## 2 Theory and Analysis  
To accomplish the stable radiation-pressure ion acceleration process, we employ a combination target consists of a solid foil and a near-critical-density plasma. An ultrashort, intense circularly polarized (CP) laser pulse is incident from the left side onto the combination target. In the stage of laser-foil interaction, we begin with some primitive theoretical analysis based on when a nanoscale solid foil is illuminated by an ultrashort laser pulse. Under ideal conditions, the obtained mean ion energy can be estimated as [38]  
\[E_{i}=m_{i}c^{2}\frac{\Pi^{2}}{2(\Pi+1)} \tag{1}\]  
where \(\Pi=2m_{e}n_{e}a_{0}^{2}\tau c/\sum_{i}m_{i}n_{i}L\), \(c\) is the speed of light in vacuum, the normalized amplitude of the CP laser pulse \(a_{0}\simeq(l_{0}\lambda^{2}/2.74\times 10^{18}\) W cm\({}^{-2}\)\(\upmu\)m\({}^{2})^{1/2}\), \(n_{c}=m_{e}\omega^{2}/4\pi e^{2}\) is the critical density, \(\omega\) is the angular frequency of the laser, \(\tau\) is the duration of the laser pulse, \(L\) is the foil thickness, \(m_{e}\) is the electron mass, and \(m_{i}\) and \(n_{i}\) are the mass and the number density of the i-th ion species, respectively. For instance, in a typical simulation case discussed in the next section, we employ a 30 fs CP laser pulse at normalized amplitude \(a_{0}=100\) (\(I_{0}\simeq 2.74\times 10^{22}\) W cm\({}^{-2}\)). Here, a laser wavelength \(\lambda=1\)\(\upmu\)m is assumed and \(T_{0}=\lambda/c\) is the laser period. The foil is composed of carbon and hydrogen ions with \(n_{e}=200n_{c}\) and a thickness \(L=0.085\lambda\). Substituting these parameters into Equation (1), we can easily get a mean proton energy \(E_{p}\simeq 1.35\) GeV. As a matter of fact, the RPA process is prematurely terminated as long as the target becomes transparent due to the transverse instabilities,the target deformation, and the plasma heating [39]. Furthermore, in the intense laser interaction with the foil, the longitudinal electric field induced by charge separation travels close to the speed of light as the foil target becomes transparent. Meanwhile, the amplitude of the longitudinal electric field, which is induced by charge separation, will drop rapidly. Therefore, the obtained proton energy is much lower than the prediction by Equation (1). Our two-dimensional (2D) particle-in-cell (PIC) simulation with the above laser and target parameters results in a broad spectrum with mean proton energy \(E_{p}\simeq 500\) MeV, which corresponds to a sub-relativistic mean proton speed \(v_{p}\simeq 0.73c\). To date, the corresponding optimum foil thickness, and/or the proton energy scaling with the fitted laser temporal profile have been verified [40; 41] by the code EPOCH [42] of particle-in-cell simulations. To maintain an efficient RPA process, a near-critical-density (NCD) plasma is applied to suppress the plasma heating by continuously replenishing cold electrons. More importantly, a longitudinal electric field with large amplitude is excited as the ultra-intense laser pulse propagates in relativistically transparent regime. The considerable amplitude of this excited electric field can be roughly approximated by \(E_{max}\approx a_{0}\omega m_{e}c/e\)[43]. This electric field will boost the speed of the protons dramatically. For simplicity, we assume that the local accelerating field felt by the protons is \(E_{x}(x)\sim E_{max}/2\), the Lorentz factor of the protons \(\gamma_{p}=1/(1-v_{p}^{2}/c^{2})^{1/2}\) can be deduced from \(d\gamma_{p}/dx\approx\pi a_{0}m_{e}/\lambda m_{p}\)[26]. For \(a_{0}=100\) and \(v_{p}\simeq 0.73c\) (the corresponding \(E_{p}\simeq 500\) MeV) obtained in the laser-foil interaction stage, the sub-relativistic protons will be accelerated rapidly to relativistic speed in tens of microns. The necessity of a near-critical-density plasma can be analyzed by a conservative Hamiltonian of ion motion [43].