Critical Angle Calculator

Index
Index
Critical Angle (θc)
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Error: n1 must be greater than n2 for TIR to occur.

How it works?

Total Internal Reflection (TIR) occurs when light attempts to travel from a medium with a higher refractive index to one with a lower refractive index (e.g., Glass to Air). At a specific "Critical Angle," the light can no longer escape the denser medium and is reflected entirely back inside.

$$ \theta_c = \arcsin \left( \frac{n_2}{n_1} \right) $$ Snell's Law Limit

Conditions:

  • n_1 > n_2 : Light must travel from the denser medium into the rarer medium.
  • \(\theta_i \ge \theta_c\) : The incident angle must be equal to or greater than the critical angle for TIR to happen.

Why calculate the Critical Angle?

Key Advantages of TIR
  • 100% Reflection: TIR is more efficient than any mirror (no absorption loss).
  • Fiber Optics: The mechanism that traps light inside the core of a fiber.
  • Sensing: Used in rain sensors and fingerprint scanners to detect surface contact.
  • Prisms: Enables image inversion in binoculars and cameras without coatings.

The "Perfect Mirror" Effect

Normally, when light hits a boundary (like glass-to-air), some reflects and some refracts out. However, if the light hits the boundary at a steep enough angle—past the Critical Angle—Refraction becomes mathematically impossible.

At this point, nature forces 100% of the energy to reflect back into the dense medium. This phenomenon, Total Internal Reflection (TIR), allows us to guide light over thousands of kilometers in fiber optic cables with minimal loss.

1. Fiber Optics

Optical fibers work because the Core has a slightly higher index (n₁) than the Cladding (n₂). This creates a critical angle. Any light entering the fiber within the acceptance cone hits the wall at an angle > θc and is trapped, bouncing down the line indefinitely.

2. Gemology & Diamonds

Why do diamonds sparkle? Diamond has a very high refractive index (2.42), resulting in a small critical angle (~24°). This means light that enters a cut diamond is likely to bounce around inside via TIR and exit through the top face, creating brilliance.

3. Prism Binoculars

Binoculars use Porro prisms to flip the image right-side up. These prisms don't use silvered mirror coatings; they rely entirely on TIR to reflect the image. This results in a brighter image and a more durable optical system.

4. Automotive Rain Sensors

Modern cars use an IR LED bouncing inside the windshield via TIR. When rain hits the glass, the water (n=1.33) changes the critical angle condition at that spot, allowing light to escape. The sensor detects this drop in reflection and triggers the wipers.

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Related Engineering Tools

Snell's Law

Calculate angles of refraction before the critical angle is reached.

 Numerical Aperture

The acceptance cone of a fiber is directly derived from the critical angle.

 Fresnel Reflection

Calculate reflection percentage for angles less than the critical angle.