2 Model analysis

Figure 1 shows the X-ray polarization converter based on linear TIR. The X-ray laser pulse is obliquely incident on the surface of a solid-density plasma with a refractive index less than unity. TIR occurs^[ Reference Carniglia and Brownstein ^{32 ,} Reference Liu, Chen, Parrott, Ung, Xu and Pickwell-MacPherson ^{33 ]} if the incident angle ${\theta}_{\mathrm{i}}$ exceeds the critical angle ${\theta}_{\mathrm{c}}=\arcsin \left({n}_2/{n}_1\right)=\arcsin \left({n}_\mathrm{p}\right)$ , where ${n}_1=1$ and ${n}_2={n}_{\mathrm{p}}<1$ are the refractive indices of the vacuum and plasma, respectively. This process is governed by the Fresnel equations, which describe the amplitude and phase relationships of the reflected and transmitted waves at interfaces. Under the boundary continuity condition (i.e., matching of electromagnetic field components at the interface), TIR induces not only incident and reflected waves but also evanescent waves penetrating the plasma medium. This surface wave acts as an energy buffer, temporarily storing and redistributing energy flux to maintain conservation during the TIR^[ Reference Milosevic ^{34 ]}. The $x$ and $y$ components of the laser’s Poynting vector of the surface wave are as follows:

(1) $$\begin{align}{S}_{\mathrm{P}x}&=-\frac{k_\mathrm{t}}{2w}\left(\frac{\sin {\theta}_{\mathrm{i}}}{n_{\mathrm{p}}}\right){D}_0^2\left(1+\cos \left(2\left({k}_\mathrm{t}\frac{\sin {\theta}_{\mathrm{i}}}{n_{\mathrm{p}}}x-\omega t\right)\right)\right),\end{align}$$

(2) $$\begin{align}{S}_{\mathrm{P}y}&=\frac{k_\mathrm{t}}{2w}\sqrt{\frac{\sin^2{\theta}_{\mathrm{i}}}{n_{\mathrm{p}}^2}-1}\;{D}_0^2\sin 2\left({k}_\mathrm{t}\frac{\sin {\theta}_{\mathrm{i}}}{n_{\mathrm{p}}}x-\omega t\right),\end{align}$$

(3) $$\begin{align}{D}_0&={e}_z{E}_0^\mathrm{t}\exp \left(-{k}_\mathrm{t}{\left(\frac{\sin^2{\theta}_{\mathrm{i}}}{n_{\mathrm{p}}^2}-1\right)}^{\frac{1}{2}}y\right),\end{align}$$

Figure 1

(a) Schematic of the proposed scheme. (b) Total internal reflection at the plasma surface. Here, E denotes the electric field, while the superscripts i, t and r denote the incident, refracted and reflected light, respectively. The subscripts s and p denote the s- and p-polarization components of the laser.

where ${k}_\mathrm{t}$ is the wave vector in the $x$ -direction, $\omega$ denotes the laser angular frequency and ${E}_0^\mathrm{t}$ represents the electric field amplitude of the refracted light. Within half a laser period, the energy of the surface wave enters the plasma and is stored in a thin layer near the interface. In the other half of the period, this energy is released and becomes the energy of the reflected wave. However, the average value is zero within the same period. Therefore, during TIR, the instantaneous energy flow in the direction perpendicular to the interface within the penetration depth is not zero, but the average energy flow is zero. That is, the conversion from incident to reflected waves is mediated by the surface wave’s energy oscillation, leading to a phase shift in the reflected polarization states.

In linear TIR, we have $\sin {\theta}_{\mathrm{i}}>{n}_{\mathrm{p}}$ , $\cos {\theta}_{\mathrm{t}}=i\sqrt{\sin^2{\theta}_{\mathrm{i}}/{n}_{\mathrm{p}}^2-1}$ , where ${\theta}_{\mathrm{i}}$ and ${\theta}_{\mathrm{t}}$ represent the incident and refracted angles, respectively. By substituting the Fresnel equations and performing algebraic manipulations, the following relations can be readily obtained^[ Reference Jackson ^{35 ]}:

(4) $$\begin{align}\frac{E_{\mathrm{s}}^{\mathrm{r}}}{E_\mathrm{s}^\mathrm{i}}={e}^{-i{\delta}_\mathrm{s}},\kern1em \tan \frac{\delta_\mathrm{s}}{2}=\frac{\sqrt{\sin^2{\theta}_{\mathrm{i}}-{n}_{\mathrm{p}}^2}}{\cos {\theta}_{\mathrm{i}}},\end{align}$$

(5) $$\begin{align}\frac{E_\mathrm{p}^\mathrm{r}}{E_\mathrm{p}^\mathrm{i}}={e}^{-i{\delta}_\mathrm{p}},\kern1em \tan \frac{\delta_\mathrm{p}}{2}=\frac{\sqrt{\sin^2{\theta}_{\mathrm{i}}-{n}_{\mathrm{p}}^2}}{n_{\mathrm{p}}^2\cos {\theta}_{\mathrm{i}}},\end{align}$$

where ${E}_\mathrm{s}^\mathrm{i}$ , ${E}_\mathrm{p}^\mathrm{i}$ , ${E}_\mathrm{s}^\mathrm{r}$ and ${E}_\mathrm{p}^\mathrm{r}$ are the electric field amplitudes of the s-polarized (perpendicular to the incident plane) and p-polarized (parallel to the incident plane) components of the incident and reflected light, respectively. It can be seen that the electric field magnitudes of the p- and s-polarized components of the reflected light are the same as that of the incident laser, or $\mid {E}_\mathrm{s}^\mathrm{r}\mid =\mid {E}_\mathrm{s}^\mathrm{i}\mid$ , $\mid {E}_\mathrm{p}^\mathrm{r}\mid =\mid {E}_\mathrm{p}^\mathrm{i}\mid$ . However, both undergo phase shifts ${\delta}_\mathrm{s}$ and ${\delta}_\mathrm{p}$ relative to incident light, and the phase difference $\delta ={\delta}_\mathrm{p}-{\delta}_\mathrm{s}$ is given by the following:

(6) $$\begin{align}\tan \left(\frac{\delta }{2}\right)=\tan \left(\frac{\delta_\mathrm{p}-{\delta}_\mathrm{s}}{2}\right)=\frac{\cos \left({\theta}_{\mathrm{i}}\right)\sqrt{\sin^2\left({\theta}_{\mathrm{i}}\right)-{\left({n}_{\mathrm{p}}\right)}^2}}{\sin^2\left({\theta}_{\mathrm{i}}\right)}.\end{align}$$

In this model, the X-ray laser propagates in a vacuum and undergoes TIR at the plasma surface, enabling conversion of the polarization state. For X-rays, the refractive index in the solid-density plasma is given by ${n}_{\mathrm{p}}=\sqrt{1-{n}_\mathrm{e}/{n}_\mathrm{c}}$ . For a relatively short-wavelength X-ray laser, ${n}_\mathrm{e}<{n}_\mathrm{c}$ and ${n}_{\mathrm{p}}<1$ , where ${n}_\mathrm{e}$ denotes the electron density, and the critical density ${n}_\mathrm{c}$ is given by ${n}_\mathrm{c}={m}_\mathrm{e}{\omega}_0^2/4\pi {e}^2={n}_0/{\lambda}_{\mathrm{nm}}^2$ . Here, ${m}_\mathrm{e}$ is the electron mass, ${\omega}_0$ is the central angular frequency and $e$ is the elementary charge. The parameter ${n}_0\approx 1.1\times {10}^{33}\;{\mathrm{m}}^{-3}$ , and ${\lambda}_{\mathrm{nm}}$ is the laser wavelength normalized to nanometers (nm). Letting $\alpha \equiv \tan \left(\frac{\delta }{2}\right)$ and substituting the expression of ${n}_{\mathrm{p}}$ in Equation (6), we get the following:

(7) $$\begin{align}{\lambda}_{\mathrm{nm}}=\sqrt{\frac{n_0}{n_\mathrm{e}}}\times \sqrt{1-{\sin}^2{\theta}_{\mathrm{i}}\left(1-{\alpha}^2{\tan}^2{\theta}_{\mathrm{i}}\right)}.\end{align}$$

This equation describes the relation among the laser wavelength, the incident angle and the phase shift. The value of $\delta$ can range from $0$ to $\pi /2$ , so that $0<\alpha \le 1$ , where $\alpha =1$ refers to the conversion from linear to circular polarization. Equation (7) suggests an upper limit of the laser wavelength as ${\lambda}_{\mathrm{nm}}<\sqrt{\frac{n_0}{n_\mathrm{e}}}$ , which corresponds to the condition of ${n}_\mathrm{e}<{n}_\mathrm{c}$ . For example, for a solid-density Al plasma with ${n}_{\mathrm{e}}=7.8\times {10}^{29}\;{\mathrm{m}}^{-3}$ , this gives an upper limit of $37.5\;\mathrm{nm}$ . Namely, for an X-ray laser with ${\lambda}_{\mathrm{nm}}<37.5\;\mathrm{nm}$ , conversion from linear to elliptic (including circular) polarization can be achieved.

It is interesting to note that for oblique incidence, the effective critical density for laser reflection is reduced by ${n}_\mathrm{c}^{\prime }={n}_\mathrm{c}\cos {\theta}^2$ , where $\theta$ is the incident angle^[ Reference Kruer ^{36 ]}. This indicates that laser reflection could occur for ${n}_\mathrm{e}>{n}_\mathrm{c}^{\prime }$ . For the case of TIR, the critical angle for TIR can be written as ${\theta}_\mathrm{c}=\arcsin \left(\sqrt{1-\frac{n_\mathrm{e}}{n_\mathrm{c}}}\right)$ . That is, the critical angle corresponds to ${n}_\mathrm{e}={n}_\mathrm{c}\cos {\theta}_\mathrm{c}^2={n}_\mathrm{c}^{\prime }$ . When the incident angle $\theta$ exceeds the critical angle ${\theta}_\mathrm{c}$ , TIR occurs. This actually also corresponds to the condition of ${n}_\mathrm{e}>{n}_\mathrm{c}^{\prime }$ . Thus, these two descriptions actually are equivalent.

3 Particle-in-cell simulations

To verify the above scheme, we have performed a two-dimensional (2D) PIC simulation using EPOCH code^[ Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers ^{37 ]}. Firstly, we demonstrated the feasibility of polarization state conversion, where linearly polarized (LP) X-ray laser pulses can be converted into circularly polarized (CP) or elliptically polarized (EP) laser pulses. The LP X-ray laser pulse is of incident angle ${\theta}_{\mathrm{i}}=41.3{}^{\circ}$ (far exceeding the critical angle of about $18.4{}^{\circ}$ for the parameters given here), wavelength of ${\lambda}_0=36\;\mathrm{nm}$ , (full width at half-maximum) duration of $\tau =2.12\;\mathrm{fs}$ and radius of $5.6{\lambda}_0$ . The incident light is initially LP and consists of both p-polarized and s-polarized components with identical phase and amplitude. The laser intensity is $I=8\times {10}^{18}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ , so that relativistic amplitude of laser ${a}_0={eE}_0/{m}_\mathrm{e}{\omega}_0c=0.087$ , where ${E}_0$ is the laser amplitude and $c$ is the speed of light in vacuum. The laser pulse is focused on the target, which is a fully ionized solid aluminum plasma slab of electron density ${n}_{\mathrm{e}}=7.8\times {10}^{29}\kern0.1em {\mathrm{m}}^{-3}$ , or approximately $0.9{n}_\mathrm{c}$ . The initial temperature of the electrons is $1\;\mathrm{keV}$ . Such plasma can be experimentally generated by using a double-laser irradiation scheme^[ Reference Beier, Allison, Efthimion, Flippo, Gao, Hansen, Hill, Hollinger, Logantha, Musthafa, Nedbailo, Senthilkumaran, Shepherd, Shlyaptsev, Song, Wang, Dollar, Rocca and Hussein ^{38 ]}, which has led to very homogeneous, micrometer-scale, solid-density plasma with a well-defined structure and temperature ranging from $0.25$ to $2.5\;\mathrm{keV}$ . In this case, the assumption of an ideal plasma is applicable, and the ionization effect of the X-ray laser can also be neglected.

The simulation domain extends from $0$ to $120{\lambda}_0$ in the $x$ -direction and from $-20{\lambda}_0$ to $100{\lambda}_0$ in the $y$ -direction. The simulation uses an ultra-high-resolution 2D grid configuration with $7680\times 7680$ grid points, a grid spacing of ${\lambda}_0/64$ (corresponding to 64 points per wavelength) and a high-density particle distribution ( $10$ particles per grid point). The surface wave skin depth is ${l}_{\mathrm{skin}}={\lambda}_0/\left(2\pi {n}_{\mathrm{p}}\sqrt{\sin^2{\theta}_{\mathrm{i}}/{n}_{\mathrm{p}}^2-1}\right)\approx 0.94{\lambda}_0$ . The resolution of the simulation is thus much smaller than ${l}_{\mathrm{skin}}$ , ensuring that the physical processes in the plasma boundary layer are adequately resolved.

Figure 2(a) shows the electric field distribution of the incident X-ray laser, and Figure 2(b) shows the electric field distribution at the moment of laser–plasma interaction. It can be observed that surface waves moving along the interface exist within the plasma medium. Figure 2(c) shows the electric field distribution of the reflected X-rays. The reflectivity reaches $99.97\%$ , indicating that almost all incident light is reflected. This high reflectivity not only verifies the effectiveness of the polarization conversion mechanism based on TIR, but also holds significant implications for practical applications.

Figure 2

Electric field distributions in the laser–plasma interaction. Electric field of the incident laser pulse at (a) $t=60{T}_0$ and (b) at $t=82{T}_0$ , as it reaches the surface. (c) Electric field distribution of the circularly polarized reflected wave at $t=120{T}_0$ . Electric field on the laser axis of (d) the incident laser and (e) the reflected pulse. In the above, the laser incidence angle is $41.3{}^{\circ}$ . (f) Electric field of the reflected pulse for $0{}^{\circ}$ incidence angle. In all the lower panels, the 3D field vector (red), profiles of the two orthogonal field components ${E}_\mathrm{s}$ (green) and ${E}_\mathrm{p}$ (blue) and the projection of ${E}_\mathrm{s}-{E}_\mathrm{p}$ (orange) are displayed.

The LP incident laser waveform, as illustrated in Figure 2(d), exhibits an electric field comprising two orthogonal components (p- and s-components) ${E}_\mathrm{p}^{\mathrm{i}}$ and ${E}_\mathrm{s}^{\mathrm{i}}$ with identical amplitudes and phases ${\phi}_\mathrm{p}^{\mathrm{i}}={\phi}_\mathrm{s}^{\mathrm{i}}$ . Figure 2(e) displays the reflected laser electric field from the target surface, showing that the polarization state has been converted into left-handed circular polarization with Stokes parameter $P=-0.95\pm 0.05$ . The closer the absolute value of $P$ is to 1, the higher the degree of circular polarization. Specifically, $P>0$ indicates right-handed circular polarization, while $P<0$ indicates left-handed circular polarization. Due to the TIR effect induced by oblique incidence, the phase difference between the reflected field components ${E}_\mathrm{p}^{\mathrm{r}}$ and ${E}_\mathrm{s}^{\mathrm{r}}$ shifts from $0$ to $\pi /2$ approximately. To further verify this phenomenon, Figure 2(f) shows the reflected laser electric field under normal incidence ( $\theta =0{}^{\circ}$ ). Since TIR did not occur, the laser undergoes refraction and reflection at the interface, resulting in the reflected optical electric field being much smaller than the incident optical electric field, and the polarization state of the reflected laser pulse remains linear.

When the incident angle exceeds the critical angle for TIR, the refracted wave is converted into a surface wave. This surface wave then ultimately transforms into the reflected wave, and it is precisely this conversion sequence that gives rise to the change in the phase difference of the reflected wave. To visualize this dynamic process, Figure 3(a), derived from 2D PIC simulations, illustrates the s-component electric field of the surface wave ${E}_\mathrm{s}^{\mathrm{t}}$ at various instants. When $t=82{T}_0$ , where ${T}_0$ represents the characteristic period of the incident laser, the moment of interaction between the laser and the plasma surface occurs. At this juncture, ${E}_\mathrm{s}^{\mathrm{t}}$ reaches its maximum value. As time progresses, it is evident that the amplitude of the surface wave gradually diminishes, and during this period, the surface wave undergoes a gradual transformation into the reflected wave, completing the TIR process.

Figure 3

(a) The s-polarized electric field of the surface wave ${E}_\mathrm{s}^{\mathrm{t}}$ at different moments. (b) The transmission coefficient of ${E}_\mathrm{s}^{\mathrm{t}}$ at different incident angles.

Figure 3(b) shows the transmission coefficients of ${E}_\mathrm{s}^{\mathrm{t}}$ under different incident angles. As the incident angle increases, the electric field amplitude of the surface wave gradually weakens, and this phenomenon is consistent with the following theoretical formula:

(8) $$\begin{align}\mid {t}_\mathrm{s}\mid =\left|\frac{E_\mathrm{s}^\mathrm{t}}{E_\mathrm{s}^\mathrm{r}}\right|=\frac{2\cos {\theta}_{\mathrm{i}}}{\sqrt{1-{n}_{\mathrm{p}}^2}},\end{align}$$

where ${t}_\mathrm{s}$ is the transmission coefficient and ${E}_\mathrm{s}^\mathrm{t}$ and ${E}_\mathrm{s}^\mathrm{r}$ represent the electric fields of the surface wave (transmitted wave) and the reflected wave, respectively. This formula, grounded in the fundamental principles of electromagnetic wave propagation at the plasma–vacuum interface, provides a theoretical framework for predicting the behavior of surface waves. The close match between the simulation results and the theoretical formula, as shown in Figure 3(b), further validates the accuracy of our theoretical model.

Next we demonstrate that this plasma-based polarization modulator is tunable and capable of achieving a controllable phase shift between two orthogonal field components. Specifically, in TIR process, the phase difference between p- and s-polarized components varies with the incident angle, as indicated in Equation (6), which endows this principle with continuous tunability. Figure 4(a) shows the simulation and theoretical results of the Stokes parameter of the reflected laser under different incident angles. It can be seen that the simulation results agree well with the theoretical results, and demonstrate the tunability of the polarization degree in the present scheme.

Figure 4

(a) The Stokes parameter P under different incident angles. (b) Reflection efficiencies for different normalized laser amplitudes ${a}_0$ . (c) The laser intensity distribution of the reflected wave when the normalized laser amplitude ${a}_0=0.087$ . (d) The laser intensity distribution of the reflected wave when the normalized laser amplitude ${a}_0=0.87$ . To display the distribution of the transmitted laser, the magnitude of the colorbar is reduced by two orders of magnitude.

We further consider the influence of laser amplitude ${a}_0$ . The normalized amplitude ${a}_0= eE/{m}_\mathrm{e}\omega c$ , as a core Lorentz invariant, holds the following physical significance: when ${a}_0\ge 1$ , the laser intensity enters the fully relativistic regime. In the near-threshold region ( $0.1<{a}_0<1$ ), the plasma target structure can already be modified through the density perturbation mechanism (see Figure 5(d)). As shown in Figure 4(b), although the overall conversion efficiency gradually decreases with increasing ${a}_0$ , it remains above $95\%$ across the studied intensity range. Furthermore, we verified that even under specific extreme parameters with a wavelength of 10 nm and a laser intensity of ${10}^{22}\ \mathrm{W}/{\mathrm{cm}}^2$ (the corresponding laser power is 1 TW), the reflection efficiency still reaches $96.37\%$ , confirming that TIR can be stably maintained within this high-efficiency regime. It is noted that for ${a}_0=0.087$ (Figure 4(c)), no detectable light intensity is observed behind the target, confirming that the laser energy is predominantly retained via TIR. In contrast, when ${a}_0=0.87$ (Figure 4(d)), a distinct transmitted signal emerges behind the target, demonstrating that a high-intensity laser can breach TIR constraints and penetrate the plasma target directly.

Figure 5

(a) Electron density and electric field variation at the interface for ${a}_0=0.87$ . (b) Electric field of surface wave versus ${a}_0$ . (c) Electron density distribution for ${a}_0=0.087$ . (d) Electron density distribution for ${a}_0=0.87$ .

The physical origin of these transmission disparities lies in the dynamics of surface plasmon polaritons (SPPs). During TIR, surface waves propagate along the vacuum–plasma interface, inducing oscillating electric fields that generate surface charges. Collective oscillations of these charges form SPPs^[ Reference Zhang, Zhang and Xu ^{39 ]}. As the laser intensity increases, the electric field of the surface wave grows, amplifying electron oscillations and promoting the excitation of SPPs. Figure 5(a) demonstrates the synchronous evolution of plasma electron density perturbations (depicted by the red curve) and the electric field along the interface (represented by the blue curve), capturing the critical phase of laser–plasma interaction. The SPPs cause the electric field of the total-reflection-induced surface wave at the interface to be slightly lower than the theoretical value, as shown in Figure 5(b), where the amplitude of the electric field of the surface wave can be theoretically expressed as follows:

(9) $$\begin{align}{E}_{\mathrm{surf}}=\frac{2i\cos {\theta}_{\mathrm{i}}\sqrt{n_{\mathrm{p}}^2-{\sin}^2{\theta}_{\mathrm{i}}}}{n_{\mathrm{p}}^2\cos {\theta}_{\mathrm{i}}+i\sqrt{n_{\mathrm{p}}^2-{\sin}^2{\theta}_{\mathrm{i}}}}\cdot \frac{m_\mathrm{e}c{\omega}_0}{e}\cdot {a}_0. \end{align}$$

Figures 5(c) and 5(d) show that plasma electron density responses vary significantly with ${a}_0$ . At low amplitudes ( ${a}_0=0.087$ , Figure 5(c)), electron density perturbations remain minimal, and the interface remains smooth. At high amplitudes ( ${a}_0=0.87$ , Figure 5(d)), intense SPP excitation induces the formation of protrusive structures at the plasma–vacuum interface, accompanied by drastic electron density redistribution. These structures alter the laser incidence angle and plasma density gradient, allowing partial laser transmission around TIR conditions. Quantitatively, increasing ${a}_0$ from 0.087 to 0.87 decreases the reflection efficiency from 99.97% to 96.31%, directly validating the SPP-induced transmission enhancement. These findings deepen our understanding of laser–plasma interactions and offer insights for optimizing experiments where TIR efficiency matters.

5 Concluding remarks

We present a novel approach to achieve polarization conversion of high-power, high-intensity X-ray lasers via oblique reflection from a solid-density plasma target. Leveraging the refractive index of solid-density plasma (which is less than unity in the X-ray regime), TIR occurs when the X-ray laser is incident at angles exceeding the critical angle. This configuration enables compact and efficient polarization conversion. Particle-in-cell simulations confirm this plasma-based polarization modulator, demonstrating not only its high damage threshold and efficiency (up to 95% conversion efficiency) but also its tunability through adjustment of the incident angle. This combination of compactness, robustness, efficiency and adjustability positions the scheme as a versatile tool with potential applications in high-energy-density physics, advanced X-ray laser systems and related research frontiers.