Even Light Can Bend

From the post "Success Is In The Way You Handle The Question" dated 13 Jun 2014,

$n=1+\cfrac { C }{ \lambda^{ 4/3 }_{o} }$

A sample plot with $C=1$, n=1+1/x^(4/3) is given below,

It is possible to manipulate the expression $n(\lambda)$,

$n=1+\cfrac { C }{ \lambda ^{ 4/3 }_{ o } }$

$n^{ 2 }=1+\cfrac { C^{ 2 } }{ \lambda ^{ 8/3 }_{ o } } +2\cfrac { C }{ \lambda ^{ 4/3 }_{ o } }$

Consider,

$2\cfrac { C }{ \lambda _{ o }^{ 4/3 } } =\cfrac { 1 }{ (\lambda _{ o }^{ 2 })^{ 2/3 } } =2\cfrac { C }{ (\lambda ^{ 2 }-D_{ 1 })^{ 2/3 } } =2\cfrac { C(\lambda ^{ 2 }-D_{ 1 })^{ 1/3 } }{ \lambda ^{ 2 }-D_{ 1 } }$

when we consider the Taylor expansion of ,

$(\lambda^2-D_1)^{1/3}$ about $\lambda=0$,

$(\lambda^2-D_1)^{1/3}=(-1)^{1/3}D_1^{1/3}-\cfrac{(-1)^{1/3}}{3D_1^{2/3}}\lambda^2-\cfrac{(-1)^{1/3}}{9D_1^{5/3}}\lambda^4+...$

Ignoring higher terms of $\lambda^4$,

$2\cfrac { C }{ \lambda _{ o }^{ 4/3 } } =\cfrac { B_1\lambda^2 }{ \lambda ^{ 2 }-D_{ 1 } }+\cfrac{A_1}{\lambda ^{ 2 }-D_{ 1 }}$

Consider,

$\cfrac { C^{ 2 } }{ \lambda ^{ 8/3 }_{ o } } =\cfrac { C^{ 2 } }{ (\lambda _{ o }^{ 2 })^{ 4/3 } } =\cfrac { C^{ 2 } }{ (\lambda ^{ 2 }-D_{ 2 })^{ 4/3 } } =\cfrac { C^{ 2 }(\lambda ^{ 2 }-D_{ 2 })^{ -1/3 } }{ \lambda ^{ 2 }-D_{ 2 } }$

The Taylor expansion of

$(\lambda^2-D_2)^{-1/3}$ about $\lambda=0$,

$(\lambda^2-D_2)^{-1/3}=-(-1)^{2/3}D_2^{-1/3}-\cfrac{(-1)^{2/3}}{3D_2^{4/3}}\lambda^2-2\cfrac{(-1)^{2/3}}{9D_2^{7/3}}\lambda^4+...$

Ignoring higher terms of $\lambda^4$,

$\cfrac { C^{ 2 } }{ \lambda ^{ 8/3 }_{ o } } =\cfrac { B_2\lambda^2 }{ \lambda ^{ 2 }-D_{ 2} }-\cfrac{A_2}{\lambda ^{ 2 }-D_{ 2 }}$

When,

$D_{ 1 }=C_{ 1 }$

$D_{ 2 }=C_{ 2 }$

and if we consider only the real parts, we obtain,

$n^{ 2 }=1+\cfrac { B_{ 1 }\lambda^2 }{ \lambda ^{ 2 }-C_{ 1 } } +\cfrac { B_{ 2 }\lambda^2 }{ \lambda _{ o }^{ 2 }-C_{ 2 } }+Err$

where $Err$ is an error term.

If we let,

$Err=\cfrac { B_{ 3}\lambda^2 }{ \lambda ^{ 2 }-C_{3 } } =\cfrac{A_1}{\lambda ^{ 2 }-C_{ 1 }}-\cfrac{A_2}{\lambda ^{ 2 }-C_{ 2 }}+expansion\,\,error$

because,

$\cfrac { A_{ 1 } }{ \lambda ^{ 2 }-C_{ 1 } } -\cfrac { A_{ 2 } }{ \lambda ^{ 2 }-C_{ 2 } } =$

$\cfrac { A_{ 1 }(\lambda ^{ 2 }-C_{ 2 })-A_{ 2 }(\lambda ^{ 2 }-C_{ 1 }) }{ (\lambda ^{ 2 }-C_{ 1 })(\lambda ^{ 2 }-C_{ 2 }) } =\cfrac { (A_{ 1 }-A_{ 2 })\lambda ^{ 2 }-(A_1C_{2 }-A_2C_{ 1 }) }{ \lambda ^{ 4 }-(C_{ 1 }+C_{ 2 })\lambda ^{ 2 }+C_{ 1 }C_{ 2 } }$

Ignoring $\lambda^4$,

$\cfrac { \cfrac { (A_{ 2 }-A_{ 1 }) }{ (C_{ 1 }+C_{ 2 }) } \lambda ^{ 2 }-\cfrac { A_2C_{1}-A_1C_{ 2 } }{ C_{ 1 }+C_{ 2 } }  }{ \lambda ^{ 2 }-\cfrac { C_{ 1 }C_{ 2 } }{ C_{ 1 }+C_{ 2 } }  } \approx \cfrac { B_{ 3 }\lambda ^{ 2 } }{ \lambda ^{ 2 }-C_{ 3 } }$

which is valid when $(A_2C_1-A_1C_2)/(C_1+C_2)$ is small compared to $n^2$ and $B_3\gt\gt A_2C_1-A_1C_2$.

Quite arbitrarily,

$n^{ 2 }=1+\cfrac { B_{ 1 } \lambda^2}{ \lambda ^{ 2 }-C_{ 1 } } +\cfrac { B_{ 2 }\lambda^2 }{ \lambda ^{ 2 }-C_{ 2 } }+\cfrac { B_{ 3}\lambda^2 }{ \lambda ^{ 2 }-C_{3 } }$

which is Sellmeier experimental fit for $n$.

Even light can bend; given a dose of insanity everything else bends.  Obviously insanity is in $Err$.