Abstract

Planar photonic crystal waveguide structures have been modelled using the finite-difference-time-domain method and perfectly matched layers have been employed as boundary conditions. Comprehensive numerical calculations have been performed and compared to experimentally obtained transmission spectra for various photonic crystal waveguides. It is found that within the experimental fabrication tolerances the calculations correctly predict the measured transmission levels and other major transmission features.

© 2004 Optical Society of America

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References

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Appl. Phys. Lett.

S. Fan, �??Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,�?? Appl. Phys. Lett. 80, 908-910 (2002).
[CrossRef]

H. Benisty et al., �??Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,�?? Appl. Phys. Lett. 76, 532-534 (2000).
[CrossRef]

M. Loncar et al., �??Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,�?? Appl. Phys. Lett. 80, 1689-1691 (2000).
[CrossRef]

Comput. Phys. Commun.

A. J. Ward and J. B. Pendry, �??A program for calculating photonic band structures, Green�??s functions and transmission/ reflection coefficients using a non-orthogonal FDTD method,�?? Comput. Phys. Commun. 128, 590-621 (2000).
[CrossRef]

IEEE J. Quantum Electron.

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, �??AlGaAs-based two-dimensional photonic crystal slab with defect waveguides for planar lightwave circuit applications,�?? IEEE J. Quantum Electron. 38, 760-769 (2002).
[CrossRef]

H. Benisty et al., �??Models and measurements for the transmission of submicron-width waveguide bends defined in two-dimensional photonic crystals,�?? IEEE J. Quantum Electron. 38, 770-785 (2002).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosburger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, �??Optical study of two-dimensional InP-based photonic crystals by internal light source technique,�?? IEEE J. Quantum Electron. 38, 786-799 (2002).
[CrossRef]

IEEE Trans. Ant. Prop.

S. D. Gedney, �??An anisotropic perfectly matched layer �?? absorbing medium for the truncation of FDTD lattices,�?? IEEE Trans. Ant. Prop. 44, 1630-1639 (1996).
[CrossRef]

IEEE Trans. Microw. Theory Technol.

F. L. Teixeira and W. C. Chew, �??On causality and dynamic stability of perfectly matched layers for FDTD simulations,�?? IEEE Trans. Microw. Theory Technol. 47, 775-785 (1999).
[CrossRef]

IEEE-LEOS Summer Topical Meetings 2003

S. J. McMab and Y. A. Vlasov, �??SOI 2D photonic crystals for microphotonic integrated circuits,�?? Holey fibers and photonic crystals, (Digest of IEEE-LEOS Summer Topical Meetings, Vancouver, 2003) pp. 79-80.

J. Appl. Phys.

Y. Sugimoto et al., �??Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,�?? J. Appl. Phys. 91, 922-929 (2002).
[CrossRef]

J. Comput. Phys.

P. G. Petropoulos, L. Zhao and A. C. Cangellaris, �??A reflectionless sponge layer absorbing boundary condition for the solution of Maxwell�??s equations with high-order staggered finite difference schemes,�?? J. Comput. Phys. 139, 184-208 (1998).
[CrossRef]

J. Lightwave Technol.

J. Mod. Opt.

A. J. Ward and J. B. Pendry, �??Refraction and geometry in Maxwell�??s equation,�?? J. Mod. Opt. 43, 773-793 (1996).
[CrossRef]

J. Opt. Soc. Am. B

Microw. Opt. Technol. Lett

W. C. Chew and W. H. Weedon, �??A 3-D perfectly matched medium from modified Maxwell�??s equations with stretched coordinates,�?? Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Opt. Commun.

K. Yamada et al., �??Improved line-defect structures for photonic-crystal waveguides with high group velocity,�?? Opt. Commun. 198, 395-402 (2001).
[CrossRef]

Opt. Express

Opt. Lett

M. Thorhauge, L. H. Frandsen and P. I. Borel, �??Efficient Photonic Crystal Directional Couplers,�?? Opt. Lett. 28, 1525-1527 (2003).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. B

T. Ochiai and K. Sakoda, �??Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,�?? Phys. Rev. B 63, 125107-125113 (2001).
[CrossRef]

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

M. Qiu and S. He, �??Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,�?? Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

Phys. Rev. E

M. Agio and C. M. Soukoulis, �??Ministop bands in single-defect photonic crystal waveguides,�?? Phys. Rev. E 64, 055603-055606 (2001).
[CrossRef]

Phys. Rev. Lett.

S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Other

V. P. Bykov, Radiation of atom in a resonant environment, (World Scientific, Singapore, 1993).

A. Taflove and S. C. Hagness, Computational electrodynamics: The finite-difference time-domain method, (Artech House, Boston, 2000).

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic crystals: Molding the flow of light, (Princeton University Press, 1995).

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Figures (6)

Fig. 1.
Fig. 1.

Transmission spectra for a PhCW calculated for (a) three different spatial resolutions in 2D (b) two different spatial resolutions in 3D.

Fig. 2.
Fig. 2.

2D Band diagram (left) shown for modes of different parities in a W1 PhCW. Even guided modes are shown in red, odd modes in blue, and slab modes in black. Transmission spectrum (right) shown for excitation with even modes.

Fig. 3.
Fig. 3.

Scanning electron micrographs of (a) a straight PhCW of length 10 µm, and (b) a PhCW containing two modified 60° bends (details shown in zoom), which are separated by a 20Λ long straight PhCW.

Fig. 4.
Fig. 4.

(a) 3D FDTD transmission spectra for different lengths of PhCW. (b) The measured (gray) and calculated (dashed black) transmission spectra for a 10 µm PhCW.

Fig. 5.
Fig. 5.

The measured (gray) and calculated (dashed black) propagation losses for the TE polarization.

Fig. 6.
Fig. 6.

Measured (gray) and calculated (dashed black) bend loss in two consecutive 60° bends for the TE polarization.

Equations (50)

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Δ t + E ( r , t ) = ε ( r ) 1 q × H ( r , t ) , Δ t H ( r , t ) = Q H μ ( r ) 1 q + × E ( r , t ) ,
ε ij = ε q x q y q z q i q j q 0 , μ ij = μ q x q y q z q i q j q 0 ,
ε ˜ = ε Λ ˜ , μ ˜ = μ Λ ˜ ,
Λ ˜ = ( s y s z s x 0 0 0 s x s z s y 0 0 0 s x s y s z ) ,
Λ ˜ ( r , ω ) = Λ ˜ x ( x , ω ) Λ ˜ y ( y , ω ) Λ ˜ z ( z , ω ) ,
Λ ˜ x = ( 1 s x 0 0 0 s x 0 0 0 s x )
E i ( t + t ) = 1 1 + σ j t { E i ( t ) + 1 ε ii [ q × H ( t ) ] i + σ i t ε ii n = 0 [ q × H ( t n t ) ] i }
E j ( t + t ) = 1 1 + σ i t { E j ( t ) + 1 ε jj [ q × H ( t ) ] j + σ j t ε jj n = 0 [ q × H ( t n t ) ] j }
E k ( t + t ) = E k ( t ) σ i σ j t 2 n = 0 E k ( t n t ) + ( × H ( t ) ) k ε kk 1 + ( σ i + σ j ) t + σ i σ j t 2
H i ( t ) = 1 1 + σ j t ×
{ H i ( t t ) Q H μ ii [ q × E ( t ) ] i σ i t μ ii Q H n = 0 [ q × E ( t n t ) ] i }
H j ( t ) = 1 1 + σ i t ×
{ H j ( t t ) Q H μ jj [ q × E ( t ) ] j σ j t μ jj Q H n = 0 [ q × E ( t n t ) ] j }
H k ( t ) = H k ( t t ) σ i σ j t 2 n = 0 H k ( t t n t ) Q H ( × E ( t ) ) k μ kk 1 + ( σ i + σ j ) t + σ i σ j t 2 .
ε ˜ xx = ε xx ( 1 + i σ z ω ) , ε ˜ yy = ε yy ( 1 + i σ z ω ) , ε ˜ zz = ε zz ( 1 + i σ z ω ) 1 ,
E x ( t + t ) E x ( t ) = ( ε ˜ 1 ) xx [ q × H ( t ) ] x = 1 ε xx ( 1 + i σ z ω ) 1 [ q × H ( t ) ] x .
[ q × H ] x = i t ε xx ω + ( 1 + i σ z ω ) E x
E x ( t + t ) = E x ( t ) + 1 ε xx ( 1 + i σ z ω ) 1 [ q × H ( t ) ] x
= 1 + σ z t 1 + σ z t E x ( t ) + 1 ε xx ( 1 + i σ z ω ) 1 [ q × H ( t ) ] x
= 1 1 + σ z t E x ( t ) + σ z t 1 + σ z t E x ( t ) + 1 ε xx ω ω + i σ z [ q × H ( t ) ] x
1 1 + σ z t E x ( t ) + [ q × H ( t ) ] x ε xx ( 1 + σ z t ) × [ i ω σ z ω + ( ω + i σ z ) + ω ( 1 + σ z t ) ω + i σ z ]
= 1 1 + σ z t ( E x ( t ) + [ q × H ( t ) ] x ε xx ) .
E z ( t + t ) = E z ( t ) + 1 ε zz ( 1 + i σ z ω ) [ q × H ( t ) ] z
= E z ( t ) + 1 ε zz [ q × H ( t ) ] z + i σ z ε zz · i t 1 e i ω t [ q × H ( t ) ] z
= E z ( t ) + 1 ε zz [ q × H ( t ) ] z + σ z t ε zz · n = 0 e in ω t [ q × H ( t ) ] z
= E z ( t ) + 1 ε zz [ q × H ( t ) ] z + σ z t ε zz · n = 0 [ q × H ( t n t ) ] z .
ε ˜ = ( ε 11 1 + i σ y ω 1 + i σ x ω 0 0 0 ε 22 1 + i σ x ω 1 + i σ y ω 0 0 0 ε 33 ( 1 + i σ x ω ) ( 1 + i σ y ω ) ) .
E x ( t + t ) E x ( t ) = ( ε ˜ 1 ) xx [ q × H ( t ) ] x
= 1 ε xx ( 1 + i σ x ω ) ( 1 + i σ y ω ) 1 [ q × H ( t ) ] x .
E x ( t + t ) = 1 1 + σ y t ×
{ E x ( t ) + 1 ε xx [ q × H ( t ) ] x + σ x t ε xx · n = 0 [ q × H ( t n t ) ] x }
E z ( t + t ) E z ( t ) = ( ε ˜ 1 ) zz [ q × H ( t ) ] z
= 1 ε zz ( 1 + i σ x ω ) 1 ( 1 + i σ y ω ) 1 [ q × H ( t ) ] z .
( 1 + i σ x ω ) ( 1 + i σ y ω ) ( E z ( t + t ) E 2 ( t ) ) =
( 1 + i ( σ x + σ y ) ω σ x σ y ( ω ) 2 ) ( E z ( t + t ) E z ( t ) ) .
i ( σ x + σ y ) ω ( E z ( t + t ) E z ( t ) ) = i ( σ x + σ y ) ( i t ) 1 e i ω σ t ( E z ( t + t ) E z ( t ) )
= ( σ x + σ y ) t k = 0 e ki ω t ( E z ( t + t ) E z ( t ) )
= ( σ x + σ y ) t ×
( k = 0 E z ( t + t k t ) k = 0 E z ( t k t ) )
= ( σ x + σ y ) t E z ( t + t ) ;
σ x σ y ( ω ) 2 ( E z ( t + t ) E z ( t ) ) = σ x σ y ω i t 1 e i ω t ( E z ( t + t ) E z ( t ) )
= i t σ x σ y ω E z ( t + t )
= ( i t ) 2 σ x σ y 1 e i ω t E z ( t + t )
= t 2 σ x σ y k = 0 E z ( t + t k t )
= t 2 σ x σ y ( E z ( t + t ) + k = 0 E z ( t k t ) ) .
1 ε zz [ q × H ( t ) ] z = E z ( t + t ) E z ( t ) + t ( σ x + σ y ) E z ( t + t ) +
t 2 σ x σ y ( E z ( t + t ) + k = 0 E z ( t k t ) ) .
E z ( t + t ) = E z ( t ) σ x σ y t 2 k = 0 E z ( t k t ) + ( × H ( t ) ) z ε zz 1 + ( σ x + σ y ) t + σ x σ y t 2 .
ε ˜ = ( ε xx ( 1 + i σ y ω ) ( 1 + i σ z ω ) 1 + i σ x ω 0 0 0 ε yy ( 1 + i σ x ω ) ( 1 + i σ z ω ) 1 + i σ y ω 0 0 0 ε zz ( 1 + i σ x ω ) ( 1 + i σ y ω ) 1 + i σ z ω ) .
ε zz E z t + ε zz ( σ x + σ y ) E z + ε zz σ x σ y 0 t E ˜ z ( t ) dt = ( × H ) z .

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