# Poynting Vector

Definition: a vector indicating the strength and direction of energy flow of an electromagnetic wave

German: Poynting-Vektor

Formula symbol: <$S$>

Author: Dr. RĂ¼diger Paschotta

The *Poynting vector* has been introduced into the theory of electromagnetism by John Henry Poynting in 1884 and used it e.g. in conjunction with *Poynting's theorem* concerning the conservation of energy. It is defined as the cross product of electric field vector <$E$> and magnetic field vector <$H$>:

In its *microscopic* version, not using any model of material data, the Poynting vector can be written as

with the magnetic flux density <$B$>. This version can be considered to be most general, as it is not based on any material model which would introduce additional assumptions.

The presented vectors are all real and time-dependent vectors. Within each cycle, the magnitude of the Poynting vector oscillates between zero and twice the maximum value. In many cases, the Poynting vector is averaged over a full optical oscillation cycle for monochromatic light, which delivers half of the peak value.

The Poynting vector can be shown to indicate the direction of energy flow of an electro-magnetic wave, and with its magnitude also the optical intensity, meaning the radiant flux per unit area (→ radiometry). The time-averaged version rather than the oscillating version is often associated with optical intensity.

A similar equation may be set up for phasors (complex vectors), but with an additional constant factor (depending on the exact definition of phasors). Such a Poynting vector will generally be complex. The net energy flow is then indicated by its real part, while its imaginary part is related to reactive power (oscillating back and forth during each optical cycle).

Curiously, the Poynting vector can be non-zero even for static electromagnetic fields.

## Poynting Vector in Optics

The Poynting vector is a general concept in electromagnetism, and is often applied in optics and photonics, in addition to other fields such as radio waves.

In a homogeneous and isotropic optical material (or in vacuum), the Poynting vector is always perpendicular to the wavefronts of light, i.e., directed parallel to the wave vector. In birefringent media, however, it can be somewhat tilted against the wave vector, which implies some spatial walk-off. This phenomenon is relevant, for example, for nonlinear frequency conversion with critical phase matching in nonlinear crystal materials.

For total internal reflection at an optical interface, the Poynting vector has no component in the direction of that interface.

### Bibliography

[1] | J. H. Poynting, “On the transfer of energy in the electromagnetic field”, Philosophical Transactions of the Royal Society of London. 175, 343 (1884), DOI:10.1098/rstl.1884.0016 |

See also: optical intensity, wavefronts, spatial walk-off

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