Optics 024 · Zoom without losing focus

Parfocal Zoom Challenge

Can moving lens groups change the focal length while the image stays on the film? Yes, if a second group, the compensator, follows a curved cam set by the zoom group’s position. In a 1976 Canon design the focal length then runs from 7.24 to 68.8 mm with the image fixed on the film. Leave the compensator behind and the image drifts about a millimetre off the film; a straight cam misses by up to 0.79 mm.

Zoom a four-group lens from 7.24 to 68.8 mm by sliding one group. A second group must follow its cam to keep the image on the film; off it, the picture blurs.

Focal length of the whole lens $\text{—}$

Angle of view across the image circle $\text{—}$

Image behind the film (negative: in front of it) $\text{—}$

Blur circle of a distant point on the film $\text{—}$

Compensator position on the cam, from the front group $\text{—}$

The other sharp compensator position $\text{—}$

Magnification of the zoom group $\text{—}$

Magnification of the compensator $\text{—}$

Entrance pupil diameter $\text{—}$

F-number $\text{—}$

The lens is Example 1 of Canon’s US patent 3,972,591 (M. Suwa, 1976) as four thin lenses in air: group I of 61.153 mm, the zoom group of −15.120 mm and the compensator of 35.716 mm, with the patent’s zoom ratio of 9.504 (68.79 mm over 7.238 mm) and its touch point at 0.8 of the zoom group’s travel. The groups are combined with the paraxial ray-transfer matrices of Kogelnik and Li (1966); real groups have thickness, principal planes apart from their glass, and aberrations, all left out. The patent gives no single focal length for the relay, group IV, so a 58.1 mm relay 92 mm behind group I is chosen to give the patent’s 7.238 mm wide-end focal length; the film then lies 119.3 mm behind group I. The iris sits at the relay and is sized for the patent’s F/1.4; the image circle, 7.41 mm across, is set by the patent’s 54.2° angle at the wide end. The scene is at infinity, and focusing on near objects with group I is not modelled. Blur is geometric: a point is spread over a uniform disc set by the iris, with no diffraction, aberrations, vignetting or film grain. An image within a micrometre of the film is called sharp; the patent gives no depth of focus. The drawn off-axis light comes from halfway to the picture’s edge, because light from the edge at the wide end would meet group I outside the drawn lens. Group diameters are not in the patent and are drawn just wide enough for the drawn light. The lens is drawn to scale, lengths and heights alike.

Physics tutorial

Keep a zoom lens in focus with a cam

BackgroundA zoom lens changes its focal length by moving lens groups along the axis. Move only one group and the image slides off the film as well, so a second group has to move with it. In Canon’s 1976 design used here, the front group and the relay at the back stay still, a negative zoom group slides in a straight line, and a positive compensator follows a curved groove, the cam, cut in a turning barrel.

Why it mattersFilm and television zoom lenses must stay sharp while the operator zooms during a shot. Electron microscopes face the same task with magnetic lenses: they change magnification by changing lens currents and must keep the specimen in focus while doing so.

Start with the essentials

Focus question: Can moving lens groups change the focal length while the image stays on the film?
One-sentence intuition: The two moving groups must together carry the front group’s image to the same point, which fixes the sum of their object-to-image distances. Choose the zoom group’s magnification $m_{2}$ and that leaves two compensator magnifications whose product is 1, $m_{3}$ and $1/m_{3}$ . The cam follows one of them; the focal length $f_{1}m_{2}m_{3}m_{4}$ grows ninefold while the image never leaves the film.

Core mathematical model

How far a lens carries its image

L=f\left(2-m-\frac{1}{m}\right)

A thin lens of focal length $f$ working at lateral magnification $m$ puts its image a distance $L$ beyond its object. The expression is unchanged when $m$ is replaced by $1/m$ : swapping object and image gives the same separation. For the zoom group, $f_{2}=-15.12\,\mathrm{mm}$ , and at the middle of the zoom $m_{2}=-0.645$ gives $L_{2}=-63.4\,\mathrm{mm}$ , a virtual image in front of its object.

The parfocal condition

m_{3}+\frac{1}{m_{3}}=2-\frac{4(f_{2}+f_{3})-L_{2}}{f_{3}}

Groups II and III together must carry the front group’s focus to the relay’s fixed object point, so $L_{2}+L_{3}$ stays at one value. The patent makes the two solutions touch where both groups work at $-1$ , which fixes that value at $4(f_{2}+f_{3})=82.4\,\mathrm{mm}$ . The equation is a quadratic in $m_{3}$ whose two roots multiply to 1: at mid-zoom, $-0.751$ and $-1.33$ .

Focal length of the whole lens

f=f_{1}\,m_{2}\,m_{3}\,m_{4}

A distant scene comes to focus at $f_{1}=61.153\,\mathrm{mm}$ behind the front group, and each later group magnifies that image. The relay here works at $m_{4}=0.530$ . At the wide end $m_{2}=-0.406$ and $m_{3}=-0.550$ , giving 7.24 mm; at the long end $-1.58$ and $-1.35$ , giving 68.8 mm. The same number is $-1/C$ of the whole lens’s ray-transfer matrix, which the Lab checks.

The blur circle

b=D\,\frac{|\delta|}{s}

When the image forms a distance $\delta$ from the film, the cone of light from the iris, $D=19.5\,\mathrm{mm}$ across, to the image point, $s$ away, is cut by the film in a disc of diameter $b$ . This is geometric blur only; diffraction and aberrations are left out.

Common difficulties

The compensator also zooms

Typical misconceptionThe zoom group alone sets the focal length, and the compensator only focuses.

Better mental modelThe focal length is $f_{1}m_{2}m_{3}m_{4}$ . Along the cam, $m_{2}$ changes 3.88 times and $m_{3}$ 2.45 times, which together make the ratio of 9.50. The compensator’s path is the one that also keeps the image on the film.

Two positions, not one

Typical misconceptionFor each zoom setting there is only one compensator position that is in focus.

Better mental modelBecause $L$ is the same for $m_{3}$ and $1/m_{3}$ , there are two, until they meet at $m_{3}=-1$ . The second one is just as sharp but gives another focal length; the cam cannot reach it.

A straight cam is not close enough

Typical misconceptionIf the compensator is right at both ends of the zoom, moving it in a straight line between them is good enough.

Better mental modelThe needed path is curved. A straight cam here leaves the image up to 0.788 mm off the film, a blur circle of 0.58 mm, far larger than a sharp picture on this 7.41 mm image circle allows.

The film does not move

Typical misconceptionTo stay in focus while zooming, the film or sensor has to move.

Better mental modelIn this design the front group, the relay and the film are fixed; only the two middle groups move. The cam makes their combined object-to-image distance stay the same, so the relay always receives its object at the same place.

Run the experiment

01
Watch the lens on its cam
Keep the default preset, On the cam: sharp in mid-zoom.
What to observe: The zoom group sits 22.6 mm behind the front group and the compensator 81.0 mm behind it. The focal length is 15.7 mm, the angle of view 26.5°, and the light from the middle of the scene meets in one point on the film.
02
Zoom along the cam
Drag the zoom group from one end of its travel to the other.
What to observe: The compensator follows its groove in the barrel, moving 28.4 mm forwards while the zoom group moves 27.6 mm back. The focal length runs from 7.24 mm to 68.8 mm and the simulated picture magnifies, but the blur circle stays a point.
03
Leave the compensator behind
Load Compensator left behind: the picture blurs.
What to observe: The compensator stays where the cam puts it at the wide end, 88.2 mm, while the zoom group is in the middle. The focal length is 13.2 mm, but the image forms 0.993 mm behind the film and every point blurs to a 0.685 mm disc: a varifocal lens, which must be refocused after zooming.
04
Try a straight cam
Load A straight cam: sharp only at the ends.
What to observe: A straight groove joining the cam’s two ends puts the compensator at 74.0 mm. The image lands 0.682 mm in front of the film. The process row shows this cam is sharp only at the two ends and misses by up to 0.788 mm near a zoom-group position of 19.2 mm.
05
Find the second sharp position
Load The second sharp position: a longer focal length.
What to observe: With the compensator at 60.3 mm the image is on the film again, at a focal length of 27.8 mm instead of 15.7 mm. Its magnification is $-1.33$ , the reciprocal of the cam’s $-0.751$ .
06
Go to the touch point
Load Where the two sharp positions meet.
What to observe: At a zoom-group position of 30.9 mm, both groups work at $m=-1$ and the two sharp positions meet at 72.1 mm, with a focal length of 32.4 mm. The cam passes from one solution curve to the other here, so the compensator keeps moving the same way and the focal length keeps growing.
07
Refocus by hand
Switch off Follow the cam, move the zoom group, then use the compensator slider to bring the blur circle back to a point.
What to observe: There are two places that work. One lies on the cam; the other, on the purple curve in the cam chart, gives a different focal length.

Parfocal Zoom Challenge

Keep a zoom lens in focus with a cam

Core mathematical model

How far a lens carries its image

The parfocal condition

Focal length of the whole lens

The blur circle

Common difficulties

The compensator also zooms

Two positions, not one

A straight cam is not close enough

The film does not move

Run the experiment

Watch the lens on its cam

Zoom along the cam

Leave the compensator behind

Try a straight cam

Find the second sharp position

Go to the touch point

Refocus by hand

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