AcademyGeometric Optics
Academy
Spherical Refraction
Level 1 - Physics topic page in Geometric Optics.
Principle
A spherical refracting surface forms images because refraction changes ray convergence.
Notation
\(n_1\)
index on incident side
1
\(n_2\)
index on refracted side
1
\(s\)
object distance from surface
\(\mathrm{m}\)
\(s'\)
image distance from surface
\(\mathrm{m}\)
\(R\)
surface radius of curvature
\(\mathrm{m}\)
\(m\)
lateral magnification
1
Method
Derivation 1: Paraxial refraction formula
Small-angle refraction at a spherical boundary gives a linear relation among object distance, image distance, and curvature.
Surface equation
\[\frac{n_1}{s}+\frac{n_2}{s'}=\frac{n_2-n_1}{R}\]
Plane limit
\[R\rightarrow\infty\Rightarrow\frac{n_1}{s}+\frac{n_2}{s'}=0\]
Magnification
\[m=-\frac{n_1s'}{n_2s}\]
The sign of \(R\) depends on whether the center of curvature lies on the outgoing side of the surface.
Rules
Spherical refraction
\[\frac{n_1}{s}+\frac{n_2}{s'}=\frac{n_2-n_1}{R}\]
Refraction magnification
\[m=-\frac{n_1s'}{n_2s}\]
Plane refraction
\[s'=-\frac{n_2}{n_1}s\]
Examples
Question
Light goes from air to glass
\[(n=1.50)\]
at a spherical surface with \[R=0.30\,\mathrm{m}\]
An object is \[0.60\,\mathrm{m}\]
away. Find \[s'\]
Answer
\[\frac{1.00}{0.60}+\frac{1.50}{s'}=\frac{0.50}{0.30}\]
\[\frac{1.50}{s'}=0\]
The image is at infinity.Checks
- Use one sign convention consistently.
- Plane refraction changes apparent depth without focusing to a real point.
- The formula is paraxial; large angles require Snell's law ray tracing.