Abstract

We propose coupled-mode analysis of a two-dimensional photonic crystal add–drop filter based on ring resonators, which can be applicable to photonic integrated circuits. Mechanism of this proposed add–drop filter is analogous to that of ring resonators resonance, which involves interaction of waveguides and resonators. Simulation results of this filter obtained from the finite-difference time-domain method are consistent with those from the coupled-mode theory. Total transmission up to 99% is verified by both methods.

© 2008 Optical Society of America

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References

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  3. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2004).
  4. S. G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Mater. 51, 5823-5835 (2003).
    [CrossRef]
  5. M. Djavid, F. Monifi, A. Ghaffari, and M. S. Abrishamian, “Hetrostructure wavelength division demultiplexing using photonic crystal ring resonators,” Opt. Commun. 281, 4028-4032 (2008).
    [CrossRef]
  6. K. Oda, N. Tokato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728-736 (1991).
    [CrossRef]
  7. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
    [CrossRef]
  8. A. Sharkawy, S. Shi, and D. W. Prather, “Multichannel wavelength division multiplexing with photonic crystals,” Appl. Opt. 40, 2247-2252 (2001).
    [CrossRef]
  9. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960-963 (1998).
    [CrossRef]
  10. Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express 13, 2596-2604 (2005).
    [CrossRef] [PubMed]
  11. B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59-63 (2004).
    [CrossRef]
  12. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
    [CrossRef]
  13. Z. Qiang, W. Zhou, and R. A. Soref, “Optical add-drop filters based on photonic crystal ring resonators,” Opt. Express 15, 1823-1831 (2007).
    [CrossRef] [PubMed]
  14. V. Dinesh Kumar, T. Srinivas, and A. Selvarajan, “Investigation of ring resonators in photonic crystal circuits,” Photonics Nanostruct. Fundam. Appl. 2, 199-206 (2004).
    [CrossRef]
  15. A. Ghafari, M. Djavid, F. Monifi, and M. S. Abrishamian, “A numeric analysis of photonic crystal tunable add-drop filters based on ring resonators,” Presented at the 20th Annual Meeting of the IEEE Lasers and Electro-optics Society (LEOS, 2007), Lake Buena Vista, Fla., USA, 21-25 Oct. 2007.
  16. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
    [CrossRef]
  17. K. Hwang and G. Song, “Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948-1957 (2005).
    [CrossRef] [PubMed]
  18. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  19. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), pp. 35-253.
  20. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919-933 (1973).
    [CrossRef]

2008 (1)

M. Djavid, F. Monifi, A. Ghaffari, and M. S. Abrishamian, “Hetrostructure wavelength division demultiplexing using photonic crystal ring resonators,” Opt. Commun. 281, 4028-4032 (2008).
[CrossRef]

2007 (1)

2005 (2)

2004 (2)

V. Dinesh Kumar, T. Srinivas, and A. Selvarajan, “Investigation of ring resonators in photonic crystal circuits,” Photonics Nanostruct. Fundam. Appl. 2, 199-206 (2004).
[CrossRef]

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59-63 (2004).
[CrossRef]

2003 (1)

S. G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Mater. 51, 5823-5835 (2003).
[CrossRef]

2001 (1)

1999 (2)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

1998 (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960-963 (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. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

1994 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

1991 (1)

K. Oda, N. Tokato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728-736 (1991).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

Abrishamian, M. S.

M. Djavid, F. Monifi, A. Ghaffari, and M. S. Abrishamian, “Hetrostructure wavelength division demultiplexing using photonic crystal ring resonators,” Opt. Commun. 281, 4028-4032 (2008).
[CrossRef]

A. Ghafari, M. Djavid, F. Monifi, and M. S. Abrishamian, “A numeric analysis of photonic crystal tunable add-drop filters based on ring resonators,” Presented at the 20th Annual Meeting of the IEEE Lasers and Electro-optics Society (LEOS, 2007), Lake Buena Vista, Fla., USA, 21-25 Oct. 2007.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

Chu, S. T.

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

Dinesh Kumar, V.

V. Dinesh Kumar, T. Srinivas, and A. Selvarajan, “Investigation of ring resonators in photonic crystal circuits,” Photonics Nanostruct. Fundam. Appl. 2, 199-206 (2004).
[CrossRef]

Djavid, M.

M. Djavid, F. Monifi, A. Ghaffari, and M. S. Abrishamian, “Hetrostructure wavelength division demultiplexing using photonic crystal ring resonators,” Opt. Commun. 281, 4028-4032 (2008).
[CrossRef]

A. Ghafari, M. Djavid, F. Monifi, and M. S. Abrishamian, “A numeric analysis of photonic crystal tunable add-drop filters based on ring resonators,” Presented at the 20th Annual Meeting of the IEEE Lasers and Electro-optics Society (LEOS, 2007), Lake Buena Vista, Fla., USA, 21-25 Oct. 2007.

Fan, S.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960-963 (1998).
[CrossRef]

Foresi, J.

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

Ghafari, A.

A. Ghafari, M. Djavid, F. Monifi, and M. S. Abrishamian, “A numeric analysis of photonic crystal tunable add-drop filters based on ring resonators,” Presented at the 20th Annual Meeting of the IEEE Lasers and Electro-optics Society (LEOS, 2007), Lake Buena Vista, Fla., USA, 21-25 Oct. 2007.

Ghaffari, A.

M. Djavid, F. Monifi, A. Ghaffari, and M. S. Abrishamian, “Hetrostructure wavelength division demultiplexing using photonic crystal ring resonators,” Opt. Commun. 281, 4028-4032 (2008).
[CrossRef]

Haus, H. A.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960-963 (1998).
[CrossRef]

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

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), pp. 35-253.

Hwang, K.

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Mater. 51, 5823-5835 (2003).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960-963 (1998).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

S. G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Mater. 51, 5823-5835 (2003).
[CrossRef]

Khan, M. J.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Kim, J. E.

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59-63 (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. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Little, B. E.

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

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

Min, B. K.

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59-63 (2004).
[CrossRef]

Monifi, F.

M. Djavid, F. Monifi, A. Ghaffari, and M. S. Abrishamian, “Hetrostructure wavelength division demultiplexing using photonic crystal ring resonators,” Opt. Commun. 281, 4028-4032 (2008).
[CrossRef]

A. Ghafari, M. Djavid, F. Monifi, and M. S. Abrishamian, “A numeric analysis of photonic crystal tunable add-drop filters based on ring resonators,” Presented at the 20th Annual Meeting of the IEEE Lasers and Electro-optics Society (LEOS, 2007), Lake Buena Vista, Fla., USA, 21-25 Oct. 2007.

Oda, K.

K. Oda, N. Tokato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728-736 (1991).
[CrossRef]

Park, H. Y.

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59-63 (2004).
[CrossRef]

Prather, D. W.

Qiang, Z.

Qiu, M.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2004).

Selvarajan, A.

V. Dinesh Kumar, T. Srinivas, and A. Selvarajan, “Investigation of ring resonators in photonic crystal circuits,” Photonics Nanostruct. Fundam. Appl. 2, 199-206 (2004).
[CrossRef]

Sharkawy, A.

Shi, S.

Song, G.

Soref, R. A.

Srinivas, T.

V. Dinesh Kumar, T. Srinivas, and A. Selvarajan, “Investigation of ring resonators in photonic crystal circuits,” Photonics Nanostruct. Fundam. Appl. 2, 199-206 (2004).
[CrossRef]

Toba, H.

K. Oda, N. Tokato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728-736 (1991).
[CrossRef]

Tokato, N.

K. Oda, N. Tokato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728-736 (1991).
[CrossRef]

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960-963 (1998).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

Zhang, Z.

Zhou, W.

Acta Mater. (1)

S. G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Mater. 51, 5823-5835 (2003).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

J. Lightwave Technol. (2)

K. Oda, N. Tokato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728-736 (1991).
[CrossRef]

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

Opt. Commun. (2)

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59-63 (2004).
[CrossRef]

M. Djavid, F. Monifi, A. Ghaffari, and M. S. Abrishamian, “Hetrostructure wavelength division demultiplexing using photonic crystal ring resonators,” Opt. Commun. 281, 4028-4032 (2008).
[CrossRef]

Opt. Express (3)

Photonics Nanostruct. Fundam. Appl. (1)

V. Dinesh Kumar, T. Srinivas, and A. Selvarajan, “Investigation of ring resonators in photonic crystal circuits,” Photonics Nanostruct. Fundam. Appl. 2, 199-206 (2004).
[CrossRef]

Phys. Rev. B (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960-963 (1998).
[CrossRef]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Other (3)

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2004).

A. Ghafari, M. Djavid, F. Monifi, and M. S. Abrishamian, “A numeric analysis of photonic crystal tunable add-drop filters based on ring resonators,” Presented at the 20th Annual Meeting of the IEEE Lasers and Electro-optics Society (LEOS, 2007), Lake Buena Vista, Fla., USA, 21-25 Oct. 2007.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), pp. 35-253.

Supplementary Material (2)

» Media 1: AVI (3582 KB)     
» Media 2: AVI (3038 KB)     

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Figures (4)

Fig. 1
Fig. 1

(a) Add–drop filter with ring resonator. (b) Optical power transmission characteristic of this filter.

Fig. 2
Fig. 2

Single-frame excerpts from video recordings of electric field intensity of add–drop filter, which are achieved by FDTD at (a) λ 1 = 1520 nm Media 1⟩ and (b) λ 2 = 1582 nm Media 2⟩.

Fig. 3
Fig. 3

The structure of our add–drop filter, which is modeled for theoretical analysis.

Fig. 4
Fig. 4

Optical power transmission characteristic of the filter, which is achieved by CMT.

Equations (8)

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

A ( t ) e j ω 0 t e t τ 0 e t τ e .
κ i = 1 τ e , i = 1 , 2 , 3 , 4 .
{ s + 1 = 1 s + 2 , + 3 , + 4 = 0 } .
{ τ 0 = 2 Q 0 ω 0 τ e = 2 Q e ω e } .
{ d A d t = ( j ω 0 1 τ 0 2 τ e ) A + κ 1 e j β d 2 s + 1 + κ 2 e j β d 2 s + 2 + κ 3 e j β d 2 s + 3 + κ 4 e j β d 2 s + 4 s 1 , 2 = e j 2 β d s + 2 , + 1 κ 2 , 1 * e j β d 2 A s 3 , 4 = e j 2 β d s + 4 , + 3 κ 4 , 3 * e j β d 2 A } .
{ T 2 = s 2 s + 1 2 = 1 1 τ e 1 j ( ω ω 0 ) + 1 τ 0 + 2 τ e 2 T 4 = s 4 s + 1 2 = 1 τ e 1 τ e 4 j ( ω ω 0 ) + 1 τ 0 + 2 τ e 2 }
β d = m π + π 2 .
1 τ e 1 + 1 τ e 2 = 1 τ e 3 + 1 τ e 4 = 2 τ e .

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