Abstract

For our recently designed continuous-wave and single-frequency ring laser with intra-cavity isolator, we have formulated a rate-equation theory which accounts for two sources of mutual back-scattering between the clockwise and counterclockwise modes, i.e. induced by side-wall irregularities and due to inversion-grating-induced spatial hole burning. With this theory we first confirm that for a ring laser without intra-cavity isolation, from sufficiently large pumping strength on, the inversion-grating-induced bistable operation (i.e. either clockwise or counterclockwise) will overrule the back-reflection-induced coupled-mode operation (i.e. both clockwise and counterclockwise). We then analyze the robustness of unidirectional operation in case of intra-cavity isolation against the intra-cavity back-reflection mechanism and grating-induced mode coupling and derive for this case an explicit expression for the directionality in the presence of external optical feedback, valid for sufficiently strong isolation. The predictions posed in the second reference remain unaltered in the presence of the mode coupling mechanisms here considered.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Unidirectional, widely-tunable and narrow-linewidth heterogeneously integrated III-V-on-silicon laser

Jing Zhang, Yanlu Li, Sören Dhoore, Geert Morthier, and Gunther Roelkens
Opt. Express 25(6) 7092-7100 (2017)

Square-wave oscillations in semiconductor ring lasers with delayed optical feedback

Lilia Mashal, Guy Van der Sande, Lendert Gelens, Jan Danckaert, and Guy Verschaffelt
Opt. Express 20(20) 22503-22516 (2012)

Experimental investigation on feedback insensitivity in semiconductor ring lasers

Song-Sui Li, Vincenzo Pusino, Sze-Chun Chan, and Marc Sorel
Opt. Lett. 43(9) 1974-1977 (2018)

References

  • View by:
  • |
  • |
  • |

  1. B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
    [Crossref]
  2. T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
    [Crossref]
  3. M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
    [Crossref]
  4. A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).
  5. G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
    [Crossref]
  6. M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
    [Crossref]
  7. C. R. Doerr, N. Dupuis, and L. Zhang, “Optical isolator using two tandem phase modulators,” Opt. Lett. 36(21), 4293–4295 (2011).
    [Crossref] [PubMed]
  8. T. T. M. van Schaijk, D. Lenstra, K. A. Williams, and E. A. J. M. Bente, “Analysis of the operation of an integrated unidirectional phase modulator,” Accepted for ECIO 2018.
  9. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  10. D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
    [Crossref]
  11. D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22(7), 8143–8149 (2014).
    [Crossref] [PubMed]
  12. M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
    [Crossref]
  13. T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express 15(20), 12941–12948 (2007).
    [Crossref] [PubMed]
  14. D. Lenstra and E. A. J. M. Bente, “Bistable operation of a monolithic ring laser due to hole-burning-induced inversion grating,” Accepted for ECIO 2018.
  15. I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
    [Crossref]

2018 (1)

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

2014 (4)

A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22(7), 8143–8149 (2014).
[Crossref] [PubMed]

B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

2013 (1)

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

2012 (1)

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

2011 (1)

2010 (1)

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

2007 (1)

2003 (1)

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

2002 (1)

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Balle, S.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Bente, E. A. J. M.

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

Colet, P.

Danckaert, J.

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Doerr, C. R.

Donati, S.

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

Dontsov, A. A.

A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).

Dupuis, N.

Ermakov, I. V.

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

Giuliani, G.

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

Javaloyes, J.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Laybourn, P. J. R.

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

Lenstra, D.

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22(7), 8143–8149 (2014).
[Crossref] [PubMed]

Mechet, P.

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

Melati, D.

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

Melloni, A.

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

Mezösi, G.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Miglierina, M.

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

Mirasso, C. R.

Mizumoto, T.

B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

Morichetti, F.

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

Morthier, G.

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

Pérez, T.

Pérez-Serrano, A.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Scirè, A.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express 15(20), 12941–12948 (2007).
[Crossref] [PubMed]

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

Sorel, M.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

Stadler, B. J. H.

B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

Strain, M. J.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Van der Sande, G.

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express 15(20), 12941–12948 (2007).
[Crossref] [PubMed]

van Schaijk, T. T. M.

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

Verschaffelt, G.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Williams, K. A.

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

Yousefi, M.

Zhang, L.

Appl. Phys. Lett. (2)

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Commun. Nonlinear Sci. Numer. Simul. (2)

A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

IEEE J. Quantum Electron. (3)

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

IEEE Photonics J. (1)

B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

J. Opt. (1)

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Other (2)

T. T. M. van Schaijk, D. Lenstra, K. A. Williams, and E. A. J. M. Bente, “Analysis of the operation of an integrated unidirectional phase modulator,” Accepted for ECIO 2018.

D. Lenstra and E. A. J. M. Bente, “Bistable operation of a monolithic ring laser due to hole-burning-induced inversion grating,” Accepted for ECIO 2018.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1 Sketch of a feedback insensitive ring laser subjected to EOF. Lasing in a single longitudinal mode is ensured by the filter and by the cavity itself, while the isolator ensures unidirectional laser operation. EOF is modeled by a point-reflection that reflects a fraction R of the optical power. τ is the feedback delay time.

Tables (2)

Tables Icon

Table 1 Parameter listing.

Tables Icon

Table 2 Variables listing.

Equations (35)

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

E cw ̇ = 1 2 ( 1+iα )ξ( N 0 E cw + N 1 * E ccw )+ K   e i φ bs E ccw ;
  E ccw ̇ = 1 2 ΔΓ E ccw + 1 2 ( 1+iα )ξ( N 0 E ccw + N 1   E cw ) + K   e i φ bs E cw +                                 + 1 2 γ e i φ fb E cw (tτ)
N 0 ̇ =ΔJ N 0 T 0 ( Γ+ξ N 0 )Iξ( N 1 E cw E ccw * +c.c.);
  N 1 ̇ = N 1 T 1 ( Γ+ξ N 0 ) E cw * E ccw   ξ N 1 I.
A φ ccw φ cw ,
M N 1 e iA .
N 0 ̇ =ΔJ N 0 T 0 ( Γ+ξ N 0 )I2ξ I cw I ccw ReM;
M ̇ =i A ̇ M M T 1 ( Γ+ξ N 0 ) I cw I ccw ξMI.
I cw ̇ =ξ N 0 I cw +2 K   I cw I ccw cos( φ bs +A )+ξ I cw I ccw {ReM+αImM};
I ccw ̇ =ΔΓ I ccw +ξ N 0 I ccw +2 K   I cw I ccw cos( φ bs A )+ ξ I cw I ccw { ReMαImM }++γ I cw ( tτ ) I ccw cos( φ fb + φ cw ( tτ ) φ ccw );
A ̇ = 1 2 ξ{ I cw I ccw ( αReM+ImM ) I ccw I cw ( αReMImM ) }+K{ I cw I ccw sin( φ bs A )              I ccw I cw  sin( φ bs +A )}+ γ 2 I cw ( tτ ) I ccw sin( φ fb A+ φ cw ( tτ )  φ cw ).
ϵ(t) I ccw I cw .
   ϵ   ̇ + ΔΓ 2 ϵ= 1 2 ξ{ 1 ϵ ( ReMαImM )ϵ( ReM+αImM ) }ϵ+K{ 1 ϵ cos( φ bs A )                 ϵ cos( φ bs +A )}ϵ+ 1 2 γ I cw ( tτ ) I cw cos( φ fb A+ φ cw ( tτ )  φ cw ).
M T 1 ΓIϵ/(1+ ϵ   2 ).
Δω= 1 2 ξαM( 1 ϵ ϵ ) 1 2 αξ T 1 ΓI (1 ϵ 2 )   /(1+ ϵ   2 ).
K{ 1   ϵ sin( φ bs A )ϵ sin( φ bs +A ) }= W BS sin( Ψ BS A ),
Ψ BS Arctan(( 1+ ϵ 2 )cos φ bs , (1 ϵ 2 )sin φ bs );
  W BS K ϵ 2 + ϵ 2 +2cos2 φ bs ,
A ̇ =Δω+ W BS sin( Ψ BS A )+ W fb sin( Ψ fb A),
W fb γ 2ϵ ,
Ψ fb   φ fb Δ ω op τ,
Δ ω op   φ cw ̇ = φ ccw ̇ ,
A ̇ =Δω+  C eff sin( Ψ eff A ),
C eff   W BS 2 + W fb 2 +2 W BS W fb cos( Ψ BS Ψ fb )   ;
Ψ eff Arctan( W BS cos Ψ BS + W fb cos Ψ fb ,  W BS sin Ψ BS + W fb sin Ψ fb .
C eff >|Δω|.
A= Ψ eff +Arcsin( Δω C eff ).
ϵ ̇ =( ΔΓ 2 Δω α )ϵ+K{cos( φ bs  A ) ϵ 2 cos( φ bs  +A )}+ γ 2 cos( Ψ fb A ),
ϵ ˙ = Δω α ϵ ξ T 1 ΓI 2 1 ϵ 2 1+ ϵ 2 ϵ,
ϵ=0,
ϵ=1.
ϵ ± = ΔΓ 2   1± 1+ 16Kcos( φ bs  +A ) { Kcos( φ bs  A )+ γ 2 cos( Ψ fb A ) } Δ Γ 2   2Kcos( φ bs  +A) .
ϵ= 2Kcos( φ bs  A )+γcos( Ψ fb A) ΔΓ ,
ΔΓ max{ | Δω |,4 K(K+ γ 2 ) }.
ϵ= 2K+γ ΔΓ 1 Kγ (K+ γ 2 ) 2 [1cos( φ bs  Ψ fb )] ,

Metrics