Refractive index (n)

The refractive index of a medium is

$$n \;=\; \frac{c}{v_p},$$

where $c$ is the speed of light in vacuum and $v_p$ is the phase velocity in the medium. For real materials, $n$ is generally complex: $\tilde n = n' + i n''$. The real part $n'$ describes phase propagation; the imaginary part $n''$ describes absorption via the relation $\alpha = 4\pi n'' / \lambda_0$.

For non-magnetic dielectric materials ($\mu_r = 1$):

$$n \;=\; \sqrt{\epsilon_r},$$

where $\epsilon_r$ is the relative permittivity at optical frequencies.

Typical values at 1550 nm:

Material	$n$
Vacuum	1.000000
Dry air (1 atm, 20 °C)	1.000270
Water	1.318
Fused silica (SiO$_2$)	1.444
Silicon nitride (Si$_3$N$_4$, stoichiometric)	2.00
Aluminum oxide (Al$_2$O$_3$)	1.746
Silicon (Si)	3.476
Indium phosphide (InP)	3.166
Gallium arsenide (GaAs)	3.376
Lithium niobate (LiNbO$_3$, ordinary)	2.211
Germanium (Ge)	4.275
BK7 glass (visible $\lambda$)	1.515
Polymer cladding (typical)	1.45 – 1.55

Wavelength dependence (dispersion). Refractive index varies with wavelength, described empirically by the Sellmeier equation:

$$n^2(\lambda) \;=\; 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i},$$

where $B_i$ and $C_i$ are material-specific Sellmeier coefficients. The wavelength derivative determines the chromatic dispersion of the material.

Temperature dependence. The thermo-optic coefficient $dn/dT$ governs how index changes with temperature:

Material	$dn/dT$ at 1550 nm
Silicon	$+1.86 \times 10^{-4}$ K$^{-1}$
Silicon nitride	$+2.5 \times 10^{-5}$ K$^{-1}$
Fused silica	$+1.1 \times 10^{-5}$ K$^{-1}$
InP	$+2.0 \times 10^{-4}$ K$^{-1}$
Polymer (typical)	$-1$ to $-2 \times 10^{-4}$ K$^{-1}$

Silicon's large positive thermo-optic coefficient drives the thermal sensitivity of silicon photonic resonators ($\sim 80$ pm/K wavelength shift for typical SOI ring resonators) and enables thermo-optic phase shifters as the dominant active element on silicon photonic PICs.

Polymer claddings with negative $dn/dT$ are used in athermal waveguide designs to partially compensate the silicon thermal sensitivity.

Refractive index also depends on carrier density (plasma dispersion effect in semiconductors), electric field (electro-optic effect), optical intensity (Kerr effect), and mechanical strain (photoelastic effect) — these are the basis of essentially all active optical modulation.