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

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by  Richard R. Grote

Single-Mode Optical Waveguides on Native High-Refractive-Index Substrates
Richard R. Grote, Lee C. Bassett
∗
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering,
University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA 19104, USA and
∗Corresponding author: [email protected]
High-refractive-index semiconductor optical waveg-
uides form the basis for modern photonic integrated
circuits (PICs) , but the conventional methods of
achieving optical confinement require a thick lower-
6 refractive-indexsupportlayerthatimpedeslarge-scale
1 co-integration with electronics. To address this chal-
0
lenge,wepresentageneralarchitectureforsingle-mode
2
waveguidesthatconfinelightinahigh-refractive-index

n material on a native substrate. Our waveguide con-
a
sists of a high-aspect-ratio fin of the guiding material
J
surroundedbylower-refractive-indexdielectricsandis
6
compatible with standard top-down fabrication tech-

] niques. The proposed waveguide geometry removes
s
the need for a buried-oxide-layer in silicon photonics,
c
i as well as the InGaAsP layer in InP-based PICs and
t
p will allow for photonic integration on emerging mate-
o rialplatformssuchasdiamondandSiC.
.
s PICs arerapidly beingdeveloped for high-refractivein-
c dexmaterialsthatallowfortightopticalconfinement,small
i FIG.1. Thefinwaveguide. Anexampleofafinwaveguideon
s on-chip bend radii, and strong light-matter interactions.
y a diamond substrate designed for λ = 637nm. The geometry
For example, high-performance PICs in both silicon [1]
h supportsasinglemodewhennf > nH > nL. Theprofileofthe
and InP [2, 3] platforms are playing an increasingly im-
p confinedopticalmodeshownincross-sectionhasbeencalculated
[ portant role in data applications with the potential to en- numerically.

able exascale computing [4] and on-chip core-to-core op-
1
tical communication [5]. Similarly, wide-band gap semi-
v
9 conductors, such as diamond [6–13] and SiC [14], have 19]. For InP-based PICs, optical confinement is limited
3 emergedaspromisingmaterialsforaplethoraofnewPIC bythelowindexcontrastbetweenInPandInGaAsP[2,3].
2 applications. Among these are non-linear optics [12–14] Here, we propose a new type of waveguide optimized
1
andintegratedquantuminformationprocessing[8–11,15], forhigh-indexsubstratesthatutilizesstackeddielectriclay-
0
which is enabled by the presence of spin defects with de- ers to confine light in the top of a fin of high-index ma-
.
1 sirablequantumproperties[16,17]. terial. An example of a SiO /Si N stack on a diamond
0 2 3 4
Common to all of these applications is a need for low- fin/substrate at a wavelength of λ = 637nm is shown in
6
1 propagation-loss single-mode waveguides that can be fab- Fig. 1. Although the refractive index of both the buffer
: ricated on a high-refractive-index substrate in a scalable and confinement layers (n = n 1.45 and n =
v L SiO2 ≈ H
fashion. Whileahighrefractiveindexisbeneficialforop- n 2.0, respectively) are lower than that of the fin
Xi ticaldesign,italsorequiresaburiedlower-refractive-index anSdi3Ns4ub≈strate (n = n 2.4), the proposed design
f diamond
≈
r layerandthetransferorgrowthofthinfilmsofhigh-index achieves confinement by engineering the effective index,
a
material[1–3,6–8,10,12–14,18],free-standingstructures resultinginanopticalmodeconfinedwithinthehigh-index
[9, 11, 19], or pedestals [20] to minimize optical power material (diamond in the case of Fig. 1). This waveguide
leakage from the waveguide into the substrate. These ap- mode can propagate without leaking power into the un-
proacheslimitthedevicerobustness,uniformity,andscala- derlying substrate, while the waveguide itself can be fab-
bilityrequiredforthedevelopmentofdensePICsonwide- ricated using conventional top-down lithography, etching,
bandgapsemiconductors. dielectric deposition, and planarization techniques. Our
Even on mature PIC platforms optical confinement proposed architecture obviates the need for a buried low-
presents significant technological challenges. In silicon index layer, providing a pathway towards large-area, scal-
photonics, the buried-oxide-layer thickness required for ablePICsonnativesubstrates.
optical confinement is much larger than the optimum for Figures 2a–b illustrate an intuitive physical picture that
VLSI electronics, making co-integration difficult [5, 18, accounts for optical confinement in the fin waveguide ge-
2
(a) (b) can be formed from the slab-waveguide effective indicies
Fin waveguide Effective index model
in xˆ. In Fig. 2b, the two slab waveguides comprised of
n /n/n (Slab 1) and n /n/n (Slab 2) from Fig. 2a
H f H L f L
w
are replaced by homogeneous layers in xˆ with the ef-
h Slab 1 h fective indices of the lowest-order supported modes nx
eff,1
y y and nx , respectively, where the effective index, nx =
Slab 2 eff,2 eff,i
z x z x βix/k0,isequaltotheslabwaveguidepropagationconstant
in zˆ, βx, of Slab i divided by the free space wavenumber,
n > n > n nx > max{ nx , n } i
f H L eff,1 eff,2 H k = 2π/λ. The dispersion curves of the two slabs as
0
(c) a function of the normalized waveguide width, w/λ, are
xˆ confinement shown in Fig. 2c with example refractive index values of
2.5 n n = 2.5,n = 2.0,andn = 1.5.
f f H L
Forbidden Thistreatmentiskeytounderstandingthenatureofcon-
finement in the fin structure: a fin mode exists when the
nx eff,1 etuffreec,tinve i,ndseaxtisofifesthethtewoc-odnidmiteinosni:onanllxy con>finend stru>c-
eff eff,1 eff
neff2.0 nH max{nxeff,2,nH},asindicatedbythegreenandblueshaded
regionsinFig.2c. Whenthisconditionisnotmet,thecon-
xneff,2 Radiation modes fimnoeddesmaonddesbeacroemdeegleenaekrya,taeswinitdhicaatceodnbtiynuthuemgorafyrahdaitacthioend
region in Fig. 2c. As a consequence of the effective in-
dex confinement in the fin structure, only a single-mode
1.5 n
L in xˆ is supported for the refractive index values chosen in
0.00 0.25 0.50 0.75 1.00
Fig.2c. Higher-ordermodescanbeconfinedforadifferent
w/λ
choice of material indices, but only if higher-order modes
(d)
ofSlab1arecontainedintheblueorgreenshadedregion
yˆ confinement
ofFig.2c,i.e.,theyhaveaneffectiveindexlargerthanthe
2.5
lowest-ordermodeofSlab2.
2.4 Forbidden n II Thesupportedmodesofthetwo-dimensionally-confined
o structure are found by solving for the modes of the yˆ-
0.45 gi
2.3 e confinedslabwaveguideinFig.2b. Theresultingfinmode
0.35 Multimode R dispersion as a function of normalized waveguide height,
neff2.2 n I h/λ, (Fig.2d)hastwodistinctregionsthatdependonthe
2.1 0.25 gio finwidth,w:
0.15 e
2.0 w/λ = 0.05 R RegionI: nxeff,2 < nH;w < wsymm (greenshading),
Radiation modes RegionII: nx > n ;w > w (blueshading).
1.9 eff,2 H symm
0.0 0.5 1.0 1.5 2.0 The boundary between these two regions occurs when
h/λ nx = n at a width that we label w = w , which
eff,2 H symm
is indicated by a vertical red line in Fig. 2c and the red
FIG. 2. Generalized fin waveguide dispersion. (a) Cross- dispersion curve (wsymm/λ = 0.25) in Fig. 2d. For
sectionofthefinwaveguidegeometry.Dashedlinesmarkregions anasymmetricslabwaveguide,thehigher-refractive-index
thatareapproximatedbyeffectiveindicesin(b). (c)Dispersion cladding determines both the cut-off condition and the ef-
ofthefinwaveguideasafunctionofw/λ. Theregioninwhich fective mode width. In Region I, the properties of the
modesareconfined(blueandgreenshading)isboundbytheef- asymmetricwaveguidearedeterminedbyn ,whileinRe-
H
fective indices of the slab waveguides in (a). (d) Dispersion of gion II, this role is taken by nx . The change in cut-
the fin-waveguide as a function of h/λ. Semi-analytical curves eff,2
offconditionbetweenthetworegionscausestheinflection
areplottedforspecificvaluesofw/λ,andpointsaretheresultof
pointinthethecut-offheight(boundaryforthemultimode
fully-vectorialnumericalsimulations.
region in Fig. 2d) at w . While the fin waveguide can
symm
only be single or few mode in width, it can be multimode
inheight,asindicatedinFig.2d.
ometry. By treating the two-dimensional cross-section of The dispersion curves provide the allowable geometry
the zˆ-invariant waveguide dielectric topology in Fig. 2a and wavelength at which confined modes are supported.
as two stacked slab waveguides with horizontal (xˆ) con- Withintheseconstraints,themodearea,A ,andconfine-
eff
finement, a slab waveguide with vertical (yˆ) confinement ment factor, Γ, which quantifies the overlap between the
3
(a) confinementlayer,relaxingtherequirementsonthebuffer
Region I w= w Region 2 layer thickness for low leakage at the expense of reduced
symm
0.05λ Γ. Conversely, in Region II Amin increases with w/λ, Γ
0.90λ approachesunity,andthemodeextendswithinthefininto
0.25λ
λ 0.30λ λ the buffer layer. Waveguides designed in this region may
7 9
1. 1. bedesirableforhigh-powerapplications. Thetightestcon-
y y y finement(smallestA )occursatw , whichalsocor-
min symm
z x z x z x respondstothemaximumgroupindex,Ng = c/vg,where
cisthevacuumspeedoflightandv isthemodalgroupve-
g
10.0 4.0 locity(seeSupplementarySection1),makingw = wsymm
an ideal design criterion for applications in non-linear or
quantum optics. The waveguide properties in Figs. 2 and
7.5 3.0
3 have been calculated for the lowest-order mode with
2λ λ the dominant electric field component along xˆ. Discus-
A/min5.0 2.0 h/min sainoanlyotfichailgahnedronrudmerermicoadlecsaalcnudlaftuiortnhserardeetpariolsviodfeoduirnsSemupi--
plementarySection1.
2.5 1.0 To illustrate the potential of the fin waveguide, we ex-
plore geometries in two important material platforms for
PICs: diamondandsilicon. Thediamondwaveguideisde-
0.0 0.0
signedwithanSiO bufferlayer,aconformal200nm-thick
2
0.0 0.5 1.0 1.5
Si N confinementlayer,andSiO overcladdingforsingle-
w/λ 3 4 2
modeoperationatλ =637nm. Theoperatingwavelength
(b)
corresponds to the nitrogen-vacancy center zero phonon
line[8],whichisusedtoachievecoherentspin-lightinter-
1.00
actions[21]anddistributedentanglement[22]betweendi-
amondspins. Wedesignthewaveguideforthetightestcon-
0.75 finementwithw = w asdiscussedabove. Thewaveg-
symm
uidedispersioncurveandgroupindexisshowninFig.4a,
alongwiththewaveguidedimensionsandcalculatedmode
Г0.50 intensity profile at λ = 637nm in the inset. We calcu-
late that for a buffer layer thickness exceeding 1.0µm the
0.25 propagationlossduetosubstrateleakageis<0.15dB/cm,
whichissmallenoughthatscatteringduetofabricationim-
perfectionsinarealisticdevicewouldbeexpectedtodom-
0.00 inate. The bending loss for a bend radius of 10µm is de-
0.0 0.5 1.0 1.5 terminedtobe<0.06dBper90◦ bendwithabufferlayer
w/λ thickness of 2.5µm, corresponding to an unloaded Q ex-
ceeding30,000fora20µm-diameterringresonator.
FIG. 3. Optical confinement of the fin waveguide. (a) Mini- Similarly, the silicon waveguide depicted in the in-
mummodeareaasafunctionofw/λandcorrespondingh /λ.
min set to Fig. 4b is designed for minimum mode area at
Modeprofilesatthethreevaluesofw/λmarkedbydashedlines
λ = 1.55µm for telecommunications applications. With
are plotted above. The smallest achievable mode area occurs
a buffer layer thickness exceeding 1.5µm, the propa-
whennH = nxeff,2 atwsymm/λ =0.25. (b)Theconfinementfac- gation loss due to substrate leakage is calculated to be
tor,Γ,measuresthegeometricoverlapbetweentheopticalmode
<0.1dB/cm. Thebendinglossforabendradiusof10µm
andthehigh-indexmaterial.
is determined to be < 0.1dB per 90 bend with a buffer
◦
layer of 2.5µm, corresponding to an unloaded Q exceed-
optical mode and the guiding material, provide useful de- ing 10,000 for a ring resonator. Additional details regard-
signmetricsformaximizinglight-matterinteractionsinfin ingthedesignandmodelingoffinwaveguidesareprovided
waveguides (see Supplementary Section 1 for definitions inSupplementarySection2.
oftheseparameters). InFig3a,wecalculateA forase- The design strategy described in this paper provides a
eff
ries of w/λ, and plot the minimum, A , along with the straightforward pathway towards the implementation of
min
corresponding height, h /λ. Mode intensity profiles for a full PIC architecture based on monolithic fin waveg-
min
threevaluesofw/λarealsoshowninFig.3a,andthecor- uides. The required aspect ratios of the fins in Fig. 4 can
respondingΓatA isplottedinFig.3b. be achieved through dry etching [23, 24], and the dielec-
min
In Region I most of the field penetration occurs in the tricstackcanbefabricatedusingstandarddeposition,pla-
4
Diamond fin fornativehigh-refractive-indexsubstrates. Thismethodis
(a)
compatiblewithstandardfabricationprocessesandallevi-
2.4 2.8
200 nm atestheneedforaburiedlow-indexlayer,providingapo-
tential route for CMOS-compatible co-integration of sili-
2.3 2.7 conphotonicswithVLSIelectronics. ForInP-basedPICs,
thefinwaveguidedesignprovidesanalternativetotheIn-
eff 2.2 350 nm 200 nm 2.6 Ng GaAs guiding layer, providing much higher confinement
n
andasmallermodearea. Furthermore,thegeometrycanbe
adapted for any high-index substrate material, which will
2.1 2.5
lead to rapid development of PICs on emerging materials
platformsforquantuminformationprocessingandsensing
2.0 2.4 applications.
0.5 0.6 0.7 0.8
λ (µm)
ACKNOWLEDGMENTS
Si fin
(b)
2.8 4.2
We thank F. Aflatouni, J. B. Driscoll, S. A. Mann, A.
210 nm
L. Exarhos, and D. A. Hopper for helpful discussions and
2.6 4.0
carefulreadingofthemanuscript. Weacknowledgefinan-
cialsupportforthisresearchfromtheUniversityofPenn-
eff 2.4 540 nm 200 nm 3.8Ng sylvania.
n
2.2 3.6
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Single-Mode Optical Waveguides on Native High-Refractive-Index Substrates: Supplementary
Information
Richard R. Grote, Lee C. Bassett
∗
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering,
University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA 19104, USA and
∗Corresponding author: [email protected]
(Dated:January7,2016)
WAVEGUIDEDISPERSIONDERIVATIONAND reduces to two separable equations representing two or-
DEFINITIONS thogonal polarizations. We represent these polarizations
by r = xˆ for the horizontal (E )-polarization, and r = yˆ
6 x
1 Thesupportedmodesofthezˆ-invariantrefractiveindex forthevertical(Ey)-polarization. Wefollowtheapproach
0 profile, n(x,y), in Fig. 2a of the main text can be solved of[2]tosolvefortheslabwaveguideeffectiveindices. The
2 bycastingMaxwell’sequationsasaneigenvalueproblem: indicesofthexˆconfinedslabwaveguidesshowninFig.2a

n ofthemaintextare:
a
J Aˆ m = β2 m (S.1)
| i m| i
6 n = n , x > w
s] wk0he=re2βλπmis=thke0fnremeeffsispathceepwraovpeanguamtiobnerc,onnmesfftainsttohfemefofedcetimve, Slab1: n1(x) = ncf,,1 H w2 > x2 > −w2 (S.5)
c refractive index of mode m and the operator Aˆ takes the nc,1 = nH, −w2 > x
i
pt followingformwhenprojectedontothepositionbasis: nc,2 = nL, x > w2
o Slab2: n (x) = n, w > x > w (S.6)
cs. Aˆ= ∇2t +k02n2(x,y). (S.2) 2 ncf,2 = nL, −2 w2 > x −2
i Theeigenvectorsrepresentthetransverseelectricfields:
s Thefieldisassumedtobesinusoidalintheguidingregion,
y
h and exponentially decaying outside. The phase constants
p Aˆ Em = β2 Em (S.3) forthefieldineachregionaredefinedas:
| t i m| t i
[
where

1
v γcx,i = k0 (nxeff,i)2−n2c,i cladding (S.7)
9 Em
23 |Emt i = (cid:20)Exym(cid:21). (S.4) kfx,i = k0qn2f −(nxeff,i)2 guidingregion (S.8)
q
1 Withtwo-dimensionalconfinementthetwotransversefield where i = 1,2, corresponding to Slab 1 and Slab 2, re-
0
componentsaremixedbyboundaryconditionsforthetan- spectively. TheeffectiveindexofSlabi,nx ,isfoundby
. eff,i
1 gentialelectricfieldsandperpendiculardisplacementfields
matching the phase constants at the boundaries to find the
0 at the dielectric interfaces contained in n2(x,y). The
followingeigenvalueequations:
6
modes of the two-dimensionally confined structure can-
1
notbecalculatedanalytically; however, solutionstoequa-
:
v tions (S.3) can be found using numerical approaches such k
i k w = (p +1)π 2tan 1 x,i , r = yˆ
X as finite-difference method (FDM) [1]. An alternate ap- x,i i − − γcx,i!
proach is to use the effective index method [2, 3] to find
r (S.9)
a an approximate solution by treating equation (S.1) as two
separableproblemsinxˆandyˆ. Weusetheeffectiveindex n 2 k
k w = (p +1)π 2tan 1 c,i x,i , r = xˆ
mmeattheosdolouutitolinnsedtoitnhethfienfowllaovweginugidseedctiisopnertsoiofinn,danadppvreorixfiy- x,i i − − (cid:18) nf (cid:19) γcx,i!
(S.10)
ourcalculationswithFDM[1].
where p = 0,1,2,... is the mode index for Slab i. The
i
phase constants are related to β by equations (S.7) and
Effectiveindexmethod
(S.8),andsolutionstothetranscendentalequations(S.9)or
(S.10) provide the propagation constant for mode p . The
i
In the one-dimensional analysis for xˆ, n2(x,y) cut-offconditionformodep canbeexpressedintermsof
→ i
n2(x), Aˆ Aˆ = ∂2 + k2n2(x), and equation (S.3) aminimumwaveguidewidth:
i → x ∂x2 0 i
2
where n > n > n . The effective index of Slab 2,
(a) f s c
nx ,dependsonw/λ,thuswedefinethe“substrate”and
xˆ confinement eff,2
“cladding”indicesinthefollowingway:
2.5 n
f
Forbidden
ny = max nx ,n (S.13)
s eff,2 H
nxeff,1 nyc = min (cid:8)nxeff,2,nH(cid:9) (S.14)
eff2.0 n Usingthesedefinitions,the(cid:8)phasecon(cid:9)stantsoftheasym-
n H metricwaveguidearedefinedasfollows:
nxeff,2
Radiation modes
γy = k (ny )2 (ny)2 (S.15)
c 0 eff − c
q
1.5 n γy = k (ny )2 (ny)2 (S.16)
L s 0 eff − s
0.00 0.25 0.50 0.75 1.00
q
w/λ k = k (nx )2 (ny )2 (S.17)
y 0 eff,1 − eff
q
(b) whichresultinthefollowingeigenvalueequations:
yˆ confinement
2.5 kyh = (q+1)π tan 1 kfy
Forbidden f − − γy
2.4 n II (cid:18)ksy(cid:19)
0.35 o tan 1 f , r = xˆ
2.3 gi − − γy
0.25 Re (cid:18) c (cid:19) (S.18)
eff2.2 Multimode
n 2
22..01 0.1 5 w/λ = 0.05 Region I kfyh = (q+1)π−tan−1 nnxefysf,1!2(cid:18)γksfyy(cid:19)
ny ky
Radiation modes tan 1 c f , r = yˆ
1.9 − − nxeff,1! (cid:18)γcy(cid:19)
0.0 0.5 1.0 1.5 2.0
(S.19)
h/λ
whereq = 0,1,2,...isthemodeindexforyˆ-confinement.
The limits on allowable height for single mode operation
FIG.S1. Dispersionplotsforthelowest-order 0,0,yˆ modeof
| i are found by the cut-off condition for the lowest and first
thefinwaveguideinFig.2aofthemaintext.
order modes of the asymmetric waveguide in Fig. S3d,
whichdefinetheasymmetryparameteray:
Sincwecutht-oeffs,plaib=w2avpegλnuπ2fidp−eisni2Hn,xˆ arpeis=ym0m,1e,tr2ic,,..t.here(Sis.1n1o) ay = (cid:16)(n(nnxenfxeysff,ycf1),12)2−(cid:17)−(4n(nyc(nys()n2)xef2ysf,,1)2)2−−(n(nycys)2)2, rr == xyˆˆ (S.20)
cut-off for the lowest order mode. Using the solutions Thecut-offformode 0,q,r occursataheightof:
| i
for the slab waveguides in Fig. 2a of the main text, a yˆ-
confined asymmetric slab waveguide (Fig. 2b of the main λ tan 1√ay +qπ
−
text)canbeconstructedwiththefollowingindices: hcut-off,q = , q = 0,1,2,...
2π(cid:0) (nxeff,1)2−(nys)(cid:1)2
q (S.21)
n , y > h
H 2
n(y) = nx , h > y > h (S.12) Although pure polarization state solutions of the two-
 eff,1 2 −2 dimensionally confined modes do not exist, we use the
nx , h > y
eff,2 −2 mode numbers of the approximate effective index method
solutionsaswaveguideeigenvaluelabels:

Using the nomenclature from [2], the asymmetric
waveguide can be parameterized by a “substrate” with re-
fractive index n and “cladding” with refractive index n , m p,q,r . (S.22)
s c
| i ≈ | i
3
All of the calculations in the main text have been per-
(a)
formed for the 0,0,xˆ mode. The dispersion curves for
the 0,0,yˆ mo|de of thie structure in Fig. 2a of the main Diamond fin
| i
textareshowninFig.S1.
102 S
IntheexampleconsideredinFig.2ofthemaintext,the y
finwaveguideonlysupportsasinglehorizontalmode(p = y m
0w)i.dFthoroafhdiigffheerer-notrcdheorimceoodfesniLnatnhdenguHi,dsiuncghltahyaetrtihsescmuat-loleffr cm) 10-2 z x 2.5 µ
/
tbhuatnthwespym=m,0amddoidtieonfoarlthhoeribzuofnftearlrmegoidoensaclawnabyesspurpopvoidretesda, (dB 10-6 Sy 0.5 µm
α
lowerlimitforn ofconfinedmodesasshowninFig.2c
eff y
ofthemaintext.
x
z
10-10
0.5 1.0 1.5 2.0 2.5
Powerflow
Buffer layer thickness (µm)
ThecomplexPoyntingvectorisdefinedas[4]:
(b)
1 Si fin
S = E H∗. (S.23)
−2 ×
103 S
y
The zˆ-component of the Poynting vector represents the
mode intensity profiles plotted in the main text and is de- y m
) µ
finedas: Re{Sz} = Re{S ·zˆ}. Thetime-averagedpower cm 100 z x 2.5
flowinzˆcanalsobefoundfromthecomplexPoyntingvec- /
B
tor: d Sy 0.5 µm
( 10-3
α
y
P = Re S dA (S.24)
z z x
{ } 10-6 z
Z
whereAisarea. Forallcalculationsthemodesarenormal- 0.5 1.0 1.5 2.0 2.5
izedsuchthatPz =1W. Buffer layer thickness (µm)
For substrate loss calculations, there is power flow in
yˆ, which is visualized by the yˆ-component of the Poynt-
ing vector, Re Sy = Re S yˆ , plotted in the insets of FIG.S2. (a),(b)Propagationloss,α,versusbufferlayerthick-
{ } { · }
Fig.S2. Thetime-averagedpowerflowinyˆisdefinedina ness at the design wavelength. Solid lines and points repre-
similarmannertoequation(S.24). senttwodifferentcalculationmethods:absorbingboundarycon-
ditions (blue line) and coupled-mode theory (red points). (in-
set)Poyntingvectorinthey-direction,Re Sy ,forbufferlayer
{ }
thicknessesof0.5µmand2.5µm.
Modeareaandconfinementfactor
For small perturbations the effect of waveguiding on
light-matterinteractioncanbeapproximatedas[5,6]:
Re S dA
Γ = nf { z} (S.26)
Re S dA
∆n R { z}
∆n N F (S.25)
eff ≈ − n g andisplottedinFig.3boRfthemaintext.
Theratioofthegroupvelocity,v ,tophasevelocity,v ,
g p
where∆n isthechangeinthewaveguideeffectiveindex
eff canberelatedtoelectricandmagneticfieldenergiesinthe
duetoaperturbation∆n,whichdependsonboththegroup
followingway[7]:
index,N ,andthefractionofmodeenergycontainedinthe
g
perturbedregion,F. Theperturbationcanbecomplexval- v
g
uedandcanrepresentlight-matterinteractionssuchasab- v = Ft−Fz (S.27)
sorption, gain, ormaterialnonlinearity. Sincewearecon- p
cernedwithperturbationstothewaveguidecore,F canbe where F is the fraction of mode energy contained in the
t
approximated by the confinement factor, Γ, which is de- transversefields,E ,E ,H ,H ,andF isthefractionof
x y x y z
finedas: mode energy contained in the longitudinal fields, E ,H .
z z
4
(a)
TABLEI.Sellmeierequationparameters
Diamond fin
Material i Ai λi[µm] Ref. ×105
Diamond 1 0.3306 0.175 [9] 0.20 10.0
2 4.3356 0.106
)
B
1 10.6684 0.3015 d
Si 2 0.0030 1.1347 [4] d (0.15 7.5
3 1.5413 1104 en
Si3N4 11 20..86993692 00..1036986470 [10] 0º b0.10 5.0 Q
9
SiO2 2 0.4079 0.1162 [4] er
3 0.8975 9.8962 p
s 0.05 2.5
s
o
L
Aswasnotedin[8],N isinverselyproportionaltotheef-
g 0.00 0.0
fectivemodearea,A ,asrelatedthroughtheenergycon-
eff 10 20 30 40 50
tainedinthelongitudinalfields. Thus,A ,canbeusedas
eff Bend radius (µm)
aproxyforN ,andtheminimummodearea,A ,corre-
g min
spondstothemaximumNg. (b)
The effective mode area, A , plotted in Fig. 3a of the Si fin
eff
×105
maintexthasbeencalculatedastheareaofanellipsewith
0.20 2.8
axes defined by the modal effective width, w and effec-
eff
tiveheight,h ,asfollows: B)
eff
d
π nd (0.15 2.1
A = w h (S.28) e
eff 4 eff eff b
where weff and heff are calculated using the definitions in 90º 0.10 1.4 Q
SupplementarySection: er
p
s 0.05 0.7
s
2 o
w = w+ (S.29) L
eff γx
c,1 0.00 0.0
1 1
10 20 30 40 50
h = h+ + (S.30)
eff γsy γcy Bend radius (µm)
FIG.S3. BendinglossandQ-factorasafunctionofbendradius
DIAMONDANDSILICONFINWAVEGUIDEDESIGNS for(a)diamondfinand(b)siliconfin.
Substrateleakage
(PML) boundary conditions are added to the simulation
Thesubstrateleakageasafunctionofbufferlayerthick- cell. Theseboundaryconditionsallowforabsorptionatthe
nessshowninFig.S2hasbeencalculatedusingtwodiffer- simulation cell edge with minimal reflections. The FDM
entmethods:coupled-modetheory(redpoints)andabsorb- solver can find complex eigenvalues [1], which can be re-
ingboundaryconditions(blueline). Forthecoupled-mode latedtothepropagationlossinthefollowingmanner:
theorycalculationstheunperturbedmodeiscalculatedus-
ing FDM without the high index substrate. Coupling of 4πIm n
α = { eff} (S.32)
theunperturbedmodetothesubstrateisthencalculatedby eff
λ
usingthefollowingoverlapintegral[2,3]:
ωε Bendingloss
α = E ∆ε E = 0 E ∆n2(x,y)EdA
eff ∗
h | | i 4 ·
Z
(S.31) BendinglossofthestructuresinFig.S3iscalculatedin
where ∆n2(x,y) = n2 and the overlap region extends cylindricalcoordinateswithFDMusingthemethodin[11],
f
fromthebottomofthebufferlayerto inyˆ, andfrom wherethelossper90 bendasafunctionofbendradiusis
◦
−∞
to+ inxˆ. shown. Abuffer layerthickness oft = 2.5µmhas been
B
−∞ ∞
For the second method, the high-index substrate is in- used for both waveguides. The bending-loss limited Q-
cludedintheFDMcalculation,andperfectlymatchedlayer factor of a ring resonator is also shown in Fig. S3, where
5
Q = ω/γ,ω = ck isthecenterfrequencyandγ = αeffc. [3] A.YarivandP.Yeh,Photonics: opticalelectronicsinmod-
ThevaluesinFig.S03a,barecalculatedatλ =637nmNangd erncommunications(OxfordUniv.Press,NewYork,2007),
6thed.
λ =1.55µm,respectively.
[4] B. E. Saleh and M. C. Teich, Fundamentals of photonics,
vol.22(JohnWiley&Sons,NewYork,2007),2nded.
[5] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D.
MaterialSellmeierequations
Meade,Photoniccrystals:moldingtheflowoflight(Prince-
tonuniversitypress,2011).
ThecalculationsinthemaintextuseSellmeierequations [6] R. R. Grote, “Nanophotonics for optoelectronic devices:
to model the various material refractive indices. The pa- Extrinsic silicon photonic receivers and organic photo-
rametersforeachmaterialaregivenintableIandareused voltaics,” dissertation, Columbia University in the city of
inthefollowingequation[4]: NewYork(2014).
[7] P.-R.Loh,A.F.Oskooi,M.Ibanescu,M.Skorobogatiy,and
S. G. Johnson, “Fundamental relation between phase and
A λ2
n2 = 1+ i (S.33) group velocity, and application to the failure of perfectly
λ2 λ2 matchedlayersinbackward-wavestructures,”Phys.Rev.E
i − i
X 79,065601(2009).
For the designs in Fig. 4 of the main text, the dia- [8] J.B.Driscoll,X.Liu,S.Yasseri,I.Hsieh,J.I.Dadap,and
mond material stack has the following refractive indices R.M.Osgood,“Largelongitudinalelectricfields(ez)insil-
at 637nm: n = 2.41, n = 2.01, and n = icon nanowire waveguides,” Opt. Express 17, 2797–2804
diamond Si3N4 SiO2
1.46asdeterminedfromSellmeierequationsforeachma- (2009).
terial. The silicon material stack has the following refrac- [9] R.MildrenandJ.Rabeau,Opticalengineeringofdiamond
tive indices at λ = 1.55µm: n = 3.48, n = 1.98, (JohnWiley&Sons,2013).
Si Si3N4 [10] T. Ba˚a˚k, “Silicon oxynitride; a material for grin optics,”
andn = 1.44.
SiO2 Appl.Opt.21,1069–1072(1982).
[11] W. W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, and W.-P.
Huang,“Full-vectorialwavepropagationinsemiconductor
opticalbendingwaveguidesandequivalentstraightwaveg-
uideapproximations,”IEEEJ.LightwaveTechnol.16,910
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