The mathematics of reach

You may have heard that a 300mm lens on micro four thirds gives the same reach as a 600mm lens on full-frame. This is of course false in most cases and is based on a fundamental misunderstanding of how cameras work. However, to understand this, you have to understand what “reach” is. Actually, the topic is a bit complex.

To understand this phenomenon, we should first understand that we don’t get any additional reach by cropping. Cropping makes a subject in the total frame look bigger, but won’t give you anything additional that wasn’t there before. In short, cropping does not increase your reach.

What people usually mean by reach is “pixels on your subject”. Actually, you also have to be careful here because not all pixels are created equal. Imagine the following scenario: you are shooting a bird with a 24MP full-frame camera but the light is very dim so you’re using ISO 50,000. Yes, you can see the bird, but the end result is fairly noisy. Let’s say it’s one of your noisiest images.

Would switching to a 45MP full-frame camera with the same lens and settings change anything? Probably not. Yes, you’ll get more pixels on your subject, but since there is so much noise, those pixels aren’t going to be providing any detail.

How to think about reach

The only thing that reach depends on is your lens and your pixel density. However, as I said, not all pixels are created equal. For example, let’s say I created a 300MP full-frame camera. Yes, I’d get a bit more reach with it, but after a point, those really small pixels aren’t going to do much good in most real-world scenarios.

In other words, there’s a law of dimishing returns when it comes to reach: going from 24MP to 45MP gives you more reach. You’ll get even more by going to 100MP, but after 100MP, going to 200MP might not be that advantageous because those pixels won’t give you any additional information.

In other words, you need to think about reach in terms of information, or how much detail you will actually resolve. Because there are so many variables that affect this, the best way to think about reach is compare two lenses and two pixel densities. And this is why it’s incorrect to say that a 300mm on micro four thirds gives the same reach as a 600mm lens on full-frame. For, if you had a full-frame camera with the same pixel density as a micro four thirds camera, then a 300mm lens on micro four thirds gives the exact same reach as a 300mm lens on that specific full-frame camera. On the other hand, a 600mm lens will give a lot more.

The bottom line is this: sensor size has nothing to do with reach, only pixel density and focal length do. On the other hand, the larger sensor you have, the closer you can get with the same framing with a given focal length, thereby increasing your image quality. And of course, the quality of your lens will determine how valuable that reach really is.

How to calculate reach

Reach, in terms of pixels on target and quality of pixels, must be calculated based on pixel density. Since a ratio of focal lengths tells you the linear crop factor, to calculate equivalent focal lengths, we need to use the pixel pitch, which is defined to be the length of a single pixel. It’s easy to do that: take the length of your sensor and divide it by the horizontal resolution of your camera. So that you don’t have to do it, here’s a table of the most common pixel pitches based on sensor formats and resolutions:

With this, table, it is very easy to calculate the equivalent focal lengths so that you’ll get the exact same number of pixels on your subject. If $P_1$ is the pixel pitch of camera one and $P_2$ is the pixel pitch of camera two, then the actual crop factor is defined as:
$${\rm crop factor} = \frac{P_1}{P_2}.$$ It’s that simple, and we just need to apply this crop factor to both the focal length and the aperture. The reason why we apply it to the aperture is so that we take into account the quality of the pixels due to noise degradation. Let’s do three examples.

Example 1. A popular lens for micro four thirds is the Olympus 300mm f/4. What lens would you need on full-frame to get the same number of pixels on your target as this fine lens? Well, we need to know which full-frame camera and which micro four thirds camera, so let’s take the 45.7MP full-frame camera as this is by far the most popular resolution these days for wildlife shooters. For the micro four thirds camera, let’s take the 20MP one because that is the most popular micro four thirds resolution today. The crop factor going from full frame needs to be calculated with $P_1 = 4.348$ and $P_2 = 3.337$, so the crop factor is $P_1/P_2 = 1.3029…$ or about 1.3

Now, just multiply this number by the focal length and aperture: we get that a 300mm f/4 lens on a 20MP micro four thirds camera gives the exact same reach as a 391mm f/5.2 lens on a 45.7MP full-frame camera. Any lens which is better than 391mm f/5.2, by which I mean any lens which has at least 391mm and at least an aperture of f/5.2, will give more reach than the 300mm f/4 lens on the 20MP micro four thirds sensor. So in particular, the Nikon 400mmm f/4.5 lens gives more reach. As you can see, a 300mm f/4 lens does not come anywhere close to a 600mm lens on a 45.7MP full-frame camera.

It is true that a 300mm lens on a micro four thirds camera will give the same field of view as a 600mm lens on a full-frame camera, but the 600mm full-frame combination (in this particular example of resolutions) will give so many more pixels on your target and hence give you more actual reach (and much more image quality).

Example 2. What lens do you need on a 20MP micro four thirds camera in order to give the same reach as a 600mm f/4 lens on the Sony a7R V, a 61MP full-frame camera? Again, we caluclate the crop factor. In this case, $P_1 = 3.756$ and $P_2 = 3.337$, so the crop factor is a bit smaller this time: it’s 1.13.

We divide the 600mm f/4 by this factor to get a 531mm f/3.5. You’ll need at least 531mm.

Aperture and other caveats

Notice that I included aperture in my calculations. Of course, if the light is bright and sufficient, then aperture matters less purely in information-theoretical terms. When the light becomes dim, the equivalent aperture becomes more and more necessary, but if you’ve got good light, then you can get away with smaller apertures. In ideal light, only the focal length really matters in terms of information-resolving power. Of course, the aperture calculation is still crucial for depth of field.

Of course, that assumes equivalent performance, too. In other words, when doing these calculations, I am assuming that the lenses are sharp. For example, you could technically put a 2X teleconverter on a bad 300mm lens to make it 600mm, but I’d rather use the Olympus 300mm f/4 on a 20MP micro four thirds camera than a 300mm lens with a 2X TC on a 45.7MP full-frame camera.

To put it another way, these calculations only make sense for roughly equivalent lenses in terms of sharpness. Sometimes I shoot with an old adapted 500mm lens on my Panasonic G9. Even though this combinations technically gives more pixels on my target than my Nikon 500mm f/5.6 PF lens on my Nikon Z6, I’d rather take the Nikon Z6 option any day.

I would not get too obsessed about pixels on your subject. Yes, sometimes this value is important when you are trying to set people right when they mistakenly think a 300mm f/4 will perform the same as a 600mm f/4 on full-frame, but there are many other factors to consider. For example, with a full-frame camera, you might not get as many pixels on your target as the newest OM SYSTEM 150-600mm lens on a micro four thirds camera, but you will usually get much better light performance on the occasions where you can actually get close and get an ideal framing without cropping.

Finally, should you use this information to choose systems? Yes and no. It can provide some useful context, but there is something far more important to consider: lens ecosystems. Although reach calcuations can tell you about equivalent lenses, usually those equivalent lenses do not exist! That’s why I stick with Nikon: because they have lenses that no other manufacturers have like the 600mm f/6.3 PF, the 500mm f/5.6 PF, the 800mm f/6.3 PF, the 600m f/4 1.4TC, and others. If you go through the above calculations, you’ll get some equivalent lenses on other systems that simply do not exist.

Are there any advantages to smaller sensors?

Smaller sensors like micro four thirds do typically have smaller pixel pitches (the length of an individual pixel). Even the Sony a7R V with 61MP still has a pixel pitch of 3.756µm, which is larger than the 3.337µm pixel pitch of a 20MP micro four thirds sensor. Therefore, a 300mm lens on the Panasonic G9 will give very slightly more pixels on target than a 300mm lens on the Sony a7R V for example. However, the difference is not great: 300mm on 20MP micro four thirds is the same as 337mm on the Sony a7R V.

The true problem with micro four thirds is that they don’t have many quality lenses beyond 300mm, the only exception being the Olympus 150-400mm f/4.5 1.25TC lens. Using our calculations, 400mm f/4.5 gives the same number of pixels on target as a full-frame 45.7MP camera with a 521mm lens, and you can get a much cheaper 500mm lens for a full-frame camera: the Nikon 500mm f/5.6 PF. Keeping in mind that in dim light, you have to content with a 521mm f/5.9, it is actually much cheaper to get a Nikon Z8 with a 500mm f/5.6PF.

In fact, the Olympus 150-400mm f/4.5 has an MSRP of 7500USD, which is 2700USD more than Nikon’s 600mm f/6.3PF, and the latter will give you so much better image quality on the Z8. Of course, the Olympus 150-400mm f/4.5 is a fine lens and has the advantage of zoom, but if you’re a bird photographer who primarily cares about the long end of 500mm or more, then micro four thirds doesn’t make too much sense.

Now, again, that doesn’t mean micro four thirds isn’t a good bird system. It’s a pretty good one, and I would never contend with that. However, there is one thing I do maintain: if you are going after the best image quality and you are primarily interested in supertelephoto prime lenses rather than zooms and you want creamy, shallow depth of field, it’s simply much easier to do that with primes on a full-frame camera (or even APS-C) than anything that exists for micro four thirds.

(Of course, I would change my statement if micro four thirds had equivalent lenses. If it had a 400 f/4 or even 400 f/2.8, then it would be a far more versatile system.)

So, which format?

Beyond simple reach calculations, almost any format can get you good shots. But I do think that if you’re interested in birds, the most versatile format is high-resolution full-frame, assuming you don’t mind the cost or the weight, and you are actually going to get decent lenses like supertelephoto primes or quality zooms. APS-C is a close second when it can use that gorgeous glass. Of course, I love my micro four thirds camera and it can produce world-class images too, but in general I would not use it as a primary birding format.