1. Introduction

There has been extensive study of high quality two dimensional photonic crystal waveguides in various materials being reported (for example see [1–6]). The technology is now considered a serious candidate for compact true all optical photonic chips. Of the studied materials, silicon (Si) promises a major revolution, as it has done in electronics, with its excellent potential for integrated photonics and optoelectronics continuing to increase [7]. Triangular lattices of holes in Si structures with a row of holes removed to define the supporting waveguide have demonstrated transmission bandgaps extending well beyond the telecommunications windows (>1300-1600nm) [8]. However, these values are generally for one polarisation eigenstate – the quasi- TE state. The quasi TM state in contrast has been more difficult to tame within more complex circuits, including bends, although using various designs optimised through topology based algorithms dual TE and TM propagation over significant spectral range is possible [9,10]. Topology optimisation begins with the crystal structure as the starting point but has focussed on reducing loss rather than specifically being concerned with bandgap transmission, effectively working out the optimum optical impedance matching solution regardless of the bandgap features. Consequently, the complex starting point leads to complex non-periodic solutions that raise interesting fabrication challenges, suggesting that alternative solutions based on regular straight waveguides with optimally chamfered bends, for example, may work equally as well. Overall, propagation loss remains a significant issue for much more complex and extensive circuits although in free membrane structures losses <0.1dB/mm are being reported [8]. Most loss is often due to the step index guidance determined by the vertical confinement and the higher index contrast in membrane structures partly accounts for the improved loss figures. On the other hand, significant challenges remain in implementing more sophisticated devices and circuits, which would have to be suspended in air. For this reason conventional solid supported bandgap structures and devices remain critically important. Although loss in all structures also arises from surface roughness, which is sensitive to the fabrication process, techniques are increasingly available for reducing such roughness [11]. In this paper, we focus on mapping the spectra of both quasi TE and TM eigenstates of typical Si-on-SiO₂ (SOI) photonic crystals and compare the results with recent work on membrane versions. Much more extensive control of the polarisation is employed and we study both input and output states to explore polarisation mixing.

2. The photonic crystal waveguide

The waveguides were fabricated using a commercial SOI-wafer as platform. Electron-beam lithography (JEOL-JBX9300FS) and inductively coupled plasma (ICP) etching were applied to define the photonic crystal (PhC) structure and coupling elements into the 320 nm top silicon layer.

Figure 1(a) shows a scanning electron microscope (SEM) image of a typical fabricated structure. Adiabatically tapered straight waveguides from 4μm to 0.7μm allow coupling into and out of the PhC waveguide. In order to couple light into the tapers themselves, lensed end fibres are used to focus light to a ~3 μm spot onto the waveguide facet. From the SEM images we measured the triangular lattice constant (pitch), Λ = 370 nm, the hole radius r = (125±5)nm, which gives for this PhC r/Λ = 0.34, and the Si waveguide thickness, h = 320 nm. The PhC waveguide is created by removing a row of holes along the Γ–K direction similar to that reported in reference [8]. The length of the investigated PhC waveguides is 50μm. The notation followed throughout is identical to that used in reference [12].

 

Fig. 1. (a) SEM image of 50um long photonic crystal waveguide (DUT); (b) Schematic of setup used to characterise polarisation of the photonic crystal waveguide.

PC – polarisation controller; P – polariser; SC – supercontinuum source; LF – lensed fibre; DUT – device under test; D – driver for piezo control of XYZ; XYZ – positioner; OSA – optical spectrum analyser.

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3. Characterisation setup

Figure 1(b) is a schematic of the setup employed. Given the nature of the experiments, a detailed description is necessary as measured outcomes are influenced by parameters such as fibre single mode cut-off, and the number and position of polarisers, for example. The output of the quasi-CW supercontinuum (SC) source is slightly polarised with a small preference found for the TM eigenstate.

Instead of using polarisation preserving fibre, we use polarisers (P) and polarisation controllers (PC). Each polarisation controller manages the polarisation entering and leaving the polarisers and the device under test (DUT) – they are basically a 1/4:1/2:1/4 waveplate configuration. The output of the supercontinuum is unpolarised. The PC between the SC and the first P is used to rotate the polarisation so that there is maximum signal transmission through the polariser at the chosen polarisation – this optimisation is done for both eigenstates when each measurement is undertaken. The consecutive PC, likewise, is adjusted to optimize the alignment with the DUT for the one polarisation state – the input polarisation is recognised as TE when the spectrum transmission through the PhC waveguide has the quasi-TE waveguide eigenstate band edge depth maximised. This approach avoids the need for regular bulk optic characterisation to determine explicitly the polarisation eigenstate. The external PC and P at the output before the optical spectrum analyser (OSA) allow measurement of the polarisation coming out of the waveguide and can be scanned through both eigenstates to further increase the contrast between TE and TM and to see if there has been any device-generated cross coupling or mixing. When measuring pure TE only the first PC is both optimised on the signal transmission in the bandgap whilst simultaneously maximising the contrast of the bandgap edge. With this setup and careful optimisation we have measured the cut-off edge of the quasi-TE mode at longer wavelengths to extend more than 40dB. In contrast, the optimised TM eigenstate shows no equivalent band edge in the same region. This careful alignment is necessary since the system is sensitive to any perturbation in polarisation – all measurements were repeated several times and in several similar waveguides to ensure reliability and consistency in measurements.

Consequently, either TE or TM polarised light is launched into the tapered waveguides that couple light directly into the photonic crystal waveguides. A straight ridge waveguide used as reference is also characterised. The spectra are recorded using the OSA. The photonic crystal waveguide spectra are subtracted from the reference spectra obtained from the straight waveguides so that the contributions from coupling and the tapered sections are minimised, leaving the photonic crystal waveguide spectra.

4. Results and Discussion

Spectra for TE-TE, TE-TM, TM-TM and TM-TE configurations, established using the input and output polarisers respectively, are measured. The processed spectra are shown in Fig. 2(a)–(d). What can be observed immediately is the input fibre single mode cut-off, λ _cut-off ~ 1275nm. Given the complex nature of the coupling into higher order modes, particularly as the number increases at shorter wavelengths, it is not unexpected that variations in signal and profile will be observed. This complexity can be seen in practice and the signal rise seen in some of the spectra at λ _cut-off needs to be interpreted with caution.

Figure 2(a) shows the expected transmission cut-off around 1500nm for the fundamental quasi- TE eigenstate. For this sample, the contrast is below the noise floor, i.e. >30dB. The steepness and depth of the edge reflects the maturity and quality of the fabrication process, which we found reproducible over all six 50μm-long samples produced on the one chip. With careful alignment of the polarisation, and more input power the contrast was measured to be as high as 40dB. The loss below 1200nm has to be treated with care since it is most likely due to coupling to higher order modes that can have orthogonal (TM) polarisation to the aligned fundamental mode polarisation. It is also in this region where coupling within the crystal to slab modes may occur. Slight variations in polarisation led to dramatic variations in this multimode region.

At longer wavelengths there is a small spectral dip, B. In order to better understand the origin of this band, a full 3D Finite-Difference Time-Domain (FDTD) simulation using commercially available software from CrystalWave [13] was carried out. As well, the 3D band diagram was obtained using freely downloadable code available from MIT [14] (spatial resolution = Λ/16). The results are presented in Fig. 3 – the FDTD simulations are overlain on the matching experimental data. It is clear from the band diagram there are no substantive crossings of various modes, including either the odd TE-like modes, with the fundamental quasi- TM mode at longer wavelengths. At shorter wavelengths the fundamental quasi- TM mode crosses the even fundamental quasi- TE at frequency 0.245 and the odd quasi- TE at 0.28, although the lack of an efficient coupling mechanism makes cross coupling between them very unlikely. Further, from the band diagram most relevant crossings occur at much shorter wavelengths and do not show significant convergence suggesting that cross coupling between such modes is unlikely to be significant (despite vertical asymmetry) in our sample. Comparing with the x- odd TE-like mode cross coupling reported in [8], experimentally, there is no clear evidence of such cross coupling for either odd TE-like mode in our structure. This may in part be that such crossings are at significantly shorter wavelengths given the lower effective index.

 

Fig. 2 a) Transmission spectra with both input and output P set on TE, b) Transmission spectra with input P set to launch TE and output P set to measure TM; c) Transmission spectra with both input and output P set on TM; d) Transmission spectra with input P set to launch TM and output P set to measure TE. A – optical fibre cut-off ~1270nm, B – attributed to TM stopgap; C – resonant spike.

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Fig. 3. On the left is the calculated band diagram (resolution = Λ/16) for the photonic crystal waveguide; on the right hand side is the corresponding FDTD transmission simulation superposed on the measured transmission spectra of TE-TE and TM-TM of the PhC waveguide. Note that the graphs show normalized frequency for comparison with standard bandgap calculations rather than wavelength (λ = 0.37/Λ).

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On the other hand, the observed dip at longer wavelengths, (B), coincides well with the start of the TM mode in our structure – the small stopgap can then be explained as an opening at the TM band edge arising from the vertical asymmetry, or optical anisotropy, of the structure. Both the experimental transmission spectra and FDTD calculated spectra for both quasi TE and TM fundamental modes are in qualitative agreement above λ _cut-off.

We also observe this dip in the quasi- TE spectra, which should, at first glance, not be the case. To account for this, we suggest that the vertical (out-of-plane) asymmetry of the structure leads to coupling of quasi- TE light into quasi- TM light – that is quasi- TM light is still present in the spectra. The band diagram in Fig. 3 is complex but shows numerous coincidences between TE and TM light where this may occur throughout the spectra below the light line. This is verified experimentally in Fig. 3(b) where the quasi- TE light is filtered out leaving behind a quasi- TM spectra – we can initially conclude, therefore, that this long wavelength dip is the small TM stopgap opened up by asymmetry at the fundamental TM mode band edge.

Although it appears that asymmetry together with polarisation dispersion is a necessary precondition for the TE-TM cross coupling, the significant degree to which this occurs suggests something more fundamental – one explanation is the role of the finite height of the high index contrast crystal structure which, because it does not extend to infinity, greatly enhances any asymmetry. Another, however, is based on greater scattering of the quasi TE eigenstate by the asymmetric crystal lattice, the origins of which we will discuss in more detail below. Notably, for a perfectly symmetric structure no TM bandgap or stopgap is expected [15].

We commented above that Fig. 2(b) shows the output spectra after filtering of the TE eigenstate by rotating the output polariser 90°. As expected, below ~1270nm the signal increases indicating that the higher order modes supported at these wavelengths within the optical fibres are not aligned on one polarisation eigenstate. Further, the profile may be complicated by coupling into crystal lattice modes and other cladding modes [8]. Close to λ _cutoff there is also what appears to be a small band appearing which may or may not be related to similar phenomena associated with coupling to the first higher order mode of the optical fibres.

On the other hand, at longer wavelengths in the single mode domain the band edge is suppressed and a resonant spike (C) appears where the transmission band edge of the quasi TE mode was – this suggests that the band edge dispersion is phase matching the difference between quasi TE and TM modes and converting some of the light to TM. By adjusting polarisation slightly the spike can be increased or decreased – the observed peak intensity is limited by the resolution of the supercontinuum source. An FDTD simulation modified to examine for any light cross coupled from TE (E_x field) to TM (E_y) field is shown in Fig. 4. Qualitative agreement is observed indicating that cross coupling is expected and readily established. Analogous behavior has been observed and utilized in a number of conventional fibre and waveguide structures that employ the band edge of 1-D Bragg gratings in planar waveguides [16–18] and in 1D periodic nanopillar waveguides [19] as the dispersive element to couple into various waveguide modes. Indeed, a more sophisticated application of this process based on cantor structured grating nanopillar waveguides has been proposed for novel selectable lasing [20]. Related dispersion phenomena are therefore expected in photonic crystal waveguides although to our knowledge this is the first time it has been observed. It is conceivable, for example, that this property can be exploited to obtain single polarisation DFB lasing in a photonic crystal cavity.

 

Fig. 4. FDTD simulation of relative conversion of E_x (TE) into E_y (TM) for the structure shown in figure 1. The conversion is normalised to the initial input E_y.

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Also observed in Fig. 2(b) is the narrower transmission gap ~1370nm – this is associated directly with the TM transmission band that, for these waveguides, lies within the TE transmission bandgap, which is shown in Fig. 2(c). This amount of the quasi TM mode profile leaking through is unexpected – despite significant care adjusting and varying the input coupling and repeated measurements, over numerous samples, including over several lengths, we found the quasi TE mode appears to have a significant TM component. However, when the output polariser is rotated to 45° the TE profile appears to swamp this signal as shown in the non-normalised data in Fig. 5(a). Whilst the source of mixing is likely to be dispersion, we will comment on a likely mechanism for this later.

Fig. 3(c) shows the transmission spectra for the quasi TM eigenstate after the contributions from the tapers and fibres are removed. Over the range covering the region occupied by the TE transmission bandgap there is little propagation of light except for two regions that represent small transmission windows in essentially a large stopgap region. These bands coincide with those observed in the TE coupled system described above – the one at or just below λ _cut-off for single mode behaviour in the optical fibres can be explained by coupling into the higher order mode which has not been optimised for the particular polarisation chosen for longer wavelengths. Fig. 2(d) shows the spectra when the output polariser is oriented to transmit TE light. The presence of the partial TM stopgap is again observed in both cases as in the TE spectra, indicating TM light has not been fully suppressed. In contrast to the quasi TE profile, there is no broadband TE light leaking through, no matter the adjustments of the output filtering – this indicates that TM light is not coupled into TE light by the structure. An explanation for this apparent violation of reciprocity is offered below. A length dependence of the strength of this dip with the photonic crystal length (5, 10, 25 and 50μm) is also observed.

There are no further surprises –the signal decreases substantially overall but the spectra remains the same. On the other hand, when the polariser is rotated to 45°, a much stronger resonant band is obtained indicating that some quasi TM light has been converted to quasi TE light over this narrow spectral region. Coincidentally, a much stronger TM stopgap at longer wavelengths is seen. Based on these observations, we can conclude that the dispersion is sufficient for these phenomena to be observed and therefore the quality of the periodicity of the fabricated structure is high. In other words there is minimal intrinsic in-plane asymmetry or non-periodicity or chirp within the crystal lattice itself.

 

Fig. 5. (a) TE data with output polariser on 45°; (b) TM date with output polariser 45 ; (c) Difference between TE and TM transmission spectra.

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In order to investigate the degree of mixing, data was collected for each input polarisation with an output polarisation of 45°. The results are shown in Figure 5(a)-(c). As expected within the resolution of the supercontinuum source there is an enhancement of the resonance at the edge of the bandgap for the TM light as some light is cross-coupled into TE. TE light swamps any conversion to TM. Also observed is the feature associated with the partial TM bandgap, indicating TM light is present in the TE spectra despite washing out the resonance.

To compare the performance of (quasi) TE and TM, the TM spectra were subtracted from the TE spectra, as shown in Fig. 5(c). Anything above zero indicates higher transmission in mainly TE light whereas anything below zero indicates higher transmission in TM light. At zero both TE and TM are transmitted equally. From this plot it is apparent that there is an unpolarised transmission window centred ~1370nm. This suggests close to 100% unpolarised transmission over a small spectral range within the waveguide similar to that described for structures with non-circular holes where a complete photonic bandgap has been described [21,22]. Outside of this window, between 1200 and 1500nm TE polarised light is preferred with an extinction ratio over TM polarised light varying between 10 and 20dB. In contrast, above 1500nm there is a region where the extinction ratio of TM is 30dB.

The presence of this bandgap seems surprising given that the quasi TM modes (the odd modes with respect to a horizontal mirror plane) of a 2D photonic crystal slab waveguide are strongly and non-uniformly shifted to shorter wavelengths beyond the bandgap of the quasi TE modes [21] compared to the 3D structure. Or where the 2D lattice extends to infinity, where the eigenstates are, or are close to, degenerate and therefore overlap of the TE and TM bandgaps are possible. We note that the complete bandgap described for non-circular holes can only arise because they introduce asymmetry in the band structure of the lattice enabling more mixing between TE and TM light. Further, in real systems there are quasi TE and TM fundamental modes defined by an effective plane of the waveguide. As predicted and demonstrated by Tanaka et al. [23,24] it is possible to have quasi TE and TM fundamental mode mixing if there is vertical asymmetry in the structure as a whole and should not be limited to in-plane asymmetric hole shapes alone. They have treated this effect as an unwanted loss since TE is converted to TM which is assumed to be lossy – the reason why many devices are designed for TE only polarisation. Similarly, this mixing is substantially enhanced in our structures by the vertical asymmetry arising from the differences between top and bottom layers. If the focus is not solely on TE-only propagation, then there is a complete bandgap for this structure. An important point to note that in contrast to asymmetric holes, this does not affect the 2-D periodicity and dispersion of the crystal lattice, which are the key feature that potentially allow such structures to have unique device functionality. Additionally, the enhanced fabrication tolerances created by non-circular holes becomes unnecessary.

Finally, the steep edge obtained for the so-called TE bandgap has suggested a new interpretation of the effects involved – the relationship between the effective index determined by the wavelength and hence the effective modal k vector, which is related to an “effective angle of incidence” of the fundamental mode to the periodic interface, suggests that this edge is, to first order approximation, an effective critical angle cutoff for total internal propagation of TE light. That is, wavelengths below this are transmitted whilst beyond this they are reflected. To examine this more closely unpolarised light was coupled into the waveguide - we monitored the unpolarised as well as the TE and TM spectra at the output. Both the TE and TM spectra are subtracted from the unpolarised spectra. Fig. 6(a) summarises these results. Fig. 6(b) shows the results redrawn in linear fashion against 1/λ. which is proportional to k = 2πn/λ. If we consider that k also describes the effective angle of the incident mode, there is a simple qualitative resemblance to the TE and TM profiles of light incident as a function of angle at an interface. Indeed, the minimum TE value will correspond to that wavelength which matches the Brewster condition for transmission through such an interface (hence why TE is so lossy). From such an analysis the effective index of the mode outside the bandgap structure is estimated to be n ~ 3.33 – this accounts for the deviation of the Brewster angle obtained from 1/λ ~ 0.66 (Fig. 8(b) and is close to that obtained by simulation. We therefore propose that this so-called band edge is in fact the critical edge determining step index-like guidance. This offers a simple qualitative explanation of why the transmission window for TE is large. The maximum difference between TE and TM before this point defines the wavelength for which there is an effective Brewster condition. This relationship is more pronounced than that suggested by Bassett for circular Bragg fibres [24,25] where the situation of a radial periodic structure can differentiate between the two eigenstates –consequently, for single polarisation propagation the most appropriate regime to operate is just outside the so-called TE bandgap above 1500nm. At this point the signal contrast between the two eigenstates is >30dB. Beyond this cut-off more complex resonant effects introduce stopgaps where additional loss is observed.

 

Fig. 6. (a) TE and TM transmission normalised against unpolarised light through the photonic crystal waveguide; (b) When plotted against inverse wavelength, characteristic features of polarisation scattering from an interface are observed up to the band edge. Resonant effects beyond the edge are also visible.

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The above model also explains why there is significant quasi TM light when coupling only into quasi TE light – the scattering component of TE back into the waveguide is asymmetric and therefore can couple directly into the quasi TM eigenstate. On the other hand, TM light which has its electric vector perpendicular to the lattice structure, does not experience the same degree of scattering, even with some asymmetry present, and is instead transmitted – there is no mechanism within the crystal waveguide to enable it to efficiently couple into the quasi TE eigenstate. This apparent bias in coupling also arises from the 3-D nature of the crystal lattice – therefore, introducing asymmetry in the vertical offers a unique approach to creating complete bandgaps that is distinct to that offered by introducing asymmetry within the crystal structure itself. It also circumvents the need for an infinite extension of the crystal lattice.

5. Conclusions

A comprehensive broadband polarisation characterisation of a conventional SOI photonic crystal waveguide has been undertaken. The partial bandgap of the quasi- TM mode is observed as a small dip in the spectrum beyond the TE band edge. We also report resonant cross coupling between quasi- TE and TM fundamental modes and an inherent spectral directionality with TE light substantially containing TM light but not the other way around. This is explained by a simple interpretation based on polarisation scattering and the inherent nature of the 3-D structure with asymmetry. In addition, the observation of polarisation cross coupling at the bandgap edge, supported by FDTD simulation, and the large, steep bandgap depths of the quasi TE state indicates the quality of the periodicity of the fabricated structures is high despite some asymmetry being present in the vertical direction of conventional photonic crystal waveguides. This behaviour also suggests novel devices that better exploit the differences in dispersion between the two eigenstates. Consequently, there are significant advantages of having high finesse structures with control of asymmetry introduced through the vertical plane rather than more demanding changes in the crystal lattice itself. Consistent with this, we have reported a complete photonic bandgap in such a structure with a crystal lattice made up of circular holes. More excitingly, such asymmetries are a natural result of strain between materials - this is of particular relevance given the apparent relationship recently reported between strain and electro-optic coefficient [27]. Whilst strain has usually been treated as something undesirable, it is increasingly playing a key role in enabling important advances in integrated photonics, thus having an optical parallel with recent trends in using strained silicon to improve electronic performance [28].