Complex beams -such as Laguerre-Gaussian, Bessel, and radially-polarized- can present a strong field component in the direction of propagation (here z). In the case of Bessel beams the vector potential is [1]
A(r)=A0[eσJℓ(qrr)eiℓφ−iσezqrqzJℓ+σ(qrr)ei(ℓ+σ)φ],
[in the Coulomb gauge E=−∂tA] where σ=±1 indicates circular polarization, and eσ=ex+iσey. For Laguerre-Gaussian is [2]
E(r)=E0[ik(αex+βey)u−(α∂xu−β∂yu)ez],
where u∝Llp(r)exp[−(r/w0)2]exp(iℓφ) and the constants {α,β} give the polarization of the field; in particular the circular polarization is σ=i(αβ∗−α∗β). The radially-polarized beam can be easily deduced from the above fields, since it is a superposition of TL with opposite orbital and spin angular momenta [3].
It is interesting that Bessel and LG modes present a strong longitudinal field component for anti-parallel fields - having opposite spin and orbital angular momenta- close to the phase singularity: qrr≪1. For Bessel, the field is approximated using that Jm(x)=(x/2)m/m!, which shows that Az∝(qrr)ℓ+σ and is thus stronger than A⊥ only when Sgn(σ)≠Sgn(ℓ) (antiparallel field). The same happens to the LG beam, that reduces close to the singularity to (ℓ>0)
Ex(r)∝irℓeiℓφEy(r)∝−σrℓeiℓφEz(r)∝(σ−1)rℓ+σei(ℓ+σ)φ,
where for parallel beams (σ=1) Ez=0. Finally, when a radially-polarized beams is focused it evolves a strong component in the direction of propagation, as Fig. 1 suggests.
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Figure 1: By focusing a radially-polarized beam the electric
field is tilted (to "follow the rays"), and develops a strong component in z.
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References
[1] G. F. Quinteiro and P. I. Tamborenea, Theory of the optical absorption of light carrying orbital angular momentum by semiconductors, EPL 85, 47001 (2009).
[2] Loudon, Rodney. Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics. Physical Review A 68, no. 1 (2003): 013806.
[3] Quinteiro, G.F., Reiter, D.E. and Kuhn, T., 2017. Formulation of the twisted-light–matter interaction at the phase singularity: Beams with strong magnetic fields. Physical Review A, 95(1), p.012106.
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