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

Recently, Vertical Cavity Surface Emitting Semiconductor Lasers (VCSEL) have become focus of an increasing number of experimental and theoretical investigations1. Having extremely low lasing thresholds and operating on a single longitudinal laser mode, VCSELs are promising for becoming applied in future optical communication systems, optical interconnects or in optical recording and storage devices. The prospect of employing coherently phase-coupled VCSEL-arrays makes them even more attractive as high-power laser sources or in optical switching devices. Indeed, ultra high frequency oscillations with frequencies up to 240 GHz have recently been observed and attributed to dynamical variations of multiple transverse modes2. There, the reasons for the simultaneous amplification of various transverse modes was associated with mode competition and frequency beating for lower (10–30 GHz) and the ultra-high (~ 200 GHz) oscillation frequencies, respectively. Other phenomena related to multi-transverse mode behavior are the onset of self-pulsations3 or an extremely narrow emission line-splitting4. In the time averaged regime, near-field optical measurements reveal the simultaneously observed spatial and spectral variations of the transverse laser modes in VCSELs5. Taken these experimental observations, what is the mechanism for the formation of dynamically varying transverse modes in VCSELs? Can the effects be harnessed and used for the conception ultra-fast spatio-temporal switching devices based on coupled VCSEL-arrays?

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, 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 and applied to the description of broad-area lasers7. 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. In practice, however, the polarization of the emitted light is frequently stable under usual cw operating conditions9. Indeed, our analysis of the mutual influence of multiple anisotropies (gain-, loss- and frequency-anisotropy) and quantum fluctuations on the emission properties of VCSELs13 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)
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)]

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)
tN(x,t)=TDfTN(x,t)γnrN(x,t)+Λ(x,t)+G(x,t)W(x,t)

of the intracavity optical field E(x,t) and, via the charge carrier density N(x,t), with the ambipolar transport (diffusion coefficient Df ) of charge carriers. The linear absorption α(x) and the built-in refractive index structure η(x) reflect the geometry of the laser14 and T2 is the transverse Laplacian. Λ and W are the macroscopic rates of current injection and spontaneous emission, respectively, Kz denotes the optical wavenumber, dcv the optical dipol matrix element, nl the refractive index of the active layer, ε 0 the permittivity of free space, and l the diffraction length. The microscopic interactions are related to the macroscopic properties through the nonlinear polarization

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

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)]
G(x,t)=χ′′ħE(x,t)212ħIm[E(x,t)Pnl*(x,t)],

with the imaginary parts χ˜ ″ and χ″ of the spectrally resolved and integrated linear susceptibilities, respectively6. Note that due to the explicit consideration of the dynamics of the polarization, both linear and nonlinear spatial and spectral variations of the gain and the induced refractive index are automatically included. In the active layer of a VCSEL, the charge carriers microscopically interact with each other and with the phonons of the semiconductor crystal. The resulting normalization of the intracavity field and the transition energy by many-body interactions - mainly screening of the Coulomb interaction and scattering - is accounted for by the internal fields u and Δu 6, as well as the renormalization of the semiconductor band-gap energy ħω¯(k). Note that ω¯(k) = ωT (k) - ω denotes the detunig of the transition frequency ωT (k) from the cavity-determined reference frequency ω of the intracavity field E. In (1a) and (1b) the dephasing and relaxation dynamics are accounted for by microscopically determined carrier-carrier and carrier-phonon scattering rates for electrons (τe1(k, N)), holes (τp1(k,N)), and the polarization (τp1(k,N))6. The injection rate Λ e,h (k, x,t) models the spatially structured electric injection of charge carrier from the contacts into the active region14. Spontaneous emission, being related to the quantum nature of the light field, leads to a recombination term in the equations of motion for the distribution functions ~ Γsp(k)fe (±k, x,t)fh (∓k, x, t), the upper sign referring to electrons and the lower sign referring to holes, with the spontaneous recombination coefficient Γ sp (k)14. Due to the strong influence of the laser cavity of a VCSEL, a considerable proportion β of the spontaneous radiation is re-directed into the lasing mode. The non-radiative recombination mechanisms reduce the efficiency of the carrier-light coupling and are modeled by a constant rate γnr . The relevant boundary conditions at the edges x,y=W2 represent absorption (αw ) of the optical field E as ∇ T E(x) = ±αw E(x) and surface-recombination (vsr ) of charge carriers reading DfT N(x) = ±vsr N(x), where the total width of the laser structure W = ∑i (wi + si ) + wc includes the width of the electronic contacts wi and separations si between the lasers, as well as the width wc pertaining to the cladding layers.

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

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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, 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]

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Animation 2 demonstrates the simultaneous relevance of spectral and spatial holeburning processes: Through stimulated emission, the optical field depletes at a certain spatial and spectral location the distribution of charge carriers. This spectral “hole” represents the local line-width of the laser light. Its width is – via the dephasing rate – τp1 - determined by the microscopic carrier-carrier and carrier-phonon scattering rates and mediated through the internal field u(k,x,t) and the interband polarization pnl (k,x,t). The spectral “refill” of such a hole which occurs on ultra-short time scales below 100 femtoseconds is, in turn, governed by the scattering processes with their characteristic time scales τe,h . At the same time, transport of charge carriers leads to a spatial redistribution of the various spectral properties. Light emitted from a VCSEL with multiple transverse modes thus carries both, the characteristic spatial and spectral properties of the active semiconductor medium. As a consequence, the linewidth of a VCSEL displays spatial and temporal variations. Next to this spatio-spectral dynamics of the nonlinear gain inside the VCSEL, the optical properties of the VCSEL cavity - i.e. the waveguiding and nonlinear induced refractive index structure - vary dynamically and resemble in their spectral and spatial dependence the microscopic properties. As in the electrically pumped VCSEL electrons and holes are perpetually resupplied via carrier injection, the combined nonlinear dynamic variations of gain and effective refractive index structure result in ultrafast dynamical variations of the transverse optical modes in space and spectrum. Clearly, this interplay would not be possible is small VCSELs and not be included in theories which only consider either the macroscopic spatial dynamics or are based on the assumption of spatially homogeneous spectra.

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]

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Animation 3 gives an impression of the spatio-temporal intensity dynamics of a strongly coupled (s = 5 μm) large-aspect ratio VCSEL-array (w = 30 μm). Next to the characteristic structures of the intensity (c.f. Fig. 1) the dynamics of the transverse modes of one VCSEL is clearly influenced by the coupling between adjacent lasers and results in correlated mode pulsations resembling rotating wheels. Indeed, the transverse modes seem to “dance” within the quasi-two-dimensional active region. Temporal integration then reproduces the typical experimentally observed mode distributions of wide VCSELs in good agreement with spectrally resolved measurements5,4.

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]

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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).

References

  • View by:
  • |

  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).

Other (15)

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]

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]

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]

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]

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).

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]

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]

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]

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]

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]

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]

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]

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]

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

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

Supplementary Material (3)

» Media 1: MOV (347 KB)     
» Media 2: MOV (566 KB)     
» Media 3: MOV (369 KB)     

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Figures (4)

Fig. 1.
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.

Fig. 2.
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]

Fig. 3.
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]

Fig. 4.
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]

Equations (10)

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t f e , h ( k , x , t ) = g ( k , x , t ) + Λ e , h ( k , x , t ) τ e , h 1 ( k ) [ f e , h ( k , x , t ) f eq e , h ( k , x , t ) ]
Γ sp ( k ) f e ( k , x , t ) f h ( k , x , t ) γ nr f e , h ( k , x , t )
t p nl ( k , x , t ) = [ i ω ¯ ( k ) + τ p 1 ( k ) ] p nl ( k , x , t ) + β Γ sp ( k ) f e ( k , x , t ) f h ( k , x , t )
+ 1 Δ U nl ( x , t ) 1 U ( x , t ) [ f e ( k , x , t ) + f h ( k , x , t ) ]
n l c t E ( x , t ) = i 2 1 K z T 2 E ( x , t ) ( γ m + α ( x ) 2 + ( x ) ) E + i n l 2 ε 0 P nl ( x , t )
t N ( x , t ) = T D f T N ( x , t ) γ nr N ( x , t ) + Λ ( x , t ) + G ( x , t ) W ( x , t )
P nl ( x , t ) = d cv * V k p nl ( k , x , t ) ,
g ( k , x , t ) = χ ˜ ( ω ; k , x , t ) 2 ħ E ( x , t ) 2
1 2 ħ Im [ d cv E ( x , t ) p nl * ( k , x , t ) Δ U ( x , t ) p * ( k , x , t ) ]
G ( x , t ) = χ′′ ħ E ( x , t ) 2 1 2 ħ Im [ E ( x , t ) P nl * ( x , t ) ] ,

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