1. Introduction

Vertical-cavity surface-emitting lasers (VCSELs) have many characteristics that make them very promising for the next generation of optical networks. They have low threshold current, high power conversion efficiency, high modulation bandwidth, and emit a single-longitudinal-mode at wavelengths of interest for data links and optical fiber technologies [1, 2]. They are compact, have a circular output profile that allows direct fiber coupling, and are easily packaged into two dimensional arrays. VCSELs use DBR mirrors to form a small but highly resonant Fabry Perot cavity. Nowadays photonic-crystal structures are being employed to improve the optical confinement and the mirror reflectivity, resulting in even lower thresholds and smaller cavities with higher Q values [3, 4, 5].

The main drawbacks of VCSELs come from polarization and transverse-mode instabilities [6, 7], and various control methods have been demonstrated, such as the use of a sub-wavelength surface grating locally etched near the optical axis [8]. Because of the polarization-sensitive effective refractive index of the grating structure, one linear polarization has higher reflectivity than the orthogonal one, and because the grating is centrally located, the fundamental transverse mode has higher reflectivity than the higher-order modes. Moreover, sub-wavelength gratings have also great potential for the development of tunable VCSELs [9, 10].

VCSELs can be directly modulated at high speeds and a lot of effort has focused on achieving a small-signal wide modulation bandwidth. Under strong amplitude current modulation nonlinear effects arise (such as period doubling, chaos, multistability, etc. [11, 12, 13]). In VCSELs, the polarization and transverse-mode competition greatly enhance the complexity of the nonlinear dynamics [14, 15, 16, 17, 18]. Here we propose a novel way to exploit the nonlinearities to generate fast optical pulses using asymmetric triangular modulation, with a modulating signal that is, on average, below the static threshold (the threshold for cw operation).

To the best of our knowledge VCSELs with asymmetric triangular modulation have not been studied so far. Most of previous studies considered sinusoidal modulation; triangular (symmetric) modulation was studied experimentally and numerically in [19, 20, 21], focusing on the influence of the modulation frequency on the polarization-resolved L-I hysteresis curve.

Our work is motivated by a recent experimental and theoretical study using a Nd ³⁺:YVO₄ diode-pumped laser [22], where asymmetric modulation was applied to the power delivered by the pumping diode laser. It was shown that an asymmetric triangular signal with a slow raising ramp can lead to the emission of pulses, even when the laser is operated, on average, below threshold. In contrast, a signal with a fast raising ramp and the same averaged value does not lead to pulse emission, the intensity remains at the noise level during all the modulation cycle.

In [22] the modulation period was of the order of tens of µs; here we show that a similar effect can be observed in VCSELs but with much faster modulating signals. The simulations are done using the spin-flip model [23, 24] that has proven to be successful in describing VCSEL dynamics [25, 26, 27]. We show that under suitable modulation parameters subnanosecond pulses on two orthogonal linear polarizations can be obtained even through the injection current is, on average, below the cw threshold. We interpret the results as due to the nonlinear interplay of the optical field and the carrier density in the active medium.

The method proposed here for the generation of optimal pulses with low average injection current via asymmetric modulation is based on exploiting the nonlinear light-matter interation, and from this point of view, is similar to the methods proposed in [28, 29], for optimizing the performance of directly modulated semiconductor lasers by shaping the current input. Appropriate square-shaped current inputs allow to control the laser time-evolution in the plane (photon density, carrier density). In [28] the aim was to avoid dynamical memory effects that arise because even if the observable intensity has returned to it stationary value after a current waveform was applied, the unobservable carrier density may not have reached its stationary value. By suppressing dynamical memory effects the laser output is not influenced by previously communicated information, which improves its performance in digital data communication systems. In [29] adequate square-shaped injection current inputs were demonstrated numerically and experimentally, to switch on a semiconductor laser without relaxation oscillations (first a large pump value was applied to speed up the switch-on, followed by a lower value, temporarily below threshold, tailored to eliminate just the right amount of the accumulated carriers, whose excess would otherwise cause damped relaxation oscillations).

This paper is organized as follows: Section 2 presents the model employed, Section 3 presents the results of the simulations, characterizing the influence of various asymmetric modulation parameters, and Section 4 presents a summary and the conclusions.

2. Model

The spin-flip rate equations for the orthogonal linearly polarized slowly-varying amplitudes, E_x and E_y, the total carrier density, N=N ₊+N_-, and the carrier difference, n=N ₊-N _-(N ₊ and N _- being populations with opposite spin) are [23, 24]:

where k is the field decay rate, γN is the decay rate of the total carrier population, γs is the spin-flip rate which accounts for the mixing of carrier populations with different spins, α the linewidth enhancement factor, γa and γp are linear anisotrophies representing dichroism and birefringence, β_sp is the noise strength, ξ_x,y are uncorrelated Gaussian white noises and µ(t) is the normalized injection current parameter: the static cw threshold is at µ_th,s=1.

 

Fig. 1. Time traces of the intensities of the orthogonal linear polarization: I_x (red), I_y (blue), and the injection current µ(t) (dashed) for an asymmetry parameter (a) α_a=0.8, (b) 0.6 and (c) 0.2. (d) Detail of a pulse in Fig.1(a). (e) Color plot of the average pulse total amplitude, 〈A_T〉, for a fixed modulation amplitude, Δµ=1. (f) and (g) Time averaged intensities, 〈I〉, and pulse amplitudes, 〈A〉, respectively (x polarization (red), y polarization (blue) and total intensity (black)). (h) Normalized standard deviation, σ/〈A〉, of the pulse amplitude vs. the asymmetry parameter, α_a. The modulation amplitude is Δµ=1, the period is T=3 ns. The DC value µ₀=0.37 is fixed in captions (a)–(d) and (f)–(g) and is varied in (e).

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The current is modulated with an asymmetric triangular signal of amplitude Δµ, rising from µ ₀ a time interval T₁ and falling back to µ₀ a time interval T₂. One modulation cycle is: µ(t)=µ₀+Δµ(t/T₁) for 0≤t≤T₁, µ(t)=µ ₀+Δµ[1-(t-T₁)/T₂] for T₁≤t≤T₁+T₂. The average current, µave=µ0+Δµ/2, is independent of the modulation period, T=T1+T2. The asymmetry of the modulation is characterized by the parameter α_a=T₁/T with 0≤α_a≤1.

3. Results

The equations were simulated with typical VCSEL parameters [24]: k=300 ns^-1, α=3, γN=1 ns^-1, γa=0.5 ns-1, γp=50 rad/ns, γs=50 ns^-1, and β_sp=10^-6 ns^-1.

Current modulation leads to the emission of optical pulses even when, on average, the injection current is below the cw threshold. The intensity is emitted in irregular pulses in both linear polarizations. Figures 1 (a)–(d) display time traces of I_x=|E_x|² and I_y=|E_y|² for three modulation asymmetries and the same average current value, µ_ave=0.87<1. The modulation period and amplitude are chosen such that the laser emits only one pulse per modulation cycle.

It can be observed that with slow-rising and fast-decreasing ramps, Fig. 1(a), the laser emits larger pulses than with a more symmetric signal, Fig. 1(b). In contrast, with fast-rising and slow-decreasing ramps, Fig. 1(c), the laser does not turn on, in agrement with Ref.[22].

 

Fig. 2. (a)–(c) Time averaged intensities [x polarization 〈I_x〉 (red), y polarization 〈I_y〉 (blue) and total intensity 〈I_T〉 (black)] vs. average current, µ_ave, for different modulation amplitudes (a) Δµ=1.0, (b) 0.5, and (c) 0.15. (d) Color plot of average total intensity 〈I_T〉. The asymmetry parameter α_a=0.8 and the period T=3 ns are fixed.

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Figure 1(f) displays the time averaged intensities, 〈I_x〉, 〈I_y〉 and 〈I_T〉=〈I_x+I_y〉; Fig. 1(g) displays the time averaged pulse amplitudes, 〈A_x〉, 〈A_y〉 and 〈A_T〉 (when there is more than one pulse per modulation cycle, we calculate the average amplitude of the largest pulse). The amplitudes are one order of magnitude larger than the intensities because the laser emits sharp pulses and is off during most of the modulation cycle. Figure 1(h) displays the dispersion of the amplitude of the pulses, characterized in terms of the standard deviation normalized to the mean value. There is an optimal modulation asymmetry, typically α_a≅0.8, for which the averaged intensity and the averaged pulse amplitude reach their maximum value, and for this asymmetry the dispersion of the pulse amplitude reaches its minimum value.

For the optimal asymmetry a pulse is emitted just at the end of the modulation cycle, as can be seen in Fig. 1(d). This is also in good agreement with the observations of [22], and can be interpreted, as in [22], as due to the nonlinear interplay of the photons and the carriers in the active region. Two mechanisms trigger the emission of a pulse: spontaneous emission and the radiation left by the previous pulse. When the radiation left by the previous pulse is absorbed by the carriers during the fall part of the cycle, spontaneous emission is the dominant mechanism for triggering the next pulse in the next cycle. On the contrary, when the radiation left by the previous pulse has not been completely absorbed, it dominates over spontaneous emission for triggering the next pulse.

The “effective” lasing threshold depends on the asymmetry of the modulation as can be seen in Fig. 1(e), that displays a color-coded 2D plot of the averaged amplitude of the pulses of the total intensity, 〈A_T〉, vs. α_a and µ₀, for fixed modulation amplitude [Δµ is the same as in Figs. 1(f)–(h), thus, Fig. 1(f)–(h) are an horizontal scan in Fig. 1(e)]. In the vertical axis of Fig. 1(e) µ_ave is plotted instead of µ₀ (µ_ave=µ₀+Δµ/2) to show explicitly that the laser emits pulses even when the average current is below the cw threshold. 〈A_T〉 increases with µ_ave, and the relation is nonlinear, as will be discussed below. For the optimal asymmetry that leads to maximum pulse amplitude with µ_ave<1 (α_a≅0.8), there is also the largest effective threshold reduction. In contrast, for modulations on average above the cw threshold, the optimal asymmetry to obtain maximum pulse amplitude is such that αa<0.5 [yellow region in the top-left corner of Fig. 1(e)], i.e., the modulation has a fast rising ramp followed by a slow decreasing one.

The “effective” threshold depends also on the modulation parameters µ₀ and Δµ. Figures 2(a)–(c) show the averaged intensities, 〈I_x〉, 〈I_y〉 and 〈I_T〉, for α_a=0.8 and three values of Δµ. In each caption Δµ andα_a are kept fixed while µ₀ varies, but in the horizontal axis we plot µ_ave instead of µ₀ to show that, for large Δµ and small µ₀, there is laser emission with µ_ave<1. 〈I_T〉 increases with µ_ave, and for large Δµ, Figs. 2(a), 2(b), the relation is nonlinear; kinks appear which are due to the emission of additional pulses in each modulation cycle.

 

Fig. 3. (a)–(c) Time averaged pulse amplitudes [x polarization 〈A_x〈 (red), y polarization 〈A_y〉 (blue) and total amplitude 〈A_T〉 (black)] vs. average current, µ_ave, for different modulation amplitudes (a) Δµ=1.0, (b) 0.5, and (c) 0.15. (d) Color plot of the average total intensity, 〈I_T〉. Parameters are as in Fig. 2.

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The variation of the effective threshold with both, µ₀ and Δµ, is illustrated in Fig. 2(d), that presents a color-coded 2D plot of 〈I_T〉. Also here the horizontal axes displays µ_ave instead of µ₀, and Figs. 2(a)–(c) correspond to horizontal scans in 2(d). In the bottom-left corner of Fig. 2(d), Δµ and/or µ₀ are too small, and the laser does not turn on, the black color representing the intensity at the noise level. We observe a smooth turn-on: as Δµ and/or µ₀ increase, 〈I_T〉 gradually increases.

Figure 3 displays the time averaged pulse amplitude, 〈A_T〉, for the same parameters as Fig. 2. It can be noticed that near the effective threshold 〈A_T〉 increases nearly linearly with µ_ave, while for larger µ_ave, 〈A_T〉 saturates but 〈I_T〉 continues increasing with µ_ave, as seen in Fig. 2. This is due to the fact that the laser emits more than one pulse per modulation cycle.

4. Conclusion

The dynamics of a VCSEL driven by asymmetric triangular current modulation was studied numerically using the spin-flip model. When the injection current is, on average, below the cw threshold irregular optical pulses in two orthogonal linear polarizations can be generated by using large amplitude modulation of period of a few nanoseconds. For an optimal modulation asymmetry, with a slow rising ramp followed by a fast decreasing one, the effective threshold reduction is about 20%, the pulse amplitude is maximum and the dispersion of the pulse amplitude is minimum. In contrast, when the averaged current value is above the static threshold, the optimal modulation asymmetry that leads to maximum pulse amplitude has a fast rising ramp followed by a slow decreasing one.