New multi-mode delay differential equation model for lasers with optical feedback

Abstract

In this paper, we discuss the relations between the spatially-distributed traveling wave, Lang–Kobayashi, and a new multi-mode delay differential equation models for Fabry–Perot type semiconductor diode lasers with an external optical feedback. All these models govern the dynamics of the slowly varying complex amplitudes of the optical fields and carrier density. To compare the models, we calculate the cavity modes determined by the threshold carrier density and optical frequency of the steady states in all three models. These calculations show that the Lang–Kobayashi type model is in good agreement with the traveling wave model only in the small feedback from the external cavity regimes, whereas newly derived multi-mode delay differential equation model remains correct even at moderate and large optical feedback regimes.

Appendix: Normalization of the TW model

Before normalization, the TW model within the considered Fabry–Perot-type diode laser (longitudinal coordinate ${\tilde{z}}\in [-{\tilde{L}},0]$) reads as in Bandelow et al. (2001):

$$\begin{aligned} \left( \frac{n_g}{c_0}\partial _{{\tilde{t}}}\pm \partial _{{\tilde{z}}}\right) {\tilde{E}}_\pm&= \left( \frac{(1+i\alpha _H)g({\tilde{n}}-n_{tr})}{2} - i\delta -\frac{\alpha }{2}\right) {\tilde{E}}_\pm - \frac{\tilde{{\bar{g}}}}{2}\left( {\tilde{E}}_\pm - {\tilde{P}}_\pm \right) , \\ \frac{d}{d{\tilde{t}}}{\tilde{P}}_\pm&= \tilde{{\bar{\gamma }}} {\tilde{E}}_\pm + \left( i\tilde{{\bar{\omega }}} - \tilde{{\bar{\gamma }}}\right) {\tilde{P}}_\pm , \\ \frac{d}{d{{\tilde{t}}}} {\tilde{n}}&= \frac{I}{e{\bar{L}}\sigma } - \frac{{\tilde{n}}}{\tau _N} - \frac{c_0}{n_g{\tilde{L}}}{\mathfrak {R}}\int _{-{\tilde{L}}}^0 \left( {\tilde{E}}, g({\tilde{n}} - n_{tr}){\tilde{E}} -\tilde{{\bar{g}}}\left( {\tilde{E}} - {\tilde{P}}\right) \right) d {\tilde{z}}, \end{aligned}$$

whereas the boundary conditions for the optical fields are

$$\begin{aligned} {\tilde{E}}_+(-{\tilde{L}},{\tilde{t}})&= -r^*_f {\tilde{E}}_-(-{\tilde{L}},{\tilde{t}}),\quad {\tilde{F}}_i({\tilde{t}}) = K e^{i\phi } {\tilde{F}}_e({\tilde{t}} - \tilde{\tau }), \\ {\tilde{F}}_e({\tilde{t}})&= t_r{\tilde{E}}_+(0,{\tilde{t}}) -r^*_{r}{\tilde{F}}_i({\tilde{t}}),\quad {\tilde{E}}_-(0,{\tilde{t}}) = r_{r} {\tilde{E}}_+(0,{\tilde{t}}) +t_r{\tilde{F}}_i({\tilde{t}}). \end{aligned}$$

The model parameters $n_g$, g, $n_{tr}$, $\alpha$, $\delta$, I, $\sigma$, and $\tau _N$ are the group velocity index, the differential gain, the carrier density at the transparency, the field losses, the detuning from the central frequency, the injected current, the cross-section area of the active zone, and the linear carrier recombination time, respectively. $c_0$ and e are the physical constants denoting the speed of light in vacuum and the electron charge. Finally, $\tilde{\phantom {a}}~$ indicates original non-normalized variables, coordinates, and parameters. In this paper, the detuning $\delta$ is such that

$$\begin{aligned} \delta = \frac{\alpha _{H}G}{2}+ \frac{\arg (-r_f^*r_r)}{2{\tilde{L}}} + \frac{\tilde{{\bar{g}}} \tilde{{\bar{\gamma }}} \tilde{{\bar{\omega }}}}{2(\tilde{{{\bar{\gamma }}}}^2+\tilde{{{\bar{\omega }}}}^2)}, ~~ G = \alpha - \frac{\ln |r_fr_r|}{{\tilde{L}}} + \frac{\tilde{{\bar{g}}}\tilde{{{\bar{\omega }}}}^2}{\tilde{{{\bar{\gamma }}}}^2+\tilde{{{\bar{\omega }}}}^2} = g(n_{th} - n_{tr}), \end{aligned}$$

where $n_{th}$ and G are the threshold carrier density and threshold gain of the solitary diode laser, so that G represents full (radiative and non-radiative) losses of the laser in the absence of the feedback from the external cavity. This special choice of the detuning factor $\delta$ implies the zero relative optical frequency of the solitary laser.

The normalization of the TW model is made by scaling the coordinates and the dynamical variables according to

$$z:=G {\tilde{z}}, \,\, t:=\frac{c_0G}{n_{g}} {\tilde{t}}, \,\, n:=\frac{{\tilde{n}}-n_{th}}{2 (n_{th} - n_{tr})}, \,\, {\varPsi }:= \sqrt{\frac{c_0 g \tau _N}{2n_g}} \, \tilde{{\varPsi }},$$

where ${\varPsi }$ denotes $E^+$, $E^-$, $P^+$, $P^-$, $F_e$ or $F_i$. The parameters of the normalized TW model (3) are related to the parameters of the original model as

$$\begin{aligned} L&= G\,{\tilde{L}},\quad \tau = \frac{c_0G}{n_{g}}\, \tilde{\tau },\quad \xi _0 = \left( i\delta -\frac{\alpha +(1+i\alpha _H)G}{2}\right) {\tilde{L}}, \quad {\bar{g}} = \frac{\tilde{{\bar{g}}}}{G},\quad \\ {\bar{\omega }}&= \frac{n_{g}}{c_0G}\,\tilde{{\bar{\omega }}},\quad {\bar{\gamma }} = \frac{n_{g}}{c_0G}\,\tilde{{\bar{\gamma }}}, \quad \epsilon = \frac{n_{g}}{c_0G\tau _N},\quad J =\frac{I-I_{th}}{2(I_{th}-I_{tr})}, \quad I_{j} = \frac{e{\tilde{L}}\sigma n_{j}}{\tau },~~ \quad j=th,tr. \end{aligned}$$