Free field theory: Why is it said that different Fourier modes in case of a free field (say, real Klein-Gordon field) are independent of each other?

Interacting field theory: How exactly does the addition of non-linear term in the Lagrangian makes the Fourier modes couple to each other?

Free field theory: Why is it said that different Fourier modes in case of a free field (say, real Klein-Gordon field) are independent of each other?

Interacting field theory: How exactly does the addition of non-linear term in the Lagrangian make the Fourier modes couple to each other?

Why is it said that different Fourier modes in case of a free field (say, real Klein-Gordon field) are independent of each other?

How exactly does the addition of non-linear term in the Lagrangian makes the Fourier modes couple to each other?

Free field theory: Why is it said that different Fourier modes in case of a free field (say, real Klein-Gordon field) are independent of each other?

Interacting field theory: How exactly does the addition of non-linear term in the Lagrangian makes the Fourier modes couple to each other?