__Focal Length and Magnification__

Like all enthusiasts, photographers have their own vocabulary, and so they often use special terms like “aperture”, “sharpness”, “focal length”, etc. It seems that we particularly like this term: “focal length”. We like the term so much that we use it all the time...

Because all digital SLRs do not use sensors of the same size, we (photographers) often use a new concept: "equivalent focal length" (actually, photo-gear manufacturers invented the term, but we’ve quickly adopted it). Since then, sentences have become common, like the following:

– "On a Nikon D90, a 105mm lens has an “equivalent focal length” of a 160mm lens on a full-frame Nikon D700".

Nevertheless, the concept of *angle of view* would be much more appropriate here (and for a lot of people, even easier to understand).

But we do like to use “focal length”…

Similarly, we often hear:

– “My 70-200mm, when set on 200 mm and on minimum focus, has an "effective focal length" of only 180 mm”.

As a matter of fact, it might be “safer” to say:

– “My 70-200mm, when set on 200 mm and on minimum focus, provides the same *magnification* as an old no-floating-element 180mm lens”.

The calculation of the focal length is not so simple! And, as we’ll see further, it may also come with surprises...

Actually, adjectives like "equivalent" or "effective" may only be there to hide our embarrassment in front of a concept which remains,

for most of us, quite difficult to understand, and even really mysterious for some... After all, who really knows what happens inside of a zoom lens when the zoom ring is rotated? Looking at things with a bit of exactness, one realizes how much the words “focal length”

might be false friends...

But it is true that, like the relative aperture, the focal length is a fundamental characteristic of any photographic lens. Why is it so important? Because when the object is infinitely distant, the focal length is what determines the dimensional relationship between the object and its image on the sensor. For closer objects, things are a bit different, but the focal length remains a very important parameter…

__Table of contents__:

1 – Systems having focal points.

1.1 – The focal length.

1.2 – The angle of view.

1.3 – The lateral magnification.

1.3.1 – Case of inert lenses.

1.3.2 – Case of lenses including moving elements.

1.4 – Rear lens converters.

1.4.1 – Operation and angular magnification.

1.4.2 – Evolution of the focal length when focusing.

1.4.2.1 – Case of inert lenses.

1.4.2.2 – Case of lenses including moving elements.

1.4.3 – Conclusion.

2 – Afocal systems.

2.1 – Operation and angular magnification.

2.2 – Front lens converters.

2.3 – Other afocal-system applications.

3 – References.

__1 – Systems having focal points.__

__1.1 – The focal length.__

One can rather easily visualizes the physical quantity hiding behind the words "focal length" when it comes to a single thin lens, but it becomes much more difficult when referring to photographic lenses... This probably comes from the fact that it’s impossible to measure this quantity with
a simple ruler, as can be done to the outer length or diameter of any lens. However, when photographing an object, it is this characteristic length, so difficult to materialize, which determines the dimensions of the image formed on the camera’s sensor.

The focal length of a lens can be determined with the relationship:

f’ = h’ / tan(Theta) *

Where

- f’ is the focal length,
- h’ is the image height of an infinitely distant object,
- Theta is the angular extent of the infinitely distant object.

* *To be rigorous, f’ is the limiting value of the ratio h’/ tan(Theta) when h’ and Theta tend toward zero.*

But this is certainly a bit too “mathematical” for we photographers who want to keep "close" to the lens...

Fortunately, there is another definition: the focal length is the distance between one of the principal points (first or second) and the corresponding focal point (usually, only the “second” points are used for this purpose) This definition is more accessible; easier to visualize.

Therefore, once the location of at least one couple of these points is known (second principle point and second focal point or the “first” points), the calculation of the focal length is very easy. All these special points (plus the first and second nodal points) are the “cardinal points” of the optical system.

__Note__

- For photographic lenses used in the air (which is almost always the case) nodal point and principal points are strictly coincident.

Ray tracing is a good way to determine the location of the cardinal points of a lens. Figure 01 shows the result of this kind of computation applied to a very common real photographic lens: a 50mm. Like any other elaborate photographic lens, it is made of several lenses. All together they constitute an optical system of axial symmetry, i.e. all the elements surfaces are figures of rotation about a common axis: the __optical axis__.

When a bundle of rays parallel to the optical axis goes through all the lens’ elements, it emerges as a converging beam (Figure 01, mouse out). The common focus on the axis is the __second focal point__ (F’). It is called "second" because it is located behind the lens, on the image side. One can extend the incoming rays toward the right and the emerging rays toward the left until they intersect. The points of intersection define a surface called the __second principal plane__. Actually, on spherically corrected lenses this surface is not a plane, it’s a sphere centered on the second focal point (F’); but it is called so because in the paraxial region (tiny area close to the optical axis) it can be considered as a plane. The __second principal point__ (H’) is located where the optical axis strikes the second principal plane. The distance between the second principal point (H') and the second focal point (F') is the effective __focal length__ (f’). On the example bellow, this distance equals 50 mm. The distance from the vertex of the last surface to the second focal point (F') is the __back focal length__.

__Fig. 01__: *Optical system of a real photographic lens (50mm f/1.8).*

__Mouse out__*: **Second principal point, second focal point and focal length f’.*

__Mouse over__*: **First principal point, first focal point and focal length f.*

When light rays are entering the optical system from the rear (Figure 01, mouse over) one can define the position of the “first” cardinal points.

The distance between the first and second principal points H and H’ (null space, algebraic value considering the direction of propagation of light) is an important quantity: it may be considered as the “optical thickness” of the system.

In some cases, in order to simplify calculations, it may be interesting to replace the entire optical system of a photographic lens by a single thin convergent lens of same effective focal length (Figure 02). By positioning the center of the single lens precisely at the location of the second principal point (H') of the photographic lens, and as long as the incident light rays meet certain conditions (close to the optical axis, small angle of incidence, monochromatic light), the two systems can be considered equivalents.

__Fig. 02__: *Replacing the entire optical system of a photographic lens by a single thin lens
having same focal length may be a good way to simplify some calculations.*

The location of the cardinal points F, H, H’, F’ depends on the type of lenses (wide angle, telephoto, zoom, etc.) and on their configurations (focusing, with or without a rear lens converter, etc.). The first principal point H can be on the right as well as on the left of the second principal point H’, and one or the other, or even both of them may be outside of the physical boundaries of the photographic lens.

For example, the second principal point (H’) of a retrofocus wide-angle lens, like the AF-Nikkor 20mm f/2.8D, is located behind it, between the lens and the sensor (Figure 03). And its first focal point (F) is inside the lens. Therefore, when used in the reverse way, this lens creates no real image of an infinitely distant object.

__Fig. 03__: *Cardinal points* of the AF-*Nikkor 20mm f/2.8D set on infinity focus.*

All the cardinal points of telephoto lenses set on infinity focus are generally located outside their physical boundaries. The Reflex-Nikkor 500mm f/8, like other catadioptric telephoto lenses, is an extreme case: its first focal point (F) is located over 2 meters (7 ft.) in front of the image plane (Figure 04).

__Fig. 04__: *Cardinal points of the Reflex-Nikkor 500mm f/8 set on infinity focus.*

A photographic lens has only one pair of principal points H and H’ and one pair of focal points F and F’. The focal length (f’) is always determined with respect to H’ and F’. The four cardinal points F, H, H’, F’ largely define the optical system of a lens.

__Fig. 05__: *Cardinal points of the Zoom-Nikkor AF-S VR 70-300mm f/4.5-5.6G set on infinity focus.*

__Notes__

- When talking about lenses used in the air, the focal length f’ (H’F’) is always equal to f (HF), in absolute value.
- A photographic lens has no “optical center” (this expression should only be used when talking about very thin single lenses).

To accurately determine the location of the cardinal points of a photographic lens in any configuration, it is necessary to know its optical-system definition. One can find this information in the patents filed by the major lens manufacturers. For example, the following data table completely defines the optical system of the Zoom-Nikkor AF-S VR 70-300mm f/4.5-5.6G previously seen (patent available here).

__Fig. 06__: *Optical-system definition of the Zoom-Nikkor AF-S VR 70-300mm f/4.5-5.6G.*

One can find in this table all that is needed to compute rays tracing, cardinal points location and more:

- r, radii of curvature of the surfaces,
- d, distances between the vertex of the surfaces,
- vd, Abbe number of the glasses (reference wavelength = 587.56 nm)
- n, refractive index of the glasses (wavelength = 587.56 nm),
- S, position of the stop (not always given in the patents)
- D0, distance from the object to the vertex of the first surface.

The other values are not essential (and not always published in the patents) but they may help to check one’s own calculations:

- f, focal length,
- FNO, f number,
- 2w, angle of view,
- Bf, back focal length,
- R, distance from object to sensor,
- M, magnification.

This example shows a quite complete data table. Unfortunately, every patent does not offer such complete and accurate tables!

*Back to Table of contents.*

__1.2 – The angle of view.__

First of all, it is important to distinguish the specific angle of view of a lens (alone) from the angle of view of a lens-plus-sensor system. Both are not necessarily equal.

The specific angle of view of a lens depends more on its optical-system design than on its effective focal length. So, there we go out of the scope of this article. One single example to support this assertion: Figure 07 (below) shows the optical diagrams of two lenses having both strictly identical angles of view, although the focal length of one is twice that of the other. A simple scale effect (x2) differentiates both lenses.

__Fig. 07__: *Scale effect. The angle of view of a lens is not necessarily related to its focal length.*

Of course, if such lenses are used alternately on the same camera, i.e. on the same sensor, the angle of view of the lens-plus-sensor system will be twice as small as with the longest focal-length lens (shaded light beam on Figure 07). This leads to the concept of angle of view of a lens-plus-sensor system. As this theme has already been developed in a large number of articles and presentations, I'll stick to one simple illustration: Figure 08...

On top, the well-known Zoom-Nikkor 80-200mm f/2.8D ED AF is associated with a 24x36 mm sensor (infinity focus). When the focal length is set on 80 mm (effective focal length = 80.9 mm) the angle of view is slightly greater than 30°.

At the center, the same lens in the same configuration is used in combination with a 16x24 mm sensor. Of course, the specific angle of view of the lens is unchanged, but the sensor size is such that only the incoming light beams whose angle of incidence is less than or equal to 10° can reach it. So in this case, when the focal length is set on 80 mm, the angle of view is 20°.

__Fig. 08__: *Angle of view of a lens-plus-sensor system.*

Bottom. To get the same angle of view (20°) with a 24x36 mm sensor, the focal length must be set on 120 mm (120.8 mm exactly).

So, a focal length of 120 mm on a 24x36 mm sensor provides the same angle of view as a focal length of 80 mm on a 16x24 mm sensor. This is where the concept of "equivalent focal length" comes from.

*Back to Table of contents.*

__1.3 – The lateral magnification__.

When talking about photographic lenses, the magnification that matters the most is the __lateral__ (or __transverse__) __magnification__ (g).

The lateral magnification (g) is the ratio of the image height to the object height. Therefore:

- g < -1 if the image is smaller than the object;
- g = -1 if the image is as large as the object;
- g > -1 if the image is larger than the object.

The minus sign means that the image is upside down in comparison to the object (it is omitted in most lenses specifications).

For example, Figure 09 illustrates the Micro-Nikkor AF 60mm f/2.8D when set on minimum focus. Then, every object in focus is reproduced at the same size on the sensor. The lateral magnification of the lens in this configuration is g = -1 (and its focal length is f’ = 49.4 mm).

__Fig. 09__: *Cardinal points of the Micro-Nikkor AF 60mm f/2.8D set on minimum focus (g = -1).*

For the “mathematicians”, Figure 10 (below) shows how to represent diagrammatically the optical system of a photographic lens. Using the similarity of the blue shaded triangles in the object space and red shaded triangles in the image space, this construction allows making out two simple relationships defining the lateral magnification (among other things).

__Fig. 10__: *Construction of the image A'B' of a plane object AB.*

The focal length, by itself, does not allow determining the lateral magnification of a photographic lens when the object-to-image distance (AA') is different from infinity. In such cases, the knowledge of at least another distance is required. The distance A’F’ is here particularly appropriate since it allows using of the simple ratio g = A’F’ / f'. The distance A’F’ is the extension of the back focal length when the lens goes from infinity focus to any closer focus distance.

__Reminder (Figure 11)__

When the object is infinitely distant, the center of the image plane (A’) is coincident with the second focal point (F’), then A’F’ = 0. When the object comes closer, its image moves gradually away (on the right) from the second focal point (F’): A’F’ increases. Object and image always move in the same direction. To “focus” is to accurately place the camera sensor on A’, at the right distance from the second focal point (F’). To do so, one can use several ways:

- to move the sensor away from the optical system, or to move the entire optical system away from the sensor without changing its internal characteristics (F’ and the other cardinal points come along just as they are);
- to change the characteristics of the optical system so as to change the location of its cardinal points (including F’);
- to combine both previous ways.

__Fig. 11__: *Image behavior with regard to the object distance.*

*Back to Table of contents.*

When the relative position of each single lens does not change with respect to the others, the cardinal points are fixed with regard to the optical system, no matter what the focus distance is. Let’s call this kind of system: “inert optical system”. Understand “inert” in the sense that there are no moving (floating) elements with regard to the others. To focus, such a system must shift __as a whole__, and its cardinal points go with it __as a whole__ too (none of them move in comparison to the others).

Figure 12 (below) shows an example of an inert-optical-system lens: a real 50mm f/1.8 (effective focal length f’ = 51.6 mm). All the elements of this photographic lens are strictly fixed with respect to each other. Focusing is achieved by moving forward the optical system as a whole, away from the sensor. During the process, the cardinal points move along, just as they are. When the lens is set on minimum focus (object-to-sensor distance = 450 mm), the displacement of the entire system is 7.7 mm. Therefore, the magnification is:

g = A’F’ / f’ = -7.7 / 51.6 = -0.149

One can say g = -1 / 6.7, which is equivalent.

__Fig. 12__: *50mm f/1.8. *

__Mouse out__*: **Infinity focus.*

__Mouse over__*: **Minimum focus (0.45 m).*** **

The optical system of the Nikkor 180mm f/2.8 ED Ais belongs to the same category (Figure 13). When set on minimum focus (1800 mm), the optical system shifts 24.1 mm forward. Then, the lateral magnification is:

g = A’F’ / f' = -24.1 / 180 = -0.134

or g = -1 / 7.47

__Fig. 13__: *Nikkor-ED **180mm f/2.8 Ais. *

__Mouse out__*: **Infinity focus.*

__Mouse over__*: **Minimum focus.*** **

__Conclusion__

- Since the cardinal points F, H, H’ and F' of this kind of lenses (inert lenses) are fixed with respect to the optical system, their focal length (f’) and null space (HH’) are constant.

- With such photographic lenses, the magnification can be easily calculated using the value of the back-focal-length extension that occurs when focusing (precisely corresponding to the distance A’F’).

*Back to Table of contents.*

__1.3.2 – Case of lenses including optical systems with moving elements.__

The relative movement of one or more components of an optical system, with respect to the others, leads to alter the location of all its cardinal points. Therefore, depending on the configuration of the lens, focal length (f') and null space (HH’) take different values.

Most modern lenses fall into this category:

- zooms (of course),
- front, internal or rear focusing lenses,
- lenses including close range correction systems (CRC),
- inert lenses when associated to a rear conversion lens (because the optical system of the lens is not fixed with regard to the converter).

The magnification relationship (g = A’F’ / f') connecting the focal length and the distance A’F’ remains the same and is still valid, yet its calculation is a bit more complicated because:

- the distance A’F’ cannot be deducted from the possible extension of the back focal length (back focal length may even stay strictly constant while focusing);
- the focal length should be determined after each new focus setting.

Thus, the lateral magnification can only be calculated by:

- comparison of the sizes of the object and its image,
- a more complex computation, provided that the optical definition of the lens is known.

Figure 14 shows the optical system of the Micro-Nikkor AF-S VR 85mm f/3.5G set on infinity focus (mouse out) and minimum focus (g = -1, mouse over). In the latter case, the focal length decreases by almost 19%, (down to f’ = 71.5 mm) and the value of the null space (HH') becomes very weak (in that specific configuration, this lens can be considered as a thin convergent single lens: its focal length is almost equal to a quarter of the distance AA’).

__Fig. 14__: *Micro-Nikkor AF-S VR 85mm f/3.5G. *

__Mouse out__*: **Infinity focus.*

__Mouse over__*: **Minimum focus.*** **

The evolution of the focal length of zoom lenses such as the Zoom-Nikkor AF-S VR 70-200mm f/2.8G, along with their focusing range, attract much comment. Indeed, the maximum magnification of this lens is only g = -0.165 (maximum focal length, closest focus distance). However, the magnification of a 200mm inert lens set on the same focus distance reaches g = -0.188. From this observation, many photographers concluded that the focal length of this lens decreases together with the focus distance. But in reality this is not correct. As a matter of fact, it is the opposite that occurs: the effective focal length increases when focusing is set on close objects.

Then, why is the magnification lower than expected?

Because, in this lens, focusing is carried out by the rear part of the front unit (see Figure 15), and the forward shift of that sub-unit has two consequences:

- for a given focus distance, it does not move forward the second focal point (F’) as much as in the case of an inert optical system; thus, on minimum focus (1500 mm - 5 ft.), A’F’ = -33.4 mm whereas a 200mm inert lens would have A’F’ = -37.6 mm;
- it increases the focal length from 196 mm (infinity focus) to almost 203 mm (minimum focus).

__Fig. 15__: *Zoom-Nikkor AF-S VR 70-200mm f/2.8G set on 200 mm.*

__Mouse out__*: **Infinity focus.*

__Mouse over__*: **Minimum focus (1.5 m).*

__Note__

- Regarding its successor (70-200 VRII), focusing is not carried out in the same way, and the focal length actually decreases when focusing on close objects.
- These points are treated on my page Telephoto zooms.

__Conclusion__

Regardless of the optical system of any photographic lens, and whatever its configuration may be, it is always possible to determine the position of its cardinal points and therefore its null space, focal length and magnification. Relationships linking these parameters apply in all cases.

At relatively high magnification values, real photographic lenses should not be compared to single thin convergent lenses, because it would very likely lead to (important) mistakes in the calculations of both focal length and magnification.

*Back to Table of contents.*

__1.4 – Rear lens converters.__

Rear lens converters (also called “teleconverters” or simply “converters”) are very common accessories used to increase the magnification of a photographic lens.

Some rear lens converters can also be found inside very famous macro lenses, as built-in units (see further).

Converters used in association with interchangeable photographic lenses are inserted between the lens and the camera body. They are called "rear lens converters", as opposed to "front lens converters" that may be attached at the front of some point-and-shoot camera lenses or camcorder lenses (see further).__1.4.1 – Operation and angular magnification.__

All interchangeable rear lens converters are divergent optical systems that capture the image at the back of the photographic lens to re-form it on the sensor, enlarging it. As divergent optical systems, their focal points (first and second) are located in the reverse way (Figure 16, bellow).

When a bundle of rays parallel to the optical axis enters from the left (front of the converter), the emerging light beam is divergent; thus, the second focal point (F’) is located in the front (in the object space); no real images are created in such conditions. F’ is the virtual image of an on-axis infinitely distant object point which can only be seen through the converter, from the rear.

A bundle of rays entering front the opposite side allows locating the first focal point (F) in the back of the converter (in the image space).

__Fig. 16__: *Optical system and cardinal points of the rear lens converter Nikon TC-20.*

So, how does a converter work when fixed at the rear end of a real photographic lens?

Sticking with the Nikon TC-20, we may consider three cases where a bundle of light rays incoming from the left (where the photographic lens is supposed to be) becomes gradually more and more convergent (Figure 17)…

- Top of the picture (reminder) the rays are parallel to the optical axis, then the emerging light beam is divergent: no real image created (as seen above).

- 1st case. Incident rays converge toward any point located beyond the first focal point (F) of the system; then the emerging beam remains divergent: no real image created.

- 2nd case. Incident rays converge precisely toward the first focal point (F) of the system; then the emerging light beam is parallel to the optical axis (as seen above), and again: no real image created.

- 3rd case. The incoming light beam converges toward any point X located before the first focal point (F); then the emerging light beam converges toward another single point X’: the point X’ is the image of the point X (we say that X’ and X are conjugate points).

__Fig. 17__: *Behavior of a convergent light beam going through the rear lens converter Nikon TC-20.*

Let's have a look at this very last case (see Figure 18, bellow)...

There are two particular conjugate points C and C’ such that every incident light ray directed toward point C and making an angle u about the optical axis before entering the optical system, emerges from the system striking the axis at point C’ making an angle u’ whose value is exactly half of u.

For this couple of points, the so-called “angular magnification” (G) of the system is then:

G = u' / u = 1 / 2 = 0.5

This angular magnification is a fundamental characteristic of the rear lens converter. The reciprocal of the angular magnification (1 / G) is the lateral magnification (g) of the system:

g = 1 / G = 1 / 0.5 = 2

The lateral magnification (g) of any rear lens converter is engraved on its body in the form of a “multiplier coefficient” (for example: “2x” for g = 2, or “1.4x” for g = 1.4).

__Fig. 18__: *Conjugate planes of the Nikon TC-20 (2x).*

Points C and C’ define two planes, P and P’, perpendicular to the optical axis. Every point belonging to P has a corresponding image in P’. The distance Delta P between these two planes is another fundamental characteristic of the rear lens converter. Positioning the optical system of the converter behind a photographic lens so that the image from the lens is coincident with the plane P (transferring the flange focus distance FFD), we get a new image in the plane P’. The location of the final image determines the position of the rear flange of the converter (transferring the flange focus distance FFD again). The distance between both flanges of the rear lens converter is then strictly equal to the distance Delta P.

Comparing the images CD and C’D’, we can check the converter-lateral-magnification value:

g = C’D’ / CD = 2

So, the final image is twice as large as the image from the photographic lens, and positioned in the same way (g is positive).

Obviously, the sensor can only receive the central portion of the image from the photographic lens, once enlarged by the converter. Thus, the angle of view of the lens-plus-converter couple is equal to the product of the angle of view of the lens by the angular magnification (G) of the converter.

In the same manner, the angular magnification of a rear lens converter Nikon TC-14 would be G = 0.714, and its lateral magnification:

g = 1 / 0.714 = 1.4.

Figure 19 (below) illustrates the effect of a rear lens converter Nikon TC-20 on a light beam passing through a Nikkor AF-S VR 300mm f/2.8G. As seen above, the converter divides by two the apex-angle of the cone of illumination. In the process, it:

- doubles the distance between the second principal point (H’) and the second focal point (F’); thus, the focal length increases from f’ = 293.7 mm to f’ = 587.4 mm;
- makes the relative aperture of the lens-plus-converter system twice smaller than that of the photographic lens alone (f/5.8 instead of f/2.9).

__Fig. 19__: *Nikkor AF-S VR 300mm f/2.8G.*

__Mouse out__*: **Alone.*

__Mouse over__*: **Associated with the rear lens converter Nikon TC-20 (2x).*

Figure 20 illustrates the position of the second principal point (H’) of the Micro-Nikkor AF 200mm f/4D alone and associated with the converters TC-14 or TC-20 (infinity focus).

__Fig. 20__: *Second principal point (H’) of the Micro-Nikkor AF 200mm f/4D with and without converters.*

*Back to Table of contents.*

__1.4.2 – Evolution of the focal length when focusing.__

Associating a photographic lens with a rear lens converter creates a new optical system that always includes floating elements, even when an inert-optical-system lens is used (in such a case, the lens becomes the moving part with respect to the converter, which remains static).

Well, whoever says “floating elements”, says “change in the position of the cardinal points” which inevitably leads to a change in focal length. So, let's see how the focal length of a lens-plus-converter combination changes during focusing...

__1.4.2.1 – Case of inert lenses.__

The combination of two optical systems having focal points (for example, a photographic lens and a rear lens converter) forms a new system whose focal length (f’) is given by the following equation:

f’ = f1’ . f2’ / -Delta

Let’s call it the “association relationship”, where f1’ and f2’ are the focal lengths of the two sub-systems, and Delta, or “optical gap”, is the distance between the second focal point of the sub-system #1 (F1’) and the first focal point of the sub-system #2 (F2). *Use algebraic values considering the direction of light propagation, i.e. positive from left to right, and negative in the reverse direction.*

Figure 21 (bellow) shows a very simple optical system consisting of a single convergent lens, as an objective, and a single divergent lens, as a rear lens converter:

- focal length of the objective, f1’ = 50 mm.
- focal length of the converter, f2' = -50 mm.
- optical gap F1’F2, variable.

__Fig. 21__: *Evolution of the focal length of a lens-plus-converter system*

*with respect to the distance between them (space).*

The objective is movable, so its second focal point (F1’) moves along. The converter is perfectly static, so its first focal point (F2) remains still as well. When the lens moves, the space between both elements varies, creating an identical variation in the optical gap (Delta). Since the latter is the denominator of the association relationship, any variation of the space between the two elements results in a variation of the focal length of the entire system in the opposite way. The graph (Figure 21, on the right) shows the variation of the focal length of the whole system with regard to the space between both elements.

This simplistic setup is perfectly representative of what actually happens when focusing an inert-optical-system photographic lens associated with a rear lens converter. When the object gets closer, the lens moves away from the sensor in order to focus. In the process, it also moves away from the converter that remains fixed, and the focal length of the whole system decreases.

Obviously, the focal length of the lens-plus-converter system is twice the one of the lens alone only for a single value of the space between both elements. In this example, given the thickness of the lenses, this value is 22.8 mm; with infinitely thin lenses it would have been 25 mm (since 50 x -50 / 100 = -25).

MC = 73.9 / 51.6 = 1.43.

__Fig. 22__: *50mm f/1.8 lens.
Focal length with and without TC-14A-type converter.*

Set on minimum focus, the whole 50mm lens optical system moves 7.7 mm forward. In the process, the focal length of the lens-plus-converter system decreases from f’ = 73.9 mm to f’ = 66 mm (Figure 23).

__Fig. 23__: *50mm f/1.8 lens plus TC-14A-type converter.
Focal length with respect to the focus distance (from infinity to 0.45 m).*

Figure 24 compares the optical system of the photographic lens alone with the lens-plus-converter system when both of them are set on minimum focus (maximum magnification). In both cases the image is created in A’, and the magnification of each system can be calculated.

__Fig. 24__: *Comparison of the maximum magnifications of a 50mm f/1.8 lens
with and without converter.*

In this configuration (minimum focus), the ratio of both magnifications with / without converter is (as expected):

MC = g2 / g1 = -0.213 / -0.149 ≈ 1.43

While the ratio of both focal lengths with / without converter is:

f’2 / f’1 = 66 / 51.6 ≈ 1.28.__Note__

- The multiplier coefficient of a rear lens converter applies to the focal length of the lens only when it is set on infinity focus; once the lens is no longer set on infinity focus, the multiplier coefficient applies to the magnification.

To find an inert-optical-system 300mm lens, we have to go back in time... Figure 25 compares the focal length’s behavior of the Nikkor ED 300mm f/4.5 with and without the converter Nikon TC-300 (2x).

__Fig. 25__: *Focal length evolution of an inert-optical-system 300mm lens
with respect to the focus distance, with and without rear lens converter *

*Nikon TC-300*

*.*

Without any rear lens converter, the focal length of this 300mm is, of course, perfectly constant with respect to the focus distance. While the focal length of the lens-plus-converter combination is only 489.5 mm on minimum focus (4 m - 13 ft.) and increases gradually with the focus distance until it reaches 600 mm on infinity focus. However, when set on minimum focus, the magnification of the lens alone is g = -0.095 and the magnification of the combination is g = -0.190 (that is to say 2 x -0.095, as expected).

*Back to Table of contents.*

__1.4.2.2 – Case of lenses with optical systems including moving elements.____ __

Let’s take, for example, the Micro-Nikkor AF 200mm f/4D IF-ED (Figure 26). On this lens, moving a group of elements between the front unit and the aperture stop carries out the focusing. During the process, the front part of the front unit shifts slightly forward and back again (same position on infinity focus as on minimum focus, see Figure 26).

The computation of the cardinal points position in both extreme focus configurations shows that the displacement of the floating elements induces a substantial reduction of the lens focal length, from f' = 200 mm down to f' = 102 mm.__Fig. 26__: *Cardinal points of the Micro-Nikkor AF 200mm f/4D IF-ED set on infinity focus
and on minimum focus (g = -1).*

Unlike what takes place with a zoom, the focal length variation is here directly related to the focus distance. To maintain the image in the sensor plane while the object comes closer, the second focal point (F’) of the whole system must shift forward. Decreasing the system focal length is a very efficient way to do so (among others). This working principle may be compared to the previous case where the same result was achieved by the forward shift of the whole lens (§ 1.3.1).

Let's take our simplistic optical system again to highlight the effect of the focal length variation of a lens placed in front of a converter (Figure 27). This time, the space between the objective and the converter remains constant, but the single convergent lens (objective) is replaced by a series of lenses with different focal lengths. The graph shows that when the objective focal length varies, the whole-system focal length changes in the same direction, but much faster.__Fig. 27__: *Evolution of the focal length of a lens-plus-converter system*

*with respect to the focal length of the lens.*

This is perfectly logical, since any reduction of the objective focal length (f1') leads to an equal increase of the optical gap (Delta). Now, the position of these two factors in the “association relationship” (as seen earlier, f’ = f1’ . f2’ / -Delta) is such that both variations act in the same way on the whole system focal length (f').

Thus, in the association of an internal-focusing lens and a rear lens converter, two phenomenons combines to strongly reduce the whole-system focal length when focusing on close distances:

- the reduction of the focal length of the lens (due to internal focusing),
- the virtual (or real) increase of the space between the lens and the converter.

__Fig. 28__: *Maximum magnification of the Micro-Nikkor AF 200mm f/4D IF-ED
associated with Nikon TC-14-type and TC-20-type.*

__Note__

- It is very obvious that the multiplier coefficient of the rear lens converter does not apply to the lens focal length any more, once it is no longer on infinity focus configuration. Indeed, the focal length of the Micro-Nikkor AF 200mm f/4D IF-ED plus Nikon TC-20 (2x) association when set on minimum focus is only f' = 65.6 mm (which is quite far away from what might be expected).

The Micro-Nikkor AF 200mm f/4D IF-ED is a relatively long-focal-length lens allowing high magnification. In this sense, it is an extreme case (that's why I chose it to illustrate my point). However, all internal focusing telephoto lenses show an important focal length evolution when combined with a rear lens converter.

__Fig. 29__: *Focal length and magnification variation curves of the Nikkor AF-S 600mm f/4D
compared with those of the Nikkor AF-S VR 300mm f/2.8G alone or associated different converters.*

On infinity focus:

- only the old Nikkor ED 300mm f/4.5 associated with the TC-300 (2x) actually reaches 600 mm;
- the focal length of the Nikkor AF-S VR 300mm f/2.8G associated with a converter TC-20 (2x) is almost equal to that of the Nikkor AF-S 600mm f/4D (respectively f' = 587.4 mm and f' = 587.7 mm).

Gradually, as the focus distance becomes shorter, the focal length of the five studied cases decreases in different ways (for different reasons):

- Nikkor ED 300mm f/4.5 with TC-300 (2x); inert-optical-system lens moving away from the converter; moderate variation of the focal length;
- Nikkor AF-S 600mm f/4D; internal-focusing lens inducing a decrease of the focal length when the object comes closer; moderate variation of the focal length;
- Nikkor AF-S VR 300mm f/2.8G with TC-20 (2x) or TC-14 (1.4x); combination of the two previous effects (internal focusing and use of a converter); important variation of the focal length.

On short focus distance setting, the focal length curves of the Nikkor AF-S VR 300mm f/2.8G alone and associated with the TC-14 and TC-20 are quite scattered, but the magnification curves (on the right) are in full harmony with the multiplier coefficients of the converters.

The magnification curves of the Nikkor AF-S 600mm f/4D alone and Nikkor AF-S VR 300mm f/2.8G with TC-20 (2x) converter are nearly identical despite the large difference of their focal lengths when focused on close distance. For example, at 6 m (19.6 ft.) the focal length of Nikkor AF-S 600mm f/4D is almost 12% higher than that of the Nikkor AF-S VR 300mm f/2.8G associated with the TC-20 (2x), while their magnifications are almost equal (deviation 1.7%). The explanation is given by the following illustration...__Fig. 30__: *Location of the cardinal points of the Nikkor AF-S 600mm f/4D
compared with those of the Nikkor AF-S VR 300mm f/2.8G associated with the TC-20,
both focused on the same distance (6 m -19.6 ft.).*

On the same focus distance the focal lengths of both lenses are very different, as well as the distances A’F’. One difference compensates the other to provide almost equal lateral magnifications.

The combination of a lens (master lens) and a rear lens converter as a built-in unit has often been used in the design of macro lenses. This type of association allows a high magnification potential with a relatively small moving range (and low weight) of the focusing elements. Figure 31 (bellow) shows a simplified section of the Micro-Nikkor AF 60mm f/2.8D. This lens is made up of a 50mm master lens including floating elements (CRC system) and a 1.2x multiplier coefficient rear lens converter. The focusing is carried out in a conventional manner by the forward shift of the master lens (the front part of the lens moving faster than the rear one to maintain good aberration corrections at high magnification). The use of a converter allows to reach the magnification g = -1 with a shift of the front lens elements less than 43 mm.__Fig. 31__: *Working of the Micro-Nikkor AF 60mm f/2.8D.*

The Micro-Nikkor 105mm f/2.8 Ais (Figure 32) adopts a similar optical-system principle. The master lens is a 75mm f/2 and the converter multiplier coefficient is 1.4x. As previously seen, increasing the space between sub-units #1 and #2 of the master lens induces a slight increase of its focal length. On minimum focus, this combination reaches the magnification g = -0.5 with a shift of the front lens elements less than 25 mm.

__Fig. 32__: *Optical system of the Micro-Nikkor 105mm f/2.8 Ais.*

The Micro-Nikkor AF 105mm f/2.8D (Figure 33) is more different from its predecessor than it seems. Although its master lens is still a 75mm f/2, the aperture stop is now placed at the center, and the 1.4x converter includes only three elements instead of four. In addition, the CRC system works by decreasing the space between sub-units #1 and #2 of the master lens, which tends to reduce its focal length instead of increasing it. This allows it to reach the magnification g = -1 with no excessive increase in the forward shift of the master lens (less than 36 mm).

__Fig. 33__: *Optical system of the Micro-Nikkor AF 105mm f/2.8D.*

*Back to Table of contents.*

Regardless of the optical system of the lens, the focal length of a lens-plus-rear lens converter combination decreases as the focus distance becomes shorter.

The multiplier coefficient of a rear lens converter applies to the focal length of the lens only on infinity focus setting. In any other cases, it applies to the lateral magnification.

The multiplier coefficient of a rear lens converter applies to the f-number set on the lens (see the page “Pupilles et ouvertures”, translated soon).

__Note__ *(Just for the pleasure of knowledge)*

It is possible to design rear lens converters with angular magnification greater than G = 1 (multiplier coefficient lower than 1x). Such attachments are convergent systems. Although, the focal length of the lens-plus-converter combination is then smaller than that of the lens alone, this does not make a wider-angle combination, on the contrary. The main usefulness of such converters is to provide a shorter focal length without reducing the entrance pupil of the system. Thus, the relative aperture may be greatly increased (usually more than f/1). The drawbacks of such associations are:

- a small image circle,
- a short back focal length.

__Fig. 34__: *Mr. M. J. Herzberger’s rear lens converter (multiplier coefficient: 0.41x).*

*Back to Table of contents.*

__2 – Afocal systems.__

Afocal systems are widespread. Lots of optical instruments are afocal systems (for example, binoculars and telescopes when set on infinity focus) or they incorporate such systems (some wide-angle lenses, some telephoto lenses and some zooms). Front lens converters also are sheer afocal systems.

__2.1 – Operation and angular magnification.__

An afocal system is a system whose focal points are infinitely distant (in other words, a no-focal-point system); therefore, it has no focal length.

All that is necessary to make a simple afocal system are two lenses: for example, a convergent one (positive) and a divergent one (negative). Figure 35 shows that, if the second focal point (F1') of the first lens (convergent 1) is coincident with the first focal point (F2) of the second one (divergent 2), then the optical gap (Delta) is zero. As the latter is the denominator of the “association relationship” (f’ = f1’ . f2’ / -Delta), when it tends toward zero the focal length of the entire system tends toward infinity.

Once F1' and F2 are superimposed, the space between both lenses is set. A shorter space would produce a divergent system. On the contrary, a longer space would produce a convergent system (that’s how telephoto lenses are made up).__Fig. 35__: *Composition of a Galilean afocal system.*

When a bundle of parallel rays goes through an afocal system, it emerges as a bundle of parallel rays too. Then, the image of an infinitely distant object is formed at infinity. Used individually on a camera body, such a system wouldn't be very useful. Nevertheless, its properties make it particularly interesting in numerous cases...

__Property 1__

Any incoming light beam making an angle u about the optical axis emerges from the system with an angle u' different from u (Figure 36). As previously seen, the angular magnification (G) of the system is:

G = u' / u

This angular magnification (G) is a fundamental characteristic of afocal systems. For example, the angular magnification of 8x30 binoculars is G = 8 (30 being the entrance pupil's diameter, in millimeters).__Fig. 36__: *Properties of Galilean afocal systems.*

One can also determine the angular magnification (G) of the afocal system with the ratio:

G = | f1’ / f2’ |

where f1' and f2' are the respective focal lengths of the front and rear lenses of the afocal system.

__Property 2.__

The diameter of the emerging light beam (D') is different from that of the incoming light beam (D). The lateral magnification (g) of the system is:

g = D' / D

Unlike systems with focal points, the lateral magnification of afocal systems is constant and independent of the object distance.

As previously seen, the lateral magnification (g) is the reciprocal of the angular magnification (1 / G). Thus, taking again as an example the 8x30 binoculars, one can easily determine its exit pupil's diameter (D'):

Angular magnification, G = 8

Entrance pupil's diameter, D = 30 mm

D’ = D . g

So, D’ = D / G = 30 / 8 = 3.75 mm

__Property 3.__

Afocal systems are theoretically reversible: they can be used with the convergent lens on the front and divergent lens on the rear (positive-negative), or vice versa (negative-positive). In the process, the magnification values (G and g) reverse.

Once put on the front of a normal lens, an afocal system can deeply transform its characteristics, so they are often used as front lens converters. Such converters are widely used by photo or video amateurs to extend the possibilities of lenses that cannot be taken apart from the camera or camcorder. As they are simple afocal attachments, they should not be confused with rear lens converters, which are sheer divergent systems (with focal points).

*Back to Table of contents.*

Figures 37 (bellow) illustrates how our simple afocal attachment can change the focal length of a 50 mm photographic lens, depending on whether the front lens is positive or negative. I sized the afocal-system elements to get an angular magnification G ≈ 2 when the positive lens is on the front (lateral magnification g ≈ 0.5). Consequently, when the negative lens is placed on the front, the angular magnification is G ≈ 0.5 (lateral magnification g ≈ 2).

As one can see, when the **convergent** lens is on the front (mouse out), the focal length of the whole system is f’ = 100 mm. So, that would be a front teleconverter. When the **divergent** lens is on the front, the focal length of the combination is reduced to 25 mm. So, that would be a front wide-angle converter.

__Fig. 37__: *A very simple reversible front lens converter.
Mouse out: telephoto converter (convergent lens on the front).
Mouse over: wide-angle converter (divergent lens on the front).*

Some reversible front lens converters were studied during years 1950-60 (WE Schade, 1956). Some of them allowed switching from wide-angle to telephoto position and vice versa, simply rotating them, without taking the converter apart from the lens. The intermediate position (positive and negative lenses up and down on both sides of the optical axis) allowed using the lens without the converter.

Nowadays, these accessories are sophisticated optical instruments, and the good front lens converters are no longer made up of two simple lenses but of two compound units, each of them including several lenses to correct aberrations. Besides, they are now optimized for use either as tele-converters, or as wide-angle converters, and so they are no longer reversible in practice. Low-dispersion glass may even be used (for example, elements #2 and #3 of the Nikon TC-E3ED).__Fig. 38__: *Front tele-converter Nikon TC-E3ED (3x).*

__Fig. 39__: *Front tele-converter Nikon TC-E3ED on zoom 8-24mm set in telephoto configuration.*

__Fig. 40__: *Front wide-angle converter Nikon WC-E63 (0.63x).*

__Fig. 41__: *Front wide-angle converter Nikon WC-E63 on zoom 8-24mm set in wide-angle configuration.*

The multiplier coefficient of a front lens converter applies to the lens focal length regardless of the focus distance. If properly sized, a front lens converter does not affect the relative aperture of the lens (whereas, by their very principle, rear lens converters do change the aperture of the lens in all cases).

*Back to Table of contents.*

__2.3 – Other afocal-system applications.__

Galilean afocal systems, such as those described above, are not only used to make front lens converters: they are also often used as built-in units inside the optical systems of some telephoto or wide-angle lenses.

For example, Figure 42 illustrates a telephoto lens (Nikkor AF-S VR 300mm f/2.8G IF-ED) including a positive-negative afocal unit placed in front of a master lens. The focal lengths of the different lens units are:

- front part of the afocal system (positive unit), f1’ = 150 mm,
- rear part of the afocal system (negative unit), f2' = -57 mm,
- master lens, f3' = 111.7 mm.

The angular magnification of the afocal system is G = | f1’ / f2’ | = 2.63.

The focal length of the whole system is f' = f3’ . G = 111.7 x 2.63 ≈ 293.7 mm.

__Fig. 42__: *Telephoto lens **Nikkor AF-S VR 300mm f/2.8G IF-ED (afocal system + master lens)**.*

The optical systems of the Micro-Nikkor AF 200 mm f/4D IF-ED (Figure 26) and AF-S Nikkor 600mm f/4D (Figure 30) are built on the same principle (not so easy to see on the 600mm lens because when the focus distance is as close as 6 m, as it is shown, the rear part of the afocal system is very close to the small master lens).

Figure 43 (bellow) illustrates a wide angle lens including a negative-positive quasi-afocal system placed in front of a master lens: the Nikkor 28mm f/2 Ais.

The focal length of the master lens is f3’ ≈ 42.9 mm, and the theoretical angular magnification of the afocal unit is G ≈ 0.667. The effective focal length of the whole system is then f’ ≈ 42.9 x 0.667 ≈ 28.6 mm.__Fig. 43__: *Wide-angle lens **Nikkor 28mm f/2 Ais (quasi-afocal system + master lens)**.*

__Fig. 44__: *Telezoom lens Zoom-**Nikkor 180-600mm f/8 ED Ais (afocal system + master lens)**.*

So far, we've only seen afocal systems used in front position; yet they can be placed in an intermediate position too (mostly inside zoom lenses). Such built-in afocal attachments are called “extenders”, as they extend the focal length range of the zooms they are associated with. A switching mechanism allows including the extender inside the optical system of the zoom, or retracting it. Extenders have been very common for years on professional camcorder zooms, but the release of the Canon 200-400mm f/4 shows that they might be used more regularly on SLR zooms in the future.

An extender alters the focal length of the lens in exactly the same way as a front lens converter. The focal length of the combination is the product of the lens focal length by the angular magnification (G) of the extender. Generally, the extender fits into the master lens of a zoom, in a space between two sub-units where the light rays are parallel (in the example bellow, the master lens is on the right of the aperture stop).

Figure 45 shows a 19 x 9.5mm professional video zoom set on the focal length of 100 mm (mouse out). This lens comes with a 2x multiplier-coefficient extender. When the extender is switched in (mouse over), the diameter of the light beam passing through the central space of the master lens is reduced by half. Then the second principal point (H') is thrown forward twice as far: the focal length is doubled.__Fig. 45__: *Canon 19 x 9.5**mm f/1.85-2.85 **zoom lens **with 2x extender**.*

*Back to Table of contents.*

__3 – References__

• Optical system – USP 2,186,605 – M. J. Herzberger – 1937.

• Wide angle photographic lens – USP 3,736,049 – Yoshiyuki Shimizu, Kawasaki-shi, Kanagawa-ken – 1973.

• Optical system for the magnification varying portion of an ultra-telephoto type zoom lens – USP 3,743,384 – Soichi Nakamura – 1973.

• Modified Gauss type photographic lens – USP 4,139,265 – Sei Matsui – 1979.

• Rear conversion lens – USP 4,154,508 – Soichi Nakamura – 1979.

• Lens system capable of short distance photography – USP 4,392,724 – Yoshinari Hamanishi – 1983.

• Rear conversion lens – USP 4,514,051 – Yoshinari Hamanishi – 1985.

• Catadioptric telephoto lens – USP 4,666,259 – Yutaka Iizuka – 1987.

• Retrofocus type wide-angle lens – USP 4,690,517 – Daijiro Fujie – 1987.

• Lens system capable of close-up photographing – USP 4,986,643 – Keiji Moriyama – 1991.

• Rear conversion lens – USP 5,253,112 – Kenzaburo Suzuki, Yoshinari Hamanishi – 1993.

• Telephoto lens system allowing short-distance photographing operation – USP 5,402,268 – Wataru Tatsuno – 1995.

• Internal focusing telephoto lens – USP 5,745,306 – Susumu Sato – 1998.

• Lens capable of short distance photographing with vibration reduction function – USP 5,751,485 – Kenzaburo Susuki – 1998.

• Front tele-converter and front tele-converter having vibration-reduction function – USP 6,424,465 – Kenzaburo Suzuki – 2002.

• Wide converter lens – USP 6,504,655 B2 – Atsushi Shibayama – 2003.

• Zoom lens system – USP 6,693,750 B2 – Susumu Sato – 2004.

• Zoom lens and photographing system – USP 6,965,481 B2 – Yasuyuki Tomita, Shinichiro Yakita – 2005.

• Zoom lens system – USP 7,158,315 B2 – Atsushi Shibayama – 2007.

• Imaging lens, optical device thereof, and method for manufacturing imaging lens – USP 2009 / 0190220 A1 – Haruro Sato – 2009.

• Optique géométrique (Imagerie et instruments) – Bernard Balland – 2007.

• Lens design Fundamentals – Rudolf Kingslake – 1978.

• A history of the photographic lens – Rudolf Kingslake – 1989.

• Modern Optical Engineering – Warren J. Smith – Fourth edition 2008.

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*PT, July 25, 2010.*

*Updated February 24, 2011. *

*Many thanks to Kelly Bellis for the review of my English.*

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Autres sujets :

Telephoto zooms (in English)

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