# Mathematical-Physics of propagating modes in planar waveguides

## TL;DR

This paper provides a detailed physical and mathematical analysis of the formation of propagating transverse modes in planar dielectric waveguides, highlighting the roles of cosine and sine functions in describing stationary and quasi-stationary modes.

## Contribution

It introduces a mathematical-physics approach to understanding mode formation in planar waveguides, emphasizing the physical interpretation of mode functions.

## Key findings

- Pure stationary mode described by cosine function
- Quasi-stationary mode described by sine function
- Physical interpretation of mode formation at wavelength scale

## Abstract

In this paper we present a detailed physical analysis of the formation of the propagation transverse modes in planar dielectric waveguides using a mathematical-physics approach. We demonstrate physically that, at the wavelength scale, the pure stationary mode inside planar waveguide is described by the cosine function. Meanwhile, the sine function yields a quasi-stationary periodic mode.