Light · two surfaces
Lenses
A lens is two refracting surfaces close together. Thicker in the middle, it bends parallel light together to a focus. Thinner in the middle, it spreads light as if from a point in front of it. Here every ray is traced through both real spherical surfaces, so you see what a simple formula hides: the rays from the rim do not focus where the rays near the axis do.
The lens
The light
Measured
The lensmaker’s formula
Light travels left to right. A surface’s radius R is positive when its centre of curvature lies to the right of it. So a biconvex lens has R1 > 0 and R2 < 0. For a lens of index n and centre thickness t, the focal length f is
The focal length is measured from the lens’s principal plane. The back focal distance, from the last surface to the focus F′, is what you measure with a ruler. Power is 1/f in dioptres when f is in metres: a +2.5 reading glass has f = 400 mm. For the default lens here (crown glass, R = ±100 mm, 12 mm thick) f = 98.8 mm and the back focal distance is 94.7 mm.
Two flaws of a simple lens
Spherical aberration. A sphere is easy to grind but is not the shape that sends every ray to one point. Rays through the rim are bent too much and cross the axis short of the focus, so the image of a star is a small blur, not a point. Make the lens wider and watch the crossing points spread along the axis. Opticians reduce the flaw by sharing the bending between surfaces (turn a plano-convex lens with its curved side to the parallel light), by stopping down the aperture, or by grinding an asphere.
Chromatic aberration. The index is higher for blue, so blue focuses closer to the lens than red. For the default lens the blue F light focuses 1.49 mm before the red C light. Choose white light and look at the coloured fringe at the focus. An achromatic doublet cures most of it: a converging crown-glass lens cemented to a weaker diverging flint-glass lens, whose larger dispersion cancels the crown’s while leaving some converging power.
The ideal thin lens
The ideal thin lens is a model with no thickness and no aberrations. Every ray from one point meets again at one point. It is what the familiar formula describes, with u the object distance in front of the lens, v the image distance behind it (negative when the image is virtual) and magnification −v/u. Choose it, put a point source on the axis, and compare it with a real lens of the same focal length.
Reading stones, polished domes of rock crystal or glass laid on a page, are older than spectacles. Spectacles appeared in northern Italy in the late 1280s: in a sermon preached in Florence in February 1306, the Pisan friar Giordano said the art of making them had been found less than twenty years before. The achromatic doublet was made by Chester Moor Hall in the 1730s and patented by John Dollond in 1758. Today a camera lens of a dozen elements is designed by tracing many thousands of rays, exactly as this bench does, only in three dimensions.
These are rays, the geometrical-optics picture: light is treated as lines that bend and split at surfaces. Diffraction and interference, which need the wave picture, are not shown. The bench is a flat cross-section, so a lens here is really a cylinder of glass seen end-on. That is exact for rays in the plane of the drawing. Air is taken as n = 1, and the glass and water indices are given relative to air, so the angles are right. Colours on screen are an approximation of the spectrum.