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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 9, SEPTEMBER 2003 1243

Accurate Modeling of Line-Defect Photonic CrystalWaveguides

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau

AbstractAn accurate three-dimensional method to calculatethe Bloch modes of photonic crystal (PhC) waveguides is proposed.Good agreement with available experimental and numerical datais obtained. The originality of the method lies in the fact that theBloch modes are seen as the electromagnetic fields associated tothe complex poles of an in-plane transversal scattering matrix. Incomparison with previous approaches, the computational domaindiscretized is smaller and a higher accuracy for the losses of PhCwaveguides is achieved.

Index TermsIntegrated optics, optical waveguide, photoniccrystal (PhC), waveguide computation.

ALINEAR ROW of defects in a photonic crystal (PhC) pro-

duces PhC Bloch modes within the photonic band gap.

Because the manufacture of three-dimensional PhC with sub-

micronic feature sizes is extremely challenging, PhC waveg-

uides etched into slab heterostructures have recently been the

subject of intense research [1][3]. These waveguides are ex-

pected to have unique properties not provided by conventional

dielectric waveguides. For example, guided modes with slow

group velocity [2] can be realized, thus allowing enhanced pro-

cesses induced by light-matter interaction such as amplification

or wavelength conversion. In order to design these new active

devices, it is essential to be able to accurately model line-de-

fect PhC waveguides and, in particular, their propagation losses.Fig. 1(a) shows a top view of the PhC waveguide studied in this

work, obtained by removing a single row of holes in a two-di-

mensional triangular lattice of air holes etched into a slab het-

erostructure. This line-defect is surrounded on both sides by

a finite number of rows. Although the propagation losses

of line-defect waveguides are an important issue, there have

been few reports dealing with their calculation [2], [4][9]. The

main reason is that the calculation of the attenuation is a dif-

ficult numerical challenge. Mode solver techniques developed

for translation-invariant waveguides in integrated optics cannot

be straightforwardly applied to compute the leakage of modes

in periodic structures. In addition, the PhC waveguide mod-

eling requires a huge computational domain for sampling the

lengthy transverse direction of the waveguide, the -direction in

Fig. 1(a). In this letter, a rigorous method that bypasses this dis-

Manuscript received November 12, 2002; revised April 28, 2003.C. Sauvan, P. Lalanne, J. C. Rodier, and J. P. Hugonin are with the Labora-

toire Charles Fabry de lInstitut dOptique, Centre National de la RechercheScientifique, Universit Paris Sud, F-91403 Orsay Cedex, France (e-mail:christophe.sauvan@iota.u-psud.fr).

A. Talneauis withthe Laboratoirede Photoniqueet de Nanostructures,CentreNational de la Recherche Scientifique, F-91460 Marcoussis, France.

Digital Object Identifier 10.1109/LPT.2003.816123

Fig. 1. Computational domains used to model single-line-defect PhCwaveguides. (a) Top view and (b) side view for a waveguide with N = 7rows of holes on each side of the defect. The bold rectangles delimitate thecomputational domain. (c) Transverse section of the computational domain tobe discretized with the present approach. Pseudoperiodic boundaries and PMLare used to delimitate the computational domain, respectively, along the z - andthe x -directions.

cretization problem is proposed. The net benefit is an increased

accuracy.

To illustrate the originality of the present work, we first re-

view previous approaches. These approaches may be separated

into two groups. The first group refers to methods relying on

a full three-dimensional discretization of the computational

domain. Finite-difference time-domain techniques [2], [4], [7]

and finite-element techniques in the frequency domain [5] have

been used for the computation of PhC modes, with pseudope-

riodic conditions in the -direction and absorbers in the -

and -directions. The second group refers to semi-analytical

methods relying on a recursive computation of some transfer

matrix with a two-dimensional discretization. A frequency-do-main approach [6] was developed to compute the PhC modes

as the eigenstates of the matrix defined along the -direction

[see Fig. 1(a)]. In [8] and [9], the PhC waveguide is artificially

periodized along the -direction (super-cell technique) in order

to generate a biperiodic grating composed of parallel waveg-

uides separated by a few hole rows. In [8], the PhC modes are

computed by a finite-basis expansion and diffraction losses

are obtained approximately by treating the coupling to leaky

modes by use of Fermis golden rule. In [9], the PhC modes

are computed as the complex poles of a transversal scattering

1041-1135/03$17.00 2003 IEEE

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1244 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 9, SEPTEMBER 2003

matrix [along the -direction, see Fig. 1(b)] which links the

electromagnetic field amplitudes in the claddings. In grating

theory, this technique, called polology [10], has been used to

successfully study diffraction anomalies or resonances in the

past. Although they elegantly suppress the need for virtual

absorbers in the claddings, these approaches introduce an

artificial coupling between adjacent waveguides. This coupling

reflects in the computed attenuation, which exhibits sharp andunphysical spectral variations which have to be smoothed by

some hand-driven averaging procedure [8], [9] over different

super-cell widths. In all these approaches [2], [4][9], we note

that the lengthy -direction has to be discretized.

Let us consider the transversal scattering matrix along the

in-plane -direction [see Fig. 1(a)]. To calculate this matrix, one

is facing a diffraction problem by a one-dimensional grating in

an optical waveguide with a periodicity along the -direction.

The incident medium and the substrate correspond to the

unetched heterostructure on the left and right sides of the PhC

[see Fig. 1(a)]. Letus assumethat the grating is illuminated from

the incident medium by the fundamental guided mode of

the unetched heterostructure. We denote by the -componentof the in-plane wave vector of the incident mode. We have re-

cently developed numerical tools to solve such in-plane diffrac-

tion problems in integrated optics [11][13] using Fourier-ex-

pansion techniques and absorbers in the transverse -direction.

Within the approach, the scattering matrix linking the out-

going and ingoing modal amplitudes in the planes and

of Fig. 1(a) is computed recursively by approximating the real

continuous profile by a stack of slices with piecewise-constant

permittivities. Thus, the integration of Maxwells equations in

the -direction is done analytically and the modes in every slice

are computed exactly. Details concerning the Fourier-expansion

technique used to solve Maxwells equations can be found in

[13]. Fig. 1(c) shows the cross section of a typical slice used for

the computation. The PhC modes are searched by allowing for

complex values of and looking for the complex poles of the

scattering matrix. The effective indexes of every PhC mode

are then given by , where represents

the attenuation of the PhC modes. To our knowledge, this work

is the first one to report on a polology approach for the diffrac-

tion of guided-waves by gratings integrated in planar waveg-

uides. Because the analytical integration is performed along the

lengthy transverse -direction, the computational domain to be

discretized is considerably reduced in comparison with those

used in previous works. Additionally, we note that absorbers

along the -direction are not required with the present approach.They have been implemented in the approaches of [2], [4], [6],

and[7] and, although they arerequiredfor settling rigorously the

electromagnetic problem, they are not implemented in [5], [8],

and [9]. For the sake of performance, the absorbers are imple-

mented in this work as perfectly matched layers (PMLs) using

a complex-coordinate stretching [14].

The method has been tested against experimental results ob-

tained for a PhC waveguide etched into an InP heterostructure

and against available numerical results obtained fora PhC wave-

guide etched into an air-membrane. Fig. 2(a) provides a com-

parison with experimental data obtained for a single-row PhC

waveguide etched into an InP heterostructure composed of a

Fig. 2. Validation of the method. (a) Comparison with experimental dataobtained for a one-line-defect PhC waveguide etchedinto a heterostructure withInP claddings. (b) Comparison with other numerical data for an air membrane.Thin curve: data obtained with the method in [8]. Triangles: computed data in[6]. Bold curve: computed data with the present method. The discrepancy fork > 0 : 2 8 3 between the present results and the triangles is due to the weakeraccuracy of the method in [6]. The vertical solid and dashed lines represent the

wavevector values for which the dispersion relation of the PhC mode crosses,respectively, the air light line and the upper band edge.

500-nm-wide core (refractive index 3.36) with InP claddings

(refractive index 3.17). The cover thickness is 200 nm. Scanning

electron microscope photographs have revealed that the holes

are 3 m deep with a radiusof 150 nm, and thatthe triangular

lattice constant is 450 nm. Measurements of the PhC wave-

guide attenuation were performed using a fiber-to-fiber setup

under transverse-electric (TE) polarization for different wave-

guide lengths. More details concerning the fabrication and the

characterization can be found in [15]. For this heterostructure

configuration, the fundamental PhC mode is operating above

the light line of the claddings. Thus, as it propagates along therow-defect, it is attenuated and leaks mainly in the substrate. As

shown in Fig. 2(a), the general trend of the measured attenua-

tion, i.e., the increase of the attenuation with the wavelength,

is well reproduced by the numerical predictions. However, uni-

formly over the spectrum, the predictions are 2030 dB/mm

lower than the experimental data. We believe that this system-

atic deviation is likely to result from intrinsic factors (like hole

roughness or disorder) not taken into account in the modeliza-

tion.

After comparison with experimental data, let us confront our

numerical predictions with available computational data. We

consider for that purpose a PhC waveguide etched in a semicon-ductorair membrane, a system which has attracted much atten-

tion recently [1][3] because it supports a nonleaky PhC mode

for small frequencies. The refractive index of the semiconductor

slabis 3.4, the triangularlatticeconstant is 390 nm, the slab

thickness is , and the hole radius is . The dispersion

relation of the fundamental PhC mode for such membrane pa-

rameters is well known (see [16, Fig. 2(b)]). Above the air light

line, the fundamental PhC mode looks like a refractive one

and leaks in the air claddings. Below the light line, it becomes

truly guided and possesses a small group velocity which van-

ishes for . The attenuation [bold curve in Fig. 2(b)] of

the fundamental PhC mode for a defect surrounded by

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SAUVAN et al.: ACCURATE MODELLING OF LINE-DEFECT PhC WAVEGUIDES 1245

rows was computed over the entire bandgap region. A compar-

ison with other computational data obtained for the same system

is shown in Fig. 2(b). The thin curve represents data provided by

Andreani using the approximate method described in [8]. Trian-

gles represent data from [6]. Note that these data, obtained for

220 nm in [6], have been rescaled by a factor 220/390.

A very good agreement is obtained between the three methods

in the wavevector region between 0.1 and 0.283, which corre-sponds to the frequency region above the light line.

The prediction of attenuation for modes operating below the

light line ( ) deserves attention since it may appear

surprising at first sight and has not been discussed in the liter-

ature. Below the light line, for a line-defect surrounded by two

semi-infinite PhC, the mode is truly guided and the attenuation

is theoretically null. However, for a finite number of rows as

we consider here, the PhC mode leaks into the unetched wave-

guide sections previously called the incident medium and the

substrate. The mechanism responsible for this leakage is a

tunneling of the mode through the finite thickness of the PhC.

This tunneling has been confirmed by calculations performed

for larger values. As shown in Fig. 2(b), smaller attenuationsare observed for and ; for , the attenua-

tion varies between 10 and 10 dB mm and does not show

up in the figure. The attenuation variation below the light line,

showing a peak for , is understood by considering the

band diagram of the fundamental PhC mode. For this specific

value, the energy difference between the PhC mode and the

lower bandgap edge is small and minimal (see [16, Fig. 2(b)]).

Since the penetration depth into a PhC increases as one ap-

proaches the bandgap edge, the tunneling effect is favored for

. As one departs from this value, the PhC mode fre-

quency movestoward the midgap frequency (see [16, Fig. 2(b)])

and the tunneling through the finite PhC thickness decreases.

In this letter, a numerical method to calculate the Bloch

modes of PhC waveguides embedded into heterostructure slabs

has been proposed. Within the approach, the Bloch modes

are seen as the fields associated to the complex poles of an

in-plane scattering matrix of a grating diffraction problem in

an optical waveguide. Because the approach is semi-analytical

in the lengthy transverse direction of the waveguide, the com-

putational domain that has to be discretized is rather small and

a good accuracy for the attenuation losses induced by radiation

into the cladding or by tunneling through the finite thickness

of the PhC is obtained. The approach developed in this work

is general and other numerical methods, not based on modal

Fourier expansion, could benefit from this work.

ACKNOWLEDGMENT

The authors would like to thank L.C. Andreani for providing

them with some of the numerical data used for the compar-

ison, andthe authors acknowledgethe reviewers fortheirhelpful

comments.

REFERENCES

[1] S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, Demonstra-tion of highly efficient waveguiding in a photonic crystal slab at the 1.5 m wavelength, Opt. Lett., vol. 25, no. 17, pp. 12971299, 2000.

[2] M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, andI. Yokohama, Extremely large group-velocity dispersion of line-de-fect waveguides in photonic crystal slabs, Phys. Rev. Lett., vol. 87, p.253902, 2001.

[3] M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P.Pearsall, Waveguiding in planar photonic crystals, Appl. Phys. Lett.,vol. 77, no. 13, pp. 19371939, 2000.

[4] B. DUrso, O. Painter, J. OBrien, T. Tombrello, A. Yariv, andA. Scherer, Modal reflectivity in finite-depth two-dimensionalphotonic-crystal microcavities, J. Opt. Soc. Amer. B, vol. 15, pp.11551159, 1998.

[5] G. R. Hadley, Out-of-plane losses of line-defect photonic crystalwaveguides, IEEE Photon. Technol. Lett., vol. 14, pp. 642644, May

2002.[6] P. Lalanne, Electromagnetic analysis of photonic crystal waveguides

operating above the light cone, IEEE J. Quantum Electron., vol. 38,pp. 800804, July 2002.

[7] Y. Dsires, T. Benyattou, R. Orobtchouk, A. Morand, P. Benech, C.Grillet, C. Seassal, X. Letartre, P. Rojo-Romeo, and P. Viktorovitch,Propagationlosses of thefundamental mode in a singleline-defect pho-tonic crystal waveguide on an InP membrane, J. Appl. Phys., vol. 92,no. 5, pp. 22272234, 2002.

[8] L. C. Andreani and M. Agio, Intrinsic diffraction losses in photoniccrystal waveguides with line defects, Appl. Phys. Lett., vol. 82, pp.20112013, 2003.

[9] M. Le Vassor dYerville, Modlisation de Cristaux Photoniques Bidi-mensionnels de Hauteur Finie, PhD dissertation (in French), Universitde Montpellier II, 2002.

[10] R. Petit, Electromagnetic Theory of Gratings. Berlin, Germany: Sp-inger-Verlag, 1980.

[11] P. Lalanne and E. Silberstein, Fourier-modal methods applied towaveguide computational problems, Opt. Lett., vol. 25, no. 15, pp.10921094, 2000.

[12] P. Lalanne and H. Benisty,Out-of-plane losses of two-dimensionalpho-tonic crystals waveguides: Electromagnetic analysis, J. Appl. Phys.,vol. 89, no. 2, pp. 15121514, 2001.

[13] E. Silberstein, P. Lalanne, J. Hugonin, and Q. Cao, Use of grating the-ories in integrated optics, J. Opt. Soc. Amer. A, vol. 18, no. 11, pp.28652875, 2001.

[14] W. C. Chew and W. H. Weedon, A 3D perfectly matched medium frommodified Maxwells equations with stretched coordinates, MicrowaveOpt. Technol. Lett., vol. 7, no. 13, pp. 599604, 1994.

[15] A. Talneau, M. Mulot, S. Anand, and P. Lalanne, Compound cavitymeasurement of transmission and reflection of a tapered single-linephotonic-crystal waveguide, Appl. Phys. Lett., vol. 82, no. 16, pp.25772579, 2003.

[16] A. Chutinan and S. Noda, Waveguides and waveguide bends in two-

dimensional photonic crystal slabs, Phys. Rev. B, vol. 62, no. 7, pp.44884492, 2000.

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