Calculate Angle of Refraction

Online calculator and formulas for calculating the optical angle of refraction

Angle of Refraction Calculator

Snell's law of refraction

Calculates the angle of refraction (θ₂) or angle of incidence (θ₁) when light transitions between two media with different refractive indices.

dimensionless
First medium (e.g. air: 1.00)
dimensionless
Second medium (e.g. water: 1.33)
°
Angle to the normal of the interface
°
Angle to the normal of the interface
Result

Example Calculation

Example: Light from air to water
Problem:

Light falls from air (n₁ = 1.00) onto water (n₂ = 1.33) with an angle of incidence of 30°. What is the angle of refraction?

Given:
  • Refractive index air: n₁ = 1.00
  • Refractive index water: n₂ = 1.33
  • Angle of incidence: θ₁ = 30°
  • Find: Angle of refraction θ₂
Solution:

Snell's law:

\[n_1 \cdot \sin(θ_1) = n_2 \cdot \sin(θ_2)\]
\[\sin(θ_2) = \frac{n_1 \cdot \sin(θ_1)}{n_2}\]
\[\sin(θ_2) = \frac{1.00 \cdot \sin(30°)}{1.33}\]
\[\sin(θ_2) = \frac{1.00 \cdot 0.5}{1.33} = 0.3759\]
\[θ_2 = \arcsin(0.3759) ≈ 22.1°\]
Physical Interpretation

Refraction toward normal: Light is refracted toward the normal when transitioning from air to water, because water is optically denser than air (n₂ > n₁).

Result: The angle of refraction (22.1°) is smaller than the angle of incidence (30°), which corresponds to the expected behavior.

Formulas for angle of refraction

Snell's law of refraction describes the refraction of light at the interface between two optical media with different refractive indices.

Snell's law of refraction

The fundamental law of light refraction at interfaces.

\[n_1 \cdot \sin(θ_1) = n_2 \cdot \sin(θ_2)\]
n₁, n₂ = Refractive indices
θ₁, θ₂ = Angles to normal [°]
Calculate angle of refraction

Rearrangement to calculate the angle of refraction.

\[θ_2 = \arcsin\left(\frac{n_1 \cdot \sin(θ_1)}{n_2}\right)\]
Application of arcsine for angle calculation.
Calculate angle of incidence

Rearrangement to calculate the angle of incidence.

\[θ_1 = \arcsin\left(\frac{n_2 \cdot \sin(θ_2)}{n_1}\right)\]
Reverse calculation of the angle of incidence.
Critical angle for total reflection

Critical angle for total internal reflection (n₁ > n₂).

\[θ_c = \arcsin\left(\frac{n_2}{n_1}\right)\]
For θ₁ > θc, total internal reflection occurs.
Important Refractive Indices
Vacuum: n = 1.00000
Air: n ≈ 1.00029
Water: n = 1.33
Ethanol: n = 1.36
Window glass: n ≈ 1.52
Flint glass: n ≈ 1.6-1.9
Diamond: n = 2.42
Quartz glass: n = 1.46

Detailed description of light refraction

Physical Fundamentals

Light refraction occurs when light passes from one medium to another and changes its direction of propagation. Snell's law of refraction describes this process quantitatively through the refractive indices of the media involved.

Usage Instructions

Select with the radio buttons whether the angle of refraction or angle of incidence should be calculated. Enter the refractive indices and the known angle and click "Calculate".

Application Areas

Optics and Photonics

Lens design, prism optics, laser optics, fiber optics. Foundation for optical systems and instruments.

Material Science

Determination of refractive indices, material characterization, quality control in the glass industry.

Medical Technology

Endoscopy, microscopy, laser therapy. Optical diagnostic procedures and imaging systems.


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