The geometry of a typical laser-plasma experiment where a point source of protons is used to image electric and magnetic field structures of size 2a in a plasma of interest. The proton source and the plasma are separated by a distance l.
A qualitative sketch of possible angular distributions of the proton beam vs. the polar angle ψ (measured from the beam axis). Dotted line: an isotropic distribution typical of D 3He fusion sources; solid line: the expected smooth angular distribution of a short pulse laser-accelerated source; its width should be significantly larger than the angle a/l subtended by the object; dashed line: a too-narrow distribution that would not allow proper imaging of the object.
A three-dimensional prolate Gaussian blob with a shaded contour at the 0.5ϕ 0 equipotential surface. The moderately ellipsoidal blob has an aspect ratio b/a = 2, and the axis of the blob is tilted about the x 0-axis by an angle θ = 45° with respect to the z axis. The size of the blob is grossly exaggerated for better visibility. In reality a, b ≪ l.
Four spherical Gaussian blobs of defocusing (positive) electric potential as seen in the object (a) and image (b), (c) planes for two moderate values of μ, a normalized potential that is defined in Sec. IV. The blob radius a varies ±50%, resulting in a corresponding variation in μ. The nonlinear nature of proton imaging and caustic formation can be seen in the complex and distorted shapes where the deflections from multiple objects overlap. These shapes, which become more exaggerated as μ increases, cannot be created by a linear superposition of spherically symmetric deflections. The nonlinear magnification caused by the α ≠ 0 term in Eqs. (1) and (2) is clearly visible in the curved distortion at the upper right margin of image (c).
One-dimensional mappings between the object plane r 0 and the image plane r for spherically symmetric electrostatic potentials. (a) Defocusing potential; (b) Focusing potential.
The intensity distribution in the image plane for a positive (defocusing) potential that is insufficient for caustic formation, μ = 0.25.
The position in the object plane (normalized to the radius a) of a caustic from a spherical Gaussian blob, plotted parametrically in r 0 vs. the dimensionless potential μ. For a defocusing potential, the caustic appears as a ring of a finite radius, which then splits in two closely spaced rings. For a focusing potential μ < 0, the caustic appears as a point, which then grows into a ring. The second caustic stays at the center (a point focus). No caustics occur in the shaded region where the magnitude of μ is too small. Gray scale images for these focusing and defocusing cases are shown in Figs. 9(e), 9(f), and 9(c), respectively, projected to the image plane.
Normalized magnification parameter N vs the dimensionless potential μ, plotted parametrically in the object plane coordinate r 0. A focusing potential corresponds to negative values of μ.
Characteristic images of W = 10 MeV protons formed by single spherically symmetric (a = b) Gaussian blobs of electric potential. Strong variations in intensity can be observed even without a caustic being formally generated. Shown here is a progression of proton images from the weak to the strong deflection regimes, for positive ((a)–(c)) and negative ((d)–(f)) electric fields. At low normalized potentials (μ < 1) the behavior is linearly defocusing (focusing) for positive (negative) electric fields. Caustics form in the nonlinear intermediate regime (1< μ < 2). Panel (c) actually has two concentric caustic rings, situated very close to each other (cf. Fig. 7). Panel (f) illustrates strong mesh twisting as proton beamlets cross over each other, as can be seen in the multi-branched nature of the line profile. (The gray scale image intensities are always formed from the sum of all branches.) In all cases a = 100 μm, l = 7 mm, and L = 100 mm. The nominal magnification M = 14.2, but this is only an approximation. For example, the size of the image feature in panels (a) through (c) varies, even though all are formed from the same object.
The function F (solid line) and its derivative F′ (dashed line), for flat-top potentials with sharp edges, as a function of the distance ξ to the boundary, normalized to the width parameter δ. Caustics appear at the extrema of F′ when ϕ0 is sufficiently large, as defined in Eqs. (45) and (46).
Sequential rotation of a very prolate (aspect ratio b/a = 10) Gaussian electric potential blob from the end-on view (θ = 0°) to the side-on view (θ = 90°) for defocusing potentials μ = +0.12 (a) and μ = +0.31 (b), as seen in the image plane for W = 10 MeV protons. Due to the elongated shape, caustics form in (b) for θ = 0° and θ = 7.5° even though μ is small; compare with Fig. 9(a). The θ = 90° views approximate an infinite cylinder with a radially Gaussian profile, for the narrow field of view shown here. The deflection in this case is nearly one-dimensional. Compare with the magnetic blobs shown in Fig. 18. In both cases a = 100 μm, l = 7 mm, and L = 100 mm.
Characteristic shapes of caustics in the object plane for electrostatic Gaussian blobs of (a) spherical, (b) very prolate (aspect ratio b/a = 10), and (c) moderately oblate (b/a = 0.25) structure. Caustics from focusing (μ < 0) and defocusing (μ > 0) potentials have similar shapes for the spherical structure but undergo a rotation for the prolate structure, and split from one into two for the oblate structure. The nominal potential required to form a caustic for the non-spherical blobs is about 5 times lower due to the potential enhancing behavior of Eq. (56) for a ≠ d.
(a) A planar electrostatic shock propagating with velocity vs in the +y 0 direction through a column of plasma. (b) Line profiles in y 0 (for x 0 = z 0 = 0) for the shock density n (with approximate error-function shape) as it jumps from n 0 to n max, the electrostatic potential ϕ (with true error-function shape), and the y-component of the electric field E y = −∂yϕ. (c)–(e): Proton images for a planar shock with θ = 0°, 0.5°, and 1°, respectively. Caustics only appear very close to θ = 0°. In all cases ϕ0 = 200 V, δ = 15 μm, G n = 2, a = 2000 μm, l = 7 mm, and L = 100 mm.
The ratio a/d (normalized to a/δ) as a function of the tilt angle θ for a/δ = 100. For shocks with a transverse extent a that is much greater than the shock thickness δ, the ratio a/d drops to half of its peak value in only one degree. This can be enough to prevent caustic formation, as illustrated in Fig. 13(e).
A magnetic flux rope of circular Gaussian cross section in the object plane (a) and the resulting images for W = 10 MeV protons with (b) B 0 = 30 T and (c) B 0 = 60 T. A weak intensity increase is visible (b) below the critical magnetic field for caustic formation (B 0 ≈ 32 T); above this value two caustics appear (c). In both cases a = 100 μm, l = 7 mm, and L = 100 mm.
The geometry of a cylindrical shock: (a) The magnetic field distribution with a narrow transition and approximately two-fold compression of the magnetic field; (b) A top view of the shock (a cross section by an x-z plane), with the shock transition shaded. Only for the rays passing through the transition can one expect a rapid variation of deflection angle. One of these rays is shown by a dashed-dotted line.
Caustics can be formed for the rays shown here that pass tangentially next to a segment of the shock front. These rays will form a quasi-filamentary structure in the image plane.
Illustration of a prolate Gaussian magnetic blob and its characteristic proton images. (a) In the object plane, a three-dimensional current j (shown here is an xz slice of the current streamlines) creates a “cocoon” of azimuthal magnetic field. Images formed by W = 10 MeV protons are shown for sequential rotation of a highly elongated (b/a = 10) magnetic blob with peak field strength B 0 = +5 T (b) and +15 T (c). (A “negative” magnetic field would signify a current reversal.) The images are similar to those from electric blobs (see Fig. 11) near θ = 0°: once again, due to the elongated shape, caustics form in (c) for θ = 0° and θ = 7.5° even though B is small. In contrast with electric blobs, the magnetic deflection fades away near θ = 90° due to the cosθ factor in Eq. (102). In both cases a = 100 μm, l = 7 mm, and L = 100 mm.
Illustration of the geometry of the higher order corrections.
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