Abstract

We investigate the influence of vectorial degrees of freedom on stationary and dynamic behavior of the amplitude-polarization parameters of vertical-cavity surface-emitting lasers. It is demonstrated that these lasers can exhibit complex dynamics, including fully developed amplitude-polarization chaos, even when a single polarization pattern is excited. The appearance of chaotic dynamics is attributed to the vectorial mechanism of destabilization, which can act on the dynamics independently of the conventional scalar mechanism. This property provides background for new measurement methods.

© 2000 Optical Society of America

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  1. N. B. Abraham and G. M. Stephan, eds., Special issue on polarization effects in lasers and spectroscopy, J. Quant. Semiclass. Opt. Part B 10, 1–308 (1998).
    [CrossRef]
  2. N. B. Abraham, M. D. Matlin, and R. S. Gioggia, “Polarization stability and dynamics in a model for a Zeeman laser that goes beyond third-order Lamb theory,” Phys. Rev. A 53, 3514–3528 (1996); M. D. Matlin, R. S. Gioggia, N. B. Abraham, P. Glorieux, and T. Crawford, “Polarization switch in a Zeeman laser in the presence of dynamical instabilities,” Opt. Commun. 120, 204–222 (1995); N. B. Abraham, E. Arimondo, and M. San Miguel, “Polarization state selection and stability in a laser with a polarization isotropic resonator: an example of no lasing despite inversion above threshold,” Opt. Commun. OPCOB8 117, 344–356 (1995).
    [CrossRef] [PubMed]
  3. C. J. Chang-Hasnain, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Polarization characteristics of quantum-well vertical-cavity surface-emitting lasers,” Electron. Lett. 27, 163–165 (1991); K. D. Choquette, K. L. Lear, R. E. Leibenguth, and M. T. Asom, “Polarization modulation of cruciform vertical-cavity laser diodes,” Appl. Phys. Lett. 64, 2767–2769 (1994).
    [CrossRef]
  4. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Deerfield Beach, Fla., 1991).
  5. 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]
  6. X. Martin-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]
  7. M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, “Physical insight into the polarization dynamics of semiconductor vertical-cavity lasers,” Phys. Rev. A 57, 2080–2090 (1998).
    [CrossRef]
  8. A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Decay rate of the electromagnetic field polarization in a vector laser,” J. Quant. Semiclass. Opt. Part B 10, 107–114 (1998).
    [CrossRef]
  9. A. Kul’minskii, Yu. Loiko, and A. Voitovich, “The effect of the vectorial degree of freedom on the dynamics of class-A lasers,” Opt. Commun. 167, 235–259 (1999).
    [CrossRef]
  10. A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Polarization chaos in a vector nonautonomous class-A laser,” J. Opt. B. 1, 294–298 (1999).
    [CrossRef]
  11. R. C. Jones, “New calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–492 (1941); “New calculus for the treatment of optical systems. VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685 (1948); “New calculus for the treatment of optical systems. VIII. Electromagnetic theory,” J. Opt. Soc. Am. A JOAOD6 46, 126–131 (1956).
    [CrossRef]
  12. A. P. Voitovich and V. N. Severikov, Lasers with Anisotropic Resonator (Nauka i Tehnika, Minsk, Belarus, 1988, in Russian).
  13. P. Paddon, E. Sjerve, A. D. May, M. Bourouis, and G. Stephan, “Polarization modes in a quasi-isotropic laser: a general anisotropy model with applications,” J. Opt. Soc. Am. B 9, 574–589 (1992).
    [CrossRef]
  14. H. J. Raterink, H. V. Stadt, C. H. Velzel, and G. Dikstra, “Development of a ring laser for polarimetric measurements,” Appl. Opt. 6, 813–820 (1967).
    [CrossRef] [PubMed]

1999 (2)

A. Kul’minskii, Yu. Loiko, and A. Voitovich, “The effect of the vectorial degree of freedom on the dynamics of class-A lasers,” Opt. Commun. 167, 235–259 (1999).
[CrossRef]

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Polarization chaos in a vector nonautonomous class-A laser,” J. Opt. B. 1, 294–298 (1999).
[CrossRef]

1998 (3)

N. B. Abraham and G. M. Stephan, eds., Special issue on polarization effects in lasers and spectroscopy, J. Quant. Semiclass. Opt. Part B 10, 1–308 (1998).
[CrossRef]

M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, “Physical insight into the polarization dynamics of semiconductor vertical-cavity lasers,” Phys. Rev. A 57, 2080–2090 (1998).
[CrossRef]

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Decay rate of the electromagnetic field polarization in a vector laser,” J. Quant. Semiclass. Opt. Part B 10, 107–114 (1998).
[CrossRef]

1997 (1)

X. Martin-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]

1995 (1)

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]

1992 (1)

1967 (1)

Abraham, N. B.

N. B. Abraham and G. M. Stephan, eds., Special issue on polarization effects in lasers and spectroscopy, J. Quant. Semiclass. Opt. Part B 10, 1–308 (1998).
[CrossRef]

X. Martin-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]

Bourouis, M.

Dikstra, G.

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]

Hendriks, R. F. M.

M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, “Physical insight into the polarization dynamics of semiconductor vertical-cavity lasers,” Phys. Rev. A 57, 2080–2090 (1998).
[CrossRef]

Kul’minskii, A.

A. Kul’minskii, Yu. Loiko, and A. Voitovich, “The effect of the vectorial degree of freedom on the dynamics of class-A lasers,” Opt. Commun. 167, 235–259 (1999).
[CrossRef]

Kul’minskii, A. M.

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Polarization chaos in a vector nonautonomous class-A laser,” J. Opt. B. 1, 294–298 (1999).
[CrossRef]

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Decay rate of the electromagnetic field polarization in a vector laser,” J. Quant. Semiclass. Opt. Part B 10, 107–114 (1998).
[CrossRef]

Loiko, Yu.

A. Kul’minskii, Yu. Loiko, and A. Voitovich, “The effect of the vectorial degree of freedom on the dynamics of class-A lasers,” Opt. Commun. 167, 235–259 (1999).
[CrossRef]

Martin-Regalado, X.

X. Martin-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]

May, A. D.

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]

Paddon, P.

Prati, F.

X. Martin-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]

Raterink, H. J.

San Miguel, M.

X. Martin-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]

Severikov, V. N.

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Polarization chaos in a vector nonautonomous class-A laser,” J. Opt. B. 1, 294–298 (1999).
[CrossRef]

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Decay rate of the electromagnetic field polarization in a vector laser,” J. Quant. Semiclass. Opt. Part B 10, 107–114 (1998).
[CrossRef]

Sjerve, E.

Stadt, H. V.

Stephan, G.

Stephan, G. M.

N. B. Abraham and G. M. Stephan, eds., Special issue on polarization effects in lasers and spectroscopy, J. Quant. Semiclass. Opt. Part B 10, 1–308 (1998).
[CrossRef]

van Exter, M. P.

M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, “Physical insight into the polarization dynamics of semiconductor vertical-cavity lasers,” Phys. Rev. A 57, 2080–2090 (1998).
[CrossRef]

Velzel, C. H.

Voitovich, A.

A. Kul’minskii, Yu. Loiko, and A. Voitovich, “The effect of the vectorial degree of freedom on the dynamics of class-A lasers,” Opt. Commun. 167, 235–259 (1999).
[CrossRef]

Voitovich, A. P.

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Polarization chaos in a vector nonautonomous class-A laser,” J. Opt. B. 1, 294–298 (1999).
[CrossRef]

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Decay rate of the electromagnetic field polarization in a vector laser,” J. Quant. Semiclass. Opt. Part B 10, 107–114 (1998).
[CrossRef]

Woerdman, J. P.

M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, “Physical insight into the polarization dynamics of semiconductor vertical-cavity lasers,” Phys. Rev. A 57, 2080–2090 (1998).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

X. Martin-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]

J. Opt. B. (1)

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Polarization chaos in a vector nonautonomous class-A laser,” J. Opt. B. 1, 294–298 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Quant. Semiclass. Opt. Part B (2)

A. M. Kul’minskii, A. P. Voitovich, and V. N. Severikov, “Decay rate of the electromagnetic field polarization in a vector laser,” J. Quant. Semiclass. Opt. Part B 10, 107–114 (1998).
[CrossRef]

N. B. Abraham and G. M. Stephan, eds., Special issue on polarization effects in lasers and spectroscopy, J. Quant. Semiclass. Opt. Part B 10, 1–308 (1998).
[CrossRef]

Opt. Commun. (1)

A. Kul’minskii, Yu. Loiko, and A. Voitovich, “The effect of the vectorial degree of freedom on the dynamics of class-A lasers,” Opt. Commun. 167, 235–259 (1999).
[CrossRef]

Phys. Rev. A (2)

M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, “Physical insight into the polarization dynamics of semiconductor vertical-cavity lasers,” Phys. Rev. A 57, 2080–2090 (1998).
[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]

Other (5)

R. C. Jones, “New calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–492 (1941); “New calculus for the treatment of optical systems. VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685 (1948); “New calculus for the treatment of optical systems. VIII. Electromagnetic theory,” J. Opt. Soc. Am. A JOAOD6 46, 126–131 (1956).
[CrossRef]

A. P. Voitovich and V. N. Severikov, Lasers with Anisotropic Resonator (Nauka i Tehnika, Minsk, Belarus, 1988, in Russian).

N. B. Abraham, M. D. Matlin, and R. S. Gioggia, “Polarization stability and dynamics in a model for a Zeeman laser that goes beyond third-order Lamb theory,” Phys. Rev. A 53, 3514–3528 (1996); M. D. Matlin, R. S. Gioggia, N. B. Abraham, P. Glorieux, and T. Crawford, “Polarization switch in a Zeeman laser in the presence of dynamical instabilities,” Opt. Commun. 120, 204–222 (1995); N. B. Abraham, E. Arimondo, and M. San Miguel, “Polarization state selection and stability in a laser with a polarization isotropic resonator: an example of no lasing despite inversion above threshold,” Opt. Commun. OPCOB8 117, 344–356 (1995).
[CrossRef] [PubMed]

C. J. Chang-Hasnain, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Polarization characteristics of quantum-well vertical-cavity surface-emitting lasers,” Electron. Lett. 27, 163–165 (1991); K. D. Choquette, K. L. Lear, R. E. Leibenguth, and M. T. Asom, “Polarization modulation of cruciform vertical-cavity laser diodes,” Appl. Phys. Lett. 64, 2767–2769 (1994).
[CrossRef]

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Deerfield Beach, Fla., 1991).

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Figures (9)

Fig. 1
Fig. 1

Stability diagram of various polarization states for μ=2.4 on the plane (ωP, γP). The x (y) mode is stable above (below) the HBx (HBy) curve outside the left (right) circle. The EP modes are stable inside vertically hatched domains. Inside horizontally hatched domains the stable dynamics is time dependent. Dashed–dotted curve, variation of ωP and γP for magnitudes of Δ in Figs. 24. Dashed lines are given for reference. See text for other details.

Fig. 2
Fig. 2

Phase diagram on plane (Δ, μ). Amplitude anisotropy is assumed to be absent (p=1). The stability domain of the x mode lies below curve PBx (dashed arrows). The y mode is stable at the right of line HBy. The EP states are stable in the vertically hatched domain. In the domain labeled dynamical states, stable solutions are time dependent.

Fig. 3
Fig. 3

One-parameter diagram of ellipticity as a function of phase anisotropy, showing some dynamic states when the y mode (LPy) becomes unstable (right-hand edge of the dashed–dotted curve in Fig. 1). Lower inset, symmetry-breaking (PB), period-doubling (PD), and torus (TR) bifurcations that affect symmetric (Ps) and asymmetric (Pa±) periodic attractors. Upper inset, domain with period-doubling cascade PD of asymmetric periodic solution Pa+, which leads to asymmetric chaotic attractor CHa. CHs, domain of a fully developed symmetric chaotic attractor. Horizontal arrows, directions of scanning of the phase anisotropy in the domain of existence of two asymmetric attractors, Pa+ and Pa-.

Fig. 4
Fig. 4

Same as in Fig. 3 but for the x mode (left-hand edge of the dashed–dotted curve in Fig. 1). The domain with an imperfect symmetry-breaking bifurcation PBi is magnified in the inset. In this domain of phase space there are four stable asymmetric periodic attractors, Pa1, 2±.

Fig. 5
Fig. 5

Spectra of the ellipticity for full [Eqs. (6), left-hand column] and simplified [Eqs. (14), right-hand column] models for (a), (e), period-one and (b), (f) period-two asymmetric attractors and (c), (g) asymmetric and (d), (h) symmetric chaotic attractors for the following values of Δ (rad): (a) 0.062, (b) 0.067, (c) 0.06805, (d) 0.069, (e) 0.056, (f) 0.057, (g) 0.059 (h) 0.06.

Fig. 6
Fig. 6

Same as in Fig. 3 but for the simplified model [Eqs. (14)].

Fig. 7
Fig. 7

Same as in Fig. 4 but for the simplified model [Eqs. (14)]. PD1–PD3, period-doubling cascades.

Fig. 8
Fig. 8

First return maps of the (a), (c) asymmetric CHa± and (b), (d) symmetric CHs chaotic attractors of full (left) and simplified (right) models for (a) Δ=0.06805, (b) Δ=0.069, (c) Δ=0.0585, (d) Δ=0.06 rad. CHa- attractors are circled.

Fig. 9
Fig. 9

Chaotic time series of (a), (e) carrier difference n; (b), (f) ellipticity ξ; (c), (g) Poincaré sections, and (d), (h) spectra of the ellipticity for full (left) and simplified (right) models for the same value of the phase anisotropy, Δ=-0.18 rad.

Equations (39)

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E˙+=-κE+-iω0E++κ(1+iα)(N+n)E+,
E˙-=-κE--iω0E-+κ(1+iα)(N-n)E-,
N˙=-γ(N-μ)-γN(|E+|2+|E-|2)-γn(|E+|2-|E-|2),
n˙=-γsn-γN(|E+|2-|E-|2)-γn(|E+|2+|E-|2),
E˙=cnL{Aˆr E-E}cnLBˆr E.
Aˆrcir=εx+εy2 1001+εx-εy2 0110.
E˙=c(γP+iωP)nL -111-1E.
E˙+=-κ(1+iα)E+-(γP+iωP)E++(γP+iωP)E-+κ(1+iα)(N+n)E+,
E˙-=-κ(1+iα)E--(γP+iωP)E-+(γP+iωP)E++κ(1+iα)(N-n)E-,
N˙=-γ(N-μ)-γN(|E+|2+|E-|2)-γn(|E+|2-|E-|2),
n˙=-γsn-γN(|E+|2-|E-|2)-γn(|E+|2+|E-|2).
I˙/2I=κ(N-1)-κn tanh 2β-γP(1-cos 2Φ/cosh 2β),
N˙=-γ(N-μ)-γI(N-n tanh 2β),
n˙=-γsn-γI(n-N tanh 2β),
β˙=-ωP sin 2Φ cosh 2β-γP cos 2Φ sinh 2β-κn,
Φ˙=-γP sin 2Φ cosh 2β+ωP cos 2Φ sinh 2β+κnα,
φ˙=-ωP+ωP cos 2Φ cosh 2β-γP sin 2Φ sinh 2β+κα(N-1).
Φx=0,βx=0,nx=0,Nx=1,
Ix=μ-1,φ˙x=0;
Φy=±π/2,βy=0,ny=0,
Ny=1+2γP/κ,Iy(μ-1)-2µγP/κ,
φ˙y=2(γPα-ωP).
2κ(Nx,y-1)2κIx,y-γNx,y-γ(1+Ix,y),
2γP2ωP-κ2ωP2γPκα2γIx,yNx,y0-(γs+γIx,y),
λ1,2x, y=-1/2[γ(1+Ix,y)-2κ(Nx,y-1)]±1/2{[γ(1+Ix,y)-2κ(Nx,y-1)]2-8κγ(1+Ix,y-Nx,y)}1/2.
λ3+λ2a1+λa2+a3=0,
a1=γIx,y+γs±4γP,
a2=4γP2+4ωP2+2κγIx,yNx,y±4γP(γIx,y+γs),
a3=4(γP2+ωP2)(γIx,y+γs)±4κγIx,yNx,y(γP+ωPα).
4(γP2+ωP2)(γIx,y+γs)±4Nx,y Ix,yκγ(γP+ωPα)=0.
ωP+(μ-1)γκ2[(μ-1)γ+γs]α2
+γP+(μ-1)γκ2[(μ-1)γ+γs]2
=(μ-1)2γ2κ24[(μ-1)γ+γs]2(1+α2).
a1a2-a3=0.
n˙=-γsn-γIa(n-Na tanh 2β),
β˙=-ωP sin 2Φ cosh 2β-γP cos 2Φ sinh 2β-κn,
Φ˙=-γP sin 2Φ cosh 2β+ωP cos 2Φ sinh 2β+κnα,
Na=1+n tanh 2β+γPκ 1-cos 2Φcosh 2β,
Ia=μ-NaNa-n tanh 2β.

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