Exceptional-Point-Based Optical Amplifiers

The gain-bandwidth product is a fundamental figure of merit that restricts the operation of optical amplifiers. Here, we introduce a design paradigm based on exceptional points, which relaxes this limitation and allows for the building of a new generation of optical amplifiers that exhibits a better gain-bandwidth scaling. Additionally, our results can be extended to other physical systems such as acoustics and microwaves.

Figure 1

The schematics of the proposed optical amplifier (OA) based on a microring resonator working at an exceptional point (EP). The input $s_1$ couples into the microring resonator (coupling rate $\gamma$) and is amplified by the gain $g$. The clockwise mode $a_{\mathrm{cw}}$ couples into the counterclockwise mode $a_{\mathrm{ccw}}$, while the opposite is not true because of the mirror at the drop port. The output $s_5$ is amplified in this process. Here, $r$ is the magnitude of the field-reflection coefficient of the mirror and $\alpha$ is the decay rate due to radiation and material losses.

Figure 2

(a) Amplification enhancement for an EP-based OA (as compared with the standard DP-based OA) as a function of their identical bandwidths as measured in units of $\gamma$. (b) The same as in (a) but for bandwidth enhancement as a function of the identical amplification. The insets in (a) and (b) show the relations between the material gain values that are necessary to achieve identical bandwidths or identical amplifications, respectively (see the text). Finally, the squares indicate the parameters used in the full-wave simulation (see Sec. 3). These plots are obtained for $r=1$.

Figure 3

A schematic diagram of the proposed photonic structure used in our FDTD simulations. The geometric and material parameters used in the simulations are as follows: waveguide width $w=0.25\mu \mathrm{m}$ (for both the straight and the ring waveguides), ring radius $R=5\mu \mathrm{m}$, and edge-to-edge distances between the ring and waveguides $d=0.15\mu \mathrm{m}$. To implement the mirror, we assume a thin layer of silver with a thickness of 100 nm. (In practice, there are several design principles for building a mirror; for instance, by using photonic crystals [23, 24, 25] or a section of different material [26].) The material refractive index is $n_1=3.47$ [corresponding to semiconductor materials such as silicon or (Al,Ga)As] and the background index is taken to be $n_2=1.44$. These values have been used before in DP-based microring amplifiers [4, 27]. Finally, we model the applied gain by considering a gain curve with a finite bandwidth (for more details, see Appendix pp1).

Figure 4

Full-wave FDTD simulations for EP- and DP-based amplifiers operating close to $\lambda =1.55\mu \mathrm{m}$ and having (a) identical bandwidth and (b) identical maximum amplification, respectively. The solid and dashed red lines correspond to outputs at ports $s_5$ and $s_4$, respectively, while in the DP case (blue line) the output port is $s_3$. The superior performance in terms of higher amplification in (a) and larger bandwidth in (b) is evident in both cases. The operating points of both scenarios correspond to the square dots in Figs. 2 and 2, respectively. Excellent agreement between the FDTD results and the TCMT is observed in both cases. The details of the design parameters used in our simulations are listed in the caption of Fig. 3 and in the text.

Figure 5

The absolute value of the complex electric field distributions associated with (a) a DP-based and (b) and an EP-based amplifier for the resonant frequency when they both have an equal gain [i.e., corresponding to the case of Fig. 4]. The inset in (b) highlights the interference pattern between the CW and CCW components in the latter case. The legend colors represent the value of the electric field normalized by the value of the input field. Note that the plotted fields correspond to the steady-state solution under external excitations and not to the eigenmodes of the system.

Figure 6

The transmission of the DP- and EP-based resonators without any material gain. In the amplifiers discussed in the text, we introduce a material gain, based on the Lorentz model, to the microring resonator.

Figure 7

(a) A cascaded amplifier can achieve the same functionality as the structure in Fig. 1 with $r\exp (i\varphi )=1$, as confirmed in (b) using FDTD.

Figure 8

The S-bend ring can provide a unidirectional coupling between the CW and CCW modes. This structure is studied with scattering matrices $S_j$ ($j=1,2,3,4$) in the four coupling regions (dashed lines).

Figure 9

(a) A $\mathcal{P}\mathcal{T}$-dimer-based optical amplifier having an exceptional point at $g=\kappa$. (b) A comparison between the structure in (a) and those in Fig. 1 (with and without a mirror) for identical $\gamma$ and assuming $\alpha =0$, when $g_{\mathrm{PT1}}=1.36\gamma$ (dashed green line) and $g_{\mathrm{PT2}}=2.5\gamma$ (solid green line). Clearly, one can increase the amplification of the $\mathcal{P}\mathcal{T}$ amplifier while at the same time maintaining the same bandwidth. From a practical perspective, however, increasing the amplification requires stronger coupling $\kappa$ (i.e., smaller separation between the two rings), which is limited by the fabrication tolerance.