structural:scholar_edition
Computer vision and optics

The concept of a tensor of order two is often conflated with that of a matrix. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix.

The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:

Here χ (1) {\displaystyle \chi ^{(1)}} is the linear susceptibility, χ (2) {\displaystyle \chi ^{(2)}} gives the Pockels effect and second harmonic generation, and χ (3) {\displaystyle \chi ^{(3)}} gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter.