Angular momentum of a strongly focused Gaussian beam

Focusing circularly-polarized light by a high-numerical-aperture lens generates a spin-dependent optical vortex in the focal field.

“(ii) Focusing of circularly-polarized light by a high-numerical-aperture lens results in the generation of a spin-dependent optical vortex (i.e., helical phase producing orbital angular momentum) in the focal field.”

For nonparaxial beams, the simple association of SAM and IOAM with polarization and vortex is not valid; the SAM and IOAM become (3), and the total intrinsic AM is preserved, so part of the SAM is transferred to the IOAM (spin-to-orbital AM conversion).

“For nonparaxial beams, consisting of circularly-polarized plane waves with the wavevectors forming a cone with an opening angle θ (like in Fig. B2b), the SAM and IOAM become 8,27,96- 98”

The SAM-IOAM coupling in nonparaxial optical fields is largely independent of how the field was generated; dipole Rayleigh scattering by a small particle generates a similar conical distribution and spin-to-orbital AM conversion.

“The SAM- IOAM coupling in nonparaxial optical fields is largely independent of how the field was generated. Instead of high-NA focusing, one can consider dipole Rayleigh scattering by a small particle, which generates a similar conical distribution of the outgoing wavevectors, Fig. 2d.”

The mean radius of a focused vortex beam is determined by its σ-dependent IOAM value: R ~ L_z / k sin θ; thus a beam with parallel SAM and IOAM becomes more strongly focused than a similar beam with anti-parallel SAM and IOAM.

“Namely, the mean radius of a focused vortex beam is determined by its σ -dependent IOAM value: R ~ Lz / k sin θ 8,27 . Due to this effect, a beam with parallel SAM and IOAM becomes more strongly focused than a similar beam with anti-parallel SAM and IOAM.”

The spin-to-orbital AM conversion is closely related to the geometric phase between azimuthally-distributed wavevectors in the beam spectrum.

“Importantly, this effect is closely related to the geometric phase between the azimuthally-distributed wavevectors k in the beam spectrum (Fig. B2b) 8,27 .”

Due to the off-diagonal elements, incoming circularly-polarized light with helicity σ acquires an oppositely-polarized component with helicity -σ and vortex factor, and also a longitudinal z-component with vortex; for small apertures, the longitudinal component plays the leading role in AM conversion.

“Due to these elements, the incoming circularly-polarized light with helicity σ acquires an oppositely-polarized component with helicity -σ and vortex factor b exp 2 i σφ ( ) , and also a longitudinal z -component with the vortex 2 ab exp i ( σφ ) . For small apertures, the longitudinal component plays the leading role in the AM conversion.”

The polarization evolution in focusing represents parallel transport on the k-sphere of directions and is described by a unitary matrix (4), whose off-diagonal elements contain azimuthal vortex factors responsible for AM conversion.

“Notably, this polarization evolution (described by the DebyeWolf approach 100 ) represents parallel transport on the k -sphere of directions (Fig. B2b) from the North pole θ = 0 (incoming light E ) to θ ≠ 0 (focused field ′ E ). In the global basis of the circular ( x + i σ y )-polarizations and longitudinal z -component, this 3D rotational transformation of the electric field is described by the”

Part of the orbital AM becomes helicity-dependent, so a helicity-dependent vortex should appear even in beams with l=0.

“Part of the orbital AM becomes helicity -dependent, i.e., a helicity-dependent vortex should appear even in beams with  = 0 (Fig. 2b) 22- 24,27 .”