From the post "Wave Front and Wave Back" dated 18 May 2014, a photon was conceptualized as a particle in helical motion,
\(\cfrac{x_{v1}}{cos(\alpha_1)}=\cfrac{x_{v2}}{cos(\alpha_2)}\)
\(x_{v2}=x_{v1}\cfrac{cos(\alpha_2)}{cos(\alpha_1)}\)
where \(x_{v1}\), \(x_{v2}\) are the radii of circular motion in medium 1 and 2 respectively.
and
\(\cfrac{\lambda}{n_1sin(\alpha_1)}=\cfrac{\lambda}{n_2sin(\alpha_2)}\)
\(sin(\alpha_2)=\cfrac{n_1}{n_2}sin(\alpha_1)\)
So,
\(x_{v2}=\cfrac{x_{v1}}{cos(\alpha_1)}\sqrt{1-\left(\cfrac{n_1}{n_2}\right)^2sin^2(\alpha_1)}\)
when the particle enters into to less dense medium,
\(n_2\lt n_1\)
\(1-\left(\cfrac{n_1}{n_2}\right)^2sin^2(\alpha_1)\lt 0\)
in which case, \(x_{v2}\) is complex and is rotated by \(90^o\) clockwise at the point of ncident,
\(x_{v2}=i.\cfrac{x_{v1}}{cos(\alpha_1)}\sqrt{\left|1-\left(\cfrac{n_1}{n_2}\right)^2sin^2(\alpha_1)\right|}\)
and \(\alpha_2\) is totally internally reflected. When
\(1-\left(\cfrac{n_1}{n_2}\right)^2sin^2(\alpha_1)=0\)
\(sin(\alpha_1)=sin(\alpha_c)=\cfrac{n_2}{n_1}\)
where \(\alpha_c\) is the critical angle. Unfortunately, the formula is valid only up to \(\alpha_c\). For incident angle greater than \(\alpha_c\), we know that the ray is reflected,
\(x_{v2}={x_{v1}}\)
\(1=\cfrac{cos(\alpha_2)}{cos(\alpha_1)}\)
\(\alpha_1=\alpha_2\)
both angles measured from the normal on medium \(n_1\).
This derivation for total internal reflection considers the relative speeds of the particle in the two mediums alone; \(B\) fields are not involved. Since, both loops are perpendicular to the ray \(\alpha\) only in the limiting case of \(\theta\to90^o\), the following adjustments are necessary to the values of \(\alpha\) for each of the loop as illustrated,
\(\alpha_{adj}=\alpha+90^o-\theta\)
and
\(\alpha_{adj}=\alpha-90^o+\theta\)
which indicate that the two loops can be separated (circular polarization\(\to\)linear polarization) when,
since \(\alpha\lt90^o\)
\(\alpha-90^o+\theta\lt\alpha_c\)
\(\alpha\lt\alpha_c+90^o-\theta\)
and
\(\alpha+90^o-\theta\gt\alpha_c\)
\(\alpha\gt\alpha_c-90^o+\theta\)
where \(\alpha_{2}^{'}\) has been totally internally reflected. When \(\theta\to90^o\), the range of \(\alpha\) collapses to a single value \(\alpha_c\), as \(\alpha_{adj}\to\alpha\).
