On gauge invariance and results for spin and orbital angular momenta and helicity densities in optical fields

•Spin (sn) and orbital (ln) angular momenta and helicity (hn) densities are defined.•Two types of vector potentials A and C are newly introduced here.•Gauge invariances of sn, ln and hn are first achieved without any gauge fixing.•Gauge invariance of ln results in all quantization of sn, ln and hn.•The gauge invariance needs two types of gauge functions and scalar potentials. The spin (sn) and orbital (ln) angular momenta and helicity (hn) densities of the electromagnetic fields are defined by introducing two types of electric (C) and magnetic (A) vector potentials defined as D = ∇ × C and B = ∇ × A for the electric (D) and magnetic (B) induction fields. The gauge invariances of sn, ln and hn for the optical field are achieved without gauge fixing by canceling out the contributions from four nablas ∇ acting on two gauge functions (GFs) (u and υ) and two scalar potentials (SPs) (Φ and ψ) which are generated by introducing A and C. As a result, it is found that sn, ln and hn can be quantized without gauge fixing, where θ and α are phase differences between A and C and between ∇u and ∇Φ or ∇υ and ∇ψ, respectively. In the fundamental aspect of gauge invariance, it was elucidated that the gauge invariance of ln works to make the phase differences θ and α to ± π/2 and 2nπ/ℓ, respectively. As its applied aspect, it is presumed to be useful in studying the behavior of the entanglement of electromagnetic waves. A new interrelationship between four nablas acting on two GFs and two SPs is established near the beam waist in paraxial approximation.