Poynting vector

An electromagnetic wave carries energy, and the Poynting vector describes its flow: the power per unit area crossing a surface perpendicular to the vector,

in watts per square metre. In a plane wave in an isotropic medium it points along the direction of propagation. Optical fields oscillate at hundreds of terahertz, far faster than any detector responds, so what is measured is the time average, the intensity or irradiance.

Intensity and field amplitude

For a plane wave with electric field amplitude $E_0$ in a medium of refractive index $n$, the time-averaged Poynting vector has magnitude

This converts between intensity, which is measured, and field, which enters the physics of nonlinear optics, damage and atom-light interaction. An intensity of 1 W/cm² in vacuum corresponds to a peak field of about 2.7 kV/m; direct sunlight, about 1361 W/m² above the atmosphere, to about 1.0 kV/m; and 10¹⁴ W/cm², reached at the focus of amplified femtosecond pulses, to 2.7 × 10¹⁰ V/m, about 5% of the field that binds an electron in a hydrogen atom, 5.1 × 10¹¹ V/m. Field scales as the square root of intensity, so the Kerr index change $n_2 I$ and other nonlinear effects are written equivalently in terms of either.

Where the direction matters

In anisotropic crystals the Poynting vector is not parallel to the wave vector, so a beam's energy walks off from the direction of its wavefronts, an effect that limits nonlinear conversion in birefringent phase matching. In an evanescent wave under total internal reflection, the time-averaged Poynting vector normal to the surface is zero: energy flows along the surface but not across it. In waveguides, integrating the Poynting vector's axial component over the cross-section gives the power in a mode, and the effective area used for nonlinear calculations is $(\int I\,dA)^2/\int I^2\,dA$ over the mode. Light also carries momentum, $\mathbf{S}/c^2$ per unit volume in vacuum, which gives radiation pressure and optical trapping forces.

Measurement

The Poynting vector is not measured directly; a power meter integrates it over the detector area, and a beam profiler maps its time average across a beam. Its direction is inferred from where energy goes, for example from the displacement of a beam in a birefringent crystal.

References: J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Ch. 6 and 7; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 1.