Complete transparency with three active-passive-coupled optical resonators

Abstract

The phenomena of induced transparency, with the typical examples of electromagnetically induced transparency (EIT) in atomic media and coupled optical resonators, have attracted tremendous interest since their discoveries. Due to the limitations of the involved elements, however, near-100\% transmissions were reported under highly demanding experimental conditions for atomic and other media. Based on a structure of three linearly coupled optical resonators, an active one carrying a possibly arbitrary optical gain and two passive ones simply with dissipation, we demonstrate that a transmitted light field can become completely transparent through the structure, which displays all properties similar to those of EIT. Manifested by a destructive interference to annihilate the intracavity field in the resonator directly coupled to the input, this complete transparency exists for any feasible power of the transmitted field and all realizable coupling strengths of the dark resonator with the input and the neighboring resonator, as long as the inter-cavity coupling for two other resonators is adjustable over a suitable range. A free control on the transparency window size and output field intensity can be realized by tuning two inter-cavity couplings without modifying the built-in system parameters.

Figures (6)

• Figure 1: (a) The schematic structure of three linearly coupled microresonators. Resonator $2$ carries a gain medium without showing the external pump used for the field amplification. The actual position of the optical fiber is within the plane of the microresonators. There will be a totally vanishing intracavity field $\hat{a}_1$, once resonator $2$ and $3$ satisfy the condition in Eq. (\ref{['g-condiitons']}). Then, the transmitted field will have the $100$% transmission at the center of a narrow frequency window, as shown by the example in the inset, which is obtained with $\kappa_{1}=2\kappa_{ex}=10\,\mathrm{MHz}$, $\kappa_{2}=0.2\,\mathrm{MHz}$, $\kappa_{3}=5\,\mathrm{MHz}$, and $J_1=2\,\mathrm{MHz}$, and $J_2=1\,\mathrm{MHz}$ for three identical microresonators. (b) An example of the evolving intracavity fields after including the optical gain saturation effect. The parameters are $\varepsilon_p=10^4\,\sqrt{\mathrm{MHz}}$ (equivalent to $P=12.8\, \mathrm{\mu W}$ at $\lambda=1550$ nm), $\kappa_{1}=2\kappa_{ex}=10\, \mathrm{MHz}$, $\kappa_{2,0}=0.2\, \mathrm{MHz}$, $I_S=10^8$, $\kappa_3= 5\, \mathrm{MHz}$, $J_1=20\, \mathrm{MHz}$, and $J_2=1\, \mathrm{MHz}$. The saturated gain rate after the stabilization is $\kappa_{2}=0.195\, \mathrm{MHz}$, and the cavity damping rates are chosen to be on the levels of those in some recent experiments mc1mc2mc3.
• Figure 2: (a) The distribution of the absolute susceptibility, $|\varepsilon_T|=\sqrt{2\kappa_{ex}}|A_{1}|/\varepsilon_{p}$, at $x=0$, with respect to two parameters $\sqrt{\kappa_2\kappa_3}$ and $J_2$. Here, we fix the rest parameters at $\kappa_1=2\kappa_{ex}=20\, \mathrm{MHz}$, $\kappa_3=5\, \mathrm{MHz}$, and $J_1=10\, \mathrm{MHz}$. The calculation is based on Eq. (5) but with three identical resonance frequencies. (b) The distribution of $\text{Re}(\varepsilon_T)$ with respect to $x$ and $J_1$. Here, the rest of the parameters are set to be $\kappa_1=2\kappa_{ex}=20\, \mathrm{MHz}$, $\kappa_2=0.05\, \mathrm{MHz}$, $\kappa_3=20\, \mathrm{MHz}$, and $J_2=1\, \mathrm{MHz}$. The calculation is based on Eq. (9).
• Figure 3: (a1)-(a2) The examples of $\text{Re}(\varepsilon_T)$ and $\text{Im}(\varepsilon_T)$, as well as the time delays of the transmitted fields, in the regime of $\kappa_{2}<\kappa_{3}$. We use $\kappa_1=2\kappa_{ex}=20\,\mathrm{MHz}$, $\kappa_2=0.01\,\mathrm{MHz}$, $\kappa_3=20\,\mathrm{MHz}$, $J_1=0.01\,\mathrm{MHz}$ in (a), and $J_2=\sqrt{\kappa_{2}\kappa_{3}}$, so that the transparency window size is in the order of $\mathrm{Hz}$ and the time delays approach the order of second. (b1)-(b2) The other examples in the regime of $\kappa_{2}>\kappa_{3}$. The system parameters are set to be $\kappa_1=2\kappa_{ex}=10\,\mathrm{MHz}$, $\kappa_2=1\,\mathrm{MHz}$, $\kappa_3=0.1\,\mathrm{MHz}$, and $J_2=\sqrt{\kappa_{2}\kappa_{3}}$. In (b1) we have $J_1=4\,\mathrm{MHz}$.
• Figure 4: (a) The comparison between the output/input ratios due to three different couplings $J_2$, which are, respective smaller than (the red curve), equal to (the blue curve), and larger than (the black curve) the value $J_{2,c}$ of the $100\%$ transmission at the transparency window center. Represented by these three samples, the output can be continuously amplified or reduced by tuning the coupling $J_2$ across the point $J_{2,c}$, while the built-in optical gain is unchanged. (b) The corresponding dispersion curves. Here, the fixed system parameter are chosen as $\kappa_1=2\kappa_{ex}=20\,\mathrm{MHz}$, $\kappa_2=0.06\,\mathrm{MHz}$, $\kappa_3=6\,\mathrm{MHz}$, and $J_1=1 \,\mathrm{MHz}$.
• Figure S-1: (a) Examples of the evolution of the field amplitude $|A_1|$ for different saturation intensities $I_S$, while the other parameters of the setup are fixed. Here, the initial gain rate is $\kappa_{2,0}=0.1\,\mathrm{MHz}$, and $J_{1}=4\,\mathrm{MHz}$. (b) Examples of the evolution of the field amplitude $|A_1|$ for different initial gain rates $\kappa_{2,0}$, while the other parameters of the setup are fixed. Here, the gain saturation intensity is fixed at $I_S=10^7$, and $J_{1}=4\,\mathrm{MHz}$. (c) Examples of the evolution of the field amplitude $|A_1|$ for different inter-cavity couplings $J_1$, while the other parameters of the setup are fixed. Here, we have the initial gain rate $\kappa_{2,0}=0.1\,\mathrm{MHz}$ and the gain saturation intensity $I_S=10^7$. In all these examples, the fixed system parameters are $\kappa_{1}=2\kappa_{ex}=2\,\mathrm{MHz}$ and $\kappa_{3}=1\,\mathrm{MHz}$. The pump field amplitude is fixed at $\varepsilon_p=10^4\,\sqrt{\mathrm{MHz}}$.