Cauchy Dispersion Equation: Phase Refractive Index n(λ) & Group Index ng(λ)

Why femtosecond laser pulses travel slower than c / n(λ) inside crown glass!

Across transparent visible and near-infrared windows far from UV resonance poles, Augustin-Louis Cauchy’s (1836) power series models normal chromatic dispersion with sub-10^-4 precision using three ellipsometric coefficients: n(λ) = A + B / λ^2 + C / λ^4 (with λ in micrometers)! Because B > 0 and C >= 0, shorter blue/violet wavelengths experience higher refractive index n(λ) and bend more sharply than red light!

For ultrafast laser optics and white-light interferometry, envelope wavepackets do NOT propagate at phase velocity c / n(λ)—they propagate at group velocity v_g = c / n_g(λ), where differentiating Cauchy’s series gives n_g(λ) = n(λ) - λ * (dn/dλ) = A + 3 * B / λ^2 + 5 * C / λ^4! For borosilicate glass (A = 1.5046, B = 0.00420 μm^2, C = 0.00005 μm^4) at green λ = 0.550 μm (550 nm): phase index n = 1.51903, while group index jumps nearly +0.030 higher to n_g = 1.54898!

Cauchy Phase Refractive Index n(λ) = A + B / λ^2 + C / λ^4

Evaluates n(λ) to 5 decimal places for wavelength B5 in micrometers (e.g., 0.55 for 550 nm).

=IF(OR(B2 < 1, B5 <= 0), "Invalid Cauchy Inputs", ROUND(B2 + B3 / (B5 ^ 2) + B4 / (B5 ^ 4), 5))

Ultrafast Pulse Group Refractive Index n_g(λ) = A + 3B / λ^2 + 5C / λ^4

Applies the exact analytical derivative -λ*(dn/dλ) = +2B/λ^2 + 4C/λ^4 to compute envelope group index n_g(λ).

=IF(OR(B2 < 1, B5 <= 0), "Invalid Cauchy Inputs", ROUND(B2 + 3 * B3 / (B5 ^ 2) + 5 * B4 / (B5 ^ 4), 5))

Further reading

• Microsoft Support — ROUND function