Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation

The emission characteristics of a vertical-cavity surface-emitting laser (VCSEL) coupled to an external cavity with a diffraction grating as a frequency-selective element are theoretically analyzed. We introduce envelope functions for the set of external-cavity modes based on the loci of modes with extremal gain or frequency in the proper parameter space. Replacing the set of discrete stationary solutions by these envelope functions, simple analytical expressions are derived for the existence of bistability between a lasing state strongly affected by the feedback and a state close to the solitary laser emission (in particular the nonlasing state) and for the frequency of the VCSEL in the grating-controlled regime. It is shown how the initial jump of the laser intensity during abrupt turn-on can be maximized. By a control of the feedback change, the width of the hysteresis loop can be increased significantly. The scheme under consideration can be useful in all-optical photonic switching applications.

Figure 1

(Color online) Sets of the steady states in the domains (a) $(\mathrm{\Delta }{\omega }_0,\Omega )$ and (b) $(\mathrm{\Delta }{\omega }_0,N)$. Dark gray lines in the printed version (red and green lines for $x$- and $y$-polarized modes in the on-line version) denotes lasing modes (above threshold). Black lines denote ${\mathbf{Ew}}^{\mathbf{l}}$ and ${\mathbf{En}}^{\mathbf{l}}$ envelopes, open circles $(엯)$ and crosses $(\times )$ denote limit and threshold points in the envelope approximation. $\mathrm{\Delta }{\omega }_0^{\mathrm{th}}=30\pi {\mathrm{ns}}^{-1}$, ${\omega }_0\tau =0$, ${\omega }_m=0$ other parameters are given in the text.

Figure 2

(Color online) Evolution of the averaged intensities ($⟨\mid E_x\mid {⟩}^2$ and $⟨\mid E_y\mid {⟩}^2$) and of the averaged frequencies ($⟨{\Omega }_x⟩$ and $⟨{\Omega }_y⟩$) for $E_x$- and $E_y$-polarized components (averaging time is $200\mathrm{ns}$ with sampling step $0.08\mathrm{ns}$) versus (b) injection current or (a) frequency $\mathrm{\Delta }{\omega }_0$. The thick solid and dashed curves denote the results obtained from numerical simulations for the $x$ and $y$ component, respectively (in the online version the curves for the $x$ and $y$ component are black and blue, respectively). The thin lines denote the analytically obtained steady-state solutions ($x$: light gray in print version; green in online version). ${\beta }_n={10}^{-5}$, the other parameters are the same as for Fig. 1.

Figure 3

(a) Bistability boundaries in the space spanned by the normalized full detuning $w^m$ and the normalized feedback strength $\Lambda$ for the double reflection from the grating (thick lines) with their asymptotes (thin lines) and for an Lorentzian filter (dashed lines). (b) The normalized initial detuning $\Delta$ and the normalized threshold current $m$ vs $w^m$ for the case when the laser threshold coincides with the bistability boundaries of (a) in accordance with Eqs. (A45, A46) for the parameters of Fig. 1.

Figure 4

The normalized threshold injection current $m$ vs the normalized initial detuning $\Delta$ (solid line) and the corresponding normalized output laser frequency $w$ (dashed line) for $\Lambda =4.08$ $(\lambda =0.8)$ in accordance with Eqs. (A35, A36); the other parameters are the same as for Fig. 1.

Figure 5

(Color online) Bistability domain in the space spanned by the normalized feedback strength $\Lambda$ and the normalized full detuning $w^m$ (thick line), and threshold curves (thin lines) corresponding to different normalized initial detuning $\Delta$ in the same parameter space. The curves $1,2,\dots ,9$ correspond to $\Delta =[-6.45,-4.82,-3.19,-1.56,-0.1333,0.83,1.69,2.89,3.32]$. (For comparison, the frequency $\mathrm{\Delta }{\omega }_0^{\mathrm{th}}=30\pi {\mathrm{GHz}}^{-1}$ from Fig. 1 corresponds to the normalized detuning $\Delta =-2.45$.) The other parameters are the same as for Fig. 1. The inset shows an enlargement around the point of self-intersection of curve 8.

Figure 6

(Color online) The same as Fig. 2, but with additional phase control in accordance with Eq. (A24).

Figure 7

(Color online) The frequency $\Omega$ and intensity $\mid E{\mid }^2$ of external cavity modes (thin lines) and their stability versus frequency $\mathrm{\Delta }{\omega }_0$ (a) or vs $\mu$ (b) with additional phase control for the parameters of Fig. 6. Crosses $(\times )$ and circles $(엯)$ denote the unstable and stable modes correspondingly.

Figure 8

Transient dynamics. (a) The amplitude of $x$-polarized component $\mid E_x\mid$, (b) the total population inversion $N$, and (c) the instant frequency ${\Omega }_x$ vs time. The initial conditions are at the mode with maximal gain. The parameters are $\mu =0.995$ and ${\beta }_n={10}^{-8}$; the other parameters are the same as for Fig. 2. The small-amplitude oscillations for large times are an asymptotically stable state.

Figure 9

The envelopes $\mathbf{En}$ in [(a) and (b)] the domain $(w^m,n)$ and in (c) the domain $(w^m,w)$ for $\lambda =2.1$ (thick lines) calculated by Eqs. (A22, A23). The thin line in (a) and (b) is the threshold line for $\Delta =0$ [Eqs. (A8, A14)]. Other parameters are given in the text. Open circles $(엯)$ denote limit points calculated by Eq. (A27), the triangles gives the limit point [Eq. (A43)] arising due to the envelope metamorphoses, and squares denote the points where ${\mathbf{En}}^{\mathbf{l}}$ and ${\mathbf{En}}^{\mathbf{u}}$ are joined due to the metamorphoses.