Maximum Electric/Magnetic field in vacuum

The photoelectric effect suggests that the energy of electromagnetic field for monochromatic light should be a function of the frequency \(\omega\). Since waves are not generally monochromatic, to generalize this, we look for an expression of \(\omega\) in terms of the electromagnetic field (electric/magnetic) itself: since $$\textbf{E}(z,t)=E_0\sin(kz-\omega t)\hat{\textbf{x}}, \ \ \textbf{B}(z,t)=\frac{1}{c}E_0\cos(kz-\omega t)\hat{\textbf{y}},$$ we find $$\omega=\frac{1}{E_0}\frac{\Arrowvert\frac{\partial\textbf{E}}{\partial t}\Arrowvert}{\sqrt{1-(\frac{E}{E_0})^2}},$$

thus $$u\propto\Arrowvert\frac{\partial\textbf{E}}{\partial t}\Arrowvert,$$ instead of $$u\propto E^2.$$

The occurrence of \(E_0\) is an interesting problem. There is nothing left in the theory to fix this constant, while it is the maximum of electric/magnetic fields in vacuum. On the other hand, the factor $$\frac{1}{\sqrt{1-(\frac{E}{E_0})^2}}$$ from the photoelectric effect suggests that as the first step towards unifying the two expressions, we take $$u=\frac{\epsilon_0 E_0^2}{\sqrt{1-(\frac{E}{E_0})^2}}\approx \epsilon_0 E_0^2+\frac{1}{2}\epsilon_0 E^2,$$ as the corrected expression of electromagnetic energy density.