Adler Synchronization of Spatial Laser Solitons Pinned by Defects

Defects due to growth fluctuations in broad-area semiconductor lasers induce pinning and frequency shifts of spatial laser solitons. The effects of defects on the interaction of two solitons are considered in lasers with frequency-selective feedback both theoretically and experimentally. We demonstrate frequency and phase synchronization of paired laser solitons as their detuning is varied. In both theory and experiment the locking behavior is well described by the Adler model for the synchronization of coupled oscillators.

Figure 1

Locked phase differences $\Phi$ of pinned LCS for different frequency detunings (controlled by the potential depths $n_1$ and $n_2$) from integration of Eq. (1) (dots, LCS separation of 5.3 soliton widths) and the model of Ref. 23 (triangles, LCS separation of 4 soliton widths). The solid line refers to the Adler equation (3). The inset shows the near-field profile of the intensity of two interacting LCS. Such a profile is almost constant across the Adler locking region.

Figure 2

Far field fringes (a)–(c) averaged over $2\mu \mathrm{s}$, and optical spectra (d)–(f) for a time window of $5\mu \mathrm{s}$, for $\Delta \omega /\Delta {\omega }_{\mathrm{th}}=0$ (a,d), 0.99 (b,e), and 2.0 (c,f) obtained from simulations of the model of Ref. 23. In (d,e) the LCS spectral peaks (dashed and solid lines) overlap.

Figure 3

Schematic diagram of a self-imaging cavity coupling a VCSEL to a frequency-selective element. The dotted lines correspond to the centers of the bundle of rays emitted by the soliton. The tilt angle $\Theta$ is exaggerated for clarity of display. The focal lengths of the intracavity lenses $L_1$, $L_2$ are $f_1=8\mathrm{mm}$ and $f_2=50\mathrm{mm}$, the total cavity length $L\approx 12\mathrm{cm}$ with a 1.23 GHz free spectral range. $L_3$ images the near field of the VCSEL into the detection arm. The lower inset of the VCSEL aperture displays the near field of two interacting LCS.

Figure 4

Fringe visibility (black) and fringe phase (green curves, gray) as a function of the tilt angle that changes the difference between the feedback phases of the LCS. This difference is converted to a frequency scale by multiplying it by the free spectral range of the external cavity thus providing the change of the relative detuning between the two LCS in the external cavity. The zero of this detuning scale is arbitrary. The solid and dashed green (gray) curves are obtained for scanning the tilt back and forth. The fringe phase is obtained from the phase of a cosine-wave fitted to far field profiles like those in the upper row of Fig. 5. Other parameters: Temperature $69°\mathrm{C}$, current $I=373\mathrm{mA}$.

Figure 5

Upper row: Cut through far field intensity distribution orthogonal to fringe orientation. Lower row: Optical power spectra. Left column (a,d) for detunings around 12 MHz, locked with a phase of $\pi$; center column (b,e) around 18 MHz, near the end of the locking region, locked with a phase of $1.5\pi$; right column (c,f) around 22 MHz, unlocked, no clear fringes.