Abstract

A rigorous semi-analytic approach to the modelling of coupling, guiding and propagation in complex microstructures embedded in two-dimensional photonic crystals is presented. The method, which is based on Bloch mode expansions and generalized Fresnel coefficients, is shown to be able to treat photonic crystal devices in ways which are analogous to those used in thin film optics with uniform media. Asymptotic methods are developed and exemplified through the study of a serpentine waveguide, a potential slow wave device.

© 2004 Optical Society of America

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References

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Appl. Phys. Lett. (2)

A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, �??Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,�?? Appl. Phys. Lett. 79, 2690-2692 (2001).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, �??Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,�?? Appl. Phys. Lett. 74, 1370-1372 (1999).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (2)

L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, �??Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part 2, Method,�?? J. Opt. Soc. Am. A. 17, 2177-2190 (2000).
[CrossRef]

L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, �??Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part 1, Method,�?? J. Opt. Soc. Am. A. 17, 2165-2176 (2000).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys.: Condens. Matter (1)

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, 1233�??1254 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. B (2)

T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M.Kamp, and A. Forchel, �??Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,�?? Phys. Rev. B 66, 041303(R) (2002).
[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 (4)

Z. Wang, S. Fan, �??Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,�?? Phys. Rev. E 68, 066616 (2003).
[CrossRef]

Z.-Y. Li, and K.-M. Ho, �??Light propagation in semi-infinite photonic crystals and related waveguide structures,�?? Phys. Rev. E 68, 155101 (2003).

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, �??Photonic band structure calculations using scattering matrices,�?? Phys. Rev. E 64, 046603 (2001).
[CrossRef]

G.H. Smith, L.C. Botten, R.C. McPhedran, and N.A. Nicorovici, �??Cylinder gratings in conical incidence with applications to woodpile structures,�?? Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, �??High transmission through sharp bends in photonic crystal waveguides,�?? Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

S.Fan, P.R. Villeneuve, and J.D. Joannopoulos, �??Channel Drop Tunnelling through Localized States,�?? Phys. Rev. Lett. 80, 960-963 (1998).
[CrossRef]

Other (8)

C. M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, (Kluwer, Dordrecht 2001).

S. Chen, R. C. McPhedran, C. M. De Sterke, and L. C. Botten, �??Optimal tapers in photonic crystal waveguides,�?? submitted to Appl. Phys. Lett.

W. H. Press, B. P. Flannery, S. A. Teulolsky and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).

M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, 1962).

L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan and T. N. Langtry, �??Bloch mode scattering matrix methods for modelling extended photonic crystal structures,�?? in preparation.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

P. Yeh, Optical waves in layered media, (Wiley, New York, 1988), Ch. 6.

S. Wolfram, The Mathematica Book, 3rd Ed., (Wolfram Media / Cambridge University Press, 1996).

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Figures (4)

Fig. 1.
Fig. 1.

A typical three segment photonic crystal device showing the component regions M1, M2 and M3, three lateral supercells of the model and a constituent grating bounded by dashed lines.

Fig. 2.
Fig. 2.

Two equivalent serpentine waveguide geometries (assuming no tunneling through the guide ends). Both are characterized by the period Dy and the double and single guide lengths, L 1 and L 2.

Fig. 3.
Fig. 3.

(a) Band diagram for a serpentine tine waveguide with L 1=L 2=5d, where q is the Bloch coefficient along the waveguide, and Dy =L 1+L 2. (b) Transmission through an FDC with same parameters (dashed), one period of the serpentine waveguide (dotted) and two periods (solid). Note that for frequencies below ωd/(2πc)=0.3064, the double guide cavity only supports a single, odd mode, and thus the analytic result of (19) does not apply.

Fig. 4.
Fig. 4.

(a) Band diagram for a serpentine waveguide with L 1=L 2=7d. The solid curve is calculated with the full numerical simulation while the dashed curve is calculated using the approximation (19) L 1=7.5d, L 2=6.7d (b) Transmission through a FDC with L=7d (dashed), 2 periods of the serpentine guide (dotted) and 3 periods of the serpentine guide.

Equations (22)

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𝓣 𝓯 = μ 𝓯 where 𝓣 = ( T R T 1 R R T 1 T 1 R T 1 ) , 𝓯 = ( f f + ) ,
𝓣 T 𝓠 𝓣 = 𝓠 , where 𝓠 = [ 0 Q Q 0 ] .
𝓣 = 𝓕 𝓕 1 with 𝓕 = [ F F + F + F ] , = [ Λ 0 0 Λ 1 ] , Λ = diag ( μ i ) .
𝓕 T 𝓠 𝓕 = where = ( 0 I I 0 )
𝓕 H 𝓣 p 𝓕 = 𝓣 m where 𝓣 p = ( I r i I e i I e I r ) and 𝓣 m = ( I m i I m ¯ i I m ¯ I m ) ,
R ij = ( F i ) 1 ( I R j R i ) 1 ( R j R i ) F i ,
T ij = ( F j ) 1 ( I R i R j ) 1 ( I R i 2 ) F i ,
r = R 12 δ + T 21 Λ L c + , c = T 12 δ + R 21 Λ L c + ,
c + = R 23 Λ L c , t = T 23 Λ L c ,
R = R 13 = R 12 + T 21 Λ L R 23 Λ L ( I R 21 Λ L R 23 Λ L ) 1 T 12 ,
T = T 13 = T 23 Λ L ( I R 21 Λ L R 23 Λ L ) 1 T 12 .
S 12 2 = I , where S 12 = ( R 12 T 21 T 12 R 21 )
R 13 = T 12 1 ( R 21 + Λ L R 23 Λ L ) ( I R 21 Λ L R 23 Λ L ) 1 T 12 ,
= T 21 1 ( I Λ L R 23 Λ L R 21 ) 1 ( R 21 + Λ L R 23 Λ L ) T 21 .
S 13 H S 13 = I 13 , where S 13 = ( R 13 T 31 T 13 R 31 ) , I 13 = ( I 1 0 0 I 3 ) ,
w 1 T = w 3 T = [ 1 0 0 0 ] , w 2 T = [ 1 0 0 0 0 1 0 0 ] .
T ˜ 13 = T ˜ 23 Λ ˜ L ( I R ˜ 21 Λ ˜ L R ˜ 23 Λ ˜ L ) 1 T ˜ 12 ,
R 13 = ρ f = cos 2 ( Δ β L ) exp ( 2 i β ¯ L ) 1 + sin 2 ( Δ β L ) exp ( 2 i β ¯ L ) , T 13 = τ f = i exp ( i β ¯ L ) sin ( Δ β L ) ( 1 + exp ( 2 i β ¯ L ) ) 1 + sin 2 ( Δ β L ) exp ( 2 i β ¯ L ) ,
𝓣 s = ( τ s ρ s 2 τ s ρ s τ s ρ s τ s 1 τ s ) ,
𝓣 s = 𝓣 f 2 , where 𝓣 f = ( τ f ρ f 2 τ f ρ f τ f ρ f τ f 1 τ f )
μ + 1 μ 1 2 = τ f 2 ρ f 2 + 1 τ f , i. e . cos ( q D y 2 ) = ( 1 τ f ) ,
cos ( q D 2 ) = sin ( Δ β L 1 ) sin ( β ¯ L 1 + β L 2 ) + cos 2 ( Δ β L 1 ) sin ( 2 β ¯ L 1 + β L 2 ) 2 sin ( Δ β L 1 ) cos ( β ¯ L 1 ) .

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