Refraction, Index, and Total Internal Reflection

Learning Goals

• Interpret refractive index as a ratio of vacuum speed to phase speed in a material.
• Relate frequency, wavelength, and speed as light crosses a stationary boundary.
• Apply Snell's law with every angle measured from the local normal.
• Predict whether a ray bends toward or away from the normal before calculating.
• Use refraction to analyze apparent depth and parallel-sided slabs.
• Derive the critical-angle condition and identify both requirements for total internal reflection.
• Explain how a higher-index core and lower-index cladding guide light in an optical fiber.

1. Refractive Index Encodes Light Speed

The refractive index of a material is defined by

Here \(c\approx3.00\times10^8\,\mathrm{m/s}\) is the vacuum speed of light and \(v\) is the phase speed in the material. For ordinary transparent media in this course, \(n\ge1\). A larger \(n\) means a smaller phase speed at the stated wavelength.

Refractive index is not mass density. Two materials can have similar densities but different optical indices, and \(n\) can vary with wavelength, temperature, composition, and direction in anisotropic crystals.

2. Frequency Stays Continuous; Speed and Wavelength Change

A source fixes the light's frequency. At a stationary boundary, the electromagnetic fields must oscillate together along the interface, so the transmitted wave has the same frequency as the incident wave. Because

a change in speed changes wavelength:

Entering a higher-index medium decreases speed and wavelength, but not frequency. Consequently photon energy \(E=hf\) does not change merely because the photon enters a different stationary medium.

3. Snell's Law Predicts the Refracted Direction

At a planar boundary between media \(1\) and \(2\),

Both angles are measured from the normal. A useful physical prediction comes before the calculator:

• If \(n_2>n_1\), then \(\theta_2<\theta_1\): the ray bends toward the normal.
• If \(n_2\theta_1\): the ray bends away from the normal.
• If \(\theta_1=0^\circ\), the ray does not change direction even if its speed changes.

A reflected ray generally exists at the same boundary and obeys \(\theta_r=\theta_1\); Snell's law describes the transmitted ray.

4. Why Snell's Law Has Sines

Consider equally spaced wavefronts reaching a boundary at an angle. During the same time interval, the part already in the second medium advances at \(v_2\) while the part still in the first advances at \(v_1\). Matching the wavefront phase along the shared boundary produces

Rearranging gives Snell's law. This wavefront construction explains both the direction of bending and the reversibility of the path.

5. Turn Angle Data into a Material Measurement

For fixed media, solve Snell's law as

A graph of \(\sin\theta_2\) versus \(\sin\theta_1\) should be linear through the origin with slope \(n_1/n_2\). A fit to several angle pairs is more reliable than calculating an index from one measurement.

6. Parallel-Sided Slabs: Direction Restored, Position Shifted

A ray entering a parallel glass slab bends at the first surface and bends back at the second. If the external medium is the same on both sides, the emerging ray is parallel to the incident ray but laterally displaced. For slab thickness \(t\),

This expression assumes parallel faces and measures \(t\) along the surface normal. It explains why looking through a tilted window can shift an object's apparent position without permanently rotating the outgoing ray.

7. Apparent Depth at a Flat Interface

Light from an underwater object bends away from the normal when it enters air. An observer extends those rays backward in straight lines and locates a virtual image closer to the surface. For viewing nearly along the normal,

This small-angle relation is an approximation. At large viewing angles, the apparent position depends on the observer's direction and the exact Snell-law geometry.

8. Critical Angle from Snell's Law

Suppose light travels from a higher-index medium \(n_1\) toward a lower-index medium \(n_2\). As \(\theta_1\) increases, the refracted angle grows. At the critical incidence angle, the refracted ray would travel along the boundary, so \(\theta_2=90^\circ\):

The formula itself warns about direction. If \(n_1\le n_2\), the ratio is at least one and there is no critical angle for that direction of travel.

9. The Two Conditions for Total Internal Reflection

1. Light must travel from the higher-index medium toward the lower-index medium: \(n_1>n_2\).
2. The incident angle in the higher-index medium must satisfy \(\theta_1>\theta_c\).

At exactly \(\theta_c\), the ideal refracted ray travels along the boundary; this limiting case is not yet the usual “greater than critical” TIR condition. For \(\theta_1>\theta_c\), geometric optics has no propagating transmitted ray, though an evanescent field extends a short distance into the second medium.

12. Optical Fibers Guide Light

An optical fiber uses a core of index \(n_{\mathrm{core}}\) surrounded by cladding with slightly smaller index \(n_{\mathrm{clad}}\). Rays that strike the core–cladding boundary above its critical angle remain confined by repeated TIR:

Cladding provides a controlled boundary even when fibers touch, protects the core surface, and limits coupling between neighboring fibers. Excessive bending can reduce the internal incidence angle below critical, allowing light to escape.

13. Dispersion and Color

In many transparent materials, refractive index depends on wavelength: \(n=n(\lambda)\). Different colors therefore travel at different phase speeds and refract by different angles. A prism separates colors because its nonparallel faces prevent the second refraction from simply restoring the original direction.

When a problem supplies one index without mentioning wavelength, treat it as the effective index for the light being studied. Do not assume every color follows exactly the same path when dispersion matters.

14. Data Collection and Uncertainty

Use a semicircular block to reduce unintended refraction at one surface: aim the ray through the curved surface along a radius so it reaches that surface normally. At the flat face, measure incidence and refraction angles from the normal.

Collect several pairs, graph \(\sin\theta_2\) versus \(\sin\theta_1\), and use the slope to determine an index ratio. Include uncertainty from beam width, protractor resolution, surface alignment, and identifying the center of the ray. Near the critical angle, small angle errors can change whether a faint transmitted beam is observed.