Refractive Index and the Sellmeier Equation: Why Light Slows Down in Glass

Light travels at 299,792,458 meters per second in a vacuum. In glass, it slows to about 197,000,000 m/s. In diamond, it crawls at 124,000,000 m/s. The ratio of the vacuum speed to the speed inside a material is the refractive index, and it's the single most important number in optical engineering.

What Refractive Index Means

The refractive index n of a material is the ratio of the speed of light in vacuum to the speed in that material:

where $c=2.998\times {10}^8$ m/s is the vacuum speed and v is the phase velocity inside the material. A higher n means light moves slower. Some examples at 550 nm (green light):

When light crosses an interface between two materials with different refractive indices, it bends. This is refraction, described by Snell's law: $n_1\sin {\theta }_1=n_2\sin {\theta }_2$. A lens works by curving the interface so that refraction focuses light to a point. A thin-film coating works by choosing layer materials with specific n values so that reflections from each interface interfere constructively or destructively.

Why does light slow down? It doesn't literally slow down between atoms. The incoming electromagnetic wave drives the electrons in each atom to oscillate, and those oscillating electrons re-radiate secondary waves. The superposition of the original wave and all the re-radiated waves produces a combined wave that travels slower than c. You can think of each atom as a tiny antenna that receives the wave, delays it slightly, and re-transmits. In a dense material with many atoms per unit volume, the cumulative delay is larger, giving a higher n. The key physics: the electrons respond more strongly when the light frequency is close to a natural resonance frequency of the atom. Near a resonance, the delay is large and n increases sharply. This frequency dependence is the origin of dispersion.

Dispersion: Why n Changes with Wavelength

The refractive index of every material depends on wavelength. This is called dispersion. For transparent materials in the visible range (far from any absorption resonance), n decreases with increasing wavelength. Blue light (400 nm) has a higher n than red light (700 nm). This is normal dispersion, and it's why a prism separates white light into a rainbow: blue light bends more.

For BK7 glass:

The total change across the visible spectrum is Δn = 0.018 (from 1.531 at 400 nm to 1.513 at 700 nm). That seems small, but it has real consequences. A quarter-wave SiO₂ layer designed for 550 nm is 38% thicker than a quarter-wave at 400 nm (because both n and the target wavelength change). This is why no single-layer coating works perfectly across the full visible spectrum, and why broadband coating design is an optimization problem.

Optical engineers quantify dispersion with the Abbe number:

where $n_d$, $n_F$, and $n_C$ are the refractive indices at the d-line (587.6 nm), F-line (486.1 nm), and C-line (656.3 nm). BK7 has $V_d$ = 64.1, meaning relatively low dispersion. Dense flint glasses like SF11 have $V_d$ around 25, meaning high dispersion. Crown glasses have high Abbe numbers (low dispersion); flint glasses have low Abbe numbers (high dispersion).

The Sellmeier Equation

The standard mathematical model for refractive index dispersion is the Sellmeier equation. It comes directly from the resonance model of atomic electrons: each term in the equation corresponds to an absorption resonance of the material, and the overall n(λ) is the sum of contributions from all resonances.

The three-term Sellmeier form is:

where λ is the wavelength (in micrometers), $B_1,B_2,B_3$ are the oscillator strengths, and $C_1,C_2,C_3$ are related to the resonance wavelengths ($C_i={\lambda }_i^2$, where ${\lambda }_i$ is the resonance wavelength of the i-th oscillator).

For BK7 glass (Schott catalog):

The first two terms ($C_1$ and $C_2$) correspond to UV absorption resonances at ${\lambda }_1=\sqrt{C_1}$ = 0.0775 µm (77.5 nm) and ${\lambda }_2=\sqrt{C_2}$ = 0.1415 µm (141.5 nm). These are electronic transitions in the UV. The third term ($C_3$) corresponds to an infrared resonance at ${\lambda }_3=\sqrt{C_3}$ = 10.18 µm, which is a lattice vibration (phonon) absorption. Between these resonances (roughly 0.3 to 2.5 µm), the Sellmeier formula is accurate to better than 10⁻⁵ in n.

Worked Example: BK7 at 632.8 nm

What is nfor BK7 at 632.8 nm (HeNe laser)? λ = 0.6328 µm, so λ² = 0.40044.

The first two terms (UV resonances) contribute nearly all the refractive index. The IR term is small and negative, slightly reducing n. You can verify this in the Photizon material database: look up BK7, enter 632.8 nm, and the quick lookup gives n = 1.5151.

An older model, the Cauchy equation ($n=A+B\mathrm{/}{\lambda }^2+C\mathrm{/}{\lambda }^4$), is simpler but has no physical basis and breaks down outside the visible range. At 1550 nm, a Cauchy fit to BK7 is wrong by 0.005. For coating design, always use Sellmeier.

Complex Refractive Index: When Materials Absorb

Transparent materials like glass and SiO₂ have a real refractive index: light passes through with negligible absorption. Metals and semiconductors absorb light, and the absorption is captured by adding an imaginary part to the refractive index:

where n is the real part (determines the phase velocity, same as before) and k is the extinction coefficient (determines absorption). The intensity of light traveling through an absorbing material decays exponentially:

This is the Beer–Lambert law. The quantity $\alpha =4\pi k\mathrm{/}\lambda$ is the absorption coefficient, and $1\mathrm{/}\alpha$ is the penetration depth (distance where intensity drops to 1/e = 37% of its initial value).

Worked Example: Gold at 550 nm

Gold at 550 nm has n = 0.424 and k = 2.472.

Absorption coefficient: $\alpha =4\pi \times 2.472\mathrm{/}(550\times {10}^{-9})=5.65\times {10}^7{\text{m}}^{-1}$

Penetration depth: $1\mathrm{/}\alpha =17.7\text{nm}$

After just 20 nm of gold, the intensity drops to 32% of its starting value. This is why a 100 nm gold film is completely opaque. It's also why metal mirrors work: light barely penetrates the metal surface before being reflected, so only a thin layer is needed.

For comparison, silicon at 550 nm has k= 0.040. Its penetration depth is 1.1 µm, about 60 times deeper than gold. Silicon is partially transparent at visible wavelengths and becomes fully transparent in the infrared below its bandgap at 1.1 µm.

Where Do the Numbers Come From?

When you look up SiO₂ in the Photizon material database, you'll see "Malitson" as the default dataset. That's from a 1965 paper by I.H. Malitson at the National Bureau of Standards, measuring fused silica from 0.21 to 6.7 µm. It's still the gold standard 60 years later. For metals, the most-cited source is Johnson & Christy (1972), who measured the optical constants of Au, Ag, Cu, and other metals. Researchers still use their data as the default today.

Different datasets give different values for the same material, because the refractive index depends on how the material was prepared. A TiO₂ thin film deposited by sputtering has a different density and crystal structure than a bulk single crystal, and therefore a different n. This is why the Photizon database offers multiple datasets per material, and why you should always cite the specific source when reporting optical constants.

"TiO₂, n = 2.65" is ambiguous. "TiO₂, n = 2.648 at 550 nm (Devore 1951, ordinary ray)" is unambiguous and reproducible.

Putting It All Together

Here's why refractive index matters for everything we do at Photizon:

• Anti-reflection coatings work by choosing a layer with $n=\sqrt{n_{\text{substrate}}}$, so a quarter-wave layer produces zero reflectance at the design wavelength. MgF₂ (n = 1.38) on BK7 (n = 1.52) gets close to this ideal.
• High-reflector mirrors use alternating layers of high-n and low-n materials. The larger the ratio $n_H\mathrm{/}{n}_L$, the wider the stop band and the fewer pairs you need. TiO₂/SiO₂ gives $n_H\mathrm{/}{n}_L$ = 1.81.
• Dispersion limits coating bandwidth. A quarter-wave layer at 550 nm is not quarter-wave at 400 nm or 700 nm, because n changes with wavelength. This is why broadband coatings need optimized (non-quarter-wave) thicknesses.
• Metal mirrors absorb because k > 0. The penetration depth (17.7 nm for gold at 550 nm) sets the minimum film thickness for an opaque mirror.

The Sellmeier equation also has a temperature dependence (dn/dT), which matters for precision etalons and interferometry. That's a topic for a future tutorial.

Try exploring these relationships in the Photizon material database: look up any material, check its n at your wavelength of interest, and use the cross-links to load it directly into the thin-film simulator or Fresnel calculator.

References

1. M.N. Polyanskiy, "Refractive index database," Sci. Data 11, 94 (2024). The database behind our material browser.
2. I.H. Malitson, "Interspecimen comparison of the refractive index of fused silica," J. Opt. Soc. Am. 55, 1205 (1965). The standard SiO₂ dispersion data.
3. P.B. Johnson and R.W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370 (1972). The most-cited metal optical constants.
4. E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 3. Clear undergraduate treatment of dispersion and the Sellmeier equation.
5. H.A. Macleod, Thin-Film Optical Filters, 5th ed. (CRC Press, 2017), Ch. 2. Dispersion models in the context of coating design.

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