Fig. 1: Dually modulated photonic crystal for laser beam scanning.

a, b Schematic diagram of dual modulation. a Dually modulated photonic crystal whose lattice-point position vectors \({\bar{\mathbf{r}}}_{m,n}^{}\) and sizes \(\bar S_{m,n}^{}\) have been simulateously modulated from their original values \({\mathbf{r}}_{m,n}^0\) and \(S_0^{}\) by amounts \(\Delta {\mathbf{d}} \cdot {\mathrm{sin}}({\mathbf{k}} \cdot {\mathbf{r}}_{m,n}^0)\) and \(\Delta S \cdot {\mathrm{sin}}({\mathbf{k}} \cdot {\mathbf{r}}_{m,n}^0)\), respectively, where integers \(m\) and \(n\) specify the lattice point, \(\Delta {\mathbf{d}}\) is the position modulation vector, \(\Delta S\) is the size modulation amplitude, and \({\mathbf{k}}\) is a diffraction vector that diffracts the four fundamental waves (\({\mathbf{R}}_1,{\mathbf{R}}_2,{\mathbf{R}}_3\), and \({\mathbf{R}}_4\) in d) inside the air light cone. a is the photonic-crystal lattice constant and \({\uplambda}\) is the wavelength in free space. b The original photonic crystal before modulation. c Band diagram of a square-lattice photonic crystal, and a magnified view of the four band-edges A, B, C, and D at the M₁-point, which lies below the air light line and is used for lasing. d Schematic diagram of light diffraction in reciprocal lattice space at the M₁-point. Diffraction of the four fundamental waves (\({\mathbf{R}}_1,{\mathbf{R}}_2,{\mathbf{R}}_3\), and \({\mathbf{R}}_4\)) by \({\mathbf{k}}\) leads to the generation of vector \({\mathbf{K}}\), where \({\mathbf{K}} = \left( {\frac{{2{\uppi}}}{\lambda }} \right)\,({\mathrm{sin}}\theta \,{\mathrm{cos}}\phi,\,{\mathrm{sin}}\theta \,{\mathrm{sin}}\phi)\) with the emission direction in free space given by polar angle \({\theta}\) and azimuthal angle \({\phi}\) and the wavelength in free space given by λ. Note that both positive and negative vectors (\({\mathbf{K}}\) and –\({\mathbf{K}}\)) are generated because the modulation induces not only positive but also negative \({\mathbf{k}}\) vectors. Thus the emission occurs simultaneously in two directions, (\(\theta , \phi\)) and (\(\theta , \phi + 180^\circ\)). \({\mathbf{G}}_{1,0}\) and \({\mathbf{G}}_{1,1}\) are reciprocal lattice vectors.