Alex Knight-Percival

Sarah Jhumka

Martyn Hussey

Andrew James Murray
email: andrew.murray@manchester.ac.uk

Super-elastic electron scattering from the laser-excited 4 1 P 1 state of Calcium at low incident energy

Keywords: No. 34.80 Super-elastic scattering, calcium, Atomic Collision Parameters, Stokes parameters

Atomic Collision Parameters have been measured for electron impact excitation of calcium using the superelastic scattering method, at incident electron energies equivalent to ~10eV and ~12eV. The parameters P lin , L ⊥ and γ were derived for the 4 1 P 1 state, and the related Stokes parameters determined. The results are compared to previously published calculations from four different theories: a relativistic distorted wave calculation, an R-matrix calculation, an R-matrix theory using B-splines, and a Convergent Close Coupling theory.

Introduction.

Derivation of the Atomic Collision Parameters (ACP's) from electron excitation of atoms [1,2] provides the most sensitive test of different scattering theories leading to excitation of atomic targets. In these experiments, an electron of incident energy E inc c collides with an atomic target so as to excite the atom to a state with internal energy E P (see figure 1a). The outgoing electron of energy E out c = E inc c -E P scatters from the target through an angle θ e . If the target is left in a P-state (as is usual in most experiments to date) the atomic charge cloud may be aligned at an angle γ with respect to the incident electron trajectory, or the collision may impart angular momentum to the target. In the former case the alignment is described by a parameter P lin , which determines the relative length of the charge cloud to the width. γ then describes the direction of the major axis of the charge cloud, as shown in figure 1a. In the latter case, the angular momentum vector is orthogonal to the scattering plane, and has an expectation value given by the parameter L ⊥ . A further parameter ρ 00

A is also defined that relates to the possibility of spin-flip of the electron during the interaction.

The atomic collision parameters P lin ,γ , L ⊥ , ρ 00 A ( ) can be determined in several ways.

Coincidence techniques can be used to measure the polarization of photons emitted orthogonal to the scattering plane from the excited atom as it relaxes to a lower state, the photons being time-correlated to electrons that scatter from the excited atom. These measurements determine the polarization using Stokes parameters P 1 , P 2 , P 3 ( ), which form a complete set that fully describe the polarization of light. P 1 and P 2 reveal the linear components of the radiation, and can be directly related to P lin and γ through the relationships:

P 1 = P lin cos 2γ P 2 = P lin sin 2γ (1) where both P 1 and P 2 are here determined with respect to the direction of the incident electron. P 3 is related to the parameter L ⊥ through the relationship:

P 3 = -L ⊥ (2)
The parameter ρ 00

A is determined by measuring the polarization P 4 of photons emitted in the scattering plane. In this case the relationship is more complex and is given by: P 4 = 1-3ρ 00

A + P lin 1 -ρ 00 A ( ) cos 2γ 1+ ρ 00 A + P lin 1-ρ 00 A ( ) cos 2γ (3)
The coincidence technique is a well established method that allows the ACP's to be fully determined. However, a problem with this technique is low yield, due to the high probability that the photon correlated with the detected electron is emitted in a direction away from the photon detector. In this case no coincidence is observed. The low coincidence yield from these experiments has hence limited the technique to small scattering angles, where the count rates are relatively high.

A contrasting technique that also determines the ACP's is the super-elastic scattering method, pioneered by Hertel and co-workers [3,4]. This method uses time-reversal arguments to provide an equivalent set of measurements. In this case, the experiment starts with a photon from a laser beam of equal energy to that emitted by the atom in the coincidence measurement. This photon is set to have well defined polarization, and is directed in the opposite direction to that emitted in coincidence studies. The electron source (the gun) is moved to the position of the electron analyser, and is adjusted to produce electrons of energy

E inc s = E out c
. The electron analyser moves to the position of the source, and is set to detect electrons of energy E out s = E inc c . Since the detected electrons now have a larger energy than E inc s , they can only arise from de-excitation of the laser-excited atom, and so are superelastically scattered. Figure 1b shows the geometry adopted in these experiments.

The key advantage of the super-elastic method is that the laser beam is always directed orthogonal to the scattering plane, eliminating the restrictions suffered by the coincidence technique described above. Further, since time correlation techniques are no longer required, it is only necessary to measure the rate of super-elastically scattered electrons at a given scattering angle θ e as a function of the polarization of the laser beam. The count rates hence are far higher than for coincidence studies, yielding better overall statistical accuracy. Since the rates are higher, it is also possible to determine the ACP's over a much larger range of scattering angles [5].

The super-elastic technique also has limitations, principally due to the restricted number of targets that can be excited by existing laser radiation. These targets are mainly limited to alkali and alkali-earth atoms [see for example, 5-15], although new methods are under development that will allow a much larger number of atoms to be excited. A further complexity arises due to the high intensity of the laser radiation, which leads to Rabi-cycling and optical pumping of the atoms under study. Optical pumping becomes significantly more complex for targets with hyperfine structure (such as alkali atoms), and so an extensive experimental and theoretical study has been undertaken to determine these effects [16][17][18]. It has been shown that the laser interaction leads to an optical pumping parameter that can be decoupled from the electron collision, allowing the ACP's to be determined from the superelastic measurements. For atoms with hyperfine structure, the uncertainty in the optical pumping parameter leads to an additional uncertainty in the derived ACP's. For atoms with no hyperfine structure (such as calcium), the ACP's can be directly related to the super-elastic measurements.

A number of different collision theories have been applied to the study of the ACP's. At relatively high incident energies, models such as the distorted wave Born approximation (DWBA) and their relativistic derivatives (RDW) have proven accurate in describing the interactions for a range of targets [19]. Other models with proven success are the convergent close coupling (CCC) methods [20] and R-matrix techniques [21,22]. For the calcium target under study here, the RDW model is known to be accurate at incident energies above 25eV [19], however agreement between theory and experiment was seen to significantly deteriorate below this energy. This is not surprising as the RDW model is expected to become less accurate as the energy is lowered. By contrast, CCC and R-matrix calculations show excellent agreement with experiment for all energies down to 20eV [20][21][22], and further predict results at energies as low as 10eV. At 10eV, differences are seen in the Stokes parameters predicted by the different models, and so the motivation of the experiments detailed here was to provide data to test these predictions.

Experimental results are hence presented for super-elastic scattering from calcium at incident energies E inc c ~10eV and ~12eV . The parameters P lin ,γ and L ⊥ were directly determined using laser radiation from a frequency doubled Ti:Sapphire laser operating at ~423nm. ρ 00 A was not measured. From these results the Stokes parameters are inferred using equations 1 and 2, and the results compared to predictions from CCC [20], R-matrix [21], RDW [19] and B-Spline R-matrix [22] calculations at 10eV. This paper is hence divided into four sections. Following this introduction the apparatus is described, and the techniques used to obtain the experimental data and derived ACP's are detailed. The experimental results are then presented, and are compared to predictions from the different models at 10eV. Conclusions and a summary of future work are then given.

Experimental techniques.

Figure 2 depicts the apparatus used in these experiments. An electron gun produces a beam of well defined momentum that passes through the interaction region created by an atomic beam and laser beam. The incident electron energy was set to ~7.1eV and ~9.1eV for an equivalent coincidence experiment energy of ~10eV and ~12eV. The energy resolution of the apparatus was 1.2eV. The energy of the electron beam was calibrated at low beam currents ( ~100nA ) by detecting the 19.3eV elastic resonance in helium [23]. The super-elastic data were taken at significantly higher beam currents ( ~6µA ), which resulted in an additional uncertainty in the energy of the experiment due to surface charging of calcium deposition on elements inside the vacuum chamber (this has been observed for calcium previously [6]). The effect of these deposits are difficult to estimate, but were expected to lead to an overall uncertainty in the electron energy of ~2eV.

The electron detector is of a standard hemispherical design and uses a 3-element zoom lens to image electrons from the interaction region onto the entrance of the energy analyser. The residual energy of the analyser was set 2.93eV above that of the gun to detect electrons superelastically scattered from the interaction region. The electron detector was moved around the interaction region so as to measure the super-elastic signal at different scattering angles θ e . This is a variation on the 'standard' super-elastic experiment which moves the electron gun rather than the detector. This alternative arrangement produces the same results as for the conventional measurements.

The calcium atomic beam was produced from a well collimated oven resulting in an atomic beam of narrow Doppler profile (~70MHz FWHM) [24]. Atoms that passed through the interaction region were collected by a liquid nitrogen cold trap, so as to minimise deposition onto other surfaces inside the spectrometer [25]. The vacuum pressure inside the chamber was ~2 x 10 -7 torr during operation.

The laser beam used in these studies was produced from a new system in the laboratory. This was a Spectra Physics Matisse Ti:Sapphire ring laser pumped by a 15W Millenia laser, and a Wavetrain external cavity frequency doubler which produced radiation at ~423nm. The output power from the Wavetrain was set to 150mW during operation, which is sufficient to excite more than 45% of the atoms in the interaction region to the 4 1 P 1 state [23]. Fluorescence from this interaction was monitored, and the polarization of this radiation was found to be >99.5%, confirming that radiation trapping was not significant in these studies.

The Spectra Physics laser was far more stable in frequency compared to the Coherent MBR-110 laser system used in previous work [5][6][7]. It was however still necessary to stabilise the laser for long periods of time, and this was accomplished by observing fluorescence from the interaction region inside the spectrometer so as to ensure the laser remained on resonance. At the energies used here the super-elastic yield could be very low (often <1Hz at higher scattering angles), and so it was necessary to continuously operate the experiment for several weeks to produce reliable data. The improved stability of the new Spectra Physics laser system ensured that this was possible.

A difficulty with low super-elastic yield was that the data accumulation time had to be long, and the background resulting from the elastic peak then subtracted. This was accomplished by invoking a computer controlled laser beam shutter into the experiment, so that the true signal could be derived. Accumulation times of several minutes were required for each data point at the higher scattering angles, resulting in several days operation to complete a set of data at a given angle. The results presented in this paper hence took around eight months to complete.

As noted previously [5][6][7] the technique adopted in Manchester does not measure the Stokes parameters, but rather determines the ACP's directly from measurement. This results in higher precision, in particular for the alignment angle γ θ e ( ). To calculate P lin θ e ( ) and γ θ e ( )

the polarization vector ê of the incident laser beam was set linear by a Glan-laser polarizer, and was initially positioned parallel to the incident electron beam direction using a zero-order λ 2 plate. The polarization vector was then rotated by an angle ε through a total of 720°in steps of 10°by rotating the λ 2 plate through 360°. A function of the form:

S θ e ε ( )= A θ e + B θ e cos 2 ε + φ θ e ( ) (3)
was fitted to the data to establish A θ e , B θ e and φ θ e . P lin θ e ( ) is then calculated using the expression:

P lin θ e ( )= B θ e 2A θ e + B θ e (4)
γ θ e ( ) is derived directly from the phase angle φ θ e and scattering angle θ e using the expression:

γ θ e ( )= θ e -φ θ e + nπ ( 5
)
where n takes on integer values set by the geometry of the experiment (ie on which side of the detection plane the analyser is located). γ θ e ( ) is further constrained to lie between ± π 2 for direct comparison with theory.

To determine L ⊥ θ e ( ), circularly polarized incident radiation was required. A zero-order λ 4 plate was hence inserted after the λ 2 plate and the resulting laser radiation retro-reflected through the Glan-laser polarizer using a mirror. The λ 4 plate was then rotated to minimise the transmitted signal, so as to ensure the correct orientation of the optic axis with respect to that of the λ 2 plate. The λ 2 plate was then rotated through multiple angles of 45°to produce left-and right-hand circularly polarized radiation at the interaction region. The correct handedness of the radiation was determined by carrying out super-elastic experiments at 55eV, where the sign of L ⊥ is well known [7]. L ⊥ θ e ( ) was then calculated using the expression:

L ⊥ θ e ( )= S RHC θ e ( )-S LHC θ e ( ) S RHC θ e ( )+ S LHC θ e ( ) (6)
A final parameter P tot θ e ( )= P lin 2 θ e ( )+ L ⊥ 2 θ e ( ) can be deduced that defines the degree of polarization, which is a measure of the coherence of the scattering process. For a fully coherent interaction P tot = 1 .

Experimental results and comparison to theory.

3.1.1 12eV equivalent energy.

Figure 3 shows the experimental results for the higher incident energy with E inc c ~12eV ,

where the Stokes parameters P 1 , P 2 ( ) have been calculated from P lin and γ using equation 1.

Since P 3 = -L ⊥ , only L ⊥ is shown. P tot is also depicted, calculated from P lin and L ⊥ .

The results for P lin (figure 3a) indicate that minima exist at angles θ e ~65°, 100°and 125°, the minimum at the lower scattering angle being deepest at around 0.27. The largest value of The parameter L ⊥ (figure 3e) is of a relatively simpler form, and shows only a single large dip at θ e ~105°, where L ⊥ ~-1 . A similar magnitude occurs at θ e ~60°-70°, where L ⊥ ~+1 . In both cases, the charge cloud is fully oriented, with the bound electrons orbiting in opposite directions. The parameter P tot (figure 3f) lies mostly around unity, indicating that the interaction is fully coherent. There is a small decrease in this parameter at the highest scattering angles, which indicates a small loss of coherence in the interaction.

The derived Stokes parameters P 1 , P 2 ( ) using equation ( 1) are shown in figures 3c and 3d. P 1 displays two main features at θ e ~70°,110°where local maxima are observed. At the higher scattering angle P 1 ~1. By contrast, P 2 shows two broad maxima at θ e ~60°,100°. Since the Stokes parameters do not directly relate to the charge cloud structure, they are less descriptive than P lin and γ . They are presented here as theoretical colleagues often only provide the calculated Stokes parameters. This is an historical artefact that has arisen from earlier coincidence studies, which measure the Stokes parameters rather than the ACP's. Since the super-elastic technique provides data of much higher precision than coincidence studies and can directly determine P lin ,γ and L ⊥ , it would be useful if future theoretical results could also directly provide the ACP's from their calculations.

Comparison between data at 12eV & 10eV.

Figure 4 shows a comparison of the results at ~12eV against those at ~10eV for the parameters P lin ,γ and L ⊥ . The results at 10eV show a simpler structure than for 12eV, although there are similarities between the data as might be expected. P lin at 10eV shows a shallower minimum in the forward direction than at 12eV for a scattering angle θ e ~45°, with P lin ~0.55 . The dip at the higher scattering angle of θ e ~130°is of similar size to that at 12eV. It is less clear if a dip exists at intermediate angles due to the poorer statistics at this energy. The observed minima in P lin at 10eV are broader than at 12eV, and have moved apart.

The γ parameter at 10eV also is seen to be relatively constant with γ ~-60°at the lower scattering angles. The charge cloud angle does not however show the rapid change observed at 12eV when θ e ~100°, but rather increases to γ ~0°at θ e ~110°, after which it too reduces in angle as θ e increases. There is evidence of a small dip around θ e = 65°at 10eV, but there are not enough data points to make a definite conclusion about this feature.

L ⊥ shows a simpler structure at 10eV (figure 4(f)) compared to 12eV (figure 4(e)), with a broad feature appearing at θ e ~85°. The range of L ⊥ is reduced from +0.80 at θ e ~125°to -0.53 at θ e ~85°for this energy. The width of the minimum is larger at 10eV than at 12eV.

Overall the results for the ACP's at the two energies are similar, but reflect the significant changes that can occur as a function of a small change in the incident beam energy. The results contrast significantly to those obtained at higher energies as in [6], which tend to show additional structures with correspondingly narrower features.

Comparison between data at 10eV, 12eV & theory.

Figure 5 and figure 6 presents the data at ~10eV and ~12eV compared to the results from theoretical calculations at 10eV. Comparison is made to both sets of experimental data, due to the uncertainty in the incident electron energy arising from contact potentials in the spectrometer for the high incident electron beam currents used here, as discussed in section 2.

These potentials lead to an estimated uncertainty in the calibrated energy of ~2eV. The relative energy between the data sets should however be accurate at ∆E = 2eV , since the operating conditions of the spectrometer were the same at each energy.

Four different calculations are depicted, taken from data already published for the Stokes parameters. The models include a relativistic Distorted wave calculation (RDW) [19], an Rmatrix calculation (RM) [21], an R-matrix calculation using B-Splines (BSR) [22] and a convergent close coupling calculation (CCC) [20]. The R-matrix calculations use different numbers of states to converge (24 for the RM method, and 39 for the BSR method). Since the published data only provide Stokes parameters, the associated atomic collision parameters have been re-derived from the theory to allow comparison with experimental data, as detailed above.

An immediate observation from this comparison is that no theory agrees well with the experimental data for ~10eV, apart from in overall structure (figure 5). Each theory predicts different positions of the peaks and troughs, although there are similarities in the overall number of features predicted. As an example, the CCC, RM and BSR calculations all show peaks and troughs at similar scattering angles for P 1 , P 2 and L ⊥ = -P 3 (figures 5c, 5d and 5e)

although their magnitude varies significantly between different models. The RDW calculation shows large differences compared to all other calculations, with the same number of features but with magnitudes and positions of the observed features quite different to other models.

The Stokes parameters P 1 , P 2 , P 3 = -L ⊥ derived from experiment also show broadly similar structures to the calculations, but the position and magnitude of the features do not agree with any model at this energy. For L ⊥ = -P 3 the RDW model is in closest agreement with the data, however for P 1 the RDW model predicts a minimum at θ e ~70°compared to a maximum in other calculations. P 1 is better represented by the other three models. No model predicts the data for P 2 , apart from at scattering angles θ e > 105°where agreement is reasonable. The CCC, BSR and RM models do not reproduce the position of the minimum for L ⊥ , and overestimate the magnitude of this parameter at the lower scattering angles.

Since the Stokes parameters (figures (c) -(e)) are not in good agreement with experiment, the ACP's derived from each model are also not in good agreement, as seen for P lin and γ (figures 5a, 5b). For P lin no model agrees well with the data. For the γ parameter the CCC and BSR models predict a local minimum at θ e ~50°-60°but both predict this minimum to be at γ ~±90°, in contrast to the data which finds γ ~-60°over a wide range of scattering angles. Both models predict the charge cloud to reverse direction at θ e ~65°-70°which is not observed. The RM model predicts a minimum at θ e ~68°with γ ~+9°, before reversing direction at θ e ~80°, in complete disagreement with the data. The RDW model is in poorest agreement with the data overall, apart from at the higher scattering angles with θ e > 105°w here all models predict the results reasonably well.

The data indicate that there is some loss of coherence in the interaction, as given by P tot (figure 5f). In particular there is an observed reduction from P tot = 1 at θ e ~45°that is predicted by the RDW calculation. The calculated loss of coherence observed at the higher angles is not predicted.

Given the poor agreement between theory and experiment for the data at ~10eV, and since the absolute energy of the experiment has an uncertainty ~2eV, the results for 12eV are also plotted against theory in figure 6. There is now far better agreement between the data and the predictions. The RDW model still does not agree with experiment. The R-Matrix model is in much better agreement, apart from for γ where it deviated strongly from the data at the lower scattering angles. The RM model predicts P lin and L ⊥ well, although the minimum in P lin at lower scattering angles is wider and deeper than the data.

The most accurate models are the R-matrix using B-Splines (BSR) and the convergent Close

Coupling theories (CCC). The BSR does not obtain the correct position for the dips in P lin or L ⊥ , however the magnitudes of these features are well produced. The results for γ are in good agreement with the data, but once again there appears to be an angular shift in the features compared to experiment. The CCC model accurately predicts the positions of all features in the data, and is reasonably accurate in predicting the magnitudes. In particular, the agreement between the CCC model and data for γ is particularly impressive, with all features accurately reproduced. This model appears to give the closest overall prediction of the data.

It appears from the close agreement between the calculations at 10eV and the data taken at 12eV, that there is an offset in the absolute value of the experimental energy of ~2eV. This is almost certainly due to charging of calcium deposits on surfaces inside the spectrometer. This is an effect that has been observed previously for calcium [5][6][7], however it plays a more critical role in these experiments at low energies due to the higher beam currents that are required to produce statistically meaningful data.

It is worth noting that since the overall energy resolution of the experiment is ~1.2eV, the most accurate comparison with theory requires the calculations to integrate their results over this energy resolution. This is particularly important when the parameters change significantly with energy, as is observed here. It is unlikely that resonance are significant at these energies, however all calculations need to carefully consider the effects of this energy resolution particularly at these low energies, to provide a robust comparison with the experimental results.

Conclusion & Future Work.

In this study we have presented new experimental data for electron impact excitation of calcium at low energies, measured using the super-elastic scattering technique. Data have been taken for P lin , γ and L ⊥ at equivalent incident energies of ~10eV and ~12eV. The results have been compared to calculations using R-matrix, CCC and relativistic distorted wave models. It has been found that an offset of ~2eV in the data is likely, since the results at ~12eV agree well with close coupling theories for 10eV, whereas there is little agreement with the data at the lower energy. It will be interesting to see if calculations performed at 8eV agree with the data at the lower energy presented here. For a proper comparison to be made, all models will need to carefully consider the energy resolution of the spectrometer, since the derived parameters are seen to vary significantly as a function of this energy.

The super-elastic method is the only technique that can provide high precision data over a wide range of scattering angles for comparison to theory. It is therefore important to develop this technique for use with other targets, in particular those which have importance in practical applications such as in plasma research and lighting. The super-elastic technique is however limited to targets that can be excited by existing laser radiation. Recent innovations in high resolution lasers has allowed a much wider range of wavelengths to be produced, and so the opportunity now arises to explore these interactions for many other targets. This will provide new and precise data to further refine the models describing the scattering process, with a goal to establishing a universal model that will predict electron excitation of all targets.

P

  lin is ~0.93 at θ e ~115°, indicating a high degree of alignment at this angle. The alignment angle γ (figure3b) only slowly changes from γ ~-75°at the lower scattering angles from θ e = 30°to θ e = 85°with a small dip to γ = ±90°at θ e = 60°, indicating that the charge cloud is aligned orthogonal to the electron beam direction at this scattering angle . The charge cloud reverses direction from this point, γ increasing to γ ~-70°at θ e = 70°, at which point it reverses direction again and once more decreases. The charge cloud is then observed to continuously rotate around the detection plane as the scattering angle increases from θ e = 70°( note that results at γ = ±π 2 are equivalent). The charge cloud angle becomes relatively static at γ ~-55°for scattering angles 130°≤ θ e ≤ 145°.

Figure 1 .

 1 Figures.

Figure 2 .Figure 3 .Figure 4 .

 234 Figure 2. The super-elastic spectrometer, showing the electron gun, analyser and oven. The laser beam enters through a window on the top flange, and is directed orthogonal to the scattering plane, as set by a tracer laser diode inside the spectrometer.

Figure 5 . 1 Figure 6 .

 516 Figure 5. Data at 10eV nominal energy plotted against different theoretical models calculated for an incident energy of 10eV. Poor agreement is found between theory and experiment. For details see text.

Acknowledgements

The Engineering and Physical Science Research Council is acknowledged for providing funding to carry out this research, for providing part of a studentship to SJ, and for providing a postdoctoral research position for MH.


M Radziunas
email: radziuna@wias-berlin.de

K.-H Hasler
email: hasler@fbh-berlin.de

B Sumpf

Tran Quoc

H Wenzel
email: wenzel@fbh-berlin.de

Mode transitions in DBR semiconductor lasers: Experiments, simulations and analysis

Keywords: numbers: 42.55.Px, 42.65.bc, 42.60.Pk, 42.60.Mi

The performance of a multisection DBR semiconductor laser emitting around 1.06 µm is experimentally and theoretically investigated. A thermal-induced change of the refractive index implied by an increase of the injection current yields nearly-periodic transitions between neighboring cavity modes. These transitions are explained by means of a modal analysis and and by numerical simulations based on the traveling wave model.

Introduction

Wavelength stabilized single-lateral mode semiconductor lasers are required for many applications such as frequency conversion, free-space communication, spectroscopy and metrology. One possibility to achieve wavelength stabilization is through the integration of a Bragg grating into the semiconductor chip. The resulting distributed feedback (DFB) or distributed Bragg reflector (DBR) lasers emit several hundreds of milliwatts and are ideally suited for the above mentioned applications [1,2].

DFB lasers are typically single-section devices with the Bragg grating extending over the whole amplifying cavity. In order to facilitate operation in the same longitudinal mode over a large power range, the front facet is anti-reflection coated to suppress side modes. However, this leads to a high sensitivity to feedback from external surfaces and does not prevent possible transitions between the two DFB resonance modes.

DBR lasers consist of a reflector section containing the Bragg grating and one or several amplifying sections. Fig. 1 shows a schematics of a DBR laser. In order to form a resonator, the front facet has a reflectivity of R ≥ 1% with the value depending on the length of the amplifying sections. Due to the fact that all sections are separately driven, DBR lasers exhibit periodic undulations in the power-current characteristics and temporal instabilities owing to longitudinal mode hopping. On the other hand, compared to DFB lasers they are more robust to external perturbations and can be more cost-effectively fabricated due to the utilization of surface Bragg gratings [3,4,5]. Therefore, it is of great practical interest to reduce the modal instabilities which often hinder the applicability of DBR lasers.

The main aim of our paper is to use a theoretical analysis of the mode-hopping phenomena in DBR lasers to gain a better insight into the underlying physics. Our mathematical model is given by the traveling wave (TW) equations for the complex slowly varying amplitudes of the counter-propagating optical fields, coupled to equations for the induced polarization functions and the excess carrier density [6]. Following [7,8], we relate the mode-hopping in DBR lasers with temperature induced shifts of the DBR stopband and of full cavity resonances. In the considered TW model, these thermal effects are represented by a linear nonlocal dependence of the refractive index on the inhomogeneous injection current [9].

Our explanation of the mode transitions is based on observations and analysis of instantaneous longitudinal optical modes [START_REF]Optoelectronic Devices: advanced simulation and analysis[END_REF] and their dynamics in edge-emitting multisection lasers [9,[START_REF] Bauer | [END_REF]12]. According to our inhomogeneous heating model, the injection currents affect the refractive indices of separate laser parts differently and thereby imply a relative shift of the cavity modes with respect to the reflectivity of the DBR section. This relative shift is responsible for the transitions between modes, observable both in optical experiments and in simulations.

Our paper is organized as follows. In Section 2 we give a short description of the laser device and present some measured characteristics. After introducing the mathematical model in Section 3, we give a definition of optical modes and analyze the relationship between them as well as their dependence on model parameters in Section 4. Section 5 is devoted to the discussion of numerical simulation results. Finally, some conclusions are drawn in Section 6.

Experiments

The DBR lasers under investigation consist of a 3 mm long amplifying section S A and a 1 mm long DBR section S BR (see Fig. 1). The DBR section consists of a sixth order Bragg grating with a period of about 1 µm that was defined by an i-line wafer stepper and dry etched into the surface of epitaxial layer structure [3]. It can not be electrically contacted. The effective reflection coefficient of the Bragg grating is about 55%. The front and rear facets were coated with dielectric layers to obtain reflectivities of 1% and ≈ 0.01%, respectively. Fundamental lateral mode operation is ensured by the use of a 5 µm wide ridge waveguide. The active region consisting of three quantum wells with a thickness of 7 nm is located asymmetrically in a super large optical cavity and extends over the whole cavity. More details on the layer structure can be found in [13] and [14]. Experimental characteristics of an example DBR laser driven in continuous-wave mode are presented in Figs. 2 and3. All these figures indicate nearly-periodic changes of the lasing states as a function of the current I A injected into the amplifying section. Fig. 2 shows the emitted optical power at the front and rear facets versus I A . The threshold current is 90 mA. At the front facet the power-current characteristic is nearly linear with a slope efficiency of 0.75 W/A up to the power of 600 mW, except at periodically reappearing jumps. We note, that while during these jumps the power emitted at the front facet increases (panel a), the power emitted at the rear facet decreases (panel b). A nonsmooth increase of the power before each jump (most visible at the rear facet of the device) can be explained in terms of a non-stationary laser emission.

This non-stationary lasing involving two or more optical modes can be also recognized in Fig. 3 where an evolution of the optical spectrum with the increase of I A is shown. Here, each injection current "period" determined by a pair of neighboring vertical white lines contains the following typical operation regimes.

• At the low-current side of each period the laser shows a single wavelength emission.

• With an increase of I A several neighboring spectral peaks start to grow indicating the formation of periodic field intensity pulsations [START_REF] Bauer | [END_REF]. • At the high-current side of each period an abrupt change of the spectrum and field intensities occur. • After such a jump the laser operates once more a single wavelength now at the neighboring short-wavelenght optical mode.

The step-like red shift of the spectral peaks within each period is due to a limited spectral resolution of our measuring equipment. A ∼ 45 pm separation of the neighboring spectral peaks corresponds to the spacing of the adjacent laser modes. The periodicity of the mode jumps, the averaged slow and periodic fast red shifts of the lasing wavelength (indicated by the slanted white lines in Fig. 3) as well as an increase or decrease of the field emission from both laser facets during the mode jumps will be discussed below in this paper.

Traveling wave model

To simulate and analyze the dynamics of the DBR laser we use the traveling wave (TW) model [6]. It governs the temporal-spatial evolution of the complex slowly varying counter-propagating optical fields E ± (z, t) (scaled so that

|E| 2 = |E + | 2 +|E -| 2
is the local photon density) and the induced polarization functions p ± (z, t):

ng c0 ∂ t E ± = (∓∂ z -iβ(n, I)-D) E ± -iκE ∓ + F ± sp , b.c. : E + (0, t)=r 0 E -(0, t), E -(L, t)=r L E + (L, t), DE ± = ḡ 2 (E ± -p ± ), ∂ t p ± = γ(E ± -p ± )+ iωp ± . (1)
Here, t denotes time, z ∈ [0, L] corresponds to the longitudinal propagation direction, F ± represents the spontaneous emission, r 0 and r L are the complex reflectivities of the field amplitudes at the front and rear facets. The parameters ḡ, γ and ω denote the amplitude, the half width and the peak frequency of Lorentzian approximation of the material gain profile [6], n g is the group index, and κ is the field coupling coefficient which is different from zero only in the DBR section S BR . The field equations ( 1) are coupled to the rate equations for the real excess carrier density n(z, t):

∂ t n(z, t) = J(z) eσ -An + Bn 2 + Cn 3 - -c0 ng e ν=± E ν * (g(n) -2D) E ν , J(z) z∈S k = I k l k , k∈ {A, BR}. (2)
Here, J(z) is the injection current density, l k and I k are the length and the pump current of the k-th laser section S k , σ is the cross-section area of the active zone, A, B and C are three coefficients of the cubic function describing the rate of non-radiative and radiative spontaneous recombination.

The complex field propagation factor β is defined as

β = δ(n, I) + i(g(n)-α) 2 , δ = δ s + δ n (n)+ δ T (I), g(n) = g n tr ln max{n,n * } ntr , δ n (n) = ñ (n -n tr ) , δ T | z∈S k = δ T,k (I) = 2πng λ 2 0 r ν r k I r , r,k∈ {A, BR}. (3)
Here, the parameters α and δ s represent the internal absorption and the static detuning. g and ñ are the differentials of the carrier dependent effective logarithmic gain and linear index change functions g(n) and δ n (n) at the transparency carrier density n tr . n * is the reference carrier density for limiting the absorption in the logarithmic gain function. The coefficients ν r k of the linear thermal detuning function δ T (I) determine the impact of the injection currents I r on the refractive index change within each laser section S k [9]. λ 0 , c 0 and e are the central wavelength, the speed of light in vacuum and the electron charge, respectively. The values of the laser parameters used in our simulations and given in Table 1 were obtained in different ways. For example, the coupling coefficient and the internal absorption were obtained by a fit of the power-current characteristics at front and rear facets, see Fig. 6. In doing so we neglected the fact that the internal quantum efficiency is less than 100%. However, from an analysis of the length-dependence of the slope efficiency of similar broad-area devices we obtained values greater than 95% such that the internal absorption is only slightly overestimated. The cross-section of the active zone was calculated taking a total thickness of 21 nm of the three quantum wells and an effective width of the ridge waveguide of 10 µm to account for lateral current spreading and carrier diffusion.

Owing to the high effective duty cycle of the sixth order grating (> 80%), the radiation losses are low and the imaginary part of the coupling coefficient can be neglected [3]. Other parameters (e.g. transparency carrier density, differential gain and index) were obtained from a microscopic gain model [15] or taken from the literature [START_REF] Piprek | Semiconductor Optoelectronic Devices: Introduction to Physics and Simulation[END_REF]. The impact of the waveguide structure is accounted for in the effective modal gain and index functions, with optical confinement factor of 0.022.

Mode analysis

Before switching to simulations of the TW model ( 1)-(3) let us discuss the origin of the experimentally observed (see Figs. 2 and3) periodically reappearing state transitions with the increase of the injection current. For this reason we perform an analysis of the instantaneous optical modes [START_REF]Optoelectronic Devices: advanced simulation and analysis[END_REF] which determines the different continuous wave states observed experimentally.

For any fixed carrier distribution n(z) the optical field equations (1) and the assumption E(z, t) = Θ(β(n), z)e iΩ(β(n))t give rise to a spectral problem defined by the following equations:

-∂ z -iD(β, Ω) -iκ -iκ ∂ z -iD(β, Ω) Θ + Θ -= 0, Θ + (β, 0) = r 0 Θ -(β, 0), Θ -(β, L)=r L Θ + (β, L), D(β, Ω) = β(n) + ng c0 Ω -χ(Ω), (4)
where

χ(Ω) = i ḡ 2 i(Ω-ω)
γ+i(Ω-ω) is a frequency (wavelength) dependent contribution to the material gain dispersion. The instantaneous, i.e., carrier dependent optical compound cavity modes are determined by the pairs [Θ(z), Ω], where the eigenfunction Θ and the eigenvalue (complex frequency) Ω solve the spectral problem (4). The real and imaginary parts of Ω are the optical angular frequency and the damping of the mode, respectively. A continuous wave (cw) state of the system (1)-( 3) is defined by an optical mode [Θ(β(n), z), Ω(β(n))] with a real eigenvalue Ω. Here, the carrier distribution n(z) fulfills the stationary carrier rate equation (2).

Let us consider some cw state of our TW model determined by functions n(z), Θ s (β(n), z) and (real) mode frequency Ω s (β(n)). In general, a change of laser parameters (e.g., injection currents) implies some changes of this cw state. Below we present two simple parameter tuning mechanisms that maintain the carrier and photon distributions n(z) and |Θ s (β(n), z)| 2 of this state.

The first mechanism makes use of a uniform change of the propagation factor β in all sections by the same (real) factor ∆ β . It follows from (4) that such modification preserves all eigenfunctions Θ(β, z) and shifts the corresponding eigenvalues Ω(β) of the spectral problem (4): ‡

Θ(β + ∆ β , z) = Θ(β, z), Ω(β + ∆ β ) = Ω(β) - c 0 n g ∆ β .
Since the mode frequencies Ω change by a real factor, the cw state frequency Ω s remains real and the (modified) s-th mode [Θ s (z), Ω s ] defines a cw state again. This mode frequency tuning mechanism determines the slow basic red shift ∂ I λ s of the lasing wavelength (see Fig. 3). ‡ We neglect small changes of the material gain dispersion for closely located frequencies, i.e., assume that ∂ ξ χ(ξ)| ξ=Ω = 0.

The second mechanism makes use of a shift of the detuning

δ k def = 1 l k S k δ(n, I, z)dz, k ∈ {A, BR}
in the amplifying section S A alone. The field coupling coefficient κ in this section vanishes and the eigenfunctions Θ are given by

Θ ± (β, z) = Θ ± (β, 0)e ∓i R z 0 D(β(s),Ω) ds , 0 ≤ z ≤ l A .
The shift of the factor

ϕ = l A δ A /π (5)
by any integer number m preserves the complex ratio between the mode components Θ + and Θ -at the amplifier/reflector interface z = l A , and, therefore, the distributions Θ ± (z) within the DBR section S BR (up to the multiplier (-1) m ) are also unchanged.

Consequently, an integer shift of the phase factor ϕ preserves the modal frequencies Ω and the eigenfunction intensity distributions |Θ ± (z)| 2 . As it will be shown below, such change of ϕ is responsible for the periodically reappearing lasing mode transitions and for the fast red shift ∂ I λ f of the lasing wavelength. To get a deeper understanding of the periodic mode transitions we have performed a modal analysis [START_REF]Optoelectronic Devices: advanced simulation and analysis[END_REF] of the DBR laser under investigation. The evolution of the complex frequencies Ω computed with some fixed n(z) and tuned phase factor ϕ is summarized in Figs. 4 and5. Black symbols in Fig. 4(a) show the location of these eigenvalues in the complex wavelength/damping plain. The black curves connecting these eigenvalues represent their shift with changing ϕ.

The eigenvalues moving along the curve are replacing their next neighbor after a 1-periodic shift of this factor. Note also three separated eigenvalues which move 1-periodic in ϕ around some small closed loops. These separated loops are implied by the field coupling in the DBR section and can be connected to the main curve or shifted to the large damping values by an appropriate choice of the coupling strength or the field loss parameters. For more details on the formation of such loops see Refs. [START_REF]Optoelectronic Devices: advanced simulation and analysis[END_REF][START_REF] Bauer | [END_REF].

A mode with zero damping determines the cw state of the device. The other 5-6 modes located within the stop band of the DBR can be excited by the spontaneous emission noise and, therefore, can be recognized in the optical spectra and the spectral mapping diagrams. To emphasize the mode selection by the Bragg grating we consider also the modes of a reference Fabry-Perot (FP) cavity with its length set equal to the total cavity length of the DBR laser (see gray bullets and lines in the same diagram). Another important feature of the DBR section is shown in Fig. 4(b). Here, the black symbols represent the wavelength separation of a mode from its closest shortwavelength neighbor, which is not necessarily located on the same curve of Fig. 4(a). In contrast to the reference FP cavity (grey bullets and lines), where a ∼ 39 pm mode separation is strictly related to the field round-trip in the device, the mode separation in the DBR laser depends on the mode wavelength and the value of the phase factor ϕ (see the different positions of the black bullets and the variation of the thin black curve). The minima of the mode wavelength separation for selected phases ϕ occur at the sides of the DBR stop band and are related to the separate small closed loops (shorter-wavelength side of Fig. 4(a)) or the sharp peaks of the open black curve (longer-wavelength side of the same diagram). At the shorter-wavelength side a pair of eigenvalues belonging to some small closed loop and the open curve can have the same wavelength but different damping. At the longer-wavelength side the minimal wavelength separation is achieved when a pair of eigenvalues are located on the opposite sides of the same sharp peak of the open curve. We note, however, that being outside of the stop band these modes are well damped and are not relevant for the operation of the DBR laser.

The separation of the least damped modes at the stop band of the DBR (∼ 47 pm in our case) is larger than the ∼ 39 pm mode separation of the reference FP cavity. This is in a nice agreement with the experimentally observed ∼ 45 pm mode separation visible in Fig. 3. The increase of the mode separation within the stop-band of the DBR can be explained by the decrease of the effective DBR (and the compound cavity) length for the monochromatic fields with the corresponding optical frequencies.

Figure 5 gives another representation of the spectral analysis from Fig. 4. Here we show how the damping (panel a) and the wavelength (panel b) of several most important DBR stop-band modes change with the phase parameter ϕ. Due to periodicity of this parameter, the curves leaving these two diagrams at the right side (ϕ = 1) reenter the diagrams from the left side (ϕ = 0).

While the black curves in panel (a) represent only a small change in the main mode damping, the nearly straight and almost parallel curves in panel (b) indicate an almost linear increase of the modal (and lasing) wavelength with the changing factor ϕ which is nearly proportional to the injection current into amplifying sections. This linear increase of the modal wavelengths corresponds to the difference between experimentally observed periodically reappearing fast red shift and slow red shift of the lasing wavelengths (see Fig. 3).

The transitions between the modes seen in Fig. 3 occur when the damping of the neighboring mode becomes smaller than the zero damping of the mode corresponding to the cw operation of the laser. A similar situation (even though we assumed a carrier distribution n that is time invariant) can be recognized in Fig. 5(a). The exchange between the dominant modes represented by bullets and squares is expected to occur close to the value of ϕ indicated by the vertical dashed line. The grey lines in Fig. 5 represent a reference FP cavity where the damping of the modes is almost independent on the mode number or phase factor ϕ (see panel a). The mode wavelengths are equidistant and change linearly with the factor ϕ (see panel b). The mode separation is different from that of the DBR laser (see discussion above). Consequently, the expected (fast) red shift of the wavelengths in this FP cavity is smaller than that one of the DBR laser.

Taking into account the fact that the detuning δ changes in all laser sections simultaneously, we conclude:

• The slow red shift of the lasing wavelength

∂ I λ s ≈ λ 2 0 /2πn g ∂ IA δ BR (6)
observed experimentally occurs due to the changing detuning δ BR . • The periodicity of the mode transitions with the increasing injection current in the amplifier section can be approximated by

I per = 1/∂ IA φ, φ = ϕ -l A δ BR /π. (7)
That is, the periodic mode transitions occur due to the nearly linear increase of the relative index change function φ. • The fast red shift of the lasing wavelength can be approximated by

∂ I λ f ≈ ∂ I λ s + ∆ λ /I per , (8)
where ∆ λ denotes the separation between the wavelengths of the two dominant modes involved in the transition.

The mode analysis presented reproduces the experimentally observed mode transitions well and allows us to estimate some of the coefficients ν r k entering the thermal detuning model δ T in (3). In order to determine ν r k we first estimate the slow wavelength shift ∂ I λ s and the mode transition period I per from Fig. 3, and assume that the impact of electronic detuning δ n to the changes of the full detuning δ is small to give the relationship:

∂ Ir δ k ≈ ∂ Ir δ T,k = 2πn g λ 2 0 ν r k , k,r ∈ {A, BR}.
Using this relationship in combination with ( 6) and ( 7) allows us to estimate ν A A and ν A BR .

Due to the fact that the DBR section is unbiased (I BR = 0), the injection current I BR has no influence to the thermal detuning function δ T , and, therefore, to the shift of the mode wavelengths or to the jumps between different lasing states. Thus, we can assume ν BR A = ν BR BR = 0 which completes the definition of the thermal detuning function δ T .

Simulations

In this section we present our simulation results and discuss some parameter fitting procedures. Figure 6 shows simulated and experimental power-current characteristics of the considered DBR laser. Several basic features such as

• periodically occurring undulations of the emitted power, • transitions characterized by a sudden increase (decrease) of the power emitted from the front (rear) facets and • a non-monotonous growth of the power emitted from the rear facet (typical for non-stationary lasing states) before state transitions are properly reproduced by our simulations. The slope efficiency at both facets of the laser is comparable to experiments and was obtained by a careful choice of the field coupling coefficient κ in the DBR section and the internal absorption. The lasing threshold was fitted mainly by the choice of the absorption limiting density n * . The suggested limitation of the gain function for small carrier densities is in agreement with simulations of a microscopic gain model which reveal an initial slight increase and then a decrease of the absorption with increasing carrier density, in dependence on the wavelength [START_REF] Wenzel | [END_REF]. For n * = 0 the modeled field absorption α-g(n) in the unpumped DBR section in the below threshold regime is un-physically high, which would lead to an increase of the theoretical threshold current by, approximately, 10%. The influence of a non-zero n * at small injection currents is illustrated by comparing thick black lines (absorption limited model) and thin gray (non-limited model) lines in Fig. 7 In Fig. 8 the change of the simulated optical spectrum with the increase of I A is presented. As seen in experiments (see figure 3), we observe here periodically occurring jumps to a short-wavelength neighboring mode in the optical spectra, a multiple peak optical spectra before and a single mode emission after the state transitions. The mode separation in simulations is ∼ 47 pm, which is comparable to the experiments and is significantly larger than the mode separation in a similar FP-type laser. Finally, in both theoretical and experimental cases the wavelength shift within a single "period" of injection current is nearly linear.

Although our simulations successfully reproduce most experimental observations, Figs. 3 and The second of these effects in our simulations is due to the fast growth of n in the DBR section at small above-threshold currents as seen in the lower black curve in Fig. 7(a). For these injection currents, a (blue) carrier density driven shift in the refractive index δ n (n) in the DBR section plays a significant role. It counteracts and even exceeds the thermal red shift δ T (I) in the DBR section, that in turn results in a general blue wavelength shift visible for small I A in Fig. 8.

Another effect of this non-vanishing δ n (n) is represented by differences in the periods of the mode transitions. The derivation of the heating parameters ν from the experimentally estimated slow red shift ∂ I λ s ≈ 20nm/A and the transition period I per ≈ 95 mA in Section 4 was made by neglecting the contribution of δ n . For small above-threshold currents the electronic detuning δ n (n) in the amplifying sections changes only weakly, while in DBR section it decays significantly. This in turn implies an additional positive shift of φ(I) which can be even larger then the thermal one: see, for illustration, the slopes of the thin dashed and thick black lines in Fig. 7(c) representing a pure thermal and a full detuning, respectively. The growth of the function φ(I) at lower I A is much larger than at high I A . Since a current interval between a consequent mode switching corresponds to the change of φ(I) by 1, these intervals are much shorter for small injection currents. The elimination of the general blue shift and the increase of the mode-jumping periods at small injection currents could be achieved by a modification of the carrier dependent index change function δ n (n) so that ∂ n δ n ≈ 0 at small n.

For increased amplifier currents the intensity of the optical field entering DBR section grows. The balance of the carrier rate equation (2) in this section is kept by increasing the gain (decreasing the loss) function g(n) < 0, i.e., by increasing the carrier density, which for high injection levels slowly approaches the transparency density n tr (see Fig. 7a). Due to a slow change of n at larger currents the contribution of δ n is by an order of magnitude smaller than that of δ T , and the ≈ 84 mA simulated mode transition period here is only ∼10% smaller than initially assumed one.

Conclusions

We investigated experimentally and numerically the electro-optical behavior of a multisection DBR semiconductor laser with an active region extending over all sections. The laser emits an optical power of 600 mW at a current of 900 mA. The power-current characteristics show nearly periodically reappearing jumps, where the power at the front facet increases and the power at the rear facet decreases. With an increase of the injection current the laser device exhibits an average slow shift as well as periodically reappearing fast red shifts of the lasing wavelength. The corresponding periodic jumps to adjacent short-wavelength cavity modes are connected with pulsations of the field intensity.

The modal analysis and numerical simulations reveal the mechanisms of the periodic mode jumps, the observed dynamic regimes and the slow or fast shifts of the lasing wavelength. The periodicity of the mode transitions with an increase of injection current is caused by a difference in the refractive index tuning rates in DBR and amplifying sections. The observed 3 stage sequence (1) cw operation, (2) mode-beating pulsations and (3) abrupt change of the lasing wavelength is a direct consequence of the change of gain and damping of the modes with the change of refractive indices. Here, mode-beating pulsations and subsequent mode jumping are due to the vanishing damping of the neighboring mode which eventually becomes the new dominant mode.

The average slow shift of the lasing wavelength arises from the dependencies of the refractive index in the DBR section on the local carrier density and temperature. At low power the increase of the carrier density due to optical pumping can dominate over the thermal detuning and lead to an average blue shift of the lasing wavelength. At higher power the carrier density in optically well-pumped DBR section becomes saturated, which reduces the impact of the electronic detuning with increasing injection current. In this high-power regime, the variation of the total detuning in DBR section is now dominated by the increase of the temperature due to cross-heating from the electrically pumped sections, leading to an average red shift of the lasing wavelength.

The fast shift is determined by the separation of the neighboring cavity modes, by the length of the mode-transition period and, finally, by the slow wavelength shift rate. By choosing proper dependencies of the refractive index in each section on the currents injected into the same and the other section good agreement between experiment and simulation was obtained.

Many applications require stable lasing of a single longitudinal mode under different operational conditions (pulsed, modulated, cw) over a wide range of parameters such as currents and temperatures. For the case of cw operation considered in this paper one would be interested in an increase of the spacing between the mode jumps and a decrease of the dynamic mode transition regions. According to our analysis, the first task could be achieved by decreasing the length of the amplifier, by other modifications of the device leading to an appropriate change of heating coefficients ν, or by a simultaneous tuning of the injection currents in the DBR [START_REF] Hu | Optical Components and Materials III[END_REF] or some additional passive [START_REF] Kitaoka | [END_REF] or active sections. The reduction of the dynamic mode transition regions can be achieved by, e.g., an optimization of front facet reflectivity, coupling coefficient and length of DBR section: see a related study of DFB lasers in Ref. [START_REF] Bauer | [END_REF]. Both these optimization procedures, however, are subject to a further study.

Figure 1 .

 1 Figure 1. Scheme of a DBR semiconductor laser consisting of an amplifying section S A and a DBR section S BR .

Figure 2 .∂Figure 3 .

 23 Figure 2. Power-current characteristics of the DBR laser measured at the front (a) and rear (b) facets.

Figure 4 .

 4 Figure 4. Spectral analysis of the DBR laser (black) and the reference Fabry-Perot cavity (gray). Symbols: complex eigenvalues (a) and wavelength separation of the eigenvalue to its closest short-wavelength neighbor (b). Lines: changes of the above mentioned quantities with the changing phase factor ϕ.

Figure 5 .

 5 Figure 5. Changes of the damping (a) and the wavelength (b) of several of the most important modes (indicated by symbols) with the changing phase factor ϕ. The arrows and the vertical dashed line indicate the dominant modes with the lowest damping and the position where transition between these modes can be expected. Gray lines represent the reference Fabry-Perot cavity.

Figure 6 .

 6 Figure 6. Power-current characteristics of the DBR laser at the front (a) and rear (b) facets. Black: theory. Gray: experiments.

Figure 7 .

 7 Figure 7. Simulated mean carrier density n (a), gain/loss function g(n) (b) and the relative index change function φ from equation (7) (c) as a function of the injection current I A . The dashed line in panel (c) is the function φ with corrections due to the electronic detuning δn neglected. Black and grey: absorption limited and non-limited (n * = 0) gain models.

  (b). The simulated carrier density values, however, in both these approaches are approximately the same (see panel (a) of the same figure).

Figure 8 .

 8 Figure 8. Representation of the simulated optical spectra (mapping) and the emitted power (light curve) at the front facet of the laser as a function of the increased injection current.

  8 also show us several discrepancies. Compared to experiments, in simulations we have • a much more regular periodicity of mode transitions, and • a less pronounced general slow red shift for larger (> 400 mA) and a blue shift for smaller (< 400 mA) currents. The first discrepancy is due to our assumption of the linear thermal detuning model. In practice, different experimental mode transition periods correspond to different thermal detuning factors ν in (3).

Table 1 .

 1 Parameters used in simulations.

		Description	Unit	S A	S BR
	l	section length	mm	3	1
	κ	coupling coefficient	1/cm	0	9 .6
	I	injection current	A	[0 -0.9] 0
	ν A •	thermal detuning coef.	nm/A	0.75	0.2
	ν BR •	thermal detuning coef.	nm/A	0	0
	ng	group refractive index		3.6
	α	internal absorption	1/cm	2.7
	δs	static detuning	1/cm	-10
	ntr	transparency carrier density	1/m 3	1.3 • 10 24
	n *	absorption limiting density	1/m 3	0.95 • 10 24
	g	effective differential gain	m 2	7.26 • 10 -21
	ñ	effective differential index	m 2	-7.26 • 10 -21
	σ	cross-section of active zone	µm 2	0.21
	A	recombination parameter	1/ns	0.4
	B	recombination parameter	m 3 /s	1 • 10 -16
	C	recombination parameter	m 6 /s	1 • 10 -41
	ḡ	Lorentzian gain amplitude	1/cm	50
	2γ	Lorentzian FWHM	1/ps	100
	ω	Lorentzian peak detuning	1/ps	2
	λ 0	central wavelength	µm	1.064
	r 0	amplitude reflection (front facet)		-0.1
	r L	amplitude reflection (rear facet)		0.01

Acknowledgments

The work of M.R. was supported by DFG Research Center Matheon "Mathematics for key technologies: Modelling, simulation, and optimization of real-world processes". H. W. thanks C. Schwanke for support in the numerical simulation.


P Ascher

L Audirac

N Adimi

B Blank

C Borcea

B A Brown

I Companis

F Delalee

C E Demonchy

F De Oliveira

Santos

J Giovinazzo

S Grévy

L V Grigorenko

T Kurtukian-Nieto

S Leblanc

J.-L Pedroza

L Perrot

J Pibernat

L Serani

P Srivastava

J.-C Thomas

Direct Observation of Two Protons in the Decay of 54 Zn

Keywords: numbers: 23.50.+z, 27.40.+z, 29.40.Cs, 29.40.Gx

The two protons emitted in the decay of 54 Zn have been individually observed for the first time in a Time Projection Chamber. The total decay energy and the half-life measured in this work agree with the results obtained in a previous experiment. Angular and energy correlations between the two protons are determined and compared to theoretical distributions of a three-body model. Within the shell model framework, the relative decay probabilities show a strong contribution of the p 2 configuration for the two-proton emission. After 45 Fe, the present result on 54 Zn constitutes only the second case of a direct observation of the ground state two-proton decay of a long-lived isotope.

The study of exotic decays is an efficient tool to explore the structure of nuclei at the proton drip-line. In particular, the emission of two protons from the ground state of a radioactive nucleus has been searched for since 1960, when two-proton (2p) radioactivity was predicted by Goldansky [1] for nuclei beyond the proton drip line, for which one-proton emission is energetically prohibited. This new nuclear decay mode was observed first in the decay of 45 Fe in two independent experiments [2,3] and later also in 54 Zn [4] and possibly in 48 Ni [5]. In these experiments, the ions of interest were deeply implanted in silicon detectors in which the decay was observed. Therefore, only the total decay energy, the half-life, and the absence of β particles from the competing decay by βdelayed-particle emission could be clearly established. In addition, the observation of the daughter decay helped to unambiguously assign the observed decay to 2p radioactivity. These experimental results were found in reasonable agreement with predictions from different theoretical models [6], like the R-Matrix theory [7], the Shell Model embedded in the Continuum (SMEC) [8] or the three-body model [9,10].

However, in none of these experiments, the two protons were identified separately, while the main physics question in the context of 2p radioactivity is how the two protons emitted are correlated in energy and in angle. An answer to that would enable us to investigate the decay dynamics of 2p radioactivity and thus reveal details of nuclear structure at the limits of stability. In particular, these studies should reveal the single-particle ordering and other details of the wave function.

In an experiment performed in 2005 at GANIL, emis-sion of two protons in the decay of 45 Fe was observed directly for the first time with a Time Projection Chamber (TPC) [11]. The purpose of this detection set-up is to reconstruct the proton tracks in three dimensions. In another experiment performed at MSU [12,13], the 2p emission in the decay of 45 Fe was observed with an Optical Time Projection Chamber in which images of ionizing particle trajectories are recorded optically. In this latter work, high statistics data allowed the authors to obtain a first meaningful comparison with a model including the three-body dynamics of the process [9,10].

In this Letter, we report on an experiment where emission of two protons in the decay of 54 Zn was observed with the TPC. Angular and energy correlations have been determined. These results allowed a first comparison with theoretical predictions of the three-body model and the nuclear shell model.

The 54 Zn nuclei were produced by quasi-fragmentation of the projectile at GANIL. A primary 58 Ni 26+ beam with an energy of 74.5 MeV/nucleon and an average intensity of 3.5 µA was fragmented in a nat Ni target (200 µm). The 54 Zn fragments were selected by a magnetic-rigidity, energy-loss, and velocity analysis by means of the LISE3 separator including an achromatic energy degrader (500 µm of beryllium).

Two silicon detectors located at the end of the spectrometer allowed to identify individually the fragments by means of an energy-loss and time-of-flight analysis. The identification parameters were first determined for frequently produced nuclei. Then, the parameters were extrapolated to unambiguously identify the nuclei produced with low probability. Details of this procedure can be found in [14].

The main setting of the spectrometer was optimized for the production and transmission of 54 Zn. During a two weeks experiment, 18 54 Zn nuclei have been produced, as expected according to extrapolations of the cross sections of 55,56 Zn [15] and ion beam optical calculations. These ions were finally implanted in the TPC [16].

This detector is based on the principle of a timeprojection-chamber. Ions of interest are implanted in a gas volume (argon 90% -methane 10%) at 750 mbar, where the radioactive decay takes place. The electrons, produced by the slowing down of either the incoming ions or the decay products, drift in the electric field of the TPC towards a set of four gas electron multipliers (GEM) where they are multiplied and finally detected in a two dimensional strip detector. The analysis of energy signals allows to reconstruct the tracks of the particles in two dimensions; the drift-time analysis gives the third one. Details can be found in [16].

Among the eighteen 54 Zn implantation events, only thirteen could be correlated in time and space with decays. Five decay events were lost due to the data acquisition dead time and the short half-life of 54 Zn. For two decay events, no information about the energy was obtained because the protons emitted did not stop in the active volume of the chamber. For the first event, the range of the protons emitted was very long (more than 8 cm). Such a long range is only compatible with a betadelayed proton emission. For the second event, due to the large momentum acceptance of the LISE3 spectrometer and the large range distribution of 54 Zn, the ion was implanted at the entrance of the TPC. The protons were emitted backwards and left the active volume. The other eleven decay signals could be analyzed in detail except one, for which a spurious response of one set of strips did not allow us to extract the relevant information. From this information, the branching ratio for 2p emission is determined as BR = 92 +6 -13 %. The time difference between an implantation event and its subsequent decay event allowed a determination of the half-life of 54 Zn. This spectrum is shown in the left part of Figure 1. The half-life is determined as 1.59 +0.60 -0.35 ms. Measurements of charge signals from the GEMs give access to the total decay energy. The right part of Figure 1 shows the signal extracted from the third GEM, the decay energy being thus determined to 1.28±0.21 MeV. As can be seen from Table I, all these decay observables are in agreement with those determined in [4]. If we combine the present experimental BR and half-life with the previous values, we obtain BR = 90 +5 -10 % and T 1/2 = 1.78 +0.66 -0.76 ms, leading to a 2p partial half-life of T 2p 1/2 = 1.98 +0.73 -0.41 ms. Observables related to the measurements of individual protons were also determined. As an example, an implantation event spectrum is presented in Figure 2 ion enters with an angle of 45 o in the chamber and stops at a given (X 0 ,Y 0 ) position. The implantation signals are fitted with a Gaussian folded with a straight line. This function is a good approximation of the Bragg peak corresponding to the energy loss of the charged particles inside the gas chamber. It allowed to determine the implantation position of the ion (top part), which coincides with the emission position of the two protons (middle part). Their tracks in X and Y are fitted with the same function as for the implantation signals: the sum of two foldings of a straight line and a Gaussian, with the main constraint that the energy of a proton is the same along the X and Y direction. Figure 3a shows an example of a two-proton emission in two dimensions.

The energy fraction distribution of the individual protons as determined from the energy signals is plotted in Figure 4 and is found in good agreement with the predictions of the three body model. As expected in a simultaneous emission, the two protons share the decay energy equally in order to favor the barrier penetration. This theoretical approach [9,10] is the only model of 2p radioactivity which takes into account correlations between the two protons.

In a final step, the third component Z of the tracks was obtained. The bottom of Figure 2 shows the time spectra corresponding to the same event. Only the spectrum on the Y dimension is analyzed because the protons were emitted along the X strips. The spectrum is fitted by a straight line for each proton, giving the third component of each proton track. This drift-time analysis will be described in detail in a subsequent paper. Briefly, due to the short range of the protons, the determination of ∆z with the time signals is not very accurate. Therefore, the time signals were only used to determine whether the proton goes upwards or downwards. Then, we use the theoretical range r of the protons [17] to determine ∆z with ∆z 2 = r 2 -∆x 2 -∆y 2 , with ∆x and ∆y given by the energy signal analysis. The theoretical range was calculated by taking the energy sharing of the protons, as determined from the energy spectra analysis, and the sum energy, as determined from the previous measurements [4].

Among the ten events, seven have been fully reconstructed in the three dimensional space. For the three remaining, the time signals did not allow to determine if the protons went up or down. Therefore, we have for these events two possible angles between the two protons: the first one if the protons are emitted in the same hemisphere (up or down), the second one if they are emitted in different hemispheres. Figure 3b shows an example of a two-proton emission reconstructed in three dimensions.

The complete analysis of these decay events allowed to provide angular correlations between the protons. The upper part of Figure 5 shows each experimental angle, represented by a Gaussian reflecting the angular resolution. The middle part shows the angular distribution obtained. Seven events are represented in the histogram; the three other events, not fully reconstructed, are represented by full lines for the first possibility and by dashed lines for the second. In all cases, these three events are located below 90 o . Within the three body model, the angular distribution spectra are calculated for different 2 configurations of the two emitted protons. The corresponding spectra (bottom part of Figure 5) show a double-hump structure, with one broad peak centered around 50 o and a smaller one at about 145 o . When the p 2 contribution becomes negligible, the second hump does not survive. Considering that the experimental distribution shows a double-hump structure, the results can be compared with the model predictions by looking at the ratio between the first and the second hump. From an interpolation of the theoretical ratios, we obtain an experimental p 2 contribution of 30 +33 -21 %. This number can be compared to a shell model wave function decompo- sition over the same p 2 and f 2 contributions using the GPFX1A Hamiltonian [18] which yields about 80% for the p 2 contribution.

In the following, we will combine the shell model and the three-body model to determine theoretical half-lives. While the three-body model is adapted to treat the dynamics of the 2p emission, the shell model is more appropriate to describe the nuclear structure. Therefore, we use the spectroscopic factors of the shell model and the partial half-lives from the three-body model to determine the relative decay probabilities of the two 2 configurations, and thus to compare the experimental and the theoretical half-life. Two-proton removal amplitudes of a pair of protons have been calculated using the GPFX1A Hamiltonian [18] and are found to be 0.3159, 0.3121, 0.6539, 0.2631 for the (0f 7/2 ) 2 , (0f 5/2 ) 2 , (1p 3/2 ) 2 and (1p 1/2 ) 2 configurations, respectively. In LS coupling, the S=L=0 removal amplitudes are 0.443 and 0.686 for (0f) 2 and (1p) 2 configurations, respectively. Combining the half-lives calculated by the three-body model for pure configurations (45 ms and 0.91 ms for (0f) 2 and (1p) 2 , respectively) with the shell model removal amplitudes, we obtain the "shell model corrected" partial half-lives T 1/2 (0f 2 ) = 45/0.443 2 = 230 ms and T 1/2 (1p 2 ) = 0.91/0.686 2 = 1.9 ms. The total half-life of the 2p emission is therefore calculated by adding the two partial decay amplitudes coherently, with 1/T 2p 1/2 = 1/1.9+ 1/230, giving T 2p 1/2 = 1.6 ms. This value is in excellent agreement with the experimental value obtained in this work (T 2p 1/2 = 1.98 +0.73 -0.41 ms). From the above values, we can now derive the relative decay probability through the (0p) 2 configuration which is P(p 2 ) = (1/1.9)/((1/1.9)+(1/230)) = 0.99. This means that almost all the decay strength goes through the (0p) 2 configuration of the wave function. An analysis of the experimental data from 45 Fe and 54 Zn in the frame work of the models used here will be the subject of a future publication [19].

In summary, we observed directly for the first time the two protons emitted in the decay of 54 Zn with a TPC. Half-life and Q-value were determined and were found in good agreement with previous experiments and theoretical models. Energy and angular distributions could be obtained and allowed a first rough comparison with theoretical models giving information about nuclear structure. However, to establish a detailed picture of the decay process, higher statistics of implantation-decay events are needed, which can be obtained in a future experiment. In parallel, improvements of theoretical model predictions are essential to elucidate the decay mechanism which governs two-proton radioactivity.

3 FIG. 1 :

 31 FIG.1: Left: Decay-time distribution obtained in the decay of 54 Zn, giving rise to a half-life of 1.59 +0.60 -0.35 ms. Right: Energy spectrum of the 54 Zn decay events determined with the charge signals of one GEM. The total decay energy E2p = 1.28±0.21MeV is estimated from eleven events in the peak (see text for more details).

FIG. 2 :

 2 FIG.2: Example of energy loss spectra obtained with the XY strip detector and associated with the implantation of a 2p-emitter in the TPC. Top: Signals obtained for the implantation of 54 Zn: the ion enters in the chamber and stops at a given (X0,Y0) depth. The arrows indicate the direction of the incoming beam. The solid line is the fit of the implantation profile whereas the vertical line indicates the determined implantation position. Middle: The decay of 54 Zn takes place at the stopping point of the implantation event described above. The two protons are clearly identified. Their tracks are determined by fitting the decay signal with a sum of two foldings of a straight line and a Gaussian. The vertical full line corresponds to the starting point of the trajectories determined from the implantation profile of 54 Zn whereas the dashed lines are the two stopping points of the protons trajectories. Bottom: Corresponding time signals. The spectrum on the Y dimension is fitted by a straight line for each proton track, giving the directions of each individual proton.

FIG. 3 :FIG. 4 :

 34 FIG.3: a) Two-dimensional view of a 54 Zn decay event as reconstructed from the XY strip detector. The color corresponds to the energy loss detected by the strips. b) Same decay event reconstructed in the three-dimensional space.

FIG. 5 :

 5 FIG.5: Top: Experimental angles between the two protons in the three dimensional space. Each event is convoluted with a Gaussian representing the angular resolution. Middle: Experimental angular distribution between the two protons. Seven events are represented in the histogram. The dashed and full lines represent two possible angles for three not fully reconstructed events. Bottom: The three lines are the theoretical predictions of the three-body model, each line corresponding to different weights of the p 2 configuration. These model distributions are folded with a Gaussian function representing the detector angular resolution.

TABLE I :

 I Comparison of the experimental decay observables with the values obtained in a previous experiment.

We would like to thank the whole GANIL and, in particular, the LISE staff and the DAQ group for their support during the experiment. The time projection chamber was financed in part by the Conseil régional d'Aquitaine. BAB obtained support from NFS via grant PHY-0758099. Fruitful discussion with N. Smirnova are greatfully acknowlegded.