Table Of ContentOptical resonators based on Bloch surface waves
Matteo Menotti1 and Marco Liscidini1
1Department of Physics, University of Pavia, Via Bassi 6, I-27100 Pavia, Italy∗
compiled: January29,2015
5
Afewrecentworkssuggestthepossibilityofcontrollinglightpropagationattheinterfaceofperiodicmultilayers
1
supportingBlochsurfacewaves(BSWs),butopticalresonatorsbasedonBSWsareyettodemonstrate. Herewe
0
discussthefeasibilityofexploitingguidedBSWsinaringresonatorconfiguration. Inparticular,weinvestigate
2
the main issues related to the design of these structures, and we discuss about their limitations in terms of
n quality factors and dimensions. We believe these results might be useful for the development of a complete
a BSW-based platform forapplication rangingfromoptical sensingto the studyof the light-matter interaction
J inmicroandnanostructures.
8 OCIScodes: (230.5750)Resonators;(240.6690)Surfacewaves;(250.5300)Photonicintegratedcircuits
2 http://dx.doi.org/10.1364/XX.99.099999
]
s
c
i
t 1. Introduction High optical quality multilayers can be grown by
p
Light confinement near the surface of a photonic struc- molecularbeamepitaxy,butthey arealsocommercially
o
. ture is crucial in a number of applications involving the available from thin-film companies for many materials,
s light-matter interaction. For instance, it is appealing from semiconductor to oxides. More recently, the gen-
c
i for the realization of optical sensors, in which one re- eralfeaturesofguidedBSWhavebeencharacterizedthe-
s lies onthe interactionbetween the electromagneticfield oreticallyanddemonstratedexperimentally[12–14]. All
y
h and the analytes under investigation. In this respect, these results seem to suggest the opportunity for a gen-
p surface plasmon polaritons are probably the most stud- eration of optical devices based on BSWs. This could
[ ied and utilized. Here, the interactionbetweenthe light be a very powerful approach for the development of in-
and the free charges of the metal is exploited to achieve tegrated optical sensors, but also a new and versatile
1
v light confinement at a metallic surface and thus locally platform to probe interaction between photonic modes
5 increase the electromagnetic field intensity [1]. This ef- and electronic nanostructures in classical and quantum
2 fect is used, for example,to increasethe intensity ofthe optics, for instance in fundamental studies of the light-
0 Raman signal of a molecule or the efficiency of a fluo- matterinteractioninthestrong-couplingregime[15–19].
7 rescent marker bonded to a target molecule [2, 3]. Yet, All of this requires the demonstration of optical res-
0 light confinement near the surface can also be exploited onators, which are fundamental building blocks of any
.
1 to control its propagation in the plane through a nano- photonic platform, but are still lacking for BSWs struc-
0 structurizationofthesurface,whichallowsonetocreate tures.
5
waveguides and resonators with minimal fabrication re- In this work, we investigate the feasibility of realiz-
1
: quirements [4]. ing a ring resonator on the surface of periodic dielec-
v A promising strategy for such control and enhance- tric stacks supporting BSWs. In this structure the ver-
i
X ment of light is based on photonic crystal (PhC) struc- tical confinement is provided by exploiting the surface
tures supporting Bloch surface waves (BSWs). These state, and the lateral confinement is defined by a nano-
r
a are evanescent electromagnetic field modes that prop- fabrication of the surface, which could be done using
agate along the interface between a periodic dielectric either standard lithographic approaches or low-cost im-
stackandanhomogeneousmedium[5,6]. Theirconfine- printing techniques. In Ref. [13] Liscidini showed that
mentreliesontotalinternalreflectionfromthehomoge- this structure is extremely flexible and can support a
neousmediumandreflectionwithinaphotonicgapfrom variety of guided modes depending on the structure pa-
the multilayer. Although more complicatedthan metal- rameters and constituent materials. Very recently, Wu
lic systems, PhC-baseddevices are highly customizable, et al. reported on the fabrication and characterisation
and they do not suffer from the absorption losses that ofbentwaveguidessupportingBSWs,demonstratingthe
plague metallic structures [7–11]. feasibility of this geometry for more complex structures
[14].
Here we shall consider TiO /SiO multilayers and
2 2
Polymethyl-methacrylate(PMMA)ridgestoworkinthe
∗ [email protected] visible spectrum, for which the problem of designing
2
small and integrated resonators is typically more de-
manding than in the infrared wavelength range due to
the lack of transparent materials that can guarantee a
largerefractiveindexcontrast. Inthiscaseonecanwork
by etching a PhC crystalcavityin a membrane,but the
resultingstructure isusually veryfragile[20,21]. Alter-
natively, there have been demonstrated Hydex or SiN
high-qualityringresonatorsembeddedinasilicamatrix
[22, 23], but in this case the region in which one experi-
ences the largest light-matter interaction is not directly
accessible.
(b) z
While most of the theoretical tools necessary for the n ,h
PMMA
design of multilayers are already available, the design x
y n ,d
and optimization of BSW waveguides and ring res- TiO2 first
w
onatorsischallenging. Ontheonehand,asthemultilay-
n ,d
ered structure can be characterizedby numerous layers, SiO2 SiO2
even a hundred, whose thicknesses can be as small as a
n ,d
fewtensofnanometer,FDTDapproachescanbelargely TiO2 TiO2
timeconsuminganddifficulttoimplementwhenthesim-
ulation cell is particularly large and high spatial resolu-
tionisrequired,asinthisabove-mentionedcase. Onthe
otherhand,thetypicalsizeofthesimulationintheplane
of the multilayer,severalmicrometers,makesit difficult
to describe the structure also in the reciprocal space by
using Fourier Modal methods. Along the years, several
strategiestosolvefortheconfinedmodesinbentwaveg-
uideshavebeenproposed: basedonanumericalsolution
of Maxwell equations in terms of an eigenvalue problem Fig. 1. (Color online) (a) Sketch of the PhC ridge ring res-
[24], expansion in Hankel wavefunctions in the cladding onator. (b) Cross section of thecorresponding PhC ridge.
[25],conformalmappingofthebentwaveguide[26],per-
turbative approaches [27], beam propagation methods
[28], finite element discretization [29], etc...
2. Structure and theoretical approach
The structure we have in mind is a ring resonatorof ra-
Here,thedifferentmechanismsatthe baseofthecon-
diusRfabricatedonthetopofamultilayersupportinga
finementoflightintheverticaldirectionandintheplane
BSW(seeFig. 1(a)). Thelightconfinementintheverti-
ofthestructuresuggesttofollowastrategybasedmainly
caldirectionisduetothesurfacemode,whilethelateral
on effective index approaches, which reduce the dimen-
confinement is given by the PhC ridge, whose section is
sionality of the problem and thus are able to quickly
shown in Fig. 1(b). This device could be fabricated
explore several configurations in the parameter space.
using different kinds of dielectric materials, from semi-
In particular, our goal is to understand what are the
conductors to oxides, in principle to operate at a given
mainparametersthatlimitthequalityfactorachievable
workingpointinthewholespectralregion,withaband-
in an ideal BSW-based ring resonator when scattering
widthstronglydependentonthegeometricalparameters
losses due to fabrication imperfection can be neglected.
and material choice. The structure we have in mind is
Beside demonstrating the feasibility of a ring resonator
a good compromise between micro-disk resonators [30],
approach,theseresultswillserveasaguideinthedesign
whicharetypicallyrealizedinhigh-refractiveindexcon-
and development of a complete BSW-based platform.
trast platforms, and micro-pillar resonators [31], which
The paper is structured as follows: in Sec. 2 we arelessdemandingintermsofrefractiveindexcontrast,
describe the structure under investigation and the ap- buttheyarealsomoredifficulttofabricate. Thefeatures
proach to the calculation of the mode quality factor of of our structure are expected to be intermediate with
a PhC ridge ring resonator; in Sec. 3 we focus on the respectto the above-mentionedsystems: qualityfactors
analysisofthe intrinsic lossesofBSWwaveguidemodes largerthanthosetypicallyobservedinmicro-pillars(but
supportedbyaPhCridgeonafinitemultilayer;inSec. 4 smaller than what observed in micro-disks) along with
wedealwiththe bending lossesassociatedwiththe typ- flexibility in terms of fabrication and materials. In the
ical lateral confinement that can be obtained for modes following we will focus on a specific example to provide
in PhC ridges; in Sec. 5 we outline a general strategy a description of the main features of this resonator.
to design BSW-based ring resonators. Finally, in Sec. 6 Themultilayerisperiodic,withtheunitcellcomposed
we draw our conclusions. of two layers with thicknesses d = 0.085 µm and
TiO2
3
Fig. 1(a)). The resonant frequencies are obtained by
1
the usual condition [34]
1 0.9
2π
0.8 kL≡ n 2πR=2πm, (1)
mode
λ
0
0.7
0
where n is the realpartof the effective index of the
0.6 mode
)
m mode in the bent waveguide, R is the ring radius, and
( 0.5 m is an integer. In the absence of the xz-plane mir-
z -1
0.4 ror symmetry, a rigorous TE/TM classification of the
guided modes is no longer possible. However, in waveg-
0.3
uides with a rectangular cross section, the modes hav-
-2
0.2 ing the E component dominant over E and E [35]
r φ z
0.1 can be referred to as TE-like, and in the following we
shall always consider these class of solutions. Finally,
-3 0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 thisstructureischaracterizedbyacontinuousrotational
symmetry andtherefore,by choosingcylindricalcoordi-
y ( m)
nates (r,φ,z), the angular dependence of the fields can
Fig. 2. (Color online) Intensity profile in the yz-plane of be writtenin the formeimφ, with m integer. This prop-
a TE-polarized guided BSW calculated at λ0 = 0.630 µm. erty turns out to be extremely useful in the numerical
The PhC is made of N = 10 periods of a TiO2/SiO2 unit
description of the structure, as it allows one to reduce
cell (dTiO2 =0.085 µm and dSiO2 =0.128 µm) and the first the dimensionality of the electromagnetic problem, for
TiO2 layer is truncated with dfirst = 0.010 µm. The ridge
material is PMMA with a rectangular section (0.800 µm× example in FDTD calculations.
0.220 µm). The profile is obtained by means of a FDTD Our goal is the design of a ring resonator working
numerical simulation. at λ0 and the computation of the mode quality factor,
which can be used to measure the performances of the
device. In the case of a BSW-based ring resonator, this
dSiO2 =0.128 µm, respectively. The multilayer has a fi- task is all but simple, as an analytical approach is not
nitenumberN ofperiodsandistruncatedwiththefirst feasible, and even a brute force 3D numerical calcula-
layer made of TiO2 with thickness dfirst = 0.010 µm. tion, for example by using a FDTD method, can be
The structure is designed to operate in the visible spec- extremely time-consuming on a standard personal com-
tralrangeatλ0 =630nm(about1.97eV)withnTiO2 = puterormedium-sizedserver. Thus,intheviewofopti-
2.58534 and nSiO2 = 1.54270. The choice of these ma- mizing the structure, which depends on a large number
terials for the multilayer is convenient from a techno- of parameters (unit cell composition, multilayer trunca-
logical point of view, for they are commonly employed tion, height and width of the ridge, ring radius, etc...),
in the fabrication of optical filters by sputtering, thus one should rather look for an approximated but faster
this plan structure is commercially available. Finally, strategy.
we consider a PMMA (nPMMA = 1.48914) ridge with The structure and the mode profile in Fig. 2 seem
height h = 0.220 µm and width w = 0.800 µm on the to suggest a possible path towards the solution of the
top of the multilayer. problem. Indeed,thestrongfieldconfinementoflightin
The structure shown in Fig. 1(b) is symmetric the ridge waveguide indicates that the resonant modes
upon reflection with respect to the xz-plane. Conse- properties can be found by decomposing the 3D quest
quently, the guided modes can be classified according in two independent 1D and 2D problems. First, we de-
to the eigenvalues of the mirror operator σˆxz in TE scribethefieldconfinementduetothesurfacestateinan
(Transverse-Electric, σˆxz = −1) and TM (Transverse- effective multilayer,then we find the resonantmodes by
Magnetic, σˆxz = +1) modes. In the following we shall modellinganeffective2DsystemusingaFDTDmethod,
focusonlyonthefundamentalTEmode,whoseintensity which allows us to take into account also for the depen-
profile, calculated by FDTD [32, 33], is plotted in Fig. denceofn ontheringradiusR. Similarapproaches
mode
2. This mode is a perturbation of the BSW supported have been used to study lower-order [36] and whisper-
by the bare multilayer, and light is guided within the inggallerymodes[31]inmicro-pillarresonators,demon-
ridge likewise in a rib waveguide. It should be noticed strating surprising effectiveness even for R of a few µm.
that, unlike truly guided BSW, this mode can exist for Very recently, the same strategy has been successfully
any choice of the ridge height, with its properties and applied to describe light confinement and propagation
modal volume depending strongly on h [13]. Here the of BSWs in bent waveguides [14].
ridge height has been chosen to guarantee a good lat- It should be noticed that the mode quality factor Q
eralconfinement and, at the same time, keepthe modal depends on the lightleakage through the substrate, due
volume as small as possible. to the finite number N of periods in the multilayer,and
One can obtain a ring resonator by bending the the bending losses, which, for a given ridge, are a func-
straight waveguide onto itself in a circular shape (see tion of the ring radius R. One can start by considering
4
thatQmeasurestheenergylossexperiencedbytheelec- guidingregionsshape)andcalculatingineachregionthe
tromagneticmode per cycle and, assumingthat the two confinedmodeprofilesandtheireffectiveindices. Subse-
loss mechanisms are independent, can write quently, the structure in eachslice is replacedby an ho-
mogeneouslayerwitharefractiveindexn correspond-
eff
1 1 1
ing to that of a proper confined mode. Therefore, one
= + , (2)
Q Q⊥ Qk can treat the most significant light confinement mecha-
nism in each section exactly, and obtain an effective 1D
whereQ⊥ isthequalityfactordeterminedbytheout-of- structure along the complementary direction, where the
plane losses and Qk is the quality factor limited by the less intense localizationof the field takes place. This al-
in-plane losses. Thus, first we deal with the description lows the computation of the 2D field profile, the mode
oflightleakageinthesubstrate,thenwecomputeQk. It dispersion relation, and the propagationlosses.
should be noticed that here we are neglecting extrinsic One should note that the EIM predictions, although
losses, such as scattering at the interface between lay- approximated,aregenerallyveryclosetothe valuescal-
ersduetosurfaceroughnessorfabricationimperfection. culatedbymoretime-consuming2Dnumericalmethods.
In a real device these contributions can be important, However,the goodness of this approach depends on the
depending on the accuracy in the fabrication processes. structure geometry and field distribution.
3. Multilayer design and Q Referring to Fig. 2, it is clear that in our case the
⊥
field confinement is strong in both y and z direction.
Here we consider a sufficiently large ring radius R so
Forthis reason,weexpectthatdividing thestructurein
that the mode field profile and propagation wavevector
the verticalor in the horizontaldirection (although this
are slightly affected by the curvature, i.e. the effective
implies different approximations) should produce quite
index n in (1) is close to that of a straight waveg-
mode
similar and accurate results. In the following we will
uide (see also Sec. 4), and the vertical and horizontal
study the properties of the confined modes supported
dynamic can be decoupled. For this reason, the quality
by our structure employing both the strategies.
factorQ canbecomputedbydeterminingthepropaga-
⊥
tion losses of the corresponding guided BSW (GBSW)
3.A. Horizontal EIM
in a straight ridge waveguide having the cross-section
showninFig. 1(b). Inparticular,ifoneassumesasemi- Here we exploit the strengthof the verticalconfinement
infinite PhC, the mode is truly guided as there are no of the field, due to the PBG mechanism on the PhC
propagation losses. On the contrary, in a more realis- side and the TIR phenomenon at the PMMA/Air in-
tic situation, when the multilayer has a finite number terface, and hence we divide the structure along the z
N of periods, the GBSW is characterized by a finite direction. Following the EIM described above, one sub-
propagation length, even when scattering losses can be stitutesthePMMAridgewithaneffectivehomogeneous
neglected, aslightcantunnel troughthe multilayerinto layerofthicknesshandrefractiveindexneff,slab(ω),cor-
the substrate. respondingtothatofthefundamentalTMguidedmode
Theanalysisoflightconfinementintheyz planefora supported by a symmetric PMMA slab waveguide of
straight waveguide could be performed by using FDTD width w in air. Note that here TM refers to the plane
calculations or other 2D numerical approaches. Yet, in of the PMMA slab waveguide. This is equivalent to the
the spirit of suggesting a reliable but also fast approach horizontal cross section of the PMMA ridge (see Fig.
that would allow one to rapidly explore the parame- 3(a)), thus we shall refer to this method as the hori-
ter space of the structure, we adopt an effective index zontal EIM (HEIM). The effective index can be writ-
method (EIM) by separating the field confinement in ten as neff,slab = κeff,slabc/ω, where κeff,slab is the zero
the y and z direction. It should be noticed that this of the T22,slab(k,ω) element of the slab transfer ma-
approximationhasalreadybeen provedtobe reliableto trix. Finally, one can calculate the dispersion relation
study analogous structures [14, 37–40]. More recently, of the GBSW by finding the guided modes of this effec-
theaccuracyofthisstrategyforthedescriptionofguided tive multilayer, searching for the solutions of the equa-
modes in PhC ridges has been investigated in Ref. [13], tion T22,mul(k,ω) = 0. In general, these are solutions
but the issue of propagation losses has not been dis- of the kind κGSBW = β +iγ, where β = (ω/c)nGBSW
cussed. and γ = 1/(2Lprop), with Lprop the mode propagation
The EIM is conveniently employed to decouple the length.
field confinement in 2D electromagnetic problems, pro- In Fig. 3(b) we show the real part of the dispersion
vided that light is well confined in the ridge at least in curve corresponding to our parameters. As expected,
one dimension. Thus a 2D problem is reduced to the the mode dispersionrelation is within the PBG and be-
composition of two 1D tasks, easily solved by means of low the air light line. In particular, for λ0 = 630 nm
the transfer matrix method (TMM). we have β = 13.5164 µm−1 and nGBSW = 1.3553; the
First, the EIM fictitiously collapses the guiding 2D corresponding mode intensity profile is plotted in Fig.
structure in the direction where the most intense light 3(c).
confinement takes place. This is obtained by dividing For a sufficiently large N (typically more than 5, de-
the structure in 1D multilayers (in accordance with the pending on the refractive index contrast of the materi-
5
9
(a) neff,slab C 10
TE zz 108 TLiMneMa rin f it 108
7
10
xx 7
yy 10
) 6
m 10
6
( 10 Q
L 5
10
5
10
4
10
4
10
3
10
(b) (c) 5 6 7 8 9 10 11
4
Air light line 1 N
GBSW Fig. 4. (Color online) Propagation length L as a function
of the number of periods N in the multilayer, computed by
)3 0 meansoftheHEIM.Ontherightaxisthecorrespondingper-
V
(e pendicular quality factor Q⊥ (when Fig. 1(b) is the section
y m) of a ring resonator) is reported.
g
er -1 (
n2 z
E
also write Q =L /λ=β/(4πγ). In Fig. 4 we show
⊥ prop
-2 Q calculatedfromtheGBSWcomplexwavevector. As
⊥
expected,wehaveanexponentialgrowingofQ withN.
1 ⊥
It should be noticed that in our structure one can ob-
-3 tain propagation distances of about 10 m with only 10
10 12 14 16 18 20 0 1 periods. In particular, this increasing is well fitted by
-1 2
Wavevector ( m ) |Ey| (a. u.) a curve of slope (2αΛ) as suggested by Eq. (3). Thus,
our results show that a fast and reliable optimization of
Fig. 3. (Color online) (a) Sketch of the dimensionality re-
themultilayercanbedoneveryeasilybycomputingthe
duction of the electromagnetic problem byusing theHEIM.
imaginarypartoftheBlochwavevectorasafunctionof
(b) Real part of theGBSW dispersion relation. The yellow-
the sole unit cell composition.
shaded regions represent the PBGs. The dispersion relation
is calculated considering the modal dispersion in the first In Fig. 5 we plot the imaginary part of the Bloch
effective layer. (c) Ey field intensity in the truncated multi- wavevector as a function of β at λ0 = 630 nm for
layer. our structure parameters. We also indicate the work-
ing point corresponding to our final choice of the multi-
layer truncation and the ridge parameters. It should be
als), the dispersion relation is independent of the num- noticed that this point does not coincide with the curve
ber of periods of the multilayer. On the contrary, the maximum,whichwouldguaranteethefastestfielddecay
mode propagation length depends exponentially on N, inthemultilayer. Indeed,althoughoneusuallyoperates
for Lprop is directly related to the imaginary part α of in the region of large α, the BSW ring resonator is de-
the Bloch wavevector, which controls the exponentially signed to minimize also the bending losses in the ring,
damping of the electric field in the multilayer. In gen- which of course depend on the ridge parameters. Thus,
eral, one has in general, the optimal parameters do not necessarily
leadtothe fastestgrowingofQ withN,due toacom-
1 ⊥
Lprop ∝ exp[2NαΛ], (3) promise between vertical and lateral light confinement.
2
where α is 3.B. Vertical EIM
1 Tr[T ] The field intensity profile reported in Fig. 2 shows
Λ
α= arccosh , (4) that a strong confinement of the GBSW takes place
Λ (cid:18) 2 (cid:19)
also in the y direction, due to the TIR mechanism at
inwhichonehasappliedtheBloch-Floquettheoremtoa the PMMA/Air interfaces. Thus, one can obtain re-
generic PhC with period Λ and transfer matrix T [34]. sults similar to those shown in the previous section fol-
Λ
Since the quality factor of the resonator can also be lowing the EIM prescription and dividing the structure
viewedasthe number offield oscillationsin the ringbe- vertically in two different regions, named bare, where
forethefieldintensityisreducedbyafactor1/e,wecan the multilayer cladding is simply air, and loaded, in
6
5 (a) neff,loaded
HEIM n n
-1 m)4.4 TE zz eff,bare eff,bare
4 (4.2
VEIM
xx
4.103.0 13.5 14.0 yy
1 ) 3 Wavevector ( m-1)
- m
(
2
1
0=0.630 m (b) 3
Air light line
0 Bare
4 6 8 10 12 14 16 18 20 Loaded
-1
Wavevector ( m ) V)
e
Fig. 5. (Color online) Imaginary part of the Bloch wavevec- y (
tor in a PhC with TiO2/SiO2 unit cell (dTiO2 = 0.085 µm, erg2
dSiO2 = 0.128 µm), as a function of the wavevector. The En
curve is calculated at λ0 = 0.630 µm. The working point
correspondingtoourchoiceofthemultilayertruncationand
the ridge geometry, calculated both with theHEIM and the
VEIM, is reported.
1
8 10 12 14 16
which the multilayer has an additional layer of height
-1
h corresponding to ridge region (see Fig. 6(a)). We Wavevector ( m )
searchforthe TE surfacestates supportedby eachmul-
Fig.6. (Coloronline)(a)Sketchofthedimensionalityreduc-
tilayer, and we calculate the corresponding effective in-
tion oftheelectromagnetic problem byusingtheHEIM.(b)
dices n and n at λ . The straight waveg-
eff,bare eff,loaded 0 Real part of the dispersion curves for the bare and loaded
uide is finally approximated by a vertical slab of index
modes. The yellow-shaded region represents thePBG.
n with cladding of index n . Finally, one
eff,loaded eff,bare
searchesforthefundamentalTMmode,whosewavevec-
tor is κ =β+iγ. Note that here TM refers to the
GBSW
plane of the effective slab. We shall refer to this second
effectiveindexapproachasthevertical EIM (VEIM,see 8
also Ref.[13]). 10 C 108
TMM in
The dispersion curves associated to the bare and Linear fit
7
loaded multilayers are reported in Fig. 6(b), and 10 7
10
the effective indices calculated at λ = 630 nm for
0
N = 10 are n = 1.1225 + 2.0602 · 10−7i and 6
n =1.38e5ff3,b+ar2e.2347·10−9i,respectively. There- m) 10 106
suelfft,ilnoagdGedBSWwavevectorisκGBSW =(13.4332+1.2205· L ( 5 Q
10−7) µm−1. 10 105
In Fig. 7 we show Q calculated by means of the
⊥
4
VEIMasafunctionofthenumberofperiodsinthePhC. 10 4
10
Similarly to the HEIM case, Q increasesexponentially
⊥
withN,andtheslopeisveryclosetothatshowninFig. 3
10
4. Fora directcomparison,we reportthe workingpoint 5 6 7 8 9 10 11
obtainedwiththeVEIMinFig. 5alongwiththe HEIM
N
one. The two methods show a good agreement in the
Fig. 7. (Color online) Propagation length L as a function
prediction of both the propagation constant β and the
of the number of periods N in the multilayer, computed by
losses of the GBSW.
meansoftheVEIM.Ontherightaxisthecorrespondingper-
pendicular quality factor Q (when Fig. 1(b) is the section
4. Ring design and Q ⊥
k of a ring resonator) is reported.
Here, the calculation of the in-plane quality factor Q
k
is performed only by means of a combination of VEIM
7
and FDTD simulation in cylindrical coordinates, which
isanalogoustotheapproachadoptedinRef. [14],where 1.370
Straight (VEIM)
it is experimentally validated. First, the structure is di-
Straight (HEIM)
vided in bare and loaded multilayer regions (see Fig. 1.365
Ring (VEIM + Meep)
8(b)), with refractive indices n and n , re-
eff,bare eff,loaded x
e
spectively. This allows us to approximate the 3D struc- d 1.360
n
ture with a 2D ring resonator (see Fig. 8(a)). e I
v 1.355
(b) cti
e
Eff 1.350
n
eff,loaded
1.345
1.340
Air 5 6 7 8 9 10 11 12 13 14
n R ( m)
eff,bare
Fig.9. (Coloronline)EffectiveindexoftheGBSWcalculated
by means of the VEIM and 2D FDTD as a function of the
ring radius R. The values calculated with the HEIM and
theVEIM for thecorresponding straight waveguide are also
Fig. 8. (Color online) Top view of the circularly bent effec- shown.
tiveslab waveguide. Thelightblueregionrefractiveindexis
neff,loaded =1.3853 while thesurroundingmediumischarac-
terized by neff,bare =1.1225. N
12 11 10 9 8 7 6 5 4
The computation of Q aroundλ is performed using
k 0
Meep, a free FDTD simulation software package devel- 8
10
oped at MIT [41]. This code is characterised by the
possibility of exploiting the continuous rotational sym-
7
metry of the ring by working in cylindrical coordinates 10
andsolvingthe angulardependence ofthe fields analyt-
ically, according to eimφ, with m integer. This reduces 6
10
the calculation to a simple 1D simulation, with large Q Q
benefits in terms of the computational effort. 5
10
This approach takes into account the modification of
V
tItheismiondteereeffsteicntgivteoincdoemxpdauree ttohethreeswualtvsegcuaildceulbaetneddinfogr. 104 EIM HEIM
the straightridgeby means ofboth the HEIMor VEIM
3
with the value of the mode effective index in the case 10
5 6 7 8 9 10
of a bent waveguide. The comparison is shown in Fig.
9. As expected,for largerR,the effective FDTD results R ( m)
approach the mode index calculated with the VEIM. It
Fig.10. (Coloronline)LateralqualityfactorQ asafunction
k
should be noticed that in the range of R considered in
ofthebendingradiusRcalculatedbymeansoftheVEIMand
our work all the three values are very close, with differ- FDTD numerical simulations. The result is compared with
ences corresponding to an error in the determination of theperpendicular quality factors Q obtained in section 3.
⊥
theresonancefrequencylessthanthefreespectralrange
ofthe ringresonator. This confirms the goodnessofour
approximation for the parameters under consideration torsinsuchasmallringresonatorexceedsthosethatare
and highlights the possibility of determining the reso- experimentally observed in structures characterized by
nance frequency (but not the losses) by means of the a stronger refractive index contrast[42] or larger radius
analysis of the sole straight waveguide, even in the case R [43]. Naturally, these theoretical values are obtained
of relatively small R. by neglecting scattering losses and fabrication imper-
In Fig. 10 we show Q as a function of the ring ra- fections, which reduce the quality factor dramatically.
k
dius. Asexpectedfromastandardtreatmentofbending However,the qualityfactorscalculatedherearesohigh,
losses,Q increasesaboutexponentiallywithR. Wealso that even if scattering losses (and every other extrinsic
k
showQ versusN, so thatone canimmediately see the loss mechanism) will affect the resonator considerably,
⊥
essentialrequirementsinordertomeetagivenQtarget. the actual device would still display good light confine-
It should be noticed that the value of the quality fac- ment properties. For instance, let us consider a ring
8
resonator with N = 10 periods and radius R = 9 µm. erties of the mode supported by the PhC ridge. In this
For this structure the quality factor is Q ∼ 107 and communicationwehavedealtwithamodecharacterised
evenallowingthescatteringlossestolowerthisvalueby by a strong field confinement in the ridge region. How-
three orders of magnitude, we would still obtain a ring ever,aPhCridgecansupportalargervarietyofmodes,
resonatorwithQ∼104withinafootprintof∼250µm2. which all could be used for the realization of BSW ring
resonators (see e.g. [13]).
5. Design strategy and final remarks
With respect to the case investigated in this paper,
The design of a BSW-based ring resonator to achieve a
the position of the bare and loaded modes inside the
target quality factor is critically dependent on the con-
PBG at a given λ is crucial in the optimisation of
stituent materials, as well as on the PhC truncation, 0
both Q and Q . On the one hand, the best vertical
and on the ridge section. For this reason, it is useful ⊥ k
confinement of the surface state is obtained when both
to outline a basic strategy for designing and optimizing
modesarecloseto the PBGcenter(where αis highest).
BSW-based resonators.
On the other hand, one looks also for a large contrast
In the previous sections we have considered a specific
∆n = n − n , which guarantees better
TiO /SiO BSW ring resonator working at λ = 630 eff eff,loaded eff,bare
2 2 0 lateral confinement and small bending radii. These two
nm. Although the final structure depends critically on
conditions point in opposite directions in the parameter
the constituent materials and target wavelength, it is
space, thus one looks for a compromise between lateral
possible to outline a general design strategy to obtain
and vertical confinement. In our case, without any par-
structurestoworkinadifferentspectralregionandchar-
ticular applicationin mind, we havedesigned our struc-
acterised by a target quality factor and modal volume.
ture to maximize the effective index mismatch by mak-
The latter is essentially determined by the exponential
ing the bare mode effective index close to the air light
decay of the electromagnetic field in the multilayer and
line; the loaded effective index is engineered to be as
the ring radius. Despite the main goal of this work is
close as possible to the PMMA light line while main-
nottheoptimizationofastructureforaspecificapplica-
taining at the same time the vertical waveguide single
tion, one can be guided in the design of new BSW ring
mode. This results in a very compact BSW ring res-
resonators by a few simple rules.
onator, characterisedby large quality factors.
As we have seen, a BSW ring resonatoris made up of
The last step is the structure trimming by using
severallayersanddifferentmaterials,anditischaracter-
a more accurate fully-3D simulation, for example by
izedbyarathercomplexgeometrythatmakesthestruc-
means of FDTD. Yet, depending on the size of the fi-
ture modelling quite challenging. Indeed, it requires to
nal structure, this might be hard, if not impossible, on
perform several numerical simulations (looped over all
a standard personal computer.
the variables) to cover the vast parameters space. In
this regard, the EIM-based approach we have proposed
6. Conclusions
here is actually a valuable tool, suitable for this pur-
pose. Itallowstoobtaintherelevantphysicalproperties In this work we proposed an original resonator consist-
of the confined electromagnetic modes introducing min- inginaphotoniccrystalridgesupportingaguidedBSW
imal errors while requiring a very small computational and bent in a circular shape. We have studied its opti-
cost. Yet, it should be noticed that, even in this case, cal properties, including the dependence of the quality
one cannot explore such a large parameter space at the factoron both the ring size andmultilayer composition.
same time. We outlined a very general strategy that could be used
Once the target wavelength is chosen, one starts by to design and optimize BSW ring resonators made of
optimising the unit cell of the multilayer that will sup- different materials and operating in different spectral
port the entire structure. Given the materials compat- regions. This approach is based on assuming vertical
ible with the most accessible fabrication technique and andin-plane opticallossesasindependent, andcomput-
the specific application one has in mind, the best strat- ing the corresponding perpendicular quality factor Q⊥
egy is looking for strong refractive index contrast and and in-plane quality factor Qk by means of TMM and
transparency. Indeed,this willleadtoalargePBGthat effective 2D FDTD calculations. As an example, we de-
offers flexibility in the design of the PhC ridge and res- signed and analyzed a BSW ring resonator constituted
onator. ThiswillalsodeterminehighQ swithsmallN, of a PMMA ridge waveguide on a truncated periodic
⊥
for the largeris the index mismatch in the unit cell, the TiO2/SiO2 multilayer, operating at λ0 ≈ 0.630 µm,
faster is the field exponential decay in the multilayer. having a theoretical value of Q exceeding 107, and a
For the choice of the layers’ thickness, a good thumb- device footprint of ∼250 µm2.
ruleisconsideringquarter-wavestack,wherethis condi- Given the strong interest in BSWs for the realization
tion has to be satisfied at the working point below the of integratedopticalsensorsand as a tool to investigate
light line corresponding to the upper cladding material the light-matter interaction at the fundamental level,
and not at normal incidence as it is usually done in dis- and the recent experimental demonstration of guided
tributed Bragg reflectors. BSWsinbentwaveguides,webelievetheseresultsmight
The second step is choosing the multilayer truncation beofgreatinteresttothatpartofthecommunitywork-
and ridge parameters, which determine the main prop- ing in integrated optics.
9
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