Nagayama et al.

FIG. 4. The spatial lineouts are extracted from Fig. 3 at Mg Heα (magenta) and its nearby continuum (green). Mg Heα bound-bound line transmission spatial lineouts (blue) are extracted by dividing the magenta by the green. The “half-moon” boundary and the x-axis are defined based on the transmission −9◦ +9◦ spatial lineouts, and xBL and xBL are defined based on the continuum peaks.

III. RESULTS

Parallax is systematically applied to ten Fe opacity shots performed under different sample configurations.6 There are three different CH configurations and multiple different Fe thicknesses for each configuration. There is one shot where the sample is raised by 1.5 mm from its nominal location. For each shot, parallax is applied to the available bound-bound lines that are strong enough to define the “half-moon” boundary from their line-transmission spatial profiles. The number of usable lines depends on their areal density, Stark line width, and the spectral range of the spectrometers used. For each shot, the mean h and its standard deviation are computed from parallax results of two to seven Fe and Mg lines. Parallax results from Fe and Mg lines agree with each other. The validity of this uncertainty estimate was also verified from the shots with four spectrometers, two each at +9◦ and −9◦ .

Rev. Sci. Instrum. 85, 11D603 (2014)

Figure 5 summarizes the measured h as a function of Te inferred from Mg K-shell spectroscopy.6 We confirm a strong anti-correlation between h and Te (Pearson correlation coefficient = −0.91). To investigate this point synthetically, we use a 3D view factor code VISRAD9 and a calibrated ZPDH intensity image from one of our experiments to calculate the heating radiation at the sample as a function of the sample distance from the ZPDH radiation source. The details of the calculation will be discussed elsewhere. The blue curve in Fig. 5 shows the resultant radiation brightness temperature, TB , as a function of h. This result suggests that the radiation source heats the sample to different temperatures due to source radiation geometric dilution. The radiation brightness temperature is systematically higher than Te due to the complex heating mechanism involving radiation transport and hydrodynamics and beyond the scope of this article. Figure 5 also shows that, for similar Te , shot-to-shot variation in the inferred h is larger than the individual measurement uncertainties due to the 3D radiation transport effects of the backlighter. While Eq. (1) is derived assuming an instantaneous point backlighter, the actual backlighter emission is a result of the radiation transport through the 3D ZPDH plasma, which spatially varies over a few ns duration. Thus, the variation in the inferred h comes from the irreproducibility in the evolution of 3D ZPDH plasma and the resultant irreproducibility in the line-of-sight dependent effects on the measurements. We found that h was anti-correlated to Te and confirmed that the sample reached a different temperature due to the geometric dilution of the radiation source. The parallax results are important (i) to better understand our platform and further optimize SNL Z opacity experiments and (ii) to better understand the sample heating and accurately evaluate how close our sample is to local thermal equilibrium. ACKNOWLEDGMENTS

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the (U.S.) Department of Energy (DOE) under Contract No. DE-AC0494AL85000. 1 D.

FIG. 5. There is a strong correlation between the measured electron temperature, Te , and the measured source-to-sample distance, h. The blue curve is a modeled radiation brightness temperature as a function of h.

Mihalas, Stellar Atmospheres, A Series of Books in Astronomy and Astrophysics, 2nd edition (W. H. Freeman, 1978). 2 T. Perry et al., Phys. Rev. E 54, 5617 (1996). 3 J. E. Bailey et al., Phys. Plasmas 16, 058101 (2009). 4 J. Bailey et al., Phys. Rev. Lett. 99, 265002 (2007). 5 G. A. Rochau et al., Phys. Plasmas 21, 056308 (2014). 6 T. Nagayama et al., Phys. Plasmas 21, 056502 (2014). 7 T. J. Nash et al., Rev. Sci. Instrum. 81, 10E518 (2010). 8 J. E. Bailey et al., Rev. Sci. Instrum. 79, 113104 (2008). 9 J. J. MacFarlane, J. Quant. Spectrosc. Radiat. Transfer 81, 287 (2003).

REVIEW OF SCIENTIFIC INSTRUMENTS 85, 11D604 (2014)

X-ray tests of a two-dimensional stigmatic imaging scheme with variable magnificationsa) J. Lu,1,b) M. Bitter,2 K. W. Hill,2 L. F. Delgado-Aparicio,2 P. C. Efthimion,2 N. A. Pablant,2 P. Beiersdorfer,3 T. A. Caughey,4 and J. Brunner4 1 Key Laboratory of Optoelectronic Technology and System of Ministry of Education, Chongqing University, Chongqing 400030, China 2 Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA 3 Physics Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA 4 Inrad Optics, 181 Legrand Avenue, Northvale, New Jersey 07647, USA

(Presented 4 June 2014; received 31 May 2014; accepted 2 July 2014; published online 22 July 2014) A two-dimensional stigmatic x-ray imaging scheme, consisting of two spherically bent crystals, one concave and one convex, was recently proposed [M. Bitter et al., Rev. Sci. Instrum. 83, 10E527 (2012)]. The Bragg angles and the radii of curvature of the two crystals of this imaging scheme are matched to eliminate the astigmatism and to satisfy the Bragg condition across both crystal surfaces for a given x-ray energy. In this paper, we consider more general configurations of this imaging scheme, which allow us to vary the magnification for a given pair of crystals and x-ray energy. The stigmatic imaging scheme has been validated for the first time by imaging x-rays generated by a micro-focus x-ray source with source size of 8.4 μm validated by knife-edge measurements. Results are presented from imaging the tungsten Lα1 emission at 8.3976 keV, using a convex Si-422 crystal and a concave Si-533 crystal with 2d-spacings of 2.21707 Å and 1.65635 Å and radii of curvature of 500 ± 1 mm and 823 ± 1 mm, respectively, showing a spatial resolution of 54.9 μm. This imaging scheme is expected to be of interest for the two-dimensional imaging of laser produced plasmas. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4890248] I. INTRODUCTION

X-ray 2D imaging systems with high spatial resolution and high spectral resolution that are suitable for recording and analyzing the compression of the imploded core of fuel capsules are very much needed in inertial confined fusion (ICF) research. X-ray monochromatic backlighting imaging of high-density plasmas, using spherically bent crystals, is one of the most effective target diagnostic methods in the ICF diagnostics.1–5 Spherically bent crystals offer many advantages for the imaging of plasmas: they provide a much higher luminosity than pin-hole cameras and high spatial resolution over a large field of view (FOV); and they can also be placed at large distances from the plasma where they are not at risk of being destroyed. However, there is also the disadvantage that they must be operated at Bragg angles near 90◦ in order to minimize the astigmatism, regardless of whether the image is formed by self-emitted radiation or backlighting of an object with an external x-ray source. Such restrictions on Bragg angles along with the fixed interplanar spacing (2d-spacing) of crystals limit the wavelength range that can be used for these diagnostics. A modification of the traditional x-ray monochromatic backlighting scheme is the x-ray crystal imaging microscope (XCIM),6–10 where the Bragg angles can deviate substantially a) Contributed paper, published as part of the Proceedings of the 20th

Topical Conference on High-Temperature Plasma Diagnostics, Atlanta, Georgia, USA, June 2014. b) Author to whom correspondence should be addressed. Electronic mail: [email protected] 0034-6748/2014/85(11)/11D604/4/$30.00

from 90◦ , e.g., 76.7◦ .11 It was, however, clearly pointed out by Labate et al.12 that a nearly monochromatic beam with a bandwidth of several mÅ, propagating in the meridional direction, can only be obtained if the source is placed on the Rowland circle. While in some applications,11 the X-ray source needs to be inside the Rowland circle to collect more photons, with the cost of smaller imaged area on the object. In such configurations, there is always a compromise between intensity and FOV. In a previous paper,13 we proposed a novel, twodimensional, stigmatic x-ray imaging system, which consists of two concentric, convex and concave, spherically bent crystals. This stigmatic x-ray imaging system, which is also designed for a monochromatic light source, has the unique properties that the Bragg condition is simultaneously fulfilled at each point on two crystal surfaces, and it allows choosing a wide range of Bragg angles. It also has an adjustable magnification, which is uniform in all directions. In principle, this imaging system could therefore be used for x-ray imaging of large, spatially extended objects as well as small objects, such as laser-produced plasmas, where spatial resolution of the order of microns would be required. We also presented tests with visible light, which verified this new imaging concept. In this paper, more general configurations of this imaging scheme, with theoretically arbitrary magnifications are considered. We present results from first tests of this imaging scheme with x-rays. The main goals of these x-ray tests were to verify that the Bragg condition is indeed satisfied at each point on the two crystal surfaces and to measure the achievable spatial resolution.

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which is the angle of a rotation of the ray pattern about an axis through the common center, M, perpendicular to the drawing plane in Fig. 1. We define α as positive for a counterclockwise rotation and negative for a clockwise rotation. By such a rotation, the object moves from O to O and the image moves from N to N . The magnification is then given by M=−

cos(2θ1 + α) . cos(2θ2 + α)

(2)

For a given x-ray energy and matched pair of crystals, one can therefore achieve arbitrary magnification for different applications by simply varying the rotation angle, α. B. Optical alignment

FIG. 1. Stigmatic imaging scheme with a rotation angle that implements variable magnifications with given Bragg angles. The object and image position move after rotating.

II. EXPERIMENT A. Concept

The concept of this x-ray imaging system consists of one concave spherically bent crystal, C1 , and of one convex spherically bent crystal, C2 , as illustrated in Fig. 1. Only x-rays with a certain wavelength λ, which satisfies the Bragg condition on both crystals, can be reflected. To eliminate the astigmatism, it is necessary that the Johann aberrations14 for the two crystals are exactly equaled and overlapped each other. These conditions are satisfied, if the two spherically bent crystals are concentric and if the radii of curvature, R1 and R2 , of the two crystals and the 2d-spacings of the Bragg reflecting crystal lattice planes are properly matched, such that R1 cos(θ 1 ) = R2 cos(θ 2 ), where θ 1 and θ 2 are Bragg angles of concave and convex crystal for wavelength λ. To assure that the two crystals and the image are on the same side of the object, the Bragg angles, θ 1 and θ 2 should be >45◦ and