Lasing thresholds and photon statistics in high-$\beta$ buried multiple quantum well photonic crystal nanocavity lasers

We investigate the lasing thresholds and photon statistics of buried multiple quantum well (MQW) photonic crystal (PhC) lasers with a high spontaneous emission coupling coefficient $\beta$ and a large carrier transparency number $N_0$. Using measured input-output curves and delay-dependent second-order photon correlations $g^{\left(2\right)}\left(\tau \right)$, we estimate that our population inversion threshold is near the kink threshold of the input-output curve, which indicates that the buried MQW PhC lasers operate close to the “lasers without inversion” (intensity jump without inversion) regime proposed by Yamamoto and Björk. With respect to static photon statistics $g^{\left(2\right)}\left(0\right)$, the super-Poissonian photon statistics $g^{\left(2\right)}\left(0\right)>1$ are observed even one order of magnitude above the kink threshold. The super-Poissonian photon statistics are specific to the class-B lasers where the photon decay rate ${\gamma }_c$ is much larger than that of carriers ${\gamma }_{\parallel }$ and the relaxation oscillation occurs. We also propose that the pump power needed to maximize the damping time of the relaxation oscillation could be the noise threshold at which the photon statistics change from thermal to Poissonian.

Figure 1

(a), (d) Steady-state photon $\overline{n}$ and (b), (e) carrier number $\overline{N}$ as a function of pump power normalized by the kink threshold $P_{\mathrm{th}}^{\mathrm{kink}}$, which are based on the steady-state solutions of the rate equations. (c), (f) Calculated $g^{\left(2\right)}\left(0\right)$ based on stochastic numerical simulations of the master equation. (a), (b), (c) are plots for low-$\beta$ lasers with low-$Q$ cavities where $\beta ={10}^{-3}, 1/{\gamma }_c=1.0$ ps, $1/{\gamma }_{\parallel }=3.0$ ns, and $N_0=5.0\times {10}^5$, thus, $\xi =0.17$. (d), (e), (f) are for high-$\beta$ lasers with large-$N_0$ and high-$Q$ cavities where $\beta =0.8, 1/{\gamma }_c=50$ ps, $1/{\gamma }_{\parallel }=3.0$ ns, and $N_0=1.0\times {10}^4$, thus $\xi =133$. The blue, green, black, and red vertical dashed lines, respectively, represent the inversion $P_{\mathrm{th}}^{\mathrm{inv}}$, quantum $P_{\mathrm{th}}^{\mathrm{qu}}$, kink $P_{\mathrm{th}}^{\mathrm{kink}}$, and gain-loss balance $P_{\mathrm{th}}^{\mathrm{bal}}$ thresholds. In the low-$\beta$ lasers (a), (b), (c), the black and red lines overlap $P_{\mathrm{th}}^{\mathrm{kink}}\simeq P_{\mathrm{th}}^{\mathrm{bal}}$, whereas in the high-$\beta$ lasers (d), (e), (f) the blue and red lines overlap $P_{\mathrm{th}}^{\mathrm{inv}}\simeq P_{\mathrm{th}}^{\mathrm{bal}}$. The yellow vertical dashed lines in (c), (f) represent the noise threshold $P_{\mathrm{th}}^{\mathrm{noise}}$ defined and explained in Sec. 5. The two horizontal dotted lines in (a), (d) represent the photon number of unity and ${\tilde{\beta }}^{-1/2}$. The horizontal dotted lines in (b), (c) represent the carrier transparency numbers $N_0$. The inset in (e) is an enlarged figure and plots the carrier number simulated with the birth-death master equation (blue curve) and with the rate equations (black curve), which shows the blue curve reaches $N_0$ with a smaller pump power than the black line.

Figure 2

$G_0$ and $S_0$ values defined in Eq. (18) and (19), respectively. The parameters for (a) and (b) are the same as in Figs. 1–1 and 1–1, respectively. The unit of $y$ axis is the cavity decay rate ${\gamma }_c$. The bottom figure in (b) is the enlarged figure of the top one, which clearly shows the cross of $G_0$ and $S_0$. For any pump power $G_0+S_0={\gamma }_c$ is satisfied. The colored vertical dashed lines represent various thresholds and are defined in Fig. 1.

Figure 3

Scanning electron microscope (SEM) image and structure of a PhC nanocavity laser with an MQW (top). Schematic of experimental setup (bottom).

Figure 4

Measured output photon number (input-output curve) (a) and linewidth (b) as a function of pump power. In (a), the output photon number in the $y$ axis is estimated from the fitting of the measured light output intensity and pump light intensity curve with Eq. (9). The linewidth measurement is limited by the resolution of the spectrometer, which is represented by the blue shaded area. The pump power is normalized by the kink threshold of the input-output curve, which is represented by the vertical dashed lines. The blue solid line in (a) is a fitting with Eq. (9), which gives $\tilde{\beta }=0.019$. The inset in (b) is the emission spectrum at $P/P_{\mathrm{th}}^{\mathrm{kink}}=0.9$.

Figure 5

Second-order photon correlation function $g^{\left(2\right)}\left(\tau \right)$ measurements performed with a Hanbury Brown-Twiss (HBT) interferometer and superconducting single photon detectors (SSPDs). $g^{\left(2\right)}\left(0\right)$ (a), the oscillation frequency ${\omega }_r$, (b) and the damping time $1/{\gamma }_r$ (c) of $g^{\left(2\right)}\left(\tau \right)$ are shown as a function of the pump power normalized by the kink threshold. The blue solid lines (a) and (b), (c) are the results of a stochastic simulation of the birth-death master equation and the small signal analysis, respectively. For these simulations and plots, $1/{\gamma }_c=25$ ps, $1/{\gamma }_{\parallel }=4.5$ ns, $\beta =0.16$, and $N_0=10000$ are used. These parameters lead to $\xi =8.7$. Examples of the measured (d) and simulated (e) second-order photon correlation functions.

Figure 6

Fixing $1/{\gamma }_c=25$ ps, $1/{\gamma }_{\parallel }=4.5$ ns, and $\tilde{\beta }=0.019$, we plot (a) the steady-state carrier number, (b) $g^{\left(2\right)}\left(0\right)$, (c) the oscillation frequency ${\omega }_r$, and (d) the damping rate ${\gamma }_r$ of $g^{\left(2\right)}\left(\tau \right)$ for four different combinations of $N_0$ and $\beta$. The colored lines represent $N=5000$ ($\beta =0.04$), 10000 (0.16), 15000 (0.39), and 20000 (0.54), which respectively correspond to $\xi =1.0$, 8.7, 33, and 60. While (a) and (b) are based on a stochastic simulation of the master equation, (c) and (d) are plotted with the analytical solution of a small signal analysis.

Figure 7

(a) Simulated input-output curve and (b) carrier number, (c) $g^{\left(2\right)}\left(0\right)$, (d) damping time ${\gamma }_r$ of $g^{\left(2\right)}\left(\tau \right)$, and (e) the Fano factor for $1/{\gamma }_c=25$ ps, $1/{\gamma }_{\parallel }=4.5$ ns, $\beta =0.16$, and $N_0=10000$. These parameters are the same as those used in Fig. 5. The simulations are based on the birth-death master equation. The blue line in (b) is the damping time of the relaxation oscillation with the small signal analysis Eq. (27), whereas the blue filled circle is based on the direct fitting of simulated $g^{\left(2\right)}\left(\tau \right)$. The blue, green, black, and red vertical dashed lines, respectively, represent the inversion $P_{\mathrm{th}}^{\mathrm{inv}}$, quantum $P_{\mathrm{th}}^{\mathrm{qu}}$, kink $P_{\mathrm{th}}^{\mathrm{kink}}$, and gain-loss balance $P_{\mathrm{th}}^{\mathrm{bal}}$ thresholds. The yellow vertical dashed line is the noise threshold $P_{\mathrm{th}}^{\mathrm{noise}}$ defined in Eq. (30). The two horizontal dotted lines in (a) represent photon numbers of unity and ${\tilde{\beta }}^{-1/2}$. The horizontal dotted line in (b) represents the carrier transparency number $N_0=10000$.

Figure 8

$G_0$ and $S_0$ values are plotted based on the parameters used in Fig. 7. $G_0$ and $S_0$ are defined respectively in Eq. (18) and (19). The definition of the colored vertical dashed lines is the same as in Fig. 7.

Figure 9

Simulations for low-$\beta$ ($\beta =0.001$) (a) class-A and (b) class-B lasers without a carrier transparency number ($N_0=0$). Top figures: carrier number $\overline{N}$ (black line) and photon $\overline{n}$ (blue line). Middle figures: $g^{\left(2\right)}\left(0\right)$ (black line) and the Fano factor (blue line). Bottom figures: the damping time (black line or filled circles) and oscillation frequency of $g^{\left(2\right)}\left(\tau \right)$ (blue line). In the bottom figures, the black line is the result of the small signal analysis [Eq. (27)], while the black filled circles are obtained by the fitting of simulated $g^{\left(2\right)}\left(\tau \right)$. The black and yellow vertical dashed lines respectively represent the kink $P_{\mathrm{th}}^{\mathrm{kink}}$ and noise $P_{\mathrm{th}}^{\mathrm{noise}}$ thresholds. The lifetimes are $1/{\gamma }_c=1$ ns, $1/{\gamma }_{\parallel }=0.1$ ns for (a), while $1/{\gamma }_c=1$ ps, $1/{\gamma }_{\parallel }=1$ ns for (b).

Figure 10

Same as Fig. 9 but for low-$\beta$ ($\beta =0.001$) (a) class-A and (b) class-B lasers with a carrier transparency number $N_0={10}^3$. The black, blue, and yellow vertical dashed lines, respectively, represent the kink $P_{\mathrm{th}}^{\mathrm{kink}}$, inversion $P_{\mathrm{th}}^{\mathrm{inv}}$ and noise $P_{\mathrm{th}}^{\mathrm{noise}}$ thresholds. The lifetimes are $1/{\gamma }_c=1$ ns, $1/{\gamma }_{\parallel }=0.1$ ns for (a), while $1/{\gamma }_c=1$ ps, $1/{\gamma }_{\parallel }=1$ ns for (b). These parameters lead to $\tilde{\beta }=9.1\times {10}^{-5}$ and $\xi =10$ for (a), while $\tilde{\beta }=0.001$ and $\xi =0.001$ for (b).

Figure 11

Simulations for high-$\beta$ ($\beta =0.8$) (a) class-A and (b) class-B lasers without a carrier transparency number ($N_0=0$). Top figures: carrier number $\overline{N}$ (black line) and photon $\overline{n}$ (blue line). Middle figures: $g^{\left(2\right)}\left(0\right)$ (black line) and the Fano factor (blue line). Bottom figures: the damping time (black line or filled circles) and oscillation frequency of $g^{\left(2\right)}\left(\tau \right)$ (blue line). In the bottom figures, the black line is the result of the small signal analysis [Eq. (27)], while the black filled circles are obtained by the fitting of simulated $g^{\left(2\right)}\left(\tau \right)$. The black and yellow vertical dashed lines respectively represent the kink $P_{\mathrm{th}}^{\mathrm{kink}}$ and noise $P_{\mathrm{th}}^{\mathrm{noise}}$ thresholds. The lifetimes are $1/{\gamma }_c=1$ ns, $1/{\gamma }_{\parallel }=0.1$ ns for (a), while $1/{\gamma }_c=1$ ps, $1/{\gamma }_{\parallel }=1$ ns for (b).

Figure 12

Same as Fig. 11 but for $\mathrm{high}-\beta$ ($\beta =0.8$) (a) class-A and (b) class-B lasers with a carrier transparency number $N_0={10}^3$. The black, blue, and yellow vertical dashed lines, respectively, represent the kink $P_{\mathrm{th}}^{\mathrm{kink}}$, inversion $P_{\mathrm{th}}^{\mathrm{inv}}$, and noise $P_{\mathrm{th}}^{\mathrm{noise}}$ thresholds. The lifetimes are $1/{\gamma }_c=1$ ns, $1/{\gamma }_{\parallel }=0.1$ ns for (a), while $1/{\gamma }_c=1$ ps, $1/{\gamma }_{\parallel }=1$ ns for (b). These parameters lead to $\tilde{\beta }=5.0\times {10}^{-4}$ and $\xi =8000$ for (a), while $\tilde{\beta }=0.7$ and $\xi =0.8$ for (b).