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Published in IET OptoelectronicsReceived on 25th August 2012Revised on 12th February 2013Accepted on 17th April 2013doi: 10.1049/iet-opt.2012.0057

T Optoelectron., 2013, Vol. 7, Iss. 3, pp. 63–70oi: 10.1049/iet-opt.2012.0057

ISSN 1751-8768

Efficient design of optical delay lines based onslotted-ring resonatorsNabeil A. Abujnah1, Rosa Letizia2, Mohamed Farhat O. Hameed3,4, Salah S.A. Obayya4,

Maher Abdelrazzak5

1Department of Electrical and Electronic Engineering, Faculty of Engineering, University of Azzaytuna, Libya2Engineering Department, Lancaster University, Lancaster LA1 4YR, UK3Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura 35516,

Egypt4Centre for Photonics and Smart Materials, Zewail City of Science and Technology, Sheikh Zayed District, 6th of October

City, Giza, Egypt5Department of Electronics and Communications Engineering, Faculty of Engineering, Mansoura University, Mansoura

35516, Egypt

E-mail: sobayya@zewailcity.edu.eg

Abstract: In this study, a novel design of coupled-resonator optical waveguide delay line based on slotted ring resonators(SCROW) is proposed and analysed. The coupling efficiency between the resonators, which can be controlled either bychanging the slot position inside the cavity or by varying the separation distance between the rings, plays an essential role incontrolling both group velocity and delay time of the suggested SCROW. The simulation results are obtained using multi-resolution time domain scheme in conjunction with uniaxial perfectly matched layer boundary conditions. By introducing aslot inside the microring resonator of the suggested structure, a quality factor of about 5 × 103 and high delay time areobtained with small ring size.

1 Introduction

The progress in integrated optical technology maysignificantly have an effect on the future rate ofdevelopment of optical networks. It is known that carryingout the process of buffering via electronics can lead tobottlenecks in high-speed optical networks [1, 2]. Thereforeoptical buffering and storage can open new era in the futurefor all optical packet-switched networks and computersystems to prevent traffic contention [2]. Optical delay linesplay an essential role in avoiding traffic contentionparticularly when multiple packets are simultaneouslydestined for the same output port [1, 3]. Different schemeshave been proposed in order to construct optical delay lines,among them, fibre loops [4]. However, devices based onthis scheme do not provide compact solution. Recent yearshave seen spectacular progress in the development ofoptical pulse delays based on integrated optic cavities suchas microrings, microdisks and photonic crystalsmicrocavities.In this context, Yanik et al. [5–6] have shown that dynamic

tuning procedures are essential to stop light pulses. Thedesign they proposed is based on photonic crystal coupledmicrocavities. However, it involves the employment ofmany microcavities which are bulky and could reduce thelevel of integration of optical devices [5–11]. More simplearchitectures, incorporating a pair of silicon ring resonators

with dynamically tuned resonance frequencies, have beensuggested for capturing and delaying portions of pulses[12]. Recently published work has shown that the use ofnanocavities based on photonic crystals has dynamicpotential for controlling the quality factor [13]. In addition,a single resonator structure that can completely capture thelight pulses in macroscopic systems was presented in [14].By means of tuning the reflectivity of one of the cavitymirrors, the cited work showed that long pulses can bearbitrarily captured with no reflection. However, suchtuning process cannot be typically implemented in on-chipapproaches, where one generally tunes the cavity resonantfrequency instead. Otey et al. [15] demonstrated that theuse of cascaded resonators is capable of capturing the lightpulses by totally compressing the system bandwidth withoutreflection and the stopped light can then be released.Slotted waveguide optical resonators can be used in many

applications such as chemical and bio-chemical sensors [16].In this regard, microring resonator-based sensors aredependent on wavelength interrogation scheme to reducenoise and enhance the sensitivity [17]. In this paper, thecoupling between slotted miocrocavity ring resonators(SMRRs) is used here for the first time, to the best of theauthors’ knowledge, to propose a novel design of opticaldelay lines. Owing to the electric field discontinuity acrossthe material interfaces, it has been shown that the SMRRsare capable of supporting strongly confined light within

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Fig. 1 Electric and magnetic field expansion coefficients as placedinside 2D MRTD unit cell

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low-refractive index materials by means of total internalreflection (TIR) mechanism at a level that cannot beaccomplished by using conventional waveguides [18]. Inthis study, the effects of the slot position inside the ring,slot filling material such as SiO2 and air are investigated.The simulation results reveal that the suggested SMRR canprovide a good level of delay time with less number ofrings compared with the results reported in [19]. The delaytime would be adjusted by changing the wavelength of thesource laser, and taking advantage of the dispersioncharacteristics of the delay medium. Clearly, theshortcoming of this approach is the necessity to change thesource wavelength when tuning is needed.Two-dimensional (2D) multiresolution time domain

(MRTD) scheme based on the expansion in terms ofCohen–Daubechies–Feanveau (CDF) scaling functions inspace is employed in order to study the characteristics ofthe proposed device [20]. The MRTD has proven itself tobe more sophisticated method than the most popular utilisedfinite-difference time domain (FDTD) for the analysis ofoptical microcavities [21]. The MRTD has high level ofaccuracy without putting any heavy burden even whenmodelling complex problems with large computationalvolumes. In addition, the MRTD has less numericaldispersion because of the high order discretisation in space.Therefore fewer points per wavelength can be employed todiscretise the computational domain. The uniaxial perfectlymatched layers (UPML) scheme has been implemented atthe boundary of the geometrical problem to truncate thecomputational domain with high accuracy. The numericalanalysis shows that the slot position inside the cavity cancontrol the coupling coefficients and hence the qualityfactor and delay time. However, the impact of the slotfilling material is rather small.This paper is organised as follows. Following this

introduction, the basic formulation of the MRTD withUPML for the analysis of delay lines based on slotted ringresonators (SCROW) is presented in Section 2. In Section 3the simulation results obtained from the analysis of theproposed structure are presented in detail. Finally,conclusion will be drawn.

2 Analysis

2.1 Multiresolution time domain method

For the transverse electrical mode (TE) problem in a 2Dstructure in x–z-plane, where y-axis is the homogeneousdirection and x-axis is the propagation direction, only thecomponents Ey, Hx and Hz are non-zero. With respect to theunit cell shown in Fig. 1, the electromagnetic fields areexpanded as a combination of scaling function in space andHaar function in time using [20] (see (1))

where n, i and j are the discrete indexes in time and in space,respectively, Φ is the scaling function representing the

Ey(x, z, t) =∑+1

n,i,j=−1n+(1/2)E

y,fi+(1/2),j+(1

n+(1/2)Ey,fi+(1/2),j+(1/2) = n−(1/2)E

y,fi+(1/2),j+(1/2) +

Dt

101i+(1/2),j+(1/2)

(

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sampling in space, h is the Haar function andn+(1/2)E

y,wi+(1/2),j+(1/2) is the expansion coefficient for Ey

component on which the iterative process of update isapplied on. The field expansions are inserted into thedifferential equations derived from Maxwell’s equations.Then they are sampled using the dual of the biorthogonalscaling function f̃m (with m = i, j) as test functions,according to the method-of-moments.The discretised equation for the principal component nE

y,fi,j

can be obtained as follows [20] (see (2))

where µ0, ɛ0 are the permeability and permittivity of the freespace, respectively, ε is the relative permittivity of themedium and Δt is the time step. In addition, Δx and Δz arethe spatial increments in the direction of x and z,respectively. The ‘stencil size’ Ls stands for the effectivesupport of the basis function that determines the number ofexpansion coefficients considered in the summation and it isequal to 5 for CDF(2,4), whereas a(l ) represents theconnection coefficients whose values are numericallycalculated and reported for the case of CDF(2,4) in theliterature [20, 22].To ensure the numerical stability of the MRTD scheme, the

time interval Δt has to be smaller than a certain limit, asfollows

Dt ≤ sD

c0(3)

with

s = 1��2

√ ∑Ls−1l=0 a(l)

∣∣ ∣∣ (4)

/2)fi+(1/2)(x)f j+(1/2)(z)hn+(1/2)(t) (1)

∑Ls−1

l=−Ls

a(l) − 1

Dz nHx,f

i+(1/2),j+l+1 +1

Dx nHz,fi+l+1,j+(1/2)

( ))(2)

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where Δ = Δx = Δz, c0 is the speed of the light, and the Courantnumber s represents the stability factor which is convenientlyfixed to 0.1 through the paper [20]. The UPML is veryefficient and robust way to terminate the computationaldomain with artificial absorber layers with very-lowreflection. The UPML has been successfully incorporatedinto the MRTD scheme following the derivation andparameters in [20, 23].

2.2 Coupled -resonator optical waveguides(CROWs)

The optical resonators are found to be ideal platforms forstoring the light in physical small sizes. A chain of coupledresonators as shown in Fig. 2 is a new type of waveguide inwhich light propagates because of the coupling betweenadjacent resonators. In particular, coupled microresonatorsmay offer a new scheme for controlling the group velocityof the optical pulses in a compact way on a chip [24]. Thearchitecture of a CROW incorporating an array of directlycoupled ring resonators with the same geometrical lengthbetween the two bus waveguides is described in detail byPoon et al. [24]. The input light pulses propagate inmicroring CROWs by the coupling between theneighbouring resonators. In this case, the light pulses spendmost of their time circulating within each resonator.Consequently, large group delay can be achieved. Thegeneral characteristics of the CROWs such as dispersionrelation and band structure can be described by thecoupling between the adjacent resonators and input buswaveguides, in terms of the free spectral range, and thequality factor Q. In addition, the performance of theCROWs can be affected primarily by the parameters that setthe resonators [24].To design a CROW based on miocrocavity ring resonators

(MRRs), it is essential to understand and control the couplingof the light between the MRRs and input and output buswaveguides. When a chain of directly coupled ringresonators are coupled to linear waveguides which serve asinput/output ports, the system behaves like tunable andfrequency dependent time delay. Poon et al. [24] presenteda theoretical framework model to analysis coupled resonatoroptical waveguides using transfer matrix method. Theseaccurate analytical formulae are found to be valid to anykind of resonators. From the coupling efficiency point ofview, these formulae are function of the couplingcoefficients which can be calculated numerically by one ofthe accurate existing numerical techniques. As described in[24], in the limit of weak coupling k≪1, the dispersion

Fig. 2 Illustration of finite CROW

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relation can be derived as

v(K) = V 1+k| |mp

cos KL( )[ ]

(5)

whereΩ is the resonance frequency of an uncoupled resonatorin radians per second, κ is the coupling coefficient, K is theBloch wave vector, Λ is the periodicity of the structure,m = (Ω neff R/c) is the azimuthal modal number, R is thering radius, c is the velocity of light in vacuum and neff isthe effective refractive index of the ring. The maximumgroup velocity at the centre of the CROW transmissionband where ω =Ω and KΛ = π/2, can be calculated usingthe following formula [24]

vg(V)∣∣∣ ∣∣∣ = c|k|L

pR neff (V)(6)

The delay time of a propagated pulse through the CROW isdetermined at the centre of the CROW band by the distancetraversed in the CROW, and the group velocity which canbe expressed by the following equation [24]

td =pRneff (V)

c

∑Ni=1

1

ki∣∣ ∣∣ (7)

where N is the number of the rings. It is evident from (7) thatthe CROW performs as a customary waveguide with groupvelocity c/neff but with effective length of [24]

Leff =ct

neff= pR

∑Ni=1

1

ki∣∣ ∣∣ (8)

The quality that describes the ratio of the group velocity infree space to the group velocity in the CROW is theslowing factor, S which can be determined by [24]

SV = pneff (V)

2 k| | (9)

The total loss from the input to the output of the CROW canbe obtained by the following equation [24]

atot = aL∑Ni=1

1

2 ki∣∣ ∣∣ (10)

where αL is the intensity attenuation coefficient of the ring, Lis the ring circumference, exp(−a)

tot is the net power attenuationcoefficients of the CROW, and exp− αL is the powerattenuation in the waveguide of the constituent resonators.The quantitative benchmark to determine the quality of thedelay line is called figure of merit (FOM) which can beexpressed as [24]

FOM = 2|k|aL

(11)

To enhance the performance of the suggested CROW, a noveldesign of coupled-resonator optical waveguide delay linebased on SCROW is proposed and analysed. In order topromote the electrical field confinement and improve theresonance effects, micrometric-scale low-index slot isinserted into the bent waveguide which builds the ring. The

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Fig. 3 To validate the simulation results

a Schematic diagram of 2D microcavity ring resonator of diameter d, width WR and gap g between the straight waveguide and the ring, where ncore = 3.2, ncl = 1and Wg = 0.3 μm. However, the microcavity ring resonator in Fig. 2b has a single slotted microring resonator

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main advantage of this slotted configuration is that thecoupling coefficients can be controlled by simplyengineering the slot microring. The slotted microcavityresonating at resonance frequencies is able to trap thesefrequencies from the primary waveguide where they cantake some time propagating within the cavity by means ofTIR and then drop them into a secondary waveguide. Inparticular, by chaining a row of side-coupled resonators, thedelay efficiency will be shown to be greatly improved. Apreliminary study of the SCROW is carried out in thispaper to investigate numerically the resonance frequencyand the coupling efficiency of the wave travelling across theslotted ring waveguide. In this work, the MRTD techniqueis used to simulate a single stage of SCROW with N = 1.Assuming the coupling coefficients are identical betweenother rings, the calculated coupling coefficients for theSMRRs can be mathematically obtained [24] to predict theperformance of the suggested delay lines.

Fig. 4 Variation of the coupling coefficient with the frequency atdifferent values of the gap size, g, using the MRTD and FDTD[21] methods for straight waveguide of width Wg = 0.3 μmcoupled to 5 mm-diameter ring microcavity with WR = 0.3 μm

3 Results

To validate the simulation results, initially the conventionalCROW structure shown in Fig. 3a [21] has beenconsidered. Fig. 3a shows a schematic diagram of 2Dmicrocavity ring resonator of diameter d and width WR. TheMRR is coupled to parallel straight waveguides, of widthWg, to represent the input and output couplers. In thisstudy, a gap g is set between the straight waveguide, andthe ring. In addition, ncore and nclade represent the refractiveindices of the core and cladding regions, respectively, asshown in Fig. 3a.To cover the frequency range of interest, the entire structure

was excited by 200 THz Gaussian pulse of the formexp− t−t0( )/T0( )2 modulated in time by a sinusoidal functionwith the shape of the fundamental mode profile of thewaveguide where t0 and T0 are the time delay and the timewidth of the Gaussian pulse fixed at 80 and 20 fs,respectively. The coupling coefficient is defined as thepercentage of the power coupled into the ring from theinput waveguide. Fig. 4 shows the variation of the couplingcoefficient κ from the left waveguide of width Wg = 0.3 μmto the ring resonator at different values of the gap g, 0.191,0.218 and 0.245 μm using FDTD [21] and MRTD methods.In this investigation, the frequency f0 = 200 THz, WR =0.3 μm, d = 5 mm, ncore = 3.2 and ncl = 1. In addition, theobservation cross-sections of interest are at the source endof the input waveguide Pin and in the ring one-quarter of

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the way around from the input waveguide Pr. Therefore thesimulation is terminated before the first round-trip of thefield inside the ring is completed to determine the couplingefficiency of the MRR. It is revealed from Fig. 4 that theMRTD results are in excellent agreement with those of theFDTD method [21]. Therefore the MRTD method isaccurate for the analysis of this class of devices. Inaddition, the execution times of the MRTD and theconventional FDTD methods are equal to 77 and 129 min,respectively. Moreover, the mesh size used by the MRTDand the conventional FDTD methods are equal to 27.25 and13.60 nm, respectively. Therefore the MRTD scheme canachieve the same level of accuracy of the FDTD methodwith shorter execution time (by a factor of approximately2). In addition, the MRTD is an efficient alternativenumerical scheme to the commonly used FDTD method forthe design of MRR structures.

Fig. 3b shows a schematic diagram of 2D microcavity ringresonator of diameter d and a gap g between the straightwaveguide of width Wg and the ring. In this study, 5µm-diameter single slotted microring resonator with totalwidth Wtotal = 709 nm is used and analysed. The SMRR ismade of silicon of refractive index ncore = 3.2 and ismicro-structured to exhibit a slot of width Wslot = 130 nm.The SMRR is coupled to two identical straight siliconwaveguides of refractive index ncore = 3.2 with width Wg =0.3 µm. The entire device is placed in air background with

IET Optoelectron., 2013, Vol. 7, Iss. 3, pp. 63–70doi: 10.1049/iet-opt.2012.0057

Fig. 5 MRTD-computed coupling coefficient, κ, as a function of the frequency for 5 μm-diameter SCROW

a At different slot position, η, 0.5 and 0.7 where g| = 245 nm andb At different gap size, g, 191, 218 and 245 nm, where η = 0.7

Fig. 6 Schematic diagram of the suggested SCROW

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refractive index nair = 1. The structure is discretised into auniform mesh with 440 × 440 cells (Δx, z = Δ = 27.25 nm)and temporal step size of Δt = 9.0896 × 10− 18 s to ensurethe code stability. In addition, 20-cell UPML is employedat the computational boundaries to absorb the reflectedpower. It is important to note that the chosen Δ is abouttwice the value required by previous analysis of MRRstructures by means of FDTD [21]. This choice will allowaccurate and yet efficient numerical analysis of this kind ofstructures with a significant saving in terms of CPU runningtime and memory usage.The slot location in the ring is expressed by the asymmetry

parameter, η, such that the inner width of the high-index ringlayer is ηw and that of the outer high-index ring layer is (1− η) w, where w = (Wtotal–Wslot). As shown in Fig. 3b,different cross-sections are chosen in order to record thetime domain variation of incident, and reflected fields. Bymeans of fast Fourier transform of the recordedtime-dependent fields, the coupled power at port Pr iscalculated and normalised to the fundamental spectrumpeak in input port Pin. The coupling efficiency plays anessential role in estimating the amount of the coupledpower into and from the ring. The coupling efficiency canbe controlled by several parameters such as the slot positioninside the ring and the gap size. In order to calculate thecoupling efficiency, 25 000 time steps were executed withthe MRTD based on CDF of order (2, 4).Fig. 5a shows the MRTD-computed coupling coefficient κ

for TE polarisation (electric field perpendicular to the plane ofthe ring) with two different values of η, 0.5 and 0.7. In thisstudy, two slot filling materials, SiO2 and air are also testedin order to compare their performances. It is found that theamount of the coupled power from the input waveguide tothe SiO2 filled SMRR for different slot positions followsquite closely the variation obtained for the air filled SMRR.It is also evident from Fig. 5a that, the coupling level atfixed slot position, η, decreases with increasing thefrequency. The variation of the coupling efficiency, κ, withthe frequency at different values of g, 191, 218 and 245 nmis shown in Fig. 5b. It is revealed from this figure that at agiven frequency, κ decreases as the gap size increases. Inaddition, at a fixed physical gap size, the amount ofthe coupled power decreases rapidly with increasing thefrequency. It is also evident from this figure that thepercentage of the coupling coefficients at 200 THz rangesbetween 0.629% and 2.461% for the air SCROW and0.631% and 2.32% for the SiO2 SCROW. This is achievedfor only 54 nm change in the physical gap size, which ispositively evaluated to the desired range of coupling

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efficiency. Typically 0.5% to 3% is required to achievegood transmission and large extinction ratio.Having studied the performance of the SMMR, the

reported SCROW structure shown in Fig. 6 that consists ofa sequence of directly coupled slotted microcavity ringresonator is next considered. In this study, each microringof diameter d = 5 µm and width Wtotal = 709 nm is made ofsilicon of refractive index ncore = 3.2 and is micro-structuredto exhibit a slot of width Wslot = 130 nm. The SMRRs arecoupled to two identical straight silicon waveguides ofrefractive index ncore = 3.2 with Wg = 0.3 µm wide. Theminimal separation between the bus waveguides and therings is taken as g = 245 nm. The entire device is placed inair background with refractive index nair = 1. The designissue of slowing the light and building delay lines withSCROW at η = 0.7 is addressed here for the first time, tothe best of the authors’ knowledge. In this investigation,two different materials, SiO2 and air are used for filling theslot. The proposed structure is selected to enable directcomparison with the work published in [24]. Following thesame procedure of the previous example, the couplingefficiency at resonance has been calculated for single stageof SCROW (N = 1).

Fig. 7 shows the variation of the coupling coefficients withthe resonance wavelength in the SCROW configuration whereSiO2 is used to fill the slots with η = 0.7. In this study, theMRTD technique is used to simulate a single stage ofSCROW (N = 1). It is clear that the amount of the coupledpower at the resonance rapidly increases towards higherwavelengths. It is also evident from this figure that thepercentage of coupling ranges between 0.4145 and 1.353%in the studied wavelength range.

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Fig. 8 shows the variation of maximum group velocity as a

function of resonant wavelength. It can be seen from thisfigure that at higher resonant wavelengths, the maximumgroup velocity is maximised while it decreases at lowerresonance. This is because of the increased coupledefficiency occurring at high resonant wavelengths. It can berevealed from Fig. 8 that the maximum group velocityranges between 2.917 × 105 m/s and 9.062 × 105 m/s in thestudied wavelength range. However, as shown in Fig. 9, theslowing factor decreases rapidly at high resonantwavelengths because of the high coupling efficiency. Inaddition, the slowing factor is found to be in the range ofvalues from 331 to 1029 in the studied wavelength range asshown in Fig. 9.

Fig. 8 Variation of the group velocity with the resonantwavelength λres at η= 0.7 for SCROW filled with SiO2

Fig. 9 Variation of the slowing factor with the resonantwavelength λres at η= 0.7 for SCROW filled with SiO2

Fig. 7 Variation of the coupling coefficients κ with the resonantwavelength λres at η= 0.7 for SCROW filled with SiO2 where N = 1

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The delay time variation with the coupling efficiency of theSiO2 filled SCROW at different number of rings N at η = 0.7is also investigated and shown in Fig. 10. In this study,different number of rings from 4 to 16 with a step of 2 isused. It is evident from Fig. 10 that the delay time rapidlyincreases by increasing the number of rings. However, thedelay time decreases by increasing the coupling efficiency.Figs. 11 and 12 show the variation of the coupling

coefficients κ and group velocity with the resonantwavelength obtained by the MRTD from 5-µm-diameterslotted microring resonator filled with air at η = 0.7 andN = 1. It is found that the SMRR filled with air showsslightly lower coupling efficiency than the SMRR filledwith SiO2 at fixed resonance frequency within the range

Fig. 10 Variation of the delay time with the coupling efficiency fordifferent N at η= 0.7 for SCROW filled with SiO2

Fig. 11 Variation of the coupling coefficients κ with the resonantwavelength λres at η= 0.7 for SCROW filled with air

Fig. 12 Variation of the group velocity with the resonantwavelength λres at η= 0.7 for SCROW filled with air

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Fig. 13 Variation of the slowing factor with the resonantwavelength λres at η= 0.7 for SCROW filled with air

Fig. 14 Variation of the delay time with the coupling efficiency fordifferent N at η= 0.7 for SCROW filled with air

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0.4109% ≤ κ≤ 1.349%. It can also be seen from Fig. 12 thatthe maximum group velocity ranges between 2.666 × 105 m/sand 8.905 × 105 m/s in the frequency window of interest.Fig. 13 shows the variation of the slowing factor with theresonant wavelength. As shown from this figure, theslowing factor decreases by increasing the resonantwavelength. In addition, the slowing factor ranges from 337to 1125 in the studied wavelength range, which is slightlylarger than that of the SiO2 filled SCROW at fixedfrequencies.The effect of the delay time on the coupling coefficients is

also investigated. Fig. 14 shows the variation of the delaytime of the SCROW filled with air with the couplingcoefficients at different number of resonators. It is revealedfrom this figure that the amount of delay time rapidlyincreases by increasing the number of rings which is similarto SiO2 filled SCROW. However, the SCROW filled with

Table 1 Comparison for CROW delay lines consisting ofN = 10 resonators

Resonator type neff R(µm)

κ(%)

Q Netloss(dB)

Delay(ps)

FOM

SMRR filledwith SiO2

2.908 2.5 1.38 4892 35 55 1.248

SMRR filledwith air

2.895 2.5 1.35 5043 34 56 1.256

semi-conductorMRR [24]

3 10 1 5000 33 31 1.300

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air allows for slightly higher delay times at fixed values ofN and κ. Table 1 holds a direct comparison of 10resonators delay line composed of various types ofresonators in different material system at 1.55 µm. Asevidenced by the comparison in Table 1, applicationrequirements, such as acceptable losses, and materialsystem, dictates the type of resonator that will be the mostsuitable. In order to achieve long delay without too muchattenuation, resonators with high quality factor are required.Therefore from Table 1, it is observed that the proposeddesign, based on air slot ring resonators, outperforms thecounterpart designs, including the structure reported in [24].

4 Conclusion

A number of key issues in designing delay lines have beenaddressed in this work. The specific case of using SCROWfor optical delay lines has been here proposed. Couplingefficiency of the SMRRs has been numerically investigatedby 2D MRTD formulation based on CDF (2, 4) scalingfunction and rigorous UPML boundary conditions. Figureof merits such as group velocity, slowing factor and delaytime have been determined by analytical expressions thatare rigorously employed to analyse and optimise theperformance of the optical delay lines based on suchconfiguration. In addition, comparison betweenconventional optical delay lines and the newly suggestedconfiguration in terms of achievable delay time and losseshas shown that the delay line based on the coupledSCROW can achieve higher delay time with overall smallerring size with unvaried losses.

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70& The Institution of Engineering and Technology 2013

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IET Optoelectron., 2013, Vol. 7, Iss. 3, pp. 63–70doi: 10.1049/iet-opt.2012.0057