The paraxial wave equation in the Cartesian coordinate system has a series of higher-order solutions known as the complex-argument Hermite-Gauss beams. This series of eigenfunctions is a complete set. These higher-order Gaussian beams are described by the mode numbers m and n in the x and y directions, respectively. The fundamental Gaussian beam is the lowest-order (m = 0, n = 0) mode in this set. In this chapter, a treatment of the complex-argument Hermite-Gauss beams is presented. As for the other previously introduced paraxial beams, the reactive power of the complex-argument Hermite-Gauss beams vanishes. The higher-order point source in the complex space required for the full-wave generalization of the complex-argument Hermite-Gauss beams is derived. The basic full complex-argument Hermite-Gauss wave generated by the complex space source is determined. The real and reactive powers of the basic full complex-argument Hermite-Gauss waves are evaluated. As for the other basic full Gaussian waves, the reactive power is infinite. 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 a fixed kw₀, the real power decreases as the mode order increases.

Chapter Contents:

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

Inspec keywords: eigenvalues and eigenfunctions; electromagnetic waves; wave equations; Gaussian distribution

Other keywords: full-wave generalization; paraxial beams; basic full complex-argument Hermite-Gauss wave; complex-argument Hermite-Gauss beams; complex space; higher-order point source; reactive power; paraxial wave equation; cartesian coordinate system; eigenfunctions; higher-order Gaussian beams

Subjects: Probability theory, stochastic processes, and statistics; Linear algebra (numerical analysis); Electromagnetic waves: theory; Algebra, set theory, and graph theory; Other topics in statistics; Waves and wave propagation: general mathematical aspects; Electromagnetic wave propagation