Single-mode optical waveguides on native high-refractive-index substrates

Photonic integrated circuits (PICs) are rapidly being developed for high-refractive index materials that allow for tight optical confinement, small on-chip bend radii, and strong light-matter interactions. For example, high-performance PICs in both silicon¹ and InP^2,3 platforms are playing an increasingly important role in data applications with the potential to enable exascale computing⁴ and on-chip core-to-core optical communication.⁵ Similarly, wide-band gap semiconductors, such as diamond^6–13 and SiC,¹⁴ have emerged as promising materials for a plethora of new PIC applications. Among these are non-linear optics^12–14 and integrated quantum information processing,^8–11,15 which is enabled by the presence of spin defects with desirable quantum properties.^16,17

Common to all of these applications is a need for low-propagation-loss single-mode waveguides that can be fabricated on a high-refractive-index substrate in a scalable fashion. While a high refractive index is beneficial for optical design, it also requires a buried lower-refractive-index layer and the transfer or growth of thin films of high-index material,^{1–3,6–8,10,12–14,18} free-standing structures,^9,11,19 or pedestals²⁰ to minimize optical power leakage from the waveguide into the substrate. These approaches limit the device robustness, uniformity, and scalability required for the development of dense PICs on wide-bandgap semiconductors.

Even on mature PIC platforms, optical confinement presents significant technological challenges. In silicon photonics, the buried-oxide-layer thickness required for optical confinement is much larger than the optimum for VLSI electronics, making co-integration difficult.^5,18,19 For InP-based PICs, optical confinement is limited by the low index contrast between InP and InGaAsP.^2,3

Here, we propose a new type of waveguide optimized for high-index substrates that utilizes stacked dielectric layers to confine light in the top of a fin of high-index material. An example of a SiO₂/Si₃N₄ stack on a diamond fin/substrate at a wavelength of λ = 637 nm is shown in Fig. 1. Although the refractive index of both the buffer and confinement layers (n_L = n_SiO2 ≈ 1.45 and n_H = n_Si3N4 ≈ 2.0, respectively) is lower than that of the fin and substrate (n_f = n_diamond ≈ 2.4), the proposed design achieves confinement by engineering the effective index, resulting in an optical mode confined within the high-index material (diamond in the case of Fig. 1). This waveguide mode is a localized eigenmode that can propagate without leaking power into the underlying substrate, while the waveguide itself can be fabricated using conventional top-down lithography, etching, dielectric deposition, and planarization techniques. Though optical confinement has been achieved by effective index variations in many contexts, including high-Q cavities formed by photonic crystal waveguides²¹ and photonic crystal fibers,²² our application of this approach to vertically stacked slab waveguides obviates the need for a buried low-index layer, providing a pathway towards large-area, scalable PICs on native substrates.

Figures 2(a) and 2(b) illustrate an intuitive physical picture that accounts for optical confinement in the fin waveguide geometry using the effective index method,²³ the results of which are confirmed using a fully vectorial finite difference method.²⁴ By treating the two-dimensional cross section of the $z ˆ$-invariant waveguide dielectric topology in Fig. 2(a) as two stacked slab waveguides with horizontal (⁠$x ˆ$⁠) confinement, a slab waveguide with vertical (⁠$y ˆ$⁠) confinement can be formed from the slab-waveguide effective indices in $x ˆ$⁠. In Fig. 2(b), the two slab waveguides comprised of n_H/n_f/n_H (Slab 1) and n_L/n_f/n_L (Slab 2) from Fig. 2(a) are replaced by homogeneous layers in $x ˆ$ with the effective indices of the lowest-order supported modes $n eff, 1 x$ and $n eff, 2 x$⁠, respectively, where the effective index, $n eff, i x = β i x / k 0$⁠, is equal to the slab waveguide propagation constant in $z ˆ$⁠, $β i x$⁠, of Slab i divided by the free space wavenumber, k₀ = 2π/λ. The dispersion curves of the two slabs as a function of the normalized waveguide width, w/λ, are shown in Fig. 2(c) with example refractive index values of n_f = 2.5, n_H = 2.0, and n_L = 1.5.

This treatment is a key to understanding the nature of confinement in the fin structure: a fin mode exists when the effective index of the two-dimensionally confined structure, n_eff, satisfies the condition: $n eff, 1 x > n eff >max { n eff, 2 x , n H }$⁠, as indicated by the green and blue shaded regions in Fig. 2(c). When this condition is not met, the confined modes are degenerate with a continuum of radiation modes and become leaky, as indicated by the gray hatched region in Fig. 2(c). In essence, vertical confinement of the fin waveguide mode is achieved by total internal reflection at the interface between slab waveguides rather than the interface between homogeneous media. As a consequence of the effective index confinement in the fin structure, only a single mode in $x ˆ$ is supported for the refractive index values chosen in Fig. 2(c). Higher-order modes can be confined for a different choice of material indices, but only if higher-order modes of Slab 1 are contained in the blue or green shaded region of Fig. 2(c), i.e., they must have an effective index larger than the lowest-order mode of Slab 2.

The supported modes of the two-dimensionally confined structure are found by solving for the modes of the $y ˆ$-confined slab waveguide in Fig. 2(b). The resulting fin mode dispersion as a function of normalized waveguide height, h/λ [Fig. 2(d)], has two distinct regions that depend on the fin width, w,

The boundary between these two regions occurs when $n eff, 2 x = n H$ at a width that we label w = w_symm, which is indicated by a vertical red line in Fig. 2(c) and the red dispersion curve (w_symm/λ = 0.25) in Fig. 2(d). For an asymmetric slab waveguide, the higher-refractive-index cladding determines both the cutoff condition and the effective mode width. In Region I, the properties of the asymmetric waveguide are determined by n_H, while in Region II, this role is taken by $n eff, 2 x$⁠. The change in cutoff condition between the two regions causes the inflection point in the cutoff height [boundary for the multimode region in Fig. 2(d)] at w_symm. While the fin waveguide can only be single or few mode in width, it can be multimode in height, as indicated in Fig. 2(d).

The dispersion curves provide the allowable geometry and wavelength at which confined modes are supported. Within these constraints, the mode area, A_eff, and confinement factor, Γ, which quantifies the overlap between the optical mode and the guiding material, provide useful design metrics for maximizing light-matter interactions in fin waveguides.²⁵ In Fig 3(a), we calculate A_eff for a series of w/λ and plot the minimum, A_min, along with the corresponding height, h_min/λ. Mode intensity profiles for three values of w/λ are also shown in Fig. 3(a), and the corresponding Γ at A_min is plotted in Fig. 3(b).

In Region I most of the field penetration occurs in the confinement layer, relaxing the requirements on the buffer layer thickness for low leakage at the expense of reduced Γ. Conversely, in Region II A_min increases with w/λ, Γ approaches unity, and the mode extends within the fin into the buffer layer. Waveguides designed in this region may be desirable for high-power applications. The tightest confinement (smallest A_min) occurs at w_symm, which also corresponds to the maximum group index, N_g = c/v_g, where c is the vacuum speed of light and v_g is the modal group velocity, making w = w_symm an ideal design criterion for maximizing light-matter interactions.²⁵ The waveguide properties in Figs. 2 and 3 have been calculated for the lowest-order mode with the dominant electric field component along $x ˆ$⁠, commonly referred to as the quasi-TE mode. Discussion of the lowest order quasi-TM mode (dominant electric field component along $y ˆ$⁠), higher order modes, and further details of our semi-analytical and numerical calculations are provided in the supplementary material.²⁵ 

To illustrate the potential of the fin waveguide, we explore geometries in two important material platforms for PICs: diamond and silicon. The diamond waveguide is designed with an SiO₂ buffer layer, a conformal 200 nm-thick Si₃N₄ confinement layer, and SiO₂ overcladding for single-mode operation at λ = 637 nm. The operating wavelength corresponds to the nitrogen-vacancy center zero phonon line,⁸ which is used to achieve coherent spin-light interactions²⁶ and distributed entanglement²⁷ between diamond spins. We design the waveguide for the tightest confinement with w = w_symm as discussed above. The waveguide dispersion and group index are shown in Fig. 4(a), along with the waveguide dimensions and calculated mode intensity profile at λ = 637 nm. For a buffer layer thickness exceeding 1.0 μm (total fin height 1.35 μm, corresponding to an aspect ratio >6.75:1) the calculated propagation loss due to substrate leakage is <0.15 dB/cm, which is small enough that scattering due to fabrication imperfections in a realistic device would be expected to dominate (see below). The bending loss for a radius of 10 μm is determined to be <0.03 dB per 90^∘ bend with a buffer layer thickness of 1.0 μm, corresponding to an unloaded Q exceeding 60 000 for a 20 μm-diameter ring resonator.

Similarly, the silicon waveguide depicted in Fig. 4(b) is designed for minimum mode area at λ = 1.55 μm for telecommunications applications. With a buffer layer thickness exceeding 1.5 μm (fin aspect ratio >9.75:1), the propagation loss due to substrate leakage is calculated to be <0.1 dB/cm. The bending loss for a radius of 10 μm is determined to be <0.1 dB per 90^∘ bend with a buffer layer of 1.5 μm, corresponding to an unloaded Q exceeding 10,000 for a 20 μm-diameter ring resonator.

The predictions above ignore scattering losses due to sidewall roughness, which is a consequence of the dry etching techniques commonly used to fabricate waveguides. In comparison to conventional silicon waveguides, we find that the maximal-confinement fin architecture exhibits similar propagation loss due to sidewall roughness compared to a 500 nm × 250 nm channel waveguide clad with air or SiO₂. Results of these calculations, additional details of fin waveguide modeling, and a design example for SiC are provided in the supplementary material.²⁵ 

Figure 4(c) depicts a process flow for fabricating fin waveguides on an arbitrary high-index substrate. All of the steps are well-established, top-down fabrication techniques that can be applied to a wide range of materials. The required aspect ratios of the fins in Figs. 4(a) and 4(b) can be achieved through anisotropic inductively coupled plasma reactive ion etching (ICP-RIE), for which aspect ratios exceeding 10:1 in diamond²⁸ and 50:1 in Si²⁹ have been demonstrated. High-aspect-ratio fins are already employed in InP-based PICs,³⁰ where the fin waveguide design can provide an alternative to the conventional InGaAsP guiding layer, leading to higher confinement and smaller mode area.

The dielectric stack can be fabricated using standard lithography, deposition, and planarization techniques [Fig. 4(c)], following a similar process flow to that used for high-aspect-ratio (>10:1) CMOS FinFETs.³¹ Indeed, all of the required steps are silicon-CMOS compatible, providing exciting opportunities for the realization of electronic-photonic co-integration and novel device structures. For example, high-speed active devices in CMOS-compatible silicon PICs could be achieved through alignment of the fin waveguide with vertical p-i-n junctions.³² Although the examples in Figs. 4(a) and 4(b) use a SiO₂/Si₃N₄ dielectric stack, the fin waveguide can be designed for any pair of materials with n_H > n_L. Further details about the fabrication methods, alternative dielectric materials, and CMOS-compatible implementations are provided in the supplementary material.²⁵ 

One challenge with the proposed architecture is the incorporation of devices that are typically multimode, such as Y-branches and grating couplers, since higher-order modes in $x ˆ$ are leaky. We envision a solution to this challenge in the form of supermode devices, where multimode propagation is achieved by coupled arrays of single-mode waveguides.

In summary, we have proposed a new waveguide design for native high-refractive-index substrates. This method is compatible with standard fabrication processes and alleviates the need for a buried low-index layer, providing a potential route for CMOS-compatible co-integration of silicon photonics with VLSI electronics. Furthermore, the geometry can be adapted for any high-index substrate material, which will lead to rapid development of PICs on emerging materials platforms for nonlinear photonics, quantum information processing, and sensing applications.

We thank F. Aflatouni, J. Driscoll, S. Mann, A. Exarhos, and D. Hopper for helpful discussions.