Basic full real-argument Laguerre-Gauss wave

Abstract

The paraxial wave equation in the cylindrical coordinate system, in the same manner as the complex-argument Laguerre-Gauss beams, has another series of higher-order solutions known as the real-argument Laguerre-Gauss beams [1]. These higher-order Gaussian beams, which form a complete set of eigenfunctions, are characterized by the radial mode number n and the azimuthal mode number m. The fundamental Gaussian beam is the lowest-order (n = 0, m = 0) mode in this set. For n = 0, the real-argument Laguerre-Gauss beams become identical to the complex-argument Laguerre-Gauss beams. Therefore, for the realargument Laguerre-Gauss beams as well, the higher-order hollow Gaussian beams treated in Chapter 8 correspond to n = 0. In this chapter, the various aspects of the real-argument Laguerre-Gauss beams are discussed. As for the complex-argument Laguerre-Gauss beams, as well as for the real-argument Laguerre-Gauss beams, the reactive power is zero. The source in the complex space required for the full-wave generalization of the real-argument Laguerre-Gauss beams is derived. The basic full real-argument Laguerre-Gauss wave generated by the complex space source is determined. The real and the reactive powers of the basic full higher-order wave are obtained. The source in the complex space is shown to be a series of higher-order point sources that are similar to those obtained for the complex-argument Laguerre-Gauss beams. Consequently, as is to be expected, the reactive power is infinite, as for the basic full complex-argument Laguerre-Gauss wave. The general characteristics of the real power are investigated. The real power increases, approaching the limiting value of the paraxial beam as kw0 is increased. For sufficiently large and fixed kw0, in general, the real power decreases as the mode order increases.