| Use 3T & 4T close-up lenses | Use PK ext. tubes | Use telezooms with close-up lenses | Focusing distances and WDs of Nikkor lenses |

| Extension tubes | Close-up lens | Depth of Field | Angle of View |

By reading the following technical notes, you should learn how to use your gear in order to take close-up pictures, even if you do not own a macro lens.

First of all, let's define the reproduction ratio, R:

Therefore, the reproduction ratio (or magnification) indicates how large will be the image on the film with respect to the actual dimensions of the subject. If we take a picture of a 1 in. diameter coin and if its diameter on the film is 1/2 in., then the reproduction ratio is 1:2 or 0.5.

Common (non-macro) lenses can focus at distances which are more or less 5 to 15 times their focal length. A "normal" 50 mm lens focuses at 45 cm. Modern 300 mm lenses focus at 2-2.5 meters. An ultra-wideangle 17 mm lens will focus at 25 cm. This occurrence is simply due to the fact that it is not easy to correct the optical aberrations in an extremely wide range of focusing distances. As a consequence, macro lenses have more complex optical designs and are more expensive than "normal" lenses having the same focal length.

Nevertheless, we should remember that:

• each lens (macro or not) has its best performance within a given focusing distance range;
• good close-up pictures can be taken with most of "normal" lenses (even with zooms) providing that we accept a lack of sharpness at the image corners.

The laws of geometrical optics teaches us that the reproduction ratio, R, increases on increasing the distance, t, between the lens and the film and/or decreasing the focal length, F, according to the following relationship:

Macro lenses usually permit a large increase of t by acting on the focus helicoid, although macro telephotos often adopt internal focusing (IF) mechanisms which ensure similar results. The above relationship also indicates that, when a lens is focused at infinity, t = F. In fact, a subject at infinity will be reproduced on the film with a nil size, i.e. R = 0 and t/F = 1.

If we own a good quality "normal" lens, we can increase t in order to focus at shorter distances and to get higher magnification ratios. Let's consider a 50 mm lens which can focus from 45 cm to infinity. The AI Nikkor 50/1.8 has a R value equal to 1/6.84 (= 0.146) at 45 cm. Which are the R values we can obtain when this lens is coupled to a PK-13 (27.5 mm) extension tube?

When focused at infinity, the lens has a t value equal to its focal length, 50 mm. With the extension tube, the new value of t is t' = 50 + 27.5 = 77.5 mm. Therefore

At 45 cm, the t value of the lens is expected to be a little bit larger than at infinity:

We have to add 27.5 mm (the length of the extension tube) to this value to get the new t' value:

Now we can calculate the new R value when the lens is coupled to the PK-13 and focused at its minimum distance:

Therefore, with a 27.5 mm long extension tube coupled to a "normal" 50 mm we can take close-up pictures with reproduction ratios in the 0.55-0.7 range.

Let me describe a further example which shows which kind of calculations we have to perform when our lens has a floating elements design and the focal length changes when we rotate the focusing ring.

Let's consider the AF Sigma 300/4 Apo Tele Macro, which employs an internal focusing (IF) mechanism. This lens can focus at 1.2 m without any accessory, thus reaching a 1:3 reproduction ratio. Which is the maximum magnification ratio attainable when a PK-13 extension tube is mounted between the lens and the camera body?

Again, when the lens is focused at infinity, t = F = 300 mm. When the lens is focused at 1.2 m, R = 0.333 (= 1:3). We cannot determine, on the basis of the above equations, the value of t at the minimum focusing distance. As a matter of fact, in the following equation

we have two unknown quantities: t and F. In fact, due to the IF mechanism, the "real" focal length is expected to shorten. How can we evaluate the focal length of our "300" when it is focused at 1.2 m?

The following relationship can help us:

where D is the focusing distance (i.e. the distance between the film and the subject), and F and R are the focal length and the reproduction ratio, respectively.

Therefore, we can evaluate the "real" F when R =1:3 (= 0.333) and D = 1200 mm (1.2 meters):

Now we know the focal length at the minimum focusing distance and, therefore, we can calculate the value of t:

Surprisingly!! t does not change! This is the effect of the IF mechanism which does not change the length of the lens and - consequently - t remains constant.

When the AF Sigma 300/4 Apo Tele Macro is used with a PK-13 extension tube and focused at the closest distance, we get:

A real macro performance, with a reproduction ratio quite close to 1:2.

Another question arises. Which is the focusing distance with the tube? The above mentioned relationship can provide the answer.

With this lens and PK-13 we can take pictures of shy animals with R ~ 1:2 and at about 1 m! By using also a good quality 1.4X teleconverter, R = 1:1.6 can be achieved at the same distance.

The optical properties of a close-up lens are described by its diopters and the focal length, F'. The diopters of a close-up lens are equal to the reciprocal of F' (in meters, m). Therefore, a 2 diopters lens has a focal length equal to 1/2 m, or 500 mm.

Close-up lenses reduce the focal length of our objective and increases, consequently, the reproduction ratio.

I caught myself looking at the results I obtained using a telezoom plus a close-up lens. In particular, I was successful in taking sharp pictures of dragonflies and butterflies with an AF Nikkor 75-300/4.5-5.6 zoom equipped with Nikon 5T (focal length, F' = 667 mm) or 6T (F' = 334 mm) close-up achromatic lenses.

Let's consider a 200 mm "normal" lens with a Nikon 4T close-up lens. The focal length, F', of 4T (3 diopters) is 334 mm. The overall focal length, OFL, of the coupled lenses is given by the following relationship:

When the lens if focused at infinity, t = F = 200 mm; therefore

Again, we can evaluate the focusing distance:

 If we attach the same close-up lens to a 105 mm, focused at infinity, we get

and

If we use a Nikon 3T close-up lens (F' = 667 mm) we get:

and

 

Therefore, the following general rule holds:

As a further example, I have calculated the reproduction ratios that can be obtained by coupling a Canon 500 D achromat (2 diopters, F' = 500 mm) with the 80-200/2.8 zoom and the AF Sigma 300/4 Apo Tele Macro.

The results are summarised in the following table:

The depth of field depends on the following quantities:

• the reproduction ratio (R);
• the aperture (f);
• the diameter (d) of the circle of confusion.

Therefore, when somebody says "the DOF of wideangles is larger than telephotos", don't believe him (her). As a matter of fact, the above sentence is correct if the distance between the film and the subject is the same. In this case, the size of the image on film is larger when a telephoto lens is used. Consequently, the reproduction ratio is larger and DOF is smaller. But if we shoot a butterfly and obtain the same size of the subject on the film, the DOF is also the same independently of the focal length. The only difference in taking close-up pictures with a 50 mm or a 200 mm lens will be the angle of view (AOV). The quantitative relationship which correlates DOF with the above quantities is:

Usually, 1/30 mm is considered to be an acceptable size of the circle of confusion; the above relationship can therefore be written as

In this case, DOF is in mm. The following table gives DOF as a function of the reproduction ratio and the aperture.

The angle of view, a, depends on the focal length, F (mm), and the reproduction ratio, R:

In the above equation, d is the diagonal (mm) of the picture (43.27 mm in the case of 24x36).

At infinity, R = 0 and the angle of view depends on the focal length only:

The following table shows the variation of the AOV with the focal length and the reproduction ratio:

The data show that at 1:1 the angle of view is one half of the one at infinity. It is worth noting that this is true when the focal length of the lens does not change with the focusing distance. In the case of modern macro lenses, however, the floating elements' design causes a decrease of F at the near limit. For example, at the minimum focusing distance, the actual focal length of the AF Micro-Nikkor 70-180 is around 90 mm when the zoom ring is set at 180 mm. As a consequence, the angle of view increases when the lens is focused at close distances. In fact, the AOV at infinity is 14° and it is 16° at 0.75 X (R = 1:1.33).