Squared vector field terms $mathbf{E}^2$ and $mathbf{H}^2$?

Consider the simple case of electromagnetic irradiation of a homogeneous isotropic dielectric, neglecting the dispersion of the refractive index. Assuming a transparent medium, the spatial density of forces acting on the dielectric in a static external electromagnetic field can be given as

$$mathbf{f} = – nabla p – nabla epsilon dfrac{langle mathbf{E}^2 rangle}{8 pi} – nabla mu dfrac{langle mathbf{H}^2 rangle}{8 pi} + nabla left[ left( rho dfrac{partial{epsilon}}{partial{p}} right)_T dfrac{langle mathbf{E}^2 rangle}{8 pi} + left( rho dfrac{partial{mu}}{partial{rho}} right)_T dfrac{langle mathbf{H}^2 rangle}{8 pi} right] + dfrac{epsilon mu – 1}{4 pi c} dfrac{partial}{partial{t}}langle [ mathbf{E} times mathbf{H}] rangle.$$

$p$ is the pressure in the medium (for a given density $rho$ and temperature $T$ in zero field.
$epsilon$ and $mu$ are the permittivity and magnetic permeability.
$c$ is the speed of light.
The angular brackets denote averaging over a time period far greater than the characteristic alternation period of light.

And my understanding is that $mathbf{E} times mathbf{H}$ is the Poynting vector.

What I don’t understand is the squared field terms $mathbf{E}^2$ and $mathbf{H}^2$. These field terms are vector fields, and so my understanding is that it is not mathematically valid to take a vector field (or any other vector) to an exponent. So what is meant by $mathbf{E}^2$ and $mathbf{H}^2$ in this context?

I would greatly appreciate it if people would please take the time to explain this.