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Category Physics & Science

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Refractive Index n₁ (Incident Medium) Range: 1.0000 – 4.0000

Refractive Index n₂ (Refracting Medium) Range: 1.0000 – 4.0000

Angle of Refraction θ₂ (°) Range: 0° – 90° (exclusive)

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About

Snell's Law governs the relationship between the angle of incidence θ₁ and the angle of refraction θ₂ at the boundary between two optical media with refractive indices n₁ and n₂. A miscalculated incidence angle in fiber-optic design causes signal loss. In lens engineering, it produces aberration. In gemology, it means a poorly cut stone with no brilliance. This calculator solves for any unknown variable in Snell's equation, computes the critical angle for total internal reflection (TIR), and derives the Brewster angle θ_B for polarization. It assumes planar wavefronts at a flat interface and monochromatic light. Dispersion effects (wavelength-dependent n) are not modeled.

The interactive ray diagram renders the normal, incident ray, reflected ray, and refracted ray at their computed geometric angles. When the incidence angle exceeds the critical angle and n₁ > n₂, the tool flags total internal reflection and suppresses the refracted ray. Pro tip: refractive index values listed in handbooks are measured at the sodium D-line (589.3 nm). At other wavelengths, expect deviations of 0.5 - 2%.

Formulas

The fundamental relationship governing refraction at a planar interface is Snell's Law:

n₁ ⋅ sin(θ₁) = n₂ ⋅ sin(θ₂)

Solving for the angle of incidence:

θ₁ = arcsin(n₂ ⋅ sin(θ₂)n₁)

Solving for the angle of refraction:

θ₂ = arcsin(n₁ ⋅ sin(θ₁)n₂)

The critical angle for total internal reflection exists only when n₁ > n₂:

θ_c = arcsin(n₂n₁)

Brewster's angle, at which reflected light is fully polarized:

θ_B = arctan(n₂n₁)

Where n₁ = refractive index of medium 1 (incident side), n₂ = refractive index of medium 2 (refracted side), θ₁ = angle of incidence measured from the surface normal, θ₂ = angle of refraction measured from the surface normal, θ_c = critical angle, θ_B = Brewster's angle.

Reference Data

Medium	Refractive Index (n)	Wavelength	State
Vacuum	1.0000	All	(vacuum)
Air (STP)	1.0003	589 nm	(g)
Water	1.3330	589 nm	(l)
Ice	1.3090	589 nm	(s)
Ethanol	1.3610	589 nm	(l)
Glycerine	1.4730	589 nm	(l)
Crown Glass	1.5200	589 nm	(s)
Flint Glass	1.6600	589 nm	(s)
Fused Silica	1.4585	589 nm	(s)
Polycarbonate	1.5850	589 nm	(s)
Acrylic (PMMA)	1.4900	589 nm	(s)
Sapphire	1.7700	589 nm	(s)
Diamond	2.4170	589 nm	(s)
Cubic Zirconia	2.1700	589 nm	(s)
Silicon	3.4200	1200 nm	(s)
Germanium	4.0000	2000 nm	(s)
Salt (NaCl)	1.5440	589 nm	(s)
Olive Oil	1.4670	589 nm	(l)
Carbon Disulfide	1.6280	589 nm	(l)
Zircon	1.9230	589 nm	(s)
Rutile (TiO₂)	2.6100	589 nm	(s)

Frequently Asked Questions

When light travels from a denser medium to a less dense medium (n₁ > n₂) and the incidence angle θ₁ exceeds the critical angle θc, 100% of the light is reflected back into the first medium. This is total internal reflection (TIR). The refracted ray ceases to exist. Fiber-optic cables exploit TIR by maintaining incidence angles above the critical angle throughout the fiber core.

Snell's Law requires that the argument of arcsin stays within [−1, +1]. If (n₁ · sin(θ₁)) / n₂ exceeds 1.0, no real refraction angle exists. This physically means total internal reflection occurs. The calculator detects this condition and reports it explicitly rather than returning a mathematically undefined result.

Refractive indices are temperature-dependent. For liquids, n typically decreases by approximately 0.0004 per °C rise. Water at 20 °C has n = 1.3330 but at 80 °C drops to approximately 1.3290. For precision work in interferometry or fiber optics, use temperature-corrected values from the Sellmeier equation rather than handbook constants.

At Brewster's angle θB = arctan(n₂/n₁), the reflected and refracted rays are perpendicular (separated by exactly 90°). The reflected beam is 100% s-polarized (electric field parallel to the surface). Photographers use polarizing filters calibrated near Brewster's angle to eliminate glare from water and glass surfaces. For air-to-glass (n = 1.52), θB ≈ 56.7°.

No. This tool assumes isotropic media with a single refractive index per material. Birefringent crystals like calcite (n_o = 1.658, n_e = 1.486) split incident light into ordinary and extraordinary rays, each obeying Snell's Law independently. For birefringent analysis, you need to run two separate calculations with the respective refractive indices.

The diagram is geometrically exact for the computed angles at a planar interface. It correctly renders the incident ray, normal line, reflected ray (θ_reflected = θ₁), and refracted ray at angle θ₂. It does not model beam width, diffraction, wavefront curvature, or Fresnel reflectance coefficients. The reflected ray intensity is shown at constant opacity for visual clarity, not scaled to the actual Fresnel reflection percentage.