Coupling between photonic-crystal defect microcavities is observed to result in a splitting not only of the mode wavelength but also of the modal loss. It is discussed that the characteristics of the loss splitting may have an important impact on the optical energy transfer between the coupled resonators. The loss splitting — given by the imaginary part of the coupling strength — is found to arise from the difference in diffractive out-of-plane radiation losses of the symmetric and the antisymmetric modes of the coupled system. An approach to control the splitting via coupling barrier engineering is presented.
©2008 Optical Society of America
Photonic confinement in optical microcavities provides a powerful tool for controlling both the characteristics of light and the features of light-matter interaction . Moreover, coupling of microcavities that allows for photon transfer introduces new grounds for the development of fast lasers , delay , routing , and non-linear  optical lines, bistability-based ultrafast generators, switchable lasers , optical memories [7, 8] and other elements of future integrated photonics circuits. The microcavity coupling can also form a basis for the interaction of spatially separated single quantum objects [9, 10] and correlated polariton systems  for applications in quantum information science and studies in quantum physics. As generally known, coupling results in a frequency splitting and in the formation of supermodes. In some circumstances, however, the supermodes also exhibit a splitting in the modal loss, which may prevent an efficient energy transfer. Here we demonstrate such a coupling between photonic-crystal (PhC) cavities, providing an evidence for the coupled-state formation and the loss splitting, and discuss approaches for loss equalization and its impact on the photon transfer.
The direct coupling of optical microcavities has been demonstrated in various cavity systems including microdisks [7, 12], distributed Bragg reflector (DBR)-based pillars [13, 14], microspheres , microrings  as well as in PhC-based few-cavity photonic molecules [17–21] and large-area arrays [2, 3, 5]. Frequency matching has been the main prerequisite in designing such microphotonics devices. However, loss associated with particular confined cavity modes is also an essential parameter, especially in the case of high-Q structures such as PhC defect cavities. Possible differences in the losses of the coupled supermodes would not only influence the interaction with charge carriers confined in different cavities (altering light-matter interaction e.g. via Purcell effect or influencing coupled-cavity laser performance) but would also affect the efficiency of the photon transfer, as it is shown here.
2. Loss splitting and energy transfer
Consider first a generalized system of two coupled oscillators (e.g. confined photons in our case) of free frequencies ω1 and ω2, and damping (loss) parameters γ 1 and γ 2. In the regime of linear coupling, the complex angular eigen-frequencies (frequencies ΩS,A and linewidths ΓS,A) of the system are given by 
where g is the coupling strength and the symmetric (S)/antisymmetric (A) supermodes correspond to the +/- signs, respectively. In the case of dissimilar cavities (i.e. ω1≠ω2), the supermode field distributions become increasingly localized to either one of the coupled parts with decreasing g. Hence, the energy (photon) cannot be exchanged efficiently between the cavities. However, Eq. (1) also shows that, apart from the splitting in frequencies (that is ΩS≠ΩA), the coupled system may exhibit also a splitting in the modal losses with ΓS≠ΓA if the coupling strength is a complex number containing a non-vanishing imaginary part Im(g). Then, even if the free frequencies are well tuned (ω1≈ω2), the dissimilar supermode losses ΓS and ΓA may prevent the energy transfer. This can be confirmed simply by considering the time evolution of an optical field initially localized at one of the coupled cavities. The intensity at either cavity then evolves with time as
This expression shows that in the case of equal modal losses ΓS=ΓA, the energy can be fully transferred between the cavities, limited only by the finite decay time 2π/ΓS. However, in the most general case where g=Re(g)±Im(g), the loss splitting occurs such that ΓS≠ΓA and a complete energy transfer is hindered due to the lack of a perfect destructive interference between the symmetric and the antisymmetric supermodes at phase matching (see Fig. 1 for illustration).
3. Observation of wavelength and loss splitting in PhC microcavities
3.1 Sample preparation and integrated quantum-wire system in PhC cavities
To probe the coupling of two optical microcavities, we used a system  consisting of two PhC L3-defect cavities each of which contains a monolithically embedded short, site-controlled quantum wire  (QWR) [Fig. 2(a)]. The PhC-QWR integration technology is based on substrate pre-patterning , which allows for a predefined nucleation site for the QWR. The lateral confinement in the QWRs ensures an efficient trapping of the excited carriers near the peaks of the optical field of the microcavity, thus yielding an efficient exciton-photon interaction. The micro-photoluminescence (micro-PL) spectrum of a “bare” InGaAs/GaAs QWR heterostructure, located outside the optical cavity, consists of two peaks [Fig. 2(b)]: one is due to recombination in the QWR (at ~923nm), the other (at ~886nm) — due to the quantum wells (QWs) located on both sides of the wire [see Fig. 2(a)]. The fabrication  starts by growing a ~1-µm Al0.7Ga0.3As layer and a 200-nm GaAs membrane layer on a (100) GaAs substrate using low-pressure metal-organic vapor phase epitaxy (MOVPE). A 1-µm pitch, ~170-nm deep V-groove grating (lined along the [01–1] direction is then fabricated using electron beam lithography (implemented on a JEOL JSM 6400 scanning electron microscope) and wet etching. Next, the patterned substrate is MOVPE-regrown with a 5-nm In0.15Ga0.85As layer sandwiched between thin GaAs barriers such that a ~10-nm thick, crescent-shaped InGaAs QWR is formed at the bottom of each groove, near the center of the 265-nm (once regrown) GaAs membrane layer. After the regrowth, the QWRs are characterized using the micro-PL spectroscopy, and PhC structures are accordingly designed to resonantly match the QWR low-temperature (10K) PL peak [Fig. 2(b)]. The PhC design implements intercavity photonic tunnel barriers consisting of one or more missing holes in a longitudinal configuration [Figs. 2(c), 2(d)]. In the experiment discussed here, the PhC lattice constant was a≈210 nm and the radius of the holes corresponds to r/a≈0.255, in order to match the QWR ground-subband emission to the L3-cavity fundamental mode at the frequency within the TE0 PhC bandgap. The spatial alignment was implemented by the electron-beam lithography using conventional alignment marks, positioning the QWR coaxially with the cavities and centering it with the accuracy of ~40 nm . The PhC holes were drilled using Cl2/SiCl4 reactive ion etching. Finally, the membrane was released by selectively undercutting the Al0.7Ga0.3As layer in HF solution.
3.2 Optical characterization method
The optical characterization was done by the low-temperature (10K) micro-PL. The excitation was performed non-resonantly using a 532-nm frequency-doubled Nd:YAG laser beam focussed by a 50x objective lens with NA=0.55. The signal was collected by the same objective lens, dispersed by a 450-mm single-grating spectrometer (resolution around 0.06 nm) and acquired by a N2-cooled CCD array.
Serving as internal light sources, the QWRs integrated into each cavity can be pumped optically independently using the micro-PL set-up that provides a ~1-µm spot size. Such a photoexcitation of the system allows for the selective cavity excitation [Fig. 2(e)]. This feature makes it possible to probe the extent of localization of the cavity modes, as further discussed below, and remove ambiguities in the conclusions on the presence or absence of the cavity coupling in experimental observations. Indeed, if the coupled-cavity structure is instead pumped uniformly, the measured spectrum can contain two peaks that may be referred as to the coupled states, while the modes actually originate from the decoupled (detuned) cavities.
Figures 3(a), 3(d) displays the measured micro-PL spectra for the single- and triple-hole barrier, coupled-cavity structures, each obtained using several locations of the pumping beam. In both cases, two cavity modes can be excited near 910 or 920nm. However, whereas these two modes have the same relative intensity in the case of the single-hole barrier, they exhibit strong dependency on the pumping location for the triple-barrier structure. In particular, for the triple-barrier structure, pumping near the left- or right-cavity yields excitation of predominantly either one of the spectral lines M1 or M2 [Fig. 3(d)].
This behavior is explained with the aid of a three dimensional (3D) finite-difference time-domain (FDTD) simulation of the coupled-cavity structures. The 3D FDTD numerical modelling is based on the standard Yee algorithm ; and the Padé-Baker approximation is used for the Fourier transforms of the signals [26, 27], which yields high spectral resolution for a moderate number of iterations. The simulations implement the actual patterns of the PhC structures extracted from scanning electron microscope (SEM) images, hence accounting for the fabrication-induced disorder in hole size, shape and position. The simulated spectrum of the single-hole barrier system [Fig. 3(b)] clearly shows a pair of modes, separated by 2.75 nm (6138 GHz), in a good agreement with the observed splitting (1.7 nm, 3803 GHz). The corresponding near-field patterns [Fig. 3(e)], confirm that these modes are indeed the “symmetric” (MS) and the “antisymmetric” (MA) coupled modes of the system. The field distributions show virtually no localization at either of the coupled cavities, indicating that the coupling in this case is strong enough to overcome the disorder-induced detuning. The delocalization is consistent with the observation in the PL spectra of only small changes in the relative intensity of the doublet’s peaks at different locations of the pump.
By contrast, the near-field distributions of the two modes confined by the structure with a triple-hole barrier show pronounced localization at either cavity [Fig. 3(e)]. Here, the intercavity coupling is significantly lower than for the single-hole case, and thus the same degree of disorder leads to mode localization. As a result, pumping with the optical beam positioned over either cavity yields excitation of the mode localized at that region.
A distinct feature apparent in the calculated and measured spectra of Fig. 3 is that the coupled-cavity modes are split not only in the resonant wavelength, but also in their losses. For the single-hole barriers, the loss splitting is manifested by measured Q-factors of 3200 and 1150 for the antisymmetric and the symmetric modes, respectively. On the other hand, the virtually uncoupled M1 or M2 modes of the three-hole barrier structures exhibit similar Q-factors of 1400 and 1520. Since —for the single-hole barrier— the fields of the eigenmodes are completely delocalized [Fig. 3(c)], the coupling term g in this case is much larger than the detuning. Therefore, the measured splitting can provide a good estimation of g, yielding Re(g)=1901 GHz and Im(g)=571 GHz. Using these values, frequency and loss detuning curves can be calculated from Eq. (1), which is shown in Fig. 4. Note that Eq. (1) necessitates the asymmetric splitting in Q-values, fully consistent with the experiment. Note also that the cavity loss parameter approaches Q=1700 for very large detuning, consistent with the measured values for the (virtually uncoupled) modes of the three-hole barrier structure. The fact that the Q-values in the latter case are lower (1400 and 1520) is explained by the frequency overdamping in the weak-coupling regime  implying that each cavity should have Q lower than that for the unperturbed case. The presence of such very weak coupling in the triple-hole barrier can be noticed in the field distributions [Fig. 3(e)].
In order to further verify the experimental observations, we performed 3D FDTD simulations for intentionally detuned cavities with ω2=ω1+Δ. In this case in order to produce comparable intensities, the modes were excited resonantly by the corresponding field distributions in the central plane of the membrane. These field distributions were obtained from the 2D finite-difference computation. The finite detuning Δ was achieved in the simulations by a slight modification of the PhC holes surrounding one of the cavities (note the cavity at right position in Fig. 5). Figure 5 displays the calculated spectra and near-field patterns for Δ≈1854 GHz (0.8 nm); the results reproduce the features observed in the PL spectra and the simulations based on the actual structures, further confirming the interpretation of the coupling behaviour. Note that for a larger cavity detuning of Δ≈2 nm, the simulations yield localized modes even for the single-barrier structure, providing estimation for the maximum detuning tolerated in this geometrical configuration.
4. Diffractive losses of the supermodes in the PhC-microcavity coupled system
Insight into the physical mechanisms responsible for the loss splitting can be gained by recalling that the cavity losses of the PhCs discussed here are governed by the out-of-plane diffraction. It strongly depends on the PhC geometry and is described by the light cone of the cladding region [29, 30]. This is different than in the case of other cavity types, e.g., microdisks for which Q difference of coupled modes ascribed to scattering at imperfections was observed . The relation of the loss splitting to the PhC diffractive losses can be visualized by inspecting the Fourier transform (FT) of a near-field pattern within a reference plane located just above the PhC membrane. Figure 6(a) shows the distributions in the reciprocal (kx,ky) space computed for ideal (disorder-free) PhC coupled-cavity structures with parameters similar to those used in the experiments. For clarity, the cases of single- and five-hole barriers are compared, the latter being an example of a “long” barrier; a single cavity is also shown, for reference. To aid visualization, Fig. 6(b) illustrates directly the out-of-plane radiation patterns in the real space. Quantitatively, the diffractive losses of each mode were estimated by the ratio of the integral intensity within the light cone to the integral intensity within the entire reference-plane:
(Only the electric component (E) of the field is considered here, since time-averaged electromagnetic energy is distributed equally between E and H). Compared to the corresponding Q-values of the modes extracted directly from the 3D FDTD temporal response [insets in Fig. 6(b)], the results of the light-cone analysis confirm much higher radiation losses for the symmetric supermode in the “short”-barrier case. The radiation-loss difference for the “long” barrier is much less pronounced resulting in much smaller loss splitting. Note, that due to formation of the supermode amplitudes, the symmetric mode will always have larger amplitude in the barrier region. However, for sufficiently long barriers, due to tight confinement provided by the PhC, both supermodes do not overlap significantly with the barrier, which equalizes their losses (see Fig. 6(b), 5-hole barrier), and the Q-factors and the radiation patterns then resemble those of an individual L3 cavity.
5. Coupling barrier engineering and the time-evolving photon transfer
Geometrical adjustments of PhC cavity terminations affect dramatically the mode diffraction within the light-cone . Therefore barrier engineering in between the two coupled cavities —affecting strongly the loss in the coupled system— should be possible. Having direct implications on the photon transfer, the loss splitting and coupling performance can therefore be designed. To this end, we performed numerical modelling based on disorder-free cavities, first varying discretely the barrier length by adding an integer number of holes [Fig. 7(a)], and secondly, changing adiabatically the single-hole barrier [Fig. 7(b)]. In the first case, in addition to the expected rapid decrease of the coupling strength manifested by reduced spectral splitting, we observe a pronounced splitting of losses for short barriers and a periodic flipping  in the spectral positions of the coupled-system states. If the barrier is removed, the coupled modes degenerate into the fundamental and the first higher-order states of the L7 cavity. Most importantly, one can notice from the spectra [Fig. 7(a)] that the loss splitting may assume different “sign” (e.g., compare the Qs of the L7 cavity and the single-hole structure). Thus, at some intermediate geometry between the two the reversal of the loss splitting sign and an equalization of the supermode loss is expected. This is indeed confirmed by adiabatically increasing the single-hole barrier [see Fig. 7(b)].
The computed trends display crossing points ΩA=ΩS and ΓS=ΓA. Note that mainly the symmetric mode is affected, consistent with the arguments presented in the previous section. In particular, at ΓS=ΓA (QS=QA≈5000) one encounters the coupling regime with a complete back-and-forth transfer of photons [see Fig. 7(c) “optimized” and 3D-FDTD real-time simulation in Fig. 7(d)]. In addition, larger splitting of wavelength reduces the transfer time. Conversely, in the “experimental” case [Fig. 7(c)], the splitting of loss [QA=3200, QS=1150, deduced from Fig. 3(a)] results in a damping of the transfer oscillations (damping of oscillations in the |MA+MS|2 intensity curves with time). This damping follows in a symmetric way, that is, per transfer cycle, the intensity in the cavity 1 doesn’t decrease all the way down to zero (note that in the Fig. 7(c) the intensity curve at its first minimum doesn’t reach the time axis) and the intensity in the cavity 2 doesn’t raise up accordingly. Consistent with Eq. (2), if the overall Q-values were higher, due to the loss splitting the energy oscillations would eventually tend to a stationary uniform energy redistribution over the two cavities, as for example is the case of the L7 cavity (QA=5150 and QS=24500).
In summary, we demonstrated the photonic coupling of two closely spaced PhC membrane defect cavities and showed that the coupled modes split not only in wavelength but also in cavity loss. Such loss splitting is given by the imaginary part of the coupling strength and explained as arising from the difference in radiative losses of the symmetric and the antisymmetric modes of the system determined by the PhC out-of-plane diffraction. Since these modes beat together to transfer the photons back and forth between the cavities by coherent superposition, such loss difference may prohibit the complete transfer. Therefore the loss splitting needs to be taken into account in structures which rely on the photon transfer processes, e.g. few laterally cavity-mode coupled QDs, coupled QD polaritons  or transfer of single photons into coupled waveguides . Means for photonic barrier engineering were introduced and shown to be effective in controlling the loss splitting, and hence the photon transfer, that may apply to such systems.
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