Abstract

We identify and analyze a class of leaky modes in two coupled identical photonic bandgap (PBG) slab waveguides. These leaky modes originate from the modes of the array structure that separates the two waveguides and may introduce additional losses to the coupled PBG waveguides when operating as a directional coupler. We derive an approximate formula to show explicitly how the leakage losses of such modes decrease exponentially with an increase in the thickness of the guiding layers. With numerical examples, we show that the leakage losses vary over a wide range, depending on the waveguide parameters. These leaky modes precede the guided modes according to the ranking of the mode indices and thus affect the orders of the guided modes. Their presence provides a simple physical explanation of the unusual phenomenon that the orders of the symmetric and antisymmetric guided modes of two coupled PBG waveguides depend on the number of the high-index elements in the array structure between the waveguides.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2007

2006

2004

2003

A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional couplers,” IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

T. P. White, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. 28, 2452-2454 (2003).
[CrossRef] [PubMed]

2002

A. Sharkawy, S. Shi, D. W. Prather, and R. Soref, “Electro-optical switching using coupled photonic crystal waveguides,” Opt. Express 10, 1048-1059 (2002).
[PubMed]

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]

2001

1992

K. S. Chiang, “Coupled-zigzag-wave theory for guided waves in slab waveguide arrays,” J. Lightwave Technol. 10, 1380-1387 (1992).
[CrossRef]

1990

G. Lenz and J. Salzman, “Bragg reflection waveguide composite structures,” IEEE J. Quantum Electron. 26, 519-531 (1990).
[CrossRef]

1977

Asatryan, A. A.

Boscolo, S.

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]

Botten, L. C.

Cheng, S.

Chiang, K. S.

J. Li and K. S. Chiang, “Guided modes of one-dimensional photonic bandgap waveguides,” J. Opt. Soc. Am. B 24, 1942-1950 (2007).
[CrossRef]

K. S. Chiang, “Coupled-zigzag-wave theory for guided waves in slab waveguide arrays,” J. Lightwave Technol. 10, 1380-1387 (1992).
[CrossRef]

Chien, F. S.

Cuesta, F.

A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional couplers,” IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

de Sterke, C. M.

Hansen, R. A.

Hong, C. S.

Hsieh, W.

Hsu, Y.

Huang, M.

Huttunen, A.

T. Koponen, A. Huttunen, and P. Torma, “Conditions for waveguide decoupling in square-lattice photonic crystals,” J. Appl. Phys. 96, 4039-4041 (2004).
[CrossRef]

Koponen, T.

T. Koponen, A. Huttunen, and P. Torma, “Conditions for waveguide decoupling in square-lattice photonic crystals,” J. Appl. Phys. 96, 4039-4041 (2004).
[CrossRef]

Koshiba, M.

Lenz, G.

G. Lenz and J. Salzman, “Bragg reflection waveguide composite structures,” IEEE J. Quantum Electron. 26, 519-531 (1990).
[CrossRef]

Li, J.

Marti, J.

A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional couplers,” IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

Martinez, A.

A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional couplers,” IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

McPhedran, R. C.

Midrio, M.

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]

Prather, D. W.

Salzman, J.

G. Lenz and J. Salzman, “Bragg reflection waveguide composite structures,” IEEE J. Quantum Electron. 26, 519-531 (1990).
[CrossRef]

Sharkawy, A.

Shi, S.

Shih, T.

Someda, C. G.

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]

Soref, R.

Torma, P.

T. Koponen, A. Huttunen, and P. Torma, “Conditions for waveguide decoupling in square-lattice photonic crystals,” J. Appl. Phys. 96, 4039-4041 (2004).
[CrossRef]

White, T. P.

Wu, Y.

Yariv, A.

Yeh, P.

Appl. Opt.

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]

G. Lenz and J. Salzman, “Bragg reflection waveguide composite structures,” IEEE J. Quantum Electron. 26, 519-531 (1990).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional couplers,” IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

J. Appl. Phys.

T. Koponen, A. Huttunen, and P. Torma, “Conditions for waveguide decoupling in square-lattice photonic crystals,” J. Appl. Phys. 96, 4039-4041 (2004).
[CrossRef]

J. Lightwave Technol.

K. S. Chiang, “Coupled-zigzag-wave theory for guided waves in slab waveguide arrays,” J. Lightwave Technol. 10, 1380-1387 (1992).
[CrossRef]

M. Koshiba, “Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. 19, 1970-1975 (2001).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

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Figures (6)

Fig. 1
Fig. 1

(a) Refractive index profile of two coupled PBG slab waveguides. (b) Refractive index profile of the coupler with the PBG structures removed.

Fig. 2
Fig. 2

Normalized dispersion curves of the leaky TE and TM modes of two coupled PBG slab waveguides with d 1 Λ = 0.5 , d g Λ = 1.45 , and n 1 2 n 2 2 = 2.25 for the cases (a) N = 2 and (b) N = 3 .

Fig. 3
Fig. 3

Variations of the attenuation coefficient α Λ with (a) the normalized frequency V at d g Λ = 1.45 and (b) the relative thickness d g Λ at V = 4.0 for the first N leaky modes of two coupled PBG slab waveguides for the cases N = 2 and N = 3 .

Fig. 4
Fig. 4

Field distributions of the first N leaky modes of two coupled PBG slab waveguides with d 1 Λ = 0.5 , d g Λ = 1.45 , and n 1 2 n 2 2 = 2.25 , calculated at V = 4.0 , for the cases (a) N = 2 and (b) N = 3 .

Fig. 5
Fig. 5

Normalized dispersion curves of the guided TE and TM modes of two coupled PBG slab waveguides with d 1 Λ = 0.5 , d g Λ = 1.45 , and n 1 2 n 2 2 = 2.25 for the cases (a) N = 2 and (b) N = 3 , where the B line marks the Brewster-incidence condition for the TM polarization.

Fig. 6
Fig. 6

Field distributions of the guided modes of two coupled PBG slab waveguides with d 1 Λ = 0.5 , d g Λ = 1.45 , and n 1 2 n 2 2 = 2.25 , calculated at V = 4.5 , for the cases (a) N = 2 and (b) N = 3 .

Equations (20)

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φ ( x ) = { a 1 , j exp [ γ 2 ( x j Λ + Λ ) ] + b 1 , j exp [ γ 2 ( x j Λ + Λ ) ] for ( j 1 ) Λ < x ( j 1 ) Λ + d 2 2 a 2 , j exp [ i k 1 ( x j Λ + Λ 2 ) ] + b 2 , j exp [ i k 1 ( x j Λ + Λ 2 ) ] for ( j 1 ) Λ + d 2 2 < x j Λ d 2 2 a 3 , j exp [ γ 2 ( x j Λ ) ] + b 3 , j exp [ γ 2 ( x j Λ ) ] for j Λ d 2 2 < x j Λ } ,
( a 1 , j b 1 , j ) = 1 2 i ( 1 + k 1 2 r 12 2 γ 2 2 ) 1 2 ( e γ 2 d 2 2 exp [ i ( ϕ k 1 d 1 2 ) ] e γ 2 d 2 2 exp [ i ( ϕ k 1 d 1 2 ) ] e γ 2 d 2 2 exp [ i ( ϕ + k 1 d 1 2 ) ] e γ 2 d 2 2 exp [ i ( ϕ + k 1 d 1 2 ) ] ) ( a 2 , j b 2 , j ) ,
( a 2 , j b 2 , j ) = 1 2 ( 1 + r 12 2 γ 2 2 k 1 2 ) 1 2 ( e γ 2 d 2 2 exp [ i ( ϕ k 1 d 1 2 ) ] e γ 2 d 2 2 exp [ i ( ϕ + k 1 d 1 2 ) ] e γ 2 d 2 2 exp [ i ( ϕ k 1 d 1 2 ) ] e γ 2 d 2 2 exp [ i ( ϕ + k 1 d 1 2 ) ] ) ( a 3 , j b 3 , j ) ,
( a 1 , j b 1 , j ) = ( T 11 T 12 T 21 T 22 ) ( a 1 , j + 1 b 1 , j + 1 ) , j = 1 , 2 , 3 , , N 1 ,
T 11 = e γ 2 d 2 [ cos k 1 d 1 + 1 2 ( r 12 γ 2 k 1 k 1 r 12 γ 2 ) sin k 1 d 1 ] ,
T 12 = 1 2 ( r 12 γ 2 k 1 + k 1 r 12 γ 2 ) sin k 1 d 1 ,
T 21 = 1 2 ( r 12 γ 2 k 1 + k 1 r 12 γ 2 ) sin k 1 d 1 ,
T 22 = e γ 2 d 2 [ cos k 1 d 1 1 2 ( r 12 γ 2 k 1 k 1 r 12 γ 2 ) sin k 1 d 1 ] .
( A 1 B 1 ) = ( T 11 T 12 T 21 T 22 ) N ( A 2 B 2 ) .
χ = cos k 1 d 1 cosh γ 2 d 2 + 1 2 ( r 12 γ 2 k 1 k 1 r 12 γ 2 ) sin k 1 d 1 sinh γ 2 d 2 .
( T 11 T 12 T 21 T 22 ) N = 1 sin K Λ ( T 11 sin N K Λ sin ( N 1 ) K Λ T 12 sin N K Λ T 21 sin N K Λ T 22 sin N K Λ sin ( N 1 ) K Λ ) .
( A 1 B 1 ) = C 1 ( exp ( i K Λ ) T 11 T 12 ) ,
( A 2 B 2 ) = C 2 ( T 12 exp ( i K Λ ) T 11 ) ,
A 1 = A 1 exp [ γ 2 ( d g d 2 ) ] , A 2 = A 2 exp [ γ 2 ( d g d 2 ) ]
B 1 = B 1 exp [ γ 2 ( d g d 2 ) ] , B 2 = B 2 exp [ γ 2 ( d g d 2 ) ] .
T 11 sin N K Λ sin ( N 1 ) K Λ = e 2 γ 2 ( d g d 2 ) { [ exp ( i K Λ ) T 11 ] ( ± sin K Λ T 12 sin N K Λ ) } .
n effi G exp ( 2 γ 2 d g )
G = sin K Λ ( ± sin K Λ T 12 sin N K Λ ) exp ( 2 γ 2 d 2 ) K Λ [ ( N 1 ) cos ( N 1 ) K Λ N T 11 cos N K Λ ] T 11 sin N K Λ .
α Λ = 4.34 V ζ [ b + 1 ( n 1 2 n 2 2 1 ) ] 1 2
ζ [ 2 n effr G ( n 1 2 n 2 2 ) ] exp [ 2 V b 1 2 ( d g Λ ) ] .

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