Abstract

Influences of polarization-selected mutual injection on polarization switching behaviors of vertical-cavity surface-emitting lasers (VCSELs) are investigated numerically. For the polarization-preserved (PP) injection, the x mode is selected and is injected mutually into the two lasers. The two lasers show nearly identical polarization-switching (PS) behaviors, even if frequency detuning is nonzero. For the polarization-rotated (PR) injection, the x mode is selected and rotated 90° before being injected into the two lasers. When frequency detuning is considered, the two lasers show totally different PS properties, and mutual injection will switch to unidirectional injection with varying parameters. In both the cases of PP and PR injection, the partitions that correspond to different emission states and the switching points of the two VCSELs are mapped in the parameter space of injection rate and bias current. Furthermore, for each different emission state, the output dynamics and synchronization performance are evaluated in the time domain.

© 2007 Optical Society of America

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  1. S. F. Yu, "Nonlinear dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 35, 332-341 (1999).
    [CrossRef]
  2. X. F. Li, W. Pan, B. Luo, and D. Ma, "Multi-transverse-mode dynamics of vertical-cavity surface-emitting lasers with external optical injection," J. Opt. Soc. Am. B 23, 1292-1301 (2006).
    [CrossRef]
  3. J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Emcux, "Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers," Opt. Commun. 201, 129-137 (2002).
    [CrossRef]
  4. B. Nagler, M. Peeters, J. Albert, G. Verschaffelt, K. Panajotov, H. Thienpont, I. Veretennicoff, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Polarization-mode hopping in single-mode vertical-cavity surface-emitting lasers: theory and experimental," Phys. Rev. A 68, 013813 (2003).
    [CrossRef]
  5. M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, "Investigation of polarization properties of vesels subject to optical feedback from an extremely short external cavity--part I: theoretical analysis," IEEE J. Quantum Electron. 42, 89-101 (2006).
    [CrossRef]
  6. Z. G. Pan, S. Jiang, M. Dagenais, R. A. Morgan, K. Kojima, M. T. Asom, R. E. Leibenguth, G. D. Guth, and M. W. Focht, "Optical injection induced polarization bistability in vertical-cavity surface-emitting lasers," Appl. Phys. Lett. 63, 2999-3001 (1993).
    [CrossRef]
  7. T. H. Russell and T. D. Milster, "Polarization switching control in vertical-cavity surface-emitting laser," Appl. Phys. Lett. 70, 2520-2522 (1997).
    [CrossRef]
  8. A. Valle, L. Pesquera, and K. A. Shore, "Polarization selection and sensitivity of external cavity vertical-cavity surface-emitting laser diodes," IEEE Photonics Technol. Lett. 10, 639-641 (1998).
    [CrossRef]
  9. C. Masoller and M. S. Torre, "Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 483-489 (2005).
    [CrossRef]
  10. C. Masoller and N. B. Abraham, "Low-frequency fluctuation in vertical-cavity surface-emitting semiconductor lasers with optical feedback," Phys. Rev. A 59, 3021-3031 (1999).
    [CrossRef]
  11. M. C. Soriano, M. Yousefi, J. Danckaert, S. Barland, M. Romanelli, G. Giacomelli, and F. Marin, "Low-frequency fluctuations in vertical-cavity surface-emitting lasers with polarization selective feedback: experiment and theory," IEEE J. Sel. Areas Commun. 10, 998-1005 (2004).
  12. Y. Hong, J. Paul, P. S. Spencer, and K. A. Shore, "Bias-current dependence of anticorrelation of polarization dynamics in vertical-cavity surface-emitting lasers with long external cavity," Appl. Phys. Lett. 89, 081123 (2006).
    [CrossRef]
  13. I. Gatare, K. Panajotov, and M. Sciamanna, "Frequency-induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection," Phys. Rev. A 75, 023804 (2007).
    [CrossRef]
  14. Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, "Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback," IEEE J. Quantum Electron. 41, 619-624 (2005).
    [CrossRef]
  15. M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, "Polarization fluctuations in vertical-cavity surface-emitting lasers," Phys. Rev. A 58, 4191-4205 (1998).
    [CrossRef]
  16. J. Mulet, C. R. Mirasso, and M. San Miguel, "Polarization resolved intensity noise in vertical-cavity surface-emitting lasers," Phys. Rev. A 64, 023817 (2001).
    [CrossRef]
  17. M. Sciamanna, K. Panajotov, H. Thienpont, I. Veretennicoff, P. Mégret, and M. Blondel, "Optical feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers," Opt. Lett. 28, 1543-1545 (2003).
    [CrossRef] [PubMed]
  18. M. W. Lee, Y. Hong, and K. A. Shore, "Experimental demonstration of VCSEL-based chaotic optical communication," IEEE Photonics Technol. Lett. 16, 2392-2394 (2004).
    [CrossRef]
  19. M. Sciamanna and K. Panajotov, "Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers," Phys. Rev. A 73, 023811 (2006).
    [CrossRef]
  20. I. Gatare, M. Sciamanna, J. Buesa, H. Thienpont, and K. Panajotov, "Nonlinear dynamics accompanying polarization switching in vertical-cavity surface-emitting lasers with orthogonal optical injection," Appl. Phys. Lett. 88, 101106 (2006).
    [CrossRef]
  21. N. Fujiwara, Y. Takiguchi, and J. Ohtsubo, "Observation of the synchronization of chaos in mutually injected vertical-cavity surface-emitting semiconductor lasers," Opt. Lett. 28, 1677-1679 (2003).
    [CrossRef] [PubMed]
  22. R. Vicente, J. Mulet, C. R. Mirasso, and M. Sciamanna, "Bistable polarization switching in mutually coupled vertical-cavity surface-emitting lasers," Opt. Lett. 31, 996-998 (2006).
    [CrossRef] [PubMed]
  23. R. Vicente and C. R. Mirasso, "Dynamics of mutually coupled VCSELs," Proc. SPIE 5349, 331-338 (2004).
    [CrossRef]
  24. W. L. Zhang, W. Pan, B. Luo, X. F. Li, X. H. Zou, and M. Y. Wang, "Polarization switching of mutually coupled vertical-cavity surface-emitting lasers," J. Opt. Soc. Am. B 24, 1276-1282 (2007).
    [CrossRef]
  25. K. D. Choquette, R. P. Schneider Jr., K. L. Lear, and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Areas Commun. 1, 661-666 (1995).
  26. M. San Miguel, Q. Feng, and J. V. Moloney, "Light-polarization dynamics in surface-emitting semiconductor lasers," Phys. Rev. A 52, 1728-1739 (1995).
    [CrossRef] [PubMed]
  27. B. Ryvkin, K. Panajotov, A. Georgievski, J. Danckaert, M. Peeters, G. Verschaffelt, H. Thienpont, and I. Veretennicoff, "Effect of photon-energy-dependent loss and gain mechanisms on polarization switching in vertical-cavity surface-emitting lasers," J. Opt. Soc. Am. B 16, 2106-2113 (1999).
    [CrossRef]
  28. T. Heil, I. Fischer, and W. Elsässer, J. Mulet, and C. R. Mirasso, "Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers," Phys. Rev. Lett. 86, 795-798 (2001).
    [CrossRef] [PubMed]
  29. M. Sondermann, H. Bohnet, and T. Ackemann, "Low-frequency fluctuations and polarization dynamics in vertical-cavity surface-emitting lasers with isotropic feedback," Phys. Rev. A 67, 021802 (2003).
    [CrossRef]
  30. Y. Hong, P. S. Spencer, and K. A. Shore, "Suppression of polarization switching in vertical-cavity surface-emitting lasers by use of optical feedback," Opt. Lett. 29, 2151-2153 (2004).
    [CrossRef] [PubMed]

2007 (2)

I. Gatare, K. Panajotov, and M. Sciamanna, "Frequency-induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection," Phys. Rev. A 75, 023804 (2007).
[CrossRef]

W. L. Zhang, W. Pan, B. Luo, X. F. Li, X. H. Zou, and M. Y. Wang, "Polarization switching of mutually coupled vertical-cavity surface-emitting lasers," J. Opt. Soc. Am. B 24, 1276-1282 (2007).
[CrossRef]

2006 (6)

R. Vicente, J. Mulet, C. R. Mirasso, and M. Sciamanna, "Bistable polarization switching in mutually coupled vertical-cavity surface-emitting lasers," Opt. Lett. 31, 996-998 (2006).
[CrossRef] [PubMed]

X. F. Li, W. Pan, B. Luo, and D. Ma, "Multi-transverse-mode dynamics of vertical-cavity surface-emitting lasers with external optical injection," J. Opt. Soc. Am. B 23, 1292-1301 (2006).
[CrossRef]

Y. Hong, J. Paul, P. S. Spencer, and K. A. Shore, "Bias-current dependence of anticorrelation of polarization dynamics in vertical-cavity surface-emitting lasers with long external cavity," Appl. Phys. Lett. 89, 081123 (2006).
[CrossRef]

M. Sciamanna and K. Panajotov, "Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers," Phys. Rev. A 73, 023811 (2006).
[CrossRef]

I. Gatare, M. Sciamanna, J. Buesa, H. Thienpont, and K. Panajotov, "Nonlinear dynamics accompanying polarization switching in vertical-cavity surface-emitting lasers with orthogonal optical injection," Appl. Phys. Lett. 88, 101106 (2006).
[CrossRef]

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, "Investigation of polarization properties of vesels subject to optical feedback from an extremely short external cavity--part I: theoretical analysis," IEEE J. Quantum Electron. 42, 89-101 (2006).
[CrossRef]

2005 (2)

C. Masoller and M. S. Torre, "Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 483-489 (2005).
[CrossRef]

Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, "Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback," IEEE J. Quantum Electron. 41, 619-624 (2005).
[CrossRef]

2004 (4)

M. C. Soriano, M. Yousefi, J. Danckaert, S. Barland, M. Romanelli, G. Giacomelli, and F. Marin, "Low-frequency fluctuations in vertical-cavity surface-emitting lasers with polarization selective feedback: experiment and theory," IEEE J. Sel. Areas Commun. 10, 998-1005 (2004).

M. W. Lee, Y. Hong, and K. A. Shore, "Experimental demonstration of VCSEL-based chaotic optical communication," IEEE Photonics Technol. Lett. 16, 2392-2394 (2004).
[CrossRef]

R. Vicente and C. R. Mirasso, "Dynamics of mutually coupled VCSELs," Proc. SPIE 5349, 331-338 (2004).
[CrossRef]

Y. Hong, P. S. Spencer, and K. A. Shore, "Suppression of polarization switching in vertical-cavity surface-emitting lasers by use of optical feedback," Opt. Lett. 29, 2151-2153 (2004).
[CrossRef] [PubMed]

2003 (4)

M. Sciamanna, K. Panajotov, H. Thienpont, I. Veretennicoff, P. Mégret, and M. Blondel, "Optical feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers," Opt. Lett. 28, 1543-1545 (2003).
[CrossRef] [PubMed]

N. Fujiwara, Y. Takiguchi, and J. Ohtsubo, "Observation of the synchronization of chaos in mutually injected vertical-cavity surface-emitting semiconductor lasers," Opt. Lett. 28, 1677-1679 (2003).
[CrossRef] [PubMed]

M. Sondermann, H. Bohnet, and T. Ackemann, "Low-frequency fluctuations and polarization dynamics in vertical-cavity surface-emitting lasers with isotropic feedback," Phys. Rev. A 67, 021802 (2003).
[CrossRef]

B. Nagler, M. Peeters, J. Albert, G. Verschaffelt, K. Panajotov, H. Thienpont, I. Veretennicoff, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Polarization-mode hopping in single-mode vertical-cavity surface-emitting lasers: theory and experimental," Phys. Rev. A 68, 013813 (2003).
[CrossRef]

2002 (1)

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Emcux, "Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers," Opt. Commun. 201, 129-137 (2002).
[CrossRef]

2001 (2)

J. Mulet, C. R. Mirasso, and M. San Miguel, "Polarization resolved intensity noise in vertical-cavity surface-emitting lasers," Phys. Rev. A 64, 023817 (2001).
[CrossRef]

T. Heil, I. Fischer, and W. Elsässer, J. Mulet, and C. R. Mirasso, "Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers," Phys. Rev. Lett. 86, 795-798 (2001).
[CrossRef] [PubMed]

1999 (3)

B. Ryvkin, K. Panajotov, A. Georgievski, J. Danckaert, M. Peeters, G. Verschaffelt, H. Thienpont, and I. Veretennicoff, "Effect of photon-energy-dependent loss and gain mechanisms on polarization switching in vertical-cavity surface-emitting lasers," J. Opt. Soc. Am. B 16, 2106-2113 (1999).
[CrossRef]

S. F. Yu, "Nonlinear dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 35, 332-341 (1999).
[CrossRef]

C. Masoller and N. B. Abraham, "Low-frequency fluctuation in vertical-cavity surface-emitting semiconductor lasers with optical feedback," Phys. Rev. A 59, 3021-3031 (1999).
[CrossRef]

1998 (2)

A. Valle, L. Pesquera, and K. A. Shore, "Polarization selection and sensitivity of external cavity vertical-cavity surface-emitting laser diodes," IEEE Photonics Technol. Lett. 10, 639-641 (1998).
[CrossRef]

M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, "Polarization fluctuations in vertical-cavity surface-emitting lasers," Phys. Rev. A 58, 4191-4205 (1998).
[CrossRef]

1997 (1)

T. H. Russell and T. D. Milster, "Polarization switching control in vertical-cavity surface-emitting laser," Appl. Phys. Lett. 70, 2520-2522 (1997).
[CrossRef]

1995 (2)

K. D. Choquette, R. P. Schneider Jr., K. L. Lear, and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Areas Commun. 1, 661-666 (1995).

M. San Miguel, Q. Feng, and J. V. Moloney, "Light-polarization dynamics in surface-emitting semiconductor lasers," Phys. Rev. A 52, 1728-1739 (1995).
[CrossRef] [PubMed]

1993 (1)

Z. G. Pan, S. Jiang, M. Dagenais, R. A. Morgan, K. Kojima, M. T. Asom, R. E. Leibenguth, G. D. Guth, and M. W. Focht, "Optical injection induced polarization bistability in vertical-cavity surface-emitting lasers," Appl. Phys. Lett. 63, 2999-3001 (1993).
[CrossRef]

Appl. Phys. Lett. (4)

Z. G. Pan, S. Jiang, M. Dagenais, R. A. Morgan, K. Kojima, M. T. Asom, R. E. Leibenguth, G. D. Guth, and M. W. Focht, "Optical injection induced polarization bistability in vertical-cavity surface-emitting lasers," Appl. Phys. Lett. 63, 2999-3001 (1993).
[CrossRef]

T. H. Russell and T. D. Milster, "Polarization switching control in vertical-cavity surface-emitting laser," Appl. Phys. Lett. 70, 2520-2522 (1997).
[CrossRef]

Y. Hong, J. Paul, P. S. Spencer, and K. A. Shore, "Bias-current dependence of anticorrelation of polarization dynamics in vertical-cavity surface-emitting lasers with long external cavity," Appl. Phys. Lett. 89, 081123 (2006).
[CrossRef]

I. Gatare, M. Sciamanna, J. Buesa, H. Thienpont, and K. Panajotov, "Nonlinear dynamics accompanying polarization switching in vertical-cavity surface-emitting lasers with orthogonal optical injection," Appl. Phys. Lett. 88, 101106 (2006).
[CrossRef]

IEEE J. Quantum Electron. (4)

S. F. Yu, "Nonlinear dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 35, 332-341 (1999).
[CrossRef]

Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, "Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback," IEEE J. Quantum Electron. 41, 619-624 (2005).
[CrossRef]

C. Masoller and M. S. Torre, "Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 483-489 (2005).
[CrossRef]

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, "Investigation of polarization properties of vesels subject to optical feedback from an extremely short external cavity--part I: theoretical analysis," IEEE J. Quantum Electron. 42, 89-101 (2006).
[CrossRef]

IEEE J. Sel. Areas Commun. (2)

M. C. Soriano, M. Yousefi, J. Danckaert, S. Barland, M. Romanelli, G. Giacomelli, and F. Marin, "Low-frequency fluctuations in vertical-cavity surface-emitting lasers with polarization selective feedback: experiment and theory," IEEE J. Sel. Areas Commun. 10, 998-1005 (2004).

K. D. Choquette, R. P. Schneider Jr., K. L. Lear, and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Areas Commun. 1, 661-666 (1995).

IEEE Photonics Technol. Lett. (2)

A. Valle, L. Pesquera, and K. A. Shore, "Polarization selection and sensitivity of external cavity vertical-cavity surface-emitting laser diodes," IEEE Photonics Technol. Lett. 10, 639-641 (1998).
[CrossRef]

M. W. Lee, Y. Hong, and K. A. Shore, "Experimental demonstration of VCSEL-based chaotic optical communication," IEEE Photonics Technol. Lett. 16, 2392-2394 (2004).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Emcux, "Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers," Opt. Commun. 201, 129-137 (2002).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (8)

M. San Miguel, Q. Feng, and J. V. Moloney, "Light-polarization dynamics in surface-emitting semiconductor lasers," Phys. Rev. A 52, 1728-1739 (1995).
[CrossRef] [PubMed]

M. Sondermann, H. Bohnet, and T. Ackemann, "Low-frequency fluctuations and polarization dynamics in vertical-cavity surface-emitting lasers with isotropic feedback," Phys. Rev. A 67, 021802 (2003).
[CrossRef]

B. Nagler, M. Peeters, J. Albert, G. Verschaffelt, K. Panajotov, H. Thienpont, I. Veretennicoff, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Polarization-mode hopping in single-mode vertical-cavity surface-emitting lasers: theory and experimental," Phys. Rev. A 68, 013813 (2003).
[CrossRef]

C. Masoller and N. B. Abraham, "Low-frequency fluctuation in vertical-cavity surface-emitting semiconductor lasers with optical feedback," Phys. Rev. A 59, 3021-3031 (1999).
[CrossRef]

M. Sciamanna and K. Panajotov, "Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers," Phys. Rev. A 73, 023811 (2006).
[CrossRef]

M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, "Polarization fluctuations in vertical-cavity surface-emitting lasers," Phys. Rev. A 58, 4191-4205 (1998).
[CrossRef]

J. Mulet, C. R. Mirasso, and M. San Miguel, "Polarization resolved intensity noise in vertical-cavity surface-emitting lasers," Phys. Rev. A 64, 023817 (2001).
[CrossRef]

I. Gatare, K. Panajotov, and M. Sciamanna, "Frequency-induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection," Phys. Rev. A 75, 023804 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

T. Heil, I. Fischer, and W. Elsässer, J. Mulet, and C. R. Mirasso, "Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers," Phys. Rev. Lett. 86, 795-798 (2001).
[CrossRef] [PubMed]

Proc. SPIE (1)

R. Vicente and C. R. Mirasso, "Dynamics of mutually coupled VCSELs," Proc. SPIE 5349, 331-338 (2004).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Schematic of a system of mutually coupled VCSELs with filtered PP (without HWP) and PR (with HWP) injections. XMP, x-mode polarizer; HWP, half-waveplate; BS, beam splitter; SL1, laser 1; SL2, laser 2.

Fig. 2
Fig. 2

Polarization-resolved L–I curves, under PP injection. Solid (dotted) curve corresponds to XP (YP) mode. (a) Δ f A = 0 ; (b) Δ f A = 5 GHz .

Fig. 3
Fig. 3

Contour lines of NPD in the parameter space of injection rate and bias current, under PP injection. In (a) and (b), the solid, dotted and dashed (plus, triangle, and circle) curves correspond to NPD = 1 , 1 , and 0 respectively, for Δ f A = 0 ( Δ f A = 5 GHz ) .

Fig. 4
Fig. 4

Output powers in the time domain, under PP injection, with Δ f A = 5 GHz and k = 20 ns 1 . (a) J = 0.98 J th ; (b) J = 1.2 J th .

Fig. 5
Fig. 5

Output powers and their correlation index as a function of time shift with Δ f A = 5 GHz and k = 20 ns 1 . (a) and (b) J = 0.98 J th ; (c) and (d) J = 1.2 J th .

Fig. 6
Fig. 6

Output powers in the time domain, under PP injection, with Δ f A = 5 GHz , k = 20 ns 1 , and J = 1.8 J th . (a) XP mode; (b) YP mode.

Fig. 7
Fig. 7

Correlation indexes relating to Fig. 6. Solid curve, XP mode; dotted curve, YP mode.

Fig. 8
Fig. 8

Polarization-resolved L–I curves, under PR injection. Solid (dotted) curve corresponds to XP (YP) mode. (a) laser 1; (b) laser 2.

Fig. 9
Fig. 9

Contour lines of NPD in the parameter space of injection rate and bias current, under PR injection. The solid and dash-dotted curves (plus and triangle) correspond to NPD = 1 and 1 respectively, for laser 1 (laser 2).

Fig. 10
Fig. 10

Output power in the time domain, under PR injection. The solid (dotted) curves correspond to x mode (y mode). In (a) and (b), J = 1.2 J th and k = 20 ns 1 ; in (c) and (d) J = 1.35 J th and k = 6 ns 1 .

Fig. 11
Fig. 11

Output powers in the time domain, under PR injection, with J = 1.2 J th and k = 10 ns 1 . (a) XP mode; (b) YP mode.

Equations (6)

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

d F X 1 , X 2 d s = ( 1 + i α ) M X 1 , X 2 F X 1 , X 2 + σ η F X 2 , X 1 ( s ϵ ) exp [ i ( ± θ A ϕ X 2 , X 1 ) ] ,
d F Y 1 , Y 2 d s = ( 1 + i α ) M Y 1 , Y 2 F Y 1 , Y 2 + ( 1 σ ) η F X 2 , X 1 ( s ϵ ) exp [ i ( θ B , C ϕ X 2 , X 1 ) ] ,
T d M X 1 , X 2 d s = P X 1 , X 2 M X 1 , X 2 μ [ ( 1 + 2 M X 1 , X 2 ) F X 1 , X 2 2 ( 1 + 2 M Y 1 , Y 2 ) F Y 1 , Y 2 2 ] ,
M X = μ ( M Y + 0.5 ) 0.5 ,
g X = g Y + g 0 ( 1 J J 0 ) ,
NPD = P X P Y P X + P Y ,

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