Refraction at Spherical Surfaces
When light travels from one transparent medium into another — say, from air into glass — it bends at the boundary. If that boundary is curved rather than flat, something remarkable happens: the bending is no longer uniform across the surface, and rays that were originally diverging or converging are redirected so that they converge to, or appear to diverge from, a single point called an image. This is the principle behind lenses, the human eye, and virtually every optical instrument ever built. Refraction at a spherical surface is the fundamental building block of all such devices. A spherical refracting surface is simply the curved interface between two media of different refractive indices, whose shape is part of a sphere. Think of the front surface of a glass paperweight, or the curved surface of a fish's eye. Each point on the surface has a normal that points toward the centre of curvature of the sphere. Because Snell's law acts along each local normal, rays at different heights from the axis are bent by slightly different amounts, producing a focusing effect. The geometry is elegantly captured by a single refraction formula that relates object distance, image distance, and the radius of curvature. Understanding refraction at a single spherical surface is essential before studying lenses, because a lens is simply two such surfaces back-to-back. Mastering the sign convention, the derivation, and the physical meaning of the formula gives you a powerful toolkit: you can predict where images form for convex and concave surfaces, identify real versus virtual images, and understand how the eye focuses light on the retina — all from one equation.
- Setting Up the Geometry
- Applying the Cartesian Sign Convention
- Deriving the Refraction Formula
- Interpreting the Formula
- First and Second Focal Points
Setting Up the Geometry
Place the spherical surface so that its pole (the point where the surface crosses the principal axis) is at the origin. The centre of curvature C lies on the principal axis at distance R from the pole. Medium 1 (refractive index n₁) is on the incoming side; medium 2 (refractive index n₂) is on the transmitted side. An object O sits in medium 1 at distance u from the pole, measured along the axis.
Applying the Cartesian Sign Convention
All distances are measured from the pole. Distances in the direction of the incident light (left to right) are positive; distances against the incident light are negative. So u is negative if the object is to the left of the surface (real object), v is positive if the image is to the right (real image), and R is positive if the centre of curvature is to the right of the surface (convex surface as seen by incoming light).
Deriving the Refraction Formula
Consider a ray from O hitting the surface at height h above the axis (with h small — the paraxial approximation). The angle of incidence i and angle of refraction r are both measured from the normal at the point of incidence. Using the small-angle approximation (sin θ ≈ tan θ ≈ θ for angles in radians) and Snell's law (n₁ sin i = n₂ sin r), along with the geometry of the triangle formed by O, the point of incidence, and C, one can show after careful algebra that: n₁/u + n₂/v is related to the curvature. The result is the standard formula shown in the equation block.
Interpreting the Formula
The formula tells you the image position v given any object position u. If v comes out positive, the image is on the transmission side — a real image, formed by actual converging rays. If v is negative, rays only appear to diverge from that point — a virtual image. A concave surface (R < 0 for light travelling left-to-right into a denser medium) tends to produce virtual images, while a convex surface into a denser medium tends to converge rays and produce real images sufficiently far from the surface.
First and Second Focal Points
The first focal point F₁ is the object position that produces an image at infinity (parallel transmitted rays). Setting 1/v = 0 gives the first focal length f₁ = n₁ R / (n₂ − n₁). The second focal point F₂ is where parallel incident rays converge; setting 1/u = 0 gives f₂ = n₂ R / (n₂ − n₁). Note that f₁ and f₂ are different in magnitude unless n₁ = n₂, unlike mirrors. This asymmetry is fundamental to how compound optical systems are designed.
The Key Equation
- n_1 — Refractive index of medium 1 (dimensionless): The refractive index of the medium containing the incoming light (e.g. air ≈ 1.00)
- n_2 — Refractive index of medium 2 (dimensionless): The refractive index of the medium into which light is transmitted (e.g. glass ≈ 1.5)
- u — Object distance (metres (m)): Distance from the pole of the surface to the object, negative for a real object to the left of the surface
- v — Image distance (metres (m)): Distance from the pole to the image; positive means a real image on the transmission side
- R — Radius of curvature (metres (m)): Radius of the spherical surface; positive if the centre of curvature is on the transmission side (convex surface)
Worked Example
Problem. A glass sphere of refractive index n₂ = 1.50 has a convex front surface with radius of curvature R = +20 cm. A small object in air (n₁ = 1.00) is placed 60 cm to the left of the surface. Find the position of the image formed by refraction at this single surface.
Answer. The image forms 36 cm to the right of the surface (inside the glass), and is a real image because v is positive.
Origins · 1 minWillebrord Snellius and how this was discovered
Willebrord Snellius (also known as Snell) discovered the law of refraction empirically around 1621, showing that the ratio of the sines of the angles of incidence and refraction is constant for a given pair of media. This law — Snell's Law — is the cornerstone of every calculation involving refraction, including the formula for refraction at spherical surfaces. Without it, predicting how curved interfaces focus light would be impossible. Snellius never published his discovery himself; it became widely known only after René Descartes independently derived and published it in 1637. Historians later found Snell's manuscript, confirming he had reached the same result over a decade earlier.
Related Concepts
- Snell's Law and Refractive Index — The refraction formula for spherical surfaces is derived directly from Snell's Law applied at each point of the curved interface, so a thorough understanding of Snell's Law is a prerequisite.
- Thin Lenses and the Lens Formula — A thin lens consists of two spherical refracting surfaces; the lensmaker's equation is derived by applying the single-surface refraction formula twice in succession.
- Reflection at Spherical Mirrors — Spherical mirrors obey an analogous formula (the mirror equation) and use the same Cartesian sign convention, making comparison with spherical refracting surfaces highly instructive.
- Total Internal Reflection — When light travels from a denser to a rarer medium at a curved surface, total internal reflection can occur instead of refraction, setting a practical limit on the use of curved refracting surfaces.