# Why do we use 'modes' in quantum optics?

In this material, it gives the definition of modes:

The modes are basically defined by the properties of coherence and orthogonality: modes are orthogonal solutions of the wave equation.

Since the modes are eigenfunctions in this material, why do we need to use a new term 'mode' instead of the original one, i.e., eigenfunctions?

• Does the introduction in your link not answer your question? But I agree that the term is somewhat unneccessary and it regularly confuses people who are not familiar with optics. May 26, 2021 at 14:43
• I learned the definition from this material. But I don't know the necessity of the term. There already exists an answer in this post. May 26, 2021 at 15:42

In math: label two eigenfunctions by $$a$$ and $$b$$, a single photon with a single eigenfunction might be written as $$|1\rangle_a\otimes |0\rangle_b$$ or $$|0\rangle_a\otimes |1\rangle_b$$. The eigenfunction language can't really distinguish between, say, $$|1\rangle_a\otimes |0\rangle_b$$ and $$|2\rangle_a\otimes |0\rangle_b$$. Moreover, it does not do well at describing $$|1\rangle_a\otimes |1\rangle_b$$. So we either add to the definition of eigenfunctions, or we define a new term that lets us write $$|\psi\rangle=\sum_{i,j}\psi_{i,j}|i\rangle_a\otimes |j\rangle_b.$$