Nonlinear transformation optics and engineering of the Kerr effect

The concept of transformation optics is extended to nonlinear electrodynamics. It is shown that transformation optics favors implicit constitutive relations in terms of energy densities $\mathit{D}·\mathit{E}$ and $\mathit{B}·\mathit{H}$ rather than ${\mathit{E}}^2$ and ${\mathit{H}}^2$. The Kerr nonlinearity is studied in detail and the transformation-optics-based engineering of self-interaction effects is discussed. As a specific example we introduce transformation optics applied on a self-focusing field configuration.

Figure 1

Illustration of the laser beam in free space (A) and in a Kerr medium without transformation (B). Vertical lines indicate the region where the transformation will be applied in the second step; in this plot they are presented for illustrative purposes only.

Figure 2

The effect of transformations with $\alpha =2$ (A) and $\alpha =0.25$ (B). The transformation is applied to the central region indicated by the two vertical lines only; the left and right parts remain unchanged compared to the Kerr medium in Fig. 1.

Figure 3

(A) The untransformed Kerr medium with $\chi =1.9\times {10}^{-9}\frac{\text{Cm}}{{\text{V}}^3}$. (B) $\alpha =0.01$ is chosen in the central region indicated by the two vertical lines, such that the focal point freezes and builds a concentrated flux tube.