A Propagation modes and linear systems

AI Thread Summary
The discussion centers on the definition of waveguide modes in linear systems as presented in "Fundamentals of Photonics," highlighting that modes are invariant inputs characterized by eigenvalues. The connection to the eigenvalue problem in the Helmholtz equation is questioned, particularly regarding the relationship between input and output in linear operators. It is noted that while a mode satisfies a specific relationship with the operator, the Helmholtz equation presents a different formulation that complicates this direct analogy. The inquiry emphasizes the need for clarity on how the Helmholtz equation fits into the framework of linear systems and modes. Understanding this relationship is crucial for applying these concepts in photonics.
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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?
 
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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.
 
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