Question 9.7: REFRACTIVE AND GROUP INDEX OF Si AT 1550 nm Silicon photonic...
REFRACTIVE AND GROUP INDEX OF Si AT 1550 nm Silicon photonic endeavors to integrate various photonic functionalities such as light guiding, light modulation, detection, etc., into the silicon microelectronics. Calculate the refractive and group index of Si at 1550 nm, one of the main communication wavelengths.
The 1550 nm wavelength is equivalent to a photon energy in eV of
h f=h c / \lambda=\left(6.626 \times 10^{-34}\right)\left(3 \times 10^{8}\right) /\left(1550 \times 10^{-9}\right)\left(1.602 \times 10^{-19}\right)=0.800 eV
Using the Cauchy dispersion relation for Si 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(-2.04 \times 10^{-8}\right)(0.800)^{-2}+3.4189+\left(8.15 \times 10^{-2}\right)(0.800)^{2}+\left(1.25 \times 10^{-2}\right)(0.800)^{4} \\ &=3.4711 \end{aligned}
We can obtain the group index through Equation 9.23. We can change this equation from wavelength λ dependence to photon energy hf dependence by using h f=h c / \lambda . From straightforward calculus, the result is
N_{g}=n-\lambda \frac{d n}{d \lambda}=n+(h f) \frac{d n}{d(h f)}
Differentiating the Cauchy relation and substituting it into the above, we obtain
N_{g}=-n_{-2}(h f)^{-2}+n_{0}+3 n_{2}(h f)^{2}+5 n_{4}(h f)^{4}
Substituting hf = 0.800 eV we find
\begin{aligned} N_{g} &=-\left(-2.04 \times 10^{-8}\right)(0.800)^{-2}+3.4189+3\left(8.15 \times 10^{-2}\right)(0.800)^{2}+5\left(1.25 \times 10^{-2}\right)(0.800)^{4} \\ &=3.5756 \end{aligned}
N_{g} is about 3 percent smaller than n. Sometimes the empirical expression for n is not as easy to differentiate analytically as above, in which case we can simply find N_{g} by numerically differentiating n by calculating n and n + \delta n at two very closely spaced wavelengths λ and λ + δλ.
| 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} | |