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We investigated experimentally 1D and 2D arrays of coupled L3 photonic crystal cavities. The optical modes of the coupled cavity arrays are fed by a site-controlled quantum wire light source. By performing photoluminescence measurements and relying on near-field calculation of the cavitiy modes, we evidence optical coupling between the cavities as well as supermode delocalization. In particular, for small cavity separations, fabrication induced disorder effects are shown to be negligible compared to optical coupling between cavities.
© 2013 Optical Society of America
The study of cavity quantum electrodynamics (cQED) using solid-state implementations has been motivated by potential applications in quantum information processing [1] and quantum computing [2]. In this context, the coupling of semiconductor quantum emitters to nanocavities has been studied extensively, leading to the realization of quantum information circuits building blocks such as all-optical switching gates [3] or single quanta selection devices [4]. These systems exploit the interaction between a quantum emitter and a single optical cavity. However, the need for scalability in quantum circuits has motivated the extension of this study to multiple cavities. Coupled cavity arrays (CCAs) are promising candidates for realizing quantum networks [5, 6], enabling the transfer of information between spatially separated quantum objects via transmission of confined photons. They are also an ideal platform to pursue the implementation of quantum simulators [7, 8], motivated by the observation of quantum phase transitions with ultra-cold atoms in optical lattices [9]. The simplest CCA system comprising two nominally identical cavities, has been demonstrated with micropillars [10–12], microdisks [13] and photonic crystal (PhC) structures [14–19]. Optical coupling in larger scale CCAs is more difficult to realize since fabrication-induced optical disorder prevents the realization of identical cavities. This, in turn, results in mode localization, which effectively prevents optical coupling between sufficiently remote cavities. To counteract disorder effects, cavity configurations with large enough optical coupling are necessary. Nevertheless, ultra slow light pulse propagation has been achieved in large scale linear CCAs coupled to waveguides [20, 21], and more recently, 2D arrays of coupled cavities have also been investigated [22]. However, these experiments employed either passive optical excitation or randomly positioned emitters to probe the optical modes. While the first approach is less suited for the realization of on-chip devices, the second one renders the identification of delocalized coupled modes difficult. Indeed, if light emitters are inserted in all the cavities of the CCA, frequency shifts of individual cavities caused by disorder [23] rather than cavity coupling can explain the observation of multiple optical modes in the spectrum of the array. A more direct method of identifying mode delocalization consists in observing the field distribution by performing optical near-field measurements [24].
We present in this article the first demonstration of coupling between a localized quantum wire (QWR) emitter and arrays of three and five coupled PhC membrane cavities. By performing photoluminescence (PL) spectral measurements, we show that short site-controlled QWRs [25], located in one cavity of the coupled array, are sufficient to excite all the supermodes of the CCA. This renders the identification of delocalized optical modes straightforward, removing the need to rely on statistical arguments or near field measurements. Furthermore, the study of the optical modes is made easier by using QWRs rather than quantum dots (QDs). The broader emission spectrum of the QWRs facilitates spectral matching with the supermodes of the arrays, thus requiring less control over their spectral characteristics.
The investigated samples were produced using metalorganic vapor phase epitaxy (MOVPE) of V-groove QWRs, conventional e-beam lithography with accurate alignment, inductively coupled plasma (ICP) and reactive ion etching (RIE) for PhC cavity definition. First, vertical stacks of five In0.15Ga0.85As/GaAs QWRs were grown on a (100)-oriented GaAs “membrane” wafer consisting of a sacrificial Al0.7Ga0.3As layer overgrown by a 200 nm thick GaAs layer in which an array of 10 μm-pitch V-groove grating was defined. After QWR growth, the planarized samples had a total membrane thickness of 260–265 nm. Using high-precision alignment (∼40 nm accuracy), PhC cavity patterns were produced in a polymethyl methacrylate layer (PMMA), transferred into a SiO2 hard mask and then etched into the GaAs layer using an optimized BCl3/N2 ICP recipe [26] to produce straight cylindrical PhC holes. The GaAs membrane was released by etching the sacrificial Al0.7Ga0.3As layer using a 4% HF: H2O solution at slightly elevated temperature (∼30°C) in order to exclude the formation of cracks. Before characterization, the samples were etched by a 1 mol citric acid solution [27] in order to remove GaAs oxide and residues of SiO2.
Optical characterization of the CCA arrays was conducted using a standard micro-PL setup. The samples were placed inside a He-flow cryostat and excited optically with a Ti:sapphire laser at 700 nm wavelength under continuous wave operation. A single microscope objective was used to focus the laser beam on the sample surface (spot diameter of 1 μm) and collect the luminescence. The objective was placed above the cavity of the array containing the QWRs, to excite them non-resonantly. The field of view of our PL setup (∼10 μm) allowed us to collect the light emitted by the whole CCA structure. A half-wave plate and a linear polarizer were placed in the detection path for polarization-resolved measurements.
The two CCA designs that were fabricated and studied are shown in Figs. 1(a) and 1(b). The cavities are diagonally positioned to increase the overlap between the evanescent tails of their localized optical field. This configuration optimizes their optical coupling [28] and helps counteract the impact of optical disorder on mode localization. The V-groove QWR stacks [Fig. 1(c)] are inserted only in the central cavity of each array in order to correlate the observed mode spectra with their spatial patterns. The isolated (without cavity) QWR PL spectrum [Fig. 1(d)], linearly resolved in polarization along the vertical (V) and horizontal (H) directions as indicated in Fig. 1(b), shows polarization anisotropy consistent with mixing between heavy-hole and light-hole bands induced by the two-dimensional quantum confinement [29]. The measured degree of polarization (DOP) of the QWR PL, defined as
where IV and IH are the collected intensities corresponding to the V and H directions, reaches approximately −0.5 at the center of the QWR line. The QWR PL lineshape displays sharp features that are the signature of localized excitons [30, 31]. Figure 2 shows the electric field distributions of the first three supermodes (M01, M02 and M03) of the CCA array formed of three L3 photonic crystal cavities. The near-field distributions were computed using a 3D finite-difference time-domain (FDTD) method for a PhC pitch of 200 nm, a hole radius of 48.9 nm and a membrane thickness of 265 nm. The PhC structures are assumed to be ideally uniform. Results are shown for two different inter-cavity separations (distance between the centers of two adjacent cavities). Figures 2(a)–2(f) [2(g)–2(l)] correspond to an inter-cavity separation of 1.4 μm [0.8 μm] which is equivalent to the cavities being separated by 3 rows [1 row] of PhC holes along the y direction. The x- and y- components of the electric field in the central plane of the PhC membrane are displayed for each inter-cavity separation. The x and y directions are identical to, respectively, the V and H directions defined in Fig. 1. As a guide to the eye, the cavities and QWRs positions are highlighted by white lines in Figs. 2(a) and 2(g). The corresponding calculated mode energies and energy separations are listed in Table 1. The delocalized envelope functions of the three modes closely resemble those derived from coupled mode theory [32,33]; note, in particular, the node in the envelope function of mode M02 that spatially overlaps with the central cavity in the 3-rows configuration [Fig. 2(b)], which implies that this mode cannot be excited with a light source localize at that cavity’s position. However, structural disorder is expected to alter the envelope function shapes, causing increasing localization at either one of the cavities with increasing disorder and/or increasing distance between the cavities (lower coupling).
Fig. 1 Schematics of 1D (a) and 2D (b) arrays of coupled PhC L3 nanocavities on a slab membrane; and (c) cross-sectional view of the stacked QWRs light sources integrated in the central cavity of the PhC arrays. (d) Measured spectrum of isolated QWRs (no cavities) linearly resolved in polarization along the directions V and H indicated in (b). The corresponding degree of polarization (DOP) is shown in grey (inset).
Fig. 2 Computed near-field distributions for the modes of the 3 coupled cavities. (a)–(f) Cavities separated by 3 rows : (a)–(c) y component of the field for M01, M02 and M03 ; (d)–(f) x component of the field for M01, M02 and M03. (g)–(l) Cavities separated by 1 row : (g)–(i) y component of the field for M01, M02 and M03 ; (j)–(l) x component of the field for M01, M02 and M03. For clarity reasons, the positions of the cavities are indicated in figures (a) and (g). The white horizontal line indicates the position of the QWR light source. The directions x and y are analogue to the directions V and H defined in Fig. 1(b).
Table 1. Calculated mode energies associated to the modes of Fig. 2 for three coupled cavities separated by 3 rows (top) and 1 row (bottom). The energy separations Δ12 and Δ32 are the energy differences between the modes M02 and M01, and M03 and M02.
Incorporating a site-controlled light source in a CCA allows to unambiguously determine via PL spectral measurements whether or not the cavities are coupled and if the supermodes are delocalized. Figures 3(a) and 3(b) display the low temperature (10K) PL spectra for the three-cavity CCAs with 3 rows [see Figs. 2(a)–2(f)] and 1 row [see Figs. 2(g)–2(l)] cavity separations, respectively. Scanning electron microscope (SEM) images of the structures are shown in inset. In Fig. 3(a), Purcell enhancement of the QWRs emission [25] is clearly observed in the form of three thin peaks (quality factor ∼ 3800) at distinct energies, suggesting that the QWRs in the central cavity emit into three confined optical modes. Indeed, as explained in [25], the lifetime of the QWR emission coupled to a cavity mode is reduced as a consequence of Purcell effect [34]. This translates in an increased intensity of the QWR emission spectrally overlapping with the cavity mode. Note that the DOP of these mode peaks is opposite in sign to that of the QWR background emission, including the sharp PL lines of the QWR that arise from localized exciton recombination. These cavity modes correspond to the three supermodes of the array M01, M02 and M03, as will be shown more quantitatively below. In case of complete localization of the modes at different cavities, only the mode localized in the center cavity, incorporating the localized QWR light source, would have a signature in the PL. The observation of three cavity peaks in the PL spectrum is a proof that mode delocalization and thus optical coupling among cavities occurs.
Fig. 3 Measured PL spectra of three diagonally coupled L3 cavities, resolved in linear polarization along the directions V and H; QWR light source inserted in center cavity only: (a) spectra for 3 rows separation, (b) spectra for 1 row separation. The DOP is indicated for both spectra. Insets: SEM image of the corresponding structure. The dark line indicates the position of the QWRs. (c) Calculated M02 Ey field distribution for 3-row cavity separation; the PhC hole size follows a normal distribution with standard deviation of σd = 3 nm to mimic fabrication induced disorder. (d) Near-field intensity of the M02 mode integrated along the QWR within the central cavity as a function of σd.
It might seem surprising that we observe coupling between the QWRs and the supermode M02, which, for an ideal structure, has a vanishing field intensity at the central cavity. This observation is explained by the effect of fabrication-induced structural disorder. To get a qualitative picture of this effect on the mode distributions, we artificially introduced disorder in our numerical simulation by changing randomly the PhC hole sizes, following a normal distribution characterized by an average r = 48.9 nm and a standard deviation σd. Figure 3(c) displays the calculated y-component of the M02 field distribution for σd = 3 nm. Compared to the case of vanishing disorder [Fig. 2(b)], it is apparent that the disorder significantly increases the electric field amplitude in the central cavity. In addition, we calculated the integrated intensity of the M02 mode along the QWRs position within the central cavity as a function of σd [Fig. 3(d)]. For each σd, this value was averaged over 150 repetitions of the simulation. We observe a clear enhancement of the central cavity near-field intensity as σd is increased.
Figure 3(b) shows the PL spectra of a CCA with 1 row separation, where the inter-cavity coupling is expected to be larger than for 3 rows separation. The three delocalized supermodes can be clearly identified here as well, and their spectral separation is much greater than for the 3 rows separation. This is in qualitative agreement with the calculations of the mode splitting in a three-element coupled system which scales with the coupling strength as . Note the different polarization of mode M03 in this case, believed to occur due to the stronger optical coupling in this case.
Further evidence for the occurrence of delocalized modes in the three-element CCAs was obtained by observing similar PL spectra for 17 other nominally identical structures on the same sample. For each of these structures, we recorded the mode energies EM01, EM02, EM03 which are displayed as a function of the mean energy (EM01 + EM03)/2 in Figs. 4(a) and 4(b). From these measurements, we computed the mode separations Δ12 = EM02 − EM01 and Δ23 = EM03 − EM02. The fabrication-induced disorder present in real CCAs prevents a straightforward estimation of the coupling strength g from mode splitting measurements. Indeed, in the case of two coupled cavities, the mode separation can be expressed as
where Δ0 is the energy difference between the two cavities induced by disorder [22]. Although Δ0 is difficult to estimate, the relative weight between the contributions of the disorder and the coupling strength to the total mode splitting can be derived from a statistical analysis. Indeed, as explained in [22], these contributions can be estimated by considering the ratio between the mean mode separation Δ̄ of nominally identical CCAs and its corresponding standard deviation σ. In particular, a ratio σ/Δ̄ ≪ 1 indicates a dominant contribution of the coupling strength to the mode separation. Figures 4(c) and 4(d) list the average mode separations and corresponding standard deviations as well as the mode splittings computed with 3D FDTD simulations for the 3 rows separation and 1 row separation configurations. For the smaller inter-cavity separation, we get a ratio σ/Δ̄ ∼ 0.1 which indicates a strong inter-cavity coupling and a relatively negligible disorder effect. This is confirmed by comparing the average mode separations to the mode splitting of the system without disorder calculated by 3D FDTD. As expected, the FDTD simulations reproduce the measured mode splitting since the disorder effect in that case is negligible. For the 3 rows configuration, the CCA no longer satisfies the condition σ/Δ̄ ≪ 1, meaning that the contribution of the coupling strength is now comparable to the contribution of the disorder to the total mode splitting. As a result, the measured average mode separation is now more important than the value obtained from the FDTD simulation of the system without disorder. However, we show here that even when the fabrication-induced disorder is comparable to the inter-cavity coupling strength, mode delocalization still occurs. In other words, the formation of supermodes in a CCA is surprisingly robust to disorder.Fig. 4 Measured energies of the M01, M02 and M03 supermodes as a function of their mean energy value (dotted lines) for three cavity CCAs with a cavity separation of 3 rows (a) and 1 row (b). The vertical yellow lines indicate the set of points belonging to the same structure. The mean value of the supermodes energy separation Δ̄ as well as its standard deviation σ are indicated in (c) and (d) for respectively the 3 rows and 1 row separations. Δ̄ is compared to the values computed using 3D FDTD.
One-dimensional CCAs are useful because of their relative simplicity and use in applications such as slow light propagation. However, to achieve the necessary scalability required to implement quantum networks, CCAs of higher dimensionality have to be fabricated. We investigated also the five-cavity 2D cross-shaped CCAs [Fig. 1(b)], with the QWRs inserted at the center cavity alone, by performing PL-measurements. Figure 5(a) shows the PL spectra of a structure for which we could identify 5 modes fed by the QWR emission. An SEM image of the structure is shown in inset. We observe the Purcell enhancement of the QWR emission for 5 distinct energies, together with a change in the far-field polarization of the luminescence clearly visible in the DOP [see top panel of Fig. 5(a)]. Polarized mode emission, a consequence of the Purcell effect, has been observed previously in QD-cavity systems [35, 36]. The QWR emission that is resonant with the cavity modes acquires their polarization properties, explaining the difference in polarization between the coupled and uncoupled QWR emission. This adds further proof that the QWR light source feeds the first 5 delocalized modes of the 2D CCA: M01, M02, M03, M04 and M05. Their calculated near-field distributions are shown in Figs. 5(b)–5(f). In Table 2, the energy spacing between the modes is reported and compared with the calculated mode spacing. The difference between the calculated and experimental mode separations is explained by the finite amount of structural disorder present in the fabricated CCA. The average difference between the calculated and experimental values amounts to 3.08 meV and is comparable to the difference obtained for the 3 coupled cavities with 3 rows separation [Fig. 4(c)]. Indeed, the optical inter-cavity coupling and fabrication-induced disorder are nominally the same for these two designs, thus we expect similar contributions of the coupling strength and disorder to the total mode splitting. However, despite this fabrication-induced disorder, the QWR light source of the central cavity feeds the 5 CCA modes, indicating that cavity modes are delocalized. The coupling of QWRs to the M02, M03 and M04 is also explained by an increase of the near-field intensity in the central cavity of the array caused by structural disorder. The experimental demonstration of coupling between a single site-controlled light source and the supermodes of a 2D CCA is an important step towards the realization of larger scale cavity arrays incorporating site-controlled emitters such as QDs in each cavity. Such advanced emitter-cavity configurations could enable solid-state quantum simulators in the future [37].
Fig. 5 (a) PL spectra linearly resolved in linear polarization along the directions defined in Fig. 1 for a structure consisting of 5 coupled cavities (refer to inset for design). Inset: SEM image of the structure. The dark horizontal line indicates the nominal position of the QWR light source. (b)–(f) Calculated intensity distributions for the M01, M02, M03, M04 and M05 supermodes.
Table 2. Calculated and experimental mode splitting for the five-cavity 2D CCA.
In summary, we reported the fabrication of 1D and 2D CCAs with an embedded site-controlled QWR light source. The QWR light source was placed only in one of the cavities of the CCA structure. This design allowed us to distinguish between structures for which optical coupling was sufficient to allow for mode delocalization from structures for which fabrication-induced disorder dominated over the optical coupling, using simple PL characterization. We studied photonic crystal CCAs with different cavity separation and showed that mode delocalization persists even when the disorder is comparable to the optical coupling strength. Incorporating site-controlled QWR light sources in CCA structures is a first step towards the integration of site-controlled QDs in multiple cavity systems, which will provide the optical nonlinearities [38] required for quantum simulations.
Acknowledgments
All the FDTD simulations presented in this article were performed with Fredrik Karlsson’s FDTD computer code. We thank Alexey Lyasota for acquiring the SEM images presented in this article. This work was supported by the Swiss National Science Foundation.
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