How To Calculate Focal Length Of A Lens

Before the 1590s, simple lenses dating back as far as the Romans and Vikings allowed limited magnification and simple eyeglasses. Zacharias Jansen and his father combined lenses from simple magnifying glasses to build microscopes and, from there, microscopes and telescopes changed the world with geometrical optics. Understanding the focal length of lenses was crucial to combining their powers.

Different focal lengths will change the field of view and the perspective of the object you are observing. It mostly depends on the distances between the object, lens, and viewing piece, but the index of refraction of the glass and other conditions do also affect the observations. With a camera lens, the sensor size, effective focal length, and pixels of resolution all factor into the final image as well. Longer focal length will often lead to greater magnification with a sharper image than digital zoom, while shorter focal length will lead to a wider field of view.

There are two basic types of lenses: convex and concave. Convex lenses are thicker in the middle than on the edges and cause light rays to converge to a point. Concave lenses are thicker on the edges than in the middle and cause light rays to diverge. These operate on very similar principles to a concave mirror and a convex mirror, but they permit light to pass through and get refracted instead of reflecting off. These mirrors are often used in combination with lenses for telescopes and cameras.

Convex and concave lenses come in different configurations. Plano-convex lenses are flat on one side and convex on the other while bi-convex (also called double-convex) lenses are convex on both sides. Plano-concave lenses are flat on one side and concave on the other side while bi-concave (or double-concave) lenses are concave on both sides.

A combined concave and convex lens – known as a concavo-convex lens – is more commonly called the positive (converging lens) meniscus lens. This lens is convex on one side with a concave surface on the other side, and the radius of curvature on the concave side is greater than the radius of the convex side.

Another type, a combined convex and concave lens called a convexo-concave lens is more commonly called a negative (diverging lens) meniscus lens. This lens, like the concavo-convex lens, has a concave side and a convex side, but the radius on the concave surface is less than the radius on the convex side.

The focal length of a lens ‌f‌ is the distance from a lens to the focal point ‌F‌. Light rays (of a single frequency) traveling parallel to the optical axis of a convex or a concavo-convex lens will meet at the focal point which is on the same axis as the center of the lens.

A convex lens converges parallel rays to a focal point with a positive focal length. Because the light goes through the lens, positive image distances (and real images) are on the opposite side of the lens from the object. The image will be inverted (up-side down) relative to the actual image.

A concave lens diverges parallel rays away from a focal point, has a negative focal length and forms only virtual, smaller images. Negative image distances form virtual images on the same side of the lens as the object. The image will be oriented the same direction (right-side up) as the original image, just smaller.

Finding focal length uses the focal length formula and requires knowing the distance from the original object to the lens ‌u‌ and the distance from the lens to the image ‌v‌. The lens formula says that the inverse of the distance from the object plus the distance to the image equals the inverse of the focal distance ‌f‌. The equation, mathematically, is written:

\(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\)

Sometimes the focal length equation is written as:

\(\frac{1}{f} = \frac{1}{o} + \frac{1}{i}\)

where ‌o‌ refers to the distance from the object to the lens, ‌i‌ refers to the distance from the lens to the image and ‌f‌ is the focal length.

The distances are measured from the object or the image to the pole of the lens.

When an object distance is very large, the ‌u‌ term becomes negligible because 1/‌u‌ converges to approximately 0. For these objects that are very far away we can approximate:

\(f \approx v\)

Focal Length Examples

To find the focal length of a lens, measure the distances and plug the numbers into the focal length formula. Be sure all measurements use the same measurement system.

‌Example 1‌: The measured distance from a lens to the object is 20 centimeters and from the lens to the image is 5 centimeters. Completing the focal length formula yields:

\(\frac{1}{20}+\frac{1}{5}=\frac{1}{f}\)
\(\text{or}\; \frac{1}{20}+\frac{4}{20}=\frac{5}{20}\)
\(\text{Reducing the sum gives }\frac{5}{20}=\frac{1}{4}\)

The focal length is therefore 4 centimeters.

‌Example 2‌: The measured distance from a lens to the object is 10 centimeters and the distance from the lens to the image is 5 centimeters. The focal length equation shows:

\(\frac{1}{10}+\frac{1}{5}=\frac{1}{f}\)
\(\text{Then}\; \frac{1}{10}+\frac{2}{10}=\frac{3}{10}\)

Reducing this gives:

\(\frac{3}{10}=\frac{1}{3.33}\)

The focal length of the lens is therefore 3.33 centimeters.

The field of geometrical optics is very complicated, and it extends far beyond focal length. The principal plane of the lens, refractive index, wavelength, and many other formulas and properties all contribute greatly to the process of optical observation.

Khan Academy has a very extensive introduction to geometric optics that is a great place to start!