Prism Dispersion Calculator

**General deviation angle of a triangular prism:** δ = θ₁ + θ₂ − A Where: - **δ**: total deviation angle (angle between incident and emergent rays) - **θ₁**: angle of incidence at first surface (from normal) - **θ₂**: angle of emergence at second surface (from normal) - **A**: apex angle of the prism **Snell's law applied at each surface:** sin θ₁ = n × sin θ_refracted_at_first_surface n × sin θ_at_second_surface = sin θ₂ With the geometric constraint: θ_refracted_at_first + θ_at_second = A **Minimum deviation condition:** Minimum deviation occurs when the ray passes symmetrically (θ₁ = θ₂, and the ray inside the prism is parallel to the base). δ_min = 2 × arcsin(n × sin(A/2)) − A Or rearranged to solve for n: n = sin((A + δ_min)/2) / sin(A/2) **Worked example: 60° equilateral prism, n = 1.50, normal-incidence symmetric case** A = 60°, n = 1.50, A/2 = 30°. δ_min = 2 × arcsin(1.50 × sin 30°) − 60° = 2 × arcsin(0.75) − 60° = 2 × 48.59° − 60° = 37.18° So a yellow ray (n = 1.50) deflects 37.2° at minimum deviation. **Angular dispersion:** dδ/dλ = (dδ/dn) × (dn/dλ) For minimum deviation: dδ/dn = 2 × sin(A/2) / cos((A + δ)/2) For our example: dδ/dn = 2 × 0.5 / cos(48.6°) = 1.0 / 0.661 = 1.513 Combined with dn/dλ from the material (typically −0.05 per μm for glass in visible): dδ/dλ = 1.513 × (−0.05/μm) = −0.076 rad/μm = −76 mrad/μm = −4.3°/μm For visible light (400–700 nm = 0.4–0.7 μm), total dispersion ≈ −4.3 × 0.3 = −1.3° spread across visible spectrum. Way less than a grating gives. **Refractive index of common prism glasses at different wavelengths:** | Glass | n at 486 nm (F line, blue) | n at 589 nm (D line, yellow) | n at 656 nm (C line, red) | |---|---|---|---| | BK7 | 1.522 | 1.517 | 1.514 | | SF2 (flint) | 1.671 | 1.648 | 1.637 | | SF11 (high flint) | 1.799 | 1.785 | 1.778 | | Fused silica | 1.463 | 1.458 | 1.456 | | Sapphire | 1.774 | 1.768 | 1.765 | **Abbe number (V) — chromatic dispersion metric:** V = (n_D − 1) / (n_F − n_C) For BK7: V = (1.517 − 1)/(1.522 − 1.514) = 0.517/0.008 = **64.7** (low dispersion crown) For SF11: V = (1.785 − 1)/(1.799 − 1.778) = 0.785/0.021 = **37.4** (moderate flint) Lower Abbe number = higher chromatic dispersion. Achromat doublets pair high-V crown (positive lens) with low-V flint (negative lens) to cancel chromatic dispersion. **Snell's law deviation derivation at the first surface:** When light enters glass: - sin θ₁ = n sin θ₁' - θ₁' = arcsin(sin θ₁ / n) Geometric: θ₂' = A − θ₁' (since the angles add up to the prism apex) - n sin θ₂' = sin θ₂ - θ₂ = arcsin(n sin θ₂') Total deviation: δ = θ₁ + θ₂ − A.