Designing coupled-resonator optical waveguides based on high-q tapered grating-defect resonators

Transcription

1 Desinin coupled-resonator optical waveuides based on hih-q tapered ratin-defect resonators Hsi-Chun Liu * and Amnon Yariv Department of Electrical Enineerin, California Institute of Technoloy, Pasadena, California 925, USA * hliu@caltech.edu Abstract: We present a systematic desin of coupled-resonator optical waveuides (CROWs) based on hih-q tapered ratin-defect resonators. The formalism is based on coupled-mode theory where forward and backward waveuide modes are coupled by the ratin. Althouh applied to stron ratins (periodic air holes in sinle-mode silicon-on-insulator waveuides), coupled-mode theory is shown to be valid, since the spatial Fourier transform of the resonant mode is enineered to minimize the couplin to radiation modes and thus the propaation loss. We demonstrate the numerical characterization of stron ratins, the desin of hih-q tapered ratin-defect resonators (Q>2 6, modal volume =.38 (λ/n) 3 ), and the control of inter-resonator couplin for CROWs. Furthermore, we desin Butterworth and Bessel filters by tailorin the numbers of holes between adjacent defects. We show with numerical simulation that Butterworth CROWs are more tolerant aainst fabrication disorder than CROWs with uniform couplin coefficient. 22 Optical Society of America OCIS codes: (4.478) Optical resonators; ( ) Coupled resonators; ( ) Photonic crystals; (3.748) Wavelenth filterin devices. References and links. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, Coupled-resonator optical waveuide: a proposal and analysis, Opt. Lett. 24(), 7 73 (999). 2. F. N. Xia, L. Sekaric, and Y. Vlasov, Ultracompact optical buffers on a silicon chip, Nat. Photonics (), 65 7 (27). 3. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, Continuously tunable byte delay in coupled-resonator optical waveuides, Opt. Lett. 33(2), (28). 4. F. N. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, Ultra-compact hih order rin resonator filters usin submicron silicon photonic wires for on-chip optical interconnects, Opt. Express 5(9), (27). 5. A. Melloni, F. Morichetti, and M. Martinelli, Four-wave mixin and wavelenth conversion in coupledresonator optical waveuides, J. Opt. Soc. Am. B 25(2), C87 C97 (28). 6. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, Transmission and roup delay of microrin coupledresonator optical waveuides, Opt. Lett. 3(4), (26). 7. M. Notomi, E. Kuramochi, and T. Tanabe, Lare-scale arrays of ultrahih-q coupled nanocavities, Nat. Photonics 2(2), (28). 8. T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, Planar photonic crystal coupled cavity waveuides, IEEE J. Sel. Top. Quantum Electron. 8(4), (22). 9. H. A. Haus and Y. Lai, Theory of cascaded quarter wave shifted distributed feedback resonators, IEEE J. Quantum Electron. 28(), (992).. S. Nishikawa, S. Lan, N. Ikeda, Y. Suimoto, H. Ishikawa, and K. Asakawa, Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects, Opt. Lett. 27(23), (22).. A. Martínez, J. García, P. Sanchis, F. Cuesta-Soto, J. Blasco, and J. Martí, Intrinsic losses of coupled-cavity waveuides in planar-photonic crystals, Opt. Lett. 32(6), (27). 2. P. Velha, E. Picard, T. Charvolin, E. Hadji, J. C. Rodier, P. Lalanne, and D. Peyrade, Ultra-hih Q/V Fabry- Perot microcavity on SOI substrate, Opt. Express 5(24), (27). 3. A. R. M. Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, Ultra hih quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI), Opt. Express 6(6), (28). # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9249

2 4. E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. G. Roh, and M. Notomi, Ultrahih-Q onedimensional photonic crystal nanocavities with modulated mode-ap barriers on SiO 2 claddins and on air claddins, Opt. Express 8(5), (2). 5. Q. M. Quan, P. B. Deotare, and M. Loncar, Photonic crystal nanobeam cavity stronly coupled to the feedin waveuide, Appl. Phys. Lett. 96(2), 232 (2). 6. Y. Akahane, T. Asano, B. S. Son, and S. Noda, Hih-Q photonic nanocavity in a two-dimensional photonic crystal, Nature 425(696), (23). 7. A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University Press, 27). 8. H. C. Liu and A. Yariv, Synthesis of hih-order bandpass filters based on coupled-resonator optical waveuides (CROWs), Opt. Express 9(8), (2). 9. J. W. Mu and W. P. Huan, Simulation of three-dimensional waveuide discontinuities by a full-vector modematchin method based on finite-difference schemes, Opt. Express 6(22), (28). 2. V. Van, Circuit-based method for synthesizin serially coupled microrin filters, J. Lihtwave Technol. 24(7), (26). 2. C. Ferrari, F. Morichetti, and A. Melloni, Disorder in coupled-resonator optical waveuides, J. Opt. Soc. Am. B 26(4), (29). 22. S. Mookherjea and A. Oh, Effect of disorder on slow liht velocity in optical slow-wave structures, Opt. Lett. 32(3), (27). 23. H. C. Liu, C. Santos, and A. Yariv, Coupled-resonator optical waveuides (CROWs) based on tapered ratindefect resonators, in CLEO (San Jose, USA, 22). 24. H. C. Liu, C. Santis, and A. Yariv, Coupled-resonator optical waveuides (CROWs) based on ratin resonators with modulated bandap, in Slow and Fast Liht (Toronto, Canada, 2), p. SLTuB2.. Introduction A coupled-resonator optical waveuide (CROW) consists of a sequence of weakly coupled resonators in which liht propaates throuh the couplin between adjacent resonators []. Both the bandwidth and the roup velocity are dictated solely by the inter-resonator couplin strenth. Such a unique waveuidin mechanism has found applications such as optical delay lines, optical buffers, optical bandpass filters, interferometers, and nonlinear optics [2 5]. CROWs can be realized with various types of resonators, such as microrins [2 6], photonic crystal resonators [7, 8], and waveuide-ratin resonators [9, ]. CROWs based on waveuide-ratin resonators are attractive for their natural implementation in waveuides. Gratin structures are defined on waveuides to chane the roup velocity of liht, requirin no additional desin for the couplin between waveuides and CROWs. The buildin block of ratin CROWs is a ratin-defect resonator where an artificial defect is introduced in a waveuide ratin. The defect cavity supports a mode with a resonant frequency inside the ratin band ap. The modal field is centered at the defect and evanesces exponentially in the ratin. A ratin CROW consists of a sequence of defects where adjacent resonators couple to each other via the evanescent field in the intervenin ratin. The couplin strenth depends on the product of the ratin strenth and the spacin between adjacent defects. Gratin CROWs based on the approximation of weak ratins can be analyzed with coupled-mode equations where two counter-propaatin modes are connected by the ratin [9]. When the ratin is stron, such as periodic air holes in a silicon waveuide, the lenth of each ratin-defect resonator can be as short as a few microns. Hih density of resonators is important for optical buffers since the delay-bandwidth product is proportional to the number of resonators. CROWs based on such small resonators have been experimentally demonstrated in silicon waveuides []. However, the major limitation was the intrinsic propaation loss due to radiation []. Hihly confined modes lead to lare spatial Fourier components which are phase-matched with the lossy radiation modes. The resultin low quality factor of the resonators (Q<) leads to power decay time constant of approximately picosecond, limitin applications such as optical delay lines. Because of the couplin to the hiher-order (radiation) modes, coupled-mode equations which consider only forward and backward uided modes are no loner valid. Consequently, the desin of ratin resonators based on stron ratins usually relies on three-dimensional simulation of the entire structures. In this paper, we propose to reduce the propaation loss of ratin-defect CROWs by desinin hih-q ratin-defect resonators as the buildin blocks. Hih-Q ratin resonators # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 925

3 have been demonstrated both theoretically and experimentally. Two major approaches are respectively based on taperin the ratin near the defect [2, 3] and spatially modulatin the ratin (period or hole radius) without a physical defect section [4, 5]. Both approaches aim to define a smooth modal field. Gaussian field distributions are well-known functions which reatly reduce the couplin to radiation modes [5, 6]. We present a systematic desin of hih-q tapered ratin-defect resonators where 4 or 6 periods on each side of the defect are tapered. We start with a numerical characterization of ratins with different hole radii. For a iven taper profile, we can determine the modal field based on the calculated ratin strenth as a function of hole radius and perform spatial Fourier transform of the mode. We determine the optimal taper profile which minimizes the couplin to radiation modes and confirm the results with numerical simulation of quality factor. When CROWs are based on hih-q resonators, the couplin to radiation modes is neliible, so the coupled-mode equations are valid. The coupled-mode formalism which we will present in Section 2 is useful for the analysis and desin of ratin-defect resonators and CROWs. After showin the systematic desin of hih-q tapered resonators in Section 3, we will demonstrate the control of inter-resonator couplin for CROWs in Section 4 and filter desin based on tailorin the couplin coefficients alon a CROW in Section 5. Filter desin not only optimizes the transmission and dispersive properties of CROWs but also improves the tolerance of CROWs aainst fabrication disorder, as will be discussed in Section 6. Finally, it is worth mentionin that we desin the resonators and CROWs to resonate at the Bra wavelenth of the ratin in order to ensure that the resonant wavelenth will not chane with the number of holes. If the resonant wavelenth is not at the Bra wavelenth, especially near the band ede, an extra phase section will be required when cascadin resonators for CROWs, as will be shown in the appendix. 2. Coupled-mode formalism for ratin CROWs A Bra ratin is a periodic perturbation to a waveuide. A ratin with a period Λ couples counter-propaatin modes with a propaation constant β if the phase-matchin condition is satisfied, i.e. 2π/Λ = 2β. The coupled-mode equations relatin the amplitudes of the forward mode a and the backward mode b are iven by [7] da = iδ a+ iκ * ( z) b dz, db = iδ b iκ ( z) a dz where δ β β B is the detunin from the Bra condition, β B π/λ, and κ (z) is the couplin coefficient of the ratin. The absolute value and phase of κ (z) represent the strenth and phase of the ratin respectively. κ (z) is a constant for a uniform ratin. If the ratin strenth is tapered, κ (z) varies alon the ratin. For a ratin structure distributed between z = and z = L and an input a() = from z =, the eneral approach of solvin the transmission and the field distribution is as follows: (i) Set the boundary condition at the output as a(l ) = and b(l ) = (no input from z = ). (ii) Propaate a and b from z = L back to z = analytically or numerically, usin Eq. (). (iii) Divide the resultin a(z) and b(z) by a() to recover the input amplitude a() =. A ratin-defect resonator is formed by insertin an artificial defect section in a ratin, as shown in Fi. (a). The defect section is a cavity where liht is lonitudinally confined by the () # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 925

4 (a) L defect L (b) L L2 3 L L N + κ ( z ) z κ ( z ) z 2 E z 2 E Fi.. Schematic drawins, couplin coefficients, and field intensity of (a) a ratin-defect resonator and (b) a ratin-defect CROW. z Bra ratins at frequencies inside the band ap ( δ <κ ). If the defect lenth is λ/(4n) (a quarter-wave-shifted (QWS) defect; n is the effective index of the waveuide mode), the phase of κ (z) is shifted by π, and the defect mode resonates at the Bra frequency (δ = ). The distribution of the modal enery, a 2 + b 2, is centered at the defect and evanesces exponentially in the ratin, as shown in Fi. (a). The modal fields (a and b) are proportional to exp( κ z), where z is the distance from the defect. A ratin-defect CROW consists of a sequence of defects, where adjacent defect modes interact with each other via their evanescent fields, as shown in Fi. (b). κ (z) alternates between κ and κ. The inter-resonator couplin is determined by the spacin between defects, denoted as L. For a ratin structure consistin of only QWS defects (i.e. a real κ (z)), the field distribution at δ = for an input a() = can be derived as and L where ( a( L ) cos h κ ( ') ' z dz ) L ( κ z ) a( z) = a( L )cos h ( z ') dz ' (2a) L ( κ z ) b( z) = ia( L )sin h ( z ') dz ', (2b) = is the transmitted amplitude. We consider an interresonator spacin L and L = L N + = L/2 at the boundary, which uarantees L ( z dz ) cos h κ ( ') ' = and thus unity transmission a(l ) =. The enery stored in the ratin, L 2 2 ( ) /, can be derived as L sin h( κ L) / κ L / v E = a + b dz v stored, where v is the roup velocity of the waveuide mode. Since the input power a() 2 is, the roup velocity in the ratin-defect CROW is iven by v L κ L = = (3), CROW v. Estored sin h( κ L) The slowin factor is a function of κ L. The time-domain inter-resonator couplin coefficient, denoted as κ, can be obtained by solvin the frequency splittin of two coupled defect resonators separated by L, which results in κ = κ v exp( κ L). (4) # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9252

5 with the assumption of exp( κ L) [9, 8]. The roup velocity at resonant frequency is thus v, = 2κ L= v 2κ L exp( κ L), which, in the limit of exp( κ L), is the same CROW as Eq. (3). Finally, we consider a sinle ratin-defect resonator, as shown in Fi. (a). The resonator is coupled to input and output waveuides via ratins of lenth L. The loss rate of the mode amplitude due to couplin to each waveuide is iven by τ = κ v exp( 2 κ L) [8]. The total loss rate is τ = 2τ e, and the quality factor of the resonator is e ω exp(2 κ L) Q= ωτ =. (5) 2 4κ v Q is an exponential function of L. Fiures 2(a) and 2(b) show the spectra of transmission and roup delay of -resonator ratin-defect CROWs with L = 2 µm and L = 3 µm respectively. We choose a roup index of 4 and a weak ratin with κ =./µm. The lenths of the first and last ratin sections, L and L N +, are chosen to be L/2 to match the CROW section to the waveuides [8]. Accordin to Eq. (4), the couplin coefficients of the two CROWs differ by a factor of exp(κ L) = e, which arees with the bandwidths and the roup delay shown in Fi. 2. (a).8 (b) 2 Transmission Delay (ps) f (GHz) f (GHz) Fi. 2. Spectra of (a) transmission and (b) roup delay of N = ratin-defect CROWs with inter-defect spacin L = 2 µm (blue) and L = 3 µm (reen). κ =./µm. 3. Hih-Q tapered ratin-defect resonators in silicon waveuides 3. Numerical characterization of Bra ratins Fiure 3(a) shows a Bra ratin in a sinle-mode silicon-on-insulator waveuide which is 49 nm wide and 22 nm thick. The ratin is composed of periodic air holes, which are etched throuh the silicon layer. For weak ratins, the evaluation of κ is usually based on perturbation theory An overlap interal of the perturbation ε(x,y) and the modal fields E(x,y) [7]. However, this method is not accurate for stron ratins, where the modal fields are stronly perturbed. The propaation constant of the waveuide mode is also stronly perturbed by the ratin. Therefore, for each hole radius r, the ratin period Λ needs to be determined for a iven Bra wavelenth (57 nm throuhout this paper). We desin ratin-defect resonators and CROWs to resonate at the Bra wavelenth for three reasons. First, the analysis based on coupled-mode equations is the simplest (δ = in Eq. ()). Second, the ratin strenth is maximal at the Bra wavelenth, which enables the shortest possible device lenth. Last, when cascadin resonators for CROWs, additional waveuide sections between adjacent resonators will be required if the resonant frequency is not at the Bra frequency (we show this in the appendix). # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9253

6 Given a ratin with a hole radius r and a period Λ, we can determine its Bra wavelenth and κ by simulatin the transmission and reflection of the ratin. For a ratin with a constant κ between z = and z = L and an input a() =, the phase of the reflected mode b() can be derived from Eq. () as θ r = π/2 sin (δ/κ ) if L is sufficiently lon. The Bra wavelenth (δ = ) can be obtained at the condition θ r = π/2. κ can be determined from the transmitted power at Bra wavelenth, a(l) 2 = /cosh 2 (κ L). We simulate ratin structures in silicon waveuides usin a 3D mode-matchin method (MMM) [9]. Air holes are divided into lonitudinal z-invariant sections. In each section, the total field is expressed as a superposition of the local eien-modes (both forward and backward modes), which are solved usin a finite-difference full-vectorial mode solver. Fields in adjacent sections are related by a scatterin matrix which obeys continuity of the tanential components of electric and manetic fields. The scatterin matrix of the entire ratin structure is obtained by cascadin the scatterin matrix of each section. MMMs are especially efficient for periodic structures. Once the scatterin matrix of one unit cell is obtained, the entire ratin structure can be constructed quickly. One major difference between our simulation and conventional MMMs is that in order to account for every component of radiation loss when simulatin hih-q resonators, we use a complete set of modes. Therefore, we have to find a balance between accuracy and computation cost. This method is efficient compared to other simulation methods, especially when the ratin structure is lon. Fiure 3(b) shows the calculated κ (r) and Λ(r) for a Bra wavelenth of 57 nm. Since the area of holes is proportional to r 2, κ (r) is quadratic at small radii. At larer radii, κ (r) becomes linear and the slope starts decreasin, since κ corresponds to the first-order Fourier component of the ratin. On the other hand, the perturbation of the propaation constant corresponds to the constant term of the Fourier components, so Λ(r) is nearly a quadratic curve. Note that for a hole radius of nm, κ is.49/µm, which is 6% of the propaation constant and thus corresponds to a very stron ratin. (a) a() Λ a(l) (b).5 45 b() r κ (/µm) Period (nm) Radius (nm) Fi. 3. (a) Schematic drawin of a stron ratin in a silicon waveuide and its cross-section. (b) Simulated κ and ratin period as functions of hole radius 3.2. Desin of hih-q tapered ratin-defect resonators Gratin-defect resonators in stron ratins inevitably incur radiation losses. Althouh the resonant mode consists of only forward and backward waveuide modes, a and b, the spatial Fourier components which are phase-matched and thus couple to the radiation modes lead to radiation loss. Fiure 4(a) shows the modal field of a QWS resonator. The amplitude oscillation is due to the interference between the forward and backward modes, and the envelope decays exponentially with the distance from the defect (at z = ). As a result, the spatial Fourier transform consists of two lorentzian functions which are centered at the propaation constants of the forward and backward modes, as shown in Fi. 4(b). The Fourier components whose frequencies lie within the continuum of radiation modes ( n clad <k z /k <n clad ) lead to radiation # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9254

7 (a) Electric field z (µm) (b) Spatial Fourier spectrum -3-6 radiation modes k /k z Fi. 4. (a) Field distribution of a QWS resonator mode. κ =.75/µm (b) Spatial Fourier transform of the QWS resonator mode. loss. The loss associated with a hihly localized mode is especially lare, since the lorentzian functions are broad. To reduce the radiation loss, we can enineer the modal field so as to minimize its spatial Fourier components within the radiation continuum. For example, if the envelope of the field is a Gaussian function, the Fourier transform consists of two Gaussian functions, which decay much faster than lorentzian functions. In other words, the smoother Gaussian envelope leads to narrower functions in the spatial frequency domain. The modal field is controlled by the z ratin strenth κ (z), since the envelope is proportional to exp κ ( ') ( '), z d z where z is the distance from the defect. If Gaussian distribution is desired, κ ( z) should be linear. A iven profile of ratin strenth κ (z) can be realized by the choice of the hole radii. Fiure 5(a) shows a tapered ratin-defect resonator where the 6 nearest unit cells on each side of the defect are tapered. Both the radii and the periods are varied to ensure the same Bra wavelenth, 57 nm. The defect lenth d is chosen to be λ/(4n eff ) = 62.5 nm, where n eff is the effective index of the waveuide mode. We choose r = nm for the reular holes, which corresponds to Λ = 43 nm and κ =.49/µm accordin to Fi. 3(b). For the tapered ratin, we assin κ,i = κ [i/(n t + )] α for the i-th hole, where n t is the number of tapered holes. The radius r i and period Λ i of each unit cell are determined based on the curves in Fi. 3(b). κ (z) is a step function which is constant within each unit cell. If α =, κ (z) is approximately linear and leads to a Gaussian field distribution. We consider ratin-defect resonators with 4 and 6 tapered holes respectively. The objective is to find an α which minimizes the radiation loss. Fiure 5(b) shows the field distribution on one side of the defect for α =,.55, and based on their κ (z). To estimate the radiation loss, we interate the spatial Fourier spectrum over the radiation continuum. Fiure 5(c) shows the portion of enery in the radiation continuum, η rad, as a function of α for 4 and 6 tapered holes respectively. Taperin the ratin reduces η rad by more than 3 orders of manitude. The effect of 6 tapered holes is better than that of 4 tapered holes. The minimum of η rad occurs at α =.48 and.55 for 4 and 6 tapered holes respectively. This result shows that tapered ratins with α~.5, correspondin to a field distribution of approximately exp[ ( z) 3/2 ], are better than linear tapers with Gaussian distribution. Their field distributions are shown in Fi. 5(b), and the spatial Fourier spectra are shown in Fi. 5(d). Compared to α =, while the spectrum of α =.55 is larer at hiher frequencies, it is an order of manitude smaller within the radiation continuum. As a result, the taper profile with α =.55 constitutes an optimal desin. # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9255

8 (a) Λ Λ Λ Λ 6 Λ 5 Λ 4 Λ 3 Λ 2 Λ d (b) Electric field (arb. unit) r r r r 6 r 5 r 4 r 3 r 2 α= α=.55 α= r (c) η rad n t =6 n t =4 (d) Spatial Fourier spectrum z (µm) -3-6 (f) radiation modes α= α=.55-9 α= k z /k 7 (e) Q α Q i =2.6x n Quality factor n t =4 n t = α Fi. 5. (a) Schematic drawin of a tapered ratin-defect resonator with 6 tapered holes. (b) Field distribution (one side of the defect) of tapered ratin-defect resonators with α =,.55, and, respectively. (c) Enery portion in the continuum of radiation modes of ratin-defect resonators as a function of α. (d) Spatial Fourier spectra of the modal fields for α =,.55, and. (e) Quality factor as a function of number of holes on each side of the defect. α =.55, n t = 6. (f) Quality factor as a function of α. Dashed lines show η rad. # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9256

9 To verify the results obtained from the Fourier analysis, we simulated resonators with various α usin 3D MMMs. The quality factor Q was obtained from the linewidth of resonance in the transmission spectrum. Q consists of external Q (Q e ) due to the couplin to waveuides and intrinsic Q (Q i ) due to radiation loss: = +, (6) Q Q ( n) Q e where n is the number of holes on one side of the defect. In order to find Q i, we increase n until Q is saturated at Q i, as shown in Fi. 5(e). The quality factor as a function of α for 4 and 6 tapered holes is plotted in Fi. 5(f). We plot η rad (multiplied by a constant so that it is equal to Q at α = ) on the same fiure for comparison. The curves of Q aree closely with η rad. The maximum of Q occurs at α =.5 and.55 for 4 and 6 tapered holes respectively. The hihest Q for 6 tapered holes is at α =.55. It is an order of manitude hiher than the theoretical Q of ratin-defect resonators desined in the literature [2, 3]. The radii of the tapered holes are 4.5, 54.6, 64.8, 72.9, 82.9, 92. nm, and the periods are 346.3, 357.7, 367.9, 378.3, 389.7, 4.3 nm. The modal volume is.38 (λ/n Si ) 3, which is smaller than those of D photonic crystal resonators resonatin at frequencies near the ratin band ede [4, 5]. Further increasin the number of tapered holes will result in a hiher Q. However, the resultin smaller holes may not be practical. 4. CROWs based on tapered ratin-defect resonators Gratin CROWs are formed by cascadin the hih-q ratin-defect resonators desined in Section 3. Fiure 6 shows the first two resonators of a ratin CROW. The inter-resonator couplin is controlled via the number of holes between neihborin defects, denoted as m. m includes the number of tapered holes (2n t ) and the number of reular holes (n re ). In the appendix, we will show that when cascadin two symmetric ratin resonators with external quality factors Q and Q 2 respectively, the couplin coefficient is iven by 2 i ω κ =. (7) 4 Q Q If the resonant frequency is not at the Bra wavelenth, an additional waveuide section between two resonators is required for appropriate couplin in CROWs (see appendix). We have shown in Eq. (5) that Q is proportional to exp(2κ L). For a tapered resonator, the relation is modified as n L n t Q exp 2 κ ( z) dz = exp 2 κ Λ = exp 2 κ Λ exp 2 n κ Λ, (8), i i, i i re i= i= which breaks down into the contribution of each hole in the tapered reion and the reular ratin, respectively. Therefore, Q can be written as 2n re, Q= Q a (9) where Q is the quality factor of a resonator with only the tapered reion (no reular hole) and a exp(κ Λ) =.849. We fit the curve in Fi. 5(e) (α =.55, n t = 6) with Eq. (9) and obtain Q = 548 and a =.848. If we cascade two resonators with n re, and n re,2 respectively, we obtain the inter-resonator couplin coefficient iven by ω ω n κ = = a re, () 4 Q Q 4Q 2 # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9257

10 m m 2 tapered reion reular holes tapered reion Fi. 6. Schematic drawin of the first two resonators of a ratin-defect CROW. where n re = n re, + n re,2 is the total number of reular holes between two defects. For the first or last resonators, the external loss rate to the waveuides is iven by (also shown in the appendix) ω ω 2n = = a re, () τ 4Q 4Q e where n re is the number of reular holes in the first or last ratin section. To match between the CROW and the waveuides, we require κ = /τ e [8]. Therefore, the number of holes in the first and last sections is one half of the number of holes in the middle sections, i.e. m = m N+ = m/2. Fiures 7(a) and 7(b) show the spectra of transmission and roup delay of N = CROWs for m = 2, 4, and 6 respectively (n re =, 2, and 4). The band center is at nm. Both the bandwidth and the roup delay are dictated by m. The bandwidth is equal to 4κ and the roup delay at the band center is iven by N/(2κ). Addin two holes between defects results in a factor of a 2 = 3.45 in κ, which arees with simulation. Note that the maximal transmission for m = 2 is only.955. This is due to the stron index contrast between the waveuide and the ratin section, which scatters liht to the radiation modes. The stron index contrast can be reduced by taperin the ratin at the input [3]. For larer m, the radiation loss increases due to the loner delay. The transmitted power can be written as exp( ωτ/q i ), where τ is the roup delay. The delay for m = 6 is 6.3 ps, which leads to a transmitted power of.943. Includin the scatterin loss at the input, the transmission drops to about.9, which arees with the simulation. (a) Transmission m=2 m=4 m=6 (b) 25 Group delay (ps) m=2 m=4 m= Wavelenth (nm) Wavelenth (nm) Fi. 7. Spectra of (a) transmission and (b) roup delay of -resonator ratin-defect CROWs with m = 2, 4, and 6 respectively. 5. Filter desin based on ratin CROWs Hih-order bandpass filters with optimized transmission and dispersive properties can be realized in CROWs if the couplin coefficients are allowed to take on different values. For example, Butterworth filters exhibit maximally flat transmission, while Bessel filters possess maximally flat roup delay. For a desired filter response, the couplin coefficients which determine the transfer function of CROWs can be derived [8, 2]. Table lists the couplin # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9258

11 Table. Couplin Coefficients of N = Butterworth and Bessel CROWs Filter type ( τ e, κ, κ 2,, κ N, τ e2 ) / B N = Butterworth (3.96,.876,.883,.63,.533,.56,.533,.63,.883,.876, 3.96) N = Bessel (3.478, 2.3,.932,.63,.35,.333,.652,.772,.56, 2.29, 3.745) Table 2. Numbers of Reular Holes of N = Butterworth and Bessel CROWs Filter type n re N = Butterworth (.495,.854, 3.78, 3.625, 3.896, 3.98, 3.896, 3.625, 3.78,.854,.495) N = Bessel (.426,.726, 2.99, 3.67, 4.82, 4.659, 3.568, 3.295, 2.787,.589,.366) coefficients of N = Butterworth and Bessel filters respectively. These couplin coefficients are normalized to a chosen bandwidth parameter B. In ratin-defect CROWs, the couplin coefficients are translated to the numbers of holes based on Eqs. () and (). However, these numbers of holes are in eneral non-inteers. Table 2 lists the numbers of reular holes correspondin to the two filters in Table. The bandwidth parameter B is chosen as 2 κ(n re = 4) so that its bandwidth is equal to those of CROWs with uniform couplin and n re = 4 (m = 6 in Fi. 7). Since κ is an exponential function of n re, we can add an arbitrary n re in order to chane the bandwidth without havin to rederive all the n re. A non-inteer n re can be realized by an inteer number n int = n re of identical holes which are equivalent to a fraction γ = n re/ n re of a reular hole. For example, 3.6 reular holes are equivalent to 4.9 holes. This can be seen in n a re = exp n κ Λ = exp n κ ( r) Λ( r). re int (2) Therefore, we need to determine the hole radius r whose κ (r)λ(r) is equal to γκ Λ. This can be done by interpolatin the curve of κ Λ versus r. For example, the radius of a γ =.9 hole is 9. nm. If n re is neative, we can reduce the hole sizes startin from the outermost tapered holes. An alternative way is to choose a resonator with fewer tapered holes, such as the resonators with n t = 4 desined in Section 3. Fiures 8(a) and 8(b) show the spectra of transmission and roup delay of an N = Butterworth CROW and an N = Bessel CROW, respectively. The values of transmission at the band center are both.96, indicatin a small intrinsic loss due to the lare roup delay in addition to the scatterin loss at the input. Substractin the scatterin loss which corresponds to a transmission of.955, the intrinsic Q can be obtained from the intrinsic loss and the roup (a) Transmission (b) 5 5 Group delay (ps) Transmission Group delay (ps) Wavelenth (nm) Wavelenth (nm) Fi. 8. Spectra of transmission and roup delay of (a) an N = Butterworth ratin CROW and (b) an N = Bessel ratin CROW. # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9259

12 delay and is determined to be and.97 6 for Butterworth and Bessel CROWs, respectively. Therefore, the tailorin of couplin coefficients does not derade the Q of the resonators. In practice, the Q of the resonators may derade due to imperfection of fabrication. With larer intrinsic loss, the transmission spectrum of Butterworth CROWs may be distorted since the loss is proportional to the roup delay. A predistortion technique can be applied to pre-compensate for the distortion provided that the Q can be estimated and is uniform over the CROW [8]. Since the roup delay of Bessel CROWs is flat within the bandwidth, the transmission spectrum is not distorted by uniform resonator loss. 6. Effect of fabrication disorder on ratin-defect CROWs The major limitation in the experiment of CROWs has been the unavoidable fabrication imperfection which leads to disorder in the resonant frequency of each resonator and the couplin coefficients. The disorder distorts the CROW response and limits the minimum bandwidth which CROWs can be desined with. The yield of CROWs drops as the number of resonators is increased or as the CROW bandwidth is decreased. The effect of disorder on CROWs has been investiated in the literature [2, 22]. In this section we analyze the disorder effect on ratin-defect CROWs. Due to the ultra-small modal volume of the ratin-defect resonators desined in this paper, the shift of resonant wavelenth due to deviation of hole radii is relatively lare. Fiure 9(a) shows the wavelenth shift correspondin to nm chane of radius for each hole startin from the one nearest to the defect. Since the mode is concentrated near the defect, the resonant wavelenth is more sensitive to the deviation of the first three holes. If the standard deviation of each hole radius is δr = nm, the standard deviation of the resonant wavelenth, considerin holes on both sides of the defect, is δλ =.8 nm. Dependin on the fabrication quality (δr), the minimum CROW bandwidth is limited by δλ =.8 δr. Fiures 9(b) and 9(c) show the simulated transmission spectra of -resonator CROWs with uniform couplin coefficient in the presence of disorder in resonant frequency, δω. The CROW bandwidth is 2B, and δω are.5b and.25b respectively. The disorder leads to oscillations in the passband. Considerin the center half of the bandwidth, the averaed oscillation amplitudes are 2. db and 4. db, respectively. The disorder of couplin coefficient is relatively small. (a) (b) (c) Wavelenth shift (nm) Hole number (d) Transmission (db) ω/b (e) Transmission (db) ω/b Transmission (db) Transmission (db) ω/b ω/b Fi. 9. (a) Shift of resonant wavelenth due to nm chane of radius for each hole startin from the one nearest to the defect. (b-e) Simulated transmission spectra of -resonator CROWs with disorder in resonant frequencies. (b,c) Uniform couplin coefficients. (d,e) Butterworth filters. (b,d) δω =.5B. (c,e) δω =.25B. # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 926

13 For δr = nm, δκ/κ = 3%. For the bandwidth of interest, the distortion due to δκ is neliible compared to the distortion due to δω. The oscillations in the passband can be reduced by applyin Butterworth filter desin. Fiures 9(d) and 9(e) show the simulated spectra of Butterworth CROWs with the same bandwidth and the same δω as in Fis. 9(b) and 9(c). The averae oscillation amplitudes are reduced to 4.4 db and. db, respectively. The disorder is a perturbation to an ideal CROW and can be taken as scatterers in the CROW. For CROWs with uniform couplin coefficient, the boundaries between the CROW and the waveuides cause reflection and form a cavity. Disorder in uniform CROWs can be thouht of as scatterers in a cavity which can cause lare oscillations. In a Butterworth CROW, the couplin coefficients are tailored to adiabatically transform between the CROW and the waveuides, thereby removin the cavity and reducin the amplitude of oscillations. Because of the hiher wavelenth sensitivity of ratin-defect resonators compared to those of larer resonators such as rin resonators, ratin-defect CROWs desined in this paper are more practical for larer bandwidth applications (for example, in our experimental results [23],, 3, and nm). Althouh larer bandwidth corresponds to smaller delay, the roup velocity is still small considerin the lenth of ratin-defect resonators. If the ratin is chosen to be weaker, such as shallower holes, the wavelenth sensitivity will become smaller due to the larer modal size. Therefore, the lenth of each resonator and the wavelenth sensitivity appear to a trade-off in the desin of ratin-defect CROWs. 7. Conclusion We have demonstrated a systematic approach to desin hih-q tapered ratin-defect resonators, control the inter-resonator couplin, and desin hih-order ratin CROW filters. The formalism based on coupled-mode theory is valid in stron ratins, with the help of 3D simulations for the characterization of ratins. The optimized Q of is an order of manitude hiher than the theoretical Q of ratin-defect resonators desined in the literature. Based on these hih-q resonators, CROWs which are shorter than 6 µm exhibit a roup delay of more than ps while maintainin a transmission of.9. The control of interresonator couplin via the number of holes provides a convenient way of desinin coupledresonator structures. Furthermore, we demonstrated the desin of tenth-order Butterworth and Bessel filters which possess maximally flat transmission and roup delay, respectively. Besides flat transmission, Butterworth CROWs are more robust aainst fabrication disorder compared to CROWs with uniform couplin coefficient. The ratin-defect CROWs desined in this paper are attractive for their small footprints, hih quality factors, and their natural couplin to input and output waveuides. The coupledmode formalism developed in this work can be further applied to other types of stron ratin structures to minimize the couplin to radiation modes and reliably calculate the transfer function of ratin structures when the couplin to radiation modes is neliible. Appendix. Inline couplin of resonators Inline resonators are, by definition, fabricated, cascaded, and coupled in a sinle waveuide. The objective of this appendix is to derive the inter-resonator couplin coefficient as a function of individual quality factors and the lenth of the couplin waveuide. Fiure (a) shows a symmetric inline resonator, i.e. the couplin to the waveuides is equally stron on both sides. In a symmetric ratin-defect resonator, the number of holes on both sides of the defect is equal. The time-domain coupled-mode equations of the structure in Fi. (a) are # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 926

14 (a) (b) a ω sin s r s + τ e a τ i τ e i s e θ + sout i τ s e θ s e τ 2 e 2 a 2 ω a a d 2 a a ω ' κ a 2 ω 2 ' Fi.. Schematic drawins and the correspondin ratin structures of (a) a symmetric resonator and (b) inline couplin of two resonators. da 2 = ( iω ) a iµ s dt τ τ s r out i in e = iµ a s = s iµ a where a is the resonator mode amplitude, s in, s r, and s out are the input, reflected, and transmitted mode amplitudes respectively, ω is the resonant frequency, /τ i and /τ e are the intrinsic loss and the external loss to each waveuide respectively, and µ is the waveuideresonator couplin. It can be shown that µ = 2τ e usin conservation of enery. The quality factor of the resonator is iven by Q = ωτ/2, where /τ = /τ i + 2/τ e is the total loss rate. In the reime where intrinsic loss is neliible (the linear reion in Fi. 5(d)), Q = ωτ e /4. Therefore, we obtain /τ e = ω/(4q) if Q is iven. In Fi. (b), we consider two inline resonators cascaded in a waveuide. The interresonator couplin is via the couplin waveuide of lenth d. The coupled-mode equations of the two resonators and the couplin waveuide are iven by da i = ( iω ) a iµ s e dt da dt 2 iθ = ( iω ) a iµ s e. in θ τ e τ e2 iθ s+ = s e iµ a iθ s = s+ e iµ 2a2 The notations are shown in Fi. (b). θ is the phase accumulated in the propaation. The resonant frequencies and the external losses of the two resonators can be different in eneral. The intrinsic losses and the couplin to the other resonators or waveuides have been inored and can be added to the equations. Combinin the last two equations of Eq. (4), s + and s - can be expressed as linear combinations of a and a 2, and Eq. (4) can be rewritten as coupledmode equations of two directly coupled resonators (also shown in Fi. (b)): da cotθ cscθ = i( ω ) a i a2 dt τ e τ eτ e2. da2 cotθ cscθ = i( ω ) a i a dt 2 2 τ e2 τ eτ e2, (3) (4) (5) # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9262

15 The couplin leads to detunin of resonant frequencies and a couplin coefficient which depend on the round-trip phase of the couplin waveuide cavity, 2θ: cotθ ω,2 = τ e,2. cscθ κ = τ τ e e2 When desinin CROWs, we require identical resonant frequencies. If the frequency detunin ω is nonzero, the resonators in a CROW may experience different frequency detunin. For example, the frequencies of the first and last resonators are less detuned since they only couple to one resonator, while the other resonators have two neihbors. Therefore, we require ω =, which corresponds to 2θ = π, a totally destructive interference in the couplin cavity. 2θ = π also leads to a minimal couplin coefficient. The cavity round-trip phase includes the phase of reflection from the ratin and the propaation phase in the cavity. At the Bra wavelenth, the reflection phase θ r = π/2 sin (δ/κ ) is π/2. Therefore, the round-trip phase with d = is already π. This is an important reason why we choose to work at the Bra wavelenth. If the resonance is near the band ede, an additional couplin waveuide of lenth d is required to satisfy a round-trip phase of π [24]. The couplin coefficient for the ratin-defect resonators in this paper is thus iven by which proves Eq. (7). Acknowledments 2 2 (6) ω κ = =, (7) τ eτ e 4 Q Q The authors thank Christos Santis for helpful discussions. This work was supported by National Science Foundation and The Army Research Office. # $5. USD Received 7 Feb 22; revised 3 Mar 22; accepted 2 Apr 22; published 6 Apr 22 (C) 22 OSA 9 April 22 / Vol. 2, No. 8 / OPTICS EXPRESS 9263