Propagation modes and linear systems

[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?

 

[Lodeg]

I want to clarify my question. In fact, a linear system is caracterised by a linear operator H shch that
A_o = H A_i, where A_i and A_o are respectively the input and output. A mode of this linear system should satisfy
A_o = λ A_i, so that H A_i = λ A_i.
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.