Short Presentations

Summary:

01 - Depth of field.
02 - Nikkor 135mm f/2 Ais and 135mm f/2.8 Ais.
03 - Reflex-Nikkor 500mm f/8 Ai.
04 - What is the maximum aperture consistent with the Nikon F-mount?
05 - Micro-Nikkor 55mm f/2.8 (Ais and AF).
05a - Effect of the CRC system of both Micro-Nikkor 55mm f/2.8 (Ais and AF) on their focal length.
06 - AF Micro-Nikkor 200mm f/4D IF-ED.
07 - AF-S DX Micro-Nikkor 85mm f/3.5G ED VR.
08 - Working of a single-lens-reflex camera.
09 - Anatony of a modern lens.
09a - Synchronized motions.
09b - The piezoelectric motor with progressive wave.

to be continued...

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Depth of Field.

There are lots of articles on the concept of depth of field, so here is just a simple animation after a few words of introduction…

The concept of depth of field is based on the tolerance of our eyes to blurriness, i.e. their ability to liken, in some circumstances, a spot to a point. If our eyes did not show any tolerance to blurriness, only the objects located exactly in the focus plane would appear “sharp” on the picture; therefore it would be impossible to create a two-dimensional image from a three-dimensional scene with a classic camera.

A sharp image consists of picture elements that the resolution of the eyes cannot discern. Thus, considering an image observed from a certain distance, it is possible to define the maximum dimension of a spot that is likened to a point by the eyes. Beyond this dimension, points are actually seen as spots, and the image they form appears blurred.

Referenced to the sensor of the camera (with regard to the magnification), this maximum spot size determines the diameter of the circle of confusion C. The animation below shows how this circle of confusion sets the depth of focus dF at the center of the image for a given f-number N.

Since the angular aperture of the light cone depends only on the f-number, the same goes for the depth of focus; one can easily demonstrate that the depth of focus dF = 2 . C . N (so the depth of focus does not depend on the lens focal length).

The points located at the limits of the depth of focus, on the image side, are conjugated to the points that materialize the “limits” of the depth of field, on the object side. Therefore, the image of any object located within these “limits” appears sharp on the image plane. Since the size of the spot on the sensor varies continuously as the object-point moves away from the focus plane, the sharp to blur transition on the image is progressive.

Fig. 01: Depth of focus - Depth of field.

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Nikkor 135mm f/2 Ais and 135mm f/2.8 Ais.

According to US Patent 4,057,330 (1977), example # 2, and US Patent 4,062,630 (1977), example # 5. Inventor (in both cases): Mr. Sei Matui.
The patent data are given for f' = 100 units; here, they are transposed to obtain f’ = 135 mm.

Optically, both of these objectives are not real telephoto lenses (positive-negative); their optical system is related to the Ernostar-type lenses.

Fig. 02: Nikkor 135mm f/2.8 Ais.
Mouse out: set to infinity focus.
Mouse over: set to minimum focus (1.3 m).

Fig. 03: Nikkor 135mm f/2 Ais.
Mouse out: set to infinity focus.
Mouse over: set to minimum focus (1.3 m).

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Reflex-Nikkor 500mm f/8 Ai.

This type of lens has an annular-shaped entrance pupil. When the lens is set to infinity focus, the half angular aperture of the outer side of the emerging cone is 4.28° which corresponds to a relative aperture of N ≈ 6.7 (f/6.7). This is the relative aperture value to be used for depth of field calculations (geometrical aperture).

Fig. 04: Reflex-Nikkor 500mm f/8 Ai.
Mouse out: set to infinity focus.
Mouse over: set to minimum focus (1.5 m)
.

The nominal f-number (Nnom = 8) is actually a corrected and rounded value taking into account the central blind part of the emerging cone.

The half angular aperture of the blind part of the emerging cone is 2.47° which corresponds to a relative aperture of N’ = 11.6 (f/11.6). Given the weakness of the aperture angles of both cones (outer and blind), the corrected f-number of the lens can be calculated from the diameter of a circular surface of equal area to that of the annular surface of the entrance pupil, as follows…

Outer diameter of the entrance pupil: Øout = f' / N = 499 / 6.7 ≈ 74.5 mm.
Corresponding area: Sout ≈ 4356.5 mm2.

Diameter of the blind part of the entrance pupil: Øin = f' / N' = 499 / 11.6 ≈ 43.0 mm.
Corresponding area: Sin ≈ 1453.4 mm2.

Area of equivalent circular surface: Seq = Sout – Sin ≈ 2903.1 mm2.
Corresponding diameter: Øeq = √ (4 * Seq / π) ≈ 60.8 mm.

Corrected relative aperture: Ncor = f' / Øeq = 499 / 60.8 ≈ 8.2.

Nota bene:

Fig. 05: Reflex-Nikkor 500mm f/8 Ai.
Mouse out: entrance pupil of the axial light beam.
Mouse over: entrance pupil of the most oblique light beam.

Pictures taken with this kind of lens show the annular shape of the entrance pupil in the blur annular-shaped spots of luminous out-of-focus objects. At the periphery of the image, these rings appear rather as closed crescents. For objects located on this side of the focus plane, the thick part of the crescents is facing the center of the image, while it’s facing in an outward direction when the objects are located beyond the focus plane.

Reference: US Patent 4,666,259 (1987), Mr. Yutaka Iizuka - Example # 1.

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What is the maximum aperture consistent with the Nikon F-mount?

The Nikon F-mount was introduced in 1959 with the release of the 35 mm SLR Nikon F.

The dimensions of the bayonet and the minimum back focus distance compatible with the operation of the reflex mirror determine the maximum angular aperture of the emerging light cone of the lenses of a SLR System. In the Nikon SLR System, this angle is close to 54°. Thus, regardless of electrical and/or mechanical connection accessories that may reduce even more the light beam cross-cutting, the minimum theoretical relative aperture is about N = 1.1 (f/1.1).

Fig. 06: Nikon F bayonet.
Mouse out: bayonet free of connection accessories.
Mouse over: bayonet with electrical connections.

However, considering the size of the rear optical elements of a 50 mm f/1.2 and the space required for their mechanical supports, it is very likely that the practical limit is closer than N ≈ 1.2, i.e. an emerging cone with an angular aperture of 49.2° (2x24.6°).

Fig. 06bis: Angular aperture the emerging light cone of a 50mm f/1.2.

The Nikkor-N 5cm f/1.1 was introduced in 1956 for S-mount rangefinder cameras. It was “the second lens in history faster than f/1.2”. The short back focus distance allowed by this type of camera (no reflex mirror) makes it much easier to design very fast lenses with small mount diameter.

Fig. 06ter: optical system of the Nikkor-N 5cm f/1.1.

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Micro-Nikkor 55mm f/2.8 Ais and AF Micro-Nikkor 55mm f/2.8.

According to US Patent 4,260,223 (1981), Mr. Yoshinari Hamanishi - Example # 1.
The patent data are given for f' = 100 units; here, they are transposed to obtain f’ = 55 mm.

Both of these lenses share the same optical system (related to the double-Gauss type). Although originally designed to support reproduction ratios up to 1:1, this optical system has been mechanically restrained at the ratio of 1:2 until the advent of the autofocus in 1986.

The Micro-Nikkor 55mm f/2.8 Ais was released in December 1979 (the United States patent application was filed only 4 months earlier).

A CRC system (Close-Range Correction) maintains good performance over the full range of focus distance and particularly at high reproduction ratio. On double-Gauss type objectives, close-range correction is achieved by varying the space between the front and the rear sub-units as the lateral magnification increases (7.3 mm on infinity focus to 12 mm on minimum focus). Actually, the front sub-unit moves forward faster than the rear one as the focus plane comes closer.

Fig. 07: Micro-Nikkor 55mm f/2.8 Ais.
Mouse out: set to infinity focus.
Mouse over: set to minimum focus (0.25 m).

The AF Micro-Nikkor 55mm f/2.8 was released in September 1986.

This lens allows using the full magnification range for which its optical system was originally designed. The maximum reproduction ratio (1:1) is reached at a focus distance slightly less than 23 cm (about 9 inches). As the CRC system works on a longer range, the space between the front and the rear sub-units increases to up to 16.4 mm when the lens is set to minimum focus.

Fig. 08: AF Micro-Nikkor 55mm f/2.8.
Mouse out: set to infinity focus.
Mouse over: set to minimum focus (0.229 m).

A 27.5 mm extension ring (Nikon PK-13) is needed to reach the reproduction ratio of 1:1 with the manual focus version. But in this case, as the CRC system cannot exceed the configuration required for the ratio of 1:2 (maximum ratio of the lens alone), the performance of the lens at the extreme edges of the image is not optimal (astigmatism and field curvature). The AF model, with its CRC system operating continuously up to the reproduction ratio of 1:1 should perform better.

Fig. 09: Comparison between both lenses set to lateral magnification g = -1.
Mouse out: Micro-Nikkor 55mm f/2.8 Ais + extension ring PK-13 (27.5 mm).
Mouse over: AF Micro-Nikkor 55mm f/2.8.

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Effect of the CRC system of both Micro-Nikkor 55mm f/2.8 (Ais and AF) on their focal length.

Like all double-Gauss type lenses, these Micro-Nikkors consist of two positive sub-units placed on both sides of the aperture stop. Varying the distance between the two sub-units as the focus distance shortens reduces the aberrations over the full range of reproduction ratio. However, as one characteristic of the optical system changes, the focal length of the whole system changes too. Let’s see how much?

A double-Gauss type lens can be likened to a combination of two positive sub-units forming a new system which focal length f’ is given by the following equation:

f’ = f1' . f2' / -Delta

where f1' and f2' are the respective focal lengths of each sub-unit, and Delta is the “optical gap” (distance between the second focal point F1’of sub-unit # 1 and the first focal point F2 of sub-unit # 2). Use algebraic values considering the direction of light propagation, i.e. positive from left to right, and negative in the reverse direction.

Figure 10 below show that any increase in the space between the two sub-units induces a decrease of same value of the optical gap Delta. Considering the equation above, when Delta gets shorter the focal length of the whole system f’ increases, and vice and versa.

When the Micro-Nikkor 55mm f/2.8 Ais is set to minimum focus (reproduction ratio 1:2), the space between the two sub-units is about 12 mm, so the focal length of the lens is about 57 mm (instead of 55 mm on infinity focus).

Optically, the only thing that changes between both “mouse out” and “mouse over” images below is the space between both of the sub-units.

Fig. 10: Micro-Nikkor 55mm f/2.8 Ais.
Effect of the increase in the space between both sub-units on the focal length of the whole system.
Mouse out: lens set to infinity focus.
Mouse over: lens set to minimum focus (0.25 m).

When the AF Micro-Nikkor 55mm f/2.8 is set on minimum focus (reproduction ratio 1:1), the space between the two sub-units is about 16 mm, so the focal length of the lens is about 59 mm (instead of 55 mm on infinity focus).

Fig. 11: AF Micro-Nikkor 55mm f/2.
Effect of the increase in the space between both sub-units on the focal length of the whole system.
Mouse out: lens set to infinity focus.
Mouse over: lens set to minimum focus (0.229 m).

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AF Micro-Nikkor 200mm f/4D IF-ED.

According to US Patent 5,402,268 (1995), Mr. Wataru Tatsuno - Example # 1.
Calculations are made on the basis of a variation of the space d14 as a function of g in accordance with a polynomial of degree 2 (a0 = 31.9737, a1 = 18.4347, a2 = -13.5390).

By its own principle, this lens requires no close-range correction (CRC) when set to minimum focus (496 mm). The non-linear movement (back and forth) of the three front elements compensates mainly the over-corrected spherical aberration at intermediate focus distances. The position of these elements on both extreme focus distances (infinity and minimum) is exactly the same. The correction peaks at the reproduction ratio of about 1:1.8.

Fig. 12: AF Micro-Nikkor 200mm f/4D IF-ED.

Distinctive feature of this lens: the focal length of the whole optical system varies almost linearly with regards to the lateral magnification (see graph on Figure 13, below).
 
Another distinctive feature of this lens: when focused on close objects, the aperture stop actually closes down (about one f-stop when set to minimum focus). Why?

When the focus plane gets closer, the front elements of the lens tend to limit the diameter of the incoming light beam. Thus, when set to the minimum focus distance (496 mm / 19.5 inches), the front elements diameter determines the minimum value N = 5.5 of the relative aperture of the emerging light beam. For the aperture stop retains its functions, a mechanism gradually tightens its opening as the focus distance decreases, keeping the iris in contact with the light beam. Without this mechanism, it is the front elements that would determine the pupils of the lens, and the aperture stop would lose its ability to accurately adjust the aperture.

In the Figure 13 (below), the aperture stop is shown fixed at N = 4.0 so as to highlight the phenomenon of detachment of the light beam. The Micro-Nikkor AF-S VR 105mm f/2.8G shows the same feature, but less pronounced.

Fig. 13: AF Micro-Nikkor 200mm f/4D IF-ED - Lens operation.

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AF-S DX Micro-Nikkor 85mm f/3.5G ED VR.

According to the US patent 2009/0190220 A1 (2009), Mr. Haruo Sato - Example # 5.
Calculations are made on the basis of a variation of the space d7 as a function of g in accordance with a polynomial of degree 2 (a0 = 0, a1 = -20.87309, a2 = -5.11258).

The operating principle of this new type of macro lens is very different from the previous models of double-Gauss type + built-in rear converter (AF 60mm f/2.8D, 105mm f/2.8D AF), or afocal / pseudo-afocal telephoto lenses (AF 200mm f/4 IF-ED, see above).

Fig. 14: AF-S DX Micro-Nikkor 85mm f/3.5G ED VR.

This objective consists of five distinct sub-units G1 to G5 (see figure 15 below). Focusing is achieved by the synchronized movement in opposite directions of the two sub-units G2 and G3 located on both sides of the aperture stop. Sub-unit G4 is assigned to optical stabilization (VR). The reproduction ratio of 1:1 is reached at the minimum focus distance of 236 mm (9.3 inches); the subject is then at 145 mm (5.7 inches) from the front element.

When the focus plane gets closer, the rearward movement of the first mobile sub-unit G2 strongly modifies the optical power of the half part of the lens located in front of the aperture stop (from -3.4 diopters on infinity focus, to +7.4 diopters on minimum focus). Thus, the angular aperture of the light cone passing through the aperture stop varies only from -5.3° (infinity focus) to -11.2° (minimum focus), when the aperture stop is fully open (N = 3.6). To maintain constant back focus despite this angular aperture variation, the optical power of the rear part of the lens (located behind the aperture stop) must increase slightly as the subject gets closer. This is the role of the second mobile sub-unit G3 whose synchronized forward movement changes the focal length of the rear part of the lens from 57.2 mm (infinity focus) to 49.4 mm (minimum focus).

Fig. 15: AF-S DX Micro-Nikkor 85mm f/3.5G ED VR - Lens operation.

During focusing, the focal length of the whole optical system varies with regards to the lateral magnification in a more complex manner than it used to do with the previous models whose focal length decreases more or less steadily as the focus distance decreases. In this particular case, the focal length of the lens is equal to or greater than the nominal value over most of its focusing range: from 0.33 m (13 inches,
g = -0.6) to infinity.

The Micro Nikkor AF-S 60mm f/2.8G ED and the Micro Nikkor AF-S VR 105mm f/2.8G ED operate under the same principle. This focusing system including two mobile sub-units may also be used on longer focal length lenses. Thus, US Patent 2011/0109979 (May 2011) presents several optical systems of the same type whose focal length is between 160 mm and 220 mm; all of them can reach the reproduction ratio of 1:1. For instance, the following figure shows a 200 mm f/2.5 providing the reproduction ratio of 1:1 at the focus distance of 0.486 m.

Fig. 15bis : Micro Nikkor 200mm f/2.5 ED (project).
Mouse Out : set to infinity focus.
Mouse over: set to minimum focus (0.486 m).

According to US Patent 2011/0109979 (May 2011), Mr. Mitsuaki Wada - Example # 1.
Calculations are made on the basis of a variation of the space d11 as a function of g in accordance with a polynomial of degree 2 (a0 = 0,
a1 = -28.43686, a2 = 2.33644). Back focus values (BF) given in the patent (example #1) are incorrect. Actually, BF = 50,833 mm.

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Working of a single-lens-reflex camera.

The animation below is essentially based upon the kinematics of the Nikon F5.
Interval of time between two frames: about 1333 µs.
Actual full cycle: 108 ms for a 4 ms exposure time (1/250 s). Blinds and mirrors acceleration and stabilization phases not shown.
The partial resetting of the front blind at the end of the cycle allows covering the boundary areas between the blades of the rear blind.
Lens: 50 mm f/1.8 lens (US patent n° 4,514,051), aperture set to f/16 for the shooting.
Viewfinder: adjustable power viewfinder set to refocus the image formed on the focusing-screen at a point of about -1 diopter (adjustable from -3 to +1 diopter).
Focal length of the entire viewfinder f'v = 70,42 mm; focal length of the eyepiece f’e = 78,03 mm (US patent n° 4,664,485).

Fig. 16: Working of a single-lens-reflex camera (study of the reflections on the mirrors).

A video made at 4300 frames/s on a Nikon F5 can be reached clicking on the image below.
Actual sequence time: 104 ms.
Exposure time of the Nikon F5: 1/2000 s.

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Anatomy of a modern lens.

With their very fast autofocus system, their optical stabilization unit, and their very sophisticated optical system, most of the modern camera lenses are very complex instruments. The illustration below shows such a lens. Here, focusing is achieved by simultaneously moving two lens-groups (on the opposite direction), each group being located on either side of the aperture unit.

When focusing is carried out manually, both orange-colored and red-colored parts rotate around the optical axis, while only the red-colored parts rotate when the motor achieves focusing (autofocus).

Fig. 17: Simplified section of a modern lens (Micro-Nikkor AF-S VR 105mm f/2.8G).

1 - Manual focus detection commutator;
2 - Stator of the ring-shaped piezoelectric motor;
3 - Rotor of the ring-shaped piezoelectric motor;
4 - 1rst mobile lens-group guidance groove;
5 - Coupling key between rotor and magnetic tape ring;
6 - Fixed tube;
7 - Driving pin of the cam ring;
8 - Focus distance reading window;
9 - Focus distance index scale;
10 - Focus distance encoding commutator;
11 - Iris operating lever (truncated);
12 - 2nd mobile lens-group guidance groove;
13 - Gyro meter (pitch);
14 - Mobile part of the stabilization unit;
15 - Fixed part of the stabilization unit;
16 - Connecting plug (with the camera body);
17 - Power supply commutator of the piezoelectric motor;
18 - 1rst mobile lens-group;
19 - Manual focus ring;
20 - Aperture unit;
21 - Magnetic tape ring (cam ring rotation control);
22 - Aperture unit fixing screw;
23 - Magnetic tape reading unit;
24 - Cam ring;
25 - Cam coupling the aperture and the position of the 2nd mobile lens-group;
26 - Lock pin of the mobile part of the stabilization unit;
27 - 2nd mobile lens-group;
28 - Suspension spring of the mobile part of the stabilization unit;
29 - Ball (rolling of the mobile part on the fixed part of the stabilization unit);
30 - Stabilization unit electronic board;
31 - Electronic board (gyro meters, magnetic tape, focus, stabilization, motor power supply, etc.).

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Synchronized motions

Although each of the two groups of optical elements ensuring focusing is located on either side of the aperture unit (fixed), one single cam ring allows the synchronization of the two opposite movements. The aperture unit is secured to the fixed tube by two screws passing through grooves machined into the cam ring (only one is shown to simplify the drawing).

The animation below shows how the grooves machined into the cam ring cause the displacement of the two lens-groups when the motor drives the two drive pins (only one is shown to simplify the drawing). Whatever the focus distance, the shape of the cam ring grooves determines the relative position of each of the two lens-groups. Therefore their machining must be performed with high accuracy.

Fig. 18: Synchronization of the movements of the two mobile lens-groups on either side of the aperture unit.

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The piezoelectric motor with progressive wave

In most modern interchangeable lenses, moving elements ensuring the focus is usually entrusted to a piezoelectric motor with progressive wave. The lens presented above (Micro-Nikkor AF-S VR 105mm f/2.8G) uses an annular (ring-shaped) piezoelectric motor. This type of motor was developed in 1982 by T. Sashida and was marketed for the first time in June 1986 by the Japanese society Shinsei Co. (associated with the manufacturer Fukoku Co.).

The piezoelectric motor with progressive wave is particularly suitable for photographic lenses motorization. Its main advantages are:

To highlight the performance of their lenses equipped with such motors, each manufacturer uses its own naming: Canon uses “USM” for Ultra Sonic Motor, Nikon uses “SWM” for Silent Wave Motor, etc.

Operating principle

A piezoelectric motor operates a two-stage energy conversion process: at first, electrical energy is converted into oscillating bending motion by the piezoelectric elements of the stator, then, these ultrasonic vibrations (above 20 kHz) are converted into unidirectional rotating movement of the rotor by friction on the stator. In this case, both rotor and stator are ring-shaped (see Figure 19, below).

The rotor is pressed against the stator by an item not shown in the illustration (preload force). A thin layer of friction (slider), glued to the underside of the rotor ensures a good friction coefficient between rotor and stator. The preload force and the friction between the rotor and the stator determine the passive holding torque of the motor (without power).

Fig. 19: Annular piezoelectric motor with progressive wave.

The stator consists of an elastic stator ring and piezoelectric ceramics (synthetic) glued to its underside. The piezoelectric ceramic has the particularity to deform when an electrical field is applied (reverse piezoelectric effect). Accordingly, when subject to an alternating voltage of high frequency, the piezoelectric ceramic vibrates. If the excitation frequency is equal to the natural frequency of the stator (depending on its shape and type of material used), it resonates. The amplitude of the mechanical oscillations can reach 2 to 3 µm.

The piezoelectric ceramic elements consist ​​of two units, respectively fed by a distinct sinusoidal voltage generating two standing waves. A spatial phase shift (a quarter wavelength) and a temporal phase shift (Ø ± π / 2) generate the traveling wave. Unlike a standing wave, the wave travels around the circumference of the stator ring in the manner of a sea wave (see Figure 20, below). The sign of the temporal phase shift determines the propagating direction of the traveling wave and, therefore, the rotating direction of the rotor. Teeth machined in the upper part of the stator ring amplify its driving effect, while slots collect material particles detached by the friction of the rotor on the stator.

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Fig 20: Standing wave, progressive wave (travelling wave).

The animation associated with Figure 21, below, shows how the progressive wave traveling along the stator in a given direction leads the rotor to spin in the opposite direction. As the wave passes by, each point on the surface of the stator moves alternately in the same direction as the wave (in the low point of wave), then in the opposite direction (on the wave peak), in a retrograde elliptical motion*. On contact with the peaks, the rotor is then driven by friction in the opposite direction of the propagating wave.

* In the illustration, the trajectory of a surface point (on the left) is plotted in red. Taking into account the exaggerated proportions between wavelength, oscillations amplitude and rotor thickness, the trajectory of the point does not appear elliptical. In this example, the ratio of the oscillation amplitude on the thickness of the stator is quite huge (≈ 1/4). In reality, this ratio is much smaller (≈ 1/1000).

Fig. 21: Rotor drive by friction on the wave peaks of the stator.

The rotational speed of the rotor is proportional to the oscillations amplitude, the stator thickness and the propagating speed of the wave around the stator.

However, these motors show some drawbacks:

Ref.:
Piezoelectric actuators and ultrasonic motors – Kenji Uchino (1997)
Modélisation et commande du moteur piézoélectrique à onde progressive – Matteo Bullo (2005).

 

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Other subjects :

Fisheyes
Focal length and Magnification
Telephoto zooms