Modes of structurally chiral lasers

We employ coupled wave theory to enumerate the lasing modes of structurally chiral lasers. The elliptical modes are shown to be fundamentally distinct from those of a scalar distributed feedback laser. High threshold modes are shown to lase with the opposite chirality as the active medium, in contrast to their low-threshold counterparts that lase with the same chirality as the active medium. The lasing mode structure suggests the intriguing possibility of dynamically changing the polarization handedness of a chiral laser, as well as the possibility of lasing within the forbidden band-gap region. These observations arise from the fundamental interplay between the distributed chirality-preserving reflections within the active medium and the localized chirality-reversing reflections at the medium's boundaries.

Figure 1

Right-handed SCM with surrounding media of indices $n_1$ at $z=0$ and $n_2$ at $z=L$. The transverse principal axes are aligned with the coordinate axes at $z=0$ and rotate around the $z$ axis with period $L_p$. We show that such right-handed SCMs can support both right-handed and left-handed lasing modes.

Figure 2

The four round-trip light paths for circularly polarized light: (a) $a_{LL}=r_1{r}_2{e}^{ikL}{t}_c$, (b) $a_{LR}=r_1{r}_c$, (c) $a_{RL}=e^{2i(k+p)L}{r}_1{r}_2^2{r}_c$, and (d) $a_{RR}=e^{ikL}{r}_1{r}_2{t}_c$. Vertical lines represent the SCM boundaries and curved arrows represent the light paths. The letters denote circular polarization state.

Figure 3

Lasing modes for a chiral laser as (a) a surface plot and (b) a contour plot. The lasing modes appear as poles that occur in two groups, one group with high-gain thresholds $\left(0_L,1_L,2_L,3_L\right)$ and a group with low-gain thresholds $\left(1_R,2_R,3_R\right)$.

Figure 4

Movement of the lasing modes with respect to varying $\kappa L$. The solid lines trace the evolution of the modes of Fig. 3 (red dots) as $\kappa L$ is reduced to zero (black crosses). The dashed lines connect modes of constant $\kappa L$. The superimposed ellipses represent the lasing output polarization when $\kappa L=4$.

Figure 5

Intracavity intensity envelopes for the seven modes in Fig. 3 of the total (green solid line), LCP (red dashed line), and RCP (blue dash-dotted line) electric fields for a SCM cavity.

Figure 6

Evolution of chiral modes as the facet reflectivies are reduced by the given factor: (a) $r$, (b) $0.1\times r$, (c) $0.01\times r$, and (d) $0\times r$. (a) corresponds to Fig. 4, while in (d) the purely RCP modes correspond exactly to the scalar DFB modes enumerated by Kogelnik and Shank [11].