A Roszkiewicz

W Nasalski

Unidirectional SPP excitation at asymmetrical two-layered metal gratings

Keywords: 73, 20, Mf, 42, 79, Dj

A new configuration of a reflective one-dimensional surface-relief metallicgrating structure for unidirectional excitation of surface plasmon polaritons (SPPs) is proposed and numerically explored for plane wave normal incidence. The structure is embedded between two different dielectric media and composed of two layers, each of them consisted of periodically placed rectangular metal stripes. It is shown that even a small horizontal shift between these two layers, or a change of dielectric contrast of the grating fillings, may redirect energy flow propagation in a vicinity of the grating structure. From these two cases of counter propagating energy flows only one shows the field behaviour typical for SPP excitation. This evident directivity phenomenon exists together with high concentration of the electromagnetic field at the periodic structure. The configurations analysed may be useful in designing optical devices like photodetectors or, in general, in any case where efficient control of energy propagation directivity is of primary importance.

Introduction

Plasmonics is one of the most rapidly developing branches in modern science, with tremendous progress in all its theoretical and experimental aspects [START_REF] Girard | Near fields in nanostructures[END_REF]. A great effort has been recently made in developing efficient methods for controlling surface plasmon polaritons (SPPs) [START_REF] Lee | The use of plasmonics in light beaming and focusing[END_REF] and designing new plasmonic devices with growing number of potential applications in several aspects of science and technology [START_REF] Roszkiewicz | Surface plasmon polariton applications in selected branches of modern science and technology[END_REF]. Those include, among others, spectroscopic chemical and biological sensors used in industrial process control, monitoring environmental pollution and planet surfaces, remote sensing with optical fibre sensors, as well as production of efficient solar cells and light-emitting diodes with high efficiency. Due to the strong wavelength discrimination effect, SPPs allow also producing wavelength-tunable filters. The ability of SPP to overcome the Rayleigh criterion together with great enhancement of electromagnetic field in near vicinity of a metal surface allows visualization of nanoobjects of dimensions much lower than an incident wavelength. In optical communication and data storage SPPs are considered as a new type of optical information carrier.

For further research and development of plasmonics, devising photonic elements similar to those used in conventional optics, but working with the help of SPPs, are necessary. For this goal, special photonic elements have been developed, like, for example, bent waveguides with high transmission efficiency, microgratings working as Bragg waveguides with orthogonal bends, beam splitters, Bragg mirrors, interferometers and SPP semiconductor lasers. In this paper we present another photonic structure which strongly interrelates its geometrical parameters with directivity of the SPP propagation.

Grating geometry and numerical modelling

Surface plasmon polaritons, being electromagnetic waves coupled to density waves of free electrons in metals, are surface waves that propagate along the interface between two media with their dielectric constants of opposite signs [START_REF] Raether | Surface Plasmons on Smooth and Rough Surfaces and on Gratings[END_REF]. They can be excited only with a TM polarized wave and, as having larger wave vector components in the horizontal direction than those of incident light, it happens only when this wave vector is increased in the direction of SPP propagation. Commonly used configurations for this purpose are glass prisms or diffraction gratings. In order to achieve SPP propagation in one desirable direction, an oblique incidence regime should be applied. However, in several configurations, due to complexity of the optical system, it is necessary to excite SPPs by normally incident beams. This also can be accomplished by using of diffraction gratings.

When light of frequency ω impinges on a one-dimensional (1D) diffraction grating at an angle θ i , the horizontal (parallel to the grating vector) components of diffracted orders are: k x = ωnsin(θ i )/c ± mK, where n is the refractive index of the medium through which the grating is illuminated, m is an integer denoting the diffraction order and c is the speed of light in vacuum, respectively. K = 2π/Λ is reciprocal 1D lattice vector, where Λ denotes the grating period. However, a significant waste of energy occurs when a symmetric grating is used due to equal coupling strengths of SPP propagating in the two opposite directions. This can be prevented by using a non-symmetric grating [START_REF] Bonod | Unidirectional excitation of surface plasmons by slanted gratings[END_REF][START_REF] Kleemann | Area-coded effective medium structures, a new type of grating design[END_REF][START_REF] Bai | Asymmetrical excitation of surface plasmon polaritons on blazed gratings at normal incidence[END_REF]. The authors of [START_REF] Bonod | Unidirectional excitation of surface plasmons by slanted gratings[END_REF] analyzed two aluminium gratings. One was slanted rectangular grating and the second had smooth sinusoidal slanted shape. Although both gratings were shown to be able to efficiently lead SPPs in one direction, they are relatively difficult to manufacture, especially in the visible region, due to the requirement of maintaining high accuracy, e.g. in preserving slant angle.

Asymmetry of the SPP coupling is used also in designing directional beaming effects [START_REF] Kim | Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric surface gratings[END_REF][START_REF] Kim | Tunable directional beaming from subwavelength metal slits with metal-dielectric composite surface gratings[END_REF]. Offaxis directional beaming originates from different coupling efficiences of different gratings placed on both sides of a subwavelength slit in an optically thick metal film. By tuning gratings' parameters like a grating period in the case of a full-metallic grating or a refractive index in the case of a metallic grating covered by a dielectric layer, it is possible to adjust precisely an angle of the beam propagation direction. Similar breaking the configuration symmetry was utilized in designing slanted grating directional couplers [START_REF] Wang | Compact slanted grating couplers[END_REF]. Devices of such a type couple the light coming from a vertical fibre directly into a specially designed core and further, without any additional optics, into a horizontal waveguide.

We propose a different asymmetrical reflection configuration of the diffraction grating for unidirectional excitation of SPPs. By manipulating with asymmetry of the structure it is possible to change the direction of a local energy flow that is manifested by the sign change of the local Poynting vector near the surface. We will show that it is possible to achieve distinct and measurable effects resulted from the asymmetry of the periodic structure. The structure proposed in this paper could be manufactured with standard photolithography or focused ion beam milling processes. In this configuration the precise control of depth variation within the grating period is not so crucial as for the slanted gratings mentioned in [START_REF] Bonod | Unidirectional excitation of surface plasmons by slanted gratings[END_REF]. In that case evaporation of subsequent layers must be repeated several times with different masks in order to approximate a desired smooth shape of the grating.

The problem under consideration is depicted in figure 1. Normally incident plane wave of wavelength λ and TM polarization impinges on one-dimensional (1D) asymmetrical planar periodic structure. The structure is composed of two metal subgratings, one on the top of another and shifted in relation to each other. By d x we denote a shift between centres of metallic stripes in both gratings. A single grating layer is composed of periodic metal and dielectric areas. A complex valued refractive index of metal is described by Drude model: [START_REF] Girard | Near fields in nanostructures[END_REF] with parameters for gold: ω p = 1.38•10 16 s -1 and γ = 1.07•10 14 s -1 . Dielectrics placed between metal stripes in the upper and lower grating layers have the refractive indices n 4 and n 5 , respectively. Dielectric substrate has the refractive index n 3 . The filling factors (areas occupied by metal in each layer) are denoted as f 1 and f 2 , respectively and the period of the grating is Λ. The upper medium is air with n 1 = 1, although may be replaced, in general, by any constant value n 1 , when the ratio n 1 /n 3 is preserved. Therefore, this configuration should be considered as the metal grating embedded between In numerical calculations the Rigorous Coupled-Wave Analysis [START_REF] Moharam | Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings[END_REF][START_REF] Moharam | Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach[END_REF] (also called the Fourier Modal Method) with factorization rules described in [START_REF] Li | Use of Fourier series in the analysis of discontinuous periodic structures[END_REF] are applied. This frequency-domain method is widely used in grating diffraction analysis. It is based on the Fourier expansion of the relative permittivity function in the grating region. Maxwell equations in periodic region are solved for electric and magnetic fields decomposed into Floquet-Bloch modes. Eigenvalues and eigenmodes associated with grating modes are computed for each grating layer, invariant in the z-direction.

Those solutions are matched with each other and with solutions in the superstrate and the substrate by boundary conditions imposed on tangential field components. Eventually, the set of equations is solved for transmitted, reflected and grating mode amplitudes.

The energy flow (denoted by arrows on each fourth subplot of figures 2-5) is expressed by the time-averaged Poynting vector: <S> = Re(E x H*)/2, where E and H are electric and magnetic field vectors and * denotes complex conjugation. The Poynting vector in each figure must be considered as a resultant energy flow of the incident wave and the reflected wave, where the reflected wave may acquire one from the two (positive or negative) propagation directions along the x-axis. The two waves propagating in the two opposite directions possess, in general, different complex amplitudes. A dispersive material (gold in our case) has the non-zero frequency-dependent absorption coefficient that increases significantly in abnormal dispersion regions. The relation between real and imaginary parts of susceptibility and the refractive index and the extinction coefficient are given by: [START_REF] Lee | The use of plasmonics in light beaming and focusing[END_REF] where n and γ are the refractive index and the extinction coefficient, respectively, and k 0 is a freespace wave-number. Kramers-Krönig relations relate n and γ for real (χ 1 ) and imaginary (χ 2 ) parts of susceptibility for a sufficiently wide range of frequencies.

In numerical simulations we assume λ = 632.8nm, n 1 = 1, n 3 = 1.5, Λ = 600nm, h 1 = h 2 = 60nm, f 1 = f 2 . The height of the grating should be precisely tailored in order to obtain the directionality effect, as will be described below. Values of refractive index of dielectric media are not considered as absolute values but rather as contrast values with respect to the upper dielectric medium. For simplicity we assume that n 1 = 1 in the incident medium. Different configurations of the two-layer structure, with different directions of the field energy flow, will be presented in the next section.

χ χ γ i k i n + + = -

Asymmetry of the grating configuration and field distribution

The grating period is chosen smaller than the wavelength λ in order to make all diffraction orders, except the zero-order one, evanescent with almost completely suppressed energy losses. When the reflectivity and transmittivity reach their minimum possible value, the electromagnetic field obtains its maximum value near the surface. In order to achieve the propagation of SPP in only one direction, amplitudes of specularly reflected waves have to be significantly decreased and the energy flow has to be transferred into the first diffraction order, due to the coupling of the one from many evanescent waves to SPP. Such a wave field should match the following conditions: the equality of frequency of the incident wave and SPP, as well as the equality of the incident wave vector horizontal component and the wave number of the SPP. We preliminary optimized the filling factors f 1 , f 2 of both layers following the procedure similar to that described in [START_REF] Bonod | Unidirectional excitation of surface plasmons by slanted gratings[END_REF]. The resultant filling factors are f 1 = f 2 = 0.4. The fourth subplots of figures 2-5 shown below present the main result of numerical simulations, that is the magnitude and direction of the Poynting vector. The three remaining subplots displaying the electromagnetic field components E x , H y and E z , indicate the asymmetry of the field distributions.

Influence of horizontal shifts in the grating geometry

We first consider two structures characterized by the refractive indices n 4 = 1 and n 5 = 1.5, with the horizontal shift change between the upper and lower layers within the range between 0 and 0.5Λ. We plotted two configurations with different shifts: d x = 0.2Λ (figure 2) and d x = 0.375Λ (figure 3). In both figures the plane boundary between two dielectric fillings is placed in between two grating sublayers.

In both cases the asymmetrical distribution of the Poynting vector magnitude in the vicinity of the structure is clearly seen. There is a small leakage of radiation into the substrate and a distinct directionality of energy flow near the surface. The shift difference between both configurations is relatively small but, nonetheless, it changes completely the direction of energy propagation. Such a small shift difference causes a measurable change in the propagation conditions and therefore in the direction of energy propagation in the vicinity of the surface. It can be easily seen that the component of electromagnetic field that changes its sign at a certain value of d x , reversing therefore the sign of the Poynting vector, is H y . Resonance conditions at the upper and lower boundary are not equal due to the presence of media with different refractive indices and, therefore, the SPPs are excited only at one boundary.

Grating period has to be properly chosen with respect to the wavelength. Although it is not shown for brevity, we verified that too short or too large grating period with respect to the wavelength does not allow guiding the electromagnetic field along the structure in the form of SPP, since the coupling conditions between incident evanescent order and the SPP are not fulfilled. This means that for a given structure, even asymmetric, the choice of wavelength range is limited by the structure ability to change the direction of SPP propagation. Additionally, if grating is too thin or too thick, the directivity effect is also suppressed. This is the reason for choosing the height of the whole grating equal to 120 nm. In particular, the values of half or twice the chosen height (60 nm or 240 nm) are too small or too large to enable relevant change in direction of energy propagation in the described configuration. Only the narrow range of grating heights allows achieving the opposite energy flow directions for the shifts between 0 ≤ d x ≤ 0.5Λ.

There are also some general restrictions regarding the filling factor. It cannot be too small in order to prevent direct transmission of light between the metal stripes into the substrate. On the other side, we have also numerically verified that too wide metal stripes cause creation of additional curls above the grating, due to reflection of significant amount of electromagnetic field from metal areas. The reflected field interferes with the incident light and form an interference pattern. This phenomenon smears out the directionality and smoothness of the energy flow.

Influence of the dielectric medium inside the grating regions

Subsequently, we analyze the field behaviour above the structure specified by a vertical shift of the dielectric boundary inside the grating region. Two cases are presented. In the first case, the grating filled with dielectric of the refractive index higher than incident medium n 4 = n 5 > n 1 is considered (figure 4). In the second case, the grating with dielectric medium of the refractive index equal to the index of incident medium n 4 = n 5 = n 1 is presented (figure 5). In both cases it is possible to find a particular value of horizontal shift (d x ) at which the strong directional effect is obtained and significant asymmetry in the field distributions occurs. Most of the field energy density, flows along the surface. Disturbances above the structure are small and the flow over the grating is not very much disturbed. In this case the change of field energy directivity is imposed by the vertical shift of dielectric media, not by the horizontal mutual shift (d x ) of metallic stripes, as it was shown in the previous subsection.

An interesting comparison can be made when one structure is filled with different dielectric media in both periodic layers and the second structure of the same fillings in both periodic layers. The presence of the dielectric of the higher refractive index changes radically the direction of the energy flow in comparison with the structure filled with completely the same dielectric medium, certainly in spite of the substrate. In the metallic structure exposed to dielectric with n 4 = n 5 = n 1 = 1 the energy propagation direction is the same as in case of slanted metallic grating placed on a metal substrate [START_REF] Bonod | Unidirectional excitation of surface plasmons by slanted gratings[END_REF]. However, by adding a higher refractive index material between the metal stripes the sign of the Poynting vector changes and the energy flow becomes oppositely directed. The difference in those cases is, as in the previous cases, connected with change of sign of the H y component. It seems that the difference between the refractive indices in the grating layers n 4 and n 5 is crucial for obtaining the one from the two opposite directions of the energy flow in the two, nearly the same, configurations. They differ only slightly in d x . The analysis of the E x and H y components shows that the field is tightly confined in the vicinity of the metal stripes, and therefore is stronger in cases when the stripes are embedded in dielectrics with n 4 , n 5 > 1, in opposition to the case of n 4 = n 5 = n 1 = 1. Propagation constants of the SPPs moving in the opposite directions (+/-first diffraction orders) are equal in the case of normal incidence, despite of the asymmetry of the grating. It appears that decisive in the unidirectional propagation of the SPPs are the different coupling strengths of both SPPs and, in consequence, different amplitudes of the both propagating modes [START_REF] Bonod | Unidirectional excitation of surface plasmons by slanted gratings[END_REF].

Action of the structures with horizontal symmetry

Finally, we present two grating configurations that are symmetrical with respect to the y-z plane (figure 6), that is the structures specified by d x = 0.5Λ. The refractive indices in the first configuration are: n 4 = 1, n 5 = 1.5, in opposition to the second one: n 4 = n 5 = 1.5. In both configurations the curls, in centres of which the energy flow is zero, emerge in the distribution of the Poynting vector. Those curls are originating from the interference of incoming, reflected and diffracted fields, together with the propagating SPPs. This interference is destructive and decreases the magnitude of the Poynting vector. As the structure is symmetrical with respect to the y-z plane, no direction along x-axis is preferred and no horizontal energy flow near the grating structure is observed. Moreover, the replacement of n 4 in the first configuration by n 1 in the second one, changes the sign of the magnetic field component H y , but still maintains the shape of the interference pattern over the whole structure.

Additional remarks

The results presented above have been discussed in the context of the energy flow direction of the total field. However, apparently similar or almost symmetric results shown for the Poynting vector evaluated over the really asymmetric grating structure may suggest different physical phenomena, which are the origins of the field propagation in the two opposite directions. In order to investigate physical reasons of such behaviour of asymmetric two-layered structure, we plot in addition the local reflection coefficient for one of the mentioned above geometries of the structure. Similar charts could be plotted for other configurations, although, for clarity, we present here only one set of the plots for one configuration.

The plots in figure 7 are calculated for the case of the material and geometrical parameters related to these of figures 2 and 3 (i.e. n 4 = 1, n 5 = 1.5). They present the local reflection coefficients of the +1 diffraction order, the -1 diffraction order and all diffraction orders together for E x and H y components of electromagnetic field. For this purpose the local reflection coefficient is defined at some point above the structure as the total field reduced by the incident field and normalized to the incident field. We plotted the local reflection coefficients just above the structure, at the points 2nm over the middle of upper metal stripes, where the energy flow has the smoothest profile. These plots refer to numerical simulations illustrating the change of the Poynting vector sign in the presence of small structure modifications. It appears that in cases of too thin or too thick metal stripes the energy flow redirection cannot be achieved. Suppression of the directivity effect in the case of too thin gratings may be explained by the weak influence of thin metal stripes on the incident wave and the quite high transmittivity of the structure. In this case the metal stripes are not sufficiently optically thick and opaque. On the other hand, in the case of too thick gratings, the influence of the lower subgrating on the incident field may be too weak to enable the efficient redirection of the energy flow. The explanation of the numerical results is given here on the grounds of the examples of the plane wave incidence upon the infinite reflecting structures. Even this idealized geometry allows us to achieve a few interesting observations. They indicate that the change of the energy flow direction is possible by introduction of small differences either in the structure geometry or in values of the refractive index of the dielectric filling. The excitation efficiency of this energy flow redirection can be estimated in this idealized view of this phenomenon as well, as it is shown in figure 8. In this figure the normalized (to the incident field amplitude) amplitudes of the reflected from and transmitted through the structure optical field are drawn. The differences between the left and the right field peaks are consistent with the direction of the energy flow preferred in the two cases described in figures 2 and 3 and subsequently in figure 7. Further analysis and more complete interpretation of these phenomena will be possible by taking into account a more realistic case of finite geometry of the problem. This means the cases of beams with finite cross-sections incident on spatially finite periodic SPPs guiding structures. In such cases we will be able to take into account the finite excitation area of SPPs, precisely calculate their excitation efficiency, propagation length and other quantitative parameters specifying more realistically the problem under consideration.

Conclusion

We have demonstrated that a simple two-layered structure, composed of two gratings with rectangular metal stripes embedded in dielectric media, can be used to control electromagnetic field behaviour in the vicinity of this structure. By a specific choice of the geometric configuration of the structure one may determine a desired direction of the energy flow. It appears that, by tailoring properly the horizontal asymmetry of the structure by shifting the gratings with respect to each other, it is possible to redirect the energy flow into one of the two opposite directions. Moreover, the grating shifts applied for this purpose may be small. The directivity change of the energy flow direction is accompanied by a sign change of the magnetic field component H y at a certain value of the shift d x . The same directivity change can be also obtained at a specified vertical position of a contrast boundary between two dielectric filling media. The advantage of this configuration is that it is easy to produce with the photolithography or focused ion beam techniques, which are not time or cost consuming. Moreover, in the case of metal stripes embedded in the two different dielectrics, the field components are tightly confined to the metal structure.

The geometry analyzed may find several applications in photonics. It can serve as an optical device for controlling direction of electromagnetic energy flow. Maintaining the advantages of normal incidence, it opens, at the same time, the possibility of manipulation with the direction of energy propagation. Similar simple asymmetric grating configurations with further optimization can serve as optical concentrators. By breaking the symmetry of the grating and thus forming a subwavelength slit between two grating systems with the reversed shift, and properly adjusting the phase differences of SPPs incoming from both sides, a strong concentration of light in the middle of the structure can be obtained. Such configurations can serve as elements of photodetectors, where the radiation has to be concentrated within a small active area below the metallization region.

  media. The height of the whole grating structure is h, the heights of individual grating layers are h 1 and h 2 , respectively.

Figure 1 (

 1 Figure 1 (colour online). Geometry of a two-layered grating diffraction problem. Two onedimensional surface-relief metallic diffraction gratings with different dielectric media in the gaps between the metal stripes are stacked and shifted in relation to each other.

Figure 2 (

 2 Figure 2 (colour online). Grating configuration with n 4 = 1, n 5 = 1.5 and d x = 0.2Λ and all nonzero components of electromagnetic field, normalized to the incident wave, in the vicinity of diffraction grating: (a) E x , (b) H y and (c) E z and (d) the Poynting vector magnitude |S| with the direction of the energy flow denoted by arrows.

Figure 3 (

 3 Figure 3 (colour online). Grating configuration with n 4 = 1, n 5 = 1.5 and d x = 0.375Λ and all nonzero components of electromagnetic field, normalized to the incident wave, in the vicinity of diffraction grating: (a) E x , (b) H y and (c) E z and (d) the Poynting vector magnitude |S| with the direction of the energy flow denoted by arrows.

Figure 4 (

 4 Figure 4 (colour online). Grating configuration with n 4 = 1.5, n 5 = 1.5 and d x = 0.4Λ and all nonzero components of electromagnetic field, normalized to the incident wave, in the vicinity of diffraction grating: (a) E x , (b) H y and (c) E z and (d) the Poynting vector magnitude |S| with the direction of the energy flow denoted by arrows.

Figure 5 (

 5 Figure 5 (colour online). Grating configuration with n 4 = 1, n 5 = 1 and d x = 0.25Λ and all nonzero components of electromagnetic field, normalized to the incident wave, in the vicinity of diffraction grating: (a) E x , (b) H y and (c) E z and (d) the Poynting vector magnitude |S| with the direction of the energy flow denoted by arrows.

Figure 6 (

 6 Figure 6 (colour online). Two symmetrical grating configurations (d x = 0.5Λ) with n 4 = 1, n 5 = 1.5 (upper subplots (a) and (b)) and n 4 = n 5 = 1.5 (lower subplots (c) and (d)) and the component of the magnetic field H y (left column (a) and (c)) and the Poynting vector magnitude |S| with the direction of the energy flow denoted by arrows (right column (b) and (d)).

Figure 7

 7 Figure 7 indicates that the geometry of the structure really favours the SPP excitation only in one direction. However, this probably does not exclude the change of Poynting vector sign and therefore

Figure 7 (

 7 Figure 7 (colour online). Local reflection coefficient plotted versus the horizontal shift d x for the grating configuration as in figures 2 and 3 (n 4 = 1, n 5 = 1.5); an imaginary part (a) and a magnitude (c) of E x , together with a real part (b) and a magnitude (d) of H y , of the local reflection coefficient of +1 diffraction order (blue dotted line), -1 diffraction order (green dashed line) and of all diffraction orders (red solid line).

Figure 8 (

 8 Figure 8 (colour online). Normalized amplitudes of the reflected (blue solid line) and transmitted (green dashed line) optical field measured above and below the two-layered structure for the shift d x =0.2Λ and the energy flow in the +x direction (a) and for d x =0.375Λ and the energy flow in the -x direction (b).


S W Epp
email: epp@asg.mpg.de

J R Crespo López-Urrutia

M C Simon

T Baumann

G Brenner

R Ginzel

N Guerassimova

V Mäckel

P H Mokler

L Schmitt

B L Schmitt

H Tawara

J Ullrich

X-ray laser spectroscopy of highly charged ions at FLASH

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Introduction

With an average density of only 10 -25 kg/m 3 , the universe contains nevertheless tremendous amounts of hot, and in principle observable, baryonic matter [1]. In most circumstances atoms will be found in ionic states ranging from those missing only a few electrons to the ones with most of them stripped off, up to bare nuclei. These highly charged ions (HCI) [2] are present in stars and their atmospheres, in supernovae, in near-stellar clouds, in shocks and jets from active galactic nuclei, in stellar winds and in various other astrophysical environments. Their spectral features provide us with direct and unique information on the composition and the state of matter of even the most distant known objects. Gaining insight into the spectra is therefore indispensable for understanding and developing astrophysical and cosmological models. Huge efforts have been undertaken to measure their emission spectra with increasing resolution, culminating in a series of X-ray satellite missions in the past decade like XMM-Newton or Chandra. Moreover, distinct quantitative knowledge of photon emission and absorption by HCI is crucial for modelling earthbound plasmas such as those used in fusion research [2] or for exploring the equation of state (EOS) of warm dense matter (WDM) [3]. HCI are, in comparison to neutral atoms, relatively simple atomic systems. Neglecting the constituents of the nucleus, only a few particles -for spectroscopy at least one bound electron is needed-are interacting, and therefore the electronic level structure become less complex than in atoms. Furthermore, in HCI one can control this complexity by choosing the number of bound electrons. As an example, exploring hydrogen-, helium-, or lithium-like systems with the same nuclear charge is one way of varying electron-electron correlation contributions in the system. In this respect, HCI are an ideal test bench for forefront atomic structure theory, with important fundamental implications. In HCI the strength of various electromagnetic effects rises in comparison with their equivalents in neutral atoms. It is not surprising that this increase scales up with Z, the atomic number of the ion, since it is the field created by the charge Ze (e being the elementary charge) in which the last bound electron (1s in U 91+ ) experience electric and magnetic field strengths of up to 10 21 V/m and 10 8 T. At such high field strengths, the semi-classical Bohr radius is already smaller than the Comptonwavelength. Effects caused by non-trivial quantum vacuum, e. g., vacuum polarization (VP) and selfenergy (SE) [4], exert significant influence on the interaction between the electrons and the nucleus. In the above example, field strengths close to the Schwinger limit (8 x 10 18 V/m), in which the energy gained by an electron within the distance of a Compton-wavelength is sufficient to create real electron-positron pairs, cause very large radiative contributions. Scaling laws are usually given in the form Z κ , where the exponent κ is dependent on the origin and characteristic of the specific electromagnetic interaction. While electron-nucleus interactions within the ion scale with κ=3, interelectronic correlations grow with κ=2. Hence, in HCI the relative influence on the total binding energy of interelectronic correlation goes down while Z rises. Relativistic contributions to the binding energy, for example, scale with Z 4 , and are therefore strongly boosted in HCI compared to the case of neutral atoms. Transition probabilities can show an even stronger Z dependence, the M1 transition matrix element for example is rising with Z 11 . The linewidth of inter-combination transitions between multiplets of different total spin S scales as Z 10 . Since the lifetimes scale in a reciprocal way, the corresponding optical 'forbidden' transitions become observable. Such lines were found in the emission spectrum of the sun already in 1869, but their explanation had to wait until the 1930's [5]. Newer approaches try to exploit the advantageous Z 5 dependency of contributions to the transition wavelength due to Z boson exchange inducing parity non-conservation [6], or to detect minute drifts in time of fundamental physical constants as the finestructure constant α [7]. Although highly abundant in the universe, HCI do not occur naturally on earth, as temperatures needed for their production easily exceed 10 5 K. However, HCI up to even H-like uranium have become available in a few laboratories, either with accelerators [8] feeding ion storage rings [9,10], or by means of electron beam ion traps (EBITs) [11,12,13]. A great variety of spectroscopic methods have been applied [14] to cover the broad emission spectrum of HCI, from the infrared up to the hard X-ray range. Spectroscopy of neutral atoms and molecules was revolutionized by the advent of lasers, with impressive examples, such as the ∆ν⁄ν=1.8⋅10 -14 (or even better) accuracy demonstrated in measurements of the absolute 1S-2S transition frequency ν in atomic hydrogen [15], an improvement compared to conventional spectroscopy by several orders of magnitude. In spite of the broad range of laser spectroscopy applications to neutral (cf. [15,16]) and singly-charged species (cf. [17,18]), these techniques have been applied to HCI in the visible and near ultraviolet spectral range only in a few exceptional cases, such as the hyperfine splitting of heavy H-like ions [19], and the fine structure splitting of some light ions [20,21,22] in which the electronic structure of the HCI allows for an optical transition. However, photon emission by HCI is naturally most intense in the X-ray region. The absence of laser spectroscopic data in this region was a consequence of the complete lack of intense X-ray laser sources, and of the fact that HCI were only available as very tenuous beams. The lack of appropriate lasers meant that X ray wavelength measurements in HCI had to use conventional wavelength or energy dispersive methods as crystal or grating X-ray spectrometers [13,23], or solid-state detectors [24,25]. Despite intense efforts invested in instrumentation development over several decades, their absolute limitations with regard to accuracy, stability, and efficiency remained an insurmountable barrier to further experimental progress. As a consequence, the most accurate wavelength measurements in HCI with a nuclear charge Z around 20 or higher did not exceed 2 parts-per-million (ppm) [23] in general or 15 ppm in Li-like ions [26], systematically limited by instrumental resolving power and statistics. Also recently Lestinsky and coworkers performed a storage ring based energy measurement on Li-like scandium (Z=21) with an overall accuracy down to 5 ppm [27]. Due to these limitations, in accuracy and resolution, the status of HCI spectroscopy resembled that of optical spectroscopy prior to the availability of tunable lasers.

In this report we summarize our work extending laser spectroscopy, a key technology for the last 50 years in fields like biology, chemistry, physics and various others, to ground-state transitions in HCI and, moreover, into the soft X-ray range. The goal is to develop a technique of utmost precision in ion spectroscopy by combining two advanced technologies, namely a state-of-the-art EBIT -providing a target of trapped highly charged ions-and the first soft X-ray laser. With the highly intense and bright radiation of FLASH we were able to excite one-photon transition up to 64 eV energy difference so far.

During our two week-long campaigns at FLASH in the years 2006 and 2008 we focused on the ground-state transitions in Li-like highly charged ions with Z>20. These ions have three remaining electrons and a fully occupied n=1 shell, while the 2s sub-shell is half filled by one electron. Not coincidentally, those systems are among the ones measured with the highest accuracies in the field of HCI and thus most challenging to compare with, and beyond that, they are also of enormous interest to test predictions of strong field quantum electrodynamics (QED) [4,[28][29][30][31]. The transitions we are most interested in are the two ground-state transitions 2 2 S 1/2 -2 2 P 1/2 and 2 2 S 1/2 -2 2 P 3/2 present in all Li-like ions (figure 1 (c)). Fortunately, at least for the 2 2 S 1/2 -2 2 P 1/2 transition almost all transition energies along the Li-like iso-electronic sequence from Z=15 (≈20 eV photon energy) to the heaviest elements in the periodic table up to Z=75 (≈200 eV) can be addressed by FLASH using its fundamental mode. This set of data on different nuclear charge Z can be of crucial importance to understand QED and other contributions to the transition energy such as nuclear size effects. With an outer electron in a s-state Li-like systems show many parallels with hydrogen or H-like ions.

With the exception of positronium, hydrogen or H-like ions have the simplest structure among bound systems. This simplicity was the only reason the minuscule Lamb-shift [32] could be discovered as the energy difference between the 2s 1/2 and the 2p 1/2 levels, which should not exist according to Dirac's theory stating the degeneracy of states with the same angular momentum. The explanation of this energy shift strongly promoted the theory of QED. Within its framework, calculating the vacuum polarisation and especially the self-energy contributions became possible in the one-loop representation [33], shown as a Feynman graph in figure 1 (b). In principle, for arbitrary accuracy on the QED part of any transition, theorists need to calculate an infinite number of such graphs. Since their complexity rapidly increases with the loop-order, the calculation of just a single diagram becomes more and more challenging. Steady progress over decades has made weak-field QED the best tested scientific theory, with impressive confirmation by experiment in the case of atomic hydrogen [34], but for strong-field QED both theory and experiment remain still away from that certitude by many orders of magnitude. A calculation of an individual Feynman graph is usually carried out by series expansion with Zα as one of the expansion parameters (with α≈1/137 the fine-structure constant). In weak-field QED, Zα<<1 (Z=1 for atomic hydrogen) is a small parameter, and the expansion converges rapidly. In a HCI, however, Z is much larger, and the series frequently does not converge at all. Therefore bound strong-field QED requires a different approach using all-order techniques [4]. An obvious approach to test QED with HCI might be investigating the Lamb-shift 2s 1/2 -2p 1/2 in different H-like systems. There have been direct measurements [35][36][37][38] of this transition and also related indirect determinations using measurements of the 2s 1/2 -2p 3/2 transition [20,[39][40][41]. In one particular case of H-like sulphur this was performed by applying laser techniques [42], but only with comparatively light elements. The upper limit, so far, is Ar 17+ (Z=18), and the accuracy did not exceed 10,000 ppm [38]. Experimentally difficulties arose from the fact that the initially long-lived 2s 1/2 level loses its longevity with the 11 th power of Z. Already at Z=18, the 2s 1/2 decays to the 1s 1/2 ground state in less than 3 ns. As a consequence, absorption techniques starting from the 2s 1/2 level suffer from dilute targets due to the rapid decay. Furthermore, emission spectroscopy with suitable spectrometers is hindered by the unfavourable branching ratio between the suppressed 2s 1/2 -2p 1/2 and the dominant 2s 1/2 -1s 1/2 decay channels. These problems had brought Lamb-shift investigations to nearly insurmountable experimental limits. QED effects do not exclusively apply to the original 2s 1/2 -2p 1/2 Lamb shift, but exerts its influence in all electronic energy levels. The 2s 1/2 -2p 1/2 shift in hydrogen is the remainder of an unequal shift of two levels. In order to speak of a shift, a reference point needs to be defined, and this is in this case the energy eigenvalue found by solving the Dirac-equation. As a rule of thumb, in hydrogen or H-like ions the electronic shells with smaller principal quantum number n are influenced more strongly by QED, so for instance 1s more than 2s and 2p more than 3p. Additionally, s levels are shifted stronger than p levels, and so lies 2s higher than 2p. Therefore, the 1s electron has the largest shift with respect to its Dirac value, causing a smaller total binding energy. Thus, measurements of the 1s Lamb shift in Hlike ions with different Z up to U 91+ have contributed significantly to QED tests (see [9,43] and references therein). This strong scaling in the nuclear charge Z can be illustrated by comparing the values of the shift with 0.000034 eV in hydrogen, 1.145 eV in Ar 17+ and 467 eV in H-like uranium [4].

As already mentioned, Li-like systems form a term level diagram similar to that of hydrogen-like systems, see figure 1 (a) and figure 1 (c). It is immediately noticeable that, however, the 2 2 S 1/2 level is the most tightly bound one, contrasting to the hydrogenic case. In Li-like systems this is simply a consequence of the shielding due to the two 1s electrons which affects the 'penetrating orbits' of s electrons far more strongly than those of p electrons. We see that the largest fraction of the energy splitting 2 2 S 1/2 -2 2 P 1/2 in Li-like ions is not related to QED corrections but rises up from humbler roots. Nonetheless, this transition is still the one having the highest possible QED contribution to the total energy difference. From neutral Li with 1.85 eV transition energy and a -0.000031 eV total QED contribution (17ppm) [44] to that transition, it scales up rapidly to 48.600 eV and -0.503 eV QED part (10000 ppm) in case of Li-like Fe 23+ to finally 280.645 eV transition energy and a -42.8 eV QED in Li-like U 89+ (152000 ppm) [45,29]. So, even if the measurable absolute QED shifts of the optical electron in Li-like ions is smaller than that of H-like ions of same Z, this transition in Li-like ions nevertheless offers a higher relative QED contribution. This is a great experimental advantage for spectroscopy, in particular since it is an intra-shell transition with n=2 and not an inter-shell transition involving the 1s level, as needed for H-like ions. In other words, in the latter the significant QED contribution is hidden within the very large energy difference between the 1s level and any other one. For example at least 3318 eV are needed in H-like Ar 17+ (having a 1 eV QED part) to excite the 1s electron, and 102.17 keV in U 91+ that has 467 eV QED (plus nuclear size) contributions [46]. A complete discussion of the influence of QED on bound electrons definitively requires investigation of He-like ions (see [23,[47][48][49] and references therein) and Be-like ions [50], which have an increasing complexity. The electronic sub-shells of those systems are fully closed, therefore their description, even for the outer electrons, is influenced by inner-shell interactions. With combined theoretical and experimental efforts, a stepwise understanding of the few-electron HCI would appear a sensible approach to understanding the QED of many-electron systems, and finally, that of inner-shell electrons in atoms.

Experimental setup

The Heidelberg FLASH-EBIT

An EBIT [51] produces and stores ions in a cylindrical volume with the dimensions of a human hair of a few centimetres length. It is different to most other devices used for producing and investigating HCI in its 'table-top' character. The first EBIT was commissioned 1986 at Lawrence Livermore National Lab by Levine and co-workers [52]. Scince then a dozen of such machines now operate around the world. A scheme of FLASH-EBIT is shown in figure 2. A near monoenergetic electron beam emerging from the cathode of an electron gun is used to sequentially ionize atoms from their neutral to any desired charge state. The beam electrons are accelerated to the energy required for ionization by an appropriate potential difference between the gun and the trap electrodes. Typical values are of the order of some keV depending basically on the desired charge state and the ion species. To efficiently produce H-like uranium, for example, a kinetic energy of 200 keV is needed. The high current electron beam (up to 550 mA) is additionally compressed in diameter to achieve a very high current density, in some cases beyond 10000 A/cm 2 , or nearly one electron per barn per second. For this purpose EBITs are equipped with a non-uniform radially symmetrical (with respect to the electron beam axis) magnetic field produced by a solenoid or a pair of Helmholtz coils, which increases from zero (at the cathode of the gun) to up to 6 T at the trap region. As a consequence of the conservation of angular momentum in the cyclotron motion of the electrons, a 3 millimetre diameter beam starting at near-zero magnetic field is compressed to diameters of 60-100 µm by the strong magnetic field at the trap centre.

The electron beam provides another essential ingredient for an electron beam ion trap, namely the stationary electrostatic field created by its negative space charge. In this way, a narrow cylindrical cloud of ions is radially confined within the beam or around it (figure 3 (a)). Typically, there is a complex dynamic interplay between the beam and the ions, caused by various processes such as electron impact ionization, radiative and dielectronic recombination, electronic excitation and radiative decay, plasma heating and others. This combination of interactions of the various charged particles produces ion clouds where most of the ions are typically within a diameter of 2-3 times larger than that of the electron beam, in the present experiments of around 200 to 300 µm. The extension of the ion cloud along the electron beam, strictly speaking the trap length, is determined by the settings of several ring shaped drift tubes on both sides of the central trap region to appropriate positive potentials. This stack of nine electrodes allows thus for adjusting the trap length. In all experiments within the context of this report the trap length was set to be as short as possible, i. e., around 55 mm. Since highly charged ions are extremely sensitive to charge transfer in collisions with the neutrals of the residual gas, a vacuum level below 10 -11 mbar is established by cryo-cooling down to 4-5 K the whole electrode stack by means of a two-stage 3 K coldhead, which also cools the superconducting coils (see figure 3). In this respect the FLASH-EBIT was the first cryo-cooled EBIT, not employing a helium cryostat but instead taking advantage of a closed-cycle refrigerator. Low maintenance, reduced cost of operation and longer experimental stability are some of the rewards of this setup. The device is on one hand capable of running at acceleration voltages up to 80 kV. On the other hand, it is also possible to operate it at acceleration voltages of only 50 V, which is very low for such devices [54]. This allows the experimenter to produce, trap and investigate a huge variety of different ions and charge states in a targeted way, presently from N 3+ to He-like Ba 54+ or even higher. The beam intensity is space-charged limited; at acceleration voltages of more than 5 kV we have reached already currents of more than 550 mA. This makes the device the one with the highest current worldwide. At low acceleration potentials (100 V) it still supports currents of a few mA.

The entire apparatus was designed to be as compact as possible without compromising performance, and was built to exploit radiation facilities worldwide. It fits with its periphery, ancillary equipment and electronics in a modified shipping container. The EBIT itself is typically operational within 2-4 days after shipping. Cutting-edge performance is usually regained within some 7 to 10 days later and is a price paid for venting the experiment partly (although with dry nitrogen) for shipping.

Prior to experiments at FLASH the EBIT ran already successfully at BESSY in Berlin on photoionization of ions [53,54] and will have beamtime in the near future at LCLS (Linear Collider Light Source) in Menlo Park, USA. Theoretically, a focus dimension RMS hor. × RMS ver. of 100-150 µm × ∆S [0-2000 µm] with ∆S the adjusted slit width could be expected [55]. During our measurements we found a spot approximately double the size in vertical direction which corresponds with results in [56]. A significant reduction of the spot size has since been reported [57]. The monochromator should by design reduce the intrinsic energy bandwidth of FLASH of about E/dE ≈ 100 to a level up to E/dE ≤ 30,000. However, one has to consider the direct relationship between bandwidth and resolution, on one hand, and the final photon flux of photons at the interaction region on the other hand. The monochromator scheme trades off photon flux for resolution. Free-electron laser (FEL) seeding schemes should in future improve the current situation. During the here reported runs, we tried to use the maximum resolution available at sufficient flux (2006: E/dE ≈ 2000, 2008: E/dE≈ 3000). As displayed in figure 4 (a), we used a linear configuration in which the photon beam entered the EBIT from the back first passing the electron collector (where the electron beam is dumped), continuing on its way co-linearly with the electron beam towards the ion cloud (target) in the centre of the trap region. Finally, the photon beam was dumped on the electron gun cathode (see figure 2). In principle it should be possible to utilize the photo-electron signal derived from the cathode as a timing trigger for the photon-ion interaction. However, efforts to use this signal to receive additional shot-to-shot intensity information did not achieve convincing results.

Principle of measurement.

So far, all of our measurements on HCI soft X-ray laser spectroscopy were performed in a single photon excitation scheme. The wavelength of the FLASH laser beam was adjusted by means of the monochromator to excite the transition under investigation. In the case of a short lived upper level, as it is the case for electric dipole transitions, its rapid decay leads to the emission of fluorescence radiation. In the simplest scenario, this radiation has the same wavelength as the exciting radiation. In a more general scheme, the upper level B could also decay to an intermediate level C by emission of photons with longer wavelength. However, the emission of this fluorescence radiation occurs on the whole perpendicularly to the FLASH beam axis. The situation is depicted in figure 3 (a). The Fluorescence radiation was detected in a non-dispersive single photon counting mode. Due to the lack of wavelength resolution in this detection scheme, we applied a time coincidence condition with the FLASH pulse to discriminate 'true' fluorescence photons from the strong, electron impact excited radiation background of the EBIT.. The ion cloud and the photon beam were brought to overlap. Both had a comparable diameter of 200-300 µm (figure 3 (a)). The FLASH beam consisted of photon pulse-trains of 30 single pulses, see figure 5 (a), with a 1 µs separation between them. The pulse trains itself had a 5 Hz repetition structure summing up to 150 (nonequidistantly distributed) single pulses every second. Each pulse contained approximately 2•10 12 photons within the typical length of 20-80 fs. The total flux amounts to roughly 3•10 14 photons/s. As mentioned, fluorescence occurs immediately after each single photon pulse has interacted with the ions. In this spectral range, electric dipole allowed transitions have decay times in the nanosecond range or below, and thus practically all fluorescence photons from a singe pulse are recorded together, since detection time resolution was limited to a few nanoseconds. However, multi hits are not expected since interaction rate times detection efficiency lies well below one fluorescence photon per The IGLET detects X-ray photons in the energy range 1 to 160 keV and permits analysis of the contents of the trap, exploiting the characteristic radiation of differently charged ions. A further diagnostic tool is the grating instrument, of which a typical spectrum is displayed in figure 6. It shows the wavelength interval from 12-33 nm with a resolution up to 1000. This allows the user to crosscheck our laser spectroscopic measurements. In our experience, a maximum in fluorescence yield for our laser spectroscopy measurement is directly connected with a maximum visibility of the same line (electron beam excited) in the grating spectrometer, which operates in a time integrating mode with exposures lasting several minutes. We stress however that the fluorescence yield is too small to become visible on the grating spectrometer, no matter if the FLASH radiation is 'on' or 'off'. A strong line in the grating spectrum simply indicates an optimized ion production process and thus a maximum amount of the desired ions in the trap.

Results

First run in 2006

This first beamtime period was dedicated to demonstrate the proof-of-principle of our scheme for extending laser-spectroscopy with its enhanced accuracy for the first time to transitions in ions in the soft X-ray spectral region. The results appeared in [59]. As target we chose Li-like iron (Fe 23+ ), for which we intended to excite the already mentioned ground state transition 2 2 S 1/2 -2 2 P 1/2 , see figure 1 (b). Iron was an ideal candidate for experiments for various reasons. It is abundant both on earth and in outer space, and it is a key element in the solar corona [60] in particular in charge states from 9+ to 23+. Iron is from a spectroscopist's point of view one of the most intense and visible elements used in EBITs. Fortunately, the 2 2 S 1/2 -2 2 P 1/2 transition energy of 48.6 eV (wavelength λ=24.5 nm) was within the range FLASH was capable to deliver stably with high photon flux at that time. With a 2 2 P 1/2 -level lifetime of 0.6 ns, fluorescence follows immediately excitation, and our detection scheme discussed above could be applied effectively.

To introduce iron into the trap we injected vapour of iron pentacarbonyl (Fe[CO] 5 ) through the twostage gas inlet (figure 3 (b)). The minimum electron energy needed to ionize beryllium-like Fe 22+ to our desired state Fe 23+ is 1960 eV. However, the electron impact ionization cross-section at ionization threshold is zero, and reaches a maximum, as a rule of thumb, at two to three times higher values. At electron energies of 4900 to 5600 eV, however, production of the He-like charge state Fe 24+ is almost equally probable, since its production threshold lies with 2050 eV only slightly above that of the Lilike species. To be able to maximize the Fe 23+ target density, we ran the EBIT in a mode in which Helike iron is resonantly recombined into the Li-like state at a high rate. This is possible by exploiting a dielectronic recombination resonance [61] at around 4660 eV, more specifically two close-lying Helike KLL resonances. At that electron energy setting, the energy released by radiative recombination of a free electron from the beam into a 2p level of the He-like ion matches exactly the energy required for inner-shell excitation of a 1s electron to the 2p level. The intermediate Li-like states populated thereby relax radiatively to the Li-like ground state by characteristic emission of a 6.64 keV photon.

Figure 7 shows a two-dimensional density plot with the rate of photons emitted by the ions with energies between 2-9 keV, which are detected by our IGLET detector as a function of the electron acceleration voltage applied between the electron gun and the trap electrodes. Clearly visible is the resonant character of dielectronic recombination (black rectangle labelled Fe KLL). The electron energy was tuned to resonance with the lowest energy transition of the four intense resonances observed. Due to strong space charge effects, the acceleration voltage times the elementary charge e lies always higher than the actual electron energy, from which follows that the resonance appears in this representation at values higher than the theoretical 4660 eV. From the data in figure 7 we estimate the number of Li-like ions in the trap lying between one and two million at electron beam currents of more than 500 mA, and thus achieving average ion target densities of more than 10 10 per cm 3 within the trapped ion cloud. The most cumbersome task during this first campaign was to bring the FLASH laser beam with its diameter of 200 µm to overlap with the ion cloud of a similar diameter. The light path of FLASH is fixed in space and cannot be adjusted by the user. In the EBIT itself, the relative alignment of the superconducting magnet axis and the electron beam are optimized carefully for minimized electron beam loss. With the magnet coils fixed to the EBIT supporting frame, and all other elements of the apparatus adjusted to it, moving ten million Fe ions in space involves a mechanical effort increased by 23 orders of magnitude. Therefore, we had to adjust the whole EBIT with a total mass of more than 1,500 kg on a sub-millimetre basis until we fixed the open five free orientation parameters to align those two cylindrical volumes. To ease alignment, we first used a brighter, unmonochromatized beam (monochromator in 0 th order) to find fluorescence. Afterwards we sought and found fluorescence with the monochromatic beam, and extracted excitation energy information. Figure 8 shows our main result, the accumulated number of photons detected during a single, final continuous run of approximately 33 minutes in which we repetitively scanned the monochromator over the resonance.

The relative arrival time of those photons with respect to the 1 µs gate is displayed as the ordinate in the two-dimensional representation in figure 8.

As we can see, fluorescence photons appear concentrated in a region well defined with respect to wavelength and arrival time. This result tells us two things: Firstly, counts depend on the wavelength set at the monochromator, in contrast to the wavelength-independent background by the continuously running EBIT (as high as 50,000 counts per second, three orders of magnitude more intense than the fluorescence) as well as by FLASH radiation scattering on vacuum parts or (elastically) on trapped ions in the trap. Secondly, photons counted on that spot structure arrive within a small 70 ns time window within each 1 µs gate, which proves their correlation to the FLASH photons in an additional way, thus allowing for even stricter photon background discrimination. The projection of the resonance structure (as contained within the two horizontal white lines) onto the photon energy axis is represented by the white data points (right scale), and delivers the resonance energy evaluation of the 2 2 S 1/2 -2 2 P 1/2 transition. A Gaussian fit to the data provides a centre-of-mass energy E 0 = (48.6127±0.0011stat.) eV, or conversely a wavelength λ 0 = (25.5045 ± 0.0006 stat .) nm. The resolution of this measurement is given by E 0 /FWHM=1350 (full-width-at-half-maximum), and corresponds approximately to the monochromator resolution of 2000 in connection with the chosen 700 µm exit slit. One has to keep in mind that the radial extension of the ion cloud acts practically as an additional slit and can thus increase the resolving power in comparison with the nominal exit slit width. The total true fluorescence count within a 5σ-interval beneath the red fit curve in figure 8 is only 370 within the 2000 s of run time. Those 370 counts have been isolated from a background of total 10 8 photons by the techniques discussed above. Although our measurement was carried out within a comparatively short run we were able to reach a statistical accuracy similar to the total uncertainty level of the so far best value [62] achieved with grating spectroscopy of E lit. = (48.5997±0.0009 abs. ) eV. Unfortunately, for causes unrelated to this novel technique, our wavelength measurement was limited by the systematic uncertainty in the calibration of the monochromator, which was not better than ±0.0150 eV at this time. This adds a systematic error of that magnitude to our final result E 0 = (48.6127±0.0011 stat. ±0.0150 sys. ) eV.

Second run in 2008

The plan for our ten 12 hour shifts in the 2008 campaign was to resume work where it was left in 2006, namely the clear evidence of a wavelength dependent fluorescence signal on the 2 2 S 1/2 -2 2 P 1/2 transition in Fe 23+ , and from this point to focus in obtaining an absolute wavelength determination. For this task it was mandatory to calibrate and characterize the PG2 monochromator. To our knowledge there has not been a single instance for which a soft X-ray monochromator calibration with our desired accuracy at synchrotron light sources was reported. This is most likely consequence of the lack of appropriate calibration sources or techniques, as we will be discuss in more detail below.

Our experimental setup remained principally unmodified to the description given above, but for the addition of a small and, in principle, straightforward, beam monitor that nonetheless served very well to achieve overlap between the FLASH beam and the ion cloud target. This arrangement consists of a Ce:YAG scintillator crystal which is inserted through a slotted aperture into the central trap electrode by a push-pull manipulator (figure 3 (c)). Challenges were posed to design by the cryogenic, 4K temperature of this drift tube, the high voltage applied to it, the need to rotate the crystal after insertion through the narrow aperture, its exposure to and charging-up by the electron beam, and the long distance from which the extremely weak scintillation had to be imaged. The scintillator itself is a disc of 10 mm diameter with a thickness of one millimetre, and can be adjusted after insertion into the central electrode such that one side faces the electron beam and the other side the FLASH beam. A Ce:YAG crystal emits fluorescence light in the optical range when irradiated by either soft X-ray photons or bombarded by charged particles. By means of a fast, electron multiplication intensified CCD camera (Andor LUCA), our setup allows for imaging the fluorescence spots induced either by the FLASH photon beam or by the electron beam, see figure 3 (d). With the position of the electron beam recorded by the camera, which is mounted on the EBIT supporting frame, we have an ideal proxy for the ion cloud axis. Overlap of the electron beam position, and hence the theoretical centre of the ion cloud, and the FLASH beam axis, or strictly speaking the EBIT axis is then established by observing the photon-beam fluorescence position while adjusting the EBIT supporting frame. Additionally, this setup enables one to directly monitor the beam shape of FLASH while, e. g., changing the exit slit width. We investigated the 2 2 S 1/2 -2 2 P 1/2 as well as the 2 2 S 1/2 -2 2 P 3/2 transition in Li-like Fe 23+ and the 2 2 S 1/2 -2 2 P 1/2 transition in Li-like Cu 26+ during this run. In the case of 2 2 S 1/2 -2 2 P 1/2 in Li-like iron we tested the possibility to achieve a calibration by using known photo-ionization lines of neutral neon.

3.2.1. 2 2 S 1/2 -2 2 P 1/2 transition in Fe 23+ and photoionization of Ne. With the ability to visualize the FLASH beam as well as the ion cloud position we were able in 2008 to swiftly return to the situation at the end of our 2006 run. Figure 9 shows two subsequent measurements on the 2 2 S 1/2 -2 2 P 1/2 transition in Fe 23+ . The first dataset took over 4600 s, the second dataset 4000 s exposure time. Both measurements show around 700 true fluorescence counts, and the achieved resolving power is E/FWHM=2000 on both, with a FWHM of 22 meV. It is noteworthy that the line centers, which are, E 1 =(48.57350 ± 0.00039 stat. ) eV and E 2 =(48.57403 ± 0.00043 stat. ) eV respectively, match perfectly within their sub meV uncertainties, and thus prove stable conditions over the 2.5 hours of measurement, resulting in an average of E 0 =(48.57374 ± 0.00029 stat. ) eV. This demonstrates a statistical accuracy of 6 ppm, a fourfold improvement in comparison with the 23 ppm achieved in 2006. However these results cannot constitute absolute values while monochromator energy (or wavelength) scale is not independently calibrated at a better level, which is by itself a rather challenging task. One approach to achieve a moderately accurate absolute calibration is the use of known photoionization absorption lines in neutral neon, where a group of resonances lie between 48.7 and 49.0 eV. Photons with energy matching absorption lines generate three different types of signals which could be exploited: one can detect the (i) photoelectrons emitted from the atoms while scanning the wavelength, or analogously the (ii) photoion yield, and, since beam photons are absorbed by the ionization process, (ii) the FLASH beam intensity after passing the neon gas absorber (which was a photon beam attenuator by design). Thus, the photon beam flux downstream the gas attenuator can be measured as a function of the photon energy and the positions of the absorption lines can be compared to their literature values. In principle, this could be best done by inserting the neon gas directly into the area surrounding our interaction point in the central trap drift tube. But in order to achieve measurable absorption the gas pressure needed would be so high that we would spoil the ultra high vacuum of our trap for days. Therefore, it would be more convenient using an absorption cell inserted in the PG2 beamline right before the EBIT entrance port. Unfortunately, this absorption cell was not operational due to damage to its MCP detector, and there was no way to overcome this problem within our user beamtime. We had to follow the inconvenient way of using the 15 m long beam attenuator of FLASH located upstream to the point where the FLASH beam is distributed to various beamlines, flooding it with Ne to base pressures up to 150 mbar. This location lies upstream from the monochromator, implying that the FLASH light first interacted with the neon, ionizing it, and then passes the monochromator. By scanning it accross the interval from 48.7 to 49 eV, where the usable absorption lines lie, we had to measure the intensity of FLASH as a function of the photon energy. We did that by using our Ce:YAG crystal in its position within the central trap-electrode in two different ways. First, we recorded the fluorescence intensity by means of our intensified CCD camera, and second, we used in parallel our silicon-mirror setup to counting soft x-ray photons scattered at the scintillator crystal. Figure 10 (b) shows three photo absorption resonances we could record by this arrangement. A fundamental drawback of this approach is that one has to measure a small intensity decrease on top of a large signal, with poor signal-to-noise ratio, thus requiring several hours to record the curves shown. Nevertheless, the centroid determination of all three peaks has an uncertainty in all cases of roughly 1 meV or even better. We compared their values with those reported in [63] and [64,65] respectively, which are given with absolute accuracies of 1 meV. Those reference data are shown in figure 10 (a). A least-square minimization of the resonance energy positions in comparison with the reference values yields a shift of (20.95 ± 0.50 stat.. ) meV with a remarkably small uncertainty, thus indicating perfect agreement within error bars between the two calibration measurements. Inexplicably, inclusion of this shift as correction in our measured transition energy E 0 leads to a calibrated result of E 0,cal =(48.55278 ± 0.00075 stat. ± 0.001 sys. ) eV. Comparing this with the literature value given in section 3.1 indicates that our experimental value is off by almost 45 meV (actually the measured shift of about 21 meV points in the 'wrong' direction). Since the two measurements, namely that of the Li-like transition, and the photoionization calibration, were carried out with utmost caution, we have no explanation but that some systematic, unknown deviation has affected our measurement. It is not clear what origin this problem might have. It could result from stability problems since it is necessary to keep the FLASH photon beam parameters, such as its pointing, or the source-point position, constant for almost half a day. Other possible cause could be the fact that we do not use the position of the ion cloud itself (which acts as an exit slit under certain circumstances) as the reference in both measurements. In the case of the photoionization, all light passing the exit slit of the monochromator is scattered by the crystal or causes it to scintillate; in contrast, in the case of the photoexcitation of the 2 2 S 1/2 -2 2 P 1/2 transition, the ion cloud acts as a second slit, which may or may not be perfectly centered on the mechanical exit slit. This can lead to an asymmetric response, and hence to a constant shift, if the (at the exit slit) angularly divergent but geometrically focussed photons of different energies spread out on their way to the interaction point. Nevertheless, it is worth mentioning that photoionization lines can not be used to calibrate on the level of the small statistical uncertainty demonstrated here, since they are at their best good to 1 meV in this spectral range. Lack of adequate calibration sources and their inherently asymmetric Fano lineshape is limiting the suitability of these photoionization lines as secondary standards in general. energy range than usual was scanned in order to include the designated reference lines. The measurement time is about 3600 s, but each wavelength data point has an statistical significance obviously lower than that of the measurements in iron described above. When we switched from iron to copper, it was not possible to optimize the trapping parameters to the same level we reached with the former element. Unfortunately, this copper compound was less suitable than the iron carbonyl molecule for efficient trap loading due to the low fraction of the element in the sum molecular weight.

E 0 =(55.177 ± 0.003 stat. ) eV, respectively. The blue data curve is taken from the spectrum simultaneously recorded with the soft-x-ray grating spectrometer (1D representation). This data also shows a weak -but clearly visiblespectral line at an energy position of 54.6 eV, and an additional strong spectral line at around 54.4 eV not reproduced in the laser excitation data, although it reaches half the intensity of the strong line at 55.2 eV. This line can be identified as a transition (2s 2 2s 2 -2s 2 2s2p) from the ground state in Be-like Cu 25+ . Since this is an intercombination line between two different multiplets, the transition probability is about 40 times smaller than that of the 2 2 S 1/2 -2 2 P 1/2 peak at 55.2 eV. At this level of statistics it should remain invisible in the photoexcitation spectrum. The fact that is shows up in the emission spectrum taken with the grating spectrometer results from direct electron impact excitation in the EBIT, and is not due to FLASH photons. Electron impact excitation is subject to different selection rules than electric dipole transitions. Moreover, the upper level can be populated by cascades from higher levels easily excited by the electron beam and not accessible to resonant laser excitation. Once populated, decay of this state to the ground level proceeds in the EBIT, no matter what the transition probability is, because (i) there are no competing radiative decay channels, and (ii) collisional quenching by electrons plays only a minor role. We also found a spectral line in our grating spectrum at 71.55 eV, which we assign to the (1s2s) 3 S 1 -(1s4p) 3 P 2 transition in He-like C 4+ , thus supporting the assumption that the line seen at 54.6 eV belongs to that carbon ion. However, there is always some possibility that this line could be an unknown line in a highly-charged silicon or fluorine ion, but we can certainly exclude lines from C, O, Fe (still present as a residue in the trap) or Cu. The upper (1s3p) 3 P 2 level of the reference line in C 4+ is a triplet state (J=0,1,2) where each individual state within the multiplet is excited with equal probability. The central transition energy of this triplet is (1) E C4+ = (54.57147 ± 0.00009) eV [START_REF] Yu | NIST Atomic Spectra Database Available[END_REF]. Taking this value as a calibration for the energy axis in Fig. 12, the 2 2 S 1/2 -2 2 P 1/2 transition in Cu 25+ takes a value of (2) E Cu = (55.134 ± 0.005) eV, what has to be compared to the literature value [START_REF] Yu | NIST Atomic Spectra Database Available[END_REF] of (55.156 ± 0.001) eV. The two values disagree by four standard deviations. Alternatively, by reversing the original thought, and taking the copper line as a reference instead, we find for the line at 54.6 eV a value of (1) E new =(54.593 ± 0.005) eV fitting to F 3+ line at (54.595 ± 0.009) eV. We see here that a higher statistical significance would be required to arrive at a conclusive interpretation. However, we succeeded in performing laser spectroscopy on two different ion species at the same time, a significant and necessary step toward high accuracy absolute measurements frequency comparisons of soft X-ray transitions in highly charged ions.

Conclusions and outlook

In this work resonant one-photon laser spectroscopy of an atomic transition by means of a freeelectron laser was demonstrated, with a considerable foray into the soft X-ray region. Furthermore, the study was carried out on a electric dipole transition of a highly charged ion with large QED contributions. Figure 13 gives a comprehensive illustration of prior and present laser spectroscopy work on HCI with regard to transition energy. For reference, some selected UV and VUV results on neutral species are included. Using FLASH , lines such as the 2 2 S 1/2 -2 2 P 3/2 transition at 64.5 eV in Fe 23+ were excited, and thus laser spectroscopy in HCI at 10 times higher photon energy than hitherto reported [19]. With the 2 2 S 1/2 -2 2 P 1/2 transition in Cu 26+ we investigated with laser spectroscopy the second highest charge state after Bi 82+ [19]. The uncertainties in previous works [19], [42] lie at 160 ppm and 200 ppm, respectively, partly as a consequence of the extremely dilute targets and harsh environments in which HCI are usually produced. In this respect, our technique clearly points toward new frontier, in view of the 6 ppm statistical uncertainty of the 2 2 S 1/2 -2 2 P 1/2 transition in Fe 23+ [40,37,41,42,[START_REF] Dinneen | [END_REF]68,19,69,21,70,22,20]) and some selected examples in neutral atoms (black data points chronologically [71,72,16,73,15]). With the upcoming new generation of light sources like FLASH in Hamburg, LCLS (Linac Coherent Light Source) in Menlo Park, USA, and XFEL (X-Ray Free-Electron-Laser) the field of laser spectroscopy will open to completely new wavelength ranges. Here, HCI are suitable both as attained in 2.5 hours of measurement time. Given the comparatively low resolution settings at the monochromator of not more than E/∆E=3000, great improvements can in principle be soon expected, with the photon flux loss easily being compensated by improvements of the FLASH performance expected after the major shutdown in 2009/2010, which will bring the facility closer to its initial design parameters [START_REF]TESLA Technical Design Report[END_REF]. However, using a resolving power of E/∆E>100,000 as reported [START_REF] Follath | [END_REF] for a similar monochromator at 3rd generation synchrotron light sources might prove fruitful only if the pointing stability of FLASH is improved.

the near future, establishing absolute energy and frequency standards based in electronic transitions in the soft X-ray spectral range will draw our full attention. If we ask ourselves how much energy, e. g., a "48.6" eV photon carries, we will immediately see that the answer is finally connected to its frequency and thus to the definition of the second. This physical constant, however, is still linked to a Cs hyperfine transition lying in the microwave region. Our work is far off this spectral range (figure 13), and presently no frequency chains exist which extend into the spectral range of FLASH. The frequency-comb technique, recently shown to reach into the VUV region [89], could provide a suitable approach for measurements of the 1S-2S interval of He + [90] at energies as high as 20.3 eV. Unfortunately, it was not possible to use existing experimental data to cross-calibrate our measurements. Considering for instance the experimental accuracies reported for the 2 S 1/2 -2 2 P 1/2 transition in different Li-like ions, as seen in figure 14, it becomes evident that our present statistical accuracy limit of 6 ppm already exceeds the precision of the best conventional measurements, and thus a calibration with data obtained by those seems inadequate. Therefore, in future we have to link our data to more fundamental transitions in light H-like and He-like highly charged ions as Be, B, C, N, O, Figure 14. Theoretical [29,74,30,75] as well as experimental results for the Li-like isoelectronic 1s 2 2s 2 S 1/2 -1s 2 2p 2 P 1/2 transition for various elements with different nuclear charge Z. Shown from the top down: absolute energy, main QED contributions to the transition, main sources of uncertainty to the theoretical error budget, total theoretical uncertainty for some Z [45,76], and finally total uncertainty of this work (statistical error) as well as of other reported experiments [77,49,78,79,28,62,[80][81][82][83][84][85][86]10,13]. The spectral tunability of FLASH indicated on top allows for systematic measurements across the periodic table.

F and Ne, and laser spectroscopy has to be applied on two different species of ions simultaneously. Our Li-like Cu 26+ measurement is a first step in this direction. The statistical accuracy achieved here is ten times smaller than the current uncertainty of theory for the 2 S 1/2 -2 2 P 1/2 transition Fe 23+ [30]. Therefore, sensitivity to three-loop contributions to the interelectron interaction (black curve in figure 14) and potentially to the dominant contributions of the pure H-like self-energy correction in order O(α 2 ) (purple curve) appears within reach. However, to take advantage of these experimental advances the main uncertainties from the relativistic recoil of the electron and the screening of the H-like self-energy (a third order effect) have to be reduced by better calculations. Obviously, theory will benefit from higher experimental accuracy in these transitions, which have the largest relative QED contributions in atomic physics. The spectral range of FLASH (in its fundamental) allows for investigations of the 2 S 1/2 -2 2 P 1/2 transition across almost the whole periodic table. Systematic studies for different values of Z as well as on various isotopes will be of crucial importance to determine nuclear size contributions rapidly rising with Z (figure 14), thus checking different models of the nuclear charge distribution. The use of a laser ion source for filling the trap will enable those studies. As figure 13 illustrates, FLASH, and other upcoming FELs open an unexplored photon energy range to laser spectroscopy. Higher photon energies will not only allow for resonant excitation of fundamental ground state transitions as 1S-nP in H-like ions aiming at the 1S Lamb-shift, but also for photoionization experiments in HCI. Moreover, the short FEL pulses make femtosecond-scale lifetime studies possible [91,92] by pump-probe schemes using one or two-colour schemes. Steep scaling laws as a function of Z for the lifetimes of levels depopulated by multipole transitions (M1, E2, 2E1, M2...) facilitates those studies with HCI. Lifetimes reflect spatial aspects of the wavefunction to which transition energies are less sensitive, especially to correlation terms between the initial and final states, which are nearly unexplored in the domain of relativistic, QED-sensitive electronic transitions of HCI.

Figure 1 .

 1 Figure 1. (a) Energy scheme of a hydrogen-like ion versus (c) that of a Li-like ion. In (b), Feynman graphs of the one-loop self-energy (top) and vacuum polarization (bottom) responsible for the major contributions to the Lamb shift.

Figure 2 .

 2 Figure 2. Longitudinal section of FLASH-EBIT and photography of the device out of a similar perspective (upper right). The electron beam is indicated in red colour. It emerges from the electron gun, passes the drift tube assembly, where ions are produced and trapped, and is dumped on the collector. The apparatus has a length of 2.5 m and weighs approximately 1,500 kg.

2. 2 .Figure 3 .

 23 Figure 3. (a) Longitudinal section view of the trap assembly within the superconducting magnet section. Ions (green, centre) are produced by successive electron impact ionization caused by a highly compressed, intense and energetic electron beam (red) guided through the trap. The negative space charge potential of the compressed electron beam is able to confine radially the ion cloud. Its length (about 55 mm) is defined by potentials applied to the different trap electrodes. The FLASH photon beam (yellow) enters from the left and illuminates collinearly the ion cloud. (b) Radial section at the position of the ion cloud. Various detectors installed at 90 collect radiation emitted by the ions. (c) scintillator (1) with its holder (2) which is inserted into the trap region to verify (d) the position of the electron beam (left) and of the soft X-ray beam (right) by scintillation in the visible range. (c) (d)

Figure 4 .

 4 Figure 4. (a) Photograph of the set up during our beamtime showing the last 12 m of the beam path after the mirror M3. (b) Schematic drawing of the light path of the PG2 beamline from the undulator to the experiment in a side and top view. The angles α and β of the grating (see inset) can be set and determine the dispersion (resolution) at the slit. Adapted from [55].(a) (b)

Figure 5 .

 5 Figure 5. (a) FLASH pulse structure. Each pulse train consist out of 30 single pulses separated by ∆t SP =1 µs. The pulse trains repeat on a 5 Hz frequency. (b) Fluorescence (blue) of Fe 23+ ions from excitation of the ions with the un-monochromatized FLASH radiation (0 th order). No fluorescence is detected for either running on an empty trap (green) or stopping the injection of iron into the trap (black), thus clearly proofing the origin of the fluorescence at this early stage.(a) (c)

Figure 6 .

 6 Figure 6. Spectrum recorded by our flat field grating spectrometer in the wavelength range from 13 to 33 nm (from 95 eV down to 37 eV). Two iron lines we excited also by FLASH radiation are indicated.

Figure 7 .

 7 Figure 7. Electron recombination photons recorded as a function of the electron beam acceleration voltage and their energy. The most prominent structures belong to the Fe KLL resonances.

Figure 8 .

 8 Figure 8. 2 2 S 1/2 to 2 2 P 1/2 resonance data. 2D plot (smoothed, 5 meV × 0.016 µs binning) of the fluorescence signal as a function of the photon arrival time relative to the FEL pulse (left y-axis, absolute offset of time scale arbitrary) versus photon energy (x-axis). The total measuring time was 1987 seconds. The resonance contains 370 true counts over a 5σ interval after baseline subtraction. Inset: Projection of the two dimensional representation onto the photon energy axis yielding the number of photons (white squares) per 5 meV photon energy interval (right y-axis) as a function of the photon energy (see text). Red line: Gaussian fit to the data.

Figure 9 .

 9 Figure 9. Two subsequent measurements of the Fe 23+ 2 2 S 1/2 -2 2 P 1/2 transition energy. Onedimensional data extracted from two-dimensional data similar to figure 8. The results of the Gaussian fits to determine the centre-of-mass energy are plotted within the graphs. (a) Run of 4600 s. (b) Run of 4000 s.(a) (b)

Figure 10 .

 10 Figure 10. Absorption lines of neon between 48.7 eV and 48.95 eV. From left to right the resonances are: 2p 4 ( 3 P)3s( 2 P 3/2 )np 3/2 1 P o and 2p 4 ( 3 P)3s( 2 P 1/2 )np 1/2 1 P o belonging to the same Rydberg series, and 2p 4 ( 1 D)3s( 2 D)3p 1 P o from another one. (a) Experimental data taken from [63-65]; (b) Experimental absorption data from this work with Gaussian fit (green). Error bars are estimates in both cases.

  where both the measurement line and a reference line are present at the same time, which we could realize in Li-like Cu 26+ for the first time.

3. 2

 2 .3. 2 2 S 1/2 -2 2 P 1/2 transition in Cu26+ . Measuring simultaneously a transition of interest and the calibration references would seem an ideal way to attain highly accurate absolute results. A class of very appropriate candidates for reference lines are H-like and He-like light ions, such as Be, B, C, N, O, and to a lesser extent F and Ne. These highly abundant species have naturally stimulated the physicists' interest in their spectra since decades, and thus a comprehensive body of data exists[START_REF] Yu | NIST Atomic Spectra Database Available[END_REF]. Moreover, they allow already (or are likely to do it in the near future) for sufficiently precise electronic structure calculations, and hence even the theoretical energy levels of these fundamental transitions can provide references with adequately small uncertainties. The 2 2 S 1/2 -2 2 P 1/2 transition energy in Li-like Cu lies at about 55.2 eV, close to the (1s2s) 3 S 1 -(1s3p) 3 P 2 transition in He-like C 4+ at 54.5715 eV. The lower state of the latter is not the ground state but a long-lived metastable state (lifetime τ=20 ms in C 4+ ), and the lowest possible excited state of this ion. It is analogous to the well known metastable state of the triplet configuration of the isoelectronic neutral helium atom. This metastability is crucial because absorption technique like laser spectroscopy require a well populated lower level, thus either ground state or a metastable state, for photon absorption to happen. Volatile copper compounds are extremely rare and choices limited, we vaporized a rather complicated copper organometallic compound (CH 3 ) 3 SiC≡CSi(CH 3 ) 3 •[CF 3 COCH=C(O-)CF 3 ]Cu (CAS 137039-38-4) to bring Cu into the trap. It has a molecular weight of 480 u, and carries a single copper atom only, but contains also the elements H, C, O, F, and Si in different proportions, which were intended to serve for calibration. Fig 12 shows a measurement of the Cu 26+ 2 2 S 1/2 -2 2 P 1/2 line in which a broader

Figure 11 .

 11 Figure 11. Measurements of the Fe 23+ 2 2 S 1/2 -2 2 P 3/2 transition energy. One-dimensional data extracted from two-dimensional data similar to figure 8. The data was recorded in 1 hour of measurement time. Parameters of the Gaussian fit (red) are plotted.

Figure 12 .

 12 Figure 12. Measurements of the Cu 26+ 2 2 S 1/2 -2 2 P 1/2 transition energy. One-dimensional data extracted from two-dimensional data similar to figure 8. The data (black) shows two peaks fitted by a Gaussian (red) and their centroid. The lower energy scale (black) is uncalibrated. The blue curve shows data from our grating spectrometer for comparison; its energy scale (upper scale) is shifted on basis of the strongest line.

Figure 13 .

 13 Figure 13. Transition energies accessible by laser spectroscopy in multiply or highly charged ions (magenta data points chronologically [40, 37, 41, 42, 67, 68, 19, 69, 21, 70, 22, 20]) and some selected examples in neutral atoms (black data points chronologically [71, 72, 16, 73, 15]). With the upcoming new generation of light sources like FLASH in Hamburg, LCLS (Linac Coherent Light Source) in Menlo Park, USA, and XFEL (X-Ray Free-Electron-Laser) the field of laser spectroscopy will open to completely new wavelength ranges. Here, HCI are suitable both as

  Presently, neither calculations nor experiments on Li-like HCI reach a one ppm level. Although a hypothetical deviation, if found at higher accuracy levels, might not necessarily lead to 'new' or 'not yet expected' physics, one can argue that for a theory as uniquely fundamental as QED is, it becomes mandatory to predict accurate energy values for transitions in systems like H-like, He-like, or Li-like HCI. If nothing else to benchmark the mathematical tools, since these are not only applied to these few cases, but find use in all realms of physics and advanced chemistry. Naturally, verification by comparison to accurate experimental results is called for.

  3.2.2. 2 2 S 1/2 -2 2 P 3/2 transition in Fe23+ . With a photon energy of around 64 eV, the 2 2 S 1/2 -2 2 P 3/2 transition in Li-like iron is the most energetic transition we have resolved this far by laser spectroscopy. Fig.11shows a run over 3600 s yielding E 0 = (64.54658 ± 0.00050 stat. ). We achieve 8 ppm accuracy in the determination of the line center within a hour beamtime. It is interesting to compare the energy difference between the 2 2 S 1/2 -2 2 P 3/2 and the 2 2 S 1/2 -2 2 P 1/2 transitions, namely ∆ exp. = . 15,9729 eV, with the most accurate reported value[START_REF] Yu | NIST Atomic Spectra Database Available[END_REF] ∆ lit. =15.9659 eV. Both values disagree by 0.007 eV only, which has to be viewed in the context of the 0.045 eV shift of the first of the two inferred if the Ne absorption calibration was correct. Obviously, relative changes of the monochromator settings over the comparatively large scanning distance from 48.6 to 64.5 eV still lie well within reasonable bounds. This finding would favour our second scheme for absolute calibration,

Acknowledgements

We gladly acknowledge help from: S. Düsterer, B. Farts, J. Feldhaus, V. Kocharyan, M. Kuhlmann J. R. Schneider, E. A. Schneidmiller, S. Toleikis, R. Treusch, M.V. Yurkov, and the whole FLASH-team (DESY). We also thank to M. Wellhöfer, M. Martins and W. Wurth, all from Hamburg University, as well as to C. Kaiser, K. Bechberger, N. Müller, and T. Busch (MPI-K) for their technical support. Funding was provided through the MIDFEL-Initiative.