Abstract

We present a systematic design of coupled-resonator optical waveguides (CROWs) based on high-Q tapered grating-defect resonators. The formalism is based on coupled-mode theory where forward and backward waveguide modes are coupled by the grating. Although applied to strong gratings (periodic air holes in single-mode silicon-on-insulator waveguides), coupled-mode theory is shown to be valid, since the spatial Fourier transform of the resonant mode is engineered to minimize the coupling to radiation modes and thus the propagation loss. We demonstrate the numerical characterization of strong gratings, the design of high-Q tapered grating-defect resonators (Q>2 × 106, modal volume = 0.38·(λ/n)3), and the control of inter-resonator coupling for CROWs. Furthermore, we design Butterworth and Bessel filters by tailoring the numbers of holes between adjacent defects. We show with numerical simulation that Butterworth CROWs are more tolerant against fabrication disorder than CROWs with uniform coupling coefficient.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett.24(11), 711–713 (1999).
    [CrossRef] [PubMed]
  2. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics1(1), 65–71 (2007).
    [CrossRef]
  3. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett.33(20), 2389–2391 (2008).
    [CrossRef] [PubMed]
  4. 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. Express15(19), 11934–11941 (2007).
    [CrossRef] [PubMed]
  5. A. Melloni, F. Morichetti, and M. Martinelli, “Four-wave mixing and wavelength conversion in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B25(12), C87–C97 (2008).
    [CrossRef]
  6. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett.31(4), 456–458 (2006).
    [CrossRef] [PubMed]
  7. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics2(12), 741–747 (2008).
    [CrossRef]
  8. T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002).
    [CrossRef]
  9. 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]
  10. S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett.27(23), 2079–2081 (2002).
    [CrossRef] [PubMed]
  11. A. Martínez, J. García, P. Sanchis, F. Cuesta-Soto, J. Blasco, and J. Martí, “Intrinsic losses of coupled-cavity waveguides in planar-photonic crystals,” Opt. Lett.32(6), 635–637 (2007).
    [CrossRef] [PubMed]
  12. P. Velha, E. Picard, T. Charvolin, E. Hadji, J. C. Rodier, P. Lalanne, and D. Peyrade, “Ultra-high Q/V Fabry-Perot microcavity on SOI substrate,” Opt. Express15(24), 16090–16096 (2007).
    [CrossRef] [PubMed]
  13. A. R. M. Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express16(16), 12084–12089 (2008).
    [CrossRef] [PubMed]
  14. E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. G. Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO2 claddings and on air claddings,” Opt. Express18(15), 15859–15869 (2010).
    [CrossRef] [PubMed]
  15. Q. M. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett.96(20), 203102 (2010).
    [CrossRef]
  16. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003).
    [CrossRef] [PubMed]
  17. A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University Press, 2007).
  18. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express19(18), 17653–17668 (2011).
    [CrossRef] [PubMed]
  19. J. W. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express16(22), 18152–18163 (2008).
    [CrossRef] [PubMed]
  20. V. Van, “Circuit-based method for synthesizing serially coupled microring filters,” J. Lightwave Technol.24(7), 2912–2919 (2006).
    [CrossRef]
  21. C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B26(4), 858–866 (2009).
    [CrossRef]
  22. S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett.32(3), 289–291 (2007).
    [CrossRef] [PubMed]
  23. H. C. Liu, C. Santos, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on tapered grating-defect resonators,” in CLEO (San Jose, USA, 2012).
  24. H. C. Liu, C. Santis, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on grating resonators with modulated bandgap,” in Slow and Fast Light (Toronto, Canada, 2011), p. SLTuB2.

2011

2010

2009

2008

2007

2006

2003

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003).
[CrossRef] [PubMed]

2002

T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002).
[CrossRef]

S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett.27(23), 2079–2081 (2002).
[CrossRef] [PubMed]

1999

1992

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]

Akahane, Y.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Asakawa, K.

Asano, T.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Blasco, J.

Brown, D. H.

T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002).
[CrossRef]

Charvolin, T.

Cuesta-Soto, F.

De La Rue, R. M.

Deotare, P. B.

Q. M. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett.96(20), 203102 (2010).
[CrossRef]

DeRose, G. A.

Ferrari, C.

García, J.

Hadji, E.

Haus, H. A.

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]

Huang, W. P.

Ikeda, N.

Ishikawa, H.

Johnson, N. P.

Karle, T. J.

T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002).
[CrossRef]

Kawasaki, K.

Krauss, T. F.

T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002).
[CrossRef]

Kuramochi, E.

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]

Lalanne, P.

Lan, S.

Lee, R. K.

Liu, H. C.

Loncar, M.

Q. M. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett.96(20), 203102 (2010).
[CrossRef]

Martí, J.

Martinelli, M.

Martínez, A.

Melloni, A.

Mookherjea, S.

Morichetti, F.

Mu, J. W.

Nishikawa, S.

Noda, S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Notomi, M.

Oh, A.

Peyrade, D.

Picard, E.

Poon, J. K. S.

Quan, Q. M.

Q. M. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett.96(20), 203102 (2010).
[CrossRef]

Rodier, J. C.

Roh, Y. G.

Rooks, M.

Sanchis, P.

Scherer, A.

Sekaric, L.

Song, B. S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Sorel, M.

Steer, M.

T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002).
[CrossRef]

Sugimoto, Y.

Tanabe, T.

Taniyama, H.

Van, V.

Velha, P.

Vlasov, Y.

Wilson, R.

T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002).
[CrossRef]

Xia, F. N.

Xu, Y.

Yariv, A.

Zain, A. R. M.

Zhu, L.

Appl. Phys. Lett.

Q. M. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett.96(20), 203102 (2010).
[CrossRef]

IEEE J. Quantum Electron.

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 J. Sel. Top. Quantum Electron.

T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Nat. Photonics

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics2(12), 741–747 (2008).
[CrossRef]

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics1(1), 65–71 (2007).
[CrossRef]

Nature

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Other

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

H. C. Liu, C. Santos, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on tapered grating-defect resonators,” in CLEO (San Jose, USA, 2012).

H. C. Liu, C. Santis, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on grating resonators with modulated bandgap,” in Slow and Fast Light (Toronto, Canada, 2011), p. SLTuB2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Schematic drawings, coupling coefficients, and field intensity of (a) a grating-defect resonator and (b) a grating-defect CROW.

Fig. 2
Fig. 2

Spectra of (a) transmission and (b) group delay of N = 10 grating-defect CROWs with inter-defect spacing L = 200 μm (blue) and L = 300 μm (green). κg = 0.01/μm.

Fig. 3
Fig. 3

(a) Schematic drawing of a strong grating in a silicon waveguide and its cross-section. (b) Simulated κg and grating period as functions of hole radius

Fig. 4
Fig. 4

(a) Field distribution of a QWS resonator mode. κg = 0.75/μm (b) Spatial Fourier transform of the QWS resonator mode.

Fig. 5
Fig. 5

(a) Schematic drawing of a tapered grating-defect resonator with 6 tapered holes. (b) Field distribution (one side of the defect) of tapered grating-defect resonators with α = 0, 0.55, and 1, respectively. (c) Energy portion in the continuum of radiation modes of grating-defect resonators as a function of α. (d) Spatial Fourier spectra of the modal fields for α = 0, 0.55, and 1. (e) Quality factor as a function of number of holes on each side of the defect. α = 0.55, nt = 6. (f) Quality factor as a function of α. Dashed lines show ηrad−1.

Fig. 6
Fig. 6

Schematic drawing of the first two resonators of a grating-defect CROW.

Fig. 7
Fig. 7

Spectra of (a) transmission and (b) group delay of 10-resonator grating-defect CROWs with m = 12, 14, and 16 respectively.

Fig. 8
Fig. 8

Spectra of transmission and group delay of (a) an N = 10 Butterworth grating CROW and (b) an N = 10 Bessel grating CROW.

Fig. 9
Fig. 9

(a) Shift of resonant wavelength due to 1 nm change of radius for each hole starting from the one nearest to the defect. (b-e) Simulated transmission spectra of 10-resonator CROWs with disorder in resonant frequencies. (b,c) Uniform coupling coefficients. (d,e) Butterworth filters. (b,d) δω = 0.5B. (c,e) δω = 0.25B.

Fig. 10
Fig. 10

Schematic drawings and the corresponding grating structures of (a) a symmetric resonator and (b) inline coupling of two resonators.

Tables (2)

Tables Icon

Table 1 Coupling Coefficients of N = 10 Butterworth and Bessel CROWs

Tables Icon

Table 2 Numbers of Regular Holes of N = 10 Butterworth and Bessel CROWs

Equations (18)

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

da dz =iδa+i κ g * (z)b db dz =iδbi κ g (z)a ,
a(z)=a( L g )cosh( z L g κ g (z')dz' )
b(z)=ia( L g )sinh( z L g κ g (z')dz' ),
v g,CROW = L g E stored = v g κ g L sinh( κ g L) .
κ= κ g v g exp( κ g L).
Q= 1 2 ωτ= ωexp(2 κ g L) 4 κ g v g .
1 Q = 1 Q e (n) + 1 Q i ,
κ= ω 4 Q 1 Q 2 .
Qexp[ 2 0 L κ g (z)dz ]=exp[ 2 i=1 n κ g,i Λ i ]=exp[ 2 i=1 n t κ g,i Λ i ]exp[ 2 n reg κ g Λ ],
Q= Q 0 a 2 n reg ,
κ= ω 4 Q 1 Q 2 = ω 4 Q 0 a n reg ,
1 τ e = ω 4Q = ω 4 Q 0 a 2 n reg ,
a n reg =exp[ n reg κ g Λ ]=exp[ n int κ g (r)Λ(r) ].
da dt =(iω 1 τ i 2 τ e )aiμ s in s out =iμa s r = s in iμa ,
d a 1 dt =(i ω 1 1 τ e1 ) a 1 i μ 1 s e iθ d a 2 dt =(i ω 2 1 τ e2 ) a 2 i μ 2 s + e iθ s + = s e iθ i μ 1 a 1 s = s + e iθ i μ 2 a 2 .
d a 1 dt =i( ω 1 cotθ τ e1 ) a 1 i cscθ τ e1 τ e2 a 2 d a 2 dt =i( ω 2 cotθ τ e2 ) a 2 i cscθ τ e1 τ e2 a 1 .
Δ ω 1,2 = cotθ τ e1,2 κ= cscθ τ e1 τ e2 .
κ= 1 τ e1 τ e2 = ω 4 Q 1 Q 2 ,

Metrics