Abstract

In this paper we study the optical bistability in a double coupler fiber ring resonator which consists of an erbium doped fiber amplifier (EDFA) in half part of the fiber ring and a quantum dot doped fiber (QDF) saturable absorber in the other half. The bistability is provided by the QDF section of the ring resonator. The EDFA is employed to reduce the switching power. The transmitted and reflected bistability characteristics are investigated. It is shown that the switching power for this new bistable device is less than 10 mW.

© 2012 Optical Society of America

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  1. A. R. Bahrampour, S. S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set-reset flip-flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
    [CrossRef]
  2. A. R. Bahrampour, M. Karimi, M. J. Abolfazli Qamsari, H. Rooholamini Nejad, and S. Keyvaninia, “All-optical set reset flip flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
    [CrossRef]
  3. W. F. Sharfin and M. Dagenais, “High contrast 1.3 μm optical AND gate with gain,” Appl. Phys. Lett. 48, 1510–1512 (1986).
    [CrossRef]
  4. B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
    [CrossRef]
  5. V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
    [CrossRef]
  6. M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
    [CrossRef]
  7. A. M. Kaplan, G. P. Agrawal, and D. N. Maywar, “All-optical flip-flop operation of VCSOA,” Electron. Lett. 45, 127–128 (2009).
    [CrossRef]
  8. D. N. Maywar, G. P. Agrawal, and Y. Nakano, “All-optical hysteresis control by means of cross-phase modulation in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 18, 1003–1013 (2001).
    [CrossRef]
  9. S. Djabi, H. Boudoukha, and S. Meguellati, “Analytical model for optical bistability in laser with saturable absorber,” J. Nonlinear Opt. Phys. Mater. 20, 389–395 (2011).
    [CrossRef]
  10. S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a trimodal laser containing a saturable absorber,” ARPN J. Eng. Appl. Sci. 2, 14–21 (2007).
  11. C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. 283, 2957–2960 (2010).
    [CrossRef]
  12. F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).
  13. F. Zhou, Y. Liu, Z. Y. Li, and Y. Xia, “Analytical model for optical bistability in nonlinear metal nanoantennae involving Kerr materials,” Opt. Express 18, 13337–13344 (2010).
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    [CrossRef]
  16. J. M. Ward, D. G. O’Shea, B. J. Shortt, and S. N. Chormaic, “Optical bistability in Er–Yb codoped phosphate glass microspheres at room temperature,” J. Appl. Phys. 102, 023104 (2007).
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  17. X. Ren, J. Song, Y. Guo, G. Yang, and Y. Huang, “Interesting nonlinear effects in Er+3–Yb+3 co doped fibers,” Optoelectronics, Proceedings Sixth Chinese Symposium (2003).
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    [CrossRef]
  19. Z. Zang and Y. Zhang, “Low-switching power (<45  mW) optical bistability based on optical nonlinearity of ytterbium-doped fiber with a fiber Bragg grating pair,” J. Mod. Opt. 59, 161–165 (2012).
    [CrossRef]
  20. Z. Zang and Y. Zhang, “Analysis of optical switching in a Yb3+-doped fiber Bragg grating by using self-phase modulation and cross-phase modulation,” Appl. Opt. 51, 3424–3430 (2012).
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  21. Z. Zang and W. Yang, “Theoretical and experimental investigation of all-optical switching based on cascaded LPFGs separated by an erbium-doped fiber,” J. Appl. Phys. 109, 103106 (2011).
    [CrossRef]
  22. H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17, 17630–17635 (2009).
    [CrossRef]
  23. H. Zhang, Q. Bao, D. Tang, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95, 141103 (2009).
    [CrossRef]
  24. G. Sobon, J. Sotor, J. Jagiello, R. Kozinski, M. Zdrojek, M. Holdynski, P. Paletko, J. Boguslawski, L. Lipinska, and K. M. Abramski, “Graphene oxide vs. reduced graphene oxide as saturable absorbers for Er-doped passively mode-locked fiber laser,” Opt. Express 20, 19463–19467 (2012).
    [CrossRef]
  25. Y. D. Jeong, J. S. Cho, Y. H. Won, H. J. Lee, and H. Yoo, “All-optical flip-flop based on the bistability of injection locked Fabry-Perot laser diode,” Opt. Express 14, 4058–4063 (2006).
    [CrossRef]
  26. P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear SOA-based fiber ring resonator,” Proc. SPIE 7797, 779712 (2010).
    [CrossRef]
  27. L. Wei, S. Song, and Y. N. Wang, “Influence of nonlinear absorption effects on optical bistability in semiconductor ring resonators,” Opt. Laser Technol. 37, 432–437 (2005).
    [CrossRef]
  28. J. Shao, S. Li, Q. Shen, Z. Wu, Z. Cao, and J. Gu, “Experiment and theoretical explanation of optical bistability in a single erbium-doped fiber ring laser,” Opt. Express 15, 3673–3679 (2007).
    [CrossRef]
  29. L. Luo and P. L. Chu, “Optical bistability in a coupled fiber ring resonator system with nonlinear absorptive medium,” Opt. Commun. 129, 224–228 (1996).
    [CrossRef]
  30. C. Li, N. Dou, and P. P. Yupapin, “Milliwatt and nanosecond all-optical switching in a double-coupler ring resonator containing an EDFA,” J. Opt. A 8, 728–732 (2006).
    [CrossRef]
  31. N. Dou and C. Li, “Optical bistability in fiber ring resonator containing an EDFA,” Opt. Commun. 281, 2238–2242 (2008).
    [CrossRef]
  32. F. Pang, X. Su, H. Guo, J. Yan, J. Wang, X. Zeng, Z. Chen, and T. Wang, “A PbS quantum dots fiber amplifier excited by evanescent wave,” Opt. Express 18, 14024–14030 (2010).
    [CrossRef]
  33. C. Cheng and H. Zhang, “Characteristics of bandwidth, gain and noise of a PbSe quantum dot-doped fiber amplifier,” Opt. Commun. 277, 372–378 (2007).
    [CrossRef]
  34. C. Jiang, “Ultrabroadband gain characteristics of a quantum-dot-doped fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15, 140–144 (2009).
    [CrossRef]
  35. K. Kang, K. Daneshvar, and R. Tsu, “Size dependence saturation and absorption of PbS quantum dots,” Microelectron. J. 35, 629–633 (2004).
    [CrossRef]
  36. J. F. Philipps, T. Topfer, H. Ebendorff-Heidepriem, D. Ehrt, R. Sauerbrey, and N. F. Borrelli, “Diode-pumped erbium-ytterbium-glass laser passively Q-switched with a PbS semiconductor quantum-dot doped glass,” Appl. Phys. B 72, 175–178 (2001).
    [CrossRef]
  37. J. E. Raynolds and M. LoCascio, “Semiconductor nanocrystal based saturable absorbers for optical switching applications,” MRS Proc. 737, E4.5 (2002).
  38. A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).
  39. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  40. P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology (Academic, 1999).
  41. K. Y. Ko, M. S. Demokan, and H. Y. Tam, “Transient analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 6, 1436–1438 (1994).
    [CrossRef]
  42. C. Cheng, “A multiquantum-dot-doped fiber amplifier with characteristics of broadband, flat gain, and low noise,” J. Lightwave Technol. 26, 1404–1410 (2008).
    [CrossRef]
  43. B. Pedersen, A. Bjarklev, O. Lumholt, and J. H. Povlsen, “Detailed design analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 3, 548–550 (1991).
    [CrossRef]

2012

Z. Zang, “Numerical analysis of optical bistability based on fiber Bragg grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun. 285, 521–526 (2012).
[CrossRef]

Z. Zang and Y. Zhang, “Low-switching power (<45  mW) optical bistability based on optical nonlinearity of ytterbium-doped fiber with a fiber Bragg grating pair,” J. Mod. Opt. 59, 161–165 (2012).
[CrossRef]

Z. Zang and Y. Zhang, “Analysis of optical switching in a Yb3+-doped fiber Bragg grating by using self-phase modulation and cross-phase modulation,” Appl. Opt. 51, 3424–3430 (2012).
[CrossRef]

G. Sobon, J. Sotor, J. Jagiello, R. Kozinski, M. Zdrojek, M. Holdynski, P. Paletko, J. Boguslawski, L. Lipinska, and K. M. Abramski, “Graphene oxide vs. reduced graphene oxide as saturable absorbers for Er-doped passively mode-locked fiber laser,” Opt. Express 20, 19463–19467 (2012).
[CrossRef]

2011

Z. Zang and W. Yang, “Theoretical and experimental investigation of all-optical switching based on cascaded LPFGs separated by an erbium-doped fiber,” J. Appl. Phys. 109, 103106 (2011).
[CrossRef]

S. Djabi, H. Boudoukha, and S. Meguellati, “Analytical model for optical bistability in laser with saturable absorber,” J. Nonlinear Opt. Phys. Mater. 20, 389–395 (2011).
[CrossRef]

A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

2010

C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. 283, 2957–2960 (2010).
[CrossRef]

F. Zhou, Y. Liu, Z. Y. Li, and Y. Xia, “Analytical model for optical bistability in nonlinear metal nanoantennae involving Kerr materials,” Opt. Express 18, 13337–13344 (2010).
[CrossRef]

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear SOA-based fiber ring resonator,” Proc. SPIE 7797, 779712 (2010).
[CrossRef]

F. Pang, X. Su, H. Guo, J. Yan, J. Wang, X. Zeng, Z. Chen, and T. Wang, “A PbS quantum dots fiber amplifier excited by evanescent wave,” Opt. Express 18, 14024–14030 (2010).
[CrossRef]

2009

C. Jiang, “Ultrabroadband gain characteristics of a quantum-dot-doped fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15, 140–144 (2009).
[CrossRef]

H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17, 17630–17635 (2009).
[CrossRef]

H. Zhang, Q. Bao, D. Tang, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95, 141103 (2009).
[CrossRef]

A. M. Kaplan, G. P. Agrawal, and D. N. Maywar, “All-optical flip-flop operation of VCSOA,” Electron. Lett. 45, 127–128 (2009).
[CrossRef]

A. R. Bahrampour, S. S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set-reset flip-flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

2008

A. R. Bahrampour, M. Karimi, M. J. Abolfazli Qamsari, H. Rooholamini Nejad, and S. Keyvaninia, “All-optical set reset flip flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

N. Dou and C. Li, “Optical bistability in fiber ring resonator containing an EDFA,” Opt. Commun. 281, 2238–2242 (2008).
[CrossRef]

C. Cheng, “A multiquantum-dot-doped fiber amplifier with characteristics of broadband, flat gain, and low noise,” J. Lightwave Technol. 26, 1404–1410 (2008).
[CrossRef]

2007

C. Cheng and H. Zhang, “Characteristics of bandwidth, gain and noise of a PbSe quantum dot-doped fiber amplifier,” Opt. Commun. 277, 372–378 (2007).
[CrossRef]

J. M. Ward, D. G. O’Shea, B. J. Shortt, and S. N. Chormaic, “Optical bistability in Er–Yb codoped phosphate glass microspheres at room temperature,” J. Appl. Phys. 102, 023104 (2007).
[CrossRef]

J. Shao, S. Li, Q. Shen, Z. Wu, Z. Cao, and J. Gu, “Experiment and theoretical explanation of optical bistability in a single erbium-doped fiber ring laser,” Opt. Express 15, 3673–3679 (2007).
[CrossRef]

S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a trimodal laser containing a saturable absorber,” ARPN J. Eng. Appl. Sci. 2, 14–21 (2007).

2006

Y. D. Jeong, J. S. Cho, Y. H. Won, H. J. Lee, and H. Yoo, “All-optical flip-flop based on the bistability of injection locked Fabry-Perot laser diode,” Opt. Express 14, 4058–4063 (2006).
[CrossRef]

C. Li, N. Dou, and P. P. Yupapin, “Milliwatt and nanosecond all-optical switching in a double-coupler ring resonator containing an EDFA,” J. Opt. A 8, 728–732 (2006).
[CrossRef]

2005

L. Wei, S. Song, and Y. N. Wang, “Influence of nonlinear absorption effects on optical bistability in semiconductor ring resonators,” Opt. Laser Technol. 37, 432–437 (2005).
[CrossRef]

2004

K. Kang, K. Daneshvar, and R. Tsu, “Size dependence saturation and absorption of PbS quantum dots,” Microelectron. J. 35, 629–633 (2004).
[CrossRef]

2003

M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

2002

J. M. Harbold, F. Ö. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
[CrossRef]

J. E. Raynolds and M. LoCascio, “Semiconductor nanocrystal based saturable absorbers for optical switching applications,” MRS Proc. 737, E4.5 (2002).

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

2001

D. N. Maywar, G. P. Agrawal, and Y. Nakano, “All-optical hysteresis control by means of cross-phase modulation in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 18, 1003–1013 (2001).
[CrossRef]

J. F. Philipps, T. Topfer, H. Ebendorff-Heidepriem, D. Ehrt, R. Sauerbrey, and N. F. Borrelli, “Diode-pumped erbium-ytterbium-glass laser passively Q-switched with a PbS semiconductor quantum-dot doped glass,” Appl. Phys. B 72, 175–178 (2001).
[CrossRef]

2000

1998

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).

1996

L. Luo and P. L. Chu, “Optical bistability in a coupled fiber ring resonator system with nonlinear absorptive medium,” Opt. Commun. 129, 224–228 (1996).
[CrossRef]

1994

K. Y. Ko, M. S. Demokan, and H. Y. Tam, “Transient analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 6, 1436–1438 (1994).
[CrossRef]

1991

B. Pedersen, A. Bjarklev, O. Lumholt, and J. H. Povlsen, “Detailed design analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 3, 548–550 (1991).
[CrossRef]

1986

W. F. Sharfin and M. Dagenais, “High contrast 1.3 μm optical AND gate with gain,” Appl. Phys. Lett. 48, 1510–1512 (1986).
[CrossRef]

Abolfazli Qamsari, M. J.

A. R. Bahrampour, M. Karimi, M. J. Abolfazli Qamsari, H. Rooholamini Nejad, and S. Keyvaninia, “All-optical set reset flip flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

Abramski, K. M.

Absil, P. P.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

Aggarwal, I. D.

Agrawal, G. P.

A. M. Kaplan, G. P. Agrawal, and D. N. Maywar, “All-optical flip-flop operation of VCSOA,” Electron. Lett. 45, 127–128 (2009).
[CrossRef]

D. N. Maywar, G. P. Agrawal, and Y. Nakano, “All-optical hysteresis control by means of cross-phase modulation in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 18, 1003–1013 (2001).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Ahmad, H.

A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

Aitken, B. G.

J. M. Harbold, F. Ö. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
[CrossRef]

Al-Khateeb, W. F.

A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

Bahrampour, A. R.

A. R. Bahrampour, S. S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set-reset flip-flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

A. R. Bahrampour, M. Karimi, M. J. Abolfazli Qamsari, H. Rooholamini Nejad, and S. Keyvaninia, “All-optical set reset flip flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

Bao, Q.

H. Zhang, Q. Bao, D. Tang, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95, 141103 (2009).
[CrossRef]

Bao, Q. L.

Becker, P. C.

P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology (Academic, 1999).

Bjarklev, A.

B. Pedersen, A. Bjarklev, O. Lumholt, and J. H. Povlsen, “Detailed design analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 3, 548–550 (1991).
[CrossRef]

Boguslawski, J.

Borrelli, N. F.

J. F. Philipps, T. Topfer, H. Ebendorff-Heidepriem, D. Ehrt, R. Sauerbrey, and N. F. Borrelli, “Diode-pumped erbium-ytterbium-glass laser passively Q-switched with a PbS semiconductor quantum-dot doped glass,” Appl. Phys. B 72, 175–178 (2001).
[CrossRef]

Boudoukha, H.

S. Djabi, H. Boudoukha, and S. Meguellati, “Analytical model for optical bistability in laser with saturable absorber,” J. Nonlinear Opt. Phys. Mater. 20, 389–395 (2011).
[CrossRef]

S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a trimodal laser containing a saturable absorber,” ARPN J. Eng. Appl. Sci. 2, 14–21 (2007).

Cao, Z.

Chen, Z.

Cheng, C.

C. Cheng, “A multiquantum-dot-doped fiber amplifier with characteristics of broadband, flat gain, and low noise,” J. Lightwave Technol. 26, 1404–1410 (2008).
[CrossRef]

C. Cheng and H. Zhang, “Characteristics of bandwidth, gain and noise of a PbSe quantum dot-doped fiber amplifier,” Opt. Commun. 277, 372–378 (2007).
[CrossRef]

Cheng, X. S.

A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

Cheong, S. W.

Cho, J. S.

Chormaic, S. N.

J. M. Ward, D. G. O’Shea, B. J. Shortt, and S. N. Chormaic, “Optical bistability in Er–Yb codoped phosphate glass microspheres at room temperature,” J. Appl. Phys. 102, 023104 (2007).
[CrossRef]

Chu, P. L.

L. Luo and P. L. Chu, “Optical bistability in a coupled fiber ring resonator system with nonlinear absorptive medium,” Opt. Commun. 129, 224–228 (1996).
[CrossRef]

Costanzo-Caso, P. A.

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear SOA-based fiber ring resonator,” Proc. SPIE 7797, 779712 (2010).
[CrossRef]

Dagenais, M.

W. F. Sharfin and M. Dagenais, “High contrast 1.3 μm optical AND gate with gain,” Appl. Phys. Lett. 48, 1510–1512 (1986).
[CrossRef]

Daneshvar, K.

K. Kang, K. Daneshvar, and R. Tsu, “Size dependence saturation and absorption of PbS quantum dots,” Microelectron. J. 35, 629–633 (2004).
[CrossRef]

Demokan, M. S.

K. Y. Ko, M. S. Demokan, and H. Y. Tam, “Transient analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 6, 1436–1438 (1994).
[CrossRef]

Djabi, M.

S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a trimodal laser containing a saturable absorber,” ARPN J. Eng. Appl. Sci. 2, 14–21 (2007).

Djabi, S.

S. Djabi, H. Boudoukha, and S. Meguellati, “Analytical model for optical bistability in laser with saturable absorber,” J. Nonlinear Opt. Phys. Mater. 20, 389–395 (2011).
[CrossRef]

S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a trimodal laser containing a saturable absorber,” ARPN J. Eng. Appl. Sci. 2, 14–21 (2007).

Dou, N.

N. Dou and C. Li, “Optical bistability in fiber ring resonator containing an EDFA,” Opt. Commun. 281, 2238–2242 (2008).
[CrossRef]

C. Li, N. Dou, and P. P. Yupapin, “Milliwatt and nanosecond all-optical switching in a double-coupler ring resonator containing an EDFA,” J. Opt. A 8, 728–732 (2006).
[CrossRef]

Ebendorff-Heidepriem, H.

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[CrossRef]

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J. F. Philipps, T. Topfer, H. Ebendorff-Heidepriem, D. Ehrt, R. Sauerbrey, and N. F. Borrelli, “Diode-pumped erbium-ytterbium-glass laser passively Q-switched with a PbS semiconductor quantum-dot doped glass,” Appl. Phys. B 72, 175–178 (2001).
[CrossRef]

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M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
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A. R. Bahrampour, S. S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set-reset flip-flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

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M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
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A. R. Bahrampour, S. S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set-reset flip-flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
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Guo, H.

Guo, Y.

X. Ren, J. Song, Y. Guo, G. Yang, and Y. Huang, “Interesting nonlinear effects in Er+3–Yb+3 co doped fibers,” Optoelectronics, Proceedings Sixth Chinese Symposium (2003).

Hamida, B. A.

A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

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J. M. Harbold, F. Ö. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
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F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).

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A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

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V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
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X. Ren, J. Song, Y. Guo, G. Yang, and Y. Huang, “Interesting nonlinear effects in Er+3–Yb+3 co doped fibers,” Optoelectronics, Proceedings Sixth Chinese Symposium (2003).

Hwang, H. Y.

Ibanescu, M.

M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
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V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
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F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).

Ilday, F. Ö.

J. M. Harbold, F. Ö. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
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M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
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V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
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M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
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Keyvaninia, S.

A. R. Bahrampour, M. Karimi, M. J. Abolfazli Qamsari, H. Rooholamini Nejad, and S. Keyvaninia, “All-optical set reset flip flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
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A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

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Lee, H. J.

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F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).

Lenz, G.

Li, B.

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
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C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. 283, 2957–2960 (2010).
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C. Li, N. Dou, and P. P. Yupapin, “Milliwatt and nanosecond all-optical switching in a double-coupler ring resonator containing an EDFA,” J. Opt. A 8, 728–732 (2006).
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Li, Z. Y.

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H. Zhang, Q. Bao, D. Tang, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95, 141103 (2009).
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M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
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B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
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A. R. Bahrampour, S. S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set-reset flip-flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

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A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

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Nejad, H. Rooholamini

A. R. Bahrampour, M. Karimi, M. J. Abolfazli Qamsari, H. Rooholamini Nejad, and S. Keyvaninia, “All-optical set reset flip flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

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J. M. Ward, D. G. O’Shea, B. J. Shortt, and S. N. Chormaic, “Optical bistability in Er–Yb codoped phosphate glass microspheres at room temperature,” J. Appl. Phys. 102, 023104 (2007).
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P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology (Academic, 1999).

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Pang, F.

Pedersen, B.

B. Pedersen, A. Bjarklev, O. Lumholt, and J. H. Povlsen, “Detailed design analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 3, 548–550 (1991).
[CrossRef]

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J. F. Philipps, T. Topfer, H. Ebendorff-Heidepriem, D. Ehrt, R. Sauerbrey, and N. F. Borrelli, “Diode-pumped erbium-ytterbium-glass laser passively Q-switched with a PbS semiconductor quantum-dot doped glass,” Appl. Phys. B 72, 175–178 (2001).
[CrossRef]

Povlsen, J. H.

B. Pedersen, A. Bjarklev, O. Lumholt, and J. H. Povlsen, “Detailed design analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 3, 548–550 (1991).
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F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).

Raynolds, J. E.

J. E. Raynolds and M. LoCascio, “Semiconductor nanocrystal based saturable absorbers for optical switching applications,” MRS Proc. 737, E4.5 (2002).

Ren, X.

X. Ren, J. Song, Y. Guo, G. Yang, and Y. Huang, “Interesting nonlinear effects in Er+3–Yb+3 co doped fibers,” Optoelectronics, Proceedings Sixth Chinese Symposium (2003).

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F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).

Sanghera, J. S.

Sauerbrey, R.

J. F. Philipps, T. Topfer, H. Ebendorff-Heidepriem, D. Ehrt, R. Sauerbrey, and N. F. Borrelli, “Diode-pumped erbium-ytterbium-glass laser passively Q-switched with a PbS semiconductor quantum-dot doped glass,” Appl. Phys. B 72, 175–178 (2001).
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J. M. Ward, D. G. O’Shea, B. J. Shortt, and S. N. Chormaic, “Optical bistability in Er–Yb codoped phosphate glass microspheres at room temperature,” J. Appl. Phys. 102, 023104 (2007).
[CrossRef]

Siahmakoun, A.

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear SOA-based fiber ring resonator,” Proc. SPIE 7797, 779712 (2010).
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P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology (Academic, 1999).

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Sobon, G.

Soljacic, M.

M. Soljačić, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in non-linear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
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X. Ren, J. Song, Y. Guo, G. Yang, and Y. Huang, “Interesting nonlinear effects in Er+3–Yb+3 co doped fibers,” Optoelectronics, Proceedings Sixth Chinese Symposium (2003).

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L. Wei, S. Song, and Y. N. Wang, “Influence of nonlinear absorption effects on optical bistability in semiconductor ring resonators,” Opt. Laser Technol. 37, 432–437 (2005).
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B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
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K. Y. Ko, M. S. Demokan, and H. Y. Tam, “Transient analysis of erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 6, 1436–1438 (1994).
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H. Zhang, Q. Bao, D. Tang, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95, 141103 (2009).
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Tang, D. Y.

Topfer, T.

J. F. Philipps, T. Topfer, H. Ebendorff-Heidepriem, D. Ehrt, R. Sauerbrey, and N. F. Borrelli, “Diode-pumped erbium-ytterbium-glass laser passively Q-switched with a PbS semiconductor quantum-dot doped glass,” Appl. Phys. B 72, 175–178 (2001).
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F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).

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V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
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Wang, T.

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L. Wei, S. Song, and Y. N. Wang, “Influence of nonlinear absorption effects on optical bistability in semiconductor ring resonators,” Opt. Laser Technol. 37, 432–437 (2005).
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B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
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J. M. Ward, D. G. O’Shea, B. J. Shortt, and S. N. Chormaic, “Optical bistability in Er–Yb codoped phosphate glass microspheres at room temperature,” J. Appl. Phys. 102, 023104 (2007).
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L. Wei, S. Song, and Y. N. Wang, “Influence of nonlinear absorption effects on optical bistability in semiconductor ring resonators,” Opt. Laser Technol. 37, 432–437 (2005).
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J. M. Harbold, F. Ö. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
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Wu, J. F.

C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. 283, 2957–2960 (2010).
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Xia, Y.

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C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. 283, 2957–2960 (2010).
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Yang, G.

X. Ren, J. Song, Y. Guo, G. Yang, and Y. Huang, “Interesting nonlinear effects in Er+3–Yb+3 co doped fibers,” Optoelectronics, Proceedings Sixth Chinese Symposium (2003).

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B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
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C. Li, N. Dou, and P. P. Yupapin, “Milliwatt and nanosecond all-optical switching in a double-coupler ring resonator containing an EDFA,” J. Opt. A 8, 728–732 (2006).
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A. W. Naji, B. A. Hamida, X. S. Cheng, M. A. Mahdi, S. Harun, S. Khan, W. F. Al-Khateeb, A. A. Zaidan, B. B. Zaidan, and H. Ahmad, “Review of erbium-doped fiber amplifier,” Int. J. Phys. Distrib. Logist. Manag. 6, 4674–4689 (2011).

Zakeri, S. S.

A. R. Bahrampour, S. S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set-reset flip-flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
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Z. Zang and Y. Zhang, “Low-switching power (<45  mW) optical bistability based on optical nonlinearity of ytterbium-doped fiber with a fiber Bragg grating pair,” J. Mod. Opt. 59, 161–165 (2012).
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Appl. Opt.

Appl. Phys. B

J. F. Philipps, T. Topfer, H. Ebendorff-Heidepriem, D. Ehrt, R. Sauerbrey, and N. F. Borrelli, “Diode-pumped erbium-ytterbium-glass laser passively Q-switched with a PbS semiconductor quantum-dot doped glass,” Appl. Phys. B 72, 175–178 (2001).
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Appl. Phys. Lett.

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Figures (8)

Fig. 1.
Fig. 1.

Schematic configuration of fiber ring resonator containing an erbium doped fiber amplifier (EDFA) and quantum dot doped fiber (QDF). Fiber couplers are noted by FC.

Fig. 2.
Fig. 2.

(a) Theoretical bistable response. (b) Numerically simulated optical hysteresis loop obtained for the transmitted power under the conditions t 1 = t 2 = 0.97 , P p = 0.5 W , N t = 5 × 10 22 / m 3 , N t = 1 × 10 22 / m 3 , G = 1.0025 , P sat Er = 0.3203 W , P sat QD = 0.0414 .

Fig. 3.
Fig. 3.

Effect of pump power on the transmitted and reflected bistability under the condition t 1 = t 2 = 0.97 , N t = 1 × 10 22 / m 3 (a, c)  N t = 5 × 10 22 / m 3 (b, d)  N t = 5 × 10 23 / m 3 in the steady state. δ is the switching power and ρ is the contrast in the switch-up process.

Fig. 4.
Fig. 4.

Effect of (a, c) transmittance parameter of fiber coupler 1 and (b, d) fiber coupler 2 on the transmitted and reflected bistability under the conditions P p = 0.01 W , N t = 5 × 10 22 / m 3 , N t = 1 × 10 22 / m 3 (a, c)  t 2 = 0.97 and (b, d)  t 1 = 0.97 . δ is the switching power and ρ is the contrast in the switch-up process.

Fig. 5.
Fig. 5.

Effect of total density of erbium ions on the transmitted and reflected bistability under the conditions t 1 = 0.9 , t 2 = 0.99 , P p = 0.05 W , N t = 1.5 × 10 22 / m 3 . δ is the switching power and ρ is the contrast in the switch-up process.

Fig. 6.
Fig. 6.

Effect of total density of quantum dots on transmitted and reflected bistability under the conditions t 1 = 0.9 , t 2 = 0.99 , P p = 0.05 W , N t = 1 × 10 24 / m 3 . δ is the switching power and ρ is the contrast in the switch-up process.

Fig. 7.
Fig. 7.

Variation of (a) and (c) pump and (b) and (d) signal along the fiber ring resonator for various input powers under conditions t 1 = 0.9 , t 2 = 0.99 , Pump = 0.037 W , N t = 2 × 10 24 / m 3 , N t = 1.5 × 10 22 / m 3 , P sat QD = 0.0414 W .

Fig. 8.
Fig. 8.

Transmitted hysteresis loop obtained by considering and without considering the constant pump power approximation under conditions t 1 = 0.9 , t 2 = 0.99 , Pump = 0.037 W , N t = 2 × 10 24 / m 3 , N t = 1.5 × 10 22 / m 3 .

Tables (1)

Tables Icon

Table 1. Optical Parameters of Double Coupler Fiber Ring Resonator Containing an EDFA and a QDF Saturable Absorber

Equations (27)

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N ( z , t ) t = ( W 12 + R 12 ) ( N t N ( z , t ) ) ( W 21 + A 21 ) ( N t + N ( z , t ) ) ,
q s ( z , t ) z + n c q s ( z , t ) t = ( Γ s ( σ es + σ as ) 4 N ( z , t ) + Γ s ( σ es σ as ) 4 N t α 2 ) q s ( z , t ) .
N 2 ( z , t ) t = N 2 ( z , t ) τ + σ as Γ s q s ( z , t ) 2 S s ω s ( N t N 2 ( z , t ) ) σ es Γ s q s ( z , t ) 2 S s ω s N 2 ( z , t ) ,
= q s ( z , t ) z + n c q s ( z , t ) t = ( Γ s σ as 2 ( N t N 2 ( z , t ) ) + Γ s σ es 2 N 2 ( z , t ) α 2 ) q s ( z , t ) ,
E ( 0 ) t 1 E ( L ) = i χ 1 E in ,
E ( L + 2 ) = t 2 E ( L 2 ) .
q s ( 0 ) = t 1 q s ( L ) + χ 1 q in ,
q s ( L + 2 ) = t 2 q s ( L 2 ) ,
d P s d z = g 0 1 + P s P sat Er P s α P s ,
d P s d z = α 0 1 + P s P sat QD P s α P s ,
P sat Er = S s ω s S p ω p τ σ ap Γ p P p + S p ω p τ Γ s ( σ as + σ es ) ,
P sat QD = S s ω s τ Γ s ( σ as + σ es ) .
g 0 = Γ s N t τ σ es σ ap Γ p P p σ as S p ω p τ σ ap Γ p P p + S p ω p ,
α 0 = Γ s N t σ as .
P s ( z ) = G ( z ) 2 P s ( 0 ) , 0 z L 2 ,
ln ( P ( z ) P ( L + 2 ) ) + P ( z ) P ( L + 2 ) = α 0 ( z L 2 ) , L + 2 z L ,
P s ( L + 2 ) = t 2 2 G 2 P s ( 0 ) ,
P s ( L ) = P s ( 0 ) 2 χ 1 P in P s ( 0 ) t 1 2 ,
P out t = χ 2 2 G 2 P s ( 0 ) .
P in = 1 4 χ 1 2 ( A ( P out t χ 2 G ) 3 + B ( P out t χ 2 G ) t 1 2 P sat QD + ( P out t χ 2 G ) 2 ) 2 ,
P out t ± = χ 2 2 G 2 2 A ( B 3 A t 1 2 P sat QD ± ( B 3 A t 1 2 P sat QD ) 2 4 A B t 1 2 P sat QD ) .
( B 3 A t 1 2 P sat QD ) 2 4 A B t 1 2 P sat QD > 0.
B 3 A t 1 2 P sat QD A > 0 ,
B A > 0.
α 0 > 16 ( 1 t 1 2 t 2 2 G 2 ) α c .
P p ( z , t ) z + n c P p ( z , t ) t = ( Γ P σ ap ( N t N ( z , t ) ) 2 + α ) P p ( z , t ) .
d P P d z = σ ap Γ p N t P p 1 + P s P sat Er S p ω p S s ω s τ σ es Γ s P s + S s ω s τ σ ap Γ p P p + S p ω p .

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