We performed a theoretical analysis of the method used to extract the PM from the output of a DM-OIL semiconductor laser by AM compensating for the typical homodyne detection method. Our analysis was based on Display Formula
$EML(t)=AML\u2009cos(\omega MLt+\varphi ML),$(1)
Display Formula$EOIL(t)=AOIL\u2009cos(\omega MLt+\varphi OIL),$(2)
Display Formula$\varphi OIL=\varphi ML+\Delta \varphi ,$(3)
where $EML(t)$ and $EOIL(t)$ are the time-varying optical field expressions of the master and injection-locked lasers, respectively; $AML$, $\omega ML$, and $\varphi ML$ are the field amplitude, angular frequency, and initial phase of the master laser, respectively; and $AOIL$, $\varphi OIL$ are the amplitude and phase of the OIL laser, respectively. The $\varphi OIL$ term can be expressed as $(\varphi ML+\Delta \varphi )$, where $\Delta \varphi $ is the phase shift of the OIL laser. The $AOIL$ and $\varphi OIL$ terms can be calculated using the standard coupled-rate equation that describes the behavior of an OIL laser^{17}Display Formula$dS(t)dt={g[N(t)\u2212Ntr]\u2212\gamma p}S(t)+2\kappa SMLS(t)\u2009cos[\varphi (t)\u2212\varphi ML],$(4)
Display Formula$d\varphi (t)dt=\alpha 2{g[N(t)\u2212Ntr]\u2212\gamma p}\u2212\kappa SMLS(t)sin[\varphi (t)\u2212\varphi ML]\u2212\Delta \omega ,$(5)
Display Formula$dN(t)dt=J(t)\u2212\gamma nN(t)\u2212g[N(t)\u2212Ntr]S(t),$(6)
where $S(t)$, $\varphi (t)$, and $N(t)$ are the photon number, phase, and carrier number of an injection-locked slave laser, respectively, and $Ntr$ is the transparency carrier number of a free-running slave laser, which is defined as $Ntr=Nth\u2212\gamma p/g$, where $Nth$, $\gamma p$, and $g$ are the threshold carrier number, photon decay rate, and linear gain, respectively. The term $\Delta \omega (=2\pi \Delta f)$ is the angular frequency difference between the master and free-running slave lasers, $\alpha $ is the linewidth enhancement factor of the laser, $J(t)$ is the number of electrons in the DC bias of the slave laser, $\kappa $ is the field coupling ratio between the master and free-running slave lasers, and $\gamma n$ is the carrier decay rate. The phase shift $\Delta \varphi $ of the injection-locked laser can be derived as Display Formula$\Delta \varphi =sin\u22121{\u2212\Delta \omega \kappa 1+\alpha 2SOILSML}\u2212tan\u22121\alpha ,$(7)
where $SOIL$ is the steady-state photon number given by Display Formula$SOIL=Sfree,SL\u2212(\gamma n/\gamma p)\Delta N01+(g\Delta N0/\gamma p),$(8)
where $\Delta N0$ is the steady-state carrier number. From Eq. (7), $\Delta \varphi $ can be modulated by either varying $SOIL/SML$ or $\Delta \omega $, which are closely related to, or identical to, the definitions of the injection-locking parameters $\Delta f$ and $R$. The modulated phase shift is in the range Display Formula$\u2212\pi 2\u2264\Delta \varphi \u2264cot\u22121\alpha .$(9)