1. Introduction

High-performance guided-wave photonic devices implemented using high-throughput and scalable fabrication methods are desired in many applications from optical communications to biosensing [1,2]. The simplest waveguide structure is a slab waveguide, which consists of a 1D stack of low and high index materials but cannot confine light in 2D. Therefore, integrated photonics are usually implemented using photonic ridge waveguides, formed by a high refractive-index core surrounded in 2D by lower refractive index materials as the substrate and the cladding. In these waveguides, light is confined in the waveguide core in the transverse plane, perpendicular to the direction of propagation, and does not spread laterally as in a slab waveguide.

One of the key challenges of realizing scalable photonic devices is to devise repeatable, reliable, and machine-independent fabrication processes. An early approach used silicon etching to form a trench, which was then filled with dielectric material to take advantage of the highly smooth silicon sidewalls [3,4]. Today, integrated photonic waveguides are primarily realized using planar microfabrication processes, where the waveguide core is defined using lithography and reactive ion etching (RIE) to form a ridge structure. A major source of propagation loss in photonic waveguides is light scattering due to sidewall roughness arising from fabrication imperfections. For example, the performance of our recent Parylene photonic platform [5], in which photonic waveguides are realized in Parylene C on a PDMS substrate, was mainly limited by the etched sidewall roughness. To alleviate the issue of sidewall roughness when etching the outline of the Parylene C waveguide core, a two-step process was developed to first etch the outline of the waveguide core and then deposit a thin layer of the same polymer to smoothen the rough sidewalls [6]. Using this technique, we could reduce the propagation loss from more than 30 dB/cm to less than 5 dB/cm.

Addressing the issue of sidewall roughness due to etching would be a boon to high-performance photonics, and research on this front continues to produce new results [7–9]. To remove the need for etching the outline of the ridge waveguide into the core layer, etchless fabrication of silicon photonic waveguides has been previously introduced [10–12], whereby the waveguide core was isolated from the rest of the silicon-on-insulator thin film using an oxidation process. In this method, the waveguide architecture is still a ridge but surrounded by SiO₂ from the sides and the bottom. Similarly, an innovative etchless fabrication process for polydimethylsiloxane (PDMS) polymer waveguides uses ultraviolet exposure to locally increase the refractive index of the polymer to define the waveguide core region [13]. Leveraging the specific material properties, these methods have enabled implementation of waveguides that provide 2D confinement of light without the need to etch the waveguide outline. Previous techniques imposed a 2D refractive index or thickness variation on the core layer after deposition using an RIE or etchless chemical process. Here, we demonstrate an etchless waveguide architecture that is formed by the geometry of the deposition itself, without any subsequent processing steps. This is a critical innovation since it removes the need to optimize the etching process for the core material to maintain sidewall quality. The point here is not merely simplifying the fabrication process, but instead to offer a design that enables the confinement of optical modes in 2D without etching the waveguide core, completely removing concerns of the sidewall quality.

The idea of exploiting material deposition characteristics combined with substrate topography to form pedestal optical waveguides with confined modes has been explored before. In these designs, the core is made thicker than the surrounding region either through exploiting the mechanics of growth in convex corners [14] or via undercutting [15]. In our design, we are modeling a core film thickness that remains the same across the width of the structure and the mechanism of optical confinement is based on the interplay of modes at the corner bends.

Previous work has explored the stratified trench waveguide geometry, where conformal deposition of the core material in a triangular or trapezoidal-shaped silicon mold forms the waveguide core [16–20], so named by the shape of the trench cross-section. These previous investigations have primarily focused on a triangular V-shaped trench composed of traditional high-index contrast materials such as SiN and SiO­₂. Our work extends this previous work to discuss the generalization to low-index contrast polymer materials, and visible-range operation, which is relevant for biology applications. Additionally, we focus exclusively on a wide-trapezoidal trench, with 45-degree sidewall angles and a rectangular trench with 90-degree sidewall angles, which have implications for microfabrication and system integration, which will be discussed in this paper.

Here, we show a waveguide design where a stratified material stack of continuous layers can confine an optical mode in 2D without any change in the waveguide core thickness or refractive index along its width. This is a fundamentally new waveguide design and a core novelty of the presented work. As opposed to conventional 2D photonic waveguides such as ridge waveguides or optical fibers that confine light based on total internal reflection at the boundary of the waveguide core and cladding, our novel waveguide architecture confines light along the lateral direction because of its trench geometrical structure (Fig. 1(a)). Light is confined at the bottom horizontal segment of this stratified trench waveguide in 2D with low light leakage because the mode does not efficiently couple to the sidewall modes (Fig. 1(b)). In all the analyses in this paper, we assume the thickness of the waveguide substrate and cladding to be 1 µm, and only vary the thickness of the waveguide core. Four key parameters affect the performance of the trapezoidal trench waveguide design: W, the width of the trench, D, the depth of the trench, θ, the angle of the sidewall, and T, the thickness of the thin-film waveguide core layer (Fig. 1(c)). We assume that the waveguide core is made of Parylene C with a refractive index of n = 1.639, and the cladding and substrate are made of PDMS with n = 1.4. We show that this design provides etchless modal confinement (Fig. 1(b)) even for a low refractive index contrast between the core and cladding materials (Δn = 0.239) with a low propagation loss of 0.145 dB/cm. All simulations are performed at a wavelength λ = 470 nm.

 

Fig. 1. a) 3D model of the trapezoidal stratified trench waveguide cross-section b) Fundamental mode profile showing a confined optical mode at the bottom of the trench. The outline illustrates the shape of the waveguide core. Axis scale is in µm. c) Schematic illustration of the mold-based waveguide cross-section and design parameters, depth (D), width (W), sidewall angle (θ), core thickness (T), and cladding thickness (T_Clad) in trapezoidal stratified trench waveguide, where mode confinement is achieved via introducing corners into the waveguide. In this study, we fix cladding thickness to 1 µm for simplicity.

Download Full Size | PDF

Although we do not show experimental results in this paper, from a practical point of view, this waveguide design simplifies the fabrication process and sidesteps sidewall roughness from etching. This design can be implemented in different material platforms, especially using polymers, for which innovative fabrication techniques are needed to realize flexible or biocompatible integrated photonic devices [21–23]. The most common method of patterning polymers is to use O₂ plasma etching to break the polymer chain through oxidation and physical removal of reaction products [24,25]. The O₂ plasma etching process usually results in very rough sidewalls, [6] which contribute to a large propagation loss. The waveguide architecture presented in this paper can be realized based on a completely etchless process in which conformal layers can be deposited into a trench mold, which is compatible with Parylene photonics and other materials that can be deposited conformally. Therefore, implementing this waveguide does not require etching the waveguide outline and obviates the need to mitigate the issue of sidewall roughness.

2. Results

2.1 Principle of operation: confined modes

A waveguide structure supports one or more confined optical modes that transmit light with minimal loss from the input facet to the output facet. These optical modes can be obtained by solving the Wave Equation:

2.2 Mechanisms of 2D light confinement

To explain the confinement mechanism, we consider a 90-degree rectangular stratified trench waveguide, shown in Fig. 2(a). The 90-degree rectangular stratified trench waveguide can be approximated as the combination of multiple horizontal and vertical segments of slab waveguides. Although the waveguide thickness does not change, the bends in the waveguide structure break up the single slab mode. The cross-section is structured in distinct segments separated by the corners (i.e., the top, bottom, and sides of the channel). The confined guided optical modes of the stratified trench waveguide are more complicated than a simple combination of modes of its individual segments, but here we present an intuitive description of the mode structure based on the coupling of optical modes between the individual segments.

 

Fig. 2. a) Schematic cross-section of the 90-degree rectangular stratified trench waveguide structure showing how segments can be approximated as slab waveguides. b) Juxtaposition of the segment A and segment C slab equivalents, showing the electric field decay outside of the core. c) Juxtaposition of the segment A and segment B slab equivalents showing the perpendicular polarizations of the TE₀ modes. Inset images show the log-scale slab mode E-field intensity profile.

Download Full Size | PDF

Considering each segment of the waveguide independently, the optical mode of each region can be approximated as a slab waveguide mode. It should be noted that the modal structure of the horizontal segment at the bottom of the trench (segment A) and the horizontal segment on top (segment C) are similar. With identical polarizations and propagation constants, optical power could potentially couple between these two segments. The coupling of light from the bottom segment to the top horizontal segment would compromise the 2D confinement and result in the leakage of light to the sides. Here, we show that with the proper design of the waveguide geometry, the vertical sides of the trench can minimize the coupling between the two horizontal sections, thus contributing to a 2D confinement of light in the bottom horizontal segment.

Using a perturbation analysis method, the coupling between any two modes of the waveguide segments, e.g., modes m and n can be obtained as,

In these expressions, $\Delta n^{2}_{n}(x,y)$ is the refractive index perturbation caused by the top segment waveguide to the bottom segment waveguide. Also, ${\kappa _{mm}}$ and ${\kappa _{nn}}$ are small corrections to the propagation constants of each mode due to the dielectric perturbation of the other waveguide.

The increased power transfer between different segments of the waveguide structure as a function of geometrical parameters can be evaluated by either an increase in $\kappa $ or decrease in $\delta $. For example, the field strength of the confined slab fundamental mode of segment A at the bottom of the trench, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{0A}$, exponentially decays outside of the core:

Therefore, the coupling between the top and bottom segments decays by increasing the depth of the trench, D (Fig. 2(b)), because

The coupling between the horizontal slab mode at the bottom of the trench and adjacent sidewall slab modes is another route for power leakage. Consider the TE₀ mode of the horizontal slab waveguide structure in segment A, which has only an E_x component. The equivalent TE₀ mode in the 90-degree sidewall, segment B only has an E_y component (Fig. 2(c)), due to the rotated frame of reference:

Therefore, these two modes are orthogonal, i.e., $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} {_{0A}} \bot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} {_{0B}}\; \; $, which makes the coupling coefficient zero, since $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} {_{0A}}\cdot\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{0B}^* = 0$.

In the full 2D structure, unlike the slab mode equivalent approximation, pure TE modes do not have a perfect field polarization. For example, the TE-like mode of the bottom segment A, has a small ${E_y}$ component. Therefore, in the 2D case, we would expect a small coupling between segments A and B.

Another potential optical leakage path is via coupling between the horizontal slab TE₀ mode at the bottom of the trench and the TM₀ mode of the vertical sidewall. These modes have the same polarization, due to the 90-degree sidewall orientation. This creates a leakage pathway through the coupling of the optical mode of the bottom of the trench to the vertical sidewall and eventually to the top segment. This leakage path is primarily limited by the phase mismatch $2\delta $. In a 200 nm-thick slab waveguide, the phase mismatch between the TE₀ and TM₀ modes is significant: 0.56 µm^-1. However, as the thickness of the core becomes larger, the propagation constants converge as the modes become more confined in the core, reducing the phase mismatch. Thus, our model predicts increased coupling efficiency, leading to power leakage from the bottom of the trench as the thickness of the waveguide increases. This behavior is contrary to a more traditional ridge waveguide structure, where increased core thickness typically increases the modal confinement. In section 2.4, we will demonstrate that the simplified model prediction of lower losses in lower thickness waveguides is corroborated by finite-difference eigenmode (FDE) simulation of the full 2D structure. Intuitively, this leakage path will be small for small waveguide core thicknesses (i.e., subwavelength) due to the large phase mismatch between the TE₀ and TM₀ modes, as well as the low field overlap between the vertical and horizontal waveguide segments.

The fundamental mode structure of the 200 nm-thick 90-degree stratified trench waveguide computed using FDE simulation is shown in Fig. 3. In the model system, the only predicted route for loss is the coupling between the TE₀ mode of the bottom segment to the TM₀ mode of the sidewall segment, which is weak because of the phase mismatch when the thickness of the waveguide is small enough. We should note that in the full 2D waveguide structure, the modes of the structure are not pure TE or TM and we expect to have hybrid modes. Nevertheless, the rigorous simulations of the full 2D structure corroborate the existence of confined modes with very low loss for the structures presented in the following sections.

 

Fig. 3. a) Fundamental mode of a 90-degree stratified trench waveguide. b) Schematic cross-section of the waveguide structure showing the layers formed by conformal deposition into a rectangular mold. Due to the conformal deposition, the top corners are rounded. c) |E_x²| component of the fundamental mode. The field is primarily contained in the horizontal segments. c) ) |E_y²| component of the fundamental mode. The field is primarily contained in the vertical segments. Axis scale is in µm.

Download Full Size | PDF

The mechanism of the light confinement that was explained for the stratified trench waveguide with vertical sidewalls can be extended to other waveguides with angled sidewalls. The only difference is that when the sidewall segment B is not orthogonal to the bottom segment A, then the TE₀ optical mode of the bottom segment A would not be orthogonal to the TE₀ mode of the slanted sidewall segment B, thus creating a leakage pathway, which can be mitigated by minimizing the field overlap. Using this intuition, a waveguide structure can be designed which supports guided modes confined to the bottom of the trench without any core thickness variation along the width.

2.3 Low-loss guided modes in trapezoidal stratified trench waveguides

Here, we demonstrate low-loss (0.145 dB/cm) guided modes in a trapezoidal stratified trench waveguide structure with 45-degree trench sidewalls (Fig. 4). This practical design can be realized, for example, by sequential conformal deposition of polymer layers on a Si mold with a trapezoidal trench geometry. We choose an exemplar geometry formed by a 4 µm trench width and 4 µm trench depth, and 45-degree sidewalls. We then model conformal depositions of 1 µm PDMS substrate, 200 nm Parylene C core, and 1 µm PDMS cladding. These dimensions are chosen as a result of the parameter sweeps discussed in sections 2.4 and 2.5. FDE analysis shows the existence of several guided modes at the bottom segment of the stratified trench waveguide.

 

Fig. 4. a) Fundamental mode of a 45-degree trapezoidal stratified trench waveguide. b) Line plot of electric field intensity across the mode cross section with a schematic illustration of the waveguide geometry c) |E_x²| component of the fundamental mode, which is allowed in both the horizontal slab region and the 45-degree sidewalls. d)) |E_y²| component of the fundamental mode is weaker in the horizontal slab regions. Axis scale is in µm.

Download Full Size | PDF

The fundamental confined mode of the optical waveguide is shown in Fig. 4(a), which has an effective index of n_eff = 1.52. This mode is predominantly guided in the horizontal center section of the waveguide at the bottom of the trench (Fig. 4(b)). The magnitudes of the horizontal electric field component (|E_x²|) and the vertical electric field (|E_x²|) are shown in Fig. 4(c) and (d). Most of the guided mode power is concentrated in the E_x component of the electric field, which is mainly confined in the horizontal segment of the waveguide at the bottom of the trench. There is a rather weak E_y component, which is mainly confined in the diagonal sections of the waveguide core. Here, the materials are modeled to be lossless, so the only loss mechanism is due to optical power leakage from the central waveguide trench into the sidewalls and out of the simulation domain. Moreover, a perfectly matched-layer (PML) boundary condition [27] is used around the outer regions of the simulation domain to properly model the semi-infinite extension of the top horizontal sections of the waveguide. Even in fully confined ridge waveguides, the finite simulation domain can result in the measurement a nonphysical loss as an artifact of evanescent field overlap with the PML. The leakage loss reported here is not simply an artifact of the simulation domain, it was robust to the PML placement. Varying the simulation domain from 14-28 µm wide resulted in a change in the computed loss by ±0.15% for a trench waveguide with 4 µm trench width, 4 µm trench depth, and 200 nm Parylene C, so the resulting loss values are highly converged. Thus, the calculated optical loss is a direct indication of the amount of modal confinement and power leakage in the waveguide structure. The amount of power propagating outside of the simulated region (20 µm wide) is small, i.e., 0.145 dB/cm propagation loss. Unlike a slab waveguide, the stratified trench waveguide does not support pure TE or TM modes. We classify the fundamental mode as “TE-like” based on the dominant polarization.

2.4 Effect of waveguide thickness

The modal structure and propagation loss of the stratified trench waveguide depends on the thickness of the core layer. Using the same size trench mold geometry (4 µm deep and 4 µm wide), we modeled the conformal deposition of a 1 µm PDMS substrate and a range of core thicknesses between 200 nm and 800 nm. The relationship between the core thickness, the propagation loss, and the effective index of the fundamental mode is shown in Fig. 5(a).

 

Fig. 5. a) Fundamental TE mode loss and effective index vs core thickness (T). Inset images show the mode field profile. Scale bar indicates 4 µm. b) |E_x²| and |E_y²| components of 0.8 µm-thick waveguide structure, showing increased field overlap in the horizontal and diagonal segments compared to the 0.2 4 µm-thick counterparts shown in Fig. 4 c,d. Axis scale is in µm.

Download Full Size | PDF

The relationship between confinement and core thickness for our waveguide structure is the opposite of traditional ridge or rib waveguides. In a traditional waveguide, a thicker core results in a higher level of confinement for the fundamental mode, i.e., less exposure to the cladding material and higher effective index. Instead, as the waveguide core thickness is increased in the trapezoidal stratified trench waveguides, the modal confinement is decreased, while at the same time, the effective index of the fundamental mode is increased, indicating that more of the modal field is propagating in the high-index slab region outside of the trench. As discussed in section 2.2, by increasing the waveguide core thickness, the confined optical field in the bottom horizontal segment of the waveguide couples to the sidewall modes and leaks into the slab region outside of the trench (Fig. 5(b)), thus increasing the propagation loss.

The fundamental mode losses increase drastically with increasing core thickness. At a thickness of 0.8 µm, the losses of the stratified trench waveguides due to leaky modes is 4.23 dB/cm, which is comparable to our Parylene C ridge waveguides with optimized smoothing techniques [6], indicating that this is the limit of the regime where stratified trench waveguides confer a performance advantage for the Parylene C/PDMS material platform.

This simulation suggests that the thickness of the waveguide core should be minimized, and that the stratified trench waveguide conformal coating confinement strategy is not tenable beyond a certain thickness. For our exemplar waveguide structure, we choose a 200 nm thickness as a good tradeoff between propagation loss and feasibility of fabrication, i.e., thin layers become increasingly difficult to repeatably fabricate and attain high quality films.

2.5 Effect of trench depth

The modal structure and propagation loss of the stratified trench waveguide also depends on the depth of the trench. To understand the impact of the trench mold geometry on the waveguide performance, we modeled the conformal deposition of identical layers (i.e., 1 µm PDMS substrate and 200 nm Parylene C core) into various trench molds with a width of 4 µm and varied depth (1 to 5 µm). The relationship between the trench depth, the propagation loss, and the effective index of the fundamental mode is shown in Fig. 6.

 

Fig. 6. Fundamental TE mode loss and effective index vs trench depth (D). Inset images show the mode field profile. Scale bar indicates 4 µm.

Download Full Size | PDF

Increasing the depth of the trench significantly decreases the fundamental mode loss. At a depth of 4 µm, the losses of the stratified trench waveguides due to leaky modes is 0.145 dB/cm, which decreased further to 0.021 dB/cm at a depth of 5 µm. By contrast, the effective index of the fundamental TE mode does not significantly vary across the range of trench depths and is primarily determined by the core thickness.

These results show that the deeper trench geometries yield better modal confinement. Since the depth and width of the sidewall are related by the sidewall angle, deeper trenches inherently limit the routing density of waveguides. Therefore, there is a fundamental tradeoff between density and propagation loss. For our exemplar waveguide structure, we choose a 4 µm depth as a good tradeoff between propagation loss and the trench depth. Ultimately, the choice of geometry will depend on the requirements of the specific application.

2.6 Waveguide array crosstalk

In a photonic platform, it is desirable to integrate several waveguides in an array with minimal crosstalk between them. While the optical mode is strongly confined in the bottom segment of the trapezoidal stratified trench waveguides, still a small fraction of the optical mode leaks to the top slab waveguide (Fig. 5(b)). Therefore, if multiple of such waveguides are densely integrated next to each other, there is a chance that light can be coupled between adjacent waveguides due to crosstalk. To quantify this crosstalk effect, we have simulated two adjacent optical waveguides as shown in Fig. 7(a). In this simulation, light is propagating in the fundamental mode of the “input waveguide”, and the “probe waveguide” is monitored for the coupled optical power. The waveguide trenches are placed beside each other in our exemplar configuration. Each waveguide trench is 4 µm deep, which corresponds to a 4 µm wide sidewall due to the 45-degree angle, and 4 µm wide, resulting in a total waveguide pitch of 12 µm. At this spacing, we observe a negligible crosstalk of < -60 dB over a coupling length of 5 cm (Fig. 7(b)).

 

Fig. 7. a) Schematic image of parallel waveguide cross-section. b) Crosstalk simulation showing power coupled into the adjacent waveguide (Top view)

Download Full Size | PDF

3. Discussion

In this work, we have discussed the mechanism of effective 2D light confinement in trench optical waveguides without any change in the waveguide thickness along the cross-section. Three possible leakage mechanisms that contribute to the propagation loss were discussed to demonstrate that by choosing a deep enough trench and maximizing the phase mismatch between the modes of the horizontal segment and the sidewalls, light can be confined in 2D in the bottom segment of the optical waveguide. We justified the counter-intuitive effect of increasing the waveguide thickness decreasing the light confinement due to the increase in mode coupling between the horizontally polarized modes of the bottom segment and the sidewalls of the trench optical waveguide.

As predicted by our model system, light is mainly confined in the bottom trench of the waveguide, and only a small amount of leakage into the sidewalls and upper regions of the waveguide structure is observed. Therefore, the conditions to ensure effective confinement of light in the bottom segment A are (i) large enough depth of the trench to prevent direct coupling of the optical mode in the bottom segment A to the top segment C; and (ii) small enough core thickness to prevent leakage through the coupling between the TE₀ mode of the bottom segment A and the TM₀ mode of the sidewall segment B.

This design promises to simplify the fabrication process for realizing integrated photonic structures by removing the need for an etch step to define the waveguide outline. The trapezoidal trench geometry with 45-degree sidewall angles enables co-fabrication of the waveguides with micromirrors that provide out-of-plane input and output coupling, thus enabling dense system-level integration. Furthermore, the design is generalizable to any material platform which can be deposited in a mold via a conformal process. Without the need to optimize etch processes for new materials, the stratified trench waveguide design promises to open high-performance photonic capabilities for new exotic material systems.

Although this method would present a greatly simplified fabrication process that does not require etching to define the waveguide core, the optical performance would still be dependent on the deposition quality. Deposition of Parylene C waveguide core material using a chemical vapor deposition (CVD) method is highly conformal and uniform layers can be achieved across 4” wafers with <10% thickness uniformity [28]. We have recently shown that thickness variations due to spin-coating across topography can be mitigated by diluting PDMS in hexane and optimizing the spinning speed and acceleration [5]. There are other techniques such as spray-coating PDMS that can potentially result in more uniform and conformal films [29]. Ultimately, the wide range of polymer materials that can be deposited via CVD [30] could be incorporated into this architecture. Experimentally, deposition over a trench topography may not result in the uniform deposition modeled here, including thinner deposition on trench sidewalls, depending on the technique used [31]. The unique contribution of this work is to demonstrate the feasibility of confining modes in waveguides without thickness variation of the core. In future work, thickness variation in the sidewall region can be explored to achieve even stronger confinement through a combination of the etchless and traditional rib waveguide confinement.

In previous work, we demonstrated that Parylene C and PDMS polymers can be used to realize Parylene photonic waveguides with low propagation losses of less than 5 dB/cm [5]. These ridge waveguides were realized by etching the outline of the Parylene C polymer waveguide core on a PDMS substrate. The primary loss mechanism in these waveguides was determined to be scattering due to sidewall roughness, and the low losses were only achieved through a highly optimized post-fabrication smoothing process [6]. Here, we show that these previous limitations can be exceeded by polymer deposition in a carefully designed Si wafer mold. We demonstrate that a dense array of such etchless waveguides (12 µm pitch) can be designed with low losses (0.145 dB/cm) to exceed the state of the art while promising to simplify the fabrication process.

Although we did not demonstrate experimental results in this paper, the 45-degree trench geometry was chosen, in part, because it is feasible to fabricate a master mold via wet anisotropic etching of silicon using potassium hydroxide (KOH). This fabrication method provides several advantages to traditional masking and etching of the waveguide core outline. The wet etching process is low-cost, high-throughput, and robust [32]. Additionally, with the controlled addition of surfactants, the process is capable of producing highly smooth mirror-like surfaces along the Si crystal planes [33,34].

The <110> Si crystal plane can be used to produce 45-degree mold sidewalls, which can be used as micromirrors for out-of-plane input/output coupling from the ends of the waveguide [5,34]. Compact micromirrors have been demonstrated for efficient broadband input/output coupling with robust fabrication tolerances compared to more traditional grating couplers [35]. Since the mirror surfaces would be formed by the mold itself, the mirrors and waveguides will be self-aligned. In addition to a simplified fabrication process, the trench mold design promises to facilitate dense system-level integration and packaging of light sources and photodetectors with the waveguides.

Our simulation analysis corroborates the predictions of the intuitive models and offers a rigorous method to explore the design space for the trench photonic waveguides with various trench angles and dimensions. In this paper, we demonstrated confined optical modes in 90-degree stratified trench waveguides to explain the mechanisms of 2D light confinement without changing the thickness or refractive index of the thin film that forms the waveguide core. We also discussed the simulation results for a 45-degree stratified trench waveguide that can be implemented practically by depositing conformal layers of polymer thin films onto a silicon mold to realize an etchless integrated photonic waveguide structure.

4. Materials and methods

4.1 Geometry modeling

Modeling of conformal deposition was done using OpenSCAD 2019.05, based on CGAL 5.0. First, a polyhedron is constructed using the parameters of the silicon mold, i.e., the trench width, depth, and sidewall angle. We assume that the silicon crystal planes are perfectly flat. We then model conformal definition using the computational geometry technique of the Minkowski sum [36]. We model the conformal deposition of a layer of thickness T by taking the Minkowski sum of the substrate with a sphere of radius T, then the Boolean difference is taken between the resulting entity and the substrate to yield only the conformal layer. This method is repeated for each layer of the polymer waveguide on top of the silicon mold: waveguide substrate, waveguide core, and waveguide cladding.

4.2 Modal simulation

The waveguide modal structure was simulated using commercial software (Lumerical MODE Solutions 2019b, Lumerical Inc., USA). A Finite-Difference Eigenmode (FDE) simulation with a 5 nm mesh step size was used along with a perfectly matched layer (PML) boundary. The number of PML layers was increased until the loss values of the structure showed convergence. This occurred at n = 1024 layers with a thickness of 5 nm. A region of 20 µm × 12 µm was simulated around the waveguide core. Simulations were performed at a wavelength of λ = 470 nm. This wavelength is chosen since it is widely used in biology experiments to excite fluorophores and opsins [37,38].

4.3 Propagation loss

Waveguide core and cladding materials are assumed to have a real refractive index, i.e, to have zero absorption coefficient. Using the PML boundary condition, the waveguide power can extend beyond the simulation boundary. Any power that extends outside of the simulation region is counted as loss. Thus, slab modes in the waveguide show very high loss, whereas modes that are confined to the waveguide core do not.

4.4 TE fraction

Modes are classified as TE-like or TM-like based on the “TE Fraction,”

4.5 Crosstalk simulation

The crosstalk simulation is performed using a bidirectional eigenmode expansion (EME) solver (Lumerical MODE Solutions 2019b, Lumerical Inc., USA). Two parallel waveguides are placed with adjacent trenches (12 µm spacing from core to core = 2D + W). These waveguides are simulated over a propagation length of 5 cm. Input optical power in the “input waveguide” is normalized to 1 and the proportion of optical power in the adjacent “probe waveguide” is measured. Over the 5 cm propagation length, < 0.001% of the power is coupled from the first waveguide to the adjacent one. Therefore, the crosstalk due to the coupling of modes is negligible.