Tunable virtual gain in resonantly absorbing media

Virtual gain refers to the simulation of real light amplification using radiation with exponentially decaying amplitude so that its complex frequency corresponds to the scattering pole. We study theoretically virtual gain in a two-level resonant medium for different regimes of light-matter interaction depending on the radiation intensity. We show that virtual gain at the pole can be most clearly observed for low intensities, when the medium is absorbing, in contrast to the saturated medium at high intensities. The efficiency of virtual gain can be tuned with the light intensity and controlled dynamically through the population inversion of the medium. Our results show that resonantly absorbing media paradoxically mimic a gainlike response, which admits a number of related phenomena and methods to mold both optical signals and material properties without relying on instability-prone gain media.

Figure 1

Schematic representation of the situation studied. Exponentially decaying radiation is impinging on the layer of a resonantly absorbing medium. The color gradient inside the medium shows the growing intensity of light corresponding to virtual gain.

Figure 2

(a) Transmitted light intensity and (b) inversion dynamics at the entrance of the medium for different values of Rabi frequency. The parameters are $T_1=1$ ns, $T_2=1$ ps, ${\omega }_L=3.7\times {10}^{11}{\mathrm{s}}^{-1}, L=\lambda =1\mu \mathrm{m}$, and $n=3$.

Figure 3

Virtual gain in the absorbing medium under low-intensity radiation ${\mathrm{\Omega }}_{\text{max}}=0.01/T_2$. Light intensity profiles are shown for different values of decay time $\tau$: (a) $\tau =0.0001$ ps and (b) $\tau =0.1$ ps. (c) and (d) Radiation distributions at the switching-off moment ($t_{\text{max}}=20$ ps) and a short time later ($t_{\text{max}}+\mathrm{\Delta }t=20.01$ ps). (e) Ratio $R$ as a function of $\mathrm{\Delta }t$ for different $\tau$. (f) Ratio $R$ at different time instants as a function of the decay time $\tau$.

Figure 4

Virtual gain in the thick layer ($L=10\lambda =10\mu \mathrm{m}$) of the absorbing medium under low-intensity radiation ${\mathrm{\Omega }}_{\text{max}}=0.01/T_2$. (a) and (b) Radiation distributions at the switching-off moment ($t_{\text{max}}=20$ ps) and a short time later ($t_{\text{max}}+\mathrm{\Delta }t=20.1$ ps). (c) Ratio $R$ as a function of $\mathrm{\Delta }t$ for different decay times $\tau$. (d) Ratio $R$ at different time instants as a function of $\tau$.

Figure 5

Case of a saturated medium under high-intensity radiation ${\mathrm{\Omega }}_{\text{max}}=3/T_2$. (a) and (b) Light intensity profiles for different values of decay time. (c) and (d) Radiation distributions at the switching-off moment ($t_{\text{max}}=20$ ps) and a short time later ($t_{\text{max}}+\mathrm{\Delta }t=20.006$ ps).

Figure 6

Case of a passive medium under low-intensity radiation ${\mathrm{\Omega }}_{\text{max}}=0.01/T_2$. (a) and (b) Radiation distributions at the switching-off moment ($t_{\text{max}}=20$ ps) and a short time later ($t_{\text{max}}+\mathrm{\Delta }t=20.006$ ps). (c) Ratio $R$ as a function of $\mathrm{\Delta }t$ for different $\tau$.

Figure 7

Tunable virtual gain under high-intensity radiation ${\mathrm{\Omega }}_{\text{max}}=3/T_2$. (a) Dynamics of population inversion for different switching-off moments $t_{\text{max}}$. (b) Ratio $R$ as a function of $\mathrm{\Delta }t$ for different $t_{\text{max}}$. (c) and (d) Radiation distributions at the switching-off moment $t_{\text{max}}$ and a short time later (at $t=t_{\text{max}}+\mathrm{\Delta }t$ marked in the plot). (e) and (f) Light intensity profiles for different switching-off moments $t_{\text{max}}$. (g) Ratio $R$ as a function of inversion at $t_{\text{max}}$ calculated at the instant $t=1.02t_{\text{max}}$. The decay time is $\tau ={10}^{-4}$ ps.

Figure 8

Testing RWA applicability. Light intensity profiles at $\tau =0.1$ ps are shown for different values of the Lorentz frequency: (a) ${\omega }_L=3.7\times {10}^{10}{\mathrm{s}}^{-1}$ and (b) ${\omega }_L=3.7\times {10}^{11}{\mathrm{s}}^{-1}$.

Figure 9

Virtual gain in the layer not satisfying the pole condition ($L=1.1\lambda =1.1\mu \mathrm{m}$) under low-intensity radiation ${\mathrm{\Omega }}_{\text{max}}=0.01/T_2$. (a) and (b) Light intensity profiles for different values of decay time $\tau$. (c) Ratio $R$ as a function of $\mathrm{\Delta }t$ for different $\tau$. (d) and (e) Radiation distributions for different $\tau$ at the switching-off moment ($t_{\text{max}}=20$ ps) and a short time later ($t=t_{\text{max}}+\mathrm{\Delta }t$).