Observation of optical feedback dynamics in single-mode terahertz quantum cascade lasers: Transient instabilities

We provide an experimental evidence of transient instabilities (TIs) in a terahertz (THz) quantum cascade laser (QCL) under optical feedback, in contrast to the widely accepted claim that THz QCLs are ultrastable against feedback. The TIs appear as periodic oscillations in emitted power or terminal voltage of the laser with an increasing oscillation frequency as feedback increases. The absence of relaxation oscillations and low linewidth enhancement factor in THz QCLs makes them a platform uniquely suitable for exploring external-cavity-related dynamics in semiconductor lasers. This work opens a pathway to a THz sensing and imaging modality based on these TIs, which has much reduced complexity compared to existing approaches using laser feedback interferometry.

Figure 1

Schematic of a THz QCL under OF, where $n_{\text{ext}}$ and $L_{\text{ext}}$ are the refractive index and length of the external cavity, respectively, and $R$ is the reflectivity of the mirror. $P_{\text{out}}\left(t\right)$ and $V\left(t\right)$ are the emission power and the terminal voltage of the single-mode THz QCL, respectively.

Figure 2

Simulated (a) time traces of the emitted power from the THz QCL $P_{\text{out}}\left(t\right)$ and (b) emission spectra offset from the single-mode frequency at 2.752 THz $P_{\text{out}}\left(\nu \right)$ at varying feedback coupling coefficient $\kappa$ with $L_{\text{ext}}$ of 0.8 m and $\alpha$ of $-0.1$, where the blue curves show the results with no OF. The changes of the locations of the central mode (CM), first side mode (SM) (SM1), second SM (SM2), and third SM (SM3) are grouped by dashed circles, respectively, in the emission spectra. The mode spacing between the adjacent modes is equal and is defined as $\mathrm{\Delta }f$.

Figure 3

Simulated amplitude of the emission power wave forms at 350 ns as a function of $\mathrm{\Delta }{L}_{\text{ext}}$ from 0.8 m with $\kappa =-4.0\mathrm{dB}$ (blue crosses), which has a similar trend of the self-mixing wave form obtained from solving the excess phase equation with the same value of $\kappa$ (red curve). Typical wave form of $P_{\text{out}}\left(t\right)$ along $\mathrm{\Delta }{L}_{\text{ext}}$ at points a, b, c, d, e are also shown in inset a, b, c, d, e, respectively.

Figure 4

Simulated mode frequency splitting from CM (red) to adjacent SM1 (blue), SM2 (black), and SM3 (magenta) with the OF level $\kappa$. The crosses indicate the results from solving RREs, while the solid curves are the results from the excess frequency equation.

Figure 5

Simulated mode spacing $\mathrm{\Delta }f$ as a function of $\kappa$ for (a) $L_{\text{ext}}$ = 0.8 m (red crosses and curve), 1.6 m (blue crosses and curve), and 3.2 m (black crosses and curve) where $\alpha =-0.1$; and for (b) $\alpha =-0.1$ (red crosses and curve), 1.0 (purple crosses and curve), and 2.0 (green crosses and curve), where $L_{\text{ext}}$ = 0.8 m. The crosses indicate results from solving the RREs, while the solid curves are the results from the excess frequency equation. Dashed lines indicate the external cavity resonant frequencies $f_{\text{EC}}$ for each $L_{\text{ext}}$.

Figure 6

Simulated dependence of $\mathrm{\Delta }f$ on $L_{\text{ext}}$ and $\kappa$ with $\alpha =-0.1$, where $L_{\text{ext}}$ is varying from 0.2 to 3.2 m and $\kappa$ is varying from $-39.04$ to 0.96 dB. The color bar indicates the value of $\mathrm{\Delta }f$ (MHz). The red curve indicates $f_{\text{EC}}=c/\left(2n_{\text{ext}}{L}_{\text{ext}}\right)$ for varying $L_{\text{ext}}$, which matches very well with simulated $\mathrm{\Delta }f$ by solving RREs at varying $L_{\text{ext}}$ with the strongest OF ($\kappa$=$-0.96\mathrm{dB}$).

Figure 7

Simulated dependence of OF dynamics on $\alpha$ with $\kappa =-4.0\mathrm{dB}$: (a) time traces of $P_{\text{out}}\left(t\right)$ for a hypothetical QCL with $\alpha =-0.1$, 1, 2, 3, and 4; (b) the corresponding emission spectrum offset to the emission frequency $P_{\text{out}}\left(\nu \right)$ with $\alpha =-0.1$, 1, 2, 3, and 4. The results are offset for clarity, the vertical values can only be used for relative variance comparisons.

Figure 8

Experimentally observed time traces of TIs (left column) and corresponding spectra (right column) in a pulse-mode THz QCL when (a) $L_{\text{ext}}$ = 0.8 m, with an increasing OF strength when the round-trip transmission through the polarizer was set at $-2.68\mathrm{dB}$ (orange), $-1.58\mathrm{dB}$ (magenta), and $-1.38\mathrm{dB}$ (blue), respectively; (b) $L_{\text{ext}}$ = 1.0 m, with an increasing OF strength when the round-trip transmission through the polarizer was set at $-7.68\mathrm{dB}$ (orange), $-7.32\mathrm{dB}$ (magenta), and $-6.63\mathrm{dB}$ (blue), respectively; ($\mathrm{c})L_{\text{ext}}$ = 1.5 m, with an increasing OF strength when the round-trip transmission through the polarizer was set at $-2.50\mathrm{dB}$ (orange), $-1.71\mathrm{dB}$ (magenta), and $-0.5\mathrm{dB}$ (blue), respectively; (d) simulated oscillation frequency $\mathrm{\Delta }f$ as a function of $\kappa$ with $\alpha =-0.0001$ for $L_{\text{ext}}$ = 0.8 m, 1.0 m, and 1.5 m. The peak frequencies in (a) correspond to red dots at points i, ii, and iii when $L_{\text{ext}}$ = 0.8 m. The peak frequencies in (b) correspond to purple dots at points iv, v, and vi when $L_{\text{ext}}$ = 1.0 m. The peak frequencies in (c) correspond to green dots at points vii, viii, and ix when $L_{\text{ext}}$ = 1.5 m.