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Dispersion Formulas

Each optical material (glasses, plastics, gases) have a different refraction index for each wavelength.

Instead of keeping long tables, it’s possible to describe the behaviour of optical materials by formulas.

here are the main formulas used :

1: Sellmeier (preferred)
n^2-1=C_1 + \frac{C_2 \lambda^2}{\lambda^2-C_3^2} + \frac{C_4 \lambda^2}{\lambda^2-C_5^2} + \frac{C_6 \lambda^2}{\lambda^2-C_7^2} + \frac{C_8 \lambda^2}{\lambda^2-C_9^2} + \frac{C_{10} \lambda^2}{\lambda^2-C_{11}^2} + \frac{C_{12} \lambda^2}{\lambda^2-C_{13}^2} + \frac{C_{14} \lambda^2}{\lambda^2-C_{15}^2} + \frac{C_{16} \lambda^2}{\lambda^2-C_{17}^2}

2: Sellmeier-2
n^2-1=C_1 + \frac{C_2 \lambda^2}{\lambda^2-C_3} + \frac{C_4 \lambda^2}{\lambda^2-C_5} + \frac{C_6 \lambda^2}{\lambda^2-C_7} + \frac{C_8 \lambda^2}{\lambda^2-C_9} + \frac{C_{10} \lambda^2}{\lambda^2-C_{11}} + \frac{C_{12} \lambda^2}{\lambda^2-C_{13}} + \frac{C_{14} \lambda^2}{\lambda^2-C_{15}} + \frac{C_{16} \lambda^2}{\lambda^2-C_{17}}

3: Polynomial
n^2 = C_1 + C_2 \lambda^{C_3} + C_4 \lambda^{C_5} + C_6 \lambda^{C_7} + C_8 \lambda^{C_9} + C_{10} \lambda^{C_{11}} + C_{12} \lambda^{C_{13}} + C_{14} \lambda^{C_{15}} + C_{16} \lambda^{C_{17}}

4: RefractiveIndex.info
n^2 = C_1 + \frac{C_2 \lambda^{C_3}}{\lambda^2-{C_4}^{C_5}} + \frac{C_6 \lambda^{C_7}}{\lambda^2-{C_8}^{C_9}} + C_{10} \lambda^{C_{11}} + C_{12} \lambda^{C_{13}} + C_{14} \lambda^{C_{15}} + C_{16} \lambda^{C_{17}}

5: Cauchy
n = C_1 + C_2 \lambda^{C_3} + C_4 \lambda^{C_5} + C_6 \lambda^{C_7} + C_8 \lambda^{C_9} + C_{10} \lambda^{C_{11}}

6: Gases
n-1 = C_1 + \frac{C_2}{C_3-\lambda^{-2}} + \frac{C_4}{C_5-\lambda^{-2}} + \frac{C_6}{C_7-\lambda^{-2}} + \frac{C_8}{C_9-\lambda^{-2}} + \frac{C_{10}}{C_{11}-\lambda^{-2}}

7: Herzberger
n = C_1 + \frac{C_2}{\lambda^2-0.028} + C_3 (\frac{1}{\lambda^2-0.028})^2 + C_4 \lambda^2 + C_5 \lambda^4 + C_6 \lambda^6

8: Retro
\frac{n^2-1}{n^2+2} = C_1 + \frac{C_2 \lambda^2}{\lambda^2-C_3} + C_4 \lambda^2

9: Exotic
n^2 = C_1 + \frac{C_2}{\lambda^2-C_3} + \frac{C_4 (\lambda-C_5)}{(\lambda-C_5)^2 + C_6}

refractive index

refractive index = \frac{speed\ of\ light\ in\ vacuum}{speed\ of\ light\ in\ the\ current\ medium\ (eg.\ glass)} =: \frac{c}{v}

With c = 2.99792458 \cdot 10^8 \frac{m}{s} , roughly 300000km per second

In other media than vacuum the light is slower. Therefore, the smallest refraction index is 1.
The speed of light in the medium is v = \frac{c}{v}.
Therefore the speed of light in a medium is factor “refractive Index” slower than the speed of light c in vacuum.
Glass slows down light of different wavelengths (“colors”) by different factors.
Results are different refractive indices, “depending on the color (wavelength) of light .”
The higher the refraction index, the stronger the change of direction at the boundary between different media
As media have different refraction index for blue, green red light, red/green/blue light takes a different path through a lens!

[table]Medium,typical refractive index
Vacuum,1
Air,1.000293
Helium,1.000036
Hydrogen,1.000132
Carbon dioxide,1.00045
Water at 20 °C,1.333
Ethanol at 20 °C,1.36
Olive oil at 20 °C,1.47
Ice,1.31
PMMA (= acrylic = plexiglas),1.49
Window glass,1.52
Polycarbonate (Lexan™),1.58
Flint glass (typical),1.62
Sapphire,1.77
Cubic zirconia,2.15
Diamond,2.42
Moissanite,2.65, -[/table]

After a reflection light propagates from right to left, and it’s velocity can be regarded as negative. Using velocity instead of speed in the above equation, the index of refraction can also be regarded as negative.