Ion traps with enhanced optical and physical access
Robert Maiwald, Dietrich Leibfried, Joe Britton, J. C. Bergquist, Gerd Leuchs, D. J. Wineland
IIon traps with enhanced optical and physical access
Robert Maiwald ∗ and Gerd Leuchs Institute of Optics, Information and Photonics, University Erlangen-Nuremberg, 91058 Erlangen, Germany
Dietrich Leibfried, † Joe Britton, J. C. Bergquist, and D. J. Wineland
Time and Frequency Division, National Institute of Standards and Technology, Boulder, CO 80305, USA(Dated: 19th March 2009)Small, controllable, highly accessible quantum systems can serve as probes at the single quantumlevel to study multiple physical e(cid:27)ects, for example in quantum optics or for electric and mag-netic (cid:28)eld sensing. The applicability of trapped atomic ions as probes is highly dependent on themeasurement situation at hand and thus calls for specialized traps. Previous approaches for iontraps with enhanced optical access included traps consisting of a single ring electrode [1, 2] or twoopposing endcap electrodes [2, 3]. Other possibilities are planar trap geometries, which have beeninvestigated for Penning traps [4, 5] and rf-trap arrays [6, 7, 8]. By not having the electrodes liein a common plane the optical access in the latter cases can be substantially increased. Here, wediscuss the fabrication and experimental characterization of a novel radio-frequency (rf) ion trapgeometry. It has a relatively simple structure and provides largely unrestricted optical and physicalaccess to the ion, of up to 96% of the total 4 π solid angle in one of the three traps tested. We alsodiscuss potential applications in quantum optics and (cid:28)eld sensing. As a force sensor, we estimatesensitivity to forces smaller than 1 yN Hz − / . PACS numbers: 37.10.Ty, 42.50.Ct
I. BASIC GEOMETRY
The basic electrode geometry is shown in Fig. 1 and isformed by two concentric cylinders over a ground plane.The design provides straightforward indexing and assem-bly of the trap electrodes, with large solid angle accessto the ion. Four additional electrodes were placed on acircle between the grounded plane and the rf electrode tobreak the rotational symmetry of the rf pseudopotentialabout the vertical axis and to compensate for stray elec-tric (cid:28)elds in order to minimize ion rf-micromotion in thetrap [9].Three di(cid:27)erent traps were built adjacent to each otheron the same test set-up. These traps range from a conser-vative design with larger trap depth, higher motional fre-quencies and a smaller accessible solid angle, to a weakertrap with greater optical access. This change in prop-erties is achieved by varying the protrusion height ∆ h of the central grounded electrode with respect to the rfelectrode (Table I).The degeneracy of motional frequencies in the radialdirection was lifted by applying potentials on the orderof 0.1 to 1 V to the compensation electrodes A − D . Thiscreated a static quadrupole (cid:28)eld de(cid:28)ning the principalradial axes of the trap along the lines connecting com-pensation electrode A with D and B with C. Thus theaxes were oriented at angles of about 45 ◦ relative to thetwo cooling beams (Fig. 1b). In addition, the entiretrap assembly was tilted by about 7.5 ◦ with respect to ∗ Electronic address: [email protected] † Electronic address: [email protected] the direction de(cid:28)ned by the laser beams, ensuring thatthe vertical axis of the traps was not orthogonal to thewavevectors of the cooling beams. In this way all threenormal modes of the ion were su(cid:30)ciently Doppler cooledby a single laser beam.
II. TRAP CONSTRUCTION
The trap electrodes are made of stainless steel rodsand hypodermic tubing. The dimensions indicated inFig. 1 are as follows: OD _ rf = 710 µ m , ID _ rf =535 µ m , OD _ cGND = 205 µ m , ID _ cGND = 100 µ m , OD _ COMP = 150 µ m , h _ rf = 1110 µ m . Structuralintegrity and insulation is provided by a combination ofalumina and Macor spacers. Electrical connections weremade by resistive welding of gold ribbons to the trap elec-trodes and to gold traces (thickness 5-10 µ m) that weresilk-screened onto alumina. The outer ground plane wasalso formed by this silk-screening process. All insulatingsurfaces have been recessed or positioned to prevent adirect line of sight to the ion position. This suppressesdistortions of the trapping (cid:28)eld caused by charged insu-lating surfaces. The potentials applied to the individualDC electrodes (central ground and compensation elec-trodes A − D ) are passively (cid:28)ltered by RC low-pass (cid:28)l-ters with R = 1 kΩ and C = 2 nF . The components aremounted on a connection board inside the vacuum closeto the trap.The assembled trap package is attached to the connec-tion board (Fig. 2) which is in turn is mounted inside acopper tube that forms part of a rf resonator that suppliesthe rf potential. The traps, Mg ovens, and copper tubereside inside a quartz envelope with extrusions and (cid:29)at a r X i v : . [ qu a n t - ph ] M a r radial coordinate [mm] a x i a l c oo r d i na t e [ mm ] GNDcGNDrfCOMP ADB C (b) top view: (c) h Δ h Ω schematic notdrawn to scale GND gold planegold leadto rf inputto COMP voltagegold lead to cGNDcontrolpotential Mgvapor alumina insulatoralumina insulatorMacor insulator gold lead
ID_cGND OD_cGNDID_rfOD_rf OD_COMPh_rf c oo li ngbea m s i on i z a t i onbea m (a) Figure 1: (a) Simulations of the trap rf pseudopotential in-dicate smaller trap potential depths and binding frequenciescompared to those of more traditional trap designs of similarsize, rf-drive frequency and amplitude. Shown is an examplecalculation of the pseudo-potential (in eV) for electrode con-(cid:28)guration Mg + properties, while neglecting the compen-sation electrodes. Isoline separation is 25 meV and the axialcoordinate is measured from the grounded plane. (b) Place-ment of the electrodes. The central ground electrode (cGND)is surrounded by the rf electrode (rf) and the grounded plane(GND). These electrodes provide the primary trapping po-tential. In addition, four symmetrically placed compensationelectrodes (COMP) provide (cid:28)ne adjustments to the overallpotential. The distance h between the ion and the centerelectrode varies with ∆ h , the height di(cid:27)erence between thecenter electrode and the rf electrode. The accessible solid an-gle Ω is also illustrated. The inset shows the position of thetrap electrodes with respect to the laser beams for coolingand ionization and the direction of the principal radial axesof the trap. The direction from which the neutral Mg vaporenters the trapping region is indicated. (c) The layered as-sembly showing insulating planes with conducting gold leadsand laser-cut holes to house the electrodes. windows for laser beams and for imaging of the trappedions. The vacuum system is completed by a 20 liter/s iongetter pump, a Ti-sublimation pump and an ionizationpressure gauge. The system was pumped to 5 × − Paand baked-out at 210 ◦ C for 8 days. After cooling thesystem to room temperature and several applications ofTi-sublimation, a base pressure of about 3 × − Pa (atthe gauge) was achieved.Helical resonators were driven by a low-noise signalgenerator to generate the rf trap potentials. They hadloaded Q -factors ranging from 300 to 430 and were drivenwith input powers in the range of 22 to 35 dBm. This re-sulted in rf amplitudes (inferred from simulations and theresulting trap frequencies) in the range of 290 to 460 V. III. EXPERIMENTAL SETUP AND TRAPCHARACTERIZATION A Mg oven produced a vapor of neutral atoms thatwas directed at the traps in the horizontal direction ofFig. 1b. To load ions into the traps, the neutral atomsinside the con(cid:28)nement region were then photoionized.All experiments were performed at a magnetic (cid:28)eld of B (cid:39) m m
10 µm10 µm10 µm
Figure 2: The three test traps with di(cid:27)ering rf electrodeprotrusion height ∆ h (increasing from front to back) of thecentral electrode can be seen. Visible in the upper right-handside of the (cid:28)gure is also the ceramic board with copper traces,that accommodates surface mounted components constitutinglow-pass (cid:28)lters for the DC electrodes. Two parallel gold wiresconnecting the rf-electrode can be seen in the lower left-handside of the (cid:28)gure. The insets show di(cid:27)erent experimentallyobserved ion con(cid:28)gurations (from top to bottom): a singletrapped ion, two ions revealing the orientation of the weakesttrap axis, and an ion crystal in an approximately cylindricalpotential that allows the outer ions to rotate about the trapvertical axis. ( λ = 285 nm, 2 to 4 mW, 50 µ m waist) was resonant withthe s S ↔ s p P transition in neutral Mg [10]. Asecond photon either at 285 nm or at 280 nm (below)promotes the electron from the s p P state to thecontinuum, producing a Mg + ion in the con(cid:28)nementregion. For cooling, a second laser beam ( λ = 280 nm, 1mW, 30 µ m waist) was tuned about 400 MHz below the s S / ↔ s p P / transition of Mg + for initialcooling. A third, much weaker beam ( λ = 280 nm, 2-10 µ W), with intensity below saturation and tuned belowthe s S / ↔ s p P / transition by half of the nat-ural linewidth ( (cid:39)
20 MHz), subsequently cooled the ionsto near the Doppler limit. At that point photon scat-tering and cooling was dominated by the closely detunedbeam. In the case of sudden ion heating, for example dueto a collision with background gas, the 400 MHz detunedbeam could e(cid:30)ciently recool the ion to a point where thenear-resonant beam would take over.The ion was detected by collecting (cid:29)uorescence with ahigh numerical aperture (NA (cid:39) × − Pa reached in our system, the mean ion lifetimewith laser cooling was around 30 s in this particular case.In all three traps, we observed ion lifetimes in excessof three hours under continuous laser cooling. Withoutcooling light, lifetimes were greater than 10 s. The sec-ular frequencies of ion motion along all three axes weredetermined by applying additional sinusoidal drive po-tentials to the compensation electrodes or the centralelectrode. If the drive frequency is resonant with a secu-lar frequency, ion motion is excited to large amplitudes,causing a sharp drop of laser induced (cid:29)uorescence.Operating parameters for each of the three traps aresummarized in Table I together with observed trap fre-quencies, rf-drive voltage and trap depth as inferred frommeasured trap frequencies and numerical simulations foreach trap. Accessible solid angle ignores the compen-sation electrodes and outer ground plane. This seemsreasonable because in future implementations, these elec- trodes could be made smaller and/or recessed below therf electrode. This has not been done so far to facilitateconstruction of the basic electrode structure.To compensate micromotion due to electric stray (cid:28)elds,we varied the potential on all four compensation elec-trodes and the central tube while changing the powerlevel of the rf drive. Compensation was achieved whenthe ion position remained stationary for di(cid:27)erent rf driveamplitudes [9]. In addition, we could minimize micromo-tion along the direction of the cooling beam by maximiz-ing the scattering signal close to resonance. By loweringthe rf drive after loading we determined a minimum (cid:16)sta-ble(cid:17) trapping rf power level which seemed to be limitedonly by background gas collisions. A trap loaded in con-(cid:28)guration
IV. LARGE SOLID ANGLE PHOTONABSORPTION AND EMISSION
The compact design permits placing the trap inside ametallic parabolic mirror, which can also serve as an rfground electrode (Fig. 3a). By moving the trap struc-ture, an ion can be placed at the focal point of the mir-
Table I: These typical operating parameters were derivedfrom measurements on single trapped Mg + ions. The maindi(cid:27)erence between the three traps is the protrusion length ∆ h of the center electrode beyond the rf electrode (Fig. 1b).Due to this variation, di(cid:27)erent trap frequencies and accessiblesolid angles are obtained. Some parameters (denoted by (cid:78) )are di(cid:30)cult to measure directly and were inferred from themeasured trap frequencies and the numerical simulation ofthe trapping potentials. Radial AD ( BC ) indicates a normalmode direction along the line connecting compensation elec-trodes A and D ( B and C ), see also Fig. 1b. The observeddistances h were 10-20 % smaller than predicted by simula-tions. This seems reasonable since the simulations neglectedthe rf-grounded compensation electrodes, which if included,would reduce the predicted values of h .electrode con(cid:28)guration trap ∆ h µ mrf drive voltage U (cid:78)
290 460 400 Vrf drive frequency Ω rf / (2 π ) AD BC (cid:78)
71 178 195 meVobserved distance h
168 244 290 µ maccessible solid angle Ω / (4 π )
71% 91% 96% ror for e(cid:30)cient (cid:29)uorescence collection. Moreover, parallellight beams directed along the mirror optical axis are fo-cused directly onto the ion. By appropriately shapingthe transverse mode pattern of the incident light [11, 12]the (cid:28)eld at the focus can produce a linear dipole ex-citation pattern aligned with the axis of symmetry ofthe parabolic mirror, which could lead to very e(cid:30)cientphoton-ion coupling [13]. With electrode con(cid:28)guration π . For a linear electric dipole aligned alongthe mirror axis, this geometry would lead to a collectione(cid:30)ciency of 94 %. Conversely, light sent onto the ionin a dipolar pattern would provide a near perfect atom-to-photon coupling to a linear dipole transition, possiblyworking down to the single-photon level [14, 15]. To thisend one could use ions with even-numbered charge whichallow for J=0 to J=1 transitions [13].This scheme might also provide a signi(cid:28)cant improve-ment in the e(cid:30)ciency of remote entanglement of trappedions as described in Refs. [16, 17, 18]. As an exam-ple, the entanglement scheme used in [18] is based on(cid:28)ltering the emission of the atom on F’=0,1 to F=0,1transitions to only overlap photons on a beamsplitter re-sulting from π -transitions. Limitations of the experimentin [18] were caused by restrictions in collection solid angle(0.02 % of 4 π ) as well as coupling of the emission pat-tern that was imaged through a multi-element lens intoa single mode (cid:28)ber (e(cid:30)ciency 20 %). A parabolic collec-tion mirror could potentially improve the situation be-cause ideally it transforms the orthogonal mode patternsof photons emitted on π or σ -transitions while preservingtheir orthogonality [19]. Orthogonality is also preserved (a) sample holder on translation stagedifferently treated surface samples1 2 3 (b) f z Figure 3: (a) Placement of the ion in the focus f of aparabolic mirror with depth z to maximize photon-ion cou-pling. (b) Scanning of di(cid:27)erent surfaces with the ion as asensitive probe. in good approximation by the remaining elements of themode shaping envisioned in [11, 12]. Therefore it mightbe possible to not only couple photons from π -transitionswith near unit e(cid:30)ciency to a single mode (cid:28)ber, but themode converter will also act as a (cid:28)lter blocking out un-desired photons from σ -transitions. Ideally one wouldexpect to boost the production rate of entangled pairsby more than × over the values reported in [18].E(cid:30)cient coupling could also be obtained through a res-onant interaction of an ion with a cavity [20, 21]. Inthis method, to achieve a coupling e(cid:30)ciency of 90 %, aminimum cooperativity of 4.5 is required. Currently, thisis di(cid:30)cult to achieve for many ions that have ultraviolettransition wavelengths and additional complications maybe encountered with mirror charging [20, 21]. V. ION TRAP SURFACE SENSOR
The open geometry of such traps also suggests applica-tions as a probe of (cid:28)elds near a surface. Surfaces of inter-est could be brought close to the ion, as in Fig. 3b. By ei-ther scanning the ion trap over the surface or translatingthe surface itself, one could map out the electromagneticor force (cid:28)elds in proximity of the surface. Modeling theion trap in the presence of a ground plane located hor-izontally above the ion in Fig. 1 indicates that a stablequadrupole minimum is retained until the distance to thesurface is approximately equal to the distance of the ionto the center electrode. In the traps described here, thiswould limit the ion to distances of about 170 µ m fromthe surface, a limit that could be reduced by miniatur-izing the trap. The ion serves as a stylus probe tip thatis extremely sensitive to forces oscillating at its motionalfrequencies. These frequencies can be tuned over at leasttwo decades from approximately ω/ (2 π ) = 100 kHz to 10MHz. In addition, information on the force (cid:28)eld direc-tion can be extracted by utilizing all three nondegeneratemodes of motion of the ion. With the same apparatus,static and time-dependent magnetic (cid:28)elds can be mea-sured by observing (cid:28)rst and second order shifts of theion on narrow internal transitions.To estimate the sensitivity to oscillating force (cid:28)eldswe assume that the ion, initially cooled to its motionalground state, is driven in resonance with a motional modefor a duration t = 1 / ∆ b , where ∆ b is the approximatemeasurement bandwidth. The amplitude α of the coher-ent state grows as [22] α = F z (cid:126) t, (1)where F is the amplitude of the driving force, z = (cid:112) (cid:126) / (2 mω ) is the size of the ion’s harmonic oscillatorground state wave function, m is the ion’s mass, and (cid:126) is Planck’s constant divided by 2 π . To yield a detectablesignal, this coherent excitation has to be comparable tothe excitation due to motional heating that grows ac-cording to (cid:104) n n (cid:105) = (cid:104) ˙ n (cid:105) t where (cid:104) ˙ n (cid:105) is the ion’s heating rate[23, 24]. Heating rates in the range of 0.2 quanta/msto 2 quanta/ms have been observed in traps with sim-ilar electrode-to-ion distances for a motional frequency ω/ (2 π ) (cid:39) MHz [10, 25]. Since the geometry discussedhere minimizes the amount of material close to the iona heating rate of 1 quantum per millisecond should bea realistic estimate. (For small cryogenic traps, heatingrates on the order of 1 quantum per second have beenobserved [24].) Assuming a signal-to-noise ratio of one,we require (cid:104) n n (cid:105) = (cid:104) n c (cid:105) = | α | and therefore F √ ∆ b (cid:39) (cid:112) (cid:104) ˙ n (cid:105) (cid:126) z . (2)For Mg + , (cid:104) ˙ n (cid:105) = 1/ms and ω/ (2 π ) = 1 MHz, thisimplies a force sensitivity of 0.46 yN/ √ Hz (1 yN = 1yocto-Newton = − N), several orders of magnitudebelow the smallest forces detectable with atomic forcemicroscopes or micro-mechanical cantilevers [26]. Thisforce sensitivity corresponds to an electric (cid:28)eld sensitiv-ity of 2.9 ( µ V/m)/ √ Hz . For a cryogenic ion trap where (cid:104) ˙ n (cid:105) = 1/s the sensitivity would be increased by approxi-mately a factor of 30.To detect the magnetic (cid:28)eld at the ion’s position wemay excite a narrow transition (for example a hyper-(cid:28)ne transition) with well known (cid:28)eld dependence. Themethod would be limited by quantum projection noise[27]. On a Zeeman-shifted transition with a magneticmoment di(cid:27)erence of 1 Bohr magneton ( ∆ ν/ ∆ B = 14MHz/mT) when probed by the Ramsey method with freeprecession time T R , the magnetic (cid:28)eld resolution ∆ B is ∆ B ( τ ) = 12 π (∆ ν/ ∆ B ) √ T R τ = 1 . × − T / (cid:112) τ / s , (3)where τ is the averaging time and the last expression as-sumes T R = 1 s. By actively compensating the surface(cid:28)eld with external coils that lead to a known (cid:28)eld ge- ometry, we could detect not only the modulus but alsothe direction of the local magnetic (cid:28)eld. While magnetic(cid:28)elds can be sensed closer to the surface utilizing nitrogenvacancy centers [28, 29] and with higher signal-to-noiseratio with a large number of cold neutral atoms [30], apotential advantage of the ion sensor could be the com-bined sensitivity to electric and magnetic (cid:28)elds.In principle the electric (cid:28)eld in all space can be recon-structed from the (cid:28)eld in one plane, but this inversionis an ill-posed problem for spatial features much smallerthan the ion-to-feature distance. Therefore the lateralspatial resolution when scanning the surface will be lim-ited by the attained signal-to-noise ratio and is roughlyequal to the distance to the surface if the signal-to-noiseratio is of order 1.An interesting application of the ion sensor in quantuminformation processing with trapped ions is to use it forstraightforward comparisons of heating rates of di(cid:27)erentsurfaces [3, 23, 24]. Typically, to compare heating ratesfrom di(cid:27)erent electrode surfaces, separate traps com-posed of di(cid:27)erent materials have been built and tested.This leads to uncertainties due to variations in the trapgeometry, the exact steps of materials processing, surfacecontamination due to cleaning agents, and the bake-outprocedure. Most of these variables could be eliminated,and the testing of di(cid:27)erent materials could be acceleratedby use of one ion sensor on a variety of material samplesdeposited on the same carrier surface (Fig. 3b). Acknowledgments
This work was supported by IARPA and the NISTQuantum Information Program. We thank C. Ospelkausand S. Ospelkaus for comments on the manuscript. Thispaper is a contribution by the National Institute of Stan-dards and Technology and not subject to U.S. copyright. [1] N. Yu, W. Nagourney, and H. Dehmelt, J. Appl. Phys.69, 3779 (1991).[2] C. Schrama, E. Peik, W. Smith, and H. Walther, OpticsCommunications 101, 32(cid:21)36 (1993).[3] L. Deslauriers, S. Olmschenk, D. Stick, W. K. Hensinger,J. Sterk, and C. Monroe, Phys. Rev. Lett. 97, 103007(2006).[4] S. Stahl, F. Galve, J. Alonso, S. Djekic, W. Quint,T. Valenzuela, J. Verdu, M. Vogel, and G.Werth, Europ.Phys. J. D 32, 139 (2004).[5] J. R. Castrejoan-Pita, H. Ohadia, D. R. Crick, D. F. A.Winters, D. M. Segal, and R. C. Thompson, J. Mod. Opt.11, 1581 (2007).[6] J. Chiaverini, R. B. Blakestadt, J. Britton, J. Jost,C. langer, D. Leibfried, R. Ozeri, and D. J. Wineland,Quantum Inf. Comput. 5, 419 (2005).[7] S. Seidelin, J. Chiaverini, R. Reichle, J. J. Bollinger,D. Leibfried, J. Britton, J. H. Wesenberg, R. B. Blakestad, R. J. Epstein, D. B. Hume, et al., Phys. Rev.Lett. 96, 253003 (2006).[8] C. E. Pearson, D. R. Leibrandt, W. S. Bakr, W. J. Mal-lard, K. R. Brown, and I. L. Chuang, Phys. Rev. A 73,032307 (2006).[9] D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M.Itano, and D. J. Wineland, J. Appl. Phys. 83, 5025(cid:21)5033(1998).[10] R. J. Epstein, S. Seidelin, D. Leibfried, J. H. Wesenberg,J. J. Bollinger, J. M. Amini, R. B. Blakestad, J. Britton,J. P. Home, W. M. Itano, et al., Phys. Rev. A 76, 033411(2007).[11] S. Quabis, R. Dorn, M. Eberler, O. Gl(cid:246)ckl, andG. Leuchs, Optics Communications 179, 1 (2000).[12] N. Lindlein, R. Maiwald, H. Konermann, M. Sonder-mann, U. Peschel, and G. Leuchs, Laser Physics 17,927(cid:21)934 (2007).[13] M. Sondermann, R. Maiwald, H. Konermann,[1] N. Yu, W. Nagourney, and H. Dehmelt, J. Appl. Phys.69, 3779 (1991).[2] C. Schrama, E. Peik, W. Smith, and H. Walther, OpticsCommunications 101, 32(cid:21)36 (1993).[3] L. Deslauriers, S. Olmschenk, D. Stick, W. K. Hensinger,J. Sterk, and C. Monroe, Phys. Rev. Lett. 97, 103007(2006).[4] S. Stahl, F. Galve, J. Alonso, S. Djekic, W. Quint,T. Valenzuela, J. Verdu, M. Vogel, and G.Werth, Europ.Phys. J. D 32, 139 (2004).[5] J. R. Castrejoan-Pita, H. Ohadia, D. R. Crick, D. F. A.Winters, D. M. Segal, and R. C. Thompson, J. Mod. Opt.11, 1581 (2007).[6] J. Chiaverini, R. B. Blakestadt, J. Britton, J. Jost,C. langer, D. Leibfried, R. Ozeri, and D. J. Wineland,Quantum Inf. Comput. 5, 419 (2005).[7] S. Seidelin, J. Chiaverini, R. Reichle, J. J. Bollinger,D. Leibfried, J. Britton, J. H. Wesenberg, R. B. Blakestad, R. J. Epstein, D. B. Hume, et al., Phys. Rev.Lett. 96, 253003 (2006).[8] C. E. Pearson, D. R. Leibrandt, W. S. Bakr, W. J. Mal-lard, K. R. Brown, and I. L. Chuang, Phys. Rev. A 73,032307 (2006).[9] D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M.Itano, and D. J. Wineland, J. Appl. Phys. 83, 5025(cid:21)5033(1998).[10] R. J. Epstein, S. Seidelin, D. Leibfried, J. H. Wesenberg,J. J. Bollinger, J. M. Amini, R. B. Blakestad, J. Britton,J. P. Home, W. M. Itano, et al., Phys. Rev. A 76, 033411(2007).[11] S. Quabis, R. Dorn, M. Eberler, O. Gl(cid:246)ckl, andG. Leuchs, Optics Communications 179, 1 (2000).[12] N. Lindlein, R. Maiwald, H. Konermann, M. Sonder-mann, U. Peschel, and G. Leuchs, Laser Physics 17,927(cid:21)934 (2007).[13] M. Sondermann, R. Maiwald, H. Konermann,