Berry Curvature Dipole-Induced Chiral Terahertz Gain and Lasing Threshold in Bulk Tellurium

TL;DR

The paper investigates Berry curvature dipole–driven, non-Hermitian electro-optic gain in bulk $n$-doped Tellurium to realize polarization-selective terahertz amplification and lasing. By analyzing three configurations of DC bias and wave propagation, it derives dispersion relations, polarization eigenstates, and gain thresholds, identifying regimes where one mode amplifies (negative dissipated power) while the other dissipates. Lasing conditions are then solved for a simple Fabry–Perot cavity, revealing bias thresholds below the material’s breakdown and highlighting how Lorentz resonances around a few terahertz shape the mode spectrum and enable higher-order mode lasing at reduced biases. Collectively, the results establish bulk Te as a practical, electrically tunable, polarization-controlled THz laser medium with potential for compact, chiral photonic devices and topological light–matter interactions in 3D materials.

Abstract

We investigate the use of Berry curvature dipole in $n$-doped Tellurium as a mechanism for achieving terahertz amplification and lasing by applying a DC electric field. When the electrical bias and wave vector are aligned along the trigonal $c$-axis, the right-handed circularly polarized mode experiences amplification at relatively low bias, while the left-handed mode is attenuated. Furthermore, when the electrical bias and wave vector are orthogonal to the $c$-axis, the structure supports elliptically polarized eigenmodes that also exhibit gain under suitable bias conditions, where the degree of ellipticity is tunable by the applied bias. We also investigate lasing conditions for a Fabry-Perot cavity incorporating biased Te as an active medium. Due to the resonance in the dielectric permittivity of Tellurium, there are discrete lasing intervals. Our results show that bulk chiral Tellurium could be used as an electrically tunable, polarization-selective gain medium for micrometer-scale terahertz lasers, with lasing achievable at bias fields below the material's breakdown threshold, paving the way towards new terahertz devices.

Figures (16)

• Figure 1: Schematic illustration of DC-biased tellurium interacting with optical fields. (a) Molecular packing of right-handed Tellurium showing the crystallographic $a$-, $b$- and $c$-axes relative to the laboratory-frame $x$, $y$, $z$ coordinate system asendorf1957space. (b) Case I: Both the optical wave propagation direction and the applied bias are aligned along the $z$ direction. (c) Case II: The bias is applied along $z$, while the optical wave propagates along $x$. (d) Case III: both the optical wave propagation direction and the applied bias are aligned along the $x$ direction. In all cases, $L$ is the length of the cavity along which the lasing occurs.
• Figure 2: (a) Dissipated power density for Case I, showing results for mode 1 (solid curves) and mode 2 (dashed curves). Negative values indicate optical amplification. For a bias of $E_0 = 5 \times 10^4 \;\mathrm{V/m}$, amplification of mode 1 occurs for $f > 12.95\;\mathrm{THz}$. Increasing the bias to $E_0 = 10^5 \;\mathrm{V/m}$ broadens the amplification range to $1.05\ \mathrm{THz} < f < 1.27\ \mathrm{THz}$ and for $f > 7.65\;\mathrm{THz}$. (b) Real and imaginary parts of the wavenumber $k_1$ for different values of the static electric bias. The zero crossings of $\alpha_1$ align exactly with the amplification regions shown in (a). (c) Real and imaginary parts of the wavenumber $k_2$ for different values of the static electric bias. Both $\beta_2$ and $\alpha_2$ remain positive across all frequencies, indicating that this mode is always decaying.
• Figure 3: (a) Dissipated power density for Case II, showing results for mode 1 (solid curves) and mode 2 (green dashed curves). Negative values indicate optical amplification. For a bias of $E_0 = 2\times10^5\;\mathrm{V/m}$, mode 1 exhibits amplification in the range $0.50\ \mathrm{THz} < f < 1.49\ \mathrm{THz}$. Increasing the bias to $E_0 = 4\times10^5\;\mathrm{V/m}$ expands the amplification range to $0.45\;\mathrm{THz} < f < 1.95\;\mathrm{THz}$ and for $f > 5.06\ \mathrm{THz}$. The dissipated power for mode 2 remains strictly positive and is independent of the applied static bias. (b) Real and imaginary parts of the wavenumber $k_1$ for different values of the static electric bias. (c) Real and imaginary parts of the wavenumber $k_2$, which do not depend on $E_0$.
• Figure 4: Polarization eigenstates for Case II: (a) Amplifying mode (mode 1) at three representative frequencies, showing elliptical polarization. Results are plotted for two static electric field biases, $E_0 = 4 \times 10^5\;\mathrm{V/m}$ (solid curves) and $E_0 = 6 \times 10^5\;\mathrm{V/m}$ (dashed curves). (b) Decaying mode (mode 2) exhibits linear polarization aligned along the $z$-axis.
• Figure 5: (a) Dissipated power density for Case III, showing results for mode 2 (solid curves) and mode 1 (dashed curves). Negative values indicate optical amplification. For a bias of $E_0 = 4 \times 10^5 \;\mathrm{V/m}$, mode 2 exhibits amplification in the range $0.9\;\mathrm{THz} < f < 1.6\;\mathrm{THz}$. Increasing the bias to $E_0 = 6 \times 10^5 \;\mathrm{V/m}$ expands the amplification range to $0.7\;\mathrm{THz} < f < 1.8\ \mathrm{THz}$ and for $f > 6\;\mathrm{THz}$. (b) Real and imaginary parts of the wavenumber $k_1$ for different values of the static electric bias. Both $\beta_1$ and $\alpha_1$ remain positive across all frequencies, indicating that this mode is always decaying. (c) Real and imaginary parts of the wavenumber $k_2$ for different values of the static electric bias.