# Telephoto zoom lenses

Text: Marc Levoy

If long focal length lenses were built using a single thin lens, with object and image distances given by the Gaussian lens formula, then a 250mm lens focused on a subject 1 meter away would need to be placed 333mm (13 inches) from the sensor. But you can buy a Tamron 18-250mm zoom lens that, even when extended to 250mm focal length, measures only 6 inches long. How is this possible? The secret lies in a clever arrangement of convex and concave lenses that together are called a telephoto lens. Since many telephoto lenses also let you change their focal length, including the Tamron product just mentioned, it's worth folding that functionality into our explanation. Formally, the Tamron is called a telephoto zoom lens.

#### How a zoom lens works

Start by clicking on the "Close-up Filter" check box. A close-up filter is a weak convex lens you can attach to the end of any lens, by screwing it into the filter threads. Its purpose is to shorten the object distance, for example to turn a regular lens into a macro lens without the need to buy a separate macro lens. We're going to use a close-up filter here so that the in-focus plane (where the blue and red bundles of rays individually converge to two points in object space) fits inside our applet frame. For the rest of this discussion, try to pretend that this filter doesn't exist; it just makes the visualization easier to understand.

Now click on the "Equivalent Thin Lens" check box. A thin green lens should appear. Imagine that your long focal length lens consists solely of this lens. For the moment, ignore the two lenses to its right. The red bundle of rays start from the in-focus plane on the left edge of the applet, diverge for a while, pass through the green lens (remember that we're ignoring the close-up filter), then bend and follow the green lines, reconverging at the red circle on the sensor (vertical gray bar). The blue bundle of rays does the same thing, converging at the blue circle. Since the red and blue circles lie at the two ends of the sensor, the angle subtended by the central rays of the red and blue bundles where they strike the green lens represents the field of view. To complete our analysis, the object distance is the distance from the green lens to the left side of the applet, and the image distance is the distance from the green lens to the sensor. The focal length of the green lens is neither of these distances, but is related to them through the Gaussian lens formula.

Try moving the focal length slider. This changes the focal length of the green lens. Note that it gets thicker and thinner as you do this, reflecting what would be required to actually change the focal length of a single-lens system like this. As the focal length increases, the field of view (angle between the red and blue bundles) decreases, as you would expect. You can also move the sensor size slider to change the field of view. This arrangement is called a zoom lens. Here's where it gets interesting. Notice that as you adjust the focal length, the applet keeps the in-focus object plane and the sensor stationary. We do this by solving a system of two simultaneous equations: (1) the Gaussian lens formula, with the focal length fixed at the value you set using the slider, (2) the sum of object distance and image distance must equal the distance from the left edge of the applet to the sensor, which is fixed by the design of the applet. This arrangement, where the optics stays focused at the same object distance (a.k.a. subject distance) while you change the focal length, is called an optically compensated zoom lens.

#### How a telephoto zoom lens works

One problem with this design is that the green lens is far from the sensor. If built this way, it would yield a physically long lens, as explained in the introduction. Another problem, of course, is that there's no way to make glass lenses change shape (get thicker and thinner) once they've been fabricated. To address both problems, we move to a multi-element design. Unclick the "Equivalent Thin Lens" check box. Now the red and blue bundles continue spreading out as they pass the place where the green lens was, strike the convex lens, bend inwards towards the optical axis (central horizontal line), strike the concave lens, and bend outwards again, converging to the sensor at the same points struck by the green rays. In other words, these two optical arrangements - the green lens alone or the convex-concave lens combination - have the same effective focal length. As a result, they make the same picture. Why would you prefer the second arrangement over the first? Look how much closer the convex-concave lens combination is to the sensor than the green lens was. This is a more compact design. It's called a telephoto lens. Try changing the focal length. The two lenses move, and the field of view changes. So it's a telephoto zoom lens. But the in-focus object plane and sensor also remain stationary. So it's a optically compensated telephoto zoom lens. It's interesting to see how the two lenses move; they don't move together. Explaining how we compute their motion is beyond the scope of this applet; we do it using ray transfer matrices. Briefly, any system of thin lenses and air gaps can be modeled as a 2 x 2 matrix describing how that system bends and shifts rays of light. By constructing and equating the matrices for an ideal thin lens and a telephoto zoom lens system, we can derive equations that make one system optically equivalent to the other. In a commerical lens these motions are encoded into curved slots in the sides of the lens barrel, as suggested by the patent application drawing at left.

Finally, try moving the "Focus" slider. Now the location of the in-focus plane changes in object space; it is no longer fixed at the left edge of the applet. Look how the two lenses move; this time they do move together. More slots in the lens barrel. By the way, this is not the only possible design for a telephoto zoom lens. In fact most commercial lenses have many more lens elements. However, our applet gives the basics, and to our knowledge you can't make a simpler arrangement than the one we've shown here.