Geometric Optics Deep Dive: Bending Light and Making Images

Refraction

The change in direction of a light ray when it crosses a boundary between two different media because light travels at different speeds in different materials.

Wavefront model (of refraction)

A way to visualize refraction: part of a wavefront entering a slower medium first slows first, causing the wavefront to pivot; rays (perpendicular to wavefronts) therefore change direction at the boundary.

Normal

The line drawn perpendicular to a surface at the point where a ray hits; angles in Snell’s law are measured from the normal, not from the surface.

Refractive index (n)

A unitless measure defined by n = c/v, where c is the speed of light in vacuum and v is the speed in the medium; higher n means lower speed in that medium.

Snell’s Law

The law of refraction: n1 sin(θ1) = n2 sin(θ2), relating incident and refracted angles to the refractive indices of the two media.

Angle of incidence (θ1)

The angle between the incoming (incident) ray and the normal to the surface.

Angle of refraction (θ2)

The angle between the transmitted (refracted) ray and the normal to the surface.

Bend toward the normal

What happens when light enters a higher-index (slower) medium (n2 > n1): the refracted angle decreases (θ2 < θ1).

Bend away from the normal

What happens when light enters a lower-index (faster) medium (n2 < n1): the refracted angle increases (θ2 > θ1).

Frequency (across a boundary)

The frequency of light does not change when crossing between media; it is set by the source.

Wavelength change (v = fλ)

Because v = fλ and f stays constant at a boundary, a decrease in speed (higher n) causes wavelength λ to decrease, and an increase in speed causes λ to increase.

Apparent depth

An effect where underwater objects appear closer to the surface because rays refract at the water–air boundary and your brain traces them back as straight lines.

Total internal reflection (TIR)

When light in a higher-index medium hits a boundary to a lower-index medium at a large enough incident angle and reflects entirely back into the original medium (no transmitted refracted ray).

Critical angle (θc)

The incident angle (from the normal) in the higher-index medium for which the refracted ray would be at 90°; given by sin(θc) = n2/n1 (with n1 > n2).

Evanescent wave

A non-propagating electromagnetic field that can extend a short distance into the lower-index medium even during total internal reflection.

Fiber-optic cable

A light-guiding cable that confines light in its core by repeated total internal reflections, typically using a core index slightly higher than the cladding.

Thin lens

A lens approximated as having negligible thickness compared with object and image distances, allowing simplified image-formation equations (thin lens equation).

Converging (convex) lens

A lens that bends parallel incoming rays to meet at a focal point; it has positive focal length (f > 0) in the usual sign convention.

Diverging (concave) lens

A lens that spreads parallel incoming rays as if they came from a focal point on the incident side; it has negative focal length (f < 0) in the usual sign convention.

Focal length (f)

The distance from the lens to the focal point for rays that are initially parallel to the principal axis.

Thin lens equation

The relationship between focal length, object distance, and image distance: 1/f = 1/do + 1/di.

Sign convention ("real is positive" for lenses)

Common AP convention: f > 0 for converging and f < 0 for diverging; do > 0 for real objects; di > 0 for real images and di < 0 for virtual images.

Magnification (m)

Image-to-object size ratio: m = hi/ho = −di/do; |m| gives size change and the sign indicates orientation.

Real image

An image formed where rays physically converge (typically di > 0); for a single converging lens with a real object outside f, the real image is usually inverted (m < 0).

Virtual image

An image formed where rays only appear to originate when traced backward (typically di < 0); it is on the same side of the lens as the object and is typically upright (m > 0) for a single lens.