Lenses, the eye and optical instruments

A camera forms an image on a sensor, while a magnifying glass shows enlarged letters without a screen. The same converging lens can do both. Trace where the rays meet as the object moves.

Send a beam parallel to a lens's axis through it. If refracted rays converge, the lens is converging; if they spread out, it is diverging. A convex glass lens in air usually converges light, but shape is not enough: the relative refractive index of glass and medium matters. In another medium, the lens may act differently. Its name describes an observable effect on rays that we can then calculate.

For a converging lens, rays parallel to the axis meet at the principal focus. For a diverging lens, their backward extensions meet at a virtual focus; the rays themselves never passed there. Focal length is measured from the optical centre of a thin lens. Stronger bending of the parallel beam means a nearer focus. This is a lens property; the image position of an arbitrary nearby object must still be found separately.

Choose the object's top point and draw two rays with known paths. A ray parallel to the principal axis passes through the far focus after a converging lens. A ray through the optical centre of a thin lens changes direction negligibly. Their intersection gives the image of that point. We selected two useful rays from many as a construction method; the lens does not transmit only two thin lines of light from the point.

If the actual rays from a point converge after the lens, place a screen there to obtain a real image. If rays diverge, extend them backward with dashed lines. Where the extensions meet is the apparent source position: a virtual image. The eye can see it by receiving the diverging beam, but a screen at that apparent position cannot catch it. The distinction depends on actual light paths, not on how convincing a picture seems.

Start with an object beyond 2F of a converging lens. The central ray and the ray through the focus meet on the far side between F and 2F, below the axis for the object's top. The image is therefore real, inverted and smaller. Move the object to 2F: the image lies at 2F and has equal size. As the object approaches, its image moves away; the focus is not the fixed location of every image.

Move the object farther in, between F and 2F. The rays now meet beyond 2F, giving an enlarged image that remains real and inverted. Once the object crosses the focus and is closer to the lens, emerging rays diverge. Their extensions form an upright, enlarged virtual image on the object's side. This is how a magnifying glass works: object position changes both size and how the image is formed.

What if an object is exactly at a converging lens's focus? Rays from each point emerge as a parallel beam, so no finite screen image forms. As the object approaches the focus from farther away, the image recedes ever farther. For a real object, a diverging lens consistently forms an upright, smaller virtual image between lens and near focus. Its emerging rays do not actually intersect beyond the lens.

Use signs to distinguish cases within one formula. For a real object, d > 0. Focal length F > 0 for a converging lens and F < 0 for a diverging lens. Image distance f > 0 for a real image and f < 0 for a virtual one. These are algebraic distances under our convention. Negative f is not a negative physical length: its sign encodes image type and side.

Thin-lens ray geometry gives 1/F = 1/d + 1/f. Use the same distance unit and keep the sign convention. For a very distant object, 1/d is nearly zero, so f is near F: the focus definition returns. At d = F, 1/f = 0 and there is no finite image. The formula is approximate: the lens is thin and rays stay near the axis; wide beams and thick glass require refinements.

Similar triangles formed by the central ray give Γ = |f|/d. This is the positive, dimensionless image-to-object size ratio. Γ = 2 means twice the size; Γ = 0,5 means half. The absolute value removes the sign of algebraic f; ray paths determine whether the image is upright or inverted. Γ = 1 means equal sizes, but does not by itself specify orientation.

Two lenses have different focal lengths: which converges light more strongly? A shorter focal length means stronger ray bending. Optical power expresses this as D = 1/F, with F necessarily in metres. Its unit is the dioptre: 1 дптр = 1 м⁻¹. Halving positive F doubles D; both are negative for a diverging lens. Using centimetres without conversion causes a hundredfold error.

The cornea and crystalline lens act as the eye's lens system, forming a real, inverted image on the retina. Objects can be near or far, but the retina does not follow each image. The lens changes curvature and optical power to focus: this is accommodation. Our model helps explain focusing, although a real eye is more complex than one thin lens.

In uncorrected short-sightedness, a distant object's image forms in front of the retina. A diverging lens reduces convergence, helping move the focus onto the retina. In long-sightedness, focus can lie behind it; a converging lens strengthens convergence. A camera's real image must likewise fall on its sensitive sensor. Focusing mechanisms differ, but the sharpness condition is the same: the required rays must meet on the receiver.

A converging lens has F = 10 см. An object is at d = 30 см. Find the image position and magnification.

F = 10 см, so 2F = 20 см. At 30 cm, the object lies beyond 2F: expect a real, inverted, reduced image between 10 and 20 cm.

Find image distance: 1/f = 1/F − 1/d = 1/10 − 1/30 = 1/15 см⁻¹. Thus f = 15 см, with a positive sign.

Γ = |f|/d = 15/30 = 0,5: the image is half the object's size.

15 cm does lie between F and 2F. Positive f agrees with actual rays meeting beyond the lens, where a screen can be placed.

The image lies 15 cm beyond the lens: real, inverted and half-size (Γ = 0,5). A screen there shows a sharp image in this model.

Moving the object changes image position, size and type; the focus is the special case for parallel rays.

Actual rays form a real image; their extensions form a virtual one. Check this before calculating.

The lens formula needs consistent signs, while D = 1/F requires focal length in metres.

The ray model has limits: next we explain fringes, edges and colour.