Abstract

We present analysis and design of microring resonators integrated with reflective elements to obtain custom wavelength-selective devices. We introduce a graphical method that transforms the complicated design problem of the integrated structure into a simple task of designing a reflective element possessing an appropriate reflection profile. Configurations for obtaining a comb mirror, a single peak mirror, an ultranarrow band transmission filter, and a sharp transition mirror are presented as examples.

© 2010 Optical Society of America

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References

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  1. J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
    [CrossRef]
  2. I. Chremmos, and N. Uzunoglu, “Reflective properties of double-ring resonator system coupled to a waveguide,” IEEE Photon. Technol. Lett. 17, 2110–2112 (2005).
    [CrossRef]
  3. G. T. Paloczi, J. Scheuer, and A. Yariv, “Compact microring-based wavelength-selective inline optical reflector,” IEEE Photon. Technol. Lett. 17, 390–392 (2005).
    [CrossRef]
  4. J. Scheuer, G. T. Paloczi, and A. Yariv, “All optically tunable wavelength-selective reflector consisting of coupled polymeric microring resonators,” Appl. Phys. Lett. 87, 251102 (2005).
    [CrossRef]
  5. O. Schwelb, “Band-limited optical mirrors based on ring resonators: analysis and design,” J. Lightwave Technol. 23, 3931–3946 (2005).
    [CrossRef]
  6. Y. Chung, D.-G. Kim, and N. Dagli, “Reflection properties of coupled-ring reflectors,” J. Lightwave Technol. 24, 1865–1874 (2006).
    [CrossRef]
  7. V. Van, “Dual-mode microring reflection filters,” J. Lightwave Technol. 25, 3142–3150 (2007).
    [CrossRef]
  8. C. Vázquez, and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281, 4910–4916 (2008).
    [CrossRef]
  9. H. Sun, A. Chen, and L. R. Dalton, “A reflective microring notch filter and sensor,” Opt. Express 17, 10731–10737 (2009).
    [CrossRef] [PubMed]
  10. Y. M. Kang, and L. L. Goddard, “Semi-analytic modeling of microring resonators with distributed Bragg reflectors,” in 9th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), pp. 123–124 (2009).
  11. Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2009).
    [CrossRef]
  12. B. E. Little, S. T. Chu, and H. A. Haus, “Second-order filtering and sensing with partially coupled traveling waves in a single resonator,” Opt. Lett. 23, 1570–1572 (1998).
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  13. O. Schwelb, and I. Frigyes, “All-optically tunable filters built with discontinuity-assisted ring resonators,” J. Lightwave Technol. 19, 380–386 (2001).
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  14. T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
    [CrossRef]
  15. D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structures with intracavity reflections,” IEEE J. Sel. Top. Quantum Electron. 15, 1420–1426 (2009).
    [CrossRef]
  16. Y. M. Kang, “Semi-analytic simulations of microring resonators with scattering elements,” M.S. thesis, University of Illinois, Urbana, IL (2010), http://hdl.handle.net/2142/15975.
  17. R. Grover, “Indium phosphide based optical micro-ring resonators,” Ph.D. thesis, University of Maryland, College Park, MD (2003), http://hdl.handle.net/1903/261.
  18. D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting in high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. 20, 1835–1837 (1995).
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  19. B. E. Little, J.-P. Laine, and S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Opt. Lett. 22, 4–6 (1997).
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  20. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
    [CrossRef]
  21. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002).
    [CrossRef]
  22. Z. Zhang, M. Dainese, L. Wosinski, and M. Qiu, “Resonance-splitting and enhanced notch depth in SOI ring resonators with mutual mode coupling,” Opt. Express 16, 4621–4630 (2008).
    [CrossRef] [PubMed]
  23. J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
    [CrossRef]
  24. L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, (Wiley-Interscience, 1995).
  25. V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
    [CrossRef]
  26. D. Goldring, U. Levy, and D. Mendlovic, “Highly dispersive micro-ring resonator based on one dimensional photonic crystal waveguide design and analysis,” Opt. Express 15, 3156–3168 (2007).
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2010 (1)

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

2009 (4)

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2009).
[CrossRef]

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structures with intracavity reflections,” IEEE J. Sel. Top. Quantum Electron. 15, 1420–1426 (2009).
[CrossRef]

H. Sun, A. Chen, and L. R. Dalton, “A reflective microring notch filter and sensor,” Opt. Express 17, 10731–10737 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (2)

2006 (1)

2005 (4)

O. Schwelb, “Band-limited optical mirrors based on ring resonators: analysis and design,” J. Lightwave Technol. 23, 3931–3946 (2005).
[CrossRef]

I. Chremmos, and N. Uzunoglu, “Reflective properties of double-ring resonator system coupled to a waveguide,” IEEE Photon. Technol. Lett. 17, 2110–2112 (2005).
[CrossRef]

G. T. Paloczi, J. Scheuer, and A. Yariv, “Compact microring-based wavelength-selective inline optical reflector,” IEEE Photon. Technol. Lett. 17, 390–392 (2005).
[CrossRef]

J. Scheuer, G. T. Paloczi, and A. Yariv, “All optically tunable wavelength-selective reflector consisting of coupled polymeric microring resonators,” Appl. Phys. Lett. 87, 251102 (2005).
[CrossRef]

2004 (1)

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

2002 (1)

2001 (1)

2000 (1)

1998 (1)

1997 (1)

1995 (1)

1993 (1)

V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[CrossRef]

Alexandropoulos, D.

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structures with intracavity reflections,” IEEE J. Sel. Top. Quantum Electron. 15, 1420–1426 (2009).
[CrossRef]

Arbabi, A.

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2009).
[CrossRef]

Chen, A.

Chen, D.-R.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Chremmos, I.

I. Chremmos, and N. Uzunoglu, “Reflective properties of double-ring resonator system coupled to a waveguide,” IEEE Photon. Technol. Lett. 17, 2110–2112 (2005).
[CrossRef]

Chu, S. T.

Chuang, Z.-M.

V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[CrossRef]

Chung, Y.

Coldren, L. A.

V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[CrossRef]

Dagli, N.

Dainese, M.

Dalton, L. R.

Frigyes, I.

Goddard, L. L.

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2009).
[CrossRef]

Goldring, D.

Gorodetsky, M. L.

Hare, J.

Haroche, S.

Haus, H. A.

He, L.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Ilchenko, V. S.

Jayaraman, V.

V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[CrossRef]

Kang, Y. M.

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2009).
[CrossRef]

Kim, D.-G.

Kippenberg, T. J.

Laine, J.-P.

Lefèvre-Seguin, V.

Levy, U.

Li, L.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Little, B. E.

Liu, F.

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

Mendlovic, D.

Ozdemir, S. K.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Paloczi, G. T.

J. Scheuer, G. T. Paloczi, and A. Yariv, “All optically tunable wavelength-selective reflector consisting of coupled polymeric microring resonators,” Appl. Phys. Lett. 87, 251102 (2005).
[CrossRef]

G. T. Paloczi, J. Scheuer, and A. Yariv, “Compact microring-based wavelength-selective inline optical reflector,” IEEE Photon. Technol. Lett. 17, 390–392 (2005).
[CrossRef]

Poon, J. K. S.

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

Pryamikov, A. D.

Qiu, M.

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

Z. Zhang, M. Dainese, L. Wosinski, and M. Qiu, “Resonance-splitting and enhanced notch depth in SOI ring resonators with mutual mode coupling,” Opt. Express 16, 4621–4630 (2008).
[CrossRef] [PubMed]

Raimond, J.-M.

Sandoghdar, V.

Scheuer, J.

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structures with intracavity reflections,” IEEE J. Sel. Top. Quantum Electron. 15, 1420–1426 (2009).
[CrossRef]

J. Scheuer, G. T. Paloczi, and A. Yariv, “All optically tunable wavelength-selective reflector consisting of coupled polymeric microring resonators,” Appl. Phys. Lett. 87, 251102 (2005).
[CrossRef]

G. T. Paloczi, J. Scheuer, and A. Yariv, “Compact microring-based wavelength-selective inline optical reflector,” IEEE Photon. Technol. Lett. 17, 390–392 (2005).
[CrossRef]

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

Schwelb, O.

Spillane, S. M.

Su, Y.

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

Sun, H.

Tian, Y.

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

Tong, Y.

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

Uzunoglu, N.

I. Chremmos, and N. Uzunoglu, “Reflective properties of double-ring resonator system coupled to a waveguide,” IEEE Photon. Technol. Lett. 17, 2110–2112 (2005).
[CrossRef]

Vahala, K. J.

Vainos, N. A.

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structures with intracavity reflections,” IEEE J. Sel. Top. Quantum Electron. 15, 1420–1426 (2009).
[CrossRef]

Van, V.

Vázquez, C.

C. Vázquez, and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281, 4910–4916 (2008).
[CrossRef]

Wang, J.

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

Wang, T.

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

Weiss, D. S.

Wosinski, L.

Xiao, Y.-F.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Yang, L.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Yariv, A.

J. Scheuer, G. T. Paloczi, and A. Yariv, “All optically tunable wavelength-selective reflector consisting of coupled polymeric microring resonators,” Appl. Phys. Lett. 87, 251102 (2005).
[CrossRef]

G. T. Paloczi, J. Scheuer, and A. Yariv, “Compact microring-based wavelength-selective inline optical reflector,” IEEE Photon. Technol. Lett. 17, 390–392 (2005).
[CrossRef]

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

Zhang, Z.

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

Z. Zhang, M. Dainese, L. Wosinski, and M. Qiu, “Resonance-splitting and enhanced notch depth in SOI ring resonators with mutual mode coupling,” Opt. Express 16, 4621–4630 (2008).
[CrossRef] [PubMed]

Zhu, J.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

J. Scheuer, G. T. Paloczi, and A. Yariv, “All optically tunable wavelength-selective reflector consisting of coupled polymeric microring resonators,” Appl. Phys. Lett. 87, 251102 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structures with intracavity reflections,” IEEE J. Sel. Top. Quantum Electron. 15, 1420–1426 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

I. Chremmos, and N. Uzunoglu, “Reflective properties of double-ring resonator system coupled to a waveguide,” IEEE Photon. Technol. Lett. 17, 2110–2112 (2005).
[CrossRef]

G. T. Paloczi, J. Scheuer, and A. Yariv, “Compact microring-based wavelength-selective inline optical reflector,” IEEE Photon. Technol. Lett. 17, 390–392 (2005).
[CrossRef]

J. Lightwave Technol. (4)

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

Nat. Photonics (1)

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Opt. Commun. (2)

T. Wang, Z. Zhang, F. Liu, Y. Tong, J. Wang, Y. Tian, M. Qiu, and Y. Su, “Modeling of quasi-grating sidewall corrugation in SOI microring add-drop filters,” Opt. Commun. 282, 3464–3467 (2009).
[CrossRef]

C. Vázquez, and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281, 4910–4916 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Opt. Quantum Electron. (1)

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2009).
[CrossRef]

Other (4)

Y. M. Kang, “Semi-analytic simulations of microring resonators with scattering elements,” M.S. thesis, University of Illinois, Urbana, IL (2010), http://hdl.handle.net/2142/15975.

R. Grover, “Indium phosphide based optical micro-ring resonators,” Ph.D. thesis, University of Maryland, College Park, MD (2003), http://hdl.handle.net/1903/261.

Y. M. Kang, and L. L. Goddard, “Semi-analytic modeling of microring resonators with distributed Bragg reflectors,” in 9th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), pp. 123–124 (2009).

L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, (Wiley-Interscience, 1995).

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Figures (10)

Fig. 1.
Fig. 1.

A schematic diagram of a microring resonator integrated with a reflective element characterized by the scattering matrix S. The reflective element couples two modes propagating in opposite directions; superscripts + and − are used to denote the fields propagating in the direction of the arrows and the opposite direction, respectively.

Fig. 2.
Fig. 2.

Contour plots of the reflectance of the integrated microring for different values of τ 2 and α 2 on the Θ-r domain. Note the changes in the color code scales for different values of α 2 and τ 2.

Fig. 3.
Fig. 3.

(a) Contour maps of critical reflection coefficient and (b) the resultant reflectance from the integrated microring on the τ 2α 2 plane.

Fig. 4.
Fig. 4.

Contour plot and reflectance spectra for a comb reflector. (a) The plot of |a 1 (Θ,r)|2 overlaid with the reflection profile of the reflective element r(Θ), represented by the dashed line. One can expect a periodic peak reflection at Θ = 2. (b) Reflectance spectra of the FP-MRR for various values of α 2. In each case, the FP reflection coefficients are set to their corresponding critical value r = rc (0).

Fig. 5.
Fig. 5.

Schematic diagrams of single-peak reflector configurations using a DBR grating in the microring.

Fig. 6.
Fig. 6.

Single peak reflector contour plot and its resultant reflectance spectra. (a) Reflection profile of the reflective element that can be used to realize a single peak reflector. (b) Reflectance spectra of the single-peak DBR-MRR configurations.

Fig. 7.
Fig. 7.

Schematic diagram of a DBR-E-MRR. Each DBR mirror element is defined symmetrically by the dashed lines, which gives the extra length Λ for the ring portion.

Fig. 8.
Fig. 8.

An ultranarrow transmission filter configuration obtained from the effective mirror model. (a) The overlaid reflection profile of the etalon. (b) Resultant transmission response. Note that under the lossless condition, the sum of reflection and transmission power at any point is unity.

Fig. 9.
Fig. 9.

The response of ultranarrow transmission filter evaluated numerically by a TMM. (a) The overlaid reflection profile. (b) Resultant transmission response. (inset1) Transmission response as a function of λ. Note the domain is the same as the main figure but that the horizontal scales are different due to nonlinear compression. (inset2) Phase response Θ(λ).

Fig. 10.
Fig. 10.

A sharp transition mirror configuration.

Equations (30)

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

S = ( S 11 S 12 S 21 S 22 ) .
S = ( ire i ψ t t ire i ψ ) e i ϕ ,
b 1 + = τ α t ( 1 + τ 2 ) e i ( θ + ϕ ) + α 2 τ e i 2 ( θ + ϕ ) 1 2 α t τ e i ( θ + ϕ ) + α 2 τ 2 e i 2 ( θ + ϕ )
a 1 = i α r ( 1 τ 2 ) e i ( θ + ϕ ψ ) 1 2 α t τ e i ( θ + ϕ ) + α 2 τ 2 e i 2 ( θ + ϕ ) ,
ϕ = S 12
θ = β L
Θ = ( θ θ 0 ) + ( ϕ ϕ 0 )
r c ( Θ ) = ( 1 α 2 τ 2 ) 2 + 4 α 2 τ 2 sin 2 Θ 1 + α 2 τ 2
a 1 c = α ( 1 τ 2 ) 1 α 2 τ 2 .
r c ( 0 ) = 1 α 2 τ 2 1 + α 2 τ 2 ,
A = a 1 c 2 r c 2 ( 0 ) = α 2 ( 1 τ 2 ) 2 ( 1 + α 2 τ 2 ) 2 ( 1 α 2 τ 2 ) 4 ,
S 11 2 = 4 R int sin 2 β d ( 1 R int ) 2 + 4 R int sin 2 β d
r = 2 R int 1 + R int .
S 12 = ( 1 R int ) e i β d 1 R int e i 2 β d .
r ( Θ ) = { 0 for Θ = 2 m π , m 0 , r c ( 0 ) for Θ = 0 .
S 11 2 N R int sin Θ g Θ g ,
p = Θ g Θ
0 < p L g L + L g 1 .
r ( Θ ) = 2 N R int sin ( p Θ ) p Θ ,
L g = L = 1 2 L t or L g = L t , L = 0 .
S g = ( ± r g it g it g ± r g ) e i ϕ g ,
S 12 = t g 2 e i 2 ϕ g e i β d 1 r g 2 e i 2 ϕ g e i 2 β d
S 12 = arctan [ Γ tan ( β d + ϕ g ) ] + ϕ g + π ,
r 2 = S 11 2 = 4 R g sin 2 ( β d + ϕ g ) ( 1 R g ) 2 + 4 R g sin 2 ( β d + ϕ g ) ,
r ( ϕ ϕ g ) = 2 R g 1 + R g sin ( ϕ ϕ g ) ,
r g = tanh ( N ln 1 ± r int 1 r int )
ϕ g = 2 ( β β 0 ) L e
L e = 1 2 m e Λ ( 1 1 + r int 2 1 2 m e )
m e = tanh ( N ln 1 ± r int 1 r int ) tanh ( ln 1 ± r int 1 r int ) ,
ϕ ϕ g ( d + 2 L e ) Γ ( d + 2 L e ) Γ + 2 L e + L + Λ Θ + ϕ 0 .

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