Stability analysis for bad cavity lasers using inhomogeneously broadened spin-1/2 atoms as a gain medium

Bad cavity lasers are experiencing renewed interest in the context of active optical frequency standards, due to their enhanced robustness against fluctuations of the laser cavity. The gain medium would consist of narrow-linewidth atoms, either trapped inside the cavity or intersecting the cavity mode dynamically. A series of effects like the atoms finite velocity distribution, atomic interactions, or interactions of realistic multilevel atoms with auxiliary or stray fields can lead to an inhomogeneous broadening of the atomic gain profile. This causes the emergence of unstable regimes of laser operation, characterized by complex temporal patterns of the field amplitude. We study the steady-state solutions and their stability for the metrology-relevant case of a bad cavity laser with spin-1/2 atoms, such as ${}^{171}\mathrm{Yb}$, interacting with an external magnetic field. For the stability analysis, we present an efficient method, which can be applied to a broad class of single-mode bad cavity lasers with inhomogeneously broadened multilevel atoms acting as a gain medium.

Figure 1

Illustration of the stability analysis by means of tracing the variation of $arg(\mathfrak{D}\left(\lambda \right))$. Left panel shows $\lambda$ complex plane with roots (red dots) and poles (blue diamonds) of the function $\mathfrak{D}\left(\lambda \right)$ (26), and the contour over which to trace the phase (thick green curve). Inset shows a scaled view of the region around the point $\lambda =0$. Right panel shows the dependence of $arg\left(\mathfrak{D}\right)$ on $\mathrm{Im}\left(\lambda \right)$ along the contour. Here two eigenvalues have positive real parts (which indicates instability), and the change of the phase over the contour is equal to $-4\pi$.

Figure 2

Level structure of individual atoms and notations used for levels, frequencies, and detunings: ${\omega }_c$ and $\omega$ are the frequencies of the cavity mode and the lasing field; ${\omega }^j$ is the averaged frequency of the lasing transitions of the $j\mathrm{th}$ atom; ${\delta }_z={\omega }_c-\omega$ is the differential Zeeman shift, and ${\mathrm{\Delta }}_j={\omega }^j-{\omega }_c$ is the individual frequency detuning of the $j\mathrm{th}$ atom. We consider only a $\pi$-polarized cavity field; therefore, the relative energy shift between the levels $|g_{-1/2}^j\rangle$ and $|g_{1/2}^j\rangle$ is not significant.

Figure 3

Characteristics of the center-line solutions. (a) Steady-state output power $P=\hslash \omega |\langle \widehat{c}\rangle {|}^2\kappa$ versus the incoherent pumping rate $w$ for different values of the inhomogeneous broadening ${\mathrm{\Delta }}_0$ [styled and colored curves labeled by the values of ${\mathrm{\Delta }}_0$ (in ${\mathrm{s}}^{-1}$)]. Differential Zeeman shift ${\delta }_z=100$. The solid black curve, representing the output power of the laser with ${\mathrm{\Delta }}_0={\delta }_z=0$, is given as a reference. (b) Peak output power $P_{\max }$ versus ${\mathrm{\Delta }}_0$ for different values of ${\delta }_z$. (c) Repumping rate $w_m$ maximizing the output power $P$ as a function of ${\mathrm{\Delta }}_0$ for different values of ${\delta }_z$. The dashed horizontal line indicates $w_m={\mathrm{\Gamma }}_c/2$ at ${\mathrm{\Delta }}_0={\delta }_z=0$; see Ref. [2] for details.

Figure 4

Characteristics of the frequency-detuned solutions. Frequency detuning $\delta =\omega -{\omega }_c$ (top) and output power $P=\hslash \omega |\langle \widehat{c}\rangle {|}^2\kappa$ (bottom) corresponding to the frequency-detuned solutions for ${\delta }_z=100{\mathrm{s}}^{-1}$ and ${\delta }_z=10{\mathrm{s}}^{-1}$ (left and right pairs of plots respectively) at different values of ${\mathrm{\Delta }}_0$ (different curves on the same plot) as a function of the pumping rate $w$ ($x$ axis). Curves $\delta \left(w\right)$ are labeled by the values of ${\mathrm{\Delta }}_0$ (in ${\mathrm{s}}^{-1}$), the same style-color encoding is valid for the power plot with the same ${\delta }_z$. Insets show lower threshold $w_{th}$ to lasing as a function of ${\mathrm{\Delta }}_0$. Vertical dotted lines on the upper plots and horizontal dotted lines on the insets represent the fundamental lower threshold level $w=\gamma$.

Figure 5

Domains of existence and stability of various steady-state solutions in the $(w,{\mathrm{\Delta }}_0)$ plane for (a) ${\delta }_z=10{\mathrm{s}}^{-1}$, and (b) ${\delta }_z=100{\mathrm{s}}^{-1}$. The zero-field solution exists everywhere but is stable only in the unshaded domain where no other solution exists. Other domains are I: line-centered solution exists and is stable, II: line-centered solution exists but is unstable, III: line-centered and frequency-detuned solutions exist but are unstable, IV: frequency-detuned solutions exist but are unstable. The inset is an enlarged view of the area near the lower lasing threshold for ${\delta }_z=10{\mathrm{s}}^{-1}$. (c)–(h) Time evolution of the intracavity photon number ${\left|E\right|}^2$ according the numerical simulation for ${\delta }_z=100{\mathrm{s}}^{-1}$ and different values of $w$ and ${\mathrm{\Delta }}_0$ (given as plot labels; also indicated as red dots in panel (b) with respective labels (c)–(h). The seed value of ${\left|E\right|}^2$ was set to ${10}^{-6}$ for this simulation.

Figure 6

General pumping scheme (hyperfine structure not shown) for a 804 nm bad cavity laser using ${}^{176}\mathrm{Lu}$ ions. Dashed lines denote the most relevant spontaneous decays, solid lines correspond to both spontaneous and laser-induced transitions (wavelengths are indicated).

Figure 7

Characteristics of the steady-state solutions for inhomogeneously broadened two-level laser with incoherent pumping. (a) Output power $P$ versus the incoherent repumping rate $w$ for atomic ensembles without (solid black curve) and with inhomogeneous broadening ${\mathrm{\Delta }}_0$ [colored curves labeled by values of ${\mathrm{\Delta }}_0\left({\mathrm{s}}^{-1}\right)]$ at ${\gamma }_R=0$. (b) Peak output power $P_{\max }$ and (c) repumping rate $w_m$ maximizing the output power versus ${\mathrm{\Delta }}_0$ for different values of the incoherent dephasing rate ${\gamma }_R$. Curves $w_m\left({\mathrm{\Delta }}_0\right)$ are labeled by the values of ${\gamma }_R$ (in ${\mathrm{s}}^{-1}$); the same style-color encoding is valid for the plot of $P_{\max }\left({\mathrm{\Delta }}_0\right)$ in panel (b) with the same ${\gamma }_R$. The dashed horizontal line indicates $w_m={\mathrm{\Gamma }}_c/2$.

Figure 8

Domains of existence and stability of the steady-state solutions for the two-level incoherently pumped laser with inhomogeneous broadening ${\mathrm{\Delta }}_0$ in the $(w,{\mathrm{\Delta }}_0)$ plane for different values of incoherent dephasing rate ${\gamma }_R$: (a) ${\gamma }_R=0$, (b) ${\gamma }_R=10{\mathrm{s}}^{-1}$, (c) ${\gamma }_R=70{\mathrm{s}}^{-1}$. Insets are enlarged views of the stability domains at small ${\mathrm{\Delta }}_0$ near the lower lasing threshold. Parameters of the cavity and the atomic ensemble are the same as for Fig. 7.