I am studying evanescent field and diffraction limit and I have one question.
Given a field $ U(x,y,0)$ we can decompose into 2D plane waves.
$U(x,y,0)= \int \int dk_x dk_y \tilde{U}(k_x,k_y) e^{+i(k_x x + k_yy)}$
if we want to have $U(x,y,z_0)$ we can propagate each wave to the plane $z_0$ in this way.
$U(x,y,z_0)= \int \int dk_x dk_y \tilde{U}(k_x,k_y) e^{+i(k_x x + k_yy)} e^{i k_z z_0}$
This is clear to me.
It is not clear why $k_z = \sqrt{ (\frac{2\pi}{\lambda})^2 - k_x^2 -k_y^2 }$.
Why suddenly the plane waves should be monochromatic with wavelength $\lambda$ ?