Snell’s Law: How Light Changes Direction at a Boundary

A straw looks bent in water because light does not travel through water and air at the same speed. Snell’s law makes that visual fact calculable. It predicts the direction of a transmitted ray at one interface—not its brightness, a lens’s focal length, or the quality of an image formed by many surfaces.

For a ray crossing a smooth interfacen₁ sin θ₁ = n₂ sin θ₂

Both θ angles are measured from the normal: the perpendicular to the boundary, not from the surface itself.

A teal light ray crosses from air into a blue glass block, bending toward a dashed vertical normal while an optical engineer observes — When light enters the higher-index block, its transmitted ray is closer to the normal. The normal is the reference line that makes the angle convention unambiguous.

What each symbol carries

n₁, n₂Refractive index

Dimensionless. Each value belongs to a stated material, wavelength, temperature, and optical condition.

θ₁Incident angle

Angle from the normal on the side where the ray arrives. Use radians in radian-based software calls.

θ₂Transmitted angle

Angle from the normal in the second medium. It is only defined for a propagating transmitted ray.

c / vSpeed relation

For the simple phase-index picture, n = c/v; both speeds have units m s⁻¹.

The dimensional check is deliberately plain: n and sin θ are dimensionless, so both sides are dimensionless. The physics is less plain. A larger refractive index corresponds, in the simple model, to a lower phase velocity. If n₂ > n₁, then sin θ₂ < sin θ₁ and the ray bends toward the normal. Swap the index step and it bends away.

Why the sines appear

One derivation starts with the physical path’s stationary travel time. Let a ray cross a flat boundary at a variable horizontal point x; its start and end are perpendicular distances a and b from that boundary, with total horizontal separation D. The travel time is:

Travel timet(x) = √(x² + a²)/v₁ + √((D − x)² + b²)/v₂

Stationary pathdt/dx = 0 → sin θ₁/v₁ = sin θ₂/v₂

Use n = c/vn₁ sin θ₁ = n₂ sin θ₂

Huygens’s construction reaches the same result from wave fronts: in equal time intervals, one side of a front advances a different distance after crossing the interface, yet the new front must stay continuous. Both explanations assume a sharp, stationary boundary and a ray or wave-front description that is sensible at the relevant scale.

Solve it in the useful direction

Transmitted angleθ₂ = sin⁻¹((n₁/n₂) sin θ₁)

Incident angleθ₁ = sin⁻¹((n₂/n₁) sin θ₂)

Unknown n₂n₂ = n₁ sin θ₁ / sin θ₂

Critical angleθ_c = sin⁻¹(n₂/n₁) for n₁ > n₂

Every inverse-sine argument must lie between −1 and 1. Above 1 is not a calculator rounding problem. In the lossless ray model it says that no propagating transmitted ray can satisfy the boundary condition. At normal incidence both angles are zero, so index-solving forms divide by zero and cannot recover an index ratio from that one ray.

Worked example 1: air into glass

Take n₁ = 1.000 for air, n₂ = 1.500 for a stated ordinary-glass condition, and θ₁ = 30.0°, measured from the normal:

Rearrangesin θ₂ = (1.000/1.500) sin 30.0°

Evaluate= 0.666667 × 0.500000 = 0.333333

Angle and checkθ₂ = 19.471°; (1.500)(0.333333) = 0.500000

So the refracted angle is 19.5° to three significant figures. It is smaller than 30.0°, as the higher-index destination requires. The equality check also gives n₁ sin θ₁ = (1.000)(0.500000) = 0.500000. The law gives direction only: real glass also reflects some light at the surface.

Worked example 2: water into air and the critical angle

Use n₁ = 1.333 for water and n₂ = 1.000 for air at the stated wavelength and conditions. First, find the boundary angle:

Critical angleθ_c = sin⁻¹(1.000/1.333) = sin⁻¹(0.750188) = 48.61°

Test 55.0° incidencesin θ₂ = (1.333)(sin 55.0°) = (1.333)(0.819152) = 1.09193

Physical result1.09193 > 1, so no real θ₂: total internal reflection

At 48.61°, the refracted ray would run along the boundary at 90° to the normal. At 55.0°, there is no propagating refracted ray in this ideal interface model; that is total internal reflection. Snell’s law has not failed—its domain check has told you which optical regime applies.

A diver sends three beams toward a water surface: one exits, one travels along the surface, and one reflects back underwater — For light traveling from water toward air, increasing the incident angle leads from refraction to the grazing critical case and then to total internal reflection.

Error checks worth doing before you trust a result

Use the normal, not the surface

A ray 20° to the surface is 70° to the normal. Substituting 20° changes the model.

Keep labels with media

n₁ belongs to the incident side and n₂ to the transmitted side; do not swap one without the other.

Match angle units to the tool

Most code-library sine functions expect radians. A degree value passed directly produces a wrong numerical sine.

Do not equate angle with intensity

Reflectance, transmission, absorption, and polarization require more than Snell’s direction relation.

Where a single index is too simple

The scalar formula is most reliable at a locally flat, smooth boundary between homogeneous isotropic media. In a dispersive material, refractive index varies with wavelength and temperature; the National Institute of Standards and Technology measures exactly those dependencies. A prism therefore needs n(λ), not one rounded glass number for every color.

A prism separates a white beam into teal, gold, and terracotta paths as a materials scientist compares optical sample cards — Different wavelengths can require different refractive indices. The familiar prism effect is a reminder that “the refractive index” needs conditions attached.

In birefringent crystals, polarization and propagation direction can select different indices and refracted directions. In graded-index materials, rays bend continuously rather than at one boundary. Rough, patterned, absorbing, nonlinear, moving, or time-varying interfaces may require scattering, diffraction, complex-index, or frequency-shift treatments. Near the critical angle, beam width, surface quality, and index uncertainty matter too.

Snell’s law is therefore a precise local tool. It can say where an ideal transmitted ray points once you define the media and the angle convention. It cannot, by itself, say how much light arrives there, whether the material data match the sample, or what an optical system will resolve.