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

Supplementary Material for: Single-Mode Optical Waveguides on Native High-Refractive-Index Substrates

Richard R. Grote and Lee C. Bassett∗ Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA 19104, USA (Dated: June 14, 2016) 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.

1. WAVEGUIDE DISPERSION DERIVATION A. Effective index method AND DEFINITIONS In the one-dimensional analysis forx ˆ, n2(x, y) → The supported modes of thez ˆ-invariant refractive in- 2 ˆ ˆ ∂2 2 2 ni (x), A → Ax = ∂x2 + k0ni (x), and equation (S.3) dex profile, n(x, y), in Fig. 2a of the main text can be reduces to two separable equations representing two or- solved by casting Maxwell’s equations as an eigenvalue thogonal polarizations. We represent these polarizations problem: by r =x ˆ for the horizontal (Ex)-polarization, and r =y ˆ for the vertical (Ey)-polarization. We follow the ap- Aˆ |mi = β2 |mi (S.1) proach of [2] to solve for the slab waveguide effective m indices. The indices of thex ˆ confined slab waveguides m shown in Fig. 2a of the main text are: where βm = k0neff is the propagation constant of mode 2π m m, k0 = λ is the free space wavenumber, neff is the ˆ  effective refractive index of mode m and the operator A n = n , x > w takes the following form when projected onto the position  c,1 H 2 Slab 1: n (x) = n , w > x > − w (S.5) basis: 1 f 2 2  w nc,1 = nH, − 2 > x  w ˆ 2 2 2 nc,2 = nL, x > A = ∇t + k0n (x, y). (S.2)  2  w w Slab 2: n2(x) = nf, 2 > x > − 2 (S.6) The eigenvectors represent the transverse electric fields:  w nc,2 = nL, − 2 > x

Aˆ |Emi = β2 |Emi (S.3) The field is assumed to be sinusoidal in the guiding re- t m t gion, and exponentially decaying outside. The phase con- where stants for the field in each region are defined as:

m q m Ex x x 2 2 |E i = . (S.4) γc,i = k0 (n ) − nc,i cladding (S.7) t Em eff,i y q kx = k n2 − (nx )2 guiding region (S.8) With two-dimensional confinement the two transverse f,i 0 f eff,i field components are mixed by boundary conditions for the tangential electric fields and perpendicular dis- where i = 1, 2, corresponding to Slab 1 and Slab 2, respectively. The effective index of Slab i, nx , is found by placement fields at the dielectric interfaces contained in eff,i n2(x, y). The modes of the two-dimensionally confined matching the phase constants at the boundaries to find structure cannot be calculated analytically; however, so- the following eigenvalue equations: lutions to equations (S.3) can be found using numerical approaches such as finite-difference method (FDM) ! −1 kx,i [1]. An alternate approach is to use the effective index kx,iw = (pi + 1)π − 2 tan x , r =y ˆ method [2, 3] to find an approximate solution by treating γc,i equation (S.1) as two separable problems inx ˆ andy ˆ. We (S.9) use the effective index method outlined in the following 2 ! −1 nc,i kx,i section to find approximate solutions to the fin waveguide kx,iw = (pi + 1)π − 2 tan x , r =x ˆ dispersion, and verify our calculations with FDM [1]. nf γc,i (S.10)

where pi = 0, 1, 2, ... is the mode index for Slab i. The ∗ Corresponding author: [email protected] phase constants are related to β by equations (S.7) and 2

(a) xˆ confinement refractive index ns and “cladding” with refractive index nc, where nf > ns > nc. The effective index of Slab 2, 2.5 nf x neff,2, depends on w/λ, thus we define the “substrate” Forbidden and “cladding” indices in the following way:

x n eff,1 y x ns = max neﬀ,2, nH (S.13)

eff y x 2.0 nH n nc = min neﬀ,2, nH (S.14)

x eff,2 n With these deﬁnitions, the phase constants of the Radiation modes asymmetric waveguide are deﬁned as follows:

q 1.5 nL y y 2 y 2 γc = k0 (neff) − (nc ) (S.15) 0.00 0.25 0.50 0.75 1.00 q y y 2 y 2 w/λ γs = k0 (neff) − (ns ) (S.16) q x 2 y 2 (b) yˆ confinement ky = k0 (neff,1) − (neff) (S.17) 2.5 Forbidden which result in the following eigenvalue equations: 2.4 0.35 y 2.3 y −1 kf kf h = (q + 1)π − tan y

0.25 Region II γs eff 2.2 Multimode y n −1 kf − tan y , r =x ˆ 2.1 γc 0.15 (S.18) w/λ = 0.05 2.0 Region I 2 y ! y Radiation modes y −1 ns kf kf h = (q + 1)π − tan x y 1.9 neff,1 γs 0.0 0.5 1.0 1.5 2.0 2 y ! y h/λ −1 nc kf − tan x y , r =y ˆ neff,1 γc FIG. S1. Generalized fin waveguide dispersion for the (S.19) lowest-order |0, 0, yî (lowest order quasi-TM) mode. The waveguide geometry is shown in Fig. 2a of the main text. where q = 0, 1, 2, ... is the mode index fory ˆ-confinement. The limits on allowable height for single mode operation are found by the cut-off condition for the lowest and first (S.8), and solutions to the transcendental equations (S.9) order modes of the asymmetric waveguide in Fig. S4d, or (S.10) provide the propagation constant for mode p . i which define the asymmetry parameter ay: The cut-off condition for mode pi can be expressed in terms of a minimum waveguide width: y 2 y 2  (ns ) −(nc ) x 2 y 2 , r =x ˆ y  (neff,1) −(ns ) λπpi a = nx 4 y 2 y 2 (S.20) eff,1 (ns ) −(nc ) wcut-off,pi = , pi = 0, 1, 2, ... (S.11)  y x 2 y 2 , r =y ˆ p 2 2 nc (n ) −(ns ) 2 nf − nH eff,1 Since the slab waveguides inx ˆ are symmetric, there is The cut-off for mode |0, q, ri occurs at a height of: no cut-off for the lowest order mode. Using the solutions for the slab waveguides in Fig. 2a of the main text, ay ˆ- √ λ tan−1 ay + qπ confined asymmetric slab waveguide (Fig. 2b of the main hcut-off,q = , q = 0, 1, 2, ... (S.21) q x 2 y 2 text) can be constructed with the following indices: 2π (neff,1) − (ns )

 Although pure polarization state solutions of the two- n , y > h  H 2 dimensionally confined modes do not exist, we use the  x h h n(y) = neff,1, 2 > y > − 2 (S.12) mode numbers of the approximate effective index method  x h neff,2, − 2 > y solutions as waveguide eigenvalue labels: Using the nomenclature from [2], the asymmetric waveguide can be parameterized by a “substrate” with |mi ≈ |p, q, ri . (S.22) 3

All of the calculations in the main text have been performed for the |0, 0, xî (lowest order quasi-TE) mode. R Re {Sz} dA The dispersion curves for the |0, 0, yî (lowest order quasi- nf Γ = R (S.26) TM) mode of the structure in Fig. 2a of the main text Re {Sz} dA are shown in Fig. S1. In the example considered in Fig. 2 of the main text, and is plotted in Fig. 3b of the main text. the fin waveguide only supports a single horizontal mode The ratio of the group velocity, vg, to phase velocity, vp, can be related to electric and magnetic field energies (p = 0). For a different choice of nL and nH, such that the cut-off width of higher-order modes in the guiding layer in the following way [7]: is smaller than wsymm, additional horizontal modes can be supported, but the p = 0 mode for the buffer region vg = Ft − Fz (S.27) always provides a lower limit for neff of confined modes vp as shown in Fig. 2c of the main text. where Ft is the fraction of mode energy contained in the transverse fields, Ex,Ey,Hx,Hy, and Fz is the fraction of B. Power flow mode energy contained in the longitudinal fields, Ez,Hz. As was noted in [8], Ng is inversely proportional to the effective mode area, A , as related through the energy The complex Poynting vector is defined as [4]: eff contained in the longitudinal fields. Thus, Aeff, can be used as a proxy for Ng, and the minimum mode area, 1 Amin, corresponds to the maximum Ng. S = − E × H∗. (S.23) 2 The effective mode area, Aeff, plotted in Fig. 3a of the main text has been calculated as the area of an ellipse Thez ˆ-component of the Poynting vector represents the with axes defined by the modal effective width, weff and mode intensity profiles plotted in the main text and is de- effective height, heff, as follows: fined as: Re {Sz} = Re {S · zˆ}. The time-averaged power flow inz ˆ can also be found from the complex Poynting π vector: Aeff = weffheff (S.28) 4 where w and h are calculated using the definitions in Z eff eff section 1: Pz = Re {Sz} dA (S.24) where A is area. For all calculations the modes are nor- 2 weff = w + x (S.29) malized such that Pz = 1 W. γc,1 For substrate loss calculations, there is power flow in 1 1 yˆ, which is visualized by they ˆ-component of the Poynt- heff = h + y + y (S.30) γs γc ing vector, Re {Sy} = Re {S · yˆ}, plotted in the insets of Fig. S3. The time-averaged power flow iny ˆ is defined in a similar manner to equation (S.24). 2. FIN WAVEGUIDE DESIGNS FOR SPECIFIC MATERIALS SYSTEMS

C. Mode area and conﬁnement factor A. Silicon carbide design

For small perturbations the effect of waveguiding on A design for a silicon carbide fin waveguide with an light-matter interaction can be approximated as [5, 6]: SiO2 buffer layer, a conformal 200 nm-thick Si3N4 confinement layer, and SiO2 overcladding is shown in Fig. S2. ∆n Silicon carbide has recently emerged as a promising ma- ∆n ≈ − N F (S.25) terial for integrated non-linear photonics [9]. We note eff n g that this design is for cubic silicon carbide [12] with an where ∆neff is the change in the waveguide effective in- isotropic refractive index; however, the fin waveguide ge- dex due to a perturbation ∆n, which depends on both ometry can also be applied to birefringent silicon carbide the group index, Ng, and the fraction of mode energy polytypes. contained in the perturbed region, F . The perturbation can be complex valued and can represent light-matter interactions such as absorption, gain, or material nonlin- B. Substrate leakage earity. Since we are concerned with perturbations to the waveguide core, F can be approximated by the confine- The substrate leakage as a function of buffer layer ment factor, Γ, which is defined as: thickness shown in Fig. S3 has been calculated using 4

two diﬀerent methods: coupled-mode theory (red points) 2.5 3.1 210 nm and absorbing boundary conditions (blue line). For the coupled-mode theory calculations the unperturbed mode 2.4 3.0 is calculated using FDM without the high index substrate. Coupling of the unperturbed mode to the sub- 2.3 450 nm 200 nm 2.9 strate is then calculated by using the following overlap g eff

N integral [2, 3]: n 2.2 2.8 ωε Z α = hE|∆ε|Ei = 0 E∗ · ∆n2(x, y)E dA (S.31) 2.1 2.7 eﬀ 4

2 2 2.0 2.6 where ∆n (x, y) = nf and the overlap region extends 0.5 0.6 0.7 0.8 0.9 1.0 from the bottom of the buffer layer to −∞ iny ˆ, and from −∞ to +∞ inx ˆ. λ (µm) For the second method, the high-index substrate is in- cluded in the FDM calculation, and perfectly matched FIG. S2. SiC design example. Fin waveguide designed layer (PML) boundary conditions are added to the simu- with SiC for visible and near-infrared wavelengths. lation cell. These boundary conditions allow for absorption at the simulation cell edge with minimal reflections. (a) The FDM solver can find complex eigenvalues [1], which Diamond fin can be related to the propagation loss in the following 102 manner: Sy

y 4πIm{neﬀ} 10-2 x αeﬀ = (S.32) z 2.5 µm λ

Sy 0.5 µm 10-6 C. Bending loss α (dB/cm) y Bending loss of the structures in Fig. S4 is calculated z x 10-10 in cylindrical coordinates with FDM using the method in [10], where the loss per 90◦ bend as a function of bend 0.5 1.0 1.5 2.0 2.5 radius is shown. A buﬀer layer thickness of tB = 2.5 µm Buffer layer thickness (µm) has been used for both waveguides. The bending-loss lim- ited Q-factor of a ring resonator is also shown in Fig. S4, (b) Si fin where Q = ω/γ, ω = ck0 is the center frequency and αeffc γ = N . The values in Fig. S4a,b are calculated at 3 S g 10 y λ = 637 nm and λ = 1.55 µm, respectively.

y 0 10 x D. Material dispersion models z 2.5 µm S y 0.5 µm The calculations in the main text use Sellmeier equa- 10-3 α (dB/cm) tions to model the various material refractive indices. y The parameters for each material are given in table S1 x and are used in the following equation [4]: 10-6 z

0.5 1.0 1.5 2.0 2.5 2 X Aiλ n2 = 1 + (S.33) Buffer layer thickness (µm) λ2 − λ2 i i FIG. S3. Substrate leakage. The propagation loss, α, ver- For the designs in Fig. 4 of the main text, the di- sus buffer layer thickness at the design wavelength is shown amond material stack has the following refractive in- for (a) diamond fin, and (b) silicon fin. Solid lines and dices at 637 nm: ndiamond = 2.41, nSi3N4 = 2.01, and points represent two different calculation methods: absorb- n = 1.46 as determined from Sellmeier equations for ing boundary conditions (blue line) and coupled-mode the- SiO2 ory (red points). (inset) Poynting vector in the y-direction, each material. The silicon material stack has the following refractive indices at λ = 1.55 µm: n = 3.48, Re {Sy}, for buffer layer thicknesses of 0.5 µm and 2.5 µm. Si nSi3N4 = 1.98, and nSiO2 = 1.44. 5

(a) 6 Diamond fin ×10 4.5 210 nm 0.08 2.0

tB = 1.0 µm 540 nm 500 nm tB = 2.0 µm 0.06 1.5 3.5 250 nm Q 0.04 1.0 (dB/cm) s 2.5 α 500 nm 0.02

0.5 250 nm

Loss per 90º bend (dB) 1.5 0.00 0.0 1.4 1.5 1.6 1.7 10 20 30 40 50 λ (µm) Bend radius (µm) FIG. S5. Scattering loss. Propagation loss due to sidewall (b) Si fin ×105 roughness is compared for a Si ﬁn waveguide and a Si channel 0.20 2.8 with both air and SiO2 cladding as a function of wavelength.

tB = 1.5 µm Sidewall roughness is modeled with a standard deviation of

tB = 2.5 µm σs = 2 nm and a correlation length of Lc = 5 nm. 0.15 2.1

E. Scattering loss

0.10 1.4 Q

Scattering loss due to sidewall roughness in a Si fin 0.05 0.7 waveguide and a 500 nm × 250 nm Si channel waveguide with both air and SiO2 cladding is modeled using Loss per 90º bend (dB) the method given in Ref. [14] with an exponential au- 0.00 0.0 tocorrelation characterized by a sidewall roughness stan- 10 20 30 40 50 dard deviation of σs = 2 nm and a correlation length of Bend radius (µm) Lc = 5 nm. The lowest order |0, 0, xî (quasi-TE) mode has been used for both types of waveguides. The results FIG. S4. Bending loss. The loss per 90◦ bend and Q-factor of this calculation are shown in Fig. S5. of a ring resonator as a function of bend radius are shown for To a first order approximation, the scattering loss co- (a) diamond fin and (b) silicon fin. efficient, αs, 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 N for The design in Fig. S2 uses Cauchy’s equation, n = 3 4 the fin waveguide, SiO or air for the the channel waveg- A + (λ /λ)2, with values of A , λ given in table S1 to 2 1 1 1 1 uide) [14, 15]. While the mode intensity profile overlap model the dispersive refractive index of silicon carbide. 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 TABLE S1. Sellmeier equation parameters waveguides shown in Fig. S5 do not provide an exhaustive

Material i Ai λi (µm) Ref. comparison between the two types of waveguides; how- Diamond 1 0.3306 0.175 [11] ever, they do illustrate that the fin waveguide geometry 2 4.3356 0.106 does not exhibit a substantial difference in propagation 1 10.6684 0.3015 loss due to sidewall scattering as compared to a standard Si 2 0.0030 1.1347 [4] channel waveguide. 3 1.5413 1104 SiC a 1 2.5538 0.1849 [12] Si N 1 2.8939 0.13967 [13] 3 4 F. Group velocity dispersion 1 0.6962 0.06840 SiO2 2 0.4079 0.1162 [4] 3 0.8975 9.8962 The waveguide dispersion parameter, which encom- passes effects of both waveguide and material dispersion, a Modeled using Cauchy’s equation is defined as [3]: 6

Fin Modulator NMOS Transistor (a) Diamond fin -1.0

-1.5

-2.0 p (ps/nm-m)

λ -2.5 D -3.0 0.5 0.6 0.7 0.8 λ (µm) n n n (b) Si fin p -4.7

Si SiO2 Si3N4 Poly-Si TiSi2 W Cu -4.9 FIG. S7. CMOS implementation. Schematic of a ﬁn -5.1 waveguide modulator co-integrated with an NMOS transis- (ps/nm-m) λ tor. Devices are not to scale. D -5.3 1.4 1.5 1.6 1.7 λ (µm) G. Fabrication feasibility

(c) SiC fin A potential fabrication ﬂow for the ﬁn waveguide is shown in Fig. 4c of the main text. Candidate processes -2.0 for each of the fabrication steps in Fig. 4c are as follows:

-3.0 1. Lithography: Electron beam [20] or deep-UV lithography [20, 21] are suitable for achieving the

(ps/nm-m) -4.0 required waveguide dimensions. λ D -5.0 2. Anisotropic etch: Inductively-coupled plasma 0.5 0.6 0.7 0.8 0.9 1.0 reactive-ion etches (ICP-RIE) have been developed λ (µm) that provide high-aspect ratio structures in many high-refractive-index materials. Example etches for FIG. S6. Group velocity dispersion (GVD). GVD pa- a few of the materials discussed here are outlined rameter as a function of wavelength for the a diamond, b in Table 2 F. silicon, and c silicon carbide fin waveguides in Fig. 4a,b of the main text, and Fig. S2. 3. Low index growth: Optical quality SiO2 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 flow- 2 1 2 d neff able oxides, other low index polymer layers [22], Dλ = − λ (S.34) cλ dλ2 or chalcogenide glasses [23]. and has been plotted in Fig. S6 for the waveguides shown 4. Planarization: Planarization of the low-index in Fig. 4 of the main text and Fig. S2. layer can be achieved by chemical-mechanical pol- ishing (CMP) or local oxide growth and anisotropic etching [20, 21]. TABLE S2. Anisotropic etches of high-refractive index materials 5. Low index etch: Anisotropic etching of the low Material Chemistry Aspect ratio Ref. index layer can be performed using ICP-RIE or Si F or Cl2-based > 50 : 1 [16] capacitively-coupled RIE [20, 21]. InP Cl2/N2/Ar > 8 : 1 [17] Diamond O2 > 10 : 1 [18] 6. High index growth: The high-index confinement SiC SF6/O2 > 7 : 1 [19] layer can be grown by PECVD or LPCVD of Si3N4 (n ≈ 2.0), sputtering of Al2O3 (n ≈ 1.8) or AlN (n ≈ 2.2), spin casting other high index polymer 7

layers such as SU-8 (n ≈ 1.5), or deposition of Hy- photonic-electronic co-integration. The proposed im- dex (n = 1.5 to 1.9) [24]. plementation alleviates the need for silicon-on-insulator (SOI) with a thick buried-oxide-layer, providing a path- Each step of the fabrication flow is compatible with way for implementing optical interconnects on VLSI a conventional CMOS process used for VLSI electron- chips. A schematic of a fin waveguide co-integrated with ics [20, 21], making the fin waveguide a candidate for an NMOS transistor is shown in Fig. S7.

[1] A. B. Fallahkhair, K. S. Li, and T. E. Murphy, IEEE J. [13] T. B˚a˚ak,Appl. Opt. 21, 1069 (1982). Lightwave Technol. 26, 1423 (2008). [14] F. Grillot, L. Vivien, S. Laval, and E. Cassan, IEEE J. [2] H. Nishihara, M. Haruna, and T. Suhara, Optical inte- Lightwave Technol. 24, 891 (2006). grated circuits (McGraw-Hill Book Company, New York, [15] F. Payne and J. Lacey, Optical and Quantum Electronics 1989). 26, 977 (1994). [3] A. Yariv and P. Yeh, Photonics: optical electronics in [16] B. Wu, A. Kumar, and S. Pamarthy, J. Appl. Phys. 108, modern communications, 6th ed. (Oxford Univ. Press, 051101 (2010). NY, 2007). [17] S. Park, S.-S. Kim, L. Wang, and S.-T. Ho, IEEE J. [4] B. E. Saleh and M. C. Teich, Fundamentals of photonics, Quantum Electron. 41, 351 (2005). 2nd ed., Vol. 22 (John Wiley & Sons, NY, 2007). [18] B. J. Hausmann, M. Khan, Y. Zhang, T. M. Babinec, [5] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and K. Martinick, M. McCutcheon, P. R. Hemmer, and R. D. Meade, Photonic crystals: molding the flow of light M. Lonˇcar, Diamond and Related Materials 19, 621 (Princeton university press, 2011). (2010). [6] R. R. Grote, Nanophotonics for Optoelectronic Devices: [19] J. Choi, L. Latu-Romain, E. Bano, F. Dhalluin, Extrinsic Silicon Photonic Receivers and Organic Photo- T. Chevolleau, and T. Baron, Journal of Physics D: Ap- voltaics, dissertation, Columbia University in the city of plied Physics 45, 235204 (2012). New York (2014). [20] J. D. Plummer, M. Deal, and P. B. Griffin, Silicon VLSI [7] P.-R. Loh, A. F. Oskooi, M. Ibanescu, M. Skorobogatiy, technology: fundamentals, practice, and modeling (Pren- and S. G. Johnson, Phys. Rev. E 79, 065601 (2009). tice Hall, NJ, 2000). [8] J. B. Driscoll, X. Liu, S. Yasseri, I. Hsieh, J. I. Dadap, [21] S. Wolf, Microchip manufacturing (Lattice press, CA, and R. M. Osgood, Opt. Express 17, 2797 (2009). 2004). [9] J. Cardenas, M. Yu, Y. Okawachi, C. B. Poitras, R. K. [22] J. Halldorsson, N. B. Arnfinnsdottir, A. B. Jonsdottir, Lau, A. Dutt, A. L. Gaeta, and M. Lipson, Opt. lett. B. Agnarsson, and K. Leosson, Opt. express 18, 16217 40, 4138 (2015). (2010). [10] W. W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, and W.- [23] B. J. Eggleton, B. Luther-Davies, and K. Richardson, P. Huang, IEEE J. Lightwave Technol. 16, 910 (1998). Nature Photon. 5, 141 (2011). [11] R. Mildren and J. Rabeau, Optical engineering of dia- [24] D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, mond (John Wiley & Sons, 2013). Nature Photon. 7, 597 (2013). [12] P. T. Shaffer, Appl. Opt. 10, 1034 (1971).