Observation of an Exceptional Point in a Chaotic Optical Microcavity

We present spectroscopic observation of an exceptional point or the transition point between mode crossing and avoided mode crossing of neighboring quasieigenmodes in a chaotic optical microcavity of a large size parameter. The transition to the avoided mode crossing was impeded until the degree of deformation exceeded a threshold deformation owing to the system’s openness also enhanced by the shape deformation. As a result, a singular topology was observed around the exceptional point on the eigenfrequency surfaces, resulting in fundamental inconsistency in mode labeling.

Figure 1

Eigenfrequencies of $l=2$ (in orange) and $l=4$ (in blue) modes as a function of the deformation $\eta$ and the internal parameter $n$. The magnified view shows the high resolution spectra taken with a spectrometer of 0.05 nm resolution. Green curves are Lorentzian fits.

Figure 2

(a) Magnitude $\Delta {\nu }_{14}$ of AMC between two uncoupled states, $l_0=1$ and $l_0=4$ modes, as a function of $\eta$. (b) Decay rates ${\gamma }_1$ and ${\gamma }_4$ of $l_0=1$, 4 uncoupled states. The dotted line for ${\gamma }_1$ is an extrapolation beyond spectral visibility.

Figure 3

CMF spectra for the case of (a) $\eta =0.125$, (b) $\eta =0.102$, and (c) $\eta =0.078$. Orange and blue arrows indicate $l_0=1$ and $l_0=4$ modes, respectively. The output coupling efficiency of $l_0=1$ modes is too small for these modes to be observed in (a)–(c). Red arrows indicate where $l_0=1$ and 4 modes appear to be overlapped. Deformation $\eta$ was fine-tuned within 0.005 to maximize the peak. (d) Pump power is increased above a lasing threshold for $l_0=1$ mode at 661 nm with $\eta =0.120$. $l_0=1$ modes are clearly seen. (e) Observed output coupling efficiency (black squares) compared to $2{ϵ}_{+}$ (red squares). Slight reduction in $2{ϵ}_{+}$ itself is mostly due to the decrease of decay rates of the involved modes. The black dotted line is a spline fit for visual guidance.

Figure 4

(a) The observed spectrum for $\eta =0.187$. (b) The relative frequencies of observed quasieigenmodes with respect to the reference frequency of the same $n$ as a function of $n$. (c) Eigenfrequency surfaces of two coupled states $l=2$ (orange dots) and $l=4$ modes (blue dots) in $n\mathrm{\text{−}}\eta$ parameter space show a complex-square-root-like topology with a branch-point singularity at the EP, $(n_0,{\eta }_0)=(175,0.125)$.