What is minimum deviation?

Minimum deviation occurs when the light ray passes symmetrically through the prism (angle of incidence equals angle of emergence). At minimum deviation, δ_min = 2×arcsin(n×sin(A/2)) − A. The deviation is least sensitive to small angular errors at this point, making it ideal for measurement.

What causes dispersion in a prism?

Dispersion occurs because the refractive index varies with wavelength. Shorter wavelengths (blue) have higher refractive indices in most glasses and are deviated more than longer wavelengths (red). This produces the rainbow spectrum at the prism output.

Why do prisms create rainbows?

Different colors of light have slightly different refractive indices in glass. When white light passes through the prism, each color refracts by a slightly different amount, separating spatially after the prism. The spread is visible as a rainbow spectrum. The classic 60° equilateral prism gives about 1° of spread across the visible range.

How are prisms different from diffraction gratings?

Prisms refract light continuously through glass; gratings diffract by interference at periodic structures. Gratings have much higher dispersion and resolving power; prisms have higher throughput and no order overlap. Modern spectroscopy mostly uses gratings; prisms remain useful in telescopes, binoculars, and specialty applications.

What is the Abbe number?

Abbe number V = (n_D − 1)/(n_F − n_C), where n at three standard wavelengths (yellow, blue, red). Low V = high dispersion (flint glass). High V = low dispersion (crown glass). Achromat doublets pair high-V and low-V to cancel chromatic aberration.

How do roof prisms in binoculars work?

Light enters the binocular objective, totally internally reflects off two flat faces of a "roof" prism that makes a 90° angle (often paired with a Schmidt or Pechan correction prism), then exits to the eyepiece. The TIR replaces a mirror with no reflection loss and provides image erection without an additional element.

Can you use a prism as a beam splitter?

Yes — a cube beamsplitter is two right-angle prisms cemented at their hypotenuses with a partial-reflective coating between them. The geometry preserves beam alignment perfectly (unlike a plate beamsplitter that introduces parallel beam shift) and is widely used in interferometers, holography, and microscopy.

Prism Dispersion Calculator

Determine the deviation angle of light passing through a prism, including minimum deviation conditions and angular dispersion. Useful for spectroscopy and prism design.

A prism splits white light into its component colors because refractive index varies slightly with wavelength — a phenomenon called "dispersion." Blue light (shorter wavelength) refracts more than red light (longer wavelength), so the same prism deflects different colors by different amounts, producing the familiar rainbow spectrum. Newton's classic experiments with prisms in the 1670s established that "white" light was actually a mixture of colors, refuting Aristotle's theory that color was a property added to light by the medium.

This calculator computes the deviation angle (how much light bends from its original direction) for a triangular prism, given the prism's apex angle, its refractive index, and the angle of incidence. It also handles the special "minimum deviation" geometry where light passes symmetrically through the prism — important because that's where measurements are taken in classical prism spectrometers, since deviation is least sensitive to small angular errors at that point.

Modern spectroscopy mostly uses diffraction gratings (higher resolution, broader wavelength range), but prisms still have advantages: no order overlap, higher light throughput, and elegantly simple geometry. Prisms appear in telescopes (Amici roof prisms in binoculars), spectrometers (Schmidt-Cassegrain "atmospheric dispersion correctors"), pulse stretchers/compressors for ultrafast lasers (prism pairs), and prismatic eyeglass lenses (for double vision correction).

Inputs

Prism Angle A (°)

Refractive Index n

Incident Angle (°)

dn/dλ (per μm)

Rate of change of refractive index with wavelength (negative for normal dispersion)

Results

Deviation Angle

37.38°

Min Deviation

37.18°

Angular Dispersion

-0.0756 °/μm

Prism Dispersion Results

Parameter	Value
Deviation Angle δ	37.3813°
Minimum Deviation δ_min	37.1808°
Emergence Angle i₂	52.3813°
Internal Angle r₁	28.1255°
Internal Angle r₂	31.8745°
Angular Dispersion dδ/dλ	-0.075593 °/μm
Prism Angle A	60°
Refractive Index	1.5

Last updated: May 29, 2026

Formula

**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.

How to use this calculator

Enter the prism apex angle (60° for equilateral; 30° for thin prisms used as wedges).
Enter the refractive index at your wavelength of interest (1.5 for common crown glass).
Enter the angle of incidence at the first surface (from normal).
The calculator returns the deviation angle and the minimum deviation if you adjust to the symmetric case.
For dispersion: enter dn/dλ in per μm; the calculator returns angular dispersion per nm or per μm.
For minimum deviation measurement (classical spectrometer technique), find the angle at which deviation stops changing as you rotate the prism — that's the symmetric minimum.

Worked examples

Equilateral prism on a sunny day

**Scenario:** A 60° BK7 glass prism (n = 1.517 at yellow) deflects sunlight at minimum deviation. Find the deviation angle. **Calculation:** δ_min = 2 × arcsin(1.517 × sin 30°) − 60° = 2 × arcsin(0.759) − 60° = 2 × 49.34° − 60° = 38.7°. **Result:** Yellow light deflects 38.7° from its original direction. Blue (n ≈ 1.522) deflects about 0.5° more; red (n ≈ 1.514) about 0.5° less. Total visible spectrum spread ≈ 1°. Not much for spectroscopy, but visually striking as a rainbow.

Spectrometer at minimum deviation

**Scenario:** Sodium D-line (589.0 and 589.6 nm) in a spectrometer with a 60° SF2 prism (n ≈ 1.648). Can the prism resolve the doublet? **Calculation:** dn/dλ for SF2 ≈ −0.046/μm. dδ/dn at minimum deviation = 2 sin(30°)/cos((60+δ)/2). δ_min ≈ 43.6°, so cos((60+43.6)/2) = cos(51.8°) = 0.619. dδ/dn = 1.0/0.619 = 1.615. dδ/dλ = 1.615 × (−0.046)/μm = −0.074 rad/μm = −7.4 × 10⁻⁵ rad/nm. For 0.6 nm doublet: angular separation = 4.4 × 10⁻⁵ rad = 9 arcsec. **Result:** ~9 arcsec angular separation, requiring a telescope with at least 50× magnification to comfortably resolve. With a good spectrometer eyepiece (or modern CCD camera), the sodium D-doublet is easily resolved with a single 60° flint prism — historically how Joseph Fraunhofer first identified the lines in the early 1800s.

Prism pair pulse compression

**Scenario:** A 60° BK7 prism pair compresses an ultrafast laser pulse. The first prism disperses the spectrum; the second reverses it. What's the path-length difference between red and blue? **Calculation:** Across visible (400–700 nm), Δn ≈ 0.008. Angular dispersion at minimum deviation ≈ 1.5 × (dn/dλ). For Δλ = 300 nm: angular separation between blue and red ~1.2° = 21 mrad. Over a 30 cm prism separation, transverse spatial separation ≈ 21 × 0.3 = 6.3 mm — but the path length difference (axial) is much smaller: typically 0.1–1 mm of glass-equivalent path. **Result:** Prism pairs introduce negative group delay dispersion (GDD), useful for compensating positive GDD accumulated in optical fiber or other dispersive elements. Adjusting the prism separation tunes the compression. Used in mode-locked femtosecond lasers (Ti:Sapphire) to achieve transform-limited pulses.

When to use this calculator

**Use prism math for:**

- **Classical spectroscopy**: Fraunhofer-style spectrometers, instructional spectroscopes. - **Telescope eyepiece design**: roof prisms (Amici), Porro prisms for image erection. - **Periscope and gun-sight prisms**: Bauernfeind, Schmidt, and Abbe-type prisms. - **Ultrafast laser pulse compression**: prism pairs for chirped-pulse compression (often paired with grating pairs). - **Wavelength selection in dye lasers**: birefringent or Lyot filter alternatives. - **Light-piping in optical fibers**: prismatic couplers at fiber ends. - **Prismatic eyeglass lenses**: correcting double vision (diplopia) via small angular deflection. - **Cube beamsplitters**: two prisms cemented at hypotenuses with partial-mirror coating.

**Prism shapes and uses:**

- **Equilateral (60°)**: classical dispersing prism for spectroscopy. - **Right-angle (90°)**: TIR-based beam redirector (binocular roof prism, Porro for image erection). - **Wedge (small angle)**: tiny deflection for fine pointing, prismatic eyeglass correction. - **Pentaprism (5 sides)**: viewfinder in SLR cameras, 90° deviation regardless of incidence angle. - **Roof prism**: combined inversion and erection in compact binoculars. - **Penta prism (Schmidt)**: 90° deviation, image not inverted.

**Why minimum deviation is special:**

- Sensitive measurement: δ is least sensitive to angular error of incidence when at minimum. - Symmetric geometry: refraction angle is the same at both surfaces. - Used in goniometric spectrometers: rotate prism and find the angle at which spectrum stops changing — that's the minimum deviation orientation.

**Prism vs grating for spectroscopy:**

| Property | Prism | Grating | |---|---|---| | Dispersion | Modest, nonlinear | High, ~linear | | Resolving power | Low (R ~ 10³–10⁴) | High (R ~ 10⁴–10⁶) | | Wavelength range | Limited by glass transmission | Wide | | Order overlap | None | Issue | | Throughput | Single beam, ~95% transmission | Multiple orders, ~30–60% per order | | Cost | Often cheaper for small prisms | Mid-range to expensive |

Modern instruments mostly use gratings. Prisms appear in legacy designs, telescopes, lasers, and specialized applications.

Common mistakes to avoid

Using minimum deviation formula at non-symmetric incidence. δ_min applies only when angles are equal at both surfaces.
Forgetting that n varies with wavelength. A single n value gives only one color's deviation; for dispersion, need n(λ).
Confusing prism angle (apex A) with angle of incidence. Different angles.
Trying to use simple formula for highly dispersive glass at extreme angles. Real prism behavior near total internal reflection is more complex.
Assuming linear dispersion. Prism angular spread per wavelength is NOT linear; it concentrates near the violet end of the visible spectrum.
Forgetting Fresnel reflection losses at each surface (~4% per uncoated air-glass interface). For unpolarized light through a prism, only ~92% transmits even at minimum deviation.
Specifying a very small apex angle and expecting big dispersion. Dispersion scales roughly with A; a 5° wedge has 1/12 the dispersion of a 60° equilateral.

Frequently Asked Questions

Sources & further reading

SponsoredShop Top Deals on AmazonSupport CalcMountain — browse top-rated products at no extra cost to you.