S. Xiao and M. Qiu Vol. 24, No. 5 /May 2007/J. Opt. Soc. Am. B 1225
Optical microcavities based on surface modes intwo-dimensional photonic crystals andsilicon-on-insulator photonic crystals
Sanshui Xiao
Department of Microelectronics and Applied Physics, Laboratory of Optics, Photonics and Quantum Electronics,Royal Institute of Technology (KTH), Electrum 229, 16440 Kista, Sweden and MIC-Department of Micro and
Nanotechnology, NanoDTU, Technical University of Denmark, Building 345 East, DK-2800 Kongens Lyngby, Denmark
Min Qiu
Department of Microelectronics and Applied Physics, Laboratory of Optics, Photonics and Quantum Electronics,Royal Institute of Technology (KTH), Electrum 229, 16440 Kista, Sweden
Received July 3, 2006; revised October 27, 2006; accepted December 15, 2006;posted January 11, 2007 (Doc. ID 72634); published April 17, 2007
. INTRODUCTIONhotonic crystals (PhCs), which have periodic refractive
ndex modulations, are a new kind of artificialaterial.1–3 The most characteristic feature of a PhC is
he existence of a photonic bandgap where photons cannotropagate into the structure. By introducing different ar-ificial defects into photonic bandgap structures, manyunctional components can be implemented, e.g., ultras-all channel add and drop filters.4–6 In the past few
ears, much work has been devoted to the study of micro-avities in two-dimensional (2D) PhC slabs, which exhibithigh quality factor �Q�, as well as a small modal volume
V�. The quality factor of the microcavities in 2D PhClabs was thought to be limited by vertical loss, however,t has been shown that these structures can achieve aigh Q value if an appropriate design is used to reducehe vertical radiation.5,7–10 Based on waveguide-mode-gaponfined theory, several high-Q microcavities in PhCsave been realized, including a hexagonal cavity termi-ated by mode-gap waveguides and a double-hetero-PhCavity where the lattice constant was changed at thenterfaces.5,7 Recently, extremely high-Q microcavitiesre realized by a local waveguide width modulation,here the intrinsic Q values can reach up to 7�107.10
Surface waves are propagating electromagnetic waves,hich are bound to the interface between materials and
ree space.3 For dielectric interfaces, they usually do notxist on dielectric materials. However, due to the exis-ence of surface photonic bandgaps, it has been shownhat dielectric PhCs may support surface waves for someases, e.g., a truncated or deformed structure at the inter-ace. Many of the high-Q microcavities in 2D PhC slabs
roposed previously are based on the waveguide-mode-ap confined theory. Although surface-mode microcavitiesave been studied,11,12 they both focused on 2D PhC sym-etric slabs. In the present paper, we will study high-Qicrocavities based on surface modes in 2D PhCs and
ilicon-on-insulator (SOI) PhCs.
. MICROCAVITY REALIZED IN A TWO-IMENSIONAL SQUARE PHOTONICRYSTALirst we consider a 2D square PhC with dielectric rods inir. The permittivity of the rods is �=11.56, and the ra-ius of the rods is R=0.2a, where a is the lattice constant.urface defects at the PhC–air interface are introducedy reducing the radius of rods to Rd=0.15a, which is illus-rated in the inset of Fig. 1. The dispersion relation forhe TM surface modes is calculated using the 2D finite-ifference time domain (FDTD) method13 and is shown inig. 1, where the shadow regions are the projected bandtructure for TM slab modes. It can be seen from Fig. 1hat such a surface structure only supports one surfaceode. One nearly zero group-velocity surface mode can be
ound for k0=� /a and �0=0.3151�a /��. Suppose an opticalicrocavity is composed by the 2D square PhC, as shown
n Fig. 2(a). The gray rods, acting as reflecting mirrors inhe y direction, are the same as those interior rods. Theength of the cavity is denoted by L. Surface modes can gohrough the central surface but are terminated by the re-ecting mirror. The structure at hand can be considereds a conventional Fabry–Perot resonator whose resonantavelengths are given by � =2� /k , with the wave vector
n n
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1226 J. Opt. Soc. Am. B/Vol. 24, No. 5 /May 2007 S. Xiao and M. Qiu
n= �N�−��� /L, where n is the mode number and �� ishe phase shift associated with reflection from the bound-ry. For the PhC under study, the surface state exists onlyn a limited frequency interval, 0.2725�a /��0.3151 (seehe dispersion relation for the surface state in Fig. 1). Itollows from this dispersion relation, for a fixed cavityength L, that only some of modes with different waveectors are situated inside this interval, which will stay inhe cavity for a long time, i.e., become a surface resonantode. For L=10a, there are five surface resonant modes
elow the light line, namely, k1=6� /L, k2=7� /L, k38� /L, k4=9� /L, and k5=10� /L, if we assume �� islose to �. However, our simulation results show that theuality factor of the resonant mode corresponding to k5� /a, i.e., nearly zero group-velocity surface mode, is anrder of magnitude larger than those of cavity modes withther k vectors. Therefore, in this paper, we will alwaysocus on the resonant modes based on nearly zero group-elocity surface modes in PhCs.
ig. 1. (Color online) Surface band structure for the (10) surfacef a 2D square PhC of dielectric rods in air, where the radius ofods is 0.20a (a is the lattice constant), the dielectric constant ofhe rods is 11.56, and the radius of the rods at the surface is.15a. The inset shows the corresponding structure.
ig. 2. (Color online) (a)Schematic of a microcavity composed by2D square PhC. The gray rods, acting as reflecting mirrors in
he y direction, are the same as those interior rods. Length of theavity is denoted by L. (b) Resonant mode in a cavity of length=5a. Here we show the electric field cross section of the surface
esonant mode. (c) Resonant mode for L=11a.
Resonant modes are analyzed by 2D FDTDimulations,13 combining a boundary treatment of per-ectly matched layers. We excite the resonant mode with aaussian pulse and then monitor the radiative decay of
he field. The frequencies � and quality factors Q of theesonant modes are calculated using a combination ofDTD techniques and Padé approximation with Baker’slgorithm. In Fig. 2(b), we show the electric field crossection of a high-Q surface resonant mode in a cavity ofength L=5a. The mode has an angular frequency �0.3058�a /�� and quality factor Q=1.06�104. Similarly,
or the cavity length of L=11a, Fig. 2(c) shows a modeith �=0.3123�a /�� and Q=3.73�104. From the two fieldatterns, we infer that these two surface resonant modesre based on the same surface mode in the PhC. It can belearly seen from Figs. 2(b) and 2(c) that the resonantodes are governed by wave vector k0=� /a, which are re-
ated to the nearly zero group-velocity surface mode in thehC. All results presented above show that the high-Qesonant mode indeed has its origin in the nearly zeroroup-velocity surface mode in the PhC.
The results of Q and � of the surface resonant modes inhe square PhC microcavity with different cavity lengthL� and reflecting mirror width �W� are shown in Fig. 3.ote that we only consider the resonant mode, related to
he nearly zero group-velocity surface mode in the PhC.ith an increasing of the cavity length, the quality factor
or W=1a becomes larger, and the frequency of the sur-ace resonant mode converges to �0=0.3151�a /��, corre-ponding to that for the nearly zero group-velocity surfaceode. For such a microcavity, with no absorption by theaterial, Q is mainly determined by the reflection loss at
he interface between the surface defect rods and the mir-or rods in the y direction. To decrease the reflection loss,e also study the quality factor with a changing of theidth �W� of the reflecting mirror in the x direction, which
s also shown in Fig. 3. The frequencies of the resonantode or the microcavities with different width of reflect-
ng mirror [shown in Fig. 3(b)] do not change much, andonverge to �0 as the cavity length increases. Q increasesignificantly when enlarging the width of the reflectingirrors from W=1a to 2a. However, the changes to Q are
ig. 3. (Color online) (a) Quality factor �Q� is plotted as a func-ion of the cavity length �L� and the width �W� of the reflectingirrors of the microcavity in a 2D square PhC. (b) Resonant fre-
uency as a function of the cavity length and the width of the re-ecting mirrors of the microcavity.
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S. Xiao and M. Qiu Vol. 24, No. 5 /May 2007/J. Opt. Soc. Am. B 1227
uch smaller when increasing W from 2a to 4a. When theavity length increases, Q still becomes larger and willuctuate but the increases are in average. We note thathe high-Q microcavity only exists when there are surfaceodes at the interface. Our calculations show that the Q
alue significantly varies when changing the radius of theefect rods �Rd�. As an example, for a specific length �L11a� and width �W=3a� of the cavity, Q increases from.3�105 to 4.3�105, then to 2.4�106 when the radius ofhe defect rods increases from 0.14a to 0.16a in steps of.1a.
. MICROCAVITY REALIZED IN A TWO-IMENSIONAL TRIANGULAR PHOTONICRYSTALet us next consider a 2D triangular PhC with air holesxtending through a high-index ��=11.56� infinite-heightielectric material, which is shown in the inset of Fig.(a). The holes have a radius of 0.30a, and the distanceetween the right boundary (PhC–air interface) and theenters of the first right holes is d=�3a /2. The dispersionelation for such a triangular PhC TE surface mode alonghe K direction is shown in Fig. 4(a), where the shadowegions are the projected band structure for TE slabodes. One can see from Fig. 4(a) that there exist three
urface mode curves in the triangular PhC. For simplicity,e only consider the angular frequency between 0.2�a /��nd 0.25�a /��, where there is only one surface mode in aarge region. Two nearly zero group-velocity surface
odes (mode A with k1=� /a and mode B with k20.7� /a) can be found in our considered frequency region.s depicted above, we expect that there will be two high-resonant modes based on two nearly zero group-velocity
urface modes in a microcavity composed by the triangu-ar PhC. Now imagine an optical cavity composed by thehC, as shown in Fig. 4(b). The reflecting mirrors in the yirection are introduced by enlarging the radius �Rd� ofurface holes. Using the method mentioned above, we dond two high-Q surface resonant modes in the cavity.owever, here we only consider the high-Q resonantode based on the surface mode with k1=� /a.The quality factor and frequency of the resonant mode
s a function of cavity length and width of the reflectingirror are shown in Fig. 5. The curves of circles, squares,
nd diamonds represent the results for Rd=0.31a, 0.32a,nd 0.33a, respectively. The dotted curve in Fig. 5(b) rep-esents the position of frequency �0=0.2341�a /��, relatedo the nearly zero group-velocity surface mode with k1� /a in the PhC. Our results show that, when the cavity
ength increases, the frequency of the resonant mode con-erges to that of the nearly zero group-velocity surfaceode in the PhC. However, Q values fluctuate as the cav-
ty length L increases, as shown in Fig. 5(a). As describedn Ibanescu’s work,14 Q should be enhanced when 2kL+2�� is a multiple of 2�. Due to the complicated re-ection interfaces, as seen in Fig. 4(b), the reflectionhase shift �� is not always close to � for the mode. Wexpect, according to the resonant condition, that it is pos-ible to improve the Q value by slightly varying the sepa-ation of the reflecting mirrors, which is in agreementith our calculations. As an example, for the case of R
d
0.33a with the cavity length of 14a, the Q value is onlypproximately 2.8�104. By a careful design, the Q valueill reach approximately 2.0�105 when L=14.25a. Peaksf Q appear when is close to a multiple of 2�. If the cav-ty length is further increased, Q values will again fluctu-te but in average increasing. From Fig. 5(a), one also cannow that the quality factor generally decreases as theadius of the reflecting holes increases. As the radius ofeflecting holes increase, judged from the structure shownn Fig. 4(b), it can be naturally understood that the scat-ering loss will also be enhanced, i.e., the quality factorill decrease.
. MICROCAVITY REALIZED IN A SILICON-N-INSULATOR-BASED PHOTONICRYSTALoth microcavities proposed above are based on 2D PhCs,
n which there only exists one kind of mechanism for con-ning light in the plane, i.e., surface-mode-gap confinedheory. For most of the PhC microcavities in the litera-ure, the total quality factor is mainly limited by the ver-
ig. 4. (Color online) (a) Surface band structure for the K sur-ace of a 2D triangular PhC of air holes, see inset, extendinghrough a high-index ��=11.56� infinite-height dielectric. The ra-ius of holes is 0.3a and the distance between the centers of firstight holes and the right boundary of the dielectric is d=�3a /2.b) Schematic of a microcavity composed by a 2D triangular PhC.he reflecting mirrors in y direction are introduced by enlarginghe radius of surface color-filled holes.
ig. 5. (Color online) (a) Quality factor is plotted as a function ofhe cavity length and the radius �Rd� of the reflecting holes of theicrocavity in a 2D triangular PhC. (b) Resonant frequency as a
unction of the cavity length and the radius of the reflecting holesf the microcavity.
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ical loss in the third direction. It has been shown that anxtremely small mode-volume high-Q cavity can be real-zed by a 2D PhC membrane structure. It is our interesthether it is possible to realize a high-Q microcavity in aOI-based PhC structure.Here, we consider a 2D triangular lattice of air hole
hCs patterned on a SOI structure. The height of the sili-on layer is 0.6a, and the substrate (in this case, siliconioxide) is assumed to be of infinite thickness. Except forhese, the in-plane structure and the surface structure athe interface are exactly the same as the triangular caseiscussed above, i.e., Fig. 4(b). First consider the PhCsith the air hole height of 0.6a. The TE-like surface modelong the K direction for such a SOI structure is shownn Fig. 6, where the shadow regions are the projectedand structure for TE-like slab modes. It can be seen fromig. 6 that such a surface structure supports one surfaceode. Group velocity for the surface mode is nearly zero
or the wave vector k0=� /a. Suppose an optical microcav-ty, similar to the structure shown in Fig. 4(b), where theeflecting mirrors are introduced by enlarging the radiusf the air holes to 0.31a. Figure 7 shows the frequencynd quality factor of the resonant mode as a function ofhe cavity length. It shows that there is only one high-Qesonant mode as the cavity length varies and the reso-ant mode is related to the same zero-group-velocity sur-ace mode. From Fig. 7, one can see that Q increases ashe cavity length increases and that with increasing ofhe cavity length the frequency of the surface resonantode converges to �0=0.2518�a /��, corresponding to that
or the nearly zero group-velocity surface mode in thehC slab. For L=12a, Q is approximately 4.2�104.Most of the high-Q cavities existing in the literature
re based on 2D PhC membrane structures, which areind of fragile structures. It has turned out that Q valuesre significantly reduced for corresponding SOI-basedtructures. The SOI-based surface-mode cavity proposedbove is a promising solution, as it avoids fragile mem-rane structure and can be easily integrated withaveguiding devices using standard SOI technology.igh-Q cavity can also be easily realized, and the Q value
radually increases as the cavity length increases. Com-ared with the one-dimensional PhC cavity, although the
ig. 6. (Color online) Surface band structure for the K surfacef a triangular PhC slab with air holes in a SOI structure.
odal volume for the SOI-based surface-mode cavity isarger, the quality factor is an order of magnitude larger,hich is important for applications. Recently, we demon-
trated experimentally an optical filter based on side cou-ling between a silicon wire waveguide and a 2D PhCurface-mode cavity.15 The intrinsic Q is estimated to be000 while the theoretical value is �104.
. CONCLUSIONSn this paper, we have studied novel optical microcavitiesased on nearly zero group-velocity surface modes in 2Dhotonic crystals (PhCs) and SOI PhCs. The high qualityactor can be easily obtained for such surface-mode micro-avities. We have also studied the influence of the micro-avity length and the structure of reflecting mirrors to theesonant mode. It shows that, with increasing of the cav-ty length, the quality factor is gradually enhanced andesonant frequency converges to �0, corresponding to thator the nearly zero group-velocity surface mode in thehC.
CKNOWLEDGMENTShis work was supported by the Swedish Foundation fortrategic Research (SSF) through the INGVAR program,he SSF Strategic Research Center in Photonics, and thewedish Research Council (VR). S. Xiao acknowledgesupport from the Danish Council for Strategic Researchhrough the Strategic Program for Young Researchersgrant 2117-05-0037).
Corresponding author S. Xiao can be reached by e-mailt [email protected].
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ig. 7. (Color online) Frequencies and quality factors of theesonant mode as a function of the cavity length. The microcavitys composed by a 2D PhC slab in a SOI structure with air holeeight of 0.6a.
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