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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 2, Iss. 10 — May. 11, 1998
  • pp: 424–430
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Spatio-spectral dynamics and spontaneous ultrafast optical switching in VCSEL arrays

Ortwin Hess  »View Author Affiliations


Optics Express, Vol. 2, Issue 10, pp. 424-430 (1998)
http://dx.doi.org/10.1364/OE.2.000424


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Abstract

Microscopic simulations on the basis of semiconductor Maxwell-Bloch equations show that in the short-time spatio-temporal dynamics of large aspect vertical cavity surface emitting lasers (VC-SEL) and coupled VCSEL-arrays microscopic and macroscopic effects are intrinsically coupled. The combination of microscopic spatial and spectral dynamics of the carrier distribution functions and the nonlinear polarization of the active semiconductor medium reveal spatio-spectral hole-burning effects as the origin of ultra-fast mode-switching effects. In coupled VCSEL-arrays the simulations predict the emergence of spontaneous ultra-fast spatial switching.

© Optical Society of America

In the following, we will develop a theoretical model for spatially extended and coupled VCSELs and perform numerical simulations which deliver answers to these questions. Motivated by the indirect experimental evidence that spatial, spectral and temporal properties are of combined and simultaneous relevance to the transverse mode - dynamics of VCSELs2–5

2. O. Buccafusca, J. L. A. Chilla, J. J. Rocca, C. Wilmsen, S. Feld, and R. Leibenguth, “Ultrahigh frequency oscillations and multimode dynamics in vertical cavity surface emitting lasers,” Appl. Phys. Lett. 67, 185–187 (1995). [CrossRef]

, we numerically model the interrelations of the spatial and spectral distributions of the ultra-high frequency dynamics of VCSELs and phase-coupled VCSEL-arrays. To account for the microscopic processes which act in concert with the macroscopic spatio-temporal interactions we will base our investigation of large-aspect-ratio and coupled VCSELs on the semiconductor laser model derived in6

6. O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description,” Phys. Rev. A 54, 3347–3359 (1996). [CrossRef] [PubMed]

and applied to the description of broad-area lasers7

7. O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics,” Phys. Rev. A 54, 3360–3368 (1996). [CrossRef] [PubMed]

. In addition, the polarization of light emitted from VCSELs is generally highly sensitive to small anisotropies in the crystal structure, strain or optical anisotropies in the mirrors8–12

8. C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. V. Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, Polarization, and Transverse mode Characteristics of Vertical Cavity Surface Emitting Lasers,” IEEE J. Quantum Electron. 27, 1402–1409 (1991). [CrossRef]

. In practice, however, the polarization of the emitted light is frequently stable under usual cw operating conditions9

9. F. Koyama, K. Morito, and K. Iga, “Intensity noise and polarization stability of GaAlAs-GaAs surface emitting lasrs,” IEEE J. Quantum Electron. QE-27, 1410–1416 (1991). [CrossRef]

. Indeed, our analysis of the mutual influence of multiple anisotropies (gain-, loss- and frequency-anisotropy) and quantum fluctuations on the emission properties of VCSELs13

13. H. F. Hofmann and O. Hess, “Quantum Noise and Polarization Fluctuations in Vertical Cavity Surface Emitting Lasers,” Phys. Rev. A 56, 868–876 (1997). [CrossRef]

allows us here to resort to the assumption of a single stable polarization direction.

For spatially inhomogeneous VCSELs and VCSEL-arrays the general Maxwell-Bloch equations for the Wigner distributions fe,h (k, x, t) of electrons (e) and holes (h) and the interband polarization pnl (k, x, t)

tfe,h(k,x,t)=g(k,x,t)+Λe,h(k,x,t)τe,h1(k)[fe,h(k,x,t)feqe,h(k,x,t)]
Γsp(k)fe(k,x,t)fh(k,x,t)γnrfe,h(k,x,t)
(1a)
tpnl(k,x,t)=[iω¯(k)+τp1(k)]pnl(k,x,t)+βΓsp(k)fe(k,x,t)fh(k,x,t)
+1ΔUnl(x,t)1U(x,t)[fe(k,x,t)+fh(k,x,t)]
(1b)

couples the microscopic spatial (x = (x,y)) and spectral (k) dynamics of the Wigner distributions with the macroscopic spatio-temporal dynamics

nlctE(x,t)=i21KzT2E(x,t)(γm+α(x)2+(x))E+inl2ε0Pnl(x,t)
(2a)
tN(x,t)=TDfTN(x,t)γnrN(x,t)+Λ(x,t)+G(x,t)W(x,t)
(2b)

Pnl(x,t)=dcv*Vkpnl(k,x,t),
(3)

where V is the volume, and the microscopic and macroscopic generation rates

g(k,x,t)=χ˜(ω;k,x,t)2ħE(x,t)2
12ħIm[dcvE(x,t)pnl*(k,x,t)ΔU(x,t)p*(k,x,t)]
(4a)
G(x,t)=χ′′ħE(x,t)212ħIm[E(x,t)Pnl*(x,t)],
(4b)

In our analysis we particularly concentrate on the interrelations between microscopic and macroscopic processes in the active laser medium which we consider in the numerical simulations simultaneously and together with the macroscopic device properties. To this means the coupled system of partial differential equations is solved by direct numerical integration15

15. O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).

, where the relevant material properties and parameters employed in the simulation are detailed in Ref.14

14. O. Hess and T. Kuhn, “Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis,” Prog. Quantum Electron. 20, 85–179 (1996). [CrossRef]

. Fig. 1 displays snapshots of the spatial distribution of the intensity and density of a typical large-aspect ratio VCSEL with a transverse width w = 30 μm. The time interval between successive snapshots is 3 picoseconds.

Fig. 1. Snapshots of the intensity (left column) and charge carrier density (right column) of a large aspect-ratio (d = 30 μm) VCSEL. The time between successive snapshots is Δt = 3 ps.

From the pictures of the intensity one can clearly observe the amplification of a number of transverse mode distributions in the VCSEL and their variation from frame to frame. The corresponding spatial distribution of the charge carrier density (momentum integrated) has the shape of a volcano with a steep slope at the edges, peak values at the rim, and more gradual but dynamic variations in the middle. It displays the spatial holes burnt by the intracavity field. In turn, N(x) determines the induced waveguiding properties. Additionally, the transport of charge carriers leads to a spatial redistribution of the carriers. Due to the difference in characteristic time scales14

14. O. Hess and T. Kuhn, “Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis,” Prog. Quantum Electron. 20, 85–179 (1996). [CrossRef]

, however, it occurs on a much slower time scale than the spatial hole burning. As Fig. 1 shows, after only 3 picoseconds the intracavity intensity distribution has fundamentally changed in space by 5–10 μm. After spatial integration, the resulting signal corresponds to the high-frequency spectra observed experimentally.

The reason for this rapid spatio-temporal intensity variations lies in the dynamics of the microscopic carrier Wigner distributions which simultaneously may display ultrafast spectrally and spatially varying properties. Animation 2 visualizes the spatio-spectral dynamics of the electron Wigner distribution δfe (k,x,y 0,t) = fe (k,x,y 0,t) - feqe (k,x,y 0,t) along a spatial cut at y 0 = 0 during a characteristic time-interval of 100 ps.

Fig. 2. Animation of the spatio-spectral dynamics of the carrier Wigner distribution δfe (k,0,x,t) during a time interval of 200 ps. The horizontal axis shows the momentum (k) dependence of δfe (k,0,x,t), given in units of the inverse exciton Bohr radius a 0 = 1.295 × 10-6 cm of GaAs, i.e. a shift from e.g. ka 0 = 2 to ka 0 = 2.5 corresponds to a wavelength shift of approximately 10 nm. The vertical axis depicts the spatial dependence of δfe and is given in units of μm. [Media 1]

How are the microscopic spatio-spectral effects in the VCSEL effected by the coherently phase coupling in a VCSEL-array? To limit the complexity of the discussion, we here confine ourselves to an array of four (2 × 2) round gain-guided VCSELs. Depending on the separation s between the lasers, the array will be strongly or weakly coupled through the evanescent optical waves and charge carriers which may diffuse within and between the lasers.

Fig. 3. Animation of the spatio-temporal intensity dynamics of a strongly coupled VCSEL-array consisting of four round large-aspect ratio VCSELs. Each VCSEL has a transverse width w = 30μm and is separated from its neighbor at a distance s = 5μm. The total time sequence shown in the animation displays a typical time-period of 100 ps. [Media 2]

In vivid contrast to animation 3, animation 4 demonstrates that reducing the transverse width of the lasers from w = 30 μm to w = 5 μm leads to a suppression of higher transverse modes. In spite of identical pumping, however, the lasers show spontaneous ultra-fast spatial optical switching in the free-running condition displayed in animation 4. It is the combination of local spectral holeburning with the diffractive interaction of the evanescent optical fields together and the diffusive transport of charge carriers which leads to a dynamical interplay of gain, self-focusing effects, and dispersion. As a consequence, the VCSELs individually display ultrafast spatially homogeneous pulsations. The array configuration allows coherent intracavity field-coupling between adjacent lasers and thus leads to phase-coupled oscillations of the array, where adjacent lasers oscillate in anti-phase with respect to their neighbor.

Fig. 4. Self-induced ultrafast spatio-temporal switching of four round small (w = 5μm) VCSELs being separated from each other at a distance s = 5μm. The total time sequence corresponds to a characteristic time-period of 100 ps. [Media 3]

In conclusion, the microscopic simulation of the short-time spatio-temporal dynamics of large aspect VCSELs and coupled VCSEL-arrays shows combined spatial and spectral dynamics of the charge carrier Wigner distributions. It is the interplay of diffraction and phase-coupling of the optical field with the spatio-spectral hole-burning which in a broad area VCSEL determines the spatio-temporal variation of the transverse field modes. In coupled VCSEL-arrays the simulations predict ultra-fast spontaneous ultra-fast spatial optical switching as a consequence of the spatio-spectral transport of the carriers and the interband-polarization.

Acknowledgment

I sincerely would like to thank Y. Yamamoto for his hospitality at Stanford University.

References

1.

C. J. Chang-Hasnain, “Vertical cavity surface-emitting laser arrays,” in Diode Laser Arrays, D. Botez and D. R. Scrifres, eds., (Cambridge University Press, Cambridge, 1994), pp. 368–413. [CrossRef]

2.

O. Buccafusca, J. L. A. Chilla, J. J. Rocca, C. Wilmsen, S. Feld, and R. Leibenguth, “Ultrahigh frequency oscillations and multimode dynamics in vertical cavity surface emitting lasers,” Appl. Phys. Lett. 67, 185–187 (1995). [CrossRef]

3.

D. G. H. Nugent, R. G. S. Plumb, M. A. Fischer, and D. A. O. Davies, “Self-pulsations in vertical-cavity surface emitting lasers,” Electron. Lett. 31, 43–44 (1995). [CrossRef]

4.

J. E. Epler, S. Gehrsitz, K. H. Gulden, M. Moser, H. C. Sigg, and H. W. Lehmann, “Mode behavior and high resolution spectra of circularly-symmetric GaAs/AlGaAs air-post vertical cavity surface emitting lasers,” Appl. Phys. Lett. 69, 2312–2314 (1996). [CrossRef]

5.

I. Hörsch, R. Kusche, O. Marti, B. Weigl, and K. J. Ebeling, “Spectrally resolved mode imaging of vertical cavity semiconductor lasers by scanning near-field optical microscopy,” Appl. Phys. Lett. 79, 3831–3833 (1996).

6.

O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description,” Phys. Rev. A 54, 3347–3359 (1996). [CrossRef] [PubMed]

7.

O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics,” Phys. Rev. A 54, 3360–3368 (1996). [CrossRef] [PubMed]

8.

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. V. Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, Polarization, and Transverse mode Characteristics of Vertical Cavity Surface Emitting Lasers,” IEEE J. Quantum Electron. 27, 1402–1409 (1991). [CrossRef]

9.

F. Koyama, K. Morito, and K. Iga, “Intensity noise and polarization stability of GaAlAs-GaAs surface emitting lasrs,” IEEE J. Quantum Electron. QE-27, 1410–1416 (1991). [CrossRef]

10.

D. Vakhshoori, “Symmetry considerations in vertical-cavity surface-emitting lasers: Prediction of removal of polarization isotropy on (001) substrates,” Appl. Phys. Lett. 65, 259–261 (1995). [CrossRef]

11.

K. D. Choquette, J. P. Schneider, K. L. Lear, and R. E. Leibenguth, “Gain-dependent polarization properties of vertical-cavity lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 661–666 (1995). [CrossRef]

12.

A. K. J. van Doorn, M. P. van Exter, and J. P. Woerdman, “Elasto-optic anisotropy and polarization orientation of vertical-cavity surface-emitting semiconductor lasers,” Appl. Phys. Lett. 69, 1041–1043 (1996). [CrossRef]

13.

H. F. Hofmann and O. Hess, “Quantum Noise and Polarization Fluctuations in Vertical Cavity Surface Emitting Lasers,” Phys. Rev. A 56, 868–876 (1997). [CrossRef]

14.

O. Hess and T. Kuhn, “Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis,” Prog. Quantum Electron. 20, 85–179 (1996). [CrossRef]

15.

O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.2010) Lasers and laser optics : Diode laser arrays
(140.5960) Lasers and laser optics : Semiconductor lasers
(250.0250) Optoelectronics : Optoelectronics
(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

ToC Category:
Research Papers

History
Original Manuscript: February 13, 1998
Revised Manuscript: December 31, 1997
Published: May 11, 1998

Citation
Ortwin Hess, "Spatio-spectral dynamics and spontaneous ultrafast optical switching in VCSEL arrays," Opt. Express 2, 424-430 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-10-424


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References

  1. C. J. Chang-Hasnain, "Vertical cavity surface-emitting laser arrays," in Diode Laser Arrays, D. Botez and D. R. Scrifres, eds., (Cambridge University Press, Cambridge, 1994), pp. 368-413. [CrossRef]
  2. O. Buccafusca, J. L. A. Chilla, J. J. Rocca, C. Wilmsen, S. Feld, and R. Leibenguth, "Ultrahigh frequency oscillations and multimode dynamics in vertical cavity surface emitting lasers," Appl. Phys. Lett. 67, 185-187 (1995). [CrossRef]
  3. D. G. H. Nugent, R. G. S. Plumb, M. A. Fischer, and D. A. O. Davies, "Self-pulsations in vertical-cavity surface emitting lasers," Electron. Lett. 31, 43-44 (1995). [CrossRef]
  4. J. E. Epler, S. Gehrsitz, K. H. Gulden, M. Moser, H. C. Sigg, and H. W. Lehmann, "Mode behavior and high resolution spectra of circularly-symmetric GaAs/AlGaAs air-post vertical cavity surface emitting lasers," Appl. Phys. Lett. 69, 2312-2314 (1996). [CrossRef]
  5. I. Hoersch, R. Kusche, O. Marti, B. Weigl, and K. J. Ebeling, "Spectrally resolved mode imaging of vertical cavity semiconductor lasers by scanning near-eld optical microscopy," Appl. Phys. Lett. 79, 3831-3833 (1996).
  6. O. Hess and T. Kuhn, "Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description," Phys. Rev. A 54, 3347-3359 (1996). [CrossRef] [PubMed]
  7. O. Hess and T. Kuhn, "Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics," Phys. Rev. A 54, 3360-3368 (1996). [CrossRef] [PubMed]
  8. C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. V. Lehmen, L. T. Florez, and N. G. Stoffel, "Dynamic, Polarization, and Transverse mode Characteristics of Vertical Cavity Surface Emitting Lasers," IEEE J. Quantum Electron.27, 1402-1409 (1991). [CrossRef]
  9. F. Koyama, K. Morito, and K. Iga, "Intensity noise and polarization stability of GaAlAs-GaAs surface emitting lasrs," IEEE J. Quantum Electron. QE-27, 1410-1416 (1991). [CrossRef]
  10. D. Vakhshoori, "Symmetry considerations in vertical-cavity surface-emitting lasers: Prediction of removal of polarization isotropy on (001) substrates," Appl. Phys. Lett. 65, 259-261 (1995). [CrossRef]
  11. K. D. Choquette, J. P. Schneider, K. L. Lear, and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995). [CrossRef]
  12. A. K. J. van Doorn, M. P. van Exter, and J. P. Woerdman, "Elasto-optic anisotropy and polarization orientation of vertical-cavity surface-emitting semiconductor lasers," Appl. Phys. Lett. 69, 1041-1043 (1996). [CrossRef]
  13. H. F. Hofmann and O. Hess, "Quantum Noise and Polarization Fluctuations in Vertical Cavity Surface Emitting Lasers," Phys. Rev. A 56, 868-876 (1997). [CrossRef]
  14. O. Hess and T. Kuhn, "Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis," Prog. Quantum Electron. 20, 85-179 (1996). [CrossRef]
  15. O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).

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