Abstract

We propose a novel coupled resonator optical waveguide (CROW) structure that is made up of a waveguide loop. We theoretically investigate the forbidden band and conduction band conditions in an infinite periodic lattice. We also discuss the reflection- and transmission- spectra, group delay in finite periodic structures. Light has a larger group delay at the band edge in a periodic structure. The flat band pass filter and flat-top group delay can be realized in a non-periodic structure. Scattering matrix method is used to calculate the effects of waveguide loss on the optical characteristics of these structures. We also introduce a tunable coupling loop waveguide to compensate for the fabrication variations since the coupling coefficient of the directional coupler in the loop waveguide is a critical factor in determining the characteristics of a loop CROW. The loop CROW structure is suitable for a wide range of applications such as band pass filters, high Q microcavity, and optical buffers and so on.

© 2014 Optical Society of America

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References

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  1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
    [Crossref]
  2. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
    [Crossref] [PubMed]
  3. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
    [Crossref] [PubMed]
  4. S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
    [Crossref] [PubMed]
  5. P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
    [Crossref] [PubMed]
  6. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
    [Crossref]
  7. 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]
  8. X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
    [Crossref]
  9. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000).
    [Crossref] [PubMed]
  10. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21(9), 1665–1673 (2004).
    [Crossref]
  11. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2006).
    [Crossref]
  12. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
    [Crossref] [PubMed]
  13. A. Yariv and P. Yeh, Optical waves in crystals: propagation and control of laser radiation (Wiley, 1984).
  14. P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67(4), 423–438 (1977).
    [Crossref]
  15. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011).
    [Crossref] [PubMed]
  16. J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
    [Crossref]

2012 (1)

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

2011 (1)

2008 (2)

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

2006 (1)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2006).
[Crossref]

2004 (3)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[Crossref] [PubMed]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21(9), 1665–1673 (2004).
[Crossref]

2003 (2)

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref] [PubMed]

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

2000 (1)

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000).
[Crossref] [PubMed]

1999 (1)

1997 (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

1977 (1)

Akahane, Y.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref] [PubMed]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

Asano, T.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref] [PubMed]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Chutinan, A.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000).
[Crossref] [PubMed]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Fang, Q.

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

Feng, S.

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

Hong, C. S.

Imada, M.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[Crossref] [PubMed]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000).
[Crossref] [PubMed]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Kwong, D. L.

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

Lee, R. K.

Liow, T. Y.

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

Liu, H. C.

Lo, G. Q.

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

Luo, X.

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Noda, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[Crossref] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref] [PubMed]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000).
[Crossref] [PubMed]

Ogawa, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[Crossref] [PubMed]

Okano, M.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[Crossref] [PubMed]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

Poon, A.

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

Poon, J. K. S.

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

Scherer, A.

Scheuer, J.

Sekaric, L.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2006).
[Crossref]

Song, B. S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref] [PubMed]

Song, J.

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

Song, J. F.

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

Tao, S. H.

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Vlasov, Y.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2006).
[Crossref]

Xia, F.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2006).
[Crossref]

Xu, Y.

Yariv, A.

Yeh, P.

Yoshimoto, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[Crossref] [PubMed]

Yu, M.

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

Yu, M. B.

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

IEEE Photon. Technol. Lett. (2)

X. Luo, J. Song, S. Feng, A. Poon, T. Y. Liow, M. Yu, G. Q. Lo, and D. L. Kwong, “Silicon high-order coupled-microring-based electro-optical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24(10), 821–823 (2012).
[Crossref]

J. F. Song, S. H. Tao, Q. Fang, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Thermo-optical enhanced silicon wire interleavers,” IEEE Photon. Technol. Lett. 20(24), 2165–2167 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Nat. Photonics (2)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2006).
[Crossref]

Nature (4)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref] [PubMed]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Science (2)

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[Crossref] [PubMed]

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

Other (1)

A. Yariv and P. Yeh, Optical waves in crystals: propagation and control of laser radiation (Wiley, 1984).

Supplementary Material (4)

» Media 1: MOV (347 KB)     
» Media 2: MOV (1010 KB)     
» Media 3: MOV (320 KB)     
» Media 4: MOV (383 KB)     

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Figures (16)

Fig. 1
Fig. 1 (a) Sketch map of a L-CROW with a repeat element of a waveguide loop structure. (b) Loop element. Light is coupled from the left hand side, and four paths are shown by the dotted line. The left dotted line depicts the reflected light path, and the right dotted line depicts the transmitted light path.

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Fig. 2
Fig. 2 Illustration of band structure. The grey regions denote the conduction bands while the white colors regions denote the forbidden bands.

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Fig. 3
Fig. 3 (a) Reflectivity spectrum with different lattice numbers. (b) Transmissivity spectrum with different lattice numbers. Black dashed lines are infinite lattice. Reflection phase and differential term vs. detune oscillation phase with different lattice numbers. (c) and (d) are reflectivity and transmissivity phases. (e) and (f) are phase and differential term, respectively. (f) is averaged by loop lattice number. t = 0.1. The evolutions of real- and imaginary- part with optical phase detune (see Media 1) gives a better understanding of the reflection and transmission’s amplitude and phase variation.

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Fig. 4
Fig. 4 Relation of delay time with reflectivity. Blue and red color curves denote reflection and transmission respectively.

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Fig. 5
Fig. 5 (a) 6 order Butterworth filter, transmission and delay with phase detune. (b) 6 order Bessel filter, transmission and delay with phase detune. Blue and red color curve denote transmission and delay respectively.

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Fig. 6
Fig. 6 Illustrative comparison of loop cavity and ring resonator. (a) Loop cavity and (b) Add/drop ring resonator.

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Fig. 7
Fig. 7 Transmission spectrum of two loop F-P cavity.

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Fig. 8
Fig. 8 Schematic of loop lattices with defect.

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Fig. 9
Fig. 9 Characteristics comparison of entire lattice and with defect lattice. (a), (c) and (e) are for entire lattice. (b), (d) and (f) are defect lattice. (a) and (b) shows the transmission spectrum. (c) and (d) shows the optical energy distribution in lattice with different phase detune. The energy density distribution evolves from one state in (c) to another state in (d) as the phase changes from 0 to π/2 (see Media 2). (e) and (f) shows the optical energy distribution at Δθ = 0. The total loop lattice number is 20, and self coupling coefficient t = 0.1.

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Fig. 10
Fig. 10 Comparison of differential term of with/without defect. Blue curve is without defect. Red curve is with defect.

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Fig. 11
Fig. 11 Schematic of scattering layers and directions.

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Fig. 12
Fig. 12 Reflection spectrum with different absorption. Imaginary parts of phases are chosen as 0:0.002:0.01. Blue color curve denote lossless case. (a) Reflectivity. (b) Phase of reflectivity. (c) Delay time. (d) Zoom in of delay time. Media 3 shows the real- and imaginary- part evolution.

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Fig. 13
Fig. 13 Reflection characteristics comparison with different absorption in defect loop lattice. Imaginary parts of the phases are chosen as 0:0.002:0.01. Blue color curve denote the lossless case. (a) Reflectivity. (b) Phase of reflectivity. A zoom in of conduction band is inserted. (c) Zooming in of conductance band. The phase changing is in 2π range. (d) Zoom in of forbidden band. The phase change is in π range.

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Fig. 14
Fig. 14 Transmission spectrum with different absorption in defect loop lattice. Imaginary parts of phases are chosen as 0:0.002:0.01. Blue color curve denote the lossless case. (a) Transmission spectra. (b) Zoom in transmission spectra at oscillation frequency. (c) Delay time. (d) Zoom in of delay time. Media 4 shows the real- and imaginary- part with phase detune for θi = 0.002.

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Fig. 15
Fig. 15 Sketch map of tunable loop reflector. (a) A basic loop element. (b) The loop region with heater. (c) Directional coupler replaced by MZI.

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Fig. 16
Fig. 16 Transmission and reflection with the phase difference and self-coupling coefficients. (a) Transmission. (b) Reflection.

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Tables (1)

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Table 1 Cross coupling coefficients of loop filter

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Equations (31)

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{ r e =2itκexp( iθ ) t e =( t 2 − κ 2 )exp( iθ )
{ t=cos( φ DC ) κ=sin( φ DC )
( a ⇀ b ↼ )=T( a ′ ⇀ b ′ ↼ )
T=( T 1,1 T 1,2 T 2,1 T 2,2 )= 1 cos( 2 φ DC ) ( exp( −iθ ) −isin( 2 φ DC ) isin( 2 φ DC ) exp( iθ ) )
Λ ± = cos( θ )±i cos 2 ( 2 φ DC )− cos 2 ( θ ) cos( 2 φ DC )
cos 2 ( 2 φ DC )≥ cos 2 ( θ )
cosϑ= cos( θ ) cos( 2 φ DC )
Nπ−[ 2 φ DC ]>θ>Nπ+[ 2 φ DC ]
( a N b N )= T N ( a 0 b 0 )
T N = U N−1 T− U N−2 I
{ t N = 1 T 1,1 U N−1 − U N−2 r N = T 2,1 U N−1 T 1,1 U N−1 − U N−2
{ | t N | 2 = 1 1+ | T 2,1 | 2 | U N−1 | 2 | r N | 2 = | T 2,1 | 2 | U N−1 | 2 1+ | T 2,1 | 2 | U N−1 | 2
{ | t env | 2 = sin 2 ϑ re sin 2 ϑ re + | T 2,1 | 2 | r env | 2 = | T 2,1 | 2 sin 2 ϑ re + | T 2,1 | 2
Φ r =ΔΦ+ Φ t
tan( Φ t )= sin( θ ) U N−1 cos( 2 φ DC ) T N
ΔΦ=angle [ T 21 U N−1 ]= π 2 sgn[ tan( 2 φ DC ) sin( Nϑ ) sin( ϑ ) ]
τ=− d Φ r dω =− T c d Φ r dθ
d Φ r dθ | Δθ→0 = tan( Nϑ ) tan( ϑ )  or | tan( Nϑ ) sin( 2 φ DC ) |
d Φ r dθ | Δθ→0 N→∞ = 1 | sin( 2 φ DC ) |
∂ Φ r ∂θ | ϑ→Mπ =N 1+ tan 2 ( 2 φ DC ) ( 1+2 N 2 ) /3 1+ tan 2 ( 2 φ DC ) N 2
κ loop = ( 1+ κ ring ) /2
| t 2 | 2 = 1 1+4 cos 2 ( θ ) sin 2 ( 2 φ DC ) cos 4 ( 2 φ DC )
Q∝ 1 Δ θ 3dB = | sin( 2 φ DC ) | cos 2 ( 2 φ DC ) = 2tκ ( 1−2 t 2 ) 2
| a c | 2 + | b c | 2 = ( 1+| r b 2 | )| t f 2 | | 1− r b r f exp( 2i θ c ) | 2
| a c | 2 + | b c | 2 = ( 1+| r b 2 | )| t f 2 | ( 1−| r b || r f | ) 2
( b ↼ a ′ ⇀ )=( r e t e t e r e )( a ⇀ b ′ ↼ )=S( a ⇀ b ′ ↼ )
( b ↼ 1 a ⇀ ′ 2 )= S ( 1,2 ) ( a ⇀ 1 b ↼ ′ 2 ),( b ↼ 2 a ⇀ ′ 3 )= S ( 2,3 ) ( a ⇀ 2 b ↼ ′ 3 ),( b ↼ 1 a ⇀ ′ 3 )= S ( 1,3 ) ( a ⇀ 1 b ↼ ′ 3 )
( b ↼ 1 a ⇀ 2 )= S ¯ ( 1,2 ) ( a ⇀ 1 b ↼ 2 )
S ¯ ( 1,2 ) =( 1 0 0 exp( i θ 2 ) ) S ( 1,2 ) ( 1 0 0 exp( i θ 2 ) )
S ( 1,3 ) =[ ( S ¯ 1,1 ( 1,2 ) 0 0 S 2,2 ( 2,3 ) )+( 0 S ¯ 1,2 ( 1,2 ) S 2,1 ( 2,3 ) 0 ) ( 1 − S ¯ 2,2 ( 1,2 ) − S 1,1 ( 2,3 ) 1 ) −1 ( S ¯ 2,1 ( 1,2 ) 0 0 S 1,2 ( 2,3 ) ) ]
{ r tunable =2ikt( t 2 P 2 −P ) t tunable =−2 ( kt ) 2 P 2 +P−1

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