Single-Mode Optical Waveguides on Native High-Refractive-Index Substrates

High-refractive-index semiconductor optical waveguides form the basis for modern photonic integrated circuits (PICs) , but the conventional methods of achieving optical confinement require a thick lower-refractive-index support layer that impedes large-scale co-integration with electronics. To address this challenge, we present a general architecture for single-mode waveguides that confine light in a high-refractive-index material on a native substrate. Our waveguide consists of a high-aspect-ratio fin of the guiding material surrounded by lower-refractive-index dielectrics and is compatible with standard top-down fabrication techniques. The proposed waveguide geometry removes the need for a buried-oxide-layer in silicon photonics, as well as the InGaAsP layer in InP-based PICs and will allow for photonic integration on emerging material platforms such as diamond and SiC.

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 SiO 2 /Si 3 N 4 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 SiO 2 ≈ 1.45 and n H = n Si 3 N 4 ≈ 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 waveguides 21 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 theẑ-invariant waveguide dielectric topology in Fig. 2(a) as two stacked slab waveguides with horizontal (x) confinement, a slab waveguide with vertical (ŷ) confinement can be formed from the slab-waveguide effective indices inx. 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 inx with the effective indices of the lowest-order supported modes n x eff,1 and n x eff,2 , respectively, where the effective index, n x eff,i = β x i /k 0 , is equal to the slab waveguide propagation constant inẑ, β x i , of Slab i divided by the free space wavenumber, k 0 = 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 x eff,1 > n eff > max{n x eff,2 , 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 inx 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 theŷ-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, Region I : n x eff,2 < n H ; w < w symm (green shading), Region II : n x eff,2 > n H ; w > w symm (blue shading). The boundary between these two regions occurs when n x eff,2 = 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 x eff,2 . 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 alongx, commonly referred to as the quasi-TE mode. Discussion of the lowest order quasi-TM mode (dominant electric field component alongŷ), 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 SiO 2 buffer layer, a conformal 200 nm-thick Si 3 N 4 confinement layer, and SiO 2 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 interactions 26 and distributed entanglement 27 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 SiO 2 . 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 diamond 28 and 50:1 in Si 29 have been demonstrated. High-aspectratio 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 SiO 2 /Si 3 N 4 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 inx 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. Details of the approximate and numerical methods used for analyzing the fin waveguide structure are provided, along with additional calculations of substrate leakage loss, bending loss, and scattering loss. A proposed fabrication flow is provided.

WAVEGUIDE DISPERSION DERIVATION AND DEFINITIONS
The supported modes of theẑ-invariant refractive index profile, n(x, y), in Fig. 2a  where β m = k 0 n m eff is the propagation constant of mode m, k 0 = 2π λ is the free space wavenumber, n m eff is the effective refractive index of mode m and the operatorÂ takes the following form when projected onto the position basis:Â The eigenvectors represent the transverse electric fields: With two-dimensional confinement the two transverse field components are mixed by boundary conditions for the tangential electric fields and perpendicular displacement fields at the dielectric interfaces contained in n 2 (x, y). The modes of the two-dimensionally confined structure cannot be calculated analytically; however, solutions to equations (S.3) can be found using numerical approaches such as finite-difference method (FDM) [1]. An alternate approach is to use the effective index method [2,3] to find an approximate solution by treating equation (S.1) as two separable problems inx andŷ. We use the effective index method outlined in the following section to find approximate solutions to the fin waveguide dispersion, and verify our calculations with FDM [1]. * Corresponding author: lbassett@seas.upenn.edu

A. Effective index method
In the one-dimensional analysis forx, n 2 (x, y) → , and equation (S.3) reduces to two separable equations representing two orthogonal polarizations. We represent these polarizations by r =x for the horizontal (E x )-polarization, and r =ŷ for the vertical (E y )-polarization. We follow the approach of [2] to solve for the slab waveguide effective indices. The indices of thex confined slab waveguides shown in Fig. 2a of the main text are: The field is assumed to be sinusoidal in the guiding region, and exponentially decaying outside. The phase constants for the field in each region are defined as: where i = 1, 2, corresponding to Slab 1 and Slab 2, respectively. The effective index of Slab i, n x eff,i , is found by matching the phase constants at the boundaries to find the following eigenvalue equations: Since the slab waveguides inx are symmetric, there is no cut-off for the lowest order mode. Using the solutions for the slab waveguides in Fig. 2a of the main text, aŷconfined asymmetric slab waveguide (Fig. 2b of the main text) can be constructed with the following indices: Using the nomenclature from [2], the asymmetric waveguide can be parameterized by a "substrate" with refractive index n s and "cladding" with refractive index n c , where n f > n s > n c . The effective index of Slab 2, n x eff,2 , depends on w/λ, thus we define the "substrate" and "cladding" indices in the following way: n y s = max n x eff,2 , n H (S.13) n y c = min n x eff,2 , n H (S.14) With these definitions, the phase constants of the asymmetric waveguide are defined as follows: which result in the following eigenvalue equations: where q = 0, 1, 2, ... is the mode index forŷ-confinement. The limits on allowable height for single mode operation are found by the cut-off condition for the lowest and first order modes of the asymmetric waveguide in Fig. S4d, which define the asymmetry parameter a y : The cut-off for mode |0, q, r occurs at a height of: Although pure polarization state solutions of the twodimensionally confined modes do not exist, we use the mode numbers of the approximate effective index method solutions as waveguide eigenvalue labels: |m ≈ |p, q, r . (S.22) All of the calculations in the main text have been performed for the |0, 0,x (lowest order quasi-TE) mode. The dispersion curves for the |0, 0,ŷ (lowest order quasi-TM) mode of the structure in Fig. 2a of the main text are shown in Fig. S1.
In the example considered in Fig. 2 of the main text, the fin waveguide only supports a single horizontal mode (p = 0). For a different choice of n L and n H , such that the cut-off width of higher-order modes in the guiding layer is smaller than w symm , additional horizontal modes can be supported, but the p = 0 mode for the buffer region always provides a lower limit for n eff of confined modes as shown in Fig. 2c of the main text.

B. Power flow
The complex Poynting vector is defined as [4]:

C. Mode area and confinement factor
For small perturbations the effect of waveguiding on light-matter interaction can be approximated as [5,6]: where ∆n eff is the change in the waveguide effective index due to a perturbation ∆n, which depends on both the group index, N g , and the fraction of mode energy contained in the perturbed region, F . The perturbation can be complex valued and can represent light-matter interactions such as absorption, gain, or material nonlinearity. Since we are concerned with perturbations to the waveguide core, F can be approximated by the confinement factor, Γ, which is defined as: 26) and is plotted in Fig. 3b of the main text. The ratio of the group velocity, v g , to phase velocity, v p , can be related to electric and magnetic field energies in the following way [7]: where F t is the fraction of mode energy contained in the transverse fields, E x , E y , H x , H y , and F z is the fraction of mode energy contained in the longitudinal fields, E z , H z . As was noted in [8], N g is inversely proportional to the effective mode area, A eff , as related through the energy contained in the longitudinal fields. Thus, A eff , can be used as a proxy for N g , and the minimum mode area, A min , corresponds to the maximum N g . The effective mode area, A eff , plotted in Fig. 3a of the main text has been calculated as the area of an ellipse with axes defined by the modal effective width, w eff and effective height, h eff , as follows: where w eff and h eff are calculated using the definitions in section 1:

A. Silicon carbide design
A design for a silicon carbide fin waveguide with an SiO 2 buffer layer, a conformal 200 nm-thick Si 3 N 4 confinement layer, and SiO 2 overcladding is shown in Fig. S2. Silicon carbide has recently emerged as a promising material for integrated non-linear photonics [9]. We note that this design is for cubic silicon carbide [12] with an isotropic refractive index; however, the fin waveguide geometry can also be applied to birefringent silicon carbide polytypes.

B. Substrate leakage
The substrate leakage as a function of buffer layer thickness shown in Fig. S3 has been calculated using two different methods: coupled-mode theory (red points) and absorbing boundary conditions (blue line). For the coupled-mode theory calculations the unperturbed mode is calculated using FDM without the high index substrate. Coupling of the unperturbed mode to the substrate is then calculated by using the following overlap integral [2,3]: where ∆n 2 (x, y) = n 2 f and the overlap region extends from the bottom of the buffer layer to −∞ inŷ, and from −∞ to +∞ inx.
For the second method, the high-index substrate is included in the FDM calculation, and perfectly matched layer (PML) boundary conditions are added to the simulation cell. These boundary conditions allow for absorption at the simulation cell edge with minimal reflections. The FDM solver can find complex eigenvalues [1], which can be related to the propagation loss in the following manner:

C. Bending loss
Bending loss of the structures in Fig. S4 is calculated in cylindrical coordinates with FDM using the method in [10], where the loss per 90 • bend as a function of bend radius is shown. A buffer layer thickness of t B = 2.5 µm has been used for both waveguides. The bending-loss limited Q-factor of a ring resonator is also shown in Fig. S4, where Q = ω/γ, ω = ck 0 is the center frequency and γ = α eff c Ng . The values in Fig. S4a,b are calculated at λ = 637 nm and λ = 1.55 µm, respectively.

D. Material dispersion models
The calculations in the main text use Sellmeier equations to model the various material refractive indices. The parameters for each material are given in table S1 and are used in the following equation [4]: For the designs in Fig. 4 of the main text, the diamond material stack has the following refractive indices at 637 nm: n diamond = 2.41, n Si3N4 = 2.01, and n SiO2 = 1.46 as determined from Sellmeier equations for each material. The silicon material stack has the fol- The design in Fig. S2 uses Cauchy's equation, n = A 1 + (λ 1 /λ) 2 , with values of A 1 , λ 1 given in table S1 to model the dispersive refractive index of silicon carbide. Sidewall roughness is modeled with a standard deviation of σs = 2 nm and a correlation length of Lc = 5 nm.

E. Scattering loss
Scattering loss due to sidewall roughness in a Si fin waveguide and a 500 nm × 250 nm Si channel waveguide with both air and SiO 2 cladding is modeled using the method given in Ref. [14] with an exponential autocorrelation characterized by a sidewall roughness standard deviation of σ s = 2 nm and a correlation length of L c = 5 nm. The lowest order |0, 0,x (quasi-TE) mode has been used for both types of waveguides. The results of this calculation are shown in Fig. S5.
To a first order approximation, the scattering loss coefficient, α s , is proportional to both the overlap of the mode intensity profile with the waveguide sidewall, as well as the square of the index contrast between the guiding medium (Si in our case) and the cladding (Si 3 N 4 for the fin waveguide, SiO 2 or air for the the channel waveguide) [14,15]. While the mode intensity profile overlap with the vertical sidewall is larger for the fin waveguide the index contrast is lower, resulting in marginally higher scattering loss in the fin waveguide at shorter wavelengths and lower loss at higher wavelengths. We note that the waveguides shown in Fig. S5 do not provide an exhaustive comparison between the two types of waveguides; however, they do illustrate that the fin waveguide geometry does not exhibit a substantial difference in propagation loss due to sidewall scattering as compared to a standard channel waveguide.

F. Group velocity dispersion
The waveguide dispersion parameter, which encompasses effects of both waveguide and material dispersion, is defined as [3]: and has been plotted in Fig. S6 for the waveguides shown in Fig. 4 of the main text and Fig. S2.

G. Fabrication feasibility
A potential fabrication flow for the fin waveguide is shown in Fig. 4c of the main text. Candidate processes for each of the fabrication steps in Fig. 4c are as follows: 1. Lithography: Electron beam [20] or deep-UV lithography [20,21] are suitable for achieving the required waveguide dimensions.
2. Anisotropic etch: Inductively-coupled plasma reactive-ion etches (ICP-RIE) have been developed that provide high-aspect ratio structures in many high-refractive-index materials. Example etches for a few of the materials discussed here are outlined in Table 2 F. 3. Low index growth: Optical quality SiO 2 can be grown by plasma-enhanced chemical vapor deposition (PECVD) or low-pressure chemical vapor deposition (LPCVD) [20,21]. Alternatively, spin casting can be used for the application of flowable oxides, other low index polymer layers [22], or chalcogenide glasses [23]. 4. Planarization: Planarization of the low-index layer can be achieved by chemical-mechanical polishing (CMP) or local oxide growth and anisotropic etching [20,21].
5. Low index etch: Anisotropic etching of the low index layer can be performed using ICP-RIE or capacitively-coupled RIE [20,21]. 6. High index growth: The high-index confinement layer can be grown by PECVD or LPCVD of Si 3 N 4 (n ≈ 2.0), sputtering of Al 2 O 3 (n ≈ 1.8) or AlN (n ≈ 2.2), spin casting other high index polymer