Improved Complex-Envelope Alternating-Direction-Implicit Finite-Difference-Time-Domain Method for Photonic-Bandgap Cavities

440 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 1, JANUARY 2007

Improved Complex-EnvelopeAlternating-Direction-Implicit

Finite-Difference-Time-Domain Methodfor Photonic-Bandgap Cavities

Domenico Pinto and S. S. A. Obayya, Senior Member, IEEE

Abstract—In this paper, an improved complex-envelopealternating-direction-implicit finite-difference time-domain(CE-ADI-FDTD) method has been presented for the analysis ofphotonic-bandgap cavities. The improvement relies on a differentapproach of the perfectly matched-layer absorbing-boundarycondition in order to avoid the formation of instability, as reportedin the literature. The high numerical precision and efficiencyobtained are clearly demonstrated through the agreement of theresults obtained using CE-ADI-FDTD and their counterpartsobtained using other rigorous approaches reported in theliterature.

Index Terms—Alternating-direction-implicit finite-differencetime-domain (ADI-FDTD), complex-envelope (CE) approximation,photonic-crystal (PhC) cavities, quality factor, resonant mode.

I. INTRODUCTION

PHOTONIC-BANDGAP (PBG) materials, which are alsoknown as photonic crystals (PhCs), have attracted great

research attention because of their capability to control thepropagation of the electromagnetic waves at the range of opticalfrequency. In fact, because of their structure, PhCs have oneor more ranges of optical frequency, termed the “photonicbandgap,” in which no electromagnetic wave is allowed topropagate inside the PhCs. This property is due to the peri-odicity of the refractive index along one or more directions.This pattern gives rise to a band structure for photons that isanalogous to the band structure for an electron when it prop-agates in a semiconductor [1]. For this reason, PBG materialshave been used to realize a wide range of devices, such as sharpbends [2], filters [3], [4], cavities [5], [6], lasers [7], [8], andmany others. For instance, a waveguide is obtained from a PhCwhen a path of defect inside the pattern is created. On theother hand, a PhC cavity is obtained using the same strategy bycreating a central defect in the lattice. In this way, it is possibleto “trap” a mode inside the cavity formed by the defect in thelattice so that localized resonant modes are created.

In the literature, various modeling techniques have beenproposed for the simulation of optical devices. These tech-niques include the beam-propagation method [9]–[11], the

Manuscript received April 4, 2006; revised September 20, 2006.The authors are with the Institute of Advanced Telecommunications,

University of Wales Swansea, SA2 8PP Swansea, U.K. (e-mail:[email protected]).

Digital Object Identifier 10.1109/JLT.2006.886712

finite-difference time-domain method (FDTD) [12], [13], andthe finite-element time domain (FETD) [14]. However, theFDTD stands as one of the most popular techniques as itcan handle a very wide range of photonic devices. Being anexplicit technique that relies on field update at each step withoutrendering to any matrix inversion makes it possible to modelultracomplicated structures on desktop computers. The maindrawback of conventional FDTD is the need for high compu-tational resources due to the Courant criterion, which boundsthe time-step size in order to avoid numerical instability. Sometechniques have been proposed in literature to circumvent theconstrain due to the Courant criterion. Among these techniques,the method proposed by Rao et al. in [15], known as complex-envelope alternating-direction-implicit finite-difference time-domain (CE-ADI-FDTD), has been shown to be very attractivefor photonic simulations in terms of low numerical dispersionand reduction of required computational resources. In fact, theCE-ADI-FDTD presented in [15] can get results more accu-rate than the ADI-FDTD algorithm utilizing same time step.This leads to the possibility to employ larger time steps and,consequently, less computational burden without compromis-ing accuracy. However, this technique sometimes may sufferfrom numerical instability near the end of the simulation ofsome photonic devices as reported in [15]. This may limit thepotential usefulness of the technique particularly if instabilityappears in early stage of the simulation before obtaining the re-quired time-domain information. Alternatively, a new approachis proposed here for the formulation of the CE-ADI-FDTD,which eliminates the stability problem due to the employmentof perfectly matched layer (PML), even for large values of theCourant number, which is defined as the time step used onthe simulation over the maximum time step that is possible touse with a conventional FDTD method fixed by the Courantlimit. The developed method relies on applying an alternatingforward and backward difference to the calculation of thederivatives of the coefficient of the PML equations, allowingbetter performance in terms of the absorption of impingingwaves on PML. This idea has been proposed in [16] in thecontext of conventional ADI-FDTD for microwave devices.However, the proposition of CE-ADI-FDTD with this new typeof PML for analysis of photonic devices is suggested herefor the first time, to the best of the authors’ knowledge. Theproposed CE-ADI-FDTD method is used to model the PhC

0733-8724/$25.00 © 2007 IEEE

PINTO AND OBAYYA: IMPROVED CE-ADI-FDTD METHOD FOR PHOTONIC-BANDGAP CAVITIES 441

Fig. 1. Schematic diagram of the simulated 5 × 5 dielectric rods PhC cavitywith PML boundary conditions.

cavity, and the time variation of the electromagnetic field isrecorded at a certain reference point. Then, upon using thefast Fourier transform (FFT) of these time-domain data, thespectral response will be utilized to identify the resonancewavelength and quality factor of the resonant mode. ThisCE-ADI-FDTD code has been extensively used in order to ver-ify its stability properties and its accuracy through the analysisof PhC cavities, which is still a topic in which many researchgroups are widely investigating [17]. The effects of the PhCarrangement, the number of dielectric rods, and the defect onthe resonance wavelength and quality factor of the fundamentallocalized resonant mode have been investigated and the resultsobtained with the proposed CE-ADI-FDTD are shown to be inexcellent agreement with those reported in literature.

This paper is organized as follows. Following this in-troduction, a brief mathematical treatment of the proposedCE-ADI-FDTD method is described in Section II, includ-ing how to incorporate the PML boundary conditions. Then,detailed simulation results are presented and explained inSection III. Conclusions are then drawn.

II. FDTD ANALYSIS

The geometry of the two-dimensional (2-D) cavity problemin hand is shown in Fig. 1, in which the length of the rodsis considered infinitely extended. With respect to this cavitystructure, it can support only resonant TE modes [1], whereTE modes have electric-field component that is parallel to thelength of the dielectric rods while magnetic-field componentsare in the plane of the Fig. 1. Under the scalar approximation,the following 2-D equations can be derived:

∂Hx

∂t=

1µrµ0

(−∂Ez

∂y

)(1)

∂Hy

∂t=

1µrµ0

(∂Ez

∂x

)(2)

∂Ez

∂t=

1εrε0

(∂Hy

∂x− ∂Hx

∂y

)(3)

where εr and ε0 are the relative permittivity and the free-spacepermittivity, respectively, and µr and µ0 are the relative perme-ability and the free-space permeability, respectively. Applyingthe slow-varying envelope approximation, the electromagnetic-field components can be expressed as

Hx(x, y, z, t) =Hxa(x, y, z, t)ejωct (4)

Hy(x, y, z, t) =Hya(x, y, z, t)ejωct (5)

Ez(x, y, z, t) =Eza(x, y, z, t)ejωct (6)

where Eza, Hxa, and Hya are the envelopes of the fast-varying field components Ez , Hx, and Hy , respectively, and ωc

represents the angular-carrier frequency. Substitution of (4)–(6)into (1)–(3) yields [15]

∂Hxa

∂t+ jωcHxa =

1µrµ0

(−∂Eza

∂y

)(7)

∂Hya

∂t+ jωcHya =

1µrµ0

(∂Eza

∂x

)(8)

∂Eza

∂t+ jωcEza =

1εrε0

(∂Hya

∂x− ∂Hxa

∂y

). (9)

The discretization in space is performed based on the unit cellof the Yee space lattice allowing the grid to be nonuniform,as shown in Fig. 2, while discretization in time is obtained byfollowing the procedure in [18]. The space cells are nonuniformso as to allow more accurate and flexible representation of thephotonic device. Applying this procedure to (7)–(9), the follow-ing set of equations for the first-half time step are derived as

Hxa|n+1/2i,j+1/2 =αxh|i,j+1/2Hxa|ni,j+1/2

− βxh|i,j+1/2

(Eza|ni,j+1 − Eza|ni,j

)(10)

Hya|n+1/2i+1/2,j =αyh|i+1/2,jHya|ni+1/2,j

+ βyh|i+1/2,j

(Eza|n+1/2

i+1,j − Eza|n+1/2i,j

)(11)

Eza|n+1/2i,j =αe|i,jEza|ni,j + βxe|i,j

×(Hya|n+1/2

i+1/2,j − Hya|n+1/2i−1/2,j

)

+ −βyh|i,j(Hxa|ni,j+1/2 − Hxa|ni,j−1/2

)(12)

while for the second-half time step, the following set ofequations are obtained:

Hxa|n+1i,j+1/2 =αxh|i,j+1/2Hxa|n+1/2

i,j+1/2

− βxh|i,j+1/2

(Eza|n+1

i,j+1 − Eza|n+1i,j

)(13)

Hya|n+1i+1/2,j =αyh|i+1/2,jHya|n+1/2

i+1/2,j

+βyh|i+1/2,j

(Eza|n+1/2


)(14)

Eza|n+1i,j =αe|i,jEza|n+1/2

i,j

+ βxe|i,j(Hya|n+1/2

i+1/2,j − Hya|n+1/2i−1/2,j

)

+−βyh|i,j(Hxa|n+1

i,j+1/2 − Hxa|n+1i,j−1/2

)(15)


Fig. 2. Electric- and magnetic-field components placed in a 2-D nonuniform Yee’s unit cell.

where

αxh|i,j+1/2 =αyh|i+1/2,j =αe|i,j =4−jωc∆t

4+jωc∆t(16a)

βxh|i,j+1/2 =2∆t

(4 + jωc∆t)µrµ0∆yj(16b)

βyh|i+1/2,j =2∆t

(4 + jωc∆t)µrµ0∆xi(16c)

βxe|i,j =2∆t

(4 + jωc∆t)εrε0hxi(16d)

βye|i,j =2∆t

(4 + jωc∆t)εrε0hyj(16e)

where ∆t is the time step, ∆xi and ∆yj are the discretizationsteps along x and y directions, respectively, and hxi and hyj ,as also shown in Fig. 2, are defined as

hxi =∆xi + ∆xi−1

2i = 2, 3, . . . , Nx (17a)

hyj =∆yj + ∆yj−1

2j = 2, 3, . . . , Ny (17b)

where Nx and Ny are the total number of cells of thecomputational domain along x- and y-direction, respectively.The updating process of the first-half time step starts with theexplicit update of (10) in order to obtain the new values ofthe magnetic-field component Hxa. As shown, (11) cannot beexplicitly solved. Substituting (12) into (11) and solving the de-rived equation for Hya, a system of equations is derived whosecoefficients form a tridiagonal matrix, which can be efficientlysolved in order to obtain the new values of the magnetic-fieldcomponent Hya inside the computational domain. Once themagnetic-field component Hya has been calculated, Eza canbe explicitly updated using (12). A similar procedure can befollowed on second-half time step for (13)–(15).

To simulate the extension of the computational domain toinfinity, PML absorbing-boundary conditions are used. In con-ventional FDTD, PMLs have been extensively used becauseof their excellent absorption properties that give a robust wayto terminate the computational domain. In [15], PMLs havebeen incorporated in the CE-ADI-FDTD method, which hasbeen used to simulate integrated photonic devices. However,

problems with numerical stability of the CE-ADI-FDTD algo-rithm have been reported due to the accumulation of reflectioncoming from the PML in the computational domain [15].Here, a different PML approach is proposed. Applying thePML scheme to (7)–(9), the following set of equations can beobtained for the first-half time step:

Hxa|n+1/2i,j+1/2 = αxh|i,j+1/2Hxa|ni,j+1/2

− βxh|i,j+1/2

(Eza|ni,j+1 − Eza|ni,j

)(18)

Hya|n+1/2i+1/2,j = αyh|i+1/2,jHya|ni+1/2,j

+βyh|i+1/2,j

(Eza|n+1/2


)(19)

Ezxa|n+1/2i,j = αxe|i,jEzxa|ni,j

+ βxe|i,j(Hya|n+1/2

i+1/2,j − Hya|n+1/2i−1/2,j

)(20)

Ezya|n+1/2i,j = αye|i,jEzya|ni,j − βye|i,j

×(Hxa|ni,j+1/2 − Hxa|ni,j−1/2

). (21)

The coefficients of the PML equations are calculated usingeither a forward or backward differencing approximation inorder to collocate the field component on the left-hand side ofthe equation at the same time step of the field component on theright-hand side of the same equation. This procedure yields thefollowing coefficients for the first-half time step:

αxh|i,j+1/2 =4 −

(jωc + 2 σ∗

y

µrµ0

)∆t

4 + jωc∆t(22a)

βxh|i,j+1/2 =2∆t

(4 + jωc∆t)µrµ0∆yj(22b)

αyh|i+1/2,j =4 − jωc∆t

4 +(jωc + 2 σ∗

x

µrµ0

)∆t

(22c)

βyh|i+1/2,j =2∆t(

4 +(jωc + 2 σ∗

x

µrµ0

)∆t

)µrµ0∆xi

(22d)

αxe|i,j =4 − jωc∆t

4 +(jωc + 2 σx

εrε0

)∆t

(22e)


βxe|i,j =2∆t(

4 +(jωc + 2 σx

εrε0

)∆t

)εrε0hxi

(22f)

αye|i,j =4 −

(jωc + 2 σy

εrε0

)∆t

4 + jωc∆t(22g)

βye|i,j =2∆t

(4 + jωc∆t)εrε0hyj. (22h)

Similar expressions can be derived for the coefficients of thePML equations for the second-half time step:

αxh|i,j+1/2 =4 − jωc∆t

4 +(jωc + 2 σ∗

y

µrµ0

)c∆t

(23a)

βxh|i,j+1/2 =2∆t(

4 +(jωc + 2 σ∗

y

µrµ0

)∆t

)µrµ0∆yj

(23b)

αyh|i+1/2,j =4 −

(jωc + 2 σ∗

x

µrµ0

)∆t

4 + jωc∆t(23c)

βyh|i+1/2,j =2∆t

(4 + jωc∆t)µrµ0∆xi(23d)

αxe|i,j =4 −

(jωc + 2 σx

εrε0

)∆t

4 + jωc∆t(23e)

βxe|i,j =2∆t

(4 + jωc∆t)εrε0hxi(23f)

αye|i,j =4 − jωc∆t

4 +(jωc + 2 σy

εrε0

)∆t

(23g)

βye|i, j =2∆t(

4 +(jωc + 2 σy

εrε0

)∆t

)εrε0hyj

. (23h)

The arrangement proposed here for FDTD equations with thePML boundary condition leads to a stable algorithm, even witha large Courant number, as will be shown in the followingsection.

III. RESULTS

Before proceeding to the analysis of PhC cavities, the modi-fied PML boundary condition has been tested in order to verifythe effectiveness of their absorption properties that would allowavoidance of numerical instability. The structure considered forthis test is a 5 × 5 square PhC cavity consisting of dielectricrods with refractive index nrods = 3.4 in the air, as shownin Fig. 1. First, simulations have been carried out with auniform mesh and with different values of time step in orderto test the effect of the time-step value on the stability ofthe developed CE-ADI-FDTD code. The discretization stepwas fixed to 17.73 nm and ten cells of PML layer have beenused to truncate the computational domain on all sides ofthe PhC cavity, as shown in Fig. 1. Although the CE-ADI-FDTD code has been developed to rely on nonuniform mesh,however, to test the stability properties of the proposed method,a uniform mesh has been utilized only in this test. Using theCourant criterion formula and the discretization steps for x- andy-direction utilized for this simulation, the maximum time step

Fig. 3. Time variation of the envelopes of the electric field recorded at thecenter of the 5 × 5 square lattice cavity obtained with the new CE-ADI-FDTDapproach and with the approach in [15].

that is possible to use with a conventional FDTD was calculatedto be ∆tCL

∼= 0.042 fs. The source used to excite the PBGcavity is a Gaussian-shaped source in space and in time, andit is defined as

Ezs(x, y, t) = e−(

t−t0T0

)2

e−(

x−x0X0

)2

e−(

y−y0Y0

)2

(24)

where t0 and T0 are the delay and the width of the Gaussianpulse in time, respectively, x0 and X0 are the displacement andthe width of the Gaussian pulse along x-direction, respectively,and y0 and Y0 are the displacement and the width of theGaussian pulse along y-direction, respectively. For all simula-tions, T0 and t0 were set to 30 and 90 fs, respectively, x0 andy0 were set to the coordinates of the center of the PBG cavity,X0 and Y0 were fixed to a/2, where a is the lattice constantof the PhC, as shown in Fig. 1, and the angular frequency ωc

of the source was fixed to 1.256 × 1015 rad/s (λ = 1.5 µm).The soft-source technique has been used in order to insert thesource in the computational domain [18]. In this technique, thevalue of the source at each point of the computational domain,which is calculated from the analytical expression of the shapeof the source, is added to the value of the electric field at thesame point. This operation is repeated for each time step takinginto account the analytical expression of the source in time. Thetime-domain variation of the electric field at the center of thecavity was recorded. In Fig. 3, the time-domain responses ofthe envelope of the electric field obtained with the developedCE-ADI-FDTD and the approach used in [15] are plotted, bothobtained with a time-step size fixed to 20 times the Courantlimit. As clearly shown in Fig. 3, the approach used in [15] givesrise to instability at the very early stage of the simulation whilethe proposed CE-ADI-FDTD leads to the formation of thefundamental resonant mode of the cavity and looks very stable.Further simulations carried out with the same computationaldomain and utilizing the approach used in [15] have been


Fig. 4. Spectral distributions of the resonant-mode energy for the 5 × 5 squarelattice cavity obtained with different simulations using different time steps.

shown that the maximum Courant number, which gives stableresults is equal to five, while with the proposed CE-ADI-FDTDlarger Courant number can be applied obtaining at the sametime faster results and reduction of required computationalresources. An explanation to this behavior can be attributedto the superior performance in terms of absorption propertiesof the PML layers developed with the proposed approach. Theincreased absorption of the PML layers, even for a very largeCourant number, avoids the accumulation of reflection in thecomputational domain, which can iteratively add to build upinstability of the simulation, and leads to an unconditionallystable algorithm. In order to verify the effect of the time-stepsize on the numerical dispersion of the results, the FFT ofthe time-domain responses of the electric field inside the PhCcavity for all three simulations have been computed. Fig. 4shows the result of this procedure. As shown in this figure, allthe results obtained are in very good agreement with the dataobtained with a conventional FDTD simulation, from which thenormalized frequency a/λ of the fundamental resonant modeof the 5 × 5 PhC cavity has been calculated to be 0.378. Forthe simulations carried out with time steps fixed to five and tentimes the Courant limit, respectively, the shift of the calculatednormalized frequency of the resonant mode has been found tobe negligibly small. Moreover, for a Courant number of 20, theproposed CE-ADI-FDTD gives a normalized frequency of theresonant mode that is shifted by only 0.2% from that obtainedusing conventional FDTD.

Next, the fundamental TE resonant mode of the 5 × 5 cavity,shown in Fig. 1, is considered. For all simulations, nonuniformmesh has been utilized in such a way that the computational do-main grid contains more points in the center of the cavity, wherethe electromagnetic field of the resonant mode of the cavity istrapped. The structure has been discretized by a nonuniformmesh of 153 cells along x- and y-directions in such a way thatthe minimum size step considered to discretize the structurewas fixed to 17.73 nm. Furthermore, ten cells of PML have

Fig. 5. Electric field profile of the resonant mode inside the 5 × 5 squarelattice cavity.

been used to terminate the computational domain. The structureis excited with an electric-field profile given by (24). For thisstructure, the results obtained from the simulation employinga nonuniform mesh for the discretization of the computationaldomain were very similar to the results obtained from previoussimulations, where a uniform mesh was adopted. These resultsare reported in Fig. 3, in which the time variation of theenvelope of the electric field inside the cavity is shown, andFig. 4, which shows the spectral distribution of the resonant-mode-energy density. The normalized frequency a/λ can beeasily determined from Fig. 4 to be 0.3789. The electric-fielddistribution of the resonant mode in the cavity is shown inFig. 5, which is obtained after running the code for 1024 fs.The quality factor Q of the resonant mode can be calculatedfrom the time variation of the field Et as the ratio of the energystored to the energy lost after one cycle using [19]

Q = 2π|Et|2

|Et|2 − |Et+T |2 (25)

where T represents the time cycle of the resonant mode. Forthis structure, the resonant wavelength is λres = 1.548 µm, andthe time cycle is T = 5.16 fs, so that the quality factor has beenfound to be Q = 184. The mode area Amod is also calculatedusing [21]

Amod =∫

(εE∗ · E)ds

[εE∗ · E]max(26)

where ε is the permittivity of the dielectric medium in whichthe cavity is formed, [εE∗ · E]max represents the maximumenergy stored inside the cavity, and the integration is performedover the entire computational window. For this structure, themode area was found to be (0.31λ)2. These results of resonantwavelength, quality factor, and mode area, which are obtainedwith the new developed CE-ADI-FDTD code, are in excellentagreement with their counterparts reported in [19] and [20]and obtained using different FETD methods. Furthermore, the


Fig. 6. Variation of the normalized resonant frequency and quality factor ofthe resonant mode with the cavity size.

CE-ADI-FDTD can easily run on normal desktop computers.For example, it took 15 min to run the 5 × 5 cavity for a meshof 153 cells both in x- and y-directions and for 1200 timesteps on a PC (Pentium IV, 3 GHz with 1 GB of RAM). Inorder to clarify this point, it should be mentioned that for a7 × 7 PhC cavity the proposed CE-ADI-FDTD was only abouttwice time faster than the conventional FDTD algorithm, evenif the Courant number employed for the former was about 20.One should be expecting an execution time much faster thanthat, but it should be noted that the CE-ADI-FDTD algorithmrelies on the computation of complex numbers while the FDTDalgorithm is based on real numbers. Thus, the employmentof a time step much larger than the maximum fixed by theCFL limit can greatly reduce the total number of time stepsnecessary to run the simulation, but on the other hand, theuse of complex numbers increases the execution time of eachsingle time step. For this reason, the total gain in executiontime is slightly less than expected.

Fig. 6 shows the calculation of Q and the normalized fre-quency of the resonant mode for different sizes of the squarecavity. The excellent agreement between the results reportedhere, using CE-ADI-FDTD and their counterparts reported in[20]–[22] using a finite-element frequency-domain approach,can be observed. As noted in Fig. 6, for cavities bigger than5 × 5 , the normalized frequency is almost the same (≈0.378),while the Q factor is exponentially increased. Also, the mode-area variation with cavity size is shown in Fig. 7. As may beobserved from this figure, the mode area tends to be nearlyunchanged when cavity size is bigger than 3 × 3 rods.

Next, a hexagonal-cavity arrangement, shown in Fig. 8,is considered. The case of no central dielectric rod is firstsimulated. This hexagonal cavity, with and without a centralrod, is excited with a Gaussian-shaped source modulated by aGaussian pulse in time, as in the previous example. Fig. 9 showsthe spectral distribution of the energy stored in a four-ringhexagonal cavity. The narrow spectral width of that resonancecurve shows the high selectivity and quality factor of this hexag-onal cavity compared to the square case. This high selectivity

Fig. 7. Variation of the mode area of the resonant mode with cavity size.

Fig. 8. Schematic diagram of a four-ring hexagonal PhC cavity.

property of this type of cavity is confirmed by Fig. 10, whichshows the quality factor and the normalized frequency of theresonant mode for different number of rings surrounding thecavity. From this figure, it can be noted that the normalizedfrequency of the resonant mode remains nearly unchangedas the number of rings is greater than two. Furthermore, itcan be observed also the huge increase of the quality factorof the hexagonal-ring cavity owing to the arrangement of alarge number of dielectric rods. Fig. 11 shows the mode areacomputed for different cavity size using (26). As may be notedfrom this figure, the mode area decreases as the size of thecavity is increased. However, the decreasing rate of the modearea becomes smaller as the cavity size increases.

Next, a hexagonal four-ring cavity with a dielectric rodlocated in the center of the cavity is considered. The radius ofthe central rod is rrod = 0.1a, where a is the constant lattice ofthe PhC, and the refractive index of the rod is varied from 1.2to 3.4. For each cavity type, the normalized frequency of theresonant mode and the quality factor were computed. Fig. 12shows the variation of the normalized frequency and the quality


Fig. 9. Spectral distribution of the resonant mode energy for the four-ringhexagonal PhC cavity.

Fig. 10. Variation of the normalized resonant frequency and quality factor ofthe resonant mode with the number of rings.

factor with the refractive index of the central rod, calculatedfor these cavities. From this figure, it can be observed theshift of the normalized frequency of the resonant mode towarda lower frequency as the refractive index of the central rodincreases. These results can be intuitively justified as follows.The localized mode obtained by using lattice defect consistingof removing dielectric material (“air defect”) presents the nor-malized resonance frequency near the “air band” (upper edgeof the bandgap). On the other hand, localized mode obtainedby using lattice defect consisting of adding dielectric material(“dielectric defect”) presents normalized resonance frequencynear the “dielectric band” (lower edge of the bandgap) [1].This property can be used to tune the resonant mode of thecavity inside the range of the bandgap by properly choosingthe refractive index of the central rod, as shown in Fig. 12. Themaximum of the quality factor Q can be explained consideringthe frequency shift of the resonant mode along the bandgap,whose extension for this particular PhC is from 0.408 to 0.49

Fig. 11. Variation of the mode area of the resonant mode with the numberof rings.

Fig. 12. Variation of the normalized resonant frequency and quality factor ofthe resonant mode with the refractive index of the central defect rod.

in terms of normalized frequency, with the variation of therefractive index of the central rod. As shown in Fig. 12, resonantmodes whose frequency is near the edge of the bandgap showa low-quality factor, which implies relative high-energy lossesfrom the cavity. These losses can be attributed to a couplingof a part of the energy trapped in the cavity itself with modeswhose frequency is not inside the bandgap of the PhC and canpropagate inside the PhC. This coupling is less efficient as thenormalized frequency of the trapped mode is shifting from theedge of the bandgap, and this implies lower energy losses fromthe cavity and, consequently, a higher quality factor Q.

IV. CONCLUSION

A CE-ADI-FDTD method has been proposed for the sim-ulations of PhCs-based cavities. The developed algorithm hasshown to be stable, even for a very large Courant number, and


this property has been utilized to exploit the potentials of theCE-ADI-FDTD algorithm to thoroughly investigate PhC-basedresonant cavities. Resonant wavelength, quality factor, andmodal area of the resonant mode have been calculated for differ-ent types of cavities obtained by adding more rings of dielectricrods around the central defect. Furthermore, hexagonal cavitieshave also been investigated, and it has been found that by usinga central defect consisting of a rod with different value of therefractive index, it is possible to tune the wavelength of the res-onant mode. The accuracy and numerical efficiency of the pro-posed technique have been revealed through comparison withother numerical methods reported in the literature. Althoughthe accuracy of the proposed technique has been demonstratedfor 2-D simulation, these results cannot be extended to 3-DPhC cavities because the 2-D approach cannot take into accountthe effects on the cavity characteristics due to a finite thicknessof the cavity itself. For this reason, the extension of this ap-proach to full-wave analysis of 3-D PBG cavities is now underconsideration.

REFERENCES

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Domenico Pinto was born in Castellana Grotte,Italy, on April 29, 1975. He received the M.S. de-gree in electronic engineering (with specialization intelecommunication) from Politecnico di Bari, Bari,Italy, in 2004. He is currently working toward thePh.D. degree in numerical modeling of optoelectron-ics and microwave devices at the University of WalesSwansea, Swansea, U.K.

He is currently with the Institute of AdvancedTelecommunications, University of Wales Swansea.His research interests include the design and devel-

opment of numerical finite-difference-time-domain tools for the simulation oflinear and nonlinear optical devices.

S. S. A. Obayya (SM’05) was born in Shirbeen,Egypt, on May 27, 1969. He received the B.Sc. (with“excellent” grade) and M.Sc. degrees in electron-ics and communications engineering from MansouraUniversity, Cairo, Egypt, in 1991 and 1994, respec-tively. From September 1997 to September 1999,he was with the Department of Electrical, Elec-tronic, and Information Engineering, City UniversityLondon, London, U.K., as a Visiting Fellow to carryout the research part of his Ph.D. under the “jointsupervision scheme” between the City University

London and Mansoura University, from which he was received the Ph.D. degreein December 1999. In his Ph.D. dissertation, he developed a novel finite-element-based full-vectorial-beam-propagation algorithm for the analysis ofvarious photonic devices.

In September 1995, he joined the Department of Electronics and Communi-cations, Mansoura University, to work as a Teaching Assistant. From January2000 to July 2000, he worked as a Lecturer at Mansoura University, andfrom July 2000 to June 2003, he worked as a Research Fellow at the Schoolof Engineering, City University London. During his work at City UniversityLondon, he has carried out confidential research to QinetiQ, Malvern, U.K.,and also conducted research on the design of semiconductor electroopticmodulators funded by the EPSRC, U.K. In June 2003, he joined the Schoolof Engineering and Design, Brunel University, West London, U.K., to work asa “Lecturer” and was subsequently promoted to “Senior Lecturer” in May 2006.In September 2006, he joined the Institute of Advanced Telecommunications,University of Wales Swansea, Swansea, U.K., where he is currently working asa “Reader in Optical Communications” and leading a research group workingon “Modeling of Optical Devices.” His main research interests are focusedon the areas of frequency-selective surfaces, linear, nonlinear, passive, andactive photonic devices, microstrip antennas for mobile-phone networks, andradio-over-fiber systems. He has published more than 70 papers in the bestinternational journals and conferences in the areas of optics and microwaves.

Dr. Obayya is the recipient of the prize for “The Best Ph.D. Thesis”from Mansoura University, for the years 2001–2002. He was the recipientof the “State Incentive Prize for Engineering Sciences” from the EgyptianGovernment in 2005.

Documents

Improved Complex-Envelope Alternating-Direction-Implicit Finite-Difference-Time-Domain Method for Photonic-Bandgap Cavities