Spherical Refracting Surfaces: Formula and Ray Diagrams

Most lenses and mirrors are parts of spheres. A single spherical refracting surface — the boundary between two media curved like part of a ball — already can form images. Understanding one surface is the key to understanding complete lenses.

Eyes, cameras and microscopes all rely on spherical surfaces to bend light predictably. The formula for a single surface combines geometry and Snell law for rays close to the axis.

Key Concepts

Centre of curvature C and radius R: Centre of the sphere of which surface is part. R in metres, positive if centre is on refracted side in common convention.

Paraxial rays: Rays close to principal axis and making small angles, where Snell law can be approximated. Real imaging assumes these rays.

Object distance u, image distance v: Measured from pole of surface. Sign convention must be consistent. In Cartesian convention, light travels left to right, distances to right positive.

• Surface separates medium n1 (incident) and n2 (refracted)
• Convex surface toward incident light usually has positive R for image side
• Formula works for both real and virtual images

Formulas

n2 divided by v minus n1 divided by u = (n2 minus n1) divided by R. Here n1, n2 dimensionless, u, v, R in metres with sign. Assumes paraxial rays and spherical surface.

Derivation uses Snell law n1 i approx n2 r for small angles and triangle geometry.

Special case: plane surface R infinite, formula reduces to n2 divided by v = n1 divided by u, giving apparent depth.

Worked Example

A spherical surface radius 10 cm separates air n1 = 1.0 (left) from glass n2 = 1.5 (right), convex toward air. Object in air 20 cm left of surface. Find image position. Use Cartesian: u = minus 20 cm (object left, but formula uses sign where distances left negative? Let us use convention distances measured from pole, object side negative, so u = minus 20 cm, R = plus 10 cm (centre right).

1. Write formula: n2 divided by v minus n1 divided by u = (n2 minus n1) divided by R.
2. Plug: 1.5 divided by v minus 1.0 divided by minus 20 = (1.5 minus 1.0) divided by 10.
3. Left: 1.5 divided by v plus 0.05 = 0.05.
4. So 1.5 divided by v = 0, meaning v infinite.

Image at infinity means refracted rays are parallel. Object at special point creates collimated output. Check: object distance 20 cm equals (n1 R) divided by (n2 minus n1) gives aplanatic condition. A slightly nearer object would give finite image, students can recalc with u = minus 15 cm to get v = 90 cm.

Common Misconceptions

• R sign always positive: Sign depends on convention and whether centre is on refracted side. Using wrong sign flips image location.
• Formula only for lenses: It is for single surface. Lens needs applying twice, once per surface.
• Large angles obey same formula exactly: Formula is paraxial approximation. Wide rays cause spherical aberration and blur.

Real-World Applications

• Human eye cornea is main spherical refracting surface providing most focusing power
• Design of simple magnifiers and lens elements starts from single surface power (n2 minus n1) divided by R
• Apparent depth and fish-eye views explained by plane and spherical surfaces
• Optical fibre ends and ball lenses use spherical refraction

Single spherical surface formula n2 divided by v minus n1 divided by u = (n2 minus n1) divided by R predicts image with just curvature and indices. Apply it twice and you have full lens theory.