Refractive Index & Critical Angle Calculator

Refractive Index Guide

What do I need to know about Snell's Law and Refractive Index?

Snell's law: n₁ sin θ₁ = n₂ sin θ₂. Refractive index of medium 2 relative to medium 1: n = sin θ_i / sin θ_r (when going from medium 1 to medium 2, with medium 1 being air/vacuum). The refractive index of a medium: n = c / v where c = speed of light in vacuum (3×10⁸ m/s) and v = speed of light in the medium. Air: n ≈ 1.0003 (effectively 1). Water: n ≈ 1.33. Glass: n ≈ 1.5. Diamond: n ≈ 2.42. Higher n means light slows more and bends more on entering the medium.

What do I need to know about Critical Angle and Total Internal Reflection?

When light travels from a denser medium to a less dense medium, it bends away from the normal. At the critical angle θ_c, the refracted ray travels along the boundary (90° to normal). Above θ_c: total internal reflection — all light is reflected back inside the denser medium. sin(θ_c) = n₂/n₁ = 1/n for glass-to-air. Diamond (n=2.42): θ_c = arcsin(1/2.42) = 24.4°. Very small critical angle means most light is totally internally reflected — this is why cut diamonds sparkle. Optical fibres: light is repeatedly totally internally reflected along the fibre's core, allowing it to travel enormous distances with minimal loss, which is the basis of modern telecommunications.

What should I know about Apparent Depth?

Objects in water appear shallower than they really are due to refraction. Apparent depth = real depth / n. Water (n=1.33): a fish at 2m real depth appears at 2/1.33 = 1.50m. This apparent depth effect is why swimming pools look shallower than they are — a 2m deep pool appears to be approximately 1.5m deep from the surface. The effect explains the apparent bending of a straw in a glass of water — the submerged part appears displaced because light from it refracts at the water-air interface.

What should I know about Applications of TIR?

Optical fibres: core glass has higher n than cladding glass. Light entering within the acceptance cone is totally internally reflected repeatedly along the core. Monomode fibres (core 8-10μm): carry a single light mode, used for long-distance telecoms. Multimode fibres (core 50-62.5μm): multiple modes, used for shorter distances (local area networks). Endoscopes: flexible bundles of optical fibres transmit images from inside the body. Retroreflectors: cat's eyes in road markings and road signs use tiny embedded glass spheres that reflect headlight beams directly back toward the driver, making them visible at night without needing their own power source.

What should I know about Snell's Law Calculations?

Snell's law states n₁sin(θ₁) = n₂sin(θ₂), relating the refractive indices and angles (measured from the normal) on either side of a boundary between two media. For light travelling from air (n≈1.0) into glass (n≈1.5) at an incidence angle of 30°: 1.0×sin(30°) = 1.5×sin(θ₂), so sin(θ₂) = 0.5/1.5 = 0.333, giving θ₂ = sin⁻¹(0.333) ≈ 19.5°. The light bends TOWARD the normal when entering a denser medium (higher n), and the general relationship n = c/v (refractive index equals speed of light in vacuum divided by speed of light in that medium) explains why: light genuinely travels slower in a denser medium, and this speed change at an angled boundary is what causes the bending — the same reason a car's wheels turn when one side hits mud while the other stays on tarmac, slowing unevenly across the boundary.

What do I need to know about Total Internal Reflection and the Critical Angle?

When light travels from a denser medium into a less dense one (glass into air, for instance), there's a critical angle beyond which refraction becomes impossible and the light is instead completely reflected back into the denser medium — total internal reflection (TIR). The critical angle is found by setting the refracted angle to exactly 90° in Snell's law: sin(θc) = n2/n1 (less dense over denser). For glass (n=1.5) to air (n=1.0): sin(θc)=1.0/1.5=0.667, so θc≈41.8°. Any light hitting the glass-air boundary at an angle greater than 41.8° from the normal reflects entirely rather than escaping — this is the principle behind fibre optic cables (light bounces repeatedly down the fibre core via TIR, losing almost no energy at each bounce) and why diamonds sparkle so brilliantly (diamond's very high refractive index of about 2.42 gives a small critical angle of about 24°, trapping and internally reflecting most light that enters until it exits through the top facets).

What should I know about Refractive Index and the Speed of Light?

The refractive index is formally defined as n = c/v, the ratio of light's speed in a vacuum to its speed inside the medium — so a refractive index of 1.5 for glass means light travels there at only two-thirds of its vacuum speed. This slowing, not any change in frequency, is what causes refraction: as a wavefront enters a medium at an angle, the part of it that reaches the denser medium first slows down while the rest keeps moving at full speed, pivoting the entire wavefront and bending the beam's direction.

What should I know about Refractive Index and Speed of Light in Various Materials?

What should I know about Dispersion in a Prism?

A single material doesn't actually have one fixed refractive index — n varies slightly with wavelength (dispersion), which is why a glass prism splits white light into a rainbow. Typical crown glass has n≈1.513 for red light but n≈1.532 for violet light. For light entering a prism face at 40° from air: for red, sinθ=sin(40°)/1.513=0.6428/1.513=0.4248, θ≈25.1°. For violet, sinθ=0.6428/1.532=0.4196, θ≈24.8°. This small 0.3° difference at one surface compounds across a triangular prism's two refracting faces into the clearly visible spread of colours, with violet bending more sharply than red because its refractive index is consistently higher in ordinary glass.

What should I know about Lateral Displacement Through a Glass Block?

When light passes through a parallel-sided glass block, it emerges travelling in the same direction as it entered but shifted sideways — lateral displacement, d = t×sin(θ1−θ2)/cos(θ2). Worked example: light enters a 5cm-thick glass block (n=1.5) at θ1=50°. First, θ2: sinθ2=sin(50°)/1.5=0.766/1.5=0.5107, θ2≈30.7°. Displacement: d=5×sin(19.3°)/cos(30.7°)=5×0.3304/0.8599≈1.92cm. The emerging ray is parallel to the original but displaced sideways by almost 2cm — a real effect that matters in optical design whenever light passes through a window or lens cover and needs to arrive at a precise location.

What should I know about Brewster's Angle and Polarisation?

At one particular angle of incidence, called Brewster's angle, light reflecting off a surface becomes completely polarised — all the reflected light vibrates in a single plane. Brewster's angle is found from tanθ_B = n2/n1. For light reflecting off water (n=1.33) from air (n=1.0): tanθ_B = 1.33, so θ_B ≈ 53.1°. This is why polarising sunglasses are so effective at cutting glare off water and wet roads: the glare is reflected light near Brewster's angle, strongly polarised in one orientation, and a polarising filter oriented to block exactly that orientation removes most of the glare while letting ordinary, unpolarised light from the rest of the scene through largely unaffected. At Brewster's angle, the reflected and refracted rays are also exactly perpendicular to each other — a geometric relationship that follows directly from combining Brewster's law with Snell's law, and one that further confirms why the reflected light is purely polarised in a single plane at this specific angle.

What should I know about Critical Angles for Common Materials?

Each critical angle follows directly from sin(θc)=1/n: water's relatively low refractive index gives a correspondingly large critical angle, meaning light must strike the water's underside at a fairly shallow angle before totally internally reflecting — which is why looking up from underwater, the entire sky compresses into a comparatively narrow circular window directly overhead (Snell's window), with everything outside that roughly 97.6-degree cone replaced by a mirror-like reflection of the water's own underside.

What should I know about Refraction and Mirages?

Mirages are a real-world refraction effect caused by a gradient in air temperature (and therefore refractive index) near the ground, rather than a single sharp boundary between two media. Hot air near a road surface is less dense and has a slightly lower refractive index than the cooler air above it, bending light from the sky upward into the driver's line of sight at a shallow angle so it appears to come from the road surface itself, mimicking the look of a wet, reflective patch ahead — a continuously varying form of the same refraction and (near-)total internal reflection principles that govern a sharp boundary between two distinct media.