Abstract

Experiments are described that show that polarization of light in a vertical-cavity surface-emitting laser subjected to isotropic optical feedback can jump regularly back and forth between two orthogonal states when the injection current is increased. The polarization-resolved light-current characteristic curves are thus channeled. This effect occurs when the dichroism of the laser is weak enough to allow the effective (isotropic) reflectance to impose the polarization. A comparison of a model introduced by San Miguel et al. [Phys. Rev. A 52, 1728 (1995)] and the standard equations indicates that the best description is given when no spin relaxation is included.

© 1999 Optical Society of America

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References

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  1. J.-H. Ser, Y. G. Ju, J.-H. Shin, and Y. H. Lee, “Polarization stabilization of vertical-cavity top-surface-emitting lasers by inscription of fine metal-interlaced gratings,” Appl. Phys. Lett. 66, 2769–2771 (1995).
    [CrossRef]
  2. D. Sun, E. Towe, P. H. Ostdiek, J. W. Grantham, and G. J. Vansuch, “Polarization control of vertical-cavity surface-emitting lasers through use of an anisotropic gain distribution in [110]-oriented strained quantum-well structures,” IEEE J. Sel. Top. Quantum Electron. 1, 674–680 (1995).
    [CrossRef]
  3. D. Vakshoori and R. E. Leibenguth, “Experimental observation of vertical-cavity with polarization birefringence using asymmetric superlattice,” Appl. Phys. Lett. 67, 1045–1047 (1995).
    [CrossRef]
  4. P. Besnard, B. Meziane, and G. Stéphan, “Feedback phenomena in a semiconductor laser induced by distant reflectors,” IEEE J. Quantum Electron. 29, 1271–1284 (1993).
    [CrossRef]
  5. K. Peterman, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 480–489 (1995).
    [CrossRef]
  6. G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7, 87–143 (1995).
    [CrossRef]
  7. A. Klehr, U. Mertlingk, R. Müller, and R. Oberschmid, “Influence of external optical feedback on polarization switching in InGaAsP/InP lasers,” Opt. Commun. 134, 179–185 (1997).
    [CrossRef]
  8. F. Robert, P. Besnard, M. L. Charès, and G. M. Stéphan, “Switching of the polarization state of a VCSEL using polarized feedback,” Opt. Quantum Electron. 27, 805–811 (1995).
    [CrossRef]
  9. P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Theoretical modeling of vertical surface emitting lasers with polarized feedback,” Phys. Rev. A 56, 3191–3205 (1997).
    [CrossRef]
  10. P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Vertical surface emitting lasers with optical feedback,” presented at the Cardiff Conference on Semiconductor and Integrated Optoelectronics, Cardiff, Wales, March 1997.
  11. J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
    [CrossRef]
  12. G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELS,” Opt. Commun. 146, 136–140 (1998).
    [CrossRef]
  13. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
    [CrossRef]
  14. 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]
  15. 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]
  16. H. Van der Lem and D. Lenstra, “Saturation-induced frequency shift in the noise spectrum of a birefringent vertical-cavity surface emitting laser,” Opt. Lett. 22, 1698–1700 (1997).
    [CrossRef]

1998

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELS,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

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]

1997

H. Van der Lem and D. Lenstra, “Saturation-induced frequency shift in the noise spectrum of a birefringent vertical-cavity surface emitting laser,” Opt. Lett. 22, 1698–1700 (1997).
[CrossRef]

P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Theoretical modeling of vertical surface emitting lasers with polarized feedback,” Phys. Rev. A 56, 3191–3205 (1997).
[CrossRef]

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

A. Klehr, U. Mertlingk, R. Müller, and R. Oberschmid, “Influence of external optical feedback on polarization switching in InGaAsP/InP lasers,” Opt. Commun. 134, 179–185 (1997).
[CrossRef]

1995

F. Robert, P. Besnard, M. L. Charès, and G. M. Stéphan, “Switching of the polarization state of a VCSEL using polarized feedback,” Opt. Quantum Electron. 27, 805–811 (1995).
[CrossRef]

K. Peterman, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 480–489 (1995).
[CrossRef]

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7, 87–143 (1995).
[CrossRef]

J.-H. Ser, Y. G. Ju, J.-H. Shin, and Y. H. Lee, “Polarization stabilization of vertical-cavity top-surface-emitting lasers by inscription of fine metal-interlaced gratings,” Appl. Phys. Lett. 66, 2769–2771 (1995).
[CrossRef]

D. Sun, E. Towe, P. H. Ostdiek, J. W. Grantham, and G. J. Vansuch, “Polarization control of vertical-cavity surface-emitting lasers through use of an anisotropic gain distribution in [110]-oriented strained quantum-well structures,” IEEE J. Sel. Top. Quantum Electron. 1, 674–680 (1995).
[CrossRef]

D. Vakshoori and R. E. Leibenguth, “Experimental observation of vertical-cavity with polarization birefringence using asymmetric superlattice,” Appl. Phys. Lett. 67, 1045–1047 (1995).
[CrossRef]

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

P. Besnard, B. Meziane, and G. Stéphan, “Feedback phenomena in a semiconductor laser induced by distant reflectors,” IEEE J. Quantum Electron. 29, 1271–1284 (1993).
[CrossRef]

1980

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

Abraham, N. B.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Besnard, P.

P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Theoretical modeling of vertical surface emitting lasers with polarized feedback,” Phys. Rev. A 56, 3191–3205 (1997).
[CrossRef]

F. Robert, P. Besnard, M. L. Charès, and G. M. Stéphan, “Switching of the polarization state of a VCSEL using polarized feedback,” Opt. Quantum Electron. 27, 805–811 (1995).
[CrossRef]

P. Besnard, B. Meziane, and G. Stéphan, “Feedback phenomena in a semiconductor laser induced by distant reflectors,” IEEE J. Quantum Electron. 29, 1271–1284 (1993).
[CrossRef]

Chares, M.-L.

P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Theoretical modeling of vertical surface emitting lasers with polarized feedback,” Phys. Rev. A 56, 3191–3205 (1997).
[CrossRef]

Charès, M. L.

F. Robert, P. Besnard, M. L. Charès, and G. M. Stéphan, “Switching of the polarization state of a VCSEL using polarized feedback,” Opt. Quantum Electron. 27, 805–811 (1995).
[CrossRef]

Feng, Q.

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]

Gabrysch, M.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELS,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Giacomelli, G.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELS,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Grantham, J. W.

D. Sun, E. Towe, P. H. Ostdiek, J. W. Grantham, and G. J. Vansuch, “Polarization control of vertical-cavity surface-emitting lasers through use of an anisotropic gain distribution in [110]-oriented strained quantum-well structures,” IEEE J. Sel. Top. Quantum Electron. 1, 674–680 (1995).
[CrossRef]

Gulden, K. H.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELS,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Ju, Y. G.

J.-H. Ser, Y. G. Ju, J.-H. Shin, and Y. H. Lee, “Polarization stabilization of vertical-cavity top-surface-emitting lasers by inscription of fine metal-interlaced gratings,” Appl. Phys. Lett. 66, 2769–2771 (1995).
[CrossRef]

Klehr, A.

A. Klehr, U. Mertlingk, R. Müller, and R. Oberschmid, “Influence of external optical feedback on polarization switching in InGaAsP/InP lasers,” Opt. Commun. 134, 179–185 (1997).
[CrossRef]

Kobayashi, K.

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

Lang, R.

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

Lee, Y. H.

J.-H. Ser, Y. G. Ju, J.-H. Shin, and Y. H. Lee, “Polarization stabilization of vertical-cavity top-surface-emitting lasers by inscription of fine metal-interlaced gratings,” Appl. Phys. Lett. 66, 2769–2771 (1995).
[CrossRef]

Leibenguth, R. E.

D. Vakshoori and R. E. Leibenguth, “Experimental observation of vertical-cavity with polarization birefringence using asymmetric superlattice,” Appl. Phys. Lett. 67, 1045–1047 (1995).
[CrossRef]

Lenstra, D.

H. Van der Lem and D. Lenstra, “Saturation-induced frequency shift in the noise spectrum of a birefringent vertical-cavity surface emitting laser,” Opt. Lett. 22, 1698–1700 (1997).
[CrossRef]

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7, 87–143 (1995).
[CrossRef]

Marin, F.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELS,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Mertlingk, U.

A. Klehr, U. Mertlingk, R. Müller, and R. Oberschmid, “Influence of external optical feedback on polarization switching in InGaAsP/InP lasers,” Opt. Commun. 134, 179–185 (1997).
[CrossRef]

Meziane, B.

P. Besnard, B. Meziane, and G. Stéphan, “Feedback phenomena in a semiconductor laser induced by distant reflectors,” IEEE J. Quantum Electron. 29, 1271–1284 (1993).
[CrossRef]

Moloney, J. V.

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]

Moser, M.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELS,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Müller, R.

A. Klehr, U. Mertlingk, R. Müller, and R. Oberschmid, “Influence of external optical feedback on polarization switching in InGaAsP/InP lasers,” Opt. Commun. 134, 179–185 (1997).
[CrossRef]

Oberschmid, R.

A. Klehr, U. Mertlingk, R. Müller, and R. Oberschmid, “Influence of external optical feedback on polarization switching in InGaAsP/InP lasers,” Opt. Commun. 134, 179–185 (1997).
[CrossRef]

Ostdiek, P. H.

D. Sun, E. Towe, P. H. Ostdiek, J. W. Grantham, and G. J. Vansuch, “Polarization control of vertical-cavity surface-emitting lasers through use of an anisotropic gain distribution in [110]-oriented strained quantum-well structures,” IEEE J. Sel. Top. Quantum Electron. 1, 674–680 (1995).
[CrossRef]

Pesquera, L.

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]

Peterman, K.

K. Peterman, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 480–489 (1995).
[CrossRef]

Prati, F.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Regalado, J. M.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Robert, F.

P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Theoretical modeling of vertical surface emitting lasers with polarized feedback,” Phys. Rev. A 56, 3191–3205 (1997).
[CrossRef]

F. Robert, P. Besnard, M. L. Charès, and G. M. Stéphan, “Switching of the polarization state of a VCSEL using polarized feedback,” Opt. Quantum Electron. 27, 805–811 (1995).
[CrossRef]

San Miguel, M.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

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]

Ser, J.-H.

J.-H. Ser, Y. G. Ju, J.-H. Shin, and Y. H. Lee, “Polarization stabilization of vertical-cavity top-surface-emitting lasers by inscription of fine metal-interlaced gratings,” Appl. Phys. Lett. 66, 2769–2771 (1995).
[CrossRef]

Shin, J.-H.

J.-H. Ser, Y. G. Ju, J.-H. Shin, and Y. H. Lee, “Polarization stabilization of vertical-cavity top-surface-emitting lasers by inscription of fine metal-interlaced gratings,” Appl. Phys. Lett. 66, 2769–2771 (1995).
[CrossRef]

Shore, K. A.

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]

Stephan, G.

P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Theoretical modeling of vertical surface emitting lasers with polarized feedback,” Phys. Rev. A 56, 3191–3205 (1997).
[CrossRef]

Stéphan, G.

P. Besnard, B. Meziane, and G. Stéphan, “Feedback phenomena in a semiconductor laser induced by distant reflectors,” IEEE J. Quantum Electron. 29, 1271–1284 (1993).
[CrossRef]

Stéphan, G. M.

F. Robert, P. Besnard, M. L. Charès, and G. M. Stéphan, “Switching of the polarization state of a VCSEL using polarized feedback,” Opt. Quantum Electron. 27, 805–811 (1995).
[CrossRef]

Sun, D.

D. Sun, E. Towe, P. H. Ostdiek, J. W. Grantham, and G. J. Vansuch, “Polarization control of vertical-cavity surface-emitting lasers through use of an anisotropic gain distribution in [110]-oriented strained quantum-well structures,” IEEE J. Sel. Top. Quantum Electron. 1, 674–680 (1995).
[CrossRef]

Towe, E.

D. Sun, E. Towe, P. H. Ostdiek, J. W. Grantham, and G. J. Vansuch, “Polarization control of vertical-cavity surface-emitting lasers through use of an anisotropic gain distribution in [110]-oriented strained quantum-well structures,” IEEE J. Sel. Top. Quantum Electron. 1, 674–680 (1995).
[CrossRef]

Vakshoori, D.

D. Vakshoori and R. E. Leibenguth, “Experimental observation of vertical-cavity with polarization birefringence using asymmetric superlattice,” Appl. Phys. Lett. 67, 1045–1047 (1995).
[CrossRef]

Valle, A.

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]

Van der Lem, H.

van Tartwijk, G. H. M.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7, 87–143 (1995).
[CrossRef]

Vansuch, G. J.

D. Sun, E. Towe, P. H. Ostdiek, J. W. Grantham, and G. J. Vansuch, “Polarization control of vertical-cavity surface-emitting lasers through use of an anisotropic gain distribution in [110]-oriented strained quantum-well structures,” IEEE J. Sel. Top. Quantum Electron. 1, 674–680 (1995).
[CrossRef]

Appl. Phys. Lett.

D. Vakshoori and R. E. Leibenguth, “Experimental observation of vertical-cavity with polarization birefringence using asymmetric superlattice,” Appl. Phys. Lett. 67, 1045–1047 (1995).
[CrossRef]

J.-H. Ser, Y. G. Ju, J.-H. Shin, and Y. H. Lee, “Polarization stabilization of vertical-cavity top-surface-emitting lasers by inscription of fine metal-interlaced gratings,” Appl. Phys. Lett. 66, 2769–2771 (1995).
[CrossRef]

IEEE J. Quantum Electron.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

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

P. Besnard, B. Meziane, and G. Stéphan, “Feedback phenomena in a semiconductor laser induced by distant reflectors,” IEEE J. Quantum Electron. 29, 1271–1284 (1993).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

K. Peterman, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 480–489 (1995).
[CrossRef]

D. Sun, E. Towe, P. H. Ostdiek, J. W. Grantham, and G. J. Vansuch, “Polarization control of vertical-cavity surface-emitting lasers through use of an anisotropic gain distribution in [110]-oriented strained quantum-well structures,” IEEE J. Sel. Top. Quantum Electron. 1, 674–680 (1995).
[CrossRef]

IEEE Photonics Technol. Lett.

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]

Opt. Commun.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELS,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

A. Klehr, U. Mertlingk, R. Müller, and R. Oberschmid, “Influence of external optical feedback on polarization switching in InGaAsP/InP lasers,” Opt. Commun. 134, 179–185 (1997).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

F. Robert, P. Besnard, M. L. Charès, and G. M. Stéphan, “Switching of the polarization state of a VCSEL using polarized feedback,” Opt. Quantum Electron. 27, 805–811 (1995).
[CrossRef]

Phys. Rev. A

P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Theoretical modeling of vertical surface emitting lasers with polarized feedback,” Phys. Rev. A 56, 3191–3205 (1997).
[CrossRef]

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]

Quantum Semiclassic. Opt.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7, 87–143 (1995).
[CrossRef]

Other

P. Besnard, F. Robert, M.-L. Chares, and G. Stephan, “Vertical surface emitting lasers with optical feedback,” presented at the Cardiff Conference on Semiconductor and Integrated Optoelectronics, Cardiff, Wales, March 1997.

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Figures (5)

Fig. 1
Fig. 1

Experimental arrangement for control of the polarization state of VCSEL’s.

Fig. 2
Fig. 2

(a) Experimental L-I curve. The total optical intensity has a a common behavior, whereas polarization switching is observed for each polarized component when the bias current is varied. The output power of X returns to zero when the output power of Y takes maxima. Only the L-I curve of X is represented for clarity. The small modulation of the total power is due to optical feedback. Similar curves have been obtained with polarized optical feedback. (b), (c) Lower feedback strength.

Fig. 3
Fig. 3

Numerical simulation of polarization switching in L-I curves (SFM model). The optical length is fixed at 4 mm. When r3 is increased, switching is observed for higher values of the bias current. (a) C=0.1092, (b) C=0.04368.

Fig. 4
Fig. 4

Computation of the largest real part of the Lyapunov exponent for each mode (SFM model). In this simulation, L=7.5 mm. The straight lines are exponents for the laser without feedback. With feedback, the Lyapunov exponents are modulated along with the bias current. Increasing r3 increases the depth of the oscillations, so there are more jumps from one polarization to the other. Inset, exponent of mode X. Another computation is given for a small value of the spin relaxation (0.001 ps). Even for the small value of r3 shown, the laser always regularly switches from one polarization mode to the other.

Fig. 5
Fig. 5

Numerical simulation of polarization switching in L-I curves (Lang and Kobayashi model). L is the optical length. When r3 is decreased to the value shown, switching is incomplete.

Equations (13)

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

νx(y)=vx(y)th+δνδibias(ibias-ith)νxth-300[GHz/mA]ibias.
Δr2=|r2eff (x)|2-|r2eff (y)|2=-2r2sin2πδνδibias(ibias-ith)2Lcsin(πΔντ).
dNdt=P-Nτe-12ΓGN(N-N0)(Ix+Iy)-ΓGNSIm(ExEy*),
dSdt=-Sτs-ΓGN(N-N0)Im(ExEy*)-12ΓGNS(Ix+Iy),
E˙x(y)=½ΓGN(1+iα)[(N-Nth)Ex(y)iSEy(x)]±(γa+iγb)Ex(y)+κEx(y)(t-τ)exp(-i2πνx(y)),
Nx(y)s=Nth±2ΓGN{γaκcos[ωx(y)sτ]}.
Δωx(y)sτ=ωx(y)sτ-ωx(y)thτ=γbτ±αγaτ-κτ1+α2sin[ωx(y)sτ+Atan(α)].
c(a2+b2)+ΓGNJ-Nsτe(a+αb)
+λa2+b2-ΓGNJ-Nsτe-2ac
+λ2(c-2a)+λ3=0,
a={±2γa+κ[exp(-λτ)-1]cos(ωsτ)},
b={±2γb-κ[exp(-λτ)-1]sin(ωst)},
c=1τs+(P-N0/τe)(Ns-N0),

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