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We propose a novel concept for creating high-Q cavities in photonic crystal slabs (PCS). These cavities are formed by depositing a polymer layer on top of a photonic crystal membrane fabricated in a high index semiconductor slab. We show that such multilayer structures exhibit a mode-gap and can yield high-Q microcavities with quality factors of Q~106. This allows the cavity to be created by polymer processing, following the much more demanding semiconductor processing that is used to generate a uniform PCS. Depending on the polymer used, these structures can be additionally tuned using photosensitivity or the electro-optic effect.
©2007 Optical Society of America
1. Introduction
The incorporation of defects in an otherwise periodic structure allows photonic crystal slabs (PCS) to trap and guide light. Therefore the critical step in the fabrication is the incorporation of defects, waveguides and cavities, in a controllable way. However this is not an easy task, especially for high-Q cavities, as the current methods rely on extremely precise control of the holes’ size and position through nanolithographic techniques [1–3]. At present, the geometry of such a structure is finalized at the stage of fabrication, and there is very limited scope for post-processing of silicon-and high-index semiconductor-based PCSs.
There are ways to tune existing defects or even induce defects by varying the refractive index within the PCS structure. Defects can be formed without geometry perturbation by air-hole infiltration or by selective exposure to light in photosensitive-based material. The infiltration of PCS air holes with materials of refractive index larger than n=1, such as a liquid crystal (LC) or a polymer was demonstrated recently [4–8]. Moreover Intonti et al introduced a “pixel by pixel” approach for writing and rewriting PCS defect structures via fluid infiltration [9]. Another way to induce or tune a defect is to change the refractive index of the background material [10,11]. This can be realized in PCS made of photosensitive material such as chalcogenide glasses and polymers. For example the induced refractive index change of azo-polymer film was can be as large as Δn=0.1 [12] and the refractive index of chalcogenide glass can change by as much as 1 to 8%, depending on the composition [13]. The advantage of these approaches is that they can be implemented any time after fabrication.
In this paper we are particularly interested in double-heterostructure high-Q cavities. Double-heterostructures are composed of regions of slightly different PCSs in a single slab to form a cavity (see Fig. 1(a)). These structures can be formed in many different ways but they all rely on an increase of the average refractive index within the central PC2, compared to PC1. This has the effect of shifting the band structure features to lower frequencies. Therefore the waveguide, introduced across the PCS, has a lower dispersion curve within PC2 than in the surrounding PC1. Both curves are within the same photonic band gap (PBG), but there is a gap between them. If the resonant frequency falls within this mode-gap the mode propagates in PC2 and is evanescent in PC1. The part of the waveguide within PC2 then acts as a cavity due to the mode-gap effect [14]. The highest measured quality factors in PCSs were achieved using this type of cavity [15–17]. In these designs, PC2, is formed either by longitudinal [15] or lateral [16] hole displacement so that the hole density decreases, thus increasing the average refractive index. However these designs need to be finalized at the fabrication stage. There are other double-heterostructure designs which take advantage of the post-processingtechniques mentioned above. Quality factors of order Q~106 can be obtained when PC2 is generated by filling the holes with nano-porous silica in the central region of a silicon-based PCS [18]. When polymer materials or LC are used, Q=7×105 is achievable. Alternatively ultrahigh-Q cavities, Q~106, can be designed in chalcogenide-based PCS using the photosensitivity of this material [19]. The results are comparable with the best results reported to date in silicon, despite the moderate refractive index of chalcogenide glass, n=2.7, when compared with silicon and other high-index semiconductors with n≈3.4.
In this paper we introduce a multilayer design that relies on depositing a strip of material on a silicon-based PCS, which can be achieved any time after fabrication. Though a variety of materials can in principle be used, polymers are particularly convenient because of the easy integration with other optical and electronic components. Well-developed adhesion schemes permit the use of polymers on a wide range of substrates [20]. We also consider depositing a chalcogenide strip on the silicon slab. Polymers and chalcogenide glasses are photosensitive, which opens possibilities for additional post-processing. Glassy polymers are commonly used in thermo-optic devices because their fast thermal response time and their convenient thermo-optic characteristics such as high thermo-optic coefficient and low thermal conductivity [21].
2. Model and method
We consider a PCS composed of a hexagonal array of cylindrical air holes in a silicon slab with the refractive index n=3.4. The structure has holes of radius R=0.29a, where a is the lattice constant and H=0.6a is the slab thickness. Across the PCS there is a W1 waveguide, in the Γ-K direction. We consider three multilayer configurations. In the first of these, PCS I, the layer is assumed to cover the silicon on one side, see Figs 1 and 2. In the second structure, PCS II, the layer covers the silicon on both sides. In the third, PCS III, the layer covers one side and also fills the associated holes. The refractive index of the polymer is n=1.45 unless stated otherwise. In this paper we do not take polymers broad-band absorption into account. The influence of absorption on the quality factor will be addressed in future studies.
Fig. 1. (a) Top view of a double heterostructure cavity; PC2 has slightly higher average refractive index than PC1. The black holes are air holes. (b) Silicon slab with polymer strip on the top (PCS I). In our structures typically a=410 nm.
In the PCS plane the strip is defined by the width, w, as w=ma or w ′=ma-2R, where m is an integer. In the z-direction the strip extends across the entire slab. Its thickness, h, is defined as a fraction of the slab height h=H/f where f=[2.5–8]. This range corresponds to h=30nm-120 nm for a lattice constant a=410 nm and a thickness of H=246 nm. The centre of the strip is always positioned in the centre of the PCS where it is equally spaced to four of the holes.
The PBG calculations and dispersion curves of the photonic crystal waveguide are obtained using the 3D plane wave expansion (PWE) method. The quality factor is calculated using the 3D finite-difference time-domain (FDTD) method, combined with fast harmonic analysis [22]. The numerical parameters such as grid size, perfectly-matched layer (PML) width, height of the computational window strongly affect the convergence. In most calculations the PML width is 2a and the height of the computational window is 4a. The gridsize that provides satisfactory convergence depends on the quality factor. For Q~105, 28 points per period suffices, whereas 32 points per period are needed when Q~106. The resonant mode’s volume is: ∫∫∫UdV max(U) where U=ε|E|2/2 is the electric energy density.
3. Results
First we consider a bulk PCS that is infinite in the plane. We use the PWE method to obtain the PBGs and associated eigenstates of the waveguide. All structures have two guided modes below the light line in the lowest PBG-one in the middle of the bandgap and the other near the bottom. Depositing a polymer layer shifts the dispersion curves of both modes down. In Fig. 2 we show the mode-gap between the dispersion curves of the lower mode as it is a mode of interest. We plot dispersion curves of the W1 waveguide for the three PCSs of interest and compare these to the bandstructure of a silicon PCS without polymer (full circles) and those for the case we considered in [18] in which the holes are filled (filled triangles). In all calculations the layer covers the entire surface. The mode-gap extends between the dispersion curve of PC1 and any other dispersion curve of PC2. The mode-gap size depends on the polymer refractive index and the thickness of the layers. The results in Fig. 2 are obtained for a fixed layer thickness h=H/5, corresponding to h=50 nm for a lattice constant a=410 nm. This film is too thin to lead to its own band structure.
Fig. 2. Dispersion curves for W1 waveguide for the regular structure PC1 (full circles) and of PCS I (empty triangles), PCS II (full rectangles), with the holes infiltrated (full triangles), and PCS III (empty rectangles); the dashed line represents the light line.
Clearly, infiltrating the slab has more effect than depositing a strip on one or both sides of the slab. The mode-gap size Δω̃, where =ωa/2πc, is measured at the Brillouin zone edge. It ranges from Δω̃=6.8×10-4 for PCS I to Δω̃=5.5×10-3 for PCS III. At the same time the lower PBG edge shift varies from Δω̃=6.7×10-4 for PCS I to Δω̃=3.8×10-3 for PCS III. The mode-gap size varies by an order of magnitude for different configurations. Obviously it has a much stronger effect when perturbation is located inside the slab than on its surface. The largest mode-gap does not necessarily mean the optimal configurations as the relative mode-gap position within the PBG is also important [18].
Now we consider finite multilayer PCSs and calculate the quality factors using the FDTD method. First we consider a quality factor dependence on the layer width for a single strip structure only, PCS I. The width is varied as w=ma, where m is integer. The strip thickness is fixed at h=H/5 as in the example above. The results for the quality factor are shown in Fig. 3(a). Initially widening the strip increases the quality factor. The overall maximum, Q=8×105 is obtained at w=8a. Further widening of the strip decreases the quality factor but it still remains at the order of few 105. Fig. 3(a) shows that the quality factor is larger for cavity widths that are even multiples of the period. The layer’s edge is then in the high index material within the PC row adjacent to the waveguide whereas for the odd m the edge cuts across the holes, as illustrated in the inset of Fig. 3(a). For example the quality factor halves when the width changes from w=6a to w=7a.
Fig. 3. (a) Total Q versus the width m for PCS I; inset shows the cavity edge adjacent to the W1 waveguide for odd m; and (b) modal volume versus m; inset shows the major electric field component, Ex, in the plane for m=8.
This variation in Q is prevented if the strip edge is in the high index material not crossing the holes within the PC row adjacent to the waveguide. For that reason from now on we choose the strip width as w ′=ma-2R so that the edge does not cut across these holes.
The width of the cavity is large for both structures and therefore we calculate the modal volume V of the resonant mode. The ratio Q/V or Q/V 1/2 are important for many applications [3,18]. The modal volume is plotted in Fig. 3(b). The inset shows the major electric field component, Ex, at the centre of the slab for m=8. It is symmetric in the y-and anti-symmetric in the x-and z-directions. The volume increases from V=1.79 (λ/n)3 at m=5 to V=2.29 (λ/n)3 at m=10. The volume does not change significantly with the thickness of the layer. The overlap between the field and the polymer strip is small: For example it is 2.5% for the eight period wide strip.
Next we consider PCS III, a PCS with a single layer and infiltrated holes below the strip with no periodicity. The strip width is w ′=ma-2R not only to avoid the drop in the quality factor for the odd multiples of the period but also because partial air-hole infiltration is unattainable. We consider a large range of the cavity widths. Even for a narrow cavity with w ′=5a-2R, the quality factor is of order of 105. It has the maximum of Q=1×106 at w ′=12a-2R. Further widening the cavity decreases Q but it still remains around 105 as for PCS I.
Our results show that a high-Q cavity is achievable even if the holes of the PCS are filled with polymer. Hence if the polymer strip cannot be deposited without filling the holes a high-Q cavity can still be designed. For the PCS III, the optimum shifts towards wider cavities in comparison to PCS I, because the resonant mode’s relative position within the mode-gap changes with the mode-gap. The PCS II has a wider mode-gap than PCS I and therefore the optimum appears at lower frequency. In the same figure we also plot the modal volume of the resonant mode in PCS III. It changes from V=1.48(λ/n)3 at m=8 to V=2.33(λ/n)3 at m=14. At the optimum Q the modal volume is V=1.82 (λ/n)3, similar to the results for PCS I.
Fig. 5. (a) Total Q and resonant frequencies versus the layer thickness, h=H/f, for (a) PCS I, and (b) PCS II (w=8a).
We now examine how the quality factor depends on the strip thickness. Results for PCS I and PCS II are shown in Figs 5 with the insets illustrating the structures. The layer width is fixed at w=8a. In Fig. 5(a) the results are shown for PCS I. The thickness range is h=[H/6,H/2] which corresponds to h≈40-120 nm. The quality factor increases from Q=5×105 at h=H/6 to Q=9×105 at h=H/4. Further increasing the layer thickness decreases the Q though it still has values of a few times 105. Therefore PCS I is tolerant to changes in the layer thickness. The results for PCS II are shown in Fig. 5(b). The strips have the same width fixed at w=8a as for PCS I. The range of thickness is h=[H/8,H/2.5] corresponding to the h≈30-100 nm. This range differs from that for PCS I since the optimum occurs at a different value for h. The maximum of Q=5.9×106 appears at h=H/5 corresponding to h=50 nm. Note that for all values considered here the quality factor is over Q=2×106. We did not study the double-layer structure in as much detail since it would require more complex processing. The losses for all three configurations are due to the in-plane losses in the waveguide direction.
As a final point we discuss the quality factor’s dependence on the strip’s refractive index. Changing the refractive index shifts the optimum towards slightly different values. For instance for PCS I with n=1.4 and width w=8a the optimal thickness appears at h=H/3, rather than at h=H/4 for n=1.45. This shift maintains the same relative position of the mode-gap within the PBG. The thicker layer compensates for the smaller refractive index. We also considered depositing chalcogenide glass on the silicon slab. They have higher refractive indices, typically between n=2.4 and n=3 [13], than polymers [23]. Depositing a strip of the chalcogenide glass, with n=2.7, on the silicon slab results in quality factors below Q=3×104 for PCS I, and less than Q=3×105 for PCS II. These results are about one order of magnitude smaller than for the polymer-based multilayer structures.
4. Conclusions
We have introduced a novel concept for creating high-Q cavities in PCS without changing the geometry. Quality factors of order Q~106 can be obtained by depositing a polymer strip on the top surface of the silicon slab. A high-Q cavity is even achievable if the holes of the slab are filled with polymer. Therefore if a polymer strip cannot be deposited without filling the holes, a high-Q cavity can still be designed. For a symmetric structure with strips deposited on both sides the quality factor exceeds 106 without almost any optimization. These novel designs can be implemented at any time after fabrication, and depending on the kind of polymer used, can be additionally tuned using either photosensitivity or the electro-optic effect.
Acknowledgment
We acknowledge the assistance of the Australian Research Council Centres of Excellence Program. S. Tomljenovic-Hanic acknowledges the Australian Research Network for Advanced Materials for support for research at the DESE at Kyoto University.
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