Abstract

Based on the symmetry between the transfer characteristics of two coupled double-ring resonators (CDRRs) with symmetrical loss and gain distributions, the transmission and dispersion characteristics of CDRRs with gain in both rings and gain in one ring and loss in the other are analyzed systematically. It is shown that besides coupled-resonator-induced transparency and absorption as in total lossy structures, transmission spectra of inverse transparency and absorption can also exist when gain is introduced. The lasing threshold, classification of transmission response, and group velocity control of different loss and gain structures at resonance and split modes are provided.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600–602 (2002).
    [CrossRef]
  2. J. M. Choi, R. K. Lee, and A. Yariv, “Control of critical coupling in a ring resonator fiber configuration: application to wavelength selective switching, modulation, amplification and oscillation,” Opt. Lett. 26, 1236–1238 (2001).
    [CrossRef]
  3. I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
    [CrossRef]
  4. Y. Yanagase, S. Suzuki, Y. Kokubun, and S. T. Chu, “Box-like filter response and expansion of FSR by a vertically triple coupled microring resonator filter,” J. Lightwave Technol. 20, 1525–1529(2002).
    [CrossRef]
  5. D. G. Rabus, “Devices,” in Integrated Ring Resonators: The Compendium (Springer, 2007), pp. 125–209.
  6. J. Heebner, R. Grover, and T. Ibrahim, “Distributed microresonator systems,” in Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008), pp. 175–215.
  7. J. Scheuer, J. K. S. Poon, G. T. Paloczi, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing & storage of light,” Opt. Photon. News 16, 36–40 (2005).
    [CrossRef]
  8. A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
    [CrossRef]
  9. J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
    [CrossRef]
  10. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005).
    [CrossRef]
  11. H. P. Uranus and H. J. W. M. Hoekstra, “Modeling of loss-induced superluminal and negative group velocity in two-port ring-resonator circuits,” J. Lightwave Technol. 25, 2376–2384 (2007).
    [CrossRef]
  12. H. P. Uranus, L. Zhuang, C. G. H. Roeloffzen, and H. J. W. M. Hoekstra, “Pulse advancement and delay in an integrated-optical two-port ring-resonator circuit: direct experimental observations,” Opt. Lett. 32, 2620–2622 (2007).
    [CrossRef] [PubMed]
  13. J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
    [CrossRef]
  14. K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904(2007).
    [CrossRef] [PubMed]
  15. D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).
    [CrossRef]
  16. Y. Hao and M. Kong, “Symmetry between the transfer properties of micro-ring resonators with gain and with loss,” J. Mod. Opt. 57, 2182–2186 (2010).
    [CrossRef]
  17. H. Chang and D. D. Smith, “Gain-assisted superluminal propagation in coupled optical resonators,” J. Opt. Soc. Am. B 22, 2237–2241 (2005).
    [CrossRef]
  18. H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
    [CrossRef]
  19. Y. Lu, L. Xu, Y. Yu, P. Wang, and J. Yao, “Double-wavelength Fano resonance and enhanced coupled-resonator-induced transparency in a double-microcavity resonator system,” J. Opt. Soc. Am. A 23, 1718–1721 (2006).
    [CrossRef]
  20. H. Shen, J. Chen, X. Li, and Y. Wang, “Group delay and dispersion analysis of compound high order microring resonator all-pass filter,” Opt. Commun. 262, 200–205 (2006).
    [CrossRef]

2010 (1)

Y. Hao and M. Kong, “Symmetry between the transfer properties of micro-ring resonators with gain and with loss,” J. Mod. Opt. 57, 2182–2186 (2010).
[CrossRef]

2007 (4)

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904(2007).
[CrossRef] [PubMed]

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

H. P. Uranus and H. J. W. M. Hoekstra, “Modeling of loss-induced superluminal and negative group velocity in two-port ring-resonator circuits,” J. Lightwave Technol. 25, 2376–2384 (2007).
[CrossRef]

H. P. Uranus, L. Zhuang, C. G. H. Roeloffzen, and H. J. W. M. Hoekstra, “Pulse advancement and delay in an integrated-optical two-port ring-resonator circuit: direct experimental observations,” Opt. Lett. 32, 2620–2622 (2007).
[CrossRef] [PubMed]

2006 (2)

Y. Lu, L. Xu, Y. Yu, P. Wang, and J. Yao, “Double-wavelength Fano resonance and enhanced coupled-resonator-induced transparency in a double-microcavity resonator system,” J. Opt. Soc. Am. A 23, 1718–1721 (2006).
[CrossRef]

H. Shen, J. Chen, X. Li, and Y. Wang, “Group delay and dispersion analysis of compound high order microring resonator all-pass filter,” Opt. Commun. 262, 200–205 (2006).
[CrossRef]

2005 (4)

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005).
[CrossRef]

H. Chang and D. D. Smith, “Gain-assisted superluminal propagation in coupled optical resonators,” J. Opt. Soc. Am. B 22, 2237–2241 (2005).
[CrossRef]

H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
[CrossRef]

J. Scheuer, J. K. S. Poon, G. T. Paloczi, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing & storage of light,” Opt. Photon. News 16, 36–40 (2005).
[CrossRef]

2004 (1)

D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).
[CrossRef]

2003 (1)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[CrossRef]

2002 (4)

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

Y. Yanagase, S. Suzuki, Y. Kokubun, and S. T. Chu, “Box-like filter response and expansion of FSR by a vertically triple coupled microring resonator filter,” J. Lightwave Technol. 20, 1525–1529(2002).
[CrossRef]

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600–602 (2002).
[CrossRef]

2001 (1)

Bowers, J. E.

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600–602 (2002).
[CrossRef]

Boyd, R. W.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

Chang, H.

H. Chang and D. D. Smith, “Gain-assisted superluminal propagation in coupled optical resonators,” J. Opt. Soc. Am. B 22, 2237–2241 (2005).
[CrossRef]

H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
[CrossRef]

D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).
[CrossRef]

Chen, J.

H. Shen, J. Chen, X. Li, and Y. Wang, “Group delay and dispersion analysis of compound high order microring resonator all-pass filter,” Opt. Commun. 262, 200–205 (2006).
[CrossRef]

Choi, J. M.

Chu, S. T.

Dimmock, J. O.

H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
[CrossRef]

Fan, X.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

Frazier, D. O.

H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
[CrossRef]

Fuller, K. A.

H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
[CrossRef]

Gregory, D. A.

H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
[CrossRef]

Grover, R.

J. Heebner, R. Grover, and T. Ibrahim, “Distributed microresonator systems,” in Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008), pp. 175–215.

Hanumegowda, N. M.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

Hao, Y.

Y. Hao and M. Kong, “Symmetry between the transfer properties of micro-ring resonators with gain and with loss,” J. Mod. Opt. 57, 2182–2186 (2010).
[CrossRef]

Heebner, J.

J. Heebner, R. Grover, and T. Ibrahim, “Distributed microresonator systems,” in Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008), pp. 175–215.

Heebner, J. E.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

Hoekstra, H. J. W. M.

Ibrahim, T.

J. Heebner, R. Grover, and T. Ibrahim, “Distributed microresonator systems,” in Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008), pp. 175–215.

Khurgin, J. B.

Kobayashi, N.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904(2007).
[CrossRef] [PubMed]

Kokubun, Y.

Kong, M.

Y. Hao and M. Kong, “Symmetry between the transfer properties of micro-ring resonators with gain and with loss,” J. Mod. Opt. 57, 2182–2186 (2010).
[CrossRef]

Lee, R. K.

Li, X.

H. Shen, J. Chen, X. Li, and Y. Wang, “Group delay and dispersion analysis of compound high order microring resonator all-pass filter,” Opt. Commun. 262, 200–205 (2006).
[CrossRef]

Liu, B.

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600–602 (2002).
[CrossRef]

Lu, Y.

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[CrossRef]

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[CrossRef]

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[CrossRef]

Oveys, H.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

Paloczi, G. T.

J. Scheuer, J. K. S. Poon, G. T. Paloczi, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing & storage of light,” Opt. Photon. News 16, 36–40 (2005).
[CrossRef]

Park, Q. H.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Poon, J. K. S.

J. Scheuer, J. K. S. Poon, G. T. Paloczi, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing & storage of light,” Opt. Photon. News 16, 36–40 (2005).
[CrossRef]

Rabus, D. G.

D. G. Rabus, “Devices,” in Integrated Ring Resonators: The Compendium (Springer, 2007), pp. 125–209.

Roeloffzen, C. G. H.

Scheuer, J.

J. Scheuer, J. K. S. Poon, G. T. Paloczi, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing & storage of light,” Opt. Photon. News 16, 36–40 (2005).
[CrossRef]

Shakouri, A.

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600–602 (2002).
[CrossRef]

Shen, H.

H. Shen, J. Chen, X. Li, and Y. Wang, “Group delay and dispersion analysis of compound high order microring resonator all-pass filter,” Opt. Commun. 262, 200–205 (2006).
[CrossRef]

Smith, D. D.

H. Chang and D. D. Smith, “Gain-assisted superluminal propagation in coupled optical resonators,” J. Opt. Soc. Am. B 22, 2237–2241 (2005).
[CrossRef]

H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
[CrossRef]

D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).
[CrossRef]

Suter, J. D.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

Suzuki, S.

Tomita, M.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904(2007).
[CrossRef] [PubMed]

Totsuka, K.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904(2007).
[CrossRef] [PubMed]

Uranus, H. P.

Wang, P.

Wang, Y.

H. Shen, J. Chen, X. Li, and Y. Wang, “Group delay and dispersion analysis of compound high order microring resonator all-pass filter,” Opt. Commun. 262, 200–205 (2006).
[CrossRef]

White, I. M.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

Xu, L.

Yanagase, Y.

Yao, J.

Yariv, A.

J. Scheuer, J. K. S. Poon, G. T. Paloczi, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing & storage of light,” Opt. Photon. News 16, 36–40 (2005).
[CrossRef]

J. M. Choi, R. K. Lee, and A. Yariv, “Control of critical coupling in a ring resonator fiber configuration: application to wavelength selective switching, modulation, amplification and oscillation,” Opt. Lett. 26, 1236–1238 (2001).
[CrossRef]

Yu, Y.

Zhu, H.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

Zhuang, L.

Zourob, M.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600–602 (2002).
[CrossRef]

IEEE Sens. J. (1)

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J. 7, 28–35 (2007).
[CrossRef]

J. Lightwave Technol. (2)

J. Mod. Opt. (3)

J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).
[CrossRef]

Y. Hao and M. Kong, “Symmetry between the transfer properties of micro-ring resonators with gain and with loss,” J. Mod. Opt. 57, 2182–2186 (2010).
[CrossRef]

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

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

Opt. Commun. (1)

H. Shen, J. Chen, X. Li, and Y. Wang, “Group delay and dispersion analysis of compound high order microring resonator all-pass filter,” Opt. Commun. 262, 200–205 (2006).
[CrossRef]

Opt. Lett. (2)

Opt. Photon. News (1)

J. Scheuer, J. K. S. Poon, G. T. Paloczi, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing & storage of light,” Opt. Photon. News 16, 36–40 (2005).
[CrossRef]

Opt. Quantum Electron. (1)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[CrossRef]

Phys. Rev. E (1)

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904(2007).
[CrossRef] [PubMed]

Proc. SPIE (1)

H. Chang, D. D. Smith, K. A. Fuller, J. O. Dimmock, D. A. Gregory, and D. O. Frazier, “Slow and fast light in coupled microresonators,” Proc. SPIE 5735, 40–51 (2005).
[CrossRef]

Other (2)

D. G. Rabus, “Devices,” in Integrated Ring Resonators: The Compendium (Springer, 2007), pp. 125–209.

J. Heebner, R. Grover, and T. Ibrahim, “Distributed microresonator systems,” in Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008), pp. 175–215.

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 (5)

Fig. 1
Fig. 1

CDRRs with gain and loss distributing symmetrically in their counterparts. (a) Structures A and B have loss and gain in both of their rings, respectively. (b) Structures C and D have loss in one ring and gain in the other.

Fig. 2
Fig. 2

Transmission responses of CDRRs with loss or gain in both rings, t 1 = 0.999 , t 2 = 0.7 . (a) CRIT of structure A (i) and ICRIT of structure B (ii), a 1 , A = 1 / a 1 , B = 0.9999 , a 2 , A = 1 / a 2 , B = 0.9 ; (b) CRIA of A (i) and ICRIA of B (ii), a 1 , A = 1 / a 1 , B = 0.99901 , a 2 , A = 1 / a 2 , B = 0.994 .

Fig. 3
Fig. 3

Transmission responses of CDRRs with gain in one ring and loss in the other. The crosses indicate the positions of ϕ sp . t 1 = 0.999 , t 2 = 0.7 . (a) CRIA of structure C (i) and ICRIA of structure D (ii). a 1 , C = 1 / a 1 , D = 0.9999 , a 2 , C = 1 / a 2 , D = 1 / 0.99999 . (b) ICRIT of C (i) and CRIT of D (ii), a 1 , C = 1 / a 1 , D = 0.9999 , a 2 , C = 1 / a 2 , D = 1 / 0.88 .

Fig. 4
Fig. 4

Transmission spectrum of a CDRR of structure C in which ϕ sp departs from the split modes significantly. t 1 = 0.994 , t 2 = 0.96 , a 1 , C = 0.9 , a 2 , C = 1.1 .

Fig. 5
Fig. 5

Comparison of output pulses from two symmetrical CDRRs. The solid, dashed, and dashed–dotted curves indicate the pulses passing the straight waveguide only, output by structures C and D, respectively. The effective indices of the waveguide and the rings are all 3. The radii of the rings and the length of the waveguide are 300 μm and 1 cm , respectively. Other parameters are the same as in Fig. 3b.

Tables (2)

Tables Icon

Table 1 Criteria of Transmission Responses of CDRRs

Tables Icon

Table 2 Conditions for Transmission Response and Group Velocity Control of Structures C and D

Equations (28)

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

τ 1 ( ϕ 1 ) = t 1 a 1 exp ( i ϕ 1 ) 1 t 1 a 1 exp ( i ϕ 1 ) = | τ 1 | exp [ i ϕ 1 ( eff ) ] ,
τ 2 ( ϕ 1 , ϕ 2 ) = t 2 a 2 τ 1 exp ( i ϕ 2 ) 1 t 2 a 2 τ 1 exp ( i ϕ 2 ) = | τ 2 | exp [ i ϕ 2 ( eff ) ] ,
ϕ 1 ( eff ) = tan 1 [ a 1 ( t 1 2 1 ) sin ϕ 1 t 1 ( 1 + a 1 2 ) a 1 ( 1 + t 1 2 ) cos ϕ 1 ] ,
ϕ 2 ( eff ) = tan 1 [ a 2 ( t 2 2 1 ) | τ 1 | sin ( ϕ 1 ( eff ) + ϕ 2 ) t 2 ( 1 + a 2 2 | τ 1 | 2 ) a 2 | τ 1 | ( 1 + t 2 2 ) cos ( ϕ 1 ( eff ) + ϕ 2 ) ] ,
| τ 2 ( a 1 , a 2 ) | = 1 | τ 2 ( 1 a 1 , 1 a 2 ) | ,
ϕ 2 ( eff ) ( a 1 , a 2 ) = ϕ 2 ( eff ) ( 1 a 1 , 1 a 2 ) .
d n ϕ 2 ( eff ) ( a 1 , a 2 ) d ϕ 2 n = d n ϕ 2 ( eff ) ( 1 a 1 , 1 a 2 ) d ϕ 2 n ,
| τ 2 ( ϕ 1 , ϕ 2 ) | 2 = C t 2 2 + B a 2 t 2 + A a 2 2 A t 2 2 a 2 2 + B a 2 t 2 + C ,
A = a 1 2 + t 1 2 2 a 1 t 1 cos ϕ ,
B = 2 [ a 1 t 1 2 t 1 cos ϕ ( 1 a 1 2 ) + a 1 cos 2 ϕ ] ,
C = 1 + a 1 2 t 1 2 2 a 1 t 1 cos ϕ .
ϕ sp = 2 m π ± cos 1 [ t 1 ( 1 + a 1 2 ) / ( 2 a 1 ) ] .
t 1 ( ms ) = 2 a 1 1 + a 1 2 ,
t 1 ( cr ) = 2 t 2 ( 1 a 1 2 a 2 2 ) 2 a 1 t 2 ( 1 a 2 2 ) + a 2 ( 1 a 1 2 ) ( 1 + t 2 2 ) .
t 1 ( ms ) ( a 1 , a 2 ) = t 1 ( ms ) ( 1 a 1 , 1 a 2 ) ,
t 1 ( cr ) ( a 1 , a 2 ) = t 1 ( cr ) ( 1 a 1 , 1 a 2 ) .
t 1 ( cr ) t 1 ( ms ) = 2 ( 1 a 1 2 ) [ ( t 2 a 1 a 2 ) ( 1 t 2 a 1 a 2 ) ] ( 1 + a 1 2 ) [ 2 a 1 t 2 ( 1 a 2 2 ) + a 2 ( 1 a 1 2 ) ( 1 + t 2 2 ) ] .
Δ τ = a 2 ( a 1 | τ 1 , 0 | ) ( 1 t 2 2 ) ( 1 t 2 a 2 | τ 1 , 0 | ) ( 1 t 2 a 1 a 2 ) .
ϕ = 2 m π ± cos 1 ( p ± q s ) ,
p = a 1 t 2 [ ( 1 a 1 2 a 2 2 ) + t 1 2 ( a 1 2 a 2 2 ) ] ,
q = a 1 2 t 2 ( t 1 2 1 ) { t 2 [ a 2 2 t 1 2 ( 1 a 1 2 ) 2 + a 1 2 t 1 2 ( 1 a 2 2 ) 2 ( 1 a 1 2 a 2 2 ) 2 ] + a 1 a 2 t 1 2 ( 1 + t 2 2 ) ( 1 a 1 2 ) ( 1 a 2 2 ) } ,
s = 2 a 1 2 ( 1 a 2 2 ) t 1 t 2 .
d ϕ 2 ( eff ) d ϕ 2 = ϕ 2 ( eff ) ϕ 2 [ 1 + d ϕ 1 ( eff ) d ϕ 2 ] ,
ϕ 2 ( eff ) ϕ 2 = a 2 | τ 1 | ( t 2 2 1 ) [ t 2 ( 1 + a 2 2 | τ 1 | 2 ) cos ( ϕ 1 ( eff ) + ϕ 2 ) a 2 | τ 1 | ( 1 + t 2 2 ) ] [ 1 + a 2 2 | τ 1 | 2 t 2 2 2 a 2 | τ 1 | t 2 cos ( ϕ 1 ( eff ) + ϕ 2 ) ] [ t 2 2 + a 2 2 | τ 1 | 2 2 a 2 | τ 1 | t 2 cos ( ϕ 1 ( eff ) + ϕ 2 ) ] ,
d ϕ 1 ( eff ) d ϕ 2 = a 1 ( t 1 2 1 ) [ t 1 ( 1 + a 1 2 ) cos ϕ 1 a 1 ( 1 + t 1 2 ) ] [ a 1 2 + t 1 2 2 a 1 t 1 cos ϕ 1 ] [ 1 + a 1 2 t 1 2 2 a 1 t 1 cos ϕ 1 ] .
ϕ 2 ( eff ) ϕ 2 | ϕ = 0 = a 2 | τ 1 | ( t 2 2 1 ) [ t 2 ( 1 + a 2 2 | τ 1 | 2 ) cos ϕ 1 ( eff ) a 2 | τ 1 | ( 1 + t 2 2 ) ] [ 1 + a 2 2 | τ 1 | 2 t 2 2 2 a 2 | τ 1 | t 2 cos ϕ 1 ( eff ) ] [ t 2 2 + a 2 2 | τ 1 | 2 2 a 2 | τ 1 | t 2 cos ϕ 1 ( eff ) ] ,
ϕ 2 ( eff ) ϕ 2 | ϕ = ϕ sp = a 1 a 2 ( t 2 2 1 ) ( t 2 a 1 a 2 ) ( 1 t 2 a 1 a 2 ) .
A ( t ) = exp ( t 2 t d 2 ) exp ( i 2 π c t / λ 0 ) ,

Metrics