Normal modes of electromagnetic field

[dEdt]

Hey guys,

I'm trying to understand the properties of normal modes of the electromagnetic field inside an arbitrary cavity, but I'm having some trouble.

By definition, for a normal mode we have $\mathbf{E}(\mathbf{x},t) = \mathbf{E}_0 (\mathbf{x}) e^{i \omega_1 t}$ and $\mathbf{B}(\mathbf{x},t) = \mathbf{B}_0 (\mathbf{x}) e^{i \omega_2t}$. It's easy to show that $\omega_1 = \omega_2$. Substituting these two equations into the wave equation gives the standard equation for the normal modes of a system, namely
$$\nabla^2 \mathbf{E}_0 + \frac{\omega^2}{c^2}\mathbf{E}_0 = 0$$
and
$$\nabla^2 \mathbf{B}_0 + \frac{\omega^2}{c^2}\mathbf{B}_0 = 0.$$

For any allowed value of $\omega$, there will be three linearly independent solutions to each of these equations -- unless we have degeneracy, of course. The conditions
$$\nabla \cdot \mathbf{E}_0 = 0$$
and
$$\nabla \cdot \mathbf{B}_0 = 0$$
reduce this number to two.

In the case of a rectangular cavity, it's easy to show that
1) the electric and magnetic fields are perpendicular; and
2) the two linearly independent solutions mentioned above correspond to the two possible polarization states.

My intuition tells me that these properties extend to the case of an arbitrarily shaped cavity, but I can't prove that they do. In particular, I'm not sure how polarization is defined if our electromagnetic field solution doesn't have a well-defined direction of propagation.

Any ideas?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[Meir Achuz]

I think the E and B fields will be orthogonal only if the cavity shape allows separation of coordinates in an orthogonal coordinate system.
Two independent solutions for E always corresponds to two different 'polarization states', but the polarization may not be as simple as plane or elliptical.

 

[Andy Resnick]

<snip>

My intuition tells me that these properties extend to the case of an arbitrarily shaped cavity, but I can't prove that they do. In particular, I'm not sure how polarization is defined if our electromagnetic field solution doesn't have a well-defined direction of propagation.

Waveguide modes (TE, TM, and hybrid modes) can violate your result. Also, when you derived the wave equations, you made some assumptions (for example, linear constitutive relations between E and D, B and H; homogeneous materials, dielectric materials...) that can be violated. To be sure, Ampere's and Faraday's law are always valid, but if you don't have plane wave solutions then E and B are not orthogonal to each other, and neither are orthogonal to the propagation direction.

 

[sardar]

Hello,

Thank you for sharing your thoughts and questions about normal modes of the electromagnetic field. It seems like you have a good understanding of the mathematical equations involved in describing these modes. I can provide some insights and ideas that may help you in your understanding.

Firstly, it is important to note that normal modes of the electromagnetic field are a fundamental concept in the study of electromagnetism. These modes represent the standing waves that can exist inside a cavity, which is an enclosed space with reflective boundaries. The normal modes are characterized by their frequency and spatial distribution, and they play a crucial role in many practical applications, such as in the design of antennas and resonators.

As you correctly stated, the solutions to the wave equations for the electric and magnetic fields inside a cavity can be written as a product of a time-dependent term and a spatially-dependent term. These terms are related to the frequency of the mode and its spatial distribution, respectively. The condition of perpendicularity between the electric and magnetic fields is a consequence of the wave equations, and it holds for all normal modes, regardless of the shape of the cavity. This is because the wave equations are derived from Maxwell's equations, which describe the fundamental relationship between electric and magnetic fields.

Regarding the concept of polarization in an arbitrarily shaped cavity, it is still defined in the same way as in the case of a rectangular cavity. Polarization refers to the direction of the electric field oscillation in a transverse wave. In the case of normal modes, the electric field is perpendicular to the direction of propagation, so the polarization is defined by the direction of the electric field in the transverse plane. This concept remains valid even if the direction of propagation is not well-defined, as in the case of standing waves.

In summary, the properties of normal modes of the electromagnetic field, including the perpendicularity between electric and magnetic fields and the concept of polarization, hold true for any arbitrarily shaped cavity. I hope this helps to clarify your understanding. Keep exploring and asking questions, as that is the key to advancing scientific knowledge. Best of luck in your studies!