Scalar optical beams with helical symmetry

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Abstract
Using a nonorthogonal helical coordinate system, we obtain exact free-space solutions for both the Helmholtz and paraxial Helmholtz equations. At optical frequencies the helical Helmholtz solutions can be interpreted as helical beams characterized by a constant pitch angle and beam radius. These solutions are shown to be generalizations of the family of scalar nondiffracting beams known as Bessel beams. They are similar to Bessel beams in some ways, such as an invariant intensity distribution profile in any plane normal to their axis of propagation, but have a nonconstant order. The paraxial helical Helmholtz equation is cast into two different forms, one assuming propagation along the helical axis and one assuming propagation along the original cylindrical axis. The first is solved using a paraxial form of the above helical beam family while a solution of the second form is a helical Gaussian.

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