In this paper, we propose a novel optical waveguide consisting of arrays of silicon nanowires in close proximity. We show that such a structure can guide an optical mode provided the electric field is polarized along the length of the nanowires. Furthermore, such guidance can happen even if the nanowires are arranged randomly albeit at a higher scattering loss. On the other hand, high radiation losses are observed if the electric field is polarized in the transverse direction to the nanowires. We calculate the optical radiation loss for different structures using Finite Difference Time Domain (FDTD) method. We also show that the arrayed nanowire region can be approximated using an effective index bulk waveguide. The approximation allows for design and optimization of optical structures using integrated optics methodology resulting in significant savings in time and resources. The advantage of the proposed waveguide structure is that it allows for increased optical confinement while using the enhanced optical interactions of nanowire structures compared to single nanowire photonic waveguide for diameters smaller than 100 nm. For a diameter of 50 nm for the silicon nanowire, an optical confinement factor of 33 % was achieved in the proposed waveguide as opposed to 0.1 % that is achieved for a single nanowire photonic waveguide. A radiation loss of 0.12 cm−1 is achieved for nanowires of the same diameter spaced 75 nm apart. While our analysis is done on silicon nanowires at 1550 nm, the proposed structures can be extended to other materials and wavelength regimes also.
© 2010 Optical Society of America
In recent years, silicon photonics has offered low cost optoelectronic solutions for applications ranging from telecommunications, biosensing to chip-to-chip interconnections. As such various devices have been demonstrated on silicon like optical amplifiers , modulators , optical switches [3, 4] and four-wave mixers . Most of the demonstrated devices use silicon strip waveguides with small cross-sectional dimensions of approximately 450 nm × 250 nm to increase the optical confinement [6, 7]. This allows for designing waveguide bends with reduced diameters and increased diffusion of carriers in optical switches reducing the effective carrier lifetime . An etchless process in the fabrication of such waveguides has also been demonstrated in order to reduce the loss due to scattering from sidewall roughness . These devices take advantage of the increased optical fields within the core of the waveguide. Using a comprehensive theory for describing the nonlinear propagation of optical pulses through silicon waveguides including the effects of polarization, increased nonlinear effects have been calculated for waveguide geometries with high confinement . However, the fundamental material properties of the silicon do not change in this regime. Further reduction in the dimensions of the silicon core is not possible as below a certain dimension, the optical mode starts to loose confinement . An extension to these works by going to the regime of nanowires is promising because it has been observed that different optical-carrier interactions are enhanced in the nanowire regime due to the quantum confinement and increased optical intensity. Nassiopoulos et al.  and Huo et. al.  have demonstrated room temperature electroluminescence from Silicon Nanowires (SiNWs) of diameters smaller than 10 nm due to band-to-band electron-hole recombination. Cloutier et al.  have shown optical gain and stimulated emission in a periodic nanopatterned-silicon structure at cryogenic temperatures. In addition, Chen et al.  have reported stimulated emission at a bandgap energy of 1.1 eV in a nanostructured silicon p-n junction diode using current injection at room temperature. Shiri et al.  have theoretically shown that for SiNWs smaller than 10 nm in diameter, direct bandgap is achieved and the bandgap can be switched from direct to indirect by application of strain. Nonlinear effects of bulk silicon can also be enhanced using nanosized structures. It has been experimentally shown that the spontaneous Raman scattering can be enhanced by three orders of magnitude using SiNWs and silicon nanocones instead of bulk silicon provided the nanowire diameter is less than 130 nm . Furthermore, SiNWs due to their large surface area have found promising applications in biosensing area. For example, Li et al.  have demonstrated highly sensitive and sequence-specific DNA sensors using SiNWs. SiNW-based biosensors have also shown fast response and high sensitivity to glucose . Zhu et al.  have reported p-n junction diode arrays on a Si substrate which could find promising applications in nanoelectronics and optoelectronics.
An important criterion of nanowires for applications in optoelectronics devices is the guidance of light through them. Barrelet et al.  have explained an approach for guiding and manipulating light in sub-wavelength Cadmium Sulfide (CdS) nanowires with diameter of 200 nm. They have also quantitatively investigated losses through straight and bent single nanowire waveguides. Also, it has been shown that semiconducting nanowires can work as nanoscale lasers , where high refractive-index contrast between the nanowire and surrounding media provides an optical cavity for lasing. Most of these works consider nanowires of several 100 nm in diameter. However, the increased material properties in SiNWs described above are only observed for diameters less than 100 nm. A main disadvantage of SiNWs is the lack of optical confinement especially when the diameter is less than 100 nm. Using a single nanowire as an optical waveguide suffers from two main issues, namely, low coupling efficiency and low optical confinement. While the low coupling efficiency may be corrected using tapers , the devices still suffer from critical alignment tolerances making their practical applications a suspect for photonics. Figure 1 shows the calculated confinement factor for a SiNW surrounded by air as the diameter of the nanowire is changed. The free-space wavelength for this calculation is 1550 nm. The confinement factor was calculated by solving for the eigen mode of a silicon cylinder using Full-Vectorial Beam-Propagation method and calculating the optical power ratio within the silicon core. As the diameter of the nanowire is reduced below 75 nm, optical confinement factor starts to decrease appreciably and is less than 1%. The low confinement factor results in the fact that though the intrinsic properties of nanowires increase, the overall device performance is not greatly enhanced . As such, there has been some work in increasing the confinement in nanoscale devices. Plasmonic waveguides consisting of metal nanoparticles have been considered as optical waveguides [23, 24]. In principle metal nanoparticles can be attached to the surface of nanowires to increase the confinement factor. However, such waveguides suffer from increased optical loss. For example, transmission losses through nanoparticles due to resistive heating is about 6 dB/μm , while still having critical alignment requirements. Further, the increased optical interactions have been observed when the electric field is polarized along the length of the nanowires . This suggests that a single nanowire waveguide will not be able to take advantage of the increased effects as the major component of the electric field needs to be transverse to the direction of propagation, which in a single nanowire waveguide is along the length of the nanowire.
In this paper, we propose a novel optical waveguide consisting of closely packed arrays of SiNWs on a Silicon-on-Insulator (SOI) substrate. In the region where the diameter of the nanowires is much less than the free-space wavelength, the nanowires act like meta-material inclusions . We show that such a structure can guide an optical mode, provided the electric field is polarized along the length of the nanowires (the same polarization for which enhanced optical interactions are observed in nanowires). Furthermore, the guidance happens even if the nanowires are randomly arranged, albeit with an increased loss. We also show that the SiNW region can be approximated by an effective index using a weighted average method. This allows us to use conventional waveguide methods to design and optimize the waveguide structures resulting in significant savings of resources and time. The advantage of our proposed structure is that it allows us to use the enhanced optical interactions of SiNWs while achieving optical confinement and coupling efficiency of conventional optical waveguides. In our designs, we consider crystalline-silicon nanowires with diameters of tens of nanometers and lengths of 1 micron, which are achievable with current nano-fabrication techniques [27–29]. Since the nanowires can be placed randomly, maskless methods of creating SiNWs demonstrated for solar cells  can also be applied for fabricating the waveguides using low-cost fabrication.
The paper is organized as follows. In section 2, we conceptually describe our proposed device structure. In section 3, we simulate light interaction with single SiNW for different diameters and polarization to understand the conditions for which our proposed device will work. In section 4, we simulate the proposed structure for different SiNW diameters and nanowire spacing. Furthermore, we introduce the effective index method for the SiNW region.
2. Silicon nanowire optical waveguide (SNOW)
The proposed structure is shown in Fig. 2. A SOI wafer is considered as a starting substrate and etched into an array of nanowires. In the vertical direction, the refractive index difference between the silicon and insulator substrate allows for transverse guiding of the optical signal. Since the goal of our work is to increase the optical confinement factor within the SiNWs, the length of the nanowires needs to be large to have high confinement in the vertical direction. We have considered nanowires of 1 micron length which provides greater than 95% confinement in the vertical direction. In the lateral direction, if the nanowires are placed closed together, the optical mode will see an average refractive index, higher than the surrounding, thereby guiding light. The proposed structure may look similar to Photonic Bandgap (PBG) structures. However, there is no bandgap as in PBG, and the nanowire scattering points are much smaller than the ones used in PBG (at least 5 – 10 times smaller), in the range where increased optical effects have been observed in SiNWs. In order for the structure to work effectively with low radiation loss, an individual nanowire should not diffract the light. This can be observed if the phase front of an optical wave passing through them is not distorted. So it is instructive to look into how an optical wave interacts with a single nanowire and whether a condition can be achieved where the phase is not modified by the SiNW. While this may not provide a quantitative analysis, it does help to conceptualize how the proposed structure is working and the regime in which it will work well. For the calculations, free-space wavelength of 1550 nm is considered. However, the analysis can be easily extended to other wavelength regimes or different materials.
3. Light interaction with single SiNW
We are interested in how the SiNW alters the amplitude and phase front of an optical wave incident along the perpendicular-cross-sectional plan of the SiNW, shown in Fig. 3. Since we are interested in seeing how the SiNW scatters light in the x-z plane, the SiNW is considered to be infinite in the y-direction, parallel to the nanowire. A plane wave is incident on the nanowire from the z-direction. Cylindrical co-ordinates are used to solve the problem. This kind of interaction can be treated as a scattering problem because of the small diameter of the nanowire compared to the wavelength used for the analysis i.e. 1550 nm. Due to the cylindrical shape of the nanowire, the scattering problem is a canonical boundary value problem. Such a problem can be solved by separation of variable approach and the solution is dependent on the polarization of input electric field.
For the case when the electric field is polarized in the y-direction (Hy = 0 and Ey ≠ 0), the problem can be considered to be 2-dimensional. Assuming Ey ≠ 0 results in nonzero Ey, Hρ, and Hφ components. The incident magnetic vector potential is chosen as a plane wave propagating along the z direction:
By choosing cylindrical coordinates the incident field will be a periodic function of φ with a period of 2π. Due to the periodicity of incident magnetic vector potential , it can be expanded as a Fourier series,
When the incident field hits the SiNW, some part of it will be scattered, , and the other part, , will be transmitted through the SiNW. The scattered magnetic vector potential has to satisfy Helmholtz equation while ρ ≥ a and .31]:
Considering continuity of tangential electric and magnetic field at the boundary an and bn can be determined. Therefore, substituting , , and from previous equations to the following equation:
Similarly for bn, we have:
Using the mentioned equations, we solve for the total electric field distribution in amplitude and phase for different diameters ranging from 600 nm to 20 nm. The amplitude of the electric field is shown in Fig. 4 for different diameters whereas the phase distribution is shown in Fig. 5.
The wavelength of the incident light is 1550 nm within the telecom regime. For wider nanowires, some part of electric field is scattered and another part is coupled to the SiNW. This coupled part transmits through the SiNW and again at the boundary of SiNW and air experiences scattering and transmission. The coupled part sees a multi-mode interference effect to give a spatial interferometric distribution. If the nanowire diameter was further increased, whispering gallery modes  were clearly observed. The diffraction is also seen in the phase plot for the larger nanowires. As the phase is altered outside the nanowires, if the larger diameters are placed together, constructive or destructive interference can take place. PBG structures operate based on this diffraction. However, the picture interestingly changes when the nanowire diameter is less than 100 nm. This is observed in both the amplitude and phase. The change of the electric field inside and around of SiNW is small. Also the phase front is not modified by the nanowire. In fact for a diameter of 20 nm and 50 nm, there is negligible distortion of the phase of the electric field. This can be observed from Fig. 6 which plots the cross-section of the phase front at the center of the nanowire. For this figure, the magnitude of the phase front for nanowires of smaller diameters has been multiplied by a factor of 5 for ease of visualization. It is observed from the figure that the phase front is nearly flat for diameters of 50 nm and 20 nm. This suggests that for these diameters, our proposed SNOW structure will work effectively if the electric field is polarized along the length of the nanowire. It is also instructive to look into how the interaction happens when the magnetic field is polarized along the length of the nanowire (Ey = 0 and Hy ≠ 0). The incident electric vector potential is chosen like the previous case and the solution steps are also similar. Figure 7 and Fig. 8 show the amplitude and phase of the interacting magnetic field for different diameters, respectively. For this polarization it is observed that phase distortion happens even when the diameter of the SiNW is 20 nm. We believe that this is due to the discontinuity of the electric field along the boundary and the discontinuity changes as the field propagates through the nanowire. This suggests that the SNOW structure will be a poor guiding medium for this polarization of optical wave. For most of the applications, this is not an issue as the enhanced optical interactions which we want to take advantage of are also observed in the Ey ≠ 0 polarization.
4. SNOW FDTD analysis
In order to verify that the array of nanowires can guide optical signal, we simulated the structure using Finite Difference Time Domain (FDTD) method. In the first simulation, we analyzed arrays of nanowires located between two bulk-silicon waveguides. The top view of the simulated structure is shown in Fig. 9. The input waveguide is excited by a CW power at wavelength of 1550 nm. Eventually, the optical signal is coupled to an output silicon waveguide after passing through the SNOW region. The width of the all waveguides is 260 nm, in order to keep the silicon waveguide single mode. The length of the bulk Si waveguides is 1 μm and the length of SNOW part is 6 μm. The diameter of the SiNWs in the SNOW region is 20 nm and the separation between the SiNWs is 30 nm. Figure 10 shows the mode propagation when the electric field is polarized along the length of the SiNWs. Such a polarization is equivalent to the quasi-TM mode for a conventional ridge waveguide in the same orientation. As can be seen from the figure, at the intersection of the silicon waveguide and SiNWs, radiation of optical field occurs. This is expected as there is a modal change between the two waveguides. However, once coupled, the array of SiNWs is guiding the optical power. Further, a Fabry Perot (FP) cavity effect is seen in the SNOW region because of different effective indices in the two regions creating reflections at the boundary and increasing the intensity of the wave within SNOW region over multiple passes in the FDTD.
A very simple approximation can be used to calculate an effective index for the nanowire region. The approximation is modified from  which calculated the effective index for a multiple quantum well region. For the approximation it is assumed that the electric field is constant in a single cell consisting of the SiNW along with the surrounding medium. For the electric field polarized along the length of the nanowires, the TE mode calculation for quantum wells is used as the electric field is polarized along the unconfined axis of the quantum well, this being the same case for SiNW. The resulting approximation for the effective index is shown in the equation below:Figure 11 shows a FDTD simulation when the nanowire region is excited by an optical waveguide with an index equal to the effective index of the nanowire region calculated above. The two waveguides are of the same width. From the figure it is easily observed that negligible reflection and radiation are seen at the butt joint of the two waveguides, showing that the optical mode shape and propagation constants do not change and that the approximation does work well. To further evaluate this, the mode shape of optical waveguide and SNOW region (calculated after transmission of optical wave through SNOW), are also plotted in Fig. 12. A very good overlap using the proposed effective index method and for the optical mode after propagation through SNOW region observed in Fig. 12, further confirming the accuracy of the effective index calculation. Similar overlap is also seen in the vertical direction for the optical mode. This approximation can play an important role in design of optical structures using SNOW as it can allow for treating the structure only as an averaged bulk waveguide for design and optimization. Therefore, conventional integrated optics formulation can be used. The initial designs and simulations can be done using beam propagation and analytical methods before final validation with FDTD resulting in a significant time saving.
Using the effective index waveguide as the input waveguide, and measuring optical power at different regions for the SNOW waveguide, we analyzed the optical loss using the cut-back method by calculating transmitted optical power at different lengths in the SNOW region. The input to the SNOW region is the optical mode for the dielectric waveguide using effective index approximation. While the cut-back method is mainly used experimentally, it allowed us to calculate the propagation loss of the SNOW region and coupling efficiency between the two waveguides in a single simulation. Further, by measuring multiple lengths, we could check the accuracy of the analysis especially for the structures with low propagation loss. We have analyzed the dependence of loss for the two important parameters of SNOW region, i.e. the diameter of the nanowires and the pitch between the nanowires. Figure 13 shows how the loss changes as the diameter of the nanowire is increased. For this simulation, the number of SiNWs in an array and the ratio of the diameter to the pitch (1 : 1.5) were kept constant. It is observed that for the nanowire diameters less than 50 nm, the loss is lower than 0.12 cm−1, but starts to increase appreciably as the diameter increases to 100 nm. The increased loss is due to scattering of light by the nanowires. This result mirrors the understanding we got in Fig. 4 and 5 from looking at light scattering from a single nanowire. It needs to be mentioned that the loss calculation for diameters less than 50 nm is limited by the grid size. For these calculations, a grid size of 2 nm is used in all dimensions limited by the memory of the computer. Hence, the values provided here provide an upper limit on the propagation loss.
Figure 14 shows the loss as the spacing between the nanowires is increased. The diameter of the nanowires is 50 nm. We chose this diameter for the analysis as similar nanowires have been fabricated before  and this suggests that our proposed SNOW structures are feasible with current fabrication technology. From Fig. 4 and 5, it is clear that the results for smaller diameters will be better. It is seen that the loss increases once the separation increases beyond 150 nm. Since the SNOW waveguide is acting like an effective index medium, the electric field within a unit cell should be nearly constant . As the pitch is increased, the assumption that the electric field within a unit cell containing the SiNW is constant is no longer valid, thereby resulting in increased scattering loss. 150 nm corresponds to approximately to one tenth of the free-space wavelength. This suggests that beside the diameter we have an additional parameter (pitch) which can be altered to change the effective index of the SNOW region thereby optimizing the mode shape and confinement in the SiNW region while not increasing the loss appreciably.
The different calculated optical parameters for the structures are summarized in Table I. Overlap integral between the mode after propagation through SNOW region and the optical mode of the effective index waveguide is shown in the table. High value of the overlap integral (> 99%) for different pitches suggests the efficacy of the effective index approximation for the SNOW region in a wide range of pitches. The optical confinement factor for different waveguide structures is also summarized in the table. The confinement factor is the ratio of the power confined to the SiNWs in the SNOW region compared to the input power. The table also summarizes the optical confinement factor within the SiNWs for the SNOW waveguides. It is seen that the optical confinement within the SiNWs has increased appreciably in the SNOW structures compared to a single nanowire waveguide of the same diameter. For example for the diameter of 50 nm and diameter:pitch ratio of 1 : 1.5, an optical confinement factor of 33% is achieved. Compared to this, for a single nanowire of 50 nm surrounded by air, the confinement factor is ∼ 0.1%. Furthermore, even if the diameter is reduced, similar confinement factors can be achieved if the diameter:pitch ratio is kept constant. Thus, the SNOW structure allows for increased optical confinement of optical mode while the increased optical interactions of SiNWs can also be utilized in these structures. Moreover, from the effective index approximation it is clear that the confinement factor within the SiNWs is mainly dependent on the diameter:pitch ratio. Consequently, if for some applications small nanowire in diameter is required a reasonable optical confinement can be achieved by adjusting the pitch to increase the overall performance of the device while taking the advantages of nanowire properties. All the analysis has been done for SiNWs with high vertical confinement, since the goal of the work is to propose a waveguide structure where confinement is increased in the SiNWs. By decreasing the SiNWs length (typically below 300 nm), the confinement is reduced in the vertical direction and the mode will become less quasi-TM like (i.e. the Hz component will increase) which could result in higher radiation loss due to polarization coupling which may occur.
Furthermore, we wanted to see whether the SNOW region will guide light if the nanowires are arranged randomly. We did a simulation where the nanowire diameter was 20 nm and the average separation between the nanowires was 30 nm. We assumed up to 100 % change in the origin of SiNWs in all planar directions in a random distribution with equal probability over the range of the pitch. Figure 15 shows longitudinal mode propagation in this case. The results are similar to what was achieved when the SiNWs are located in an ordered fashion showing that guidance still happens. It is reiterated here that in a PBG structure with disordered scattering points no light will be guided . The optical loss in this case was 2.91 cm−1 as compared to 0.094 cm−1 for ordered nanowires of same dimensions. Again the loss measurement is the upperbound as the grid discretization is more critical in a random arrangement. While the loss has increased, it is still comparable to PBGs and is not a hindrance in many applications especially optical sensing. Unordered SNOW region will allow to use low cost fabrication schemes including maskless etching like in black silicon  and Vapor Liquid Solid (VLS) grown nanowires  provided the nanowires are vertical. However, if the SiNWs are not vertical in these processes, there will be an excess loss due to depolarization of the light with respect to the nanaowire axis.
We also simulated the optical mode propagation for the SNOW structure when the magnetic field is polarized along the length of the nanowires. This polarization corresponds to the quasi-TE mode for the conventional ridge waveguides. Figure 16 shows the mode propagation through the SNOW region for SiNW diameter of 20 nm. High radiation loss is observed in this case with a loss of ∼ 400 cm−1. This mirrors the understanding from Fig. 7 and 8 which suggest that for this polarization, the optical field is going to suffer scattering. This suggests the SNOW can be used to fabricate ultra short polarization filter within a photonic integrated circuit.
We have proposed and analyzed a novel waveguide consisting of arrays of closely placed vertical nanowires. Through FDTD simulations, we have demonstrated that such a waveguide will guide an optical wave with low loss if the nanowire diameter is less than 100 nm for a free-space wavelength of 1550 nm. This guidance happens when the electric field is polarized along the length of the nanowires but does not happen when the magnetic field is polarized in the same direction. This polarization is also the same for increased optical interactions observed in SiNWs. A minimum radiation loss of 0.12 cm−1 is achieved with a confinement factor of 33 % for the nanowires. The high confinement factor can be obtained for nanowires of even smaller diameters provided the pitch to diameter ratio is kept constant. By adjusting the pitch between the nanowires, one can tailor the mode confinement and confinement within the SiNWs. We also demonstrated that an effective index approximation can work to replace the SiNWs region for analysis and design, allowing one to use conventional waveguide design analysis.
This work was supported by the Canadian National Science and Engineering Research Council (NSERC), Ontario Centres of Excellence (OCE) and DALSA Corp.
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