\(n=1+\cfrac { C }{ \lambda^{ 4/3 }_{o} }\)
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 \(\lambda^4\),
\( \cfrac { C^{ 2 } }{ \lambda ^{ 8/3 }_{ o } } =\cfrac { B_2\lambda^2 }{ \lambda ^{ 2 }-D_{ 2} }-\cfrac{A_2}{\lambda ^{ 2 }-D_{ 2 }}\)
\( D_{ 1 }=C_{ 1 }\)
\( D_{ 2 }=C_{ 2 }\)
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 \(Err\).
