Fig. 2: Evolution of the dispersive polaritons in phonon mode A in monoclinic crystal Y₂SiO₅.

a the crystal structure of monoclinic crystal Y₂SiO₅. Vector b is perpendicular to the a–c plane, with the monoclinic angle β between a and c equal to 102.45°. b Cartesian coordinate xyz is defined to describe the optical properties in this work. Optical phonon modes A and B are the two non-perpendicular modes under our investigation. OA and OB indicate the direction of phonon modes A and B, respectively. OB_⊥ indicates the vertical direction of phonon mode B. c–f the measured near-field signal |S₂| with different incident frequencies using scattering-type scanning near-field optical microscopy (s-SNOM). The phonon polaritons (PhPs) are launched by an Au nanodisk antenna with a diameter of ~ 1 μm and thickness of 100 nm. The incident laser beam is aligned with the x-axis with p-polarization. g–j corresponding Fourier spectrum of the panel (c–f) obtained by calculating their Fourier transformation, justifying the hyperbolic dispersion of the PhPs. k–n the Fourier spectrum of dipole-launched polaritons through numerical simulations. The z-oriented dipole is placed 100 nm above the crystal surface. The green arrows indicate the transverse axis (TA) at different frequencies, which is rotating to approach OB_⊥ with increasing frequency. White curves indicate the iso-intensity contours (the points on the curve have the same intensity of the mode distribution) and the unequal lengths (the length between the original point and the furthest point of the iso-intensity contour of each petal) of the neighboring branches present the unbalanced shear response in the LSC.