The purpose of this letter is to respond to the comment of J. Wiersig [1] directed to our original paper [2] describing the reciprocal transmissions and asymmetric modal distributions in waveguide-coupled spiral-shaped microdisk resonators. While we thank the author for taking an interest in and providing an extended discussion on this subject, we must clarify here that our methodology and conclusions in [2] remain valid and are not contradicting the numerical studies presented in [1].

Our original paper [2] presented comprehensive experimental and numerical studies on waveguide-coupled spiral-shaped microdisk resonators. Based on (i) transmission measurements (Fig. 2), (ii) out-of-plane scattering measurements (Figs. 3–4), and (iii) numerical simulations (Figs. 5–9), we concluded that waveguide-coupled spiral-shaped microdisk resonators exhibit identical resonance wavelengths and quality factors (Q’s) with asymmetric modal distributions in clockwise (CW) and counterclockwise (CCW) traveling waves. The traveling waves were selectively excited by input coupling light from either an evanescently coupled waveguide at the cavity sidewall or a directly coupled waveguide seamlessly jointed at the spiral notch. The modal distributions depend on the input coupling direction and coupling mechanism. It is worth mentioning that our measurements imposed a spectral resolution of 0.02 nm, while our spectra simulations imposed a spectral resolution of 0.09 nm.

In the comment [1], the author found two non-degenerate modes displaying similar modal distributions by numerically solving eigenmodes in a spiral-shaped microdisk resonator without coupled waveguides. This is a useful result as it explicitly shows the possible existence of non-degenerate modes in spiral-shaped microresonators. Should non-degenerate modes with similar degree of mode splitting exist in our waveguide-coupled microresonators, our spectral resolutions in both experiments and numerical simulations would have forbidden us from resolving the split modes.

However, the comment [1] and our original work [2] are not necessarily contradicting. While possible non-degenerate modes can display similar modal distributions, our observations focus on the CW and CCW traveling-wave resonances which we selectively excited via waveguide coupling. Our observed field patterns are with identical resonance wavelength (as a result of reciprocity), are traveling in opposite directions (selectively launched from the different ports of the coupled waveguides), and are displaying distinguishable features (Figs. 3–4 for experiments and Figs. 6, 7, 9 for simulations). In passing, we mention that our simulations in [2] adopted radius r₀=10 µm and deformation parameter ε=0.04 as detailed in our original paper, not r₀=9.6 µm as suggested in [1].

In conclusion, we believe that the reciprocal transmissions and the corresponding asymmetric field patterns in waveguide-coupled spiral-shaped microdisk resonators reported in our original work [2] are reasonable. In particular, the spectra measurements should provide relevant experimental inputs to further understanding resonances and unidirectional lasing in spiral-shaped microdisk resonators [3]. Although we are fully aware of the fact that the side-coupled and notch-coupled waveguides can perturb the boundary condition from that of an isolated spiral-shaped microresonator, the waveguide coupling is a necessary device feature to turn the microresonator into practical on-chip devices for signal processing functionalities, as exemplified in some most recent work [4–6].