Excess quantum noise due to mode nonorthogonality in dielectric microresonators

This work presents a theory of the frequency-resolved light emission of active two-dimensional dielectric microresonators, which are characterized by a highly nonparaxial mode structure and frequently feature a position-dependent dielectric constant and nonuniform gain. The Lorentzian intensity profile is characterized by an appropriately generalized Petermann factor, a renormalized peak position, and the cold-cavity resonance lifetime $\Gamma$. The theory also delivers a relation of $\Gamma$ to the laser threshold that improves earlier phenomenological expressions even for the case of a homogeneous medium.

Figure 1

(Color) Inset: the white circles indicate the dielectric interfaces of an annular resonator with radii $R_2=0.6R_1$ and eccentricity $d=0.22R_1$. The refractive index in the inner disk is $n_2=3.3$, while in the annular region $\text{Re}{n}_1$ is fixed to 1.8. The color-coded density plot shows a TM-polarized resonator mode at its threshold, where $n_{1,0}=1.8-i0.029$ and ${\omega }_0{R}_1=6.45$. Main panel: frequency-resolved intensity close to the threshold (suitably scaled to be independent of the small detuning of $\text{Im}{n}_1$). The exact numerical result (solid black curve) is a Lorentzian whose width and height can be accurately described using the theory developed in this paper (solid red curve on top of the black curve) [Eq. (3) with parameters determined from Eqs. (4) to (7) (TM polarization)]. A discrepancy in the height is observed when the Petermann factor is determined from the homogeneous expression $K_0$ [Eq. (2) (dashed blue curved)] or when it is completely neglected (dotted blue curve, corresponding to the Schawlow-Townes result).