S Gartside

S Chaurasia

D S Tripathi

Munda

A K Rossall

M R Gartside

S Tripathi

D S Munda

N K Gupta

L J Dhareshwar

J Gaffney

S J Rose

G J Tallents

Back lighter characterisation and iron plasma opacity using K-alpha emission X-ray back lighter characterisation for iron opacity measurements using laser produced aluminium K-alpha emission

Keywords: numbers: 52.50.Jm, 52.25.Os, 52.38.Hb

published or not. The documents may come

X-ray back-lighter characterization for iron opacity measurements using laser-produced aluminium K-alpha emission

a K Rossall, L M R

Introduction

Experimental determinations of opacity for a range of plasma parameters are necessary for a wide range of applications. In astrophysics, knowledge of plasma opacity is required for radiation diffusion modelling of stars [START_REF] Rybicki | Radiative Processes in Astrophysics[END_REF]. Opacity is of importance in inertial confinement fusion, as it is required to accurately model radiative hydrodynamics [START_REF] Mίnguez | [END_REF] and hohlraum physics [3]. Methods used previously [4,5] to determine plasma opacity include the use of x-rays emitted from laser produced plasmas as back-lighters to probe heated targets. More recently, EUV lasers have been utilised to probe buried layer targets heated by a laser pulse [6]. Multiple beam, high powered lasers have usually been needed to undertake opacity measurements, due to the requirement to heat the opacity target and the x-ray back-lighter source separately [4][5][6]. Some single laser beam plasma opacity measurements have been undertaken using recorded emission from buried layer targets. Here the plasma is assumed to be in LTE and Kirchhoff's Law is used to infer opacity from the emissivity [7]. Future techniques could involve the use of free electron lasers to investigate opacity. Recent results by Nagler et al. [8] demonstrate how aluminium turns transparent when irradiated with soft x-rays.

In this work, we report plasma opacity measurements recorded using a single laser beam to both heat material and produce the back-lighter source. We use a single target comprising two layers as a heating source for the probed material, through thermal conduction, and for the probing x-ray emission source. The iron plasma conditions are estimated using 2D fluid code simulations. With this technique, it is possible to record the opacity of colder material, where emission is too weak for single beam opacity measurements utilising Kirchhoff's Law [7].

Experimental Set-up

The experiment was carried out using an Nd:Glass laser system situated at the Bhabha Atomic Research Centre (BARC). This laser provides an output energy of 8 J per pulse with a duration of 500 ps. The p-polarized laser was incident onto targets at an angle of 45 o to the target normal and focused to a spot size of 120 µm using an f/5 lens, yielding an on target peak intensity of 1.4x10 14 Wcm -2 . The targets comprised 0.8 µm thick Al, coated using a thermal evaporation plant, onto a 1 µm layer of Fe foil, supported by photo etched copper mounts with a circular target area of diameter 1 mm. The aluminium side of the targets was irradiated by the laser. Pure aluminium foils of thickness 0.8 µm, mounted in a similar manner, were used as targets for comparative purposes.

A crystal spectrometer was utilised to measure aluminium He-and Li-like lines and the solid K α emission transmitted through the aluminium and iron layers. The spectrometer was positioned at the rear side of the target along the axis of the laser at 45 o to the target normal. Time integrated spectra in the wavelength range 7 -9 Å, were recorded by dispersing the x-rays with a flat TAP crystal (2d = 25.75 Å) onto a P-11 phosphor screen followed by an image intensifier tube and a CCD camera (PixelFly QE). Scattered visible light was prevented from entering the spectrometer using B10 filters (aluminised polycarbonate, Alexander Vacuum Research, Inc.). The spectral resolution of the spectrometer was dominated by the source size with λ/∆λ ≈ 400. Four silicon photodiodes (XUV-100, OSI Optoelectronics), under reverse bias, were used to measure the time and space integrated emissivity of the aluminium plasma. Filters of 5 µm & 20 µm Al, 5 µm Ni, and 12 µm Ti were used to control the spectral range recorded (see figure 2). The diode active area is 100mm 2 , but different sized apertures were utilised to avoid detector saturation. The diodes viewed the front side of the irradiated targets.

Aluminium K α emission, at 1.5 keV, from pure 0.8 µm thick aluminium targets was used to calibrate the backward directed emission without an iron layer so that the transmission of iron heated by thermal conduction from the aluminium plasma could be recorded when shots with the iron and aluminium targets were employed. Al K α emission occurs during the laser pulse as it relies on the hot electron interaction with solid Al and so is used for our time integrated transmission measurements because the time of emission is well defined. Test measurement of transmission through unheated iron placed 7mm behind a pure aluminium target showed that the iron transmission, T, measured by this technique, is within 14% of the value (T=0.067) predicted by Henke et al. [9] (see figure 6).

Analysis

The relatively long laser pulse length of 500ps allows time for sufficient plasma expansion during the initial rise time of the pulse to provide the large under dense plasma necessary for whole beam self focussing [10]. Estimation of the ion sound speed (~8x10 4 ms -1 ) yields a scale length of ~40µm at the peak of the laser pulse. A two temperature electron distribution is observed within the Al plasma.

Source broadened spectroscopy provides an 'emission map' of the plasma and shows a central hot Back lighter characterisation and iron plasma opacity using K-alpha emission plasma surrounded by cooler material, while measurements of continuum emission show the presence of two temperature components.

Diode Array

Continuum emission from the Al plasma is measured using filtered EUV enhanced silicon inversion layer photodiodes. Assuming Maxwellian electron distributions, the continuum emission from the Al plasma can be modelled using [START_REF] Hutchinson | Principles of Plasma Diagnostics[END_REF]:

݆ሺߥሻ = ݊ ݊ ܼ ଶ ቆ ݁ ଶ 4ߨߝ ቇ ଷ 8ߨ 3√3݉ ଶ ܿ ଷ ൬ 2݉ ߨ݇ܶ ൰ ଵ ଶ ൗ exp ൬- ℎߥ ݇ܶ ൰ ݃ҧ + ܩ ߦ ݊ ଷ ߯ ݇ܶ exp ቀ χ ୧ kT ቁ + ܩ ܼ ଶ ܴ ௬  ଶ ݇ܶ 2  exp ቆ ܼ ଶ ܴ ௬  ଶ ݇ܶ ቇ ஶ ୀାଵ ൩ (1)
where m is the electron mass, R y is the Rydberg energy, ξ is the number of holes in the ground state of principal quantum number n, ݃ ഥ is the free-free Gaunt factor and is assumed to be 1, G n and G m are bound-free Gaunt factors where G m = 0 for hν < Z 2 R y /m 2 and G m =1 (or 1.4 if T > 400eV) for hν > Z 2 R y /m 2 , G n =0 for photon energies less than the ground state ionisation potential χ i , otherwise G n =1.

Analysis of the continuum emission from the Al plasma using equation 1 and comparing the ratios between diode signals indicates the requirement for two temperature regions in order to reproduce the observed signals. The ionisation level, Z, and number densities of electrons and ions, n e and n i , are determined by assuming LTE within that region. Adapting this model to allow for two temperature regions of T c and T h with relative number fraction f, and incorporating the energy dependant transmission [9] of the filters (T f ) allows for the ratio between diode channels to be determined using:

ܴ =  ܶ ଵ ሺߥሻሾܩሺܶ , ߥሻ + ݂ ଶ ܩሺܶ , ߥሻሿ݀ߥ ஶ  ܶ ଶ ሺߥሻሾܩሺܶ , ߥሻ + ݂ ଶ ܩሺܶ , ߥሻሿ ஶ ݀ߥ (2)
where

ܩሺܶ, ߥሻ = ܼሺ݇ܶሻ ି.ହ exp ൬- ℎߥ ݇ܶ ൰ ݃ҧ + ܩ ߦ ݊ ଷ ߯ ݇ܶ exp ቀ χ ୧ kT ቁ + ܩ ܼ ଶ ܴ ௬  ଶ ݇ܶ 2  exp ቆ ܼ ଶ ܴ ௬  ଶ ݇ܶ ቇ ஶ ୀାଵ ൩ (3)
and the subscripts f1, f2 refer to different filters.

Equation 2 is numerically integrated also taking into account the effect of the photodiode depletion depth. By comparing calculated ratios, R, to experimentally measured diode signal ratios, it is possible to deduce the parameters T c , T h , and f. For example, analysing the ratio between the two highest energy diode channels, R High , (5µm Ni and 12µm Ti) enables a hot electron temperature, T h , of 1keV to be deduced to within 0.1keV accuracy, as there is negligible effect on this ratio due to a change in T c or f, as shown in figure 1. Back lighter characterisation and iron plasma opacity using K-alpha emission

To determine the remaining 2 parameters (T c and f), a minimisation routine is performed using ratios from all channels to find a best fit of the model (equation 2) to the experimental data. This yields a thermal temperature of T c =140 ± 10eV and a fraction of f = 0.008 ± 0.001, consistent over multiple shots. The peak of the spectral range detected by the diodes for the filters and experimental parameters is illustrated in figure 2. A comparison of equation ( 1), with temperature T c =140eV, T h =1keV and f=0.008, is also made on figure 2 with the total emissivity as predicted by the FLYCHK code [START_REF] Chung | High Energy Density Physics[END_REF]. The model shows good agreement with continuum emission, omitting line radiation as expected. When the energy dependant filter transmission is included, the integrated diode signals as predicted by our model using equation ( 1) and FLYCHK agree to within 4%, so our omission of the small contribution to the diode signal from line emission is not significant.

TAP Crystal Spectrometer

The recorded time integrated aluminium spectrum (see figure 3 for example) predominantly comprises a series of lines originating from helium-like and lithium-like ions. The most prominent of these lines is the He-like resonance line (w), purposefully instrument saturated to enable an improved resolution of the intercombination line (x, y) and the Li-like satellites (abcd, jkl, op, qr), (the standard letter designation used here for He-like satellite lines has been tabulated by Gabriel [START_REF] Gabriel | [END_REF]). Line ratios between the Li-like and He-like lines enable the determination of plasma characteristics through comparison with the spectral modelling code FLYCHK [START_REF] Chung | High Energy Density Physics[END_REF]. Utilising the relative intensity of the He-like intercombination line and the Li-like satellites provides a useful diagnostic tool giving a measure of the electron densities and temperatures in the aluminium at the time and position of peak emission. FLYCHK does not simulate the aluminium K α emission which is produced by the hot electron interaction with solid aluminium.

Back lighter characterisation and iron plasma opacity using K-alpha emission Figure 3. Experimental aluminium spectrum (solid line) and superimposed simulation (dashed line) of temperatures T c =140 eV and T h = 1keV using FLYCHK. The fraction of suprathermal to thermal electrons is F=0.008. [START_REF] Chung | High Energy Density Physics[END_REF][START_REF] Yu | NIST Atomic Spectra Database (version 3.1.5)[END_REF].

In order to obtain a fit to the spectra, such as figure 3, using FLYCHK, a parameter scan was performed trialling a range of temperatures (50 -500eV) and densities (10 12 -10 21 cm -3 ). The simulated spectra were instrument broadened with our estimated resolution (~3eV) using a postprocessor [START_REF] Chung | Post-processor used to broaden theoretical FLYCHK spectra[END_REF] and the peak to peak ratios were deduced and compared with the experimental spectra.

A single thermal temperature component could not reproduce the observed spectra. An electron density of n c =5x10 20 cm -3 was found to best fit the spectra, consistent with the turning point density for a 45° obliquely incident 1.06µm Nd:Glass laser. As the emission is proportional to n e 2 and n e =5x10 20 cm -3 is the highest density directly heated by the laser, we assume emission from this single density is a reasonable approximation to a spatial integration of emission. A parameter scan of the hot electron temperature, T h , (0.5 -2keV) and hot to cold component fraction, f, (0.005 -0.1) was performed and found to require T h =1keV and f=0.008 for T c =140eV (see e.g. figure 4). Back lighter characterisation and iron plasma opacity using K-alpha emission (dotted line) as a function of thermal temperature. The case shown is for T h =1keV and f=0.008 (best fit). The ratio differences indicate that T c =140eV.

Spectral Line Profiles

It was found that He-like satellite spectra were best modelled using a double Gaussian line profile.

The spectral profiles are strongly source broadened (λ/∆λ ≈ 400), so the line profile represents a mapping of the spatial emission profiles of the lines, indicating two temperature regions. The superposition of source broadening effects due to a 25µm diameter plasma region of electron temperature T h =1keV and a 120µm diameter region of electron temperature T c =140eV is shown on figure 5 to fit the experimentally measured line profiles. The relative abundance of the number of hot and cold electrons used for this calculation is f=0.008, with intensity information obtained from FLYCHK. This relative abundance of hot and cold electrons is approximately consistent with the spatial dimensions of hot electrons (a circular region of radius r = 12µm) and cold electrons (a circular annulus region of radius between r =12µm and R = 60µm). Assuming uniform ablation, we would expect ݂ = గ మ గோ మ ିగ మ ≅ 0.04, but this could readily drop to f = 0.008 if the T h value is produced late in the irradiation history or at limited spatial distance from the target surface. We discuss 2D simulations of the production of T h and T c in section 4.

Figure 5. Expanded experimental spectrum as a function of distance on the CCD detector with a distance scale corresponding to the source broadening effect on the spectra. A double Gaussian profile for T c =140eV and 120µm width at the target (dotted line) and of T h =1keV and 25µm width at the target (dashed line) combine to form the emission profile observed. The K α peak is best represented by a single Gaussian (dash-dot-dot line) with a spatial width of 60µm.

Al K α Transmission Measurements

The spectrally integrated Al K α intensity recorded by the spectrometer after transmission through the iron layer was measured for different laser pulse energies and directly compared with the integrated K α signal originating from a pure Al target created using the same laser energy. A comparison of the transmitted and uninhibited spectra for a laser energy of 7.7J is shown in figure 6. Figure 7 demonstrates the change in transmission over the laser energy range investigated and shows a comparison to cold Fe transmission. It is seen that an increase in laser energy of 1.3 J causes sufficient additional heating to the iron layer through enhanced thermal conduction to produce an increase in transmission by almost a factor of 3. Al K α emission was estimated from the 2D simulations (see figure 9). We assume that the total Al K α emission, ε Kα , comes from the electron collisional excitation rate coefficient, K, with Al atoms, at E Kα the Al K α photon energy. We can write that

ߝ ఈ = ݊ ݊ ܭ ∝ ݊ න ߪሺݒሻ݃ݒ ௩ ሺݒሻ݀ݒ ∞ ට ଶா ಼ഀ (4)
where n e is the electron density for the plasma, n i is the solid Al atom density and g v is the electron distribution function. The electron distribution function can be represented by a Maxwellian

distribution ݃ ௩ ሺݒሻ݀ݒ ∝ ௩ మ ் య మ ൗ ሺݔ݁ ݒ݉- ଶ 2݇ܶ ൗ
ሻ݀ݒ . The cross section, σ(v), for K α production is assumed constant for E > E Kα [20]. Considering the relation in terms of energy and integrating gives

ߝ ఈ ∝ ݊ ቈ ܧ ఈ ඥ݇ܶ + ඥ݇ܶ ሺݔ݁ ܧ- ఈ ݇ܶ ൗ ሻ (5)
where kT e is the electron temperature. The emission of ε Kα is integrated axially and multiplied by the calculated transmission through the cells of the heated Fe layer to simulate the Al K α flux through the target at the central radial position. The opacity in the Fe layer is calculated using the code predicted temperature and density and a look-up table of IMP tabulated opacities [19]. It can be seen that the peak of the transmitted Al K α radiation is close to the peak laser irradiance on target (time '0', see figure 9). The K α transmission at peak flux through the iron layer has been calculated for h2D simulations at different laser energies and was found to agree with the experimental transmission measurements (see figure 7).

Figure 9. Flux of Al K α photons through an Fe layer at the radial centre of the laser pulse, from 2D h2d simulation of the Al-Fe targets for a 7.3 J 500 ps laser pulse assuming a focal spot width of 120 µm (dashed line). The flux is the product of the emission of Al K α from the Al layer (solid line) and transmission of Al K α through the Fe layer (dotted line).

Further simulation has revealed predicted Fe temperatures at the time of peak Al K α flux (see figure 9) range from 10 eV (back side of the target) to ≈ 70 eV (side closest to the laser) for a 7.3 J pulse and 20 eV to 150 eV for an 8.6 J pulse (see figure 10). The corresponding densities range between 0.2 gcm -3 and 1.1 gcm -3 for a 7.3 J pulse and between 0.1 gcm -3 and 0.8 gcm -3 for an 8.6 J pulse.

Figure 10. Axial temperature profiles, at the time of peak Al K α flux, for the iron layer in an ablated Al-Fe target from h2d simulation with a 500 ps pulse incident on the Al surface with a pulse energy of 7.3 J (solid line), 8 J (dotted line) and 8.6 J (dashed line).

Discussion

Experimental temperature and K α transmission data agree well with simulations using h2d and opacities from the Ionised Material Package (IMP). A two temperature spatially separate thermal distribution of electron temperatures, as observed here, has recently been reported by Colgan, et al [21].

Aluminium K α emission has been used in this experiment as there has been a significant amount of previous study into the K α spectra of aluminium, and the photon energy (1.49keV) results in photoionisation from the ground to continuum of Fe up to Fe XIX [22] so that there will be significant absorption of the radiation in iron plasmas up to electron temperatures ≈ 90eV. Once temperatures exceed ≈ 90eV, iron becomes close to transparent, so our K α transmission measurements are a good diagnostic of temperatures > 90eV.

Back lighter characterisation and iron plasma opacity using K-alpha emission

The assumption that Al K α is emitted over a different timescale to that of other spectral lines, due to the hot electron interaction, has been checked by examining the transmission of He-like satellite lines through the iron. These lines were found to be more strongly transmitted by a factor of 2, compared with K α transmission, suggesting more emission at later times when the iron transmission is greater.

The experiment has demonstrated the feasibility of performing opacity experiments using a table top laser system with modest parameters. Using this technique, we have been able to accurately characterise the Al back lighter and perform transmission measurements using a laser system with a high reproducibility (±10%) on a shot to shot basis. A number of issues need to be addressed to obtain high accuracy opacity information [23]. Simulations have demonstrated using this technique results in large temperature and density gradients in the axial and radial direction which would have to be accurately measured. Gradients in the axial direction can be reduced by introducing a plastic tamp layer on the rear side of the target, inhibiting the Fe plasma expansion, reducing the temperature gradient and providing conditions that can be validated more readily as LTE. Radial temperature gradients can be addressed by using a different heating mechanism than thermal self focussing. Using a shorter pulse length laser would prevent whole beam self focussing and could be optimised for resonance absorption to produce a higher number of hot electrons and to produce a shorter pulse back lighter. These hot electrons would increase the K α signal and propagate into the Fe layer, heating the target, producing plasma with higher radial uniformity. The shorter pulse back lighter would provide better temporal resolution. Diagnosing the plasma conditions within the Fe layer has relied on using computer codes to simulate the thermal conduction from the front side Al plasma. An independent plasma diagnostic for the Fe layer is therefore required in order to determine the plasma conditions independently of hydrodynamic simulations. This could be achieved using absolute time gated imaging, probably in the visible, of the iron target to deduce a temperature assuming emission as a black-body.

Conclusion

A method for the characterisation of a plasma back lighter and the determination of plasma opacity using a single laser beam and a layered target has been investigated. Thermal conduction from laser produced aluminium plasma has been used as a heating mechanism for an iron target layer. K α x-rays produced from the Al layer are used to measure transmission through the iron plasma. Whole-beam self focusing of the laser has reduced the effective spot size and increased the energy deposition into the iron. The temperature increase in the iron allows for opacity measurement over a temperature range ~ 10 -150 eV. Combining source broadened spectroscopy with continuum emission analysis has enabled a plasma back lighter produced by whole beam self focussing to be characterised. The experimental data is in good agreement with 2D modelling using opacities from the Ionised Material Package code for 1.5 keV photon energy. Future work involves the use of hot electron heating of the Fe layer and the inclusion of a separate plasma diagnostic to determine the Fe plasma conditions.

Figure 1 .

 1 Figure 1. Variation of diode signal ratio for the two highest energy channels (5µm Ni and 12µm Ti) as a function of the hot electron temperature. The solid line represents a cold electron temperature of T c =140eV and relative fraction of hot to cold number, f=0.008, while the dashed line represents T c =240eV and f=0.008 and the dotted line represents T c =140eV and f=0.05

Figure 2

 2 Figure2shows the photon energies where the filtering for each diode gives maximum recorded x-ray flux and illustrates how we can use the four diode signals to measure T h , T c and f. The spacing of these peaks allows the determination of the hot temperature, T h , component (from signal ratio R for the two highest energy channels), the cold temperature, T c , component (from the signal ratio R of the two lowest energy channels) and the ratio, f, the number of hot to cold electrons (from the ratio of the two intermediate channels).

Figure 2 .

 2 Figure2. Continuum emission from the laser produced aluminium plasma as predicted by equation 1, incorporating two temperature regions within the plasma (thick solid line) with a comparison of total emission as predicted by FLYCHK[START_REF] Chung | High Energy Density Physics[END_REF](thin solid line). Conditions used are T c = 140eV, T h = 1000eV and f = 0.008. The peak of the transmitted emissivity curves for the diode filters of 5µm Al (dashed line), 20µm Al (dotted line), 12µm Ti (dash-dot line) and 5µm Ni (dash-dot-dot line) are indicated.

Figure 4 .

 4 Figure 4. Plot of the difference between the satellite ratio determined via FLYCHK and our experimentally measured ratio for satellite lines qr/jkl (solid line), xy/jkl (dashed line), and xy/qr

Figure 6 .

 6 Figure 6. Experimentally observed, transmitted Al spectrum through an Fe plasma (solid line) and the spectrum from an aluminium target without iron (dotted line) for a laser energy of 7.7J.

Figure 7 .

 7 Figure 7. Transmission of K α photons (1.5 keV) as a function of laser pulse energy as determined by experiment. Experimentally measured cold Fe transmission (dashed line) is compared with cold Fe transmission predicted (dotted line) by Henke et al. [9]. The open squares show calculated transmissions from h2d simulation, with line of best fit (solid line).

Figure 8 .

 8 Photon Energy (eV)

Acknowledgements

The authors would like to thank Chris Spindloe (Rutherford Appleton Laboratory) for the target fabrication and the EPSRC for funding this work. We acknowledge the HiPER Project for funding of the HYADES and h2d codes, and Raoul Trines for the installations and maintenance of these codes.

Simulation Results

The radiation hydrodynamic code h2d [16], was used to model the ablation and heating of the Al-Fe foil target. h2d models in 2D, the ablation, ionisation and expansion of material irradiated by a laser pulse to give the plasma density, temperature and ionisation as a function of distance and time in a Lagrangian coordinate system in r,z cylindrical geometry. The code incorporates multigroup diffusion of thermal radiation and flux limited electron conduction (flux limiter = 0.1 [START_REF] Bell | [END_REF]). Laser absorption is by inverse bremsstrahlung with an energy dump of 20% at the critical density. To calculate the hydrodynamic variables in the evolving plasma, the SESAME equation of state package was used.

The SESAME tables, developed by the T-1 group at Los Alamos National Laboratory, form a computer-based library of tables giving the thermodynamic properties of materials [18]. By postprocessing the simulation output using opacity data from the IMP (Ionised Materials Package) code [19]; the transmission at the Al K α wavelength through the Fe layer was calculated.

Our 2D h2d simulations assumed a cylindrical Al-Fe target of diameter 200 µm. The effect of a laser pulse incident on the Al surface with a 120 µm focal spot width and 500 ps duration with Gaussian time and spatial profiles was simulated for the experimental conditions. Figure 8 shows evidence for a central hot plasma of ≈ 1 -1.6 keV with a surrounding cooler plasma of temperature ≈ 100 -200 eV.

These temperatures agree approximately with the 'two temperature' measurements found with our spectral fitting and diode measurements. An increase in laser energy causes an increase in the thermal self focusing effect (figure 8), causing an increase in the plasma temperature at the critical surface. This higher temperature region transfers more energy into the iron layer via thermal conduction, resulting in a higher Fe temperature, as shown in figure 10. The higher temperature results in a higher ionisation level in the Fe layer (above Fe XIX), so that direct photoionisation from the ground state of iron ions is not possible and the iron becomes transparent (see figure 7).


L Rutherford

I C Lane
email: i.lane@qub.ac.uk

J F Mccann

Doppler cooling of gallium atoms: 2. Simulation in complex multilevel systems

This paper derives a general procedure for the numerical solution of the Lindblad equations that govern the coherences arising from multicoloured light interacting with a multilevel system. A systematic approach to finding the conservative and dissipative terms is derived and applied to the laser cooling of p-block elements. An improved numerical method is developed to solve the time-dependent master equation and results are presented for transient cooling processes. The method is significantly more robust, efficient and accurate than the standard method and can be applied to a broad range of atomic and molecular systems. Radiation pressure forces and the formation of dynamic dark-states are studied in the gallium isotope 66 Ga.

Introduction

Many recipes [1][2][3][4][5] now exist in the literature for the synthesis of ultracold molecules starting from a small selection of basic ingredients: trapped, ultracold atoms [6]. The variety of molecular species is limited by the fact that, to date, only a small number of elements have the requisite electronic structure for direct laser cooling, although sympathetic cooling of molecules with ultracold buffer gases may prove an effective alternative method [7]. In order to increase the repertoire of ultracold molecules available, techniques must be developed to deal with the large swathes of the periodic table currently inaccessible to laser cooling.

The p-block elements form a particularly important part of the periodic table, but thus far the only such atoms that have been cooled and trapped are the noble gases, paradoxically the least reactive of all elements. There has been, however, some progress in laser cooling and manipulation of elements within Group 13: atomic beams of aluminium [8,9] and gallium [10] have been observed to narrow under laser irradiation and a deceleration force has been demonstrated in gallium [11] and indium atoms [12] with recent evidence of sub-Doppler transverse cooling in an indium atomic beam [13].

In general two problems need to be addressed for cooling in the p-block. The first is the tendency of all these elements to absorb light in the UV rather than the visible region when in their ground electronic states. Stable continuous-wave lasers that emit in the UV tend to be complex to operate and generate relatively low power. Sometimes it is possible to find a more convenient, laser-accessible, transition starting from a metastable state, which has been the cooling strategy in the noble gases [14,15] and in the successful Group 13 studies conducted so far. The second is the p-shell itself, with the associated fine structure. The energy splitting due to spin-orbit effects, is many times greater than the hyperfine splitting and necessitates more than one laser (frequency). The usual strategy of repumping can lead to coherent dark states that compress the cooling process [16].

Group 13 elements fortunately are untroubled by the first problem, which is more an issue with laser technology rather than fundamental physics, having absorption frequencies in the visible and near-UV (with the exception of boron). The (ns 2 , np) configuration has a simple doublet ground state, thus the presence of spin-orbit coupling in the 2 P multiplet affords the best opportunity to address the practical issue of cooling in the presence of complex electronic structure. In our first paper [17], we conducted time-dependent calculations on the interaction of 66 Ga atoms with counterpropagating laser beams, but the method involved was computationally inefficient, required the derivation of the equations of motion manually and did not easily allow the exploration of optimal experimental parameters, the information experimentalists would find most useful. In this paper, an improved method has been developed to quickly and efficiently evaluate possible cooling schemes within these complex structures.

In section 2 the mathematical model used is outlined and in section 3 the details of the computational model are discussed. The code generates each of the equations describing the evolution of the individual time-dependent density matrix elements before numerical integration to determine the temporal behaviour of the populations, coherences and cooling force for any number of (continuous-wave) laser fields. In section 4 the speed and accuracy of the numerical integration method used is tested against the standard Euler method for the (exactly solvable) two-level closed cooling system. In section 5 the transient cooling cycle in 66 Ga is calculated for a two-colour cooling scheme and in section 6 optimal parameters for maximising the cooling force are investigated. We show that the transient force is greater if N ( 2 P 1/2 ) > N ( 2 P 3/2 ), where N ( 2 P i ) is the population of each ground spin-orbit component, and that an offset between laser detunings is beneficial for maximising cooling. Though experimentally such parameters are simple to determine when a small number of laser frequencies are involved, it quickly becomes prohibitively expensive (both in cost and time) as more laser fields are added requiring optimisation. We conclude, in section 7 by demonstrating the potential of our method with a 20-level five-colour cooling scheme for 69 Ga and discuss future applications for investigating laser cooling of multi-level systems, both atomic and molecular, and incorporation of external fields that break the degeneracy of the sublevels.

Mathematical Model

We employ a standard semiclassical treatment throughout, and use well-established notation and conventions [START_REF] Chang | [END_REF]. This assists in formulating, in a systematic manner, the structure of the differential equations both mathematically and computationally.

The state of the atom is expressed in a finite set of N A eigenstates of the hyperfine Hamiltonian (H A ) which we denote by the single index, p ∈ {1, 2, . . . , N A }. This abbreviation stands for the ensemble of the hyperfine sub-level quantum numbers. Expanding this notation, we take, α p to denote the collective label for the finestructure level and uncoupled nuclear spin. That is, α p = {n p , L p , S p , J p , I p } where n p is the electron principal quantum number, (L p , S p ) are the orbital and spin angular momenta with the resultant coupling J p , and I p is the nuclear spin. The basis functions |p = |α p , F p , M p are the orthonormalized eigenstates with eigenvalues: ε p (in general not unique), such that

H A |p = ε p |p (1)
For convenience we define the corresponding atomic angular frequencies: hω p ≡ ε p . The atoms are moving at non-relativistic speeds at all times with the centre-ofmass of the atom moving with a velocity v in the laboratory frame, and thus the Galilean transformation can be employed. The internal state of the atom is expressed by the (variation of constants) expansion:

|Ψ(t) = N A p=1 c A p (t)|p (2)
Then the density operator, ρ ≡ |Ψ(t) Ψ(t)|, has the matrix representation in the time-stationary basis set:

ρ A pp (t) ≡ p|ρ A (t)|p = c A p (t)c A * p (t) (3)
The multicoloured lasers are represented by a superposition of classical monochromatic fields:

E(r, t) = 1 2 N L j=1 E j e i(kj •r-ωj t+ϕj ) + E * j e -i(kj •r-ωj t+ϕj ) (4)
with the (complex) polarizations, wave-vectors, angular frequencies and phases denoted by the usual symbols: Ê, k, ω and ϕ.

Then the time-varying perturbation can be written:

H AL (t) = -d • E(r, t) (5)
where d is the (electronic) dipole-moment operator. The dipole approximation is valid for these frequencies, and thus for an electron with coordinate r e with respect to the centre-of-mass of its atom, and thus with a position r + r e in the laboratory frame, we have:

k j • (r + r e ) ≈ k j • r = k j • vt + φ.
Then the couplings can be written in the form:

H AL pp (t) = h N L j=1 Ω j,pp e i(kj •vt-ωj t+ϕj ) + Ω * j,p p e -i(kj •vt-ωj t+ϕj ) (6)
which reveals the Doppler shift in a transparent manner. The Rabi frequencies determine the selection rules and are defined according to:

hΩ j,pp ≡ -1 2 E j • p|d|p (7)
The evolution (Liouville) equation, in the absence of dissipation, takes the form:

∂ ∂t ρ A (t) = -(i/h) H A + H AL (t), ρ A (t) (8)
Now, though a sequence of unitary transformations and approximations, the Liouville equation is converted to a stationary form. The hyperfine Hamiltonian is diagonal in this basis and it is convenient (analytically) to transform to the interaction representation,

ρ I (t) ≡ e -iH A t/h ρ A (t)e iH A t/h (9)
and with V AL (t) ≡ e -iH A t/h H AL (t)e iH A t/h , so that we have the evolution equation:

∂ ∂t ρ I (t) = -(i/h) V AL (t), ρ I (t) (10)
Now, if the pulse envelope is time-independent, the only time dependence arises from the internal and external sinusoidal oscillations; that is, the exponents of the commutator involve the Doppler-shifted detunings

δ ± j,pp ≡ ω p -ω p ± k j • v ∓ ω j (11)
where ± corresponds to absorption (+) or emission (-). At this point it is conventional, though not necessary, to make the rotating-wave approximation (RWA)

and discard the non-resonant transitions. Once this is done, a further unitary transformation is then possible to remove the time-dependence (oscillations) from the interaction entirely [START_REF] Chang | [END_REF]. This results in time-independent (stationary) equation for the evolution for the conservative terms. The inclusion of the spontaneous emission responsible for cooling, can be modelled by the Lindblad correction which is written in the form:

∂ ∂t ρ(t) = -(i/h) [H, ρ(t)] + Γ • ρ(t) (12)
where the dyadic (tensor) for spontaneous emission is denoted by Γ and expresses the dissipation (and decoherence) of the system.

Outline of computational method

The well-known master equation ( 12) provides the dynamics of the model, within the rotatingwave approximation. This equation is used to study the transient and steady-state behaviour of the system from a set of initial conditions. However the generality hides the associated linear equations, which are complex in structure and require numerical solution. We indicate a general solution for such problems.

• Interaction matrix elements

The matrix, H, in (12) represents the conservative interactions of the system, with elements H ρρ . The set of atomic hyperfine states {|p } is defined in terms of the quantum numbers, and energies. The selection rules for absorption and stimulated emission are enforced by the computer program based on this data. Then, for each (allowed) transition, the multicolour detunings are specified and thereby the nonresonant couplings (counter-rotating terms) identified and removed. The reduced matrix elements are assumed as known, so that one simply uses the Wigner-Eckart theorem [START_REF] Sakurai | Modern Quantum Mechanics[END_REF] to evaluate the coupling coefficients in terms of the reduced matrix element.

• Dissipation

The (partial) lifetimes of the excited states are assumed to be known, that is, the spontaneous decay process from any upper state to any lower state: |p → |p , is given as:

2γ pp = 4 3 | αpFp||d||α p F p | 2 |ωp -ω p | 3 (2Fp + 1)hc 3 (13)
• Units The time-scale for population transfer is of the order of nanoseconds or less due the stimulated and spontaneous emissions, and the natural unit of time is the inverse lifetime γ -1 of the system. Thus a natural energy unit is hγ, and we convert the Hamiltonian accordingly. Furthermore the detuning angular frequencies (δ j,pp ) are expressed as multiples γ, as are the Rabi frequencies of the Hamiltonian. In practice, the relevant laser-coupling parameter is the saturation intensity. That is, for each of the dipoleallowed transition the relative strength of stimulated and spontaneous emission is given by the dimensionless saturation parameter:

G j,pp = 2|Ω| 2 j,pp /γ 2 pp ( 14
)
In section 5 all forces presented are normalised forces, F/hkγ. Similarly, atomic velocities are expressed in units of k -1 γ, where k is the wavenumber of a reference laser.

• Spontaneous emission

As the spontaneous emission tensor, Γ, depends on the density matrix, ρ, it must be calculated at each time step; this involves solving the equations for the atomic density matrix elements

ρ kl = ρα a FaMa,α b F b M b = αaFaMa|ρ|α b F b M b .
To solve the equations each possible pair of ground and excited-state sublevels are considered in turn, and certain selection rules are enforced. Spontaneous emission can occur only between states obeying the usual rules:

∆F = 0, ±1 (but not F = 0 ↔ F = 0) ∆M = 0 , ±1 (15)
Each element of the matrix, Γ, can be described by one of the following four equations [START_REF] Chang | [END_REF]: where

αe 1 Fe 1 Me 1 | Γ • ρ |αe 2 Fe 2 Me 2 = -(γα e 1 Fe 1 + γα e 2 Fe 2 ) αe 1 Fe 1 Me 1 |ρ|αe 2 Fe 2 Me 2 (16a) αeFeMe| Γ • ρ |αgFgMg = -γα eFe αeFeMe|ρ|αgFgMg (16b) αg 1 Fg 1 Mg 1 | Γ • ρ |αg 2 Fg 2 Mg 2 = αe 1 ,αe
(Fg 1 Fg 2 Mg 1 Mg 2 | A|Fe 1 Fe 2 Me 1 Me 2 ) = (γα e1 F e1 ,α g1 F g1 + γα e2 F e2 ,α g2 F g2 ) × q=0,±1 (Fg 1 Mg 1 1q|Fe 1 Me 1 )(Fg 2 Mg 2 1q|Fe 2 Me 2 ) (17a) (FgMgM g | A|FeMeM e ) = 2γα eFe,αg Fg q=0,±1 (FgM g 1q|FeM e )(FgMg1q|FeMe) (17b) (FgMg1q|FeMe) = (-1) Fg -1+Me √ 2Fe + 1 Fg 1 Fe Mg q -Me (17c)
Wigner 3j symbols (17c) are calculated using the MATLAB code wigner3j.m [20], and the usual selection rules apply.

• Numerical integration Consider (12). As mentioned the naive numerical integration, in our adopted system of units, is described by the Euler formula:

ρ (∆t) = ρ(0) + ∆t × {-i[H, ρ(0)] + Γ • ρ(0)} + O( (∆t) 2 ) ( 18
)
Although this is simple to implement, in general the results are inaccurate and often unstable. The conventional modifcation is to apply the mid-point correction. However, given that the conservative term has a known evolution, we can improve on this by splitting these terms from the dissipation. Given the initial state, ρ(0), we step forward in time to ∆t, using the intermediate half-step as follows:

ρ 1 2 ∆t ≈ e -i 1 2 ∆tH ρ(0) + 1 2 ∆t Γ • ρ(0) e i 1 2 ∆tH (19) ρ(∆t) ≈ e -i∆tH ρ(0)e i∆tH + ∆te -i 1 2 ∆tH Γ • ρ( 1 2 ∆t) e i 1 2 ∆tH (20)
Clearly, formally the method is still accurate (only) to first-order in ∆t, nonetheless it is much superior in stability to the standard Euler method. This is a result of reducing the second-order error term by removing the time-independent terms in H, thus leaving the slowly-varying decay terms as the leading error. The calculation of the exponential operators, which ensure unitarity, bring some additional computational cost (though these have to be calculated only once), but in terms of numerical accuracy and efficiency the method is a significant improvement, as will be shown in section 4. The steps are repeated in the usual manner to produce the entire evolution over time. Depending on the saturation parameters, when G 1 for example, the Rabi oscillations are rapid at the beginning followed by slow damping so that small time steps are required to ensure that the integration proceeds correctly. However, at longer times, (t 1) as the steady state approaches, the characteristic oscillations are slower, and larger time steps can be taken.

Using this method, the time-evolution and steady-state of the density matrix can be studied. As well as containing information about the populations of each state, the density matrix is also used to calculate the dipole radiation force on the atoms. This force, F, is due to the interaction of the laser field with the induced atomic dipole moment and is given (in one dimension) by;

F (z, v, t) = - ∂ ∂z U (z, v, t) (21)
where U (z, v, t) = -tr(ρ d) E(z, t), E(z, t) is the electric field of the lasers along the propagation direction and d is the atomic electric dipole operator (in the atom frame). Laser cooling of translational motion requires many photon scattering events over long time-scales (microseconds). For an individual the force is a function of atom position and velocity, and explicitly on time in the case of a pulsed laser. This paper considered the force changes on ultrafast (electron) times, a few natural lifetimes, t ∼ γ -1 , rather than the much longer times of the nuclear motion, h/kBT , where kB is the Boltzmann constant. Thus, on these timescales the velocity changes are slow.

Analysis of Method

To analyse the accuracy and stability of our method, it was compared to the basic Euler method [21]. As the exact solution of the density matrix equations for a two-level atom in a closed system, with ground state, b, and excited state, a, is well known [22], this system was used to compare the two methods. Consider an atom at rest (v = 0), initially with all population in the ground state, ρaa = 0. The population of the excited state, ρaa, can be calculated as

ρaa(t) = G 2(G + 1) 1 -cos λt + 3 2λ sin λt exp(-3 2 t) ( 22
)
where G is the saturation parameter, G ≡ 2Ω 2 ab and λ = (2G -1/4) The figure shows the results of the numerical integration method presented in this work (dashed), compared to the Euler method (dot-dashed) and the exact solution (solid). The units of time are γ -1 , as explained in section 3.

Figure 1 shows the time-evolution of the excited-state population ρaa, calculated on resonance (kv = δ = 0) by both the Euler method and the method presented in this work, compared with the exact solution given by (22). It is clear that even for this simple system the new method converges significantly more accurately than the Euler method. Although the Euler method has the correct asymptotic limit, the short-time behaviour is incorrect. The errors in the Euler method become even more marked as γ is reduced. At γ = 0 (pure Rabi oscillation), and in calculations far off-resonance, the Euler method fails to converge unless step sizes are extremely small (∆t < 0.001) whereas the method presented is highly stable, even at step sizes over 100 times larger.

Comparisons for more complex systems were not possible as the extremely small stepsizes needed to counteract the instability of the Euler method meant calculations could not be carried out in realistic timescales. In contrast, the method presented here remained stable over a wide range of detunings and step-sizes. It is clear therefore that the new method provides a marked improvement on accuracy, stability and computational cost.

Previous calculations [17] on a similar system to that outlined in this paper depended on the construction, by hand, of a set of n × n density matrix equations, where n is the number of individual magnetic sublevels in the cooling scheme. For small systems this is a relatively simple task but as the number of magnetic sublevels increases, so does the number of equations needed. Soon it becomes hugely time-consuming to construct and solve these equations, with the potential for human error also increasing. The most difficult part of the master equation ( 12) to calculate is the section containing the spontaneous emission terms, Γ • ρ, which often has to take into account many different decay channels. The benefit of the method presented here is that, with the only input being a small number of quantum numbers, these terms are calculated automatically within the numerical code, and placed in a matrix. This abolishes the need for time-consuming construction of equations or matrices by hand.

Application to a Λ-system

To show the possibilities of this new method, in this paper the method presented is applied to a simple multi-level system. Consider now a Λ-system involving the lower manifolds |g , |G , and the excited state |e , as shown in figure 2. This system is assumed to correspond to the 2 P → 2 S transition in a Group 13 atom with nuclear spin I=0, specifically 66 Ga, a beta emitter and putative radiopharmaceutical for Positron Emission Tomography which can be produced with a biomedical cyclotron [23]. This scheme could also be applied to a transition to the first hyperfine excited level of a Group 1 atom with nuclear spin I=1 (eg 6 Li). As it consists of two ground states, with 2 and 4 magnetic sublevels respectively, connected through an excited state with two sublevels, it can be described simply as a 2-(4)-2 system.

G-3/2 g-1/2 g1/2 G-1/2 G1/2 G3/2 e-1/2 e1/2 !2 !1 "1 "2 #1 #2 Figure 2.
Schematic energy level diagram for 2-(4)-2 multilevel atom, corresponding to the 2 P 1/2 , 2 P 3/2 ↔ 2 S 1/2 transition in Group 13 atoms with nuclear spin I=0. Ω 1 , Ω 2 correspond to the Rabi frequencies, ω 1 , ω 2 are the laser frequencies and δ 1 , δ 2 are the laser detunings.

For the example presented, two laser frequencies are considered, ω1 and ω2, detuned by δ1 and δ2 respectively, where a negative δ corresponds to a red-detuned laser. One laser beam corresponds to the Fg = 1/2 → Fe = 1/2 transition, with the other used to pump FG = 3/2 → Fe = 1/2. The scheme considered here assumes the use of counter-propagating circularly-polarized laser fields, with a σ + -polarized beam driving the MF → MF + 1 transition, and a σ -laser driving MF → MF -1.

For simplicity it is assumed that the radiative widths for the two transitions are approximately equal, and are chosen as the reference for our units (as outlined in section 2), γ1 ≈ γ2 = γ = 1, that k1 ≈ k2 = k, and that the saturation parameters, G = G1 = G2 = 1. Consistent with (3), the Rabi frequencies are also equal, Ω = Ω1 = Ω2, where, for example, Ω1 = (G1/2) 1/2 . Tests with a range of parameters show that these assumptions have little effect on the results of the calculations.

Population Distribution

The population distribution of the system is investigated over a range of atomic velocities, along with the dipole radiation force, for both the time-dependent and steady state cases. The steady state result is obtained numerically, for any initial conditions, by timedependently solving the master equation at very long timescales, until the populations converge. Figure 3(a) shows the steady-state ground-state population distribution for a 2-(4)-2 multilevel atom, as shown in figure 2, with laser detunings δ1 = -3 and δ2 = -2.

In figure 3(a), it can be seen that as the velocity of the atom nears resonance velocities (kv = ±2, kv = ±3) there is a transfer in populations giving rise to a broad peak, as expected. The system possesses a left-right symmetry, with a change in the sign of the velocity being equivalent to a change in the sign of the quantum number MF . Sharp population structures are also observed closer to zero velocity. These resonances are due to two-photon processes producing ground-state coherences which vary rapidly at low velocities, leading to sharp variations in ground-state populations. The positions of these resonances can be predicted simply from the energy conservation law [24]. Two-photon transitions within the ground state |G do not change the energy of the atom, so (ω2 ± kv) -(ω2 ∓ kv) = 0 and the resonance occurs around zero velocity. However, in the case of two-photon transitions between states |g and |G the energy of the atom is changed; where ωi is the laser frequency used and δi is the detuning (as shown in Figure 2). These resonances, therefore, occur at velocities kv = ±(δ1 -δ2)/2, in this case at kv = ±0.5. Figure 3(b) gives an expanded view of the region in which these population shifts occurs, and as figure 3(c) shows, increasing the laser intensity leads to a broadening of these peaks. Conversely, a low saturation parameter gives rise to much narrower resonance structures. As the population distribution is highly symmetric, for clarity only the levels g 1/2 , G 1/2 and G 3/2 are shown in the expanded region. Steady-state excited populations are shown in figure 4; due to symmetry, the population distribution for each excited-state sublevel is identical. Ground-state coherences caused by two-photon processes result in the populations falling sharply to zero at these resonance points. The presence of these structures at all laser intensities clearly demonstrates the destructive atomic interference occurring in the system, with the formation of dark states.

(ω1 ± kv) -(ω2 ∓ kv) = (ω1 + δ1) -(ω2 + δ2) (23)
We now consider the formation of these resonances and dark states by studying the time-dependent evolution of the populations. Figure 5 shows how the ground-state population distribution varies with time at different velocities. In each of the figures the initial conditions are taken as the Boltzmann distribution in a gas of gallium atoms at 1500K, the typical temperature of a gallium effusive cell. Initially, 34.4% of the atoms are in each of the 2 P 1/2 states, with 7.8% in each of the 2 P 3/2 sublevels. At kv = 0, (a), a symmetry is observed as expected, resulting in three sets of overlapping populations. Due to the two-photon processes and the formation of dark states within the state |G , the population in the |g state decays to zero. As the velocity decreases slightly to kv = -0.25 (b), the six different populations can be seen to reach their steady-state distribution quickly. As the velocity is now nonzero, the symmetry of the system is broken, and the behaviour of individual populations can be observed. Figure 5(c) shows the population evolution at kv = -0.5, one of two-photon resonances between the two fine structure states. In this figure oscillations are observed, before the populations of states G 3/2 and g -1/2 rise to a maximum. All other populations tend to zero. This behaviour is notably different from that at other velocities. In figure 5(d), the internal processes can be seen clearly. At first (t < 20) population is transferred rapidly from the g -1/2 state, resulting in a gain in population, via the excited state e -1/2 , for the G 3/2 and G -1/2 sublevels. Similarly, population in the G -3/2 state decreases, resulting in a gain for the g 1/2 sublevel. After this time, the populations approach their steady-state distribution, with the g 1/2 , G 1/2 and G 3/2 states populated while the other three groundstate populations tend to zero.

Radiation Pressure Force

The force exerted on an atom due to the interaction of an induced dipole moment with a radiation field can be split into two distinct parts: a dissipative force which varies on the scale of the laser wavelength, λ, (also known as the dipole force or radiation pressure force); and a gradient force which varies on the scale of the width of the laser beam. The gradient force is conservative and does not contribute towards Doppler cooling of atoms, so is not considered in this work.

The (dissipative) radiation pressure force, using (21), can be easily calculated from the atomic density matrix, by summing the force contributed by each individual coherence. For the 2-(4)-2 level system, this is found using the following equation;

F = 2ΩIm 1 √ 3 ρ g -1 2 e 1 2 - 1 √ 3 ρ g 1 2 e -1 2 + 1 2 ρ G -3 2 e -1 2 + 1 2 √ 3 ρ G -1 2 e 1 2 - 1 2 √ 3 ρ G 1 2 e -1 2 - 1 2 ρ G 3 2 e 1 2 (24)
Calculations show that this system does not possess a cooling force in the steady state.

The absence of force is a result of coherent population trapping, leading to formation of a velocity-independent dark state which inhibits the cooling cycle. However, time dependent The state g -1/2 is depopulated quickly, with a resulting gain in population for the G -1/2 and G 3/2 sublevels. Similarly, population in the G -3/2 state decreases, resulting in a gain for the g 1/2 sublevel. After this time the populations approach their steady-state solution, with states with M F > 0 most highly populated: saturation parameter G = 1, detunings δ 1 = -3, δ 2 = -2, initial ground-state populations: ρgg = 0.344, ρ GG = 0.078. The units of time are γ -1 , as explained in section 3.

calculations show that a transient force is present, as seen in figure 6, and decays to zero as the steady state is approached. Although the ground-state populations are found to continue evolving past t = 500, this transient force has decayed to zero by around t = 40, corresponding to the peak in g 1/2 and G 3/2 populations in figure 5(d). The initial force observed (at t = 1) is found to continue oscillating at larger velocities, with a regular frequency but low amplitude, before dying away.

Optimal parameters

Analysis of individual coherences in the atomic density matrix shows that, for this system, the majority of the force is contributed by the carried out to simulate cooling if all population is transferred to the 2 P 3/2 state, for example by Stimulated Raman Adiabatic Passage. These conditions led to a much reduced initial force. This indicates that to maximize the force experienced by the atoms it is preferable to have as large a proportion of the population as possible in the 2 P 1/2 state. This would need to be taken into consideration in any experimental proposal.

Laser cooling schemes provide an extremely large parameter space, and as an example, we have investigated the dependence on laser detuning, δ. Figure 7 shows how the maximum initial force experienced by the atoms varies with laser detuning. The force is measured at resonant velocity, kv = δ1 = δ2. It can clearly be seen that increasing the red-detuning above δ = -2 does not result in a significant increase in force. For symmetry reasons the force decays to zero at kv = δ = 0.

Maximum Force

Detuning, ! It has also been observed that a small detuning difference between the two lasers is beneficial. The force is found to decay more slowly when the two detunings vary by a small amount, approximately when (δ1 -δ2) = -0.5. It is possible that this small detuning difference minimizes interference effects between the two laser beams.

Conclusion and Outlook

An improved method for calculating the transverse cooling of atoms with complex groundstate structures has been presented, and the optimal detunings determined to maximize the force on an atom such as 66 Ga. The method is efficient and can be adapted to considerably larger multilevel systems. As an example of the kind of complex system that can be considered, the Λ-shaped cooling cycle outlined above is applied to a Group 13 isotope with nuclear spin I = 3/2, for example 69 Ga. The ground state and first excited state contain 32 magnetic sublevels, requiring 1024 density matrix equations for a full treatment. Using the method presented this calculation requires the formation of only three 32 × 32 matrices. One suggested cooling cycle is a 3-5-(1-3-5)-3 scheme, involving only one of the hyperfine excited states. This requires five different coloured lasers, as shown in figure 8 An interesting atomic system for future investigation is that found in the stable isotopes of Group 13 atom thallium, 203 Tl and 205 Tl, which both have I = 1/2 and are important candidates for sensitive measurements of parity non-conservation [25].

In principle this method could also be used to model the laser cooling of molecular systems in which a suitable cooling scheme has been identified. The simplicity of the method means that electric or magnetic fields can be incorporated into the Hamiltonian matrix with little difficulty, as can time-dependent pulsed or shaped laser fields, as well as complex repumping schemes. This could be of great use in investigating methods for overcoming dark state resonances which hamper the laser cooling process.

Figure 1 .

 1 Figure 1. Population ρaa for the excited state of two-level atom in the strongcoupling regime, in which ρaa(0) = 0, G = 10, δ = 0, ∆t = 0.1, at velocity kv = 0.The figure shows the results of the numerical integration method presented in this work (dashed), compared to the Euler method (dot-dashed) and the exact solution (solid). The units of time are γ -1 , as explained in section 3.

Figure 3 .

 3 Figure 3. Ground-state populations g ±1/2 (solid), G ±1/2 (dashed) and G ±3/2 (dot-dashed) of a 2-(4)-2 multilevel atom (figure 2) in the steady state as a function of velocity v = vz. Radiative widths are equal, γ 1 = γ 2 = 1, as are laser intensities, G 1 = G 2 = 1, while detunings are δ 1 = -3, δ 2 = -2. The figure shows sharp two-photon resonances at velocities kv = ±0.5 and kv = 0. Figure (b) shows an enlarged view of (a), with G = 1, but as the population distribution is highly symmetrical the states g -1/2 , G -1/2 and G -3/2 are omitted from (b) for clarity. (c) also shows an enlarged view, in this case with G = 10, and at this higher laser intensity broadening of the two-photon population peaks is observed.

Figure 4 .

 4 Figure 4. Steady-state excited populations as a function of velocity v = vz. The populations for each of the excited states is identical, with sharp drops to zero observed as a result of two-photon resonance processes. These demonstrate destructive interference occurring with the formation of dark states; saturation parameter G = 1, detunings δ 1 = -3, δ 2 = -2.

Figure 5 .

 5 Figure 5. Evolution of ground-state populations g -1/2 , g 1/2 (solid), G -1/2 , G 1/2 (dashed) and G -3/2 , G 3/2 (dot-dashed) at (a) kv = 0, (b) kv = -0.25, (c) kv = -0.5 (two-photon resonance velocity) and (d) kv = -3 (at a resonance velocity). At kv = 0, (a), the symmetry of the system dictates that sublevels with the same modulus |F, M F | have identical populations. This symmetry is broken for non-zero velocities, as shown in (b). At two-photon resonance velocities, as illustrated by (c) kv = -0.5, markedly different behaviour is observed, particularly in the g -1/2 population, which rises to a maximum but falls in every other case. Population oscillations also occur. In subfigure (d), taken at a resonance velocity kv = δ 1 , rapid changes in population are observed at short timescales (t < 20). The state g -1/2 is depopulated quickly, with a resulting gain in population for the G -1/2 and G 3/2 sublevels. Similarly, population in the G -3/2 state decreases, resulting in a gain for the g 1/2 sublevel. After this time the populations approach their steady-state solution, with states with M F > 0 most highly populated: saturation parameter G = 1, detunings δ 1 = -3, δ 2 = -2, initial ground-state populations: ρgg = 0.344, ρ GG = 0.078. The units of time are γ -1 , as explained in section 3.

Figure 6 .

 6 Figure 6. Time dependent radiation pressure force for the 2-(4)-2 multilevel atom (figure 2) as a function of normalized velocity. Saturation parameter G = 1 and detunings δ 1 = -3, δ 2 = -2. The units of time are γ -1 , as explained in section 3.

Figure 7 .

 7 Figure 7. Variation of the maximum radiation pressure force with laser detuning, at resonance velocity, kv = δ; Saturation parameter G = 1, detuning δ = δ 1 = δ 2 .

Figure 8 .

 8 Figure 8. (a) Energy level schematic for 69 Ga. Transitions forming a possible 3-5-(1-3-5)-3 five-colour cooling cycle are highlighted. (b) Example of the evolution of selected 2 P 1/2 , 2 P 3/2 ground-state populations in 69 Ga, at resonance velocity kv = δ. Assumptions: all laser saturation parameters G i = 1, all radiative widths are equal, and chosen as the reference unit, γ i = 1, and all detunings δ i = -2. For clarity only a selection of states are shown, and are labelled according to quantum numbers |J, F, M F . Initial populations are distributed equally amongst magnetic sublevels in each fine structure ground state, with 80% in the 2 P 1/2 state, and 20% in the 2 P 3/2 . The units of time are γ -1 , as explained in section 3.

  1/2 .

	Excited State Population Population, !aa
	Time [!] Time, t

Acknowledgments

LR would like to thank the Department for Employment and Learning (NI) for funding this PhD research.


Jochen R Schneider
email: jochen.schneider@desy.de

FLASH -From Accelerator Test Facility to the First Single-Pass Soft X-ray FEL

Keywords: numbers: 41, 60 Cr, 29, 20, Ej

The development from a test facility for the TESLA project of a Linear Collider with Integrated X-ray Free-Electron Laser to the world's first soft X-ray FEL user facility is described. In the wavelength range from 6.5 to 60 nm FLASH provides short pulses (~25 fsec) containing ~10

12 photons in laterally coherent beams with brilliance about 9 orders of magnitudes higher than achieved at the best synchrotron radiation storage ring facilities today. FLASH opened a new area in photon science where matter in non equilibrium states is studied with atomic resolution in space and time.

Introduction

FLASH, the Free-Electron Laser in Hamburg, was the first FEL user facility in the spectral range of the VUV-XUV. It was initiated as a test facility for the project of the "TESLA Linear Collider with Integrated X-ray Free-Electron Laser Facility". It is fair to say, without the great engagement of the particle physics community for the linear collider project, FLASH at DESY would not exist today. In the 1990ies the photon science community at large was still rather reserved concerning FEL facilities. On the other hand a number of groups accepted the challenge and pioneered a new area for photon science with building FLASH and doing experiments at FLASH. For the first time it is now possible to study matter, including matter in extreme conditions and in non-equilibrium states, with atomic resolution in space and time. The often used buzzwords are "take movies instead of pictures". This start in a new area of science is documented by the collection of articles on science with FLASH collected in this special issue of J. Phys. B.

The paper is organized as follows: After sketching the origin of the concepts for single-pass X-ray free-electron lasers and first steps to develop the science case, the development at DESY in Hamburg, which lead to the construction of the TESLA Test Facility TTF1, is described in some detail. After presentation of layout and performance of this test facility the first experiments on clusters are described. Next the layout and performance of the second stage of expansion of the facility (see figure 1), first called TTF2 and later renamed to FLASH, are presented followed by a description of the first single-shot coherent imaging experiment performed at a FEL. The summary starts with a sketch of the upgrades of FLASH, under way or planed, followed by a short presentation of the other FELs in operation or under construction, projects in an advanced planning stage or currently discussed. Some observations concerning usage of FEL facilities conclude the paper. Beamlines, diagnostics and instrumentation at FLASH are discussed in the following paper by J.

Start at DESY -The TESLA Collaboration

The theoretical basis for X-ray free-electron lasers has been established in the 1980s by two papers by A.M. Kondratenko and E.L. Saldin [1], and R. Bonifacio, C. Pellegrini and L.M. Narducci [2]. They stimulated a lively discussion in the accelerator community in a series of workshops organized by Brookhaven National Laboratory and the Stanford Linear Accelerator Laboratory SLAC in the early 1990s. Already in 1992 detailed suggestions were made to use the existing SLAC linac equipped with low emittance guns to drive 4 nm to 0.1 nm FELs. At that time DESY shared the vision of a large part of the particle physics community to build a TeV electron-positron linear collider and pursued this project vigorously when Björn H. Wiik became Director General of the laboratory. However, there were clear indications from the funding agencies in Germany that such a large research facility should be attractive not just for the particle physics community and DESY developed within an international collaboration a Conceptual Design for a 500 GeV e + -e -Linear Collider with Integrated X-ray Laser Facility [START_REF]Conceptual Design of 500 GeV e + e -Linear Collider with Integrated X-ray Facility[END_REF]. Based on the good experience with superconducting technology at the large hadron-lepton collider HERA at DESY and the need for high luminosity at the linear collider, the challenge was accepted to realize the accelerator in superconducting RF technology in a large international effort. The so called TESLA collaboration was founded which by the end of 2002 included 50 institutions out of 12 countries. The ambitious goal was to increase the gradient of the accelerator by a factor of 5 and to reduce the cost for the cryomodules by a factor of 5, which today is achieved.

In March 2001 the TESLA collaboration presented the Technical Design Report on a Linear Collider with an Integrated X-ray Free-Electron Laser Facility at a colloquium at DESY on Scientific Perspectives and Technical Realization of TESLA which attracted more than 1100 scientists, 40% from abroad. The reaction of the science community at large and the funding agencies was positive. After an evaluation of the TESLA proposal by the German Science Council the German Government decided in February 2003 that the XFEL part of the proposal should be realized in Hamburg as a European project with Germany paying half of the cost. Construction of the European XFEL started in January 2009 and first electron beams are expected for 2014.

In the 1990s the interest of the photon science community was rather focused on hard X-ray FELs. For example, in the US in 1999 a panel on novel light sources initiated by the Department of Energy and chaired by S. Leone concluded that the most exciting potential advance in the area of innovative science is most likely in the hard X-ray region, in the range of 8-20 keV, and even higher. Today, encouraged by the scientific results obtained at the first FEL user facility FLASH at DESY, the potential of soft X-ray FELs is intensely discussed in the community and a number of new projects have been presented.

As part of the R&D for the TESLA project DESY needed to build a test facility for the development of the superconducting linear accelerator and it was decided to combine this facility with a free-electron laser in the spectral range of the VUV. It was easy to formulate a convincing scientific case, the Conceptual Design Report for the FEL was submitted in June 1995, the project was approved in October of the same year, and construction of the so called TESLA Test Facility started.

TESLA Test Facility (TTF1)

TTF1, the TESLA Test Facility in its first stage of expansion, consisted of a low emittance laser driven photocathode electron gun, a pre-accelerator, two superconducting accelerator modules, three sections of undulator structures each 4.5m long and various electron and photon beam diagnostics (see figure 2). In addition a first experiment to study clusters was installed.

In principle superconducting linear accelerators provide the possibility of CW operation. However, as shown in figure 3, at DESY a pulse train structure was chosen as the optimal bunch time pattern for the project of a Linear Collider with Integrated Free-Electron Laser. The injector is based on a laser driven Cs 2 Te photocathode which is installed in a 1.6-cell normal-conducting RF cavity operated at 1.3 GHz. The bunch compressor provides the high peak currents needed for lasing. The undulator is a permanent magnet device with a fixed 12 mm gap and a period of 2.73 cm. The electron beam is dumped in an underground solid state dump. Photon beam diagnostics included a detector to determine the pulse energy and a spectrometer to measure the spectral distribution of the X-ray pulses shot by shot.

Lasing at saturation at a wavelength of 98 nm was first reached September 10, 2001. Aftermath tunability in the range from 80 to 120 nm was demonstrated, as well as a very high degree of lateral coherence of the beam. The duration of GW radiation pulses was estimated to be between 30 and 100 femtoseconds, a tool for direct measurement of such ultra-short pulses was not available at the time. For details and parameter lists see [START_REF] Weise | Proceedings of LINAC2002[END_REF][START_REF] Rossbach | Proceedings of LINAC2002[END_REF]. The peak brilliance of the FEL radiation is ~10 30 photons/(s mrad 2 mm 2 0.1% BW) and thus about 9 orders of magnitude higher than available at the best synchrotron radiation storage rings. At the FEL one obtains about 10

12 photons in a pulse of ~25 fsec duration, i.e. as many photons as we get today at the best electron storage rings per second.

All first generation FELs are based on the principle of Self-Amplified Spontaneous Emission (SASE). Here the electron density of the bunch is modulated by a resonant process taking place in the combined presence of the periodic transverse magnetic field of the undulator and the electromagnetic radiation. The wavelength of the modulation is equal to that of the radiation produced by the undulator, and finally to the wavelength of the FEL radiation. As a consequence the FEL wavelength can be tuned by either changing the magnetic gap of the undulator or the energy of the electron beam. The modulation process starts from shot noise leading to a series of coherently modulated sections in the electron bunch, which correspond to different longitudinal modes in the beam and to spikes in the spectral distribution as well as in the time structure of the FEL pulse. As shown in figure 4 the number of longitudinal modes depends on the length of the electron bunch, which implies that in principle one can achieve single mode FEL radiation in a SASE FEL if the electron bunch can be sufficiently compressed and still reaching the peak current needed for lasing. All properties of the FEL beam fluctuate and those important for the interpretation of the experiment have to be monitored on a shot by shot basis.

As an approach to avoid the spiky structure of the pulses which is inherent to the SASE process a so called Self-Seeding Option has been proposed for the FEL at TTF2 [START_REF] Feldhaus | [END_REF]. Figure 5 shows the schematic setup (side view) of the Self-Seeding Option which should provide narrow-band FEL radiation down to about 6 nm wavelength. After the first undulator section, FEL radiation and electrons are separated. While the electrons travel through the bypass represented by the four blue dipole magnets in the upper part of figure 5, the photons propagate across the grating monochromator beamline below in order to provide a narrow-bandwidth seed that is overlayed with the electrons at the entrance of the second undulator section. Simulations show that for the FLASH parameters one can expect a spectral distribution of a single pulse instead of the ensemble of narrow spikes typical for a SASE FEL. The expected gain in peak as well as in average brilliance is about 2 orders of magnitude. The pulse duration is increased to ~300 fsec, on the other hand, this mode of operation opens very interesting spectroscopic applications where high energy resolution is needed and longer pulse duration is acceptable.

For more information we refer to http://hasyweb.desy.de/flash_booklet/flash_brochure.pdf, a brochure which describes the FLASH accelerator and the process of producing FEL radiation, as well as the beamlines, instruments and the experimental hall.

First FEL experiments on clusters

For all experiments performed at free-electron lasers it is crucial to understand the interaction of the extremely intense, ultra-short X-ray pulses with matter on the atomic and molecular level. Therefore it was rather natural to perform the first experiments at the TESLA Test Facility on clusters which are intermediate between atoms and condensed matter. The group of T. Möller studied Xenon clusters using 98 nm FEL radiation at power densities up to 7 x 10 13 Wcm -2 [7]. Figure 6 shows the principle layout of such experiments.

A pulsed beam of atoms or clusters was prepared by expanding Xe gas at high pressure through a small conical nozzle. The composition of the beam and the size of the clusters could be controlled by varying the gas pressure. The cluster beam passed through a small scimmer into the main chamber. The FEL beam was focused to a spot of 20 μm in diameter by an elliptical mirror which could handle the extremely intense VUV beam very well. No deterioration of the beam quality was observed during the experiment. Off line investigations of the mirror properties performed later did not reveal any significant radiation damage. The pulse energy was measured with a calibrated channel-plate detector, which detects a well-defined percentage of FEL light scattered from a wire. The power density was calculated assuming pulse duration of 100 fsec. Figure 7 shows single shot time-of-flight mass spectra for Xe atoms and clusters.

The most striking result is the big difference between the spectra measured at atoms and large clusters. Whereas only single charged ions are observed after irradiation of isolated atoms, atomic ions with charges up to 8+ are observed when irradiating clusters, although the photon energy of 12.7 eV is only slightly higher than the Xe ionization potential of 12.1 eV. As also shown in figure 7 this effect is strongly dependent on the power density of the photon pulses. At a power density of 7 x 10

13 Wcm -2 each atom in large clusters absorbs up to 400 eV, corresponding to 30 photons during a time of about 100 fsec, the duration of the FEL pulse. Wabnitz et al. [7] suggest that the clusters are heated up very efficiently by the VUV radiation and electrons are emitted after acquiring sufficient energy before they finally disintegrate completely by Coulomb explosion. These very first results obtained at the SASE FEL of the TESLA Test Facility stimulated a broad discussion in the community and a wealth of further experimental and theoretical work. An overview of the field will be given by T. Möller in another contribution to this special issue of J. Phys. B. Time-of-flight mass spectra of ionization products of Xe atoms and clusters after irradiation with soft X-rays of 98 nm wavelength. Left: Spectra recorded after ionization with pulses of an average power density of 2 x 10

13 Wcm -2

. The atomic spectrum (bottom trace) shows a splitting into several lines owing to the different isotopes. After irradiation of clusters, highly charged ions are detected and a strong dependence of the cluster size is observed. The mass peaks are rather broad and displaced with respect to the calculated flight times indicated by thin vertical lines (different charge states) in the uppermost part of the figure. This line shift indicates that the ions have high kinetic energies shown in the inset as a function of the charge for clusters containing N = 1,500 atoms. Right: Spectra taken at clusters with N=1,500 as a function of power density which was varied by either reducing the gain of the FEL or by moving the cluster beam out of the radiation focus. The intensity of highly charged ions increases with increasing power density [7].

Layout and performance of TTF2/FLASH

The operation of the first stage of expansion of the TESLA Test Facility, where emphasis was on development and test of the superconducting linac for the project of a Linear Collider for Particle Physics with Integrated X-ray Free-Electron Laser facility was concluded in December 2002. In the next phase, called TTF2, focus was on the construction of a soft X-ray free-electron laser user facility. Commissioning of the expanded facility started in fall 2004, first lasing at a wavelength of 32 nm was observed in January 2005. User operation officially started in August 2005.

The layout of TTF2 as in place in the fall of 2004 is shown in figure 8. Compared to the first stage of expansion of the TESLA Test Facility (figure 2) the most important changes are the increase of the electron energy from about 240 to 700 MeV by adding/replacing cryomodules, the addition of a second bunch compressor, the installation of six undulators instead of three, a collimator section to protect the undulator against Bremsstrahlung and in case of electron beam steering accidents. The electron bypass allows accelerator studies without danger of damaging the undulators. Additionally an experimental station was installed in the bypass inside the tunnel for damage and other studies using the electron beam. The performance of the facility is described in ref [8]. Later a 6

th cryomodule was added and the electron energy increased to ~1 GeV, which enabled lasing at 6.5 nm, first observed in October 2007. Ref. [9] describes in detail the operation of the facility in the wavelength range from 13.7 nm in the fundamental to 4.6 nm in the 3 rd and to 2.75 nm in the 5 th harmonic.

In 2006/2007 the spectral range of FLASH was extended into the FIR by installing a longperiod undulator and a new IR beamline [10] (see figure 9). This way intense, truly synchronized IR pulses tunable in a broad spectral range are produced e.g. for two-color pump-probe experiments. The wavelength of the fundamental of the device can be tuned between 1 μm and 200 μm (300 THz to 1.5 THz). For wavelengths longer than 20 μm pulse energies around 10 μJ are achieved. The first experiments at the beamline aimed at determining the spatial and temporal properties of individual pulses from FLASH and to unveil their temporal substructure [11].

The high peak current necessary for lasing cannot be produced in the electron gun, instead the electron bunches need to be compressed which is done in two steps. First an energy slope is imprinted on the bunch by acceleration on the falling slope of the RF field in the cavities. As a result the particles at the head of the bunch receive a smaller energy gain than those at the tail. Afterwards the particles pass through two magnetic chicanes where the electrons of larger energy travel shorter distance than the leading ones of smaller energy and thus are enabled to catch up. Due to a nonlinear effect in this process the final bunches do not possess the ideal narrow shape but consist of a leading spike of ~50 fsec duration and a long tail as shown in figure 10. The expected longitudinal electron distribution within the bunch consists of a ~50 fs long leading spike containing some 10% of the total charge with a peak current exceeding 1 kA. Lasing is possible only in this leading spike of the bunch. The duration of the FEL pulse was determined to ~25±5 fsec with a streak camera using the synchronized IR pulse of FLASH [11] and an interferometric approach [12].

Figure 11 shows the spectral peak brilliance calculated for FLASH at DESY, the Linac Coherent Light Source (LCLS) at SLAC in Stanford, and the European XFEL Facility in Hamburg in comparison with the performance of 3 rd generation synchrotron radiation facilities. Overall the gain with free-electron lasers amounts to approximately 9 orders of magnitude. It is interesting to note that the experimental results achieved at FLASH for the fundamental at 13.7 nm as well as for the 3 rd and 5 th harmonic very nicely reproduce the earlier theoretical predictions. With the higher harmonics FLASH reaches the water window. For an average pulse energy of 40 μJ of the fundamental at 13.7 nm a power of 0.25±0.1 μJ was measured for the 3 rd (4.6 nm) and 10±4 nJ for the 5 th (2.75 nm) harmonic. In other words by going to higher order radiation the number of photons per pulse is reduced by two to four orders of magnitude. These weaker beams allow e.g. spectroscopic studies of condensed matter were the full beam intensity would create space charges masking the ground state properties of the investigated system. As an example, in combination with a synchronized optical laser, femtosecond time-resolved core-level photoelectron spectroscopy experiments were performed at FLASH using 118.5±0.2 eV photons from the 3 rd harmonic [13].

To date all quantities measured at the VUV-XUV beams at the free-electron laser FLASH fit very well the theoretical predictions. Obviously the SASE process can be simulated well to great detail and as demonstrated recently at SLAC, this is also true for the spectral range of hard X-rays. with the performance of 3 rd generation synchrotron radiation facilities [9]. Blue spots show the experimental performance of FLASH at the fundamental at 13.7 nm, and at 3 rd and 5 th harmonics. The curve FLASH (seeded) refers to the proposal for self-seeding of FLASH [START_REF] Feldhaus | [END_REF].

First single-shot coherent imaging experiment at FLASH

One of the most attractive science drivers for X-ray free-electron lasers is the single-shot coherent imaging of nano-scale objects. In order to get enough signal in a single shot image about 10

12 photons focused to a spot of a few microns diameter are needed. On the other hand such high power density will destroy the sample and the question is if one can get the structural information before the target explodes. Model calculations show that in order to achieve this goal the duration of the FEL pulses should be of the order of 10 fsec [14].

H. Chapman et al. [15] have performed the proof-of-principle experiment at FLASH using intense (4×10 13 Wcm • 2 ) pulses of ~25 fsec duration which contain ~10 12 photons at 32nm wavelength. The experimental setup is shown in figure 12. The graded multilayer mirror also acts as a filter for the 32 nm FEL radiation and thus improves the signal to noise ratio on the CCD detector. The single-shot scattering picture is shown in figure 13 together with the reconstructed image. The agreement with the TEM picture of the target shown in the inset of figure 12 is excellent. After transmission of the FLASH pulse the target heats up to about 60,000 K on picosecond time scales, as a result the scattering picture taken later with a second FLASH pulse corresponds to essentially a hole in the target. Coherent single shot imaging will be discussed in much more detail in following articles.

Figure 12. The FEL beam is incident from the left and is focused to a 20 μm spot on the sample, which is a 20-nm-thick transmissive silicon nitride membrane with a picture milled through its entire thickness using an FIB (this is enlarged in the inset, and the scale bar indicates 1 μm). The direct beam passes through the sample window and exits the camera through a hole in a graded multilayer planar mirror. The diffracted light from the sample reflects from this mirror onto a CCD detector [15].

Summary and Outlook

FLASH is the world's first FEL user facility in the spectral range of the VUV/XUV and many pioneering experiments in a large variety of scientific fields have been performed to date. This includes work on • Atoms, molecules and clusters • Matter in extreme conditions, warm dense matter, radiation damage • Single shot, lens-less imaging of cells and diffraction from nano-scale crystals • Condensed matter spectroscopy and scattering • Photon beam diagnostics. To date more than 96 scientific papers have been published, almost half of them in high impact journals. For an updated list see http://hasylab.desy.de/facilities/flash/publications/selected_publications/. This list does not include the many publications documenting the enormous progress made in FEL related accelerator science and technology. Here the controlled interaction of strong laser beams with bunches of relativistic electrons is becoming more and more important. More general, as already stated in 1999 by the "Leone Panel", the state-of-the-art light source facility of the future will include a complete marriage of accelerator principles and laser art.

Worldwide three X-ray free-electron lasers are currently in operation, three FELs are under construction, seven more projects are in preparation. Upgrade plans are pursued at DESY and SLAC.

During a shutdown from September 2009 to February 2010 key components of the FLASH injector and many of the RF stations have been exchanged. A 3 rd harmonic module was installed which will create a much more homogeneous electron distribution in the bunch. The synchronization and feedback systems, as well as controls in general have been improved. The 1 st cryomodule was exchanged and an additional, i.e. the 7 th cryomodule was installed, which increases the electron energy to 1.2 GeV.

An external seeding option (HHG) has been installed at FLASH and commissioning is expected to start in the fall of 2010. Pulses of 29.6 nm wavelength and 20 fsec (FWHM) in duration produced via high-harmonics generation are overlaid with the FLASH electron bunches and amplified by passing through 10 m of undulators. The photon beam is reflected out to an experimental hutch where time-resolved pump-probe experiments will be pursued. The goal is to operate the seeded FEL parasitically to the normal FLASH operation by taking only one bunch out of each bunch train. In addition DESY decided recently to construct an extension of FLASH by adding a second undulator, which will be tunable, and a second experimental hall with 6 experimental stations. FLASH II, as the extension is called, will be a seeded FEL. A model of the facility is shown in figure 12.

In Japan a VUV FEL has been built as a prototype for the Spring-8 Compact SASE Source (SCSS) aiming for lasing at 0.1 nm in the fall of 2011. The accelerator test facility started operation for users in May 2008 and provides radiation in the wavelength range from 30 to 61 nm [16]. In the frame of an international collaboration external seeding by light produced by high-order harmonic generation in a gas was successfully demonstrated [START_REF] Lambert | FLASH I FLASH II[END_REF].

At Stanford in the US the Linac Coherent Light Source (LCLS) demonstrated very rapidly lasing at 0.15 nm in April 2009, which was again a breakthrough in FEL science and technology [START_REF] Emma | Proceedings of PAC2009[END_REF]. Because of the low emittence of the electron beam only ~80 m instead of the expected 132 m of undulator are needed to reach saturation at 0.15 nm. The now spare undulators will be used to build a so called after burner which enhances the intensity of the 2 nd harmonic to about the value of the fundamental. Operation for users started in October 2009 providing beam in the spectral range from 0.8 to 2 keV at the AMO station. The experiments benefited largely from the fact that the pulse duration at LCLS could be varied routinely between ~10 and 300 fsec by changing bunch charge and compression [START_REF] Ding | [END_REF]. Plans for further upgrades include a pair of tunable undulators for soft X-rays separated by a magnetic chicane which allow production of a two color beam for jitter free pump-probe experiments on/off resonance.

In Trieste the single pass free-electron laser FERMI@ELETTRA is under construction. This user facility will cover the wavelength range from 100 nm (12 eV) to 10 nm (124 eV) by varying the undulator gap at fixed electron beam energy. It will use a novel concept for external seeding.

The construction of the European XFEL in Hamburg started in January 2009. A linac in superconducting RF technology will serve, in the first stage of expansion, three independent freeelectron lasers covering the wavelength range from 0.1 to 6 nm. This facility combines the extreme peak brilliance characteristic for FELs with a very high average brilliance over the whole wavelength range.

The Paul-Scherrer Institute in Switzerland pursues an XFEL project aiming for wavelengths down to 0.1 nm using a 6 GeV linac in "warm technology". A 250 MeV prototype linac is currently under construction and the go-ahead for construction of the full facility, called SwissFEL, is expected for 2011.

Further FEL projects are pursued in Frascati, Italy (SPARX), Berkeley, USA (NGLS), Pohang, Korea (PAL-FEL) and in Lund, Sweden (MAX IV-FEL). Other FEL projects are discussed in the UK (NLS), France (ARC-EN-CIEL) and Poland (POLFEL).

All these projects are initiated and supported by a growing user community. The interest for performing experiments at the running FELs is increasing very rapidly. As an example, LCLS issued 3 calls for proposals for experiments targeting the available beamtime from October 2009 to December 2010. 859 scientists (no double counting) from 25 different countries have been involved in 197 proposals. About 1/3 of them got beamtime. This is a rather striking outcome taking into account that LCLS just started operation and that only three of the six LCLS experiments were offered. In average the proposals had 16 authors, a much bigger number then what is typical for proposals submitted to storage ring synchrotron radiation user facilities. These larger user groups are required for experiments at free-electron lasers because a wider range of expertise is needed. In addition the beamtime allocated for a given experiment is short, typically 5 twelve hour shifts at LCLS. As a consequence experiments have to be very well prepared and tested before start of beamtime, loss of beamtime due to technical problems during the run must be minimized.

Because beamtime at X-ray free-electron laser user facilities will always be scarce it is difficult for individual research groups to successfully run a program which largely relies on FEL beamtime. Therefore most groups combine their FEL activities with research in their home laboratories using e.g. optical lasers, or by regular submission of proposals to synchrotron radiation facilities, or both. Some groups go beyond that and form larger collaborations with a high probability that at least one of the member groups has beamtime at any of the FEL facilities available worldwide. These collaborations should have regular meetings where progress is discussed involving actively PhD students and postdoctoral fellows. At these meetings the scientific goals and the preparation of new experiments could be discussed, e.g. reports on detailed Monte Carlo type simulations of the planed experiments could be presented. A culture which makes the collaboration effective and successful on one hand, and keeps the identity and the contributions of the different groups and people visible, on the other hand, needs to be developed. Then it should also be possible to analyze data jointly within the collaboration and to publish the results together. This approach works in particle and nuclear physics and should be possible to develop on a much smaller scale in the photon science community interested in FEL work Finally, it is a great pleasure to realize that the FEL user community is very grateful to the accelerator scientists for providing such outstanding research opportunities. Thanks are also due to the funding agencies in several countries which decided to support X-ray free-electron laser work in an early stage when larger parts of the photon science community were still rather reserved with respect to FEL facilities.

Figure 1 .

 1 Figure 1. The free-electron laser facility at DESY. The TESLA Test Facility TTF1 was the start configuration with focus on development and test of the superconducting linac for the project of a Linear Collider for Particle Physics with Integrated X-ray Free-Electron Laser Facility. From 2003 on TTF1 was expanded to TTF2, an FEL user facility for the spectral range of soft X-rays including a new tunnel and a new experimental hall (in the foreground). In April 2006 the facility was renamed to FLASH: Free-Electron Laser in Hamburg

Figure 2 .

 2 Figure 2. Schematic layout of the first stage of expansion of the TESLA Test Facility, called TTF1. The total length is approximately 120 m, the electron energy ~240 MeV.

Figure 3 .

 3 Figure 3. Electron bunch time pattern of TTF with 5 Hz repetition rate and up to 800 bunches in a 800-μs-long bunch train, i.e. up to 4000 photon flashes per second could be obtained.

Figure 4 .

 4 Figure 4. Energy spectra from short (duration ~40 fsec) and long (duration ~100 fsec) FEL pulses at a wavelength of 98 nm. The number of "spikes" corresponds to the number of longitudinal modes in the radiation pulse and increases with increasing electron bunch length [5].

Figure 5 .

 5 Figure 5. Schematic setup (side view) of the Self-Seeding Option for the FEL at TTF2, which can provide narrow-band FEL radiation down to about 6 nm wavelength [6].

Figure 6 .

 6 Figure 6. Layout of the cluster experiment. The FEL beam ( =98 nm, bandwidth ~0.5 nm) is focused by a mirror to a spot size of about 20 μm in diameter where it interacts with a beam of clusters. The photon beam intensity and time-offlight mass-and/or electron spectra are measured on a shot by shot basis. In the first TTF experiments fluorescence radiation could not be measured.

Figure 7 .

 7 Figure 7.

Figure 8 .

 8 Figure 8. Schematic layout of the second stage of expansion of the TESLA Test Facility, called TTF2. The total length is approximately 250 m, the electron energy ~700 MeV.

Figure 9 .

 9 Figure 9. Arrangement of the FIR and VUV undulators in the FLASH accelerator tunnel. The FIR undulator uses the spent electron beam from the VUV undulators. The rather large IR beam is coupled out by a first mirror M1 with a hole in the centre to let pass the much narrower VUV beam [10].

Figure 10 .

 10 Figure 10. Result of a simulation of the current (solid line) and the mean energy (dashed line) distribution along the electron bunch at the entrance of the undulator. The bunch head is located at the right-hand side of the figure [11].

Figure 11 .

 11 Figure 11. Spectral peak brilliance calculated for FLASH at DESY, the Linac Coherent Light Source (LCLS) at SLAC in Stanford, and the European XFEL Facility in Hamburg in comparisonwith the performance of 3 rd generation synchrotron radiation facilities[9]. Blue spots show the experimental performance of FLASH at the fundamental at 13.7 nm, and at 3 rd and 5 th harmonics. The curve FLASH (seeded) refers to the proposal for self-seeding of FLASH[START_REF] Feldhaus | [END_REF].

Figure 13 . 14 Wcm • 2 ,

 13142 Figure 13. FLASH X-ray coherent diffraction pattern recorded for a single (4±2)×10 14 Wcm • 2 ,25±5 fsec pulse. On the right the reconstructed target is shown which agrees very well with the original shown in the inset of figure 12 [15]

Figure 12 .

 12 Figure 12. Model of the two branches of the expanded FLASH facility.