The Thin-layer Approach And Its Applications In Optics

The thin layer approach is an effective framework to study the dynamics of low-dimensional curved systems.During the past few decades,the thin-layer approach has been generalized to study various types of low-dimensional curved systems,including the charged particle in electromagnetic field,the particle described by Dirac’s equation,the particle described by the Pauli’s equation and even electromagnetic waves.Howev-er,despite the huge achievements of the thin-layer approach,there are some unsolved problem about the theory itself and the applications in optics.In chapter 2,we systematically study the fundamental framework of the thin-layer approach and show the most important results of it:the geometrical potential and the geometrical gauge potential.The geometrical potential is associated with the curvature of the low-dimensional system.The geometrical gauge potential is proportional to the intrinsic orbital angular momentum and the torsion of the low-dimensional system,where the intrinsic orbital angular momentum plays the role of effective charge.In chapter 3,we investigate the influences of different thin-layer procedure on the dynamics on a curve.We find that(a)the usual curvature-induced geometrical potential is irrelevant to the detail of the thin-layer procedure,(b)the torsion-induced geometrical potential and geometrical momentum only appears in the case of square confinement,and the geometrical gauge potential only appears in the case of circular confinement.Furthermore,by considering the spin connection and using the thin-layer approach,we deduce the effective equation for a spin-1/2 particle confined to a curved surface with the nonrelativistic limit.We obtain a pseudo-magnetic field and an effective spin-orbit interaction,where the pseudo-magnetic field is proportional to the Gaussian curvature and the effective spin-orbit interaction is determined by the Weingarter curvature tensor.We find that the pseudo-magnetic field and the effective spin-orbit interaction lead to the spin Hall effect on the curved surface.In chapter 4,by using the thin-layer approach,we derive the effective equation for the electromagnetic wave propagating along a space curve,where the electromag-netic wave is described by the transverse electric field.We find the intrinsic-orbital-angular-momentum and longitudinal-spin-angular-momentum-dependent geometrical gauge potential,where the intrinsic orbital angular momentum and longitudinal spin angular momentum of light play the role of charge.The geometrical gauge poten-tial can lead to the geometrical phases and Hall effects.Moreover,we show that the off-diagonal term in the effective equation can lead to the nonadiabatic polarization changes.In chapter 5,we consider the geometrical phase phenomena associated with the transverse spin of light.By considering the electric field with nonvanishing longitu-dinal component,we obtain an SO(3)gauge potential in the effective equation.Un-der different adiabatic approximations,the SO(3)gauge potential reduces to different SO(2)/U(1)gauge potential:the torsion-induced and curvature-induced gauge poten-tial,where the torsion-induced potential is associated with the longitudinal spin of light and the curvature-induced potential is associated with the transverse spin of light.Furthermore,we calculate the transverse-spin-dependent geometrical phase and Hall effect induced by the curvature-induced gauge potential.We believe that the present results would help the researchers to design the low-dimensional curved system to con-trol the spin-orbit interactions of light.