What is the Cauchy dispersion equation?

The Cauchy equation approximates the refractive index as n(λ) = A + B/λ² + C/λ⁴. It works well in the visible spectrum for transparent materials but is less accurate near regions of strong absorption (typically UV for glasses, IR for water and many polymers).

What is the Abbe number?

The Abbe number V_d = (n_d − 1)/(n_F − n_C) measures the dispersion of a material. n_d, n_F, n_C are refractive indices at the Fraunhofer d (587.6 nm yellow), F (486.1 nm blue), and C (656.3 nm red) lines. Higher V_d means lower dispersion. Crown glass (V > 50) is low dispersion; flint glass (V < 50) is high dispersion.

How does Cauchy compare to Sellmeier?

Sellmeier explicitly models material resonances and works across a much wider wavelength range with higher accuracy (typically ±0.001% vs Cauchy's ±0.1%). Cauchy is simpler but limited to "normal dispersion" away from absorption peaks. For precision work, use Sellmeier; for quick visible-spectrum estimates, Cauchy is fine.

What is normal vs anomalous dispersion?

Normal dispersion: refractive index decreases as wavelength increases (most materials in transparent regions). Anomalous dispersion: refractive index increases with wavelength, occurring near absorption peaks. Cauchy describes only normal dispersion; near absorption, more complex models (Lorentz oscillator, Kramers-Kronig) apply.

How do I get Cauchy coefficients for my material?

Fit published refractive index vs wavelength data with the form n = A + B/λ² + (optionally) C/λ⁴. Use RefractiveIndex.INFO or material datasheets. Most optical glass catalogs (Schott, Ohara, CDGM) publish Sellmeier coefficients directly, which can be converted to approximate Cauchy form for the visible spectrum.

Why does dispersion matter in lens design?

Different wavelengths focus at different points (chromatic aberration), causing colored fringes around high-contrast edges. Achromat doublets (crown + flint) cancel chromatic aberration at two wavelengths; apochromats correct three. Without correction, white light through a single lens spreads visibly — useful for prisms, problematic for imaging.

How is dispersion related to group velocity?

Group velocity v_g = c/n_g, where n_g = n − λ(dn/dλ). For normal-dispersion materials, n_g > n, so v_g < v_phase. This matters for ultrafast pulses, where the envelope travels at v_g while individual cycles move at v_phase. Group velocity dispersion (GVD = d²n/dλ²) causes pulses to spread temporally.

Cauchy Dispersion Calculator

Q: How does Cauchy compare to Sellmeier?

Sellmeier explicitly models material resonances and works across a much wider wavelength range with higher accuracy (typically ±0.001% vs Cauchy's ±0.1%). Cauchy is simpler but limited to "normal dispersion" away from absorption peaks. For precision work, use Sellmeier; for quick visible-spectrum estimates, Cauchy is fine.

Q: What is normal vs anomalous dispersion?

Normal dispersion: refractive index decreases as wavelength increases (most materials in transparent regions). Anomalous dispersion: refractive index increases with wavelength, occurring near absorption peaks. Cauchy describes only normal dispersion; near absorption, more complex models (Lorentz oscillator, Kramers-Kronig) apply.

Q: How do I get Cauchy coefficients for my material?

Fit published refractive index vs wavelength data with the form n = A + B/λ² + (optionally) C/λ⁴. Use RefractiveIndex.INFO or material datasheets. Most optical glass catalogs (Schott, Ohara, CDGM) publish Sellmeier coefficients directly, which can be converted to approximate Cauchy form for the visible spectrum.

Q: Why does dispersion matter in lens design?

Different wavelengths focus at different points (chromatic aberration), causing colored fringes around high-contrast edges. Achromat doublets (crown + flint) cancel chromatic aberration at two wavelengths; apochromats correct three. Without correction, white light through a single lens spreads visibly — useful for prisms, problematic for imaging.

Q: How is dispersion related to group velocity?

Group velocity v_g = c/n_g, where n_g = n − λ(dn/dλ). For normal-dispersion materials, n_g > n, so v_g < v_phase. This matters for ultrafast pulses, where the envelope travels at v_g while individual cycles move at v_phase. Group velocity dispersion (GVD = d²n/dλ²) causes pulses to spread temporally.

Use the Cauchy dispersion equation to model how refractive index varies with wavelength. Includes dispersion and absorption curves, Abbe number, and group refractive index calculations.

The Cauchy dispersion equation is a simple but effective empirical fit for how refractive index varies with wavelength in transparent materials. Augustin-Louis Cauchy developed it in 1836 — long before quantum mechanics explained dispersion microscopically — and it remains useful for back-of-envelope calculations in the visible spectrum. The two-term form n(λ) = A + B/λ² captures most of the variation; the three-term form n(λ) = A + B/λ² + C/λ⁴ gives a better fit across a broader wavelength range.

This calculator handles the Cauchy form. Enter the A, B, and C coefficients (look up published values for your material), and the calculator returns n at a specific wavelength plus a tabulated dispersion across a wavelength range. Common A values are around 1.5 for crown glass, 1.7+ for flint glass; B values are typically in the 0.003–0.02 µm² range. For high-precision work or non-visible wavelengths, the Sellmeier equation (with multiple resonance terms) gives better fits.

Cauchy dispersion is "normal" — refractive index decreases as wavelength increases (red light bends less than blue). Materials show "anomalous" dispersion near absorption peaks, where n actually increases with wavelength — but Cauchy's equation deliberately ignores those regions. For laser optics and ultrafast lasers where group velocity dispersion matters, you'd typically use Sellmeier; for visible-band lens design and educational work, Cauchy is fast and accurate enough.

Inputs

A Coefficient

Dominant refractive index term

B Coefficient (μm²)

First dispersion coefficient

C Coefficient (μm⁴)

Second dispersion coefficient

Absorption Coefficient α (cm⁻¹)

Specific Wavelength (nm)

Wavelength Range Min (nm)

Wavelength Range Max (nm)

Results

Refractive Index n(λ)

1.513223

n at 530 nm

1.514240

n at 550 nm

1.513223

n at 630 nm

1.510078

Abbe Number Vd

66.95

Group Index ng

1.539670

Dispersion Curve (n vs λ)

Transmittance vs Wavelength

Cauchy Dispersion Results

Parameter	Value
Refractive Index n(λ)	1.513223
Wavelength	550 nm
n at 530 nm	1.514240
n at 550 nm	1.513223
n at 630 nm	1.510078
Abbe Number Vd	66.95
Group Refractive Index ng	1.539670
dn/dλ	-0.048084 /μm
Absorption Coefficient	0.1 cm⁻¹
Transmittance (1 cm)	90.48%
Cauchy Formula	n(λ) = 1.5 + 0.004/λ² + 0/λ⁴

Last updated: May 29, 2026

Formula

**Cauchy dispersion equation (two-term):** n(λ) = A + B/λ² **Three-term form (more accurate):** n(λ) = A + B/λ² + C/λ⁴ Where: - **n(λ)**: refractive index at wavelength λ - **A**: constant term (typically 1.4–1.8 for visible glasses) - **B**: dispersion coefficient (units µm² when λ is in µm) - **C**: second-order dispersion (smaller correction) **Worked example: BK7 glass** Approximate Cauchy fit: A ≈ 1.509, B ≈ 0.0035 µm², C ≈ 0 µm⁴ (for visible). n at 486 nm = 0.486 µm: n = 1.509 + 0.0035/(0.486)² = 1.509 + 0.0148 = 1.524 n at 587 nm = 0.587 µm: n = 1.509 + 0.0035/(0.587)² = 1.509 + 0.0102 = 1.519 n at 656 nm = 0.656 µm: n = 1.509 + 0.0035/(0.656)² = 1.509 + 0.0081 = 1.517 (Compare to published BK7 values: 1.5224, 1.5168, 1.5143 — Cauchy fit is within 0.3%.) **Abbe number from Cauchy:** V_d = (n_d − 1) / (n_F − n_C) Where n at three Fraunhofer wavelengths: - d (Helium yellow): 587.6 nm - F (Hydrogen blue): 486.1 nm - C (Hydrogen red): 656.3 nm For our BK7 Cauchy fit: V_d = (1.519 − 1)/(1.524 − 1.517) = 0.519/0.007 = **74** (close to published 64 — Cauchy fit slightly underestimates dispersion). **Group refractive index (for pulse propagation):** n_g = n − λ × (dn/dλ) For Cauchy: dn/dλ = −2B/λ³ − 4C/λ⁵ So n_g = n + 2B/λ² + 4C/λ⁴ Group index is always larger than n for normal dispersion materials. Pulses propagate at c/n_g, slower than the phase velocity c/n. **Group velocity dispersion (GVD):** GVD = d²(1/v_g)/dω² = λ³/(2πc²) × d²n/dλ² For Cauchy: d²n/dλ² = 6B/λ⁴ + 20C/λ⁶ GVD is what spreads femtosecond pulses in glass and optical fiber. Positive GVD = "blue" components delayed; negative ("anomalous") GVD = "red" delayed. Prism pairs and grating pairs deliberately introduce negative GVD for pulse compression. **Cauchy coefficients for common materials (rough fits, visible range):** | Material | A | B (µm²) | C (µm⁴) | |---|---|---|---| | Fused silica | 1.451 | 0.003 | 0 | | BK7 crown | 1.509 | 0.004 | 0 | | SF11 flint | 1.762 | 0.012 | 0.0002 | | Sapphire | 1.764 | 0.003 | 0 | | Diamond | 2.395 | 0.013 | 0 | | Water | 1.324 | 0.003 | 0 | | Acrylic / PMMA | 1.488 | 0.003 | 0 | These are approximate Cauchy fits. For high-precision work, use published Sellmeier coefficients which are tabulated in optical glass catalogs (Schott, Ohara, CDGM). **Why Cauchy breaks down:** - Near absorption peaks (UV for most glasses): refractive index spikes; Cauchy gives nonsense. - In IR: water and other absorbers create non-Cauchy behavior. - Below 200 nm or above 2500 nm: material absorbs; index has anomalous behavior. - Strongly anisotropic crystals: requires tensor form (Cauchy is scalar).

How to use this calculator

Look up A, B, (and optionally C) for your material. For standard optical glasses, use catalog Sellmeier values converted, or use Cauchy fit values from references.
Enter the wavelength range you care about (typically visible: 380–780 nm).
The calculator returns n at the specific wavelength and a tabulated n(λ) for the range.
For higher accuracy, use the Sellmeier equation calculator instead.
For pulse dispersion calculations, use group index n_g, not phase index n.
Don't use Cauchy in regions where the material absorbs significantly (UV for most glasses, IR for water).

Worked examples

Chromatic aberration estimation for a single lens

**Scenario:** A BK7 lens with f = 100 mm at the d-line (587 nm). What's the focal length at the F (blue, 486 nm) and C (red, 656 nm) lines? **Calculation:** n_d = 1.519, n_F = 1.524, n_C = 1.517 (Cauchy fits). f scales as 1/(n−1): f_F/f_d = (1.519−1)/(1.524−1) = 0.519/0.524 = 0.990 → f_F = 99.0 mm. f_C/f_d = 0.519/0.517 = 1.004 → f_C = 100.4 mm. **Result:** Blue light focuses ~1 mm shorter than yellow; red focuses ~0.4 mm longer. This 1.4 mm spread is the chromatic aberration of a single uncorrected BK7 lens at 100 mm focal length. Achromat doublets pair crown + flint to cancel this; modern apochromat lenses use three or more elements to correct at three wavelengths.

Group velocity through fused silica

**Scenario:** A 1 ns laser pulse at 800 nm propagates through 10 cm of fused silica. How much later does it emerge compared to vacuum? **Calculation:** A=1.451, B=0.003. n(800nm) = 1.451 + 0.003/(0.8)² = 1.451 + 0.0047 = 1.456. n_g = n + 2B/λ² = 1.456 + 0.0094 = 1.465. Time in vacuum: 0.10/c = 0.33 ns. Time in silica: 0.10 × 1.465/c = 0.488 ns. Delay = 0.16 ns. **Result:** The pulse arrives ~160 ps later than light through vacuum, traveling at v_g = c/1.465. For ultrafast pulses (<100 fs), the group velocity is what matters, not the phase velocity. This timing precision matters in laser ranging, ultrafast electronics, and free-space optical communication.

Material selection by Abbe number

**Scenario:** Compare BK7 (crown, V≈64) and SF11 (flint, V≈37) for an achromat doublet. Power split if total power = 5 D? **Calculation:** Achromat condition: P_crown × V_crown = P_flint × V_flint (in magnitude). Also P_crown + P_flint = 5 D. From condition: P_flint = −P_crown × V_crown/V_flint = −1.73 × P_crown. Substituting: P_crown − 1.73 P_crown = −0.73 P_crown = 5 D → P_crown = −6.85 D... wait, that means doublet needs converging crown + diverging flint. Solve: 0.73 P_crown = 5 → P_crown = 6.85 D, P_flint = −1.85 D. **Result:** Achromat doublet: BK7 element with +6.85 D (converging), SF11 element with −1.85 D (diverging), net 5 D. Chromatic dispersion cancels because V × P balanced between the two elements. The flint element's high dispersion compensates the crown element's lower dispersion when combined with opposite powers. This achromat principle is the basis of most camera and telescope objectives.

When to use this calculator

**Use the Cauchy equation for:**

- **Quick refractive index estimates** across the visible spectrum. - **Educational and homework problems** in optics and physics courses. - **Initial lens design** before refining with Sellmeier or full optical CAD. - **Chromatic aberration estimates** for single-element or simple doublet lenses. - **Group index calculations** for ultrafast laser pulses through glass. - **Material comparison** when you need quick dispersion characteristics.

**When to use Sellmeier instead:**

- High-precision lens design. - Wavelengths outside 400–800 nm (UV or IR). - Crystals and unusual materials. - Anomalous dispersion regions near absorption peaks. - Published manufacturer data (catalogs use Sellmeier).

**Cauchy vs Sellmeier accuracy:**

For most visible-spectrum applications: Cauchy 2-term fits within ±0.3%, Cauchy 3-term within ±0.1%, Sellmeier 3-term within ±0.001%. The choice depends on precision needs.

**Practical dispersion intuition:**

- **High Abbe (V > 50, crown glass)**: low dispersion. Less chromatic aberration but slightly weaker dispersion-engineered effects. - **Low Abbe (V < 40, flint glass)**: high dispersion. More chromatic aberration but useful for separating wavelengths or correcting other elements. - **Achromat doublet**: pairs high-V with low-V to cancel chromatic aberration at two wavelengths. - **Apochromat triplet**: corrects at three wavelengths; uses three glasses including ED (extra-low dispersion) elements.

**Cauchy in fiber optics:**

Single-mode optical fibers have weak Cauchy-like dispersion in the operating band (1.31–1.55 µm). Zero-dispersion wavelength (where d²n/dλ² = 0) is around 1.31 µm for standard SMF — chosen so pulses don't spread temporally. Dispersion-shifted fibers move this zero to 1.55 µm to coincide with the EDFA gain peak.

**Pulse propagation through glass:**

Group delay: T = L × n_g / c, where L is path length. GVD-induced pulse broadening: Δτ = β₂ × L × Δω, where β₂ is the GVD parameter.

For 800 nm 100-fs pulse through 1 cm of BK7: pulse broadens to ~110 fs from the small amount of normal GVD. For ultrafast lasers, this accumulates: 10 cm of glass adds ~50 fs to a 100-fs pulse.

Common mistakes to avoid

Using Cauchy outside its valid range. Below 400 nm or above 1000 nm for most glasses, the fit breaks down.
Confusing phase index n with group index n_g. For pulse propagation, n_g matters; for steady refraction, n.
Mixing units. λ in µm if B is in µm²; otherwise units fall apart.
Forgetting that A, B, C are material-and-wavelength-range-specific. A fit valid at visible may fail in UV.
Treating Cauchy fits from one source as universal. Different references and Cauchy fits may use slightly different coefficients.
Not accounting for anomalous dispersion. Cauchy assumes normal dispersion (n decreases with λ); near absorption peaks, this fails.
Using Cauchy for very precise work where Sellmeier is needed. Cauchy is great for ~0.1% precision; Sellmeier for ~0.001%.

Frequently Asked Questions

Sources & further reading

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