Mode structure of coupled L3 photonic crystal cavities

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Introduction
The 'L3' defect was the first type of photonic crystal (PhC) nanocavity in which quality (Q) factors in excess of 10 4 were obtained experimentally [1].This nanocavity consists of a PhC membrane with a line of three holes missed out of a hexagonal lattice.High Q factors are obtained by displacing the end holes slightly outwards, which reduces the radiation losses compared to cavities with undisplaced holes.
In this article, we investigate coupled L3 cavities, which are a class of 'photonic molecules' [2].An interesting feature of photonic molecules based on coupled PhC cavities [3]- [7] compared to coupled microdisks [8,9], microspheres [10] or micropillars [11,12] is that they offer considerable design flexibility as regards their geometry.For example, an L3 photonic molecule can have a number of different orientations, including (see Fig. 1): end-to-end [6], side-to-side, 30 • diagonal, and 60 • diagonal.As we shall show here, this permits controlled tuning of the energy splitting, which could potentially be used for simultaneous resonant enhancement of two transitions [11,12].Moreover, the coupling can also be used to enhance the Q and rotate the polarization, thus offering additional possibilities for detailed mode design.In the long term, this mode control could be important for generating quantum entanglement [13] or implementing quantum-optical Josephson interferometers [14].
The paper is organized as follows.We begin by presenting numerical results for four different geometries.These simulations indicate that the 30 • diagonal geometry offers the smoothest control of the mode splitting, and so in Section 3 we present an experimental study of coupled cavities with this orientation.The conclusions are presented in Section 4.

Theoretical study
Figure 1 shows the calculated energies and quality factors of the split fundamental modes of parallel L3 cavities coupled along directions at 0 • , 30  It is apparent from Fig. 1 that the mode splitting depends strongly on the coupling geometry.For cavities coupled at 30 • , the AB mode consistently has the higher energy, while the opposite is true for the 90 • (side-to-side) geometry.However, the energy ordering switches several times for the 0 • (end-to-end) and 60 • geometries, due to oscillations in the coupling matrix element.Larger energy splittings for minimal separations are achieved from cavities coupled at greater angles, since these geometries allow greater proportions of the cavities to be in close proximity.Thus the largest splitting is calculated for the 90 • geometry.However, the splitting drops off very rapidly with separation, in contrast to the 30 • geometry, which provides the smoothest variation in the inter-cavity coupling due to the field extension along the 30 • direction.For this reason, the experimental study in Section 3 focuses on the 30 • geometry.
The high Q factor of the fundamental mode of isolated L3 cavities is caused by the suppression of the radiation losses.The E x field component has odd parity about both the x and y axes, which automatically causes field cancellation in the forward direction.This is not true for the even-parity E y component, and the high Q is caused by minimizing the integrated in-plane E y field by fine tuning of the end hole positions [18].The equivalent optimization for coupled cavities is complicated by the different relative signs for the B and AB mode fields within the two cavities.The fields at the cavity centers have opposite polarities in the AB mode, which makes cancellation of radiation in the forward direction possible.This is not the case for the B mode, and so we would, in general, expect the AB mode to have the higher Q.
The theoretical Q factors presented in Fig. 1 confirm that the AB mode does usually have the higher Q, although there are notable exceptions, especially in the 0 • geometry.This particular geometry is further complicated by the fact that, at minimal separation, the single hole separating the two cavities cannot be displaced in two directions at the same time.The calculations were thus performed with the hole undisplaced, and this results in a very large difference between the B and AB modes.It is particularly noteworthy that the Q of the B mode is significantly higher than that of the isolated cavities (≈ 40, 000) at a cavity separation of 6a.The use of coupled cavities can thus lead to an enhancement of Q, although at the expense of a larger modal volume V , such that Q/V is not enhanced [19].
In the 30 • geometry, the perturbation of the cavities' fields due to the coupling results in low Q factors for minimal separation, with the AB mode having the larger Q at small separations.This contrasts with the 60 • geometry, where the AB mode has the lower Q at the minimal separation on account of anomalously strong coupling to lossy states at the edge of the light cone.In the 90 • geometry, the AB mode consistently has the higher Q factor, as expected.

Experimental study
The theoretical studies in Section 2 indicate that the 30 • geometry offers the smoothest control of the coupling.L3 photonic molecules with this geometry were therefore fabricated by beam lithography in GaAs membrane containing a single layer of InAs dots with density of ∼ 5 × 10 10 cm −2 .ZEP520A resist was exposed with a dose of 36 µC cm −2 at 30 kV, and the pattern was transferred into the GaAs by Cl 2 -based chemically assisted ion beam etching (CAIBE).Air-bridges were then formed by selective etching with buffered hydrofluoric acid of a 1 µm-thick sacrificial Al 0.6 Ga 0.4 As layer.Photonic molecules were prepared with one, two, three, or four holes between the cavities, corresponding to separations from 2 √ 3a to 5 √ 3a, with the same d, a, f and s parameters as in Section 2. A scanning electron microscope image of a typical sample is included in Fig 3 .Photoluminescence (PL) measurements were made by using the dots as an internal light source under 633 nm excitation from a HeNe laser.The light was collected in the direction normal to the sample surface through a microscope objective with a numerical aperture of 0.42.The samples were mounted in a helium flow cryostat, at a temperature of ∼ 10 K.
Figure 3 compares a spectrum from an isolated cavity to that of dual cavities with the minimum spacing of 2 √ 3a.The modes are labelled according to the in-plane parities of their E x fields at the center of the membrane, following the notation defined in ref. [18].We place the parity along the x axis above the parity along the y axis, and add a numerical index denoting their ordering in terms of energy.In the coupled cavities, clear splittings are observed for sev-  The splitting of the fundamental mode is explored in more detail in Fig. 4(a), with the error bars representing the standard deviation for ten pairs of cavities.The experimental mode energies demonstrate excellent agreement with the theoretical results.The experimental Q factors shown in Fig. 4(b) are lower than the theoretical ones given in Fig. 1 on account of losses due to imperfect fabrication and/or background absorption.Nevertheless, the predicted reduction in Q for the minimally-spaced cavities is observed in the data.This reduction is a consequence of the perturbation to the optimized fields of the individual L3 cavities caused by the coupling.The two modes also differ in their polarization.Figure 5 shows the average experimental polarization angles (solid circles) of the fundamental modes in the 30 • geometry as a function of cavity separation.The results were obtained by measuring the intensity as a function of linear polarizer angle in detection.The inset shows a polar plot of the mode intensities for the 2 √ 3a separation, where 0 • corresponds to the x axis (i.e.parallel to the L3 cavity long-axis) and 90 • to the y axis (i.e.perpendicular to the L3 cavity axis).The main figure shows the polarization angle of each mode, determined as the axis of maximum intensity from the polar plots, as in the inset.The fundamental modes of isolated L3 cavities are polarized along the y axis due to the parities of their in-plane electric field components [18].It is therefore unsurprising that the modes of weakly-coupled pairs of cavities exhibit polarizations that lie very close to the y axis.However, as we reduce the cavity separation, the field distributions become more strongly perturbed, and the B mode gradually rotates.For the cavities with the smallest separations, the polarization of the B mode is approximately orthogonal to the coupling axis.The open circles in Fig. 5 show finite difference time domain (FDTD) simulations of the polarization 1µm above the cavity at a point in the center [21].The experimental trends are reproduced, although the quantitative agreement is not perfect.The discrepancies may indicate that the polarization is very sensitive to perturbations in the barrier associated with fabrication errors.Overall, the results in Fig. 5 show that coupled cavities may be used to engineer the mode polarization, which is an important consideration when coupling QDs to cavities [22].

Conclusions
We have performed numerical simulations of coupled L3 PhC cavities in four different geometries, and have found that the 30 • geometry shows the smoothest variation in the energy splitting of the fundamental mode with respect to cavity separation.Experimental results for this geometry are in excellent agreement with the simulations.Significant differences were observed for the polarization of the bonding and anti-bonding states, again in agreement with theory.Our results should facilitate informed decisions over the suitability of various geometries for different applications, for example, for double photonic resonances in quantum dots.

Fig. 1 .
Fig. 1.Theoretical plots of the energies and quality factors of the split fundamental modes of parallel L3 cavities coupled along lines defined relative to those of the cavities.Green lines represent bonding (B) states, red lines represent anti-bonding (AB) states.

Fig. 2 .
Fig. 2. Theoretical in-plane electric fields at the center of the slab for the bonding (a-c) and anti-bonding (d-f) states.(a,d) | E | 2 ; (b,e) E x ; (c,f) E y .Color scales run from blue (low) to red (high).In (a) and (d), blue and red represent zero and high intensity respectively, while in (b), (c), (e) and (f), blue is negative and red positive.

Fig. 3 .Fig. 4 .
Fig. 3. Comparison of an experimental PL spectrum from (a) an isolated cavity, and (b) a photonic molecule with a pair of cavities separated by 2 √ 3a.The inset shows the experimental geometry with 30 • diagonal coupling.

Fig. 5 .
Fig. 5. Experimental (solid circles) and theoretical (open circles) polarization angles for the 30 • geometry as a function of cavity separation.The inset shows the polar plot for a cavity separation of 2 √ 3a.The line at 30 • indicates the coupling axis.
[17]60 • and 90 • to the line of the defects.The results were obtained by a guided mode expansion (GME) method derived from refs[15]-[17]for GaAs structures with lattice constant a = 270nm, depth d = 130nm, fill factor f = 0.29 and end-hole displacement s = 0.15a.Figure2shows the calculated E x and E y field profiles for the 30 • geometry.(Thexand y directions are defined in the inset to Fig.3.)Since some geometries preclude labelling by strict symmetry, and the E y field is the dominant component #140134 -$15.00USD Received 23 Dec 2010; revised 25 Feb 2011; accepted 25 Feb 2011; published 11 Mar 2011 (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5671