Abstract

We demonstrated all-optical transistor operation based on the bistability principle in nonlinear distributed feedback GaInAsPInP waveguide. By increasing the pump power above a certain limit, the probe transmission that operates in the bistable regime can turn from switching to transistor mode. The unstable state is shown to play a crucial role in distinguishing the probe transmission artifacts between switching and transistor mode.

© 2007 Optical Society of America

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  1. H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
    [CrossRef]
  2. H. G. Winful, R. Zamir, and S. Feldman, "Modulational instability in nonlinear periodic structures: implications for 'gap solitons'," Appl. Phys. Lett. 58, 1001-1003 (1991).
    [CrossRef]
  3. C. M. de Sterke, "Optical push broom," Opt. Lett. 17, 914-916 (1992).
    [CrossRef] [PubMed]
  4. C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, Vol. XXXIII, E.Wolf ed. (Elsevier Science, 1994), pp. 203-260.
  5. B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
    [CrossRef] [PubMed]
  6. N. G. R. Broderick, D. Taverner, D. J. Richardson, and M. Ibsen, "Cross phase modulation effects in nonlinear fiber Bragg gratings," J. Opt. Soc. Am. B 17, 345-353 (2000).
    [CrossRef]
  7. J. Atai and B. A. Malomed, "Families of Bragg-grating solitons in a cubic-quintic medium," Phys. Lett. A 284, 247-252 (2001).
    [CrossRef]
  8. B. V. Gisin, R. Driben, and B. A. Malomed, "Bistable guided solitons in the cubic quintic nonlinear medium," J. Opt. B: Quantum Semiclassical Opt. 6, 259-264 (2004).
    [CrossRef]
  9. Yosia, S. Ping, and L. Chao, "Nonlinear switching and pulse propagation in phase-shifted cubic quintic grating," IEEE Photon. Technol. Lett. 12, 2670-2672 (2005).
    [CrossRef]
  10. S.-H. Jeong, K. Nakatsuhara, T. Mizumoto, B.-J. Ma, and Y. Nakano, "Dependence of threshold switching power on the control-light wavelength in a nonlinear distributed-feedback GaInAsP waveguide," Appl. Opt. 40, 6042-6048 (2001).
    [CrossRef]
  11. S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
    [CrossRef]
  12. K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
    [CrossRef]
  13. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, "All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry," Opt. Lett. 28, 2506-2508 (2003).
    [CrossRef] [PubMed]
  14. A. Shinya, T. Tanabe, E. Kuramochi, and M. Notomi, "All-optical switch and digital light processing using photonic crystals," NTT Technical Review 3, 61-68 (2005).
  15. Y. Yosia, S. Ping, and L. Chao, "Bistability threshold inside hysteresis loop of nonlinear fiber Bragg gratings," Opt. Express 13, 5127-5135 (2005).
    [CrossRef] [PubMed]
  16. J.-K. Seo and T. Mizumoto, "Nonlinear optical properties in GaInAsP/InP waveguides below the band-gap wavelength," Jpn. J. Appl. Phys., Part 1 45, 2612-2617 (2006).
    [CrossRef]
  17. C. M. de Sterke, K. R. Jackson, and B. D. Robert, "Nonlinear coupled-mode equations on a finite interval: a numerical procedure," J. Opt. Soc. Am. B 8, 403-412 (1991).
    [CrossRef]

2006

J.-K. Seo and T. Mizumoto, "Nonlinear optical properties in GaInAsP/InP waveguides below the band-gap wavelength," Jpn. J. Appl. Phys., Part 1 45, 2612-2617 (2006).
[CrossRef]

2005

A. Shinya, T. Tanabe, E. Kuramochi, and M. Notomi, "All-optical switch and digital light processing using photonic crystals," NTT Technical Review 3, 61-68 (2005).

Y. Yosia, S. Ping, and L. Chao, "Bistability threshold inside hysteresis loop of nonlinear fiber Bragg gratings," Opt. Express 13, 5127-5135 (2005).
[CrossRef] [PubMed]

Yosia, S. Ping, and L. Chao, "Nonlinear switching and pulse propagation in phase-shifted cubic quintic grating," IEEE Photon. Technol. Lett. 12, 2670-2672 (2005).
[CrossRef]

2004

B. V. Gisin, R. Driben, and B. A. Malomed, "Bistable guided solitons in the cubic quintic nonlinear medium," J. Opt. B: Quantum Semiclassical Opt. 6, 259-264 (2004).
[CrossRef]

2003

2002

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

2001

2000

N. G. R. Broderick, D. Taverner, D. J. Richardson, and M. Ibsen, "Cross phase modulation effects in nonlinear fiber Bragg gratings," J. Opt. Soc. Am. B 17, 345-353 (2000).
[CrossRef]

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

1996

B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

1992

1991

H. G. Winful, R. Zamir, and S. Feldman, "Modulational instability in nonlinear periodic structures: implications for 'gap solitons'," Appl. Phys. Lett. 58, 1001-1003 (1991).
[CrossRef]

C. M. de Sterke, K. R. Jackson, and B. D. Robert, "Nonlinear coupled-mode equations on a finite interval: a numerical procedure," J. Opt. Soc. Am. B 8, 403-412 (1991).
[CrossRef]

1979

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Arai, S.

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

Atai, J.

J. Atai and B. A. Malomed, "Families of Bragg-grating solitons in a cubic-quintic medium," Phys. Lett. A 284, 247-252 (2001).
[CrossRef]

Broderick, N. G. R.

Chao, L.

Yosia, S. Ping, and L. Chao, "Nonlinear switching and pulse propagation in phase-shifted cubic quintic grating," IEEE Photon. Technol. Lett. 12, 2670-2672 (2005).
[CrossRef]

Y. Yosia, S. Ping, and L. Chao, "Bistability threshold inside hysteresis loop of nonlinear fiber Bragg gratings," Opt. Express 13, 5127-5135 (2005).
[CrossRef] [PubMed]

de Sterke, C. M.

Driben, R.

B. V. Gisin, R. Driben, and B. A. Malomed, "Bistable guided solitons in the cubic quintic nonlinear medium," J. Opt. B: Quantum Semiclassical Opt. 6, 259-264 (2004).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Fan, S.

Feldman, S.

H. G. Winful, R. Zamir, and S. Feldman, "Modulational instability in nonlinear periodic structures: implications for 'gap solitons'," Appl. Phys. Lett. 58, 1001-1003 (1991).
[CrossRef]

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Gisin, B. V.

B. V. Gisin, R. Driben, and B. A. Malomed, "Bistable guided solitons in the cubic quintic nonlinear medium," J. Opt. B: Quantum Semiclassical Opt. 6, 259-264 (2004).
[CrossRef]

Hossain, S.

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

Ibsen, M.

Jackson, K. R.

Jeong, S.-H.

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

S.-H. Jeong, K. Nakatsuhara, T. Mizumoto, B.-J. Ma, and Y. Nakano, "Dependence of threshold switching power on the control-light wavelength in a nonlinear distributed-feedback GaInAsP waveguide," Appl. Opt. 40, 6042-6048 (2001).
[CrossRef]

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

Joannopoulos, J. D.

Kim, H.-C.

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Kuramochi, E.

A. Shinya, T. Tanabe, E. Kuramochi, and M. Notomi, "All-optical switch and digital light processing using photonic crystals," NTT Technical Review 3, 61-68 (2005).

Ma, B.-J.

S.-H. Jeong, K. Nakatsuhara, T. Mizumoto, B.-J. Ma, and Y. Nakano, "Dependence of threshold switching power on the control-light wavelength in a nonlinear distributed-feedback GaInAsP waveguide," Appl. Opt. 40, 6042-6048 (2001).
[CrossRef]

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

Malomed, B. A.

B. V. Gisin, R. Driben, and B. A. Malomed, "Bistable guided solitons in the cubic quintic nonlinear medium," J. Opt. B: Quantum Semiclassical Opt. 6, 259-264 (2004).
[CrossRef]

J. Atai and B. A. Malomed, "Families of Bragg-grating solitons in a cubic-quintic medium," Phys. Lett. A 284, 247-252 (2001).
[CrossRef]

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Martijn de Sterke, C.

B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Mizumoto, T.

J.-K. Seo and T. Mizumoto, "Nonlinear optical properties in GaInAsP/InP waveguides below the band-gap wavelength," Jpn. J. Appl. Phys., Part 1 45, 2612-2617 (2006).
[CrossRef]

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

S.-H. Jeong, K. Nakatsuhara, T. Mizumoto, B.-J. Ma, and Y. Nakano, "Dependence of threshold switching power on the control-light wavelength in a nonlinear distributed-feedback GaInAsP waveguide," Appl. Opt. 40, 6042-6048 (2001).
[CrossRef]

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

Nakano, Y.

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

S.-H. Jeong, K. Nakatsuhara, T. Mizumoto, B.-J. Ma, and Y. Nakano, "Dependence of threshold switching power on the control-light wavelength in a nonlinear distributed-feedback GaInAsP waveguide," Appl. Opt. 40, 6042-6048 (2001).
[CrossRef]

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

Nakatsuhara, K.

S.-H. Jeong, K. Nakatsuhara, T. Mizumoto, B.-J. Ma, and Y. Nakano, "Dependence of threshold switching power on the control-light wavelength in a nonlinear distributed-feedback GaInAsP waveguide," Appl. Opt. 40, 6042-6048 (2001).
[CrossRef]

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

Notomi, M.

A. Shinya, T. Tanabe, E. Kuramochi, and M. Notomi, "All-optical switch and digital light processing using photonic crystals," NTT Technical Review 3, 61-68 (2005).

Ping, S.

Y. Yosia, S. Ping, and L. Chao, "Bistability threshold inside hysteresis loop of nonlinear fiber Bragg gratings," Opt. Express 13, 5127-5135 (2005).
[CrossRef] [PubMed]

Yosia, S. Ping, and L. Chao, "Nonlinear switching and pulse propagation in phase-shifted cubic quintic grating," IEEE Photon. Technol. Lett. 12, 2670-2672 (2005).
[CrossRef]

Richardson, D. J.

Robert, B. D.

Seo, J.-K.

J.-K. Seo and T. Mizumoto, "Nonlinear optical properties in GaInAsP/InP waveguides below the band-gap wavelength," Jpn. J. Appl. Phys., Part 1 45, 2612-2617 (2006).
[CrossRef]

Shinya, A.

A. Shinya, T. Tanabe, E. Kuramochi, and M. Notomi, "All-optical switch and digital light processing using photonic crystals," NTT Technical Review 3, 61-68 (2005).

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, Vol. XXXIII, E.Wolf ed. (Elsevier Science, 1994), pp. 203-260.

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Soljacic, M.

Takenaka, M.

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

Tanabe, T.

A. Shinya, T. Tanabe, E. Kuramochi, and M. Notomi, "All-optical switch and digital light processing using photonic crystals," NTT Technical Review 3, 61-68 (2005).

Taverner, D.

Tsukishima, Y.

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

Wiedmann, J.

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

Winful, H. G.

H. G. Winful, R. Zamir, and S. Feldman, "Modulational instability in nonlinear periodic structures: implications for 'gap solitons'," Appl. Phys. Lett. 58, 1001-1003 (1991).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Yanik, M. F.

Yosia,

Yosia, S. Ping, and L. Chao, "Nonlinear switching and pulse propagation in phase-shifted cubic quintic grating," IEEE Photon. Technol. Lett. 12, 2670-2672 (2005).
[CrossRef]

Yosia, Y.

Zamir, R.

H. G. Winful, R. Zamir, and S. Feldman, "Modulational instability in nonlinear periodic structures: implications for 'gap solitons'," Appl. Phys. Lett. 58, 1001-1003 (1991).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

H. G. Winful, R. Zamir, and S. Feldman, "Modulational instability in nonlinear periodic structures: implications for 'gap solitons'," Appl. Phys. Lett. 58, 1001-1003 (1991).
[CrossRef]

IEEE J. Quantum Electron.

S.-H. Jeong, H.-C. Kim, T. Mizumoto, J. Wiedmann, S. Arai, M. Takenaka, and Y. Nakano, "Polarization-independent all-optical switching in a nonlinear GaInAsP-InP high mesa waveguide with a vertically etched Bragg reflector," IEEE J. Quantum Electron. 38, 706-715 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

K. Nakatsuhara, T. Mizumoto, S. Hossain, S.-H. Jeong, Y. Tsukishima, B.-J. Ma, and Y. Nakano, "GaInAsP-InP distributed feedback waveguides for all-optical switching," IEEE J. Sel. Top. Quantum Electron. 6, 143-149 (2000).
[CrossRef]

IEEE Photon. Technol. Lett.

Yosia, S. Ping, and L. Chao, "Nonlinear switching and pulse propagation in phase-shifted cubic quintic grating," IEEE Photon. Technol. Lett. 12, 2670-2672 (2005).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt.

B. V. Gisin, R. Driben, and B. A. Malomed, "Bistable guided solitons in the cubic quintic nonlinear medium," J. Opt. B: Quantum Semiclassical Opt. 6, 259-264 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys., Part 1

J.-K. Seo and T. Mizumoto, "Nonlinear optical properties in GaInAsP/InP waveguides below the band-gap wavelength," Jpn. J. Appl. Phys., Part 1 45, 2612-2617 (2006).
[CrossRef]

NTT Technical Review

A. Shinya, T. Tanabe, E. Kuramochi, and M. Notomi, "All-optical switch and digital light processing using photonic crystals," NTT Technical Review 3, 61-68 (2005).

Opt. Express

Opt. Lett.

Phys. Lett. A

J. Atai and B. A. Malomed, "Families of Bragg-grating solitons in a cubic-quintic medium," Phys. Lett. A 284, 247-252 (2001).
[CrossRef]

Phys. Rev. Lett.

B. J. Eggleton, R. E. Slusher, C. Martijn de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Other

C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, Vol. XXXIII, E.Wolf ed. (Elsevier Science, 1994), pp. 203-260.

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Figures (7)

Fig. 1
Fig. 1

Transmission spectrum of DFB waveguide in linear regime (solid curve) and probe spectrum (dashed curve).

Fig. 2
Fig. 2

Normalized input probe (solid curve) at 1566.2 nm and pump wavelength (dotted curve) at 1550 nm .

Fig. 3
Fig. 3

Schematic of experimental setup.

Fig. 4
Fig. 4

Experimental results on time average probe transmissions in the (a) switching, (b) unstable, and (c) transistor mode when the pump is in low, medium, and high regime, respectively.

Fig. 5
Fig. 5

Transient perspective of all-optical transistor operations by numerical simulations. The unstable state distinguishes the probe transmission artifacts between the switching and transistor mode.

Fig. 6
Fig. 6

Snapshots of probe transmissions in the switching (solid curve) and transistor (dotted curve) operations by setting the pump power in medium regime.

Fig. 7
Fig. 7

Unstable state (dashed line) is drawn from the experiment as the boundary to distinguish between the switching (solid curve) and transistor (dotted curve) operations.

Equations (3)

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

i A + z + i n ¯ c A + t + δ A + + κ A + Γ B { A + 2 + 2 A 2 + A p 2 } A + + R Γ P A P 2 A + = 0 , ,
i A z + i n ¯ c A t + δ A + κ * A + + Γ B { A 2 + 2 A + 2 + A p 2 } A + R Γ P A P 2 A = 0 , ,
i A P z + i n ¯ c A P t + 1 R Γ B { A + 2 + A 2 } A P + Γ P { A + 2 + A 2 + A P 2 } A P = 0 , ,

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