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International Conference on Remote Sensing of the Earth and Physics (RSE)

Initial mathematical modeling of photon radiation in fiber optic networks of various types

Ibratjon X. Aliyev, Obbozjon Kuldashov, Davronbek O. Nabiev, Akramxon A. Valixonov, Azizjon A. Aliev

Published Mar 31, 2026• 0 citations

Abstract

The article provides mathematical modeling of the transfer process in a cylindrical fiber-optic tube of a monochromatic light beam. To describe the phenomenon, the approach of gradually increasing the mathematical accuracy and complexity of the problem is used. The initial stage is a dual mathematical formalism for the Schrodinger equation with separate forms of the Hamiltonian for the Laplace and Nabla operators. A partial comparison with the formalism of Maxwell's equations and a complete mathematical visualization of the set of solutions for each of the tuples of solutions of the psi function are given. Interesting features of this approach are revealed, with the definition of some unexpected formulations that take into account the role of quantum mechanical phenomena in the process of mathematical modeling and investigation of photon beams in fiber-optic networks. The conclusion was to determine the necessity and relevance of the topic of further expansion of the study in the direction of considering the process within the framework of the formalism of increased accuracy through the Pauli, Klein-Gordon, Dirac equations, the Riemann-Silberstein formalism in an expanded topological space.

DOI: 10.1117/12.3114661

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In the traditional prescriptions, the Fermat’s principle of least time or/and the Helmholtz equation are used to derive the beam-optical Hamiltonian. Now, we know that the Maxwell’s electromagnetism gives rise to the correct theory of light. So, we developed a formalism of beam optics starting with an exact matrix-representation of Maxwell’s equations. The resulting beam-optical Hamiltonian is not only exact but also has the mathematical structure of the Dirac equation, enabling us to employ quantum methodologies. The resulting formalism of Maxwell optics successfully describes beam optics along with polarization of light.

Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.

A system of coupled quantum harmonic oscillators whose Hamiltonian conserves photon number begets a one-photon correspondence principle (OPCoP), which allows solutions to the classical linear Maxwell equations for propagation in matter to be reinterpreted as precise descriptions of one-photon states. With the help of the OPCoP, we derive the linear classical Maxwell equations from the Schrödinger equation for one-polariton state evolution. The role of the matter's initial quantum state in setting the macroscopic medium parameters is made explicit. It is shown that most of the kinds of linear Maxwell equations possible follow from this model, thus showing that the vast extant body of linear, sourceless optical waveguide theory [Optical Waveguide Theory (Chapman and Hall, 1983)] can be applied to the exact analysis of one-photon propagation in optical fibers.

We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of our investigation, we consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space\nobreakdash-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.

We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the bundle description the wavefunctions are replaced with (state) sections (covariant approach) or liftings of paths (equivalently: sections along paths) (time-dependent approach) of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation of the state sections/liftings in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Connections between the (retarded) Green functions of these equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.