## 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 investigations^{1}. 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 modes^{2}. 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-pulsations^{3} or an extremely narrow emission line-splitting^{4}. In the time averaged regime, near-field optical measurements reveal the simultaneously observed spatial and spectral variations of the transverse laser modes in VCSELs^{5}. 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 VCSELs^{2–5}, 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 in^{6} and applied to the description of broad-area lasers^{7}. 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 mirrors^{8–12}. In practice, however, the polarization of the emitted light is frequently stable under usual cw operating conditions^{9}. Indeed, our analysis of the mutual influence of multiple anisotropies (gain-, loss- and frequency-anisotropy) and quantum fluctuations on the emission properties of VCSELs^{13} 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 *f*^{e,h}
(*k*, x, *t*) of electrons (e) and holes (h) and the interband polarization *p*_{nl}
(*k*, x, *t*)

$$-{\Gamma}_{\mathit{sp}}\left(k\right){f}^{e}\left(k,\mathbf{x},t\right){f}^{h}\left(k,\mathbf{x},t\right)-{\gamma}_{\mathit{nr}}{f}^{e,h}\left(k,\mathbf{x},t\right)$$

$$+\frac{1}{\mathit{i\u0127}}\Delta {U}_{\mathit{nl}}\left(\mathbf{x},t\right)-\frac{1}{\mathit{i\u0127}}U\left(\mathbf{x},t\right)\left[{f}^{e}\left(k,\mathbf{x},t\right)+{f}^{h}\left(k,\mathbf{x},t\right)\right]$$

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

of the intracavity optical field *E*(x,*t*) and, via the charge carrier density *N*(x,*t*), with the ambipolar transport (diffusion coefficient *D*_{f}
) of charge carriers. The linear absorption *α*(x) and the built-in refractive index structure *η*(x) reflect the geometry of the laser^{14} and ${\nabla}_{T}^{2}$ is the transverse Laplacian. Λ and *W* are the macroscopic rates of current injection and spontaneous emission, respectively, *K*_{z}
denotes the optical wavenumber, *d*_{cv}
the optical dipol matrix element, *n*_{l}
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

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

$$-\frac{1}{2\mathit{\u0127}}\mathrm{Im}\left[{d}_{\mathit{cv}}E\left(\mathbf{x},t\right){p}_{\mathit{nl}}^{*}\left(k,\mathbf{x},t\right)-\Delta U\left(\mathbf{x},t\right){p}^{*}\left(k,\mathbf{x},t\right)\right]$$

with the imaginary parts $\tilde{\chi}$
″ and χ″ of the spectrally resolved and integrated linear susceptibilities, respectively^{6}. 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 Δ*u*
^{6}, as well as the renormalization of the semiconductor band-gap energy *ħω¯*(*k*). Note that *ω¯*(*k*) = *ω*_{T}
(*k*) - *ω* denotes the detunig of the transition frequency *ω*_{T}
(*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 (${\tau}_{e}^{-1}$(*k*, *N*)), holes (${\tau}_{p}^{-1}$(*k*,*N*)), and the polarization (${\tau}_{p}^{-1}$(*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 region^{14}. 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 ~ Γ_{sp}(k)*f*^{e}
(±k, x,*t*)*f*^{h}
(∓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 *γ*_{nr}
. The relevant boundary conditions at the edges $x,y=\mp \frac{W}{2}$ represent absorption (*α*_{w}
) of the optical field *E* as ∇
_{T}
*E*(x) = ±*α*_{w}
*E*(x) and surface-recombination (*v*_{sr}
) of charge carriers reading *D*_{f}
∇_{T}
*N*(x) = ±*v*_{sr}
*N*(x), where the total width of the laser structure *W* = ∑_{i} (*w*_{i}
+ *s*_{i}
) + *w*_{c}
includes the width of the electronic contacts *w*_{i}
and separations *s*_{i}
between the lasers, as well as the width *w*_{c}
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 integration^{15}, 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 *μ*m. The time interval between successive snapshots is 3 picoseconds.

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 scales^{14}, 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 *μ*m. 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 *δf*^{e}
(*k*,*x*,*y*
_{0},*t*) = *f*^{e}
(*k*,*x*,*y*
_{0},*t*) - ${f}_{\mathit{\text{eq}}}^{e}$
(*k*,*x*,*y*
_{0},*t*) along a spatial cut at *y*
_{0} = 0 during a characteristic time-interval of 100 ps.

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 – ${\tau}_{p}^{-1}$ - 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 *p*_{nl}
(*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 *τ*_{e,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.

Animation 3 gives an impression of the spatio-temporal intensity dynamics of a strongly coupled (*s* = 5 *μ*m) large-aspect ratio VCSEL-array (*w* = 30 *μ*m). 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 measurements^{5,4}.

In vivid contrast to animation 3, animation 4 demonstrates that reducing the transverse width of the lasers from *w* = 30 *μ*m to *w* = 5 *μ*m 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.

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.

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