Influence of Finite Apertures on Orthogonality and Completeness of Laguerre-Gaussian Beams

Laguerre-Gaussian (LG) beams with orbital angular momenta (OAM) are widely used in optical communication systems to increase capacities because of their orthogonality and completeness. However, the apertures of practical OAM-based multiplexing systems are always finite and inevitably affect the orthogonality and completeness. In this paper, the influences of finite apertures on the orthogonality of LG eigen beams and on the completeness of the set of LG bases are investigated. The closed-form inner product of arbitrary LG beams in finite apertures is derived. The results show that in finite apertures, the orthogonality of LG eigen beams with the same topological charge is partially lost, whereas the orthogonality of those with different integer topological charges is preserved and independent of the aperture sizes and the radial indices. The subset of LG eigen beams satisfying the root mean square radius condition is proven to be incomplete by evaluating the approximation accuracy of the truncated LG expansions of arbitrary real-order LG beams. The peaks of the root mean square error of the truncated LG expansions are greater than 0.225 under the root mean square radius condition when approximating fractional LG beams. The results can be utilized in analyzing and designing optical communication systems using LG beams with both radial and azimuthal degrees of freedom under the finite aperture limitation.