Pump-Induced Exceptional Points in Lasers

We demonstrate that the above-threshold behavior of a laser can be strongly affected by exceptional points which are induced by pumping the laser nonuniformly. At these singularities, the eigenstates of the non-Hermitian operator which describes the lasing modes coalesce. In their vicinity, the laser may turn off even when the overall pump power deposited in the system is increased. Such signatures of a pump-induced exceptional point can be experimentally probed with coupled ridge or microdisk lasers.

Figure 1

Intensity output of a laser system consisting of two 1D coupled ridge lasers, each of length $100\mu \mathrm{m}$ with an air gap of size $10\mu \mathrm{m}$ and an (unpumped) index of refraction $n=3+0.13i$. For $0<d<1$, the pump in the left ridge is linearly increased in the range $0<D<1.2$, and, for $1<d<2$, the same is done in the right ridge while keeping the left ridge at $D=1.2$. The points show the results determined from a finite-difference time-domain solution of the Maxwell-Bloch equations (using ${\gamma }_{\parallel }=0.02{\mathrm{mm}}^{-1}$). The solid lines are the solutions as given by SALT (see the text). The three different colors (red, blue, and yellow [19]) represent the results for different gain curves with a width of $2{\gamma }_{\perp }=4{\mathrm{mm}}^{-1}$ centered around $k_a=\{92.4,94.6,96.3\}{\mathrm{mm}}^{-1}$, respectively. Although the overall pump power increases monotonically over the entire range of $d$, the laser is found to turn off in the range $1.55<d<1.68$, due to the influence of an exceptional point. The insets show the electric field amplitude in the laser at the pump steps $d=\{1,2\}$ for the gain curve centered around $k_a=94.6{\mathrm{mm}}^{-1}$.

Figure 2

Contour plot of the function $f(k,d)$, which indicates the parameter regions of $k$ and $d$ where the laser in Fig. 1 is above threshold (green, inside black contour) or below threshold (red, outside black contour). The threshold condition, $f(k,d)=0$, is satisfied at the solid black contour. The EP in the center of the plot pulls the laser below the threshold in its vicinity. The frequency dependence of the solutions of the nonlinear SALT equations (dashed lines) and the corresponding laser thresholds (dots on the solid black contour) are provided for the same gain curves (shown in the top panel) as in Fig. 1. The evaluation of $f(k,d)$ involves two interconnected Riemann sheets (see Fig. 3), resulting in a cut between the two sheets (see the white line right above the EP). The two black dots at the upper edge of the plot represent the resonance frequencies of the passive cavity system.

Figure 3

(a) Real and (b) imaginary parts of the two eigenvalues ${\eta }_{1,2}(k,d)$, which are closest to the threshold in the $k$ and $d$ parameter region around the laser turn-off observed in Fig. 1. The eigenvalue surfaces display the typical structure of intersecting Riemann surfaces, centered around an EP.

Figure 4

Laser light intensity emitted from two coupled microdisks (see insets) as a function of pump parameter $d$. With the same pumping scheme as in Fig. 1 (maximum pump in each disk: $D=0.71$), the laser is also found here to turn off for increasing $d$. The disks have a radius of $45\mu \mathrm{m}$, an intercavity spacing of $10\mu \mathrm{m}$, and an index of refraction $n=3.67+0.09i$. The gain curve parameters are $k_a\sim 2.82\mathrm{THz}$ and ${\gamma }_{\perp }\sim 0.29\mathrm{THz}$. The insets show the absolute value of the electric field of the [green (dashed) line] lasing mode at the pump points $d=1$ and $d=2$.