Fourier transform of Maxwell's equations

[thaiqi]

Hello, I am unfamiliar with Maxwell's equations' Fourier transform. Are there any materials talking about it?

 

[BvU]

But you can google, right ?

e.g. in chrome I get a lot of sensible hits:

https://www.google.com/search?q=solving+Maxwell's+equations+using+Fourier+transform&oq=solving+Maxwell's+equations+using+Fourier+transform+&aqs=chrome..69i57.24167j0j7&sourceid=chrome&ie=UTF-8

 

[thaiqi]

Sorry, I am in China and cannot visit google. Besides, I don't mean solving equations using Fourier transform, but the Maxwell's equations in reciprocal(momentum) space. Better some textbooks treat it.

 

[BvU]

Can you visit

http://people.reed.edu/~wieting/essays/FourierMaxwell.pdf

 

[Dale]

Sorry, I am in China and cannot visit google.

Oh, I forgot that was a thing. That would make things difficult indeed.

 

[thaiqi]

Can you visit

http://people.reed.edu/~wieting/essays/FourierMaxwell.pdf

Thanks. I don't follow what this article said well. The books talk about it as below:

 

[vanhees71]

Well, ok. What the book obviously does is to write the Maxwell equations,
$$\vec{\nabla} \cdot \vec{E}=\rho/\epsilon_0, \quad \vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{E}=-\dot{\vec{B}}, \quad \vec{\nabla} \times \vec{B}=\frac{1}{c^2} \dot{\vec{E}} + \mu_0 \vec{j}.$$
Note that in the somwhat confusing SI units $c^2=1/(\epsilon_0 \mu_0)$.

Now they go to Fourier space wrt. to the spatial argument, i.e., they write
$$\vec{E}(t,\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 k \frac{1}{(2 \pi)^3} \exp(\mathrm{i} \vec{x} \cdot \vec{k}) \vec{\mathcal{E}}(t,\vec{k})$$
and analogously for all the other fields involved.

Then any spatial derivative is simply substituted by $\vec{\nabla} \rightarrow \mathrm{i} \vec{k}$. Then immideately get the equations you copied from the textbook.

 

[thaiqi]

Well, ok. What the book obviously does is to write the Maxwell equations,
$$\vec{\nabla} \cdot \vec{E}=\rho/\epsilon_0, \quad \vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{E}=-\dot{\vec{B}}, \quad \vec{\nabla} \times \vec{B}=\frac{1}{c^2} \dot{\vec{E}} + \mu_0 \vec{j}.$$
Note that in the somwhat confusing SI units $c^2=1/(\epsilon_0 \mu_0)$.

Now they go to Fourier space wrt. to the spatial argument, i.e., they write
$$\vec{E}(t,\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 k \frac{1}{(2 \pi)^3} \exp(\mathrm{i} \vec{x} \cdot \vec{k}) \vec{\mathcal{E}}(t,\vec{k})$$
and analogously for all the other fields involved.

Then any spatial derivative is simply substituted by $\vec{\nabla} \rightarrow \mathrm{i} \vec{k}$. Then immideately get the equations you copied from the textbook.

Thanks. This is what I need. Is it discussed in any books?