Independent fields' components in Maxwell's equations

[EmilyRuck]

In a source-free, isotropic, linear medium, Maxwell's equations can be rewritten as follows:

\nabla \cdot \mathbf{E} = 0
\nabla \cdot \mathbf{H} = 0
\nabla \times \mathbf{E} = -j \omega \mu \mathbf{H}
\nabla \times \mathbf{E} = j \omega \epsilon \mathbf{E}

If we are looking for a wave solution, traveling along the z direction, with k = k_z = \beta, that means

\displaystyle \frac{\partial}{\partial z} = e^{-j \beta z}

and the above equations (after some steps) become\displaystyle \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} = j \beta E_z
\displaystyle \frac{\partial H_x}{\partial x} + \frac{\partial H_y}{\partial y} = j \beta H_z

E_x = - j \displaystyle \frac{1}{\omega \epsilon} \frac{\partial H_z}{\partial y} + \displaystyle \frac{\beta}{\omega \epsilon} H_y
E_y = - \displaystyle \frac{\beta}{\omega \epsilon} H_x + j \displaystyle \frac{1}{\omega \epsilon} \frac{\partial H_z}{\partial x}
E_z = - j \displaystyle \frac{1}{\omega \epsilon} \frac{\partial H_y}{\partial x} + j \displaystyle \frac{1}{\omega \epsilon} \frac{\partial H_x}{\partial y}

H_x = j \displaystyle \frac{1}{\omega \mu} \frac{\partial E_z}{\partial y} - \displaystyle \frac{\beta}{\omega \mu} E_y
H_y = \displaystyle \frac{\beta}{\omega \mu} E_x - j \displaystyle \frac{1}{\omega \mu} \frac{\partial E_z}{\partial x}
H_z = j \displaystyle \frac{1}{\omega \mu} \frac{\partial E_y}{\partial x} - j \displaystyle \frac{1}{\omega \mu} \frac{\partial E_x}{\partial y}

It should be possible to express the transverse field components as functions of the longitudinal field components:

E_x = E_x (E_z, H_z)
E_y = E_y (E_z, H_z)
H_x = H_x (E_z, H_z)
H_y = H_y (E_z, H_z)

Which is equivalent to state that just two scalar functions E_z = f(x,y)e^{-j \beta z} and H_z = g(x,y)e^{-j \beta z} are actually independent. But how could it be proved? It is not evident from the equations I wrote above: they show instead that E_z, H_z appear to be functions of E_x, E_y, H_x, H_y.
The only link I could find is http://my.ece.ucsb.edu/York/Bobsclass/201B/W01/potentials.pdf: at the bottom of page 11, it shows that all the fields can be expressed in terms of two scalar functions. But it is not a direct approach, because it uses the Hertz Vector potentials.

Is there a direct approach to prove that all the 6 field components are function of just 2 of them?

 

[DuckAmuck]

You chose z as the propagation axis, so the fields along that axis won't behave like those in x and y.

 

[EmilyRuck]

You chose z as the propagation axis, so the fields along that axis won't behave like those in x and y.

This is reasonable. But why?
Anyway, there is a direct approach to express E_x, E_y, H_x, H_y as functions of E_z, H_z only.
For example, let's combine the first and the fifth equations in order to cancel H_y: we will obtain

E_x = -j \displaystyle \frac{1}{k^2 - \beta^2} \left( \omega \mu \frac{\partial H_z}{\partial y} + \beta \frac{\partial E_z}{\partial x} \right)

Where k^2 = \omega^2 \mu \epsilon. A similar result for E_y can be obtained from equations 2 and 4. The same is for the magnetic field components.