Lenses: the lens maker's formula and the thin lens formula

Lens words and ray rules

A lens is a transparent body bounded by two surfaces, at least one of them curved. A thin lens is much thinner than its radii of curvature, so all distances are measured from its optical centre. In air, a convex lens (thicker in the middle) converges light, and a concave lens (thinner in the middle) diverges it. A lens has two principal foci, one on each side. Three rays help you draw any image. (1) A ray parallel to the axis passes through the far focus (convex lens), or seems to come from the near focus (concave lens). (2) A ray through the optical centre goes on undeviated. (3) A ray through the near focus (convex), or heading for the far focus (concave), leaves parallel to the axis.

Derivation: lens maker's formula and thin lens formula

A thin lens of refractive index n₂ sits in a medium of refractive index n₁. A point object O is on the axis. Use paraxial rays and the formula for one refracting surface, twice.

1. 1
   The first surface (radius R₁) alone would form an image I₁ at distance v₁: n₂/v₁ − n₁/u = (n₂ − n₁)/R₁. ...(1)
2. 2
   I₁ acts as the object for the second surface (radius R₂). Light now goes from n₂ back into n₁. The lens is thin, so I₁ is also at distance v₁ from this surface.
3. 3
   Second surface: n₁/v − n₂/v₁ = (n₁ − n₂)/R₂. ...(2)
4. 4
   Add (1) and (2). The n₂/v₁ terms cancel: n₁/v − n₁/u = (n₂ − n₁)(1/R₁ − 1/R₂).
5. 5
   Divide by n₁ and write n₂₁ = n₂/n₁: 1/v − 1/u = (n₂₁ − 1)(1/R₁ − 1/R₂). ...(3)
6. 6
   For an object at infinity (u = −∞), the image forms at the focus, so v = f.
7. 7★
   So, 1/f = (n₂₁ − 1)(1/R₁ − 1/R₂). This is the lens maker's formula.
8. 8★
   Put this into (3): 1/v − 1/u = 1/f. This is the thin lens formula.

★ key step

How to draw the ray diagram for the derivation

• Draw a principal axis and a thin double convex lens across it. Call its left face the first surface and its right face the second surface.
• Mark the object O on the axis on the left. Mark C₁, the centre of the first surface, on the right, and C₂, the centre of the second surface, on the left.
• Draw a ray from O to the first surface. Bend it towards the axis, and continue it as a dotted line to I₁, where the first surface alone would form the image.
• At the second surface, bend the ray further so that it meets the axis at I, nearer the lens than I₁.
• Label n₁ outside the lens, n₂ inside, and R₁, R₂, u, v₁ and v. Put arrows on the rays.

Signs of R₁, R₂ and f for common lenses in air (n = 1.5)

Worked example: lens maker's formula

Activity: focal length of a convex lens and of a concave lens

Two practical experiments on an optical bench, using the no-parallax method.

1. 1
   Convex lens (u-v method): Mount the lens on the optical bench. Place the object needle beyond F. Locate the real, inverted image with the image needle by removing parallax.
2. 2
   Record u (negative) and v (positive) for six object positions. Find f = uv/(u − v) for each, and take the mean.
3. 3
   Graphs: v against u is a curve (a rectangular hyperbola). With signs, 1/v against 1/u is a straight line of slope 1 that cuts the 1/v axis at 1/f. If you plot sizes without signs, the line cuts both axes at 1/f.
4. 4
   Result: f is positive. For example, u = −30 cm and v = +60 cm give f = (−1800) ÷ (−90) = +20 cm.
5. 5
   Concave lens (using a convex lens): Form a real image I of the needle with the convex lens alone, and mark its position.
6. 6
   Place the concave lens between the convex lens and I. The image moves farther away, to I′. Locate I′ by removing parallax.
7. 7
   For the concave lens, I is a virtual object: u = +(distance to I) and v = +(distance to I′). Use f = uv/(u − v).
8. 8
   Result: f is negative. For example, u = +10 cm and v = +15 cm give f = 150 ÷ (−5) = −30 cm.