# Source of cylindrical electromagnetic waves

I have a question about waves with cylindrical wavefront. Precisely, I have read that they may be generated by a linear source, while for instance plane and spherical waves are respectively generated by an infinite distant source and a point source, respectively. It is easy to see for instance in this picture: The usual example I find for the generation of cylindrical waves is diffraction across around slit: My question is: does any linear source of electromagnetic waves generate a cylindrical wave? Or simply, we can generate it through some of all possible linear sources?

For instance, if you consider the electric field generated by a half-wave dipole antenna (which is linear), you see that (reference):

$$E_{\theta}=\frac{-i\zeta _0 I_0 }{2\pi r} \frac{\cos(\frac{\pi}{2}\cos \theta)}{\sin \theta} e^{i(\omega t -kr)}$$

This is the structure of a dipole antenna: You may see that this source is linear, but it generates a spherical waveform (which is in general generated by a point source) and not a cylindrical waveform (which would have had the square root of the radius at the denominator, as you can see here).

• Please refrain from using images for equations. Typeset equations using MathJax. – user258881 Apr 2 at 10:42

As your expression for $$E_\theta$$ suggest, this is in fact the $$\hat \theta$$ component of a plane wave (in the far field) expressed in spherical coordinates. In the far field the curvature of the wavefronts is negligible and all wavefronts can be considered as planes.
The (nice) figure you give is really in the approximation of a very very long slit. Thus you might get cylindrical wavefronts by looking at the near-field regime $$R<\lambda$$ of a very long antenna (i.e. $$\ell \gg \lambda$$). I do not know of any text that treats this case.
Here $$R$$ is the distance from the antenna to the point where you measure the field, and $$\ell$$ is the length of the antenna.