Electromagnetic scalar product in spatially bounded domains

Many physically interesting quantities of the electromagnetic field can be computed using the electromagnetic scalar product. However, none of the existing expressions for such scalar product are directly applicable when the fields are only known in a spatially bounded domain, as is the case for many numerical Maxwell solvers. In here, we derive an expression for the electromagnetic scalar product between radiation fields that only involves integrals over closed spatial surfaces. The expression readily leads to formulas for the number of photons, energy, and helicity of generic polychromatic light pulses of incoming or outgoing character. The capabilities of popular Maxwell solvers in spatially bounded computational domains are thereby augmented, for example, by a straightforward method for normalizing emitted fields so that they contain a single photon.

Figure 1

In a numerical simulation, the electromagnetic scalar products resulting in the number of photons, energy, and helicity of the pulse emitted by the object in (a) can be obtained as integrals on the $\partial D_1$ surface using Eqs. (19-d21) to (19-d21). Adjustments to the simulated emission can then be done, for example, by scaling the field so that it contains a single photon. In (b), the adjusted emitter is placed close to other objects, which interact with the emission. Integrals on the $\partial D_2$ surface provide the same quantities for the total outgoing field.

Figure 2

Energy density of the outgoing pulse from Eq. (22) at different times, plotted in the $zx$ plane with horizontal $z$ axis and vertical $x$ axis. The number of photons, helicity, and energy of the pulse can be computed with electromagnetic scalar products implemented as surface integrals on generic surfaces. The same results are obtained on spherical and cubical surfaces. The sphere centered at the origin has a radius of $2.5\mu \mathrm{m}$ and is drawn with a solid line. The cube has sides of length $5.0\mu \mathrm{m}$ and is drawn with dashed lines.