Table 1 Floquet group theory and harmonic generation selection rules in (2+1)D.
From: Selection rules in symmetry-broken systems by symmetries in synthetic dimensions
Floquet group symmetry \(\hat{X}\) | Harmonic generation selection rule |
|---|---|
\(\hat{T}\) | Linearly polarized only harmonics. They may be polarized along any axis. |
\(\hat{Q}=\hat{T}\cdot {\hat{R}}_{2}\) | Linearly polarized only harmonics. They may be polarized along any axis. |
\(\hat{G}=\hat{{{{{{\rm{T}}}}}}}\cdot {\hat{{{{{{\rm{\tau }}}}}}}}_{2}\cdot {\hat{{{{{{\rm{R}}}}}}}}_{2}\) | Linearly polarized only harmonics. They may be polarized along any axis. |
\({\hat{Z}}_{y}={\hat{{{{{{\rm{\tau }}}}}}}}_{2}\cdot {\hat{{{{{{\rm{\sigma }}}}}}}}_{{{{{{\rm{y}}}}}}}\) | Linearly polarized only harmonics, even harmonics are polarized along the reflection axis, and odd harmonics are polarized orthogonal to the reflection axis. |
\({\hat{D}}_{y}=\hat{T}\cdot {\hat{{{{{{\rm{\sigma }}}}}}}}_{y}\) | Elliptically polarized harmonics with major/minor axis corresponding to the reflection axis. |
\({\hat{H}}_{y}=\hat{T}\cdot {\hat{{{{{{\rm{\sigma }}}}}}}}_{y}\) | Elliptically polarized harmonics with major/minor axis corresponding to the reflection axis. |
\({\hat{C}}_{N}={\hat{\tau }}_{N}\cdot {\hat{{{{{{\rm{R}}}}}}}}_{N}\) | (±) circularly polarized \({Nq}\pm 1\) harmonics, \(q{\mathbb{\in }}{\Bbb{N}}\), all other orders forbidden |
\({\hat{C}}_{N,M}={\hat{\tau }}_{N}\cdot {\hat{{{{{{\rm{R}}}}}}}}_{N,M}={\hat{\tau }}_{N}\cdot {({\hat{R}}_{N})}^{M}\) | (±) circularly polarized \({Nq}\pm M\) harmonics, \(q{\mathbb{\in }}{\mathbb{N}}\), all other orders forbidden |
\({\hat{e}}_{N,M}={\hat{\tau }}_{N}\cdot {\hat{{{{{{\rm{L}}}}}}}}_{b}\cdot {\hat{{{{{{\rm{R}}}}}}}}_{N,M}\cdot {\hat{{{{{{\rm{L}}}}}}}}_{1/b}\) | (±) elliptically polarized \({Nq}\pm M\) harmonics, \(q{\mathbb{\in }}{\mathbb{N}}\), with an ellipticity b, all other orders forbidden. |
