Critical Angle Calculator

At the critical angle and beyond, light is totally internally reflected with no transmitted beam. This requires n1 > n2. Fiber optics and diamond brilliance depend on this principle.

🔍 Optics📐 θᶜ = arcsin(n₂/n₁)💡 Fiber Optics
n₁ (denser medium)
n₂ (less dense)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Critical Angleθcθc = arcsin(n2/n1)deg
Condition—n1 > n2 required—
Numerical ApertureNANA = √(n1² − n2²)—

Step-by-Step Examples

Example 1
Glass to Air

Glass n1=1.5, air n2=1.0.

  • theta_c = arcsin(1.0/1.5) = arcsin(0.667) = 41.8°
✓ 41.8° — light beyond this angle is totally reflected
Example 2
Optical Fiber

Core n1=1.48, cladding n2=1.46.

  • theta_c = arcsin(1.46/1.48) = arcsin(0.9865) = 80.6°
✓ 80.6° — light at shallow angles propagates through fiber
Example 3
Diamond

Diamond n1=2.417, air n2=1.0.

  • theta_c = arcsin(1/2.417) = arcsin(0.414) = 24.4°
  • Very small critical angle — most light trapped inside
✓ 24.4° — diamond brilliance from extensive TIR

Real-World Applications

💡
Fiber Optics
Light signals bounce along glass fibers via TIR with essentially no loss over long distances.
💎
Diamond Brilliance
Diamond low critical angle (24.4°) causes extensive TIR, creating brilliant sparkle.
🔬
Periscopes and Prisms
45-45-90 prisms use TIR to redirect light beams in binoculars and periscopes.
📻
Optical Sensors
Frustrated TIR used in touchscreens and fingerprint scanners.

Common Mistakes to Avoid

⚠️
n1 must be greater than n2

Critical angle only exists going from denser to less dense medium. If n1 < n2, there is no critical angle (no TIR possible).

⚠️
Angle measured from normal

Critical angle theta_c is measured from the surface normal, not the surface itself.

⚠️
Confusing with Brewster angle

Critical angle causes TIR; Brewster angle causes polarization. Both involve n1 and n2 but different formulas and phenomena.

Frequently Asked Questions

What is total internal reflection? ▾
When light hits a denser-to-less-dense interface at angle >= critical angle, 100% of light reflects back. No transmitted beam at all.
How do fiber optic cables work? ▾
Light enters fiber core (n~1.48) at angle less than critical angle to cladding (n~1.46). It bounces via TIR all the way down the fiber without escaping.
Why are diamonds so brilliant? ▾
Diamond has very high refractive index (n=2.417), giving a very small critical angle (24.4 degrees). Most light entering from any angle exceeds critical angle internally, causing extensive TIR and sparkle.
What is frustrated TIR? ▾
Bringing a second medium very close (within one wavelength) to the TIR interface allows some light to tunnel through — the basis of FTIR sensors and evanescent wave coupling.
What is the numerical aperture of a fiber? ▾
NA = sqrt(n_core^2 - n_cladding^2). NA determines the acceptance angle of light that can enter and propagate in the fiber. Higher NA accepts more light.

Related Physics Calculators

🔍 Brewster Angle🔍 Refractive Index🔍 Snell Law🌈 Thin Film🔍 Lensmaker Equation

Formula Explorer connections

Interpretation: This relationship connects light propagation, geometry, wavelength, refraction, interference or image formation. Assumption: Use a consistent sign convention and units. Paraxial rays, thin elements, coherent light, vacuum wavelength or ideal optical components may be assumed.

Diffraction Grating Calculator →Thin Lens Equation Calculator →Lensmaker's Equation Calculator →Physics Formula Explorer →