High resolution 3D gas-jet characterization

High resolution 3D gas-jet characterization Björn 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 Articles Fast-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 Downloaded 02 Feb 2012 to 141.35.51.130. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions REVIEW OF SCIENTIFIC INSTRUMENTS 82, 083106 (2011) High resolution 3D gas-jet characterization Björn Landgraf,1,2 Michael Schnell,1 Alexander Sävert,1 Malte C. Kaluza,1,2 and Christian Spielmann1,2 1 Institute of Optics and Quantum Electronics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, 07743 Jena, Germany 2 Helmholtz-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 interaction experiments where the shape can be non-symmetrically. With a Mach-Zehnder interferometer we measured the phase shift for different directions through the neutral density distribution of the gas jet. From the recorded interferograms it is possible to retrieve 3-dimensional neutral density distributions by tomographic reconstruction based on the filtered back projections. We report on criteria for the smallest number of recorded interferograms as well as a comparison with the widely used phase 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 as 2 × 1017 cm−3 . © 2011 American Institute of Physics. [doi:10.1063/1.3624694] I. INTRODUCTION Over the last few years we have witnessed tremendous progress in the field of laser-plasma based electron acceleration.1–3 High-power laser pulses focused into an underdense plasma excite coherent density oscillations forming a plasma wave which propagates with nearly the same velocity as the driving laser pulse. The accompanied electric field strength is in the order of several 100 GVm−1 and accelerates electrons to relativistic energies over a distance of only a few millimeters. Furthermore, the accomplishment of x-ray lasers4 and the efficient generation of high harmonic radiation5, 6 rely on the interaction of high-power laser pulses with underdense plasmas. For further optimization of the processes it is sometimes desirable to know the electron density r ) in the interaction region. For fully ionized gases (e.g., n e ( hydrogen or helium) the electron density distribution n e ( r) at the beginning of the laser-plasma interaction can be easily estimated from the measured gas density. For rotationally symmetric gas targets produced with cylindrically symmetric nozzles the gas density distribution can be deduced from a measured interferogram and an Abel inversion.7 Applying a generalized Abel inversion8 or a Gaussian-like density distribution9 it is possible to introduce an asymmetry parameter allowing to relax slightly or change the requirement of axial symmetry for obtaining meaningful results. These methods, for example, rely on the representation of the measured function as Legendre polynomials up to a limited order. Such an approach results in a first order asymmetry description. However, for completely non-rotationally symmetric target geometries10 employed to realize steeper gas density gradients11 or longer interaction lengths between laser and plasma, it is necessary to apply more advanced methods to assess the density distribution. Here we present a new arrangement suitable for characterizing arbitrarily shaped gas density distributions based on interferometric measurements followed by tomographic reconstruction. The 0034-6748/2011/82(8)/083106/6/$30.00 computing time can be significantly reduced by applying the reconstruction filtered back projection (FBP) where efficient algorithms are available.12 Using this setup, we were able to characterize gas jets produced by rectangular nozzles, which can hardly be fully characterized with conventional reconstruction methods. These highly non-rotationally symmetric jets are of great interest for electron acceleration, because for the same interaction length and gas density the overall gas load can be significantly reduced enabling operation at higher repetition rates.13 Furthermore, our newly developed method has also been proven to be suitable to characterize the gas density distribution of a jet with a razor blade or a wire in the gas stream to steepen the density gradient on one side. Such a density gradient or shock wave is subject of extensive studies for laser based electron acceleration.11, 14 With the demonstrated spatial resolution it is also possible to resolve, e.g., the effect of boundary layers15 on supersonic nozzles with rectangular symmetry. Such knowledge of the density distribution is necessary to optimize the nozzles, because the so-called isentropic core and boundary layers are influenced by the inner walls of the nozzle. The paper is organized as follows. Sec. II describes the experimental acquisition of the projections/interferograms. After reporting on the experimental setup in Sec. III we describe in Sec. IV briefly the reconstruction method and compare the results obtained with an Abel inversion and tomographic reconstruction in Sec. V followed by a brief conclusion in the last section. II. PROJECTION MEASUREMENT The density distribution of a gas jet is deduced from the measured phase shift of a light wave propagating through the jet. The intensity distribution caused by phase difference between the perturbed and the reference wave propagating through vacuum/air is called interferogram and can be recorded with a CCD camera. The phase difference φ(r0 ) of the two waves at a point r0 on the CCD detector can be 82, 083106-1 © 2011 American Institute of Physics Downloaded 02 Feb 2012 to 141.35.51.130. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions 083106-2 Landgraf et al. Rev. Sci. Instrum. 82, 083106 (2011) Mirror Beamsplitter Beamsplitter Mirror CCD Gas Density Laser Telescope Laser Nd:Yag 532nm Nozzle θ FIG. 1. Projection measured from an arbitrary density distribution shown as an isosurface plot above a rectangular nozzle. Solid arrows show the beam whose phase shift φθ (r ) is being binned on one pixel of the CCD found at position r . The image below the line-out shows a typical interferogram as recorded by the camera. written as φ(r0 ) =  k(η(l) − η0 )dl, (1) where a beam with a wave vector k propagates through media with refractive indices η0 used for the vacuum/air and η(l) for the density distribution under investigation, respectively. Recording an additional reference picture with the same setup, but without gas flow helps to improve the sensitivity of the setup, by correcting for phase distortions of other elements than the gas jet. In this way, very small phase shifts can be measured limited only by the signal-to-noise ratio introduced by random phase fluctuations and the signal bandwidth. We are now limiting the discussion to rays which propagate through a sufficient thin isotropic medium where scattering and refraction can be neglected. The measured phase shift φθ (r ) accumulated over all volume elements with the refractive index η(r) in the beam path δ(rT eθ − r0 ) is given by φθ (r0 ) =  +∞ δ(rT eθ − r0 )k(η(r) − η0 )d xd ydz, (2) −∞ rT eθ describes the projection of the position vector r onto the unity vector in projection direction eθ . It will be shown in Sec. IV that several projection angles θ are necessary in order to calculate arbitrary density distributions ρ(r). The corresponding scheme can be seen in Figure 1. III. EXPERIMENTAL SETUP The experimental setup is shown in Figure 2. As a light source we used a Q-switched and frequency doubled Nd:YAG laser delivering 4–6 ns long pulses with an energy of more than 200 mJ at 532 nm. For the described experiments the energy of the pulses is attenuated by four orders of magnitude to a level avoiding saturation of the CCD camera for a single shot exposure. The spatial beam quality needed to be improved with a spatial filter. The diameter of the pinhole was chosen to be twice the focal spot size of a diffraction limited FIG. 2. Experimental setup. The experiment is based on a Q-switched frequency doubled Nd:YAG laser followed by a spatial filter. In one arm of the Mach-Zehnder interferometer is a vacuum chamber containing the nozzle mounted on a rotary stage. The gas nozzle exit is imaged with a f/5 lens onto a CCD camera. beam with the same diameter on the focusing lens. The MachZehnder interferometer consists of AR-coated beam splitters with a surface flatness of λ/8 and retro-reflecting prisms. The object beam is launched into a vacuum chamber containing the gas nozzle mounted on a stepper motor controlled rotation stage with a resolution of 0.2 mrad. The backing pressure of the nozzle is controlled by a valve which is synchronized to the laser pulses with adjustable delay. Running the valve at the maximum repetition rate of 0.25 Hz the background pressure in the vacuum chamber is kept below 10−4 mbar. The gas jet can be probed with a delay of 0.5 to 15 ms after the valve received the opening signal depending on the type and diameter of the valve. Typically a delay of 2 ms is sufficient to observe the interferogram with less than 1% shot-to-shot fluctuations. For improving the signal-to-noise ratio, we have recorded ten independent images for each rotational angle with the maximum repetition rate. To minimize the phase offset, we used a glass plate with a thickness comparable to the vacuum chamber windows into the reference arm. The plane of the gas jet was imaged with a f/5 lens onto a CCD (Basler A102f) camera. The camera has a resolution of 1392 × 1040 pixel and the pixel width and height are 6.45 μm, respectively. The recorded interference pattern can be read out with a frame rate up to 15 frames/s at a resolution of 12 bit. The laser, the CCD camera, and the pulsed valve are synchronized allowing to follow the evolution of the gas jet with a temporal resolution given by the laser pulse length in the order of a few ns. IV. TOMOGRAPHIC RECONSTRUCTION For rotationally symmetric gas jets the density distribution can be obtained from the projection data (i.e., interferogram) using the well-known Abel inversion.7 However, even for nominally symmetric nozzles significant deviations from a cylindrical density distribution can occur due to imperfections of the nozzle orifice or a misalignment between valve and nozzle. Such deviations significantly disturb experiments, but will not be quantified with the conventional approach based on Abel inversion. The same holds for jets which are designed for having a nearly arbitrary gas density distribution. To characterize such jets, we must estimate the phase shift of a Downloaded 02 Feb 2012 to 141.35.51.130. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions Landgraf et al. Rev. Sci. Instrum. 82, 083106 (2011) single volume element and hence the local gas density distribution with the help of tomographic reconstruction. We have opted for the reconstruction based on the filtered back projection algorithm, which is well tested and implemented in several software packages such as MATLAB.16 The phase shift of a volume element can be represented as its 2D Fourier transform in polar coordinates, φz (x, y) =  2π 0  +∞ (kr , θ )kr 0 10 quadratic residuum 083106-3 −1 10 −2 10 0 × exp(i2π kr (x cos θ + y sin θ ))dkr dθ. Iterative Hann(FBP) Shepp−Logan(FBP) (3) −3 10 Subsequently the Fourier slice theorem17 can be used to insert Eq. (2) in Eq. (3) as it relates the Fourier transform of a slice with a projection. As a result, we obtain the phase shift of a volume element in a horizontal plane at height z above the nozzle exit φz (x, y) =  2π 0  +∞ P(θ )|kr | exp(i2π kr r0 )dkr dθ (4) −∞ with P(θ ) as the Fourier transform of the recorded projection and kr as the wave vector corresponding to the radius r0 = x cos θ + y sin θ . The term |kr | can be interpreted as a Ram-Lak filter function. For an experimental implementation we have to modify slightly Eq. (4). As it can be seen from 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 approximation is only allowed for optically thin plasmas (negligible absorption, refractive index ≈ 1) and/or sufficient dilute gases ensuring a linear interaction. Then, the integral has to be replaced by a sum over discrete θ . The minimum required number of discrete steps, i.e., the number of measured interferograms will be discussed in the following. For a better reconstruction and to make the solution less susceptible to numerical artifacts we have to multiply the Ram-Lak filter |kr | with a low pass filter. We compared the performance of a Hann h H (kr ) = sin2 π kr /(kmax − 1) and SheppLogan h SL (kr ) = sinc kr filter, both leave the Ram-Lak filter unchanged for low frequencies and efficiently damp higher frequencies. For testing the reconstruction algorithm we opted for a discrete delta function as a phase object. After computing the projections for different angles we reconstructed the object using the discrete form of Eq. (4). As a measure of the quality of the tomographic reconstruction we evaluated the residuum between the reconstructed and assumed object function. For different applied filters we have calculated the residuum as a function of the number of projections as shown in Figure 3. Our calculations suggest that the residuum scales roughly exponentially with the number of projections, and that applying the Hann filter, we obtain more accurate results compared to the direct FBP method under the assumption of the lack of noise. It can also be seen that, if less than 20 different directions are available, FBP is not the optimum choice. For such a low number of projections, an iterative approach based on a numerical solution of the equations governed by the different projections delivers more accurate results. 1 10 2 10 number of projections FIG. 3. As a measure of the quality of the tomographic reconstruction we evaluated the residuum between the reconstructed and assumed object function (Dirac delta function) as a function of the number of projections. To reduce the noise we have applied either a Hann or Shepp-Logan filter, which show a comparable performance. For comparison we have also calculated the residuum using an iterative algorithm, which is the preferred method, if less than 20 projections are considered. As important as the estimation of the minimum number of projections is also the prediction of the spatial resolution. For our optical setup we estimated a FWHM of the point spread function (PSF) of dmin = 1.22 λ/N A ≈ 40 μm where N A = 0.01 is the numerical aperture of our optical system. The spatial broadening of the reconstruction should be at least in the same order or better as the broadening caused by the optical system avoiding further reduction of the overall spatial resolution. The spatial resolution of the reconstruction is defined as the bandwidth for which the spatial frequencies of the object and the reconstruction are reproduced with an error margin of 10%. Equation (4) can be expressed as a system of discrete equations. The number of steps, i.e., the size of a volume 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 reconstruction, at least 45 projections evenly distributed across 180◦ are necessary to obtain a spatial resolution of 35 μm in the central region. This estimation is only valid if no systematic errors or noise reduce the quality of the tomographic reconstruction. Finally, the convolution of both PSF functions yields an overall resolution of 53 μm, which can be improved by increasing the number of projections. In this respect it is worth mentioning that the FBP algorithm is fast enough to enable online analysis of the data, even with a commercial personal computer. In order to avoid errors in the reconstruction, e.g., due to an imperfect alignment of the different projections with respect to each other, tracking of the exact position of the valve (and as a consequence of the gas jet) in the range of a few 10 μm will be necessary. This enables us to align every projection measurement with high accuracy even if the mounting of the valve on a rotation stage is less accurate. According to the Clausius-Mossotti equation the particle number density n can be calculated from the index of refraction η and the phase shift of the corresponding volume element. For weak Downloaded 02 Feb 2012 to 141.35.51.130. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions Landgraf et al. 3 neutral density [1018/cm ] Rev. Sci. Instrum. 82, 083106 (2011) 3 neutral density [1018/cm ] 083106-4 8 6 4 2 0 -2 -1 0 r [mm] 1 η= η)2 − 1 ≈ 2 η the equation αn 2ǫ0 with the polarizability α. For argon × 10−40 Cm2 /V at λ = 532 nm and 0◦ C. 6 4 2 0 -2 2 FIG. 4. Comparison of Abel inversion (thin gray) and FBP algorithm (thick black) for a highly symmetric projection of a supersonic conical nozzle with 3 mm exit diameter at 25 bar backing pressure recorded at 500 μm above the nozzle exit. perturbations η2 − 1 = (1 + can be rewritten as 8 (5) α = 1.827 V. CHARACTERIZATION OF THE METHOD Compared to previously used methods, our current approach based on FBP requires larger efforts in the measurement and reconstruction. To demonstrate the improved performance of FBP we compared FBP results with reconstructions based on the assumption of cylindrical symmetry where an Abel inversion can be applied.8 First of all we analyzed the shot-to-shot stability of our phase measurements. As a measure for the stability we calculated the cross correlations between the recorded phase patterns 500 μm above the nozzle exit. Under optimized experimental conditions, we obtained, for typically ten recorded interferograms, a so-called linear similarity parameter ranging from 0.9993 to 0.95 with an average of 0.99 (note 1.0 corresponds to identical interferograms). The same shot-to-shot stability was observed for different projection angles and density distributions. As a result we estimated the error of random fluctuations (mainly due to fluctuations in the valve opening process) to be in the order of less than 1%. The results are summarized in Figures 4 and 5. In Figure 4 we show the reconstructed density for a supersonic gas jet generated with a conical nozzle with 3 mm diameter and an Argon backing pressure of 25 bar. For the FBP we recorded 45 projections and the retrieved density shows almost perfect cylindrical symmetry. It should be noted that the average density of the isentropic core calculated by the Abel inversion differs by a factor of 1.08 from the number obtained with the FBP algorithm corresponding to an absolute difference of 0.5 × 1018 cm−3 . For the Abel inversion based reconstruction we observe huge fluctuations of the retrieved gas density which are almost missing in the results obtained with FBP. This is a clear indication of the numerically more stable reconstruction based on FBP compared to Abel inversion also for density distributions with cylindrical symmetry. -1 0 r [mm] 1 2 FIG. 5. Comparison of the retrieved gas density generated with a supersonic conical nozzle with 3 mm exit diameter at 25 bar backing pressure recorded at 500 μm above the nozzle exit using Abel inversion (thin gray) and FBP algorithm (thick black). Due to laser induced damage of the nozzle exit the jet showed some asymmetries clearly resolved by FBP. The outcome of the assumption of cylindrical symmetry is the appearance of huge density fluctuations, which are not real. The differences are even more obvious, if the gas jet is not rotationally symmetric. A comparison of the retrieved density distribution using the two different methods but this time for a non-ideal nozzle is shown in Figure 5. The nozzle in this experiment was nominally the same as before, but it has been degraded during the use in high-intensity laser-plasma interaction experiments. The laser pulses were slightly hitting the upper edge of the nozzle and laser ablation has carved dents into the nozzle resulting in an asymmetric gas flow. Again, Abel inversion slightly overestimates the density near the edges of the nozzle exit as seen in Figure 4. Moreover in the center of the nozzle the density shows huge fluctuations which are absent using FBP. Such deviations are well known for phase retrievals based on the Abel inversion, but it is almost impossible to decide which fraction of the amplitude is real and which fraction of the amplitude is attributed to numerical artifacts. As an additional test for the method we performed an experiment to estimate the spatial resolution. In this experiment we put a 100 μm thick tungsten wire into the gas flow 200 μm above a rectangular nozzle (2.4 mm × 1 mm) causing a shock wave. The wire is placed on the right side of the nozzle exit as shown in Figure 6. It was glued on the nozzle next to the opening. Numerical simulations have shown that shock waves caused by small objects in the stream of a nozzle have a spatial extent in the order of 5 μm.11 For the estimation of the PSF a shock wave generated inside the gas flow can be treated as a delta function. The spatial extent of the experimentally measured shock wave can be used as an upper limit for the FWHM of the PSF. The predicted shock wave manifest in a rather sharp local maximum of the density distribution near the edge and can bee seen in Fig. 7 for x = 1 mm. The spatial resolution can be obtained from a deconvolution of the measured width of 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 observed close to x = 1 mm, which is directly behind the wire. Note the presented line out is obtained about 200 μm above the wire. In another measurement we recorded a line out in a Downloaded 02 Feb 2012 to 141.35.51.130. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions Landgraf et al. 083106-5 Rev. Sci. Instrum. 82, 083106 (2011) 18 x 10 2 1.5 tungsten wire y [mm] 1 0.5 10 5 0 −0.5 0 −1 The asymmetry introduced by the wire is not detected at all. The retrieved density distribution is equivalent to the density distribution without a wire as obtained with FBP. These results again demonstrate that a very careful characterization of the density distribution is indispensable. Especially for laser plasma experiments it is sometimes desirable to have a steep profile, which can be experimentally not verified with the conventional methods. −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x [mm] VI. CONCLUSION FIG. 6. Topview 2D plot of the measured gas density 500 μm above the nozzle. The opening of the nozzle (2.4 × 1) mm2 is indicated as black box, a 100 μm thick tungsten wire is placed at the right side at 1 mm distance to the center, 200 μm above the nozzle exit and under an angle of ≈ 15◦ to the transversal axis. plane directly above the wire and obtained a minimum density of 2 × 1017 cm−3 , where it is expected to be zero. The finite number is attributed to the noise level and can be regarded as the minimum detectable density, which is an upper boundary of the error. Combined with the observed peak density 1 × 1019 cm−3 we can realize a dynamic range of two orders of magnitude with an error in the percent level. Finally, we applied FBP to study carefully the density distribution of a rectangular nozzle. For a wide class of rectangular nozzles the density profile along the lateral axis is determined by a sub-sonic flow, while along the longitudinal axis the jet is still super-sonic. Under this conditions a simple description of the density is possible9, 10 assuming a Gaussian distribution along the lateral axis the density can be written as   y2 (6) n(x, y, z) = A(x, z) exp − 2 σ 3 18 neutral density [10 /cm ] with A(x, z) density and σ the width. σ can be estimated from two projections with an angle of 90◦ . A(x, z) can be calculated by dividing the measured projection numbers by the value of the integral over n(x, y, z) along y. The huge drawback of this simplified approach is that asymmetric features regarding the x symmetry axis are blurred as seen in Figure 7. 10 We have successfully demonstrated tomographic characterization of neutral gas density distributions without rotational symmetry. However, the proposed method can be also 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 more accurate information. For more interferograms the filtered back projection method using either a Hann or Shepp-Logan filter is the better choice. It is also worth mentioning that the reconstruction error becomes smaller for more projections, but the spatial resolution will not be further improved below a lower limit given by the point spread function of the imaging system. With the methods it is possible to resolve μm scale features of the density distribution, such as shock waves. However, the setup has to be designed very carefully to avoid limitations arising from the imperfect alignment of consecutive 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 with other approaches the described method demonstrates that important features of non (rotationally) symmetric distributions can be severely blurred or vanish completely if assumptions for the symmetry of the density distributions are made. The new characterization method in combination with gas flow simulations allows to design and characterize sophisticated nozzle shapes for realizing suitable gas density distributions. This might prove to become a key tool for the optimization of gas jet targets for experiments on laser-driven electron acceleration from underdense plasma. Furthermore improvements of the measurement setup will be possible by using a single shot setup relying on a micro lens array. Such an array could be aligned centrically around the gas jet acquiring several projections at one shot by a multi-Mach-Zehnder setup. 8 6 ACKNOWLEDGMENTS 4 This study has been supported by Deutsche Forschungsgemeinschaft (DFG) (Grant No. TR18 A10), Bundesministerium fur Bildung, Forschung und Technologie (Germany) (BMBF) (Grant No. 05K10SJ2), and TMBWK (Grant No. B 715-08008). B.L. and M.S. acknowledge fruitful discussions with Oliver Jäckel. 2 0 -2 -1 0 x [mm] 1 2 FIG. 7. Enlarged line-outs of Figure 6 of the reconstructed gas density using the FBP algorithm (thick solid) along y = 0. As a comparison we show the retrieved density distribution for the same setup assuming a Gaussian density profile (thin gray) and for the same gas jet without an additional wire using FBP algorithm (dashed). 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