There has been extensive theoretical^1–9 and experimental^5–10 research on the polarization properties of electrically pumped vertical-cavity surface-emitting lasers (VCSELs), both CW and pulsed, down to a few nanoseconds.^{7, 10} A common finding in this important class of semiconductor lasers is strong anti-correlation in the intensity noise of modes with orthogonal linear polarizations, as well as polarization switching and a preference for linearly polarized emission at high injection currents. The theoretical model proposed by San Miguel, Feng, and Moloney¹ (the SFMmo del) has been successfully applied to these CW and ns regimes. In this model, there are two populations of electron-hole pairs with opposite spins, coupled by spin-flipping processes.

The polarization dynamics as a function of time depend critically on two parameters; the spin-flip time and the linear amplitude (gain) anisotropy time. The linear amplitude anisotropy time (~ns) is usually longer than the spin-flip time (1-100 ps) and this leads to three different temporal regimes with different polarization behavior. Most research has been done in the long-time regime in which there is time for the anisotropy to lead to a preference for linearly polarized emission. Kuksenkov et al. have performed measurements showing this preference develop over a few nanoseconds.⁷

In our measurements using optical pumping on sub-ps time scales we observe different behaviors than in previous studies carried out on longer time scales. For sub-ns time scales, the gain anisotropy has little effect, and we are able to study the intrinsic electron-spin and optical polarization dynamics of the VCSEL. Such dynamics and noise are important, as VCSELs are being modulated at rates up to 10 Gbit/s in short-haul communication systems.¹¹

We have measured the polarization dynamics of two VCSELs. The first (called “UA”), grown at the University of Arizona (NMC63), contained two 8.5 nm In_0.04Ga_0.96As quantum wells in a 3/2 wavelength planar optical cavity and was designed to be cooled to 10 K. The second VCSEL (called “SNL”), from Sandia National Labs, contained five 8 nm GaAs quantum wells in a one-wavelength cavity, and was designed to be run at room temperature. The SNL sample had oxide confinement with a 1×1 µm aperture, while the UA sample had no transverse confinement. Measurements were made using DC balanced homodyne detection with a 300 fs sampling time given by the reference pulse duration.¹² A quarter-wave plate or a half-wave plate was placed in the emitted beam allowing polarization statistics to be measured in three different polarization bases: vertical and horizontal linear, 45 degree rotated linear, and right and left circular (denoted R/L).¹³ In all the measurements described below, the two linear bases produced similar results and therefore only the vertical/horizontal basis (denoted V/H) will be discussed and will be referred to as the linear basis. The photon numbers within a sampling time in any two polarizations are n ₁ and n ₂. Correlations between n ₁ and n ₂ are characterized by the normalized two-mode second-order coherence function $g_{12}^{\left(2\right)}$=〈n ₁ n ₂〉/(〈n ₁〉〈n ₂〉). A value $g_{12}^{\left(2\right)}$=1 indicates uncorrelated fluctuations in n ₁ and n ₂, and a value above(below) 1 indicates positive(negative) correlations.

We first pumped the UA VCSEL above threshold with right circularly polarized pulses, populating only one spin state of electron-hole pairs. Figure 1(a) shows the mean emitted photon number per sampling time measured in the circular (R/L) basis (time=0 corresponds to excitation time in all figures). The mode having the polarization (left circular) corresponding to that of the pump builds up rapidly (circularly polarized light changes handedness upon reflection), while the orthogonal (right) polarization mode builds up at later times as a consequence of spin-flip collisions. At lower pump energies (not shown), the right circular mode fails to develop significant intensity due to reduced gain. When the emission consisted of both circular polarization modes, intensity fluctuations in the circular modes were largely uncorrelated as shown in Fig. 1(b). In the linear (V/H) detection basis, the mean photon number developed similarly for both polarizations.

 

Fig. 1. Mean photon numbers per sampling time 〈n〉 and noise correlations g ⁽²⁾ for UA sample. (a) and (b) Right circularly polarized pump. (a) 〈n_i 〉 for left circularly polarized mode (blue, solid line) and the right circularly polarized mode (red, dotted line). (b) Noise correlation between the two circularly polarized modes. (c) and (d) Horizontally polarized pump. (c) 〈n_j 〉 for vertically polarized mode (blue, solid line) and horizontally polarized mode (red, dotted line). (d) Noise correlation between the two linearly polarized modes.

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Measurements (not shown) were also made using a circularly polarized pump and linear detection basis. Well above threshold, the mean V/H photon numbers are found to be equal at all times and the intensity fluctuations are anticorrelated. At lower circular pumping levels, when the emission is almost purely circularly polarized, the fluctuations in the V/H polarizations were strongly positively correlated. This arises because nearly circularly polarized light can be decomposed into linear components whose amplitudes are nearly equal.

Next, we pumped the UA VCSEL with linearly polarized pulses. This is similar to electrical pumping in that equal populations of the two spin states of electron-hole pairs are created and these populations become mutually incoherent on 100 fs time scales as a result of electron-electron collisions. Figure 1(c) shows that the mean emitted photon numbers in the linear (V/H) basis are equal. [The mean photon numbers in the circularly polarized basis (not shown) were equal as well, and the intensity fluctuations were uncorrelated, as in the case with the circularly polarized pump.] Figure 1(d) shows that the fluctuations in the linearly polarized modes were anticorrelated. It is important to note again that the results were the same for the rotated (45 degree) linear basis. Measuring equal mean photon numbers in both linear bases indicates that the anticorrelation did not result from competition between ‘preferred’ linearly polarized modes aligned with the crystal axes as seen on longer time-scales, with phase and gain anisotropies thought to result from stress-induced birefringence and a frequency-dependent gain profile.1–10 Instead (we argue), the anticorrelation was due to a random orientation of the almost purely linearly polarized emission from pulse to pulse. For sake of argument assume that the average intensities of two circularly polarized modes are equal and uncorrelated. Then a random phase difference between the circularly polarized modes will lead to randomly oriented linearly polarized emission from pulse to pulse, and the projection onto any two orthogonal linear axes will result in a perfect anti-correlation between the amplitude of the projections.

The SNL VCSEL produced nearly identical results except for one important difference. Figure 2(a) shows that with a linearly polarized pump, the circularly polarized modes were anticorrelated, whereas these modes were uncorrelated in the UA VCSEL.

 

Fig. 2. Noise correlations between right and left circularly polarized modes for SNL sample with linearly polarized pump. (a) Experimental result. (b) Numerical simulations.

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To model the observed behavior, we have extended the SFM model¹ in a manner suitable for transient behavior in the ps regime. We approximate the carrier distribution functions by including a higher (hot) energy level into which carriers are initially pumped and then relax to the lower (cool) level, from which lasing occurs. Our rate equations for the circularly polarized normalized electric field amplitudes E ₊, E _- and the higher (hot) level $N_{+}^H$, $N_{-}^H$ and the lower (cool) level lasing normalized population densities N ₊, N _- associated with right and left circularly polarized electron-hole pairs, respectively, are

where κ is the cavity decay rate, g is the linear gain coefficient, α is the amplitude-to-phase coupling parameter, γ_p is the linear phase anisotropy, γ_a is the linear amplitude (gain) anisotropy, f _± are the stochastic Langevin noise terms associated with spontaneous emission into right and left circularly polarized fields, Γ is the (nonradiative plus spontaneous emission) carrier decay rate and γ_s is the spin flip rate. The last term in each of Eq. 2 and Eq. 3 provides a simple way to include the carrier cooling due to phonon emission as well as the limitation of carrier cooling due to state filling. The phonon emission rate coefficient is γ_r , and $N_{\pm }^T$ is the maximum possible population carrier density for the lower level. The 100 fs optical pump pulse p(t) is treated as instantaneously populating $N_{+}^H$ and $N_{-}^H$ with the relative values R _± given by the ellipticity of the pump polarization.

Qualitative agreement with our measurements was obtained with appropriate values for the spin-flip time and the amplitude and phase anisotropy times. Spin-flip times can vary from a few picoseconds to hundreds of picoseconds, and depend strongly on lattice temperature.14 To obtain good fits to experiments, we used a spin-flip time of (1/γ_s )=100 ps to model the low-temperature sample UA and a spin-flip time of 1 ps to model the high-temperature sample SNL. Frequency splitting due to birefringence and gain anisotropy rates is typically on the order of GHz,^{2, 4, 5} so we set the phase and gain anisotropy times to (1/γ_p )=1 ns and (1/γ_a )=1 ns. The phonon emission time was set to (1/γ_r )=10 ps.¹⁵ The cavity decay time was measured to be (1/κ)=7 ps, and the carrier decay time was set to (1/Γ)=1 ns,¹⁵ and the amplitude-to-phase coupling parameter was set to a typical value α=3.^{2, 4}

The values of the pump amplitude p(t) and the maximum carrier density $N_{\pm }^T$ were set to model pumping well above lasing threshold. With short pulses, lasing threshold is not a well-defined concept, though it can be understood as the point at which stimulated emission begins to modify the photon-number statistics. This is often determined by measuring the single-mode second-order coherence function $g_1^{\left(2\right)}$=〈:$n_1^2$:〉/(〈n ₁〉²). When $g_1^{\left(2\right)}$ begins to drop from 2 to 1, the laser is at threshold. In our measurements (not shown), $g_1^{\left(2\right)}$ typically dropped to a value of 1.3 near the peak of the pulse. Our pump parameters p(t) and $N_{\pm }^T$ were set to produce similar values.

With a spin-flip time of 100 ps and a circularly polarized pump, Fig. 3(a) and 3(b) show the simulated buildup of the orthogonal circularly polarized mode mean intensities and the uncorrelated fluctuations, similar to the corresponding measurements shown in Fig. 1(a) and (b). We also found that at low pump energies, the emission was circularly polarized, and the fluctuations in the linearly polarized modes were positively correlated as in our measurements (not shown). Simulating a linearly polarized pump, Fig. 3(c) and (d) show equal intensities for the linearly polarized modes and the strong anticorrelated fluctuations, as was measured and shown in Fig. 1(c) and (d). The qualitative agreement between the measurements and the simulations is remarkably good, especially considering the two-level approximation to the distribution functions.

 

Fig. 3. Simulated mean intensities 〈I〉=〈|E|²〉 (arb. units) and noise correlations g ⁽²⁾. (a) and (b) Right circularly polarized pump. (a) 〈I_i 〉 for left circularly polarized mode (blue, solid line) and the right circularly polarized mode (red, dotted line). (b) Noise correlation between the two circularly polarized modes. (c) and (d) Horizontally polarized pump. (c) 〈I_j 〉 for vertically polarized mode (blue, solid line) and horizontally polarized mode (red, dotted line). (d) Noise correlation between the two linearly polarized modes.

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Modelling of the effects of the temperature dependence of the spin-flip rate is shown in Fig. 2(b), for the case of linearly polarized pumping. The curve labelled 100 ps agrees with $g_{\mathit{\text{RL}}}^{\left(2\right)}$ in Fig. 1(b) for the low-temperature UA sample, while the curves labelled 1 ps and 0.1 ps model better $g_{\mathit{\text{RL}}}^{\left(2\right)}$ for the room-temperature SNL sample shown in Fig. 2(a). The anticorrelation in the latter case can be understood as gain competition between the two circularly polarized modes, mediated by the coupling, due to spin-flipping, between the two electron-hole populations. Stronger coupling leads to stronger competition, and thus, to stronger anti-correlated intensity fluctuations.

The anisotropy terms had little effect on the simulations except when we modelled a VCSEL pumped by a pulse with duration >1 ns or a VCSEL with increased γ_a . With a long pump time, we observed the subsequent increase of the intensity of one linear polarization mode and the decrease of the intensity of the other over a period of 0.1 to 1 ns due to the linear amplitude (gain) anisotropy. With a fast pump, increasing γ_a by a factor of 10 led to a significant increase/decrease in the peak intensity of the horizontally/vertically polarized mode, and the intensity noise remained anticorrelated. This suggests that a value of γ_a ≤1 ns^-1 is reasonable for our VCSELs. Increasing γ_p to 10 ns^-1 or 100 ns^-1 had no noticeable effect on the simulated intensities or intensity correlations in all polarization bases.

We conclude that when creating optical pulses from VCSELs on time scales less than about 100 ps, the polarization dynamics and noise are controlled not by linear anisotropies, as they are in CW or ns-pulse VCSELs, but rather by intrinsic spin relaxation, initial-state preparation by pumping, and in some (but not all) cases by gain competition. We expect, though, for VCSELs with larger gain anisotropy, one linearly polarized mode would begin to dominate.

We thank Hailin Wang for helpful physics discussions, Dan McAlister for early work on the experiments, and Michael Werner for helpful physics discussions and for help in developing the numerical simulations. This material is based upon work supported by the National Science Foundation under Grant No. 9876608 and Grant No. 9970106.