Abstract

We present a filter design formalism for the synthesis of coupled-resonator optical waveguide (CROW) filters. This formalism leads to expressions and a methodology for deriving the coupling coefficients of CROWs for the desired filter responses and is based on coupled-mode theory as well as the recursive properties of the coupling matrix. The coupling coefficients are universal and can be applied to various types of resonators. We describe a method for the conversion of the coupling coefficients to the parameters based on ring resonators and grating defect resonators. The designs of Butterworth and Bessel CROW filters are demonstrated as examples.

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2010 (1)

2009 (1)

2008 (3)

A. Melloni, F. Morichetti, and M. Martinelli, “Four-wave mixing and wavelength conversion in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 25(12), C87–C97 (2008).
[CrossRef]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

A. M. Prabhu and V. Van, “Predistortion techniques for synthesizing coupled microring filters with loss,” Opt. Commun. 281(10), 2760–2767 (2008).
[CrossRef]

2007 (6)

2006 (3)

2005 (2)

2004 (2)

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
[CrossRef] [PubMed]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

2003 (1)

2002 (2)

1999 (1)

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

1995 (1)

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7(12), 1447–1449 (1995).
[CrossRef]

1992 (1)

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–213 (1992).
[CrossRef]

Absil, P. P.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Adibi, A.

Agarwal, A.

Asakawa, K.

Asghari, M.

Banwell, T.

Blasco, J.

Capmany, J.

Chak, P.

Chang-Hasnain, C. J.

Chu, S. T.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Cuesta-Soto, F.

DeRose, G. A.

Domenech, J. D.

Dong, P.

Eggleton, B. J.

Feng, D. Z.

Feng, N. N.

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

García, J.

Gill, D.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–213 (1992).
[CrossRef]

Hryniewicz, J. V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Huang, Y. Y.

Ikeda, N.

Ishikawa, H.

Johnson, F. G.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Khan, M. H.

Kim, S.

King, O.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Ku, P. C.

Kuramochi, E.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

Lai, Y.

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–213 (1992).
[CrossRef]

Laine, J. P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Lan, S.

Lee, D. C.

Lee, R. K.

Li, Q.

Liang, H.

Lim, H.

Little, B. E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Luff, B. J.

Martí, J.

Martinelli, M.

Martínez, A.

Melloni, A.

Menendez, R.

Mookherjea, S.

Morichetti, F.

Muñoz, P.

Muriel, M. A.

Nishikawa, S.

Notomi, M.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

Orta, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7(12), 1447–1449 (1995).
[CrossRef]

Paloczi, G. T.

Park, D.

Park, I.

Poon, J. K. S.

Prabhu, A. M.

A. M. Prabhu and V. Van, “Predistortion techniques for synthesizing coupled microring filters with loss,” Opt. Commun. 281(10), 2760–2767 (2008).
[CrossRef]

Qi, M. H.

Qian, W.

Rooks, M.

Sanchis, P.

Savi, P.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7(12), 1447–1449 (1995).
[CrossRef]

Scherer, A.

Scheuer, J.

Seiferth, E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Sekaric, L.

Shen, H.

Sipe, J. E.

Soltani, M.

Sugimoto, Y.

Sumetsky, M.

Tanabe, T.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

Tascone, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7(12), 1447–1449 (1995).
[CrossRef]

Toliver, P.

Trakalo, M.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Trinchero, D.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7(12), 1447–1449 (1995).
[CrossRef]

Tucker, R. S.

Van, V.

A. M. Prabhu and V. Van, “Predistortion techniques for synthesizing coupled microring filters with loss,” Opt. Commun. 281(10), 2760–2767 (2008).
[CrossRef]

V. Van, “Circuit-based method for synthesizing serially coupled microring filters,” J. Lightwave Technol. 24(7), 2912–2919 (2006).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Vlasov, Y.

Woodward, T. K.

Xia, F. N.

Xiao, S. J.

Xu, Y.

Yariv, A.

Yegnanarayanan, S.

Zhu, L.

IEEE J. Quantum Electron. (1)

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–213 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7(12), 1447–1449 (1995).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

J. Lightwave Technol. (5)

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

Nat. Photonics (2)

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[CrossRef]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

Opt. Commun. (1)

A. M. Prabhu and V. Van, “Predistortion techniques for synthesizing coupled microring filters with loss,” Opt. Commun. 281(10), 2760–2767 (2008).
[CrossRef]

Opt. Express (7)

M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11(4), 381–391 (2003).
[CrossRef] [PubMed]

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
[CrossRef] [PubMed]

F. N. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15(19), 11934–11941 (2007).
[CrossRef] [PubMed]

S. J. Xiao, M. H. Khan, H. Shen, and M. H. Qi, “A highly compact third-order silicon microring add-drop filter with a very large free spectral range, a flat passband and a low delay dispersion,” Opt. Express 15(22), 14765–14771 (2007).
[CrossRef] [PubMed]

Q. Li, M. Soltani, S. Yegnanarayanan, and A. Adibi, “Design and demonstration of compact, wide bandwidth coupled-resonator filters on a siliconon- insulator platform,” Opt. Express 17(4), 2247–2254 (2009).
[CrossRef] [PubMed]

P. Dong, N. N. Feng, D. Z. Feng, W. Qian, H. Liang, D. C. Lee, B. J. Luff, T. Banwell, A. Agarwal, P. Toliver, R. Menendez, T. K. Woodward, and M. Asghari, “GHz-bandwidth optical filters based on high-order silicon ring resonators,” Opt. Express 18(23), 23784–23789 (2010).
[CrossRef] [PubMed]

J. Capmany, P. Muñoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15(16), 10196–10206 (2007).
[CrossRef] [PubMed]

Opt. Lett. (5)

Other (3)

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University Press, 2007).

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis, A Signal Processing Approach (Wiley, 1999).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic drawing of an infinite-length CROW. (b) The dispersion curve of an infinite-length CROW. (c) Schematic drawing of a finite-length CROW. (d) Comparison of a finite-length and an infinite-length CROW at the boundary. (e) Transmission spectra of 10-resonator CROWs with 1 / τ e = κ and 1 / τ e = 0.1 κ respectively.

Fig. 2
Fig. 2

Schematic drawing of a CROW filter.

Fig. 3
Fig. 3

Spectra of transmission and group delay of (a) a tenth-order Butterworth filter and (b) a tenth-order Bessel filter.

Fig. 4
Fig. 4

(a) Transmission spectra of Butterworth CROWs with 6, 10, and 20 resonators. (b) Transmission spectra of N = 10 Butterworth CROWs under random change of coupling coefficients.

Fig. 5
Fig. 5

Transmission spectra of predistorted N = 10 Butterworth CROWs with and without loss/gain for (a) 1 / τ i = 0.05 B (loss) and (b) 1 / τ i = 0.05 B (gain).

Fig. 6
Fig. 6

Schematic drawings of (a) a microring CROW filter and (b) the coupling of two adjacent rings.

Fig. 7
Fig. 7

Schematic drawings of the coupled-resonator structures and the corresponding microring resonators for the derivation of (a,b) inter-resonator coupling and (c,d) waveguide-resonator coupling.

Fig. 8
Fig. 8

(a,b) Transmission spectra and their enlarged passband spectra of N = 6 Butterworth microring CROW filters with (a) B = ωFSR ·0.005 and (b) B = ωFSR ·0.05. (c) Transmission and group delay of an N = 6 Bessel microring CROW. (d) Transmission spectra of Butterworth microring CROWs with 6 and 20 resonators respectively.

Fig. 9
Fig. 9

Distribution of refractive index, coupling coefficient, and the envelope of intensity along z of (a) a defect resonator, (b) a grating CROW, (c) two coupled defect resonators, and (d) two coupled defect resonators with external coupling to the waveguide.

Fig. 10
Fig. 10

Transmission and group delay spectra of (a) an N = 6 Butterworth grating CROW filter and (b) an N = 6 Bessel grating CROW filter.

Fig. 11
Fig. 11

Choices of zeros for R(s). (a) Minimum phase. (b) 1st and 3rd quadrants. (c) Nearly uniform distribution.

Tables (5)

Tables Icon

Table 1 Extraction of Coupling Coefficients for N = 4 Butterworth Filter

Tables Icon

Table 2 Extracted Coupling Coefficients of Butterworth and Bessel CROW Filters

Tables Icon

Table 3 Coupling Coefficients of Microring CROW Filters

Tables Icon

Table 4 Lengths of Grating Sections for Grating CROW Filters

Tables Icon

Table 5 Coupling Coefficients and Frequency Detuning of N = 7 Bessel CROW Filters with Different Choices of Zeros

Equations (26)

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

d a k d t = i Δ ω a k i κ a k 1 i κ a k + 1 , k ,
γ = Δ ω 2 κ ± i 1 ( Δ ω 2 κ ) 2 = e i K Λ ,
d a N d t = i Δ ω a N i κ a N 1 1 τ e a N
d a N d t = i Δ ω a N i κ a N 1 i κ a N + 1 .
1 τ e = i κ γ .
d a 1 d t = ( i Δ ω 1 τ e 1 ) a 1 i κ 1 a 2 i μ 1 s i n , d a 2 d t = i Δ ω a 2 i κ 1 a 1 i κ 2 a 3 ,                               d a N 1 d t = i Δ ω a N 1 i κ N 2 a N 2 i κ N 1 a N , d a N d t = ( i Δ ω 1 τ e 2 ) a N i κ N 1 a N 1 .
A a [ s + 1 τ e 1 i κ 1 0 0 0 i κ 1 s i κ 2 0 0 0 i κ 2 s s i κ N 1 i κ N 1 s + 1 τ e 2 ] [ a 1 a 2 a N ] = [ i μ 1 s i n 0 0 ] .
s o u t = i μ 2 a N = μ 1 μ 2 [ A 1 ] N , 1 s i n
s r = s i n i μ 1 a 1 = ( 1 μ 1 2 [ A 1 ] 1 , 1 ) s i n ,
T ( s ) = ( i ) N 1 μ 1 μ 2 κ 1 κ 2 κ N 1 det ( A ) ,
T ( s ) = k s N + b N 1 s N 1 + + b 1 s + b 0 ,
p N = ( s + 1 τ e 1 ) p N 1 + κ 1 2 p N 2 , p N 1 = s p N 2 + κ 2 2 p N 3 ,                           p 3 = s p 2 + κ N 2 2 p 1 , p 2 = s p 1 + κ N 1 2 , p 1 = s + 1 τ e 2 ,
T ( s ) = k p N
R ( s ) = p N μ 1 2 p N 1 p N ,
d a k d t = ( i Δ ω 1 τ i ) a k i κ k 1 a k 1 i κ k a k + 1 .
[ c 1 c 2 ] = [ t i η i η t ] [ b 1 b 2 ] ,
[ b 1 b 2 ] = e i θ ( Δ ω ) [ c 1 c 2 ] ,
η = sin ( κ f F S R ) .
η i = 2 η 1 + η .
d a ( z ) d z = i δ a ( z ) + i κ g * ( z ) b ( z ) , d b ( z ) d z = i δ b ( z ) i κ g ( z ) a ( z ) ,
L = 1 κ g ln ( κ ω g ) ,
L i = 1 2 κ g ln ( 1 / τ e ω g ) .
T ( s ) = k ( s q 1 ) ( s q 2 ) ( s q N ) ,
| T ( i ω ) | 2 = k 2 ( ω 2 + q 1 2 ) ( ω 2 + q 2 2 ) ( ω 2 + q N 2 ) ,
| R ( i ω ) | 2 = 1 | T ( i ω ) | 2 = ( ω 2 + q 1 2 ) ( ω 2 + q 2 2 ) ( ω 2 + q N 2 ) k 2 ( ω 2 + q 1 2 ) ( ω 2 + q 2 2 ) ( ω 2 + q N 2 ) .
A a [ s i δ 1 + 1 τ e 1 i κ 1 0 0 0 i κ 1 s i δ 2 i κ 2 0 0 0 i κ 2 s i δ 3 s i δ N 1 i κ N 1 i κ N 1 s i δ N + 1 τ e 2 ] [ a 1 a 2 a N ] = [ i μ 1 s i n 0 0 ] ,

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