Abstract

We consider finite-size effects in coupled cavity structures. Starting with microring resonator structures well described by transfer matrices, we obtain conditions that lead to the minimization of finite-size effects. Our approach does not require numerical optimization and requires only slight modification of design parameters guided by closed-form analytical expressions. Using a Breit–Wigner scattering formalism, we demonstrate that the scheme can be used to minimize finite-size effects in a general class of coupled cavity structures. The strength of the present technique lies in its simplicity and its applicability to a wide variety of structures described by tight-binding formalisms.

© 2006 Optical Society of America

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References

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  1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
    [CrossRef]
  2. M. Sumetsky and B. J. Eggleton, Opt. Express 11, 381 (2003).
    [CrossRef] [PubMed]
  3. P. Sanchis, J. García, A. Martínez, and J. Martí, J. Appl. Phys. 97, 013101 (2005).
    [CrossRef]
  4. M. Sumetskii, J. Phys. Condens. Matter 3, 2651 (1991).
    [CrossRef]
  5. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1983).
  6. P. Chak and J. E. Sipe, 'Time reversal of light propagation in periodic quasi-1D structures by symmetry transformation,' manuscript in preparation.
  7. G. Boedecker and C. Henkel, Opt. Express 11, 1590 (2003).
    [CrossRef] [PubMed]
  8. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, J. Opt. Soc. Am. B 21, 1818 (2004).
    [CrossRef]
  9. J. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, Opt. Express 12, 90 (2004).
    [CrossRef] [PubMed]

2005

P. Sanchis, J. García, A. Martínez, and J. Martí, J. Appl. Phys. 97, 013101 (2005).
[CrossRef]

2004

2003

1999

1991

M. Sumetskii, J. Phys. Condens. Matter 3, 2651 (1991).
[CrossRef]

Boedecker, G.

Boyd, R. W.

Chak, P.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, J. Opt. Soc. Am. B 21, 1818 (2004).
[CrossRef]

P. Chak and J. E. Sipe, 'Time reversal of light propagation in periodic quasi-1D structures by symmetry transformation,' manuscript in preparation.

Eggleton, B. J.

García, J.

P. Sanchis, J. García, A. Martínez, and J. Martí, J. Appl. Phys. 97, 013101 (2005).
[CrossRef]

Heebner, J. E.

Henkel, C.

Huang, Y.

Lee, R. K.

Martí, J.

P. Sanchis, J. García, A. Martínez, and J. Martí, J. Appl. Phys. 97, 013101 (2005).
[CrossRef]

Martínez, A.

P. Sanchis, J. García, A. Martínez, and J. Martí, J. Appl. Phys. 97, 013101 (2005).
[CrossRef]

Mookherjea, S.

Paloczi, G. T.

Pereira, S.

Poon, J.

Sanchis, P.

P. Sanchis, J. García, A. Martínez, and J. Martí, J. Appl. Phys. 97, 013101 (2005).
[CrossRef]

Scherer, A.

Scheuer, J.

Sipe, J. E.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, J. Opt. Soc. Am. B 21, 1818 (2004).
[CrossRef]

P. Chak and J. E. Sipe, 'Time reversal of light propagation in periodic quasi-1D structures by symmetry transformation,' manuscript in preparation.

Sumetskii, M.

M. Sumetskii, J. Phys. Condens. Matter 3, 2651 (1991).
[CrossRef]

Sumetsky, M.

Xu, Y.

Yariv, A.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1983).

J. Appl. Phys.

P. Sanchis, J. García, A. Martínez, and J. Martí, J. Appl. Phys. 97, 013101 (2005).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Condens. Matter

M. Sumetskii, J. Phys. Condens. Matter 3, 2651 (1991).
[CrossRef]

Opt. Express

Opt. Lett.

Other

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1983).

P. Chak and J. E. Sipe, 'Time reversal of light propagation in periodic quasi-1D structures by symmetry transformation,' manuscript in preparation.

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Figures (3)

Fig. 1
Fig. 1

Top, schematic of a coupled micro - ring resonator structure. The unit cell of the structure is enclosed within the dashed rectangle. Bottom, plot of t N ( ω ) 2 for N = 10 (solid curve), R ( ω ) (dotted curve). All rings have effective index n = 3.0 , effective circumference L = 52.0 μ m , and cross-coupling coefficient σ = 0.95 .

Fig. 2
Fig. 2

Transmission and group delay for finite structures with (a), (b) 5 and (c), (d) 25 cavities, using parameters listed in Fig. 1. (a), (c) Comparison of transmission for structures with (dashed) or without (solid) AR modification. (b), (d) Comparison of delay for a structure with AR (dashed) and corresponding delay within an infinite structure (solid).

Fig. 3
Fig. 3

(top) Schematic of the finite coupled cavity structure discussed in the text. We used δ 0 = 4.2 × 10 3 , with AR elements constructed by using Eqs. (11). (a), (c) Comparison of transmission for structures with (dotted) or without (solid) AR modification. (b), (d) Comparison of delay for a structure with AR (dashed) and corresponding delay within an infinite structure (solid). In (a) and (b) ψ = π 2 such that ω ref = ω 0 ; in (c) and (d) ψ = π 4 such that ω ref = ω 0 ± 2 Γ .

Equations (14)

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M ( ω ) = 1 τ ( ω ) [ e i ϕ ( ω ) 1 τ 2 ( ω ) e i μ 1 τ 2 ( ω ) e i μ e i ϕ ( ω ) ] ,
cos ζ ¯ ( ω ) = cos ϕ ¯ ( ω ) τ ( ω ) .
R ( ω ) = i 1 τ 2 e i μ ¯ τ sin ζ ¯ + 1 τ 2 cos 2 ζ ¯ .
P ( ω ) = [ e i ω n L 4 c 0 0 e i ω n L 4 c ] , Σ = ( i κ ) 1 [ 1 σ σ 1 ] .
R ( ω ) = ( 1 ) l σ ( κ sin ψ + 1 κ 2 cos 2 ψ ) 1 ,
ψ ( ω ) = arccos [ ( ω ω 0 ) Γ ] , Γ 2 κ c n L .
P ( ω ) = [ e i θ ( ω ) 0 0 e i θ ( ω ) ] , Σ = ( i κ ) 1 [ 1 σ σ 1 ] ,
2 θ ( ω ref ) = { π odd l 0 even l } ,
σ = R = η 2 + 1 η ,
η ( ω ref ) = σ 1 1 σ 2 sin ψ ( ω ref ) .
Σ = ( i κ ) 1 [ 1 σ σ 1 ] .
σ = σ ( 1 + κ 2 ) 1 2 , σ = 4 η 2 + 1 2 η ,
Λ ( ω ) = [ ω ω 0 + i γ 2 δ 12 0 0 δ 21 ω ω 0 δ 23 0 0 0 0 ω ω 0 + i γ 2 ] ,
γ 2 = 2 δ 0 sin ψ ( ω ref ) , δ 1 , 2 = δ N , N 1 = 2 δ 0 ,

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