Fisheyes

 

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In the range of photographic lenses offered by major brands, fisheyes, with their extremely short focal length and huge angle of view, are at the opposite end of telephoto lenses that are the natural wildlife photographers’ tools. So far, I never had the opportunity to use one of these amazing lenses capable to take in an entire hemisphere, and it’s just out of simple "optics" curiosity that I decided to have a look at the way they operate (therefore, this page is just a collection of study materials tidied up to make it presentable). As a user of Nikon equipment, I naturally focused my attention towards the fisheyes of this manufacturer.

Currently, two "full frame" fisheyes are listed in the Nikon catalog:
• a 10.5 mm f/2.8 covering the 15.8 x 23.6 mm frame (Dx),
• a 16 mm f/2.8 covering the 24 x 36 mm frame.

The production of the last "circular frame" fisheyes of the brand, the imposing 6 mm f/2.8 Ais and the 8 mm f/2.8 Ais, ceased shortly before year 2000. With its 220° angle of view, the working of the 6 mm f/2.8 has long remained an enigma for me. I even thought that this imposing lens was the materialized ultimate limit of what was achievable in the matter of wide-angle lenses. My research showed me that I was wrong…

The 10.5 mm f/2.8 and the 16 mm f/2.8, both current Nikon fisheyes (Nikon documents).

 

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Summary :

Fisheyes are not like other lenses.
The AF Dx Fisheye-Nikkor 10.5 mm f/2.8G ED.
The AF Fisheye-Nikkor 16 mm f/2.8D.
The Fisheye-Nikkor 6 mm f/2.8.
A lens that could have been the “Fisheye-Nikkor 5.4 mm f/5.6”.


I  - Fisheyes are not like other lenses.

What makes fisheyes different from other lenses?

The answer to this question might be "their 180° field of view" (or even more). Well, I’d rather say that their “mapping function” is what makes them so different from the others…

Just a few words about this mapping function...
 
On the sensor of the camera, the position of the image of an object point depends on:

• the angle between the optical axis and the incoming light beam (from the object point),
• the focal length of the lens,
• the mapping function of the lens.

Figure 1 (below) shows an example of a lens operating in the “classic way”: it’s mapping function is commonly called “perspective projection” or “gnomonical projection”. Almost all lenses are designed according to this principle (or tend to satisfy it, more or less). This is also the mapping function of a pinhole.

Fig. 1 : Image obtained by perspective projection.

Theoretically, this mapping function allows reproducing all the straight lines of the scene by straight lines on the picture. The latter is then theoretically free of distortion (not to be confused with perspective distortion due to the fact that if the lines are faithfully represented, the angles between these lines are not).

However, the design of perspective projection lenses whose angle of view exceeds 110°, and fitted to SLRs (long back focal length) is quite difficult. To achieve this, one must create very short focal length optical systems, which are very sophisticated and therefore very expensive.

Mathematically, the perspective projection is expressed as follows:

R = f . tan(Theta)

where
• R is the “radial distance”, or distance between the center of the image (where the optical axis meets the sensor) and the image point;
• f is the focal length of lens;
• Theta is the angle between the optical axis and the incoming light beam (from the object point).

The operating limits of lenses that work according to this principle are quite clear: to achieve a 180° field of view (Theta = 90°) for a given R value, the focal length must tend towards zero. There, is the problem…

Figure 2 (below) illustrates how these factors act in the image formation process. The graph (on the left) shows that the radial distance (R) increases rapidly with the angle Theta (red curve, mouse out). This example uses a 15 mm perspective projection lens. Thus, when angle Theta = 45°, the corresponding radial distance is R = 15 mm. On the sensor (on the right), all 45° incident light beams describe a 15 mm radius circle that cannot fully enter the 24 x 36 mm frame.

Fig. 2 : The three main mapping functions.
Mouse out: perspective and equidistant functions.
Mouse over: equidistant and equisolid angle functions.

To overcome the limitations of the perspective projection, optical systems implementing other mapping functions have been designed. The purpose of these alternative functions is to "shrink" the images of the objects located on the edge of the field of view so as to make it possible to increase angle Theta while maintaining a realistic focal length value. In other words, with such optical systems, the magnification is not constant over the entire surface of the frame: it is lower at the periphery than what it is at the center. This is the point that makes fisheyes so different from the other lenses.

Out of several mapping functions, two functions are used most often on fisheyes:

• the equidistant function, whose mathematical expression is

R = f . Theta (Theta in radians) or R = f . Theta . pi / 180 (Theta in degrees)

• the equisolid angle function,

R = k1 . f . sin(Theta / k2)

theoretically the expression is written with k1 = k2 = 2, but in practice both coefficients are specific to each fisheye.

Though not used as often as the equisolid angle function, the equidistant function more easily allows angular measurements.

With mouse over Figure 2, we can see that, for angle Theta = 45°, a 15 mm equisolid angle projection fisheye gives a radial distance R = 11.5 mm. Therefore, on the sensor (on the right) all 45° incident light beams describe a 11.5 mm radius circle which perfectly fits the 24 x 36 mm frame. But in return, the barrel distortion is important and all the straight lines that do not intersect the optical axis are reproduced as curves.

Unlike equisolid angle projection, equidistant projection is linear. Consequently it compresses less the edges of the image.

To quantify the "distortion" of an equisolid angle fisheye, we often compare its projection curve to the one of an equidistant fisheye of same focal length.

 

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II  - The AF Dx Fisheye-Nikkor 10.5 mm f/2.8G ED.

The following illustrations are made from U.S. Patent 7,161,746 B2 data, filed by Mr. Mizuguchi (January 9, 2007). This document follows and completes U.S. Patent 6,844,991 B2 (January 18, 2005). The optical system of the lens is completely defined by the following data table (same on both patents - example 9 - figure 17).

Fig. 3 : AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.
Definition of the optical system (data in black taken from US Patent 7,161,746 B2).

Note however that the total optical length TL = 103.7 mm (bottom left) does not match the sum of the spaces (column d), added up to the back focal length Bf. Moreover, the calculations show that the value d13 = 8.776 mm is not compatible with the back focal length value Bf = 41.1 mm. It turns out that this d13 value corresponds of the optical system configuration when the focusing distance is 140 mm, but then we should have Bf = 42.1 mm (instead of 41.1 mm). So, when the lens is focused at infinity, we should have d13 = 6.467 mm. Deliberate error?

All surfaces of this lens are spherical. Only one element (# 09) is made of low dispersion glass E-FK01 (Abbe number 81.61).

With a 28.4 mm image circle diameter, this small lens covers the 15.8 x 23.6 mm frame (Nikon DX) with a 182.6° angle of view (a 180° angle of view corresponds to a 28.2 mm image circle diameter).

Fig. 4 : AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.
Simplified section (full aperture, infinity focus).
Note how different the cones of illumination openings are for Theta = 0° and Theta = 91.3°
(optical vignetting).

With a cut off hood (permanent alteration frequently referred to as a “shaved” lens), this fisheye may be interesting for 24 x 36 mm sensor camera users too. Then, the angle of view exceeds 200° (at the expense of a perceptible field curvature at the edge of the image circle). With a 200° angle of view, the image circle diameter reaches 29.5 mm. In practice, the height of the frame (24 mm) limits the angle at 142° in the vertical direction.

Fig. 5 : AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.
Image formed on a 24 x 36 mm sensor
.

The minimum focus distance of this lens is 140 mm (5.5 inches); the subject is then only 33 mm (1.3 inches) ahead of the front lens.

On this lens, focusing is carried out by a CRC (Close-Range Correction) system. When the focus distance varies from infinity to 140 mm, the entire optical system moves forward (as usual), increasing the back focal length, but here the distance of the moving elements located in front of the stop is three times larger (3.3 mm) than that of the elements located behind the stop (1 mm). This is a way to maintain a good field flatness regardless of the focus distance.

Fig. 6 : AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.
Simplified section, fully open and focused at 140 mm (with CRC system).

This CRC system is not a simple advertising gimmick. In this case, it is perfectly justified and effective. To illustrate its effectiveness, I compared the ray tracing of two light beams incoming with an angle Theta = 80°, at the minimum focus, wide open, with and without CRC.

• First case (Figure 6, above). Lens in normal configuration focused at 140 mm with CRC system. The oblique light beam (in red) correctly focuses on the image plane.

• Second case (Figure 7, below). Lens in infinity focus configuration is moved forward as a whole to achieve focusing at 140 mm (without using the CRC system). The oblique light beam (in red) focuses before the sensor: the object plane is really sharp only at the center of the frame.

Fig. 7 : AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.
Highlighting of the field curvature when CRC system is not used (focused at 140 mm).

Just a few words about the entrance pupil…

The position of the entrance pupil of a lens determines its perspective center (no-parallax point). The entrance pupil of narrow-angle lenses can be considered as a fixed point. Yet, as the entrance pupil location depends on the angle between the optical axis and the incoming light beam, it is not possible to define a single no-parallax point for very wide-angle lenses. In the case of fisheye lenses, the entrance pupil of large off-axis angle light beams is located quite further forward from the entrance pupil of an on-axis light beam.

Reverse ray tracing from the top and bottom edges of the aperture stop (in the meridional plane) allows determining the entrance pupil location with accuracy. Figure 8 (below) shows an example of this plot applied to the Nikkor 10.5 mm f/2.8. The aperture is deliberately set at f/8 to get rid of the optical vignetting effect. The entrance pupil of an incoming light beam with incident angle Theta = 80° is located 11.8 mm before the entrance pupil of the axial beam, and 16.3 mm away from the optical axis. It is also tilted 30° backwards.

Fig. 8 : Reverse rays tracing from the aperture stop, applied to the optical system
of the AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.

Figure 9 (below) and the attached animation illustrate the entrance pupil shift of the Fisheye-Nikkor 10.5 mm f/2.8 when angle Theta varies from 0° to 90°. In the process, the intersection between a ray passing through the center of the entrance pupil with the optical axis (aka “Least Parallax Point”) moves forward too.

Fig. 9 : AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.
Entrance pupil location as a function of the angle Theta (focus at infinity).

Panorama photographs require images with the least amount of error introduced by parallax in the overlapped areas. To achieve this, the camera must be rotated around the least-parallax point corresponding to the angular positions of the seams (half the swiveling angle between two consecutive shots). Michel Thoby (among others) talks a lot about this particular subject in his web site.

Figure 9a (below) and the attached animation illustrate the progression of the parallax errors occurring between the images of two objects when the lens swivels around the least-parallax point corresponding to the incident angle Theta = 70°… A and B are two objects perfectly lined up along the optical axis: before rotating the lens, the images of A and B are perfectly superimposed at the center of the sensor. As soon as the lens starts swiveling, both images move away from each other (parallax) as they shift away from the center of the frame, because light beams incoming from objects A and B do not “see” the same entrance pupil. The gap between both images reaches a maximum value, then decreases as the angle of rotation approaches 70°. At last, both images meet again when the 70° value is reached (no parallax). Beyond this value, parallax reverses and increases again until the extreme edge of the image field.

Fig. 9a : AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.
Highlighting of the parallax occurring between the images of two perfectly lined up objects.

Graphs of Figure 10 show the calculated projection, distortion, and shift of the least-parallax point of the Fisheye-Nikkor 10.5 mm f/2.8.

Fig. 10 : AF DX Fisheye-Nikkor 10.5mm f/2.8G ED.
Projection, distortion and least-parallax point location (infinity focus).

The mapping function of this lens is of the equisolid type:

R = k1.f.sin (Thêta/k2)

Michel Thoby experimentally determined k1 ≈ 1.47 and 1/k2 ≈ 0.713. These coefficients match the calculated results presented above quite well (left graph).

 

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III  - The AF Fisheye-Nikkor 16 mm f/2.8D.

The following illustrations are made from U.S. Patent 5,434,713 data, filed by Mr. Sato (July 18, 1995). The optical system of this lens is very likely the result of a development of the former system invented by Mr. Shimizu, father of the first generations of Nikon 16 mm fisheyes (U.S. Patent 3,734,600 - May 22, 1973). The table below defines the optical system of the lens (yet, there is no mention of the aperture stop's exact location).

Fig. 11 : AF Fisheye-Nikkor 16mm f/2.8D.
Definition of the optical system (data in black taken from U.S. Patent 5,434,713).

This lens has no low-dispersion glass elements (Abbe number of the least dispersive glass is 70); all surfaces are spherical. Setting the aperture stop 5.7 mm behind the exit face of the fourth element (according to the diagram Figure 1 of the patent) the 43.3 mm image circle diameter (24 x 36 mm) covers an angle of view slightly over 180 degrees (180.56°).

Fig. 12 : AF Fisheye-Nikkor 16mm f/2.8D.
Simplified section (full aperture, infinity focus).

Focusing is carried out by a CRC system of the same kind as the one of the 10.5 mm fisheye. Minimum focus is 250 mm (≈ 10 inches); the subject is then 148 mm (5.8 inches) ahead of the front lens.

Fig. 13 : AF Fisheye-Nikkor 16mm f/2.8D. Simplified section at full aperture.
Mouse out: infinity focus.
Mouse over: minimum focus (CRC system).

Figure 14 shows the calculated projection, distortion, and shift of the least-parallax point of the 16 mm fisheye (blue curves).

Fig. 14 : AF Fisheye-Nikkor 16mm f/2.8D.
Projection, distortion and least-parallax point location (infinity focus).

The mapping function of this lens is also of the equisolid type:

R = k1.f.sin (Thêta/k2) with k1 ≈ 1.77 and 1/k2 ≈ 0.57.

I pause here to highlight the differences between this equisolid 16 mm fisheye and (as an example) the venerable Nikon 15 mm f/5.6 (perspective projection). Both focal lengths are quite close but the optical system of the latter is much more sophisticated and its angle of view is much narrower (110°). Figure 15 (below) presents its projection curve: this lens is almost gnomonical over its 110° angle of view.

Fig. 15 : Nikkor 15 mm f/5.6.
Simplified section and mapping function (infinity focus).

For comparison again, Figure 16 and the attached animation illustrate the behavior of the 15 mm f/5.6 entrance pupil. The entrance pupil shift is completely different from the one of the fisheye: here, we can observe a backwards tipping.

Fig. 16 : Nikkor 15 mm f/5.6.
Entrance pupil location as a function of the angle Theta (focus at infinity).

Note that as long as angle Theta < 33°, the aperture stop is the only thing that limits the size of the incoming light beams. When angle Theta > 33°, the edges of other elements (see red arrows) proved to be more limiting than the aperture stop (optical vignetting).

Figure 16a (bellow) and the attached animation illustrate another example of entrance pupil’s behavior (AF-Nikkor 28 mm f/2.8D).

Fig. 16a  :  AF-Nikkor 28 mm f/2.8D.
Entrance pupil location as a function of the angle Theta (focus at infinity).

 

 

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IV  - The Fisheye-Nikkor 6 mm f/2.8.

Just out of curiosity...

The optical system of this lens is not only impressive because of its size —the effective diameter of the first two menisci are respectively 213 mm (8.4 inches) and 100 mm (4 inches), and the distance between the vertex of the front lens and the image plane is 208 mm (8.2 inches)—, but also because of the fact that the first three menisci are made of BK7 glass.

Fig. 17 : Fisheye-Nikkor 6 mm f/2.8.
Definition of the optical system (data in black taken from U.S. Patent 3,737,214).

The following illustration is made from U.S. Patent 3,737,214 data (example 1), filed by Mr. Shimizu (June 05, 1973).

The first version of this atypical lens was released in March 1972. With its 236 mm diameter, its weight exceeding 5 kg (11 lbs.) and its insane price, very few photographers have had the opportunity (or the luck) to use it!

Fig. 18 : Fisheye-Nikkor 6 mm f/2.8.
Simplified section, full aperture, and infinity focus.
Note the weak difference between the cones of illumination openings for Theta = 0° and Theta = 110°
(very low vignetting).

Graphs of Figure 19 (below) show the calculated projection and distortion of the fisheye 6 mm f/2.8. With just over 3% distortion at
Theta = 110°, the mapping function of this lens is almost of the equidistant projection type. On a 24 x 36 mm sensor, the circular image has a diameter slightly over 23 mm.

Fig. 19 : Fisheye-Nikkor 6 mm f/2.8.
Projection and distortion (infinity focus).

 

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V  - A lens that could have been the “Fisheye-Nikkor 5.4 mm f/5.6”.

Another curiosity...

In August 1970, Mr. Isshiki and Mr. Matsuki of the Nippon Kogaku KK filed a patent in the United States (U.S. Patent 3,524,697) for the invention of two fisheyes. The respective angle of view of these two lenses is 220° and 270°. Neither of them has been subject to mass production. Their designs are very similar; the main difference between both optical systems comes from the huge divergent meniscus capping the second specimen.

I chose to focus my attention on this second “very wide angled” fisheye for a while... As it appears in the patent, the optical definition of this lens is quite ambiguous: generally the optical systems are defined for their effective focal length or, alternatively, for a focal length of 1 (or sometimes 100). Here we have 10...

Fig. 20 : 270° Fisheye.
Definition of the optical system (data in black taken from U.S. Patent 3,524,697).

Now, with a focal length of 10 mm, this lens would cover exactly the full-size 24 x 36 mm frame. This may seem perfectly coherent. However, with a 349 mm effective diameter front meniscus (over 13.7 inches) weighting over 11 kg (24 lb. - BK7 glass), this fisheye would have been quite monstrous! Moreover, its 41.6 mm diameter rear elements would have make it impossible to be introduced into the reflex chamber of a standard SLR.

It is therefore highly likely that if this lens had been actually produced it would have been as a fisheye lens giving a circular image fitting the 24 x 36 mm frame. Transposing the patent data in this way, we obtain a 5.4 mm f/5.6 much more reasonable in size, with a 12.34 mm back focal length and rear optical elements having an effective diameter of 22.5 mm, for a 23.5 mm image circle diameter.

Fig. 21 : The 5.4 mm f/5.6 Fisheye (Nikon patent).
Simplified section of the optical system (full aperture, infinity focus).

Given the enormous angle of view, the distortion of this 5.4 mm f/5.6 fisheye is quite remarkable...

Fig. 22 : The 5.4 mm f/5.6 Fisheye (Nikon patent).
Mapping function and distortion (infinity focus).

No doubt that, with such characteristics, this fisheye could have illustrated the front page of Nikon catalogs for many years...

 

Pierre Toscani, May 5, 2010.
Revised and supplemented December 3, 2010.

Many thanks to Kelly Bellis and Michel Thoby for their constructive remarks, comments, and for the helpful review of my English.

 

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Other subjects :

Telephoto zooms (in English)
Focal length and Magnification (in English)
La mise au point (in French)
Pupilles et ouvertures (in French)
AF-S VR 300 mm f/2.8G (in French)
Fluorine et verres ED (in French)