I have an optical fiber and I also have a light source. The light was shot into the optical fiber, and the beam of light wandered around in the optical fiber, and finally shot out from the other end of the optical fiber. So I repeated the operation just now and found that the position of the second beam and the state of any spatial position at any time are the same as before. I call this a model. That is, I see its entire propagation process as a whole, it contains all aspects of this beam of light, this is a mode.
The Light is an electromagnetic field, the current method to human understanding of the world, you want the most complete description of it, with only a mathematical expression that solving electromagnetic field equations, which is on the model , the most complete expression. In addition, there are many ways to distinguish different modes, such as observing the shape of the light in the cross section of the fiber , that is, the light spot .
From the source of light, let me explain in a popular way what is a pattern. And where does the light entering the fiber come from? It is produced by a laser.
Laser production
The laser light is produced by a laser source, and there is an optical resonator in the laser source, which is a box that can be artificially fooled with light inside ( the working principle of a laser ).
There is a saying in electromagnetic field theory, "In a cavity with certain boundary conditions, the electromagnetic field can only exist in a series of discrete eigenstates."
Those who have studied linear algebra should be somewhat impressed by the so-called eigenstate. Eigenvalue is also called eigenvalue, eigenvalue, eigenvector. It's just a statement. In this small device that can generate laser light, due to the physical limitations of this small box, we can only get one or a few solutions by solving its electromagnetic field equation. Each solution is a pattern.
Definition of laser mode
Electromagnetic field angle: Each eigenstate has a fixed oscillation frequency and spatial distribution, which is called a laser mode.
Photon angle: A photon state that can be distinguished in an optical resonator, called a laser mode.
Field pattern angle: A field pattern distributed along the cross section of the square, called a laser mode.
For the convenience of analysis, we are divided into two parts to discuss the state of the laser. 1. The direction of wave propagation (longitudinal); 2. The plane perpendicular to the direction of light propagation (lateral).
PS: Remember the temporal coherence and spatial coherence, which are also divided into vertical and horizontal discussion~ ( Popular interpretation of coherent light )
Therefore, the laser mode (electromagnetic field state with different oscillation frequencies and spatial distribution) is decomposed into longitudinal mode and transverse mode . The longitudinal mode is used to represent the longitudinal electromagnetic field distribution, in the case of the horizontal mode, it is the electromagnetic field distribution on the cross section.
It is well understood that the only thing that affects the laser mode is the laser source, (more specifically, it refers to the optical resonator. In fact, the material that generates the laser also affects the frequency of the generated laser at the atomic level. So we still use the laser source to compare Rigorous) So the laser source is determined, and the laser mode (oscillation frequency & spatial distribution of electromagnetic field) is determined.
Longitudinal mode
When the laser is generated, it must be when the resonant cavity meets the standing wave condition.
The light reflects back and forth in the resonant cavity, and only when they are coherent and coherent will they become lasers (so the power will become higher). That is, when a standing wave is formed in the resonant cavity. That is, it is required that the phase difference between the two beams of light to and fro satisfy Δ ϕ = 2 π ⋅ q \Delta\phi=2\pi\cdot qΔ ϕ=2 π⋅q where q is an integer. After related derivation, the wavelength of the laser mode corresponding to different q isλ q = 2 L ′ q \lambda_q=\frac{2L^{'}}{q}λq=q2 L′Where L ′ L^{'}L' Is the length of light traveling in the cavity.
As a result, different q corresponds to different standing waves and lasers of different wavelengths. The electromagnetic fields of these standing waves have different longitudinal distributions.
Therefore, q as a parameter, which characterizes the longitudinal stable field distribution, is called the longitudinal mode of the laser. q is called the longitudinal mode number , and different longitudinal modes correspond to different q, corresponding to different light wavelengths.
PS: The q that can be solved by each resonator is about 1 0 4 10^4104~ 1 0 6 10^6 10Six , can there be so many vertical modes? of course not. In fact, each longitudinal mode has a certain bandwidth and contains many qs, and due to other conditions, not all qs can have laser output.
Horizontal mode
The transverse mode reflects the spatial distribution of the electromagnetic field on the cross section. Observing the laser from the cross section, you can see the following spots (light intensity distribution diagram of the beam cross section). 
Each transverse mode corresponds to a transverse stable field distribution.
We have two representations for the transverse mode. The difference lies in the use of different coordinate systems. When cylindrical coordinates are used, the resulting modes are called TEM mode, TE mode, TM mode, HE mode, and EH mode. When a rectangular coordinate system is used, the resulting mode is called LP (Linear Polarization) mode.
LP mode is composed of a set of transmission constant β \betaVectors whose β are very close are degenerate.
Degenerate: When two modes have the same or similar values of \betaβ, we call these two modes degenerate, because the TE mode and the TM mode are completely degenerate near the cutoff frequency. The meaning of degeneracy is that when these two modes are transmitted together, the field pattern they form together can maintain a long distance. Although the field pattern formed by them has the same stable transmission as the mode field, due to the different proportions of the two, the concept of mode is generally different.