Using a finite-difference time-domain method, we theoretically investigate the optical spectra of crossing perpendicular photonic crystal waveguides with quantum dots embedded in the central rod. The waveguides are designed so that the light mainly propagates along one direction and the cross talk is greatly reduced in the transverse direction. It is shown that when a quantum dot (QD) is resonant with the cavity, strong coupling can be observed via both the transmission and crosstalk spectrum. If the cavity is far off-resonant from the QD, both the cavity mode and the QD signal can be detected in the transverse direction since the laser field is greatly suppressed in this direction. This structure could have strong implications for resonant excitation and in-plane detection of QD optical spectroscopy.
©2012 Optical Society of America
Cavity quantum electrodynamics (cavity QED) studies the interaction between a quantum emitter and light confined in a resonant cavity [1, 2]. It gives rise to novel regimes of light-matter interaction and has important applications for quantum information processing  and highly efficiency light sources [4–7]. In particular, semiconductor quantum dots (QDs) offer an attractive material system for studying cavity QED due to their large exciton dipole moment, fixed in position and stable . Both weak  and strong coupling [9–11] between a single QD and a small volume crystal nanocavity has been observed in photoluminescence, which provides potential for solid-state single-photon sources. However, for QDs, one of main problems is that the scattering laser light is strong, hence it is difficult to differentiate the QD emission from the scattering laser with nearly the same frequency [12–14]. Currently, incoherent non-resonant excitation and photoluminescence detection are widely used to detect the emission of QDs [9–11]. Using a filter, the quantum dot emission can be discriminated from the scattered laser background easily. However, this method does not allow direct coherent manipulation of the quantum states.
To solve this problem, several methods have been proposed, and resonant quantum dots spectroscopy became a hot topic in recent years [7, 12–16]. For example, A. Muller et al. have proposed using a planar optical microcavity to suppress laser scattering . This method enabled, for the first time, the resonant excitation of a single QD in a cavity  and measurement of Mollow-triplet emission spectrum  in a semiconductor system. Another method is using non-resonant dot-cavity coupling to detect resonant quantum dots emission properties. This method comes from an intriguing phenomenon which has been observed repeatedly in many semiconductor quantum dot-cavity coupling experiments showing that there is a significant cavity mode emission in photoluminescence even when the quantum dot has a large detuning from the cavity [9, 11, 14, 15]. Though the underlying mechanism of this so called “non-resonantly coupled emission” is still not yet totally understood and aroused a heated discussion recently [16–19], it already shows the great potential of using the detuned cavity signal to readout the resonantly excited quantum dot spectroscopy [14, 15, 20].
For photoluminescence measurement, the main drawback is that the photons are emitted out of the cavity, and thus it is difficult to further manipulate them [7, 21]. To overcome this problem, finding efficient ways to couple light into and out of the microcavity-QD system through optical channel is especially important [7, 21, 22]. Recently, P. Yao et al. proposed a solution which integrates finite-size waveguides, on-chip couplers, and photonic crystal cavities together to achieve controlling the photons on-chip . In work conceptually related to ours, K. Srinivasan et al. used a fiber taper waveguide to perform direct optical spectroscopy of a system consisting of a quantum dots embedded in a microdisk . There, strong coupling was demonstrated through observation of vacuum Rabi splitting in the transmitted and reflected signals from the microdisk cavity.
In this paper, we introduce a new method based on two crossing perpendicular waveguides to achieve the in-plane resonant excitation and detection of QD. We show that not only the vacuum Rabi splitting but also the pure quantum dots signal can be detected directly through the photonic channel. The structure we employed was first designed by Johnson et al. to eliminate cross talk in waveguide intersections for constructing integrated optical circuits . The main idea is using symmetry mismatch to prevent the resonant excited mode from the input port decaying into the transverse ports. Here we show, for the first time to our knowledge, that this structure can be used as a powerful tool for detecting the resonant optical signal of an embedded QD. The QD is placed in the center rod of the intersection of the two waveguides which form a cavity (see Fig. 1 ). Since the crosstalk is reduced by the symmetry, the laser background is very weak in the transverse port, so that optical signals of the QD can be detected at this port. This method overcomes the difficulty of differentiating the QD signal from the input laser at nearly the same frequency, and achieves the in-plane optical detection at the same time.
2. Theoretical methods
We consider a two-dimensional crystal of dielectric rods in air on a square array with lattice constant a = 400nm. The radius of the rods r = 0.2a, and the dielectric constant ε = 11.56. The radius of the single rod in the center is increased to 0.33a and surrounded by the photonic crystal layers to create a resonant cavity (see Fig. 1(b)) . A finite-difference time-domain method with enhancements to archive subpixel accuracy was used to analyze the propagation properties of light in this structure . To decrease the propagation losses  and calculation time, we use a finite-size structure; the computational cell is 40 × 20 lattice constants. The spatial discretization corresponds to Δx = Δy = a/25. To truncate the computation domain, appropriate absorbing boundary conditions must be used. For traditional dielectric waveguides, it has been demonstrated that the perfectly matched layer (PML) boundary condition is robust and efficient for terminating the FDTD lattices . But for photonic crystal waveguide, even if the waveguide is terminated with PML, there still exists substantial reflection from the boundary due to the dispersion relation mismatch (on the order of 20% to 30%) [27, 29, 30]. To remove this unphysical reflection from the boundary, Koshiba et al. suggested that the original photonic structure should be continued and exist also in the PML . This so-called PC-based PML greatly reduces the reflection amplitude from the photonic crystal waveguide ends. In our setup the reflection is less than −50dB across a wide range of frequencies . Here we used a photonic crystal based convolutional perfectly matched layer (PC-based CPML) boundary condition (see the red area in Fig. 1(a)). Compared with PML, the CPML offers a number of advantages, such as independence of the host medium and added capability of effectively absorbing evanescent waves . The thickness of the PC-based CPML was chosen as d = 10a. The pulse used for excitation is sent to the input port (PI), and then the electric fields are detected at different positions getting the throughput (PT) and the crosstalk spectra (PC) (see Fig. 1(a)).
The QD is embedded in the single central rod. In experiments it has been demonstrated recently using the atomic force microscopy metrology, that cavities can be precisely positioned around a single preselected QD . We consider the case that the dipole orientation of the QD is along the z axis which is parallel to the polarization direction of electric field. The interaction of a single QD with the resonant electric field is described by two-level optical Bloch Eq , which are numerically solved by the Runge-Kutta method, and coupled with the Maxwell Eq. by the macroscopic polarization in a self-consistent approach [32, 33].
3. Results and discussions
First, we investigate the case without embedded QD. We send a 50fs broad-spectrum TM (in-plane magnetic field) Gaussian pulse to the input port (the spectrum is shown in Fig. 2(a) ) so that we can achieve the propagation information within a wide spectrum regime by only one-time simulation run. The simulated propagating electric field polarized along z direction is shown in Fig. 1(c). The detected throughput (ET) and crosstalk pulses (EC) are Fourier transformed to obtain the throughput and crosstalk spectra which are shown in Fig. 2(b) and 2(c), respectively. It can be seen that, due to large cavity’s quality factor (Q ≈3.5 × 104), the bandwidth of the throughput is very narrow and centered on the resonance frequency of the cavity. As desired for the crosstalk, the intensity is quite low (about 10−11 compared with the input spectrum) (see Fig. 2(a) and 2(c)). Since we consider a finite size photonic crystal, the detection position of the crosstalk is near to the cavity, so there is an obvious signal of the cavity at the resonant frequency of the cavity. The physical mechanism of low-crosstalk was analyzed in detail in Ref : two resonant modes exist in the cavity, each of which is even with respect to one waveguides’ mirror plane and odd with respect to the other. Each resonant state only couples to modes in just one waveguide and is orthogonal to modes in the other waveguide. As a result, the resonant modes that are excited from the input port can be prevented by symmetry from decaying into the transverse port. When increasing the Q of the resonance, the crosstalk can be decreased further.
In a next step we put the QD in the central rod of the cavity, the position of the QD with respect to the cavity mode is shown in Fig. 1(d). Essentially the QD, once it is excited, behaves as if there would be a point source in the center of the cavity. The dipole moment of the QD is aligned along z direction. Hence the emission of the QD can propagate along both the x and y direction. We first consider the case when the transition frequency of the QD is resonant with the cavity (λc = λq). The Q factor is large enough to reach the strong coupling regime. In Fig. 3(a) and 3(b), we show the throughput and crosstalk spectra, respectively. Clear vacuum Rabi splitting can be observed from both the throughput and the crosstalk spectra. In the inset of Fig. 3(b), the crosstalk spectrum is shown with a larger frequency range, it can be seen that though there is a background, the signal of vacuum Rabi splitting can still be clearly resolved.
Next, we focus on the case when the cavity mode is far off-resonance from the QD transition frequency (λq-λc = 6nm). Figure 4(a) and 4(b) show the throughput spectra when the dipole moment of the QD equals to 60D and 90D, respectively. It can be seen that only the cavity mode can be observed. Nearly no difference can be seen between the cases with and without QD (see Fig. 4(a), 4(b) and Fig. 2(b)). This is because, in x direction, the function of the photonic crystal structure we used is similar to a filter, hence nearly 100% throughput can be obtained at the cavity mode frequency . In our case, since the calculation time is not long enough, the calculated throughput intensity at the cavity mode frequency is lower than the input pulse intensity. However, even in this case, the cavity mode is still strong enough to overwhelm the weak signal of QD. As a result, in x direction, it is difficult to detect the signal of QD through optical channel when the QD is largely detuned from the cavity mode.
In the transverse port, the situation is much different. Figures 5(a) -5(c) show the crosstalk spectra, the detuning is the same as in Fig. 4. Both the cavity mode and the QD can be resolved in the crosstalk spectra. When the dipole moment of the QD is increased (see Fig. 5(b)), the signals also increased. Moreover, the increasing of the cavity mode signal is even more obvious. This indicates a clear non-resonant dot-cavity coupling which have been observed previously in the photoluminescence spectra. Here we show for the first time, due to the strong suppression of the crosstalk, that the non-resonant dot-cavity coupling can also be detected through the optical channel even for the case of large detuning. Figure 5(c) shows the case when the dephasing rate is decreased, it can be seen that the signals of the QD and the cavity change accordingly. This is consistent with previous investigation on the photoluminescence spectra that the pure dephasing is closely related to the non-resonant dot-cavity coupling [18, 19].
We obtain the cross-talk spectra by Fourier transformation of the detected electric field (EC) (). The detected cross-talk electric field EC is composed of two parts: the background (Eb) and the emission of the QD and cavity (EQ/Cavity). The special shape in cross-talk spectra (See in Figs. 2(c) and 5(a)-5(c)) is induced by the coherent superposition of these two part electric fields. The relative phase between them induces the increase and decrease of the superposed spectra respect to the background.
To verify this point, we first calculate the crosstalk field without a QD giving the background field Eb and subtract it from the crosstalk field with QD (EC) in order to obtain the pure emission field of the QD/Cavity (EQ/Cavity): EQ/Cavity = EC- Eb. The generated spectra of QD/Cavity () are shown in Fig. 5(d)-5(e) which shows a typical Lorentzian shape. Moreover, as can be seen in Fig. 5, when the dephasing rate is large (Fig. 5(a), 5(b), 5(d), 5(e)), the intensity of the cavity mode is larger than that of the QD signal. With increasing dipole moment of the QD, the cavity mode also increases accordingly. As a result, detecting the cavity mode in the crosstalk spectrum provides a potential route to detect the resonant properties of the QD. When the dephasing rate is lower, the resonant signal of the QD itself can be easily obtained directly from the crosstalk spectra.
In experimental implementations of this scheme, the system may suffer from disorder and manufacturing imperfections, which will reduce the Q factor of the resonant cavity and increase the crosstalk. To overcome the effects of the manufacturing imperfections, one might further increase the cavity size or slightly shift and optimize the dielectric rods to increase the Q factor of the cavity and decrease the crosstalk.
In conclusion, we have demonstrated that the photonic crystal waveguide intersection might be used as a powerful tool which allows both the detection of resonant quantum dot optical spectroscopy through non-resonant dot-cavity coupling, and vacuum Rabi splitting under resonant dot-cavity coupling through optical channel. In particular, in-plane excitation and detection will benefit constructing integrated photonic system coupled with solid state emitters.
This work is supported by the Deutsche Forschungsgemeinschaft DFG via the Research Training Group (GRK 1464), the Emmy-Noether Group “Computational Nanophotonics”, and the BMBF (01BQ1040).
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