We model the optical properties of L3 photonic crystal nano-cavities as a function of the photonic crystal membrane refractive index n using a guided mode expansion method. Band structure calculations revealed that a TE-like full band-gap exists for materials of refractive index as low as 1.6. The Q-factor of such cavities showed a super-linear increase with refractive index. By adjusting the relative position of the cavity side holes, the Q-factor was optimised as a function of the photonic crystal membrane refractive index n over the range 1.6 to 3.4. Q-factors in the range 3000-8000 were predicted from absorption free materials in the visible range with refractive index between 2.45 and 2.8.
© 2007 Optical Society of America
A photonic crystal (PC) is a structure in which a periodic variation in refractive index occurs at the scale of the wavelength of light in one or more directions . If the refractive index contrast of the PC is sufficiently large, it can result in the formation of a photonic band-gap (a range of frequencies in which the propagation of light is forbidden) . By introducing a defect in such photonic crystals, it is possible to strongly localise light within a small volume [1–7]. Such defects are called nano-cavities and characterised by small mode volume (V) and high Q-factor , making them attractive systems in which to study physical processes such as the Purcell effect which is characterised by the factor Q/V [7,9], exciton photon strong coupling phenomena which is characterised by Q/√V , nonlinear optical effects which are characterised by Q 2/V [8, 10], and ultra-low threshold lasers [1,2,6]. Ideally to take full advantage of these physical effects, a three dimensional photonic crystal is required [1,11]. However, fabricating such structures is technically challenging. Fabricating two dimensional photonic crystal (2D PC) slabs (a thin layer of high refractive index material in which a two dimensional PC is patterned)  is however much easier, making them an attractive alternative to three dimensional PCs. In a 2D PC slab, light is confined in lateral directions through the band-gap of the 2D PC, and vertically by total internal reflection. Introducing a nano-cavity into a 2D PC slab can be achieved in two ways; by missing one or more air holes known as a donor-type nano-cavity [3,13], or by enlarging an air hole, known as an accepter nano-cavity .
In nano-cavities constructed from materials having no intrinsic absorption, the cavity Q-factor is determined by the radiation loss from the cavity surface. Radiation losses can be reduced by using high refractive index materials (n ~ 3.4) such as Si or GaAs, or by modifying the nano-cavity design to reduce radiation losses and thereby increase the Q-factor [3,5,14–22]. Ryu et al  discussed improving the Q-factor of the hexapole mode of one missing air hole defect (H1) by optimising the size and position of the six nearest-neighbour holes around the defect. A Q-factor of 2×106 and a mode volume of 1.4(λ/n)3 was predicted from such cavities based on using materials of refractive index of 3.4. Akahane et al  investigated the Q-factor of a linear-type defect consisting of three missing air holes (L3) based on a Si membrane by adjusting the position of the first nearest-neighbour end holes of the cavity. A Q-factor of 105 and a mode volume of 0.71(λ/n)3 were predicted and a Q-factor of 45,000 was experimentally realised. Further adjustment to the positions of the second and third nearest-neighbour end holes improved the theoretical Q-factor of these cavities to 2.6×105 and the experimental Q-factor to 105 . The improvements in the Q-factor of L3 nano-cavities are attributed to the suppression of radiation losses by modifying the mode-envelope field-profile through displacing the side-holes [3,15]. The above results motivated several research groups to fabricate nano-cavities based on GaAs [5–7], InGaAlP , chalcogenide glasses , SiN  and organic materials . Yoshie et al  demonstrated experimentally a Q-factor of 20,000 from an L3 nano-cavity based on a GaAs membrane. Ruan et al  used Ge33As12Se55 chalcogenide glass of refractive index n = 2.7 to fabricate an L3 nano-cavity at 1.55 μm with a Q-factor up to 10,000. Kitamura et al  used an organic layer of Alq3 (n = 1.85) to fabricate an H1 nano-cavity at 683 nm with a Q-factor of 1000. Tanabe et al  measured a Q-factor of 3.2×105 from the hexapole mode of an H1 nanocavity in a silicon membrane. Böttger et al  studied the effect of building cavity resonators using low refractive index materials (n ~ 1.54) in a thick square lattice PC (with a hole depth of ~ 3 μm). A Q-factor of 400 was demonstrated experimentally. By modifying the radius of the air holes around the cavity resonator, it was predicted that a Q-factor of 20,000 might be achieved. In spite of these important results, it is unclear how an L3 nano-cavity Q-factor is dependent on the exact refractive index (n) of the slab material and on the relative displacement (S) of its edge holes in an L3 nano-cavity (see Fig. 1).
In this paper we model the optical behaviour of the Q-factor of an L3 nano-cavity as a function of the photonic crystal membrane refractive index (n) and the side hole displacement (S). The calculations are based on the guided mode expansion (GME) method described in detail in references  and . We use a unit cell with the nano-cavity at its center, surrounded by a photonic crystal of size (14×7√3)a.This supercell is then tiled by the application of periodic boundary conditions. The out-of-plane losses are found by assessing the overlap of the confined state with scattering states. We do not calculate in-plane losses, but in real structures, these may be reduced to negligible levels simply by the inclusion of a sufficient number of holes around the cavity. We have favourably compared results from GME with results from conventional 3D finite difference time domain (FDTD) software [26, 27], where the cavity was placed at the centre of a PC that consisted of 42 × 26 unit cells.
Our motivation for this is the construction of nano-cavities operating at visible wavelengths which contain fluorescent organic molecules. Here, our work is prompted by our recent observation of an enhanced spontaneous emission rate in a micro-pillar containing an organic dye . Clearly, as most organic materials emit light at visible wavelengths, it is necessary to construct a nano-cavity based on a dielectric material having low-loss at visible wavelengths. However it is not clear to what extent the relatively reduced refractive index of most visibly transparent materials would have on the optical properties of a nano-cavity constructed from them. Here, we show that nano-cavities can be created having a Q-factor of up to 7700 for materials such as crystalline TiO2 of n ~ 2.8 . This result suggest that high Q-factor cavities containing organic materials can be created and used in a range of applications in photonics and quantum communications.
Figure 1 shows a schematic diagram of the structure investigated, which consists of a triangular 2D PC slab having a lattice constant of a = 240 nm, a slab thickness of d = 0.6a, a hole radius of r = 0.29a, a side hole displacement of S (varying between 0 and 0.25a) and a refractive index n varying between 1.6 and 3.4. Vertically, the slab is surrounded by air. The first step in our modelling is to investigate the ability of such structures to support an optical band-gap. To do so, we calculate the band structure for both TE-like and TM-like modes. The calculations revealed that our structures can support a full TE-like band gap for n ≥ 1.6. The results of our calculations are summarised in Fig. 2. We predicted that as the refractive index of the photonic crystal membrane increases, the width of the band-gap increases and the band-gap centre energy shifts to lower energy. The results of our calculations in Fig. 2 predict a full TE-like band-gap from materials of refractive index as low as 1.6 in good agreement with the work of Kee et al .
Materials having refractive index around ~ 1.6 include polystyrene. In Fig. 2, we also indicate the typical refractive index of a series of dielectric materials having optical transparency at the PC stop band centre wavelength. The results in Fig. 2 indicate that a PC constructed from materials such as amorphous Nb2O5 and TiO2 would exhibit a full TE-like band gap of width ~ 350 meV, suggesting that they could be used to create nano-cavity structures working at visible frequencies.
Figure 3(a) shows the Q-factor of an L3 nano-cavity as a function of the photonic crystal membrane refractive index (n) for S = 0. Here the Q-factor was extracted from our GME model by relating the dissipation of energy from the nano-cavity to the total energy stored in the cavity. Experimentally the Q-factor defined as E/ΔE where E is the cavity mode energy and ΔE is its full width at half maximum. At n = 1.6 the Q-factor defined as is 300. As n increased to 3.4, the Q-factor increases super-linearly reaching a maximum value of 3900. The improvement in Q with n can be attributed to the better confinement of the optical field in the vertical direction. In all cases the fundamental cavity mode was located within the photonic crystal band-gap, with electric field distribution (Ey) localised within the cavity region and consisting of three lobes as expected . An important figure of merit for such cavities is the mode volume V, as shown in Fig. 3(b) also plotted as a function of n. All cavities can support a small mode volume of the order of (λ/n)3. From the values of Q and V, the Purcell factor (maximum enhancement in the spontaneous emission rate) is evaluated using the equation Fp = 3Qλ 3/4π 2 n 3 V. Nano-cavities with a refractive index as low as 1.6 are thus expected to have a Purcell factor of 15 as shown in Fig. 3(b), with this value increasing to 440 as the refractive index increases to 3.4. Materials such as amorphous TiO2 with a refractive index of n = 2.45  are expected to support a cavity having a Q-factor of 870, a mode volume of V = 0.72(λ/n)3 and a Purcell factor of Fp= 90. To gain confidence in our results, we calculated the Q-factor for an L3 cavity with a refractive index n = 2.45 using a 3D FDTD model. Using this alternative approach we predict a Q-factor of 940 in good agreement with that calculated using GME method (where Q =870).
The profile of the envelope field of the L3 cavity mode can by modified by adjusting the position of the side holes [3,15] which can result in an improvement of the cavity Q-factor, as a result of reducing radiation losses. We tested the effect of moving the side holes both inwards and away from the centre of the cavity. Moving the side holes toward the centre of the cavity decreases its Q-factor indicating that this procedure increases the coupling to radiation modes. However, moving the holes outwards improves the Q-factor as shown in Fig. 4. Here, the Q-factor is plotted as a function of hole displacement (S) and the PC membrane refractive index n. For large n, it is clear that Q-factor is a very strong function of S. Improvement in the Q-factor by a factor of 10 times can be achieved for n = 3.4 by displacing the side holes by S = 0.19a. From Fig. 4, one can see that for low n, the Q-factor is a weak function of the side hole displacement. For n = 2.45 enhancement in the Q-factor by a factor of 3.3 times is possible when S = 0.21a. For n = 2, the Q-factor increases by a factor of 2 times when S = 0.22a.
Figure 5(a) shows the maximum value of Q-factor (optimised for each value of n by adjusting the side holes by a distance S) as a function of n. It is apparent that the behaviour of Qmax with n has an almost exponential behaviour. An important finding from these calculations is that materials of refractive index n ~ 2.8 such as chalcogenide glasses for near infrared region and crystalline TiO2 or InGaAlP for the visible region can support nano-cavities of Q-factors up to 7700 by optimising the cavity design. We also plot in Fig. 6 the maximum Purcell factor Fpmax as a function of the photonic crystal membrane refractive index. This also exhibits an exponential behaviour with n. Materials such as amorphous TiO2 (n ~ 2.45) are expected to support cavities of a Purcell factor of 240, with this value increasing to 700 for materials such as crystalline TiO2 (n ~ 2.8) and to 3600 for materials such as Si and GaAs (n ~ 3.4). However such high Purcell factors are unlikely to be realised experimentally due to the spectral and spatial mismatch between the emitting dipole and the cavity mode, and also due to losses via leaky modes.
Figure 5(b) shows the calculated energy of the cavity mode at the optimised value of S as function of refractive index n. It can be seen that the cavity mode shifts to lower energy with increasing refractive index in a similar fashion to that predicted for the optical bandgap as shown in Fig. 2. We also find that the cavity mode shifts to lower energy with increasing S, however this shift is very small and of the order of δE ~ 1%.
In this work we concentrated on the effect of shifting the first nearest-neighbour end holes, however it will be possible to further improve the Q-factor by adjusting the positions of the second and third nearest-neighbour end holes . Recently Tomljenovic-Hanic et al , have calculated the effect of adjusting the positions of the second and third nearest-neighbour end holes on the Q-factor of an L 3 nano-cavity based on a diamond single crystal membrane. Here, the diamond membrane had a refractive index of n = 2.4 and a slab thickness of 0.91a. It was predicted that a Q-factor of 6000 could be expected when the first, second and third nearest-neighbour end holes were shifted by 0.21a, 0.025a and 0.20a respectively.
In summary, we have used a guided mode expansion method to model the refractive index dependence of the Q-factor of an L3 two-dimensional photonic crystal nano-cavity. Band structure calculations revealed that a TE-like full band-gap does exist for materials of refractive index as low as 1.6 (such as polystyrene), with a band-gap width of 75 meV. Q-factor calculations indicated that the Q-factor of an L3 nano-cavity with no side hole displacement (S = 0) behaves in a super-linear manner with photonic crystal membrane refractive index. By displacing the side holes of the cavity, the Q-factor of an L3 nano-cavity improved by a factor of 3 times using a photonic crystal membrane of refractive index of n = 2.45, and by a factor of 10 times for n = 3.4. Several absorption-free materials of refractive index n between 2.45 and 2.8 in the visible region exist such as amorphous TiO2, crystalline TiO2 and InGaAlP; these are expected to support nano-cavities of Q-factors between 3000 and 8000. Further work will address creating nano-cavities using such materials, and combining them with fluorescent organic materials.
We wish to thank the UK EPSRC for support of this research via grant EP/D064767/1 ‘Nano-Scale Organic Photonic Structures’.
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