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We propose pillar integrated silicon waveguides to exploit the entire transparent window of silicon. These geometries posses a broad and flat dispersion (from 2 to 6 μm) with four zero dispersion wavelengths. We calculate supercontinuum generation spanning over two octaves (2 to >8 μm) with long wavelengths interacting weakly with the lossy substrate. These structures have higher mode confinement in the silicon - away from the substrate, which makes them substrate independent and are promising for exploring new nonlinear phenomena and highly sensitive molecular sensing over the entire silicon’s transparency range.
© 2015 Optical Society of America
1. Introduction
Th ability to utilize silicon’s entire transparency window is an important goal for silicon photonics [1, 2]. This has been a difficult task since the conventional substrate, silica, has high material absorption due to its multiphonon edge beyond 3.5 μm [3–5], and platform such as silicon-on-sapphire are limited to ∼6 μm by the material absorption of sapphire [6–13]. The SiGe and Silicon on nitride (or SiN) can be used for longer wavelengths [14–16], however, the nonlinearity is lower due to low index difference between the core and the cladding.
Here, we propose a completely different approach to extend the operational bandwidth of silicon waveguides. We designed geometries (upright T and inverted L) as shown in Fig. 1(a, b), which support modes that are independent of the propagation losses induced by the substrate. They have a flat dispersion over a wide mid-IR range which is useful for low power operation and exploiting nonlinear phenomenon such as a supercontiuum genertaion (SCG) [17–22].
Figure 1 (a) T waveguide made of silicon (cyan) on a sapphire substrate (purple) and (b) inverted-L waveguide.
These geometries have four zero dispersion wavelengths (ZDWs) due to the unique waveguide dispersion provided by the shape of the waveguide. This is unlike the recent demonstration where a silica layer is sandwiched in the silicon slot waveguide which is not scalable to mid-IR due to high absorption losses of silica [22]. The long wavelength modes of these waveguides are well confined in the silicon causing low propagation losses due to weak interaction with the lossy substrate [23–25], and can essentially be made of any substrate with adjusted dimensions to produce desired dispersion. In this work we show by modelling more than two octave spanning supercontinuum covering silicon transparency upto 8 μm.
We discuss below in detail the factors responsible for 4 ZDWs and the dependence of dispersion on the dimensions of the waveguide T and L separately. Further we show the supercontinuum generation in both structures and compare their mode energy confinement versus wavelength.
2. Dispersion calculation in T and L waveguides
The waveguides were dispersion engineered by calculating the effective index (neff) (using finite element method - Comsol) using the sellmeier equation for bulk silicon and sapphire [26, 27]. The dispersion is calculated with, D = −(λ/c)d2neff/dλ2.
2.1. T structure
The T structure incorporates a pillar in the middle section as shown in Fig. 1(a). The dispersion of the waveguide shown in Fig. 2(a, b), have dimensions: width (w) = 2200 nm, thickness (t) = 423 nm, pillar width (Pw) = 435 nm, height (h) = 700 nm. The thickness of the sapphire substrate is 500 μm.
Figure 2 Dispersion of the T waveguide with flat dispersion (inset) and the silicon material dispersion. (a) TE mode profile at 1st ZDW and b) at 4th ZDW.
The dispersion varies from −25 to 25 ps/nm-km in the range 2.2 – 5 μm. The four zero dispersion wavelengths are at 2.55, 2.9, 3.54 and 4.9 μm out of which the profiles of the TE mode at 1st and the 4th ZDW are plotted in Fig. 2(a, b). The mode energy confinement at 1st ZDW in: silicon = 89.5%, air = 10%, sapphire = 0.5 %; and for 4th ZDW: silicon = 75%, air = 20%, sapphire = 5%.
To understand the factors affecting the dispersion profile we calculated the dispersion for separate parts of the structure. In Fig. 3(a), for the strip in air, the dispersion shifts into the normal regime due to extensive negative waveguide dispersion (mode profile at 3.6 μm is used as the pump wavelength for SCG, discussed below). The mode energy distribution for the strip suspended in air is: silicon = 81%, sapphire = 1.3%, air = 17.7%. By introducing the supporting pillar Fig. 3(b), waveguide dispersion becomes anomalous due to the mode interaction with the pillar (the mode energy in: silicon = 85% air = 15%). The inclusion of the sapphire substrate Fig. 3(c), however, shifts the waveguide dispersion towards normal region and produces 4 ZDWs (the mode energy distribution: in silicon = 84%, sapphire = 2%, air = 18%). These calculations show a silicon waveguide without a pillar or substrate − suspended in air cannot produce a wide, flat and low dispersion.
Figure 3 TE mode profile at 3.6 μm for, (a) Strip of the T waveguide (neff, 2.48) with dispersion (black dashed dot). (b) T waveguide in air (neff, 2.54) with dispersion (blue dashed line). (c) T waveguide (neff, 2.55), with dispersion (green solid line).
We have analysed the dispersion sensitivity on the waveguide dimensions, namely the width, thickness, height and pillar width [21, 22]. The D varies (at 3.6 μm) from −83 to 115 ps/nm-km for 400 nm change in the width, see Fig. 4(a). The dispersion shifts towards the normal regime with increasing width due to the reduction in the mode confinement in the silicon for the wavelengths on the dispersion curves having similar dispersion trend (shown with green arrow in Fig. 4 (a, b)). For example, the neff at 4.85 μm for +200 nm waveguide is lesser by 2 % (reduction in the mode confinement) than the neff at 4.65 μm for +100 nm waveguide (which has similar dispersion trend as around 4.85 μm for a +200 nm waveguide) [21]. Hence the neff gradually reduces from 4.3 μm for 2 μm wide waveguide (−200 nm) to 4.85 μm for 2.4 μm wide waveguide (+200 nm). Additionally, the dispersion curves converge at shorter wavelengths (∼2 μm), because the modes are well confined and have almost the same neff. The dispersion is highly sensitive to the thickness variation because the high aspect ratio of the waveguide causes an abrupt increase in neff with thickness, for example, the neff at ∼3.95 μm for −100 nm waveguide is smaller than neff at ∼5.6 μm for +100 nm waveguide, see Fig. 4(b).
Figure 4 (a) Dispersion shifts of T waveguide into the normal regime for width change from 2 to 2.4 μm at 100 nm steps. The green arrow indicates the regions on different dispersion curves having similar trend (only shown for width and thickness variations) (b) A larger dispersion shifts for thickness (t) variation of 200 nm at 50 nm steps. (c) Dispersion shifts into anomalous region with the height variation from 600 to 800 nm. (d) Dispersion shifts into the normal region for a increase in pedestal width (Pw) by 400 nm with 100 nm steps.
2.2. L structure
The L structure incorporates a pillar on the side as shown in Fig. 1(b). The dimensions for the waveguide with dispersion is shown in Fig. 5(a, b): width (w) = 2200 nm, the thickness (t) = 500 nm, width of the pillar (Pw) = 500 nm, and the height (h) = 850 nm. The dispersion varies from −29 to 29 ps/nm-km from 2.1 – 6.2 μm. The four zero dispersion wavelengths are at 2.25, 3.6, 4.74 and 5.95 μm of which the profiles of the TE mode at 1st and the 4th ZDW are plotted. The energy confinement at 1st zdw in: silicon = 93.5%, air = 5%, sapphire = 1.5 %; for 4th ZDW: silicon = 66%, air = 22.5%, sapphire = 11.5%.
Figure 5 Dispersion of L waveguide with flat dispersion (inset) and the silicon material dispersion. (a) TE mode profile at 1st ZDW and (b) at the 4th ZDW.
Just as for the T waveguide, to understand the dispersion profile we calculated the D for separate parts of the L waveguide. The dispersion for the strip in air is in the anomalous regime (mode profile at 4.1 μm is shown in Fig. 6(a), which was used as the pump wavelength for SCG as discussed below). The mode energy distribution in: silicon = 82%, sapphire = 0.9%, air = 17.1%.
Figure 6 TE mode profile at 4.1 μm for, (a) Strip of L waveguide (neff, 2.47) with dispersion (black dashed dot line). (b) L waveguide in air (neff, 2.49) with dispersion (blue dashed line). (c) L waveguide (neff, 2.5), with dispersion (green solid line).
By introducing a supporting pillar, see Fig. 6(b), the waveguide dispersion becomes highly positive at longer wavelengths causing the total dispersion to be strongly anomalous (the mode energy distribution is: silicon = 83% air = 17%). With the inclusion of sapphire substrate, Fig. 6(c), the dispersion decreases and gets flatter with 4 ZDWs (the mode energy distribution is: silicon = 82.3%, sapphire = 1%, air = 16.7%). These calculation show that it is important to have both pillar and the substrate to achieve 4 ZDW.
The dispersion sensitivity on the waveguide dimensions are shown in Fig. 7(a, b, c, d). For the width variation, the D varies (at 4.1 μm) from −43 to 28 ps/nm-km for a 200 nm change. Like the T waveguide, the dispersion shifts down towards normal regime with increasing width due to the reduction in neff for the wavelengths on the dispersion curves having similar trend. For example, neff at 4.3 μm for +100 waveguide is lesser than neff at 4 μm for −100 waveguide. The D is relatively more sensitive to the thickness variation, about 200 nm change in the thickness varies D from −296 to 154 ps/nm-km, which is due to the abrupt increase in the neff.
Figure 7 Dispersion shifts (a) into the normal regime for width change in L waveguides from 2 to 2.4 μm at 100 nm steps, (b) for thickness (t) variation of 200 nm. (c) Dispersion shift into normal (∼5.5 μm) with height variation, (d) for variation in pedestal width (Pw) from 300 to 700 nm the dispersion is modified unlike for the T waveguide in Fig. 4(d).
For 200 nm variation in height the D varies relatively less, with the dispersion shift in the opposite directions for short and long wavelengths. That is because the pillar is on the side so the mode is interacting relatively less with the pillar unlike the T structure where the modes are always overlapping with the pillar in the middle of the structure. Thus, increasing the height causes the longer wavelength modes to leak out into the air, i.e. reduction in neff (<0.4% for ±100 nm change in height). In Fig. 7(d), the abrupt change in waveguide dispersion for thinner pillars (from 0 to −200 nm) shifts the total dispersion around 4 μm into anomalous regime.
3. Supercontinuum generation in T and L waveguides and discussion
We modelled the SCG generation in the T and L structure. The nonlinear Schrodinger equation used is as follows [28, 29]:
Here, E is the electric field envelope, α(ω) is the frequency dependent propagation loss, α(ω) = α(ω0) + ∂α/∂ω(ω − ω0), and the effective area, Aeff = |∫ (E × H*).ẑdA|2/∫ |(E × H*).ẑ|2dA [30] at the pump wavelength, βm is the mth dispersion order for which we have used m = 6, R(t) includes instantaneous electronic and delayed Raman responses (negligible in silicon), and γ4pa is the four photon absorption coefficient [31, 32].
In both T and L we have seen a quite broad SCG, Fig. 8(a, b), hence we used the frequency dependence of the nonlinearity parameter, γ(ω), which is given by , where is negligible for silicon in the operating wavelength range [31,32]. To include frequency dependence of the three photon absorption, as it has higher absorption between 2–3 μm, we included 3pa(ω) [31,32]. Here, , where 3pa(ω0) is zero for both L and T structures while , where the last term is zero at ω0.
Figure 8 Supercontinuum spectra of L and T waveguides. (a) SCG for L waveguide (w = 2.01 μm, h = 700 nm, t = 465 nm, Pw = 500 nm), with dispersion profile (inset), pumped at 4.1 μm. (b) SCG for T waveguide (w = 2.2 μm, h = 700 nm, t = 423 nm, Pw = 435 nm), with dispersion profile in the inset, pumped at 3.6 μm. The orange dashed curve represents the input pulse. (c), (e), (d) & (f) are the pulse evolution in the time and spectral domain for L and T waveguide respectively.
The T waveguide (1.6 cm long with 1 dB/cm propagation loss [12, 13]) is pumped with 320 fs pulses at 3.6 μm having 1 kW peak power with Aeff = 0.95μm2.
For the L structures, the pump at 4.1 μm with 1 kW peak power was launched into a 2 cm long waveguide having 1 dB/cm propagation loss. The mode area is Aeff = 1.04μm2 and the launched pulse width is 320 fs. For the SCG modelling, the dimensions of the L structure were adjusted to achieve a dispersion shown in the inset of Fig. 8(a). This is to generate a desired anomalous dispersion in the range where our laser operates (for future experiments [13,33]). The dimensions are: w = 2.01 μm, h = 0.7 μm, t = 465 nm and pw = 500 nm.
We observe from Fig. 8(b) that the SCG is wider for T structures which gets flatter beyond 4 μm and can in theory be observed easily beyond 8 μm, whereas the SCG in L waveguides used in this study starts dropping around 8 μm. The signal level, however, is higher in the L waveguide compared to the T waveguide up to 7.5 μm. We note that it is possible to achieve supercontinuum in the near infrared regime with the same waveguides by pumping them at the wavelengths between 1st and 2nd ZDW, which is feasible with a tunable source such as an OPO [33].
The variation of the nonlinear parameter, γ, with wavelength is shown in Fig. 9. The nonlinearity is higher for the T waveguide in near-IR compared to the L due to it’s smaller Aeff. This is because the mode gets pulled towards the centred pillar of T hence narrowing the width of the mode and reducing the Aeff. Whilst, at the longer wavelengths (>4.5 μm) the Aeff increases due to the fact that the T waveguide has slightly smaller facet area than the L (∼0.1 %), and that the mode starts leaking into the substrate through the T’s pillar as is seen below.
Figure 9 The calculated γ vs wavelength for L and T waveguides is based on measured n2 [32]. The calculated Aeff for T and L is shown in inset.
We calculated the mode confinement in the T and L structure for which the SCG was calculated is shown in Fig. 10(a, b). The mode cutoff for the T waveguide is around 7.15 μm, whereas, the L waveguide has higher energy confinement in the silicon (35%) upto 7.8 μm before gradually dropping to ∼7 % at 8.4 μm. The energy confinement in both structures remain similar up to 6 μm, beyond which it starts leaking into the sapphire in the T structure. The energy in silicon of the T waveguide at 7 μm is ∼40% which is same as the energy in silicon at 7.5 μm in the L waveguide. T structures can also be designed to confine energy for longer wavelengths by using a slightly bigger waveguide, seen in Fig. 4(a), as the +200 nm waveguide has reasonably flatter dispersion which can be modified for dispersion close to zero with slight thickness increase, ∼20 nm.
Figure 10 (a) Energy confinement vs wavelength for the T waveguide and for, (b) L waveguide with dimensions used for the SCG modelling in Fig. 8.
Finally, these structures can be fabricated by depositing the silicon in two steps [34], and creating the overhang with sacrificial layer such as silica.
4. Conclusion
We present silicon waveguides for Mid-IR with wide and flat dispersion (upto 2200 nm for T and 4100 nm for L waveguide) while having 4 ZDWs. The waveguide mode interaction with the highly lossy substrate in the Mid-IR is very low, making them effectively substrate independent structures. We have studied the role of different parts of the waveguide in producing a flat dispersion and calculated the sensitivity of dispersion on the dimension. Up to two octave spanning SCG is achievable due to the flatness of the dispersion. These waveguides can be scaled for different wavelength regimes, such as the near-IR, and have potential applications in mid-IR sensing and nonlinear studies as well as a broad band light source operating over the entire silicon transparency.
Acknowledgments
This work was supported by the Australian Research Council’s (ARC) Laureate Fellowship ( FL120100029), Centre of Excellence (CUDOS, CE110001018), and Discovery Early Career Researcher Award ( DE130101033) schemes.
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