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Snell's Law

Snell's Law n₁sin(θ₁) = n₂sin(θ₂) describes light refraction at interfaces. Total internal reflection occurs above the critical angle when n₁ > n₂; Brewster's angle produces polarized transmission.

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Fiber optics use TIR to guide light—core n > cladding n traps light by total reflection. Diamonds sparkle because n=2.417 gives critical angle ~24°, causing multiple TIR events. Brewster's angle θB = arctan(n₂/n₁) gives zero p-polarized reflection. Dispersion: shorter wavelengths (blue) refract more than longer (red), creating rainbows.

Key quantities
28.9965°
θ₂
Key relation
90.0000°
θc
Key relation
55.5674°
θB
Key relation
4.4514%
R
Key relation

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Why: Snell's Law governs lenses, fiber optics, prisms, and optical coatings. TIR enables fiber optic communication; Brewster's angle is used in polarizers and laser optics.

How: Light bends toward the normal when entering a denser medium (n₂ > n₁) and away when entering a rarer medium. Critical angle θc = arcsin(n₂/n₁) exists only when n₁ > n₂.

Fiber optics use TIR to guide light—core n > cladding n traps light by total reflection.Diamonds sparkle because n=2.417 gives critical angle ~24°, causing multiple TIR events.
Sources:HyperPhysics RefractionMIT Optics

Run the calculator when you are ready.

Calculate RefractionEnter materials and incident angle to find refracted angle and Fresnel coefficients

Input Parameters

Angle from normal (0-90°)

589nm = sodium D-line

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Snell's Law Refraction Analysis
Refracted Angle: 28.9965°
Incident: 45.0000° • n₁=1.0398 • n₂=1.5167
Critical: 90.0000° • Brewster: 55.5674° • Reflectance: 4.4514%
numbervibe.com/calculators/physics/snells-law-calculator
snells-law@bloomberg:~$
REFRACTION: SHALLOW
INCIDENT ANGLE
45.00°
REFRACTED ANGLE
29.00°
CRITICAL ANGLE
90.00°
STATUS
REFRACT
●Snell's Law Results
REFRACTION

INCIDENT ANGLE

45.0000°

REFRACTED ANGLE

28.9965°

CRITICAL ANGLE

90.0000°

No TIR

BREWSTER ANGLE

55.5674°

Zero p-reflection

n₁

1.0398

n₂

1.5167

DEVIATION

16.0035°

Rs

8.2261%

Rp

0.6767%

AVG R

4.4514%

STEP-BY-STEP CALCULATION

▶ Input Parameters
Medium 1: Custom (n₁ = 1.0398)
Medium 2: Custom (n₂ = 1.5167)
Incident Angle: θ₁ = 45.0000°
Wavelength: λ = 589 nm
▶ Snell's Law Calculation
Formula: n₁ × sin(θ₁) = n₂ × sin(θ₂)
1.0398 × sin(45.0000°) = 1.5167 × sin(θ₂)
sin(θ₂) = 1.0398 × 0.7071 / 1.5167
sin(θ₂) = 0.4848
▶ Refracted Angle: θ₂ = 28.9965°→ θ₂ = 28.9965°
▶ Critical Angle Analysis
Critical Angle: θc = arcsin(n₂/n₁) = arcsin(1.5167/1.0398)
No critical angle (light going from less dense to denser medium)
▶ Brewster's Angle
θB = arctan(n₂/n₁) = arctan(1.5167/1.0398) = 55.5674°
At Brewster's angle, p-polarized light has zero reflection
▶ Fresnel Coefficients
Reflectance (s-pol): Rs = 8.2261%
Reflectance (p-pol): Rp = 0.6767%
Average Reflectance: R = 4.4514%
▶ Wave Properties in Media
Speed in Medium 1: v₁ = c/n₁ = 288.3237 km/s
Speed in Medium 2: v₂ = c/n₂ = 197.6602 km/s
Wavelength in Medium 1: λ₁ = 566.4675 nm
Wavelength in Medium 2: λ₂ = 388.3414 nm

Visualizations

For educational and informational purposes only. Verify with a qualified professional.

🔬 Physics Facts

📡

Fiber optic cables use TIR to transmit light over kilometers with minimal loss.

— Telecommunications

💎

Diamond's high n (2.417) creates low θc (~24°), trapping light for brilliance.

— Gemology

🌊

Underwater, Snell's window compresses the view above due to TIR at θ > 48.6°.

— Underwater Optics

🕶️

Polarized sunglasses use Brewster angle principle to block reflected glare.

— Optics

📋 Key Takeaways

  • • Snell's Law governs refraction: n₁sin(θ₁) = n₂sin(θ₂) — light bends when crossing material boundaries, with the bending direction determined by the relative refractive indices
  • • Total Internal Reflection (TIR) occurs above critical angle: When light travels from denser to less dense medium (n₁ > n₂) at angles greater than θc = arcsin(n₂/n₁), 100% reflection occurs — essential for fiber optics
  • • Brewster's angle produces polarized light: At θB = arctan(n₂/n₁), p-polarized light has zero reflection, creating perfectly polarized transmitted beams — used in polarizing filters and sunglasses
  • • Refractive index varies with wavelength: Dispersion causes different colors to refract differently (prism effect), with shorter wavelengths (blue) refracting more than longer wavelengths (red)

💡 Did You Know?

📡Fiber optic cables use total internal reflection to guide light over kilometers with minimal loss. The core has higher refractive index than the cladding, trapping light through TIR.Source: Telecommunications
💎Diamonds sparkle because their high refractive index (n=2.417) creates a low critical angle (~24°), causing multiple TIR events that trap light and create brilliance.Source: Gemology
🌊When you look up from underwater, objects above appear compressed into a circle (Snell's window) due to TIR at angles greater than 48.6° (water's critical angle).Source: Underwater Optics
🕶️Polarized sunglasses use Brewster's angle principle — they block horizontally polarized light reflected from surfaces like water and roads, reducing glare.Source: Optics
🌈Prisms create rainbows because different wavelengths have different refractive indices (dispersion). Blue light (shorter λ) bends more than red light (longer λ).Source: Spectroscopy
🔬Microscope immersion oil (n≈1.5) matches glass refractive index, eliminating air gaps that cause reflection and loss of light, improving resolution.Source: Microscopy

🔬 How It Works

Snell's Law Fundamentals

Snell's Law describes how light changes direction when passing between materials with different optical densities. The law states that the product of refractive index and sine of the angle (from normal) remains constant across the interface.

n₁sin(θ₁) = n₂sin(θ₂)
Refractive index × sine of angle is constant

Total Internal Reflection

When light travels from a denser medium (higher n) to a less dense medium (lower n), there exists a critical angle above which no transmission occurs. All light reflects back into the denser medium — this is total internal reflection.

Brewster's Angle

At Brewster's angle, p-polarized (parallel to plane of incidence) light has zero reflection. This produces perfectly polarized transmitted light, making it essential for polarizing filters and reducing glare.

🎯 Expert Tips

📐

Always measure angles from the normal (perpendicular to surface), not from the surface itself. This is critical for correct Snell's Law calculations.

💡

Remember TIR only occurs when n₁ > n₂ — light going from less dense to denser medium never experiences TIR, only refraction toward the normal.

🌈

Refractive index depends on wavelength (dispersion). Use the correct n value for your specific wavelength, especially for precision optics.

📡

For fiber optics design, ensure incident angles exceed the critical angle to maintain TIR and minimize signal loss over long distances.

📊 Refractive Index Comparison Table

MaterialRefractive Index (589nm)Critical Angle (to air)Typical Use
Air1.000-Reference medium
Water1.33348.6°Aquariums, pools, underwater optics
Crown Glass1.51741.2°Lenses, windows, optical components
Flint Glass1.62038.1°Prisms, achromatic lenses
Sapphire1.77034.4°Watch crystals, LEDs, high-durability optics
Diamond2.41724.4°Jewelry, cutting tools, high-index optics

❓ Frequently Asked Questions

What is Snell's Law and when is it used?

Snell's Law (n₁sin(θ₁) = n₂sin(θ₂)) describes how light bends when crossing material boundaries. It's used in lens design, fiber optics, prism calculations, and understanding how light behaves in different media.

When does total internal reflection occur?

TIR occurs when light travels from a denser medium (higher n) to a less dense medium (lower n) at angles greater than the critical angle θc = arcsin(n₂/n₁). Above this angle, 100% of light reflects back into the denser medium.

Why do diamonds sparkle so much?

Diamonds have a very high refractive index (n=2.417), creating a low critical angle (~24°). This causes multiple TIR events that trap light inside the diamond, creating the characteristic brilliance and fire.

What is Brewster's angle used for?

Brewster's angle (θB = arctan(n₂/n₁)) produces perfectly polarized transmitted light with zero p-polarized reflection. It's used in polarizing filters, laser optics, and reducing glare in sunglasses.

How does wavelength affect refraction?

Refractive index varies with wavelength (dispersion). Shorter wavelengths (blue/violet) typically have higher n values and refract more than longer wavelengths (red). This creates the rainbow effect in prisms.

Why do objects appear bent underwater?

Water has n=1.333 vs air's n=1.000. Light from underwater objects refracts at the water-air interface, making objects appear displaced and bent. Looking up from underwater, you see a compressed "Snell's window" due to TIR at angles >48.6°.

How do fiber optic cables work?

Fiber optics use TIR to guide light. The core has higher n than the cladding, so light entering at angles exceeding the critical angle reflects repeatedly along the fiber with minimal loss, enabling long-distance data transmission.

What are Fresnel coefficients?

Fresnel coefficients describe how much light reflects vs transmits at an interface. They differ for s-polarization (perpendicular) and p-polarization (parallel), explaining why reflection depends on polarization angle.

📊 Snell's Law by the Numbers

1.000
Air Refractive Index
1.333
Water Refractive Index
2.417
Diamond Refractive Index
24.4°
Diamond Critical Angle

⚠️ Disclaimer

This calculator is for educational and scientific purposes. Refractive indices vary with wavelength (dispersion), temperature, and pressure. For precision optics applications (lens design, fiber optics, optical coatings), consult material datasheets and account for all environmental factors.

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