Design of a Ultrahigh-Q Hybrid Nanocavity on aMultiheterostructure Photonic Crystal

Ashfaqul Anwar Siraji, Student Member, IEEE and M. Shah Alam, Senior Member, IEEE

AbstractWe designed an ultrahigh-Q hybrid cavity based onthe concept of multiheterostructure and space modulation. Firstwe demonstrated an analytical method of designing the desiredmultiheterostructure and then applied space modulation to formthe hybrid cavity. We studied the confinement mechanism of thehybrid cavity for different extent of the space modulation andcalculated the resonant characteristics of the cavity using finitedifference time domain method. We found that the hybrid cavitycan sustain two different modes using two different confiningmechanism with similar Q. We also investigated the disorderstability of the cavity and demonstrated that both the modes ofthe hybrid cavity are equally insensitive to position disorder.

Index TermsIEEEtran, journal, LATEX, paper, template.

I. INTRODUCTION

Formation of nanocavities with very high quality factor(Q) utilizing photonic crystal (PC) double heterostructure(DH) has been on the spotlight for the past decade. Thisis because DH nanocavities can be easily incorporated intoplanar photonic circuits and, unlike defect based cavities,these types of cavities are relatively insensitive to fabricationerrors. Generally, DH nanocavities are formed by perturbingthe photonic crystal lattice in a strip like region withinwhich the resonant mode remains confined. Welna et al. hasdemonstrated a DH cavity with improved disorder stability byengineering the dispersion of a PC line-defect waveguide [1].Mock et al. has demonstrated a DH cavity with two additionalairholes within the line defect [2]. We have demonstrateda tunable nanocavity based on rectangular lattice that cansustain both TE and TM mode [3]. Kuramochi et al. hasdesigned a DH cavity by localy modulating the width of aline defect [4]. Recently, several ultrahigh Q nanocavities havebeen demonstrated that take the idea of double heterostructureone step further and employ several photonic crystals withgradually perturbed lattice, forming a multiheterostructure.Cheng et al. has demonstrated PC cavity in GaN bulk usingfour successive heterostructure. Tanaka et al. has demonstrateda PC cavity which uses many successive heterostructuresto confine the resonant mode as gently as possible, whichled to very high Q [5]. In such multiheterostructure (MHT)nanocavities, the confinement along the line defect is carefullycontrolled to ensure gentle confinement. However, along thedirection perpendicular to the waveguide, the confinementremains abrupt.

The manuscript has been recieved on ...Ashfaqul Anwar Siraji and M. Shah Alam are with the Department

of Electrical and Electronic Engineering, Bangladesh University of Engi-neering and Technology (BUET), Dhaka-1000, Bangladesh (e-mails: aasir-aji100@yahoo.com; shalam@eee.buet.ac.bd)

In this work, we propose a hybrid nanocavity which utilizesthe concept of multiheteostructure [5] and space modulation[6]. By using MHT along the waveguide, a Gaussian envelopefor the resonant mode was obtained, whereas the confinementin the perpendicular direction is softened by space modulation.We use two dimensional(2D) finite difference time domainmethod (FDTD) with perfectly matched layer (PML) boundaryconditions to simulate the designed structure and calculate theresonant properties. In the 2D geometry, we study the TEmodes. The effect of space modulation on the MHT nanocavityis studied by varying the extant of modulation. To elucidatethe effect of space modulation on the confinement mechanism,the spatial fourier transform (SFT) of the resonant modes areused. Furthermore, we study the impact of position disorderon the hybrid cavity by calculating the resonant wavelengthswith added random position disorder.

II. DESIGN AND VERIFICATIONWe first design the MHT PC based on the workflow

presented in [5]. Although Tanaka et al. performed theircalculation in case of index guiding, the principle remainsequally valid incase of bandgap guiding. We start with a linedefect waveguide corresponding to a row of missing hole ina hexagonal lattice PC with lattice constant a, airhole radiusr = 0:35 a as shown in Fig. 1(a). The background materialis assumed to be Silicon (n = 3:4). The dispersion curve,calculated using 2D FDTD, is plotted against the real partof the wavevector along the waveguide (kx) in Fig. 1(b).The slope of this curve near the mode-edge is almost zero,indicating bandgap guiding as opposed to index guiding. Thedispersion curve is fitted by a Taylor series expansion of theterm (k 0:5), where k = kx. The fitted curve (as displyedin Fig. 1(b)) can be expressed as:

f = 0:2257 + 0:1541(k 0:5)2 + 0:7048(k 0:5)4; (1)where f is the frequency of the guided mode. Since theguiding mechanism of the waveguide is bandgap guiding,the dispersion is not quadratic unlike that presented in [5].We consider terms upto the fourth power, since the fit issufficiently close without terms of higher order. By substitutingk = 0:5 + iq in (1), we can obtain the complex dispersionrelation in the bandgap region.

f = 0:2257 0:1541q2 + 0:7048q4; (2)where q is the imaginary part of the wavevector. Now, to obtaina Gaussian resonant mode profile, the condition on q is q =Bx, where B is an arbitrary constant. From eq. (2), we obtain

fcut f = 0:1541B2x2 0:7048B4x4: (3)

(a) (b)

Fig. 1. (a) The structure of a hexagonal lattice photonic crystal waveguideformed by a row of missing holes.(b)The dispersion curve of the PCwaveguide.Here, fcut is the cutoff frequency of the waveguide and f is theresonant frequency of the cavity. Since the resonant frequencyof the cavity should be constant, the x dependent term in eq.(3) is fcut(x). Thus, fcut is different in different regions of thePC. Now, the cutoff frequency and and lattice constant of a PCis inversely proportional to each other. Hence, fcut(x)fcut(0) =

a(0)a(x) ,

where fcut(0) = 0:2257( ca ) is the cutoff frequency at x = 0(central region of the MHT) and a(0) is the lattice constantin the same region. The bound states of a cavity form nearthe stationary points of the dispersion curve [7]. Hence, wecan assume that the resonant frequency of the cavity will beclose to f = fcut(0). Furthermore, assuming small change inthe lattice constant (n) and two periods per PC, the distance(xn) of the of the nth PC from the center of the MHT isxn = (2n+ 0:5)a0. Thus, from eq. (3), we obtain

an =a0fcut(0)

fcut(0) + :154B2a20(2n+12 )

2 :705B4a40(2n+ 12 )4;

(4)which produces the lattice constant of the nth PC away fromthe center. In eq. (4), the constant B is arbitrary. We select avalue for B such that the variation in lattice constant remainswithin 4% of a0, so that our assumption of small n remainstrue.We designed a MHT nanocavity using eq. (4) and calculated

its resonant properties using 2D FDTD with fast fouriertransform (FFT). By gradually changing the lattice constant,the bandgap of the PC is gradually changed so that the guidedmode in the central PC of the MHT falls within the bandgap ofthe successive outer PCs. The normalized impulse response ofthe MHT nanocavity for TE mode is shown in Fig. 2(b). It canbe seen that the resonant peak (r) is at a = 0:24, which isvery close to the fcut(0) = 0:2257( ca ) predicted earlier. In theinset of Fig. 2(b), we show the decay of energy in the cavitywith respect to time when excited at the resonant frequency.From this, we calculate the Q to be 9 104 using the methodwe used earlier [3]. The resonant TE mode of the MHT, asshown in Fig. 2(a), is much more gently confined comparedto a double heterostructure cavity. This is confirmed by themomentum space profile of the magnetic field, which is verysharply confined, as shown in Fig. 2(c). The quality factorof this MHT nanocavity compares favourably with previousliterature as shown in Tbale I. The eq. (4) is designed toproduce a MHT nanocavity that has a Gaussian envelopealong the waveguide. In our designed cavity, the waveguideis along the x axis. The magnetic field profiles of the resonant

TABLE ICOMPARISON OF QUALITY FACTOR OF THE MHT NANOCAVITY WITH

PREVIOUSLY REPORTED VALUES.

Reference Material In plane Quality Factor

Makarova et al. [8] Si 396Kim et al. [9] GaN 550Mock et al. [2] Si 3:37 105Siraji et al. [3] BaTiO3 2800This work Si 9 104

(a) (b)

(c) (d)

Fig. 2. (a) The structure of the designed MHT nanocavity. The resonant TEmode is shown superimposed on the structure.(b) The normalized impulseresponse of the MHT cavity. The decay of energy with respect to time in thecavity when excited by the resonant frequency is shown ni the inset. (c) Themomentum space profile of the magnetic field. (d) The magnetic field of theresonant mode along the x and z axes along with corresponding least errorfit. The rms error of fit is shown with respect to n in the inset.

TE mode along the x and z axes are shown in Fig. 2(d).Using the formalism used in [3], we calculate the least errorfit of the envelopes of the magnetic fields using a formH = e(ax

n+bxn+1); where H is the magnetic field, n is theperformance parameter and a; b are fitting parameters. We firstdetermine the n that produces the least error. In the inset ofFig. 2(d), we show the rms error in the fitting the envelopes bythe mentioned form. In Table II, the vales of fitting parametera and b for the values of n in case of fitting the magnetic fieldprofile along the x axis is given. From the table, it can beconcluded that along the x axis, the envelope of the magneticfield becomes H = eax

2

, which is the desired Gaussianenvelope. However, along the z axis, the least error fit requiresH = eax

4

implying a more abrupt confinement along the zaxis.

III. FORMING THE HYBRID CAVITY

We apply space modulation along the z axis to the airholesimmediately around the MHT nanocavity. First, we applyspace modulation to the three holes in the center PC (PC0).Then, we apply space modulation to the two airholes im-mediately around the PC0(PC1). After that, we apply space

TABLE IIRMS ERROR IN FITTING THE MAGNETIC FIELD PROFILE ALONG THE X

AXIS

n n+1 error a b

0 1 0.09058 2:697 1013 0.29241 2 0.08057 4:518 107 0.093012 3 0.07908 0.09301 4:426 10113 4 0.0869 0.0291 3:812 10124 5 0.099 0.00947 7:323 1013

Fig. 3. The normalized impulse responses of the MHT cavities formed byapplying space modulation on the PC0, PC1 and both.

modulation to both PC0 and PC1. In the Fig. 3, the normalizedimpulse response of the space modulated cavity is shownfor the mentioned cases. It can be seen that the resonantwavelength shifts from a = 0:246 to

a = 0:259 with a Q

= 1:242 105 when only the PC0 is modulated. Again, theresonant wavelength shifts from a = 0:246 to

a = 0:247 with

a Q = 8:47104 when only PC1 is modulated. However, whenboth PC0 and PC1 is space modulated, two resonant peakscan be found. The peak at a = 0:261 shows Q = 1:121 105and the peak at a = 0:2588 shows Q = 1:084 105. In thiscase, two different confinement mechanism becomes activesimultaneously, as explained in the subsequent paragraphs. Asa result, two separate resonant modes of equally high Q canbe observed.In Fig. 4, the magnetic field profiles and corresponding

SFTs of the resonant modes in Fig. 3 are shown. It can beseen that the mode profiles of the space modulated cavity ismuch different from the unmodulated cavity. To investigate,we observe the spatial fourier transform (SFT) of the modeprofiles shown along side the corresponding modes. The SFTin Fig. 4(a) resembles the SFT of a donor defect cavitydemonstrated by Srinivasan et al. in [10], with spreading ofenergy in wavevectors. We have mentioned previously thatthe waveguide in the MHT can be thought of as a seriesof line defects when viewed along the z axis. When onlyPC0 is space modulated, the defects in PC0 act as a defectcavity and the heterostructure becomes irrelevant because mostof the energy is confined within the PC0. When the PC1is modulated, the heterostructure between the PC0 and PC1comes into play along with the defect cavities formed inPC1. It can be seen in Fig. 4(b) that the magnetic fieldprofile of the MHT cavity when PC1 is modulated contains

(a) (b)

(c) (d)

Fig. 4. The magnetic field profiles and corresponding SFT of the resonantTE modes when (a) PC0 is space modulated, (b) PC1 is space modulated.When both PC0 and PC1 are space modulated, two resonant modes can beobserved. The magnetic field profiles and corresponding SFTs of the modesat (c) a

= 0:2588 and (d) a

= 0:261 are shown.

sharp peaks in the middle due to the defect cavity as wellas the trailing tails due to successive heterostructures. TheSFT of this magnetic field profile contains the sharp peaksthat characterizes the confinement due to multiheterostructureand the spreading of energy that characterizes a defect cavity.Evidently, when just PC1 is modulated, the confinement is dueto both multiheterostructure and defect. Now, when both PC0and PC1 is modulated, two peaks in the normalized impulseresponse can be observed in Fig. 3. In this case, a L7 cavity isformed due to equal space modulation in PC0 and PC1. Again,this same cavity can act as a MHT cavity because of presenceof successive PCs with gradually changing lattice parameter.From Figs. 4(c) and 4(d), it can be observed that the modewith higher wavelength shows a resonant magnetic field profilewith a much gentler confinement with a Q = 1:084 105.Observing the resonant field profile, it can be said that thismode is resonant due to the presence of MHT. To verifythis, we observe the SFT of this mode which shows clearresemblance to the one displayed in Fig. 2(c). The mode withlower wavelength in Fig. 4(d) shows much more compactconfinement, with a Q = 1:21 105. From the resonant fieldprofile, it can be said that this mode is caused by the defectcavity. This can be verified by observing the SFT of this modewhich clearly resembles the mode SFT corresponding to adefect cavity shown in [10]. Thus, when both PC0 and PC1 isspace modulated, the resultant cavity can sustain two modesof similar Q with different wavelengths and different confiningmechanism. Hence, This cavity is the hybrid cavity.

IV. IMPACT OF SPACE MODULATION

A. Extent

B. Depth

It is now evident that the extent of space modulationhas clear impact on the confinement mechanism of a spacemodulated MHT cavity. To investigate the effect of the depth

(a)

(b)

Fig. 5. (a)The resonant wavelengths and Q of the cavity for increasingnumber of modulated PC layers. (i) Q of the modes caused by the defectcavity. (ii) Q of the modes caused by the MHT. (iii) The resonant wavelengthof modes caused by the defect cavity. (iv) The resonant wavelength of modescaused by the MHT. (b) The difference between the two resonant wavelengthof the space modulated MHT cavity for increasing number of modulated PClayers..

of space modulation on the resonant properties of the hybridcavity, we studied the hybrid cavity with increasing modulationdepth (D). The change in resonant wavelengths and Q againstincreasing modulation depth (as fraction of the lattice constant)is shown in Fig. 6(a). It can be seen that all the resonantcavities and quality factors decrease, although the Q factors ofthe hybrid cavity always remain higher than the unmodulatedMHT cavity. At deeper space modulation, the transition alongz axis becomes too abrupt which reduces the quality factor.Also, at higher space modulation, the two competing confine-ment mechanism becomes increasingly detuned, as evident bythe fact that the difference between the resonant wavelengthsof the cavity decreases with increasing space modulation(Fig. 6(b)). Thus, the modulation depth has a measure ofcontrol over the competition between the MHT cavity anddefect cavity. However, modulation depth has no effect on

(a)

(b)

Fig. 6. (a)The resonant wavelengths and Q of the cavity against increasingmodulation depth. (i) Q of the modes caused by the defect cavity. (ii) Q of themodes caused by the MHT. (iii) The resonant wavelength of modes causedby the defect cavity. (iv) The resonant wavelength of modes caused by theMHT. (b) The difference between the two resonant wavelength of the spacemodulated MHT cavity against modulation depth.

the mechanism themselves. In Fig. 7, the resonant modes ofthe cavity and corresponding SFTs for two modulation depthD = a=25 and D = a=10 are shown. The salient featuresof the field profiles and their SFT remain unchanged for twodifferent modulation depths.

V. DISORDER STABILITY

Most of the times, due to fabrication error some uncertaintyon the design parameters of a PC cavity is introduced. Forexample, all the air holes may not be of same radius or allthe airholes may not be in the exact position specified by thedesign. Defect cavities are specially sensitive to this type oferrors, which is quite problematic. To investigate the disorderstability of the designed space modulated MHT cavity, wecalculated the impulse response of the cavity after introducingdisorder in the positions of the airholes. We performed separatecalculations for added disorder with standard deviation ()

(a) (b)

(c) (d)

Fig. 7. The magnetic field profiles of both resonant modes of the cavityfor D = a=25 caused by (a) defect cavity and (b) MHT cavity. The sameprofiles for D = a=10 are shown in (c) and (d) respectively. In the insetcorresponding SFTs are shown.

Fig. 8. The normalized impulse response of the cavity with added disorder.Results are shown for disorders of increasing standard deviation.

of = a=10, = a=5 and = 3a=10 respectively. Theresults are shown in the Fig. 8. From the figure, it is evidentthat despite increasing disorder, the space modulated MHTcavity continues to demonstrate two separate resonant modeswhose wavelengths remain very close despite the disorder. Theresults are summarized in the Table III. It can be seen that bothresonances show remarkable robustness against the disorderdespite their different confinement mechanism.

TABLE IIIPERCENT CHANGE IN THE RESONANT WAVELENGTHS OF THE CAVITY.

Standard Deviation % 1 % 20:1a 0.62 % 0.181 %0:2a 1.78 % 3.44 %0:1a 2.51 % 1.734 %

VI. CONCLUSION

In this work, we have demonstrated an analytical designflow for designing MHT cavity with gaussian field profileeven when the waveguide dispersion is nonquadratic. Then weintroduced space modulation along the direction perpendicularto the waveguide in the MHT, which resulted in a higher Q. Weinvestigated the effect of space modulation on the MHT andfound out that if only the central PC is space modulated, thecavity acts as a donor defect cavity. When the PCs immediatelybesides the central PC are space modulated, the cavity actsas a mix between defect cavity and MHT cavity. But whenthe central PC and those immediately besides it are spacemodulated, the cavity acts as a hybrid cavity with two differentconfinement mechanisms, two different resonant wavelengthsand resonant field profiles and two similarly high Q. Themodes of this hybrid cavity becomes increasingly detunedat higher modulation depth, but their Q doesnt fall below5 104. We also investigated the disorder stability of thecavity by simulating the cavities with added position disorder.We found that both modes of the hybrid cavity are highlyinsensitive to disorder in position. Since this cavity can sustaintwo controllably coupled high Q modes with stability againstfabrication disorder, this type of cavities can be used in lasers,switching and in planar photonic circuits.

REFERENCES[1] K. Welna, S. Portalupi, M. Galli, L. OFaolain, and T. Krauss, Novel

dispersion-adapted photonic crystal cavity with improved disorder sta-bility, IEEE Journal of Quantum Electronics,, vol. 48, no. 9, pp. 11771183, 2012.

[2] A. Mock, L. Lu, E. Hwang, J. OBrien, and P. D. Dapkus, Modalanalysis of photonic crystal double-heterostructure laser cavities, IEEEJournal ofSelected Topics in Quantum Electronics,, vol. 15, no. 3, pp.892900, 2009.

[3] A. Siraji and M. Alam, A tunable photonic double heterostructure cavityon ferroelectric barium titanate, IEEE Photonics Technology Letters,,vol. 25, no. 17, pp. 16761679, 2013.

[4] E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, andT. Watanabe, Ultrahigh-q photonic crystal nanocavities realized by thelocal width modulation of a line defect, Applied Physics Letters, vol. 88,no. 4, p. 041112, 2006.

[5] Y. Tanaka, T. Asano, and S. Noda, Design of photonic crystal nanocav-ity with q -factor of 109, Journal of Lightwave Technology,, vol. 26,no. 11, pp. 15321539, 2008.

[6] A. Siraji, M. Alam, and S. Haque, Impact of space modulation onconfinement of light in a novel photonic crystal cavity on ferroelectricbarium titanate, Journal of Lightwave Technology,, vol. 31, no. 5, pp.802808, 2013.

[7] A. Mock, L. Lu, and J. D. OBrien, Spectral properties of photoniccrystal double heterostructure resonant cavities, Optics Express, vol. 16,no. 13, pp. 93919397, 2008.

[8] M. Makarova, Y. Gong, S.-L. Cheng, Y. Nishi, S. Yerci, R. Li, L. Dal Ne-gro, and J. Vuckovic, Photonic crystal and plasmonic silicon-basedlight sources, IEEE Journal of Selected Topics in Quantum Electronics,,vol. 16, no. 1, pp. 132140, 2010.

[9] D.-U. Kim, S. Kim, J. Lee, S.-R. Jeon, and H. Jeon, Free-standing gan-based photonic crystal band-edge laser, IEEE Photonics TechnologyLetters,, vol. 23, no. 20, pp. 14541456, 2011.

[10] K. Srinivasan and O. Painter, Momentum space design of high-qphotonic crystal optical cavities, Optics Express, vol. 10, no. 15, pp.670684, 2002.

Ashfaqul Anwar Siraji received the B.Sc. Eng. degree in electrical and elec-tronic engineering from Bangladesh University of Engineering and technology(BUET) in 2012. His current research interests are photonic crystal resonators,quantum phenomena in nanostructures and plasmonic nano-structures.

M. Shah Alam (SM04) received the B.Sc. Eng., M.Sc. Eng., and the Ph.D.degrees in Electrical and Electronic Engineering in 1989, 1994, and 1997,respectively. He was awarded a gold medal for outstanding performancein B.Sc. Engineering examination. He received the Japanese GovernmentScholarship from April 1991 to March 1997 for pursuing his graduatestudies, and received his Ph.D. degree from Hokkaido University, Sapporo,Japan. In 1997, he was a visiting researcher in Electrotechnical Laboratory,Tsukuba, Japan. Then in 1998, he became a lecturer in the Department ofElectrical and Electronic Engineering, Bangladesh University of Engineeringand Technology (BUET), Dhaka, Bangladesh, where he is now a Professor.During 2003-2004, he was on postdoctoral study leave from BUET andworked as a research fellow with the photonics research group in CityUniversity London, UK. His current research interests include optical fibers,photonic crystal fibers, nonlinear properties in fibers, electrooptic modulators,the application of numerical techniques to guided wave photonics problems,and microwave integrated circuits.

Dr. Alam is a senior member of IEEE, and member of BangladeshComputer society and the Institution of Engineers, Bangladesh.