Success Is In The Way You Handle The Unknown

Continuing from the previous post "Photons Like Bubbles", if we invoke Hertz Law,

$F=\cfrac{4}{3}E^{*}a_\psi^{1/2}d^{3/2}$

and assume that the medium presents a constant resistance force to all photons,

$A=a_\psi^{1/2}d^{3/2}$

where $A=\cfrac{3}{4E^{*}}F$ is a constant, we have

There is a new derivation in a later post "Success Is In The Way You Handle The Question".

$n=\cfrac{a_\psi}{a_\psi-d}$

as $n$ is the refractive index of the medium which is also the ratio of the normal components of $\psi$ just before and just after passing into the medium as derived from the previous post.

 $a_\psi(1-\cfrac{1}{n})=d$

Substitute the above in to the expression for $A$,

$A=a_\psi^{2}(1-\cfrac{1}{n})^{3/2}$

$\cfrac { 1 }{ n } =1-\cfrac { A^{ 2/3 } }{ a_{ \psi  }^{ 4/3 } }$

$n=\frac { a_{ \psi  }^{ 4/3 } }{ a_{ \psi  }^{ 4/3 }-B}$  and $\lambda=2\pi a_\psi$

where $B=A^{ 2/3  }$ is a constant.

This is not Sellmeier equation.  Furthermore, from Hertz Law,

$\cfrac{1}{E^{*}}=\cfrac{1-\nu_p^2}{E_p}+\cfrac{1-\nu_m^2}{E_m}$

where $E_p$ and $E_m$ are the elastic modulus of the photon and the medium respectively; $\nu_p$ and $\nu_m$ are the Poisson's ratios; it is not immediately clear in this analogy what exactly they are.  We could have started instead with,

$F=\cfrac{4}{3}E^{*}a_\psi^{1/2}d^{m}$

where $m$ captures the interaction between photons and the medium.  The rest being geometry for the time being.