Electrical addressing and temporal tweezing of localized pulses in passively-mode-locked semiconductor lasers

We show that the pumping current is a convenient parameter for manipulating the temporal localized structures (LSs), also called localized pulses, found in passively-mode-locked vertical-cavity surface-emitting lasers. While short electrical pulses can be used for writing and erasing individual LSs, we demonstrate that a current modulation introduces a temporally evolving parameter landscape allowing one to control the position and the dynamics of LSs. We show that the localized pulse drifting speed in this landscape depends almost exclusively on the local parameter value instead of depending on the landscape gradient, as shown in quasi-instantaneous media. This experimental observation is theoretically explained by the causal response time of the semiconductor carriers that occurs on a finite time scale and breaks the parity invariance along the cavity, thus leading to a different paradigm for temporal tweezing of localized pulses. Different modulation waveforms are applied for describing exhaustively this paradigm. Starting from a generic model of passive mode locking based upon delay differential equations, we deduce the effective equations of motion for these LSs in a time-dependent current landscape.

Figure 1

Spatiotemporal diagrams showing the addressing, i.e., the writing (a) and the erasing (b) of an LS, by an electrical pulse in the pumping current. The current values are represented in color scales, while optical pulses trajectories are represented by black lines. $J_{cw}=211\mathrm{mA}$. The amplitude and the width of the writing pulse are $A=130\mathrm{mA}$ and $2.9\mathrm{ns}$, respectively.

Figure 2

Drifting speed per roundtrip of a LS induced by a bias current variation around a reference value ($J-J_{\mathrm{ref}})$, $J_{\mathrm{ref}}=226\mathrm{mA}$. Bias current has been changed in different ways: (crosses connected by red line) by changing its steady value; (circles) by modulating it sinusoidally, as obtained from Fig. 3 and (diamonds) by modulating it with a triangular waveform, as obtained from Fig. 5.

Figure 3

Spatiotemporal diagram of pulse position evolutions (dark trace) under a sinusoidal modulation of the current around $J_{\mathrm{ref}}=226\mathrm{mA}$, at ${\nu }_{\mathrm{cav}}=66614250\mathrm{Hz}$. The current values are represented in color scales. (a) Three different situations are shown corresponding to stable equilibrium positions for $\mathrm{\Delta }=-0.25\mathrm{kHz}$ (continuous line), $\mathrm{\Delta }=-4.750\mathrm{kHz}$ (dashed line), and at $\mathrm{\Delta }=+1.750\mathrm{kHz}$ (dashed-dotted line). (b) $\mathrm{\Delta }=-14.25\mathrm{kHz}$, (c) $\mathrm{\Delta }=-9.25\mathrm{kHz}$, (d)$\mathrm{\Delta }=3.25\mathrm{kHz}$, and (e)$\mathrm{\Delta }=10.75\mathrm{kHz}$.

Figure 4

(a) The blue circles represent the localized pulse position within the cavity at each roundtrip during the trajectory represented in Fig. 3. From these data points a continuous trajectory is interpolated and represented in red. (b) The values of the current experienced by the localized pulse at each point of its trajectory. (c) Drifting speed calculated in two different manners. The blue dots correspond to the instantaneous drifting speed calculated as $v_J+v_{\mathrm{\Delta }}=0$, and using the values of the bias current experienced by the LS in (b) and where the function $v_J$ has been obtained by fitting the curve $v_J(J-J_{\mathrm{ref}})$ shown in Fig. 2 by circles. The red curve corresponds to the derivative of the interpolated trajectory in (a).

Figure 5

Spatiotemporal diagram of pulse position evolutions when the current is modulated by a triangular waveform of amplitude $A=120\mathrm{mA}$ around $J_{\mathrm{ref}}=253\mathrm{mA}$ and at a frequency closed to ${\nu }_{\mathrm{cav}}=66608500\mathrm{Hz}$. The current values are represented in color scales. (a) Four different situations are shown corresponding to stable equilibrium positions for different detuning values: $\mathrm{\Delta }=-4\mathrm{kHz}$ (dashed line), $\mathrm{\Delta }=-0.5\mathrm{kHz}$ (dot-dashed line), $\mathrm{\Delta }=1\mathrm{kHz}$ (continuous line), and $\mathrm{\Delta }=1.5\mathrm{kHz}$ (dotted line). Drifting regimes with (b) $\mathrm{\Delta }=-10.5\mathrm{kHz}$, (c) $\mathrm{\Delta }=-6\mathrm{kHz}$, (d) $\mathrm{\Delta }=2.5\mathrm{kHz}$, and (e) $\mathrm{\Delta }=+6.5\mathrm{kHz}$.

Figure 6

Spatiotemporal diagram of pulse position evolutions (dark trace) when the current is modulated by a rectangular waveform of amplitude $A=76\mathrm{mA}$ around $J_{cw}=221\mathrm{mA}$ and at a frequency close to the resonance ${\nu }_{\mathrm{cav}}=66612500\mathrm{Hz}$. The current values are represented in color scales. (a) Three different situations are shown corresponding to stable equilibrium positions for different detuning values: $\mathrm{\Delta }=-1.3\mathrm{kHz}$ (dash-dotted line), $\mathrm{\Delta }=1\mathrm{kHz}$ (dotted line), and $\mathrm{\Delta }=2.5\mathrm{kHz}$ (continuous line). Out of the locking range, drifting trajectories are depicted for detuning values of (b) $\mathrm{\Delta }=-7.5\mathrm{kHz}$, (c) $\mathrm{\Delta }=-1.9\mathrm{kHz}$, (d) $\mathrm{\Delta }=3.8\mathrm{kHz}$, and (e)$\mathrm{\Delta }=7\mathrm{kHz}$.

Figure 7

Spatiotemporal diagrams showing the evolution of a localized pulse when an electrical pulse is applied to the pumping current for two different initial positions. The current values are represented in color scales. The current pulse amplitude is $A=100\mathrm{mA}$, $J_{cw}=202\mathrm{mA}$, $\mathrm{\Delta }=1.4\mathrm{kHz}$ (left panel), and $\mathrm{\Delta }=0\mathrm{kHz}$ (right panel).

Figure 8

Spatiotemporal diagrams showing evolution of five pulse positions (dark trace) in the presence of a sinusoidal current modulation at ${\nu }_{\mathrm{mod}}=5{\nu }_{\mathrm{cav}}-4.750\mathrm{kHz}$, having an amplitude of 120 mA and $J_{cw}=275\mathrm{mA}$. The current values are represented in color scales.

Figure 9

(a) Localized pulse profile in terms of the field intensity, the carrier, and the absorption with $G_0=0.99G_{th}$. (b)–(d) Depiction of the drift velocity $\upsilon$, the pulse integrated energy $P$, and the residual frequency $\omega$ along the stable solution branch. All quantities are dimensionless. Parameters are $\left(\gamma ,\kappa ,\alpha ,\beta ,\mathrm{\Gamma },Q_0,s\right)=\left(40,0.8,1,0.5,0.04,0.3,30\right)$.

Figure 10

Numerical simulations using the EEMs. (a) Resonant harmonic potential. The LS goes to the rising front and escape of the falling front that corresponds to the stable and the unstable fixed points, respectively. (b) Strongly detuned harmonic potential. The LS motion corresponds to unlocked trajectories. Here, the LS slows down around and accelerates periodically. (c) Strongly detuned motion beyond the locking range but in a square potential. Note how the speed is simply bi-valued for the square modulation yielding a broken line trajectory. The frequency of the external modulation corresponds to the fourth harmonic ${\omega }_{\mathrm{ext}}=4/T_0$ with $T_0=15\mathrm{ns}$. The amplitude of the harmonic (a) and (b), and square (b) modulation is $A=0.04$ and $A=0.02$, respectively. The detuning between the natural LS motion and the external periodic potential is represented by different values of ${\upsilon }_{\mathrm{\Delta }}$ and reads (a) ${\upsilon }_{\mathrm{\Delta }}=-2.35\times {10}^{-3}$ (no detuning), (b) ${\upsilon }_{\mathrm{\Delta }}=-1\times {10}^{-3}$, and (c) ${\upsilon }_{\mathrm{\Delta }}=-1.3\times {10}^{-3}$. The current values are represented in color scales.

Figure 11

Numerical simulations using the EEMs. Effect of self- and mutual interactions between LSs. Notice how the drift velocity is affected by the effect of self-interactions as the only difference between (a) and (b) is the length of the cavity. Mutual interactions in a regular crystal structure presented in (c) also influence the drifting speed with respect to (a).

Figure 12

Numerical simulations using the EEMs. (a) Asymmetrical interactions between nearby LSs. As soon as an external modulation is applied (b) the LSs fall at the bottom of the induced potential. The amplitude of the sinusoidal modulation is $A=0.04$ and the current values are represented in color scales.