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 ^, n2(x; y) ! The supported modes of thez ^-invariant refractive in- 2 ^ ^ @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 ^ for the horizontal (Ex)-polarization, and r =y ^ for the vertical (Ey)-polarization. We follow the ap- A^ jmi = β2 jmi (S.1) proach of [2] to solve for the slab waveguide effective m indices. The indices of thex ^ 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 ^ 8 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 8 w ^ 2 2 2 nc;2 = nL; x > A = rt + 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 A^ jEmi = β2 jEmi (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 jE 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, re- spectively. 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 numeri- cal 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 ^ method [2, 3] to find an approximate solution by treating γc;i equation (S.1) as two separable problems inx ^ andy ^. 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 ^ 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 ^ 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 ^ neff;1 γc FIG. S1. Generalized fin waveguide dispersion for the (S.19) lowest-order j0; 0; y^i (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 ^-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 8 (ns ) −(nc ) x 2 y 2 ; r =x ^ 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 ^ p 2 2 nc (n ) −(ns ) 2 nf − nH eff;1 Since the slab waveguides inx ^ are symmetric, there is The cut-off for mode j0; 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 ^- p 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 ) 8 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 jmi ≈ jp; q; ri : (S.22) 3 All of the calculations in the main text have been per- formed for the j0; 0; x^i (lowest order quasi-TE) mode. R Re fSzg dA The dispersion curves for the j0; 0; y^i (lowest order quasi- nf Γ = R (S.26) TM) mode of the structure in Fig. 2a of the main text Re fSzg 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 ^-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 fSzg = Re fS · z^g. The time-averaged power flow inz ^ 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 fSzg dA (S.24) where A is area.