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Higher-order incidence transfer matrix method

used in three-dimensional photonic crystal

coupled-resonator array simulation

Ming Li, Xinhua Hu, Zhuo Ye, and Kai-Ming Ho

Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011

Jiangrong Cao and Mamoru Miyawaki

Canon Development Americas, Inc., 15975 Alton Parkway, Irvine, California 92618

Received July 25, 2006; revised August 27, 2006; accepted September 9, 2006;

posted September 13, 2006 (Doc. ID 73433); published November 9, 2006

The plane-wave-based transfer matrix method with rational function interpolation and higher-order plane-

wave incidence is proposed as an efficient calculation approach to simulate three-dimensional photonic crys-

tal devices. As an example, the dispersion relations and quality factors are calculated for resonant cavity

arrays embedded in a woodpile photonic crystal. An interesting ultraslow negative group velocity is observed

in this structure. © 2006 Optical Society of America

OCIS codes: 140.4780, 220.4830, 230.5750.

Photonic crystals (PCs), periodic dielectric media, can

inhibit electromagnetic wave propagation in certain

frequency ranges called photonic bandgaps.1,2Intro-

ducing point or line defects into PCs can create

highly localized defect modes within the photonic

bandgaps,3resulting in resonant cavities of high

quality factor ?Q? and low-loss waveguides.4–6Re-

cently, coupled PC cavity arrays have received much

attention because of their potential application in in-

tegrated optical circuits.7–10However, most of these

structures are based on two-dimensional (2D) PCs or

a 2D PC slab. There are few studies8on resonant cav-

ity arrays in three-dimensional (3D) PCs because of

difficulties in fabrications and numerical simula-

tions. In this Letter we theoretically study periodic

resonant arrays in 3D PCs by using the plane-wave-

based transfer-matrix methods (TMMs)11,12

higher-order plane-wave incidence and rational-

function interpolation13techniques. As an example,

both the quality factor and dispersion relation are ob-

tained very efficiently for a resonant cavity array

based on the layer-by-layer woodpile PCs. An inter-

esting ultraslow negative group velocity is observed

in this structure. To our knowledge, this is the first

time that dispersions are calculated for coupled reso-

nant cavity arrays in 3D woodpile PCs in all direc-

tions.

The 3D layer-by-layer woodpile14PC is composed of

25 layers (in the z direction) of square dielectric rods

of refractive index n=2.4 and width 0.35a0, with the

cladding material at both ends along the z direction

of the refractive index ncl. Here the lattice constant

a0 is the distance between two neighboring rods

[shown in Fig. 1(a)]. The cavities are located in the

13th layer of double periodicities of 5a0in both x and

y directions. Each cavity is created by filling a volume

of a0?a0?0.35a0 with the rod material ?n=2.4?

[shown in Fig. 1(b)]. The related reciprocal lattice

[Gx=2i?/?5a0?, Gy=2j?/?5a0?] and irreducible Bril-

louin zone of the Bloch wave vector ?qx,qy? for this pe-

with

riodic cavity array are shown in Figs. 2(b) and 1(c),

respectively. Since the resonant cavity array has

x-axis and y-axis mirror symmetry, the cavity modes

belong to the C2vgroup with four irreducible repre-

sentations (A1, A2, B1, and B2), and the character

table of the C2vgroup is listed in Fig. 2(c).15

We consider the incidence of plane waves with

wavevector[kx=qx+Gx,

−ky

k0=ncl?/c with c the speed of light and nclthe clad-

ding substrate refractive index. kx

+ky

waves, respectively [shown in Figs. 2(a) and 2(b)]. In

our TMM algorithm,11,12the Bloch boundary condi-

tion with ?qx,qy? is used for a 5?5 supercell. The

ky=qy+Gy,

kz=?k0

2−kx

2

2?1/2] upon the above resonant cavity array, where

2+ky

2?k0

2and kx

2

2?k0

2correspond to propagating and evanescent

Fig. 1.

cavity in the 13th layer): (a) 5?5 supercell of total 25 lay-

ers along the z direction with dielectric cladding of refrac-

tive index ncl(cladding is not shown); (b) top view of the

13th layer with the cavity of volume of a0?a0?0.35a0and

n=2.4; (c) irreducible Brillouin zone of ?qx,qy?. 3D woodpile

PC is composed of dielectric square rods of n=2.4 and

width 0.35a0, where a0is the distance between two neigh-

bored rods (i.e., the lattice constant).

xy-plane periodic array of 3D woodpile PC (with

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electric-field coefficient vector ?Eij,x,Eij,y? of plane

waves with Gx=2i?/?5a0?, Gy=2j?/?5a0? [named as

order ?i,j? in the following] can be transferred from

layer to layer in the z direction by a transfer matrix.

Here i,j=−N,...0,1,...N, and a favorable conver-

gence is found by using N=12 for the present 5?5

supercell. Previously13normal incidence with kx=ky

=0 [i.e., order (0,0) with qx=qy=0] was used to study

the PC cavities. However, not all the cavity modes

could be excited with (0,0) incidence with qx=qy=0

(as shown in Fig. 3) because of group symmetry con-

siderations. To excite all the cavity modes of PC cavi-

ties, we performed higher-order ?i,j? plane-wave inci-

dences with both e- and h-polarizations defined by

Eij,x

where the superscript index 0 represents the electric-

field components before entering the PC structure.

The incidence of plane waves with qx=qy=0 is first

considered. The irreducible representations that the

order ?i,j? could cover can be found through the pro-

jection operation,15and the results are shown in Fig.

2(d). Clearly, if the zero order ?i=j=0? (i.e., normal in-

cidence) is used alone, only the B1or B2representa-

tion is excited (for e- and h-polarizations, respec-

tively) and any resonant modes belonging to A1or A2

representation will not be excited. Actually for this

particular photonic crystal cavity array structure

there is indeed one resonant mode (mode A) that be-

longs to A2representation at normalized frequency

0=−1, Eij,y

0=0 and Eij,x

0=0, Eij,y

0=1 respectively,

0.43948; the other resonant mode (mode B) belongs to

B1representation at normalized frequency 0.44763.

The mode shape profiles for both modes are also cal-

culated by TMM with higher-order incidence and are

illustrated in Fig. 4(a). Based on the projection

operation,15any higher-order ?i,j? incidence with i

?1 and j?1 will cover all the representations and

both resonant modes can be excited.

To find the frequency and quality factor of the reso-

nant modes, the nearly continuous transmission

spectra are obtained accurately from the rational

function interpolation of 21 individual frequencies

ranging in the first bandgap.13Figure 3 shows the

transmission results for the plane-wave incidence of

the first four orders. Although the transmission am-

plitude for different orders and polarizations varies

dramatically, the resonant frequency and Q value for

each resonant mode are practically identical: fA

=0.43948, QA=11910 and fB=0.44763, QB=6900 (the

relative difference of f and Q for different incidences

is less than 10−5and 10−4, respectively). The excited

resonant peaks for each order ?i,j? incidence agree

well with the above group theory analysis. For ex-

ample, the group theory analysis indicates that the

(1,0) order incidence with e-polarization covers the

irreducible representations of A1and B1, and hence

mode B (belonging to B1), instead of mode A (belong-

ing to A2), appears in the calculated transmission

spectrum for the (1,0) incidence. There is also a third

resonant mode found by TMM at normalized fre-

quency ?a/?2?c?=0.4635.

The transmission spectra are calculated for plane-

wave incidence with different ?qx,qy?. We note that

Fig. 2.

[kx=qx+Gx, ky=qy+Gy, kz=?k0

cavity array (periodic in the xy-plane); solid (dashed) ar-

rows represent propagating (evanescent) waves. (b) Recip-

rocal lattice for 5?5 supercell PC cavity array; G points in-

side (outside) the dashed circle [with radius k0=ncl?/c and

center ?−qx,−qy?, shown for ?a/?2?c?=0.43, ncl=1.0, and

qx=qy=0] represent propagating (evanescent) waves. (c)

C2vgroup character table for the PC cavity array with E,

C2, ?x, and ?ysymmetry operation.15(d) Irreducible repre-

sentations for the e- and h-polarized incident plane waves

of order ?i,j?, i.e., Gx=2i?/?5a0? and Gy=2j?/?5a0?.

(a) Incidence of a plane wave with wave vector

2−kx

2−ky

2?1/2] upon the 3D PC

Fig. 3. Transmission spectra for the plane waves incidence

of qx=qy=0 and order ?i,j? upon the 3D PC cavity array

with e-polarization and h-polarization defined by Eij,x

Eij,y

A and B are also labeled. The first bandgap in the z direc-

tion [i.e., the (0, 0) incidence] opens from ?a/?2?c?=0.395

to ?a/?2?c?=0.515. There is the third resonant mode at

normalized frequency ?a/?2?c?=0.4635, which is not

shown in this figure.

0=−1,

0=0 and Eij,x

0=0, Eij,y

0=1, respectively. Resonant modes

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all the cavity modes can be excited for the case of

(qx?0, qy?0) because of the broken symmetry of

nonzero qxand qy. The frequencies of resonant cavity

modes can be extracted from the spectra, and the re-

lated dispersion relations are illustrated in Fig. 4(b).

The slopes of the dispersions are very flat, indicating

ultraslow group velocities in the woodpile PC cavity

array structure. The average group velocity (calcu-

lated by finding the slope of the straight line connect-

ing two high symmetry points) of mode A in the

?–X?X?? direction is ?g=0.0012c=3.6?105m/s ??g

=0.0048c=1.4?106m/s? and that of mode B in the

?–X? direction is ?g=0.0056c=1.7?106m/s. It is in-

teresting that a negative group velocity of ?g=

−0.0009c=−2.7?105m/s can be achieved for mode B

in the ?–X direction.

In conclusion, we have developed a higher-order

plane-wave incidence concept for the plane-wave-

based transfer matrix method with a rational-

function interpolation algorithm to efficiently simu-

late three-dimensional photonic crystal devices. As

an example, the dispersion relations and quality fac-

tors were calculated for the resonant cavity array

embedded in a layer-by-layer photonic crystal struc-

ture.An interesting ultraslow negative group velocity

is observed in this structure. One other advantage of

the transfer-matrix method is that there is no limita-

tion on the length of photonic crystal structure along

the propagation direction, making this method ideal

for waveguide simulation.16

M. Li’s e-mail address is mli@iastate.edu.

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13: (1)fullbandgap

interpolation to obtain continuous spectra, (2) no

limitation on the length of the PC along the

propagation direction.

range rational-function

Fig. 4.

cavity modes A and B at ? point ?qx=qy=0?. (b) Dispersion

relation of both cavity modes in the 3D PC cavity array.

Solid curve, ?–X–M; red dashed curve, ?–X?–M. Note:

?–X–M and ?–X?–M represent different ?qx,qy? directions

as shown in Fig. 1(c).

(Color online) (a) Electric field mode profiles for

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