Azimuthal mode decomposition of millimeter-wave integer and non-integer optical vortices generated by a spiral mirror

We investigated the characteristics of both integer and non-integer optical vortices (OVs) in the millimeter-wave regime generated by a spiral mirror. An OV, characterized by the phase term eiℓϕ, also known as a Laguerre–Gaussian (LG) beam, is a solution to the wave equation in cylindrical coordinates under the paraxial approximation in free space. This typically restricts the azimuthal mode number ℓ to integer values. However, the spiral mirror enables the generation of OVs with expected azimuthal mode numbers ℓ̄, which can take either integer or non-integer values in the reflected wave. We employed a field reconstruction method to analyze both integer and non-integer OVs with ℓ̄ generated by the spiral mirror, decomposing them into their constituent integer ℓ components. Our results showed that when ℓ̄ is specified as an integer, the generated OV primarily contains that single ℓ. In contrast, specifying a non-integer ℓ̄ results in an OV composed of a superposition of multiple integer ℓ values. In other words, non-integer OVs are superpositions of LG beams with different integer azimuthal mode numbers. These OVs are generated by the spiral mirror with a focus function, allowing for the specification of arbitrary waist sizes and focal positions, as designed for this research. Furthermore, we revealed that this focus spiral mirror can also avoid the separation of singularities that were present in OVs generated by conventional flat spiral mirrors.