Author: Leda Masi
Focal length, field angle and digital multiplication factor.
Focal length, with its implications, is one of the main themes of digital photography, a source of endless discussions.
What follows is a compendium of the information on focal length (and its consequences) that can be found extensively, but sparsely, online and in books. I've tried to bring it together into a coherent discussion, so that it can be a useful tool in this somewhat confusing field of digital photography.
Let's start with a definition and a premise.
Definition: the focal length is the distance between the optical center of the lens and the plane of the sensitive element (film or sensor) * see note 1. This value is an invariable characteristic of each lens, regardless of the camera on which it is mounted. I will return to this topic later.
Note 1. There are special measures to ensure that the optical center of the lens is not located in the physical center, but shifted along the axis, sometimes to the point of being external to the lens itself. This is used, for example, to create reflex lenses with very short focal lengths that can be mounted on the standard mount without extending into the camera, where they would block the movement of the mirror.
Introduction: The field angle of a lens depends both on its focal length and on the size of the sensitive element.
Attention! There are two variables and consequently there are no absolute "long" or "short" focal lengths, but only in relation to the size of the sensitive element.
As a lens's focal length increases, its field of view, that is, the width of the scene captured, decreases. Thus, for the same media format, lenses with short focal lengths have greater coverage than lenses with longer focal lengths.
This naturally affects the final photo, and different lenses are used to achieve different effects and angles. Typically, a "short" focal length lens is used to photograph close-up objects or to capture a large portion of space, while "longer" lenses are used to bring distant subjects closer or to isolate a subject from its context.
The above applies regardless of the storage medium, or sensor, if you will. The problem is that film SLR cameras have accustomed us to considering a sensor size of 24 x 36 mm as "standard," while digital cameras use sensors of extremely variable dimensions, from 4–5 mm up to 24 x 36 (so-called full-frame) and even larger (medium or large format backs).
What happens when the media size changes?
It happens that lenses change their field angle: at the same distance and focal length, as the size of the sensor decreases the field angle decreases (in other words it is as if the focal length increases), while supports larger than 24*36 cause a widening of the field angle (as if the focal length becomes shorter).
Let's see why this happens.
A lens produces a circular image that is projected onto a medium, usually rectangular. The rectangular portion within this circle is then imprinted on the medium. Standard commercial lenses produce image circles with a diameter equal to the diagonal of a 24mm x 36mm frame, approximately 43,3mm. Therefore, on a 35mm camera, the image captured on the film corresponds to the central portion of the circle, and the diagonal of the image corresponds to the diameter.
If the same lens projected the image onto a larger sensor (for example onto a medium format film), it would only be able to cover the central part (remember that the distance between the lens and the sensitive plane is fixed and depends only on the focal length).
If, however, the circular image is projected onto a smaller sensor, the image diagonal no longer corresponds to the diameter of the projected image, but only to its central part, corresponding to the diagonal of the sensor.
This phenomenon translates into a "closer" and "enlargement" of the photographed image; it is as if the lens had changed focal length, lengthening, and consequently reduced its field of view, or as if we had used a focal multiplier.
As you can see from the drawing, on a sensor smaller than the film (in yellow) the image captured corresponds to a part of the one captured with a 24*36, and the effect is that of a medium telephoto.
The lens actually frames the same portion of space, that is, it maintains its optical properties; what varies is the portion of space actually imprinted on the medium.
The difference in coverage angle is translated into the so-called Digital Multiplication Factor.
This is a constant obtained by dividing the diagonal of the 24*36 frame by the diagonal of the sensor in use [43,3/d]. It is indicated by (N)x and is the number by which the focal length used must be multiplied to find out what focal length it would be equivalent to on a 35mm format, or to find out what focal length to use to obtain the same coverage as a given lens on a 35mm.
Just to give a little concreteness to the discussion, let's look at the dimensions of the main sensors on the market and their multiplication factors:
Two observations stand out.
The first is that all compact cameras have “significant” multiplication factors, starting from 4 and reaching over 7. In all respects, these cameras are a category in themselves and for each of them, manufacturers usually also take care to specify the “equivalent focal length” (we will see this in the next table).
The second is that only one SLR (the Canon 1DS MKII) has a full-frame sensor: all other Canon SLRs and those of all other manufacturers (currently Pentax – Samsung, Sony – Minolta, Nikon, Fuji, Sigma) have smaller sensors, generically defined as APS size. These dimensions are not standardized (sometimes not even published), but are placed around a multiplication factor between 1,3 and 1,6.
SLR cameras with the 4/3 system (currently Olympus and Panasonic) are in a category of their own, as their sensor has a multiplication factor of approximately 2 (this makes calculations easier).
As you can easily imagine, the smaller the sensor, the greater the “stretching” effect on the focal length.
Let's look at another table: it lists the main focal lengths in relation to the different sensor formats. Each real focal length corresponds to different "apparent" focal lengths due to the multiplication factor of the different sensors.
I have highlighted the “normal” lens, the 50 mm, in green to highlight how it changes radically depending on the size of the sensor: as you can easily see, from a “normal” lens on a 35 mm camera it becomes a telephoto lens already in the 4/3 format and transforms into a very powerful telephoto lens if the size of the sensor is further reduced. In yellow, I have instead highlighted for the different formats what the focal length should be to obtain a "normal" lens; as can be seen, to obtain the coverage of a 50 mm with a 1/1.8" sensor (the one used for the illustration above) I should use a 10 mm. (which I actually did, but we'll talk about that later). It is evident that the deviation in focal length required decreases as the size of the sensor gets closer to that of the frame: so on an APS-C (1.8"?) I have to use a 35 mm to obtain (approximately) the coverage of a 50 mm, while I will have to use a 10 mm if the sensor size drops to 1/1,8".
The effects described apply to all lenses: in the case of a reflex camera, I'll already know which lens I'm using, so it's easy to find out what angle of coverage (or "equivalent focal length") it actually corresponds to depending on the sensor. Using a digital compact camera, usually equipped with a zoom lens, if you need to know the "equivalent focal length" used, just look at the photo's EXIF data and then multiply the focal length value used by the multiplication factor to find out what focal length it would correspond to on a 35 mm camera.
Let's now look at a table that relates the focal lengths, for a 35 mm format, to the coverage angles of the different lenses.
This data is interesting because it takes into account both the focal length and the sensor size. Consequently, the coverage angle can finally indicate an "absolute value" with respect to the scene being captured. A lens with a coverage angle of 46° will always be a "normal" lens, regardless of the focal length and sensor size. These must, of course, have a specific relationship to each other, but we've already discussed this and will be discussed in detail below.
I reiterate that it is the angle of coverage – and only that – that determines whether a lens will be normal, telephoto or wide angle.
The classifications in the right column are indicative and serve for clarity. I don't know if there's an official table that prescribes a unified terminology.
As mentioned, varying the size of the image storage medium changes the lens's coverage. For those who want to go into detail, the following chart shows the field angles of different lenses for the most common sensors.
Note: The formula for calculating the field angle actually covered by a certain focal length on a given sensor is: α=2*arctan(d/2f) where d is the sensor diagonal and f is the actual focal length. Just for those who still remember trigonometry and want to calculate other focal lengths that aren't listed here.
I am indebted for this table to Emanuele A. “Juza��? ( www.juzaphoto.com ) who kindly allowed me to use the table he had painstakingly calculated, thus saving me from having to redo all the calculations. The table and the related article are published on www.photorevolt.com excellent photography site to which Emanuele collaborates assiduously.
Here too I have highlighted the variation in coverage of a 50mm in relation to the different sensor formats and the coverage of the different lenses on the 24*36 format.
This focal length multiplication effect brings benefits and problems, as is easy to imagine.
A first big problem concerns wide-angle lenses: to have sufficiently wide field angles, such as those we would have for example with a 28 mm, on a smaller sensor (let's say with a multiplication factor of 1,6x, we would have to use a lens with a focal length of 17 mm (i.e. a very wide angle).
On the other hand, telephoto lenses increase their magnification, as if they were longer, or as if using a teleconverter. However, and this is the beauty of it, they maintain the same brightness as the actual lens.
We know that the brightness of a lens is a function of its diameter and its length (focal length divided by diameter); the multiplication factor changes the angle of view (as if the focal length increased), but the diameter and the "physical" length of the lens do not vary, so their ratio does not change; consequently, the brightness does not change. A 200mm lens becomes a 300mm on a sensor with a 1,5x factor. A 200mm f/2,8 lens has one weight and cost, a 300mm f/2,8 lens has a much higher weight and cost! Using my 200mm f/2,8 on such a sensor gives me the magnification effect without losing brightness and with a significant saving in terms of money and weight. Not bad, huh?
The same goes for teleconverters: these are additional tubes that "extend" the lens, achieving the same magnification effect, but the extension is physical, real, and therefore also affects the lens' brightness. Using the same lens on a digital SLR, I get the same result as using a teleconverter, but without the associated loss of brightness.
In reflex lenses “designed for digital” – that is, to cover sensors smaller than 24*36 – it is possible to use smaller lenses, with the same brightness, or have greater brightness without increasing size, weight and cost.
In general, the push toward smaller sensors, even in SLRs, stems from the possibility of building lighter and/or brighter lenses at the same cost. Resolution remains a major challenge for digital sensors, but today's lenses are capable of handling this issue adequately.
Another important difference concerns the depth of field.
We already know that depth of field is a function of, among other things, focal length.
As the focal length increases, the depth of field decreases, down to just a few mm for the longest telephoto lenses. A lens designed for 35 mm and used on a digital camera with a smaller sensor will therefore provide the same angle of view as a lens with a longer focal length, but the same depth of field as the original, since the length of the lens is a physical, objective, and immutable data.
To give an example, with the 200mm mentioned above I have the coverage and magnification effect of the 300, but with the depth of field offered by the 200.
All this is true if we are talking about digital SLRs coupled with 35mm lenses, which therefore generate an image circle of adequate size for the frame.
The lenses mounted on compact cameras, on the other hand, generally have very, very short focal lengths to provide acceptable field angles and are designed to create smaller image circles, appropriate for the small size of the sensors. They are therefore much smaller and lighter, and because of their shorter focal length, they provide a greater depth of field than would be possible with equivalent focal lengths.
This effect of increasing depth of field is partially offset by the reduction in the circle of maximum confusion: on a smaller format, this circle also becomes smaller, resulting in a decrease in depth of field (and an increase in noise, but we'll talk about that another time). This small decrease, however, is significantly less than the increase due to the short focal length, so the overall balance is positive. In conclusion, it can be stated (roughly but confidently) that depth of field is primarily dominated by the focal length-aperture combination.
This is why it's best to use quotation marks to refer to "equivalent focal length." The only "true" equivalence is in the angle of coverage; the depth of field is different, and for this reason, selective focus enthusiasts won't find satisfaction in compact cameras.
For the record, this is also one of the reasons why compact cameras don't have apertures that can "close" much: they don't serve to increase an already very large depth of field, while they easily cause diffraction phenomena with the very small cells of the sensor.
What has been said so far applies to photography in general.
But we are divers and unfortunately we also have to deal with optical phenomena peculiar to the underwater environment.
One phenomenon in particular interests us here: refraction. As Snell's law teaches us, a light ray, passing through two media of different densities, is partially refracted: it will move away from the interface normal if it passes from a denser medium to a less dense one, and it will move closer to it if it passes from a less dense medium to a denser one.
Air is a less dense medium than water, and the refractive index of water is about 1.33.
We are all familiar with the phenomenon of refraction, which causes the objects we observe to appear closer and larger when we dive underwater with a mask: everything appears larger and closer.
Refraction also causes our lenses to change “length” and to modify the field angle, specifically by reducing it by approximately 25%.
So my “normal” lens, the 50mm, when fitted with a housing and taken underwater becomes a 70mm, or rather, it has the field angle of a 70mm.
This is a physical law, which always acts when a ray of light passes through media of different densities and, with very little sensitivity towards us, it does not care about human technologies and all our problems as underwater photographers.
In the case of using a digital SLR with a housing, this naturally presents us with a further problem, to be added to the digital multiplication factor.
In fact, we will have to take into account, in addition to the reduction in field angle due to the size of the sensor, also the reduction in coverage due to refraction.
Let's look at a couple of tables. The first shows the angles covered by different lenses, on a 35mm sensor, in air and water:
The second table shows the coverage angles of the different focal lengths, on 35 mm, on a 1,8��? sensor and on the same sensor but in water.
(I have reported the calculations on a sensor with a fairly common crop factor; on the same basis everyone can do the calculations on the sensor in their possession).
As you can easily see, to have the coverage of a 50 mm on this sensor and underwater I would have to mount a 24 mm, a lens that can rightly be defined as a wide angle and certainly not a “normal” one?!
Using our old film SLR cameras we have become accustomed to managing this reduction "by eye", we have so to speak internalized it and it doesn't bother us that much.
We also know that to have good coverage with wide angles, and not lose that 25% field angle, we can use a spherical port, which restores the focal length to its original coverage, as well as reducing some geometric aberrations due to the use of wide angles and refraction.
If we have switched to a digital SLR while keeping our old equipment of lenses and ports, refraction will not scare us, we will manage it as always by taking into account exclusively the digital crop factor.
The situation is different if we're talking about long focal length, medium telephoto, or telephoto lenses. In this case, the reduction in field angle due to refraction must be added to that due to digital cropping, as we're unlikely to use spherical ports for these lenses (frankly, I don't even know if they're for sale). So, as you can see from the table, a nice 105 mm, which when used on a 1,8" sensor already reduces its field angle to roughly the same as a 135/140 mm, becomes equivalent to a 200 mm when taken underwater!
Someone pointed out to me that all this discussion could be defined, in the words of my logic professor, as "a mental masturbation." Perhaps that's true, if we think in terms of the equipment we own.
Perhaps less so if we are considering purchasing new equipment.
So, to help us make our choice, it will certainly be useful to consider the sensor size of the chosen camera not only in terms of effective megapixels and technology, but also in terms of focal length, and also to take into account, when choosing the lenses to take underwater and related accessories, the "unpleasant" Snell's law.
In short, as a friend told me: “if water produces the ignoble effect you say, then you have to calculate it.”
At this point, I would like to thank Frank Tagliaferro, a photographer and friend, for the enormous and precise work of reviewing and checking the data, as well as for the invaluable advice and information and for the delightful conversations, from which I always have so much to learn.
Bibliography:
– “Digital photography – how is it made and how does it work?” Ron White, Ed. Mondadori Informatica
– “Focal length, field angle and perspective?” Emanuele A. “Juza” on www.photorevolt.com
www.dpreview.com
- Wikipedia
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