ionization of 6p excited Cesium atoms recorded at various photon energies above the ionization threshold. The 4d photoelectron spectrum is discussed, and a preliminary analysis of a part of the Auger spectrum is presented.

Experimental:

The experiments were performed at a branch line of the Gas Phase beamline at the storage ring Elettra (Trieste, Italy). The general layout and performance of the beamline [21] and the laser setup [22] have been described previously, and so only the changes to the apparatus relevant for the present experiment will be briefly given here.

The photoelectron spectra were recorded using a commercial hemispherical photoelectron analyzer (VG). The analyzer was placed at the magic angle (55°) with respect to the synchrotron light polarization vector, in the plane defined by the polarization and the propagation direction of the light. The analyzer was used in constant pass energy mode with the resolution set to ~100 meV to record the photoelectron spectra, and to ~200 meV to record the Auger spectra. Cesium was introduced into the ionization region using a resistively heated, anti-inductively wound oven, perpendicular to the plane defined by the light beam and the analyzer.

The temperature was kept at ~90°C throughout the experiment, monitored using a thermocouple. The ionization region was surrounded by a liquid nitrogen cooled copper shield in order to minimize the contamination of the chamber and analyzer by Cesium vapor. Nevertheless small shifts in the kinetic energy scale of the analyzer due to Cesium coating were observed. To minimize the effects of this shift on the analyzer resolution multiple spectra were recorded and then, if necessary, the spectra were aligned on strong features due to ionization of ground state Cesium atoms, and summed. The overall spectral resolution of the experiment including the contribution of the beamline monochromator was determined to be ~120 meV. The transmission of the analyzer in the kinetic energy range of interest was determined by recording the Auger spectrum of Xenon [23]. The spectra of Cesium were recorded several times during the experiment. The changes in analyzer transmission were small and limited to the lowest kinetic energies used in the experiment (< 15 eV). Spectra of laserexcited atoms were obtained by subtraction, after alignment on the strongest line of ground state photoionization. The effect of the subtraction is seen in the spectra, where strong ground state features cause small oscillations in the baseline.

While the procedure outlined above proved accurate enough for the analysis of the strong XPS spectra, the analysis of the Auger spectra, in which the intensity is distributed over many transitions, was affected much more by the small shifts in the kinetic energy scale. Therefore for this part of the experiment a chopper with a modulation frequency of ~3 kHz was inserted into the laser beam. In this way the laser-on/laser-off spectra were recorded almost simultaneously making the subtraction procedure more reliable. The signal-to-noise (S/N) ratio was enhanced by using zeroorder light coming from the undulator, with the undulator gaps set to center the emission at 145 eV and the higher harmonics of the undulator efficiently cut by the beamline optics. The photon energy resolution corresponds to the width of the undulator emission (~8 eV) and so photoelectron features are washed out, while Auger peaks are not affected by photon bandwidth.

A commercial mode-locked Ti:Sapphire oscillator (Tsunami, Spectra Physics) was used to excite Cesium atoms in the interaction region. The laser beam was set counter-propagating to the synchrotron radiation. Typically ~1 W of tunable radiation was measured at the entrance window of the chamber. During the experiment it was tuned to the D 1 and D 2 (6s 2 S 1/2 6p 2 P 1/2,3/2 ) absorption lines of the Cesium atom at 894 nm and 852 nm, respectively [START_REF] Steck | Cesium D Line Data[END_REF]. As both lines are very narrow, a reference cell (glass cuvette) was used to monitor the stability of the laser wavelength by measuring the absorption of the metal vapor.

The polarization of the laser was adjusted using l/2 and l/4 waveplates. In the experiments the laser was used mode-locked and running at a frequency ~83 MHz, but not synchronized to the ring frequency [22], as the lifetime of Cesium excited states is long (>30ns [START_REF] Steck | Cesium D Line Data[END_REF]) compared to the period of the laser. The mode-locked laser was operated in the picosecond configuration with a pulse width of ~12 ps which was monitored continuously. The picosecond pulse configuration gives high power densities, so the beam was used unfocused at an estimated power density in the interaction region of ~30 kW/cm 2 . From the spectra we estimate that we saturate the transitions. The unfocussed beam approximately matches the site of the effusive beam of Cesium atoms from the furnace (~2 mm diameter), with the synchrotron beam size ~5 times smaller. Thus variations in the spatial overlap of the two beams are negligible. The signals in the photoelectron spectrum turned out to be strong enough to fine-tune the laser wavelength directly, thus possible differences in the lineshape due to the different Doppler widths in the reference cell and experimental chamber, or saturation effects were not a cause of instabilities. Assuming the same cross-sections for the 4d ionization of the ground state and the laser excited states we estimate that more than 30% of the atoms in the interaction region are excited -indicating saturation of the transition.

Results and Discussion:

The Cesium photoelectron spectrum at the 4d edge has been reported [4,17] and is relatively simple when compared to the spectrum below threshold or to other thresholds of the same atom. This is somewhat surprising, as the 4d region is characterized by the strong 4f absorption above threshold, and in other atoms like barium for example, leads to many satellite features showing the importance of electron correlation [START_REF] Snell | [END_REF]. The recent high resolution investigation by Fritzsche et al [4] has shown that also in Cesium shake-up and conjugate shake-up states are populated upon ionization, but their relative intensity is low. In the former process the dipole ionization of a core electron is accompanied by the monopole excitation of a valence electron while in the conjugate shake-up the monopole ionization of a core electron is accompanied by the dipole excitation of a valence electron to an unoccupied atomic orbital. Here we examine first the 4d ionization spectrum recorded with resolution similar to the recently published data, and then compare it with the spectra of laser excited atoms. The ionization of laser-excited Cesium generates some of the same final states observed in the ionization of ground state Cesium, so a comparison allows an assignment of the spectral features.

Ground state atoms, 4d photoelectron spectrum:

The spectra were recorded at nine photon energies between 105 eV and 145 eV. They are dominated by two strong lines due to the ionization of the two spin-orbit components of the 4d orbital. As different photon energies were used overlapping Auger peaks can be distinguished from the photoelectron spectrum. The experimental resolution was similar to that used in the recent study [4], so no new features in the spectrum were observed in our experiments. Therefore we adopt the published assignment, and only discuss the differences observed between the spectra recorded at various photon energies. Care was taken in identifying the Auger peaks, which were accounted for in the analysis. The spectrum taken at 145 eV photon energy is shown in figure 1. It was analyzed as follows: the final state energies given by Fritzsche et al [4] were used as starting parameters of the fit, and the strongest transition (to the 4d - 1 5/2 6s final state) was used to calibrate the binding energy scale. The change in line shape and the possible shift due to post-collision-interaction [26] was neglected in the analysis. This is justified by the long lifetime of the state and the high photon energy, 30-60 eV above threshold. The experimental resolution was treated as a free parameter (giving a Gaussian of ~120 meV FWHM). The lifetime width was fitted for the strongest lines in the spectrum, and then set equal for all peaks in the spectrum, at ~100 meV. With these values all peaks identified in the spectrum agree within 5 meV with the binding energy values given in reference [4]. There are, however some differences in the relative intensities, which can be attributed to the different photon energies used. Also, within our S/N (signal/noise) ratio the 8s 1/2 and 7d 3/2,5/2

(following the notation of reference [4]) final states are too weak to be observed, either for the 4d -1 5/2 group transitions or their 4d -1 3/2 counterparts. This difference is particularly clear for the 4d -1 3/2 8s 1/2 state, which lies in a region free from overlapping Auger bands.

In order to examine the dependence of the ratio between the 4d -1 3/2 and 4d -1 5/2 main lines we have recorded the XPS spectra for various photon energies in the range 105 eV … 145 eV (step 5 eV). The results are summarized in Figure 2, together with the ratios of some satellite lines and data obtained from the spectra of laser-excited atoms. For the main lines the ratio at the photon energy of 105 eV is ~0.69, and a monotonic increase to ~0.82 at the highest energy is observed. The latter value is in good agreement with the value of ~0.8 given by Fritzsche et al. [4] at 142.5 eV. Older results by Prescher et al [27] show a similar dependence on photon energy, but are systematically higher. This discrepancy may be due to the lower resolution used in that experiment, which made the evaluation of intensity of overlapping features more difficult.

The smaller values near threshold, which are closer to the statistically expected 2/3 ratio, might be 'accidental' in view of the fact that the partial ionization cross section depends on the shape resonance near threshold [4,28]. The results by Prescher et al. [27] show strong variations of the main line intensity ratio when approaching the ionization limits. Some insight can be obtained from a comparison with the 4d ionization of Xenon which has been studied in great detail both experimentally and theoretically [28]. At a similar energy above threshold the main line ratio in Xenon is well predicted by Relativistic Random Phase Approximation (RRPA) theory and very similar to what we observed for Cesium. It approaches the statistical ratio at ~200 eV above the ionization edge. The observed similarityalthough our experiments cover only a limited photon energy range -indicates that the 4d core electron ionization is only weakly influenced by the presence of valence electrons. Notably, also the main features of the 3d ionization cross section of Rubidium [4] are similar to the 3d ionization of Krypton [29].

There is a clear difference in the 7s/6s shake-up to main line ratio, which -at the photon energy of 105 eV -we determine to be ~0.085 (± 0.03) (for 4d -1 5/2 ) and ~0.076 (± 0.03) (for 4d -1 3/2 ), respectively. The corresponding values derived by Fritzsche et al [4] for ionization at 142.5 eV are ~0.12 for both, in good agreement with our spectra recorded at 145 eV photon energy. The monopole shake-up intensity increases slowly in the range of experimental photon energies, while the 4d -1 3/2 7s:4d - 1 5/2 7s ratio follows that of the main photolines. This observation reinforces the hypothesis that the shape resonance affects the monopole shake-up lines and the 4d main lines similarly [4], and core-valence interaction is small.

The conjugate shake-up probabilities can be determined for the 6p and 5d lines. Both show a small decrease with respect to the main lines with rising photon energy (see figure 2). The results are in good agreement with the published single photon energy data at 142.5 eV [4]. The trends with photon energy are similar to the ones observed in Rubidium [4], but there Multi-Configuration-Dirac-Fock (MCDF) calculations only qualitatively reproduce the experimental data, especially at lower photon energies. At 142.5 eV the calculated values [4] for Cesium are in good agreement with experiments, but energy dependent calculations are required to test the theoretical description of valence-excited core-hole states.

Laser-excited atoms, 4d photoelectron spectra:

We have recorded spectra exciting the D 1 and D 2 transitions of Cesium with linearly (parallel and perpendicular to the synchrotron light polarization vector) and circularly polarized light. Synchrotron photon energies of 105 eV and 145 eV were used, as for ground state Cesium. The spectra shown (figure 1) have been recorded using linearly polarized laser radiation, perpendicular to the synchrotron light polarization vector. Using other polarizations did not lead to any difference in the relative intensities of peaks -within our signal-to-noise ratio. For D 1 (6s 2 S 1/2 6p

2 P 1/2 ) excitation, we observed a reduction of signal by approximately a factor of two when exciting it with circularly polarized light. This is an additional indication that the transition is saturated, as the m j selection rules for circularly polarized light limit the signal. The polarization dependence of the photoelectron spectrum was checked also at a laser power reduced by a factor of 4, and no differences in the relative peak intensities were observed.

The spectrum of ground state Cesium atoms has been subtracted after rescaling the intensity of the 4d -1 5/2 6s peaks in the two spectra, and the binding energy adjusted for the laser excitation energy in each trace. Thus the features correspond directly to binding energies of core-excited states. Key features can be assigned by comparison with the ground state spectrum and published calculations. The assignment follows the calculations by Fritzsche et al [4], so the dominant peaks in the spectrum correspond to the 4d -1 6p states. As in the ground state spectrum, and also in the case of Rubidium [30] and Potassium [31] the fine-structure of the main photolines is not resolved. If the outermost 6p electron conserves its angular momentum in the D 1 excited spectrum mainly the 6p 1/2 states are accessed, while the D 2 excitation will lead to 6p 3/2 states. Thus the spin-orbit splitting in the core-excited state can be measured directly. Within the experimental resolution we see the same 4d splitting for both 6p states (2.27 eV and 2.28 eV for D 2 and D 1 excitation, respectively), indicating that core-valence interaction is not very strong. The 6p spin-orbit splitting determined for the two main lines is also very similar (0.21 eV for 4d - 1 3/2 and 0.22 eV for 4d -1 5/2 ).

Both the D 1 and D 2 excited spectra exhibit similar features. Shake-up (7p, 6d)

and also very weak shake-down (6s, 5d) transitions can be identified. The selective preparation of the state by the laser permits the identification of spin-orbit components of states which cannot be distinguished in the ground state spectrum. So, for the 7p shake-up state we observe a clear separation of the 3/2 and 1/2 components of ~74 (± 10) meV, indicating a preference for the conservation of angular momentum of the valence electron upon shake-up. The 4d spin-orbit splitting of the 4d -1 7p states is determined to be 2.27 eV, confirming that the core-valence interaction is weak. The splitting between the two components of the 7p orbital was not observed in previous studies. In the ionization of laser-excited atoms the 7p states are populated via the monopole shake-up process from selectively excited atoms, leading to higher intensities and more selectivity in the population of the final states.

While the S/N ratio of the spectra does not permit an accurate evaluation of the spin-orbit splitting of the other states, a small difference is observed for the 6d shake-up transition. Fritzsche et al [4] have determined the spin-orbit splitting of the 4d -1 5d states to be ~100 meV, while the corresponding value for the 4d orbital of these states is 2.2 eV. The corresponding values for the neutral atom are ~69 meV for 6p, ~22 meV for 7p and ~12 meV for 5d [START_REF] Sansonetti | Handbook of Basic Atomic Spectroscopic Data (version 1.1[END_REF]. This comparison indicates that while the 4d spin-orbit splitting is little affected by the outer electron, the 6p and 7p splitting are increased by a factor of ~3 for the core-ionized states. For states with the occupied 5d orbital the increase is much larger, indicating mixing between the 4d and 5d orbitals. We note that Hartree-Fock calculations [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF] predict that also other configurations (e.g. the 4f states) are located in this energy range.

When compared to the spectra of the ground state Cesium atom, the spectra of laser excited atoms show significantly more excitation to shake-up states. For example in the spectra recorded at 145 eV photon energy the intensity of the 7p final states amounts to ~28% of the main photoelectron line. This is in general agreement with theoretical predictions [2]. At 105 eV photon energy the amount of excitation is less (~18%), but still significantly higher than in the ground state spectra. A similar effect has been observed for other alkali atoms and also investigated theoretically [2].

For example in Rubidium 4p [11] and Cesium 5p [12] ionization different finestructure components were accessed selectively and identified in the laser-excited atom spectra. In the case of the 4d ionization of Cesium (similar to the 3d ionization of Rubidium [30]) all the different final states are located within the lifetime-width of the nd hole, so a direct identification of the fine structure components is impossible.

In Lithium [START_REF] Cubaynes | [END_REF] the shake-up probabilities have been examined in a systematic way by following higher excitations and their energy dependence. The increased shake-up probability in the core ionization of excited states has been attributed to an improved overlap between the initial state and final state wave-functions [START_REF] Cubaynes | [END_REF]. Unexpectedly, in the case of Lithium, the increased shake-up probability was similar for conjugate shake-ups, despite the fact that a different mechanism leads to their formation. An examination of the intensities of conjugate shake-up final states in the spectra of 6p laser excited states (final state configurations 4d -1 6d, 4d -1 7s) of Cesium does not show a similar enhancement. It has to be kept in mind, however, that here we examine spectra of a single excited state, so a more systematic investigation of higher excited states (e.g. 7p and 8p, lying in the wavelength range of commercial lasers) -both theoretical and experimental -is necessary to provide definite answers.

Auger spectra:

The 4d Auger spectrum of ground state Cesium atoms has been studied with varying resolution over many years [18,19]. The most recent high resolution study [19] concentrated on secondary Auger processes, and showed a spectrum in the kinetic energy range up to 40 eV recorded at photon energy of 140 eV. We have recorded Auger spectra for ground state and laser excited Cesium atoms at a photon energy of 145 eV. The ground state Cesium atom 4d Auger spectrum agrees well with previously published data [19] and is shown in figure 3 We briefly discuss the main features of the Auger spectra, concentrating on the similarities between the ground state and laser-excited data. A detailed assignment of the spectra requires calculations, but the main features can be interpreted by comparison with published data on ground state Cesium, resonant Auger spectra of Xenon [35] and ground and laser-excited state Auger spectra of Rubidium [30]. The binding energies of many states of the Cs 2+ ion-final states of the Auger decay -in the energy range of interest were reported in a recent high-resolution optical study [36]. The Cs 2+ 5p 4 6s configuration generates 8 distinct states which have all been observed. Using the energies reported in reference [36], together with the 4d -1 5/2,3/2 6s binding energies, all strong features in the part of the Auger spectrum shown in figure 3 can be matched exactly. Several peaks in the spectrum overlap, so their relative intensity is uncertain. The spectrum is dominated by the decay to states of the 5p 4 ( 1 S)6s and 5p 4 ( 1 D)6s configurations, while 5p 4 ( 3 P)6s states are weaker. Spectra from the 4d -1 5/2 and 4d -1 3/2 core-hole ion states can be identified by their spin-orbit separation. The assignment agrees with previously published results, and is indicated in the figure. In order to show the various contributions the spectrum has been fitted

including the dominant decays. Only the main line (4d -1 3/2 6s and 4d -1 5/2 6s to states of 5p 4 6s configuration) and the most intense shake-up (4d -1 3/2 7s and 4d -1 5/2 7s to states of 5p 4 7s configuration) decays were considered. In the fit the energies of all final states were kept fixed, with the relative intensities and a single common width treated as adjustable parameters. The uncertainties are mainly caused by the overlap of various final states. The results are included in the graph (figure 3). While still not all structures are accounted for, the main features in the spectrum are well reproduced.

Based on the results of the fit also the two main-line components' contributions can be separated, and the final states leading configurations identified. Similar to Xenon [35] the main line decay shows a different relative population distribution of the final states, with the 4d -1 3/2 hole decay to states of ( 1 S)6s configuration being stronger than for its 4d -1 5/2 counterpart.

While the Auger spectra of ground state Cesium are dominated by the decay of the 4d -1 6s states, in the case of laser-excited atoms the decaying states are mainly of the 4d -1 6p configuration. However, due to the higher shake-up intensities in the 4d ionization of laser-excited atoms, also decays from other core-ionized states can contribute significantly. The ground state Cesium Auger spectra resemble closely the resonant Auger spectra of 4d 6p excited Xe [19,35]. For the laser-excited atoms, an even closer correspondence can be expected, as the configurations involved are identical. An important difference, however, results from the excitation process. The resonant excitation in Xenon is governed by dipole selection rules and this limits the number of populated fine-structure components to J=1 states. In the 4d ionization of Cesium more states can be populated and have comparable intensity [30].

The most intense lines in the spectrum are in the kinetic energy range 30 -40 eV, where the most intense features of the ground state spectrum also appear. This energy range corresponds to final states of 5p 4 6p configuration [19,36]. This single configuration leads to 21 states, which have been identified in the resonant Auger spectrum of Xe. We calculate the positions of the 82 lines in the spectra using tabulated energies from the recent high resolution experiment [36] and the binding energies of the core hole states measured in the XPS of laser-excited atoms. changes noticeably. Similar differences have been observed in the spectrum of laser-excited Rubidium published recently [30]. In the resonant Auger spectra of Xenon [35] and Auger spectra of laser-excited Rubidium [30] states of shake-up configurations appear in the range of the np 4 (n+1)p final states. Similar complications of the spectrum can be expected in laser-excited Cesium, where already the 4d photoelectron spectra show considerable excitation of satellites. The recent high resolution study of the Cs 2+ ion indicates that many states of 5p 4 nl configuration are located in the same energy range.

Despite their high complexity the Auger spectra of laser-excited atoms permit the identification of leading configurations in the final states. The kinetic energy ranges corresponding to states built on the 1 S, 1 D and 3 P cores have been marked in the figures. Detailed theoretical studies and high S/N experimental spectra are needed to further refine the assignment.

Conclusions:

We have described results obtained using a pump-probe setup available at the Gas Phase beamline at the storage ring Elettra (Trieste). The combination of a high resolution spectrometer and selective excitation allows a detailed study of the electronic structure of core ionized states of Cesium. The spectra were assigned by comparison with core level spectra of ground state Cesium atoms, and the photoionization dynamics were examined. The results for ground state Cesium are in good agreement with recent studies. Although the electronic structure of Cesium at first appears to be simple because it consists of a closed shell and a single valence electron, in practice it is very complex, due to correlation and relativistic effects. The 4d core hole photoelectron spectra are however simpler than the 5p spectra. The 4d spin-orbit splitting is not significantly affected by the valence electron, but correlation is evident because the 6p and 7p splitting is affected by the angular momentum of the core hole, either 5/2 or 3/2. A noticeable increase of the relative intensity of the shake-up features has been observed for excited atoms. The main features in the Auger spectra of laser-excited atoms were interpreted using the recently published [36] energies of the Cs 2+ ion. In spite of the complexity many Auger features can be assigned, although some intensities are not yet explained.

Figures:

  . The D 1 and D 2 excited Auger spectra of Cesium are shown in the panels (a) and (b) of figure 4.

Figures 4 ( 1 3/ 2

 412 Figures 4 (a) and (b) show the part of the Auger spectra in the kinetic energy

  Figure 1 (color online). Comparison of ground state (bottom trace, black), D 1 (middle

Figure 2 .

 2 Figure 2. (color online) Intensity ratios between main lines (circles) and main and

  Photon Energy (eV)

Figure 3 .

 3 Figure 3. (color online) Part of the Auger spectrum of ground state Cesium recorded at 145 eV. The comb in the top of the figure indicates the energies of the Cs 2+ finalstates from reference[36]. The contribution of states of 5p 4 7s, as obtained from the fit

Figure 4 .

 4 Figure 4. (color online) Part of the Auger spectra of excited atoms recorded at 145

Acknowledgements:

We would like to thank all our colleagues at Elettra for support during the course of the experiments, and for generous loans of equipment.


Sonia Melle
email: smelle@fis.ucm.es

Oscar G Calderón

Marcelo Moreno

Group velocity coupling and synchronization-like phenomenon of two light beams based on coherent population oscillations

We report on the control of the slow and fast light propagation velocity of two light beams (at 1550 nm and 980 nm) through an erbium doped fiber amplifier by means of coherent population oscillations. We modulate the amplitude of both beams with the same frequency so that the relative phase between both modulations allows us to control their group velocities. The same group velocity, in absolute value, is achieved when beams are modulated in phase or phase opposition. The experimental results are explained by a rate equation model.

Introduction

Slow light has recently become a subject of intense research due to its potential applications for tunable delay lines which can be used for data synchronization [1]. Thus, much research has been focused on the control of the speed of light in optical fibers since these devices would be compatible with fiber-optic communication systems (see review about slow light in optical fibers [2]). Song et al. [3] and Okawachi et al. [4] demonstrated slow and fast light in optical fibers for the first time. The underlying mechanism used on those works, stimulated Brillouin scattering, consists in the interaction of two propagating waves, a pump wave and a Stokes wave, which generates an acoustic wave at the frequency difference of the pump and the Stokes fields. The slow light resonance can be placed at the desired wavelength by changing the frequency of the pump field. A related process, stimulated Raman scattering, has also been used in optical fibers to demonstrate an ultrafast all-optical controllable delay [5]. Electromagnetically induced transparency (EIT), a pioneering technique to produce slow light propagation, has been recently demonstrated experimentally in hollow-core photonic-bandgap fibers filled with acetylene [6]. This technique creates a very deep transparency window in an atomic absorption profile with almost null absorption. However, EIT medium should preserve quantum coherence, therefore dephasing processes limit this technique.

Coherent population oscillations (CPO) have been shown to be another physical mechanism which allows for the variation of group velocity in optical fibers at room temperature. The periodic modulation of the ground-state population at the beat frequency between a control and a probe field sharing a common atomic transition produces scattered light from the control field into the probe field leading to a decrease in the absorption of the probe field. This give rise to a narrow hole in the absorption profile for the probe field leading to slow light propagation [7]. The hole linewidth is proportional to the inverse of the relaxation lifetime of the excited level [8]. Note that this technique is not affected by dephasing processes, in contrast to EIT. However, the transparency depth achieved in CPO is smaller than the one obtained with EIT. Therefore, CPO could be affected by the residual absorption. By means of an additional pump to populate upper levels, the CPO induces a dip in the gain spectrum, leading to anomalous dispersion and fast light (group velocities greater than c or even negative). A modification of group velocity by CPO in an erbium doped fiber (EDF) was reported for the first time by Schweinsberg et al [9], where an amplitudemodulated 1550 nm signal co-propagates with a 980 nm pump beam. They observed a change from sub-to superluminal propagation upon increasing pump power. The delay or advancement achieved saturates with both pump and signal powers, being this a limit in the delays and advancements obtained. Larger delays/advancements have been obtained for ultra-highly EDFs [10,11,12]. By using the same experimental system, fast light pulse propagation has been studied in more detail in [13,14,15]. Furthermore, a change from superluminal to subluminal propagation solely based upon increasing the beat frequency between the weak and the control field sharing a common atomic transition has been reported [16,17]. This peculiar behavior is due to the interplay between pump absorption and pump-power broadening of the spectral hole induced by CPO.

Recently, Arrieta-Yáñez et al have developed a novel mechanism for enhancing the fractional delay and advancement obtained in EDFs by one order of magnitude [18]. They force the population oscillations by modulating with the same frequency and simultaneously both, the 980 nm pump beam and the 1550 nm signal beam. The population oscillations are amplified by the pump modulation. Relative phase between both modulations was used as a knob for changing the propagation regime from ultraslow velocities to fast-or negative-velocities. The same physical mechanism was previously used by Stepanov et al [19] to control the phase shift of a modulated signal beam by modulating an additional beam simultaneously and with the same frequency. These two beams have different carrier frequencies, both within the fundamental absorption spectrum 1470-1570 nm of Er 3+ ions.

Our aim in this work is to study the coupling of two light beams through the population oscillations and its effect in the propagation velocity of both beams. To this end, we induce a beat in both beams with the same frequency and analyze simultaneously the group velocity of both modulated beams. We use an EDF and the two light beams correspond to the usual signal (at 1550 nm) and pump (at 980 nm) beams. This method allows us to control the group velocity of both beams by means of the relative phase between both beats. We analyze under which conditions the coupling between the two beams makes that both propagate with the same group velocity, that is, the conditions that lead to a group velocity synchronization-like phenomenon.

The paper is organized as follows. The theoretical model used to study the propagation of two amplitude periodically-modulated beams through an EDF using CPO is described in section 2. The experimental setup is described in section 3. The experimental results and simulations are presented in section 4. The final conclusions are summarized in section 5. The propagation of an amplitude periodically-modulated signal through an EDF has been modeled in previous works on CPO by using both semiclassical and rate equation procedures [7,9]. We consider the erbium ions as three-level atomic systems interacting with a power-modulated 1536 nm signal which couples the transition 4 I 15/2 -→ 4 I 13/2 and a 977 nm pump laser which pump the ions onto the upper state 4 I 11/2 (see figure 1). We also assume a fast decay from the upper state 4 I 11/2 to the metastable state ( 4 I 13/2 ). We obtain the rate equation for N 1 (the population of the ground state normalized to the total ion density ρ):

Propagation equations

∂N 1 ∂t = 1 τ 1 -N 1 + β s 1 + β s P s (1 -N 1 ) - 1 1 + β s P s N 1 -P p N 1 , (1)
where the four terms on the right hand side of the equation stand for the spontaneous emission from the metastable state, the stimulated emission from the metastable state, the absorption to the metastable state, and the absorption to the upper level, respectively. Here t is the time variable. P s is the signal power normalized to the signal saturation power P sat s ≡ hω s A s /(τ (σ 21 + σ 12 )), where τ is the lifetime of the metastable state, σ 21 is the emission cross section at the signal frequency ω s , σ 12 is the absorption cross section at the signal frequency ω s , and A s is the signal mode area. The ratio between the signal cross sections is β s ≡ σ 21 /σ 12 . P p is the pump power normalized to the pump saturation power P sat p ≡ hω p A p /(τσ 13 ), where σ 13 is the absorption cross section at the pump frequency ω p , and A p is the pump mode area. η s ≡ A c /A s is the ratio between the fiber core area and the signal mode area, and η p ≡ A c /A p is the ratio between the fiber core area and the pump mode area.

The behavior of the signal and pump powers along the fiber can be described by the following propagation equations:

∂P s ∂z + n gs c ∂P s ∂t = σ 12 ρ [β s -(1 + β s )N 1 ] η s P s , (2)
∂P p ∂z + n gp c ∂P p ∂t = -σ 13 ρN 1 η p P p , (3)
where z is the distance along the fiber. Here n gs and n gp are the group indexes of the signal and pump in the fiber glass, respectively. The signal is modulated as follows: P s = P s0 + P sm cos(2πf m t), where P s0 is the average signal power, P sm is the modulation amplitude (modulation depth P sm /P s0 ), and f m is the modulation frequency. Due to this modulation, a periodic coherent oscillation of the population of the ground state occurs at this frequency (N 1 = N 1st +N 1c cos(2πf m t)+N 1s sin(2πf m t)) which produces a hole in the absorption spectrum for the modulated part of the signal. Thus, a slow light propagation of P sm takes place. The presence of the pump beam allows to turn the delay into advancement (fast light), depending on the value of the pump power. However, the magnitude of the advancement obtained remains nearly constant when certain pump level has been achieved [7,9,10,16,20].

Here, we also modulate the pump beam with the same frequency as the signal beam, being the pump power made up of an average pump power P p0 , and a sinusoidal part with amplitude P pm varying at frequency f m , i.e., P p = P p0 +P pm cos(2πf m t-ϕ). Note that the pump beam could be out of phase with the signal beam by a magnitude ϕ. Thus, the stationary part (N 1st ) and the oscillating parts (N 1c and N 1s ) of the population of the ground state will be given by:

N 1st = 1 w c 1 + β s 1 + β s P s0 , (4)
N 1c = w c P sm w 2 c + w 2 m β s 1 + β s -N 1st - N 1st P pm w 2 c + w 2 m (w c cos ϕ -w m sin ϕ) , (5)
N 1s = w m P sm w 2 c + w 2 m β s 1 + β s -N 1st - N 1st P pm w 2 c + w 2 m (w m cos ϕ + w c sin ϕ) , (6)
where w m ≡ 2πf m τ is the dimensionless modulation frequency and w c ≡ 1 + P s0 + P p0 is a dimensionless frequency which determines the width of the transparency hole created in the absorption or gain spectrum by means of CPO. This magnitude roughly measures the maximum modulation bandwidth that can experience the full slow or fast light effect [21] and it can be increased with the signal and pump powers. The expression obtained for the out of phase part of the population oscillation N 1s presents two terms: the first one, which is proportional to P sm , is responsible of the delay/advancement experienced by the modulated part of the signal. The second term, which is proportional to P pm , arises from the modulation imposed to the pump beam and incorporates explicitly the phase delay ϕ. We will show that by adjusting the value of ϕ, the delay and advancement experienced by the modulated part of the signal can be controlled. Finally, we can obtain the propagation equations for the average powers and the amplitudes of the modulation of the signal and pump beams, and for the phase shift experienced by the periodic part of the signal and pump beams due to CPO:

dP s0 dz = α s P s0 [β s -(1 + β s )N 1st ] , (7)
dP p0 dz = -α p P p0 N 1st , (8)
dP sm dz = -α s (1 + β s )P s0 N 1c + α s [β s -(1 + β s )N 1st ] P sm , (9)
dP pm dz = -α p P p0 (N 1c cos ϕ + N 1s sin ϕ) -α p P pm N 1st , (10)
dφ s dz = α s (1 + β s ) P s0 P sm N 1s , (11)
dφ p dz = α p P p0 P pm (N 1s cos ϕ -N 1c sin ϕ) , (12)
where the distance has been normalized to the fiber length L such that z ≡ z/L. α s ≡ Lσ 12 ρη s is the dimensionless signal absorption coefficient and α p ≡ Lσ 13 ρη p is the dimensionless pump absorption coefficient. We neglect the term with the temporal derivative in equations ( 2) and ( 3). This term leads to a small delay (delay due to the fiber glass refractive index dispersion) compared to the delay due to CPO. The fractional delay of the signal (time delay normalized to the time duration of the signal) at the end of the fiber can be obtained from the signal phase shift at z = L, i.e., F s ≡ φ s (z = L)/(2π). While for the pump it will be F p ≡ φ p (z = L)/(2π).

Experimental setup

The experimental setup is depicted in figure 2 and consists of an EDF in the forwardpumped configuration. The signal beam comes from a pigtailed distributed feedback laser diode operating at 1536 nm provided with a current and temperature controller which allows us to operate the laser at room temperature. This laser beam couples the transition between the erbium ground state 4 I 15/2 and the metastable state 4 I 13/2 (see figure 1). The signal beam is split into two beams: one part of the beam (1%) is sent directly to a switchable-gain amplified InGaAs photodetector, which is used as reference. The other part of the beam (99%) goes through the EDF under study and then to an identical photodetector. The EDF is pumped by a co-propagating beam coming from a pigtailed laser diode operating at 977 nm provided also with a current and temperature controller. This laser beam couples the erbium ground state with the fast decay upper pump level ( 4 I 11/2 ) (see figure 1). The pump beam is also split into two beams: one part of the beam (1%) is sent directly to a photodetector, which is used as reference. The other part of the beam (99%) goes through the EDF and then to an identical photodetector. The signal and pump beams are injected into the EDF through a wavelength division multiplexer. The pump and the signal beams at the output of the EDF are separated by means of an identical wavelength division multiplexer. Both laser powers are sinusoidally modulated with the same modulation frequency f m by using two synchronized function generators which allow us to control the relative phase ϕ between both laser powers at the input of the fiber. Four signals (the reference signal, the signal propagated through the EDF, the reference pump, and the pump propagated through the EDF) are recorded with a fast data acquisition card and transferred to a computer for analysis. The experiment is controlled with a LabView program. We compute the time delay or advancement of the signal t ds from the correlation of the reference signal and the signal propagated through the EDF. Similarly, we compute the time delay or advancement of the pump t dp from the correlation of the reference pump and the pump propagated through the EDF. Then, the fractional delay/advancement of the signal is defined as F s ≡ t ds f m , while the fractional delay/advancement of the pump is defined as F p ≡ t dp f m . We used a 0.1 m-long single mode Al 2 SiO 5 -glass-based fiber highly doped with Er 3+ ions (ion density ρ = 6.3 × 10 25 m -3 ).

Experimental results and simulations

We have carried out two different set of experiments. In the first one, we analyze the enhancement of the slow and fast light propagation effects produced when forcing the population oscillations with a modulated pump. In the second one, we study the coupling of both modulated beams.

In order to theoretically reproduce the experimental results, we numerically solve the propagation equations ( 7)- (12). We have considered the following parameter values which are consistent with values previously used in other works [9,22]: the signal cross section is σ 12 = σ 21 = 5× 10 -25 m 2 , the pump cross section is σ 13 = 3× 10 -25 m 2 , the signal mode area is A s 33 µm 2 (signal mode field diameter 6.5 µm), the pump mode area is A p 20 µm 2 (pump mode field diameter 5.0 µm), the fiber core area is A c 14 µm 2 (fiber core diameter 4.2 µm), the lifetime of the metastable state is τ 9 ms. Using these parameters, the signal saturation power is P sat s 0.5 mW and the pump saturation power is P sat p 1.5 mW. The dimensionless signal absorption coefficient is α s 1.35, and the dimensionless pump absorption coefficient is α p 1.33. We used the insertion losses in the EDF as fitting parameters to reproduce the experimental curves. Thus, we considered an insertion losses of 0.8 for the signal beam and 0.65 for the pump beam. We are not considering changes in the refractive index due to nonlinear effects such as Kerr type nonlinearity since we are not working in a high power regimen. First of all, we have analyzed the viability of controlling the propagation regime of the signal by means of the pump modulation. In figure 3 we show the dependence of the signal fractional delay F s with the relative phase between the signal and pump modulated beams ϕ. We modulate the signal power (0.14 mW) and the pump power (10 mW) with the same modulation depth (P sm /P s0 = P pm /P p0 0.5) and frequency. Different curves correspond to different modulation frequencies. The experimental results (symbols) have been plotted together with the simulated curves (lines) obtained by numerically solving the system of equations ( 7)- (12). We obtain ultra slow group velocities or very fast group velocities (even negatives) depending on the relative phase ϕ. It can be seen that the value of ϕ that leads to the maximum fractional delay or advancement shifts with the modulation frequency. For low frequencies, the maximum delay takes place around π/2 while the maximum advancement takes place around 3π/2. As the modulation frequency increases, the position of the maximum delay shifts and tends to 0 while the position of the maximum advancement tends to π. This behavior can be analytically explained by neglecting the attenuation or amplification of the signal and pump powers (undepleted solution) in equation (11). In this case, the phase shift of the signal is obtained by assuming implicitly the absence of z-dependence in the problem:

Signal delay controlled by the modulated pump

φ s (z = L) α s P s0 (w 2 c + w 2 m ) w c w m (β s P p0 -1) - P pm P sm (1 + β s + β s P s0 ) sin [ϕ + ξ m ] ,( 13
)
where ξ m ≡ arctan (w m /w c ). By analyzing equation ( 13) we obtain that the maximum fractional advancement/delay takes place at ϕ = (π/2) + nπ -ξ m (with n = 0, 1). Then, at low modulation frequencies (w m < w c ), ξ m 0, so the minimum and the maximum take place around π/2 and 3π/2, respectively; whereas at high modulation frequencies (w m > w c ), ξ m π/2, so the minimum and the maximum take place around 0 and π, respectively, in agreement with the experimental results. With the aim of better understanding the above results, we are going to take into account that the pump beam dominates the population oscillation since the pump power is much higher than the signal power, so its modulation amplitude is higher too. Thus, when only the pump beam is modulated, an oscillation will be induced in the signal beam due to the population oscillations. It is easy to analytically show that, in this particular case, the signal beam will oscillate with a phase shift of ξ m with respect to the pump beam. In fact, this phase shift is the phase difference of the population oscillations with respect to the oscillation of the pump beam. In other words, the signal beam will oscillate in phase with the population oscillations which are induced by the modulated pump. In order to experimentally check this result, we have measured the phase shift between the signal and the pump beams when modulating a 1.5 mW pump beam with a constant signal beam of 0.1 mW. In figure 4 we plot the relative phase between the signal and pump beams as a function of the modulation frequency. We observe that for low modulation frequencies, the signal oscillates in phase with the pump beam, and for high modulation frequencies, the shift between the pump beam and the signal beam is π/2. As expected, the relative phase between the signal and pump beams behaves as ξ m , as can be seen in figure 4 (solid line).

So the oscillation that the pump induces in the signal has a phase shift of ξ m with respect to the pump oscillation. When we modulate both beams simultaneously, as in figure 3, the signal oscillation will not affect to the population oscillations since the signal power and, thus, the signal modulation amplitude are smaller than the corresponding to the pump. Then, at low modulation frequencies (ξ m 0), i.e., when the oscillations of the pump and the population are in phase, the maximum delay/advancement takes place when signal and pump are initially out of phase. However, at high modulation frequencies (ξ m π/2), i.e., when the oscillations of the pump and the population are dephased by π/2, the signal will have a maximum delay or advancement when the signal oscillations are initially in phase with the pump oscillations. Thus, the maximum fractional delay (or advancement) occurs when the phase difference between the population and the signal beam is π/2 (or 3π/2) for all modulation frequencies, i.e., ϕ + ξ m = π/2 (or 3π/2). We have also analyzed the behavior of F s with the amplitude of the pump modulation P pm . Figure 5 shows the experimental (symbols) and the simulated (lines) fractional advancement of the signal F s versus ϕ when we modulate the pump and the signal powers at a low modulation frequency of f m = 20 Hz. The simulated curves have been obtained by numerically solving the system of equations ( 7)- (12). Different curves correspond to different modulation depths of the pump beam. The rest of parameters are the same than the ones used in figure 3. We see how the maximum advancement and maximum delay increase linearly with the modulation amplitude, in agreement with the analytical expression (see equation ( 13)). Note that, for the largest pump modulation depth, the shape of the curve developed by F s (ϕ) slightly deviates from a sinusoidal profile (see circles in figure 5), i.e., from the prediction given by equation (13). This analytical expression was obtained by assuming implicitly the absence of z-dependence in the problem. However, the phase shift of the signal is coupled to the amplitude of the signal modulation, in contrast to what occurs without pump modulation. Therefore, even in the undepleted case, for large amplitudes of the pump modulation we must take into account the spatial variation of the amplitude of the modulated part of the signal in order to accurately explain the behavior of the phase shift of the signal. In order to show this effect, let us consider that the spatial variation of the amplitude of the signal modulation P sm is very small so that:

P sm (z) P sm (0) + zα s L P pm cos ϕ P s0 (1 + β s + β s P s0 ) w 2 c , (14)
where we have considered a low modulation frequency, i.e., w m < w c . By using this expression we calculate the phase shift φ s at the fiber output:

φ s (z = L) ∝ -sin ϕ 1 -P pm cos ϕ α s P s0 (1 + β s + β s P s0 ) 2w 2 c P sm (0) , ( 15
)
which shows a deviation from the sinusoidal profile. This correction leads to a shift of the position of the maximum delay towards larger values of ϕ whereas leads to a shift of the position of the maximum advancement in the opposite direction. The magnitude of this shift increases as the amplitude of the pump modulation increases, in agreement with the experimental results (see figure 5). This deviation of the F s (ϕ) curve from the sinusoidal profile is due to the coupling of the phase shift φ s with the amplitude of signal modulation P sm , which in fact, arises when modulating the pump. Finally, we compare the magnitude of the signal fractional delay/advancement obtained in the presence or absence of modulation of the pump beam. To this aim, we have measured several curves of F s versus f m in the absence and in the presence of modulation of the pump beam. In the presence of modulation, we have also varied the values of the relative phase between the signal and pump modulated beams ϕ. These results are plotted in figure 6. When the pump is not modulated, the maximum fractional delay or advancement that can be achieved within the undepleted approximation is given by ±α s /(16π) which, for our experimental system, is close to ± 0.025. Note that when this approximation is not valid (high optical densities) a more complex relation of the maximum fractional delay with the fiber length is obtained [17]. These two optimum cases have been measured when we modulate the signal with an amplitude of P sm = 0.3P s0 . Optimum delay is achieved without pump, at the optimum modulation frequency w m w c , and an average signal power of 0.5 mW close to the signal saturation power P s0 1 (see squares in figure 6 (a)). Maximum advancement is achieved increasing the signal power to 2 mW and applying a high pump power of 23 mW (see circles in figure 6 (a)). In order to analyze the effect of the periodic modulation of the pump beam, we consider similar parameters to the ones used without pump modulation while we periodically modulate the pump power such that P pm = 0.7P p0 . Different curves shown in figure 6 (b)-(d) correspond to different values of the relative phase between the signal and pump modulated beams ϕ. The simulated curves have also been plotted in figure 6 (lines). We observe that the delay and advancement obtained when modulating the pump beam is around one order of magnitude larger than the one obtained without pump modulation. Furthermore, the behavior exhibited by the signal delay strongly depends on ϕ. We observe that for values of ϕ around π/2, delay is achieved for all values of f m (see figure 6 (b)), whereas for values of ϕ around 3π/2, advancement is achieved for all modulation frequencies (see figure 6 modulation frequency switches from delay to advancement (see figure 6 (c)). These results have been reproduced by the simulations.

The signal fractional advancement can be related with the refractive index of the material at the probe field frequency which usually shows a dispersive-like profile. However, curves in figure 6 show a dispersive-or absorptive-like shape depending on the value of the relative phase between the signal and pump modulated beams. In order to clarify this behavior, we analyze theoretically the evolution of the fractional advancement and the modulation gain (the ratio of the output to input modulation depths) for the signal and the pump as a function of the modulation frequency. In figure 7 we plot that dependencies for four representative values of the relative phase between the signal and pump modulated beams and for the same parameters used in figure 6. As we said before, when the pump modulation amplitude is much higher than the signal modulation amplitude, the population oscillations are mainly due to the pump beam. In that case, the dependence of the pump phase shift with the modulation frequency is given by φ p ∝ -ω m /(ω 2 c + ω 2 m ) while the dependence of the pump modulation amplitude with ω m is P pm ∝ ω c /(ω 2 c + ω 2 m ). That is, φ p shows the form of a dispersive-like curve while P pm follows the form of an absorptive-like curve which shows the hole produced by the population oscillations in the pump absorption spectrum.

Both, the dispersive and absorptive phenomena that characterize φ p and P pm , contribute to the phase shift and also to the modulation amplitude of the signal beam through the relative phase ϕ. Thus, for ϕ = 0 or ϕ = π, φ s follows a dispersive-like curve whereas P sm develops an absorptive-like curve similarly to the behavior of φ p and P pm , respectively. In particular, for ϕ = 0, φ s and P sm behaves in the same way than φ p and P pm , respectively, as can be seen in figures 7 (a) and (b). As the signal modulation gain measures the ratio of the transmittance of the modulated part of the signal and the average signal power, we see in figure 7 (b) that the modulated part of the signal P sm exhibits more gain than the average signal power P s0 . Thus, the signal delay can be explained as an antihole in the gain spectrum (gain increase) for P sm generated by the population oscillations governed by the pump. For ϕ = π, φ s and P sm behaves in a similar manner although with opposite sign (see figures 7(c) and (d)). In this case, the signal advancement is due to the hole produced in the gain spectrum by the population oscillations. More complex behaviors are obtained for ϕ = π/2 or ϕ = 3π/2. In those particular cases, there is an interchange between the roles of φ s and P sm . That is, φ s develops an absorptive-like curve following P pm whereas P sm develops a dispersive-like curve following φ p . The case of ϕ = 3π/2 is shown in figures 7 (e) and (f). We see that φ s resembles the spectral hole obtained for the pump whereas P sm looks like the dispersive-like curve developed for the pump phase shift. In this case, the modulated part of the signal P sm exhibits more gain than the average signal power P s0 and this phenomenon occurs for frequencies around ω c (see figures 7 (e) and (f)). This shifted gain peak could be seen as a hole in the gain spectrum around ω m = 0, leading to signal advancement. For ϕ = π/2, a similar result is obtained (see figures 7 (g) and (h)). In this case, the modulated part of the signal P sm exhibits less gain than the average signal power P s0 and this phenomenon occurs again for frequencies around ω c (see figures 7 (g) and (h)). This shifted hole peak in the gain spectrum could be see as an antihole around ω m = 0. Thus, the signal delay could be explained in terms of this antihole. In summary, the behavior of the signal described above can be explained in terms of the interaction between the signal and the population oscillations generated by the pump. In some way, we could say that the pump intensity profile is imprinted onto the signal one through the population oscillations.

Signal and pump delay coupling

Now, let us analyze the interplay between the delay undergone by the signal beam and the one experienced by the pump beam, i.e., we study simultaneously the group velocity of both beams and the influence of the relative phase ϕ in those velocities. We analyze under which conditions the coupling between the two beams makes that both propagate with the same group velocity, that is, the conditions that lead to a synchronization-like phenomenon. To this end, let us first analyze in more detail the relation between the analytical expression of the signal phase shift derived in equation ( 13) and the expression derived for the pump phase shift following the same approach:

φ p (z = L) α p P p0 (w 2 c + w 2 m ) w c (1 + β s ) -w m (1 + β s + β s P s0 ) + P sm P pm (β s P p0 -1) sin [ϕ -ξ m ] .( 16
)
We look for a condition which allows both beams to travel with the same group velocity independently of the average pump or signal powers (P p0 and P s0 ). To this end, we consider the case where the two beams are modulated in phase or in opposite phase, i.e., ϕ = nπ with n = 0, 1. Then, the signal and pump phase shifts remain:

φ s (z = L) α s P s0 w m (w 2 c + w 2 m ) w c (β s P p0 -1) - P pm P sm (1 + β s + β s P s0 ) (-1) n , (17)
φ p (z = L) α p P p0 w m (w 2 c + w 2 m ) w c (1 + β s ) P sm P pm (β s P p0 -1) (-1) n -(1 + β s + β s P s0 ) . (18)
An inspection of equations ( 17)-( 18) reveals that both beams suffer the same delay (in absolute value) when the ratio of the modulation depths is:

P sm /P s0 P pm /P p0 = α s (1 + β s ) α p , (19)
which for our experimental system is close to 2. Thus, φ s = φ p at ϕ = 0 and φ s = -φ p at ϕ = π. To experimentally observe these predictions, we have measured simultaneously F s and F p as a function of the modulation frequency f m for different values of ϕ. We modulate the signal power (1.8 mW) with a modulation depth of P sm /P s0 0.65 and the pump power (2.2 mW) with a modulation depth of P pm /P p0 0.3 in such a way that the ratio between the modulation depths is close to the one given by equation (19). The experimental results (symbols) and the simulated ones (lines) are shown in figure 8. F s and F p are almost equal when both beams are modulated in phase, i.e., at ϕ = 0 (see figure 8 (a)), in agreement with the theoretical prediction. In this situation, the coupling of both beams leads to similar signal and pump group velocities for all modulation frequencies. That means that both beams propagate in phase along the fiber. On the contrary, when we modulate both beams in phase opposition, i.e., at ϕ = π, F s and F p take the same value but with different sign, in agreement with the theoretical prediction, that is, while the signal suffers advancement the pump suffers a delay of the same value being the optimum modulation frequency around 100 Hz (see figure 8 (c)). More complex behaviors are observed at other values of the phase difference between both modulations, as shown in figure 8 (b) and (d). For example, at ϕ = π/2, the pump beam suffers a transition from advancement to delay as the modulation frequency increases. Note that only the case of exactly equal velocities (ϕ = 0) is stable under propagation. However, for the other values of ϕ, within the undepleted approximation, the difference between the signal and pump phase shifts will depend linearly with the fiber length although this dependence is not relevant in our experiments due to the small values of F s or F p obtained.

We have also experimentally corroborated that, when both beams are modulated in phase (ϕ = 0) or in opposite-phase (ϕ = π), they propagate with the same group velocity (in absolute value) independently of the average pump power. These results are shown in figure 9 where we plot the fractional advancement of the signal and pump as a function of the average pump power for these two relative phases ϕ and for a modulation frequency of f m = 50 Hz. As in figure 8, we use again a signal power of 1.8 mW with a modulation depth of 0.65, and a modulation depth for the pump power of 0.3. We also plot the simulated fractional advancement of the signal (dashed line) and pump (solid line). As expected, the values of F s an F p are quite similar for each pump power when both beams are modulated in phase. Furthermore, when we modulate both beams in phase opposition, the fractional delay of the pump shows the same value than the fractional advancement of the signal and vice versa. As we Figure 8. Fractional advancement of the signal (circles) and pump (squares) as a function of the modulation frequency for different relative phases between the signal and pump modulated beams ϕ. Simulated fractional advancement of the signal (dashed line) and pump (solid line). Signal power = 1.8 mW (P s0 = 3) with Psm = 0.65P s0 , and pump power = 2.2 mW (P p0 = 0.9) with Ppm = 0.3P p0 .

have shown, the in-phase propagation of both beams is a general trend that does not depend on the modulation frequency or the average pump power, so it could be used for the propagation of pulses.

The group velocity synchronization-like phenomenon described in this work could be applied to a more interesting situation where we use two amplitude-modulated light beams, both with carrier frequencies within the optical communications C band (approximately from 1530 to 1565 nm). In fact, as we mention in section 1, Stepanov et al [19] controlled the phase shift of a 1526 nm probe beam in an EDF by using a 1568 nm saturating beam. Following the same analysis used in this section, we could achieve the same group velocity for both beams when the ratio of the modulation depths satisfy the following condition:

where the superscripts (1) and (2) correspond to the different wavelengths.

ϕ π ϕ Figure 9. Fractional advancement of the signal (circles) and pump (squares) as a function of the average pump power for two different relative phases between the signal and pump modulated beams ϕ. Simulated fractional advancement of the signal (dashed line) and pump (solid line). Signal power = 1.8 mW (P s0 = 3) with Psm = 0.65P s0 , and Ppm = 0.3P p0 . fm = 50 Hz

Conclusions

In conclusion, we report the control of the slow and fast light propagation velocity of two amplitude-modulated light beams by means of CPO. These light beams correspond to the usual signal (at 1550 nm) and pump (at 980 nm) beams used in EDF amplifiers. To this end, we induce a simultaneous beat in both beams with the same frequency so that the relative phase between both beats allows us to control their group velocities. This condition could make difficult the application of the proposed mechanism for unpredictable signals, like a data stream. When the population oscillations are mainly governed by the modulated pump, an enhancement of the delay or advancement of the 1550 nm beam in one order of magnitude is observed. When the two modulated beams contribute similarly to the population oscillations, a synchronized propagation of both beams has been found. Thus, when both beams are modulated in phase, they propagate with the same group velocity along the fiber for each modulation frequency. This simultaneous propagation is obtained for all the average pump powers. Furthermore, when we modulate both beams in phase opposition, while signal suffers advancement, pump suffers delay, traveling both with the same absolute value of the group velocity. These results could be used in semiconductors which are good candidates for telecommunication applications due to their fast characteristic relaxation time scale (GHz). In fact, Antón et al [23] have shown that by modulating the injection current in a semiconductor optical amplifier to force the population oscillations they control and enhance the delay of the optical signal. These predictions have been experimentally proved in a very recent work [24].

Figure 2 .

 2 Figure 2. Experimental setup scheme. LD TEC: laser diode and temperature controller; FG: function generator; ϕ: relative phase between the pump modulation and the signal modulation; DFB LD: distributed feedback laser diode; WDM: wavelength division multiplexer; EDF: erbium doped fiber; PD: photodetector; DAQ: data acquisition card; PC: personal computer.

Figure 3 .

 3 Figure 3.Fractional advancement of the signal as a function of the relative phase between the signal and pump modulated beams for three different modulation frequencies. Experiments (symbols) and simulations (lines). Signal power = 0.14 mW (P s0 = 0.24). Pump power = 10 mW (P p0 = 4.3). Modulation depths Psm/P s0 = Ppm/P p0 0.5.

Figure 4 .

 4 Figure 4. Phase between the signal and the pump beams as a function of the modulation frequency. Experiment (symbols) and theory (line). Signal power = 0.1 mW (P s0 = 0.17). Pump power = 1.5 mW (P p0 = 0.66). Pump modulation depth Ppm/P p0 = 0.5.

Figure 5 .

 5 Figure 5. Fractional advancement of the signal as a function of the relative phase between the signal and pump modulated beams for three different pump modulation depths. Experiments (symbols) and simulations (lines). fm = 20 Hz. The rest of parameters are the same than the ones used in figure 3.

Figure 6 .

 6 Figure 6. Fractional advancement of the signal as a function of the modulation frequency: (a) for the unmodulated pump case; and (b)-(d) when modulating the pump beam and for different relative phases between the signal and pump modulated beams ϕ. Experiments (symbols) and simulations (lines). (a) Optimum delay (squares): Signal power = 0.5 mW (P s0 = 0.9) with Psm = 0.3P s0 , and pump power P p0 = 0 mW; Optimum advancement (circles): Signal power = 2 mW (P s0 = 3.4) with Psm = 0.3P s0 , and pump power = 23 mW (P p0 = 10). (b)-(d) Signal power = 0.5 mW (P s0 = 0.9) with Psm = 0.4P s0 , and pump power = 10 mW (P p0 = 4.4) with Ppm = 0.7P p0 .

Figure 7 .

 7 Figure 7. Simulated fractional advancement of the signal (solid line) and pump (dashed line) as a function of the modulation frequency for different relative phases between the signal and pump modulated beams: ϕ = 0 (a) ϕ = π (c), ϕ = 3π/2 (e), and ϕ = π/2 (g). Simulated modulation gain for the signal [(Psm/P s0 )out/(Psm/P s0 ) in ] (solid line) and pump (dashed line) as a function of the modulation frequency for ϕ = 0 (b) ϕ = π (d), ϕ = 3π/2 (f), and ϕ = π/2 (h). P s0 = 0.9 with Psm = 0.4P s0 , and P p0 = 4.4 with Ppm = 0.7P p0 .

P

Figure 1 .

 1 Three-level system for Er 3+ ions. Wavelengths for the signal and pump beams.

			decay fast	4	I 11/2
	P p			4	I 13/2
	977 nm	t	P s	1536 nm
				4 I 15/2

Acknowledgments

This work has been supported by projects MEC-FIS2007-65382 and BSCH-UCM-GR58/08: 910133-1030. We would like to thank F. Arrieta-Yáñez, M. A. Antón, F. Carreño, J. M. Ezquerro, and E. Cabrera-Granado for fruitful discussions and scientific support.


T Efthimiopoulos

N Merlemis
email: nmerlemis@physics.upatras.gr

M E Movsessian

D Pentaris

M Katharakis

Action of counter-propagating laser beams on twophoton excitation of potassium vapour

Keywords: Counter-propagating beams, amplified spontaneous emission (ASE), stimulated hyper Raman scattering (SHRS), parametric four-wave mixing (PFWM) PACS: 32.80.Wr, 42.62.Fi, 42.65.Re, 52.35.Mw

Action of counter-propagating laser beams on two-photon excitation of potassium vapour

T Efthimiopoulos, N Merlemis, M E Movsessian, D Pentaris, M Katharakis

Introduction

The counter-propagation of laser beams in a non linear resonant medium can initiate some interesting parametric processes and produce axial or conical radiation [1]. In general the two-photon excitation near the S-S or S-D transitions of the alkali atoms causes various nonlinear processes, such as amplified spontaneous emission (ASE), stimulated hyper-Raman scattering (SHRS), parametric wave mixing (PWM) and two-step four-wave mixing (FWM) [2]. The competition among them has been examined in recent years [2][3][4][5][6]. Suppression phenomena can take place in such non-linear media because of the destructive quantum interference of the competing excitation pathways [7,8]. In the following the processes involved are presented in some detail.

Figure 1 shows the energy level diagram of a potassium atom simplified to a fivelevel configuration. Two-photon excitation of the 6S 1/2 level can populate the 5P 3/2,1/2 levels by either a SHRS or by an ASE process depending on the detuning of the two-photons from the 6S 1/2 level. In addition, two-photon excitation can lead to parametric emission of two photons at wavelengths near the 6S 1/2 -5P 3/2,1/2 , 5P 3/2,1/2 -4S 1/2 or the 6S 1/2 -4P 3/2,1/2 , 4P 3/2,1/2 -4S 1/2 transitions, due to phase matching conditions. Finally, the emissions corresponding to either ASE or SHRS processes can mix with the laser photons in a four-wave mixing (FWM) process to give a fourth photon. Efficient generation of this FWM radiation is done at different wavelengths near the 5P 3/2,1/2 and 4P 3/2,1/2 levels, compared to the emission of the parametric four wave mixing [4][5][6].

The spatial and temporal characteristics of the radiation produced from the above presented processes are different but difficult to discriminate. They can be though distinguished by their spectral characteristics. For example, by detuning the excitation from exact two-photon resonance ( 1/2 1/2 4 6 S S -

∆

) one can distinguish between ASE (on resonance) and SHRS (below resonance) emission. Additionally when the excitation frequency changes, the ASE frequency remains stable for a given transition, but the frequency of the SHRS changes.

Also detuning far from resonance causes the ASE process to decrease drastically and the SHRS and the PWM processes dominate. It should be mentioned that the partially coherent ASE or the SHRS radiation could combine with the excitation laser radiation in a four-wave mixing process to produce a new radiation. This is a two-step process with the first step being the creation of the ASE or the SHRS photons and the second the mixing with the laser photons. It is understood that since the ASE process involves real population transfer and amplification there is a delay involved in the development of the radiation and consequently the produced pulse by FWM is delayed with respect to the parametrically emitted pulses. The delay depends primarily on the lifetime of the upper level which is involved in the ASE process and the amplification gain factor.

An important phenomenon occurs when there is interaction between the processes involved. For example, ASE or SHRS radiations propagate in the forward (along the direction of the laser beam) and the backward direction but it was shown [3,4] that the presence of the parametrically emitted radiation suppressed the forward components of ASE and SHRS while the backward ones were not affected. The parametric and the FWM processes are emitted only in the forward direction along the excitation laser direction.

In this paper we investigate the radiations (under the excitation of the 4S 1/2 -6S 1/2 twophoton transition of potassium atoms), which are emitted in the direction of the laser beam (forward-I FWD ) and in the opposite (backward-I BWD ) direction. We compare the results with the ones we observe when we have two counter-propagating laser beams in the medium. We

show that using counter-propagating beams we are able to modify the atomic system response and increase the effective number of atoms that participate in the overall interaction.

Experimental apparatus

Figure 2, shows the experimental apparatus used in the experiment. An oscillator/amplifier dye laser (LUMONICS Hyperdye 300) was pumped by a XeCl excimer laser (LUMONICS HYPER-X), to excite the 4S 1/2 -6S 1/2 two-photon transition of potassium. The maximum dye laser energy available was 12 mJ/pulse and it was S-polarized. The laser radiation is equally split and directed inside the potassium vapour cell from opposite directions using a set of three partially reflecting (BS1) and highly reflecting mirrors (M'), which are carefully selected to equally divide the polarized dye laser beam into two counter-propagating beams of equal intensity (solid and dashed lines respectively). Optical path lengths (not shown equal in the figure for simplicity) and optical components inserted in the paths are carefully designed, ensuring that both laser beams entering the cell have the same properties (intensity, polarization, beam cross section etc.). Both beams are aligned properly for good beam overlapping in the cell. In the case where both counter-propagating laser beams are active (bidirectional case) the generated radiation exiting the cell was the same in each direction (I BOTH ). High reflecting mirrors M and one partially reflecting beam splitter BS1 are used to take the generated emission from each side of the cell into the monochromator. The optical paths of the emitted radiations from the two sides of the cell are carefully designed and calibrated (not shown in the graph for simplicity) in order to have comparable results.

Experiments with one laser beam entering the cell (unidirectional case) were done using several beam stops were needed in order to allow only one of the two counter-propagating laser beams from entering the cell and also to allow only forward emitted radiation from the cell (I FWD , co-propagating with the laser beam) or only backward emitted radiation (I BWD , at opposite direction from the laser beam). The emitted radiation was resolved using a monochromator (Jobin Yvon H.R. 1000 M) and detected with a photomultiplier (HAMAMATSU R955). Interference filters (EDMUND SCIENTIFIC) centered at the emitted wavelengths and color-glass (MELLES GRIOT) filters were used to block the transmitted laser beam and to improve the signal to noise ratio. Averaging was done with a boxcar averager (SRS model SR250) and the signal was recorded with a computer using the SRS provided software. The spectral width of the laser pulse was 0.1 cm -1 while the pulse duration was 25 nsec. The potassium atoms were confined in a stainless steel heat pipe with a length of 17 cm and helium was used as a buffer gas. A temperature controller was used to vary the temperature in a range of 150 0 C to 400 0 C (which corresponds to an atomic density of 10 13 cm -3 to 5x10 16 cm -3 respectively). The uniformity of the temperature within the 17 cm vapour zone was found to be better than 3%, using three J type thermocouples. The laser frequency was tuned near the two-photon 4S 1/2 -6S 1/2 transition of potassium with a wave number k=13725.3 cm -1 . Measurement of the ionization signal, due to resonant multiphoton ionization of the potassium atoms when the laser frequency was tuned at the two-photon 4S 1/2 -6S 1/2 transition, was also used in order to further verify the exact position of the twophoton transition.

Results and discussion

In this section we investigate the emission spectra of potassium atoms excited by two photons near the 4S 1/2 -6S 1/2 transition. We focus our analysis on the radiations generated when the 4P 3/2 level is acting as the intermediate level, but similar results are observed for the other transitions. We set the two-photon detuning equal to

Δ 4S 1/ 2 -6S 1/2 =2ω- Ε 6S 1/ 2 -E 4S 1 /2  h
and we examine the features of the generated radiations adjusting Δ 4S 1 /2 -6S 1/2 , the cell temperature and the buffer gas pressure P bg . First we discuss the forward (along the laser beam) and the backward emitted radiation case (opposite to the laser beam). Subsequently we present the results obtained with two counter-propagating laser beams in the cell.

It is well known that, due to the phase matching conditions, all the parametric and the wave mixing processes generate coherent radiation along the direction of the driving laser beam, while the coherent SHRS and the partially coherent ASE processes give radiation in the forward (along the direction of the laser beam) and also in the backward (opposite) direction of the laser beam. In addition, quantum interference phenomena occur when a certain excitation level can be reached by more than one route of excitation (for example the 6S 1/2 level excited by two laser photons and the same level excited by the ω IR and ω 404 radiations produced simultaneously by the parametric process). The result is the reduction of the two-photon excitation. Also suppression of the SHRS and the ASE emissions in the forward (along the laser beam) direction is imposed by the presence of the parametric process as it was shown elsewhere [5][6][7][8][9], concerning the two-photon excitation to the 6S 1/2 level and the parametric generation through the 6S 1/2 -4P 3/2 -4S 1/2 levels.

The reduction of the two-photon excitation and the suppression in the forward direction of the ASE or the SHRS are cancelled when the excitation of the atoms is done by two counter-propagating beams with opposite wave-vectors due to the non existence of the parametric process in this case. The Doppler free two-photon excitation of the 6S 1/2 state by the two counter-propagating beams causes the atoms to see always a frequency of

ω L1 ω L2 =ωω- k L1 -k L2 V =2ω for k L1 =k L2 , independent of their velocity V [10].
This is true when one photon comes from one beam and the second from the other. However when both absorbed photons belong to the same beam the above reasoning is not true. This Doppler-free case could be also seen as an indirect increase of the number density of the atoms without involving other complications (level broadening, phase mismatch, pump depletion) which occur when the atomic density is increased by raising the temperature of the cell. In order to examine the phenomenon we turn our attention to the observed emissions.

Emission near the 693.9 nm (6S 1/2 -4P 3/2 transition)

The investigation was done for a narrow two-photon detuning region near the 4S 1/2 -6S 1/2 transition. The emission corresponding to the 6S 1/2 -4P 3/2 transition in potassium at 693.9 nm propagates in both the forward and the backward directions for the unidirectional case. The intensity depends on the two-photon detuning, the excitation laser intensity and the atomic density of potassium vapour.

Figure 3 shows, for the unidirectional case the intensity I FWD of the 693.9 nm emission in the forward direction and the I BWD in the backward direction respectively as a function of the two-photon detuning Δ 4S 1 /2 -6S 1/2 for a pulse intensity of I L =1.3 MW/cm 2 . The range of the two-photon detuning is within 0.15 cm -1 from resonance and the Doppler broadening is of the order of 0.01 cm -1 . On the same graph we present the intensity of the 693.9 nm emission for the bidirectional case (I BOTH ) with each beam having a pulse intensity of I L =1.3 MW/cm 2 .

In the unidirectional case the forward 693.9 nm radiation intensity I FWD is larger than the I BWD and it is mainly parametric regardless of the two-photon detuning from resonance. The backward SHRS and ASE are not affected by the quantum cancellation effect. For the bidirectional case, we observe that the 693.9 nm radiation intensity I BOTH in each direction is higher than the total sum of the two intensities I SUM =I FWD +I BWD that should be expected if the two directions were acting independently, by more than a factor of two especially off resonance, for Δ 4S 1 /2 -6S 1/2 between 0.10 and 0.15 cm -1 .

These results are due to the quantum interference cancellation, since no wave mixing is possible in the case of counter-propagating beams, and the cancellation of the Doppler effect which brings more atoms in resonance with the field independently of their velocity.

The effect is more pronounced, if the atomic density N is increased to 8x10 15 cm -3 as it is shown in figure 3(b). We observe that the increase of the two-photon detuning range for the bidirectional case is considerable. Moreover, in both atomic densities, for the two-photon detuning of 0.15 cm -1 , we recorded an increase of the intensity of the radiations by almost two orders of magnitude.

Additionally while the I FWD and the I BWD radiations in the unidirectional case have different origin (parametric in the forward direction and mainly SHRS in the backward) in the counter-propagation case both directions have radiations of the same origin (mainly SHRS).

In order to further investigate the origin of the radiation at 693.9 nm (ASE, SHRS or parametric) we checked the action of the buffer gas pressure P bg on the emission intensity in the unidirectional and bidirectional excitation cases, respectively. We expect that, due to the different characteristics of the ASE and SHRS processes, the action of the buffer gas should be different. The ASE is a two-step process and thus one could expect some increase of the ASE intensity when the buffer gas pressure is increased, since the increase of the collisions make the 6S 1/2 state broader, which helps to get more population to the upper level. On the other hand SHRS should decrease rapidly because collisions interrupt the one step coherent process. Finally the parametric emission intensity should only slightly change, because of the small impact of the buffer gas pressure P bg on the phase matching conditions, since the atoms of the buffer gas have no resonant transition close to the wavelengths studied.

In this experiment the parametric intensity is proved to be less dependent on the buffer gas pressure variation than the other processes. The results are presented in figure 4. It is evident that by increasing the buffer gas pressure up to P bg =50 mbar the forward intensity of the 693.9 nm radiation for the unidirectional case practically does not change (parametric case). On the contrary, the intensity of the backward radiation (SHRS case) decreased by two orders of magnitude and the intensity for the bidirectional case seems to follow the SHRS emission change for high buffer gas pressure. Hence we could conclude that:

i) The backward radiation has a SHRS origin and therefore its intensity depends strongly on the buffer gas pressure P bg . If it was of ASE origin we would even expect an increase of the intensity due to level broadening.

ii) The forward radiation is basically a parametric one left after the suppression of the forward SHRS by the parametric emission whose phase matching is affected to a smaller degree by the buffer gas pressure P bg . This is probably due to the smooth change of the phase matching factor with buffer gas pressure.

iii) The nature of the emission in the bidirectional case seems to be SHRS, since the pressure dependence resembles the backward SHRS of the unidirectional case.

In Table 1 we provide a summary of indicative observations for the number density of N=9x10 15 cm -3 for two values of the laser intensity and for three values of the laser detuning from the two-photon resonance. It is obvious that for the case of zero two-photon detuning, the intensity of the 693.9 nm (6S 1/2 -4P 3/2 ) emission as seen in the forward direction increases by a factor of 5 when the input pulse intensity is doubled using two counter-propagating beams of 0.5 MW/cm 2 intensity each one. The effect is stronger when the pumping laser intensity is 1 MW/cm 2 , where the emission at 693.9 nm increases by a factor of 8 in the bidirectional case, indicating the nonlinearity of the involved processes (as mentioned above in the case of the bidirectional excitation the dominant process is SHRS which is generated by a fifth-order nonlinear susceptibility). We should emphasize that for all laser intensities shown in table 1, the emission intensity I BOTH of the bidirectional set-up is much higher than the sum of the two intensities I FWD and I BWD , that should be expected if the two directions were acting independently.

Emission near the 766.5 nm (4P 3/2 -4S 1/2 transition)

The emission near 766.5 nm for the unidirectional propagation of the pumping laser, at low laser intensity, was reported by Katharakis et al. [12]. The spectrum of the radiation has two main peaks: a high energy peak (I HEP ) near 766.5 nm and a low energy peak (I LEP ) near 766.6 nm respectively. According to a proposed model [12] the two peaks of the radiation near the 4P 3/2 -4S 1/2 transition have a different origin. In particular the I HEP is a product of a parametric process along with the 693.9 nm emission. The second peak (I LEP ) is the result of a two-step process which initially involves the creation of a radiation at 404 nm (transition at 5P 3/2 -4S 1/2 ).

This coherent radiation in sequence creates by a parametric process the I LEP and the other intermediate radiations corresponding to the transitions 5P 3/2 -5S 1/2 -4P 3/2 or 5P 3/2 -3D 3/2 -4P 3/2 . In the case of low intensity excitation of the 4S 1/2 -6S 1/2 two-photon transition the emission is observed only in the direction of the laser beam which is consisted with the proposed model.

Figure 5 shows, for the unidirectional excitation, a typical emission peak intensity (forward parametric I HEP spectral peak) recorded as a function of the two-photon detuning for a potassium density of 5x10 15 cm -3 and a low excitation intensity. As it was shown [12] there is some differentiation in the relative intensities of the I HEP and I LEP components with the twophoton detuning indicating the different generation processes involved (one-step and two-step processes correspondingly).

As the laser intensity increases above a few MW/cm 2 another broad peak appears close to the I HEP component (766.5 nm), which strongly depends on the laser intensity. This spectrally broad peak propagates in both directions and it is attributed to the ASE from the relaxing population of the 4P 3/2 state. This state can be populated from the 5P 3/2,1/2 state which is populated by the backward SHRS process or the cascade process. Resent theoretical calculations by Merlemis et al [11] indicate that under similar conditions, amplification without inversion (AWI) of the emitted ASE at the 4P 3/2 -4S 1/2 transition is possible due to the existence of the strong fields in the 6S 1/2 -4P 3/2 and 5P 3/2 -4S 1/2 transitions respectively, providing a cascade or a V-type system.

Figure 6 shows the comparison of the unidirectional and bidirectional cases for the emission at 693.9 nm (6S 1/2 -4P 3/2 ) as a function of the laser peak intensity. Emission observed with counter-propagating beams has intensity much higher than that of the unidirectional case. This is true for both emissions at 693.9 nm and 766.5 nm, respectively. The second emission is smaller than the first due to the strong 4S 1/2 -4P 3/2 absorption. It is already mentioned above that for the unidirectional excitation case the parametric 4P 3/2 -4S 1/2 emission is replaced by a strong bidirectional ASE when the laser intensity exceeds a limit. In the bidirectional case the 4P 3/2 -4S 1/2 emission is again an ASE one, possibly without population inversion [11], while the 6S 1/2 -4P 3/2 is a SHRS emission. We also observe that above certain excitation intensity saturation takes place. Recent theoretical simulations of the potassium atom as a four level system indeed predicts saturation for the I FWD as the result of the redistribution of the population among the excited states [START_REF] Armyras | proceedings of the ICO-photonics[END_REF].

Conclusions

In this paper we presented the characteristics of the emissions observed in the case of the unidirectional and the bidirectional propagation of the excitation laser beam. We studied the emissions that correspond to the 6S 1/2 -4P 3/2 (693.9 nm) and the 4P 3/2 -4S 1/2 (766.5 nm) transitions, respectively. The results can be summarized as follows:

1) In the counter-propagating beam case the two-photon excitation tuning range and the intensity of the 6S 1/2 -4P 3/2 radiation are considerably increased due mainly to the effective atomic population increase because of the Doppler free excitation.

2) In the unidirectional case the forward emitted radiation at 693.9 nm is mostly parametric while the backward is SHRS, due to the suppression of the forward SHRS by the parametric one.

3) In the counter-propagating case the 6S 1/2 -4P 3/2 emission is mainly SHRS and much stronger than the emission of the unidirectional case. This is the result mainly of the Doppler free excitation, but also of the cancellation of the quantum interference effect since there is no parametric emission generated at the 4P 3/2 -4S 1/2 transition that could suppress the SHRS emission.

4) There is a saturation for both 6S 1/2 -4P 3/2 and 4P 3/2 -4S 1/2 emissions for the unidirectional and the bidirectional cases above a certain excitation threshold due probably to the population redistribution. Above 0.6 MW/cm 2 the emission appears saturated.

Table 1. Intensity of the 693.9 nm radiation at the 6S 1/2 -4P 3/2 transition observed for two different laser excitation intensities with a number density of potassium equal to N=9x10 15 cm -3 . The buffer gas pressure P bg was constant at 1.3 mbar.

1

Figure 1 .

 1 Figure 1. Energy level diagram of the potassium atom. In this five-level configuration the two-photon excitation field with frequency ω excites the atom to the state 6S 1/2 . Parametric process produces ω IR and ω 404 respectively. Absorption of two ω photons populates the 6S 1/2 state and subsequent or simultaneous one photon absorption can cause ionization. Radiation at ω IR could be also the result of an ASE or SHRS processes.

Figure 2 . 1 Figure 3 .

 213 Figure 2. The simplified experimental set-up used in this work. Partially reflecting mirrors BS1 and high reflecting mirrors M and M' are used to properly split the dye laser beam into two-counter-propagating beams and also to take emission measurements from both sides of the cell. Radiation leaking through BS2 partially reflecting mirrors is used for energy measurements as well as triggering of the systems. Interference filters and color-glass filters are used to block laser wavelengths from entering the monochromator for better signal to noise ratio and the ionization in the vapour cell is also recorded.

Figure 4 . 1 Figure 5 . 1 Figure 6 .

 41516 Figure 4. Normalized intensities of the 693.9 nm (6S 1/2 -4P 3/2 ) radiation as a function of the buffer gas pressure P bg for the unidirectional and the bidirectional (counter propagating) case of the two-photon excitation. The difference of the forward from the backward component is clear for pressures more than 10 mbar of helium. The backward (I BWD ) component changes rapidly because it is produced by a SHRS process. The I FWD emission in the unidirextional case is mostly parametric. The component I BOTH corresponds to the counter-propagating caseand its variation resembles that of the I BWD which is a SHRS emission strongly affected by the buffer gas pressure.