We present a design of a slot waveguide in which the core layer is orthogonally slotted to form a rectangular sub-core. While the overall guiding and coupling efficiency remains the same as a conventional slot waveguide, the field confinement is enhanced and appears two-dimensional. The waveguiding is controllable by selecting the intermediate index as well as various geometrical parameters. In addition, by changing different variables, the linear/nonlinear dispersion and birefringence can be tailored with extended ranges. Constant-dispersion points, where the dispersion is insensitive to size changes, are also demonstrated.
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A slot waveguide transmits light with a strong field confinement across the slot [1,2]. The confinement is based on the large discontinuity of the electric field at a high-index-contrast interface. On the low-index side of the interface and within a small fraction of the wavelength, the electric field, mostly of evanescent components, is enhanced dramatically. In the slot waveguiding configuration in which a low-index slot layer is sandwiched by high-index layers and the slot layer is sufficiently thin, the enhanced fields at both interfaces meet, interact, and form a highly enhanced beam within the slot core. This guiding mechanism is fundamentally different from the widely adapted high-index-core waveguiding based on total internal reflection (TIR), and enables light guiding at the nanometer scale . Because of the tight spacing, slot waveguides are finding applications in various compact photonic designs -. In general, the slot waveguide might be useful for various applications including on-chip communications, microscopy, spectroscopy, sensing, and data storage -.
So far, slot waveguides transmit only one-dimensional sheet-like beams and the transmission is usually polarization dependent. In order to reduce the directional and polarization dependence, a combination of two orthogonally placed slot waveguides was considered, either side-by-side or as a cross [10,11]. The two polarization states form a tightly coupled, bifurcating pair, and the influence of polarization on optical responses becomes insensitive [10,11]. On the other hand, in order to deliver subwavelength light spots, both ridged apertures and tapered slots were proposed [12,13]. Also, note that enhanced, subwavelength beaming was achieved with metallic slits and apertures, and the transmission was mediated by the propagation of surface plasmons under cavity enhancement -.
In the present work, we developed a design in which a slot waveguide can create and guide two-dimensionally confined beams. In this new design, the main configuration remains a conventional slot waveguide. Therefore, one expects that the overall guiding and coupling efficiency (both for the input and output) would be retained. The modification is that an additional confining core region is introduced. The sub-core region can be filled with materials (or simply air) of which the refractive index is lower than the core layer. In other words, the core of the slot waveguide is again slotted in the orthogonal direction. Since the waveguide has a two-dimensional geometry, there exist quasi-TM modes. Due to the fact that the sub-core is orthogonally slotted twice, there are discontinuities of the field at all boundaries. Considering the geometry, one would expect the transmitted light beam to be highly confined in the direction across the slot, as well as in the direction along the slot with a reduced degree of confinement. Based on numerical simulations, we have found that the waveguide can guide and deliver two-dimensionally and nanometrically confined light beams. The property of the waveguide depends on the intermediate slot index as well as various geometrical parameters. In addition, by changing different variables, linear/nonlinear dispersion and birefringence can be tailored with increased ranges. Finally, constant-dispersion points where the dispersion is insensitive to size changes were demonstrated.
2. Model simulation
We simulated the new design in a finite element solver (COMSOL) program. In the simulations, all the field components, i.e. Ex, Ey, and Ez are taken into account. Since Ez is insignificant and Ey is considerably larger than Ex, the waveguiding modes involved in the simulations are of quasi-TM modes. For comparison, we also conducted numerical simulations for the original slot waveguide before the sub-core slot was incorporated. The basic system is schematically shown in Fig. 1(a) where a silicon-silica-silicon slot waveguide is embedded on silica substrate. The light wavelength is assumed to be 1550nm and the geometrical parameters were chosen, i.e. the width of the waveguide 375nm, the thickness of the slot layer 21nm, the thickness of both upper and lower silicon layers 135nm. Constant refractive indices were selected as nsilicon = 3.476 and nsilica = 1.444. In parallel, we considered incorporating the sub-core in the original slot waveguide. The modified system is schematically shown in Fig. 1b where a sub-core slot of air (nair = 1) is incorporated in the center of the basic system with the same geometric parameters and refractive indices as those in Fig. 1(a), except that the slot material is replaced by silicon nitride and the corresponding refractive index is selected as nsilicon nitride = 2.1. The reason for choosing silicon nitride for the slot layer is that it has been known that the refractive indices of silicon nitride films are controllable within a wide range (1.9~2.8) by varying deposition parameters . Note that the above sizes are chosen at the nanometer scale and other sizes were also used in the present work. The influence of the size changes on the waveguiding is to be presented in the next Sections. The sub-core was assumed of air for simplicity in simulations, and the air void could be formed by chemical etching. We are also looking into the possibility to form the waveguide involving polymer growth and deposition. We show later in the present communication that, by tuning the slot index, one can optimize the field confinement. In the modified system, the width of the sub-core slot is assumed to be 50nm. Therefore, the sub-core forms a rectangular slot along the waveguide. Since the refractive indices are chosen such that ncladding > ncore > nsub-core, one has actually two orthogonally placed slot waveguides. Mediated by the tight coupling between the two slot states [10,11], nanometrically and two-dimensionally confined light beams are guided within the sub-core slot. In the following, we present the numerical results both for the original and the modified slot waveguides. In the rest of the present communication, we refer to these two waveguides as the slot waveguide and the dual slot waveguide, respectively.
The field intensity distribution is shown in different formats in Fig. 2 , both for the slot waveguide [Fig. 2(a) and 2(c) in the left column] and the dual slot waveguide [Figs. 2(b) and 2(d) in the right column]. In Fig. 3 , cross-sectional curves from the intensity distributions in Fig. 2 are plotted along two directions. For the slot waveguide, Figs. 2 and 3 show that the guided light is mostly confined in the slot layer, and the confinement is only in the direction across the slot. Since the waveguide has a finite width, the beam is, however, curved in the direction along the slot. From the field intensity distribution for the dual slot waveguide in Figs. 2 and 3, it is apparent that a two-dimensionally confined light beam is transmitted in the sub-core slot. In other words, in comparison to the slot waveguide, the dual slot waveguide supports an additional field confinement in the orthogonal direction.
Note that the point-like beam in the sub-core region is on top of the sheet-like beam in the original slot. Hence, the overall guiding and coupling efficiency of the waveguide remains the same as that of the original slot waveguide. What happens is that the field in the original slot is redistributed with a concentration in the sub-core region. The degree of the concentration would depend on the selection of the refractive indices of the materials and the width of the sub-core slot. Below, we study the influences of changing the refractive index of the slot and varying the sub-core width.
3. Index and size dependence
Recall that the degree of the field confinement in the slot waveguides depends on the index contrast. For silicon based devices such as the system studied in the present work, the index contrast between the silicon layers and the sub-core is fixed. What can be selected is the refractive index of the original slot layer. Note that, by changing the slot index, one can actually adjust the competing contributions between the original and the orthogonal confinements. We investigated the dependence of the confined beam on the refractive index in the slot. As a function of the slot index, the average peak intensity, the normalized peak height, and the full width half maximum (FWHM) of the peak are plotted in Figs. 4(a) and 4(b). Here, the average peak intensity is defined as the intensity average within the sub-core. The peak height is referred as the extra-peak above the peak formed without the sub-core and is normalized to that peak as well. From the curve in Fig. 4(a), it can be seen that the peak intensity reaches its maximum when the slot index is about two. The strongest peak occurs when contributions from the original and orthogonal confinements are balanced, which is realized by selecting an appropriate intermediate material for the slot layer. For the specific example in the present work, the optimized slot index is found to be nslot~2.1, which is in the range of the possible index of silicon nitride films . Further increasing the slot index causes a significant reduction of peak intensity. On the other hand, as shown in Fig. 4(a), peak height increases with the slot index. This implies that, with an increase in the slot index, the contribution from the original slot decreases. Figure 4(b) shows the dependence of FWHM on the slot index. In comparison with the slot waveguide (dot curve), the FWHM decreases with the slot index (square curve) for the dual slot waveguide since light is confined in both directions.
Next, we studied the dependence of the confinement on the width of the sub-core, and the numerical results are presented in Fig. 5 . In Fig. 5(a), the average intensity in the sub-core normalized to the average intensity in the slot region is plotted as a function of the sub-core width, whereas, in Fig. 5(b), the power ratio between light in the sub-core and light in the whole slot region is presented. One can see that for the specific waveguide the intensity peaks at about 50~70nm, which shows that selection of geometrical parameters would enable the transmission and confinement to be optimized. In general, when the width of the sub-core slot decreases from the optimized peak value, the field intensity increases but there is less light to pass through as the cross-section area becomes smaller. On the other hand, when the width of the sub-core slot increases, although more light passes through, the confinement actually decreases as the peak intensity is reduced. We note that the power ratio curve in Fig. 5(b) provides an important measurement of the additional field confinement stemming from the secondary slot. The curve in Fig. 5(b) reveals that the modified slot waveguide provides an additional confinement of a similar or higher degree. One finds in Fig. 5(b) that a power ratio of approximately 15% is achievable for a sub-core that is 50nm in width. This is about the position for the peak intensity to occur. (See the curve in Fig. 5a). Therefore, further increasing the width of the sub-core will bring more light in the sub-core region, but the achievable peak intensity is reduced.
4. Dispersion tailoring
Since the confinement can be realized two-dimensionally, the proposed waveguide would find applications in compact all-optical photonic devices. For example, since the field confinement is enhanced, various effects, such as linear/nonlinear dispersion and birefringence, are expected to increase significantly. In order to demonstrate the usefulness of the proposed dual slot waveguide, we researched various effects, namely, (1) the group velocity dispersion (GVD) stemming from the combined influence of the material dispersion and the waveguide modal property, (2) the nonlinear coefficient γ(λ) dispersion, and (3) the birefringence dispersion. In the calculations, standard definitions for the three dispersive functions are used -. In the simulations, the material dispersions n(λ) are introduced by Sellmeier empirical equations with properly selected fitting parameters for silicon, silica, and silicon nitride .
In Fig. 6 , for different sub-core sizes, numerical results are presented for these aforementioned dispersions, namely, the group velocity dispersion, the nonlinear coefficient dispersion, and the birefringence dispersion, respectively. For comparison, the results for the slot waveguide are also included.
In Fig. 6(a), the dispersion of the group velocity appears significantly enhanced when the sub-core width increases. Without the sub-core, the dispersion curve crosses the zero-dispersion line into the positive dispersion zone and contains two points of zero-group-velocity-dispersion (ZGVD). A close look at the dispersion curves in Fig. 6(a) reveals two special points, namely, the zero-dispersion point and the constant-dispersion point. At the zero-dispersion point, the dispersion vanishes and the waveguide does not cause any additional dispersion to the guided beams. It is well-known - that the zero-dispersion point shifts according to the effective area of the waveguide. Once the spectrum position is fixed, according to the curves shown in Fig. 6(a), one only needs to adjust the sub-core width to make sure that there would be merely one zero-dispersion point. In the case of Fig. 6, the single zero-dispersion curve is achieved when the sub-core width is 20nm. The usefulness of the single zero-dispersion curve was discussed in Ref .
Further increasing the sub-core size pushes the curve down dramatically. For the 150nm dual slot waveguide, the minimum dispersion point appears at about 3000ps/nm/km lower than the zero-dispersion line, which is considerably more significant than what was achieved with silicon-in-insulator (SOI) waveguide . In the case of dual slot waveguide, in order to achieve this range of dispersion tailoring, it is sufficient to modify only the sub-core size. On the other hand, as shown in Fig. 6(b), the nonlinear coefficient dispersion reduces and becomes more flat when the sub-core width increases. For the 150nm dual slot waveguide, the nonlinear coefficient dispersion reduces from 200 W−1m−1 to 80 W−1m−1 at the communication wavelength λ = 1550nm. The reduction and flattening should be helpful for some nonlinear optical processes. Meanwhile, as shown in Fig. 6(c), the birefringence increases especially for the shorter wavelengths. In particular, the enhanced birefringence reaches maximum (up to 0.4 for the 50nm curve) at about 1550nm in wavelength, which shows a great possibility to make use of the birefringence in photonic applications.
In Fig. 7 , more extensive dispersion curves are presented. Curves in Fig. 7(a) are partially extracted from Fig. 6(a), but the spectral curves are extended up to λ = 1700nm. Dispersion curves in Fig. 7(b) are obtained with an increased slot thickness 50nm, whereas curves in Fig. 7(c) are calculated with a fixed sub-core width 60nm while varying slot thickness. By changing the geometric parameters, we obtained significantly shifted spectra, with respect to the curves in Fig. 7(a), both toward shorter wavelengths (blue shift) and larger wavelengths (red shift). In Fig. 7(b), it shows a blue shift of the minimum dispersion point at about λ = 1350nm, whereas a red shift of the minimum dispersion point appears in Fig. 7(c) at λ = 1500nm. The curves in Fig. 7 demonstrate that, in general, it is sufficient to tailor only one geometric parameter for achieving relatively large spectral shifts.
An important observation one has from the curves in Fig. 7 is the constant-dispersion point (marked with circles) where different dispersion curves for various sizes meet each other at a common point. This implies that in these zones the dispersion is insensitive to the change of slot thickness or sub-core width, and consequently the tolerance on the fabrication variations and defects is increased. We have studied on these special points and found that the constant-dispersion point can be established and shifted according to the geometric parameters. In Fig. 7, three cases are presented. In Fig. 7(a) the constant-dispersion point occurs at about the communication frequency λ = 1650nm, whereas in Fig. 7(b) and Fig. 7(c) the constant-dispersion point occurs respectively at λ = 1520nm and λ = 1750nm. In all the three cases, the dispersion at the constant-dispersion point becomes insensitive to the change of either sub-core thickness or sub-core width.
In this article, we have presented a design of a slot waveguide in which the core layer is orthogonally slotted to form a rectangular sub-core. Based on numerical simulations we have demonstrated that the modified slot waveguide is able to guide light with confinement in two-dimension. The properties of the waveguide can be controlled by varying the intermediate index as well as the sizes. We also have found that the linear/nonlinear dispersion and birefringence can be tailored by changing various geometric parameters with extended ranges. Finally, constant-dispersion points, where the dispersion is insensitive to size changes, are demonstrated and discussed.
Yinying Xiao-Li acknowledges receipt of an Annenberg fellowship.
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