Recently, my old (and first) Sony $\alpha$-6000 camera failed in many ways:

• Its autofocus stopped working, moving the lens only slightly, I suspect hardware/motorization malfunction.
• The mode switching wheel is full of sand, my fault on bringing it to the beach hoping to shoot dramatic sunset photos.
• The manual focus wheel (which I have to use every time now since AF is dead) had an inopportune encounter with asphalt, which now impairs its rotation for more than a quarter of its course.

It is now high time to buy another camera and, why not, step up in quality.

Being an amateur fan of film rendering and after hearing a lot of great good about Fuji film cameras, I bought the last of the X100 series, the X100V.

When buying a camera (or any piece of technological asset), many sorts of technical details are thrown at you. I believe marketing teams deeply believe lambda customers are as impressed as they are in front of numbers that they can't understand. In my opinion, customers choices are more often based on specialized reviews, emotional response and/or friends/family recommendations, than on the resulting field of view of using a crop factor of 1.5 on a fixed 23 mm lens.

Nonetheless, those numbers are here to stare and stay intriguing to me. Moreover, many camera specifications are related to interesting physics (here optics) phenomenons, which is always fun to know more about.

This post is my journey trying to get a deeper understanding of what those numbers mean, for the sake of it.

Focal-length, full-frame, crop-factor, APS-C and field of view.

Inside a camera, we find at least several lenses, an aperture and a sensor.

Incoming light from the scene goes through each lens, the aperture and hits the image sensor. Even though numerous lenses are used in order to attenuate optical aberrations and distortions, the overall optical system can be thought in first order approximation as one lens with a given focal length. The focal length of a convex lens is the distance between the plane of the lens and the physical point where parallel light rays –coming from an infinite distance source– converge down to a single point: the focal point.

Focal length

The focal lengths for cameras are often given in millimeters, e.g., 35 or 23 mm. In order to retrieve the focused image from the real world, the sensor's camera should be placed where rays meet: again the focal point. One issue arises, the focal point is indeed the convergence point of incoming light, but only for infinite sources. For closer subjects, the actual point of convergence changes. This is why the concept of focus exists. Going down the rabbit-hole, what precisely is focus ?

Let's consider a camera with a lens of focal length $f$ and a focused distance lens-sensor $v$. If we are trying to focus an object at a distance $u$ from the lens, the following equation should be true for perfect focus: $$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$$ This thin-lens equation means that you can either change the subject position –by using your legs for example–, change the lens –not convenient– or change the lens-sensor distance. The latter option is what all autofocusing (AF) systems eventually end up doing, with various degrees of success.

Returning to the matter at hand, do focal lengths really changes anything ? Yes, and a lot! The greatest impact of the focal length is the change in field of view. The two are actually inversely proportional, as we will see in the next section.

Full-frame, 35 mm and APS-C

Also, a little digression: I kept noticing the value 35 mm written in many places across the web. As a matter of fact, a 35 mm camera is not a camera with a focal length of 35 mm. It is actually a type of film, also named 135 film. It is named from its film gauge height, which, you guessed it, happens to be exactly 35 mm. Back in the days when analog was not a choice but the only thing we had, the sensors in camera were not these fancy 4K array of pixels but real physical films being hit by photons. The size of the film was the sensor size. Because nothing can be simple, the film format of the 135 is actually 24 × 36 mm, which is the physical part that actually get useful light, each frame being separated by 2 mm on each side.

The impact of this old format is still strong. Today's digital cameras' sensor have either the same size or advertise their sensor size relative to the 35 mm standard. This is what “full frame” commonly refers to: it simply means that the digital camera sensor has the same size as this old film format.

But not all sensors are made the same. In more compact devices, like the X100V, the sensors are smaller. For example, the APS-C –C for classic– is a smaller format with the same aspect ratio 3:2 than the full frame format. It is only 25.1 × 16.7 mm according to the real standard, even if manufacturers do make those sensors a bit smaller or bigger in the actual cameras.

Crop factor

This leads us to the notion of crop factor, which should be easy to understand by now. When not using a full frame sensor, how can we still compare shots with others cameras, or different lenses ? Indeed, a 23 mm lens hitting an APS-C sensor will not look like a 23 mm lens hitting a full frame: the smaller sensor will miss some light and have reduced field of view. So they introduced the crop factor, which is a simple scalar that describe the ratio between the dimensions of the standard 35 mm format and another sensor's format. The crop factor is also called the focal length multiplier, as it can be used to compute the equivalent focal length for a full frame sensor.

As an example, from the specification of the X100V, we can read this line for the focal length: 23 mm (35 mm format equivalent : 35 mm). At first, this line can be intriguing because the two values 35 mm do not mean the same thing, see above. But all things considered, this means the crop factor should be around 1.5. $$35 \text{mm}/23\text{mm} \approx 1.523$$ We can double-check this on the actual sensor size, which is 23.5 mm× 15.6 mm for the X100V. Because the two sensors are supposed to have the same 3:2 ratio, we can take any dimension (height, width, or diagonal) to compute the crop factor and we should get the same result. Compared to full frame, we have a computed ratio of : $$36/23.5 \approx 1.532$$ Quite close (it is not perfect because the focal lengths advertised are not that precise).

Field of view

Fine, will all those numbers we made sure everything made sense, but how does the focal length influence the final image?

I will opt for the geometric explanation: the size of the sensor and its relative position to the lens is what dictates what can be physically seen. But here, a picture is again worth a thousand words.

Figure $1$: geometrical scene.

This little drawing shows the relation between the subject, its apparent angle $\theta$, the focal length $f$ and the sensor height $h$. The bigger $\theta$ is, the bigger we perceive the object. A bit of trigonometry leads to the following equation: $$\theta = 2 \arctan \frac{h}{2f}$$ This confirms the claim that for a smaller sensor, the field of view is reduced. Also, if the focal length is made larger while keeping the sensor in focus, this also has the effect of reducing the field of view. This is why large focal lengths also offer the greater magnification, you get the same number of pixels a.k.a. resolution but for a reduced angle of view.

Also, digressing a bit here again. The field of view concept often brings up perspective's distortion questions. You probably saw those time lapses where a subject is shot at different focal lengths, its face flattening out at large focal lengths and looking like a big noise guy when really close to the camera. This effect, sometimes called lens compression, really does not have anything to do with lenses or sensors, but with perspective. Theses time lapses are actually tricking you into thinking that higher focal lengths make people prettier and background bigger, while it is only a question of relative positioning of the camera, the subject, and the background.

In short, cropping a 23 mm shot with the same field of view than a 150 mm shot will result in an identical perspective, the 23 mm shot will simply have pretty poor resolution.

Conclusion

I scratched a bit of the surface of technical jargon that you may encounter when trying to buy a camera, and there is still a lot to cover. Anyway this post helped me formalize my understanding and I hope you gained some insights from it as well. See you next time.