where 𝑘 m ‖ is the cutoff value of the in-plane wavevector, and Θ(𝜔 k -𝜔 c ) is the Heaviside step function. The details of the derivations are provided in the Supplementary Material, Sec. VI. The DOS, 𝑔 1D (𝜔 k ), in Eq. 22 has a singularity at 𝜔 k = 𝜔 c , which is known as (the first type of) the van-Hove-type singularity . Such a concentrated peak in the 𝑔 1D (𝜔) at 𝜔 = 𝜔 c has been numerically observed in Fig. of Ref. 46. We will turn to the case of including the effects of cavity loss in the in-plane direction at the end of this section.