Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations

We analytically show that inserting a driven, two-level system inside a microcavity can improve its optical properties. In this approach, the strong dispersion induced by a pump via population oscillations increases the cavity lifetime experienced by a slightly detuned probe. We further predict that if the cavity is pumped through a resonant channel, optical absorptive or dispersive bistability can be combined with the population-oscillation-induced steep material dispersion to obtain a strong quality-factor enhancement. Moreover, differential amplification coming from the nonlinear feature of the pump transfer function can be used to drastically increase the probe transmission beyond intrinsic characteristics of the resonator. The $Q$-factor enhancement and the differential amplification can be advantageously combined with a frequency pulling effect to stabilize or readjust the microcavity resonance frequency.

Figure 1

Microcavity containing a TLS. The pump (frequency ${\omega }_P$) is nonresonantly coupled to the cavity, and $s$ (frequency ${\omega }_s={\omega }_P+\delta$) is the input probe field. The cavity mode is coupled to the reflected field $u$ via evanescent waves with a characteristic lifetime ${\tau }_e$.

Figure 2

Nonlinear microcavity and its access line. $s\left(t\right)$ is the input field consisting of a pump and a signal of which carrier frequencies are, respectively, ${\omega }_P$ and ${\omega }_s={\omega }_P+\delta$ [$\tilde{s}\left(\omega \right)$ represents the input signal spectrum]. The cavity mode $a\left(t\right)$ is coupled to $s\left(t\right)$ and to the reflected field $u\left(t\right)$ via evanescent waves with a characteristic lifetime ${\tau }_e$.

Figure 3

(a) Nonlinear transfer function $P_{\mathrm{out}}=f\left(P_{\mathrm{in}}\right)$, where $P_{\mathrm{out}}={\left|u_0\right|}^2$ of a cavity just below the bistability threshold. We have also schematically sketched the transfer function of a small modulation stemming from the beating between the pump and the probe fields. (b) For input power values close to the switching power ($P_{\mathrm{in}}=0.45$ in the example), the differential gain $|d{P}_{\mathrm{out}}/d{P}_{\mathrm{in}}|$ can be very high and can even diverge for bistable cavities.

Figure 4

(a) Probe reflection spectra for three values of $x$ ($x=0$ in dash line, $x=0.1$ in full dark line and $x=0.3$ in full light line) for a nonlinear cavity with $\eta =1.05$, ${\alpha }_H=0$ and $T_1=250{\tau }_{a0}$. (b) Intracavity normalized pump energy as a function of the input pump power. The inset is a zoom of the lowest part of the curve. (c) Probe reflection spectra, the value of $x=1.152$ has been set to obtain a $100\%$ reflection (light line). The case $\alpha =0$ (dark line) coresponds to an absorption-free linear cavity. (d) Example of a reflection spectrum with a high differential selective amplification obtained for $x=1.24$ (light line).

Figure 5

Probe reflection spectra for several values of cavity-pump detuning: (a) $\Delta /{\omega }_0=0.2/Q_0$, (b) $\Delta /{\omega }_0=0.1/Q_0$, (c) $\Delta =0$, (d) $\Delta /{\omega }_0=-0.1/Q_0$, and (e) $\Delta /{\omega }_0=-0.2/Q_0$. In all cases, $x=1.24$ and the arrow points out the cavity resonances.

Figure 6

Probe reflection spectra for $\Delta =-{\omega }_0/\left(15Q_0\right)$ and $x=1.42$ in a logarithmic scale. The inset depicts the resonance with a linear scale. The resonance full width at half maximum is ${\omega }_0/\left(3000Q_0\right)$.

Figure 7

Quality factor for the probe signal as a function of the normalized pump intracavity energy $x$ for $T_1=250{\tau }_{a0}$ (light line) and $T_1\ll {\tau }_{a0}$ (dark line). We have also plotted the value of the Q-factor of an equivalent linear cavity without absorption (dash line).

Figure 8

(a) Probe reflection spectra for $x=0$ (dash dark line) and $x=0.14$ (full line) in the case of a nonlinear cavity which physical characteristics are $\eta =3$, ${\Delta }_0=-3\sqrt{3}\eta /2$, ${\alpha }_H=25$, and $T_1=100{\tau }_{a0}$. For $x=0.14$, the cavity reflection is $100\%$. For comparison, the bright dash line represents the spectrum of the same cavity in the linear and absorption-free case with ($\alpha =0$) (dash light line). (b) Intracavity normalized pump energy as a function of the input pump power. (c) Probe reflection spectra for $x=1.4$ (light line) and $0.156$ (full line). (d) Reflectivity spectrum calculated for $x=0.14$ assuming $T_1=100{\tau }_{a0}$ (light line) or $T_1=2{\tau }_{a0}$ (dark line).