Thin Lens Calculator

Thin lens: 1/f = 1/p + 1/p'

The Gauss thin-lens equation links the focal length f, the object distance p and the image distance p': 1/f = 1/p + 1/p'. By convention, f > 0 for converging lenses (biconvex) and f < 0 for diverging lenses (biconcave). The linear magnification is A = -p'/p = i/o — negative means the image is inverted. Vergence (the unit used in eyeglass prescriptions) is V = 1/f in diopters (D), with f in meters. A normal emmetropic eye has about 60 D of refractive power. Example: an object 30 cm from a 10 cm lens gives 1/p' = 1/10 - 1/30 ⇒ p' = 15 cm, A = -0.5 (real, inverted, half size).

Applications

Eyeglasses and contact lenses (myopia uses diverging lenses with negative diopters, hyperopia uses converging with positive diopters, presbyopia after 40, astigmatism adds a cylindrical correction), progressive lenses, cameras (focal length × aperture), microscopes, telescopes (large objectives capture more light), and LASIK surgery, which reshapes the cornea to change its natural focal length.

FAQ

When is the image real or virtual? p' > 0 means a real image (formed on the opposite side, can be projected on a screen); p' < 0 means virtual (same side as the object, only seen through the lens — like a magnifying glass).

What are diopters? The unit of vergence: V = 1/f with f in meters. -2.00 D means a diverging lens with f = -50 cm (typical mild myopia).

Why does the same formula model the human eye? Because the eye behaves like a converging optical system (cornea + lens) projecting onto the retina. Refractive errors arise when the effective focal length doesn't match the eye's length.