Waveguides in three-dimensional layer-by-layer photonic crystals

Z.-Y. Li and K.-M. Ho Vol. 20, No. 5 /May 2003/J. Opt. Soc. Am. B 801

Waveguides in three-dimensional layer-by-layerphotonic crystals

Zhi-Yuan Li and Kai-Ming Ho

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

Received August 26, 2002; revised manuscript received December 12, 2002

A photonic crystal waveguide network can form in a three-dimensional layer-by-layer photonic crystal alongthree orthogonal directions. We investigate guided-mode band structures for different waveguide configura-tions by numerical calculations that combine a plane-wave expansion method with a supercell technique. Thein-plane waveguide network composed of waveguides along the (100) and (010) directions of the crystal is lo-cated in a single layer of the photonic crystal. This structural design is easy to achieve experimentally. Onecan create these in-plane waveguides either by removing one single rod or by breaking a segment in each of anarray of parallel rods. One produces the off-plane waveguide by removing some segments of rods along the(001) stacking direction of the crystal. Single-mode operation of a waveguide can be achieved by appropriateadjustment of geometrical parameters such as the location and the size of the waveguide. Intrinsic latticesymmetries in these photonic crystal waveguides have been fully employed to reduce the complexity of thenumerical solution of guided-mode band structures significantly. © 2003 Optical Society of America

OCIS codes: 230.7370, 130.2790.

1. INTRODUCTIONThe photonic crystal is a new class of material with a pe-riodic distribution of dielectric functions that can createphotonic bandgaps (PBGs) within which electromagnetic(EM) waves cannot propagate along any direction.1 Theexistence of a PBG provides a novel way to control andmold the flow of light much as conventional semiconduc-tors do to electrons. It has been suggested that photoniccrystals may be the best candidates to become platformsfor future electro-optic and all-optical ultrasmall inte-grated circuits and thus may hold the key to many poten-tial applications in science and technology.2 In recentyears much attention has been paid to this fundamentalgoal of building an optical circuit, and much progress to-ward achieving it has been made.3–23 Some of the basicfunctional elements in an ultrasmall electro-optic–all-optical circuit are photonic crystal waveguides, branches,and cavities. A photonic crystal waveguide is created bythe formation of a linear defect inside an otherwise per-fect photonic lattice. When the frequency of a guidedmode lies within the PBG, efficient confinement andpropagation of EM waves through straight lines, sharpbends, and multiple branches can be achieved.

Until now, most studies of building an optical circuit,either theoretical or experimental, were focused on func-tional elements formed inside a two-dimensional photoniccrystal and a two-dimensional photonic crystal slab3–12

because of their ease of fabrication in the optical and IRregimes by conventional lithographic techniques. How-ever, there is an intrinsic weakness in these structures:the leakage of EM waves in the vertical direction. Inprinciple, a photonic crystal waveguide is most efficient inguiding functionality in a three-dimensional (3D) latticewith a complete PBG, because the confinement of lighthere is complete and wave leakage along any direction isforbidden. A number of experimental investigations of

0740-3224/2003/050801-09$15.00 ©

waveguides in 3D photonic crystals have been performedin the microwave regime.13,14

In this paper we theoretically investigate the guided-mode behavior of a waveguide formed in a 3D layer-by-layer photonic crystal. This is a structure that possessesa complete PBG and can best be cast into a fabricationscheme that is compatible with state-of-the-art tech-niques in current semiconductor industries.15–18 Re-cently, layer-by-layer photonic crystals made from siliconand GaAs working at near-IR wavelengths were success-fully achieved, paving the way for the incorporation offunctional elements.17,18 One of the basic functional ele-ments inside a 3D lattice is a photonic crystal waveguidenetwork formed by individual waveguides extendingalong the three orthogonal principal axes of the layer-by-layer crystal. Through numerical solution of the bandstructure of guided modes, we try to find the optimumwaveguide structures that support single-mode operationwith as wide as possible bandwidths.

This paper is arranged as follows: In Section 2 webriefly describe the theoretical approach to solvingguided-mode dispersion. In Section 3 we discuss in de-tail the behavior of guided-mode dispersion in variouswaveguide structures formed inside a layer-by-layer pho-tonic crystal. Finally, we summarize this paper in Sec-tion 4.

2. THEORETICAL CONSIDERATIONS OFSTRUCTURAL SYMMETRIESA top-view schematic configuration of a layer-by-layerphotonic crystal is displayed in Fig. 1(a). We form thecrystal by stacking rectangular dielectric rods layer bylayer consecutively along the (001) direction, with aprimitive unit cell composed of two segments of rods intwo adjacent layers forming a face-centered tetragonal

2003 Optical Society of America

802 J. Opt. Soc. Am. B/Vol. 20, No. 5 /May 2003 Z.-Y. Li and K.-M. Ho

lattice. For Fig. 1(a) we have assumed that rods in thefirst layer lie along the x axis, that rods in the secondlayer lie along the y axis, and so on. Each rod has widthw and thickness h, and the rods form a square lattice withlattice spacing a in the (001) plane. The correspondingfilling fraction of rods is f 5 w/a, and the axis aspect ra-tio of the crystal is c0 /a 5 4h/a (c0 5 4h). The dielec-tric constants for the rod material and the background di-electric are 11.56 and 1, respectively, corresponding to astructure composed of silicon rods embedded in an airbackground. The conventional unit cell of this crystal iscomposed of four segments of rods (each with size a 3 w3 h) in four layers, forming a simple tetragonal lattice,and each unit cell has a size a 3 a 3 c0 . The photoniccrystal waveguide considered in this paper is naturallyselected as linear defects along the three orthogonal prin-cipal axes of the lattice, namely, the (100), (010), and (001)directions.

To reveal the optical properties of the photonic crystalwaveguides we used a plane-wave expansion method incombination with a supercell technique.19–21 The super-cell consisted of 5 3 5 conventional unit cells for all kindsof waveguide. Therefore the supercell size was a 3 5a3 5c0 for a (100) waveguide, 5a 3 a 3 5c0 for a (010)waveguide, and 5a 3 5a 3 c0 for a (001) waveguide. Weused as many as 6500 plane waves to solve the guided-mode band structure of these waveguides. As the guided

Fig. 1. Top-view schematic configuration of (a) a layer-by-layer3D photonic crystal made from rectangular dielectric rods in anair background and (b) an X waveguide created by removal of onesingle rod along the (100) direction of the photonic crystal.

mode propagates along the three principal axes of thecrystal, there are several intrinsic structural symmetries,such as mirror-reflection symmetry and central-inversionsymmetry. When they are fully investigated and ex-ploited, these structural symmetries can significantly re-duce computation effort in the solution of our electromag-netic problems.

We start from the following wave equation satisfied bymagnetic field H:

¹ 3 F 1

e~r!¹ 3 H~r!G 5

v2

c2 H~r!, (1)

where the dielectric function e(r) possesses certain sym-metries. To solve the guided-mode band structure in-volved in Eq. (1) we use a plane-wave expansion methodin which both H(r) and e(r) are expanded into a superpo-sition of plane-wave functions

H~r! 5 (K

~h1Ke1K 1 h2Ke2K!exp~iK • r!, (2)

e~r! 5 (G

eG exp~iG • r!. (3)

In Eqs. (2) and (3), G is the supercell reciprocal lattice ofthe photonic crystal, K 5 k 1 G, where k is Bloch’s wavevector. e1K and e2K are unit vectors normal to K and canbe defined as e1K 5 x 3 K 5 (0, 2Kz , Ky)/AKy

2 1 Kz2,

and e2K 5 K 3 e1K 5 (Ky2 1 Kz

2, 2KyKx , 2KzKx)/(KAKy

2 1 Kz2), where K 5 uKu. Substituting Eqs. (2)

and (3) into Eq. (1), finally we can arrive at the followingeigenequation:

HKK8ll8 S h1K8

h2K8D 5

v2

c2 S h1K

h2KD , (4)

where the Hamiltonian operator HGG8ll8 (l, l8 5 1, 2 are

superscripts for e1 and e2) is defined by

HKK8ll8 5 F e2K • e2K8 2e2K • e1K8

2e1K • e2K8 e1K • e1K8GeKK8uKuuK8u (5)

and is a Hermitian operator. In Eq. (4) the combinationof subscripts K8 and superscripts l8 means summationover them, as is usually defined in theoretical physics.

The structural symmetry is reflected both from thesymmetry of dielectric function e(r) and the propagationdirection of a guided mode characterized by Bloch’s wavevector k. In some symmetric operations, e(r) is invari-ant. In a layer-by-layer photonic crystal built from rect-angular rods arrayed in a face-centered tetragonal lattice(in some special situations a face-centered cubic lattice),the structural symmetry is much lower than the latticesymmetry owing to the low symmetry involved in a rect-angular rod. Several symmetries are present. These in-clude one central-inversion symmetry and two mirror-reflection symmetries with respect to the x axis (or theYOZ plane) and the y axis (or the XOZ plane). Nomirror-reflection symmetry with respect to the z axis orother rotation symmetry is available. In what follows,we take an example and show how these structural sym-


metries can be used to reduce the number of calculationsrequired.

Suppose that we have a mirror–reflection symmetrywith respect to the x axis (or the YOZ plane), whichmeans that e(2x, y, z) 5 e(x, y, z) and k5 (kx , ky kz) 5 (0, ky , kz). It is obvious that the Fou-rier coefficients are e11 5 e22 and e12 5 e21 , where thesubscripts 1 and 2 denote the wave vectors K15 (Kx , Ky , Kz) 5 (Gx , ky 1 Gy , kz 1 Gz) and K25 (2Kx , Ky , Kz) 5 (2Gx , ky 1 Gy , kz 1 Gz), respec-tively, with Kx . 0. In addition, we can verify that e1,1• e1,18 5 e1,2 • e1,28 , e2,1 • e2,18 5 e2,2 • e2,28 , e1,1 • e2,185 2e1,2 • e2,28 , and e2,1 • e1,18 5 2e2,2 • e1,28 . WithuK1u 5 uK2u we easily find the following symmetry rela-tions between the matrix elements of the Hamiltonian:H118

115 H228

11 , H11822

5 H22822 , H118

125 2H228

12 , and H11821

5 2H22821 . The same relations can be found between

H128ll8 and H218

ll8 , namely, H12811

5 H21811 , H128

225 H218

22 , H12812

5 2H21812 , and H128

215 2H218

21 . Besides, the Hamiltonian

H008ll8 is already diagonal [here 0 corresponds to K0

5 (0, Ky , Kz) 5 (0, ky 1 Gy , kz 1 Gz)] because e1,0• e2,08 5 e2,0 • e1,08 5 0. These symmetry relations leadus to make a unitary transformation:

F hlK0

hlK1

hlK2

G 5 SF glK0

glK1

glK2

G 5 F I 0

0 S2GF glK0

glK1

glK2

G , (6)

where hlK05 (h1K0

, h2K0)T (the superscript T denotes a

matrix transposition). S2 is defined by

S2 51

A2 F 1 0 1 0

0 1 0 1

1 0 21 0

0 21 0 1G ,

and I is a 2 3 2 unit matrix. More explicitly, the trans-formation is written as

g1K05 h1K0

, g2K05 h2K0

,

g1K15 ~h1K1

1 h1K2!/A2,

g2K15 ~h2K1

2 h2K2!/A2,

g1K25 ~h1K1

2 h1K2!/A2,

g2K25 ~h2K1

1 h2K2!/A2. (7)

With this transformation, the eigenequation

F H00ll8 H018

ll8 H028ll8

H108ll8 H118

ll8 H12ll8

H208ll8 H218

ll8 H228ll8G F hl8K08

hl8K18

hl8K28

G 5v2

c2 F hlK0

hlK1

hlK2

G . (8)

becomes

3H008

11 0 A2H01811 0 0 0

0 H00822 0 0 A2H018

21 A2H01822

A2H01811 0 H118

111 H128

11 H11812

2 H12812 0 0

0 0 H11821

1 H12821 H118

222 H128

22 0 0

0 A2H01821 0 0 H118

112 H128

11 H11812

1 H12812

0 A2H01822 0 0 H118

212 H128

21 H11822

1 H12822

4 3g1K08

g2K08

g1K18

g2K18

g1K28

g2K28

4 5v2

c2 3g1K0

g2K0

g1K1

g2K1

g1K2

g2K2

4 . (9)

Equation (9) can be further separated into two half-sizedeigenequations:

F H00811 A2H018

11 0

A2H01811 H118

111 H128

11 H11812

2 H12812

0 H11821

1H12821 H118

222 H128

22G F g1K08

g1K18

g2K18

G5

v2

c2 F g1K0

g1K1

g2K1

G , (10)

F H00822 A2H018

21 A2H01822

A2H01821 H118

112 H128

11 H11812

1 H12812

A2H01822 H118

212 H128

21 H11822

1 H12822G F g2K08

g1K28

g2K28

G5

v2

c2 F g2K0

g2K2

g1K2

G , (11)

Further consideration of symmetric relations H11812

5 H11821 and H128

125 2H128

21 yields


F H00811 A2H018

11 0

A2H01811 H118

111 H128

11 H11812

2 H12812

0 H11812

2 H12812 H118

222 H128

22G F g1K08

g1K18

g2K18

G5

v2

c2 F g1K0

g1K1

g2K1

G ,

(12)

F H00822 A2H018

21 A2H01822

A2H01821 H118

112 H128

11 H11812

1 H12812

A2H01822 H118

121 H128

12 H11822

1 H12822G F g2K08

g1K28

g2K28

G5

v2

c2 F g2K0

g1K2

g2K2

G .

(13)

After the solution of eigenvalues and eigenvectors fromEqs. (12) and (13) has been obtained, the original eigen-vector is given by

h1K05 g1K0

, h2K05 0,

h1K15 g1K1

/A2, h2K15 g2K1

/A2,

h1K25 g1K1

/A2, h2K25 2g2K1

/A2(14)

for mode 1 [corresponding to the solution of Eq. (12)] and

h1K05 0, h2K0

5 g2K0,

h1K15 g1K2

/A2, h2K15 g2K2

/A2,

h1K25 2g1K2

/A2, h2K25 g2K2

/A2(15)

for mode 2 [corresponding to the solution of Eq. (13)].Now let us see what the symmetry is behind the EMfields. For mode 1, the corresponding H field is

H~r! 5 (K0

~h1K0e1K0

1 h2K0e2K0

!exp~iK0 • r!

1 (K1

~h1K1e1K1

1 h2K1e2K1

!exp~iK1 • r!

1 (K2

~h1K2e1K2

1 hK2e2k2

!exp~iK2 • r!,

5 (K0

g1K0e1K0

exp~iK0 • r!

1 (K1

1

A2$ g1K1

@ e1K1exp~iK1 • r!

1 e1K2exp~iK2 • r!#

1 g2K1@ e2K1

exp~iK1 • r! 2 e2K2exp~iK2 • r!#%.

(16)

Inserting into Eq. (16) the definition of e1K0, e2K0

, e1K1,

e2K1, e1K2

, and e2K2, we find that

Hx~r! 5 (K1

sin~K1xx !Vx~ y, z, K1!, (17)

Hy~z !~r! 5 (K0

Uy~z !~ y, z, K0!

1 (K1

cos~K1xx !Vy~z !~ y, z, K1!, (18)

where U and V are vector functions. Now we can seethat the Hx field in mode 1 is odd at mirror-reflectionsymmetry with respect to the x axis, whereas the Hy andHz fields are even at this symmetry. Mode 2 shows acompletely inverse behavior for this mirror-reflectionsymmetry: The Hx field is an even function, Hy and Hzare odd functions with respect to the x axis.

We can see now that an x axis mirror-reflection symme-try can reduce the original eigenproblem to two half-sizedeigenproblems. Mirror-reflection symmetries with re-spect to other axes can also be utilized with the same pro-cedure as for unitary transformation. When there ismore than one symmetry, these transformations are usedsequentially, one after another, to reduce the eigenprob-lem into irreducible problems. Another type of symme-try is central-inversion symmetry. Unlike mirror-reflection symmetry, central-inversion symmetry cannotreduce the size of the eigenproblem; instead, it reduces ageneral complex Hermitian matrix into a real symmetricmatrix. Therefore the central-inversion symmetry alsosignificantly enhances the efficiency of a numerical solu-tion of the eigenproblem.

3. GUIDED MODES IN PHOTONICCRYSTAL WAVEGUIDESThroughout this paper we consider only a photonic crystalwith a filling fraction of f 5 0.25 and axis aspect ratioc0 /a 5 1.25. The band structure calculation shows thatthere is a complete PBG of 0.366–0.434(2pc/a) in thisperiodic structure, where c is the speed of light invacuum.

First we consider an in-plane waveguide network com-posed of waveguides along the (100) and (010) directions,which are termed X and Y waveguides, respectively. Thenatural way (also the easiest way experimentally) to gen-erate an X waveguide is to remove a single x-axis rod inthe first layer, as is schematically depicted in Fig. 1(b).Note that this waveguide has a central-inversion symme-try (with respect to the central axis of the rod) and amirror-reflection symmetry with respect to the y axis (or,equivalently, to the XOZ plane). The calculated guided-mode band structure is plotted in Fig. 2 by solid curves.Only modes in the bandgap are shown. The shaded re-gions denote the bulk bands of a perfect photonic crystalprojected onto the waveguide direction. To calculatethese projected bands we used a conventional unit cellcomposed of four rods and with a size a 3 a 3 c0 . Thewaveguide mode first rises monotonically from the kx5 0 point at v 5 0.375(2pc/a) to kx 5 p/a (Brillouin


zone boundary) at v 5 0.423(2pc/a) and then folds backto form a second band. Note that the second band is notmonotonic, so we can see that this X waveguide does notsupport single-mode operation of the guided mode in thefrequency regime of the second band. However, if one isinterested only in the central part of this wide guided-mode band, then single-mode operation can be achieved.We changed parameters f and c0 /a of the photonic crystaland found similar guided-mode dispersion behavior forthis X waveguide.

To construct an in-plane network of photonic crystalwaveguides it is necessary to have a Y waveguide to worktogether with the X waveguide. One can create the Ywaveguide by removing a single y-axis rod in the secondlayer, as suggested by Chutinan and Noda.22,23 The Ywaveguide created in this way is identical to an X wave-guide and thus has the same guided-mode dispersion.The X and Y waveguides form an orthogonal network, ex-cept that they shift vertically to each other by an amounth. Another way to form a Y waveguide is to remove asegment (of width wx) of each x-axis rod in the first layer,such that we have a totally in-plane waveguide network.Compared with the former scheme to create the X and Ywaveguides in two layers, this scheme for a total in-planewaveguide network may be more conveniently achievedexperimentally. Two different positions for the removedsegments can be adopted, corresponding to differentstructural symmetries. The first is shown in Fig. 3(a)and is termed a type I Y waveguide, where the centralaxis of the Y waveguide (denoted by a dashed line) is lo-cated in the YOZ mirror-reflection symmetric plane. Thesecond type is shown in Fig. 3(b) and is termed a type II Ywaveguide, where the central axis of the Y waveguide (de-noted by a dashed line) is shifted by a/4 with respect tothat of the type I Y waveguide and thus is located in acentral-inversion symmetric line.

Fig. 2. Photonic band structure of guided modes in the X wave-guide shown in Fig. 1(b). The shaded background pattern rep-resents the body’s photonic bands projected onto the waveguidedirection. The waveguide is formed in a photonic crystal madefrom silicon rods with a filling fraction f 5 0.25 and an aspect ra-tio c0 /a 5 1.25, where c0 and a are lattice spacings along the(001) and (100) directions, respectively.

The type I Y waveguide has its central axis located inthe mirror plane of the photonic crystal. This site is anatural choice; however, numerical simulation has shownthat this configuration is not the optimum structure for awaveguide, as can be seen from Fig. 4(a) for the guided-mode band structure of a Y waveguide, where the width ofthe removed segments is wx 5 0.7a. In this figure theshaded regions also denote the bulk bands of a perfectphotonic crystal projected onto the waveguide direction ( yaxis). It is clear that this Y waveguide has two principalguided modes. The first guided-mode band extends from0.366 to 0.420 2(pc/a); the second, from 0.390 to 0.428(2pc/a). Both bands are quite wide, and together theyoccupy almost the whole photonic bandgap. When wechange wx or other parameters such as f and c0 /a thedouble-mode operation of this Y waveguide remains un-changed, except for shifts of the bands and a change of thebandwidth. It seems that this double-mode characteris-tic comes from the in-plane mirror-reflection symmetrythat is intrinsic in this type of waveguide. This mirrorreflection is absent from the type II Y waveguide shown inFig. 3(b), and it is suspected that the dispersion behavior

Fig. 3. Schematic configuration of Y waveguides created in alayer-by-layer photonic crystal by removal of one wx-wide seg-ment of the x-axis rods along the (010) direction. (a) Type I Ywaveguide with its central axis located in a mirror-reflectionsymmetric plane with respect to the x axis. (b) Type II Y wave-guide with its central axis shifted by a/4 relative to that of a typeI Y waveguide and located in a central-inversion symmetric po-sition.


might be fundamentally different in the latter structure.Indeed, we have found that single-mode operation caneasily be achieved in a type II Y waveguide. Figure 4(b)shows the guided-mode band structure for a type II Ywaveguide with wx 5 0.4a. A wide single band is foundthat extends from 0.380 (2pc/a) at ky 5 0 to 0.427(2pc/a) at ky 5 p/a. This single-mode behavior makesthe type II Y waveguide superior to the type I Y wave-guide and thus a better candidate for use in an in-planewaveguide network.

We now consider the off-plane waveguide. To form a(001) waveguide along the z axis of a layer-by-layer pho-tonic crystal we may use any of a number of configura-tions. We consider only waveguides generated by re-moval of segments of rods along the (001) direction.Specifically, two different positions are selected, whoseschematic configurations are displayed in Fig. 5. We cre-ate the waveguide in Fig. 5(a) (called a type I Z wave-guide) by removing segments in rods in the first and sec-

Fig. 4. Photonic band structure of guided modes in (a) the typeI Y waveguide shown in Fig. 3(a) for wx 5 0.7a and (b) the typeII Y waveguide shown in Fig. 3(b) for wx 5 0.4a. The photoniccrystal has parameters of f 5 0.25 and c0 /a 5 1.25.

ond layers consecutively every four layers (with widths ofwx and wy , respectively), while rods in the third andfourth layers are untouched. The central axis of this Zwaveguide is located in the cross section of two mirror-reflection symmetric planes, the XOZ and YOZ planes.We create the waveguide shown in Fig. 5(b) (called a typeII Z waveguide) by removing each segment with a widthof wx in the first layer consecutively every four layers andshifting the central axis of the waveguide by a/4 relativeto that of the type I Z waveguide. Besides a mirror-reflection symmetry with respect to the y axis (or the XOZplane), this type II Z waveguide has an additional central-inversion symmetry. As occurs for the in-plane Ywaveguides, different structural symmetries will lead todifferent dispersion behaviors of guided modes.

There are two different situations for a type I Z wave-guide. One is wx Þ 0 and wy 5 0, corresponding to astructure in which only rods in the first layer are brokenwhereas rods in the second layer are untouched; the otheris wx Þ 0 and wy Þ 0, for which rods in both layers arebroken simultaneously. The guided-mode band structurefor wx 5 0.7a and wy 5 0 is displayed in Fig. 6(a); theshaded region also represents the bulk photonic bands

Fig. 5. Schematic configuration of Z waveguides created in alayer-by-layer photonic crystal by removal of some segments ofrods consecutively every four layers along the (001) direction.(a) Type I Z waveguide with its central axis located at the inter-section of two mirror-reflection symmetric planes with respect tothe x and y axes. (b) Type II Z waveguide with its central axisshifted by a/4 relative to that of a type I Z waveguide and locatedin a position with both central-inversion symmetry and mirror-reflection symmetry with respect to the y axis.


projected onto the z axis direction. Two principal guided-mode bands are present but do not overlap, so single-mode operation is achieved. However, both guided-modebands are quite narrow, and there is a gap between thetwo bands. The upper band extends from 0.399 (2pc/a)at kz 5 p/c0 to 0.411 (2pc/a) at kz 5 0. The lower bandis much narrower, spanning 0.385 (2pc/a) at kz 5 p/c0to 0.387 (2pc/a) at kz 5 0. To improve the bandwidthwe increase the widths of the removed segments. Figure6(b) shows the results for a much wider waveguide withwx 5 1.5a and wy 5 0. Now there are four guided-modebands within the photonic bandgap. The upper twobands are separated by a gap, whereas the lower twobands have a large overlap; therefore this wide waveguideoperates on a mixture of single-mode behavior (at higherfrequency) and double-mode behavior (at lower fre-quency). Compared with that of the wx 5 0.7a wave-guide, the total bandwidth is significantly increased, withthe four bands spanning 0.366–0.379, 0.371–0.381,

Fig. 6. Photonic band structure of guided modes in the type I Zwaveguide shown in Fig. 5(a) for (a) wx 5 0.7a, wy 5 0 and (b)wx 5 1.5a, wy 5 0. The photonic crystal has f 5 0.25 andc0 /a 5 1.25.

0.407–0.411, and 0.421–0.437 (2pc/a). From a naıvephysical viewpoint (such as that for a tight-bindingmodel), the bandwidth of a guided-mode band is closelyconnected to the coupling between adjacent point defects(cavities) along the linear defect chain (a waveguide). Soit might be expected that a structure with both rods bro-ken (wx Þ 0 and wy Þ 0) should have a wider guided-mode band than a structure with only one rod broken(wx Þ 0 while wy 5 0). However, this simple argumentis not supported by our numerical simulations. Figure 7shows the band structure of guided mode for a type I Zwaveguide with wx 5 wy 5 0.7a. At first glance, Fig. 7is quite similar to Fig. 6(a), except for a slight relativeshift. In fact, this waveguide supports three principalguided modes. The upper two bands are degenerate andspan 0.411 (2pc/a) at kz 5 p/c0 to 0.420 (2pc/a) at kz5 0. The single lower band is quite flat, spanning 0.399to 0.402 (2pc/a). Compared with the waveguide withwx 5 0.7a and wy 5 0 [shown in Fig. 6(a)], the latterwaveguide (wx 5 wy 5 0.7a), which is assumed to havestronger coupling, does not have an improved guided-mode bandwidth, more worse, the guiding functionalitybecomes degraded because of its double-mode behavior.

We have seen in the above discussions of (010)waveguides that a type II Y waveguide has a better guid-ing property than a type I Y waveguide because of itslower mirror-reflection symmetry. Our numerical simu-lations indicate that this is also true for a (001) wave-guide. Single-mode operation can be obtained muchmore easily in a type II Z waveguide than in a type I Zwaveguide. In addition, a type II Z waveguide allows formuch greater improvement in the bandwidth of guidedmode, which one can achieve by simply increasing thewidth of the waveguide. This can be clearly seen in theguided-mode band structures shown in Figs. 8(a) and 8(b)for a type II Z waveguide with wx values of 0.4a and 1.5a,respectively. The wx 5 0.4a waveguide is in single-modeoperation behavior, with the band extending from 0.393(2pc/a) at kz 5 p/c0 to 0.402 (2pc/a) at kz 5 0. Thebandwidth can be significantly improved by removal of

Fig. 7. Photonic band structure of guided modes in the type I Zwaveguide shown in Fig. 5(a) for wx 5 wy 5 0.7a.


wider segments from the rods. The waveguide with wx5 1.5a has three principal guided-mode bands, spanningthe frequencies 0.385–0.393, 0.412–0.416, and 0.419–0.437 (2pc/a), as shown in Fig. 8(b). With a two-timeswider guided-mode band, the waveguide still remains insingle-mode operation. This characteristic clearly indi-cates that a type II Z waveguide is a better candidate foroff-plane guiding functionality than a type I Z waveguide.

In the above discussion of both in-plane and off-planewaveguides, both single-mode operation and guided-modebandwidth have been discussed in terms of the design ofan optimum waveguide structure. We have found thatwe do not have much flexibility in designing structuralparameters to achieve optimal functionality. However, insome situations single-mode behavior is not required. Inthis case there will be much more freedom to create a pho-tonic crystal waveguide in a layer-by-layer lattice. Forinstance, suppose that we have a planar photonic crystalchip, which means that the chip has an in-plane dimen-sion far larger than the off-plane dimension. If the Zwaveguide is designed only to serve as an input–outputchannel to transport light signals into and out of an in-

Fig. 8. Photonic band structure of guided modes in the type II Zwaveguide shown in Fig. 5(b) for (a) wx 5 0.4a and (b) wx5 1.5a. The photonic crystal has f 5 0.25 and c0 /a 5 1.25.

plane waveguide network, then a single-mode waveguideis not a necessary requirement. In this situation we cancreate a wide Z waveguide that has lateral sizes far largerthan the wavelength. Then multimode even continuumbands will span the whole photonic bandgap. A largenumber of design schemes can achieve this goal. For in-stance, a simple large square hole etched into the planarchip suffices to serve as a continuum input–output chan-nel.

4. SUMMARY AND CONCLUSIONSIn summary, we have used a plane-wave expansionmethod combined with a supercell technique to investi-gate the photonic crystal waveguide network formed in-side a 3D layer-by-layer photonic crystal along the threeprincipal axes of the crystal. The intrinsic structuralsymmetries involved in these waveguides, such as mirror-reflection symmetry and central-inversion symmetry,have been fully exploited to reduce the complexity of thenumerical solution of the electromagnetic problem ofguided modes significantly.

We form a waveguide network by connecting individualin-plane waveguides and off-plane waveguides. The in-plane waveguide network, composed of waveguides alongthe (100) and (010) directions of the crystal, is located in asingle layer of the photonic crystal, a configuration that iseasy to achieve experimentally. These in-planewaveguides can be created in either of two ways, by for-mation of X or Y waveguides. We form an X waveguideby removing one single x-axis rod along the (100) direc-tion; we generate a Y waveguide by breaking a segment ineach x-axis rod along the (010) direction. The off-planewaveguide is produced by removal of one segment of oneor two rods consecutively every four layers along the (001)stacking direction of the crystal. It was found thatsingle-mode operation in both in-plane and off-planewaveguides can be achieved by appropriate adjustment ofgeometrical parameters such as the location and size ofthe waveguide. An appropriate geometric parameter canalso lead to an improved guided-mode bandwidth.

It should be noted that in this paper we have discussedonly the guiding properties of individual waveguides thatcan be grouped into a 3D waveguide network within a 3Dlayer-by-layer photonic crystal. Such an investigationwill form the basis for future research on a realistic pho-tonic crystal waveguide network, in which EM wavepropagation through the connection of differentwaveguides must be examined carefully.

ACKNOWLEDGMENTAmes Laboratory is operated for the U.S. Department ofEnergy by Iowa State University under contract W-7405-Eng-82.

Z.-Y. Li’s e-mail address is [email protected].

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Waveguides in three-dimensional layer-by-layer photonic crystals