Abstract

We generalize an earlier published analysis of critical coupling in a single-ring resonator waveguide system [Electron. Lett. 36, 321 (2000) ] to multiple-ring resonator all-pass and channel-dropping filter structures. The fundamental analytical relationships among coupling coefficient, internal loss, and oscillation frequency for critical coupling and oscillation conditions are derived for two and three rings coupled to single- and double-bus waveguides using the transfer-matrix method and Sylvester’s theorem. Furthermore, we derive the photonic band structure of an infinite periodic ring structure and show that, as the number of rings increases, the reflection band of the double-bus structure approaches the photonic bandgap.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, "Distributed and localized feedback in microresonator sequences for linear and nonlinear optics," J. Opt. Soc. Am. B 21, 1818-1831 (2004).
    [CrossRef]
  16. Y. M. Landobasa, S. Darmawan, and M. K. Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
    [CrossRef]
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2005

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on silicon chip," Nature 43, 1081-1084 (2005).

Y. M. Landobasa, S. Darmawan, and M. K. Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

S. Darmawan, Y. M. Landobasa, and M. K. Chin, "Phase engineering for ring enhanced Mach-Zehnder interferometers," Opt. Express 13, 4580-4588 (2005).
[CrossRef] [PubMed]

2004

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

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, "Distributed and localized feedback in microresonator sequences for linear and nonlinear optics," J. Opt. Soc. Am. B 21, 1818-1831 (2004).
[CrossRef]

L. Chun-Fei and B. Alireza, "Finesse-enhanced ring resonator coupled Mach-Zehnder interferometer all-optical switches," Chin. Phys. Lett. 21, 90-93 (2004).

A. Yariv, "Critical coupling and its control in optical waveguide-ring resonator systems," IEEE Photon. Technol. Lett. 14, 483-485 (2004).
[CrossRef]

V. M. Menon, W. Tong, and S. R. Forrest, "Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier," IEEE Photon. Technol. Lett. 16, 1343-1345 (2004).
[CrossRef]

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

2003

2001

2000

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, "Microring resonator arrays for VLSI photonics," IEEE Photon. Technol. Lett. 12, 323-325 (2000).
[CrossRef]

1999

1995

1993

Absil, P. P.

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

Alireza, B.

L. Chun-Fei and B. Alireza, "Finesse-enhanced ring resonator coupled Mach-Zehnder interferometer all-optical switches," Chin. Phys. Lett. 21, 90-93 (2004).

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on silicon chip," Nature 43, 1081-1084 (2005).

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on silicon chip," Nature 43, 1081-1084 (2005).

Boyd, R. W.

Casperson, L. W.

Chak, P.

Chin, M. K.

S. Darmawan, Y. M. Landobasa, and M. K. Chin, "Phase engineering for ring enhanced Mach-Zehnder interferometers," Opt. Express 13, 4580-4588 (2005).
[CrossRef] [PubMed]

Y. M. Landobasa, S. Darmawan, and M. K. Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

Choi, J. M.

Chu, S. T.

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

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, "Microring resonator arrays for VLSI photonics," IEEE Photon. Technol. Lett. 12, 323-325 (2000).
[CrossRef]

Chun-Fei, L.

L. Chun-Fei and B. Alireza, "Finesse-enhanced ring resonator coupled Mach-Zehnder interferometer all-optical switches," Chin. Phys. Lett. 21, 90-93 (2004).

Darmawan, S.

Y. M. Landobasa, S. Darmawan, and M. K. Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

S. Darmawan, Y. M. Landobasa, and M. K. Chin, "Phase engineering for ring enhanced Mach-Zehnder interferometers," Opt. Express 13, 4580-4588 (2005).
[CrossRef] [PubMed]

Forrest, S. R.

V. M. Menon, W. Tong, and S. R. Forrest, "Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier," IEEE Photon. Technol. Lett. 16, 1343-1345 (2004).
[CrossRef]

Gill, D.

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

Heebner, J. E.

Hryniewicz, J. V.

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

Huang, Y.

Johnson, F. G.

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

King, O.

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

Kokubun, Y.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, "Microring resonator arrays for VLSI photonics," IEEE Photon. Technol. Lett. 12, 323-325 (2000).
[CrossRef]

Landobasa, Y. M.

S. Darmawan, Y. M. Landobasa, and M. K. Chin, "Phase engineering for ring enhanced Mach-Zehnder interferometers," Opt. Express 13, 4580-4588 (2005).
[CrossRef] [PubMed]

Y. M. Landobasa, S. Darmawan, and M. K. Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

Lee, R. K.

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on silicon chip," Nature 43, 1081-1084 (2005).

Little, B. E.

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

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, "Microring resonator arrays for VLSI photonics," IEEE Photon. Technol. Lett. 12, 323-325 (2000).
[CrossRef]

Madsen, C. K.

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

Menon, V. M.

V. M. Menon, W. Tong, and S. R. Forrest, "Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier," IEEE Photon. Technol. Lett. 16, 1343-1345 (2004).
[CrossRef]

Mookherjea, S.

Paloczi, G. T.

Pan, W.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, "Microring resonator arrays for VLSI photonics," IEEE Photon. Technol. Lett. 12, 323-325 (2000).
[CrossRef]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on silicon chip," Nature 43, 1081-1084 (2005).

Pereira, S.

Poon, J. K. S.

Scherer, A.

Scheuer, J.

Seiferth, F.

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

Sipe, J. E.

Tong, W.

V. M. Menon, W. Tong, and S. R. Forrest, "Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier," IEEE Photon. Technol. Lett. 16, 1343-1345 (2004).
[CrossRef]

Tovar, A. A.

Trakalo, M.

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

Van, V.

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

Xu, Y.

Yablonovitch, E.

Yariv, A.

Zhao, J. H.

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

Appl. Opt.

Chin. Phys. Lett.

L. Chun-Fei and B. Alireza, "Finesse-enhanced ring resonator coupled Mach-Zehnder interferometer all-optical switches," Chin. Phys. Lett. 21, 90-93 (2004).

Electron. Lett.

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

IEEE J. Quantum Electron.

Y. M. Landobasa, S. Darmawan, and M. K. Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, "Microring resonator arrays for VLSI photonics," IEEE Photon. Technol. Lett. 12, 323-325 (2000).
[CrossRef]

A. Yariv, "Critical coupling and its control in optical waveguide-ring resonator systems," IEEE Photon. Technol. Lett. 14, 483-485 (2004).
[CrossRef]

V. M. Menon, W. Tong, and S. R. Forrest, "Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier," IEEE Photon. Technol. Lett. 16, 1343-1345 (2004).
[CrossRef]

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nature

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on silicon chip," Nature 43, 1081-1084 (2005).

Opt. Express

Opt. Lett.

Other

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

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Figures (10)

Fig. 1
Fig. 1

(a) All-pass filter with a single ring, (b) Reflection R of the all-pass filter as a function of the round-trip amplitude transmission factor a, showing critical coupling ( R = 0 ) when a = r and δ = 0 .[6]

Fig. 2
Fig. 2

Coupled RRs with one bus waveguide. Note that n is the number of RRs as well as the number of couplers.

Fig. 3
Fig. 3

Relationships between a and r for critical coupling and oscillation conditions for a single-bus CROW with one, two, or three rings.

Fig. 4
Fig. 4

Simulated output R (in contour plot) for multiple rings as a function of ring reflectivity r and round-trip phased δ for a given a. Left column: a = 0.8 . Crosses in the upper figures are the critical coupling points; lower curves are the output spectra under the critical coupling conditions. Right column: a = 1.5 . Crosses in the upper contour plots are the oscillation points, and the lower curves are the output spectra under the oscillation conditions. Note that all the curves are symmetrical about δ = 0 .

Fig. 5
Fig. 5

(a) Reflection at the resonance frequency ( δ = 2 π m ) as a function of a for one- and three-ring single-bus CROWs, showing the turning points at critical coupling and oscillation conditions. (b) Reflection spectra for the three-ring CROW for various values of a.

Fig. 6
Fig. 6

Coupled RRs with a two-bus waveguide; n is the number of RRs and Δ is the spatial period.

Fig. 7
Fig. 7

Transmission and reflection spectra of a double-bus CROW with a various number of rings ( n = 1 , 2 , 3 ) . Left: a = 1 (lossless case) and r = 0.8 , showing the oscillation in reflection and transmission. Right: The R and T spectra for a = 1.5 , showing the oscillation frequencies and their corresponding coupling conditions ( a , r ) .

Fig. 8
Fig. 8

Double-bus case: reflection and transmission at the resonance frequency as a function of a for n = 1 and 3. The point where R = 0 always occurs at a = 1 regardless of the value of r.

Fig. 9
Fig. 9

(a) Reflection and (b) transmission spectra for a three-ring double-bus CROW with r = 0.8 .

Fig. 10
Fig. 10

Left, dispersion diagram showing the real (red) and imaginary (green) part of the Bloch wave propagation constant β as a function of frequency δ. Right, the transmission ( T ) of the double-bus CROW structure with one, three, and five rings, assuming that r = 0.7 and a = 1 (lossless). Ripples occur in the transmission bands.

Equations (27)

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

ρ = b 1 a 1 = r a exp ( i δ ) 1 r a exp ( i δ ) , R = ρ 2 = r 2 2 a r cos δ + a 2 1 2 a r cos δ + a 2 r 2 ,
[ b n c n ] = [ r i κ i κ r ] [ a n d n ] ,
[ a n + 1 b n + 1 ] = [ A B C D ] [ a n b n ] = [ A B C D ] n [ a 1 b 1 ] M [ a 1 b 1 ] ,
A = a i κ exp ( i δ 2 ) , B = r a i κ exp ( i δ 2 ) ,
C = r i κ a exp ( i δ 2 ) , D = 1 i κ a exp ( i δ 2 ) .
M = [ m 11 m 12 m 21 m 22 ] = [ A U n 1 ( x ) U n 2 ( x ) B U n 1 ( x ) C U n 1 ( x ) D U n 1 ( x ) U n 2 ( x ) ] ,
ρ = b 1 a 1 = m 11 m 21 m 22 m 12 = A U n 1 ( x ) U n 2 ( x ) C U n 1 ( x ) D U n 1 ( x ) U n 2 ( x ) B U n 1 ( x ) .
A C = U n 2 ( x ) U n 1 ( x ) ,
D B = U n 2 ( x ) U n 1 ( x ) .
δ = 2 π m , a r 3 + ( 2 a 2 + a ) r 2 + ( a 2 + a 1 ) r + a 3 = 0 ;
sin 2 ( δ 2 ) = a r 3 ( 2 a 2 + a ) r 2 ( a 2 + a 3 ) r + 3 a 3 4 ( r + a 3 ) , ( a 4 2 a 2 ) r 3 + ( a 4 a 2 + 1 ) r 2 + a 4 r a 6 = 0 .
δ = 2 π m , a 2 r 3 + ( a 2 + 2 a ) r 2 + ( a 3 a 2 + a ) r 1 = 0 ;
sin 2 ( δ 2 ) = a 2 r 3 ( a + 2 ) a r 2 + ( 3 a 2 a 1 ) a r + 3 4 ( r a 3 + 1 ) , ( 2 a 4 + a 2 ) r 3 + ( a 6 a 4 + a 2 ) r 2 + a 2 r 1 = 0 .
A C = 1 A + D 1 A + D 1 A + D ,
[ c n + 1 d n + 1 ] = 1 i κ [ 1 r r 1 ] [ a n + 1 b n + 1 ] = 1 i κ [ m 11 + r m 21 m 12 + r m 22 r m 11 + m 21 r m 12 + m 22 ] [ a 1 b 1 ] .
ρ = b 1 a 1 = r m 11 m 21 m 22 r m 12 = r [ A U n 1 ( x ) U n 2 ( x ) ] C U n 1 ( x ) D U n 1 ( x ) U n 2 ( x ) r B U n 1 ( x ) ,
t = c n + 1 a 1 = i κ m 22 r m 12 = i κ D U n 1 ( x ) U n 2 ( x ) r B U n 1 ( x ) ,
r A C = U n 2 ( x ) U n 1 ( x ) .
D r B = U n 2 ( x ) U n 1 ( x ) .
δ = 2 π m , ( a 1 ) ( a 2 2 a r 2 + 1 ) = 0 ;
sin 2 ( δ 2 ) = ( 2 a 2 2 a ) r 3 + ( 3 a 3 a 2 a + 3 ) r 4 r ( a 3 + 1 ) , ( 2 a 4 + 2 a 2 ) r 2 + a 6 a 4 + a 2 1 = 0 .
δ = 2 π m , a 2 r 4 + ( a 3 2 a 2 + 3 a ) r 2 1 = 0 ;
sin 2 ( δ 2 ) = a 2 r 4 + ( 3 a 3 2 a 2 3 a ) r 2 + 3 4 ( r 2 a 3 + 1 ) , ( a 6 3 a 4 + a 2 ) r 4 + 2 a 2 r 2 1 = 0 .
[ a n + 1 b n + 1 ] = exp ( i β Δ ) [ a n b n ] ,
cos ( β Δ ) = ( A + D ) 2 = sin ( δ 2 ) κ ,
v g = d ω d β = 2 κ FSR Δ sin ( β Δ ) cos ( δ 2 ) ,
B = 2 π FSR sin 1 ( κ ) .

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