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Refraction (Snell's Law)

n1 sin(i) = n2 sin(r)

Angle of refraction (r)

19.47 deg

Understanding Refraction and Snell's Law

Definition: Refraction is the bending of light when it passes from one medium to another with different optical density. This occurs because light travels at different speeds in different materials. When light enters a new medium at an angle, one part of the wavefront changes speed before the other, causing the wave to change direction. The more optically dense a material (higher refractive index), the slower light travels through it. This fundamental phenomenon explains why a straw looks bent in water, why lenses can focus light, and why diamonds sparkle.

Mathematical Derivation of Snell's Law:

Snell's law can be derived from Huygens' principle or Fermat's principle of least time. Consider light crossing a boundary:

In medium 1: v1 = c/n1, wavelength lambda1 = lambda0/n1

In medium 2: v2 = c/n2, wavelength lambda2 = lambda0/n2

Frequency f is unchanged across boundary

sin(i)/sin(r) = (v1 x t1 x sin i)/path = n2/n1

Therefore: n1 sin(i) = n2 sin(r)

Light takes the path that minimizes travel time. At the boundary, the change in speed causes the light to bend to minimize time.

Snell's Law (Law of Refraction):

n1 sin(i) = n2 sin(r) or sin(i)/sin(r) = n2/n1

Worked Example 1: Light goes from air (n1 = 1.0) to water (n2 = 1.33) at 45 degrees incidence. Find refraction angle.

Given: n1 = 1.0, n2 = 1.33, i = 45 degrees
sin(r) = (n1/n2) x sin(i) = (1.0/1.33) x sin(45 degrees)
sin(r) = 0.752 x 0.707 = 0.532
r = sin inverse(0.532) approx 32.1 degrees

Worked Example 2: Light enters diamond (n = 2.42) from air at 60 degrees. What is the angle inside the diamond?

sin(r) = (1.0/2.42) x sin(60 degrees)
sin(r) = 0.413 x 0.866 = 0.358
r = sin inverse(0.358) approx 21.0 degrees
Light bends significantly toward the normal in diamond.

Worked Example 3: Light goes from glass (n = 1.5) to water (n = 1.33). Critical angle for glass-air is 41.8 degrees. At what angle does light enter water from glass?

If i = 30 degrees (inside glass):
n1 sin(i) = n2 sin(r)
1.5 x sin(30 degrees) = 1.33 x sin(r)
sin(r) = 0.75/1.33 = 0.564
r = sin inverse(0.564) approx 34.3 degrees

Where:

  • n1: Refractive index of the first medium (incident medium)
  • n2: Refractive index of the second medium (refracted medium)
  • i: Angle of incidence (angle between incident ray and normal)
  • r: Angle of refraction (angle between refracted ray and normal)

Refractive Index: The ratio of speed of light in vacuum to speed in the medium:

n = c/v (c = 3 x 10 to the power 8 m/s in vacuum)

Since light always slows in matter, n greater than or equal to 1 always. Vacuum has n = 1 exactly; air has n approx 1.0003 (nearly 1).

Common Refractive Indices:

  • Vacuum: 1.0 (exactly by definition)
  • Air: 1.0003 (approximately 1.0)
  • Ice: 1.31
  • Water: 1.33
  • Crown Glass: 1.5
  • Flint Glass: 1.6
  • Ruby: 1.77
  • Diamond: 2.42 (highest for common materials)

Why Does Light Bend? When light enters a different medium at an angle, one side of the wavefront changes speed first. This asymmetry causes the wavefront to pivot, resulting in bent ray direction:

  • Toward the normal: When entering optically denser medium (n2 greater than n1) - light slows down
  • Away from the normal: When entering rarer medium (n2 less than n1) - light speeds up

Critical Angle and Total Internal Reflection (TIR):

  • Only occurs when light travels from denser to rarer medium (n1 greater than n2)
  • Critical Angle: theta c = sin inverse(n2/n1)
  • Total Internal Reflection: When i greater than theta c, ALL light is reflected back into the denser medium
  • No refraction occurs; reflection is 100% efficient

Critical Angles for Common Interfaces:

  • Diamond to air: theta c approx 24.4 degrees (low critical angle = more sparkle)
  • Crown glass to air: theta c approx 41.8 degrees
  • Water to air: theta c approx 48.6 degrees
  • Flint glass to air: theta c approx 38.7 degrees

Dispersion: Different wavelengths have different refractive indices. Blue light (shorter wavelength) bends more than red light (longer wavelength). This chromatic dispersion splits white light into a spectrum through a prism. Diamond's high dispersion (difference between red and blue refractive indices) contributes to its sparkle.

Apparent Depth: Objects underwater appear closer to the surface than they actually are:

Apparent depth = Real depth / n
The index n accounts for the slower light speed in water

Real-World Applications:

  • Eyeglasses and Contact Lenses: Curved surfaces refract light to compensate for improper focus, correcting myopia, hyperopia, and astigmatism. Lens power is measured in diopters (1/f).
  • Camera Lenses: Multiple lens elements with different refractive indices correct aberrations, focus images on film or sensors, and allow zoom without moving the camera.
  • Optical Fibers (Fiber Optics): Total internal reflection guides light through curved glass fibers with minimal loss. This enables high-speed internet, medical endoscopes, and laser communications.
  • Prisms in Spectroscopy: Dispersion separates light into component wavelengths, allowing analysis of chemical composition, temperature, and motion of distant stars via spectral lines.
  • Rainbows: Refraction and internal reflection in water droplets separate white sunlight into constituent colors. Each color emerges at specific angles due to wavelength-dependent refraction.
  • Diamonds: High refractive index (2.42) and low critical angle (24.4 degrees) cause multiple internal reflections. Combined with dispersion, this creates the characteristic sparkle.
  • Binoculars and Telescopes: Prisms (using TIR) fold the optical path, allowing long focal lengths in compact instruments. Porro prisms and roof prisms are common designs.
  • Atmospheric Refraction: The setting Sun appears red and flattened because light bends through atmospheric density gradients. This same effect causes mirages in deserts and on hot roads.

Lensmaker's Formula: For thin lenses: 1/f = (n - 1)(1/R1 - 1/R2), where R is radius of curvature (positive for convex surfaces facing incident light).

Snell's Law for Spherical Surfaces: n1/u + n2/v = (n2 - n1)/R, where u and v are object and image distances from the vertex.

Common Mistakes to Avoid:

  • Using Angles with Surface: Snell's law uses angles measured FROM THE NORMAL, not from the surface. Always measure from the perpendicular.
  • Forgetting n is Relative: Refractive index is relative to vacuum (n = c/v). Don't confuse this with absolute speed.
  • Not Checking TIR Conditions: Total internal reflection only occurs when n1 greater than n2 AND i greater than theta c. Both conditions must be met.
  • Ignoring Dispersion: For white light, n varies with wavelength. Using a single n gives approximate results only.
  • Wrong Sign Convention: In lens formulas, distances are signed. Object distances are usually negative; real image distances are positive.
  • Confusing Refraction with Reflection: Refraction bends light into the second medium; TIR reflects it back. Both can occur at boundaries.
  • Assuming Symmetry: Light path is reversible. If you trace from medium 2 to 1, swap n1 and n2 in Snell's law.