Abstract

General filter architectures based on asynchronously-tuned, coupled-microring resonators are proposed for realizing optical filters with asymmetric spectral responses. Asymmetric filters enable more complex spectral shapes to be realized which can better meet the demands of more advanced applications than symmetric filters. By adjusting individual transmission zeros of the filter transfer function, the transition bands on the low and high-frequency sides of the passband can be separately optimized to achieve an optimum filter response. A method for synthesizing asymmetric spectral responses based on the energy coupling matrix will also be presented along with numerical examples of high-order asymmetric optical filters. These devices represent new microring-based architectures that can be explored for advanced applications in optical spectral shaping and dispersion engineering.

© 2007 Optical Society of America

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References

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  1. J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
    [CrossRef]
  2. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
    [CrossRef]
  3. T. Barwicz, M. Popovic, P. Rakich, M. Watts, H. Haus, E. Ippen and H. Smith, "Microring-resonator-based add-drop filters in SiN: fabrication and analysis," Opt. Express 12, 1437-1442 (2004).
    [CrossRef] [PubMed]
  4. B. E. Little, S. T. Chu, J. V. Hryniewicz and P. P. Absil, "Filter synthesis for periodically coupled microring resonators," Opt. Lett. 25, 344-346 (2000).
    [CrossRef]
  5. R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
    [CrossRef]
  6. K. Jinguji, "Synthesis of coherent two-port optical delay-line circuit with ring waveguides," J. Lightwave Technol. 14, 1882-1898 (1996).
    [CrossRef]
  7. C. K. Madsen, "Efficient architectures for exactly realizing optical filters with optimum bandpass design," IEEE Photon. Technol. Lett. 10, 1136-1138 (1998).
    [CrossRef]
  8. V. Van, "Synthesis of elliptic optical filters using mutually-coupled microring resonators," J. Lightw. Technol. 25, 584-590 (2007).
    [CrossRef]
  9. M. A. Prabhu and V. Van, "General two-dimensional coupled-cavity microring filter architectures," in IEEE Conference on Lasers and Electro-Optics, 2007, paper JTuA31.
  10. M. A. Popovic, "Sharply-defined optical filters and dispersionless delay lines based on loop-coupled resonators and ‘negative’ coupling," in IEEE Conference on Lasers and Electro-Optics, 2007, paper CThP6.
  11. H. C. Bell, "Canonical asymmetric coupled-resonator filters," IEEE Trans. Microwave Theory Technol. 30, 1335-1340 (1982).
    [CrossRef]
  12. A. E. Williams, J. I. Upshur and M. M. Rahman, "Asymmetric response bandpass filter having resonators with minimum couplings," U.S. patent 6337610 (2002).
  13. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).
    [CrossRef]
  14. A. E. Atia and A. E. Williams, "Narrow-bandpass waveguide filters," IEEE Trans. Microwave Theory Technol. 20, 258-265 (1972).
    [CrossRef]
  15. A. E. Atia, A. E. Williams and R. W. Newcomb, "Narrow-band multiple-coupled cavity synthesis," IEEE Trans. Circuits Syst. 21, 649-655 (1974).
    [CrossRef]
  16. M. H. Chen, "Singly terminated pseudo-elliptic function filter," COMSAT Technol. Rev. 7, 527-541 (1977).
  17. R. J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions," IEEE Trans. Microwave Theory Technol. 47, 433-442 (1999).
    [CrossRef]
  18. R. J. Cameron, "Advanced coupling matrix synthesis techniques for microwave filters," IEEE Trans. Microwave Theory Technol. 51, 1-10 (2003).
    [CrossRef]

2007 (1)

V. Van, "Synthesis of elliptic optical filters using mutually-coupled microring resonators," J. Lightw. Technol. 25, 584-590 (2007).
[CrossRef]

2004 (2)

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

T. Barwicz, M. Popovic, P. Rakich, M. Watts, H. Haus, E. Ippen and H. Smith, "Microring-resonator-based add-drop filters in SiN: fabrication and analysis," Opt. Express 12, 1437-1442 (2004).
[CrossRef] [PubMed]

2003 (1)

R. J. Cameron, "Advanced coupling matrix synthesis techniques for microwave filters," IEEE Trans. Microwave Theory Technol. 51, 1-10 (2003).
[CrossRef]

2002 (1)

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

2000 (2)

B. E. Little, S. T. Chu, J. V. Hryniewicz and P. P. Absil, "Filter synthesis for periodically coupled microring resonators," Opt. Lett. 25, 344-346 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

1999 (1)

R. J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions," IEEE Trans. Microwave Theory Technol. 47, 433-442 (1999).
[CrossRef]

1998 (1)

C. K. Madsen, "Efficient architectures for exactly realizing optical filters with optimum bandpass design," IEEE Photon. Technol. Lett. 10, 1136-1138 (1998).
[CrossRef]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).
[CrossRef]

1996 (1)

K. Jinguji, "Synthesis of coherent two-port optical delay-line circuit with ring waveguides," J. Lightwave Technol. 14, 1882-1898 (1996).
[CrossRef]

1982 (1)

H. C. Bell, "Canonical asymmetric coupled-resonator filters," IEEE Trans. Microwave Theory Technol. 30, 1335-1340 (1982).
[CrossRef]

1977 (1)

M. H. Chen, "Singly terminated pseudo-elliptic function filter," COMSAT Technol. Rev. 7, 527-541 (1977).

1974 (1)

A. E. Atia, A. E. Williams and R. W. Newcomb, "Narrow-band multiple-coupled cavity synthesis," IEEE Trans. Circuits Syst. 21, 649-655 (1974).
[CrossRef]

1972 (1)

A. E. Atia and A. E. Williams, "Narrow-bandpass waveguide filters," IEEE Trans. Microwave Theory Technol. 20, 258-265 (1972).
[CrossRef]

Absil, P. P.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz and P. P. Absil, "Filter synthesis for periodically coupled microring resonators," Opt. Lett. 25, 344-346 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Atia, A. E.

A. E. Atia, A. E. Williams and R. W. Newcomb, "Narrow-band multiple-coupled cavity synthesis," IEEE Trans. Circuits Syst. 21, 649-655 (1974).
[CrossRef]

A. E. Atia and A. E. Williams, "Narrow-bandpass waveguide filters," IEEE Trans. Microwave Theory Technol. 20, 258-265 (1972).
[CrossRef]

Barwicz, T.

Bell, H. C.

H. C. Bell, "Canonical asymmetric coupled-resonator filters," IEEE Trans. Microwave Theory Technol. 30, 1335-1340 (1982).
[CrossRef]

Calhoun, L. C.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

Cameron, R. J.

R. J. Cameron, "Advanced coupling matrix synthesis techniques for microwave filters," IEEE Trans. Microwave Theory Technol. 51, 1-10 (2003).
[CrossRef]

R. J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions," IEEE Trans. Microwave Theory Technol. 47, 433-442 (1999).
[CrossRef]

Chen, M. H.

M. H. Chen, "Singly terminated pseudo-elliptic function filter," COMSAT Technol. Rev. 7, 527-541 (1977).

Chu, S. T.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz and P. P. Absil, "Filter synthesis for periodically coupled microring resonators," Opt. Lett. 25, 344-346 (2000).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).
[CrossRef]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).
[CrossRef]

Gill, D.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Grover, R.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

Haus, H.

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).
[CrossRef]

Ho, P.-T.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Hryniewicz, J. V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz and P. P. Absil, "Filter synthesis for periodically coupled microring resonators," Opt. Lett. 25, 344-346 (2000).
[CrossRef]

Ibrahim, T. A.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

Ippen, E.

Jinguji, K.

K. Jinguji, "Synthesis of coherent two-port optical delay-line circuit with ring waveguides," J. Lightwave Technol. 14, 1882-1898 (1996).
[CrossRef]

Johnson, F. G.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

King, O.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz and P. P. Absil, "Filter synthesis for periodically coupled microring resonators," Opt. Lett. 25, 344-346 (2000).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).
[CrossRef]

Madsen, C. K.

C. K. Madsen, "Efficient architectures for exactly realizing optical filters with optimum bandpass design," IEEE Photon. Technol. Lett. 10, 1136-1138 (1998).
[CrossRef]

Newcomb, R. W.

A. E. Atia, A. E. Williams and R. W. Newcomb, "Narrow-band multiple-coupled cavity synthesis," IEEE Trans. Circuits Syst. 21, 649-655 (1974).
[CrossRef]

Popovic, M.

Rakich, P.

Seiferth, F.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Smith, H.

Trakalo, M.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Van, V.

V. Van, "Synthesis of elliptic optical filters using mutually-coupled microring resonators," J. Lightw. Technol. 25, 584-590 (2007).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

Watts, M.

Williams, A. E.

A. E. Atia, A. E. Williams and R. W. Newcomb, "Narrow-band multiple-coupled cavity synthesis," IEEE Trans. Circuits Syst. 21, 649-655 (1974).
[CrossRef]

A. E. Atia and A. E. Williams, "Narrow-bandpass waveguide filters," IEEE Trans. Microwave Theory Technol. 20, 258-265 (1972).
[CrossRef]

Wilson, R. A.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

COMSAT Technol. Rev. (1)

M. H. Chen, "Singly terminated pseudo-elliptic function filter," COMSAT Technol. Rev. 7, 527-541 (1977).

IEEE Photon. Technol. Lett. (3)

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

C. K. Madsen, "Efficient architectures for exactly realizing optical filters with optimum bandpass design," IEEE Photon. Technol. Lett. 10, 1136-1138 (1998).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. E. Atia, A. E. Williams and R. W. Newcomb, "Narrow-band multiple-coupled cavity synthesis," IEEE Trans. Circuits Syst. 21, 649-655 (1974).
[CrossRef]

IEEE Trans. Microwave Theory Technol. (4)

R. J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions," IEEE Trans. Microwave Theory Technol. 47, 433-442 (1999).
[CrossRef]

R. J. Cameron, "Advanced coupling matrix synthesis techniques for microwave filters," IEEE Trans. Microwave Theory Technol. 51, 1-10 (2003).
[CrossRef]

A. E. Atia and A. E. Williams, "Narrow-bandpass waveguide filters," IEEE Trans. Microwave Theory Technol. 20, 258-265 (1972).
[CrossRef]

H. C. Bell, "Canonical asymmetric coupled-resonator filters," IEEE Trans. Microwave Theory Technol. 30, 1335-1340 (1982).
[CrossRef]

J. Lightw. Technol. (3)

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).
[CrossRef]

V. Van, "Synthesis of elliptic optical filters using mutually-coupled microring resonators," J. Lightw. Technol. 25, 584-590 (2007).
[CrossRef]

J. Lightwave Technol. (1)

K. Jinguji, "Synthesis of coherent two-port optical delay-line circuit with ring waveguides," J. Lightwave Technol. 14, 1882-1898 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (3)

M. A. Prabhu and V. Van, "General two-dimensional coupled-cavity microring filter architectures," in IEEE Conference on Lasers and Electro-Optics, 2007, paper JTuA31.

M. A. Popovic, "Sharply-defined optical filters and dispersionless delay lines based on loop-coupled resonators and ‘negative’ coupling," in IEEE Conference on Lasers and Electro-Optics, 2007, paper CThP6.

A. E. Williams, J. I. Upshur and M. M. Rahman, "Asymmetric response bandpass filter having resonators with minimum couplings," U.S. patent 6337610 (2002).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of the most general coupling topology of N asynchronously-tuned microring resonators in which every microring i is assumed to be coupled to every other microring j via energy coupling coefficient µ i,j .

Fig. 2.
Fig. 2.

Examples of filter coupling topologies that are not realizable with microring resonators: (a) a triplet; (b) a quintuplet. In general a circular arrangement consisting of an odd number of microrings will lead to coupling between counter-propagating modes.

Fig. 3.
Fig. 3.

Optimum coupling topology of a 7-pole, 3-zero asymmetric microring filter.

Fig. 4.
Fig. 4.

Spectral responses at the drop port, |, H(s)| 2, and through port, | F(s)| 2, of the 7-pole, 3-zero, 25GHz asymmetric microring filter. Solid red lines are the responses of the synthesized filter; dashed blue lines are the target responses; dashed black line represents the response of a 7th-order, 25GHz Butterworth filter.

Fig. 5.
Fig. 5.

Group delay response of the 7th-order, 25GHz asymmetric microring filter.

Fig. 6.
Fig. 6.

Optimum coupling topology of a 6-pole, 2-zero asymmetric microring filter.

Fig. 7.
Fig. 7.

(a) Spectral responses at the drop port, |H(s)| 2, and through port, |F(s)| 2, of the 6- pole, 2-zero asymmetric microring filter. Solid red lines are the responses of the synthesized filter; dashed blue lines are the exact filter responses obtained from power coupling analysis. (b) Transition at the right band edge of the drop-port response of the filter.

Fig. 8.
Fig. 8.

Group delay response of the 6th-order, 10GHz asymmetric microring filter. Solid red line is the response of the synthesized filter; dashed blue line is the exact response obtained from power coupling analysis.

Fig. 9.
Fig. 9.

Broadband characteristic across one free spectral range of the 6th-order asymmetric microring filter. The insets give the closed-up views of the passbands at λ=1557.4nm and λ=1577.8nm.

Tables (2)

Tables Icon

Table 1. Columns 1 and 2: poles and zeros of the transfer function of the 7th-order asymmetric filter; Columns 3–5: poles and residues of the short-circuit admittance parameters of the equivalent electrical network.

Tables Icon

Table 2. Columns 1 and 2: poles and zeros of the transfer function of the 6th-order asymmetric filter; Columns 3–5: poles and residues of the short-circuit admittance parameters of the equivalent electrical network.

Equations (21)

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( s I + L + j M ) a = b ,
M = [ Δ ω 1 μ 1 , 2 μ 1 , 3 μ 1 , N μ 1 , 2 Δ ω 2 μ 2 , 3 μ 2 , N μ 1 , 3 μ 2 , 3 Δ ω 3 μ 3 , N μ 1 , N μ 2 , N μ 3 , N Δ ω N ] ,
s t s i = j μ o a N s i ,
s r s i = 1 j μ i a 1 s i .
H ( s ) = k = 1 N 2 ( s z k ) k = 1 N ( s p k ) = k = 0 N 2 b k s k k = 0 N a k s k = P ( s ) Q ( s ) .
M = T · Λ · T t ,
Y sc = [ y 11 y 12 y 21 y 22 ] = k = 1 N 1 s j λ k [ ξ 11 ( k ) ξ 12 ( k ) ξ 21 ( k ) ξ 22 ( k ) ] .
μ i 2 = μ o 2 = 2 k = 1 N ξ 11 ( k ) .
T 1 , k = 2 ξ 11 ( k ) μ i ,
T N , k = T 1 , k sgn { imag { ξ 12 ( k ) } } .
F ( s ) F ( s ) = 1 H ( s ) H ( s ) .
Z in ( s ) = 1 + F ( s ) 1 F ( s ) = Q ( s ) + R ( s ) Q ( s ) R ( s ) ,
Z in ( s ) = m ( s ) + n ( s ) Q ( s ) R ( s ) ,
m ( s ) = Re { a 0 + c 0 } + j Im { a 1 + c 1 } s + Re { a 2 + c 2 } s 2 + j Im { a 3 + c 3 } s 3 +
n ( s ) = j Im { a 0 + c 0 } + Re { a 1 + c 1 } s + j Im { a 2 + c 2 } s 2 + Re { a 3 + c 3 } s 3 + ,
y 11 ( s ) = y 22 ( s ) = { n ( s ) m ( s ) , if N is even , m ( s ) n ( s ) , if N is odd ,
y 12 ( s ) = y 21 ( s ) = { P ( s ) m ( s ) , if N is even , P ( s ) n ( s ) , if N is odd .
M = R ( θ r ) · M · R t ( θ r ) .
M = [ 0.8923 22.3716 22.8255 35.1739 29.4535 20.9037 0 67.7296 6.7356 7.3705 18.8095 20.2062 14.0854 55.0406 11.8001 1.1817 27.2934 35.2899 34.8664 4.2710 13.7113 25.1258 28.3751 3.1199 27.3435 61.4286 27.2107 0.8923 ] ,
M = [ 0.8923 42.2088 0 0 0 42.2088 0 4.7966 44.4642 0 0 0 42.2088 33.5104 0 0 0 0 58.6984 41.5723 0 0 35.3269 45.1662 0 2.7650 42.2088 0.8923 ] .
M = [ 0.8391 16.5149 0 0 16.5149 0 7.0767 23.4082 0 0 16.5149 6.5692 0 0 0 29.5817 6.5533 0 4.9346 16.5149 0.8391 ] .

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