Spiral-shaped microcavities have attracted considerable attention in recent years since they allow for unidirectional laser emission [1]. In polar coordinates (r,ϕ) the spiral is defined as $r\left(\varphi \right)=r_0\left(1-\frac{\epsilon }{2\pi }\varphi \right)$ with deformation parameter ε and “radius” r ₀ at ϕ=0. The radius jumps back to r ₀ at ϕ=2π creating a notch. It was generally believed that the chirality induced by the spiral shape breaks the degeneracy of clockwise (CW) and counterclockwise (CCW) traveling-waves, which is consistent with experiments on spiral-shaped microlasers. It came therefore as a surprise that this chirality affects only the spatial modal distributions but not the resonant wavelengths and quality (Q) factors of CW and CCW traveling-wave modes [2]. This conclusion has been drawn from reciprocity relations, finite-difference time-domain calculations and experiments based on the implicit assumption that the modes can be non-ambiguously classified as CW and CCW traveling-waves. In this comment, we demonstrate that this assumption is not valid.

As in Ref. [2] we consider transverse electric (TE) polarization, an effective index of refraction n=2, and shape parameters ε=0.04 and r ₀=10µm. We use a highly accurate numerical scheme, the boundary element method [3], to compute the modes H(r,ϕ)exp(-iωt) with outgoing-wave condition at r→∞ and complex-valued frequency ω. We find, for example, that the alleged pair of degenerate modes A ₁ is actually a pair of nearly degenerate modes. One of these modes has free-space wavelength λ ₁=1513.72nm and quality factor Q ₁=2434, the other one has λ ₂=1513.85nm and Q ₂=2577. Note that ε=-0.04 and r ₀=9.6µm (a minor mistake in Ref. [2]) would give λ ₁=1512.37nm and λ ₂=1512.49nm. The wavelength splitting seems to be small but it is of the order of the linewidths and therefore important for the wave dynamics, see below. More striking are Figs. 1(a) and (b) which show very similar spatial modal distributions in strong contrast to the observation in Ref. [2].

 

Fig. 1. (Color online) Calculated magnetic field intensity of mode 1 (a) and 2 (b). Distribution of angular momentum α ⁽¹⁾ _m (solid line) and α ⁽²⁾ _m (dashed) normalized to 1 at maximum: (c) absolute value squared, (d) real and (e) imaginary part. (f) Superpositions α ⁺ _m=(α ⁽¹⁾ _m+α ⁽²⁾ _m)/2 (solid) and α ^- _m=(α ⁽¹⁾ _m-α ⁽²⁾ _m)/2 (dashed, scaled by a factor of 5).

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A deeper understanding can be gained by expanding the z-component of the internal magnetic field in cylindrical harmonics H_z(r,ϕ)=∑^∞ _m=-∞ α_mJ_m(nkr)exp(imϕ) with wave number k and the mth order Bessel function J_m of the first kind [1]. Positive (negative) values of the angular momentum index m correspond to CCW (CW) traveling-wave components. Figure 1(c) shows that also the angular momentum distributions |α_m|² are almost indistinguishable. For both modes the CCW component dominates, i.e. none of the two modes can be classified as CW traveling-wave mode as opposed to the assumption in Ref. [2]. The difference between the modes is revealed by Figs. 1(d) and (e). If m<0 then both the real and the imaginary part of α_m have a different sign for the two modes. Consequently, we can construct superpositions with α ^± _m=(α ⁽¹⁾ _m±α ⁽²⁾ _m)/2 being CW and CCW traveling-waves, respectively; see Fig. 1(f). However, these superpositions are not eigenmodes of the cavity as they are composed of two modes with different frequencies and Q-factors. In Ref. [2] such kinds of superpositions are plotted due to the chosen excitation by incoming waves. The small but finite splitting λ ₂-λ ₁ leads to a beating phenomenon in the time domain which can be interpreted as scattering between CW and CCW components. The difference in intensity in Fig. 1(f) reflects the fact that the notch scatters CW traveling-waves more efficiently into CCW traveling-waves than the other way round. Note that the difference in intensity is not in conflict with the reciprocity relations as the wave equation is linear. In the case of the nonlinear laser dynamics, however, the difference in intensity ensures that CCW traveling-waves dominate, consistent with experiments [1].

In summary, we have shown that in contrast to the claims of Ref. [2] pairs of optical modes in spiral-shaped cavities have in general different wavelengths and Q-factors but similar modal distributions. Financial support from the DFG research group 760 is acknowledged.