# Propagation modes and linear systems

• A
• Lodeg
In summary, the authors of "Fundamentals of photonics" define waveguide modes as special inputs that are not altered (except for a multiplicative constant) upon passing through a linear system. These inputs are called modes or eigenfunctions, while the multiplicative constants are the eigenvalues. This definition is linked to the eigenvalue problem determined by the Helmholtz equation, where the input is the electric field and the output is -k^2E. This clarification is important because a linear system is characterized by a linear operator H that satisfies Ao = H Ai and a mode of this system satisfies Ao = λAi, where Ao and Ai are respectively the input and output.
Lodeg
In the book "Fundamentals of photonics", the authors defined waveguide modes using the notion of linear systems, where they said:

"Every linear system is characterized by special inputs that are invariant to the system, i.e., inputs that are not altered (except for a multiplicative constant) upon passage through the system. These inputs are called the modes, or the eigenfunctions, of the system. The multiplicative constants are the eigenvalues; they are the attenuation or amplification factors of the modes."

What is the link between this definition and the eigenvalue problem determined by helmholtz equation?

I want to clarify my question. In fact, a linear system is caracterised by a linear operator H shch that
Ao = H Ai, where Ai and Ao are respectively the input and output. A mode of this linear system should satisfy
Ao = λ Ai, so that H Ai = λ Ai.
However, in the case of helmholtz equation ∆E = - k2E, and we can certainly not say that the input is the electric field and the output is - k2E.

## 1. What are the different propagation modes in linear systems?

The three main propagation modes in linear systems are transverse electromagnetic (TEM), transverse electric (TE), and transverse magnetic (TM). In TEM mode, both electric and magnetic fields are perpendicular to the direction of propagation. In TE mode, only the electric field is perpendicular, while in TM mode, only the magnetic field is perpendicular.

## 2. How do propagation modes affect signal transmission in linear systems?

The propagation mode determines the behavior of the electric and magnetic fields in a linear system, which in turn affects the transmission of signals. For example, in TEM mode, there is no attenuation of the signal, while in TE and TM modes, there may be some attenuation due to the presence of dielectric materials.

## 3. What is the difference between guided and unguided propagation in linear systems?

Guided propagation refers to the transmission of signals through a physical medium, such as a cable or waveguide. In contrast, unguided propagation, also known as free space propagation, refers to the transmission of signals through the air or vacuum. Guided propagation is typically used in wired communication systems, while unguided propagation is used in wireless communication systems.

## 4. How do linear systems handle signal distortion?

Linear systems use various techniques, such as equalization and filtering, to minimize signal distortion. Equalization involves adjusting the amplitude and phase of the signal to compensate for any distortion introduced by the propagation medium. Filtering removes unwanted noise or interference from the signal, improving its quality.

## 5. What are some applications of linear systems and propagation modes?

Linear systems and propagation modes are used in a wide range of applications, including telecommunications, radar systems, satellite communication, and wireless networks. They are also essential in the design of electronic circuits and devices, such as antennas, amplifiers, and filters.

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