Fig. 1. Calculated patterns of HLG modes ${\psi}_{{n}_{1},{n}_{2}}^{(\alpha ,\beta )}(\tilde{x},\tilde{y})$ with ${n}_{1}=3$ and ${n}_{2}=8$ for various values of $\alpha $ and $\beta $ on the Poincaré sphere.

Fig. 2. Calculated patterns for the standing waves given by $\mathrm{Re}[{\psi}_{{n}_{1},{n}_{2}}^{(\alpha ,\beta )}(\tilde{x},\tilde{y})]$ (golden color) as well as $\mathrm{Im}[{\psi}_{{n}_{1},{n}_{2}}^{(\alpha ,\beta )}(\tilde{x},\tilde{y})]$ (green color), corresponding to the results in Fig. 1.

Fig. 3. Calculated results for ${\psi}_{{N}_{1},{N}_{2}}^{(\mathrm{HG})}(\tilde{x},\tilde{y})$ and ${\mathrm{\Phi}}_{{N}_{1},{N}_{2}}(\tilde{x},\tilde{y},{\varphi}_{n})$ by using Eqs. (30) and (34) with $({N}_{1},{N}_{2})=(3,4)$.

Fig. 4. Calculated results for ${\psi}_{{N}_{1},{N}_{2}}^{(\mathrm{HG})}(\tilde{x},\tilde{y})$ and ${\mathrm{\Phi}}_{{N}_{1},{N}_{2}}(\tilde{x},\tilde{y},{\varphi}_{n})$ by using Eqs. (30) and (34) with $({N}_{1},{N}_{2})=(7,8)$.

Fig. 5. Calculated results for ${\psi}_{{N}_{1},{N}_{2}}^{(\mathrm{HG})}(\tilde{x},\tilde{y})$ and ${\mathrm{\Phi}}_{{N}_{1},{N}_{2}}(\tilde{x},\tilde{y},{\varphi}_{n})$ by using Eqs. (30) and (34) with $({N}_{1},{N}_{2})=(3,12)$.

Fig. 6. Calculated results for ${\psi}_{{N}_{1},{N}_{2}}^{(\mathrm{LG})}(\tilde{x},\tilde{y})$ and ${\mathrm{\Phi}}_{{N}_{1},{N}_{2}}^{(\alpha ,\beta )}(\tilde{x},\tilde{y},{\varphi}_{n})$ by using Eqs. (43) and (45) with $(\alpha ,\beta )=(\pi /2,\pi /2)$ and $({N}_{1},{N}_{2})=(3,12)$.

Fig. 7. Calculated results for the standing waves of $\mathrm{Re}[{\psi}_{{N}_{1},{N}_{2}}^{(\mathrm{LG})}(\tilde{x},\tilde{y})]$ and $\mathrm{Re}[{\mathrm{\Phi}}_{{N}_{1},{N}_{2}}^{(\alpha ,\beta )}(\tilde{x},\tilde{y},{\varphi}_{n})]$ corresponding to the traveling wave shown in Fig. 6.

Fig. 8. Calculated results for ${\psi}_{{N}_{1},{N}_{2}}^{(\alpha ,\beta )}(\tilde{x},\tilde{y})$ and ${\mathrm{\Phi}}_{{N}_{1},{N}_{2}}^{(\alpha ,\beta )}(\tilde{x},\tilde{y},{\varphi}_{n})$ by using Eqs. (43) and (45) with $(\alpha ,\beta )=(2\pi /5,2\pi /5)$ and $({N}_{1},{N}_{2})=(4,11)$.

Fig. 9. (a) Configuration of the single lens mode converter. Two vertical lines show the positions of the beam waists produced by the spherical matching lens and by the active axis of the cylindrical lens with focal length $f$. (b) Relationship between the $xy$-Cartesian coordinate system and the ${x}^{\prime}{y}^{\prime}$-Cartesian coordinate system. The ${x}^{\prime}$ and ${y}^{\prime}$ axes are the active and inactive components of the cylindrical lens.

Fig. 10. Experimental results (first column), numerical wave patterns (second column), and phase structures (third column) for the propagation evolution of the converted beam ${\mathrm{\Psi}}_{9,0}(x,y,z;\zeta )$ with $\zeta =\pi /4$. The number in the right side denotes the size of the pattern with the unit ${\omega}_{o}/\sqrt{2}$.

Fig. 11. Experimental results (first column), numerical wave patterns (second column), and phase structures (third column) for the propagation evolution of the converted beam ${\mathrm{\Psi}}_{9,4}(x,y,z;\zeta )$ with $\zeta =\pi /4$. The number in the right side denotes the size of the pattern with the unit ${\omega}_{o}/\sqrt{2}$.

Fig. 12. Experimental results (first column), numerical wave patterns (second column), and phase structures (third coulmn) for the propagation evolution of the converted beam ${\mathrm{\Psi}}_{8,8}(x,y,z;\zeta )$ with $\zeta =\pi /4$. The number in the right side denotes the size of the pattern with the unit ${\omega}_{o}/\sqrt{2}$.

Fig. 13. Experimental results (first column), numerical wave patterns (second column), and phase structures (third column) for the propagation evolution of the converted beam ${\mathrm{\Psi}}_{9,4}(x,y,z;\zeta )$ with $\zeta =-5\pi /36$. The number in the right side denotes the size of the pattern with the unit ${\omega}_{o}/\sqrt{2}$.