Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback

Spatially self-localized states have been found in a model of vertical-cavity surface-emitting lasers with frequency-selective optical feedback. The structures obtained differ from most known dissipative solitons in optics in that they are localized traveling waves. The results suggest a route to realization of a cavity soliton laser using standard semiconductor laser designs.

Figure 1

(Color online) The scheme of the system. $F_1$ is the focal length of the first lens, and $F_2$ that of the second lens.

Figure 2

(Color online) On-axis mode properties for flat gain spectrum. (a) The amplitudes $Q$ of the steady states versus pump current $\mu$. Points $A$, $B$, $C$ are described in the text. Parameters: $\sigma =60.0$, $\kappa =300.0$, $\alpha =5.0$, $\tau =5.0$, $T=0.1845$, ${\omega }_m=-250.0$. (b) Circles are the amplitudes $Q$ versus frequency for $\mu =1$ [vertical line in (a)]. Solid curve, relative amplitude of grating feedback function.

Figure 3

(Color online) Thresholds of transverse modes in the $({\Omega }_{\perp },\mu )$ plane for different phase-amplitude couplings in the system with flat gain. (Every mode remains active for all current levels above its threshold.) ${\omega }_m=250.0$; other parameters as in Fig. 2. The gray (cyan) regions correspond to instabilities of the trivial solution determined mainly by FSF, the black to those determined mainly by the solitary VCSEL. $\alpha =\left(\mathrm{a}\right)$ 0.0; (b) 5.0 (the meaning of the rectangle is described in the caption of Fig. 4); (c) $-5.0$.

Figure 4

(Color online) (a) Identical to the part of Fig. 3b enclosed by the rectangle, but shifted to the left because here ${\omega }_m=-250$. The points $A$, $B$, $C$ are identical to the equivalent points in Fig. 2a. (b) Stability boundaries for system with spatially filtered feedback and finite gain bandwidth. The black line corresponds to solitary VCSEL modes, the light-gray (cyan) region to the modes controlled by FSF with spatial filtering. The trivial solution is unstable between lines $A$, $B$ and stable between lines $B$, $C$. (c) Dependence on transverse frequency of the relative feedback strength [gray (red) curve] and gain (black curve). Parameters as in (a) except for $\sigma =100.0$, $T_f=0.06$, ${\delta }_f=50.0$, $T_2=0.01$. The gain spectrum in (c) primarily affects the solitary VCSEL modes, for which it corresponds to $\delta =50.0$.

Figure 5

Evolution of optical field $E$ for parameters of Fig. 4b and pump current $\mu =0.8$, for which $E=0$ is an unstable state. (a) Snapshot of the real part of the field during the transient stage; (b) far-field intensity during the transient stage; (c) snapshot of the real part of the field in the asymptotic regime; (d) far field in the asymptotic regime.

Figure 6

Snapshots of localized lasing states for the same parameters as in Fig. 5, except $\mu =0.9$, for which $E=0$ is stable. (a), (d) are the real part of the field, (b), (e) are the intensities of the field $\left({\mid E\mid }^2\right)$, and (c), (f) are the far-field intensity distributions. (a), (b), (c) correspond to a localized traveling wave, (d), (e), (f) correspond to a localized standing wave (LSW). The different structures are excited using different initial conditions; see text.

Figure 7

(Color online) Evidence of the self-localization of the LTW state. Displayed is the intensity ${\mid E\mid }^2$ of the field versus the distance $r$ from its peak. Note the logarithmic intensity scale, so that the linear decrease of each of three intensity measures shows exponential localization. Solid curve is the azimuthally averaged intensity. Dotted and dashed curves are two orthogonal sections. These curves are calculated over a radial range four times larger than that displayed in Fig. 6.

Figure 8

Stable coexistence of several localized structures, two LTWs with different orientations (on the left) and one LSW. Image area is four times larger than in Fig. 6. (a) Real part and (b) amplitude of field.