Abstract

We introduce universal “anti-reflection blocks” for periodic microresonator sequences consisting of 4-port unit cells. When added to the end of a finite periodic sequence, the anti-reflection block can eliminate the reflectivity at a given frequency ωref and significantly reduce it in a frequency range around ωref. These anti-reflection blocks are universal: By adjusting only their characteristic parameters, they can be made applicable to any 4-port periodic sequence, regardless of the detailed inside structure of its unit cells, at any frequency within the photonic bands.

© 2012 Optical Society of America

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References

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  1. F. Shinobu, N. Ishikura, Y. Arita, T. Tamanuki, and T. Baba, “Continuously tunable slow-light device consisting of heater-controlled silicon microring array,” Opt. Express 19, 13557–13564 (2011).
    [Crossref] [PubMed]
  2. J. J. Cardenas, M. A. Foster, N. Sherwood-Droz, C. B. Poitras, H. L. R. Lira, B. Zhang, A. L. Gaeta, J. B. Khurgin, P. Morton, and M. Lipson, “Wide-bandwidth continuously tunable optical delay line using silicon microring resonators,” Opt. Express 18, 26525–26534 (2010).
    [Crossref] [PubMed]
  3. P. Chamorro-Posada and F. J. Fraile-Pelaez, “Fast and slow light in zigzag microring resonator chains,” Opt. Lett. 34, 626–628 (2009).
    [Crossref] [PubMed]
  4. P. P. Yupapin, “Proposed nonlinear microring resonator arrangement for stopping and storing light,” IEEE Photon. Technol. Lett. 21, 404–406 (2009).
    [Crossref]
  5. Z. S. Yang and J. E. Sipe, “Increasing the delay-time-bandwidth product for microring resonator structures by varying the optical ring resonances,” Opt. Lett. 32, 918–920 (2007).
    [Crossref] [PubMed]
  6. S. Sandhu, M. L. Povinelli, M. F. Yanik, and S. Fan, “Dynamically tuned coupled-resonator delay lines can be nearly dispersion free,” Opt. Lett. 31, 1985–1987 (2006).
    [Crossref] [PubMed]
  7. 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, 1818–1832 (2004).
    [Crossref]
  8. J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004)
    [Crossref] [PubMed]
  9. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
    [Crossref] [PubMed]
  10. Y. F. Xu, H. Huang, Y. Q. Huang, H. B. Chen, and X. M. Ren, “Optimization of finite-size effects in coupled microring resonator optical waveguide,” Acta Photonica Sinica 38, 1991–1995 (2009).
  11. P. Chak and J. E. Sipe, “Minimizing finite-size effects in artificial resonance tunneling structures,” Opt. Lett. 31, 2568–2570 (2006).
    [Crossref] [PubMed]
  12. P. Sanchis, J. Garcia, A. Martinez, and J. Marti, “Pulse propagation in adiabatically coupled photonic crystal coupled cavity waveguides,” J. Appl. Phys. 97, 013101 (2005).
    [Crossref]
  13. M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11, 381–391 (2003).
    [Crossref] [PubMed]
  14. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. Soc. Am. B 22, 2144–2156 (2005).
    [Crossref]
  15. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum well Bragg structures,” Opt. Lett. 30, 2790–2792 (2005).
    [Crossref] [PubMed]
  16. G. Boedecker and C. Henkel, “All-frequency effective medium theory of a photonic crystal,” Opt. Express 11, 1590–1595 (2003).
    [Crossref] [PubMed]

2011 (1)

2010 (1)

2009 (3)

P. Chamorro-Posada and F. J. Fraile-Pelaez, “Fast and slow light in zigzag microring resonator chains,” Opt. Lett. 34, 626–628 (2009).
[Crossref] [PubMed]

P. P. Yupapin, “Proposed nonlinear microring resonator arrangement for stopping and storing light,” IEEE Photon. Technol. Lett. 21, 404–406 (2009).
[Crossref]

Y. F. Xu, H. Huang, Y. Q. Huang, H. B. Chen, and X. M. Ren, “Optimization of finite-size effects in coupled microring resonator optical waveguide,” Acta Photonica Sinica 38, 1991–1995 (2009).

2007 (1)

2006 (2)

2005 (3)

2004 (3)

2003 (2)

Arita, Y.

Baba, T.

Binder, R.

Boedecker, G.

Boyd, R. W.

Cardenas, J. J.

Chak, P.

Chamorro-Posada, P.

Chen, H. B.

Y. F. Xu, H. Huang, Y. Q. Huang, H. B. Chen, and X. M. Ren, “Optimization of finite-size effects in coupled microring resonator optical waveguide,” Acta Photonica Sinica 38, 1991–1995 (2009).

Eggleton, B. J.

Fan, S.

S. Sandhu, M. L. Povinelli, M. F. Yanik, and S. Fan, “Dynamically tuned coupled-resonator delay lines can be nearly dispersion free,” Opt. Lett. 31, 1985–1987 (2006).
[Crossref] [PubMed]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

Foster, M. A.

Fraile-Pelaez, F. J.

Gaeta, A. L.

Garcia, J.

P. Sanchis, J. Garcia, A. Martinez, and J. Marti, “Pulse propagation in adiabatically coupled photonic crystal coupled cavity waveguides,” J. Appl. Phys. 97, 013101 (2005).
[Crossref]

Heebner, J. E.

Henkel, C.

Huang, H.

Y. F. Xu, H. Huang, Y. Q. Huang, H. B. Chen, and X. M. Ren, “Optimization of finite-size effects in coupled microring resonator optical waveguide,” Acta Photonica Sinica 38, 1991–1995 (2009).

Huang, Y.

Huang, Y. Q.

Y. F. Xu, H. Huang, Y. Q. Huang, H. B. Chen, and X. M. Ren, “Optimization of finite-size effects in coupled microring resonator optical waveguide,” Acta Photonica Sinica 38, 1991–1995 (2009).

Ishikura, N.

Khurgin, J. B.

Kwong, N. H.

Lipson, M.

Lira, H. L. R.

Marti, J.

P. Sanchis, J. Garcia, A. Martinez, and J. Marti, “Pulse propagation in adiabatically coupled photonic crystal coupled cavity waveguides,” J. Appl. Phys. 97, 013101 (2005).
[Crossref]

Martinez, A.

P. Sanchis, J. Garcia, A. Martinez, and J. Marti, “Pulse propagation in adiabatically coupled photonic crystal coupled cavity waveguides,” J. Appl. Phys. 97, 013101 (2005).
[Crossref]

Mookherjea, S.

Morton, P.

Paloczi, G.

Pereira, S.

Poitras, C. B.

Poon, J.

Povinelli, M. L.

Ren, X. M.

Y. F. Xu, H. Huang, Y. Q. Huang, H. B. Chen, and X. M. Ren, “Optimization of finite-size effects in coupled microring resonator optical waveguide,” Acta Photonica Sinica 38, 1991–1995 (2009).

Sanchis, P.

P. Sanchis, J. Garcia, A. Martinez, and J. Marti, “Pulse propagation in adiabatically coupled photonic crystal coupled cavity waveguides,” J. Appl. Phys. 97, 013101 (2005).
[Crossref]

Sandhu, S.

Scheuer, J.

Sherwood-Droz, N.

Shinobu, F.

Sipe, J. E.

Smirl, A. L.

Suh, W.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

Sumetsky, M.

Tamanuki, T.

Wang, Z.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

Xu, Y. F.

Y. F. Xu, H. Huang, Y. Q. Huang, H. B. Chen, and X. M. Ren, “Optimization of finite-size effects in coupled microring resonator optical waveguide,” Acta Photonica Sinica 38, 1991–1995 (2009).

Yang, Z. S.

Yanik, M. F.

S. Sandhu, M. L. Povinelli, M. F. Yanik, and S. Fan, “Dynamically tuned coupled-resonator delay lines can be nearly dispersion free,” Opt. Lett. 31, 1985–1987 (2006).
[Crossref] [PubMed]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

Yariv, A.

Yupapin, P. P.

P. P. Yupapin, “Proposed nonlinear microring resonator arrangement for stopping and storing light,” IEEE Photon. Technol. Lett. 21, 404–406 (2009).
[Crossref]

Zhang, B.

Acta Photonica Sinica (1)

Y. F. Xu, H. Huang, Y. Q. Huang, H. B. Chen, and X. M. Ren, “Optimization of finite-size effects in coupled microring resonator optical waveguide,” Acta Photonica Sinica 38, 1991–1995 (2009).

IEEE Photon. Technol. Lett. (1)

P. P. Yupapin, “Proposed nonlinear microring resonator arrangement for stopping and storing light,” IEEE Photon. Technol. Lett. 21, 404–406 (2009).
[Crossref]

J. Appl. Phys. (1)

P. Sanchis, J. Garcia, A. Martinez, and J. Marti, “Pulse propagation in adiabatically coupled photonic crystal coupled cavity waveguides,” J. Appl. Phys. 97, 013101 (2005).
[Crossref]

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

Opt. Express (5)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1 Schematic of a 4-port unit cell for periodic sequences or anti-reflection blocks. All the four ports are assumed to be identical waveguides.

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Fig. 2
Fig. 2 (a) Schematic of a SCISSOR unit. (b) A finite SCISSOR periodic sequence (enclosed within the dashed-line rectangular) with SCISSOR anti-reflection blocks at both ends.

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Fig. 3
Fig. 3 Band structure of the infinite periodic SCISSOR sequence around 100ωB,ps. ξ is the normalized Bloch wavevector as defined in and below Eq. (2), and the intermediate band is between 100ωB,ps and 100ωR,ps.

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Fig. 4
Fig. 4 Transmission spectrum of the 50-cell finite periodic SCISSOR sequence, without anti-reflection blocks, covering (a) intermediate band, bandgaps, and part of the upper and lower bands; (b) intermediate band and part of the bandgaps.

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Fig. 5
Fig. 5 Same as Fig. 4(b), but with SCISSOR anti-reflection blocks at both ends.

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Fig. 6
Fig. 6 (a) Schematic of a Half-CROW block. (b) Schematic of the extended Half-CROW block. (c) A finite SCISSOR periodic sequence (enclosed within the dashed-line rectangular) with extended Half-CROW anti-reflection blocks at both ends.

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Fig. 7
Fig. 7 (a) Same as Fig. 4(b), but with extended half-CROW anti-reflection blocks at both ends. (b) Comparison of the group delay δt for the periodic sequence with half-Crow anti-reflection blocks (solid) and the corresponding delay Δt within an infinite structure (dash).

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Fig. 8
Fig. 8 (a) Modulus and (b) phase of: r∞(ω) of the periodic SCISSOR sequence (dotted), β★/α of the half-CROW anti-reflection block (solid), β★/α of the SCISSOR anti-reflection block (dashed).

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Equations (26)

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( E R + E R − ) = M ( ω ) ( E L + E L − ) ,
M ( ω ) ( E + E − ) = e i ξ ( E + E − ) ,
M ( ω ) ( 1 r ∞ ) = e i ξ + ( 1 r ∞ ) ,
M A R ( ω ref ) ( 1 0 ) ∝ ( 1 r ∞ ) ,
M A R = ( α β β * α * ) ,             | α | 2 − | β | 2 = 1 ,
( α β β * α * ) ( 1 0 ) = ( α β * ) ∝ ( 1 r ∞ ) ,
β * α = r ∞ .
| β * α | = | r ∞ |
arg ( β * α ) = arg ( r ∞ ) ,
M A R ( ω ) ≡ ( α β β * α * ) = 1 2 i σ sin ( π ω ω R ) [ e i π ω ω B [ e i π ω ω R − σ 2 e − i π ω ω R ] κ 2 − κ 2 e − i π ω ω B [ σ 2 e i π ω ω R − e − i π ω ω R ] ] ,
β * α = − κ 2 e i π ω ref ω B [ − σ 2 e − i π ω ref ω R + e i π ω ref ω R ] = r ∞ ,
| − κ 2 e i π ω ref ω B [ − σ 2 e − i π ω ref ω R + e i π ω ref ω R ] | 2 = κ 4 ( 1 + σ 4 ) − 2 σ 2 cos ( 2 π ω ref ω R ) = | r ∞ | 2 ,
κ 2 = 2 | r ∞ sin ( π ω ref ω R ) | 1 − | r ∞ | 2 ( 1 − | r ∞ | 2 cos 2 ( π ω ref ω R ) − | r ∞ sin ( π ω ref ω R ) | ) ,
π ω ref ω B = arg ( − κ 2 − σ 2 e − i π ω ref ω R + e i π ω ref ω R ) − arg ( r ∞ ) .
L p s = 26 μ m ,                 π R p s = 26.01 μ m ,               σ p s = 0.96 ,             n b , p s = 3.00 ,
r ∞ = − 0.663 .
R = R p s ,                 n b = n b , p s .
κ = − 0.320 ,           L = 26.45 μ m .
M A R ( ω ) = 1 i κ [ − e i π ω ω R σ − σ e − i π ω ω R ] = [ α β β * α * ] ,
σ e i π ω ref ω R = r ∞ ,
σ = | r ∞ | .
M ′ ( ω ) = [ e in b ω c Δ l 0 0 e − in b ω c Δ l ] 1 i κ [ − e i π ω ω R σ − σ e − i π ω ω R ] = [ − e in b ω c Δ l e i π ω ω R e in b ω c Δ l σ − e − in b ω c Δ l σ e − in b ω c Δ l e − i π ω ω R ] = [ α ′ β ′ β ′ * α ′ * ] .
e − i π ω ref ω R − i 2 n b ω ref c Δ l = r ∞ | r ∞ | ,
β ′ * α ′ = r ∞ .
R = R p s ,                   n b = n b , p s .
σ = 0.663 ,             Δ l = 0.387 μ m .

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