Ultrafast Optical Switching Using Photonic Molecules in Photonic Crystal Waveguides

We study the coupling between photonic molecules and waveguides in photonic crystal slab structures using finite-difference time-domain method and coupled mode theory. In a photonic molecule with two cavities, the coupling of cavity modes results in two super-modes with symmetric and anti-symmetric field distributions. When two super-modes are excited simultaneously, the energy of electric field oscillates between the two cavities. To excite and probe the energy oscillation, we integrate photonic molecule with two photonic crystal waveguides. In coupled structure, we find that the quality factors of two super-modes might be different because of different field distributions of super-modes. After optimizing the radii of air holes between two cavities of photonic molecule, nearly equal quality factors of two super-modes are achieved, and coupling strengths between the waveguide modes and two super-modes are almost the same. In this case, complete energy oscillations between two cavities can be obtained with a pumping source in one waveguide, which can be read out by another waveguide. Finally, we demonstrate that the designed structure can be used for ultrafast optical switching with a time scale of a few picoseconds.


Introduction
To implement quantum information processing, devices such as single qubits [1,2] single-photon sources [3][4][5][6] and single-photon detectors [7][8][9] have been fabricated in recent 20 years. However, to achieve quantum photonic network using these devices [10,11] is still a major challenge. Coupling quantum nodes with quantum channels is a key step to realize photonic network. The quantum nodes and quantum channels can be implemented with optical microcavities [12] and waveguides. Photonic crystal (PhC) cavities have been widely used as quantum nodes because of their high quality factors and small mode volumes [13][14][15], and PhC waveguides can guide light with very low losses in plane [16,17]. Up to now, integration of PhC cavities and waveguides has been widely investigated to achieve photonic circuits [18][19][20][21][22].
Studies have been carried out on the coupling between waveguides and a single cavity [23][24][25] or two coupled cavities [26,27]. For example, PhC channel drop filters have been designed with coupled structures consisting of two waveguides and two coupled single-mode microcavities [26]. All-optical switching based on coupled PhC nanocavities has been demonstrated using Fano resonance [27]. In this work, we report on the energy oscillation between two strongly coupled PhC cavities, i.e. photonic molecules, and their integration with two waveguides for ultra-fast optical switching.
Photonic molecules (PM) are formed by two or more coupled optical cavities.
The coupling of optical modes of individual cavities in a PM generates a mode splitting [28]. In a PM with two cavities, linear superposition of the same mode of two isolated cavities results in a symmetric (S) and an anti-symmetric (AS) super-modes [29]. The interference between S and AS super-modes induces an energy oscillation between two cavities [30]. To excite and read out the energy oscillation, in-plane PhC waveguides have natural advantages as these can be easily coupled with PMs.
Therefore, waveguides can be used as signal input and output in PM-based photonic network. In this paper, we study the coupling between a PM and two waveguides in PhC with finite-difference time-domain (FDTD) method [31] and coupled mode theory [32]. By optimizing the quality factors of PM super-modes, a complete energy oscillation between two cavities is obtained with a pumping source in one waveguide, and the oscillating signal can be read out by another waveguide.

A PM with two PhC cavities
Firstly, we consider the coupling between two L3 cavities in a PhC slab with a triangular lattice of air holes. The slab thickness is a, where a is the lattice constant, and the dielectric constant of the slab is 12.96. The air hole radius is 0.3a. A L3 cavity is formed by removing three in-line neighboring air holes, which has been demonstrated with high quality factors and small modal volumes [13]. Placing two L3 cavities in line with two air hole barriers [as shown in Fig. 1(a)], a PM is formed due to the coupling between evanescent optical cavity modes of two individual L3 cavities.
To suppress the leakage out of plane, two air holes at the edges of PM in the x direction are shifted away from the center by 0.15a. There are 35 lattice units in both x and y directions. In the FDTD simulation, a spatial resolution of 16 pixels per lattice constant has been used. A subpixel smoothing feature [31] has been employed, so that the resolution is enough to resolve the structure modification in this work. This PhC structure has photonic band gaps in the TE-like modes with electric field in plane and magnetic field out of plane. The first photonic band gap is between the normalized frequencies of 0.22328 and 0.29659 (a/λ), where λ is the wavelength of light in vacuum.
To analyze the eigen frequencies of the coupled system, we define a 1 and a 2 as the field amplitudes of two individual cavities. With coupled mode theory we obtain:    ( 2 ) This result indicates that the coupling between two cavities generates a splitting in real frequencies. And if the square root part in Eq. (2) gives a complex number, the imaginary parts of eigen frequencies are also split.
The spectrum of the coupled system in Fig. 1(a) has been calculated using FDTD method. When two identical L3 cavities are brought together with a two-airhole barrier, the coupling of fundamental resonant modes of individual cavities generates two super-modes, which can be verified by the split peaks in Fig. 1(b). The eigen frequencies of two super-modes are 0.23248 and 0.23281 (a/λ). The different linewidths of two peaks indicate the splitting of imaginary parts of eigen frequencies.
We then calculate the field distributions and quality factors corresponding to the split eigen frequencies. In TE-like modes of cavities, electric field is the superposition of E x and E y components, while the magnetic field has only z component (H z ). To indicate super-modes clearly, we calculate H z field distribution. Figure 1 symmetric (S) modes respectively, corresponding to their odd or even field parities.
The quality factors of AS and S modes are 7880 and 33530, respectively. To explain the difference of the quality factors of AS and S modes, we employ the vertical total internal reflection mechanism [13]. The wave vector distribution of cavity mode can be obtained from the Fourier transformation of the field spatial distribution of that mode. Light with wave vector inside light cone is not confined by total internal reflection, and will leak out of the cavity. That outside of light cone can be localized in cavity. Due to different field distributions of AS and S modes, the wave vector distributions are also different, resulting in a different integral intensity in the light cone. Larger the integral intensity in the light cone, higher the leakage out of cavity and lower the quality factor, vice versa.
We then analyzed energy time-evolution in two coupled cavities. In Fig. 1 We consider the field initially localized at cavity 1 ( (1) and (2), the time evolution of the fields at two cavities can be calculated as where   and   are defined as With the parameters extracted by FDTD simulation in the case of Fig. 1  From field distributions in Fig. 1(c) and 1(d), we can infer that the energy oscillations are due to cancellation and enhancement of field amplitudes of AS and S modes [30]. A complete energy oscillation requires a perfect destructive interference between AS and S modes, which results from fine phase matching and comparable field amplitudes of two super-modes. From Eq. (4), complete energy oscillations can be achieved by equalizing the loss rates of AS and S modes, i.e.

AS S
   .

Structure optimization for complete energy oscillations in a PM
From the above discussions, complete energy oscillations between two cavities in a PM require equalized loss rates of two super-modes. We need to adjust the coupled structure in Fig. 1(a)  From the field distributions in Fig. 1(c) and 1(d), we can see that the fields located at air holes (r m ) which are surrounded by green dashed circles are different.
There are fields distributed in these air holes for S mode, whereas no field observed in the case of AS mode. It can be inferred that tuning the radius r m can modify the quality factors of AS and S modes differently and effectively, which can be used to achieve equalized quality factors. (4). Figure 3(c) shows the theoretical results with a complete energy oscillation between two cavities. In addition, the energy evolution is also calculated by FDTD method. A Gaussian source has been set at cavity 1, with a center frequency of 0.23315 (a/λ) (center of the eigen frequencies of two super-modes). A frequency width of 0.01 (a/λ) has been used, which covers both AS and S modes. Then electric field energy has been monitored at the centers of two individual cavities. The simulated results in Fig. 3(d) show complete energy oscillations, comparing with those in Fig. 2(b). Comparing the energy evolutions in the cases of r m = 0.3a [ Fig.   2(b)] and r m = 0.382a [ Fig. 3(d)], we can infer that the periods of the energy oscillations are different, which indicates the oscillation periods can be tuned. This result can also be inferred theoretically from Eq. (4).

Coupling between a PM and two waveguides
PhC waveguide is formed by removing one row of air holes in photonic lattice. This type of defect induces guided modes with frequencies in photonic band gap range [17,33], which can guide light with very low losses for a wide range of frequencies [16]. The waveguide in this work is formed by removing one row of air holes along the Γ-Κ direction (W1 waveguide) [ Fig. 4(a)]. The waveguide band diagram was calculated with a free source software MPB [34]. Figure 4(b) shows the calculated band diagram of the TE modes of W1 waveguide with same structure parameters in Fig. 1(a). The red and blue solid lines show the zeroth-order and the first-order waveguide modes, respectively. We note that only the zeroth-order mode in waveguide can spectrally match AS and S modes in PM discussed before. This spectral matching between waveguide modes and PM super-modes might induce an efficient coupling between them.
Theoretical and experimental studies have shown that an efficient coupling between waveguide and L3 cavity modes can be achieved with L3 cavity tilted to waveguide axis by an angle of 60 o , in which the field overlap between evanescent cavity mode and waveguide mode is maximized [23]. We adopt this configuration for coupled PM and waveguide structure as shown in Fig. 5(a). Two L3 cavities are aligned in line along the x-axis with two-air-hole barrier. The two air holes at the edges of PM along the x-axis are shifted by 0.15a away from center to suppress outof-plane losses. Two W1 waveguides, waveguide In and waveguide Out, are tilted with respect to the x-axis by an angle of 60 o . The waveguides are separated from PM by three air holes.
With waveguides coupled to a PM, the system can be described as where 0  denotes the resonant frequency of two individual cavities, c  is the loss rate of individual cavity without waveguide, and wg  corresponds to the loss rate into waveguide. The total loss rate of each cavity in coupled system is given by total c wg      . In this coupled system, a complete energy oscillation requires the equalization of the total loss rates of AS and S modes in the PM with considering the waveguides.
In structure of Fig. 5(a), the resonant frequencies of AS and S modes are 0.23242 and 0.23276 (a/λ) respectively. The quality factor of AS mode is 3800, and that of S mode is 6760. We then calculated the transmission spectrum with FDTD. A Gaussian source with frequency center 0.23259 (a/λ) and frequency width 0.01 (a/λ) has been set in waveguide In. The energy flux has been monitored at waveguide Out.
The transmission spectrum shows two peaks, corresponding to AS and S modes.

Structure optimization for reading out energy oscillations in a PM by a waveguide
In order to realize complete energy oscillations, we tune r m of the air holes marked by the green dashed circles in Fig. 5(c) and 5(d) to achieve equalized quality factors using the same method in section (2.2). We have shown that complete energy oscillations require equalized total loss rates of AS and S modes when excitation source is located in one cavity. In the case of exciting PM by waveguide modes, the coupling strengths between waveguide modes and two PM super-modes are also required to be same, so that AS and S modes can be excited equally by waveguide modes. In our structure, the total quality factors of AS and S modes are equalized ( It should be noted that the resonant frequency splitting can be used to adjust the period on demand, larger the resonant frequency splitting faster the energy oscillations.

Conclusion
In summary, we studied the coupling between PM and waveguides in PhC slab using FDTD calculation method and coupled mode theory. We designed a coupled PMwaveguide structure in which complete energy oscillations are achieved and the oscillating information can be transmitted to waveguides. The coupling strengths of waveguide modes with two super-modes of a PM and total quality factors of two super-modes can be adjusted to nearly equal by optimizing the radius of the air holes between the two cavities. With the optimized structure, an ultrafast energy oscillation with a period of a few picoseconds is obtained, which can be used as an ultrafast optical switch.   Fig. 1(a). The energy oscillations between two cavities are not complete as well, which is in a good agreement with the results in (a).