# Coupled mode theory - question about an equation

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• Solmyros
In summary, the conversation discusses the concept of normal modes in a waveguide with zero free charges and currents. The equations for monochromatic waves in a lossless, unperturbed waveguide are given and it is mentioned that any optical field at a given frequency can be expressed in terms of these normal modes. The conversation also mentions that a spatially dependent perturbation in the waveguide can lead to coupling and a dependence of the field amplitude on the propagation direction. The question posed is how to express the electric and magnetic fields of the waveguide in terms of their normal modes.
Solmyros
TL;DR Summary
I want to extract the equation describing the coupled mode theory. I am using a book as a reference. I have a question on how he transitioned from one step to another.
edit: Hello everyone! When I posted the question, latex equations that were visible in "preview" do not seem here. So, I upload a pdf version of the question.

We consider zero free charges and currents: ρ=J=0

$$\mathbf{\nabla} \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} = -\frac{\partial \left( \mu \mathbf{H} + \mathbf{M}\right)}{\partial t} = i \mathbf{\omega} \mathbf{\mu_0} \mathbf{H}, for \: \mathbf{M} = 0$$

$$\mathbf{\nabla} \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} = \frac{\partial \mathbf{\left( \epsilon \mathbf{E} + \Delta \mathbf{P} \right)}}{\partial t} = - i \mathbf{\omega} \mathbf{\epsilon} \mathbf{E} - i \mathbf{\omega} \Delta \mathbf{P}$$

The following two equations describe monochromatic waves of a lossless, unperturbed waveguide.
$$\mathbf{E_\nu(\mathbf{r})} = \mathbf{\mathcal{E_\nu}}(x, y)exp(i\beta_\nu z)$$

$$\mathbf{H_\nu(\mathbf{r})} = \mathbf{\mathcal{H_\nu}}(x, y)exp(i\beta_\nu z)$$

Generally, we know that normal modes can form a basis. So any optical field at a given frequency, can be expressed in terms of their expansion. And if we have a spatially dependent perturbation to the waveguide, we will have coupling and the amplitude will depend on z, where z is the propagation direction.

I am afraid I will post images here, as my latex code (did it on overleaf) does not show here.

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My question is: How can we express the electric and magnetic fields of the waveguide in terms of their normal modes?

## 1. What is coupled mode theory?

Coupled mode theory is a mathematical framework used to analyze the behavior of coupled systems, where multiple modes or oscillations are present. It is commonly used in the fields of optics, acoustics, and electromagnetics.

## 2. How is coupled mode theory applied?

Coupled mode theory is applied by using a set of equations to describe the interactions between different modes in a system. These equations can be solved to predict the behavior of the system and understand how changes in one mode affect the others.

## 3. What is the significance of coupled mode theory?

Coupled mode theory is significant because it allows for a simplified analysis of complex systems. It also provides a deeper understanding of the interactions between different modes, which can be applied to various real-world problems.

## 4. What is the equation used in coupled mode theory?

The most commonly used equation in coupled mode theory is the coupled mode equation, which describes the coupling between different modes in a system. It is typically represented as a set of differential equations.

## 5. What are some examples of systems that can be analyzed using coupled mode theory?

Coupled mode theory can be applied to a wide range of systems, including optical waveguides, resonators, and photonic crystals. It is also used in acoustic systems such as musical instruments and electromagnetic systems like antennas and transmission lines.

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