"Mode" interpretation of preliminary content 1

My purpose is very simple, to figure out what the "mode" in fiber optic communication is all about, but no book can directly address my problem. It depends on many other contents to extract the key to the problem. I came across a lot of terms and concepts that I hadn't seen before, and I slowly inquired a lot, the content is a bit too much, let's make a record.
Reference book: "Optical Waveguide Theory" edited by Wu Chongqing

introduction

Since light waves are electromagnetic waves, it is first of all an electromagnetic oscillation. Electromagnetic oscillation includes the oscillation of electric field and magnetic field. According to the custom of light field, it can be written as $$\left(\begin{array}{c}E\\ H\end{array}\right)=\left(\begin{array}{c}E\\ H\end{array}\right)(x,and,with,t)\; The$$
light field is the spatial position$(x,and,z)\; is$ a function of timettFunction of$t$ .
Q1: The relationship between electromagnetic waves and electromagnetic oscillation
A1: Electromagnetic oscillation: The phenomenon of periodic changes of electric charge and current, electric field and magnetic field over time. Electromagnetic wave: The changing electric field and the magnetic field alternately change and propagate from near to far. The process of this changing electromagnetic field propagating at a certain speed in space is called electromagnetic wave.
Electromagnetic waves can propagate outward because of the mutual transformation of electric and magnetic fields. For example, magnetism is generated on the left side of electricity, and electricity is generated on the left side of magnetism, so the energy will propagate to the left from the very beginning point.

The light field in the optical waveguide can be decomposed into longitudinal and transverse components (subscript t) $$\{\begin{array}{l}E=E_t+E_{from}\\ H=H_t+H_{from}\end{array}$$

The relationship between the longitudinal and transverse components of the light field

(I don't think this relationship is useful, just to compare with the relationship between the vertical and horizontal components in the mode field later) The relationship between the vertical and horizontal components of the light field is obtained through a series of derivations:

1. The transverse component always rotates with the distribution of the cross section, depending on the corresponding longitudinal component (E and H correspond to each other).
2. The longitudinal component always rotates with the distribution of the cross section, depending on the corresponding (E and H correspond to each other) transverse component and its own transverse component.

Regular optical waveguide

Concept of mode

An optical waveguide whose refractive index distribution is constant along the longitudinal direction is called a regular optical waveguide. Its light field can always be written as the following form $$\left(\begin{array}{c}E\\ H\end{array}\right)(x,and,with,t)=\left(\begin{array}{c}e\\ h\end{array}\right)(x,and)\cdot e^{j(\beta z-\omega t)}$$ This is equivalent to a special solution of the wave equation, which is called amode.
The wave equation is an equation satisfied by the electric field and the magnetic field, and it is implicit. To solve the wave equation is to get explicit expressions for E and H. As the above formula, after knowing its display expression, the distribution and propagation of the electromagnetic field are very clear.
Where$\beta$ is the phase shift constant, indicating that this light field has volatility (propagation from far and near). $and(x,y)$与$h(x,y)$ is themode field, which means the light field$(E,H)$ Distribution along the cross section. The reason it is called a mode field is that only a mode can have the form of the above formula.
Only the light field corresponding to the mode can be written in the mathematical form of separating variables.

To solve the wave equation of the light field in a regular optical waveguide, there are countless discrete solutions $$\left(\begin{array}{c}E\\ H\end{array}\right)=\left(\begin{array}{c}{e}_i\\ h_i\end{array}\right)(x,and)\cdot e^{j{\beta }_{i}z}$$ eachiiThe solution corresponding to$i$ is a pattern. The pattern can be understood like this

1. The mode is a special solution of the wave equation (the form of the wave equation is related to the environment in which the electromagnetic field exists) in the optical waveguide, and it satisfies the boundary conditions of the optical waveguide.
2. A mode is a field pattern distributed along the cross section of an optical waveguide.
3. The modes are sorted by the number of the roots of the separated variables in the characteristic equation (I didn’t understand what separated variables will not solve the differential equation first).
4. The linear combination of many modes constitutes the total field distribution of the optical waveguide $$\left(\begin{array}{c}E\\ H\end{array}\right)=\sum _i\left(\begin{array}{c}{a}_i{e}_i\\ b_i{h}_i\end{array}\right)(x,y)e^{j{\beta }_{i}z}$$$a_i,b_i$Indicates the relative size of the pattern.
5. The most basic physical quantity for a mode to transmit in a waveguide is its transmission constant $\beta$，$\beta$ is a function of frequency and also a function of refractive index distribution. $When\; \beta$ is a real number, it means that there is only phase shift and no attenuation during transmission. β \betaThe imaginary part of$\beta$ represents the attenuation along the optical waveguide.
6. The transmission constant β \beta of a mode$\beta$ , by the distribution form of the mode field$(e,h)$ uniquely certain; on the contrary, if$When\; \beta$ changes for some reason, the distribution form of its mode field will inevitably change.
7. The different modes of the regular optical waveguide are orthogonal.

The relationship between the longitudinal and transverse components of the mode field

Like the light field, the mode field can also be decomposed into vertical and horizontal components $$\{\begin{array}{l}e=e_t+e_{from}\\ h=h_t+h_{from}\end{array}$$Through a series of derivations, the relationship between the longitudinal and transverse components of the mode field is obtained:

1. The transverse component of the mode field can be uniquely determined by the distribution of the longitudinal component with the cross section.
2. The longitudinal component of the mode field can be uniquely determined by the distribution of the transverse component with the cross section.

An additional conclusion is that only the transverse component carries power, and the longitudinal component only serves as a guide. It shows that the regular optical waveguide has obvious properties of guiding light energy transmission.