Formula used
- n1 sin(θi) = n2 sin(θt) (Snell’s law)
- If n1 > n2, the critical angle is θc = arcsin(n2/n1)
- Total internal reflection occurs when θi > θc
- Reflectance uses Fresnel relations for s and p polarization
How to use this calculator
- Select a mode: critical angle, TIR check, or refraction angle.
- Enter n1 for the incident medium and n2 for the next medium.
- If the mode needs it, enter the incident angle measured from the normal.
- Pick degrees or radians, then press Calculate.
- Use the export buttons to save your results as CSV or PDF.
Example data table
| Case | n1 | n2 | θi (deg) | θc (deg) | Outcome |
|---|---|---|---|---|---|
| Glass → Water | 1.50 | 1.33 | 55 | 62.46 | Refraction occurs |
| Glass → Water | 1.50 | 1.33 | 70 | 62.46 | Total internal reflection |
| Water → Air | 1.33 | 1.00 | 40 | 48.75 | Refraction occurs |
| Water → Air | 1.33 | 1.00 | 55 | 48.75 | Total internal reflection |
| Air → Water | 1.00 | 1.33 | 70 | — | No TIR region |
Total Internal Reflection in Practice
1) Why a critical angle exists
When light goes from a higher refractive index to a lower one, Snell’s law predicts a larger transmitted angle. At a specific incident angle, the transmitted ray reaches 90° and skims the boundary. That incident angle is the critical angle, and beyond it the transmitted ray is no longer real.
2) Useful benchmark indices
For quick checks, common indices near visible wavelengths are: air ≈ 1.0003, water ≈ 1.333, acrylic ≈ 1.49, crown glass ≈ 1.52, and flint glass ≈ 1.62. Small variations occur with wavelength, temperature, and material grade, so treat these as practical starting values. At 1550 nm, fused silica is near 1.444, and small index steps are used to control modes and dispersion.
3) Typical critical angles
With water-to-air (1.33 → 1.00), the critical angle is about 48.75°. For glass-to-air (1.50 → 1.00), it is about 41.81°. For glass-to-water (1.50 → 1.33), it rises to about 62.46°. The calculator reproduces these quickly and consistently.
4) Fiber optics and numerical aperture
Optical fibers guide light by repeated total internal reflections at the core–cladding boundary, where ncore > nclad. A common performance metric is numerical aperture, NA = √(ncore² − nclad²), which links to the acceptance half-angle in air via sin(θa) ≈ NA.
5) Reflectance close to the limit
Near the critical angle, reflectance rises sharply and depends on polarization. s-polarized light typically reflects more strongly than p-polarized light for the same angle. In the TIR region the reflected power approaches 100%, yet an evanescent field still exists on the low-index side.
6) Evanescent wave penetration depth
Although no propagating transmitted ray appears during TIR, the field penetrates a short distance into the lower-index medium. A common estimate is dp = λ / (2π √(n1² sin²θi − n2²)). For visible light and typical indices, dp is often on the order of tens to hundreds of nanometers.
7) Practical uses in sensing
Because the evanescent field is sensitive to changes right at the boundary, TIR is used in optical sensors, ATR spectroscopy, and refractometers. A slight change in n2 shifts the critical angle, which can be measured as an intensity change at a fixed incident angle.
8) Modeling assumptions and good habits
This calculator assumes non-magnetic, isotropic media and standard Fresnel relations. For best results, keep indices positive, use angles measured from the normal, and remember dispersion: indices vary with wavelength. If you have a datasheet value at 589 nm, use that same reference when comparing materials.
FAQs
1) What condition is required for total internal reflection?
Total internal reflection requires light to travel from a higher index medium to a lower index medium, so n1 > n2. If n1 ≤ n2, a critical angle is not defined for that direction.
2) How is the critical angle calculated?
When n1 > n2, the critical angle is θc = arcsin(n2/n1). It is the incident angle at which the transmitted ray would travel exactly along the boundary (90° from the normal).
3) Why must the incident angle be measured from the normal?
Snell’s law uses angles measured from the normal because the boundary conditions are defined relative to that direction. Measuring from the surface flips the geometry and leads to wrong inputs, especially near the critical angle region.
4) Why does the calculator show 100% reflectance during TIR?
In ideal lossless media, TIR returns essentially all incident power to the original medium, so reflectance is 100%. Real systems can still have small losses due to absorption, surface roughness, or coupling to nearby materials.
5) Can total internal reflection happen from air to water?
No, not for the air-to-water direction, because air has the lower index. With n1 ≈ 1.00 and n2 ≈ 1.33, there is always a real refracted ray for any incident angle up to 90°.
6) What does polarization change in the results?
Polarization affects Fresnel reflectance below the critical angle. s and p components reflect differently at the same incident angle, so the calculator reports s, p, and average values. During TIR, both approach complete reflection.
7) How accurate are results if my indices are approximate?
Accuracy depends on your input indices. A ±0.01 change in refractive index can shift the critical angle by noticeable fractions of a degree. For precise work, use wavelength-specific, temperature-matched index data from reliable sources.