The Thin-Lens Equation and Magnification

The thin-lens equation, 1/f = 1/d_o + 1/d_i, relates the focal length f of a lens to the object distance d_o and image distance d_i, and the magnification m = −d_i ÷ d_o = h_i ÷ h_o gives the size and orientation of the image. In brief: use the sign convention that f is positive for a convex lens and negative for a concave lens, d_i is positive for a real image and negative for a virtual image, and m is negative for an inverted image and positive for an upright one. The power of a lens is P = 1/f in dioptres (D) when f is in meters.

Ray diagrams give a qualitative picture, and the thin-lens equation gives exact values. This guide states the equation and the sign convention, explains lens power and the combination of lenses in contact, compares sign conventions in a table, works through numerical examples with convex and concave lenses and a camera, and corrects common mistakes.

What is the thin-lens equation?

For a thin lens with an object at distance d_o from the lens and an image at d_i:

1/f = 1/d_o + 1/d_i

It follows from similar triangles in the ray diagram, in the paraxial approximation (rays close to the axis and at small angles to it). When the object is very far (d_o → ∞), 1/d_o → 0 and d_i = f: the image of a distant object forms at the focal plane. For a real image from a convex lens, an object distance equal to 2f gives an image distance of 2f, which is the smallest possible distance between an object and a real image (4f).

Real-image distance from 1/f = 1/d_o + 1/d_i for a convex lens with f = 20 cm. The image runs off to infinity as the object approaches the focal point and approaches the focal length for distant objects. An object at 2f = 40 cm gives an image at 40 cm (the smallest object-to-image distance, 4f = 80 cm); an object at 50 cm gives 33 cm and one at 30 cm gives 60 cm.

How is magnification found?

The (lateral) magnification is m = h_i ÷ h_o = −d_i ÷ d_o. |m| > 1 means that the image is larger than the object, and |m| < 1 means that it is smaller. The sign gives the orientation: m < 0 for an inverted image (real images of single convex lenses) and m > 0 for an upright image (virtual images). The area of the image is magnified by m². The ratio d_i ÷ d_o follows from the geometry of the central ray, which goes straight through the lens.

What is lens power?

The power of a lens is P = 1/f, with f in meters, and its unit is the dioptre (D). A convex lens with f = 0.25 m has P = +4.0 D. A concave lens with f = −0.50 m has P = −2.0 D. Opticians prescribe glasses in dioptres. For two thin lenses in contact, the powers add: P = P₁ + P₂. This is convenient because it avoids working with reciprocals.

Sign convention

Sign convention for thin lenses (real-is-positive)
Quantity	Positive when	Negative when
Focal length f	Convex (converging) lens	Concave (diverging) lens
Object distance d_o	Real object (usual case)	Virtual object (rare)
Image distance d_i	Real image, on the opposite side of the lens	Virtual image, on the same side as the object
Magnification m	Upright image	Inverted image

Worked example: calculations

Real image. A convex lens of f = 20 cm forms an image of an object 6.0 cm tall, placed 50 cm in front of it. 1/d_i = 1/20 − 1/50 = 0.0300, so d_i = 33.3 cm. m = −33.3 ÷ 50 = −0.667, so the image height is h_i = −0.667 × 6.0 = −4.0 cm (4.0 cm, inverted).
Magnifying glass. A convex lens with f = 10 cm is held 6.0 cm from a coin. 1/d_i = 1/10 − 1/6.0 = 0.100 − 0.1667 = −0.0667, so d_i = −15 cm (virtual, 15 cm behind the lens on the object side). m = −(−15) ÷ 6.0 = +2.5: upright and 2.5 times larger.
Concave lens. f = −15 cm, d_o = 45 cm: 1/d_i = −1/15 − 1/45 = −0.0889, d_i = −11.25 cm, m = +0.25. Virtual, upright, diminished.
Camera. A camera lens has f = 50 mm and photographs a person 2.0 m (2,000 mm) away. 1/d_i = 1/50 − 1/2,000 = 0.0200 − 0.0005 = 0.0195, d_i = 51.3 mm. The lens moves 1.3 mm from the infinity-focus position. m = −51.3 ÷ 2,000 = −0.0256, so a 1.8 m person gives an image 46 mm tall on the sensor.
Power and combination. A lens with f = 0.20 m has P = 5.0 D. Putting it in contact with a lens of −2.0 D gives P = 3.0 D, and f = 1 ÷ 3.0 = 0.33 m.

Where does the equation come from?

The relation between object and image distances was derived by Johannes Kepler and by Edmond Halley in the 1600s. Carl Friedrich Gauss developed the theory of the paraxial approximation in 1841, which gave the thin-lens formula its modern form. The unit dioptre was proposed by the French ophthalmologist Ferdinand Monoyer in 1872. More accurate optical design uses computer programs that trace many rays through lens systems.

Which mistakes are common?

Mistake 1: Forgetting to invert at the end

Calculate 1/d_i first, then take the reciprocal.

Mistake 2: Wrong signs for concave lenses and virtual images

Use f < 0 for concave lenses. A negative d_i means that the image is virtual.

Mistake 3: Mixing units

Use the same unit (cm or m) for f, d_o and d_i. For the power, convert f to meters.

Mistake 4: Using the equation for thick lenses and wide beams without caution

The formula is an approximation for thin lenses and rays near the axis.