Some Simple Measurements

This section is written for the benefit of the reader who does not have access to the usual optical measurement and laboratory equipment. A lens bench with collimator and a measuring microscope are essential for really accurate measurements of the imaging (i.e., gaussian) properties of a lens. However, there are a number of simple ways that an approxi-mate measurement of focal length and back focal length can be made.

The measurement of the back focal length, or BFL (see Sec. 1.2 and Fig. 1.1), is relatively easy. When using a lens bench, a target at infin-ity is provided by the collimator; the location of the focal point is determined by focusing the lens bench microscope alternately on the focal point and on the last surface of the lens. A distant object (a tree, a building, a telephone pole) makes a reasonable substitute for a colli-mator. To get a rough measurement, one simply focuses the target on a light-colored wall or a piece of typing paper as shown in Fig. 6.1,

Distant object

Lens to be measured

A wall or any smooth light surface

X +FFL

BFL + x' Image

A large distance

Figure 6.1 The back focal length (BFL) of a lens can be measured using a distant object (instead of a collimator) and measuring the lens-to-image distance with a scale or ruler. If the object is not effectively at infinity, the newtonian focus shift (x″ ⫽ ⫺f²/x)

and measures the lens-to-image distance with a scale. If the lens has spherical aberration (and most simple lenses do) your measurement will come up a bit shorter than the paraxial back focus. If there is enough light, the spherical aberration can be minimized by reducing the lens aperture with a mask.

Using an object which is not collimated (i.e., not located at infinity) will introduce a small error in your measurement. The error in locat-ing the focal point is indicated by Newton’s equation [Eq. (1.1)] as x′ ⫽

⫺f²/x, where x′ is the error, f is the focal length, and x is the distance to your target. For example, if you measure the back focus of a 2-in-focal-length lens using a target down the hall which is only 50 ft (600 in) away, the error will be x′ ⫽ 2²/600⫽ 0.007 in; this is less than the error in your crude measurement of the back focus is likely to be.

Even in cases where this error is significant, you can always calculate the error and subtract it from the measured BFL to improve the validity of the result.

One problem associated with this technique is that stray light falling on the image will make it difficult to see and focus. A card-board screen with a hole for the lens is one way around this problem;

using a light bulb as a target in a darkened room is another. Figure 6.2 shows a matchbox slipped over the scale and slid into best focus as a handy tool for making the measurement.

Lens to be measured

Machinist's steel scale Small

match box

Image

Figure 6.2 A matchbox-sized box (or a folded piece of pasteboard) slipped over a machinist’s steel scale makes a convenient way to measure the back focal length of a lens.

The measurement of the effective focal length (EFL) is considerably more difficult, because it involves the location of the principal points.

Figure 1.2 shows the principal point locations for simple lenses of var-ious shapes. A fair estimate can be made for single elements and most cemented doublets by assuming that the space between the principal points is approximately one-third of the axial thickness of the lens.

[A somewhat better estimate for singlets is t(n⫺1)/n.] For planocon-vex forms, one principal point is always located at the curved surface;

for equiconvex lenses the points are evenly spaced within the lens. As illustrated in Fig. 6.3, adding a suitable fraction of the axial thickness to the measured BFL will then get an estimate for the EFL. For an anastigmat, such as a Cooke triplet or a Tessar, the principal point locations are more difficult to estimate. Adding one-half to two-thirds of the vertex length of the lens to the BFL is about the best estimate one can make. In a complex lens the principal points are often almost coincident, and occasionally reversed.

To get a better value for the focal length one must measure a mag-nification. Magnification is simply the ratio of the image size to the object size. An illuminated object of known size is imaged and the image size is measured; admittedly, doing this accurately may be eas-ier said than done, but it can be done. The setup is shown

schemati-P1 P2 F2

BFL T

EFL T(n-1)/n T/3

Figure 6.3 One way of estimating the effective focal length (EFL) is to add a suit-able fraction of the element thickness to the measured back focal length (BFL). For a planoconvex element, the convex side should face the distant target to minimize spherical aberration. If the lens is reversed, the measured BFL will equal the EFL,

cally in Fig. 6.4. The object-to-image distance (often called the total track length) is measured. The estimated spacing between the princi-pal points is subtracted from the track length. This adjusted length is scaled to get s and s′. Dividing the adjusted track length T by (m+1), where m is the ratio of the image to object size will get s; then (T⫺s) equals s′. (Note that we use a positive sign for the magnification m in this case.) Now we solve the Gauss equation [Eq. (1.4)] to get the focal length.

If this process is carried out accurately for several different object-to-image distances, it is possible to eliminate the estimation of the principal point separation by making a simultaneous solution for the exact value.

Sample calculations

The object is a back-illuminated transparent scale, 15 in long, and the lens forms an image which is measured at 3.5 in. The magnification is thus m⫽ 3.5/15 ⫽ 0.2333. The object-to-image distance is 40 in, and the lens is 1 in thick. Assuming that the principal points are separat-ed by one-third the lens thickness, or 0.333, the adjustseparat-ed track length is 39.667 in. We get the Gauss object distance by dividing the track length by (m+1), or 1.2333, which gives us s⫽ 39.667/1.2333 ⫽ 32.162 in and s′ ⫽ 39.667⫺32.162 ⫽ 7.5045 in. Substituting s and s′ into Eq.

(1.4), and solving for f gives us the focal length. (Note that our sign convention requires s to be negative.)

h h' = m h

Figure 6.4 Setup for measuring effective focal length (EFL) by measuring the magni-fication at finite conjugates: The magnimagni-fication (here taken as the ratio of measured image size to object size—positive for an inverted image) and the track length are measured. Then the gaussian conjugates can be found by dividing the total track length (minus the principal point separation) by the magnification plus 1 to get S.

Then S″ ⫽ mS, and the Gauss equation can be solved for the focal length.

⫽ ⫹

⫽ ⫹

f⫽ 6.0847 in

Yet another way to measure EFL involves the use of a distant target whose angular size is known. The size of the image of the target is measured. Then the focal length of the lens is equal to half the image size divided by the tangent of the half angle which the object sub-tends. If the object is not at infinity, the Newton correction of f²/x can be applied (as in the discussion of BFL above). A distant building with vents, chimneys, elevator towers, etc., on its roof line can be mea-sured with a theodolite through a convenient window to serve as a target of this type. Alternatively, the measurement of a lens of known focal length can be used to determine the angle.