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I am studying the multipole expansion of electromagnetic wave radiation pattern, and it is said that any fields can be decomposed into the spherical harmonics $Y_{lm}$.

However, for $l=1$, which corresponds to a dipole, the spherical harmonics looks like a dumbbell, as shown in the attached picture, but we know that an electric dipole should have a doughnut shape radiation pattern, I don't understand why they don't match, is there any relationship between them?

 spherical harmonics dipole radiation pattern
$$ \begin{array}{cc} \left(\textbf{A}\right)~\text{spherical harmonics} & \left(\textbf{B}\right)~\text{dipole radiation pattern} \\ \hspace{300px} & \hspace{300px} \end{array} $$

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The second plot you show is a generalization of the $Y_{lm}$ - it's a vector spherical harmonic; in addition, it differs from the electrostatic case in that the radial dependence is no longer a harmonic function (i.e. a solution to the Laplace equation), and it has been replaced by a wave solution (a spherical Bessel function, a solution to the monochromatic wave equation).

You can go into more mathsy detail if you really want (e.g. this for the general formalism, or this when you specialize to $l=1$ and stop caring about formal identification as spherical harmonics), but that's the core of the difference between those two.

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  • $\begingroup$ thanks for the reply, I have been reading Jackson's book, it says the radial dependence is separated from the angular components, and both the radiation pattern and spherical harmonics are functions of angles ($\theta$,$\phi$), so I don't understand how the difference in the radial dependence solves this issue, sorry. $\endgroup$ – Zhijie Ma May 17 '17 at 10:23
  • $\begingroup$ The difference in the radial dependence doesn't much solve the issue - it's the shift to the vector spherical harmonics, $\mathbf X_{lm}=\hat{\mathbf L}Y_{lm} = \mathbf r\times\nabla Y_{lm}$ and its curl, that changes the angular dependence. $\endgroup$ – Emilio Pisanty May 17 '17 at 10:30
  • $\begingroup$ thanks, is there anyway we can visualise the vector spherical harmonics angular distributions? I really want to compare whether it is the same as the radiation patterns of an electric dipole (a donut) or quadrupole (four lobes). $\endgroup$ – Zhijie Ma May 17 '17 at 13:18
  • $\begingroup$ It's a bit trickier because you need to decide exactly what you're plotting (i.e. the radiation field is a three-dimensional vector-valued field; you can plot the norm of the field but that does not always carry all the information you need), but yes, you can take the electric field from Wikipedia, $$\mathbf E(\mathbf r,t) = A_0 \mathrm{Re}\left[i\nabla\times (h_{1}^{(1)}(kr) \mathbf X_{10}(\theta,\phi)e^{-i\omega t}\right],$$ and analyze it to your heart's content. $\endgroup$ – Emilio Pisanty May 17 '17 at 13:58

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