Isotropic Electromagnetic GLHUA Sphere
Description
Isotropic Electromagnetic GLHUA Sphere Date Written: October 01, 2025 Abstract The contents are announced in arXiv:1706.10147 [15],arXiv:1612.02857 [16] and arXiv:1005.3999 [3]. Concrete and direct application of three-dimensional positive space and negative space is the discovery of the GLHUA sphere. In (7) and (8) in [15], when R₁=0, GLHUA invisible cloak become to GLHUA Sphere body. In three-dimensional positive space, the GLHUA sphere is a sphere with a radius of R2 and a radial variation of r. Specifically, the GLHUA sphere is an isotropic electromagnetic sphere with isotropic relative dielectric parameters ε, permeability μ, and relative refractive index n, ε=μ=n=(R₂)²/r², from (7) and (8）in [15] when R₁=0. We discovered and proposed that if the electromagnetic parameters within the sphere depend only on the radial direction, there are only four types of isotropic electromagnetic spheres: the E-sphere,with ε=μ=n=-1; the FLHUA sphere,with ε=μ=n=1, free space sphere; the GLHUA sphere,with ε=μ=n=(R₂)²/r²; and the HLHUA sphere, with ε=μ=n= - (R₂)²/r². This paper studies the completely invisible electromagnetic sphere, the GLHUA sphere; the novel propagation of electromagnetic waves within the GLHUA sphere; the simultaneous propagation of two wavefronts; the intuitive propagation and transformation of quantum mechanical waveparticle duality within the GLHUA sphere; the appearance of entanglement, hawk beak, and loose entanglement of wavefronts; the complete absorption of the incident wave by the GLHUA sphere; and the simultaneous generation of an outgoing wave. The propagation of electromagnetic waves within the GLHUA sphere is neither superluminal nor time-delayed, making the GLHUA sphere completely invisible. If the electromagnetic parameters are only related to the radial direction, if r is the radial variable of the GLHUA sphere, R is the image space variable, the optical topological transformation of the induced electromagnetic parameters is d(R(r))/dr =ε=μ=(R₂)²/r², after solving the ODE equation we get topological transformation R(r) = - (R₂)²/r , from (6) in [15] when R₁=0. The above optical topological transformation maps the sphere in three-dimensional positive space one-to-one and fully onto the out of negative sphere in negative space. This induces the GLHUA sphere. Based on the transformation from positive space to negative space, R(r) = - (R₂)²/r and the sphere harmonic analysis, we obtain a solution for the propagation of electromagnetic waves through the GLHUA sphere. The novel simultaneous alternating propagation of two wavefronts explains the GLHUA sphere as a completely invisible sphere, neither exceeding the speed of light nor time delay. The GLHUA sphere is a fundamental sphere for practical invisibility science; it serves as a window into positive and negative space and the science of invisibility.