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Photo Writing is the web version of the Photo Writing mini-magazine produced by Limephoto and Emil von Maltitz since 2010. As of 2015 it is now completely online. Feel free to browse through the articles and please leave comments in the comments section if you would like to engage with us.

Wednesday, September 11, 2019

Confusing Circles and Airy Disks: A relatively simple explanation of depth of field and hyperfocal focusing

After much searching of the internet and reading of photographic textbooks (the dense kind that have little if any pictures) I discovered I’m absolutely clueless as to how to actually calculate hyperfocal distance and little better at understanding it. My goal then became to reverse this if at all possible. This tutorial then is my attempt to wade through what I saw as rather complex math and distil it into something that makes sense to other mathematical luddites like me. I’ll start the article with the simple stuff, and move on to the more complicated matters after. 


Hyperfocal focusing

To put it simply (remember I’m trying to steer clear of the technical stuff as much as possible), hyperfocal focusing is when you focus your lens such that there is a depth of field that encompasses the very near distance through to infinity. This is what creates that three dimensional look to landscape photographs that seem so bitingly sharp and crisp that we could almost step into the picture.

Most prime lenses (i.e. lenses that have only one focal point and no zoom) have a hyperfocal scale engraved or painted on the barrel of the lens. The theory is that to obtain the hyperfocal distance (defined by the Focal Encyclopedia of Photography as the “focus setting that makes the far limit of sharp focus equal to infinity) the photographer focuses on the nearest point that they require to be in focus and makes a mental note of the distance on the distance scale on the lens for this point. By using the hyperfocal scale on the lens barrel, which indicates the distance between focus points in relation to different apertures (see image) he or she is able to adjust the focus (without looking through the lens) such that the hyperfocal markers are on the infinity mark of the focus ring and either encompass or fall past the nearest required point of focus. 
Let’s say I want to photograph a scene with aloe flowers in the foreground while maintaining sharp focus through to clouds that are gathering on the far horizon. Obviously the storm clouds would be in focus at the infinity setting of the lens (the focus ring stops turning in that direction…or at any rate stops focusing – I realise that newer Canon and Nikon lenses just keep on turning). The flowers however only 1.7 metres away from where I am photographing from. Simply stopping down to f16 or even f22 still doesn’t place the elements such that both the flowers and the clouds are in focus if I focus either on the clouds or on the flowers independently. The answer is in using the hyperfocal scale. By choosing an aperture of f16 and setting the focus so that f16 hyperfocal point is in line with the infinity mark, I am able to also get the nearest distance of 1.7m into focus as well (see images). In fact I even get a little closer than 1.7m.

So what’s the problem? How many lenses, particularly zooms have hyperfocal focusing scales on them anymore. The problem is that only the most expensive lenses now cater to photographers who want to shoot using hyperfocal focusing. An easy way around this is to focus roughly a third of the way into the area that you require to be in focus. This is because, except for close-up subjects, the depth of field is assymetrical about the focused distance. Essentially the area behind the focused point is twice as deep as that in front of the focused distance. Therefore by focusing a third of the way into the area that you require to be in focus, there is a good chance that you will obtain hyperfocal focusing. 
Let’s say you want to be more exact in your calculations when you are trying to focus hyperfocally. After much delving through technical manuals I came across a reallt elegant and simple equation in a book by the British nature photographer Niall Benvie. What is particularly useful about the equation is that you can use it at home to create a small chart that can be glued to the inside of your lens cap for the hyperfocal focusing distances for your most used zooms. 

The Circle of Confusion

So why not simply stick the aperture at f22 so as to ensure a maximum depth of field and probably achieve hyperfocal focusing distance anyway? The problem lies in diffraction. Light passes through the lens and is essentially squeezed or bent through the aperture hole. When the light is bent in this way it scatters, making it harder to resolve back to its original ‘shape’. The smaller the hole that the light has to pass through, the more the light is bent and therefore the more diffraction there is. This means that f22, or f32 on some 35mm format lenses, although having the most depth of field, lose there resolving power due to diffraction. In fact, on my 35mm f2 lens (used in the images), the three sharpest apertures are f8, f11, f5.6 (in this order). F22, although not the worst, is distinctly ‘soft’.
Conventional wisdom is to rarely if ever use the lowest aperture on your lenses unless you absolutely have to. Then why are medium format and large format lenses considered sharp when images are shot at f32 and even f64. Afterall, Ansel Adams was one of the founders of the famous Group f64. The answer lies in the circle of confusion and magnification. 
When a lens resolves an image of say a point source onto the sensor or a piece of film it is essentially creating a cone of light (see image). Critical focus is achieved when the apex of the cone of light meets with the focal plane (the image sensor or film). If the apex of the cone falls short of or beyond the focal plane we can consider that the image is no longer critically sharp. The small circle that is resolved onto the focal plane is considered to the ‘circle of least confusion’ (i.e. it is perfectly sharp). It is possible, however for the apex of the cone to fall slightly before or beyond the focal plane, but still resemble a point source, or disk of light, rather than a circle. This is known as the ‘permissible circle of confusion’ (or the ‘acceptable circle of confusion’). A useful (but technically accurate) way of demonstrating this would be to shine a torch beam like that from a maglite onto a wall. By focusing its beam into a sharp point we can simulate the ‘circle of least confusion’. As we defocus the beam the disk of light enlarges until the disk suddenly becomes a circle. The defocused circle is actually a result of diffraction and is known as the ‘Airy disk’. Essentially the focused beam of light is that between the ‘circle of least confusion’ and the ‘permissible circle of confusion’ before the Airy disk appears. 

The effect of magnification

Back to why medium and large format cameras have smaller apertures. The circle of confusion is very different for different formats. Usually one views an image from a distance that is about equal to the image’s diagonal measurement (this isn’t always the case as landscape images draw us in by their size and then invite us to minutely inspect the finer details). The permissible circle of confusion is a visible phenomenon that is interpreted by our eyes and brains. If we are viewing a print of 15x20cm (with a diagonal of 25cm and therefore an optimal viewing distance of roughly 25cm), the permissible circle of confusion is 1.45mm. Remember that to obtain a 15x20cm print the original image, if from a 35mm FF sensor must be magnified 6.25 times. Suddenly the permissible circle of confusion for the sensor is a lot smaller, 0.23mm in fact. Make the image larger and the permissible circle of confusion gets even smaller. It seems to be generally established that the average person’s visual acuity resolves sees as blurred images where the circle of confusion is less than 0.2mm. So the goal for your final print is really to have a permissible circle of confusion of 0.2mm.
So how does this technically define the print size limits of our sensors or films? Let’s say that we want to print a roughly A2 sized image with its shortest side at 430mm in length. This gives us a diagonal of approximately 775mm. If we were to view the image from 775cm away, using the equation above, the circle of confusion that is required before things start getting soft or blurry is 0.45mm. A 35mm piece of film needs to be magnified 17.9 times to achieve a print size of 645x430mm. So we need a circle of confusion of .025 on the piece of film. Wonderful...we fall within our limits! Unless you want to step in to look at closer detail that is. If we wanted to obtain a circle of confusion of 0.2mm on the final print, we would have to have a permissible circle of confusion of 0.011mm on the negative, well below the achievable circle of confusion for a 35mm sensor. Hence why images go soft as they are blown up larger. If we had used a 6x4.5 medium format camera on the other hand things would be very different. The permissible circle of confusion for this format is 0.043mm. The image would only need to be enlarged 9.5 times, meaning that the circle of confusion for the final print is 0.41mm, considerably larger than the required 0.2mm.
So you can see that if you start off with a large permissible circle of confusion such as that for medium format photography you can enlarge your print to a larger size before the image starts to degrade in quality through perceived blurring of the edges. 
We have now established that different formats have different permissible circles of confusion. 35mm has a circle of confusion of 0.025mm (often rounded to 0.03mm), 6x4.5 circle of confusion is .04mm while Nikon’s APS-C format has a circle of confusion of only 0.016mm. 
To maintain the same field of view the lens must be in proportion to the actual format size. Thus when we use an 18mm lens on a 35mm FF sensor we have a very wide angle view. The same lens on a APS-C sensor only shows the equivalent of a 28mm lens on the 35mm FF sensor. This is why we get that crop effect when we use 35mm lenses on smaller frame digital cameras. It is also why a 75mm lens on a 6x4.5 camera is the equivalent of a 50mm lens on a 35mm camera and a 35mm lens on an APS-C camera. They all have the same field of view, but have very different focal lengths. Wider focal lengths have more depth of field. Thus, a 35mm camera with a 50mm set at f11 lens has more depth of field than a medium format camera with a 75mm lens set at the same aperture. The advantage for the landscape photographer with an APS-C sized sensor is that there is a full stop more depth of field in the smaller frame than in the 35mm sensor. Essentially a setting of f11 will give the same perceived depth of field as would f16 if we are using equivalent lenses on the two different formats. The disadvantage is that we cannot print as large with the smaller frame sensor. 


Although fairly long-winded I have attempted to condense a very large amount of mathematical equations into something a little simpler to understand. What we basically have above is an understanding that depth of field and perceived sharpness within the image are related to sensor size, circle of confusion, aperture choice and focal length. The equation given to obtain hyperfocal distance is the easiest I have come across so far in accurately calculating the hyperfocal focusing point for a zoom lens. 


If you would like to read more about depth of field and circles of confusion you can access the same texts that I have used in my pursuit of understanding this phenomenon.
  1. ‘Focal encyclopaedia of Photography, third edition’ (1993) by Richard Zakia and Leslie Stroebel (eds). Focal Press: Coston & London
  2. ‘Creative Landscape Photography’ (2003) by Niall Benvie. David & Charles: London
  3. ‘How to Select the F-stop’ by Q.Tuan Luong for Large format Photography (http://www.largeformatphotography.info/fstop.html)
  4. ‘Depth of Field’ by in Wikipedia (http://en.wikipedia.org/wiki/Depth_of_field)
  5. 'An Introduction to Depth of Field’ (2004) by Jeff Conrad (http://www.largeformatphotography.info/articles/IntroToDoF.pdf)


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