Page 464 - Tunable Lasers Handbook

424 Paul Zorabedian

where i;z is the reflectance of the feedback-coupling facet. and now the V; are
the longitudinal modes of the solitary gain-chip cavity in the absence of external
feedback and thus serve as constant-frequency markers. The net effect of the
gain-dependent refractive index shift is to change the dependence of the thresh-
old gain on uavelength from a sinusoid to an asymmetric sawtooth pattern.
Depending on the values of a, rf2, and the threshold versus wavelength can
be either single-valued or I-eentmrzt (Fig. 43). The condition where the threshold
versus wavelength curve is reentrant (or overhanging) corresponds to the condi-
tion of bistability [23]. In this case wavelength bands exist that occur with the
periodicity of the solitary cavity longitudinal mode spacing in which the oscilla-
tion threshold is multivalued (actually triple valued). When the bias current is
turned up from zero, the laser goes into the low-threshold state. However, if the
cavity is momentarily blocked while the bias current remains on, the oscillation
will switch to the high-threshold state when the obstruction is removed (Fig. 44).

16.2 Bistability and Axial Mode Instability
As mentioned earlier, the regions in which the threshold versus wavelength
curve is multivalued actually contain three sets of threshold states. It turns out
that the middle states are unstable and will not support steady-state oscillation.
This can be shown by performing a classical stability analysis on the rate equa-
tions and determining the conditions under which the circulating field and carrier
density return to their steady-state solutions following a small perturbation. The
existence of such classically unstable states has been experimentally correlated
with a transition from single-mode to multimode output [24]. Each intermediate-
threshold unstable mode lies between a pair of high- and low-threshold modes
along the frequency axis [144]. For a given external reflectance reSf and linewidth
broadening factor a, it can be shown that unstable states appear when the facet
reflectance I-~, exceeds a critical value rf2*, which is given by

I” il

This equation can be solved analytically and provides a boundary surface
between sets of cavity parameters for which all steady-state solutions are stable
and those for which some solutions are unstable (Fig. 45). 4n approximate
expression for the critical facet reflectance is given by

The quantity r; represents the maximum feedback-coupling facet reflectance
that can be tolerated while maintaining unconditional stability of the laser at all