High Circular Polarized Nanolaser with Chiral Gammadion Metal Cavity.

We demonstrate a circularly polarized laser with the metal-gallium-nitride gammadion nanocavities. The ultraviolet lasing signal was observed with the high circular dichroism at room temperature under pulsed optical pump conditions. Without external magnetism which breaks the time-reversal symmetry to favor optical transitions of a chosen handedness, the coherent outputs of these chiral nanolasers show a dissymmetry factor as high as 1.1. The small footprint of these lasers are advantageous for applications related to circularly polarized photons in future integrated systems, in contrast to the bulky setup of linearly-polarized lasers and quarter-wave plates.


I. Weights of Two Circularly-Polarized-Like Modes in Perturbed Counterparts under Reciprocal Perturbations of Four-Fold Symmetric Cavities
The wave equation governing the unperturbed modal electric field (0) ( ) which represents R ( ) and L ( ) of the RCP-and LCP-like modes, respectively, in a 4fold rotationally symmetric cavity can be expressed as where is the speed of light in vacuum; ̿ r (0) ( , r ) is the relative permittivity tensor corresponding to the cavity with gain and exhibits the 4-fold rotation symmetry in Eq.
(S1.1b); /2 is the matrix associated the rotation around the axis by /2; and r is the (real) resonant frequency of the two CP-like modes at the threshold. For fair comparisons, we demand that the integrals of square field magnitudes associated with the two CP-like modes in the 4-fold rotationally symmetric active region Ω a (0) (assumed to be composed of isotropic gain medium) to be identical, namely, .
(S1. 2) In the presence of a perturbation ∆ ̿ r ( , ) of relative permittivity tensor which is reciprocal [∆ ̿ r T ( , ) = ∆ ̿ r ( , )] but breaks the 4-fold rotation symmetry at a new resonance frequency , the overall permittivity ̿ r ( , ) which describes fields ( ) of perturbed modes [including the nondegenerate ones 1 ( ) and 2 ( ) originating from R ( )and L ( )] at new thresholds is ̿ r ( , ) = ̿ r (0) ( , ) + ∆ ̿ r ( , ) + ∆ r,a ( ) ̿ , where ∆ r,a is an imaginary number representing the required adjustment of relative permittivity in the active region which makes the mode self-oscillate (real ) with a constant magnitude in the perturbed environment; ( ) is an indicator function which is unity in the active region Ω a but zero elsewhere; and ̿ is the 3-by-3 identity tenor.
For simplicity, we assume that ∆ ̿ r ( , ) is only present in a finite region around the cavity. Also note that the indicator function ( ) of the active region Ω a need not be 4-fold symmetric in the presence of perturbations, in contrast to the counterpart (0) ( ) of the original active region Ω a (0) which has the 4-fold rotation symmetry. In analogy to Eq. (S1.1a), the wave equation for ( ) is written as (S1.4) We then rewrite Eq. (S1.4) in a perturbative manner. The frequency difference Δ ≡ − r and permittivity variation ∆ r,a should vary linearly with ∆ ̿ r ( , r ).
We may now expand the field ( ) as follows: (S1.6) where R and L are the zeroth-order expansion coefficients of R ( ) and L ( ), respectively; and Δ ( ) is a perturbed field of the second order in ∆ ̿ r ( , r ), that is, Substituting Eq. (S1.6) into Eq. (S1.5a), we obtain the relation: (S1.7) We then project Eq. (S1.7) into the subspace spanned by R ( ) and L ( ). For this purpose, an integration region Ω with the 4-fold rotation symmetry needs to be picked up to define the inner product. Here, we simply choose Ω as the minimal circular cylinder that completely covers the perturbation ∆ ̿ r ( , ). In this way, a 2-by-2 matrix equation for column vector = ( R , L ) T (superscript "T" means transpose) can be constructed: where Δ , (a) , and (d) are 2-by-2 matrix representations of Δ ̿ ( ), ̿ (d) ( ), and ̿ (a) ( ), respectively. Their matrix elements are defined as where indices and refer to R or L. Note that in Eq. (S1.8b), we do not use complex- which is valid for reciprocal media, the matrix is also symmetric, namely, T RL and T LR are identical. In views of these properties, the matrices Δ , (a) , and (d) can be simplified as follows: A LL (a) ), where 1 is the first Pauli matrix. With an expression of (d) in Eq. (S1.9) which is proportional to 1 , we can convert Eq. (S1.8a) into the form of eigenvalue problems by multiplying both sides with 1 : (S1.10) In principle, we need to adjust ∆ r,a to make Δ real. However, for a quick estimation, we may first drop ∆ r,a and treat Δ as a complex quantity, whose imaginary part now characterizes how fast the mode decays or grows. Furthermore, if the permittivity variation ∆ ̿ r ( , r ) perturbs the RCP-and LCP-like modes with similar magnitudes, namely, ( 1 − 2 ( LL − RR ) ). (S1.13c) We may define a perturbation-dependent phase shift Θ as Θ = LL − RR 2 . (S1.14) Substituting Eqs. (S1.13b), (S1.13c), and (S1.14) into Eq. (S1.6), we then obtain the perturbed modal profiles 1 ( ) and 2 ( ) in Eq.

II. Lasing from The Cavities with Different Periods
To ensure that the lasing action indeed originates from a single gammadion metalcavity rather than collectively from the gammadion metasurface, we characterized

III. Mode Analysis in Gammadion Metal Cavity
In addition to the degenerate CP-like modes, other cavity modes might be also present in the wavelength range around 364 nm. We need to justify that the target CP-  In Figure S2, we show the ratios Im window, but its threshold ( a = −0.103) is higher than that of CP-like modes, which is also unfavorable for lasing. Therefore, it is supported that the lasing signal in the experiment originated from the CP-like modes in gammadion metal cavities.

IV. Estimation of Dissymmetry Factor
We slightly lengthen an outer arm of the R-gammadion cavity by 3% to investigate the mixing of two CP-like modes. The horizontal field distributions of two perturbed modes (at the antinode) that are closely related to the two CP-like modes in an ideal gammadion metal cavity are depicted in Figure S3. The elongation of the arm may simply mix the two CP-like modes into a perturbed one, as indicated by the profile of -like mode in Figure S3(a). On the other hand, the effect of higher-order perturbations could also come into play so that modes with real characters ( = ±1) are mixed into perturbed ones. Such a phenomenon takes place in the case of -like mode whose horizontal field profile exhibits a hot spot in the outer arm at the bottom left, as shown in Figure S3

V. Cavity Pumping with Different Optical Polarizations
To further verify the polarization states of planar chiral nanolasers, right-handed gammadion metal cavities were optically pumped by the pulse laser with different polarization states including the linear, left-hand, and right-hand circular polarizations, as schematically illustrated in Fig. S4 (a). Figures S4(b) and (c)

VI. Estimation of Modal Volume
The FEM was applied to investigate the modal volume of CP-like modes in the gammadion metal cavities. The refractive indices of the undoped GaN and Al were obtained from the references by Palik as well as Peng and Piprek 1,2 . We setup threedimensional models for the R-and L-gammadion metal cavities to compute electric fields of the modes. The gammadion was designed to have a linewidth, width, arm length, and height of 50, 305, 200, and 500 nm, respectively. The modal volume for a mode in the GaN gammadion metal cavity, after some generalization for taking the material dispersion into account, is written as This expression is numerically evaluated for a given mode to estimate the dimensionless effective volume eff ≈ 2.56.