Engineering & Science
Sellmeier Refractive Index Dispersion Equation & Abbe Number (Vd) for Optical Glass
Calculate wavelength-dependent refractive index n(lambda) via the 3-term Schott Sellmeier equation SQRT(1 + SUM((Bi * lambda^2) / (lambda^2 - Ci))) and chromatic Abbe number Vd = (nd - 1) / (nF - nC).
Model chromatic dispersion from UV to NIR with 5-decimal Schott precision!
Designing achromatic camera doublets, fiber-optic dispersion compensators, and lithography lenses requires refractive index accuracy better than 10^-5 across ultraviolet, visible, and infrared wavelengths! Wilhelm Sellmeier (1871) modeled UV electronic and IR lattice absorption resonances via the 3-term dispersion formula: n^2(lambda) = 1 + (B1 * lambda^2) / (lambda^2 - C1) + (B2 * lambda^2) / (lambda^2 - C2) + (B3 * lambda^2) / (lambda^2 - C3), where wavelength lambda MUST be in micrometers (um) and C1, C2, C3 are in um^2!
For N-BK7 borosilicate crown glass (B1 = 1.039612, C1 = 0.006001, B2 = 0.231792, C2 = 0.020018, B3 = 1.010469, C3 = 103.56065), at the helium yellow d-line lambda = 0.5876 um, n_d = 1.51680, rising at violet mercury h-line 0.4047 um to 1.53023! Optical engineers summarize visible dispersion via Ernst Abbe's V-number Vd = (n_d - 1) / (n_F - n_C) across Fraunhofer F (486.1 nm), d (587.6 nm), and C (656.3 nm) lines: crown glasses have low dispersion (Vd >= 50, BK7 Vd = 63.80), while flint glasses have strong dispersion (Vd < 50)!
| N-BK7 Optical Metric | Wavelength(s) | Calculated Value | Optical Lens Meaning |
|---|---|---|---|
| Sellmeier Index at d-Line | lambda = 0.5876 um (Yellow) | n_d = 1.51680 | Reference crown focal power |
| Sellmeier Index at h-Line | lambda = 0.4047 um (Violet) | n_h = 1.53023 | Normal dispersion (n rises in blue/UV) |
| Abbe Dispersion Number Vd | nd=1.5168, nF=1.5224, nC=1.5143 | Vd = 63.80 | Vd >= 50 -> Low-dispersion Crown Glass |
3-Term Sellmeier Refractive Index n(lambda) (Wavelength in Micrometers)
Evaluates SQRT(1 + B1*L^2/(L^2-C1) + B2*L^2/(L^2-C2) + B3*L^2/(L^2-C3)) outside UV/IR absorption poles.
=IF(OR(B2 <= 0, (B2 ^ 2) <= B4, (B2 ^ 2) <= B6, (B2 ^ 2) = B8), "Inside Absorption Resonance", ROUND(SQRT(1 + ((B3 * (B2 ^ 2)) / ((B2 ^ 2) - B4)) + ((B5 * (B2 ^ 2)) / ((B2 ^ 2) - B6)) + ((B7 * (B2 ^ 2)) / ((B2 ^ 2) - B8))), 5))Abbe Chromatic Dispersion Number Vd = (n_d - 1) / (n_F - n_C)
Computes Abbe V-number from yellow d-line index (1.5168) and blue-to-red partial dispersion (nF - nC).
=IF(B9 <= B10, "Invalid Dispersion Order (nF > nC)", ROUND((1.5168 - 1) / (B9 - B10), 2))Try the example
Use the original values or change them to test your understanding.