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Abstract
The strong coupling between photonic nanocavities at arbitrary positions is important for the realization of photonic integrated circuits. However, the coupling between nanocavities is mainly through the evanescent field, which limits the distance between nanocavities and hinders the scalability of photonic circuits. Here, we propose a scheme to realize the strong coupling between two distant nanocavities beyond the limitations of evanescent field coupling. Two distant identical one dimensional photonic crystal cavities (1DPhCCs) more than 8 µm apart are bridged by a microring which supports whispering gallery modes (WGMs). We demonstrate that the two 1DPhCCs can be strongly coupled even though the microring is largely detuned from them. The supermodes between the two 1DPhCCs are formed while the proportions of the WGM in the microring are suppressed at large detuning. The light energy mainly oscillates between the two 1DPhCCs, leaving the WGM in the microring as a dark mode. Such a scheme can realize strong coupling between distant nanocavities without much difficulties in experiments, which provides advantages for the realization of next-generation photonic circuits.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical cavities [1] have been intensively investigated to realize optical sensors [2–4], to realize strong coupling between quantum emitters and photons [5–9], to generate single photons [10,11] and entangled photon pairs [12], to enhance nonlinearity [13–15], and so on. Photonic integrated circuits consist of coupled optical cavities are desired to push forward the applications of optical devices. So far, strong coupling between cavities has been investigated with various type of cavities [16–24]. To couple optical cavities together, evanescent field coupling plays a central role. However, the evanescent field of high-Q optical cavities decays rapidly along with distance, which requires that cavities should be located adjacently to realize their strong coupling [25]. This limitation makes the system difficult to implement separate control of individual cavities. To solve this problem, people utilize waveguide to couple distant cavities [26,27]. Nevertheless, the coupling to waveguide reduces the quality factors of cavity modes. To suppress the decay from cavity to waveguide, Yoshiya Sato et al. engineer the waveguide to satisfy mode-mismatching conditions and realize strong coupling between distant nanocavities in photonic crystal structure [28].
Here, we propose a scheme that is much easier to be realized in experiments to achieve strong coupling between two distant identical 1DPhCCs [29–33] which are bridged by a microring with diameter of 8 µm. Thanks to the extensive field distribution of the WGM [34] in the microring, the two 1DPhCCs can be coupled together through the common WGM. Surprisingly, when the resonant frequency of WGM is far detuned from the 1DPhCCs, strong coupling between the two 1DPhCCs can still be maintained, and the light energy in the microring is largely suppressed which makes the WGM as a dark mode.
Such a hybridized scheme of WGMs and 1DPhCC modes can couple two distant nanocavities strongly beyond the limitations of evanescent field coupling and significantly relaxes the experimental requirements, which provides advantages for implementing nonlocal control of quantum emitters. When a single quantum dot is located at one of the 1DPhCCs, its quantum states can be manipulated by optically adjusting the cavity modes of the other 1DPhCC which is far away from the quantum dot. Moreover, the super-modes of the strongly coupled distant nanocavities can be used to entangle two distant quantum emitters together.
2. Structure and theoretical model
We designed the structure with two identical 1DPhCCs bridged by a microring (Fig. 1(a)). The material of the structure is $GaAs$. The thickness of the structure is 0.187 µm. The outer radius of the microring is 4 µm. The inner radius of the microring is denoted by $r_{in}$ which is used to tune the resonant frequency of the microring. The width of the 1DPhCC is 0.425 µm. Cavity is formed by a five-hole tapered one dimensional photonic crystal mirror. The distance between the edges of the 1DPhCCs and microring is 0.2 µm.
Fig. 1. (a) The coupled structure with two identical 1DPhCCs bridged by a microring. The thickness of the slab is 0.187 µm. The width of the nanobeam is 0.425 µm. The 1DPhCC is formed by photonic crystal mirror. The lattice constant $a = 0.3655$ µm is linearly tapered over a five hole section to $a = 0.2805$ µm at the cavity center. The air hole radius is $r=0.28a$. The outer radius of the microring is 4 µm. The inner radius of the microring is denoted as $r_{in}$, which can be adjusted to tune the resonant frequency of the microring . The structure is symmetrical about all the three axes. And the gap between the 1DPhCCs and the microring is 0.2 µm. (b) The LDOS of the individual 1DPhCC (black line) corresponding to y-polarized dipole located at the center of the 1DPhCC. The LDOS of the individual microring (blue line) corresponding to radially polarized dipole located at the middle of the ring slab with the ring inner radius $r_{in}=3.7022$ µm. The electric field intensity of 1DPhCC (c) and microring (d).
We firstly characterize the individual optical cavities with local density of photonic states (LDOS). With an $E_y$ polarized dipole source located at the center of the individual 1DPhCC, the LDOS can be obtained with the Green’s function [22] which is calculated by finite-difference time-domain (FDTD) method (Lumerical Solutions, Inc.).
The LDOS at the center of the 1DPhCC corresponding to an $E_y$ dipole is plotted in Fig. 1(b) by black line. The LDOS at the middle of the microring slab with $r_{in}=3.7022$ µm for dipole with radial polarization is plotted in Fig. 1(b) by blue line. From the LDOS, the resonant frequencies of the individual 1DPhCC and microring are $f_a=2.34872\times 10^{14}$ Hz and $f_b=2.34866\times 10^{14}$ Hz, respectively. The corresponding quality factors are $1.02\times 10^5$ and $4.85\times 10^5$, respectively. The electric field densities of the individual 1DPhCC and microring are plotted in Fig. 1(c) and 1(d). We then analyze the eigen-frequencies and eigen-states of the system with two identical nanocavities $a_1$ and $a_2$ coupled to the same cavity $b$ [35,36]. The time evolutions of the cavity modes can be written as where $a_1$, $a_2$ and $b$ are the field amplitudes of the cavities. $\omega _n$ and $\gamma _n$ denote the resonant frequency and the loss rate of the $n$th cavity ($n=a, b$). $\kappa$ denotes the coupling strength between cavity $a$ and $b$. The detuning between cavities $a$ and $b$ is $\delta =\omega _b-\omega _a$. A time dependence of the form $e^{i\omega t}$ is assumed for the field amplitudes and one can find the eigen-frequencies of the coupled system:Fig. 2. (a) The real parts of the eigen-frequencies along with the detuning between cavities $b$ and $a$ ($\delta =\omega _b-\omega _a$). (b) The loss rates of the three eigen-frequencies. (c) The proportions of cavities $a$ and $b$ of the eigen-state $V2$. (d) The proportions of cavities $a$ and $b$ of the eigen-state $V3$.
For the normalized eigen-vectors $V2$ and $V3$, the proportions of the cavity field $a_1$, $a_2$ and $b$ along with the detuning $\delta$ are plotted in Fig. 2(c) and 2(d). For large positive detuning, the proportion of cavity field $b$ ($a$) reduces to almost 0 for $V2$ ($V3$). For large negative detuning, the proportion of cavity field $a$ ($b$) reduces to almost 0 for $V2$ ($V3$). We can infer from the proportion of cavity field that, at large positive detuning, $V2\rightarrow -\sqrt {2}/2(a_1+a_2)$ and $V3\rightarrow b$; at large negative detuning, $V2\rightarrow b$ and $V3\rightarrow \sqrt {2}/2(a_1+a_2)$. We take the large positive detuning case for example to reveal the coupling between cavities. The eigen-vectors in this case are $V1=\sqrt {2}/2(a_1-a_2)$, $V2\rightarrow -\sqrt {2}/2(a_1+a_2)$ and $V3\rightarrow b$. Here, $V1$ and $V2$ are the coupled states between cavities $a_1$ and $a_2$. If cavity $a_1$ was excited initially, energy Rabi oscillation may appear between $a_1$ and $a_2$ due to the superposition of $V1$ and $V2$ in the time evolution. Cavity $b$ is almost not excited in this process which can be considered as a dark mode. However, this dark state is indispensable to realize the strong coupling between two distant cavities beyond the evanescent field coupling.
3. Results and discussions
An $Ey$ polarized dipole source is located at the middle of the bare ring slab along the Y axis to excite the WGMs in the microring. The outer radius of the ring is set as 4 µm. And the inner radius of the ring is changed from 3.7 µm to 3.705 µm with 0.1 nm step. The LDOS of the microring according to the dipole source is calculated with Eq. (1) and is plotted in Fig. 3(a). The resonant frequency of the given WGM varies from $f_b(r_{in}=3.7 \mu m)=2.34156\times 10^{14}$ Hz ($\lambda =1.28031\mu m$) to $f_b(r_{in}=3.705 \mu m)=2.35815\times 10^{14}$ Hz ($\lambda =1.27130\mu m$) and crosses over that of the bare 1DPhCC ($f_a=2.34872\times 10^{14}$ Hz, $\lambda _a=1.27641\mu m$).
Fig. 3. (a) The LDOS in the individual microring as the inner radius $r_{in}$ changes from 3.7 µm to 3.705 µm. The dipole source is always located at the middle of the ring slab to calculate the LDOS. (b) The LDOS of the coupled 1DPhCC-microring-1DPhCC system corresponding to the change of $r_{in}$. The y-polarized dipole source is located at the center of cavity $a_1$.
We next investigate the supermodes of the coupled system shown in Fig. 1(a). An $Ey$ polarized dipole source is located at the center of cavity $a_1$. The LDOS is shown in Fig. 3(b). Triplet modes appear in the coupled system, which agrees well with Fig. 2(a). We note that the middle peak in Fig. 3(b) corresponds to the state $V1=\sqrt {2}/2(a_1-a_2)$. The $V1$ state is the superposition of cavity fields $a_1$ and $a_2$, which does not contain any field of cavity $b$. Therefore, the change of cavity $b$ hardly affects the $V1$ state. The lower and upper peaks correspond to states $V2$ and $V3$. Due to the including of cavity field $b$ in the two states, the change of microring has a great influence on $V2$ and $V3$. Anti-crossing between the two states is observed. Surprisingly, when the upper peak almost disappears at large positive detuning, the splitting between the lower two peaks still exists.
The electric field densities corresponding to the peaks in the LDOS (Fig. 3(b)) are then considered. The LDOS in the case of $r_{in}=3.7022$ µm is shown in Fig. 4(a). From Fig. 1(b), we can see that the individual cavities $a$ are nearly resonant with cavity $b$. There are three peaks corresponding to the states $V1$, $V2$ and $V3$. The electric field densities of the three states are plotted in Fig. 4(b)–4(d). For $V1$, the electric field concentrates in cavity $a_1$ and $a_2$. And there is no field in cavity $b$. For $V2$ and $V3$, the electric field distributes in the three cavities.
Fig. 4. (a) The LDOS of the coupled system with $r_{in}=3.7022$ µm. Three peaks correspond to the eigen-states $V1$, $V2$ and $V3$ expressed by Eqs. (8)–(10). (b)-(d) The electric field intensities of the three eigen-states.
In the case of $r_{in}=3.705$ µm, the individual cavities $a$ and $b$ are largely detuned ($\delta /2\pi =0.943$ THz, $\triangle \lambda =-5.11$ nm). The two peaks $V1$ and $V2$ are spectrally far apart from peak $V3$ (Fig. 5(a)). Additionally, the magnitude of peak $V3$ is much smaller than that of the other two peaks. The electric field density concentrates in cavities $a_1$ and $a_2$ with almost no field in cavity $b$ for both $V1$ and $V2$. For $V2$, the eign-state is $V2\rightarrow -\sqrt {2}/2(a_1+a_2)$ at large positive detuning. However, the electric field concentrates mainly in cavity $b$ for $V3$, which corresponds to the state $V3\rightarrow b$. We should note that the electric field density for $V3$ is two orders smaller than that of $V1$ and $V2$. It can be inferred that the states $V1=\sqrt {2}/2(a_1-a_2)$ and $V2\rightarrow -\sqrt {2}/2(a_1+a_2)$ are the superposition modes of strongly coupled cavities $a_1$ and $a_2$.
Fig. 5. (a) The LDOS of the coupled system with $r_{in}=3.705$ µm. (b)-(d) The electric field intensities of the three eigen-states.
To confirm the strong coupling between cavities $a_1$ and $a_2$, we calculated the time evolution of the light energy in the three cavities. An $Ey$ polarized dipole source (center frequency, 235 THz; bandwidth, 2.2 THz) was located at the center of cavity $a_1$ to excite it. The source pulse covers $V1$, $V2$ and $V3$ modes. To save calculating time, we calculated the time evolution of the light energy in the two dimensional plane at the middle of the structure. The energy evolutions in the three cavities are shown in Fig. 6(b). Clear Rabi oscillation between cavities $a_1$ and $a_2$ appears with a period of about 29 ps. Furthermore, the energy in cavity $b$ is very small although the source pulse covers $V3$ mode which contains large proportion of cavity $b$. The reason is that, $V3$ mode is excited minutely as we located the dipole source in cavity $a_1$ and $V3 \rightarrow b$ at large positive detuning. Snapshots of electric field density are picked during the Rabi oscillation and are shown in Fig. 6(a) $\rightarrow$ 6(c) $\rightarrow$ 6(d), which reveals the energy transfer from cavity $a_1$ to cavity $a_2$ leaving cavity $b$ almost unexcited. Therefore, we have demonstrated the strong coupling between two distant 1DPhCCs via a dark WGM.
Fig. 6. (b) Time evolution of the light energy in cavities $a_1$ and $a_2$ and $b$. (a) $\rightarrow$ (c) $\rightarrow$ (d) Picked snapshots of electric field density during the Rabi oscillation.
As the two 1DPhCCs are separated distantly, a dynamic and independent change on either 1DPhCC can be realized, which makes it possible to switch the coupling on and off [28]. When a control pulse strikes cavity $a_2$, light is absorbed by this cavity and free carriers are generated to change the refractive index, which can induce a shift of the resonant wavelength of cavity $a_2$, leaving cavity $a_1$ unaffected. Once the wavelength shift is larger than the splitting of the supermodes $V1$ and $V2$, the coupling between cavities $a_1$ and $a_2$ are switched off.
4. Conclusion
In summary, we have proposed a scheme to realize strong coupling between two distant 1DPhCCs via a dark WGM. Thanks to the extensive field distribution of the WGM in the microring, two distant identical 1DPhCCs can be coupled together via a common WGM. More interesting, strong coupling can be maintained even when the microring is largely detuned from the 1DPhCCs. At large detuning, energy oscillations between two 1DPhCCs are observed while the energy in microring is very small. Such a system relaxes the requirements of realizing strong coupling between distant optical nanocavities and thus can be implemented experimentally without much difficulties. With more microrings, it is possible to achieve strong coupling between multiple distant 1DPhCCs. The scalability of the proposed scheme paves the way for the realization of photonic integrated circuits.
Funding
National Natural Science Foundation of China (11704423).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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