Abstract

The transfer characteristics of parallel-cascaded arrays of generalized two-port microring networks with interstage phase shifts are investigated for potential applications in spectral engineering. It is found that these structures can exactly realize optical transfer functions with quadrantal symmetric zeros if the microring networks are even-symmetric. The proposed filter architectures have advantages in that the microring networks have all-positive coupling coefficients, and the array can always be reduced to the canonical form that contains only one interstage π phase-shift element. A method for exactly synthesizing cascaded symmetric microring networks based on the transfer matrix factorization technique is presented, and an example of an eighth-order generalized Chebyshev filter is given to illustrate the application of these microring lattice architectures for realizing very high-order optical transfer functions.

© 2008 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, “High order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12, 320-322 (2000).
    [CrossRef]
  2. G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248-1254 (1999).
    [CrossRef]
  3. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
    [CrossRef]
  4. 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]
  5. 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]
  6. 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]
  7. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65-71 (2007).
    [CrossRef]
  8. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447-1449 (1995).
    [CrossRef]
  9. A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20, 296-303 (2002).
    [CrossRef]
  10. V. Van, “Circuit-based method for synthesizing serially-coupled microring filters,” J. Lightwave Technol. 24, 2912-2919 (2006).
    [CrossRef]
  11. V. Van, “Synthesis of elliptic optical filters using mutually coupled microring resonators,” J. Lightwave Technol. 25, 584-590 (2007).
    [CrossRef]
  12. K. Jinguji, “Synthesis of coherent two-port optical delay-line circuit with ring waveguides,” J. Lightwave Technol. 14, 1882-1898 (1996).
    [CrossRef]
  13. C. K. Madsen, “Efficient architectures for exactly realizing optical filters with optimum bandpass design,” IEEE Photon. Technol. Lett. 10, 1136-1138 (1998).
    [CrossRef]
  14. H.-L. Liew and V. Van, “Exact realization of optical transfer functions with symmetric transmission zeros using the double-microring ladder architecture,” J. Lightwave Technol. (to be published).
  15. 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]
  16. J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15, 10196-10206 (2007).
    [CrossRef] [PubMed]
  17. V. Belevitch, Classical Network Theory (Holden-Day, 1968).
  18. A. Fettweis, “On the factorization of transfer matrices of lossless two-ports,” IEEE Trans. Circuit Theory 17, 86-94 (1970).
    [CrossRef]
  19. R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microwave Theory Tech. 47, 433-442 (1999).
    [CrossRef]

2007 (3)

2006 (1)

2004 (2)

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]

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]

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 (1)

2001 (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

2000 (2)

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “High 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]

1999 (2)

G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248-1254 (1999).
[CrossRef]

R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microwave Theory Tech. 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]

1996 (1)

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

1995 (1)

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

1970 (1)

A. Fettweis, “On the factorization of transfer matrices of lossless two-ports,” IEEE Trans. Circuit Theory 17, 86-94 (1970).
[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]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “High 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]

Barwicz, T.

Belevitch, V.

V. Belevitch, Classical Network Theory (Holden-Day, 1968).

Cameron, R. J.

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

Capmany, J.

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]

Domenech, J. D.

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Fettweis, A.

A. Fettweis, “On the factorization of transfer matrices of lossless two-ports,” IEEE Trans. Circuit Theory 17, 86-94 (1970).
[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]

Haus, H.

Ho, P.-T.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “High 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]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “High 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]

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]

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]

Lenz, G.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248-1254 (1999).
[CrossRef]

Liew, H.-L.

H.-L. Liew and V. Van, “Exact realization of optical transfer functions with symmetric transmission zeros using the double-microring ladder architecture,” J. Lightwave Technol. (to be published).

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, “High 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]

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248-1254 (1999).
[CrossRef]

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

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]

A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20, 296-303 (2002).
[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]

A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20, 296-303 (2002).
[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]

Munoz, P.

Muriel, M. A.

Orta, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

Popovic, M.

Rakich, P.

Savi, P.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

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]

Sekaric, L.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Smith, H.

Tascone, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

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]

Trinchero, D.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

Van, V.

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

V. Van, “Circuit-based method for synthesizing serially-coupled microring filters,” J. Lightwave Technol. 24, 2912-2919 (2006).
[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]

H.-L. Liew and V. Van, “Exact realization of optical transfer functions with symmetric transmission zeros using the double-microring ladder architecture,” J. Lightwave Technol. (to be published).

Vlasov, Y.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Watts, M.

Wilson, R. A.

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

Xia, F.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65-71 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

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. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

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

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

IEEE Trans. Circuit Theory (1)

A. Fettweis, “On the factorization of transfer matrices of lossless two-ports,” IEEE Trans. Circuit Theory 17, 86-94 (1970).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

J. Lightwave Technol. (5)

Nat. Photonics (1)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

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]

Other (2)

H.-L. Liew and V. Van, “Exact realization of optical transfer functions with symmetric transmission zeros using the double-microring ladder architecture,” J. Lightwave Technol. (to be published).

V. Belevitch, Classical Network Theory (Holden-Day, 1968).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic of a generalized parallel-cascaded array of N two-port microring networks; (b) example of a three-stage microring array, each stage consisting of a general feed-forward two-port microring network.

Fig. 2
Fig. 2

Examples of odd-symmetric microring networks: (a) feedback network; (b) network with coupling topology that gives rise to coupling between counterpropagating waves in the microrings.

Fig. 3
Fig. 3

Pole-zero diagram of an eighth-order generalized Chebyshev filter with six transmission zeros (black dots) located on the imaginary axis. The gray dots are the zeros of the bar–port transfer function T B ( s ) = R N ( s ) Q N ( s ) .

Fig. 4
Fig. 4

Examples of two possible configurations of even-symmetric microring networks for realizing the eighth-order generalized Chebyshev filter: (a) configuration {8} with eight direct-coupled microring resonators; (b) configuration {6, 2} consisting of a cascade of a sixth-order microring network with a second-order network. The dashed lines indicate the plane of symmetry of the device architectures.

Fig. 5
Fig. 5

Spectral responses at the cross port and bar port of the eighth-order generalized Chebyshev filter. The frequency is normalized to the cutoff frequency ω c . Gray dashed curves are the target filter responses; solid black curves are the computed responses of the cascaded microring configuration {4, 4}. The inset shows a close-up view of the passband.

Fig. 6
Fig. 6

Group delay response at the cross port of the prototype eighth-order generalized Chebyshev filter. The frequency is normalized to the cutoff frequency ω c , which is assumed to be 1 rad s . Gray dashed curve is the prescribed filter response; solid black curve is the computed response of the cascaded microring configuration {4, 4}.

Fig. 7
Fig. 7

Broadband response of the eighth-order generalized Chebyshev filter with a 50 GHz bandwidth using the cascaded microring array configuration {4, 4}. The inset shows a close-up view of the passband. Gray curves in the inset are the target filter responses; black curves are the computed responses of the microring filter with a 5 dB cm propagation loss in the waveguides.

Fig. 8
Fig. 8

Group delay response at the cross port of the eighth-order generalized Chebyshev filter with a 50 GHz bandwidth using the cascaded microring array configuration {4, 4}. Gray dashed curve is the prescribed filter response; solid black curve is the computed response of the microring filter with a 5 dB cm propagation loss in the waveguides.

Tables (2)

Tables Icon

Table 1 Alternative Cascaded Microring Array Architectures and Their Design Parameters for an Eighth-order Generalized Chebyshev Filter

Tables Icon

Table 2 Field Coupling Coefficients of the Cascaded Microring Configuration {4, 4} for an Eighth-order Filter with a 50 GHz Bandwidth

Equations (39)

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

[ a k b k ] = M k [ c k d k ] = 1 G k ( s ) [ F k ( s ) j H k ( s ) σ k j H k ( s ) F k ( s ) σ k ] [ c k d k ] ,
F k ( s ) F k ( s ) + H k ( s ) H k ( s ) = G k ( s ) G k ( s ) .
j H k ( s ) = j H k ( s ) σ k .
Λ k = [ γ k 0 0 1 ] ,
T N = S N S N 1 S 1 = S N T N 1 .
T N = 1 Q N ( s ) [ R N ( s ) j P N ( s ) γ N σ N j P N ( s ) R N ( s ) γ N σ N ] ,
P k ( s ) = γ k H k ( s ) R k 1 ( s ) σ k F k ( s ) P k 1 ( s ) ,
R k ( s ) = γ k F k ( s ) R k 1 ( s ) + σ k H k ( s ) P k 1 ( s ) ,
Q k ( s ) = G k ( s ) Q k 1 ( s ) ,
T X ( s ) = b N a 0 = j P N ( s ) Q N ( s ) ,
T B ( s ) = a N a 0 = R N ( s ) Q N ( s ) .
P k ( s ) = γ k H k ( s ) R k 1 ( s ) + F k ( s ) P k 1 ( s ) ,
R k ( s ) = γ k F k ( s ) R k 1 ( s ) H k ( s ) P k 1 ( s ) ,
where deg F k = deg G k = n k ,
deg H k n k 2 ,
deg R k 1 = deg Q k 1 = N k 1 ,
deg P k 1 N k 1 2 ,
and γ k = ± 1 .
P k 1 ( s ) = F k ( s ) P k ( s ) H k ( s ) R k ( s ) F k 2 ( s ) + H k 2 ( s ) = D k ( s ) G k ( s ) G k ( s ) ,
R k 1 ( s ) = γ k 1 H k ( s ) P k ( s ) + F k ( s ) R k ( s ) F k 2 ( s ) + H k 2 ( s ) = E k ( s ) G k ( s ) G k ( s ) ,
F k ( s ) = i = 0 n k 2 a 2 i s 2 i ,
H k ( s ) = i = 0 m k 2 b 2 i s 2 i ,
P k 1 ( s ) = i = 0 n k 2 a 2 i s 2 i P k ( s ) G k ( s ) G k ( s ) i = 0 m k 2 b 2 i s 2 i R k ( s ) G k ( s ) G k ( s ) .
s 2 i P k ( s ) G k ( s ) G k ( s ) = A 2 i ( s ) + U 2 i ( s ) G k ( s ) G k ( s ) , i = 0 , 1 , 2 , n k 2 ,
s 2 i R k ( s ) G k ( s ) G k ( s ) = B 2 i ( s ) + V 2 i ( s ) G k ( s ) G k ( s ) , i = 0 , 1 , 2 , m k 2 ,
where deg U 2 i ( s ) deg G k ( s ) G k ( s ) 2 = 2 n k 2 ,
and deg V 2 i ( s ) deg G k ( s ) G k ( s ) 2 = 2 n k 2 .
P k 1 ( s ) = i = 0 n k 2 a 2 i A 2 i ( s ) i = 0 m k 2 b 2 i B 2 i ( s ) + W ( s ) G k ( s ) G k ( s ) ,
where W ( s ) = i = 0 n k 2 a 2 i U 2 i ( s ) i = 0 m k 2 b 2 i V 2 i ( s )
i = 0 m k 2 a 2 i u 2 i , j i = 0 m k 2 b 2 i v 2 i , j = u n k , j , j = 0 , 2 , 4 , , 2 n k 2 .
M k Λ k M k 1 = Λ k ( M k 1 Λ k M k ) Λ k .
P N ( s ) = 0.037085 s 6 + 0.208456 s 4 + 0.367288 s 2 + 0.207403 ,
R N ( s ) = s 8 + 2.366164 s 6 + 1.850965 s 4 + 0.511754 s 2 + 0.028058 ,
Q N ( s ) = s 8 + 1.740980 s 7 + 3.881669 s 6 + 4.402764 s 5 + 4.782464 s 4 + 3.489324 s 3 + 2.102237 s 2 + 0.835997 s + 0.209292 .
κ i , j = μ i , j B 2 FSR , κ b = μ b B 2 FSR .
M N M N 1 M k + 1 M k ( Λ M k 1 Λ M k 2 Λ M k 3 Λ M 3 Λ M 2 Λ M 1 ) .
M N M k + 1 M k ( M k 2 Λ M k 1 ) ( M k 3 Λ M k 4 Λ M 3 Λ M 2 Λ M 1 ) .
M N M k + 1 M k M k 2 ( Λ M k 1 Λ M k 4 Λ ) M k 3 ( M k 5 Λ M 3 Λ M 2 Λ M 1 ) .
M N M k + 1 M k M k 2 ( M k 4 Λ M k 1 ) M k 3 ( M k 5 Λ M 3 Λ M 2 Λ M 1 ) .

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