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.

Chapter Contents:

• 1 Real-argument Laguerre-Gauss beam
• 1.1 Paraxial beam
• 1.2 Complex power
• 2 Real-argument Laguerre-Gauss wave
• 3 Real and reactive powers
• References

Inspec keywords: electromagnetic waves; Gaussian distribution

Other keywords: real-argument Laguerre-Gauss wave; complex space source; complex-argument Laguerre-Gauss beams; full-wave generalization; higher-order point sources; paraxial beam; higher-order wave

Subjects: Probability theory, stochastic processes, and statistics; Electromagnetic waves: theory; Other topics in statistics; Electromagnetic wave propagation