How should the concepts of transverse modes, longitudinal modes, fundamental mode, TEM, TE, and TM in lasers be understood?

TE, TM, and TEM Waves in Waveguides:

Electromagnetic waves propagating in a specific direction or through a specific path (waveguide) are called guided electromagnetic waves (guided waves). Based on Maxwell's equations, the electric field components (E, D) and magnetic field components (B, H) can be decomposed.

Introduction to Electromagnetic Wave Modes:

Electromagnetic waves have infinitely many modes. Mathematically, these are linearly independent solutions of the source-free Maxwell's equations under given conditions. For a given boundary condition, the mode refers to a specific electromagnetic field distribution pattern that can exist independently, also called the field distribution. Modes are often abbreviated as "modes" or sometimes referred to as "waveforms."

Classification of Electromagnetic Wave Modes:

Electromagnetic wave modes can be divided into resonant modes (cavity modes) and transmission modes (waveguide modes).

Transmission modes: These have specific electromagnetic field distributions in the cross-section and are studied in the context of waveguide transmission lines.
Cavity modes: These occur in resonators, where electromagnetic energy undergoes mutual conversion.

Classification of Electromagnetic Wave Transmission Modes:
Based on the relationship between the components of the electric field, magnetic field, and wave vector in the direction of propagation within a waveguide, the modes can be divided into TE modes, TM modes, and TEM modes:
4.1 TEM Mode:
Electromagnetic waves with no electric or magnetic field components in the propagation direction are called TEM waves or transverse electromagnetic waves. Their propagation mode is called the TEM mode.
4.2 TE Mode:
Electromagnetic waves with magnetic field components but no electric field components in the propagation direction are called TE waves or transverse electric waves, also known as "S waves." These are fast waves. In waveguides (closed cavity structures), the electromagnetic field components include E_y, H_x, H_z, with the propagation direction as the z -axis. The propagation mode is the TE mode.
4.3 TM Mode:
Electromagnetic waves with electric field components but no magnetic field components in the propagation direction are called TM waves or transverse magnetic waves, also known as "P waves." These are also fast waves. In waveguides (closed cavity structures), the electromagnetic field components include H_y, E_x, E_z , with the propagation direction as the z -axis. The propagation mode is the TM mode.
Number of Electromagnetic Wave Cavity Modes:
The electromagnetic waves in a cavity can be expressed as TE or TM modes based on the amplitude variation of the electromagnetic field. The indices ( m, n, p ) indicate the number of oscillations of the field strength along specific spatial dimensions. Each mode has a specific field distribution and a corresponding propagation constant.
In the equations, k = 2Π/λ , where λ is the free-space wavelength, and μ and ε represent the permeability and permittivity of the filling medium, respectively.
k is the cutoff wavenumber determined by the shape and size of the waveguide cross-section. For waveguides with non-uniform transverse media, k varies across different regions, while in uniform media waveguides, k is constant, i.e., k = 2Π/λ , where λ is the cutoff wavelength. In waveguides, when k = k (homogeneous waveguide λa = λ ), γ = 0 is the cutoff point, and only modes with k < k can propagate.
The mode with the smallest k is called the dominant mode or fundamental mode, while other modes are called higher-order modes. By appropriately selecting the waveguide dimensions for electromagnetic waves of a specific frequency, higher-order modes can be cut off, allowing only the dominant mode to propagate (i.e., single-mode transmission). For instance, rectangular waveguides typically transmit only the dominant TE10 mode, while coaxial lines and strip lines transmit only the dominant TEM mode (with k = 0 , no cutoff).
When both the dominant mode and one or more higher-order modes are allowed to propagate simultaneously, it is referred to as multi-mode transmission. If different modes within the same waveguide have identical phase constants, this is called mode degeneracy. Degenerate modes have the same phase and group velocities. For example, the TE and TM modes in rectangular waveguides are degenerate, as are the TE0n and TM1n modes in circular waveguides.
In cavities, modes with identical resonant frequencies are degenerate.

Figure 1