High resolution 3D gas-jet characterizationBjörn Landgraf, Michael Schnell, Alexander Sävert, Malte C. Kaluza, and Christian Spielmann Citation: Rev. Sci. Instrum. 82, 083106 (2011); doi: 10.1063/1.3624694 View online: http://dx.doi.org/10.1063/1.3624694 View Table of Contents: http://rsi.aip.org/resource/1/RSINAK/v82/i8 Published by the American Institute of Physics. Related ArticlesFast-electron generation in long-scale-length plasmas Phys. Plasmas 19, 012704 (2012) Femtosecond laser-plasma interaction with prepulse-generated liquid metal microjets Phys. Plasmas 19, 013104 (2012) Linearly tapered discharge capillary waveguides as a medium for a laser plasma wakefield accelerator Appl. Phys. Lett. 100, 014106 (2012) Energetics and energy scaling of quasi-monoenergetic protons in laser radiation pressure acceleration Phys. Plasmas 18, 123105 (2011) Modeling of 10 GeV-1 TeV laser-plasma accelerators using Lorentz boosted simulations Phys. Plasmas 18, 123103 (2011) Additional information on Rev. Sci. Instrum.Journal Homepage: http://rsi.aip.org Journal Information: http://rsi.aip.org/about/about_the_journal Top downloads: http://rsi.aip.org/features/most_downloaded Information for Authors: http://rsi.aip.org/authors

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REVIEW OF SCIENTIFIC INSTRUMENTS 82, 083106 (2011)

High resolution 3D gas-jet characterizationBjörn Landgraf,1,2 Michael Schnell,1 Alexander Sävert,1 Malte C. Kaluza,1,2

and Christian Spielmann1,2

1Institute of Optics and Quantum Electronics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1,07743 Jena, Germany2Helmholtz-Institute Jena, Helmholtzweg 4, 07743 Jena, Germany

(Received 1 March 2011; accepted 22 July 2011; published online 17 August 2011)

We present a tomographic characterization of gas jets employed for high-intensity laser-plasma inter-action experiments where the shape can be non-symmetrically. With a Mach-Zehnder interferometerwe measured the phase shift for different directions through the neutral density distribution of the gasjet. From the recorded interferograms it is possible to retrieve 3-dimensional neutral density distri-butions by tomographic reconstruction based on the filtered back projections. We report on criteriafor the smallest number of recorded interferograms as well as a comparison with the widely usedphase retrieval based on an Abel inversion. As an example for the performance of our approach,we present the characterization of nozzles with rectangular openings or gas jets with shock waves.With our setup we obtained a spatial resolution of less than 60 μm for an Argon density as low as2 × 1017 cm−3. © 2011 American Institute of Physics. [doi:10.1063/1.3624694]

I. INTRODUCTION

Over the last few years we have witnessed tremen-dous progress in the field of laser-plasma based electronacceleration.1–3 High-power laser pulses focused into anunderdense plasma excite coherent density oscillations form-ing a plasma wave which propagates with nearly the samevelocity as the driving laser pulse. The accompanied electricfield strength is in the order of several 100 GVm−1 andaccelerates electrons to relativistic energies over a distanceof only a few millimeters. Furthermore, the accomplishmentof x-ray lasers4 and the efficient generation of high harmonicradiation5, 6 rely on the interaction of high-power laser pulseswith underdense plasmas. For further optimization of the pro-cesses it is sometimes desirable to know the electron densityne(�r) in the interaction region. For fully ionized gases (e.g.,hydrogen or helium) the electron density distribution ne(�r )at the beginning of the laser-plasma interaction can be easilyestimated from the measured gas density. For rotationallysymmetric gas targets produced with cylindrically symmetricnozzles the gas density distribution can be deduced from ameasured interferogram and an Abel inversion.7 Applyinga generalized Abel inversion8 or a Gaussian-like densitydistribution9 it is possible to introduce an asymmetry pa-rameter allowing to relax slightly or change the requirementof axial symmetry for obtaining meaningful results. Thesemethods, for example, rely on the representation of themeasured function as Legendre polynomials up to a limitedorder. Such an approach results in a first order asymmetrydescription. However, for completely non-rotationally sym-metric target geometries10 employed to realize steeper gasdensity gradients11 or longer interaction lengths betweenlaser and plasma, it is necessary to apply more advancedmethods to assess the density distribution. Here we presenta new arrangement suitable for characterizing arbitrarilyshaped gas density distributions based on interferometricmeasurements followed by tomographic reconstruction. The

computing time can be significantly reduced by applying thereconstruction filtered back projection (FBP) where efficientalgorithms are available.12 Using this setup, we were able tocharacterize gas jets produced by rectangular nozzles, whichcan hardly be fully characterized with conventional recon-struction methods. These highly non-rotationally symmetricjets are of great interest for electron acceleration, because forthe same interaction length and gas density the overall gasload can be significantly reduced enabling operation at higherrepetition rates.13 Furthermore, our newly developed methodhas also been proven to be suitable to characterize the gasdensity distribution of a jet with a razor blade or a wire inthe gas stream to steepen the density gradient on one side.Such a density gradient or shock wave is subject of extensivestudies for laser based electron acceleration.11, 14 With thedemonstrated spatial resolution it is also possible to resolve,e.g., the effect of boundary layers15 on supersonic nozzleswith rectangular symmetry. Such knowledge of the densitydistribution is necessary to optimize the nozzles, because theso-called isentropic core and boundary layers are influencedby the inner walls of the nozzle. The paper is organizedas follows. Sec. II describes the experimental acquisitionof the projections/interferograms. After reporting on theexperimental setup in Sec. III we describe in Sec. IV brieflythe reconstruction method and compare the results obtainedwith an Abel inversion and tomographic reconstruction inSec. V followed by a brief conclusion in the last section.

II. PROJECTION MEASUREMENT

The density distribution of a gas jet is deduced from themeasured phase shift of a light wave propagating throughthe jet. The intensity distribution caused by phase differ-ence between the perturbed and the reference wave propa-gating through vacuum/air is called interferogram and canbe recorded with a CCD camera. The phase difference φ(r0)of the two waves at a point r0 on the CCD detector can be

0034-6748/2011/82(8)/083106/6/$30.00 © 2011 American Institute of Physics82, 083106-1

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083106-2 Landgraf et al. Rev. Sci. Instrum. 82, 083106 (2011)

Gas Density

Laser

Nozzle

θ

FIG. 1. Projection measured from an arbitrary density distribution shown asan isosurface plot above a rectangular nozzle. Solid arrows show the beamwhose phase shift φθ (r ) is being binned on one pixel of the CCD found atposition r . The image below the line-out shows a typical interferogram asrecorded by the camera.

written as

φ(r0) =∫

k(η(l) − η0)dl, (1)

where a beam with a wave vector k propagates through me-dia with refractive indices η0 used for the vacuum/air andη(l) for the density distribution under investigation, respec-tively. Recording an additional reference picture with thesame setup, but without gas flow helps to improve the sensi-tivity of the setup, by correcting for phase distortions of otherelements than the gas jet. In this way, very small phase shiftscan be measured limited only by the signal-to-noise ratio in-troduced by random phase fluctuations and the signal band-width.

We are now limiting the discussion to rays which prop-agate through a sufficient thin isotropic medium where scat-tering and refraction can be neglected. The measured phaseshift φθ (r ) accumulated over all volume elements with the re-fractive index η(r) in the beam path δ(rTeθ − r0) is given by

φθ (r0) =∫ +∞

−∞δ(rTeθ − r0)k(η(r) − η0)dxdydz, (2)

rTeθ describes the projection of the position vector r onto theunity vector in projection direction eθ . It will be shown inSec. IV that several projection angles θ are necessary in or-der to calculate arbitrary density distributions ρ(r). The cor-responding scheme can be seen in Figure 1.

III. EXPERIMENTAL SETUP

The experimental setup is shown in Figure 2. As a lightsource we used a Q-switched and frequency doubled Nd:YAGlaser delivering 4–6 ns long pulses with an energy of morethan 200 mJ at 532 nm. For the described experiments the en-ergy of the pulses is attenuated by four orders of magnitudeto a level avoiding saturation of the CCD camera for a sin-gle shot exposure. The spatial beam quality needed to be im-proved with a spatial filter. The diameter of the pinhole waschosen to be twice the focal spot size of a diffraction limited

LaserNd:Yag 532nm

Telescope

CC

D

Beamsplitter

BeamsplitterMirror

Mirror

FIG. 2. Experimental setup. The experiment is based on a Q-switched fre-quency doubled Nd:YAG laser followed by a spatial filter. In one arm of theMach-Zehnder interferometer is a vacuum chamber containing the nozzlemounted on a rotary stage. The gas nozzle exit is imaged with a f/5 lensonto a CCD camera.

beam with the same diameter on the focusing lens. The Mach-Zehnder interferometer consists of AR-coated beam splitterswith a surface flatness of λ/8 and retro-reflecting prisms. Theobject beam is launched into a vacuum chamber containingthe gas nozzle mounted on a stepper motor controlled rota-tion stage with a resolution of 0.2 mrad. The backing pressureof the nozzle is controlled by a valve which is synchronized tothe laser pulses with adjustable delay. Running the valve at themaximum repetition rate of 0.25 Hz the background pressurein the vacuum chamber is kept below 10−4 mbar. The gas jetcan be probed with a delay of 0.5 to 15 ms after the valve re-ceived the opening signal depending on the type and diameterof the valve. Typically a delay of 2 ms is sufficient to observethe interferogram with less than 1% shot-to-shot fluctuations.For improving the signal-to-noise ratio, we have recorded tenindependent images for each rotational angle with the maxi-mum repetition rate. To minimize the phase offset, we used aglass plate with a thickness comparable to the vacuum cham-ber windows into the reference arm. The plane of the gasjet was imaged with a f/5 lens onto a CCD (Basler A102f)camera. The camera has a resolution of 1392 × 1040 pixeland the pixel width and height are 6.45 μm, respectively. Therecorded interference pattern can be read out with a frame rateup to 15 frames/s at a resolution of 12 bit. The laser, the CCDcamera, and the pulsed valve are synchronized allowing tofollow the evolution of the gas jet with a temporal resolutiongiven by the laser pulse length in the order of a few ns.

IV. TOMOGRAPHIC RECONSTRUCTION

For rotationally symmetric gas jets the density distribu-tion can be obtained from the projection data (i.e., interfero-gram) using the well-known Abel inversion.7 However, evenfor nominally symmetric nozzles significant deviations from acylindrical density distribution can occur due to imperfectionsof the nozzle orifice or a misalignment between valve andnozzle. Such deviations significantly disturb experiments, butwill not be quantified with the conventional approach basedon Abel inversion. The same holds for jets which are designedfor having a nearly arbitrary gas density distribution. Tocharacterize such jets, we must estimate the phase shift of a

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083106-3 Landgraf et al. Rev. Sci. Instrum. 82, 083106 (2011)

single volume element and hence the local gas density distri-bution with the help of tomographic reconstruction. We haveopted for the reconstruction based on the filtered back projec-tion algorithm, which is well tested and implemented in sev-eral software packages such as MATLAB.16 The phase shift ofa volume element can be represented as its 2D Fourier trans-form in polar coordinates,

φz(x, y) =∫ 2π

0

∫ +∞

0(kr , θ )kr

× exp(i2πkr (x cos θ + y sin θ ))dkr dθ. (3)

Subsequently the Fourier slice theorem17 can be used to insertEq. (2) in Eq. (3) as it relates the Fourier transform of a slicewith a projection. As a result, we obtain the phase shift of avolume element in a horizontal plane at height z above thenozzle exit

φz(x, y) =∫ 2π

0

∫ +∞

−∞P(θ )|kr | exp(i2πkrr0)dkr dθ (4)

with P(θ ) as the Fourier transform of the recorded projec-tion and kr as the wave vector corresponding to the radiusr0 = x cos θ + y sin θ . The term |kr | can be interpreted as aRam-Lak filter function. For an experimental implementa-tion we have to modify slightly Eq. (4). As it can be seenfrom Eq. (1) the accumulated phase shift will be the same,if the beam is propagating in forward or backward direction,so only a range of 180◦ has to be considered. This approx-imation is only allowed for optically thin plasmas (negligi-ble absorption, refractive index ≈1) and/or sufficient dilutegases ensuring a linear interaction. Then, the integral has tobe replaced by a sum over discrete θ . The minimum re-quired number of discrete steps, i.e., the number of mea-sured interferograms will be discussed in the following. Fora better reconstruction and to make the solution less suscepti-ble to numerical artifacts we have to multiply the Ram-Lakfilter |kr | with a low pass filter. We compared the perfor-mance of a Hann hH (kr ) = sin2 πkr/(kmax − 1) and Shepp-Logan hSL (kr ) = sinc kr filter, both leave the Ram-Lak filterunchanged for low frequencies and efficiently damp higherfrequencies. For testing the reconstruction algorithm we optedfor a discrete delta function as a phase object. After comput-ing the projections for different angles we reconstructed theobject using the discrete form of Eq. (4). As a measure ofthe quality of the tomographic reconstruction we evaluatedthe residuum between the reconstructed and assumed objectfunction. For different applied filters we have calculated theresiduum as a function of the number of projections as shownin Figure 3. Our calculations suggest that the residuum scalesroughly exponentially with the number of projections, andthat applying the Hann filter, we obtain more accurate resultscompared to the direct FBP method under the assumption ofthe lack of noise. It can also be seen that, if less than 20 differ-ent directions are available, FBP is not the optimum choice.For such a low number of projections, an iterative approachbased on a numerical solution of the equations governed bythe different projections delivers more accurate results.

101

102

10−3

10−2

10−1

100

number of projections

quad

ratic

res

iduu

m

IterativeHann(FBP)Shepp−Logan(FBP)

FIG. 3. As a measure of the quality of the tomographic reconstruction weevaluated the residuum between the reconstructed and assumed object func-tion (Dirac delta function) as a function of the number of projections. Toreduce the noise we have applied either a Hann or Shepp-Logan filter, whichshow a comparable performance. For comparison we have also calculated theresiduum using an iterative algorithm, which is the preferred method, if lessthan 20 projections are considered.

As important as the estimation of the minimum numberof projections is also the prediction of the spatial resolution.For our optical setup we estimated a FWHM of the pointspread function (PSF) of dmin = 1.22 λ/N A ≈ 40 μm whereN A = 0.01 is the numerical aperture of our optical system.The spatial broadening of the reconstruction should be at leastin the same order or better as the broadening caused by theoptical system avoiding further reduction of the overall spa-tial resolution. The spatial resolution of the reconstruction isdefined as the bandwidth for which the spatial frequencies ofthe object and the reconstruction are reproduced with an errormargin of 10%. Equation (4) can be expressed as a system ofdiscrete equations. The number of steps, i.e., the size of a vol-ume element, must be chosen that the system of equations is(over-)determined, for a given number of projections.

Considering the modulation transfer function of the re-construction, at least 45 projections evenly distributed across180◦ are necessary to obtain a spatial resolution of 35 μm inthe central region. This estimation is only valid if no system-atic errors or noise reduce the quality of the tomographic re-construction. Finally, the convolution of both PSF functionsyields an overall resolution of 53 μm, which can be improvedby increasing the number of projections. In this respect it isworth mentioning that the FBP algorithm is fast enough toenable online analysis of the data, even with a commercialpersonal computer.

In order to avoid errors in the reconstruction, e.g., due toan imperfect alignment of the different projections with re-spect to each other, tracking of the exact position of the valve(and as a consequence of the gas jet) in the range of a few10 μm will be necessary. This enables us to align every pro-jection measurement with high accuracy even if the mountingof the valve on a rotation stage is less accurate. Accordingto the Clausius-Mossotti equation the particle number densityn can be calculated from the index of refraction η and thephase shift of the corresponding volume element. For weak

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083106-4 Landgraf et al. Rev. Sci. Instrum. 82, 083106 (2011)

-2 -1 0 1 20

2

4

6

8

r [mm]

neut

ral d

ensi

ty [1

018/c

m3 ]

FIG. 4. Comparison of Abel inversion (thin gray) and FBP algorithm (thickblack) for a highly symmetric projection of a supersonic conical nozzle with3 mm exit diameter at 25 bar backing pressure recorded at 500 μm above thenozzle exit.

perturbations η2 − 1 = (1 + η)2 − 1 ≈ 2η the equationcan be rewritten as

η = αn

2ε0(5)

with the polarizability α. For argon α = 1.827× 10−40 Cm2/V at λ = 532 nm and 0◦C.

V. CHARACTERIZATION OF THE METHOD

Compared to previously used methods, our currentapproach based on FBP requires larger efforts in the mea-surement and reconstruction. To demonstrate the improvedperformance of FBP we compared FBP results with recon-structions based on the assumption of cylindrical symmetrywhere an Abel inversion can be applied.8 First of all we an-alyzed the shot-to-shot stability of our phase measurements.As a measure for the stability we calculated the cross corre-lations between the recorded phase patterns 500 μm abovethe nozzle exit. Under optimized experimental conditions,we obtained, for typically ten recorded interferograms, aso-called linear similarity parameter ranging from 0.9993to 0.95 with an average of 0.99 (note 1.0 corresponds toidentical interferograms). The same shot-to-shot stabilitywas observed for different projection angles and densitydistributions. As a result we estimated the error of randomfluctuations (mainly due to fluctuations in the valve openingprocess) to be in the order of less than 1%.

The results are summarized in Figures 4 and 5. InFigure 4 we show the reconstructed density for a supersonicgas jet generated with a conical nozzle with 3 mm diameterand an Argon backing pressure of 25 bar. For the FBP werecorded 45 projections and the retrieved density shows al-most perfect cylindrical symmetry. It should be noted that theaverage density of the isentropic core calculated by the Abelinversion differs by a factor of 1.08 from the number obtainedwith the FBP algorithm corresponding to an absolute differ-ence of 0.5 × 1018 cm−3. For the Abel inversion based recon-struction we observe huge fluctuations of the retrieved gasdensity which are almost missing in the results obtained withFBP. This is a clear indication of the numerically more stablereconstruction based on FBP compared to Abel inversion alsofor density distributions with cylindrical symmetry.

-2 -1 0 1 20

2

4

6

8

r [mm]

neut

ral d

ensi

ty [1

018/c

m3 ]

FIG. 5. Comparison of the retrieved gas density generated with a supersonicconical nozzle with 3 mm exit diameter at 25 bar backing pressure recordedat 500 μm above the nozzle exit using Abel inversion (thin gray) and FBPalgorithm (thick black). Due to laser induced damage of the nozzle exit thejet showed some asymmetries clearly resolved by FBP. The outcome of theassumption of cylindrical symmetry is the appearance of huge density fluctu-ations, which are not real.

The differences are even more obvious, if the gas jet is notrotationally symmetric. A comparison of the retrieved densitydistribution using the two different methods but this time fora non-ideal nozzle is shown in Figure 5. The nozzle in thisexperiment was nominally the same as before, but it has beendegraded during the use in high-intensity laser-plasma in-teraction experiments. The laser pulses were slightly hittingthe upper edge of the nozzle and laser ablation has carveddents into the nozzle resulting in an asymmetric gas flow.Again, Abel inversion slightly overestimates the density nearthe edges of the nozzle exit as seen in Figure 4. Moreover inthe center of the nozzle the density shows huge fluctuationswhich are absent using FBP. Such deviations are well knownfor phase retrievals based on the Abel inversion, but it is al-most impossible to decide which fraction of the amplitude isreal and which fraction of the amplitude is attributed to nu-merical artifacts.

As an additional test for the method we performed anexperiment to estimate the spatial resolution. In this experi-ment we put a 100 μm thick tungsten wire into the gas flow200 μm above a rectangular nozzle (2.4 mm × 1 mm) causinga shock wave. The wire is placed on the right side of the noz-zle exit as shown in Figure 6. It was glued on the nozzle nextto the opening. Numerical simulations have shown that shockwaves caused by small objects in the stream of a nozzle havea spatial extent in the order of 5 μm.11 For the estimation ofthe PSF a shock wave generated inside the gas flow can betreated as a delta function. The spatial extent of the experi-mentally measured shock wave can be used as an upper limitfor the FWHM of the PSF.

The predicted shock wave manifest in a rather sharp lo-cal maximum of the density distribution near the edge andcan bee seen in Fig. 7 for x = 1 mm. The spatial resolutioncan be obtained from a deconvolution of the measured widthof this feature and the predicted width of the shock wave.A careful analysis resulted in spatial resolution better than(60 ± 10) μm. In Fig. 7 also a local minimum can be ob-served close to x = 1 mm, which is directly behind the wire.Note the presented line out is obtained about 200 μm abovethe wire. In another measurement we recorded a line out in a

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083106-5 Landgraf et al. Rev. Sci. Instrum. 82, 083106 (2011)

tungsten wire

x [mm]

y [m

m]

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0

5

10

x 1018

FIG. 6. Topview 2D plot of the measured gas density 500 μm above the noz-zle. The opening of the nozzle (2.4 × 1) mm2 is indicated as black box, a100 μm thick tungsten wire is placed at the right side at 1 mm distance tothe center, 200 μm above the nozzle exit and under an angle of ≈15◦ to thetransversal axis.

plane directly above the wire and obtained a minimum densityof 2 × 1017 cm−3, where it is expected to be zero. The finitenumber is attributed to the noise level and can be regardedas the minimum detectable density, which is an upper bound-ary of the error. Combined with the observed peak density1 × 1019 cm−3 we can realize a dynamic range of two ordersof magnitude with an error in the percent level.

Finally, we applied FBP to study carefully the densitydistribution of a rectangular nozzle. For a wide class of rect-angular nozzles the density profile along the lateral axis isdetermined by a sub-sonic flow, while along the longitudinalaxis the jet is still super-sonic. Under this conditions a simpledescription of the density is possible9, 10 assuming a Gaussiandistribution along the lateral axis the density can be written as

n(x, y, z) = A(x, z) exp

(− y2

σ 2

)(6)

with A(x, z) density and σ the width. σ can be estimatedfrom two projections with an angle of 90◦. A(x, z) can be cal-culated by dividing the measured projection numbers by thevalue of the integral over n(x, y, z) along y. The huge draw-back of this simplified approach is that asymmetric featuresregarding the x symmetry axis are blurred as seen in Figure 7.

-2 -1 0 1 20

2

4

6

8

10

x [mm]

neut

ral d

ensi

ty [1

018/c

m3 ]

FIG. 7. Enlarged line-outs of Figure 6 of the reconstructed gas density usingthe FBP algorithm (thick solid) along y = 0. As a comparison we show theretrieved density distribution for the same setup assuming a Gaussian densityprofile (thin gray) and for the same gas jet without an additional wire us-ing FBP algorithm (dashed). The comparison clearly shows that the methodbased on Gaussian distributions is not able to reveal the distortions introducedby the wire.

The asymmetry introduced by the wire is not detected at all.The retrieved density distribution is equivalent to the densitydistribution without a wire as obtained with FBP. These re-sults again demonstrate that a very careful characterization ofthe density distribution is indispensable. Especially for laserplasma experiments it is sometimes desirable to have a steepprofile, which can be experimentally not verified with the con-ventional methods.

VI. CONCLUSION

We have successfully demonstrated tomographic char-acterization of neutral gas density distributions withoutrotational symmetry. However, the proposed method can bealso applied for the reconstruction of arbitrarily shaped jets.If it is only possible to record less than 20 interferograms,reconstruction based on an iterative algorithm provides moreaccurate information. For more interferograms the filteredback projection method using either a Hann or Shepp-Loganfilter is the better choice. It is also worth mentioning that thereconstruction error becomes smaller for more projections,but the spatial resolution will not be further improved below alower limit given by the point spread function of the imagingsystem. With the methods it is possible to resolve μm scalefeatures of the density distribution, such as shock waves.However, the setup has to be designed very carefully toavoid limitations arising from the imperfect alignment ofconsecutive projections (must be better than some 10 μm)and the shot-to-shot stability from imperfections of the valve(up to 5% fluctuations of the density). In comparison withother approaches the described method demonstrates that im-portant features of non (rotationally) symmetric distributionscan be severely blurred or vanish completely if assumptionsfor the symmetry of the density distributions are made. Thenew characterization method in combination with gas flowsimulations allows to design and characterize sophisticatednozzle shapes for realizing suitable gas density distributions.This might prove to become a key tool for the optimizationof gas jet targets for experiments on laser-driven electronacceleration from underdense plasma.

Furthermore improvements of the measurement setupwill be possible by using a single shot setup relying on amicro lens array. Such an array could be aligned centricallyaround the gas jet acquiring several projections at one shot bya multi-Mach-Zehnder setup.

ACKNOWLEDGMENTS

This study has been supported by Deutsche Forschungs-gemeinschaft (DFG) (Grant No. TR18 A10), Bundesminis-terium fur Bildung, Forschung und Technologie (Germany)(BMBF) (Grant No. 05K10SJ2), and TMBWK (Grant No. B715-08008). B.L. and M.S. acknowledge fruitful discussionswith Oliver Jäckel.

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