# Mapping transient electric fields with picosecond electron bunches

Contributed by Jie Zhang, October 12, 2015 (sent for review April 12, 2015; reviewed by ChiKang Li and David Neely)

## Significance

Transient electric fields driven by intense lasers and particle beams play a key role in a number of applications from plasma-based particle accelerators to implosion dynamics of inertial fusion targets. Here, a method to map the 3D transient field structures with high resolutions both in time and space by use of picosecond electron bunches is presented. It is applied to measure the transient field evolution induced at a solid surface irradiated by a short pulse laser. This method can be applied to monitor field structures with much higher strength, which may find wide application in relevant research fields.

## Abstract

Transient electric fields, which are an important but hardly explored parameter of laser plasmas, can now be diagnosed experimentally with combined ultrafast temporal resolution and field sensitivity, using femtosecond to picosecond electron or proton pulses as probes. However, poor spatial resolution poses great challenges to simultaneously recording both the global and local field features. Here, we present a direct 3D measurement of a transient electric field by time-resolved electron schlieren radiography with simultaneous 80-μm spatial and 3.7-ps temporal resolutions, analyzed using an Abel inversion algorithm. The electric field here is built up at the front of an aluminum foil irradiated with a femtosecond laser pulse at 1.9 × 10

^{12}W/cm^{2}, where electrons are emitted at a speed of 4 × 10^{6}m/s, resulting in a unique “peak–valley” transient electric field map with the field strength up to 10^{5}V/m. Furthermore, time-resolved schlieren radiography with charged particle pulses should enable the mapping of various fast-evolving field structures including those found in plasma-based particle accelerators.### Sign up for PNAS alerts.

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As a fundamental phenomenon, transient electric fields exist widely in plasma systems, such as those driven by intense lasers and particle beams. Such fields can play an important role in the plasma evolution, which are related to various applications, including plasma-based advanced particle acceleration (1–5), inertial confinement fusion (6), high energy density physics (7), astrophysical phenomena (8), as well as shock physics (9). Because of the difficulties in experimental measurements of the fields, however, they have been studied so far mostly through theoretical models and numerical simulations. Recent advances in time-resolved electron (10–14) and proton (15–19) radiography make it possible to directly monitor the transient electric field (TEF) and transient magnetic field (TMF). Ultrashort monoenergy electron bunches are preferable in accessing the TEFs because they are readily available and compact (20–22). Previous electron probing studies have only delivered limited TEF information. For example, electron shadow imaging provides only a profile of the electric field (10–12). By using a probe beam with confined size to improve the spatial resolution, the averaged local electric field is estimated at a given probing distance only (23–25).

Here, we report a picosecond-time-resolved electron schlieren radiography (PESR) that directly maps globally the detailed 3D distribution of the TEFs in a dynamical plasma system, which is usually difficult to obtain with an optical probe. In optical schlieren photography, an optical image is modulated due to the inhomogeneity of local optical paths, which mostly results from the changes of optical refractive index induced by density variations. However, in time-resolved electron schlieren radiography, the probing particle beam is deflected by local electromagnetic fields in laser plasmas, forming a modulated pattern. The TEF and TMF can be measured separately by changing the imaging geometry of the probe beam with respect to the foil surface (13, 19). Because the TMF is very weak due to the low intensity of the pump laser beam and the electron probe propagates along the foil surface in this study, the deflection of the electrons is dominated by TEF. In our experiments, the laser plasma and related TEFs were generated by irradiating a femtosecond laser pulse onto a 2-μm-thick aluminum foil at an intensity of 1.9 × 10

^{12}W/cm^{2}.By inserting a grid to divide the probing electron beam into an array of electron beamlets, as shown in Fig. 1 and

*Methods*, the grid PESR presented here can trace the deflection of each beamlet and yield a spatially resolved schlieren-type pattern. This provides a means to map the 3D distribution of the TEF with a subhundred-micrometer spatial resolution and a several-picosecond temporal resolution, which could not be accessed in previous studies. The structural and temporal evolution of the TEF is visualized with the time-resolved electron schlieren images (Fig. 2) at different time delays (*T*) following laser irradiation obtained by the picosecond electron beamlets.Fig. 1.

Fig. 2.

Before laser irradiation (

*T*= −2 ps, Fig. 2*A*), the beamlet array is undisturbed and the image is similar to that obtained without illumination. After irradiation, the aluminum surface is ionized through multiphoton and thermal emission processes under our conditions, forming suprathermal electrons. A fraction of the hot electrons escapes from the surface, and results in a redistribution of the residual charges, most of which are neutralized in the few picoseconds following illumination (4, 18). The evolution of the escaped electrons, together with the remaining ions in the foil, contributes to the charge-separated TEF. Thus, the effect of Coulomb scattering that comes from the ions on the probing electrons can be neglected here. As the electrons in the probe beams pass through the TEF region, they are deflected. As a whole, the patterns of the electron beamlets suffer from distortion, which depend upon the time delays as shown in Fig. 2*C–H*. In the first few picoseconds of laser-induced plasma transformation, the probing electrons that pass the front of the target are strongly repelled from the surface by the escaped electrons. Thus, an electron-depleted hemispherical area is formed, visualized as a hole in the center of Fig. 2*D*. This hole expands as the escaped electrons move outward from the foil surface, as shown in Fig. 2*E*. As delay time increases in Fig. 2*G*and*H*, the displacement of the row directly over the target surface decreases because of the escaped electrons moving away from the row as well as the TEF weakening in strength.To map the field distribution, the 2D displacements of the probing electron beamlets at each time delay are obtained by comparing their positions before and after laser irradiation with those depicted in Fig. 3

*A*, which are extracted from the corresponding schlieren snapshots. As a representative, the TEF distribution at*T*= 25 ps (Fig. 4*B–D*) is extracted from the correspondent 2D displacements of beamlets by using the Abel inversion algorithm described in*Methods*. For the electric field component*E*_{x}perpendicular to the target surface, we observe a unique “peak–valley” structure on any cross-section along the symmetric axis (the*X*axis), with a demarcation line near*x*= 100 μm where the electric field reverses its direction as shown in Fig. 4*B*. If we denote*E*_{x}pointing outward as positive, then the*E*_{x}field is 1.2 × 10^{5}V/m at the “peak” (*x*= 75 μm) and −9 × 10^{4}V/m at the “valley” (*x*= 225 μm), approximately. Moreover, when looking into a cross-section of the transverse electric field component*E*_{r}parallel to the target surface, it also exhibits the structure of peak and valley. As a result, the total TEF strength as shown in Fig. 4*D*resembles a volcanic crater with negligible value in its interior. Because*E*_{r}is zero along the*X*axis, the on-axis total strength is M-shaped with its two maxima falling on the peak and the valley of*E*_{x}, respectively.Fig. 3.

Fig. 4.

Fig. 3 shows the deflections of each beamlet and the snapshot of the PESR image at

*T*= 25 ps. The electron beamlets at the sixth row are barely deflected, which indicates that the TEF strength is approximately zero. Meanwhile the electron beamlets on both sides of the sixth row in Fig. 3 are driven apart due to the opposite*E*_{x}directions they experience. The TEF goes to zero (*x*= ∼100 μm at*T*= 25 ps) because the emitted hot electrons locate there. Therefore, the average velocity of the emitted hot electrons along the*X*axis is estimated to be 4 × 10^{6}m/s. Those beamlets, passing the rear of the target, experience a near-zero TEF strength and remain in their original directions as shown by the 8th–10th rows in Fig. 3*A*because of the shields of the residual charges in the foil.The method described above can be generally extended to different probe beams. For a probing particle with charge where

*Q*, rest mass energy ${\epsilon}_{0}$, and kinetic energy ${\epsilon}_{k}$, its deflection angle $\alpha $ along its traveling path in the*Z*axis can be expressed as$$\alpha =Q\underset{-\infty}{\overset{+\infty}{{\displaystyle \int}}}\frac{E\left(z\right)dz}{{\epsilon}_{k}\left(1+\frac{{\epsilon}_{0}}{{\epsilon}_{0}+{\epsilon}_{k}}\right)\hspace{0.5em}},$$

[1]

*E*(*z*) is the electric field at the position*z*on the trace (*SI Text*,*Derivation of the Deflective Angle*). Therefore, considering a certain deflection angle (typically several milliradians), the sensitivity to the TEF is mainly determined by the kinetic energy of the probing particle regardless of the sign of its charge. For example, 15-MeV fusion-reaction–produced protons have been applied to measure TEFs up to 10^{8}V/m inside an inertial confinement fusion core (19), whereas 80-keV photoelectrons have shown their adaptability to TEFs within the 10^{5}-V/m range (24). In principle, time-resolved schlieren radiography can be scaled up to monitor field structures with a much higher strength by increasing the energy of probing particles, either electrons or protons. It is worthy to note that better resolutions of field structures in time and space are ensured by the potentially lower energy spread and shorter bunch length of electrons versus protons. In addition, the electron beams can be generated compactly using different approaches with a wider energy selection range from keV to MeV, providing broader options for transient field diagnostics.In summary, a scheme to measure the TEF evolution with both spatial and temporal resolutions, respectively, at tens of micrometers and a few picoseconds resolution has been demonstrated. In principle, such ability to capture TEF excitation shall allow for experimental detection of the accelerating-field structures of plasma-based particle accelerators that are driven by intense laser or particle beams (1, 2, 26–28). Time-resolved schlieren radiography may even serve as a means to detect electromagnetic field structures associated with some hydrodynamic instabilities, which are highly detrimental to inertial confinement fusion (29). In such case, the TEFs at a much higher magnitude of 10

^{9}V/m can be measured quantitatively by increasing the beam energy to tens of MeV.## Methods

### PESR.

The experimental configuration of the PESR shown in Fig. 1 was modified from the configuration used in previous studies (25, 30) by inserting a thin copper grid at ∼2 mm in front of the target. The probing electron beam has an FWHM diameter of 1.5 mm at the target position with a divergence angle of ∼10

^{−2}radian. The grid preimprints a periodic pattern on the electron beam and splits it into an array of electron beamlets, thus providing a 3D distribution map of the TEFs. Here the spatial resolution is the same as the 80-μm period of the grid with holes separated by ribs of 30-μm width. The resolution could be improved by applying a grid with smaller periods, although it is limited by the uncertainty of the spatial displacement of each electron beamlet, ∼12 μm in this experiment. The overall time resolution in PESR is around 3.7 ps, determined by the optical pump pulse width (*t*_{pump}= 70 fs), the probing electron bunch length (*t*_{probe}= ∼2 ps), and its interaction time with the field (*t*_{inter}= ∼3.1 ps at*T*= 25 ps). When containing ∼3 × 10^{3}electrons per pulse,*t*_{probe}of the 55-keV electron pulse with an energy spread of 10^{−4}was estimated to be ∼2 ps (31).*t*_{inter}is defined as 2*b/v*_{z}, where*b*is the impact parameter (24). At*T*= 0 ps*, b*is ∼0.2 mm, therefore*t*_{inter}equals 3.1 ps with ${v}_{z}=1.29\times {10}^{8}\text{m/s}$ at ${\epsilon}_{k}$ = 55.0 keV. At later time, the plasma field will cover a larger range due to the expansion of the ejected charge cloud, e.g., at 25 ps to 0.35 mm, the transit time of the electron beam will increase. Thus, the effective time resolution will increase gradually with time. Besides applying a shorter electron pulse, higher temporal resolution can be achieved by increasing the beam energy for shorter interaction time, at the expense of reduction in sensitivity to TEFs. Because the TEF is established instantaneously on arrival of the 70-fs pump laser pulse, we define the moment when the probing electron beam begins deflecting as the time 0,*T*= 0 ps. Each schlieren image, shown in Fig. 2, was taken by accumulating 10 ultrashort electron pulses to increase signal-to-noise ratio. The irradiated spot on the sample was refreshed every 800 laser shots to avoid potential effects of target damages on the measurement.### Data Analysis Procedure.

The TEF distribution at a particular time delay is calculated by performing Abel inversion of the electron schlieren-type radiography data (Fig. 4

*A*and*SI Text*,*Error Analysis*). The axial symmetry of the laser plasma and its associated TEF is visualized by the symmetric beamlet distribution with respect to the*X*axis at*T =*25 ps in Fig. 3*A*, which grounds the application of Abel inversion. We first calculate the components of the deflection angle ${\alpha}_{i}\left(x,y\right)$ (*i = x, y*) from the displacement of the 55.0-keV probing electron beamlets at the coordinates (*x, y*) ($Q=-e$, $m{v}_{z}^{2}=1.90{\epsilon}_{k}$). The angles ${\alpha}_{i}\left(x,y\right)$ are linked to the corresponding transient electric field $\mathit{E}\left(x,y,z\right)={E}_{x}\left(x,r\right){\mathit{e}}_{\mathit{x}}+{E}_{r}(x,r){\mathit{e}}_{\mathit{r}}$:$$\frac{\mathrm{\Delta}x}{l}\approx {\alpha}_{x}\left(x,y\right)=-2{\displaystyle \underset{y}{\overset{+\infty}{\int}}\frac{e{E}_{x}(x,r)}{m{v}_{z}^{2}}\frac{rdr}{\sqrt{{r}^{2}-{y}^{2}}}},$$

[2]

$$\frac{\mathrm{\Delta}y}{l}\approx {\alpha}_{y}\left(x,y\right)=-{\displaystyle \int \frac{ey{E}_{r}\left(x,r\right)}{mr}\frac{dz}{{v}_{z}^{2}}}=-2y{\displaystyle \underset{y}{\overset{+\infty}{\int}}\frac{e{E}_{r}(x,r)}{m{v}_{z}^{2}r}\frac{rdr}{\sqrt{{r}^{2}-{y}^{2}}}}.$$

[3]

We then compute the 3D field strengths ${E}_{x}(x,r)$ and ${E}_{r}(x,r)$ using the Abel inversion (32) as follows:

$${E}_{x}(x,r)=\frac{1}{\pi}\underset{r}{\overset{+\infty}{{\displaystyle \int}}}\frac{m{v}_{z}^{2}}{e}\frac{d{\alpha}_{x}(x,y)}{dy}\frac{dy}{\sqrt{{y}^{2}-{r}^{2}}},$$

[4]

$${E}_{r}(x,r)=\frac{r}{\pi}\underset{r}{\overset{+\infty}{{\displaystyle \int}}}\frac{m{v}_{z}^{2}}{e}\frac{d\frac{{\alpha}_{y}(x,y)}{y}}{dy}\frac{dy}{\sqrt{{y}^{2}-{r}^{2}}}.$$

[5]

The data analysis method used in this study can also be applied to proton radiography.

## SI Text

### Derivation of the Deflective Angle.

First, applying the special theory of relativity to a charged probing particle as shown in Eqs.

**S1**–**S4**, we could derive the Newtonian form of its kinetic energy as Eq.**S5**. For a 55-keV electron, $m{v}_{t}^{2}=1.90{\epsilon}_{k}$, and for a 15-MeV proton, $m{v}_{t}^{2}=1.98{\epsilon}_{k}$. (The meaning of each symbol has been listed in Table S1). The expression $m{v}_{t}^{2}$ can be directly written as $2{\epsilon}_{k}$, analogous to classical physics when the energy of electrons and protons is relatively low.$$\epsilon =m{c}^{2}=\gamma {m}_{0}{c}^{2},$$

[S1]

$${\epsilon}_{0}={m}_{0}{c}^{2},$$

[S2]

$$\gamma =\frac{1}{\sqrt{1-\frac{{v}_{t}^{2}}{{c}^{2}}}},$$

[S3]

$$\epsilon ={\epsilon}_{k}+{\epsilon}_{0},$$

[S4]

$$\text{m}{v}_{t}^{2}={\epsilon}_{k}\left(1+\frac{{\epsilon}_{0}}{{\epsilon}_{0}+{\epsilon}_{k}}\right).$$

[S5]

Table S1.

*SI Text*

Symbol | Description |
---|---|

$\epsilon $ | Total energy |

${\epsilon}_{0}$ | Rest mass energy |

${\epsilon}_{k}$ | Kinetic energy |

m_{0} | Rest mass |

m | Relativistic mass |

Q | Charge of a probing particle |

γ | Lorentz factor |

c | Speed of light |

E | Transient electric field (TEF) |

r | Radial direction in the Y–Z plane |

X axis | Normal of the target surface |

Z axis | Traveling(longitude) direction of a probing particle |

v_{z} | Initial velocity of a probing particle along the longitude direction before interaction with TEF |

t | Moment when a probing particle interacts with TEF |

α | Deflection angle of a probing particle |

v_{t} | Transient traveling velocity of a probing particle at a particular moment t |

v | Transient traveling velocity of a probing particle along the transverse direction after interaction with TEF |

l | Drifting distance of a probing particle along the longitude direction after interaction with TEF |

Second, after traveling along the

*Z*axis through the TEF area, the probing charged particle is deflected by Lorentz force with a transverse velocity*v*, as shown in Eq.**S6**. Meanwhile, the deflection angle $\alpha $ can be expressed by the transverse velocity versus the transient traveling velocity ${v}_{t}$ of the probing particle shown in Eq.**S7**. Here it should be noted that the contribution of transient magnetic fields on the deflection of the charged particle is neglected, because the electrons probing at grazing incidence are sensitive primarily to the electric fields (19). Due to the small deflection angle (mrad), the velocity change of the probing particle along the*Z*axis could be ignored, namely ${v}_{t}\approx {v}_{z}$. Similarly, the corresponding mass change could be neglected when applying the impulse-momentum theorem in Eq.**S7**.$$v=\underset{-\infty}{\overset{+\infty}{{\displaystyle \int}}}\frac{QE}{m}dt=Q\underset{-\infty}{\overset{+\infty}{{\displaystyle \int}}}\frac{E}{m}\frac{dz}{{v}_{z}},$$

[S6]

$$\alpha =\frac{v}{{v}_{z}}=Q\underset{-\infty}{\overset{+\infty}{{\displaystyle \int}}}\frac{Edz}{m{v}_{z}^{2}}=Q\underset{-\infty}{\overset{+\infty}{{\displaystyle \int}}}\frac{Edz}{{\epsilon}_{k}\left(1+\frac{{\epsilon}_{0}}{{\epsilon}_{0}+{\epsilon}_{k}}\right)\hspace{0.5em}}.$$

[S7]

### Error Analysis.

As shown in Figs. 1 and 4Then the 3D transient field was calculated with the Abel inversion from the displacement of the electron beamlets.Here we used Li et al.'s method (32), which can effectively suppress the noise amplification in calculations. Because of the rotational symmetry of the TEF, ${E}_{r}\left(x,r\right)$ can be expanded by a series of Taylor–Stieltjes polynomials of

*A*of the main text, considering the differential variable*dz*and the components of**, we rewrote Eq.***E***S7**as$${\alpha}_{x}(x,y)=-2\underset{y}{\overset{+\infty}{{\displaystyle \int}}}\frac{e{E}_{x}(x,r)}{m{v}_{z}^{2}}\frac{rdr}{\sqrt{{r}^{2}-{y}^{2}}},$$

[S8]

$${\alpha}_{y}\left(x,y\right)=-2y\underset{y}{\overset{+\infty}{{\displaystyle \int}}}\frac{e{E}_{r}(x,r)}{m{v}_{z}^{2}r}\frac{rdr}{\sqrt{{r}^{2}-{y}^{2}}}.$$

[S9]

$${E}_{x}(x,r)=\frac{1}{\pi}\underset{r}{\overset{+\infty}{{\displaystyle \int}}}\frac{m{v}_{z}^{2}}{e}\frac{d{\alpha}_{x}(x,y)}{dy}\frac{dy}{\sqrt{{y}^{2}-{r}^{2}}},$$

[S10]

$${E}_{r}(x,r)=\frac{r}{\pi}\underset{r}{\overset{+\infty}{{\displaystyle \int}}}\frac{m{v}_{z}^{2}}{e}\frac{d\frac{{\alpha}_{y}(x,y)}{y}}{dy}\frac{dy}{\sqrt{{y}^{2}-{r}^{2}}}.$$

[S11]

*r*^{2}, as in Eq.**S12**. Regarding*E*(*x,y*), ${E}_{\beta}^{\text{'}}\left(y\right)$, … ${E}_{\beta}^{\left(n\right)}\left(y\right)$ as*n+1*independent unknown variables, we can directly get the ${E}_{r}\left(x,r\right)$ by the referenced method (32). The numerical computation of the Abel inversion is implemented in MATLAB codes based on the above method.$$E\left(x,r\right)\cong E\left(x,y\right)+{\displaystyle \sum}_{j=1}^{n}\frac{{E}_{\beta}^{\left(j\right)}\left(y\right)}{j!}{({r}^{2}-{y}^{2})}^{j}.$$

[S12]

A 3D electric field trial distribution with a rotational symmetry was assumed to verify the accuracy of the MATLAB codes for the Abel inversion. For simplicity, the trial

**is composed of two Gaussian distributions, as in Eqs.***E***S13**and**S14**, where*w*is set as 200 μm,*A*= 10^{5}V/m, and $\sigma =126$ μm. Therefore, the components of**can be expressed by Eqs.***E***S15**and**S16**.$$R=\sqrt{{x}^{2}+{r}^{2}},$$

[S13]

$$\mathit{E}=A\left({e}^{-\left({\left(R-w\right)}^{2}/{\sigma}^{2}\right)}+{e}^{-\left({\left(R+w\right)}^{2}/{\sigma}^{2}\right)}\right)\frac{\mathit{R}}{\left|\mathit{R}\right|},$$

[S14]

$${E}_{x}(x,r)=\frac{\left|x\right|}{R}\times E=\frac{\left|x\right|}{\sqrt{{x}^{2}+{r}^{2}}}\times \left({e}^{-\left({\left(\sqrt{{x}^{2}+{r}^{2}}-w\right)}^{2}/{\sigma}^{2}\right)}+{e}^{-\left({\left(\sqrt{{x}^{2}+{r}^{2}}+w\right)}^{2}/{\sigma}^{2}\right)}\right),$$

[S15]

$${E}_{r}(x,r)=\frac{\left|r\right|}{R}\times E=\frac{\left|r\right|}{\sqrt{{x}^{2}+{r}^{2}}}\times \left({e}^{-\left({\left(\sqrt{{x}^{2}+{r}^{2}}-w\right)}^{2}/{\sigma}^{2}\right)}+{e}^{-\left({\left(\sqrt{{x}^{2}+{r}^{2}}+w\right)}^{2}/{\sigma}^{2}\right)}\right).$$

[S16]

Next the deflection angles were calculated analytically based on the trial distribution. Substituting Eqs.

**S15**and**S16**into Eqs.**S8**and**S9**, we obtained the deflection angles of the electron beamlets assuming the trial distribution.Finally, with the known angles we reconstructed the field distribution using the MATLAB codes for the Abel inversion and compared it with the trial distribution. To emulate the real experimental condition, discrete deflection angles were chosen to calculate the electric field by the Abel inversion program.

As shown in Fig. S1, without adding any noise to the deflection angles, the calculated electric field

*E*_{x}agrees well with the trial distribution, suggesting that the MATLAB codes used for the Abel inversion are reasonable. When adding random noise at ten percent of the maximum to all deflection angles, which is equivalent to a 10^{4}-V/m uncertainty to the field, the calculated electric field shares the trend of the trial one. The noise is not amplified after the inversed analysis, suggesting the codes are robust. Only a few typical rows of the electron beamlets in the positive*Y*direction are present here because the whole region is symmetric with respect to the*X*axis. These rows are chosen from the plane along the symmetry axis, i.e.,*X*–*Y*plane shown in Fig. 1 of the main text. The*E*_{r}component (not shown) can be obtained similarly as the*E*_{x}component.Fig. S1.

## Acknowledgments

The authors thank Mr. Yuan-Ling Huang for helpful discussion and language editing. This work was supported in part by the National Basic Research Program of China (Grant 2013CBA01500) and the National Natural Science Foundation of China under Grants 11421064, 11304199, 11220101002, and 11327902. J. Cao acknowledges the support from National Science Foundation Grant 1207252.

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**Published online**: November 9, 2015

**Published in issue**: November 24, 2015

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#### Acknowledgments

The authors thank Mr. Yuan-Ling Huang for helpful discussion and language editing. This work was supported in part by the National Basic Research Program of China (Grant 2013CBA01500) and the National Natural Science Foundation of China under Grants 11421064, 11304199, 11220101002, and 11327902. J. Cao acknowledges the support from National Science Foundation Grant 1207252.

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The authors declare no conflict of interest.

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