How is this different from the physics Snell's law calculator?

This optics version adds Brewster's angle, Fresnel reflectance coefficients for s and p polarizations, and detailed analysis of total internal reflection conditions relevant to optical system design. The physics version focuses on the basic refraction formula.

What is Brewster's angle?

Brewster's angle θ_B = arctan(n₂/n₁) is the angle at which reflected light is fully s-polarized (p-polarization has zero reflectance). At this angle, reflected and refracted rays are exactly perpendicular. Used in polarizing filters and Brewster windows in lasers.

What is the critical angle?

The angle at which refracted light just grazes the surface (refraction angle = 90°). Beyond this angle, no light passes — all of it reflects back (total internal reflection). Only exists when going from higher-n to lower-n material. For glass (n=1.5) to air, critical angle is ~41.8°.

Why does glass reflect ~4% of incident light?

At normal incidence, Fresnel reflectance R = ((n₁−n₂)/(n₁+n₂))². For air (n=1) to glass (n=1.5): R = (0.5/2.5)² = 0.04 = 4%. Cumulative across multiple lens surfaces, this becomes significant — which is why all modern optics use anti-reflection coatings.

What is total internal reflection used for?

Optical fibers (light guided by repeated TIR at the core-cladding boundary), binocular prisms (replacing mirrors with no losses), retroreflectors, decorative gemstones (diamond brilliance), endoscopes (image-guide fibers), and Total Internal Reflection Fluorescence Microscopy (TIRFM).

How does refractive index depend on wavelength?

Most materials have n that increases at shorter wavelengths (normal dispersion). For BK7 glass: n=1.514 at 656 nm (red), 1.517 at 587 nm (yellow), 1.522 at 486 nm (blue). This causes prism dispersion (rainbow) and chromatic aberration in single-element lenses.

Why do AR coatings work?

A thin quarter-wavelength coating with index √(n_air × n_glass) creates two reflected waves (from front and back surfaces of the coating) that interfere destructively, cancelling reflection. Modern multi-layer coatings extend this to broad bandwidths, dropping per-surface reflection from 4% to <0.5%.

Snell's Law for Optics

Calculate refraction at optical interfaces with detailed analysis including critical angle, Brewster's angle, total internal reflection conditions, and Fresnel reflection coefficients for s and p polarizations.

Snell's law (n₁ sin θ₁ = n₂ sin θ₂) describes how light bends when it crosses the interface between two materials with different refractive indices. This optics-focused version goes beyond the basic refraction formula to handle three critical angles you encounter constantly in optical system design: the critical angle for total internal reflection (TIR), Brewster's angle where reflected light becomes perfectly s-polarized, and the angle-dependent Fresnel reflection coefficients that determine how much light is reflected vs transmitted at each interface.

Practical optics work uses all three together. TIR is what makes optical fibers and prismatic binoculars work — light bounces inside a high-index material with no loss because it cannot escape into the lower-index surroundings. Brewster's angle is exploited in polarizing filters, laser windows, and antireflection design. And Fresnel reflection at every uncoated optical surface costs about 4% transmission per surface — which is why high-end lenses with many elements absolutely need anti-reflection coatings to remain bright.

This calculator takes the two refractive indices and the incidence angle and returns the refraction angle plus the critical angle, Brewster's angle, and Fresnel reflectance for both s and p polarizations. If the input angle exceeds the critical angle, the calculator flags total internal reflection (no transmitted ray, 100% reflection).

Inputs

Refractive Index n₁

Refractive Index n₂

Angle of Incidence θ₁ (°)

Results

Refracted Angle

48.59°

Brewster's Angle

33.69°

Reflectance (avg)

5.52%

Snell's Law Optics Results

Parameter	Value
Angle of Refraction θ₂	48.5904°
Critical Angle	41.8103°
Brewster's Angle	33.6901°
Reflectance (s-pol)	10.5773%
Reflectance (p-pol)	0.4608%
Average Reflectance	5.5190%
TIR	No
n₁	1.5
n₂	1

Last updated: May 28, 2026

Formula

**Snell's law:** n₁ × sin(θ₁) = n₂ × sin(θ₂) Where n₁, n₂ are refractive indices and θ₁, θ₂ are angles measured from the surface normal. **Solving for the refracted angle:** θ₂ = arcsin( (n₁ / n₂) × sin(θ₁) ) If the argument > 1, total internal reflection occurs (only valid when n₁ > n₂). **Critical angle (TIR threshold):** Only exists when going from denser to less dense (n₁ > n₂): θ_c = arcsin(n₂ / n₁) For glass (n=1.5) → air (n=1): θ_c = arcsin(1/1.5) = **41.81°** For water (n=1.33) → air: θ_c = **48.61°** For diamond (n=2.42) → air: θ_c = **24.4°** (why diamonds sparkle — easy TIR) **Brewster's angle (polarization angle):** θ_B = arctan(n₂ / n₁) At this angle, p-polarized reflection = 0; reflected ray is purely s-polarized. For glass: θ_B = arctan(1.5/1.0) ≈ **56.3°** (when going air → glass) **Fresnel reflection coefficients (amplitude):** For s-polarization (perpendicular to plane of incidence): r_s = (n₁ cos θ₁ − n₂ cos θ₂) / (n₁ cos θ₁ + n₂ cos θ₂) For p-polarization (parallel to plane of incidence): r_p = (n₂ cos θ₁ − n₁ cos θ₂) / (n₂ cos θ₁ + n₁ cos θ₂) **Fresnel intensity reflectance (what fraction of power is reflected):** - R_s = |r_s|² - R_p = |r_p|² - R_total = (R_s + R_p) / 2 (for unpolarized light) **Normal incidence (θ₁ = 0):** R = ((n₁ − n₂) / (n₁ + n₂))² For air → glass: R = ((1 − 1.5)/(1 + 1.5))² = (0.5/2.5)² = 0.04 = 4% For air → water: R = ((1 − 1.33)/(1 + 1.33))² ≈ 0.02 = 2% **Common refractive indices in optics:** | Material | n (visible) | |---|---| | Vacuum | 1.000 | | Air (STP) | 1.0003 | | Water (20°C) | 1.333 | | Acrylic / PMMA | 1.49 | | Crown glass (BK7) | 1.517 | | Flint glass (SF11) | 1.78 | | Sapphire | 1.77 | | Diamond | 2.42 | | Silicon (visible) | 3.5 | | Germanium (IR, 4 µm) | 4.0 | **Why n varies with wavelength (dispersion):** Refractive index increases at shorter wavelengths. For BK7 glass: - n at 486 nm (blue): 1.522 - n at 587 nm (yellow): 1.517 - n at 656 nm (red): 1.514 This causes prism dispersion (rainbow spectrum) and chromatic aberration in lenses.

How to use this calculator

Enter the refractive index of the medium light is coming from (n₁) and going into (n₂).
Enter the angle of incidence measured from the surface normal (NOT from the surface itself).
Read the refracted angle and the polarization-specific Fresnel reflectances.
If you see "Total Internal Reflection" flagged, the input angle exceeded the critical angle — no light transmits.
For laser system design, choose Brewster's angle for AR-coating-free p-polarization transmission.
For fiber optic design, ensure the angle inside the fiber stays above the critical angle for confinement.

Worked examples

Why glass surfaces reflect ~4%

**Scenario:** Light travels from air (n=1) into BK7 glass (n=1.517) at normal incidence. How much is reflected? **Calculation:** R = ((1 − 1.517)/(1 + 1.517))² = (−0.517/2.517)² = 0.0422 = 4.22%. So 95.78% of light transmits at each uncoated surface. **Result:** Every uncoated glass-air interface costs about 4% of transmitted light. A multi-element camera lens with 8 air-glass surfaces (4 elements) without anti-reflection coatings would transmit only 0.96⁸ ≈ 72% of incident light. AR coatings reduce per-surface loss to <0.5% for modern multilayer coatings, raising total transmission above 96%.

Optical fiber TIR confinement

**Scenario:** Step-index multimode optical fiber with core n=1.50, cladding n=1.48. What's the maximum acceptance angle (in air) for guided propagation? **Calculation:** Critical angle inside fiber: θ_c = arcsin(1.48/1.50) = arcsin(0.987) = 80.62° (from normal). Maximum angle from fiber axis = 90° − 80.62° = 9.38°. Outside in air, using Snell's law at the fiber entrance: NA = sin(θ_air) = n_core × sin(9.38°) = 1.50 × 0.163 = 0.245 → acceptance angle = arcsin(0.245) = 14.16°. **Result:** Light entering within a 14.2° cone (in air) couples into the fiber and propagates by TIR. NA = 0.245 is the numerical aperture — a key fiber spec. Single-mode fibers have much smaller NA (~0.1) and acceptance angle (~5.7°). Plastic optical fibers can have NA up to 0.6 (37° acceptance) for tolerant connector designs.

Brewster window in a laser

**Scenario:** A laser uses a Brewster-angled window between gain medium and external cavity to avoid wasted reflection. The window is fused silica (n=1.458). Find Brewster's angle. **Calculation:** θ_B = arctan(1.458/1.0) = arctan(1.458) = 55.6°. **Result:** Brewster's angle = 55.6°. P-polarized light passes with essentially zero reflection loss (the laser cavity self-selects p-polarization for this reason). S-polarized light suffers significant reflection and gets filtered out. This trick lets the laser operate without anti-reflection coatings on the windows — important for high-power systems where AR coatings can be damaged.

When to use this calculator

**Use Snell's law in optics for:**

- **Lens design**: refraction at every glass surface; total path through the optical system. - **Fiber optic design**: ensuring TIR confinement, computing NA and acceptance angle. - **Polarization control**: Brewster windows, polarizing beamsplitters, wire-grid polarizers (at non-Brewster angles). - **Anti-reflection coatings**: computing reflectance to design destructive-interference coatings. - **Prism design**: total internal reflection in roof prisms (binoculars) and Porro prisms. - **Underwater photography**: angle correction when looking from above water into water. - **Microscope objectives**: oil immersion (matching n to reduce surface reflections, boost NA). - **Diamond and gemstone faceting**: cuts optimized for TIR to maximize sparkle.

**Practical guidance:**

- **Anti-reflection coatings reduce R from 4% to <0.5% per surface**: cumulative gain in multi-element systems is huge (8-element zoom: 28% loss without AR, 4% with). - **Brewster's angle eliminates p-polarization reflection completely**: useful for polarizing or filtering by polarization. - **TIR is "free" — no mirror or AR coating needed**: as long as the angle stays above critical, 100% reflection. - **Critical angle depends on which way you go**: n₁ > n₂ for TIR. Air-to-glass: no critical angle (always partial reflection).

**TIR in everyday optics:**

- Binocular prisms (Porro prism): light totally reflects off uncoated glass-air surfaces twice. - Cat's-eye reflectors on roadways: TIR at the back of a plastic bead retroreflects. - Fiber optics: TIR at core-cladding interface guides light. - Diamond brilliance: high index (n=2.42) means many of the cut facets internally reflect, returning more light to the viewer.

**Angle conventions:**

- Always measured from the surface NORMAL (perpendicular to the surface), not from the surface itself. - Angle of incidence = angle of reflection (law of reflection — different from Snell's law). - "Grazing incidence" (θ → 90° from normal, light almost parallel to surface) → R → 1 (everything reflects), regardless of materials.

Common mistakes to avoid

Measuring angles from the surface instead of the normal. All formulas use angle from normal; a 30° angle from the surface is a 60° angle from the normal.
Forgetting that critical angle exists only when n₁ > n₂. Going from low-index to high-index always refracts (no TIR possible).
Mixing s- and p-polarization formulas. Make sure you're tracking the same polarization through a calculation.
Ignoring wavelength dependence. For broadband white light, n varies; chromatic dispersion at boundaries causes prism spectra and chromatic aberration.
Computing Brewster's angle but forgetting it eliminates ONLY p-polarization. S-polarized component still reflects substantially.
Using vacuum n=1.0 instead of air n=1.0003. For most calculations the difference is negligible; for precision astronomy or interferometry, it matters.
Assuming the refractive index of glass is always 1.5. Actual n varies from 1.45 (fused silica) to 1.95 (LASF flint) across optical glasses.

Frequently Asked Questions

Sources & further reading

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