Critical Angle Calculator
Solve the total-internal-reflection relation θc = arcsin(n2/n1) for any of its three unknowns — critical angle, or either refractive index — with step-by-step math, a live ray-tracing schematic, and companion fibre-optics metrics.
Total internal reflection traps light
When light travelling in a denser medium strikes a boundary with a rarer medium at an angle greater than θc, Snell's law has no real solution for the refracted angle — the sine of the refracted angle would have to exceed 1. Physically, no light crosses the boundary; 100% of the incident energy is reflected. This is the principle behind optical fibres, where light bounces along a high-index core surrounded by a low-index cladding over kilometres with almost no loss, and behind diamond's extraordinary sparkle, where θc = 24.4° traps light through many internal bounces before it escapes.
Higher index contrast means a smaller θc
The critical angle depends only on the ratio n₂/n₁, not on the absolute values. Diamond (n = 2.417) in air has θc = 24.4°, while crown glass (n = 1.52) in air has θc = 41.1°. The larger the index contrast, the smaller the critical cone inside which light can escape — which means a wider range of angles is trapped. This is why gem cutters prize high-index materials: the geometry of the cut determines which rays escape as sparkle and which are internally reflected back into the stone.
Fibre optics live on this relation
An optical fibre has a core of index n₁ surrounded by a cladding of slightly lower index n₂. The critical angle at the core–cladding boundary is typically 80–85° — very close to grazing — because the indices are close (e.g., 1.48 and 1.46). Light entering the fibre within a certain acceptance cone is guided by repeated total internal reflection. The numerical aperture NA = √(n₁² − n₂²) quantifies this acceptance cone, and the fractional index contrast Δ = (n₁ − n₂)/n₁ is typically ~0.01 for single-mode telecom fibre.
Beyond θc: the evanescent wave
At and beyond the critical angle, a small electromagnetic field still penetrates a short distance into the rarer medium. This evanescent wave decays exponentially with a penetration depth of order λ / (2π √(n₁² sin²θ − n₂²)), typically a few hundred nanometres. The wave carries no net energy away from the interface, but it can couple to another waveguide brought close by — the principle behind frustrated total internal reflection, used in fingerprint sensors, optical touch screens, and near-field scanning optical microscopy.
How to Use This Calculator
The critical-angle relation links three quantities: θc, n1, and n2. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick θc, n2, or n1 from the toggle. The calculator rearranges θc = arcsin(n2/n1) automatically for whichever variable you choose.
2. Use realistic presets
Angle presets span 15° to 80.6°; index presets cover vacuum, air, water, ethanol, fused silica, crown glass, flint glass, zirconia and diamond. The defaults — glass to air — reproduce the classic textbook value of 41.8°.
3. Respect the TIR condition
Total internal reflection is only possible when light travels in the denser medium, so n1 must be strictly greater than n2. If you try to solve for θc with n2 ≥ n1, the calculator will flag the error — there is no critical angle at that interface.
4. Read the result and verdict
The result card shows the solved quantity in degrees plus the equivalent radians and the sine value. A colour-coded verdict classifies the angle from very small (high-contrast, diamond-like) through glass-like, water-like, and fibre-optic-like.
5. Check the schematic and charts
The schematic traces the incident ray from the denser medium, the reflected ray at the same angle, and the refracted ray grazing along the interface at exactly 90°. The arc between the normal and the incident ray shows θc. The basic chart plots θc against n2 for three values of n1.
6. Switch to Advanced for depth
Advanced mode adds the θc vs. n1 chart, a sensitivity check on a +1% input change, and companion metrics including the Brewster angle, numerical aperture, acceptance cone, index contrast, and evanescent penetration depth.
What This Tool Offers
Built around the total-internal-reflection relation θc = arcsin(n2/n1), the geometric condition that traps light inside optical fibres, gemstones, and mirages.
Three-way solving
Solve for the critical angle or for either refractive index. The calculator enforces the physical constraint n1 > n2 and reports an error when no critical angle exists.
Live ray schematic
A two-medium diagram with incident, reflected, and grazing refracted rays, a shaded interface, and an arc marking θc between the normal and the incident ray.
Angle-class verdict
Each result is classified from very small (diamond-like, high-contrast) through glass, water, and up to fibre-optic-like near-grazing angles.
Multi-unit hints
Each field shows the value converted to related units — radians and sine for the angle; ratio, Brewster angle, and TIR condition for the indices.
Dual comparison charts
θc vs. n2 for three values of n1, plus an advanced θc vs. n1 chart for three values of n2 that makes the arcsin shape obvious.
Reference tables
A Snell's-law converter, a lookup grid of θc against n1, a table of common material pairs with their critical angles, and a full variable reference.
Reference Tables & Conversions
A Snell's-law converter, a θc lookup grid, and common material pairs.
Snell's-law converter at the critical angle
Critical Angle Lookup with n₂ = 1.0003 (Air)
Computed from θc = arcsin(n2/n1) in degrees.
| n₁ | ratio n₂/n₁ | θc (deg) | θc (rad) | sin θc | Material example |
|---|
Common Material Pairs & their Critical Angles
Representative values for everyday interfaces.
| Interface (denser → rarer) | n₁ | n₂ | θc | Notes |
|---|---|---|---|---|
| Diamond → air | 2.417 | 1.0003 | 24.45° | Extremely small — basis of diamond's fire and sparkle |
| Zirconia (cubic) → air | 2.15 | 1.0003 | 27.72° | Diamond simulant, also high TIR |
| Sapphire → air | 1.77 | 1.0003 | 34.40° | Watch crystals, LED substrates |
| Flint glass → air | 1.66 | 1.0003 | 37.06° | High-dispersion optical glass |
| Crown glass → air | 1.52 | 1.0003 | 41.14° | Standard optical glass — the calculator default |
| Fused silica → air | 1.46 | 1.0003 | 43.25° | Optical fibres, lenses, windows |
| Ethanol → air | 1.361 | 1.0003 | 47.30° | Liquid optics, hand sanitizer |
| Water → air | 1.333 | 1.0003 | 48.61° | Fish-eye view, mirages, pool TIR |
| Glass → water | 1.52 | 1.333 | 61.28° | Underwater windows, microscope immersion |
| Silica fibre core → cladding | 1.48 | 1.46 | 80.57° | Telecom fibre — near-grazing, extremely efficient guiding |
| Water → crown glass (impossible) | 1.333 | 1.52 | — | n₂ > n₁: no critical angle, light refracts into glass |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| θc | Critical angle | 20° – 85° | degrees or radians |
| n₁ | Refractive index of denser medium | 1.3 – 2.5 | dimensionless |
| n₂ | Refractive index of less dense medium | 1.0 – 1.5 | dimensionless |
| θB | Brewster (polarisation) angle = arctan(n₂/n₁) | 20° – 70° | degrees |
| NA | Numerical aperture = √(n₁² − n₂²) | 0 – 1.5 | dimensionless |
| Δ | Relative index contrast (n₁ − n₂)/n₁ | 0.001 – 0.3 | dimensionless |
| d_p | Evanescent penetration depth | 50 – 500 | nm |
What Does θc = arcsin(n2/n1) Measure?
The critical-angle relation connects three quantities at the heart of geometric optics: the angle above which light is trapped inside a denser medium (θc), the refractive index of that medium (n1), and the refractive index of the rarer medium on the other side of the boundary (n2). Together they describe the exact geometric condition under which a ray of light — instead of refracting out — is perfectly reflected back into its own medium. This is total internal reflection, and it is the reason optical fibres can carry signals for hundreds of kilometres, diamonds sparkle, and a calm water surface acts as a mirror when viewed from below.
Critical Angle Formula
At the critical angle, the refracted ray grazes along the interface — its angle with the normal is exactly 90°. Snell's law then gives:
which rearranges to the defining relation:
Solved for the two indices:
n₁ = n₂ / sin θc
A critical angle exists only when n₂ < n₁, i.e., when light is travelling in the denser medium toward a rarer one. If n₂ ≥ n₁, the fraction n₂/n₁ exceeds 1, arcsin is undefined, and no total internal reflection is possible.
Critical Angle Variables
| Symbol | Meaning | Notes |
|---|---|---|
| θc | Critical angle | Measured from the normal, inside the denser medium |
| n₁ | Index of denser medium | Light is travelling in this medium before reflection |
| n₂ | Index of less dense medium | Must satisfy n₂ < n₁ for TIR to occur |
| θ₁ | Angle of incidence | TIR occurs for θ₁ > θc |
| θ₂ | Angle of refraction | Equals 90° at θ₁ = θc; complex beyond θc |
| θB | Brewster angle | arctan(n₂/n₁); angle at which reflected light is fully polarised |
| NA | Numerical aperture | √(n₁² − n₂²); size of the acceptance cone in fibre optics |
How Each Variable Affects the Critical Angle
Because θc depends only on the ratio n₂/n₁, not on the individual indices, doubling both indices leaves the critical angle unchanged. Increasing n₁ with n₂ fixed reduces the ratio and therefore reduces θc — a smaller escape cone, more trapped light. Increasing n₂ with n₁ fixed raises the ratio and increases θc — a larger escape cone, less trapped light. When n₂ approaches n₁, the ratio approaches 1, arcsin approaches 90°, and the light escapes for almost any angle of incidence — this is the regime of weak-index-contrast fibre, where guiding is extremely efficient but the acceptance cone is very narrow.
The curve is the arcsin function, so the sensitivity of θc to the ratio is not uniform. Near ratio = 0 the slope is roughly 1 — small ratio changes produce proportionally small angle changes. Near ratio = 1 the slope diverges — a 1% increase in the ratio from 0.99 to 1.00 changes θc by more than 5°. This is why fibre-optic claddings must be specified to four or five decimal places: a small error in the index contrast produces a large error in the guided cone.
Critical Angle Calculation Examples
ratio = 1.0003 / 1.5 = 0.66687
θc = arcsin(0.66687) = 41.8290° = 0.7301 rad
ratio = 1.0003 / 1.333 = 0.75041
θc = arcsin(0.75041) = 48.6074°
n₂ = 1.5 · sin(41.829°) = 1.5 × 0.66687 = 1.0003
n₁ = 1.0003 / sin(24.45°) = 1.0003 / 0.41416 = 2.4155 — close to diamond's 2.417
Frequently Asked Questions
Total internal reflection (TIR) is the complete reflection of a light ray at a boundary between two media when the ray is travelling in the denser medium and strikes the boundary at an angle greater than the critical angle. Unlike ordinary partial reflection, no energy crosses the boundary — 100% of the incident light is reflected. TIR is a direct consequence of Snell's law: when the sine of the refracted angle would have to exceed 1, no real refracted ray exists, and the boundary behaves as a perfect mirror. It requires n₁ > n₂ — the light must be in the optically denser medium heading toward the rarer one.
A fish looking up at a calm water surface sees the entire sky compressed into a bright circular cone of half-angle 48.6° — the critical angle for water–air. Within that cone, light from the sky refracts down into the water and is visible. Outside the cone, the water surface acts as a mirror reflecting the dark depths below, so the fish sees a reflected image of the lake bottom instead of the sky. This "Snell's window" effect is a direct visual demonstration of the critical angle and can be photographed from any underwater viewpoint.
An optical fibre consists of a high-index core (n₁ ≈ 1.48) surrounded by a low-index cladding (n₂ ≈ 1.46). The critical angle at the core–cladding boundary is about 80.6°. Any light entering the core within the acceptance cone strikes the boundary at an angle greater than 80.6° and is totally internally reflected, bouncing along the fibre with essentially no loss beyond material absorption. Because the index difference is tiny, the guided cone is narrow — but the guiding itself is nearly perfect, which is why telecom fibres can carry signals for 100 km between amplifiers.
The Brewster angle θB = arctan(n₂/n₁) is the angle of incidence at which light reflected from a surface is perfectly polarised perpendicular to the plane of incidence. It always lies between 0° and 45° for TIR-compatible pairs and is always less than the critical angle (when a critical angle exists). At θB the reflected and refracted rays are exactly perpendicular. Both angles are determined entirely by the ratio n₂/n₁ — the same ratio that sets the critical angle — which is why the two are often quoted together in optics references. Brewster angle is used in polarising beam splitters and laser windows; critical angle governs waveguides and TIR prisms.
Yes, though weakly. Refractive index varies with wavelength — this is dispersion — so the critical angle is slightly different for red light and blue light. In crown glass, n is about 1.51 at 700 nm and 1.53 at 400 nm, giving a critical angle difference of roughly 0.4°. For most applications this is negligible, but in precision instruments — spectrometers, prism couplers, and high-resolution fibre sensors — the wavelength dependence must be taken into account. Diamond's high dispersion is part of why its sparkle is so colourful: different wavelengths escape at slightly different angles, producing the characteristic "fire".
Even in total internal reflection, the electromagnetic field does not vanish abruptly at the boundary. A small portion of the field penetrates a short distance — typically 50 to 500 nm — into the rarer medium before decaying exponentially. This is the evanescent wave. It carries no net energy away from the interface, but it can interact with anything placed within its penetration depth: another waveguide, a fluorescent molecule, or a metallic film. This is the basis of total internal reflection fluorescence (TIRF) microscopy, surface plasmon resonance (SPR) sensors, and frustrated total internal reflection (FTIR) touch screens.
Yes. TIR is a general wave phenomenon and applies to any wave encountering a boundary where its propagation speed increases. Sound waves in water reflect totally at the sea surface above about 13° from the horizontal — this is why sonar beams stay trapped near the surface in a "surface duct". Seismic waves travelling upward from a fast lower layer to a slow upper layer undergo total internal reflection at shallow angles, a key mechanism in refraction seismology used to map Earth's crust. The mathematics is identical; only the wave speed replaces the refractive index via n = c/v.
Diamond has an unusually high refractive index — n = 2.417 — because of its dense covalent carbon lattice. With n₂ = 1.0 (air), the ratio n₂/n₁ = 0.414, giving θc = 24.4°. Light entering a cut diamond bounces many times inside the stone before finding an escape angle within the narrow 24.4° cone. This trapping of light is what produces diamond's characteristic brilliance. A gem cutter's job is essentially to design facet angles that force light entering through the crown to make several total internal reflections at the pavilion before exiting back through the table as sparkle — a direct application of the critical angle.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| theta_c | Critical angle | degrees |
| n1 | Index of refraction of the denser (incident) medium | |
| n2 | Index of refraction of the less dense (transmitted) medium |
What it means
The critical angle is the angle of incidence in a denser medium (higher n) for which the refracted ray in the rarer medium grazes the interface (angle of refraction = 90°). For any incidence angle greater than θ_c, total internal reflection occurs, meaning all light is reflected back into the denser medium. This phenomenon is the basis for optical fibres, where light is guided along the core by repeated total internal reflection. It is also used in prisms, endoscopes, and in diamond cutting to exploit its brilliance. The formula θ_c = arcsin(n₂/n₁) is derived from Snell’s law. Understanding the critical angle is essential for designing fibre optics and for studying light propagation in media.
Worked example
Critical Angle – Two Examples
Real‑World| Parameter | Value |
|---|---|
| n₁ | 1.5 |
| n₂ | 1.0 |
| Parameter | Value |
|---|---|
| n₁ | 1.33 |
| n₂ | 1.0 |
Common mistakes
- Critical angle: Only defined when n₁ > n₂ (light going from denser to rarer medium).
- Arcsine: θ_c = arcsin(n₂/n₁) – ensure the ratio ≤ 1; if >1, no total internal reflection.
- Units: Result in degrees or radians – be consistent.
- Total internal reflection: Occurs when angle of incidence > θ_c.
- Media: n₁ is the incident medium, n₂ is the refracting medium.
Applications
The critical angle for total internal reflection, θ_c = arcsin(n₂/n₁), determines the angle above which light is totally reflected within a medium. This principle is the basis for optical fibres, which rely on total internal reflection to guide light with minimal loss. Engineers use it to design fibre optic cables, endoscopes, and telecommunications systems. It is also applied in prism spectrometers, in refractive index measurement, and in light‑guiding displays. By understanding the critical angle, professionals can ensure efficient light transmission and create innovative optical devices that exploit this effect.
- Design of optical fibre communication networks
- Endoscopes and medical imaging devices
- Prism spectrometers and refractometers
- Optical waveguides and couplers
- Light‑guiding displays and signage
Related calculators · Geometric Optics
Recently updated formulas
Suggest an improvement
Found a mistake? Have a better example? Let us know.
You may send up to 3 suggestions per 24 hours.