Influence of spin-dependent carrier dynamics on the properties of a dual-frequency vertical-external-cavity surface-emitting laser

The properties of a dual-frequency vertical-external-cavity surface-emitting laser (VECSEL), in which two linear orthogonal polarization modes are oscillating simultaneously, are theoretically investigated. We derive a model based on the ideas introduced by San Miguel et al. [San Miguel, Feng, and Moloney, Phys. Rev. A 52, 1728 (1995)], taking into account the spin dynamics of the carriers inside the quantum well based gain medium of the dual-frequency VECSEL. This model is shown to succeed in describing quite a few properties of the dual-frequency VECSEL, such as the behavior of nonlinear coupling strength between the modes, the spectral behavior of intensity noises of the modes, and also the correlation between these intensity noises. A good agreement is found with experimental data. The variables associated with the spin-dependent carrier dynamics can be adiabatically eliminated due to the class-A dynamical behavior of the considered laser which is based on a cm-long external cavity. This leads to a simple analytical description of the dynamics of the dual-frequency VECSEL, providing a better understanding of the physics involved.

Figure 1

Level scheme involved in the SFM model [1]. $N_{+}$ and $N_{-}$ are the population inversions for the ${\sigma }^{+}$ and ${\sigma }^{-}$ transitions, respectively. The decay rate of these population inversions is $\Gamma$. The spin-flip rate is ${\gamma }_S$.

Figure 2

Schematic representation of the dual-frequency VECSEL. The intracavity birefringent crystal (BC) spatially separates the ordinary (o) and extraordinary (e) polarized modes inside the gain medium.

Figure 3

Schematic representation of the two modes in the gain medium of the dual-frequency VECSEL. We model the two beams as two top-hat cylindrical beams with circular sections. $E_x$ ($E_y$) is the complex amplitude of the $x$-polarized ($y$-polarized) field. $N_1$ ($N_3$) is the total population inversion in the region where only the $x$-polarized ($y$-polarized) mode has a nonzero intensity. $N_2$ is the total population inversion and $n_2$ is the population inversion difference in the region where both modes are superimposed. The parameters $d$ and $\eta$, respectively, represent the spatial separation and the relative overlap between the two modes.

Figure 4

Time evolution of (a) the intracavity photon numbers $F_x$ and $F_y$ and (b) the population inversions $N_1$, $N_2$, and $N_3$ for $d=100\mu \mathrm{m}$, $w=62\mu \mathrm{m}$, $1/{\gamma }_x=6\mathrm{ns}$, $1/{\gamma }_y=6.05\mathrm{ns}$, $1/\Gamma =3\mathrm{ns}$, $\varepsilon =0.02$, $\kappa =6.3\times {10}^{-2}{\mathrm{s}}^{-1}$, $N_{0x}=N_{0y}=1.3\times {\gamma }_x/\kappa$. In this case $\eta =0.074$. (c), (d) Same as (a), (b) for $d=50\mu \mathrm{m}$, corresponding to $\eta =0.52$. (e), (f) Same as (a), (b) for $d=20\mu \mathrm{m}$, corresponding to $\eta =0.90$.

Figure 5

Same as Figs. 4 and 4 in the case of a class-B lasers. The values of the parameters are $d=50\mu \mathrm{m}$, $w=62\mu \mathrm{m}$, $1/{\gamma }_x=6\mathrm{ns}$, $1/{\gamma }_y=6.05\mathrm{ns}$, $1/\Gamma =100\mathrm{ns}$, $\varepsilon =0.02$, $\kappa =6.3\times {10}^{-2}{\mathrm{s}}^{-1}$, $N_{0x}=N_{0y}=1.3\times {\gamma }_x/\kappa$.

Figure 6

Evolution of the steady-state photon numbers $F_x$ and $F_y$ vs pumping rate $N_{0x}=N_{0y}$. The values of the parameters are $w=62\mu \mathrm{m}$, $1/{\gamma }_x=6\mathrm{ns}$, $1/{\gamma }_y=6.05\mathrm{ns}$, $1/\Gamma =3\mathrm{ns}$, $\varepsilon =0.02$, $\kappa =6.3\times {10}^{-2}{\mathrm{s}}^{-1}$. (a) $d=100\mu \mathrm{m}$. (b) $d=50\mu \mathrm{m}$. (c) $d=20\mu \mathrm{m}$.

Figure 7

Time evolution of (a), (c), and (e) the intracavity photon numbers $F_x$ and $F_y$ and (b), (d), and (f) the population inversions $N_1$, $N_2$, and $N_3$ for $d=20\mu \mathrm{m}$, $w=62\mu \mathrm{m}$, $1/{\gamma }_x=6\mathrm{ns}$, $1/{\gamma }_y=6.05\mathrm{ns}$, $1/\Gamma =3\mathrm{ns}$, $\kappa =6.3\times {10}^{-2}{\mathrm{s}}^{-1}$, $N_{0x}=N_{0y}=1.3\times {\gamma }_x/\kappa$. (a), (b) $\varepsilon =0.01$. (c), (d) $\varepsilon =0.05$. (e), (f) $\varepsilon =0.1$.

Figure 8

Evolution of ${\xi }_{xy}$ (dashed blue line) and ${\xi }_{yx}$ (full red line) vs $\delta N_0$ with $N_{0x}=N_0(1+\delta N_0)$ and $N_{0y}=N_0(1-\delta N_0)$. Thick lines: calculation from Eqs. (40) and (41) using the steady-state photon numbers $F_x$ and $F_y$ obtained from Eqs. (34) and (35). Thin lines: calculation from Eqs. (38) and (39). The values of the parameters are $d=50\mu \mathrm{m}$, $w=62\mu \mathrm{m}$, $1/{\gamma }_x=6\mathrm{ns}$, $1/{\gamma }_y=6\mathrm{ns}$, $1/\Gamma =3\mathrm{ns}$, $\varepsilon =0.02$, $\kappa =6.3\times {10}^{-2}{\mathrm{s}}^{-1}$, $N_0=1.3{\gamma }_x/\kappa$.

Figure 9

Evolution of $C={\xi }_{xy}{\xi }_{yx}$ (full line) vs $d$ obtained from Eqs. (40) and (41) using the steady-state photon numbers $F_x$ and $F_y$ obtained from Eqs. (34) and (35). Dashed line: calculation from Eq. (44). The values of the parameters are $w=62\mu \mathrm{m}$, $1/{\gamma }_x=6\mathrm{ns}$, $1/{\gamma }_y=6\mathrm{ns}$, $1/\Gamma =3\mathrm{ns}$, $\varepsilon =0.02$, $\kappa =6.3\times {10}^{-2}{\mathrm{s}}^{-1}$, $N_{0x}=N_{0y}=1.3{\gamma }_x/\kappa$.

Figure 10

(a) Relative intensity noise, (b) correlation amplitude, and (c) correlation phase spectra calculated from the present model. The values of the parameters are $w=62\mu \mathrm{m}$, $1/{\gamma }_x=6\mathrm{ns}$, $1/{\gamma }_y=6.05\mathrm{ns}$, $1/\Gamma =3\mathrm{ns}$, $\varepsilon =0.02$, $\kappa =6.3\times {10}^{-2}{\mathrm{s}}^{-1}$, ${\overline{N}}_{0x}={\overline{N}}_{0y}=1.3\times {\gamma }_x/\kappa$, ${\mathrm{RIN}}_p=-135\mathrm{dB}/\mathrm{Hz}$, and $d=100\mu \mathrm{m}$, corresponding to an overlap $\eta =0.074$. (d), (e), and (f) Same as (a)–(c) for $d=50\mu \mathrm{m}$, corresponding to $\eta =0.52$. (g), (h), and (i) Same as (a)–(c) for $d=20\mu \mathrm{m}$, corresponding to $\eta =0.90$.