Prism Light Beam Dispersion

When white light enters a prism, each wavelength experiences a slightly different refractive index, causing the light to separate into its constituent colors. This phenomenon is called chromatic dispersion.

Visible Light Wavelength Ranges

Color	Wavelength Range
Violet	380 – 450 nm
Blue	450 – 495 nm
Green	495 – 570 nm
Yellow	570 – 590 nm
Orange	590 – 620 nm
Red	620 – 750 nm

Dispersion Equations

Cauchy's Dispersion Formula relates refractive index to wavelength:

$n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4}$

where $A$ , $B$ , and $C$ are material-specific Cauchy coefficients.

Sellmeier Equation provides a more accurate dispersion model:

$n^2(\lambda) = 1 + \sum_{i} \frac{B_i \lambda^2}{\lambda^2 - C_i}$

Angular Dispersion describes how the deviation angle changes with wavelength:

$\frac{d\delta}{d\lambda} = \frac{d\delta}{dn} \cdot \frac{dn}{d\lambda}$

For a prism at minimum deviation:

$\frac{d\delta}{dn} = \frac{2 \sin(A/2)}{\sqrt{n^2 - \sin^2(A/2)}}$

Abbe Number (reciprocal relative dispersion):

$V_d = \frac{n_d - 1}{n_F - n_C}$

where $n_d$ , $n_F$ , and $n_C$ are refractive indices at the helium d-line (587.6 nm), hydrogen F-line (486.1 nm), and hydrogen C-line (656.3 nm) respectively.

Key Properties

Short wavelengths (blue/violet) experience higher refractive indices and greater deviation
Long wavelengths (red) experience lower refractive indices and less deviation
Materials with low Abbe numbers ( $V_d < 50$ ) exhibit strong dispersion (crown glass)
Materials with high Abbe numbers ( $V_d > 50$ ) exhibit weak dispersion (flint glass)

References

Original Source