Laser spectroscopic electric field measurement in krypton

A laser spectroscopic method for sensitive electric field measurements using krypton has been developed. The Stark effect of high Rydberg states of the krypton autoionizing series can be measured by a technique called fluorescence dip spectroscopy (FDS) with high spatial and temporal resolution. Calibration measurements have been performed in a reference cell with known electric field and they agree very well with numerical solutions of Schrödinger's equation for jl-coupled states. The application of this method has been demonstrated in the sheath region of a capacitively coupled radiofrequency (RF) discharge. The laser spectroscopic method allows us to add krypton as a small admixture to various low temperature plasmas.


Introduction
Electric fields play a key role for the motion and spatial distribution of charged particles in low temperature plasmas [1]. Especially in the sheaths of capacitively coupled RF discharges, electric fields induce strong heating and transport processes. The nonlinear behaviour in this region can excite collective resonances and wave phenomena. For the development of a consistent model of the sheath and its dynamic interaction with the plasma bulk, sensitive space and phase resolved electric field measurement methods are required. These methods should be independent of existing models of the investigated processes and non-invasive, as their influence on the discharge physics should be negligible.
Laser spectroscopic measurement methods for electric fields in plasmas based on the Stark effect of molecules and neutral atoms have already been developed. Some examples are: BCl [2], NaK [3], CS [4], helium [5]- [7] , hydrogen [8,9], argon [10]- [13] and recently krypton and xenon [14,15]. For discharges in those gases, one would prefer to use the respective gases also for the electric field measurements. However, it would be problematic to add molecular gases like hydrogen to another discharge. Due to the relative low dissociation degree in low temperature, low power plasmas, a relative large amount of gas is needed in order to get a sufficient neutral atom density. Furthermore, the vibrational and rotational levels in molecular gases can absorb a substantial amount of electron energy by inelastic collisions and thereby the electron energy distribution function can be strongly affected. On the other hand, rare gases can be admixed in relative small amounts as atomic probes for the electric field measurements. Calibration can easily be made without a plasma if the excitation occurs from the electronic ground state. In helium and argon, excitation out of the ground state can only be done by using vacuum UV wavelengths, thus for practical reasons it is necessary to excite out of the metastable state. In helium, a sensitivity of 50 V cm −1 has been reached [7]. Argon provides a much higher sensitivity of 3 V cm −1 [12]. Unlike atoms in the ground state, metastables have to be created in a discharge and can also be quenched easily at higher pressures. In contrast to xenon, the excitation wavelength of krypton is close to that of atomic hydrogen (205 nm), which enabled us to use our existing laser 3 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT system. Former investigations of the Stark effect in krypton have been performed at relatively high field strengths (few 100 to several 1000 V cm −1 ) [14,16] by means of laser optogalvanic spectroscopy, which limits the measurement to a line-integrated one. Furthermore, it seems not to be applicable to RF discharges.
Here, we describe the development of a more sensitive, purely optical method using krypton as a probe gas. The aim is to study the dynamics in the sheaths of radiofrequency discharges. Therefore, electric field distributions have to be measured with high spatial and temporal resolution. Fluorescence dip spectroscopy (FDS) fulfils these requirements when a pulsed laser system is used and the fluorescence is imaged onto a CCD camera. Due to the complex atomic structure, the basic data of the Stark effect have to be obtained at known electric field strengths before it can be applied as a technique for electric field measurement. In addition to the reference measurements under defined conditions, the Stark effect is calculated numerically. These calculations require, inevitably, certain assumptions and simplifications. However, combining experiment and simulation allows a verification and provides a deeper insight in the underlying relevant physical mechanisms.
In the next section, the theory of the Stark effect and the algorithm of the numerical calculation are described. Then, the spectroscopic scheme and the laser spectroscopic technique are discussed. Section 4 gives some information about the experimental set-up of the laser system, the calibration cell and the modified Gaseous Electronics Conference (GEC) reference cell. The main part of the study is presented in section 5, where detailed investigations of the Stark effect in krypton are compared to the theoretical results. In section 6, the applicability of the novel scheme in the sheath of a capacitively coupled RF plasma discharge (CCP) plasma is demonstrated. Finally, we draw conclusions and give a short outlook.

Numerical calculation of the Stark effect
The theoretical calculation of the Stark effect in krypton, that is described in the following section, is based on the calculation for argon atoms developed by Gavrilenko et al [11]. For the exact computation of the Stark effect, Schrödinger's equation for the perturbed system with the Hamiltonian H = H 0 + H S has to be solved, where H 0 is the Hamiltonian of the unperturbed system and H S = er · E = eEz is the perturbation by the electric field. Furthermore, the solution of the unperturbed system is known: For the solution of the perturbed system, we use an expression in which we express it as a linear combination of the eigenstates of the unperturbed system: This leads us to the following eigenvalue equation, which can also be expressed in terms of a matrix equation: where H S mn = ϕ m |H S |ϕ n is the matrix element of the unperturbed states with the pertubation operator. From the diagonalization of the matrix in (4), one obtains the new eigenvalues ε n , describing the energy level shifts, and the coefficients a in , quantifying the mixture of the states. For the calculation of the matrix elements H S mn , we have to consider the jl-coupling scheme in heavy rare gases [17]. When an electron from the closed outer shell is excited, the spin of the remaining, unpaired electron can have two possible orientations relative to its orbital angular momentum: j c = l 0 + s. This strongest interaction leads to two separated ionization limits (In krypton, the difference is: ν = 5370 cm −1 ). The second strongest interaction is the spin-independent part of the Coulomb interaction between the excited electron and the core electrons. The angular momentum of the core j c couples to the orbital angular momentum of the excited electron: K = j c + l. Finally, there is a weak Coulomb-exchange and spin-orbit interaction: J = K + s, where s is the spin of the excited electron. J can have (2J + 1) possible orientations relative to the quantization axis, expressed by the magnetic quantum number M, where −J M J. Energy levels are denoted by the Racah notation nl [K] J if j c = 1/2 (primed system) and nl[K] J if j c = 3/2 (unprimed system).
Applying the Wigner-Eckart theorem and decoupling the angular momenta using 6j symbols [17], one can derive the matrix element for the component q of the dipole operator D q . The special case which is important here is D q=0 = z. When the atomic states are being identified by the set of quantum numbers |ψ = |nlj c KsJM , one can express the dipole matrix elements as follows: where nl D n l = nl r n l denotes the reduced matrix element, which can be fractionized into a geometrical and a radial factor Although for high l-values, the hydrogen-like formula n, l + 1|r|nl = 3/2a 0 n n 2 − (l + 1) 2 , where a 0 is the Bohr radius, is a good approximation [11,18], the general case can be calculated by a 4-point-Lagrange interpolation [19] of the tabulated values in [20]. For allowed dipole transitions, the matrix element has to have a nonzero value. From this condition, one can obtain the following selection rules for the Stark effect (electric field in z-direction) and for excitation with parallel to the z-axis linearly polarized light, respectively From the Russell-Saunders (LS)-coupled ground state 4p 6 1 S 0 in krypton, the 5p [3/2] 2 intermediate state can be accessed by a two photon excitation, involving the ns [1/2] 1 (n 5) and nd [3/2] 1 (n 4) intermediate states [21]. When the excitation is done by a single, linearly polarized laser beam, the magnetic quantum number M = 0 of the ground state will be maintained [22]. Due to the selection rules only the ns  easily calculated, normalized to the strongest transition (table 1). The ratios of the line strengths depend on the change of the magnetic quantum number, thus on the polarization of the laser beams. The correlation of the polarization with the spectra of nd levels will be discussed later in subsubsection 5.2.4. For the computation of the Stark effect, the exact knowledge of the energies ε 0 n of the unperturbed states is essential. Rydberg states with large principal quantum numbers n can be approximated very well by a quantum defect formula: From the viewpoint of the excited electron that is predominantly far away from the core, the nucleus with charge Ze is screened by (Z − 1) electrons, resulting in an effective charge q ≈ e 'felt' by the electron. Thus the Rydberg formula can be used [23] Here, n * denotes the effective quantum number and δ lKJ the quantum defect, which depends primarily on the orbital angular momentum quantum number l, since for small l-values the shape of the electron's orbit becomes more elliptical, with a higher residence probability near the nucleus. On the other hand, quantum defects for l 3 can be neglected for our purposes. Values for quantum defects could be experimentally measured for both ns [1/2] 1 and nd [5/2] 3 Rydberg series, others were taken or derived from the literature. The values used in the calculation are listed in table 2.
Of course, the calculation can only include a finite number of states N, corresponding to a (N × N)-matrix in (4). The interaction of two different states decreases with their energetic distance; therefore, we can consider a limited number of energetic proximate states, with a similar effective principal quantum number n * . In this study, the nd and np levels are of prime interest due to their visibility in the zero and nonzero field case, respectively. Thus the set of states listed in table 3 has been used for the calculation.
When the matrix for each electric field strength has been constructed from the energy values given by (13), and the matrix elements (5), the Jacobi method [25] has been applied for the diagonalization. The results of the calculations are shown and discussed in connection with the experimental results in section 5.

Spectroscopic scheme
For the spectroscopic investigation of the Stark splitting of Rydberg states of atoms which are usually in the ground state, one has to master two basic challenges: the first is to overcome 7 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT the huge energy difference between the ground state and the Rydberg states; the second is the detection of the transition, because fluorescence photons emitted from the Rydberg states are very hard to detect due to their long lifetime and probability for autoionization in the case of j c = 1/2 (primed system). Therefore, a technique called FDS is applied that has been developed initially for measurements in hydrogen [9]. It is based on a two-photon (UV) excitation into a shortliving intermediate state from which the strong fluorescence in the visible wavelength range can easily be detected. High Rydberg states can be excited from this intermediate state by a onephoton excitation in the visible to NIR wavelength range. When the second excitation occurs instantaneously with the two-photon step, the population density of the intermediate level is being depleted before it can decay radiatively. Thus the fluorescence signal is lowered. Spectra are taken at a fixed UV wavelength by scanning the fundamental wavelength λ 2 of the second laser, while the fluorescence from the intermediate state is observed. When the excitation is in resonance with a Rydberg state, one observes a dip in the fluorescence intensity. The quantity recorded in the 8 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT spectra is the relative decrease of fluorescence: δ F ≡ (I 0 F − I F )/I 0 F . If the saturation phenomena in the second step is negligible, δ F is proportional to the excitation cross-section σ(ω 2 ) and the energy W 2 of the second laser, but independent of the population density of the intermediate level The cross-section is related to the matrix element: The excitation scheme in krypton is shown in figure 1. It is based on the two-photon absorption laser-induced fluorescence spectroscopy in krypton which has initially been developed for sensitive measurements of absolute neutral densities [21]. The two-photon excitation is carried out from the 4p 6  For the FDS, one excites to the Rydberg states of the primed system by a second laser pulse. The sensitivity for electric fields increases with the principal quantum number n. From [16], one can estimate a sensitivity of approximately 1000 V cm for the 15d levels. Thus lower states where n < 15 are not well suited for our purpose. Therefore an interval of 491.5 nm λ exc 506.2 nm for the second step excitation can be specified.

Experimental set-up
There are two set-ups connected with the dye laser system. The first one is a calibration cell for the investigation of the Stark effect at known electric fields and the second one is for space and time resolved electric field measurements in a modified GEC reference cell [26].

Laser system
The experimental set-up is shown in figure 2. The frequency-doubled output beam (532 nm) of a seeded Nd :YAG laser (Continuum, 9020), having a pulse width of 7 ns and pulse energy of 900 mJ at a repetition rate of 20 Hz, is split by a ratio of 2 : 1 to pump two tunable doublegrating dye lasers (Radiant Dyes, Narrow Scan). The first one emits a fundamental wavelength of 612.4 nm, which is frequency-tripled for the two-photon excitation (204.13 nm, 3 mJ). The oscillator of this laser has been modified compared to the original design for improving the spectral purity and stability. A λ/2 plate is inserted between the dye cell and the prism beam expander and an output mirror of reflectivity R = 30% instead of 10% is used.
The fundamental of the second dye laser (615-645 nm) is focused into a tube filled with 10 bar H 2 gas, converting it into the desired wavelength range of 490-509 nm by stimulated anti-Stokes Raman scattering (SARS) at approximately 150 µJ. A filter and dielectric mirrors block both the fundamental and Stokes component. For more sensitive measurements of changes in the fluorescence signal, an electro-mechanical shutter enables to switch the second laser beam on and off, so that drifts of the intensity (on the timescale of seconds) can be compensated. Both beams can be guided collinearly into either the calibration cell or the GEC reference cell. The laser system was calibrated by optogalvanic spectroscopy of neon lines and the absolute reproducibility of the frequency is around 0.3 cm −1 .  Figure 2. Experimental set-up of the laser system with calibration cell. A pulsed, frequency doubled Nd :YAG laser pumps two dye lasers. The first one is frequency tripled for the two photon excitation and the output of the second laser is Raman shifted for the excitation of Rydberg states. The calibration cell contains a two parallel-plate electrodes structure that ensures spatially homogeneous electric fields.

Calibration cell
The calibration cell is a vacuum chamber filled with krypton, where measurements with known electric field strength are performed. It contains a two parallel-plate electrodes structure (d = 9.8 mm) providing a homogeneous DC electric field up to 3 kVcm −1 without electrical breakdown. The laser beams pass the cell in the middle between the electrodes through quartz windows on opposite sides. The fluorescence light can leave the chamber perpendicular to the beams through a standard BK7-window and is detected by a red-sensitive photomultiplier (Hamamatsu, R943-02). An interference filter blocks scattered light from the visible laser beam.

Set-up with GEC reference cell
For phase and space resolved measurements in the sheath of a CCP, the calibration cell was replaced by a modified GEC reference cell (a standard hybride capacitively/inductively coupled plasma source introduced in 1995 [26]). The modification was a replacement of the metal cylinder surrounding the antenna (for inductive coupling) and the dielectric window by a monolithic quartz housing ( figure 3). Here, only the lower electrode is driven and the whole chamber acts as grounded electrode. Therefore, the CCP is highly asymmetric and the entire voltage drops across the sheath at the powered electrode. The voltage is measured by a high voltage probe.
The Q-switch of the Nd :YAG pump laser has been synchronized with the RF generator (f = 13.56 MHz) via frequency divider (f out = 20 Hz) and a delay generator (Stanford, DG-535). The variable delay allowed us to measure at different phases within the RF cycle. The gate of an image intensified CCD camera (Princeton Instruments) was synchronized with the laser pulse passing the GEC cell through quartz windows. The unfocused UV laser beam covers approximately 2 mm of the sheath height, and therefore excites only a fraction of 10 −4 of the atoms, thus saturation of the two photon excitation is negligible. The imaging system provides a spatial resolution of about 50 µm. The fluorescence light was accumulated on the CCD for

Rydberg series without electric field
The strong nd [5/2] 3 autoionizing Rydberg series in the zero field case has been measured for principal quantum numbers 15 n 50, and the weaker ns [1/2] 1 lines have been observed for 21 n 33. The spectrum is shown in figure 4(a). The fluorescence decrease is around 60% at n = 15, thus the transitions at lower n are almost saturated. Additional, relative broad lines have been detected at n ≈ 18.4, 23.7, 35.0. They are probably transitions to lower levels which are excited by the small fraction of the fundamental laser beam that still passes the dielectric mirrors and the filter. The energy levels obtained from the spetrum are listed in table 4. They have been fitted by the quantum defect formula ( figure 4(b)); thus the ionization energy (table 5) and quantum defects (table 6) could be determined. The width of the resonance lines shows nearly the expected a/(n * ) 3 dependence [16] ( figure  5(a)). The exponent has been determined to be −2.6 ± 0.1.

Dependence of line strengths on polarization.
The dependence of the line strengths in the zero field case on the relative polarization of the two laser beams can be qualitatively seen in figure 5

Stark effect
In the following sections, spectra of Rydberg states have been investigated at different electric field strengths. For all conditions we examined, the UV laser beam was polarized perpendicular and the VIS laser beam was polarized parallel to the electric field. At the left edge, one can only recognize the strong nd and the much weaker ns levels. With increasing field strength, they are slightly shifted to the longer wavelengths and disappear. Especially the nd levels disappear very rapidly at around 500 V cm −1 . On the other hand, the 'forbidden'np levels become visible at around 200 V cm −1 , shifting with increasing field strength very strongly and getting brighter at the same time. A manifold of the higher, quasi-degenerated states with l 3 appears above 500 V cm −1 between the nd and (n + 2)p states.   [25] 118284.6 ± 0.2 Moore (1970) [27] 118285.5 Moore (1952) [28] 118284.7  [23] 3.094 -

np levels, quadratic Stark effect.
The np states become visible when an electric field is applied, because they get admixtures of the ns and nd states due to the Stark effect.
In each spectrum, the position of maximum intensity has been measured. As already shown in the numerical simulation, the levels avoid a levelcrossing with the Stark manifold, but the intensity maximum keeps running through it. The shift can be described very well by a simple quadratic function ε np (E) = ε 0 np − 1/2α np E 2 , as shown in figure 7. Similarly to  the other unperturbed states, the extrapolated zero-field energy ε (0) np can be expressed by the quantum defect formula (13), where δ np = 2.616 ± 0.001. It turned out that the polarizability α np is proportional to (n * ) 6.86±0.06 , also very close to the hydrogen-like case where α ∝ n 7 [29] (figure 8). Comparison with the numerical calculation is shown in figure 9(a). In order to compare energy shifts, the extrapolated zero-field energies from the measurement have been identified with the calculated ones. Therefore, the measured energies had to be shifted upwards by 0.3 cm −1 . This systematic error is within the range of the reproducibility of the laser frequency (  in figure 9(b). Although the crossing of the 21p level with the Stark manifold is avoided, the line seems to cross due to the change in intensity. Because the energy gap is smaller than the line width, it is not possible to resolve the anticrossing and therefore the quadratic line shift is maintained. Considering the electric field interval where each np level could be clearly detected, a dependence of E max ∝ (n * ) −4.3±0.3 and E min ∝ (n * ) −4.5±0.6 has been observed for the upper and the lower limit, respectively (figure 12).
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Figure 10. Spectra of 20d at perpendicular polarization and different electric field strengths.

nd levels.
The most intensive lines at zero electric field are the nd [5/2] 3 levels. The spectra of 20d at perpendicular polarization and different electric field strengths are shown in figure 10. One can clearly observe the highly nonlinear wavelength shift and the decrease in intensity at the same time. The measured energy shift is shown in comparison with the numerical calculation for four different levels in figure 11. As in the case of the np levels (subsubsection 5.2.3.), the measured energies were shifted upwards by 0.3 cm −1 in order to correct the systematic error. However, in the cases of 16d , 18d and 20d there is a good agreement with the theory, but deviations can be identified for 26d . A close inspection showed that the calculations of the shifts of the nd [5/2] 3 levels are very sensitive to variations of the quantum defects, probably due to the nearby Stark manifolds that repel the nd states. In contrast to the np levels, their shift cannot be described by a simple, quadratic function. It seems that below a 'threshold' electric field strength the shift is hardly measurable. Beyond that, the shift increases rapidly and the line strength decreases at the same time. When about twice the threshold field was reached, the nd levels could not be detected any more. The dependence of the upper electric field limit on the effective principle quantum number n * can be expressed by a power function E max ∝ (n * ) −4.7±0.2 , similarly to the upper and lower limits of the np levels ( figure 12). Within the statistical uncertainties, one can probably assume a E min/max ∝ (n * ) −4.5 dependence for the electric field measurement ranges. Despite their highly nonlinear behaviour, the nd [5/2] 3 levels are useful for electric field measurements due to their line strength and visibility in the zero field case, which allows a simple measurement of the Stark shift.

Higher nl levels, linear Stark effect.
The quantum defects of the nl levels with l 3 can be neglected, therefore they are practically degenerated. In the presence of an external electric field, they exhibit a linear splitting into n − 3 equidistant lines. The fine structure components of each angular momentum state could not be resolved further due to the line width. Examples of measured spectra at n = 19 are shown in figure 13. The line comb appears at field strengths beyond approximately 500 V cm −1 , when the nearby 20d [5/2] 3 level (n * ≈ 18.7) disappears. This observation agrees also with numerical calculations of the line strengths.
The average line separation has been determined for different field strengths by fitting Lorentz profiles to all observed lines. Energy values which have been calculated from the centre wavelengths were numbered and plotted versus this number. The average line separation σ is given by the slope of the linear regression. In figure 14(a), the linear splitting for four different values of n is compared with the theory. In first approximation, the splitting is proportional to the electric field σ = p n E, where p n depends almost linearly on the principal quantum number n ( figure 14(b)). This is comparable to the linear Stark effect in hydrogen, where the separation between two adjacent lines is proportional to n [18]. A weaker dependence seems to exist for values n 15: Delsart and Keller observed a n 0.7 dependence for 8 n 15 [16]. Considering electric field measurements, the linear Stark effect promises to be the easiest and most accurate sensor, when measuring relative high electric fields and having a good signal to noise ratio of the spectra. As only energy differences have to be measured, an absolute calibration of the laser system is not necessary.

Electric field measurements in a CCP
Our motivation for the investigation of the Stark effect in krypton was to provide the basic data for a diagnostic of electric field distributions in the sheath regions of various RF discharges. A capacitively coupled pure Kr discharge in a so-called GEC reference cell was used in order to demonstrate the applicability of the technique under realistic conditions. First space and phase resolved electric field measurements were performed in a pure Kr discharge at a pressure of 10 Pa and a power of 8 W. The shift of several nd [5/2] 3 levels (n = 15 . . . 26) were measured and compared with the experimental database, if available, otherwise with the numerical calculation. Figure 15(a) shows the experimental results of the electric field measurement diagnostic. Sheath voltages have been determined by integration of the electric field distributions. For these purposes, the measured fields have been extrapolated linearly towards the powered electrode, because an approximately constant ion density and a negligible electron density can be assumed in this region.
In figure 15(b), the results of the integration are compared with values taken with a high voltage probe. The agreement is very good and shows that there are no substantial systematic errors in the measurement method. Numerical sheath models that can explain the specific field distributions are still the subject of current research.

Conclusions and outlook
A laser spectroscopic method for sensitive electric field measurements in radiofrequency discharges using krypton has been developed. The autoionizing Rydberg series has been measured and the Stark effect for different angular momentum states has been investigated in detail. The measurements have been compared with numerical calculations. The applicability of the method to problems in plasma physics has been demonstrated in a RF discharge.
At zero electric field, the ns [1/2] 1 (n = 21 . . . 33) and the nd [5/2] 3 (n = 15 . . . 50) levels have been identified and the energies have been measured, most of them for the first time. They could be described very well by a quantum defect formula. The quantum defects and the ionization energy have been determined. The (n * ) −3 dependence of the line width has been verified.
When an electric field was applied, a quadratic Stark effect could be observed for the np levels (n = 17 . . . 32). The most intensive nd (n = 16 . . . 26) states show a much more nonlinear behaviour, and the higher angular momentum states (l 3, n = 15 . . . 27) exhibit a linear Stark effect. The polarizability and the dipole moment of high, autoionizing Rydberg states in krypton is very hydrogen-like, as reported already in [16]. A sensitivity of approximately 25 V cm −1 has been reached.
There is very good agreement of the experimental results and the numerical calculations. In combination with the experimental results, the model has been improved, so that even states which have not been calibrated experimentally could be used for electric field measurement.
The applicability of the method to plasma physics has been successfully demonstrated in the sheath region of a capacitively coupled RF discharge, and high spatial and temporal resolution has been obtained. Sheath voltages which have been obtained by integrating the measured electric field distributions compare very well to measurements done by a high voltage probe. This also proves indirectly that the calibration has been performed successfully.
When the sensitivity of the detection system at the fluorescence wavelength is sufficient, krypton can be added in relatively small amounts below 0.1 Pa to various discharges and serves as a universal probe gas for electric field measurement. Heating processes, wave phenomena as well as ion dynamics can be studied.