When we look at lenses, we’re really seeing how we can control light’s path to create useful optical effects. Unlike mirrors, which reflect light, lenses allow light to pass through while bending it in predictable ways.

Types of Lenses

Converging lenses, also called convex lenses, are thicker in the middle and thinner at the edges. They bring parallel rays of light together to meet at a focal point. A magnifying glass is a great example of this in action: it can focus sunlight into a bright hot spot.

Diverging lenses, or concave lenses, work in the opposite way. They are thinner in the middle and thicker at the edges, causing parallel rays to spread apart. These rays behave as if they came from a point behind the lens.

Lenses work they way they do because of refraction. As light moves from air into the lens material and then back into air, it bends at each surface.

Lens Terminology

To make sense of how lenses form images, it helps to understand a few important terms.

The object distance is how far the object is from the lens. This is where the actual light source or object sits.

The image distance is how far the image appears from the lens. Depending on the lens type and the object’s location, the image might be real (formed by actual converging rays) or virtual (formed by rays that only appear to diverge from a point).

The focal length tells us how strongly a lens bends light. It’s the distance from the lens to the focal point, where incoming parallel rays either converge or seem to diverge from. A short focal length means the lens bends light more sharply, while a longer focal length causes a gentler bend.

The Lens Formula

The lens equation is:

1/f = 1/do + 1/di

This tells us how the focal length, object distance, and image distance are related. If you know two of them, you can calculate the third.

Why are these in reciprocals? The two main rays used to locate an image form similar triangles on each side of the lens. In those triangles, the “slope” of each ray is the height of the object or image divided by its distance from the lens. Setting the slopes equal gives a relationship that naturally involves the reciprocals 1/do, 1/di, and 1/f. Because this reciprocal of focal length neatly describes how strongly a lens bends light, and because such values add together when lenses are stacked, opticians adopted 1/f as the practical measure of lens power. We’ll talk about power more in a second.

Sign Conventions

To keep things consistent, we follow a few rules.

Light is assumed to travel from left to right. The object distance is positive if the object is on the left side of the lens, where the light comes in. The image distance is positive if the image forms on the right side, where the light exits. Focal lengths are positive for converging lenses and negative for diverging lenses. Heights above the optical axis are positive, and those below are negative.

Magnification

Magnification compares the height of the image to the height of the object. It also tells us about orientation. The formula is:

M = hi/ho = -di/do

If the magnification is positive, the image is upright. If it’s negative, the image is inverted. A magnification greater than one means the image is larger than the object, while less than one means it’s smaller.

To get a large magnified image, either the object must be placed close to the focal point, or the image must form far away from the lens.

Lens Power

In everyday practice, especially in optometry, lenses are described by their power in diopters. The formula is: D = 1/f, where f is in meters.

A high diopter means the lens bends light strongly and has a short focal length. A low diopter means the bending is more gentle and the focal length is longer. For example, a lens with a power of +2.0 diopters has a focal length of 0.5 meters. A lens with a power of -0.5 diopters has a focal length of -2 meters.

Combining Lenses

When you line up multiple lenses, the image formed by the first lens becomes the object for the next one.

To work through a multi-lens system, first calculate the image formed by the first lens. Then treat that image as the object for the second lens. If it falls on the side opposite from where the light enters the second lens, you treat the object distance as negative. You repeat this process for each additional lens in the setup.

The final image you see depends on all the lenses involved and how far apart they are. This idea is what makes complex systems like microscopes and telescopes work.

Total Magnification

If you’re using multiple lenses, the total magnification is just the product of all the individual magnifications. This is how optical systems achieve such extreme zoom!

Microscopes and Telescopes

Microscopes use two converging lenses: one near the object (the objective lens) and one near the eye (the eyepiece). The objective forms a real, magnified, inverted image. That image then becomes the “object” for the eyepiece, which magnifies it again into a virtual image. 

Telescopes work similarly but are set up for distant viewing. The objective lens (or mirror, in some types) gathers light from faraway objects and forms an image near its focal point. The eyepiece then magnifies that image. 

In both devices, the distance between lenses is carefully adjusted to bring everything into focus.

Aberrations in Real Lenses

The simple equations we use assume perfect lenses, but real lenses have flaws that cause distortions called aberrations.

Spherical aberration happens when a lens can’t focus all incoming parallel rays to a single point. Rays that pass through the outer edges of the lens focus differently than those near the center, leading to a blurred image.

This issue arises because the spherical shape of most lenses doesn’t match the ideal shape that would perfectly focus light. Different zones of the lens act with slightly different focal lengths.

Chromatic aberration occurs because light of different colours bends differently when it passes through glass. This effect causes coloured fringes around images, especially at sharp boundaries, much like a prism creating a rainbow.

It happens because the refractive index of glass depends on the wavelength of light. Blue light bends more than red light, so the two colours come into focus at different points.

Fixing Aberrations

A possible way to fix spherical aberration, is to use an aspheric lens, which has a carefully tweaked curve so every incoming ray meets at the same point. Stopping the lens down (using a smaller opening) also helps by trimming off the stubborn edge rays, and designers often stack two or more lenses whose errors cancel each other to restore sharpness.

To reduce chromatic aberration, opticians use achromatic doublets. An achromatic doublet pairs two glasses with opposite dispersion so the colours reconverge in the same place, while apochromatic groups correct three or more colours for even cleaner results. Mirrors sidestep the problem entirely by reflecting every wavelength by the same angle, which is why large telescopes rely on curved mirrors instead of oversized glass lenses.