<p> Particle acceleration in solar flares and its contribution to coronal heating are among the main&nbsp; unsolved problems in heliophysics. Accelerated electrons in a plasma radiate hard X-ray (HXR)&nbsp; emission through the well-known process of bremsstrahlung. HXR observations therefore are a&nbsp; powerful diagnostic tool, providing quantitative measurements of flare-accelerated electrons. Since&nbsp; bremsstrahlung emission depends on the density of the ambient medium, solar HXR emission is&nbsp; usually brightest from below the transition region, where the density increases rapidly towards the&nbsp; photosphere. Electron beams entering the chromosphere lose energy quickly through collisions and&nbsp; produce relatively intense HXR emission at the footpoints of magnetic field lines. Electron beams&nbsp; moving in the relatively tenuous corona suffer very few collisions, losing little energy and producing&nbsp; only faint HXR emission. Present-day HXR instrumentation does not have the sensitivity to see&nbsp; faint HXR emission from electrons traveling in the corona, nor the dynamic range to see such&nbsp; faint emission in the presence of bright HXR footpoint emission in the chromosphere. Existing&nbsp; observations therefore show us only where energetic electrons are stopped, but not where they&nbsp; are accelerated, nor along what path they escape from the acceleration site. The most sensitive&nbsp; solar HXR observations so far are provided by the Reuven Ramaty High Energy Spectroscopic&nbsp; Imager (RHESSI) (Lin et al. 2002). These measurements are obtained with a non-focusing rotation&nbsp; modulation collimator (RMC) imaging technique (Hurford et al. 2002). RMCs and other types&nbsp; of non-focusing imaging, however, have intrinsically limited dynamic range and sensitivity. HXR&nbsp; focusing optics can overcome both of these limitations (Section 1.2.2).&nbsp;</p> <p> The Focusing Optics X-ray Solar Imager (FOXSI) is a sounding rocket payload funded under the&nbsp; NASA Low Cost Access to Space (LCAS) program to test HXR focusing optics combined with&nbsp; silicon strip detectors for solar observations (Krucker et al. 2009). The FOXSI program is being led&nbsp; by the Space Sciences Laboratory at UC Berkeley in collaboration with the Marshall Space Flight&nbsp; Center (MSFC) and the Japan Aerospace Exploration Agency (JAXA). FOXSI is on schedule&nbsp; and on budget for a launch in October 2010. FOXSI will offer imaging spectroscopy and&nbsp; unprecedented HXR sensitivity and dynamic range. FOXSI will be !100 times more sensitive than&nbsp; RHESSI at 10 keV, and, for the first time, detect the non-thermal counterparts of quiet sun network&nbsp; flares (Section 1.2.4).&nbsp;</p> <p> Here we propose a continuation of the FOXSI program which includes data analysis&nbsp; and a second flight with an upgraded version of FOXSI. At moderate cost, we propose to&nbsp; enhance the effective area, in particular at higher energies (by a factor of !4 at 15 keV), by adding&nbsp; 3 more shells to the existing 7-shell optics (see Figure 9). Furthermore, our Japanese collaborators&nbsp; will provide, at no cost, newly available double-sided cadmium telluride (CdTe) detectors as&nbsp; a replacement for the Si detectors to allow us to take full advantage of the effective area at higher&nbsp; energies. A second flight will therefore not only allow us to continue testing HXR focusing&nbsp; optics for solar observations and also test newly developed CdTe strip detectors&nbsp; in flight but is also expected to provide a significant increase in scientific return. In&nbsp; this two year proposal, the first year (2011) will be used to upgrade the FOXSI payload and to&nbsp; analyze data from the first flight, while the second flight is planned for the midd