\nabla \times {\mathbf  {E}}={\hat  {{\mathbf  {k}}}}\times {\mathbf  {E}}_{0}f'\left({\hat  {{\mathbf  {k}}}}\cdot {\mathbf  {x}}-c_{0}t\right)=-{\frac  {\partial }{\partial t}}{\mathbf  {B}}