Abstract

We propose a novel synthesis method for designing flexible, tunable non-periodic filters. It is based on a building block which is presented by first time for these purposes, being the poles position tuned by means of a coupling coefficient and an amplifier gain. We present the device, the design equations and a design of a filter with flat response, 54 dB of crosstalk and less than 1.3 dB of ripple to explain and validate the method. These filters can be used as part of optical cross-connects for selecting channels avoiding very restrictive free spectral ranges. Also we check the correct operation of the device, against fabrication tolerances of coupling coefficients. Frequency dependence of the transfer function poles as a unique feature improving the crosstalk of the device spectral response is also analyzed.

© 2010 OSA

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References

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  1. K. Jinguji and T. Yasui, “Synthesis of One-Input M-Output Optical FIR Lattice Circuits,” J. Lightwave Technol. 26(7), 853–866 (2008).
    [Crossref]
  2. A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Opt. Lett. 26(12), 917–919 (2001).
    [Crossref]
  3. V. Van, “Circuit-Based Method for Synthesizing Serially Coupled Microring Filters,” J. Lightwave Technol. 24(7), 2912–2919 (2006).
    [Crossref]
  4. N. Ngo, “Synthesis of Tunable Optical Waveguide Filters Using Digital Signal Processing Technique,” J. Lightwave Technol. 24(9), 3520–3531 (2006).
    [Crossref]
  5. C. K. Madsen, “General IIR Optical Filter Design for WDM Applications Using All-Pass Filters,” J. Lightwave Technol. 18(6), 860–868 (2000).
    [Crossref]
  6. K. Jinguji, “Synthesis of Coherent Two-Port Optical Delay-Line Circuit with Ring Waveguides,” J. Lightwave Technol. 14(8), 1882–1898 (1996).
    [Crossref]
  7. K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. 25(22), 1508–1509 (1989).
    [Crossref]
  8. V. Van, “Dual-Mode Microring Reflection Filters,” J. Lightwave Technol. 25(10), 3142–3150 (2007).
    [Crossref]
  9. S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. 19(23), 1877–1879 (2007).
    [Crossref]
  10. S. Xiao, M. H. Khan, H. Shen, and M. Qi, “Silicon-on-Insulator Microring Add-Drop Filters With Free Spectral Ranges Over 30 nm,” J. Lightwave Technol. 26(2), 228–236 (2008).
    [Crossref]
  11. Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express 16(6), 4309–4315 (2008).
    [Crossref] [PubMed]
  12. J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. 282(9), 1771–1774 (2009).
    [Crossref]
  13. C. Vázquez and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281(19), 4910–4916 (2008).
    [Crossref]
  14. D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
    [Crossref]
  15. J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
    [Crossref]
  16. W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
    [Crossref]
  17. C. Vazquez, S. Vargas, and J. M. S. Pena, “Sagnac Loop in Ring Resonators for Tunable Optical Filters,” J. Lightwave Technol. 23(8), 2555–2567 (2005).
    [Crossref]
  18. H. Stoll, “Optimally Coupled, GaAs-Distributed Bragg Reflection Lasers,” IEEE Trans. Circ. Syst. 26(12), 1065–1072 (1979).
    [Crossref]
  19. Q. Xu, I. Cremmos, O. Schwelb, and N. Uzunogly, Photonic Microresonator Research and Applications (Springer New York 2010), Chap. 9.
  20. Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004).
    [Crossref] [PubMed]
  21. M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008).
    [Crossref]
  22. J. Proakis, and D. Manolakis, Digital Signal Processing, (Prentice Hall 2006).
    [PubMed]

2009 (1)

J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. 282(9), 1771–1774 (2009).
[Crossref]

2008 (5)

C. Vázquez and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281(19), 4910–4916 (2008).
[Crossref]

S. Xiao, M. H. Khan, H. Shen, and M. Qi, “Silicon-on-Insulator Microring Add-Drop Filters With Free Spectral Ranges Over 30 nm,” J. Lightwave Technol. 26(2), 228–236 (2008).
[Crossref]

Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express 16(6), 4309–4315 (2008).
[Crossref] [PubMed]

K. Jinguji and T. Yasui, “Synthesis of One-Input M-Output Optical FIR Lattice Circuits,” J. Lightwave Technol. 26(7), 853–866 (2008).
[Crossref]

M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008).
[Crossref]

2007 (3)

V. Van, “Dual-Mode Microring Reflection Filters,” J. Lightwave Technol. 25(10), 3142–3150 (2007).
[Crossref]

S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. 19(23), 1877–1879 (2007).
[Crossref]

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

2006 (2)

2005 (2)

C. Vazquez, S. Vargas, and J. M. S. Pena, “Sagnac Loop in Ring Resonators for Tunable Optical Filters,” J. Lightwave Technol. 23(8), 2555–2567 (2005).
[Crossref]

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

2004 (1)

2002 (1)

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[Crossref]

2001 (1)

2000 (1)

1996 (1)

K. Jinguji, “Synthesis of Coherent Two-Port Optical Delay-Line Circuit with Ring Waveguides,” J. Lightwave Technol. 14(8), 1882–1898 (1996).
[Crossref]

1989 (1)

K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. 25(22), 1508–1509 (1989).
[Crossref]

1979 (1)

H. Stoll, “Optimally Coupled, GaAs-Distributed Bragg Reflection Lasers,” IEEE Trans. Circ. Syst. 26(12), 1065–1072 (1979).
[Crossref]

Baets, R.

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

Beausoleil, R. G.

Byun, Y. T.

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

Dagens, B.

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

De La Rue, R. M.

M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008).
[Crossref]

Decobert, J.

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

Fattal, D.

García, J.

J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. 282(9), 1771–1774 (2009).
[Crossref]

Gnan, M.

M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008).
[Crossref]

Habara, K.

K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. 25(22), 1508–1509 (1989).
[Crossref]

Hamacher, M.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[Crossref]

Heidrich, H.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[Crossref]

Jinguji, K.

K. Jinguji and T. Yasui, “Synthesis of One-Input M-Output Optical FIR Lattice Circuits,” J. Lightwave Technol. 26(7), 853–866 (2008).
[Crossref]

K. Jinguji, “Synthesis of Coherent Two-Port Optical Delay-Line Circuit with Ring Waveguides,” J. Lightwave Technol. 14(8), 1882–1898 (1996).
[Crossref]

Khan, M. H.

Lagae, L.

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

Le Gouezigou, O.

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

Lee, S.

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

Lee, W. Y.

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

Macintyre, D. S.

M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008).
[Crossref]

Madsen, C. K.

Make, D.

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

Martí, J.

J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. 282(9), 1771–1774 (2009).
[Crossref]

Martínez, A.

J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. 282(9), 1771–1774 (2009).
[Crossref]

McNab, S. J.

Melloni, A.

Mizumoto, T.

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

Ngo, N.

Ok, S. H.

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

Okuno, M.

K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. 25(22), 1508–1509 (1989).
[Crossref]

Pena, J. M. S.

Qi, M.

Rabus, D. G.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[Crossref]

Roh, J. W.

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

Sasayama, K.

K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. 25(22), 1508–1509 (1989).
[Crossref]

Schwelb, O.

C. Vázquez and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281(19), 4910–4916 (2008).
[Crossref]

Shen, H.

Sorel, M.

M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008).
[Crossref]

Stoll, H.

H. Stoll, “Optimally Coupled, GaAs-Distributed Bragg Reflection Lasers,” IEEE Trans. Circ. Syst. 26(12), 1065–1072 (1979).
[Crossref]

Thoms, S.

M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008).
[Crossref]

Troppenz, U.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[Crossref]

Van, V.

Van Parys, W.

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

Van Thourhout, D.

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

Vargas, S.

S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. 19(23), 1877–1879 (2007).
[Crossref]

C. Vazquez, S. Vargas, and J. M. S. Pena, “Sagnac Loop in Ring Resonators for Tunable Optical Filters,” J. Lightwave Technol. 23(8), 2555–2567 (2005).
[Crossref]

Vazquez, C.

S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. 19(23), 1877–1879 (2007).
[Crossref]

C. Vazquez, S. Vargas, and J. M. S. Pena, “Sagnac Loop in Ring Resonators for Tunable Optical Filters,” J. Lightwave Technol. 23(8), 2555–2567 (2005).
[Crossref]

Vázquez, C.

C. Vázquez and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281(19), 4910–4916 (2008).
[Crossref]

Vlasov, Y. A.

Woo, D. H.

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

Xiao, S.

Xu, Q.

Yang, J. S.

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

Yasui, T.

Electron. Lett. (2)

K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. 25(22), 1508–1509 (1989).
[Crossref]

M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008).
[Crossref]

IEEE Photon. Technol. Lett. (3)

S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. 19(23), 1877–1879 (2007).
[Crossref]

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[Crossref]

W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007).
[Crossref]

IEEE Trans. Circ. Syst. (1)

H. Stoll, “Optimally Coupled, GaAs-Distributed Bragg Reflection Lasers,” IEEE Trans. Circ. Syst. 26(12), 1065–1072 (1979).
[Crossref]

IEEE Trans. Magn. (1)

J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005).
[Crossref]

J. Lightwave Technol. (8)

Opt. Commun. (2)

J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. 282(9), 1771–1774 (2009).
[Crossref]

C. Vázquez and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281(19), 4910–4916 (2008).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Other (2)

Q. Xu, I. Cremmos, O. Schwelb, and N. Uzunogly, Photonic Microresonator Research and Applications (Springer New York 2010), Chap. 9.

J. Proakis, and D. Manolakis, Digital Signal Processing, (Prentice Hall 2006).
[PubMed]

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Figures (7)

Fig. 1
Fig. 1

Schematic of a MIARR.

Fig. 2
Fig. 2

Movement of zeros of the A3/A1 output transfer function, for increasing Ki from 0.05 to 0.95. α = 2dB/cm, γM = γi = γo = 0.05, KM = 0.26, Ko = 0.05, G = 1 and |r(Ω)| = 1.

Fig. 3
Fig. 3

Block diagram of cascaded stages.

Fig. 4
Fig. 4

Spectral response of synthesized Hs[w], desired Hd[w] and gratings transfer functions /r(w)/.

Fig. 5
Fig. 5

Spectral responses of synthesized transfer functions, using the coupling coefficients theoretically calculated, and with random ± 5% variations.

Fig. 6
Fig. 6

Poles module value for two stages MIARR filter of previous section example, for each FSR. FSR 0 correspond to FSR including fG.

Fig. 7
Fig. 7

Spectral responses of MIARR stages filter, fourth order periodic delay line filter, and cascading DG with fourth order filter.

Tables (3)

Tables Icon

Table 1 List of Symbols at Fig. 1

Tables Icon

Table 2 MIARR Parameters Values per Stage; A = 1

Tables Icon

Table 3 MIARR Parameters Values per Stage; G = 1

Equations (25)

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

F R = ( 1 γ M ) ( 1 2 K M )     r ( Ω )
F T = j 2 ( 1 γ M ) ( K M ( 1 K M ) ) 1 / 2     r ( Ω )
φ G ( Ω ) = π 2 [ n b e f ( α b + k b 2 + α b 2 ) ] ( Ω Ω G c )
τ G =     [ n b e f ( α b + k b 2 + α b 2 ) ] ( 1 c )
D G ( z ) = j | r ( Ω ) | e j Δ Ω τ G = j | r ( Ω ) | z τ G τ
A 3 A 1 = ( 1 γ i ) 1 / 2 ( 1 Z c 1 z 1 )     ( 1 Z c 2 z 1 ) ( 1 Z p z 1 )     ( 1 Z p * z 1 )
Z c 1 ( z ) = ( ( 1 γ i ) ( 1 γ o ) ( 1 K o ) G ) 1 / 2 ( 1 γ M ) | r ( Ω ) | e α L r ( 1 K i ) 1 / 2 × × [ ( 2 K i ) ( K M K M 2 ) 1 / 2 + j ( ( 1 K i ) ( 1 2 K M ) 2 K i 2 ( K M K M 2 ) ) 1 / 2 ]
Z c 2 ( z ) = ( ( 1 γ i ) ( 1 γ o ) ( 1 K o ) G ) 1 / 2 ( 1 γ M ) | r ( Ω ) | e α L r ( 1 K i ) 1 / 2 × × [ ( 2 K i ) ( K M K M 2 ) 1 / 2 j ( ( 1 K i ) ( 1 2 K M ) 2 K i 2 ( K M K M 2 ) ) 1 / 2 ]
A 4 a A 1 = ( ( 1 γ i ) ( 1 γ o ) K i K o ) 1 / 2 e α L x ( 1 Z p z 1 ) ( 1 Z p * z 1 ) [ 1 2 ( 1 γ M ) ( ( 1 γ i ) ( 1 γ o ) ) 1 / 2 × × ( ( 1 K i ) ( 1 K o ) ( K M K M 2 ) G ) 1 / 2 | r ( Ω ) | e α L r z 1 ] z τ x τ
A 4 b A 1 = ( 1 γ o ) ( 1 γ M ) ( 1 γ i ) 1 / 2 ( 1 Z p z 1 )     × × ( K i K o ( 1 K o ) ) 1 / 2 ( 1 2 K M )       ​ ​ ​ | r ( Ω ) | e α ( 2 L M L x )     z 2 τ M τ x τ ( 1 Z p * z 1 )
K i < 1 + 1 + 4 Ψ 2 ψ         ;                                                                         ψ = ( K M K M 2 ) ( 1 2 K M ) 2
| Z c | = ( ( 1 γ i ) ( 1 γ o ) ( 1 K o ) G ) 1 / 2 ( 1 γ M ) | r ( Ω ) | e α L r
G 1 / 2 = 1 ( ( 1 γ i ) ( 1 γ o ) ( 1 K o ) ) 1 / 2 ( 1 γ M ) | r ( Ω ) | e α L r
φ z = ± tan 1 [ ( ( 1 K i ) ( 1 2 K M ) 2 K i 2 ( K M K M 2 ) ) 1 / 2 ( 2 K i ) ( K M K M 2 ) 1 / 2 ]
f = f G + ( k ± φ z 2 π ) F S R ;                     k Z
G 1 / 2 = 1 2 ( 1 γ M ) ( ( 1 γ i ) ( 1 γ o ) ( 1 K i ) ( 1 K o ) ( K M K M 2 ) ) 1 / 2 | r ( Ω ) | e α L r
| Z p | = ( 1 γ M ) [ ( 1 γ i ) ( 1 γ o ) ( 1 K i ) ( 1 K o ) G ] 1 / 2 | r ( Ω ) | e α L r
φ p = ± tan 1 [ ( 1 2 K M ) 2 K M ( 1 K M ) ]
G = 1 ( 1 γ M ) [ ( 1 γ i ) ( 1 γ o ) ( 1 K i ) ( 1 K o ) ] 1 / 2 | r ( Ω ) | e α L r
K i = 1 χ 1 + χ
K o = 1 χ 2
with   χ = | Z p | 2 ( 1 γ M ) 2 ( 1 γ i ) ( 1 γ o ) G | r ( Ω G + φ p · F S R ) | 2 e 2 α L r
G min > | Z p | 2 ( 1 γ M ) 2 ( 1 γ i ) ( 1 γ o ) | r ( Ω G + φ p · F S R ) | 2 e 2 α L r
K M = 1 2 ± 1 2 tan 2 ( φ p ) 1 + tan 2 ( φ p )
H d [ z ] = 0.0067 z 2 1 3.4326 z 1 + 4.5967 z 2 2.8336 z 3 + 0.6771 z 4

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