  Semiconductor lasers subject to optical feedback can behave chaotically,
which can be used as a source of randomness. The optical feedback, provided by
mirrors at a distance, determines the characteristics of the chaos and thus the
quality of the randomness. However, this fixed distance also shows itself in
the intensity, an unwanted feature called the Time Delay Signature (TDS). One
promising solution to optimize the chaotic behavior is using double optical
feedback, which we study here. In particular, we focus on the impact of the
feedback phase, a small sub-wavelength change in the position of the mirrors,
on the TDS and complexity of the system. We show that by optimizing the
feedback parameters, including the feedback phases, the TDS can be suppressed,
and that in some cases feedback phase control is necessary rather than
optional. With control of all feedback parameters, it is possible to suppress
the TDS without loss of the chaotic bandwidth. Further, we show that the system
can restabilize at strong feedback rates, and one can switch between a chaotic
and steady state by changing the feedback phase. Finally, we relate the
feedback phase sensitivity to interference between the two delayed signals.
This system is promising as one can either suppress the TDS without loss of the
chaotic bandwidth or significantly increase the chaotic bandwidth.
