Abstract

We present the interface diffraction method (IDM), an efficient technique to determine the response of planar photonic crystal waveguides and couplers containing arbitrary defects. Field profiles in separate regions of a structure are represented through two contrasting approaches: the plane wave expansion method in the cladding and a scattering matrix method in the core. These results are combined through boundary conditions at the interface between regions to model fully a device. In the IDM, the relevant interface properties of individual device elements can be obtained from unit cell computations, stored, and later combined with other elements as needed, resulting in calculations that are over an order of magnitude faster than supercell simulation techniques. Dispersion relations for photonic crystal waveguides obtained through the IDM agree with the conventional plane wave expansion method to within 2.2% of the stopband width.

© 2005 Optical Society of America

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References

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ARPACK Users??? Guide

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users�?? Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia, 1998). <a href="http://www.caam.rice.edu/software/ARPACK/">http://www.caam.rice.edu/software/ARPACK/</a>.

Electron. Lett.

T. Baba, N. Fukaya, and J. Yonekura, �??Observation of light propagation in photonic crystal waveguides with bends,�?? Electron. Lett. 35, 654�??655 (1999).
[CrossRef]

A. Adibi, R. K. Lee, Y. Xu, A. Yariv, and A. Scherer, �??Design of photonic crystal optical waveguides with singlemode propagation in the photonic band gap,�?? Electron. Lett. 36, 1376�??1378 (2000).
[CrossRef]

IEEE J. Quantum Electron.

S. Boscolo, M. Midrio, and C. G. Someda, �??Coupling and decoupling of electromagnetic waves in parallel 2-D photonic crystal waveguides,�?? IEEE J. Quantum Electron. 38, 47�??53 (2002).
[CrossRef]

L. C. Andreani and M. Agio, �??Photonic bands and gap maps in a photonic crystal slab,�?? IEEE J. Quantum Electron. 38, 891�??898 (2002).
[CrossRef]

E. Istrate and E. H. Sargent, �??Photonic CrystalWaveguide Analysis Using Interface Boundary Conditions,�?? IEEE J. Quantum Electron. 41, 461�??467 (2005).
[CrossRef]

J. Poon, E. Istrate, M. Allard, and E. H. Sargent, �??Multiple-scales analysis of photonic crystal waveguides,�?? IEEE J. Quantum Electron. 39(6), 778�??786 (2003).
[CrossRef]

IEEE Trans. Antennas Propag.

K. S. Yee, �??Numerical solution of initial boundary value problems involving maxwells equations in isotropic media,�?? IEEE Trans. Antennas Propag. 14, 302�??307 (1966).
[CrossRef]

J. Mod. Opt.

J. B. Pendry, �??Photonic band structures,�?? J. Mod. Opt. 41, 209�??229 (1994).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys.: Condens. Matter

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, �??The Wannier function approach to photonic crystal circuits,�?? J. Phys.: Condens. Matter 15, R1233�??R1256 (2003).
[CrossRef]

LAPACK Users??? Guide, 3rd ed.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users�?? Guide, 3rd ed. (SIAM, Philadelphia, 1999). <a href="http://www.netlib.org/lapack/lug/index.html">http://www.netlib.org/lapack/lug/index.html</a>.

Nat. Mater.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, �??Ultra-high-Q photonic double-heterostructure nanocavity,�?? Nat. Mater. 4, 207�??210 (2005).
[CrossRef]

Nature (London)

S. Noda, A. Chutinan, and M. Imada, �??Trapping and emission of photons by a single defect in a photonic bandgap structure,�?? Nature (London) 407(6804), 608�??610 (2000).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, �??High-Q photonic nanocavity in a two-dimensional photonic crystal,�?? Nature (London) 425(6961), 944�??947 (2003).
[CrossRef]

Opt. Commun.

K. Yamada, H. Morita, A. Shinya, and M. Notomi, �??Improved line-defect structures for photonic-crystal waveguides with high group velocities,�?? Opt. Commun. 198, 395�??402 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A 38

C. M. de Sterke and J. E. Sipe, �??Envelope-function approach for the electrodynamics of nonlinear periodic structures,�?? Phys. Rev. A 38, 5149�??5165 (1988).
[CrossRef] [PubMed]

Phys. Rev. B

E. Istrate, M. Charbonneau-Lefort, and E. H. Sargent, �??Theory of photonic crystal heterostructures,�?? Phys. Rev. B 66(075121), 075,121 (2002).
[CrossRef]

J. P. Albert, D. Cassagne, and D. Bertho, �??Generalized Wannier function for photonic crystals,�?? Phys. Rev. B 61, 4381�??4384 (2000).
[CrossRef]

O. Painter, K. Srinivasan, and P. E. Barclay, �??Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals,�?? Phys. Rev. B 68(035214), 035,214 (2003).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 60, 5751�??5758 (1999).
[CrossRef]

G. Bastard, �??Superlattice band structure in the envelope-function approximation,�?? Phys. Rev. B 24, 5693�??5697 (1981).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, �??Behavior of light at photonic crystal interfaces,�?? Phys. Rev. B 71(195122), 195,122�??1�??7 (2005).
[CrossRef]

Z.-Y. Li and K.-M. Ho, �??Light propagation in semi-infinite photonic crystals and related waveguide structures,�?? Phys. Rev. B 68, 155,101�??1 (2003).
[CrossRef]

D. M. Whittaker and I. S. Culshaw, �??Scattering-matrix treatment of patterned multilayer photonic structures,�?? Phys. Rev. B 60, 2610�??2618 (1999).
[CrossRef]

A. Chutinan and S. Noda, �??Waveguides and waveguide bends in two-dimensional photonic crystal slabs,�?? Phys. Rev. B 62, 4488�??4492 (2000).
[CrossRef]

Phys. Rev. E

Z.-Y. Li and L.-L. Lin, �??Photonic band structures solved by a plane-wave-based transfer-matrix method,�?? Phys. Rev. E 67, 046,607 (2003).
[CrossRef]

K. Busch and S. John, �??Photonic band gap formation in certain self-organizing systems,�?? Phys. Rev. E 58, 3896�??3908 (1998).
[CrossRef]

N. A. R. Bhat and J. E. Sipe, �??Optical pulse propagation in nonlinear photonic crystals,�?? Phys. Rev. E 64(056604), 056,604�??1�??16 (2001).
[CrossRef]

Phys. Rev. Lett.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, �??Existence of a photonic gap in periodic dielectric structures,�?? Phys. Rev. Lett. 65, 3152�??3155 (1990).
[CrossRef] [PubMed]

Other

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2000).

K. Sakoda, Optical Properties of Photonic Crystals, chap. 4 (Springer, 2001).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic illustration of the IDM viewing a planar triangular photonic crystal device from above. The core and cladding areas are separated by thin interface layers. The dashed rectangles mark the simulation cells used for each device region.

Fig. 2.
Fig. 2.

(a) The slab waveguide with a row of defect air holes. The structure has index 3.4 and height 0.6a with 0.29a and 0.2a radius holes in the cladding and core, respectively. (b) Dispersion relation for the slab waveguide.

Fig. 3.
Fig. 3.

(a) Schematic of the waveguides simulated illustrating the shifting of core holes from their regular lattice sites. (b) Dispersion relations for the waveguides with core radius 0.2a as the core-cladding offset is varied.

Fig. 4.
Fig. 4.

(a) Schematic of the waveguides simulated illustrating the change in core hole radius. (b) Dispersion relations for the waveguides as the core hole radius is varied with no lattice offset.

Fig. 5.
Fig. 5.

(a) Frequency and (b) propagation constant of the zero group velocity point for odd waveguide modes as a function of the core hole radius and lattice offset.

Fig. 6.
Fig. 6.

The dispersion relation for the optimized structure with a single-moded frequency range over 88.7% of the stopband width. Inset: schematic of the optimized waveguide with a three row core of radii 0.4a, 0.425a, and 0.3a.

Fig. 7.
Fig. 7.

Dispersion relation for the directional coupler and the equivalent single waveguide. Inset: schematic of the coupler with a pair of identical waveguides formed by two rows of holes of radius 0.4a separated by a row of holes of radius 0.3a.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

E B ( r ) = j α ξ j { l , m , n E jlmn exp [ i ( K j + G lmn ) · r ] } ,
E h ( r ) = m , n [ E mn + exp ( i k mn , y y ) + E mn exp ( i k mn , y y ) ] exp [ i ( K 0 , + G mn , ) · r ] ,
j ( B jmn , x + B jmn , x B jmn , z + B jmn , z ) ( ξ j B + ξ j B ) = ( E mn , x + E mn , z + ) + ( E mn , x E mn , z ) ,
j ( C jmn , x + C jmn , x C jmn , z + C jmn , z ) ( ξ j B + ξ j B ) = 1 k mn , y ( k x k z ω 2 k x '2 k z 2 ω 2 k x k z ) [ ( E mn , x + E mn , z + ) ( E mn , x E mn , z ) ] ,
B jmn ± = l E jlmn exp ( i G l,⊥ y 0 ) ,
C jmn ± = l ( K j + G lmn ) × E jlmn exp ( i G l,⊥ y 0 ) ,
( E + E ) = T ( E B + E B ) = ( T 11 T 12 T 21 T 22 ) ( E B + E B ) .
( E R + E L ) = S ( E L + E R ) = ( S 11 S 12 S 21 S 22 ) ( E L + E R ) .
E L = [ S 12 + S 11 T 12 ( T 22 S 21 T 12 ) 1 S 22 ] T 12 T 22 1 E L .
E L = ± ( 1 S 12 T 12 T 22 1 ) 1 S 11 T 12 T 22 1 E L .
S = ( e i G · r 0 , 0 0 e i G · r 0 , ) S ( e + i G · r 0 , 0 0 e + i G · r 0 , ) ,
T = ( e + i G · r 0 , 0 0 e + i G · r 0 , ) T ,
L B = 2 π β odd β even = 9.56 a ,

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