How can the spectral representation of a plane EM wave be found?

AI Thread Summary
The discussion focuses on finding the spectral representation of a plane electromagnetic wave described by E(r,t) = E(r)exp(-ikz)exp(iwt). The user seeks assistance with the Fourier Transform of this time-domain representation to obtain E(r,w). It is noted that for finite intensity fields, the spectral representation can be derived through the Fourier Transform, but the user struggles with the integration limits. A response clarifies that the spectrum results in a Dirac delta function, specifically δ(ω - ω₀), when the time-domain signal is of the form e^(-iω₀t). This highlights the relationship between time-domain signals and their spectral representations in frequency space.
Madara
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Hi,

Let E(r,t) = E(r)exp(-ikz)exp(iwt)
be a plane wave in time domain, propagating along Z direction.

I wonder how to find the spectral representation of it (i.e. E(r,w))??

I know, for a finite intensity field (i.e. |E(r,t)|^2 < infinity), we can give the spectral representation of the signal by,

E(r,w) = Fourier Transform of [E(r,t)].

But when I do the intergration in Fourier Transform between the + infinity and - infinity, I can't get a solution for above.

Can anyone help me with this?

Thanks
Madara
 
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The spectrum is a Dirac delta function in omega \delta(\omega-\omega_0)
if E(t)~e^-i\omega_0 t.
 
Thanks Clem.
 
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