Abstract

The operation of resonant channel drop filters is analyzed using coupled-mode theory. The resonator is chosen to support a single standing-wave mode, and, in the ideal case, one can realize 100% in-plane channel transfer by properly applying mirror boundaries to the waveguides. The presence of the mirrors causes the resonant frequency to shift, and the system Q factor also changes accordingly. The two variables are related by a closed curve depending on the phase introduced by the reflection and wave propagation between the two ports. When one works on different regions of the curve, the system can be tuned to work at different resonant frequencies with minimum Q-factor variations or vice versa. The mirror can be frequency selective. The same single-mode cavity can be used as a resonant mirror to terminate the waveguide. The combined system is analyzed, and we find the conditions to achieve 100% channel transfer as well as to maintain a simple Lorentzian line shape of the transmission spectra. The analysis is verified by two-dimensional (2D) finite-difference time-domain simulations in 2D hexagonal photonic crystals.

© 2006 Optical Society of America

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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. M. Qiu and B. Jaskorzynska, "A design of a channel drop filter in a two-dimensional triangular photonic crystal," Appl. Phys. Lett. 83, 1074-1076 (2003).
    [Crossref]
  4. M. Qiu, "Ultra-compact optical filter in two-dimensional photonic crystal," Electron. Lett. 40, 539-540 (2004).
    [Crossref]
  5. S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
    [Crossref] [PubMed]
  6. B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
    [Crossref] [PubMed]
  7. A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, "Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 2690-2692 (2001).
    [Crossref]
  8. B. K. Min, J. E. Kim, and H. Y. Park, "High-efficiency surface-emitting channel drop filters in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 86, 11106 (2005).
    [Crossref]
  9. 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]
  10. K. H. Hwang and G. H. 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]
  11. 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]
  12. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).
    [Crossref]
  13. 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]
  14. Y. Xu, Y. Li, E. K. Lee, and A. Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E 62, 7389-7404 (2000).
    [Crossref]
  15. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
    [Crossref]
  16. M. Qiu, F2P, "Finite-difference time-domain 2D simulator for photonic devices," http://www.imit.kth.se/info/FOFU/PC/F2P.
  17. W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
    [Crossref]

2005 (3)

2004 (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. Qiu, "Ultra-compact optical filter in two-dimensional photonic crystal," Electron. Lett. 40, 539-540 (2004).
[Crossref]

2003 (2)

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[Crossref] [PubMed]

M. Qiu and B. Jaskorzynska, "A design of a channel drop filter in a two-dimensional triangular photonic crystal," Appl. Phys. Lett. 83, 1074-1076 (2003).
[Crossref]

2001 (2)

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, "Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 2690-2692 (2001).
[Crossref]

2000 (2)

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[Crossref] [PubMed]

Y. Xu, Y. Li, E. K. Lee, and A. Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E 62, 7389-7404 (2000).
[Crossref]

1999 (1)

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]

1998 (1)

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).
[Crossref]

1996 (1)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[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]

Asano, T.

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[Crossref] [PubMed]

Chutinan, A.

A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, "Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 2690-2692 (2001).
[Crossref]

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[Crossref] [PubMed]

Fan, S.

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, and J. D. Joannopoulos, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[Crossref]

Guo, W. H.

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

Haus, H. A.

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]

Huang, Y. Z.

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

Hwang, K. H.

Imada, M.

A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, "Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 2690-2692 (2001).
[Crossref]

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[Crossref] [PubMed]

Jaskorzynska, B.

M. Qiu and B. Jaskorzynska, "A design of a channel drop filter in a two-dimensional triangular photonic crystal," Appl. Phys. Lett. 83, 1074-1076 (2003).
[Crossref]

Joannopoulos, J. D.

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, and J. D. Joannopoulos, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[Crossref]

John, S.

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

Khan, M. J.

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, "High-efficiency surface-emitting channel drop filters in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 86, 11106 (2005).
[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]

Lee, E. K.

Y. Xu, Y. Li, E. K. Lee, and A. Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E 62, 7389-7404 (2000).
[Crossref]

Li, W. J.

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

Li, Y.

Y. Xu, Y. Li, E. K. Lee, and A. Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E 62, 7389-7404 (2000).
[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]

Min, B. K.

B. K. Min, J. E. Kim, and H. Y. Park, "High-efficiency surface-emitting channel drop filters in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 86, 11106 (2005).
[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]

Mochizuki, M.

A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, "Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 2690-2692 (2001).
[Crossref]

Noda, S.

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[Crossref] [PubMed]

A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, "Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 2690-2692 (2001).
[Crossref]

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[Crossref] [PubMed]

Park, H. Y.

B. K. Min, J. E. Kim, and H. Y. Park, "High-efficiency surface-emitting channel drop filters in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 86, 11106 (2005).
[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]

Qiu, M.

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]

M. Qiu, "Ultra-compact optical filter in two-dimensional photonic crystal," Electron. Lett. 40, 539-540 (2004).
[Crossref]

M. Qiu and B. Jaskorzynska, "A design of a channel drop filter in a two-dimensional triangular photonic crystal," Appl. Phys. Lett. 83, 1074-1076 (2003).
[Crossref]

M. Qiu, F2P, "Finite-difference time-domain 2D simulator for photonic devices," http://www.imit.kth.se/info/FOFU/PC/F2P.

Song, B. S.

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[Crossref] [PubMed]

Song, G. H.

Villeneuve, P. R.

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, and J. D. Joannopoulos, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[Crossref]

Xu, Y.

Y. Xu, Y. Li, E. K. Lee, and A. Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E 62, 7389-7404 (2000).
[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.

Y. Xu, Y. Li, E. K. Lee, and A. Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E 62, 7389-7404 (2000).
[Crossref]

Zhang, Z.

Appl. Phys. Lett. (3)

M. Qiu and B. Jaskorzynska, "A design of a channel drop filter in a two-dimensional triangular photonic crystal," Appl. Phys. Lett. 83, 1074-1076 (2003).
[Crossref]

A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, "Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 2690-2692 (2001).
[Crossref]

B. K. Min, J. E. Kim, and H. Y. Park, "High-efficiency surface-emitting channel drop filters in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 86, 11106 (2005).
[Crossref]

Electron. Lett. (1)

M. Qiu, "Ultra-compact optical filter in two-dimensional photonic crystal," Electron. Lett. 40, 539-540 (2004).
[Crossref]

IEEE J. Quantum Electron. (1)

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]

IEEE Microw. Wirel. Compon. Lett. (1)

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

Nature (1)

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[Crossref] [PubMed]

Opt. Commun. (1)

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]

Opt. Express (2)

Phys. Rev. B (1)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[Crossref]

Phys. Rev. E (1)

Y. Xu, Y. Li, E. K. Lee, and A. Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E 62, 7389-7404 (2000).
[Crossref]

Phys. Rev. Lett. (3)

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]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).
[Crossref]

Science (1)

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[Crossref] [PubMed]

Other (1)

M. Qiu, F2P, "Finite-difference time-domain 2D simulator for photonic devices," http://www.imit.kth.se/info/FOFU/PC/F2P.

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Figures (12)

Fig. 1
Fig. 1

Four-port system with a resonator placed between two waveguides. The resonator supports a single standing-wave mode. The system is symmetric with respect to the planes Σ 1 , 2 , shown as the gray dashed lines.

Fig. 2
Fig. 2

Four-port resonant system with one waveguide terminated by a mirror.

Fig. 3
Fig. 3

Four-port system with mirror boundaries applied to ports 2 and 3.

Fig. 4
Fig. 4

(Color online) Normalized transmission at port 4 versus α and ω under the assumptions Q e = Q 0 10 = 2 × 10 3 and ρ = 1 .

Fig. 5
Fig. 5

(Color online) (a) Reflection (solid curve) and transmission (dashed curve) spectra of the channel with central frequency ω 0 under the assumptions Γ 1 = Γ 2 , d 1 = d 2 , ρ = 1 , and Q o Q e = 10 . (b) When α = π , the channel is totally reflected back. Light does not tunnel into the cavity and therefore undergoes no cavity loss.

Fig. 6
Fig. 6

Combined system with a single-mode resonator placed between two waveguides on the left. Port 2 (5) is connected to a resonant mirror by using the same single-mode cavity. Port 3 is terminated using an ordinary mirror that reflects all frequencies.

Fig. 7
Fig. 7

(Color online) Power transfer at port 8 versus frequency ( ω ω 02 ) ω 02 and relative detuning ( ω 02 ω 01 ) ω 02 between the two cavities under the assumptions Q e = Q 0 10 = 2 × 10 3 and ρ = 0.9 .

Fig. 8
Fig. 8

(Color online) Four-port system realized in a 2D photonic crystal. The mirror is applied by one’s terminating the waveguide with a crystal lattice. The cavity A is constructed by one’s increasing the radius of the central airhole to 0.55 L , and it provides a monopole mode that is symmetric in both in-plane directions.

Fig. 9
Fig. 9

(Color online) (a) Intensity spectrum of the transferred signal at port 4. (b) Detailed spectrum around the central frequency. The curve is plotted using Eq. (29), and the circles are data obtained from FDTD simulations.

Fig. 10
Fig. 10

(Color online) (a) System Q factor versus resonant frequencies. The curve is plotted using Eq. (33), and the circles represent the data from FDTD simulations. (b) H z field distribution when α = 2 k π + π . A point Gaussian pulse is placed inside the cavity as light source.

Fig. 11
Fig. 11

(Color online) Port 4 is terminated by a resonant mirror A 2 , which is also a single-mode cavity. The dielectric constant of the material inside hole A 2 is tuned so that the cavity mode has the same resonant frequency and decay rate into W1 as A 1 .

Fig. 12
Fig. 12

(Color online) Intensity spectra of the dropped signal at port 6 and the transferred signal at port 8. The solid curves are plotted using Eqs. (57, 58). The circles are data obtained from FDTD simulations.

Equations (61)

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

d a d t = ( j ω 0 1 τ o 1 τ e 1 τ e ) a + κ 1 exp ( j β d 2 ) S + 1 + κ 2 exp ( j β d 2 ) S + 2 + κ 3 exp ( j β d 2 ) S + 3 + κ 4 exp ( j β d 2 ) S + 4 ,
Γ = ρ exp ( j ϕ ) , 0 < ρ 1 .
s 1 = exp ( j β d ) s + 2 exp ( j β d 2 ) κ 2 * a ,
s 2 = exp ( j β d ) s + 1 exp ( j β d 2 ) κ 1 * a ,
s 3 = exp ( j β d ) s + 4 exp ( j β d 2 ) κ 4 * a ,
s 4 = exp ( j β d ) s + 3 exp ( j β d 2 ) κ 3 * a ,
β = β ,
τ e = τ e ,
κ i = 1 τ e exp ( j θ i ) , i = 1 , 2 , 3 , 4 .
S 1 = exp [ j ( θ 1 θ 2 β d ) ] Γ exp [ j ( θ 3 θ 4 β d ) + 2 + τ e τ o + j ( ω ω 0 ) τ e ,
S 2 = exp ( j β d ) { 1 1 Γ exp [ j ( θ 3 θ 4 β d ) ] + 2 + τ e τ o + j ( ω ω 0 ) τ e } ,
S 3 = exp [ j ( θ 1 θ 4 β d ) ] Γ exp [ j ( θ 3 θ 4 β d ) ] + 2 + τ e τ o + j ( ω ω 0 ) τ e ,
S 4 = exp [ j ( θ 1 θ 3 β d ) ] 1 + Γ exp [ j ( θ 3 θ 4 β d ) ] Γ exp [ j ( θ 3 θ 4 β d ) ] + 2 + τ e τ o + j ( ω ω 0 ) τ e .
s 1 2 = 1 ( 2 + τ e τ o + cos α ) 2 + τ e 2 [ ω ( ω 0 sin α τ e ) ] 2 ,
s 2 2 = ( 1 + τ e τ o + cos α ) 2 + τ e 2 [ ω ( ω 0 sin α τ e ) ] 2 ( 2 + τ e τ o + cos α ) 2 + τ e 2 [ ω ( ω 0 sin α τ e ) ] 2 ,
s 3 2 = s 1 2 ,
s 4 2 = 2 ( 1 + cos α ) ( 2 + τ e τ o + cos α ) 2 + τ e 2 [ ω ( ω 0 sin α τ e ) ] 2 .
a = τ e cos α + τ e τ o + 2 + j [ ω ( ω 0 sin α τ e ) ] τ e .
Q s = ω 0 τ e + sin α 2 1 2 + τ e τ o + cos α .
Q s = Q e 2 + Q e Q o + cos α .
d a d t = ( j ω 0 1 τ o 1 τ e 1 τ e ) a + κ 1 exp ( j β d 1 ) s + 1 + κ 2 exp ( j β d 2 ) s + 2 + κ 3 exp ( j β d 1 ) s + 3 + κ 4 exp ( j β d 2 ) s + 4 ,
s 1 = exp [ j β ( d 1 + d 2 ) ] s + 2 exp ( j β d 1 ) κ 2 * a ,
s 2 = exp [ j β ( d 1 + d 2 ) ] s + 1 exp ( j β d 2 ) κ 1 * a ,
s 3 = exp [ j β ( d 1 + d 2 ) ] s + 4 exp ( j β d 1 ) κ 4 * a ,
s 4 = exp [ j β ( d 1 + d 2 ) ] s + 3 exp ( j β d 2 ) κ 3 * a .
s 1 2 = [ cos α 1 ρ 1 ( τ e τ o + ρ 2 cos α 2 ) ] 2 + τ e 2 ρ 1 2 [ ω ( ω 0 + sin α 1 + ρ 1 ρ 2 sin α 2 τ e ρ 1 ) ] 2 ( 2 + τ e τ o + ρ 1 cos α 1 + ρ 2 cos α 2 ) 2 + τ e 2 [ ω ( ω 0 + ρ 1 sin α 1 + ρ 2 sin α 2 τ e ) ] 2 ,
s 4 2 = ( 1 + 2 ρ 1 cos α 1 + ρ 1 2 ) ( 1 + 2 ρ 2 cos α 2 + ρ 2 2 ) ( 2 + τ e τ o + ρ 1 cos α 1 + ρ 2 cos α 2 ) 2 + τ e 2 [ ω ( ω 0 + ρ 1 sin α 1 + ρ 2 sin α 2 τ e ) ] 2 ,
s 1 2 = ( cos α ρ τ e τ o ρ 2 cos α ) 2 + τ e 2 ρ 2 { ω [ ω 0 + ( 1 + ρ 2 ) sin α τ e ρ ] } 2 ( 2 + τ e τ o + 2 ρ cos α ) 2 + τ e 2 [ ω ( ω 0 + 2 ρ sin α τ e ) ] 2 ,
s 4 2 = ( 1 + 2 ρ cos α + ρ 2 ) 2 ( 2 + τ e τ o + 2 ρ cos α ) 2 + τ e 2 [ ω ( ω 0 + 2 ρ sin α τ e ) ] 2 ,
a = exp ( j β d 1 ) 1 τ e 1 + ρ exp ( j α ) ( 2 + τ e τ o + 2 ρ cos α ) + j τ e [ ω ( ω 0 + 2 ρ sin α τ e ) ] .
δ ω = 2 ρ sin α τ e = ω 0 Q e ρ sin α .
Q s = ω 0 τ e + 2 ρ sin α 2 ( 2 + τ e τ o + 2 ρ cos α ) = Q e + ρ sin α 2 + Q e Q o + 2 ρ cos α Q e 2 + Q e Q o + 2 ρ cos α .
( δ ω ω 0 Q e ) 2 + ( Q e 2 Q s Q e 2 Q o 1 ) 2 = ρ 2 .
s 1 2 = ( cos α ρ τ e τ o ρ 2 cos α ) 2 + ( 1 + ρ 2 ) 2 sin 2 α ( 2 + τ e τ o + 2 ρ cos α ) 2 + 4 ρ 2 sin 2 α ,
s 4 2 = ( 1 + 2 ρ cos α + ρ 2 ) 2 ( 2 + τ e τ o + 2 ρ cos α ) 2 + 4 ρ 2 sin 2 α .
s 1 2 = 1 cos α 2 ,
s 4 2 = 1 + cos α 2 .
d a 1 d t = ( j ω 0 1 τ o 2 τ e ) a 1 + κ 1 exp ( j β d 1 2 ) S + 1 + κ 2 exp ( j β d 1 2 ) S + 2 κ 3 exp ( j β d 1 2 ) S + 3 + κ 4 exp ( j β d 1 2 ) S + 4 ,
d a 2 d t = ( j ω 0 1 τ o 1 τ e ) a 2 + κ 5 exp ( j β d 2 2 ) s + 5 + κ 6 exp ( j β d 2 2 ) s + 6 .
s 1 = exp ( j β d 1 ) s + 2 exp ( j β d 1 2 ) κ 2 * a 1 ,
s 2 = exp ( j β d 1 ) s + 1 exp ( j β d 1 2 ) κ 1 * a 1 ,
s 3 = exp ( j β d 1 ) s + 4 exp ( j β d 1 2 ) κ 4 * a 1 ,
s 4 = exp ( j β d 1 ) s + 3 exp ( j β d 1 2 ) κ 3 * a 1 ,
s 5 = exp ( j β d 2 ) s + 6 exp ( j β d 2 2 ) κ 6 * a 2 ,
s 6 = exp ( j β d 2 ) s + 5 exp ( j β d 2 2 ) κ 5 * a 2 ,
s 7 = exp ( j β d 2 ) s + 8 ,
s 8 = exp ( j β d 2 ) s + 7 .
s ± 2 = s 5 ,
s 4 = s ± 7 .
s 1 = exp ( 2 j β d 1 ) 1 τ e ρ exp ( j ϕ ) τ e + exp [ j ( β d 1 + θ 5 θ 6 ) ] ( S 1 τ e ) + S exp [ j β ( 2 d 1 + d 2 + θ 1 θ 2 ) ] ( 1 τ e ) 2 exp [ j ( ϕ + β d 2 + θ 3 θ 4 ) ] ρ τ e S exp [ j β ( d 1 + d 2 ) ] S ( S + 1 τ e ) ,
s 6 = exp ( 2 j β d 1 ) { exp [ j ( ϕ β d 1 + θ 3 θ 4 ) ] ρ τ e + S } ( S 1 τ e ) ( 1 τ e ) 2 exp [ j ( ϕ + β d 2 + θ 3 θ 4 ) ] ρ τ e S exp [ j β ( d 1 + d 2 ) ] S ( S + 1 τ e ) ,
s 8 = exp [ j ( β d 1 + β d 2 + θ 1 θ 3 ) ] 1 τ e { 1 + ρ exp [ j ( ϕ β d 1 + θ 3 θ 4 ) ] } { 1 τ e + exp [ j β ( d 1 + d 2 ) ] S } ( 1 τ e ) 2 exp [ j ( ϕ + β d 2 + θ 3 θ 4 ) ] ρ τ e S exp [ j β ( d 1 + d 2 ) ] S ( S + 1 τ e ) ,
S = 1 τ o + 1 τ e + j ( ω ω 0 ) .
ρ = 1 ,
ϕ β d 1 Δ θ = 2 k π ,
β ( d 1 + d 2 ) = 2 k π + π ,
s 6 2 = ( τ e τ o ) 2 + τ e 2 ( ω ω 0 ) 2 ( 2 + τ e τ o ) 2 + τ e 2 ( ω ω 0 ) 2 ,
s 8 2 = 4 ( 2 + τ e τ o ) 2 + τ e 2 ( ω ω 0 ) 2 .
Q s = ω 0 τ e 2 ( 2 + τ e τ o ) = Q e 2 + Q e Q o .
s 8 2 = ( 1 + ρ ) ( S 2 + 1 τ e ) 1 τ e + S 2 ( 1 + ρ + τ e S 1 ) 2 ,
S 1 , 2 = 1 τ e + 1 τ o + j ( ω ω 01 , 02 ) .

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