What is Snell's Law Calculator?
Snell's law (also called the law of refraction) describes how a light ray bends when it crosses the boundary between two transparent media. It states that the product of the refractive index and the sine of the angle to the normal is conserved across the interface: n₁·sin θ₁ = n₂·sin θ₂. When light enters a denser medium (higher n) it slows down and bends toward the normal; when it enters a rarer medium it speeds up and bends away from the normal.
This calculator solves every variant of the law: the refracted angle from an incidence angle, the unknown incidence angle, an unknown refractive index, the critical angle that starts total internal reflection, Brewster's angle where reflected light becomes perfectly polarized, and the apparent depth that makes a pool look shallower than it is. It also reports Fresnel reflectance, deviation angle, and light's speed in each medium — with an interactive ray diagram, two reference charts, and a 15-material index table.
When to Use This Calculator
- Solving refraction problems in physics class or on exams.
- Designing lenses, prisms, fibers, and waveguide structures.
- Checking whether light will undergo total internal reflection in a medium pair.
- Estimating glare and reflectance of glass windows and screens.
- Explaining why a straw looks bent or a pool looks shallow.
- Studying polarization for photography, sunglasses, and laser setups.
Steps:
- Identify the two media and their refractive indices — the material above the boundary is n₁ (where the ray starts) and the material below is n₂.
- Measure the angle to the normal, not to the surface — angles are always taken from the dashed perpendicular line, never from the interface itself.
- Enter θ₁ and both indices in Find θ₂ mode to get the refracted angle, or switch modes to solve for the unknown value instead.
- Check the critical angle — if light travels from the denser medium (n₁ > n₂) and θ₁ exceeds θc, no ray passes through and everything reflects.
- Read the ray diagram and charts — the curve shows θ₂ rising until it stops at θc, while the Fresnel curves show how reflectance climbs from ~4% to 100% at grazing angles.
- Sanity-check with a known pair — air to water at 30° must give about 22°, and water to air above 48.6° must trigger total internal reflection.
Formula
Snell's Law:
n1 * sin(theta1) = n2 * sin(theta2)
Solving for the refracted angle:
sin(theta2) = (n1 / n2) * sin(theta1)
theta2 = arcsin( (n1 / n2) * sin(theta1) )
Critical angle (total internal reflection):
theta_c = arcsin( n2 / n1 ) [only when n1 > n2]
Brewster's angle (polarized reflection):
theta_B = arctan( n2 / n1 )
Apparent depth (looking straight down):
d' = d * (n2 / n1)
Fresnel reflectance at normal incidence:
R = ( (n2 - n1) / (n2 + n1) )^2
Light speed in a medium:
v = c / n (c = 299,792,458 m/s)
Wavelength in a medium:
lambda = lambda_0 / n
Where:
n1, n2 = refractive indices of the incident and transmitted media
theta1 = angle of incidence, measured from the surface normal
theta2 = angle of refraction, measured from the surface normal
d = real (actual) depth
d' = apparent depth as seen from the other medium
Use Cases
- Fiber-optic design — computing the numerical aperture and acceptance angle that keep light trapped inside the core by total internal reflection.
- Lens and prism engineering — predicting the bending of rays through eyepieces, camera lenses, prisms, and beam splitters.
- Geology and gemology — using critical angles to cut diamonds for maximum sparkle and to study light paths in crystals.
- Physics homework and exams — instant, verified answers for refraction, TIR, Brewster angle, and apparent-depth problems.
- Underwater imaging and diving — estimating how objects shift position and why pools look shallower than they are.
- Coating and AR-film design — reading Fresnel reflectance to estimate glare and anti-reflective coating performance.
Key Benefits
- Six modes in one tool — solve for θ₁, θ₂, n₂, the critical angle, Brewster's angle, or apparent depth without ever leaving the page.
- Instant visual feedback — the ray diagram, refraction curve, and Fresnel reflectance chart update live as you drag the angle.
- One-tap material presets — air/water, air/glass, diamond, and optical-fiber pairs prefill the exact indices for you.
- Real-world validation — the 15-material reference table shows each index, light's speed in it, and its critical angle against air.
- Free and instant — no sign-up, no ads in the way, works on phone, tablet, and desktop.
Pro Tips
- Quick memory: going into a denser medium the ray bends toward the normal (θ₂ < θ₁); going into a rarer medium it bends away (θ₂ > θ₁).
- The critical angle exists only when n₁ > n₂. If the calculator reports 'impossible', swap which medium the light starts in.
- To measure an unknown index experimentally: aim a laser across a semicircular block, find the angle where the refracted ray vanishes — that is the critical angle.
- Reflectance at normal incidence only needs one number: R = ((n₂−n₁)/(n₂+n₁))². For n = 1.5 that is (0.5/2.5)² = 4%.
- In optics fiber the core index (1.475) must exceed the cladding (1.460); the 81.8° critical angle keeps light guided for kilometers.
Common Mistakes to Avoid
- Measuring from the surface instead of the normal — θ₁ and θ₂ are always measured from the perpendicular line; a 30° ray to the surface is actually 60° to the normal.
- Applying Snell's law when going denser → rarer above θc — if θ₁ > θc there is no refracted ray; sin θ₂ > 1 is a signal for total internal reflection, not an error.
- Swapping n₁ and n₂ — the incident medium must be n₁. Swapping them inverts the bend and produces nonsense critical angles.
- Forgetting that indices are unitless ratios — refractive index has no unit; only the speeds and wavelengths use meters and nanometers.
- Using visible-angle intuition for reflectance — even at near-normal incidence glass already reflects ~4%, and reflectance climbs steeply past 60°, reaching 100% at grazing angles.
Key Terms Explained
- Refractive index (n):
- The ratio of light's speed in vacuum to its speed in the medium, n = c/v. Vacuum is exactly 1.
- Normal:
- The imaginary line perpendicular to the surface at the point where the ray strikes; all angles are measured from it.
- Critical angle (θc):
- The largest incidence angle that still produces a refracted ray when n₁ > n₂; beyond it, total internal reflection occurs.
- Total internal reflection (TIR):
- Complete reflection at a boundary — no light is transmitted, giving 100% reflectance.
- Brewster's angle (θB):
- The angle at which reflected light is perfectly polarized parallel to the surface, θB = arctan(n₂/n₁).
- Deviation angle:
- The difference |θ₁ − θ₂| — how far the ray's direction changed crossing the boundary.
Related Concepts
- Refractive index: The material property n = c/v that this calculator reads for both media — every result scales directly with it.
- Total Internal Reflection: The phenomenon created whenever θ₁ exceeds θc — the physics that carries internet traffic through optical fibers.
- Fresnel Equations: The full description of how much light reflects versus transmits at each angle, summarized by the reflectance chart.
- Apparent Depth: The everyday illusion of objects sitting higher than they are, derived from the same law at near-normal viewing.
- Polarization by Reflection: How unpolarized glare becomes polarized at Brewster's angle — the theory behind polarized sunglasses.
Example
Air → water at 30°. Light leaves air (n₁ = 1.0003) and strikes water (n₂ = 1.333) at θ₁ = 30°.
Snell's law: sin θ₂ = (1.0003 / 1.333) × sin 30° = 0.7504 × 0.5 = 0.3752
θ₂ = arcsin(0.3752) ≈ 22.02°
Deviation = 30° − 22.02° ≈ 7.98° — the ray bends toward the normal because water is optically denser.
Now reverse it — water → air at 50°: sin θ₂ = (1.333 / 1.0003) × sin 50° = 1.016 > 1, so no transmitted ray exists. The critical angle is θc = arcsin(1.0003 / 1.333) ≈ 48.61°, and at 50° the light is totally internally reflected — the physics behind fiber optics and diamond sparkle.
Interpreting Your Results
Start by comparing θ₂ with θ₁. If θ₂ < θ₁, light entered a denser medium and slowed down; if θ₂ > θ₁, it entered a rarer medium and sped up; if they are equal, both indices match and no bending occurs. Next look at the TIR status: active means the ray never crosses the boundary and 100% of the light reflects; not reached means a transmitted ray exists but you are approaching θc; impossible means the light is heading into a denser medium, so TIR can never happen in that direction. Finally, read the Fresnel reflectance: ~4% is normal glare off window glass, 20–40% is strong edge reflection, and 100% is total internal reflection.