## Abstract

In this work the design of Si / hybrid waveguides which contain a vertical infiltrated slot is studied. The case of slots infiltrated with a *χ*^{(3)} nonlinear material of relatively high refractive index (e.g. chalcogenide glasses) is specifically discussed. An optimized waveguide geometry with periodic refractive index modulation, a nonlinear figure of merit > 1 and minimum effective mode cross section is presented. Introducing a periodic refractive index variation along the waveguide allows the adjustment of the group velocity dispersion (GVD). Choosing the period accordingly, the phase matching condition for degenerate four wave mixing (GVD = 0) can be fulfilled at virtually any desired frequency and independently from the fixed optimized waveguide cross section.

© 2013 Optical Society of America

## 1. Introduction

In the last years great efforts led to a strong miniaturization of optical components, as several devices were realized on the silicon-on-insulator (SOI) platform, which is compatible to CMOS technology. Mature processing techniques can be used to fabricate wave guiding structures. The very high refractive index contrast between the Si core (*n _{Si}* = 3.5) and the oxide cladding (

*n*

_{SiO2}≈ 1.45) and air (

*n*= 1), respectively, enables a high confinement of light inside such waveguides. This enables the fabrication of small footprint devices. However, for many applications active devices exhibiting a nonlinear optical behavior are needed. Although Si shows large

_{Air}*n*

_{2}-values (

*n*

_{2}= 6 · 10

^{−18}m

^{2}/W [1], approx. 100 times stronger than SiO

_{2}[2]) it suffers from strong two photon absorption (

*β*= 6.7 · 10

^{−12}m/W [2, 3]), leading to a small figure of merit (FOM =

*n*

_{2}/

*λβ*). For Silicon the FOM is approximately 0.6 at

*λ*= 1550 nm. Generally a FOM > 1 is necessary for efficient nonlinear optical processes [4].

One way to enhance the performance of SOI based devices is to combine Si with a material showing a higher FOM such as organic materials [5], chalcogenide glasses, e.g. As_{2}S_{3}[6, 7] or silicon nanocrystals [8–10]. For this combination several geometries such as horizontal [11] and vertical [5] slot waveguides filled with nonlinear material, have been suggested. However, for nonlinear optical frequency transformation (e.g. second harmonic generation, four wave mixing) the phase matching is crucial, as well. Controlling the waveguide dispersion is, therefore, essential in designing devices using these processes. For instance for maximum performance of degenerate four wave mixing processes a group velocity dispersion (GVD) of zero is required [12].

This publication focuses on infiltrated vertical slot waveguides. Here the horizontal electric field of the waveguide mode is highly concentrated in the vertical slot, leading to an enhanced nonlinear optical performance. The suggested waveguides might be built by fabricating slot waveguide with e-beam lithography and covering those waveguides consecutively with solution processed As_{2}S_{3}[13–15].

Up to now the linear and nonlinear optical properties of waveguides were mainly tailored applying specific designs of the waveguide cross section [4, 11, 16, 17]. This generally resulted in a trade off between the aim of a a large FOM with strong field confinement and the often contradicting goal of a zero or near zero GVD. In this paper we discuss a design route which allows waveguide geometries with FOM > 1 and a maximum field concentration within the infiltrated material and a vanishing GVD at the same time. The independent adjustment of the GVD and the mode profile is achieved by varying the refractive index along the waveguide periodically. This introduces a photonic band gap for the waveguide mode leading to additional band bending in the vicinity of this band gap. Therefore the periodicity gives another degree of freedom to tailor the GVD of such hybrid waveguide systems.

## 2. Optimized waveguide cross section

#### 2.1. Basic waveguide geometry and numerical model

In Fig. 1(a) a sketch of the investigated waveguide geometry is shown. The structure consists of a silicon on insulator (SOI) strip waveguide of width *w _{wg}* with a vertical centered slot. A layer of height

*h*of an optical nonlinear material is situated on top of the substrate and is infiltrated into the slot. The half space above the waveguide is filled with air. Throughout the paper we consider the chalcogenide glass As

_{wg}_{2}S

_{3}as the infiltrated/coating nonlinear material. It is one of the most common chalcogenide glasses. Its refractive index of approx 2.34 is still quite low compared to other chalcogenide glasses and the linear absorption nearly vanishes around a wavelength of 1500 nm - our spectral range of interest. To calculate the TE-like mode frequencies and field distributions of the waveguide modes, the commercial finite element solver COMSOL was used [18]. The dispersion of the refractive indices of the materials was taken into account in these simulations using the Sellmeier Eq. (11) with the material specific coefficients presented in Table 1.

Figure 1(b) shows the field distribution of such a waveguide, as can be seen in the inset the field is enhanced in the slot region by a factor of two. This relatively moderate enhancement, compared to what is achievable with air slotted waveguides, is caused by the smaller index contrast between the silicon core and the infiltrated material.

From the distribution of the electric and magnetic fields (**e**, **h**) the complex third order nonlinear parameter *γ* of the waveguide is calculated [19]:

The material specific Kerr coefficients *n*_{2} and nonlinear absorption coefficients *β* are in general anisotropic [20, 21]. Nevertheless as the direction of Si-photonic waveguides is in practice not restricted to a specific crystallographic direction, we assume average isotropic values for *n*_{2} and *β* which are listed in Table 1. *γ* measures the strength of the 3rd order nonlinear interactions in wave guiding structures corresponding to *χ*^{(3)} in bulk materials [21]. The FOM is calculated from *γ* with FOM = Re{*γ*}/4*π* Im{*γ*}.

Closely related to the nonlinear parameter *γ*, an effective mode area *A*_{eff} can be defined. It describes the confinement of the electric field, which is responsible for the nonlinear optical response. Following the suggestion of Koos et al. [5] and others [19, 20, 22] the corresponding effective mode Area *A*_{eff} can be defined as

In this definition of *A*_{eff}, |**e**|^{4} is integrated over the areas filled with As_{2}S_{3} (*D*_{As2S3}). Thus, *A*_{eff} becomes minimal if the concentration of the field is maximized inside the strongly nonlinear optical material As_{2}S_{3} leading to waveguide geometries with compact mode volumes.

To find the waveguide cross section with minimum *A*_{eff} the width and the height of the waveguides were varied while a fixed slot width of 100 nm was assumed. It has been reported that a smaller slot width gives rise to stronger field enhancement and thus a higher nonlinear optical FOM [5,11,17]. Nevertheless, there is certainly a technological limit of the minimum slot width of vertical slots as were investigated in this paper. Therefore, we limit our slot width to a rather conservative value of 100 nm to obtain structures, which can be easily realized with standard e-beam lithography and consecutively filled with As_{2}S_{3}.

#### 2.2. Results

In Fig. 2(a) the FOM of the waveguides and its dependence on the waveguide width and height are shown. The blank space in the lower left corner indicates the parameter values, where no guided modes could be found. For wide waveguides the FOM is quite low, decreasing below unity for waveguide width *w _{wg}* > 550 nm. For these wide waveguides a large part of the mode is propagating within the two silicon rails at both sides of the slot, so that the low FOM of silicon starts to govern the FOM of the whole waveguide. On the other hand for waveguide widths approaching 300 nm a large part of the mode propagates inside the slot or leaks out beyond the silicon rails into the adjacent As

_{2}S

_{3}slab regions. For these geometries the highest FOMs of about 3 to 4 are expected (the FOM of bulk Silicon and bulk As

_{2}S

_{3}differ by a factor of 100). Finally, for ultra-narrow waveguide widths below 200 nm the waveguide modes become unbound since the radiation can couple to propagating modes inside the neighboring As

_{2}S

_{3}slab regions. The modes within the infinitely extended As

_{2}S

_{3}slab regions left and right of the waveguide therefore determine the ”light cone” for the waveguide modes.

According to this analysis waveguide widths in the range of 250 – 300 nm with modes close to the light cone appear most favorable. However, a closer look at the field distributions for these waveguides already reveals considerable mode spreading that could lead to severe cross talk between neighboring waveguides. Furthermore, the operation of such narrow waveguides close to the light cone requires a highly accurate fabrication to avoid leakage radiation. Finally, for our dispersion engineering in order to work, we need waveguide modes whose dispersion curves are well below the light line. In conclusion we do not optimize our waveguide cross section towards a maximum value of the FOM, but try to maximize the field concentration in the slot region. This still results in FOMs between 1 to 2 as discussed in the following.

To find an optimum waveguide geometry with large field concentration inside the slot, the waveguide with minimal effective mode area is determined. Figure 2(b) shows *A*_{eff} for a varying waveguide width *w _{wg}* and height

*h*. There is a maximum of 1/

_{wg}*A*

_{eff}(corresponds to a minimum of

*A*

_{eff}) at

*w*= 420 nm and

_{wg}*h*= 210 nm. The mode field distribution of the geometry with minimized

_{wg}*A*

_{eff}is shown in Fig. 1(b). Although most of the mode field intensity is confined in the slot a significant part of the mode field is already propagating in silicon giving rise to two photon absorption. Therefore the overall FOM for this geometry has a moderate value of 1.3.

## 3. Manipulation of the group velocity dispersion

For efficient nonlinear frequency transformation processes, a large FOM is not sufficient. In addition, phase matching conditions have to be fulfilled to yield efficient nonlinear amplification and generation. Since the infiltrated nonlinear material (chalcogenide glass) is amorphous only third order nonlinear optical processes, which are based on *χ*^{(3)}, are of interest here. Next we will specifically investigate the phase matching conditions for the degenerate four wave mixing process and how these can be fulfilled for the mentioned infiltrated slot waveguides.

In the case of degenerate four wave mixing two waves at *ω _{p}* and

*ω*interact to form a new wave

_{s}*ω*. For an efficient frequency conversion the frequencies and phases of the three interacting waves have to be matched according to:

_{i}For simplicity we will consider only the linear dispersion in equation (4) and neglect the nonlinear phase shift due to the Kerr effect for the moment. The consequences of a more comprehensive treatment incorporating the nonlinear change of the refractive index is shortly discussed at the end.

Condition (3) can be recast as:

while*k*(

_{i}*ω*) and

_{i}*k*(

_{s}*ω*) in (4) can be developed into Taylor series around

_{s}*ω*.

_{p}Combining Eqs. (7) and (8) with Eq. (4) and disregarding higher order terms, leads to

This means that the group velocity dispersion (GVD), which is described by the term *d*^{2}*k*/*dω*^{2}, vanishes if the phase matching condition is fulfilled. Silicon as well as As_{2}S_{3} and standard waveguides show normal dispersion, i.e., the phase index is larger for shorter wavelengths. One way to compensate this behavior is to use the anomalous dispersion of silica in the wavelength region around 1500 nm. Waveguides are then designed in such way that for longer wavelength (when the wave leaks out into the surrounding material, i.e. silica) its anomalous dispersion compensates the waveguide dispersion [23]. Another way is to find an advanced waveguide design, which shows anomalous dispersion [7, 9, 24, 25]. Yet another way is to use a slow (*a* ≫ *λ*_{0}/*n*) periodic modulation of the effective mode index along the propagation direction, with a periodicity *a*. This adds a negative contribution of 2*πm*/*a* (*m* = 1, 2, 3,...) to the phase matching condition (4). If properly designed, this allows the fulfillment of Eq. (4) in many conditions [26].

We propose a concept, which uses a small scale periodic modulation of the refractive index with a periodicity of the cladding material in propagation direction *a* < *λ*_{0}/*n*. This refractive index modulations causes a 1-D photonic bandgap, which leads to additional band bending in its vicinity. This band bending yields an anomalous contribution to the GVD that can be used to achieve zero GVD. In practice, such a structure could be achieved by exploiting the photorefractive effect of As_{2}S_{3}. It is widely reported that the refractive index of chalcogenide glasses can be changed by illumination with above bandgap radiation. In the case of As_{2}S_{3} the refractive index can be altered up to 0.1 by exposure to visible or UV light [27]. Therefore, we again focus on this material in our model system to proof our concept.

#### 3.1. Numerical Model

The model used to calculate the GVD of a waveguide infiltrated with a material with alternating refractive indices *n*_{1} ≈ 2.3 and *n*_{2} = *n*_{1} + 0.05 is depicted in Fig. 3(a). The index *n*1 corresponds to that of unexposed As_{2}S_{3} calculated by Sellmeier eq. with parameters given in Table 1. The offset of 0.05 was arbitrarily chosen as half of the maximum refractive index change reported in [27]. Again the thickness of As_{2}S_{3}-layer is the same as the height of the waveguide and the upper half space above the waveguide is assumed to be air. Such a structure might be fabricated by periodic exposure of an already deposited As_{2}S_{3} film. The eigenfrequencys *ω*(*k*) of such a periodic system for given *k* = 2*π*/*λ*-vectors were calculated in a 3D geometry using periodic boundary conditions in the propagation direction *z*[28]. The finite element software COMSOL [18] allows to take the varying refractive indices *n*_{1} of the materials for different eigenfrequencies into account. The cross section of the waveguide geometry corresponds to the optimum condition with minimum *A*_{eff} as described above. For a one-dimensional periodic index modulation a band gap around *λ*_{0}/2*n* = *a* is expected [29]. For a design wavelength of *λ*_{0} = 1550 nm this leads to *a* ≈ 330 nm. To test the flexibility of our strategy we calculated the eigenfrequencies for different lattice constants *a*.

#### 3.2. Results

The result of such a eigenfrequency calculation for *a* = 380 nm is shown in Fig. 3(b). It shows the waveguide mode which is folded back, forming a bandgap at the Brillouin zone edge. Above the gap the guided modes of the second band extend up to the shaded area where the waveguide modes are able to couple to the As_{2}S_{3}-slab modes left and right of the waveguide and become lossy (light cone). To compensate the usual normal dispersion of the material and the waveguide mode the band bending of the second band around the upper band edge will be exploited. Only there the GVD vanishes.

From *ω*(*k*) we are able to calculate the GVD parameter as

The results of the calculations for different lattice constants *a* = 300 nm, 320 nm,...,400 nm are shown on Fig. 4(a). There a clear zero crossing is observed, which shifts to longer wavelengths for larger lattice constants. A |GVD| < 0.5 ps/(nm m) is achieved within a wavelength range of Δ*λ* ≈ 10 nm. This graph also represents the huge flexibility for tuning the GVD = 0 frequency to the desired spectral region. By simply changing the period of the index modulation the condition GVD = 0 can be shifted to nearly any wavelength within the near infra-red. The cross section of the waveguide can be left unchanged. In this way the method used here allows to choose the cross section of the waveguide in such a way to optimize parameters like effective mode area or FOM and then adjust the period of the index modulation to achieve the phase matching condition (GVD = 0) at the desired wavelength.

To estimate the useful bandwidth for degenerate four wave mixing around the GVD=0 frequency we apply the condition |Δ*βL*| < *π*/2 from [21]. With this the tolerable wave-number mismatch Δ*β* = 2*k _{p}*(

*ω*) −

_{p}*k*(

_{i}*ω*) −

_{i}*k*(

_{s}*ω*) depends on the length of the waveguide

_{i}*L*. Assuming waveguide lengths between 1 and 5 mm the corresponding bandwidths were determined from the original dispersion curves and are presented in Fig. 4(b). As can be seen the bandwidth is well above 40 nm in any case and decreases more slowly with increasing waveguide length allowing even longer waveguides for narrow band four wave mixing.

As it might be difficult to prepare a chalcogenide glass layer whose thickness exactly matches the slot waveguide height, we investigated the properties of structures with different thickness mismatch Δ*h _{wg}* where the As

_{2}S

_{3}layer extends by 20 nm to 100 nm above the waveguide. This is a realistic scenario when a fabricated silicon slot waveguide is infiltrated afterward with the chalcogenide glass and no further leveling or polishing is performed. The results of these investigations are shown in Fig. 4(c). The zero point of the GVD parameter shifts to longer wavelengths for thicker cladding layer. This is expected as the field of the mode extends a bit upward out of the slot and the mode ”sees” now still the relatively high refractive index of the added As

_{2}S

_{3}with Δ

*h*. As expected the impact of Δ

_{wg}*h*on the shift of the GVD = 0 frequency levels off for larger Δ

_{wg}*h*as the mode decays quickly above the slot. However, care has to be taken that the As

_{wg}_{2}S

_{3}-layer does not become too thick, as then the frequency of the As

_{2}S

_{3}-slab modes decreases considerably and the shaded area of the “radiative modes” in Fig. 3(b) extends close to the bandgap. Ultimately, the GVD = 0 frequency would be shifted into the leaky mode regime. This places a limit to the overall allowed As

_{2}S

_{3}layer thickness in the area between 500 nm to 1 micron depending on waveguide cross section parameters and contrast of index modulation.

Finally we revisit Eq. (4) where the impact of the nonlinear phase shift Δ*β _{NL}* = 2

*P*{

_{p}Re*γ*} induced by the Kerr effect was neglected. To estimate the influence of this phase shift we calculated the phase mismatch Δ

*β*= Δ

*β*− (2

_{NL}*k*(

_{p}*ω*) −

_{p}*k*(

_{i}*ω*− Δ

_{p}*ω*) −

*k*(

_{s}*ω*+ Δ

_{p}*ω*)) for Δ

*ω*= 2

*π*1.2 · 10

^{12}Hz ≙ Δ

*λ*≈ 10nm and

*Re*{

*γ*} = 90(Wm)

^{−1}at a pump power

*P*= 20dBm [21, 26]. As a result the wavelength for which Δ

_{p}*β*= 0 shifts by 2.5 nm towards the red from the original GVD = 0 wavelength. This demonstrates that the introduced band bending due to the photonic band gap is strong enough to compensate for linear and nonlinear dispersion characteristics. Even smaller index contrasts below Δ

*n*= 0.1 are sufficient to compensate for the material and waveguide dispersion. Furthermore the idea to use a periodic index modulation to achieve the phase matching for the degenerated four wave mixing process is not limited to photo-refractive materials. A periodic modulation of waveguide width or height or slot width will have a similar effect and will create a photonic bandgap with the required band bending as it was already reported for ”nanobeams” [30]. Our strategy clearly aims to fulfill the phase matching condition in the spectral range around the telecom wavelengths (1550 nm wavelength.) Since we use the strong band bending around a photonic band edge our bandwidth is limited to about Δ

*λ*= 50 nm. In this respect our work has a certain similarity to the work on flat band slow light reported by Li et al. [31]. However, if four wave mixing should be achieved over a much wider spectral range (e.g.(

*λ*−

_{i}*λ*) > 200 nm) the quasi phase matching strategy investigated by Driscoll et al. [26] is favored. With this approach basically any wave-vector mismatch can be overcome by adding a reciprocal lattice vector of a long period grating. However, this technique becomes ineffective for four wave mixing processes of narrow bandwidth as the resulting smaller wave-vector mismatches would require gratings with periods in the cm range. This would require undesirably large optical chip sizes.

_{p}## 4. Conclusion

A design route for Si/As_{2}S_{3} waveguides with optimized waveguide cross section and tailored group velocity dispersion is presented. This method is based on a refractive index modulation along the waveguide and allows to independently optimize the effective mode area *A*_{eff} and the GVD. An FOM of about 1.3 and GVD parameter < 0.5 ps/(nm m) was achieved for a wavelength range of Δ*λ* = 10 nm. The bandwidth for degenerate four wave mixing processes amounts to Δ*λ* ≈ 50 nm for waveguide lengths of a few mm.

## 5. Appendix

In this work, the Sellmeier Eq.

with*B*and

_{i}*C*as in Table 1 has been used to calculate the refractive indices of the materials.

_{i}## Acknowledgment

We gratefully acknowledge the funding by the
Federal Ministry of Education and Research (BMBF) within the Centre for Innovation Competence SiLi-nano^{®} project 03Z2HN12.

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