Electric field due to a point dipole

[Hypochondriac]

Let's say I have a point dipole (as an approximation for an atom) at the origin and it oscillates in the z axis. The (theta component of the) electric field due to this dipole in the far field will be

E = \frac{d}{4\pi\epsilon_0}\frac{k^2\sin\theta}{r}\exp i\left(kr-\omega t\right)

where d is the dipole moment, k=2\pi/\lambda, \theta is the angle made with the z axis, r is the radial distance, \omega is the angular frequency of the oscillation and t is the time.

Due to the zero \phi dependance, i.e. the angle in the equatorial plane, there is a cylindrical symmetry. Instinct tells me that I should I have two lobes of electric field, one in the +z direction and one in -z, which will oscillate, alternatively between positive and negative. However the equation I quoted implies a doughnut shaped electric field. For a given r, E increases as \theta goes from 0 to \pi/2, then decreases from \pi/2 to \pi. I.e. I think the equation should have a \cos\theta in it instead of a \sin\theta

Where have I gone wrong? I think it's my definition of \theta, however it is always defined from the z axis.

PS. I'm pretty sure the equation is in Jackson.

 

[G01]

This is the correct form for a point dipole oscillating along z. The intensity profile for a point dipole source is indeed like a donut.

Since the dipole is oscillating along z, there is 0 intensity at \theta=0, and the intensity is maximum \theta=\pi /2 in the x-y plane. I think if you take a cross section of the donut on the z-y or z-x planes you'll find the lobes you are picturing. However, the lobes are centered in the x-y plane, not along z.

For reference, see Griffith's Eq. 11.18 and Figure 11.4