Photons Like Bubbles

For a photon we know that $\psi$ varies along a direction perpendicular to $x$ which is at light speed,

If there is no loss in $\psi$ after the impact at the point of incidence, ie $\psi$ is a constant then Snell's Law implies that the normal components of $\psi$ before and after the impact is related by

$\cfrac{\psi_{n\,i}}{\psi_{n\,r}}=\cfrac{sin\,\theta}{sin\,\alpha}=n$

that this component of $\psi$, $\psi_n$ has been reduced by a factor that is the refractive index of the material, $n$.  A simultaneous increase in the slide component $\psi_s=\psi cos\,\alpha$ along the surface of the intercepting medium, ensure that the total $\psi$ is conserved (ie. a constant) pass after the inter-surface.

$sin\theta =nsin\alpha$

$cos^{ 2 }\theta =1-n^{ 2 }sin^{ 2 }\alpha$

$sin^{ 2 }\alpha =\cfrac { sin^{ 2 }\theta  }{ n^{ 2 } }$

$cos^{ 2 }\alpha =1-\cfrac { sin^{ 2 }\theta  }{ n^{ 2 } }$

$\cfrac { cos^{ 2 }\theta  }{ cos^{ 2 }\alpha  } =\cfrac { n^{ 2 }\left( 1-sin^{ 2 }\theta  \right)  }{ n^{ 2 }-sin^{ 2 }\theta  }$

$\cfrac{\psi_{s\,i}}{\psi_{s\,r}}=\sqrt{\cfrac { 1-sin^{ 2 }\theta  }{ 1-\cfrac { sin^{ 2 }\theta  }{ n^{ 2 } }  } }$

The increase in the slide component $\psi_s$, depends on the incident angle $\theta$ and $n$.

Graphically, $\psi$ is squashed on impact.  $\psi$ is shorten perpendicular to the surface of the medium and is lengthened along the surface of the medium.

Total $\psi$ remains constant.

A changing refractive index $n$ with different wavelength of light suggests that different colored $\psi$ are deformed differently.   $\psi$ is a standing wave centered at the particle.  It has a wavelength, $\lambda_\psi$ associated with it.  At the fundamental oscillation, where there is one complete wave around a radius of $a_\psi$,

$2\pi a_\psi=\lambda_\psi$

As such, the physical size of $\psi$,

$a_\psi=\cfrac{1}{2\pi}\lambda_\psi$

is related to the wavelength of  $\psi$ and consequently the frequency of $\psi$.

Putting these facts together, they suggest that the deformation in $\psi$ depends on the initial size of $\psi$ before the impact.  So much so, refractive index $n$, depends on the size of $\psi$ and so the wavelength of $\psi$.  This way, we relate the wavelength of $\psi$ with the wavelength of light as perceived, directly.

Considering both Cauchy's and Sellmeier equations with the experimental plots of refractive index vs wavelength in general,  such deformations in $\psi$ decrease with increasing $a_\psi$.

I like bubbles too.