We report a striking $E^4$ energy dependence of transition-radiation (TR) intensity, observed at Brewster's angle in the optical region and emitted by ultrarelativistic charged particles traversing a specially engineered one-dimensional photonic crystal (PhC). This strong scaling—absent in conventional Cherenkov or TR—persists until saturation sets in at Lorentz factors $\gamma \sim \sqrt{N_s}$ (where $N_s$ is the number of unit cells). The effect originates from a Dirac conelike band structure in the dispersion relation of the PhC, analogous to that of graphene, which emerges when the slab thicknesses within each period satisfy the proportion $a_b/a_{\varepsilon }=(\sqrt{\varepsilon +b}-\varepsilon )/(b-\sqrt{\varepsilon +b})$. Remarkably, for both this periodic structure and a complementary class of disordered stacks exhibiting Brewster randomness [Z. Gong et al., Proc. Natl. Acad. Sci. USA 122, e2413336122 (2025)], the radiation problem admits an exact analytical solution at forward and backward Brewster observation angles. In the disordered case, a strong $N_s^2$ dependence of intensity on slab number $N_s$ yielding pronounced directivity either in the forward or backward direction, depending on the type of randomness, was reproduced. The radiation formation zone remains limited ($\sim \lambda /2$) even for ultrarelativistic particles—unlike x-ray TR detectors—making this system a promising platform for compact, energy-sensitive detection of relativistic particles in the optical regime.

Close

PrevNext

Reuse & PermissionsView figure in article

• 
• 
• 
• 
• 

• Electromagnetic field calculations
• Photonic crystals
• Radiation from moving charges