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Category Physics & Science

Angle of Incidence (°) Range: 0° to 90°

Refractive Index of Medium 1 (n₁)

Range: 1.0000 to 5.0000

Refractive Index of Medium 2 (n₂)

Range: 1.0000 to 5.0000

Incident Reflected Refracted Normal

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About

Incorrect refraction calculations in optical engineering lead to misaligned lenses, failed fiber-optic couplings, and flawed gemstone cuts. This calculator applies Snell's Law - n₁ ⋅ sin(θ₁) = n₂ ⋅ sin(θ₂) - to compute the refracted ray angle at a planar interface between two homogeneous isotropic media. It detects total internal reflection when the incidence angle exceeds the critical angle θ_c. The tool assumes monochromatic light and ignores dispersion effects. For polychromatic sources, each wavelength requires a separate refractive index.

Beyond the refracted angle, the calculator reports the critical angle (when applicable) and Brewster's angle θ_B = arctan(n₂ ÷ n₁), at which reflected light is fully polarized. An interactive ray diagram renders the geometry in real time. Pro tip: refractive indices are wavelength-dependent. Values listed here correspond to the sodium D-line at 589.3 nm. Using indices measured at other wavelengths introduces systematic error.

Formulas

The refracted angle is derived from Snell's Law of refraction, which governs the change in direction of a wavefront at the boundary between two media with different refractive indices.

n₁ ⋅ sin(θ₁) = n₂ ⋅ sin(θ₂)

Solving for θ₂:

θ₂ = arcsin(n₁n₂ ⋅ sin(θ₁))

The critical angle θ_c exists only when light travels from a denser to a rarer medium (n₁ > n₂):

θ_c = arcsin(n₂n₁)

When θ₁ ≥ θ_c, total internal reflection occurs and no refracted ray exists. Brewster's angle, at which reflected light is completely polarized:

θ_B = arctan(n₂n₁)

Where: n₁ = refractive index of incident medium, n₂ = refractive index of refracting medium, θ₁ = angle of incidence (measured from the normal), θ₂ = angle of refraction (measured from the normal).

Reference Data

Medium	Refractive Index (n)	Wavelength	State
Vacuum	1.0000	589.3 nm	(vacuum)
Air (STP)	1.0003	589.3 nm	(g)
Water	1.3330	589.3 nm	(l)
Ice	1.3090	589.3 nm	(s)
Ethanol	1.3610	589.3 nm	(l)
Glycerine	1.4730	589.3 nm	(l)
Olive Oil	1.4670	589.3 nm	(l)
Crown Glass	1.5200	589.3 nm	(s)
Flint Glass	1.6200	589.3 nm	(s)
Borosilicate Glass	1.5170	589.3 nm	(s)
Fused Silica (Quartz)	1.4585	589.3 nm	(s)
Polycarbonate	1.5860	589.3 nm	(s)
Acrylic (PMMA)	1.4920	589.3 nm	(s)
Diamond	2.4170	589.3 nm	(s)
Cubic Zirconia	2.1500	589.3 nm	(s)
Sapphire	1.7700	589.3 nm	(s)
Ruby	1.7600	589.3 nm	(s)
Silicon	3.4200	1200 nm	(s)
Germanium	4.0000	2000 nm	(s)
Salt (NaCl)	1.5440	589.3 nm	(s)
Sugar Solution (50%)	1.4200	589.3 nm	(l)
Carbon Disulfide	1.6280	589.3 nm	(l)
Benzene	1.5010	589.3 nm	(l)
Optical Fiber Core	1.4680	1550 nm	(s)
Optical Fiber Cladding	1.4500	1550 nm	(s)

Frequently Asked Questions

When n₁ ⋅ sin(θ₁) ÷ n₂ exceeds 1, the arcsin function has no real solution. This means the incidence angle has exceeded the critical angle and total internal reflection (TIR) occurs. No refracted ray enters the second medium. The calculator detects this condition and reports TIR instead of producing an invalid result.

Refractive indices are wavelength-dependent (chromatic dispersion). The presets in this tool use values measured at the sodium D-line (589.3 nm). For blue light (~450 nm), indices are typically 0.5 - 2% higher, and for red light (~700 nm) they are lower. If you are designing systems for specific wavelengths (e.g., telecom at 1550 nm), enter the refractive index measured at that wavelength manually.

The critical angle formula is θ_c = arcsin(n₂ ÷ n₁). For this ratio to produce a valid angle (i.e., arcsin argument ≤ 1), n₂ must be less than n₁. When light moves from a rarer to a denser medium, the refracted ray bends toward the normal and TIR is physically impossible.

Brewster's angle θ_B is the incidence angle at which reflected light is 100% polarized in the plane parallel to the surface. It is used in designing anti-glare coatings, laser cavity windows (Brewster windows), and polarizing optics. At Brewster's angle, the reflected and refracted rays are perpendicular: θ_B + θ₂ = 90°.

No. This calculator assumes isotropic media with a single refractive index. Birefringent (anisotropic) materials like calcite, quartz crystals, and liquid crystals have two refractive indices (ordinary and extraordinary). Each polarization component refracts at a different angle. For birefringent calculations, you would need to run the calculator twice with the respective ordinary and extraordinary indices.

Refractive indices decrease with rising temperature for most materials due to thermal expansion reducing density. For water, n drops by approximately 0.0001 per 1°C increase. For optical glasses, the thermo-optic coefficient (dn/dT) is typically 1 - 10 × 10⁻⁶ per °C. The preset values assume standard conditions (~20°C, 1 atm).