Question 9.4: CAUCHY EQUATION AND DIAMOND Using the Cauchy coefficients fo...
CAUCHY EQUATION AND DIAMOND Using the Cauchy coefficients for diamond in Table 9.2, calculate the refractive index at 550 nm.
At λ = 550 nm, the photon energy is
h f=\frac{h c}{\lambda}=\frac{\left(6.62 \times 10^{-34} J s \right)\left(3 \times 10^{8} m s ^{-1}\right)}{550 \times 10^{-9} m } \times \frac{1}{1.6 \times 10^{-19} J eV ^{-1}}=2.254 eV
Using the Cauchy dispersion relation for diamond with coefficients from Table 9.2,
\begin{aligned} n &=n_{-2}(h f)^{-2}+n_{0}+n_{2}(h f)^{2}+n_{4}(h f)^{4} \\ &=\left(-1.07 \times 10^{-5}\right)(2.254)^{-2}+2.378+\left(8.01 \times 10^{-3}\right)(2.254)^{2}+\left(1.04 \times 10^{-4}\right)(2.254)^{4} \\ &=2.421 \end{aligned}
The difference in n from the value in Example 9.3 is 0.08 percent, and is due to the Cauchy coefficients quoted in Table 9.2 being applicable over a wider wavelength range at the expense of some accuracy.
| Table 9.2 Sellmeier and Cauchy coefficients | ||||||
| Sellmeier | ||||||
| A_{1} | A_{2} | A_{3} | \lambda _{1} (\mu m) | \lambda _{2} (\mu m) | \lambda _{3} (\mu m) | |
| SiO_{2} (fused silica) | 0.6967490 | 0.4082180 | 0.8908150 | 0.0690660 | 0.1156620 | 9.9005590 |
| 86.5% SiO_{2}–13.5% GeO_{2} |
0.711040 | 0.451885 | 0.704048 | 0.0642700 | 0.129408 | 9.425478 |
| GeO_{2} | 0.80686642 | 0.71815848 | 0.85416831 | 0.068972606 | 0.15396605 | 11.841931 |
| Sapphire | 1.023798 | 1.058264 | 5.280792 | 0.0614482 | 0.110700 | 17.92656 |
| Diamond | 0.3306 | 4.3356 | 0.1750 | 0.1060 | ||
| Cauch | ||||||
| Range of hf (eV) | n_{−2} (eV^{2} ) | n_{0} | n_{2} (eV^{−2}) | n_{4} (eV^{−4}) | ||
| Diamond | 0.05–5.47 | −1.07 × 10^{−5} | 2.378 | 8.01 × 10^{−3} | 1.04 × 10^{−4} | |
| Silicon | 0.002–1.08 | −2.04 × 10^{−8} | 3.4189 | 8.15 × 10^{−2} | 1.25 × 10^{−2} | |
| Germanium | 0.002–0.75 | −1.0 × 10^{−8} | 4.003 | 2.2 × 10^{−1} | 1.4 × 10^{−1} | |