We show that a product of two Laguerre–Gaussian (pLG) beams can be expressed as a finite superposition of conventional LG beams with particular coefficients. Based on such an approach, an explicit relationship is derived for the complex amplitude of pLG beams in the Fresnel diffraction zone. Two identical LG beams of the duet produce a particular case of a “squared” Fourier-invariant LG beam, termed as an (LG)2 beam. For a particular case of pLG beams described by Laguerre polynomials with azimuthal numbers n − m and n + m, an explicit expression for the complex amplitude in a Fourier plane is derived. Similar to conventional LG beams, the pLG beams can be utilized for information transmission, as they are characterized by orthogonal azimuthal numbers and carry an orbital angular momentum equal to their topological charge.

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In this work, using the FullWAVE software package, numerical simulation of focusing of TM-wave with a wavelength of 633 nm was performed. It is shown that a dielectric microcylinder made of polyester with a radius of 2.1749 of wavelength coated on the shadow side with a silver film of 90 nm with a 100 nm hole, allows to get a focus with a maximum intensity of 1.37 times more (10.61 a. u), than a conventional dielectric cylinder (7.73 a. u).