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The symmetry of a Fabry-Perot-like planar cavity embedded within a three-dimensional (3D) woodpile photonic crystal prevents the observation of polarization effects. In this letter we propose a geometry to break the degeneracy of the Fabry-Perot-like cavity modes by introducing asymmetry. The introduction of a one-dimensional (1D) lattice to the centre of a planar cavity allows for distinct modes parallel (TE) and perpendicular (TM) to the layer. This hybrid 3D-1D-3D lattice structure exhibits a pronounced increase in the quality-factor, and in particular shows an increase of up to 50% more than that of a planar cavity for TM modes.
©2008 Optical Society of America
1. Introduction
The ability to manipulate light on a scale comparable to its wavelength is the key to the development of the next generation of optical devices. Photonic crystals (PCs) are such a material that can influence the propagation of photons by employing a spatially periodic dielectric environment [1, 2]. Three-dimensional (3D) PCs, in particular the woodpile lattice is attractive due to its ability to provide directional or complete photonic bandgaps, allowing for the rigorous control of photons in all directions [3]. The implementation of this lattice as a platform for the development of optical devices is however limited due to its polarization insensitivity. To realize passive elements such as waveguides [4], splitters [5] and coupler [6], and to rigorously control active optical effects such as spontaneous emission [1] require not only directional control but also effects of polarization. The implementation of a planar cavity within the centre of a woodpile lattice has been shown to produce localized defect states analogous to Fabry-Perot resonances [7]. These resonances are expected to be polarization insensitive due to there symmetry. In this paper we show that the degenerate defect mode in a simple planar cavity that exhibits no polarization effects in the band diagram can be broken by introducing a one-dimensional (1D) lattice within the planar cavity. This hybrid 3D-1D-3D lattice allows for two independent modes. The transverse-electric (TE) mode is defined parallel to the 1D lattice, and the transverse-magnetic (TM) mode is defined perpendicular to the 1D lattice, which are experimentally verified. Furthermore varying the period of the 1D lattice results in an increase in the quality factor of the defect modes.
2. Proposed hybrid 3D-1D-3D lattice
A sketch of the void-channel arrangement can be seen in Fig. 1(a) where the solid rods represent micro-void channels within a polymer host (black outline). Individual layers consist of a periodic arrangement of individual void-channels evenly spaced by and in-plane spacing (δx). Adjacent layers are perpendicular and spaced by a layer spacing (δz), while subsequent layers are offset by half and in-plane spacing. As in-plane layers are symmetrically stacked along the stacking direction of the crystal, this symmetry implies that the electromagnetic modes in this direction are degenerate for all polarizations. The band structures of woodpile photonic crystal lattices were calculated using an iterative eigensolver [8]. For calculations lattice parameters were fixed to prior experimental conditions [7] with a δx=1.4 µm, δz=1.5 µm and refractive index of 1.65. The band diagram in Fig. 1(a) is presented in normalized frequency units where a=δx. Although a complete photonic bandgap does not exist due to the low refractive index contrast [8], a sizeable stop-gap is present between the forth and fifth band centered at normalized frequency 0.305 (4.6 µm). Frequencies that fall outside the gap are shaded. A Fabry-Perot style cavity as represented in Fig. 1(b) can be introduced as a planar cavity at the centre of the lattice [7]. By displacing half of the crystal in the stacking direction by δd a localized dislocation breaks the periodicity of the surrounding lattice. Figure 1(b) show the band diagram as calculated for an appropriate super-cell [7] where the single flat band at normalized frequency 0.306 (4.58 µm) within the bandgap represents a single localized defect mode. Like the woodpile lattice without a defect (Fig. 1(a)) no polarization effects in the stacking direction is noted due to the structural symmetry in this direction. The localized defect band, like all bands calculated consists of two degenerate bands. The corresponding mode distribution can be found elsewhere [7].
Fig. 1. Band diagram calculations for woodpile lattices in the stacking direction (Γ-X’). (a) A simple woodpile lattice exhibits a stop-gap in the stacking direction between the forth and fifth band centered at normalized frequency 0.305. (b) A simple planar cavity introduced at the centre of a woodpile lattice reveals a single flat band within the stop-gap at normalized frequency 0.306. (c) The introduction of a 1D periodic lattice to the centre of the cavity in. (d) TE modes are defined perpendicular to the 1D lattice periodicity and TM modes are defined parallel to the 1D lattice periodicity.
To introduce polarization effects in the stacking direction of a woodpile lattice with a planar cavity requires the breaking of this symmetry. The introduction of a 1D lattice to the centre of the planar cavity is shown in Fig. 1(c). This periodic array introduces an asymmetric layer within the plane of the lattice and should lead to distinct effects on polarization of localized defect modes that occupy this region, which can be observed in the stacking direction as unique TE and TM modes. The band diagram for this hybrid 3D-1D-3D structure is shown in Fig. 1(c) where the periodicity of the 1D lattice (δa) is matched to the in-plane spacing of the 3D woodpile (δa=δx=1.4 µm). The stop-gap position and bandwidth is uneffected, however two localized defect modes are present with a separation of normalized frequency of 0.0023 (3.6 nm). The two modes are attributed to the breaking of the degeneracy of the planar cavity mode (Fig. 1(b)) into two distinct modes parallel to the 1D lattice (TE) and perpendicular to the 1D lattice (TM) (Fig. 1(d)). In Fig. 2 the electromagnetic energy density of TE and TM modes are calculated for a vertical plane. Overlaid on Figs. 2(a) and (b) is the lattice, while the scale bar indicates the energy density from the lowest density (black) to the highest density (white). Consistent with prior work [7], the energy is confined to the planar defect. Figures 2(a) and 2(b) are the TE and TM modes, respectively, for a hybrid 3D-1D-3D PC, the main feature to note is that the TM mode is strongly confined to the 1D lattice.
Fig. 2. The energy density of TE and TM modes calculated for a vertical plane. Overlaid in black is a trace that outline of the lattice positions. (a) The TE and (b) TM modes of a hybrid 3D-1D-3D lattice. The scale bar indicates the energy density form low density (black) to high density (white).
3. Experimental verification
To verify this theoretical prediction, experimental investigation into the polarization effects were realized by fabricating hybrid 3D-1D-3D PCs within solid polymer hosts using the femtosecond-laser direct writing method [9]. This signal step process [9] which requires no chemical post processing allows for an efficient way to rapidly produce high quality PCs with nanometer precision at infrared wavelengths. Furthermore unlike other fabrication procedures [10,11] which are limited to specific lattice geometries, the flexibility afforded by this method allows for the realization of planar defects [7] as well as the introduction of mixed 3D-1D-3D hybrid lattices. Figure 3 shows TE and TM transmission measurements made by the polarization-sensitive Fourier transform infrared spectrometer (FTIR) [12] in the stacking direction of the three crystal geometries calculated in Fig. 1. As expected no polarization dependence is measured for the woodpile PC (Fig. 3(a)) which exhibits a main stop-gap centered at wavelength 4.65 µm. Consistent with previous experimental investigations, a planar cavity of defect size δd=2.2 µm (Fig. 3(b)) revealed a defect peak within the stop-gap agreeing with the calculated prediction (Fig. 1(b)) and shows no polarization dependency.
Fig. 3. TE (black) and TM (grey) infrared transmission spectra of the PC lattices measured in the stacking direction. (a) A woodpile lattice with a transmission dip centered at wavelength 4.65 µm denoting the main stop-gap for both TE and TM polarization. (b) A woodpile lattice with a planar defect of size Δd=2.2µm showing a peak within the stop gap centered at wavelength 4.7 µm from both TE and TM modes. (c) A planar cavity with a 1D lattice of period a=1.4 µm showing two distinct TE and TM polarized modes with a peak wavelength separation of 3.3 nm. (d) A schematic of the combined Parabola and Lorentzian fitting.
The introduction of a 1D lattice to the cavity (Fig. 3(c)) with a period of δa=1.4 µm reveals two independent peaks measured for both TE (black) and TM (grey) polarizations. This confirms the prediction that the splitting of the defect modes seen in the band diagram of Fig. 1(d) is indeed polarization-dependent. Furthermore the resonance position of both modes are separated by 3.3 nm which is in excellent agreement with the calculated wavelength splitting of 3.6 nm form the band diagram in Fig. 1(c). As a consequence of introducing a 1D lattice to the centre of the planar cavity, energy that would be otherwise lost to the open edges of the cavity can be confined. This physical effect in turn results in the enhancement of the cavity quality factor (Q-factor), which is measured as the sharpening of resonance modes. Q-factors of the defect peaks were obtained by fitting the defect resonance peak of TE and TM modes using a combined Lorentzian and parabola (see Fig. 3(d)). In Fig. 3(c) both TE and TM modes show the Q-factors of 54 and 72, respectively, which correspond to an increase of 34 % and 42 % in comparison with that in the case of a planar cavity.
To investigate the dependence of the quality factor on the 1D lattice, we varied the period of the 1D lattice while the 3D woodpile lattice parameters were fixed. Figure 4 shows the Q-factor of TE (black diamonds) and TM (grey circles) as a function of the cavity periodicity of the 1D lattice. The pronounced increase in the Q-factor of greater than 50 % is observed for TM modes when the periodicity of the defect layer match or is greater than the layer spacing of the surrounding woodpile lattice. This change in the Q-factor is a result of the further modal confinement introduced by the 1D lattice as calculated in Fig. 2.
Fig. 4. The quality factor of the TE (black diamonds) and TM (grey circles) defect modes as a function of the 1D cavity periodicity. The dashed lines are a guide for the eye.
4. Conclusion
In conclusion we have proposed a hybrid 3D-1D-3D lattice which exhibits strong polarization effects otherwise unavailable to simple woodpile lattices. The breaking of the degenerate mode associated with a planar cavity embedded within a woodpile lattice leads to an effect of polarization. This feature has been experimentally verified by the measurements of the TE and TM defect modes which exhibits an increase of the quality factor by more than 50% compared with that in the planar cavity.
Acknowledgments
This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence program. CUDOS (the Centre for Ultrahigh-bandwidth Devices for Optical Systems) is an ARC Centre of Excellence.
References and links
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