Chiral modes and directional lasing at exceptional points

Abstract

Controlling the emission and the flow of light in micro- and nanostructures is crucial for on-chip information processing. Here we show how to impose a strong chirality and a switchable direction of light propagation in an optical system by steering it to an exceptional point (EP)-a degeneracy universally occurring in all open physical systems when two eigenvalues and the corresponding eigenstates coalesce. In our experiments with a fiber-coupled whispering-gallery-mode (WGM) resonator, we dynamically control the chirality of resonator modes and the emission direction of a WGM microlaser in the vicinity of an EP: Away from the EPs, the resonator modes are nonchiral and laser emission is bidirectional. As the system approaches an EP, the modes become chiral and allow unidirectional emission such that by transiting from one EP to another one the direction of emission can be completely reversed. Our results exemplify a very counterintuitive feature of non-Hermitian physics that paves the way to chiral photonics on a chip.

Keywords: asymmetric backscattering; chiral modes; directional lasing; exceptional points; whispering-gallery-mode resonator.

Fig. 1.

Experimental configuration and the effect of scatterers. (A) Illustration of a WGM resonator side-coupled to two waveguides, with the two scatterers enabling the dynamical tuning of the modes. cw and ccw are the clockwise and counterclockwise rotating intracavity fields. ${\text{a}}_{\text{cw}\left(\text{ccw}\right)}$ and ${\text{b}}_{\text{cw}\left(\text{ccw}\right)}$ are the field amplitudes propagating in the waveguides. β is the relative phase angle between the scatterers. Inset shows the optical microscope image of the microtoroid resonator, the tapered fiber waveguides (horizontal lines), and the two silica nanotips denoting the scatterers (diagonal lines on the left and right side of the resonator). (B) Varying the size and the relative phase angle of a second scatterer helps to dynamically change the frequency detuning (splitting) and the linewidths of the split modes revealing avoided crossings (Top) and an EP (Bottom). (C) Effect of β on the frequency splitting $2g,$ difference ${\gamma }_{\text{diff}},$ and sum ${\gamma }_{\text{sum}}$ of the linewidths of split resonances when relative size of the scatterers was kept fixed (SI Appendix, Figs. S3–S5). (D) Effect of β on the splitting quality factor $Q_{\text{sp}}.$ At the EP, the splitting quality factor $Q_{\text{sp}}$ approaches zero because the splitting $2g$ goes to zero (C, Top) whereas the total dissipation ${\gamma }_{\text{sum}}$ remains finite (C, Middle). Circles denote the experimental data, and the solid lines are the best fit using a theoretical model (SI Appendix, S1: Two-Mode-Approximation (TMA) Model and the Eigenmode Evolution and S2: Experimental Observation of an EP by Tuning the Size and Position of Two Scatterers).

Fig. 2.

Experimental observation of scatterer-induced asymmetric backscattering. (A and B) When there is no scattering center in or on the resonator, light coupled into the resonator through the first waveguide in the cw (A, i) [or ccw (B, i)] direction couples out into the second waveguide in the cw (A, i) [or ccw (B, i)] direction: the resonant peak in the transmission and no signal in the reflection. (A, ii and B, ii) When a first scatterer is placed in the mode field, resonant peaks are observed in both the transmission and the reflection regardless of whether the light is input in the cw (A, ii) or in the ccw (B, ii) direction. (A, iii and B, iii) When a second scatterer is suitably placed in the mode field, for the cw input there is no signal in the reflection output port (A, iii), whereas for the ccw input there is a resonant peak in the reflection, revealing asymmetric backscattering for the two input directions. Inset in B, iii compares the two backscattering peaks in A, iii and B, iii. Estimated chirality is −0.86.

Fig. 3.

Controlling directionality and intrinsic chirality of whispering-gallery modes. (A and B) Directionality D (A) and chirality α (B) of the WGMs of a silica microtoroid resonator as a function of β between the two scatterers. The directionality D given in A was obtained when the input light was injected in the cw direction. For the input in the ccw direction, the results for D are similar but with opposite sign. Note that chirality α is an intrinsic quantity that is independent of the injection direction of the input light. The solid lines are obtained from the theoretical transmission model (SI Appendix, S4: Chirality Analysis and Comparison Between the Lasing and the Transmission Models and S5: Directionality Analysis for the Biased Input Case in the Transmission Model and Eqs. S16–S20) with the parameters set as $V_1=1.50-i0.10$ and $V_2=0.608-i0.099$ for A and $V_1=1.50-i0.10$ and $V_2=1.165-i0.675$ for B.

Fig. 4.

Scatterer-induced mirror-symmetry breaking in a WGM microlaser at an EP. In a WGM microlaser with mirror symmetry the intracavity laser modes rotate both in cw and ccw directions and thus the outcoupled light is bidirectional and chirality is zero. The scatterer-induced symmetry breaking allows tuning both the directionality and the chirality of laser modes. (A) Intensity of light outcoupled into a waveguide in the cw and ccw directions as a function of β. Regions of bidirectional emission and fully unidirectional emission are seen. (B) Chirality as a function of β. Transitions from nonchiral states to unity ($\pm 1$) chirality at EPs are clearly seen. Unity chirality regions correspond to unity unidirectional emission regions in A. The solid line is obtained from the theoretical lasing model (SI Appendix, S3: Emission and Chirality Analysis for the Lasing Cavity and S4: Chirality Analysis and Comparison Between the Lasing and the Transmission Models and Eqs. S10–S12) with the parameters set as $V_1=1.50-i0.10$ and $V_2=1.485-i\mathrm{0.14.}$ (C–E) Finite-element simulations revealing the intracavity field patterns for the cases labeled C–E in A and B. Results shown in C–E were obtained for the same size factor but different β: (C) 2.628 rad, (D) 2.631 rad, and (E) 2.626 rad. $P_1$ and $P_2$ denote the locations of the scatterers. For a quantitative comparison between the experimental data and the simulations, the latter would have to be extended along the lines of ref. to include also nonlinear effects as well as possible corrections stemming from the 3D nature of the experimental setup.