Momentum density in a Divergent Beam

[Twigg]

For a divergent paraxial field like $$E = E_0 e^{-\frac{r^{2}}{w(z)^{2}}} e^{-i(kz - tan^{-1}(\frac{z}{z_{0}}))}$$

What is the direction of the momentum density of the E-field. I have two competing feelings about it. 1) The momentum density should be parallel to the Poynting vector, and since the beam is diverging it is propagating energy outwards along some combination of the r and z axes. 2) the momentum density should follow the wavevector, and the phase of the above expression does not depend on the cylindrical radial coordinate r at all, so the wavevector should be pointing exclusively along the z-axis. Is this maybe a case where the usual rule about momentum density, poynting vector and wavevector all being parallel is faulty?

 

[NFuller]

The momentum density should be parallel to the Poynting vector

Yes, since the momentum density $p$ is related to the Poynting vector $\mathbf{S}$ by
$$p=\epsilon\mu\mathbf{S}$$.

 

[NFuller]

the momentum density should follow the wavevector, and the phase of the above expression does not depend on the cylindrical radial coordinate r at all, so the wavevector should be pointing exclusively along the z-axis.

Calculate the $\mathbf{H}$ field corresponding to your $\mathbf{E}$ field. Then calculate the Poynting vector
$$\mathbf{S}=\mathbf{E}\times\mathbf{H}$$.

 

[Twigg]

Appreciate the help NFuller. I got the Poynting vector for the field, and sure enough it has non-axial components that vanish at r=0, which makes a lot of sense. I have a followup question. Does this mean that an atom moving in a red-detuned convergent beam will feel a transverse confining force that depending on the atom's transverse velocity? My logic for this is that if the momentum density direction varies in space, then the wavevector of the photons should also vary in space, and so the atom should have a transverse Doppler shift, leading to transverse cooling. Am I on the right track?